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diff --git a/manual_source/Makefile b/manual_source/Makefile new file mode 100644 index 0000000..48a0794 --- /dev/null +++ b/manual_source/Makefile @@ -0,0 +1,129 @@ +# -*- make -*- +# FILE: "/home/evmik/jobs/wm/2013_fall_Experimental_Atomic_Physics_251/manual_source/Makefile" +# LAST MODIFICATION: "Fri, 30 Aug 2013 17:33:03 -0400 (evmik)" +# (C) 2001 by Eugeniy Mikhailov, <evmik@tamu.edu> +# $Id: Makefile,v 1.3 2003/04/23 20:40:19 evmik Exp $ + +MANUSCRIPT_DIR = $(shell basename `pwd`) +SUBDIR=`find ./ -maxdepth 1 -type d |sed 1d| sed /CVS/d` + +INSTALL = install -C +CP_FILES = $(INSTALL) --mode=664 + +dest_dir = ../manual + +LATEX_OPTIONS="-interaction=nonstopmode -src-specials" + +DATE := $(shell date +%Y%m%d) + +ROOT_OF_MAIN_TEX_FILE= EIT_filter +MAIN_TEX_FILE=$(ROOT_OF_MAIN_TEX_FILE).tex + +source_files = $(wildcard *.tex) +source_basename = $(source_files:%.tex=%) + +tex_targets = $(wildcard *.tex) +pdf_targets = $(tex_targets:%.tex=%.pdf) +ps_targets = $(tex_targets:%.tex=%.ps) +dvi_targets = $(tex_targets:%.tex=%.dvi) + +default: pdf copy2dest + +INCLUDED_TEX_FILES=$(wildcard chapters/*.tex) +GENERATED_AUX_FILES=$(INCLUDED_TEX_FILES:%.tex=%.aux) + + +$(INCLUDED_TEX_FILES): + + +dvi: $(dvi_targets) + +$(dvi_targets): %.dvi : %.tex + rubber $< + + +ps: $(ps_targets) + +$(ps_targets).ps: %.ps : %.dvi + dvips -o $@ $< + + +dvipdf: $(pdf_targets) + +#home of dvipdfm utilit is at +#http://gaspra.kettering.edu/dvipdfm/ +# if one want to use hyperred in pdf and dvipdfm converter +# then \usepackage[dvipdfm]{hyperref} should be use +# but then no links in dvi would be present +# so we use \usepackage{hyperref} but make pdf with dvipdf +# which do it through dvi -> ps -> pdf conversion +#$(pdf_targets): %.pdf : %.dvi + #dvipdfmx $< + # dvipdf $< + +pdf: $(pdf_targets) + +$(pdf_targets): %.pdf : %.tex $(INCLUDED_TEX_FILES) + rubber -d $< + + + +zip: + zip -r arch.$(ROOT_OF_MAIN_TEX_FILE).`date +%F`.zip $(ROOT_OF_MAIN_TEX_FILE).tex Makefile bibliography.bib figures ol.bst osajnl.bst osajnl2.rtx osajnl2.sty `tex2figlist.sh $(ROOT_OF_MAIN_TEX_FILE).tex` + +clean_results: + rm -f $(pdf_targets) + +clean: clean_tex clean_bib clean_backups + rm -f $(ROOT_OF_MAIN_TEX_FILE).tar.gz + +real_clean: clean_all + +clean_backups: + rm -f *~ + +clean_tex: + rm -f $(tex_targets:%.tex=%.dvi) + rm -f $(tex_targets:%.tex=%.log) + rm -f $(tex_targets:%.tex=%.aux) + rm -f $(tex_targets:%.tex=%.nav) + rm -f $(tex_targets:%.tex=%.out) + rm -f $(tex_targets:%.tex=%.snm) + rm -f $(tex_targets:%.tex=%.toc) + rm -f $(tex_targets:%.tex=%.vrb) + rm -f $(tex_targets:%.tex=%.blg) + rm -f $(GENERATED_AUX_FILES) + rm -f missfont.log + rm -f *Notes.bib + +clean_bib: + rm -f $(tex_targets:%.tex=%.bbl) + +clean_for_arxive: clean_results clean_backups clean_tex + +clean_all: clean clean_results clean_backups + +$(dest_dir): + $(INSTALL) -d $(dest_dir) + +copy2dest: $(dest_dir) pdf + $(CP_FILES) $(pdf_targets) $(dest_dir)/ + +arxive_submission: $(ROOT_OF_MAIN_TEX_FILE).for_arxive.$(DATE).tar.gz + +$(ROOT_OF_MAIN_TEX_FILE).for_arxive.$(DATE).tar.gz: clean_for_arxive + cd ..; tar c --dereference --exclude=.git --exclude=*gz \ + $(MANUSCRIPT_DIR)/$(tex_targets) \ + $(MANUSCRIPT_DIR)/$(ROOT_OF_MAIN_TEX_FILE).bbl \ + $(MANUSCRIPT_DIR)/ready_plots/* \ + | gzip > $@ + mv ../$@ . + + +arch: + cd ..; tar c --dereference --exclude=.git --exclude=*gz $(MANUSCRIPT_DIR) |gzip > $(ROOT_OF_MAIN_TEX_FILE).$(DATE).tar.gz + mv ../$(ROOT_OF_MAIN_TEX_FILE).$(DATE).tar.gz . + + +upload_draft_to_web: + diff --git a/manual_source/chapters/HeNelaser.tex b/manual_source/chapters/HeNelaser.tex new file mode 100644 index 0000000..9918f80 --- /dev/null +++ b/manual_source/chapters/HeNelaser.tex @@ -0,0 +1,291 @@ +%\chapter*{Helium-Neon Laser} +%\addcontentsline{toc}{chapter}{Helium-Neon Laser} + +\documentclass{article} +\usepackage{tabularx,amsmath,boxedminipage,epsfig} + \oddsidemargin 0.0in + \evensidemargin 0.0in + \textwidth 6.5in + \headheight 0.0in + \topmargin 0.0in + \textheight=9.0in + +\begin{document} +\title{Helium-Neon Laser} +\date {} +\maketitle \noindent + \textbf{Experiment objectives}: assemble and align a 3-mW HeNe laser from +readily available optical components, record photographically the transverse mode structure of the +laser output beam, and determine the linear polarization of the light produced by the HeNe laser. + +\subsection*{Basic operation of the laser} + + The bright, highly collimated, red light beam ($\lambda = 6328 {\AA}$) from a helium-neon (HeNe) +laser is a familiar sight in the scientific laboratory, in the industrial workplace, and even at the +checkout counter in most supermarkets. HeNe lasers are manufactured in large quantities at low cost +and can provide thousands of hours of useful service. Even though solid-state diode lasers can now +provide red laser beams with intensities comparable to those obtained from HeNe lasers, the HeNe +laser will likely remain a common component in scientific and technical instrumentation for the +foreseeable future. +% +%In this experiment you will (a) assemble a 3-mW HeNe laser from readily available optical components, +%(b) align a HeNe laser cavity using two different cavity mirror configurations, (c) record +%photographically the transverse mode structure of the laser output beam, and (d) determine the linear +%polarization of the light produced by the HeNe laser. The principal goal of this experiment is for +%you to get hands-on experience with the various optical components of a working laser; however, to +%help you appreciate fully the role played by each of the components, a brief overview of the +%principles of HeNe laser operation is given here. + +\begin{figure}[h] +\centerline{\epsfig{width=\textwidth, file=HeNesetup.eps}} \caption{\label{HeNesetup.fig}Diagram of +optical and electrical components used in the HeNe laser experiment.} +\end{figure} + +The principal goal of this experiment is for +you to get hands-on experience with the various optical components of a working laser; however, to +help you appreciate fully the role played by each of the components, a brief overview of the +principles of HeNe laser operation is given here. The three principal elements of a laser are: +(1) an energy pump, (2) an optical gain medium, and (3) +an optical resonator. These three elements are described in detail below for the case of the HeNe +laser used in this experiment. +\begin{enumerate} +\item \textbf{Energy pump}. A 1400-V DC power supply +maintains a glow discharge or plasma in a glass tube containing an optimal mixture (typically 5:1 to +7:1) of helium and neon gas, as shown in Fig.~\ref{HeNesetup.fig}. The discharge current is limited +to about 5 mA by a 91-k$\Omega$ ballast resistor. Energetic electrons accelerating from the cathode +to the anode collide with He and Ne atoms in the laser tube, producing a large number of neutral He +and Ne atoms in excited states. He and Ne atoms in excited states can deexcite and return to their +ground states by emitting light spontaneously. This light makes up the bright and diffuse pink-red +glow of the plasma that is seen at even in the absence of laser action. + +The process of producing He and Ne in specific excited states is known as pumping, and in the HeNe +laser this pumping process occurs through electron-atom collisions in the discharge. In other types +of lasers, pumping is achieved by using light from a bright flashlamp or by using chemical reactions. +Common to all lasers is a process for preparing large numbers of atoms, ions, or molecules in +appropriate excited states so that a desired type of light emission can occur. + +\item \textbf{Optical gain medium}. +To achieve laser action it is necessary to have more atoms in excited states than in ground states, +and to establish what is called a \emph{population inversion}. To understand the significance of a +population inversion to HeNe laser action, it is useful to consider the processes leading to +excitation of He and Ne atoms in the discharge, using the simplified diagram of atomic He and Ne +energy levels given in Fig.~\ref{HeNelevels.fig}. The rather complex excitation process necessary for +lasing occurs +in four steps. \\ +\emph{(a)} An energetic electron collisionally excites a He atom to the state labeled $2_1S^0$ in +Fig.~\ref{HeNelevels.fig}. A He atom in this excited state is often written He*($2_1S^0$), where the +asterisk is used +to indicate that the He atom is in an excited state. \\ + +\emph{(b)} The excited He*($2_1S^0$) atom collides with an unexcited Ne atom and the two atoms +exchange internal energy, with an unexcited He atom and excited Ne atom, written Ne*(3$s_2$), +resulting. This energy exchange process occurs with high probability because of the accidental near +equality of the excitation energies of the two levels in these atoms.\\ + +\emph{(c)} The 3$s_2$ level of Ne is an example of a metastable atomic state, meaning that it is only +after a relatively long time -- on atomic that is -- that the Ne*(3$s_2$) atom deexcites to the +2$p_4$ level by emitting a photon of wavelength 6328 $\AA$. It is this emission of 6328 $\AA$ light +by Ne atoms that, in the presence of a suitable optical suitable optical configuration, +leads to lasing action. \\ + +\emph{(d)} The excited Ne*(2$p_4$) atom rapidly deexcites to the Ne ground state by emitting +additional photons or by collisions with the plasma tube deexcitation process occurs rapidly, there +are more Ne atoms in the 3$s_2$ state than there are in the 2$p_4$ state at any given moment in the +HeNe plasma, and a population inversion is said to be established between these two levels. When a +population inversion is established between the 3$s_2$ and 2$p_4$ levels of the excited Ne atoms, the +discharge can act as an optical gain medium (a light light amplifier) for light of wavelength 6328 +$\AA$. This is because a photon incident on the gas will have a greater probability of being +replicated in a 3$s_2\rightarrow 2p_4$ stimulated emission process (discussed below) than of being +destroyed in the complementary $2p_4\rightarrow 3s_2$ absorption process. + + +\begin{figure}[h] +\centerline{\epsfig{width=0.8\textwidth, file=HeNelevels.eps}} +\caption{\label{HeNelevels.fig}Simplified atomic energy level diagram showing excited states of +atomic He and Ne that are relevant to the operation of the HeNe laser at 6328~$\AA$.} +\end{figure} + +\item \textbf{Optical resonator}. As mentioned in 2(c) above, Ne atoms in the 3$s_2$ metastable +state decay spontaneously to the 2$p_4$ level after a relatively long period of time under normal +circumstances; however, a novel circumstance arises if, as shown in Fig.~\ref{HeNesetup.fig}, a HeNe +discharge is placed between two highly reflecting mirrors that form an \emph{optical cavity} or +\emph{resonator} along the axis of the discharge. When a resonator structure is in place, photons +from the Ne* 3$s_2\rightarrow 2p_4$ transition that are emitted along the axis of the cavity can be +reflected hundreds of times between the two high-reflectance end mirrors of the cavity. These +reflecting photons can interact with other excited Ne*(3$s_2$) atoms and cause them to emit 6328 +$\AA$ light in a process known as \emph{stimulated} emission. The new photon produced in stimulated +emission has the same wavelength and polarization as the stimulating photon, and it is emitted in the +same direction. It is sometimes useful for purposes of analogy to think of the stimulated emission +process as a "cloning" process for photons. The stimulated emission process should be contrasted with +spontaneous emission processes that, because they are not caused by any preceding event, produce +photons that are emitted isotropically, with random polarization, and over a broader range of +wavelengths. As stimulated emission processes occur along the axis of the resonator, a situation +develops in which essentially all Ne* 3$s_2\rightarrow 2p_4$ decays contribute deexcitation photons +to the photon stream reflecting between the two mirrors. This photon multiplication (light +amplification) process produces a very large number of photons of the same wavelength and +polarization that travel back and forth between the two cavity mirrors. To extract a light beam from +the resonator, it is only necessary that one of the two resonator mirrors, usually called \emph{the +output coupler}, has a reflectivity of only 99\% so that 1\% of the photons incident on it travel out +of the resonator to produce an external laser beam. The other mirror, called the high reflector, +should be as reflective as possible. The diameter, bandwidth, and polarization of the HeNe laser beam +are determined by the properties of the resonator mirrors and other optical components that lie along +the axis of the optical resonator. + +\end{enumerate} + + +\section*{Experimental Procedure} + +\textbf{Equipment needed}: Commercial HeNe laser, HeNe discharge tube connected to the power supply, +two highly reflective mirrors, digital camera, polarizer, photodetector, digital multimeter. + +\subsection*{Safety} +A few words of caution are important before you begin setting up your HeNe laser. \\ +First, \textbf{never} look directly into a laser beam, as severe eye damage could result. During alignment, you +should observe the laser beam by placing a small, white index card at the appropriate point in the optical path. +Resist the temptation to lower your head to the level of the laser beam in order to see where it is going. \\ +Second, \textbf{high voltage} ($\approx 1200$~V) is present at the HeNe discharge tube and you should avoid any +possibility of contact with the bare electrodes of the HeNe plasma tube. \\ Finally, the optical cavity mirrors +and the Brewster windows of the laser tube have \textbf{very delicate optical surfaces} that can be easily +scratched or damaged with a single fingerprint. If these surfaces need cleaning, ask the instructor to +demonstrate the proper method for cleaning them. + + + +\subsection*{Alignment of the laser} + +To assemble the HeNe laser and investigate its properties, proceed with the following steps. + +\begin{itemize} + +\item The discharge lamp has very small and angled windows, so first practice to align the beam of +the commercial HeNe laser through the discharge tube. To do that turn on the commercial laser, place a white +screen or a sheet of paper at some distance and mark the position of the laser spot. Now without turning the +power, carefully place the discharge tube such that the laser beam passes through both angled windows without +distortion, and hit the screen almost in the same point as without the tube. Repeat this step a few times until +you are able to insert the tube inside the cavity without loosing the alignment. Then carefully slide the tube +out of the beam and clamp it down. + +\item Set up a hemispherical resonator configuration using a flat, high reflectivity (R = 99.7\%) +mirror, and a spherical mirror with a radius of curvature of r = 0.500 m and reflectivity R = 99\%. +The focal length f of the spherical mirror is given by f = r/2 = 0.250 m. In the diagram of +Fig.~\ref{HeNesetup.fig}, the flat, highly-reflective mirror will be serving as the right end of the +cavity, and the spherical, less-reflective mirror will be serving as the left end of the cavity and +is known as the output coupler. The high reflectivity of each mirror is due to a multilayer +dielectric coating that is located on only one side of each mirror. Be sure to have the reflecting +surfaces of both mirrors facing the interior of the optical cavity. Set the distance between the two +mirrors to approximately d = 47 cm. + +\item To align the optical resonator of your HeNe laser it is easiest to use a beam of a working, +commercial HeNe laser as a guide. Direct this alignment laser beam to the center of the high reflector mirror, +with the output coupler and the HeNe discharge tube removed. With the room lights turned off, adjust the high +reflector mirror so that its reflected beam returns directly into the output aperture of the alignment laser. +Now insert and center the output coupler mirror, and also adjust it such that the reflected beam (from the back +of the mirror) returns to the alignment laser. Now insert a small white card near the front of the output +couplers very close to the laser beam but without blocking it, and locate the reflected beam from the high +reflector mirror - it should be fairly close to the input beam. Using fine adjustment screws in the high +reflector mirror overlap these two beams as good as you can. In case of success you most likely will see some +light passing through a high reflection mirror - fine-tune the position of the mirror some more to make this +light as bright as possible. +%and +%aAdjust the output coupler mirror until you observe concentric interference rings on its intracavity +%surface. It is likely that the interference rings will be converging or diverging slowly. It may be +%necessary to adjust the spacing, d, between the two mirrors to achieve perfectly circular rings. + +\item Now reinsert the HeNe plasma tube between the two mirrors of the optical cavity and adjust the +plasma tube position so that the alignment beam passes through the center of the Brewster windows of the plasma +tube. Be careful not to touch the Brewster windows or mirror surfaces during this process. With the HeNe plasma +tube in place, it should be possible to see a spot at the center of the high reflector mirror that brightens and +dims slowly. %at approximately the same rate as the diverging and converging circular interference rings +%observed earlier. + +\item Turn on the high voltage power supply to the HeNe plasma tube and (with luck) you will observe +the HeNe lasing. If lasing does not occur, make small adjustments to the plasma tube and the two +mirrors. If lasing still does not occur, turn off the high voltage supply, remove the HeNe plasma +tube, and readjust the resonator mirrors for optimal interference rings. If after several attempts +you do not achieve proper lasing action, ask the instructor for help in cleaning the Brewster windows +and resonator mirrors. + +\item Once lasing is achieved, record your alignment procedure in your laboratory notebook. %Describe +%with a well-labeled sketch the nature of the concentric rings that you observed when aligning the +%optical cavity. Determine the range of distances between the two mirrors for which lasing action can +%be maintained in the confocal resonator configuration. Do this in small steps, by increasing or +%decreasing the mirror separation distance d in small increments, and making small adjustments to the +%two mirrors to maintain laser output. +Turn off the alignment laser - you do not need it anymore. + +\end{itemize} + +\subsection*{Study of the mode structure of the laser output} + +Place a white screen at the output of your laser at some distance and inspect the shape of your beam. +Although it is possible that your beam is one circular spot, most likely you will notice some +structure as if the laser output consists of several beams. If you now slightly adjust the alignment +of either mirror you will see that the mode structure changes as well. + +As you remember, the main purpose of the laser cavity is to make the light bounce back and forth +repeating its path to enhance the lasing action of the gain medium. However, depending on the precise +alignment of the mirrors it may take the light more than two bounces to close the loop: it is often +possible for the beam to follow a rather complicated trajectory inside the resonator, resulting in +complex transverse mode structure at the output. +\begin{itemize} + +\item +Take photographs of the transverse mode structure of the HeNe laser output beam. By making small +adjustments to the mirrors and the position of the HeNe plasma tube it should be possible to obtain +transverse mode patterns. Mount your photographs in your laboratory notebook. + +\item +Adjust the mirrors such that the output mode has several maxima and minima in one direction. To +double-check that this mode is due to complicated trajectory of a light inside the resonator, very +carefully insert an edge of a white index card into the cavity, and move it slowly until the laser +generation stops. Now mover the card back and force around this point while watching the generation +appear and disappear, and pay close attention to the mode structure of the laser output. You may +notice that the complicated transverse mode pattern collapses to simpler mode when the card blocks +part of the original mode volume, forcing the generation in a different mode. Describe your +observation in the lab journal. + + +\end{itemize} + +\subsection*{Measure the polarization of the laser light} + +When a linearly polarized light beam of intensity $I_0$ passes through a linear polarizer that has +its axis rotated by angle $\theta$ from the incident light beam polarization, the transmitted +intensity $I$ is given by Malus's law: +\begin{equation} +I = I_0 cos^2\theta. +\end{equation} + +In our experiment the laser generates linearly polarized light field. This is insured by the Brewster +windows of the HeNe plasma tube: the angle of the windows is such that one light polarization +propagates almost without reflection. This polarization direction is in the same plane as the +incident beam and the surface normal (i.e. the plane of incidence). The light of the orthogonal +polarization experiences reflection at every window, that makes the optical losses too high for such +light. + +\begin{itemize} + +\item Visually inspect the discharge tube, note its orientation in the lab book. Make a rough prediction of +the expected polarization of the generated beam. + +\item Determine the linear polarization of the HeNe laser output beam using the rotatable polarizer +and photodiode detector. Make detector readings at several values of angle $\theta$ (every +$20^\circ$ or so) while rotating the polarizer in one full circle, and record them in a neat table in +your laboratory notebook. Graph your data to demonstrate, fit with the expected $cos^2\theta$ +dependence, and from this graph determine the orientation of the laser polarization. Compare it with +your predictions based on the Brewster windows orientation, and discuss the results in your lab +report. + +\end{itemize} + + +\section*{Acknowledgements} + +This lab would be impossible without help of Dr. Jeff Dunham from the Physics Department of the +Middlebury College, who shared important information about experimental arrangements and supplies, as +well as the lab procedure. This manual is based on the one used in Physics 321 course in Middlebury +College. + +\end{document} +\newpage diff --git a/manual_source/chapters/appendices.tex b/manual_source/chapters/appendices.tex new file mode 100644 index 0000000..d310873 --- /dev/null +++ b/manual_source/chapters/appendices.tex @@ -0,0 +1,154 @@ +\chapter*{Errors} + +\section*{Propagation of Random Errors} +Suppose one measures basically the same quantity twice. This might be the +number of $\gamma$-rays detected in 10 minutes with a scintillation detector. +Let $n_1$ be the number detected the first time and $n_2$ the number the +second time. Assume that the average number for many such measurements is +$\overline{n}$. We may then consider a variety of averages denoted by $<>$: +\begin{eqnarray*} +\overline{n}&=&<n>\\ +\overline{n_1}&=&<n>=\overline{n}\\ +<n_1-\overline{n}>&=&0\\ +\overline{n_2}&=&<n>\\ +\sigma_n&=&\sqrt{<(n-\overline{n})^2>} +\end{eqnarray*} + +The root-mean-square(rms) deviation from the mean, ( $\sigma$) is what is +often called the +error in a measurement. +We now determine the ``variance'' ($\sigma^2$) expected for various combinations of +measurements. One only needs to take the square root of $\sigma^2$ to obtain +the error. +\begin{eqnarray*} +\sigma^2&=&<(n_1-\overline{n}+n_2-\overline{n})^2>\\ +&=&<(n_1-\overline{n})^2+(n_2-\overline{n})^2+2(n_1-\overline{n})(n_2-\overline{n})>\\ +&=&<(n_1-\overline{n})^2>+<(n_2-\overline{n})^2>+2<(n_1-\overline{n})(n_2-\overline{n})>\\ +&=&<(n_1-\overline{n})^2>+<(n_2-\overline{n})^2>+2<(n_1-\overline{n})><(n_2-\overline{n})>\\ +\sigma^2&=&\sigma_1^2+\sigma_2^2+0 +\end{eqnarray*} +The average value of the last term is zero since the two measurements are +independent and one can take the averages of each part separately. + +With this result it is easy to get the variance in a linear combination of +$n_1$ and $n_2$. If + +\begin{displaymath} +f=a\cdot n_1 +b\cdot n_2 +\end{displaymath} + +then: +\begin{displaymath} +\sigma_f^2=a^2\sigma_1^2+b^2\sigma_2^2 +\end{displaymath} + +If the errors are small and $f$ is a function of $n_1$ and $n_2$: $f(n_1,n_2)$ +then: +\begin{equation}\label{ssgen} +\sigma_f^2=\left(\frac{\partial f}{\partial n_1}\right)^2\sigma_1^2+\left(\frac{\partial f}{\partial n_2}\right)^2\sigma_2^2 +\end{equation} +It should be clear that one can extend Eq. \ref{ssgen} to arbitrary numbers of +parameters. + +As an example of this latest form suppose $f=n_1\cdot n_2$ then: +\begin{displaymath} +\sigma_f^2=n_2^2\sigma_1^2+n_1^2\sigma_2^2 +\end{displaymath} +or +\begin{displaymath} +\frac{\sigma_f^2}{f^2}=\frac{\sigma_1^2}{n_1^2}+\frac{\sigma_2^2}{n_2^2} +\end{displaymath} + +Thus in this case the fractionial variances add. + +Note: the $\sigma_m$ the error in the mean of $n$ measurements of the +same thing is: $\sigma_m=\sigma /\sqrt{n}$. +\subsection*{Probability Distribution Functions} +\subsubsection*{Binomial} +If the probability of {\it success} in a trial is $p$ then +the probability of $n$ {\it successes} in $N$ trials is: +\begin{displaymath} +P(n)=\frac{N!}{(N-n)!n!}p^n(1-p)^{N-n} +\end{displaymath} +This distribution has a mean $\mu=Np$ and variance $\sigma^2=Np(1-p)$. +This is the starting point for figuring the odds in card games, for example. +\subsubsection*{Poisson} +The probability of $n$ events is: +\begin{displaymath} +P(n)=\frac{e^{-\mu}\mu^n}{n!} +\end{displaymath} +where is the $\mu$ is the mean value and the variance, $\sigma^2=\mu$. +This is the distribution one gets, e.g., with the number of radioactive +decays detected in a fixed finite amount of time. It can be derived from +the binomial distribution in an appropriate limit. +\subsubsection*{Normal or Gaussian Distribution} +This is the first continuous probability distribution. +\begin{displaymath} +P(x)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{\frac{-(x-\mu)^2}{2\sigma^2}} +\end{displaymath} +This function, as you might guess, has mean $\mu$ and variance $\sigma^2$. +If one makes averages of almost anything one finds that the result is +almost always well described by a Normal distribution. Both the binomial and +Poisson distributions approach this distribution in appropriate limits as +does the $\chi^2$ described below. +\subsubsection*{Chi-square distribution: $\chi^2$} +This probability density function (pdf) has the parameter: $N_f$, the number of +degrees of freedom. It is: +\begin{displaymath} +P(x)=\frac{\frac{1}{2}\left(\frac{x}{2}\right)^{(N_f/2)-1}}{\Gamma\left( +\frac{N_f}{2}\right)} +\end{displaymath} +The mean of this pdf is: $\mu=N_f$ and the variance is: $\sigma^2=2N_f$. +The pdf is of considerable use in physics. It is used extensively in the +fitting of histogrammed data. +\newpage + +\appendix{Linear Least Squares} + + +Consider a set of experimental results measured as a function of some +parameter $x$, i.e., $E(x_i)$. Suppose that these results are expected to +be represented by a theoretical function $T(x_i)$ and that $T(x_i)$ is +in turn linearly expandable in terms of independent functions $f_j(x_i)$: +\begin{displaymath} +T(x_i)=\sum_ja_jf_j(x_i) +\end{displaymath} +Suppose now one wants to find the coefficients $a_j$ by minimizing $\chi^2$, +the sum of differences between the experimental results and the theoretical +function, squared, i.e., minimize: +\begin{displaymath} +\chi^2=\sum_i\left(\sum_ja_jf_j(x_i)-E(x_i)\right)^2 +\end{displaymath} +This is found by finding: +\begin{displaymath} +0=\frac{\partial}{\partial a_k}\chi^2= +2\cdot \sum_i\left(\sum_ja_jf_j(x_i)-E(x_i)\right)\cdot f_k(x_i) +\end{displaymath} +This may be rewritten as: +\begin{equation}\label{meq} +\sum_i\left(\sum_ja_jf_j(x_i)f_k(x_i)\right)=\sum_iE(x_i)f_k(x_i) +\end{equation} +The rest is algebra. The formal solution, which can in fact be +easily implemented, is to first define: +\begin{eqnarray} +M_{j,k}&=&\sum_if_j(x_i)f_k(x_i\\ +V_k(i)&=&\sum_iE(x_i)f_k(x_i) +\end{eqnarray} +So that Eq. \ref{meq}. becomes: +\begin{displaymath} +\sum a_jM_{j,k}=V_k +\end{displaymath} +The $a_j$ may then be found by finding the inverse of $M_{j,k }$ +\begin{figure}: +\begin{displaymath} +a_j=\sum_kV_k\cdot M^{-1}_{k,j} +\end{displaymath} +Question: How does this procedure change if: +\begin{displaymath} +\chi^2=\sum_i\frac{(T(x_i)-E(x_i))^2}{\sigma(x_i)^2} +\end{displaymath} +where $\sigma(x_i)$ is the error in the measurement of $E(x_i)$? + +\centerline{\epsfig{width=\linewidth,angle=-90, file=datafg.eps}} +\caption{\label{lsqf} Data Fit to a Straight Line.} +\end{figure} diff --git a/manual_source/chapters/blackbody.tex b/manual_source/chapters/blackbody.tex new file mode 100644 index 0000000..9f6c9bb --- /dev/null +++ b/manual_source/chapters/blackbody.tex @@ -0,0 +1,339 @@ +%\chapter*{Blackbody Radiation} +%\addcontentsline{toc}{chapter}{Blackbody Radiation} +%\documentclass{article} +%\usepackage{tabularx,amsmath,boxedminipage,epsfig} +% \oddsidemargin 0.0in +% \evensidemargin 0.0in +% \textwidth 6.5in +% \headheight 0.0in +% \topmargin 0.0in +% \textheight=9.0in + +%\begin{document} +\chapter{Blackbody Radiation} +\setcounter{figure}{1} +\setcounter{table}{1} +\setcounter{equation}{1} + +%\date {} +%\maketitle \noindent + \textbf{Experiment objectives}: explore radiation from objects at certain temperatures, + commonly known as ``blackbody radiation''; make measurements testing the Stefan-Boltzmann + law in high- and low-temperature ranges; measure the inverse-square law for thermal radiation. + +\section*{Theory} + + A familiar observation to us is that dark-colored objects + absorb more thermal radiation (from the sun, for example) than + light-colored objects. You may have also observed that a good + absorber of radiation is also a good emitter (like + dark-colored seats in an automobile). Although we observe + thermal radiation (``heat'') mostly through our sense of touch, + the range of energies at which the radiation is emitted can + span the visible spectrum (thus we speak of high-temperature + objects being ``red hot'' or ``white hot''). For temperatures + below about $600^{\circ}C$, however, the radiation is emitted in the + infrared, and we cannot see it with our eyes, although there + are special detectors (like the one you will use in this lab) + that can measure it. + + An object which absorbs all radiation incident on it is known +as an ``ideal blackbody''. In 1879 Josef Stefan found an empirical +relationship between the power per unit area radiated by a blackbody and the +temperature, which Ludwig Boltzmann derived theoretically a few years later. +This relationship is the {\bf Stefan-Boltzmann law:} +\begin{equation}\label{SBl} + S =\sigma T^4 +\end{equation} +where $S$ is the radiated power per unit area ($W/m^2$), $T$ is the temperature (in Kelvins), and $\sigma=5.6703 +\times 10^{-8} W/m^2K^4$ is the Stefan's constant. + +Most hot, opaque objects can be approximated as blackbody emitters, but the most ideal blackbody is a closed volume (a cavity) with a very small hole in it. Any radiation entering the cavity is absorbed by the walls, and then is re-emitted out. Physicists first tried to calculate the spectral distribution of the radiation emitted from the ideal blackbody using {\it classical thermodynamics}. This method involved finding the number of modes of oscillation of the electromagnetic field in the cavity, with the energy per mode of oscillation given by $kT$. The classical theory gives the {\bf Rayleigh-Jeans law:} +\begin{displaymath} + u(\lambda,T) = \frac{8\pi kT}{\lambda^4} +\end{displaymath} +where $u(\lambda)(J/m^4)$ is the spectral radiance (energy radiated per unit area at a single wavelength or frequency), and $\lambda$ is the wavelength of radiation. This law agrees with the experiment for radiation at long wavelengths (infrared), but predicts that $u(\lambda)$ should increase infinitely at short wavelengths. This is not observed experimentally (Thank heaven, or we would all be constantly bathed in ultraviolet light-a true ultraviolet catastrophe!). It was known that the wavelength distribution peaked at a specific temperature as described by {\bf Wien's law:} +\begin{displaymath} + \lambda_{max}T = 2.898\times 10^{-3} m\cdot K +\end{displaymath} +and went to zero for short wavelengths. + + The breakthrough came when Planck assumed that the energy of the oscillation modes can only take on discrete values rather than a continuous distribution of values, as in classical physics. With this assumption, Planck's law was derived: +\begin{displaymath} +u(\lambda,T)=\frac{8\pi hc\lambda^{-5}}{e^{hc/\lambda kT}-1} +\end{displaymath} +where $c$ is the speed of light and $h=6.626076\times 10^{-34} J\cdot s$ is the Planck's constant. This proved to be the correct description. + + +\begin{boxedminipage}{\linewidth} +\textbf{Sometimes physicists have to have crazy ideas!} \\ +% +``\emph{The problem of radiation-thermodynamics was solved by Max Planck, who +was a 100 percent classical physicist (for which he cannot be blamed). It was +he who originated what is now known as {\it modern physics}. At the turn of the +century, at the December 14, 1900 meeting of the German Physical Society, +Planck presented his ideas on the subject, which were so unusual and so +grotesque that he himself could hardly believe them, even though they caused +intense excitement in the audience and the entire world of physics}'' + +From George Gamow, {\it ``Thirty Years that Shook Physics, The Story of Quantum Physics''}, Dover Publications, +New York, 1966.) +\end{boxedminipage} + + +\subsection*{Safety} +The Stefan lamp and thermal cube will get very hot - be careful!!! + +\section*{Thermal radiation rates from different surfaces} +\textbf{Equipment needed}: Pasco Radiation sensor, Pasco Thermal Radiation Cube, two multimeters, window glass. + +Before starting actual experiment take some time to have fun with the thermal radiation sensor. Can you detect your lab partner? What about people across the room? Point the sensor in different directions and see what objects affect the readings. \textbf{These exercises are fun, but you will also gain important intuition about various factors which may affect the accuracy of the measurements!} + + + +\begin{boxedminipage}{\linewidth} +\textbf{How does the radiation sensor work?} \\ +\vspace{0.25in} + +\includegraphics[height=2.5in]{./pdf_figs/thermopile} \\ + +\vspace{0.25in} + +Imagine a metal wire connected to a cold reservoir at one end and a hot reservoir at the other. Heat will flow between the ends of the wire, carried by the electrons in the conductor, which will tend to diffuse from the hot end to the cold end. Vibrations in the conductor's atomic lattice can also aid this process. This diffusion causes a potential difference between the two ends of the wire. The size of the potential difference depends on the temperature gradient and on details of the conductive material, but is typically in the few to few 10s of $\mu V/ K$. A thermocouple, shown on the left, consists of two different conductive materials joined together at one end and connected to a voltmeter at the other end. The potential is, of course, the same on either side of the joint, but the difference in material properties causes $\Delta V=V_1 - V_2 \neq 0$. This $\Delta V$ is measured by the voltmeter and is proportional to $\Delta T$. Your radiation sensor is a thermopile, simply a ``pile'' of thermocouples connected in series, as shown at the right. This is done to make the potential difference generated by the temperature gradient easier to detect. +\end{boxedminipage} + + +\begin{figure} +\includegraphics[height=2.5in]{./pdf_figs/bbx} +\caption{\label{bbx}Thermal Radiation Setup} +\end{figure} + +\begin{enumerate} +\item Connect the two multimeters and position the sensor as shown in Fig.~\ref{bbx}. The multimeter attached to the cube should be set to read resistance while the one attached to the infrared radiation sensor will monitor the potential (in the millivolt range). Make sure the shutter on the sensor is pushed all the way open! +\item Before turning on the cube, measure the resistance of the thermistor at room temperature, and obtain the room temperature from the instructor. You will need this information for the data analysis. +% +\item Turn on the Thermal Radiation Cube and set the power to ``high.'' When the ohmmeter reading decreases to 40 k$\Omega$ (5-20 minutes) set power switch to ``$8$''. (If the cube has been preheated, immediately set the switch to ``$8$''.) +\\ +\begin{boxedminipage}{\linewidth} +\textbf{Important}: when using the thermal radiation sensor, make each reading quickly to keep the sensor from heating up. Use both sheets of white isolating foam (with the silvered surface facing the lamp) to block the sensor between measurements. +\\ +\textbf{Sensor calibration}: To obtain the radiation sensor readings for radiated power per unit area $S$ in the correct units ($W/m^2$), you need to use the voltage-to-power conversion factor: $22~mV/mW$, and the area of the sensor $2mm\times2mm$: +\begin{displaymath} +S[W/m^2]=\frac{S[mV]}{22 [mV/mW]}\cdot 10^{-3}\cdot \frac{1}{4\cdot +10^{-6}[m^2]} +\end{displaymath} +\end{boxedminipage} +% +% +\item When the cube has reached thermal equilibrium the ohmmeter will be fluctuating around a constant value. Record the resistance of the thermistor in the cube and determine the approximate value of the temperature using data table in Fig~\ref{tcube}. Use the radiation sensor to measure the radiation emitted from the four surfaces of the cube. Place the sensor so that the posts on its end are in contact with the cube surface (this ensures that the distance of the measurement is the same for all surfaces) and record the sensor reading. Each lab partner should make an independent measurement. + +\item Place the radiation sensor approximately 5~cm from the black surface of the radiation cube and record its reading. Place a piece of glass between the sensor and the cube. Record again. Does window glass effectively block thermal radiation? Try observing the effects of other objects, recording the sensor reading as you go. + +\end{enumerate} + +%Below is a possible way to record the results of these measurements in your lab +%book. Don't forget to specify the uncertainties of your measurements! +% +% \noindent\\ +%\begin{tabular}{lr} +%Thermistor reading (room temperature)=&$\underline{\hskip 1in}\Omega$\\ +% T (room temperature)=&$\underline{\hskip 1in}K$ +%\end{tabular} +% +%\begin{tabular}{|p{13mm}|p{13mm}|p{13mm}|p{13mm}|p{13mm}|p{13mm}|p{13mm}| +%p{13mm}|} +%\hline +%\multicolumn{2}{|l|} +%{\bf Power=5.0}&\multicolumn{2}{|l|}{\bf Power=6.5} &\multicolumn{2}{|l|} +%{\bf Power=8.0}&\multicolumn{2}{|l|}{\bf Power= ``High''}\\ \hline +%\multicolumn{2}{|l|}{Th. Res. $\underline{\hskip .4in}\Omega$}& +%\multicolumn{2}{|l|}{Th. Res. $\underline{\hskip .4in}\Omega$}& +%\multicolumn{2}{|l|}{Th. Res. $\underline{\hskip .4in}\Omega$}& +%\multicolumn{2}{|l|}{Th. Res. $\underline{\hskip .4in}\Omega$}\\ \hline +%\multicolumn{2}{|l|}{Temp. $\underline{\hskip .5in}K$} +%&\multicolumn{2}{|l|}{Temp. $\underline{\hskip .5in}K$} +%&\multicolumn{2}{|l|}{Temp. $\underline{\hskip .5in}K$} +%&\multicolumn{2}{|l|}{Temp. $\underline{\hskip .5in}K$}\\ \hline +%{\bf Surface}&{\bf Sensor Reading (mV)}& +%{\bf Surface}&{\bf Sensor Reading (mV)}& +%{\bf Surface}&{\bf Sensor Reading (mV)}& +%{\bf Surface}&{\bf Sensor Reading (mV)}\\\hline +%Black&&Black&&Black&&Black&\\\hline +%White&&White&&White&&White&\\\hline +%Polished Aluminum&&Polished Aluminum&&Polished Aluminum&&Polished Aluminum&\\ +%\hline +%Dull Aluminum&&Dull Aluminum&&Dull Aluminum&&Dull Aluminum&\\ \hline +%\end{tabular} + +%Plot the measured radiated power as function of temperature for different +%surfaces. +Use your data to address the following questions in your lab report: +\begin{enumerate} +\item Is it true that good absorbers of radiation are good emitters? +\item Is the emission from the black and white surface similar? +\item Do objects at the same temperature emit different amounts of radiation? +\item Does glass effectively block thermal radiation? Comment on the other objects that you tried. +\end{enumerate} + + +\begin{figure} +\includegraphics[height=4in]{./pdf_figs/tcube} +\caption{\label{tcube}Resistance vs temperature for the Thermal Radiation Cube} +\end{figure} +\begin{figure} +\includegraphics[height=3.5in]{./pdf_figs/bbht} +\caption{\label{bbht}Lamp Connection for High-Temperature Stefan-Boltzmann +Setup} +\end{figure} + +\subsection*{Tests of the Stefan-Boltzmann Law} +\subsubsection*{ High temperature regime} +\textbf{Equipment needed}: Radiation sensor, 3 multimeters, Stefan-Boltzmann +Lamp, Power supply. + +\begin{enumerate} +\item \textbf{Before turning on the lamp}, measure the resistance of the filament of the Stefan-Boltzmann lamp at room temperature. Record the room temperature, visible on the wall thermostat and on the bench mounted thermometers in the room. + +% +%\begin{tabular}{lr} +% T (room temperature)=&$\underline{\hskip .7in}K$\\ +% Resistance of filament (room temperature)=&$\underline{\hskip .7in}$ +%\end{tabular} + +\item Set up the equipment as shown in +Fig. \ref{bbht}. VERY IMPORTANT: make all connections to the lamp when the power is off. Turn the power off before changing/removing connections. The voltmeter should be directly connected to the binding posts of the Stefan-Boltzmann lamp. NOTE: even through the power supply displays both current and voltage, it is more accurate to make these measurements using two independent multimeters. +% + +\item Place the thermal sensor at the same height as the filament, with the front face of the sensor approximately 6 cm away from the filament (this distance will be fixed throughout the measurement). Make sure no other objects are viewed by the sensor other than the lamp. +% +\item Turn on the lamp power supply. Set the voltage, $V$, in steps of one volt from 1-12 volts. At each $V$, record the ammeter (current) reading from the lamp and the voltage from the radiation sensor. Calculate the resistance of the lamp using Ohm's Law and determine the temperature $T$ of the lamp from the table shown in Fig. \ref{w_res:fig}. + +\begin{figure} +\includegraphics[height=2.5in]{./pdf_figs/w_res} +\caption{\label{w_res:fig}Table of Tungsten's Resistance as a function of temperature.} +\end{figure} + +\end{enumerate} + +Sample table for experimental data recording: + +\begin{tabular}{|p{20mm}|p{20mm}|p{20mm}|p{20mm}|p{20mm}|p{20mm}|}\hline +\multicolumn{3}{|c|}{Data($\pm$ error)}& \multicolumn{3}{|c|}{Calculations}\\ +\hline $V$((volts)&$I$(amps)&$Rad(mV)$&$R$(ohms)&$T(K)$&$T^4(K^4)$\\\hline +1.00&&&&&\\\hline \dots &&&&&\\\hline&&&&&\\\hline +\end{tabular} + +In the lab report plot the reading from the radiation sensor (convert to $W/m^2$) versus $T^4$. According to the Stefan-Boltzmann Law, the data should fall along a straight line. Do a fit and report the value of the slope that you obtain. How does it compare to the accepted value of Stefan's constant? + +Don't be alarmed if the value of slope is way off from Stefan's constant. The Stefan-Boltzmann Law, as stated in Eq.(\ref{SBl}), is only true for ideal black bodies. For other objects, a more general law is: $S=A\sigma T^4$, where A is the absorptivity. $A=1$ for a perfect blackbody. $A<1$ means the object does not absorb (or emit) all the radiation incident on it (this object only radiates a fraction of the radiation of a true blackbody). The material lampblack has $A=0.95$ while tungsten wire has $A=0.032$ (at $30^{\circ} C$) to 0.35 (at $3300^{\circ}C$). Comparing your value of slope to Stefan's constant, and assuming that the Stefan-Boltzmann Law is still valid, what do you obtain for $A$? Is it consistent with tungsten? What else could be affecting this measurement? + + +\section*{Test of the inverse-square law} +\textbf{Equipment needed}: Radiation sensor, Stefan-Boltzmann lamp, multimeter, +power supply, meter stick. +\begin{figure} +\includegraphics[height=2.5in]{./pdf_figs/bb31} +\caption{\label{bb31}Inverse Square Law Setup} +\end{figure} +A point source of radiation emits that radiation according to an inverse square +law: that is, the intensity of the radiation in $(W/m^2)$ is proportional to +the inverse square of the distance from that source. You will determine if this +is true for a lamp. + +\begin{enumerate} +\item Set up the equipment as shown in Fig. \ref{bb31}. Tape the meter stick to the table. Place the Stefan-Boltzmann lamp at one end, and the radiation sensor in direct line on the other side. The zero-point of the meter stick should align with the lamp filament (or, should it?). Adjust the height of the radiation sensor so it is equal to the height of the lamp. Align the system so that when you slide the sensor along the meter stick the sensor still aligns with the axis of the lamp. Connect the multimeter (reading millivolts) to the sensor and the lamp to the power supply. +\item With the {\bf lamp off}, slide the sensor along the meter stick. Record the reading of the voltmeter at 10 cm intervals. Average these values to determine the ambient level of thermal radiation. You will need to subtract this average value from your measurements with the lamp on. +\item Turn on the power supply to the lamp. Set the voltage to approximately 10 V. {\bf Do not exceed 13 V!} Adjust the distance between the sensor and lamp from 2.5-100 cm and record the sensor reading. \textbf{Before actual experiment think carefully about at what distances you want to take the measurements. Is taking them at constant intervals the optimal approach? At what distances you expect the sensor reading change more rapidly?} + + \item Make a plot of the corrected radiation measured from the lamp versus the inverse square of the distance from the lamp to the sensor $(1/x^2)$ and do a linear fit to the data. How good is the fit? Is this data linear over the entire range of distances? Comment on any discrepancies. What is the uncertainty on the slope? What intercept do you expect? Comment on these values and their uncertainties? + +\item Does radiation from the lamp follow the inverse square law? Can the lamp be considered a point source? If not, how could this affect your measurements? + +\end{enumerate} +% +%\item The Blackbody Spectrum Equipment: Spectrometer, computer, high +%temperature source. +%\begin{itemize} +% +%\item There are two spectra at the end of this lab. +%From Wien's law, estimate the temperature of the sources. +%\begin{displaymath} +%T_{ estimate}=\underline{\hskip .75in}K +%\end{displaymath} +%\begin{displaymath} +%T_{ estimate}=\underline{\hskip .75in}K +%\end{displaymath} +% +% +%\item On a separate graph, plot the expected spectrum +%from the Rayleigh-Jeans law and Planck's law. Which law best +%represents the spectrum acquired above? +%\end{itemize} +% +%Addendum to Blackbody Radiation Handout +%\section*{Thermal Radiation rates from Different Surfaces} +% +%\begin{itemize} +%\item Make sure the shutter on the Sensor is pushed all the +%way open! +%\item Make sure the temperature is stable when you begin readings from +%surfaces of Cube (you may have to wait at least 5 minutes between +%temperature changes). +%\item Make the readings quickly! +%\end{itemize} +% +% +% +%\section*{Inverse Square Law} +% +%Make sure that you keep the Sensor in line with the filament as you +%slide it (and you do not introduce an angle). +% + + + + +%\subsection*{The Blackbody Spectrum} +% +%In Fig. \ref{Sun} is an approximate spectrum of the sun. Determine +%the approximate temperature of the sun from this spectrum using Wien's +%law. $T=\underline{\hskip .75in}$. +%In Fig. \ref{Cobe} is the spectrum for the microwave background +%assumed to arise from the time when the photons decoupled from the +%charged particles, i.e., when most free electrons became bound. +%Determine the temperature of the microwave background. Note that what +%is plotted is waves/cm, not cm. If the present size of the visible +%universe is $13\cdot 10^9$ light years. How large was the visible +%universe when the decoupling took place. Hint: $R_u$ has experienced the +%same expansion as the wavelength and the temperature at decoupling must +%correspond to about 10 eV. +%\begin{itemize} +%\item Extra credit: what is the accepted value +%for the +%temperature of the surface of the sun? +%\item How does your extracted value +%compare? +%\item Include a copy of this spectrum in your lab report. +%On a +%separate graph, plot Planck's Law and the Rayleigh-Jeans Law for this +%same temperature. +%\item Which law does the solar spectrum appear to behave? +%\end{itemize} + + +% +% +% +%\begin{figure} +%\includegraphics[height=2.5in]{LinearSp.eps} +%\caption{\label{Sun}Approximate Sun Spectrum} +%\end{figure} +%\begin{figure} +%\includegraphics[height=2.5in]{cobespc.eps} +%\caption{\label{Cobe}Cobe: Cosmic Black Body Spectrum} +%\end{figure} +%\begin{figure} +% +%\newpage + +%\end{document} diff --git a/manual_source/chapters/ediffract.tex b/manual_source/chapters/ediffract.tex new file mode 100644 index 0000000..d03a1d5 --- /dev/null +++ b/manual_source/chapters/ediffract.tex @@ -0,0 +1,268 @@ +%\chapter*{Electron Diffraction} +%\addcontentsline{toc}{chapter}{Electron Diffraction} + +%\documentclass{article} +%\usepackage{tabularx,amsmath,boxedminipage,epsfig} +% \oddsidemargin 0.0in +% \evensidemargin 0.0in +% \textwidth 6.5in +% \headheight 0.0in +% \topmargin 0.0in +% \textheight=9.0in + +%\begin{document} +\chapter{Electron Diffraction} +\setcounter{figure}{1} +\setcounter{table}{1} +\setcounter{equation}{1} + +%\date {} +%\maketitle \noindent + + +\textbf{Experiment objectives}: observe diffraction of the beam of electrons on a graphitized carbon target, and to calculate the intra-atomic spacings in the graphite. + +\section*{History} + + A primary tenet of quantum mechanics is the wavelike + properties of matter. In 1924, graduate student Louis de + Broglie suggested in his dissertation that since light has + both particle-like {\bf and} wave-like properties, perhaps all + matter might also have wave-like properties. He postulated + that the wavelength of objects was given by $\lambda = h/p$, where where $h$ is Planck's constant, and $p = + mv$ is the momentum. {\it This was quite a revolutionary idea}, + since there was no evidence at the time that matter behaved + like waves. In 1927, however, Clinton Davisson and Lester + Germer discovered experimental proof of the wave-like + properties of matter- particularly electrons (This discovery + was quite by mistake!). They were studying electron reflection + from a nickel target. They inadvertently crystallized their + target while heating it, and discovered that the scattered + electron intensity as a function of scattering angle showed + maxima and minima. That is, electrons were ``diffracting'' from + the crystal planes much like light diffracts from a grating, + leading to constructive and destructive interference. Not only + was this discovery important for the foundation of quantum + mechanics (Davisson and Germer won the Nobel Prize for their + discovery), but electron diffraction is an extremely important + tool used to study new materials. In this lab you will study + electron diffraction from a graphite target, measuring the + spacing between the carbon atoms. + + +\begin{figure}[h] +\centering +\includegraphics[width=\textwidth]{./pdf_figs/ed1} \caption{\label{ed1}Electron +Diffraction from atomic layers in a crystal.} +\end{figure} +\section*{Theory} + Consider planes of atoms in a {\bf crystal} as shown in Fig, \ref{ed1} +separated by distance $d$. Electron "waves" reflect from each of these planes. +Since the electron is wave-like, the combination of the reflections from each +interface will lead to an interference pattern. This is completely analogous to +light interference, arising, for example, from different path lengths in the +Fabry-Perot or Michelson interferometers. The de Broglie wavelength for the +electron is given by: $\lambda=h/p$, where $p$ can be calculated by knowing the +energy of the electrons when they leave the ``electron gun'': +\begin{equation}\label{Va} +\frac{p^2}{2m}=eV_a, +\end{equation} + where $V_a$ is the accelerating potential. The condition for constructive +interference is that the path length difference for the two waves +shown in Fig. \ref{ed1} be a multiple of a wavelength. This leads to Bragg's +Law: +\begin{equation}\label{bragg} +n\lambda=2d\sin\theta +\end{equation} +where $n = 1,2,\dots$ is integer. In this experiment, only the first order +diffraction $n=1$ is observed. Therefore, the intra-atomic distance in a +crystal can be calculated by measuring the angle of electron diffraction and +their wavelength (\emph{i.e.} their momentum): +\begin{equation}\label{bragg1} +d=\frac{\lambda}{2\sin\theta} = \frac{1}{2\sin\theta}\frac{h}{\sqrt{2em_eV_a}} +\end{equation} +\noindent +where $h$ is Planck's constant, $e$ is the electronic charge, $m_e$ is the +electron's mass, and $V_a$ is the accelerating voltage. +% +% +% +% +% Knowing $\lambda$ and the angles $\theta$ for which +%constructive interference occurs, the atomic spacing $d$ can be +%extracted. + +\section*{Experimental Procedure} + +\textbf{Equipment needed}: Electron diffraction apparatus and power supply, tape, ruler. + +\subsection*{Safety} +You will be working with high voltage. Power supply will be connected for you, +but inspect the apparatus when you arrive \textbf{before} turning the power on. +In any wires are unplugged, ask an instructor to reconnect them. Also, +\textbf{before} turning the power on, identify the high voltage contacts on the +electron diffraction tube, make sure these connections are well-protected and +cannot be touched by accident while taking measurements. +\begin{figure}[h] +\centering +\includegraphics[width=6in]{./pdf_figs/ed2} \caption{\label{ed2}Electron Diffraction Apparatus.} +\end{figure} +\subsection*{Setup} + +The diagram of the apparatus is given in Fig.\ref{ed2}. An electron gun +(consisting of a heated filament to boil electrons off a cathode and an anode +to accelerate them to, similar to the e/m experiment) ``shoots'' electrons at a +carbon (graphite) target. + +The electrons diffract from the carbon target and the resulting interference +pattern is viewed on a phosphorus screen. + + The graphitized carbon is not completely crystalline but consists of crystal +sheets in random orientations. Therefore, the constructive interference +patterns will be seen as bright circular rings. For the carbon target, two +rings (an outer and inner, corresponding to different crystal planes) will be +seen, corresponding to two spacings between atoms in the graphite arrangement, +see Fig. \ref{ed3} +\begin{figure} +\centering +\includegraphics[width=4in]{./pdf_figs/ed3} \caption{\label{ed3}Spacing of +carbon atoms. Here subscripts \textit{10} and \textit{11} correspond to the +crystallographic directions in the graphite crystal.} +\end{figure} + +\subsection*{Data acquisition} +Acceptable power supplie settings: +\\\begin{tabular}{lll} +Filament Voltage& $V_F$&6.3 V ac/dc (8.0 V max.)\\ +Anode Voltage & $V_A$& 1500 - 5000 V dc\\ +Anode Current & $I_A$& 0.15 mA at 4000 V ( 0.20 mA max.) +\end{tabular} + +\begin{enumerate} +\item Switch on the heater and wait one minute for the oxide cathode to achieve thermal stability. +\item Slowly increase $V_a$ until you observe two rings to appear around the direct beam. +Slowly change the voltage and determine the highest achievable accelerating +voltage, and the lowest voltage when the rings are visible. +\item Measure the diffraction angle $\theta$ for both inner and outer rings for 5-10 voltages from that range, +using the same masking tape (see procedure below). Each lab partner should +repeat these measurements (using an individual length of the masking tape). +\item Calculate the average value of $\theta$ from the individual measurements for each +voltage $V_a$. Calculate the uncertainties for each $\theta$. +\end{enumerate} + +\textbf{Measurement procedure for the diffraction angle $\theta$} + +To determine the crystalline structure of the target, one needs to carefully +measure the diffraction angle $\theta$. It is easy to see (for example, from +Fig.~\ref{ed1} ) that the diffraction angle $\theta$ is 1/2 of the angle between the beam incident on the target and the diffracted beam to a ring, hence the $2\theta$ appearing in Fig.~\ref{ed4}. You are going to determine the diffraction angle $\theta$ for a given accelerated voltage from the simple geometrical ratio +\begin{equation} +L\sin{2\theta} = R\sin\phi, +\end{equation} +where the distance between the target and the screen $L = 0.130$~m is controlled during the production process to have an accuracy better than 2\%. {\it Note, this means that the electron tubes are not quite spherical.} + +The ratio between the arc length and the distance between the target and the +radius of the curvature of the screen $R = 0.066$~m gives the angle $\phi$ in +radian: $\phi = s/2R$. To measure $\phi$ carefully place a piece of masking tape on the tube so that it crosses the ring along the diameter. Mark the position of the ring for each accelerating voltage, and then remove the masking tape and measure the arc length $s$ corresponding to each ring. You can also make these markings by using the thin paper which cash register reciepts are printed on. + + +\begin{figure} +\centering +\includegraphics[width=4in]{./pdf_figs/edfig4} \caption{\label{ed4}Geometry of the experiment.} +\end{figure} + +\section*{Data analysis} + +Use the graphical method to find the average values for the distances between +the atomic planes in the graphite crystal $d_{11}$ (outer ring) and $d_{10}$ +(inner ring). To determine the combination of the experimental parameters that +is proportional to $d$, one need to substitute the expression for the +electron's velocity Eq.(\ref{Va}) into the diffraction condition given by +Eq.(\ref{bragg}): +\begin{equation}\label{bragg} +2d\sin\theta=\lambda=\frac{h}{\sqrt{2m_ee}}\frac{1}{\sqrt{V_a}} +\end{equation} + +Make a plot of $1/\sqrt{V_a}$ versus $\sin\theta$ for the inner and outer rings +{both curves can be on the same graph). Fit the linear dependence and measure +the slope for both lines. From the values of the slope find the distance +between atomic layers $d_{inner}$ and $d_{outer}$. + +Compare your measurements to the accepted values : $d_{inner}=d_{10} = .213$~nm +and $d_{outer}=d_{11}=0.123$~nm. + + \fbox{Looking with Electrons} + +\noindent +\begin{boxedminipage}{\linewidth} The resolution of ordinary optical microscopes +is limited (the diffraction limit) by the wavelength of light ($\approx$ 400 +nm). This means that we cannot resolve anything smaller than this by looking at +it with light (even if we had no limitation on our optical instruments). Since +the electron wavelenght is only a couple of angstroms ($10^{-10}$~m), with +electrons as your ``light source'' you can resolve features to the angstrom +scale. This is why ``scanning electron microscopes'' (SEMs) are used to look at +very small features. The SEM is very similar to an optical microscope, except +that ``light'' in SEMs is electrons and the lenses used are made of magnetic +fields, not glass. +\end{boxedminipage} + +%\begin{tabular}{lll} +%Filament Voltage& $V_F$&6.3 V ac/dc (8.0 V max.)\\ +%Anode Voltage & $V_A$& 2500 - 5000 V dc\\ +%Anode Current & $I_A$& 0.15 mA at 4000 V ( 0.20 mA max.) +%\end{tabular} +% +%\newpage +%\noindent +%Section: $\underline{\hskip 1in}$\\ +%Name: $\underline{\hskip 1in}$\\ +%Partners: $\underline{\hskip 1in}$\\ +%$\hskip 1in\underline{\hskip 1in}$\\ +%\section*{Electron Diffraction} +%Step through the electron accelerating potential from 5 kV to 2 kV in +%steps of 0.5 kV. Record $V_a$ (from needle reading on power supply) and +%measure the diameter of the two rings. In measuring the diameter, +%either try to pick the middle of the ring, or measure the inside and +%outside of that ring +%and average. Each lab partner should measure the diameter, and then average +%the result. {\bf Make sure you give units in the following tables} +% +%\begin{center} +%{\bf Final data -- fill out preliminary chart on next page first} +%\end{center} +% +%\begin{tabular}{|p{20mm}|p{20mm}|p{20mm}|p{20mm}|p{20mm}|}\hline +%$V_a$&(inner s)&$d_{inner}$&outer s &$d_{outer}$\\\hline +%&&&&\\\hline +%&&&&\\\hline +%&&&&\\\hline +%&&&&\\\hline +%&&&&\\\hline +%&&&&\\\hline +%\multicolumn{2}{|l|}{Average $d_{inner}\pm\sigma=$} +%&\multicolumn{2}{|l}{Average $d_{outer}\pm\sigma=$} +%&\multicolumn{1}{l|}{}\\\hline +%\end{tabular} +% +% +% +% +%Make a plot of $1/sqrt{V_a}$ versus $s$ for the inner and outer rings +%{both curves can be on the same graph). +% +%Compare your measurements to the known spacing below.$d_{outer}=$\\ +%True values are: $d_{inner}=.213 nm$ and $d_{outer}=0.123 nm $. +% +%\begin{boxedminipage}{\linewidth} +% +% +%Turn in this whole stapled report, including your data tables. Attach your +%plot(s) at the end. +%\end{boxedminipage} +%\newpage +%\begin{boxedminipage}{\linewidth} +%Note: start with the External Bias at 30 V. Decrease the bias if you need +%to, to increase the intensity of the rings. DO NOT EXCEED 0.2 mA on ammeter! +%\end{boxedminipage} + +%\end{document} +%\newpage diff --git a/manual_source/chapters/emratio.tex b/manual_source/chapters/emratio.tex new file mode 100644 index 0000000..04cdb4d --- /dev/null +++ b/manual_source/chapters/emratio.tex @@ -0,0 +1,267 @@ +%\chapter*{Measurement of Charge-to-Mass (e/m) Ratio for the Electron} +%\addcontentsline{toc}{chapter}{Charge-to-Mass Ratio} +%\documentclass{article} +%\usepackage{tabularx,amsmath,boxedminipage,epsfig} +% \oddsidemargin 0.0in +% \evensidemargin 0.0in +% \textwidth 6.5in +% \headheight 0.0in +% \topmargin 0.0in +% \textheight=9.0in + +%\begin{document} +\chapter{Measurement of Charge-to-Mass (\emph{e/m}) Ratio for the Electron} +\setcounter{figure}{1} +\setcounter{table}{1} +\setcounter{equation}{1} + +%\date {} +%\maketitle \noindent + +\textbf{Experiment objectives}: measure the ratio of the electron charge-to-mass ratio $e/m$ by studying the electron trajectories in a uniform magnetic field. + +\subsection*{History} + J.J. Thomson first measured the charge-to-mass ratio of the + fundamental particle of charge in a cathode ray tube in + 1897. A cathode ray tube basically consists of two metallic + plates in a glass tube which has been evacuated and filled + with a very small amount of background gas. One plate is + heated (by passing a current through it) and ``particles'' boil + off of the cathode and accelerate towards the other plate + which is held at a positive potential. The gas in between the + plates inelastically scatters the electrons, emitting light + which shows the path of the particles. The charge-to-mass + (\emph{e/m}) ratio of the particles can be measured by observing + their motion in an applied magnetic field. Thomson repeated + his measurement of \emph{e/m} many times with different metals for + cathodes and also different gases. Having reached the same + value for \emph{e/m} every time, it was concluded that a fundamental + particle having a negative charge \emph{e} and a mass 2000 times less + than the lightest atom existed in all atoms. Thomson named + these particles ``corpuscles'' but we now know them as + electrons. In this lab you will essentially repeat Thomson's + experiment and measure \emph{e/m} for electrons. + + +\section*{Theory} + The apparatus shown in Figure \ref{emfig1}. +consists of a glass tube that + houses a small electron gun. This gun has a cathode filament + from which electrons can be thermionically released (boiled + off), and a nearby anode which can be set to a potential which + is positive relative to the cathode. Electrons boiled off the + cathode are accelerated to the anode, where most are + collected. The anode contains a slit, however, which lets a + fraction of the electrons into the larger volume of the glass + tube. Some of these electrons scatter inelastically with the + background gas, thereby emitting tracer light to define the + path of the electrons. + +\begin{figure}[h] +\centering +\includegraphics[height=2.5in]{./pdf_figs/emfig1} +\caption{\label{emfig1}The schematic for the $e/m$ apparatus.} +\end{figure} + + To establish the uniform magnetic field a pair of circular + Helmholtz coils are wound and the tube centered in the volume + of the coils (see Appendix). The tube is oriented so that the beam which + exits the electron gun is traveling perpendicular to the + Helmholtz field. {\it We would like the field to be uniform, i.e., the same, over + the orbit of the deflected electrons to the level of 1\% if + possible}. + +An electron released thermionically at the cathode has on the order of 1~eV of +kinetic energy. This electron ``falls'' through the positive anode potential +$V_a$ , gaining a kinetic energy of: +\begin{equation}\label{emeq1} +\frac{1}{2}mv^2=eV_a +\end{equation} +The magnetic field of the Helmholtz coils is perpendicular to this velocity, +and produces a magnetic force which is transverse to both ${\mathbf v}$ and +${\mathbf B}$: ${\mathbf F} = e{\mathbf v \times B}$. This centripetal force +makes an electron move along the circular trajectory; the radius of this +trajectory $r$ can be found from Newton's Second Law: +\begin{equation}\label{emeq3} +m\left(\frac{v^2}{r}\right)=evB +\end{equation} +From this equation we obtain the expression for the charge-to-mass ratio of +the electron, expressed through the experimental parameters: +\begin{equation}\label{emeq4} +\frac{e}{m} = \frac{v}{rB} +\end{equation} + +We shall calculate magnetic field $B$ using the Biot-Savart law for the two +current loops of the Helmholtz coils (see Appendix): +\begin{equation}\label{emeq6} +B=\frac{8}{\sqrt{125}}\frac{\mu_0NI_{hc}}{a}. +\end{equation} +Here $N$ is the number of turns of wire that form each loop, $I_{hc}$ is the +current (which is the same in both loops), $a$ is the radius of the loops (in +meters), and the magnetic permeability constant is $\mu_0=4\pi 10^{-7} +T~m/A$). + +Noting from Eq.(\ref{emeq1}) that the velocity is determined by the potential +$V_a$ as $v=\sqrt{2eV_a/m}$, and using Eq.(\ref{emeq6}) for the magnetic field +$B$, we get: + +\begin{equation}\label{emeq7} +\frac{e}{m}= \frac{2V_a}{r^2B^2} +=\frac{125}{32}V_a\frac{1}{(\mu_0NI_{hc})^2}\frac{a^2}{r^2} +\end{equation} + +\begin{boxedminipage}{\linewidth} +The accepted value for the charge-to-mass ratio of the electron is $e/m = 1.7588196 \cdot 10^{11}$ C/kg. +\end{boxedminipage} + + +\section*{Experimental Procedure} + +\textbf{Equipment needed}: Pasco $e/m$ apparatus (SE-9638), Pasco High Voltage +Power supply (for the accelerating voltage and the filament heater), GW power +supply (for the Helmholtz coils), two digital multimeters. +\begin{figure}[h] +\centering +\includegraphics[width=0.8\columnwidth]{./pdf_figs/emratioFig2} +\caption{\label{emfig2}(a) $e/m$ tube; (b) electron gun.} +\end{figure} + +\subsection*{Pasco SE-9638 Unit:} + +The $e/m$ tube (see Fig.~\ref{emfig2}a) is filled with helium at a pressure of +$10^{-2}$~mm Hg, and contains an electron gun and deflection plates. The +electron beam leaves a visible trail in the tube, because some of the electrons +collide with helium atoms, which are excited and then radiate visible light. +The electron gun is shown in Fig.~\ref{emfig2}b. The heater heats the cathode, +which emits electrons. The electrons are accelerated by a potential applied +between the cathode and the anode. The grid is held positive with respect to +the cathode and negative with respect to the anode. It helps to focus the +electron beam. + + +The Helmholtz coils of the $e/m$ apparatus have a radius and separation of +$a=15$~cm. Each coil has $N=130$~turns. The magnetic field ($B$) produced by +the coils is proportional to the current through the coils ($I_{hc}$) times +$7.80\cdot10^{-4}$~tesla/ampere [$B (tesla) = (7.80 \cdot 10^{-4}) I_{hc}$]. A +mirrored scale is attached to the back of the rear Helmholtz coil. It is +illuminated by lights that light automatically when the heater of the electron +gun is powered. By lining the electron beam up with its image in the mirrored +scale, you can measure the radius of the beam path without parallax error. The +cloth hood can be placed over the top of the $e/m$ apparatus so the experiment +can be performed in a lighted room. + + +\subsection*{Safety} +You will be working with high voltage. Make all connections when power is off. +Turn power off before changing/removing connections. Make sure that there is no +loose or open contacts. + +\subsection*{Set up} +The wiring diagram for the apparatus is shown in Fig.~\ref{emfig3}. +\textbf{Important: Do not turn any equipment until an instructor have checked +your wiring.} +\begin{figure}[h] +\centering +\includegraphics[width=0.7\columnwidth]{./pdf_figs/emratioFig3} +\caption{\label{emfig3}Connections for e/m Experiment.} +\end{figure} + +Acceptable power supplies settings: +\begin{description} +\item[Electron Gun/filament Heater] $6$~V AC. Do not go to \unit[7]{V}! +\item[Electrodes] $150$ to $300$~V DC +\item[Helmholtz Coils] $6-9$~V DC. +\end{description} +\textbf{Warning}: The voltage for a filament heater should \textbf{never} exceed 6.3 VAC. Higher values can burn out filament. \\The Helmholtz current should NOT exceed 2 amps. To avoid accidental overshoot run the power supply at a ``low'' setting in a \textit{constant current} mode and ask the TA or instructor how to set the current limit properly. + +\subsection*{Data acquisition} +\begin{enumerate} + +\item Slowly turn the current adjust knob for the Helmholtz coils clockwise. +Watch the ammeter and take care that the current is less than 2 A. + +\item Wait several minutes for the cathode to heat up. When it does, you +will see the electron beam emerge from the electron gun. Its trajectory be +curved by the magnetic field. + +\item Rotate the tube slightly if you see any spiraling of the beam. +Check that the electron beam is parallel to the Helmholtz coils. If it is not, +turn the tube until it is. Don't take it out of its socket. As you rotate the +tube, the socket will turn. + +\item Measurement procedure for the radius of the electron beam $r$:\\ +For each measurement record: +\begin{description} +\item[Accelerating voltage $V_a$] +\item[Current through the Helmholtz coils $I_{hc}$] +\end{description} +Look through the tube at the electron beam. To avoid parallax errors, move your head to align one side the electron beam ring with its reflection that you can see on the mirrored scale. Measure the radius of the beam as you see it, then repeat the measurement on the other side, then average the results. Each lab partner should repeat this measurement, and estimate the uncertainty. Do this silently and tabulate results. After each set of measurements (e.g., many values of $I_{hc}$ at one value of $V_a$) compare your results. This sort of procedure helps reduce group-think, can get at some sources of systematic errors, and is a way of implementing experimental skepticism. + +\item Repeat the radius measurements for at least 4 values of $V_a$ and for each $V_a$ for 5-6 different values of the magnetic field. + +\end{enumerate} + +\subsection*{Improving measurement accuracy} +\begin{enumerate} + +\item The greatest source of error in this experiment is the velocity of the +electrons. First, the non-uniformity of the accelerating field caused by the +hole in the anode causes the velocity of the electrons to be slightly less than +their theoretical value. Second, collisions with the helium atoms in the tube +further rob the electrons of their velocity. Since the equation for $e/m$ is +proportional to $1/r^2$, and $r$ is proportional to $v$, experimental values +for $e/m$ will be greatly affected by these two effects. + +\item To minimize the error due to this lost electron velocity, measure radius +to the outside of the beam path. + +\item To minimize the relative effect of collisions, keep the accelerating +voltage as high as possible. (Above $250$~V for best results.) Note, however, +that if the voltage is too high, the radius measurement will be distorted by +the curvature of the glass at the edge of the tube. Our best results were made +with radii of less than $5$~cm. + +\item Your experimental values will be higher than theoretical, due to the fact that both major sources of error +cause the radius to be measured as smaller than it should be. + +\end{enumerate} + + +\subsection*{Calculations and Analysis:} +\begin{enumerate} + \item Calculate $e/m$ for each of the +readings using Eq. \ref{emeq7}. NOTE: Use MKS units for +calculations. +\item For each of the four $V_a$ settings calculate the mean +$<e/m>$, the standard deviation $\sigma$ and {\bf the standard error in the +mean $\sigma_m$.} +Are these means consistent with one another sufficiently that you can +combine them ? [Put quantitatively, are they within $2 \sigma$ of each + other ?] +\item Calculate the {\bf grand mean} for all $e/m$ readings, its +standard deviation $\sigma$ {\bf and the standard error in the grand mean +$\sigma_m$}. +\item Specify how this grand mean compares to the accepted value, i.e., how many $\sigma_m$'s is it from the accepted value ? + +\item Finally, plot the data in the following way which should, ( according to Eq. \ref{emeq7}), reveal a linear relationship: plot $V_a$ on the abscissa [x-axis] versus $r^2 B^2/2$ on the ordinate [y-axis]. The uncertainty in $r^2 B^2/2$ should come from the standard deviation of the different measurements made by your group at the fixed $V_a$. The optimal slope of this configuration of data should be $<m/e>$. Determine the slope from your plot and its error by doing a linear fit. What is the value of the intercept? What should you expect it to be? + +\item Comment on which procedure gives a better value of $<e/m>$ (averaging or linear plot). +\end{enumerate} + +\section*{Appendix: Helmholtz coils} + +The term Helmholtz coils refers to a device for producing a region of nearly uniform magnetic field. It is named in honor of the German physicist Hermann von Helmholtz. A Helmholtz pair consists of two identical coils with electrical current running in the same direction that are placed symmetrically along a common axis, and separated by a distance equal to the radius of the coil $a$. The magnetic field in the central region may be calculated using the Bio-Savart law: +\begin{equation}\label{emeq_apx} +B_0=\frac{\mu_0Ia^2}{(a^2+(a/2)^2)^{3/2}}, +\end{equation} +where $\mu_0$ is the magnetic permeability constant, $I$ is the total electric current in each coil, $a$ is the radius of the coils, and the separation between the coils is equal to $a$. + +This configuration provides very uniform magnetic field along the common axis of the pair, as shown in Fig.~\ref{emfig4}. The correction to the constant value given by Eq.(\ref{emeq_apx}) is proportional to $(x/a)^4$ where $x$ is the distance from the center of the pair. However, this is true only in the case of precise alignment of the pair: the coils must be parallel to each other! +\begin{figure}[h] +\centering +\includegraphics[width=0.6\columnwidth]{./pdf_figs/emratioFig4} +\caption{\label{emfig4}Dependence of the magnetic field produced by a Helmholtz +coil pair $B$ of the distance from the center (on-axis) $x/a$. The magnetic +field is normalized to the value $B_0$ in the center.} +\end{figure} + diff --git a/manual_source/chapters/fabry-perot.tex b/manual_source/chapters/fabry-perot.tex new file mode 100644 index 0000000..d74745f --- /dev/null +++ b/manual_source/chapters/fabry-perot.tex @@ -0,0 +1,298 @@ +%\chapter*{Fabry-Perot Interferometer and the Sodium Doublet} +%\addcontentsline{toc}{chapter}{Fabry-Perot Interferometer} +\documentclass{article} +\usepackage{tabularx,amsmath,boxedminipage,epsfig} + \oddsidemargin 0.0in + \evensidemargin 0.0in + \textwidth 6.5in + \headheight 0.0in + \topmargin 0.0in + \textheight=9.0in + +\begin{document} +\title{Fabry-Perot Interferometer and the Sodium Doublet} +\date {} +\maketitle + + +\noindent + \textbf{Experiment objectives}: Assemble and align Fabry-Perot interferometer, + and use it to measure differential wavelength for the Na doublet. + + \section*{Theory} + +\subsection*{The Fabry-Perot Interferometer} + +Any interferometer relies on interference between two or more light field. In a Fabry-Perot configuration input +light field bounces between two closely spaced partially reflecting surfaces, creating a large number of +reflections. Interference of these multiple beams produces sharp spikes in the transmission for certain light +frequencies. Thanks to the large number of interfering rays, this type of interferometer has extremely high +resolution, much better than, for example, a Michelson interferometer. For that reason Fabry-Perot +interferometers are widely used in telecommunications, lasers and spectroscopy to control and measure the +wavelengths of light. In this experiment we will take advantage of high spectral resolution of the Fabry-Perot +interferometer to resolve two very closely-spaces emission lines in Na spectra by observing changes in a +overlapping interference fringes from these two lines. +\begin{figure}[h] +\begin{center} +\includegraphics[width=0.8\linewidth]{./pdf_figs/pfig1} +\caption{\label{fpfig1}Sequence of Reflection and +Transmission for a ray arriving at the treated inner surfaces $P_1 \& P_2$.} +\end{figure} +\end{figure} + +A Fabry-Perot interferometer consists of two parallel glass plates, flat to better than 1/4 of an optical +wavelength $\lambda$, and coated on the inner surfaces with a partially transmitting metallic layer. Such +two-mirror arrangement is normally called an {\it optical cavity}. The light in a cavity by definition bounces +back and forth many time before escaping; the idea of such a cavity is crucial for the construction of a laser. +Any light transmitted through such cavity is a product of interference between beams transmitted at each bounce +as diagrammed in Figure~\ref{fpfig1}. When the incident ray arrives at interface point $A$, a fraction $t$ is +transmitted and the remaining fraction $r$ is reflected, such that $t + r = 1$ ( this assumes no light is lost +inside the cavity). The same thing happens at each of the points $A,B,C,D,E,F,G,H\ldots$, splitting the initial +ray into parallel rays $AB,CD,EF,GH,$ etc. Between adjacent ray pairs, say $AB$ and $CD$, there is a path +difference of : +\begin{equation} + \delta = BC+CK +\end{equation}%eq1 + where $BK$ is normal to $CD$. In a development +similar to that used for the Michelson interferometer, you can show +that: +\begin{equation} + \delta = 2d\cos\theta +\end{equation}%eq.2 + If this path difference produces +constructive interference, then $\delta$ is some integer multiple of , +$\lambda$ namely, +\begin{equation} + m\lambda = 2d\cos\theta %eq.3 +\end{equation}%eq.3 + +This applies equally to ray pairs $CD$ and $EF, EF$ and $GH$, etc, so that all parallel rays to the right of +$P2$ will constructively interfere with one another when brought together. + +Issues of intensity of fringes \& contrast between fringes and dark background +are addressed in Melissinos, {\it Experiments in Modern Physics}, pp.309-312. + +\subsection*{The Sodium Doublet} + + In this lab you will measure the separation between the two + famous ``sodium doublet'' lines, the two closely spaced lines + which occur at 589 $nm$ and 589.59 $nm$, respectively. This ``doublet'' + emission is evidence that the atomic electron has the property + of intrinsic angular momentum, or spin S. As you are learning + in Modern Physics 201, the discrete spectral lines in atomic + emission are due to the quantization of electron energies in + the atom. As Niels Bohr postulated, electrons in atoms are + only allowed to absorb and emit energy in discrete + quantities. When an electron moves from one orbit to another + in an atom, a well-defined amount of energy is emitted as + light at a fixed wavelength. Later in this class we will + explore the spectra of various atomic gases. +\begin{figure}[h] +\centerline{\epsfig{height=5cm, file=fpfig2.eps}} \caption{\label{fpfig2.fig}Fine Structure Splitting in sodium +giving rise to the sodium doublet lines} +\end{figure} + For many + atoms, {\bf atomic levels are further split}, for example, + by interactions of electrons with each other (Russell-Saunders + coupling), external magnetic fields (Zeeman effect), and even + the interaction between the spin of an electron and the + magnetic field created by its orbital angular momentum + (spin-orbit coupling). This is known as fine structure + splitting (FSS). The fine structure splitting for the sodium + 3P state is due to spin-orbit coupling, and is illustrated in + Figure \ref{fpfig2.fig}. The "3P" state refers to sodium's +valence electron + which has a principal quantum number of $n=3$ and an orbital + quantum number of $l=1$ (a P-state). Further, the electron has + an intrinsic spin (like a top), described by a spin quantum + number $S$, which is either +1/2 or -1/2. The electron has a + magnetic moment due to its intrinsic spin, given by $m_S$. Due to +its orbital angular momentum around a charged nucleus, it + senses a magnetic field ${\mathbf H}$. The energy of interaction of a + magnetic moment in a field is given by $E = -\mu\cdot {\mathbf H}$. +This gives + rise to the splitting and two spectral emission lines. + + +\section*{Procedure} + +\subsection*{Set Up} +\textbf{Equipment needed}: Pasco precision interferometry kit, Na lamp, +adjustable-hight platform. + +\begin{figure} +\centerline{\epsfig{width=0.7\linewidth,file=fpfig3new.eps}} \caption{\label{fpfig3.fig}The Fabry-Perot +Interferometer} +\end{figure} + +The interferometer layout is shown in Figure \ref{fpfig3.fig}. The inner spacing $d$ between two +partially-reflecting mirrors ($P1$ and $P2$) can be roughly adjusted by loosening the screw that mounts $P2$ to +its base. It is important that the plates are as closely spaced as possible. Move the plates to within $1.0 - +1.5$~mm of each other, but make sure the mirrors do not touch! + +\subsection*{Data acquisition} + +\begin{enumerate} +\item \textbf{Turn on the sodium lamp as soon as you arrive. It should warm up for about 20 minutes +before starting}. +\item Turn the micrometer close to or at 0.00. +\item Remove the diffuser sheet from in front of the lamp. Look through +plate $P2$ towards the lamp. If the plates are not parallel, you will see +multiple images of the lamp filament. Adjust the knobs on $P1$ until the images +collapse into one. At this point, you should see faint circular interference +fringes appear. +\item Place the diffuser sheet in +front of the lamp so you will now only see the fringes. Continue to adjust the +knobs on one plate (not the knobs to move the plate back and forth, but the +ones to bring it parallel) to get the best fringe pattern. It is the most +convenient to view the interference picture directly. +\item +Seek the START condition illustrated in Fig.(\ref{fpfig4.fig}), in which all bright fringes are evenly spaced. +You do this by moving the micrometer. Note that alternate fringes may be of somewhat different intensities, one +corresponding to fringes of $\lambda_1$, the other to those for $\lambda_2$. If you do not see this condition, +try moving the mirror $P2$ across the range of micrometer screw. If you still cannot find them, you can also +move the whole plate by loosening one plate and sliding it a little. +\item Practice going through the fringe conditions as shown in Fig.(\ref{fpfig4.fig}) +by turning the micrometer and viewing the fringes. Do not be surprised if you +have to move the micrometer quite a bit to go back to the original condition. +\item Find a place on the micrometer ($d_1$) where you +have the ``START'' condition for fringes shown in Fig.(\ref{fpfig4.fig}). Now +advance the micrometer rapidly while viewing the fringe pattern ( NO COUNTING +OF FRINGES IS REQUIRED ). Note how the fringes of one intensity are moving to +overtake those of the other intensity (in the manner of +Fig.(\ref{fpfig4.fig})). Keep turning until the ``STOP'' pattern is achieved +(the same condition you started with). Record the micrometer reading as $d_2$. +\emph{Remember that 1 tick mark is 1 micrometer ($10^{-6}m$). That means if you +read 1.24, your really move 124 $\mu m$.} +\end{enumerate} + +\noindent \fbox{\parbox{\linewidth}{\textbf{Experimental tip:} You may have to +``home in'' on the best START and STOP conditions. Let's say that the even +fringe spacing for the START condition ($d_1$) is not exactly in view. Now move +the micrometer, looking to see if the pattern moves toward even spacing. If so, +stop and read the micrometer for $d_1$. Move a bit more. If this second fringe +spacing looks better than the first, then accept this for $d_1$. The same +``homing in'' procedure should be used to select the reading for $d_2$. In +other words as you approach the even spacing condition of the STOP pattern, +start writing down the micrometer positions. Eventually you will favor one +reading over all the others.}} + +\section*{Analysis} + + Since the condition we are seeking above for ``START'' places + the bright fringes of $\lambda_1$ at the point of destructive + interference for $\lambda_2$, we can express this for the bull's eye + center ($\theta= 0 $) as: +\begin{equation} +2d_1=m_1\lambda_1=\left(m_1+n+\frac{1}{2}\right)\lambda_2 +\end{equation} + + Here the integer n accounts for the + fact that $\lambda_1 > \lambda_2$ , and the $1/2$ for the +condition of + destructive interference for $\lambda_2$ at the center. Since the + net action of advancing by many fringes has been to increment + the fringe count of $\lambda_2$ by one more than that of +$\lambda_1$ , + then we express the ``STOP'' condition as: +\begin{equation} +2d_2=m_2\lambda_1=\left(m_2+n+\frac{3}{2}\right)\lambda_2 +\end{equation} + Subtracting the + two interference equations gives us: +\begin{equation} +2(d_2-d_1)=(m_2-m_1)\lambda_1=(m_2-m_1)\lambda_2+\lambda_2 +\end{equation} + Eliminating $(m_2-m_1)$ + in this equation we obtain: + +\begin{equation} +2(d_2-d_1)=\frac{\lambda_1\lambda_2}{(\lambda_1-\lambda_2)} +\end{equation} + + Solving this for $\Delta \lambda = \lambda_1-\lambda_2$, and + accepting as valid the approximation that $\lambda_1\lambda_2\approx +\lambda^2$ ( where $\lambda$ is the + average of $\lambda_1$ and $\lambda_2 \approx 589.26 nm$ ), we obtain: +\begin{equation} +\boxed{\Delta\lambda=\frac{\lambda^2}{2(d_2-d_1)}} +\end{equation} + +Each lab partner should independently align the interferometer and make at least \textit{two} measurements of +``START'' and ``STOP'' positions. A sample table to record the data is shown below. \\{\large +\begin{tabular}{|p{27mm}|p{27mm}|p{27mm}|p{27mm}|} +\hline + $d_1$ $\pm \dots$ & $d _2$ $\pm \dots$& $(d_1-d_2)$ $\pm \dots$& +$\Delta \lambda(nm) $ $\pm \dots$\\ +\hline +&&&\\ +\hline &&&\\ \hline &&&\\ \hline &&&\\ \hline &&&\\ \hline +\end{tabular} +} + +\vspace{1cm} Calculate average value of Na doublet splitting and its standard deviation. Compare your result +with the established value of $\Delta \lambda_{Na}=0.598$~nm. + + +\begin{figure}[h] +\centerline{\epsfig{width=0.8\linewidth,file=fpfig4.eps}} \caption{\label{fpfig4.fig}The Sequence of fringe +patterns encountered in the course of the FSS measurements. Note false colors: in your experiment the background +is black, and both sets of rings are bright yellow.} +\end{figure} + +\end{document} + +\newpage +\noindent +Physics 251 Section:\\ +\hskip 4.5in Name:\\ +\hskip 4.5in Partners:\\ +\vskip 0.5in +\subsection*{The Fabry-Perot Interferometer} +1. Briefly describe how the Fabry-Perot interferometer gives and interference +pattern (in one or two sentences):\\ +\vskip 1.2in +2. How does the interferometer's resolving power of the fringes depend on the +reflectivity of plates, r ? That is, does the sharpness of the fringes increase +or decrease with r ? Consult Melissinos or Professor Kane's Mathview program. +(The reflectivity of the plates defines the {\it finess} of the cavity).\\ +\vskip 1in. + + +{\large +\noindent +Fill in: + +The sodium doublet lines arise because an atomic$\underline{\hskip 1.in}$ +is split into two by$\underline{\hskip 1.in}$ coupling. +The electron has +intrinsic $\underline{\hskip 1.in}$, like a top, with values of +$\underline{\hskip 1.in}$ or$\underline{\hskip 1.in}$. Because of +this, the electron has in intrinsic magnetic$\underline{\hskip 1.in}$ and +has magnetic +energy in a magnetic field given by E=$\underline{\hskip 1.in}$ +case comes from the electron's $\underline{\hskip 1.in}$ motion. + +} + +\subsection*{DATA:} +{\large +\begin{tabular}{|p{27mm}|p{27mm}|p{27mm}|p{27mm}|} +\hline + $d_1$ $\pm \dots$ & $D _2$ $\pm \dots$& $(d_1-d_2)$ $\pm \dots$& +$\Delta \lambda(nm) $ $\pm \dots$\\ +\hline +&&&\\ +\hline +&&&\\ \hline +&&&\\ \hline +&&&\\ \hline +&&&\\ \hline +\end{tabular} +} +\vskip .2in +$\Delta \lambda=$\hskip 1.5in nm\\ +\vskip .2in +Standard deviation= \hskip 1.5in nm +\newpage +\end{document} diff --git a/manual_source/chapters/faraday_rotation.tex b/manual_source/chapters/faraday_rotation.tex new file mode 100644 index 0000000..74d351f --- /dev/null +++ b/manual_source/chapters/faraday_rotation.tex @@ -0,0 +1,126 @@ +%\chapter*{Electron Diffraction} +%\addcontentsline{toc}{chapter}{Electron Diffraction} + +%\documentclass{article} +%\usepackage{tabularx,amsmath,boxedminipage,epsfig} +% \oddsidemargin 0.0in +% \evensidemargin 0.0in +% \textwidth 6.5in +% \headheight 0.0in +% \topmargin 0.0in +% \textheight=9.0in + +%\begin{document} +\chapter{Faraday Rotation} +\setcounter{figure}{1} +\setcounter{table}{1} +\setcounter{equation}{1} + +%\date {} +%\maketitle \noindent + + +\textbf{Experiment objectives}: Observe the {\it Faraday Effect}, the rotation of a light wave's polarization vector in a material with a magnetic field directed along the wave's direction. Determine the relationship between the the magnetic field and the rotation by measuring the so-called {\it Verdet constant} of the material. Become acquainted with some new tools: an oscilloscope, a function generator and an amplifier, and a new technique: phase-locking. + +\section*{Introduction} +The term polarization refers to the direction of the electrical field in a light wave. Generally, light is not polarized when created (e.g., by atomic deexcitations) but can be made so by passing it through a medium which transmits electric fields oriented in one direction, and absorbs all others. Imagine we create a beam of light traveling in the $z$ direction. We then polarize it in the $x$ direction ($\vect{E}=\vect{\hat x}E_0\cos(kz-\omega t)$) by passing it through a polarizer and then pass it through a second polarizer, with a transmission axis oriented at an angle $\theta$ with respect to the $x$ axis. If we detect the light beam after the second polarizer, the intensity is +\begin{equation} +I=I_0 \cos^{2}\theta +\end{equation} + +In 1845 Michael Faraday discovered that he could create polarized light and then rotate the direction of the polarization using a magnetic field. The field, created in our laboratory by a solenoid, points in the direction of the light beam. The Faraday effect is caused by a combination of factors: +\begin{enumerate} +\item We can describe the polarization vector in terms of a right- and left-handed components. In this description, the electric field of the right-handed component rotates clockwise around the $z$ axis as the wave travels. The left-handed component rotates counter-clockwise. +\item At the particle level, the right and left hand components correspond to photons with angular momentum $\hbar=h/2\pi$ directed parallel (right) or anti-parallel (left) to the $z$ axis. +\item When an atom is placed in a magnetic field, single atomic energy levels may divide into multiple levels, each with slightly different energies. This is called the {\it Zeeman effect} and qualitatively it occurs because the moving atomic electrons may be thought of as a current $\vect{J}$ which interacts with the external magnetic field $\vect{B}$. +\item The atomic current $\vect{J}$ contains components with different angular momenta, and those components interact differently with right and left handed photons. +\item The macroscopic effect is that a material in a magnetic field can have a different index of refraction $n$ for left and right handed light. Since the phase velocity of light $c'$ depends on $n$ as $c'=c/n$, the relative phase between the left and right handed components changes as the light travels a distance $L$ through the material. +\item That phase change causes a rotation in the polarization vector. +\end{enumerate} + +The polarization vector rotates in proportion to the length of the material, the magnitude of the magnetic field, and a material dependent constant $C_{V}$ called the {\it Verdet constant}: + +\begin{equation} \label{eq:faraday_rotation} +\phi = C_V B L +\end{equation} + +Typically $C_V$ depends on the wavelength of the light and has a value of a few $\mathrm{rad}/\mathrm{T}\cdot\mathrm{m}$. For the solenoid we'll use in this lab, the field at the center is $B=\unit[11.1]{mT/A}$, and our material, a special sort of glass, is $\unit[10]{cm}$ long. For a current of \unit[0.1]{A}, we expect a rotation a few$\times10^{-4}$ radians. This is a pretty small angle and it will require a special technique to detect. + +We are going to take polarized laser light and direct it through the glass rod, which is inserted into the center of the solenoid. The beam will then pass through a second polarizer with transmission axis at an angle $\theta$ with respect to the initial polarization of the laser. The intensity of the transmitted light will then depend on the sum of the angle $\theta$ and the additional rotation $\phi$ caused by the magnetic field: +\begin{eqnarray} +I & = & I_0 \cos^{2}(\theta+\phi)\\ + & = & I_0 \frac{1 + \cos(2\theta + 2\phi)}{2} \\ + & = & \frac{I_0}{2} + \frac{I_0}{2}\left[\cos2\theta\cos2\phi-\sin2\theta\sin2\phi \right] \\ + & \approx & I_0\left[\frac{1}{2}+\cos2\theta - \phi \sin2\theta \right] \qquad (\phi^{2}\ll 1)\\ + & = & I_0\left[\frac{1}{2}+\cos2\theta - C_V L B \sin2\theta \right] \label{eq:Ifinal} +\end{eqnarray} + +We'll see the Faraday effect by observing changes in the intensity of light as we vary the magnetic field. But, there is a problem. The term involving $\phi$ is much smaller than the other terms and it can easily be buried in noise from a photo-detector. We'll use an experimental technique, called phase-locking, to get around the problem. We will then relate the changes in the intensity to changes in the angle $\phi$ using a special calibration dataset. Since we know the corresponding changes in $B$ we can use Eq.~\ref{eq:faraday_rotation} to extract $C_V$. + +The phase-locking technique works in the following way. We'll vary the magnetic field periodically with time as a sine wave, and then observe the signal from the photodiode as a function of time. The signal will look like a large constant with a small wobble on it, along with some random noise with a similar magnitude to the wobble. However, we can subtract off the non time-varying portion of the signal, using a high pass filter. Then, since we know the period and phase of the magnetic field, we can time our observations to be exactly in sync with the magnetic field, and use this to average out the noise. What's left over is the wobble in $I$, essentially the $C_V L B \sin2\theta$ term. Knowing $B,L$ and $\theta$ we can determine $C_V$. + +\section*{Experimental Setup} +The setup consists of: +\begin{itemize} +\item A solenoid to produce the magnetic field. +\item A glass rod inside the solenoid. You'll measure $C_V$ of this material. +\item A polarized HeNe laser, to shine light through the field. +\item A second polarizer, the ``analyzer'', to define the angle $\theta$. +\item A photodiode to detect the laser light after the coil and the analyzer. +\item A function generator and amplifier to supply current to the solenoid. +\item A digital multimeter (DMM) to measure the solenoid current. +\item A digital oscilloscope to read out the photodiode. +\end{itemize} + +The experimental setup is shown in Fig.~\ref{fig:setup}. + +\begin{figure}[h] +\centering +\includegraphics[width=\textwidth]{./pdf_figs/faraday_setup} \caption{\label{fig:setup} The experimental setup.} +\end{figure} + +\section*{Experimental Procedure} + +\subsection*{Preliminaries} +\begin{description} +\item[Laser \& photodiode setup] The amplifier includes the laser power supply on the back. Plug the laser in, {\bf being careful to match colors between the cable and the power supply's connectors}. Align the laser so it travels down the center of the solenoid, through the glass rod, and into the center of the photodiode. Set the photodiode load resistor to $\unit[1]{k\Omega}$. Plug the photodiode into a DMM and measure DC voltage. + +\item[Calibration: intensity vs $\mathbf{\theta}$] We want to understand how the angle between the polarization vector of the laser light and the polarizer direction effects the intensity. Vary the angle of the analyzing polarizer and use a white screen (e.g., piece of paper) to observe how the intensity of the transmitted light changes. Find the angles which give you maximum and minimum transmission. Then, use the DMM to measure the photodiode output as a function of $\theta$, going between the maximum and minimum in $5^\circ$ steps. Tabulate this data. Graph it. What functional form should it have? Does it? + +\item[Function Generator Setup] Plug the function generator output and its trigger (a.k.a.~ pulse) output into different channels on the scope. Trigger the scope on the trigger/pulse output from the function generator and look at the function generator signal. Modify the function generator to provide a \unit[200]{Hz} sine wave with an amplitude of about \unit[1]{V}. There is no more need to touch dials on the function generator. + +\item[Amplifier setup] Now, plug the output of the function generator into the amplifier. Use a coaxial$\leftrightarrow$double-prong connector to feed the amplifier output into the scope, still triggering on the generator. Vary the amplifier dial setting and observe the change in the output. At some point, the output will become clipped, as the amplifier reaches its power output. Record this dial setting, and the amplitude of the sine wave, just as the clipping sets in. This is the maximum useful output from the amplifier. Turn the dial all the way off, and hook the amplifier up to the solenoid, with the DMM in series to measure AC current. Go back to the maximum setting and measure the current flowing through the solenoid. This is the maximum useful current. You now have a time varying magnetic field in the solenoid and you can control its magnitude with the amplifier. + +\end{description} + +\subsection*{Measuring Faraday Rotation} + +\begin{description} +\item[Choice of $\theta$] You need to pick an angle $\theta$, which may seem arbitrary. But, there is a best choice. Examine Eq.~\ref{eq:Ifinal}. Pick $\theta$ and be sure to tighten the thumbscrew. +\item[Faraday rotation] Plug the photodiode output into the scope, and set the scope so its channel is DC coupled, and make sure that the ``probe'' setting is at 1x. Turn the amplifier dial about halfway to the maximum setting you found. Observe the photodiode trace on the scope, perhaps changing the volts/div setting so you can see the trace more clearly. What is the voltage? Record it. The changing magnetic field should be causing a change in the polarization angle of the laser light, which should cause a wobble to the photodiode signal. Can you see any wobble? +\item[AC coupling] The wobble is riding atop a large constant (DC) signal. The scope can remove the DC signal by ``AC coupling'' the photodiode channel. This essentially directs the scope input through a high pass filter. Do this, and then set the photodiode channel to the \unit[2]{mV} setting. You should now see a wobble. Vary the amplifier dials setting and notice how the amplitude of the wobble changes. You are seeing the Faraday effect. +\item[Remove the noise] The signal is noisy, but now we'll really benefit from knowing the waveform that the function generator is producing. Because we trigger the scope on the function generator, the maxima and minima will, neglecting random noise, occur at the same point on the scope screen (and, in it's memory bank). The scope has a feature which allows you to average multiple triggers. Doing this mitigates the noise, since at each point on the trace we are taking a mean, and the uncertainty in a mean decreases as we increase the number of measurements $N$ as $1/\sqrt{N}$. Turn on the averaging feature by going to the ``Acquire'' menu. Observe how the averaged trace becomes more stable as you increase the number of traces being averaged. The larger the better, but $\sim$100 traces should be enough. +\item[Take measurements] You should now systematically measure the amplitude of the wobble as a function of the current in the solenoid. There should be a linear relationship, which can be fit to extract $C_V$. Take about 10 measurements, evenly separated between the smallest current for which there is a measurable wobble, and the maximum you found earlier. In each case, you want to start acquisition (Run/Stop on the scope), let the averaged signal converge onto a nice sine wave, stop acquisition and measure the amplitude of the signal using the scopes cursors. One measurement is shown in Fig.~\ref{fig:trace}. Record the negative and positive amplitudes $V_\mathrm{low}$ and $V_\mathrm{high}$ ($\unit[\pm 640]{\mu V}$ in Fig.~\ref{fig:trace}) and the peak to peak voltage $\Delta V$ ($\unit[1.28]{mV}$), along with the current in the coil -- $I_\mathrm{coil}$ . Estimate the uncertainty in your measurements. +\end{description} + +\begin{figure}[h] +\centering +\includegraphics[width=\textwidth]{./pdf_figs/faraday_scope_trace} \caption{\label{fig:trace} An example scope trace. The yellow curve is the output of the photodiode, AC coupled and averaged over 128 traces. The blue curve is the trigger output from the function generator and is being used to trigger the scope readout so that it's in phase with the changing magnetic field. The maximum and minimum amplitude of the photodiode signal is measured with the scopes cursors.} +\end{figure} + +\subsection*{Data Analysis} +A variation in the angle $\phi$ is related to a variation in the magnetic field $B$ according to: +\begin{equation}\label{eq:faraday_fit} +\Delta \phi= C_V L \Delta B +\end{equation} +We want to use this equation to extract $C_V$. Both $\phi$ and $B$ vary with time as sine waves. We'll take $\Delta \phi$ and $\Delta B$ to be the amplitude of those waves (half the peak to peak). We didn't directly measure either quantity, but we can compute them. For $\Delta B$ it's easy: $\Delta B = \unit[11.1]{mT/A}\times \sqrt{2} I_\mathrm{coil}$, where the $\sqrt{2}$ accounts for the fact that DMMs measure the root-mean-squared (RMS) value of an AC signal, not the peak value. + +Figuring out $\Delta\phi$ is a little more difficult. We need to use our calibration dataset, which relates the photodiode signal to the angle between the laser polarization and the analyzer's orientation. From that dataset, you can estimate $\mathrm{d}V/\mathrm{d}\theta$ at the value of $\theta$ you are using in your experiment. Of course a change in the polarization of the laser, $\Delta\phi$ is equivalent to holding the laser polarization constant and changing the angle of the analyzer by $-\Delta\theta$. So, we can calculate: +\begin{equation} +\Delta\phi = \left[\frac{\mathrm{d}V}{\mathrm{d}\theta}\right]^{-1} \frac{\Delta V}{2} +\end{equation} +The factor of two is because we defined $\Delta V$ as the peak to peak voltage. +%\begin{description} + +Compute $C_V$ for a few points, along with the uncertainty. Complete the analysis by fitting $\Delta\phi$ vs $\Delta B$ to a straight line and then use Eq.~\ref{eq:faraday_fit} to extract $C_V$ and its uncertainty. Do you see a nice linear relationship? What should the intercept be? Is it what you expect? Are there any outlier points, particularly at the ends of your curve? You could do the same analysis using $V_\mathrm{low}$ or $V_\mathrm{high}$... does doing so yield the same answer for $C_V$? + diff --git a/manual_source/chapters/hspect.tex b/manual_source/chapters/hspect.tex new file mode 100644 index 0000000..565c64f --- /dev/null +++ b/manual_source/chapters/hspect.tex @@ -0,0 +1,436 @@ +%\chapter*{Atomic Spectroscopy of the Hydrogen Atom} +%\addcontentsline{toc}{chapter}{Hydrogen Spectrum} +\documentclass{article} +\usepackage{tabularx,amsmath,boxedminipage,epsfig} + \oddsidemargin 0.0in + \evensidemargin 0.0in + \textwidth 6.5in + \headheight 0.0in + \topmargin 0.0in + \textheight=9.0in + +\begin{document} +\title{Atomic Spectroscopy of Hydrogen Atoms} +\date {} +\maketitle \noindent + \textbf{Experiment objectives}: test and calibrate a diffraction grating-based spectrometer + and measure the energy spectrum of atomic hydrogen. + +\subsection*{History} + + The observation of discrete lines in the emission spectra of + atomic gases gives insight into the quantum nature of + atoms. Classical electrodynamics cannot explain the existence + of these discrete lines, whose energy (or wavelengths) are + given by characteristic values for specific atoms. These + emission lines are so fundamental that they are used to + identify atomic elements in objects, such as in identifying + the constituents of stars in our universe. When Niels Bohr + postulated that electrons can exist only in orbits of discrete + energies, the explanation for the discrete atomic lines became + clear. In this laboratory you will measure the wavelengths of + the discrete emission lines from hydrogen gas, which will give + you a measurement of the energy levels in the hydrogen atom. + +\section*{Theory} + + The hydrogen atom is composed of a proton nucleus and a single +electron in a bound state orbit. Bohr's groundbreaking hypothesis, that the +electron's orbital angular momentum is quantized, leads directly to the +quantization of the atom's energy, i.e., that electrons in atomic systems exist +only in discrete energy levels. The energies specified for a Bohr atom of +atomic number $Z$ in which the nucleus is fixed at the origin (let the nuclear +mass $\rightarrow \infty$) are given by the expression: +\begin{equation}\label{Hlevels_inf} +E_n=- \frac{2\pi^2m_ee^4Z^2}{(4\pi\epsilon_0)^2h^2n^2} + = -hcZ^2R_{\infty}\frac{1}{n^2} +\end{equation} +% +where $n$ is the label for the {\bf principal quantum number} + and $R_{\infty}=\frac{2\pi m_ee^4}{(4\pi\epsilon_0)^2ch^3}$ is called the +{\bf Rydberg wave number} (here $m_e$ is the electron mass). Numerically, +$R_{\infty} += 1.0974 \times 10^5 cm^{-1}$ and $hcR_{\infty} = 13.605 eV$. + +An electron can change its state only by making a transition ("jump") from an +``initial'' excited state of energy $E_1$ to a ``final'' state of lower energy +$E_2$ by emitting a photon of energy $h\nu = E_1 - E_2$ that carries away the +excess energy. Thus frequencies of spectral emission lines are proportional to +the difference between two allowed discrete energies for an atomic +configuration. Since $h\nu = hc/\lambda$, we can write for this case: +\begin{equation} \label{Hlines_inf} +\frac{1}{\lambda}=\frac{2\pi^2m_ee^4Z^2}{(4\pi\epsilon_0)^2ch^3} +\left[\frac{1}{n_2^2}-\frac{1}{n_1^2}\right]= +R_{\infty}Z^2\left[\frac{1}{n_2^2}-\frac{1}{n_1^2}\right] +\end{equation} +Based on this description it is clear that by measuring the frequencies (or +wavelengths) of photons emitted by an excited atomic system, we can glean +important information about allowed electron energies in atoms. + +To make more accurate calculation of the Hydrogen spectrum, we need to take +into account that a hydrogen nucleus has a large, but finite mass, M=AMp (mass +number A=1 and Mp = mass of proton)\footnote{This might give you the notion +that the mass of any nucleus of mass number $A$ is equal to $AM_p$. This is not +very accurate, but it is a good first order approximation.} such that the +electron and the nucleus orbit a common center of mass. For this two-mass +system the reduced mass is given by $\mu=m_e/(1+m_e/AM_p)$. We can take this +into account by modifying the above expression (\ref{Hlines_inf}) for +1/$\lambda$ as follows: +\begin{equation}\label{Hlines_arb} +\frac{1}{\lambda_A}=R_AZ^2\left[\frac{1}{n_2^2}-\frac{1}{n_1^2}\right] \mbox{ +where } R_A=\frac{R_{\infty}}{1+\frac{m_e}{AM_p}} +\end{equation} +In particular, for the hydrogen case of ($Z=1$; $M=M_p$) we have: +\begin{equation}\label{Hlines_H} +\frac{1}{\lambda_H}=R_H\left[\frac{1}{n_2^2}-\frac{1}{n_1^2}\right] +\end{equation} +Notice that the value of the Rydberg constant will change slightly for +different elements. However, these corrections are small since nucleus is +typically several orders of magnitude heavier then the electron. + + + Fig. \ref{spec} shows a large number of observed transitions between + Bohr energy levels in hydrogen, which are grouped into series. Emitted photon + frequencies (wavelengths) span the spectrum from the UV + (UltraViolet) to the IR (InfraRed). Given our lack of UV or + IR spectrometers, we will focus upon the optical spectral lines + that are confined to the Balmer series (visible). These are + characterized by a common final state of $n_2$ = 2. The + probability that an electron will make a particular +$n_1\rightarrow n_2$ + transition in the Balmer series can differ considerably, + depending on the likelihood that the initial $n_1$ level is + populated from above in the deexcitation process. This + results in our being able to observe and measure only the following four + lines: $6 \rightarrow 2$, $5 \rightarrow 2$, $4 \rightarrow 2$, + and $3 \rightarrow 2$. + + +\begin{figure} +\includegraphics[width=0.7\linewidth]{spec.eps} +\caption{\label{spec}Spectrum of Hydrogen. The numbers on the left show the +energies of the hydrogen levels with different principle quantum numbers $n$ in +$eV$. The wavelength of emitted photon in ${\AA}$ are shown next to each +electron transition. } +\end{figure} + +In this lab, the light from the hydrogen gas is broken up into its spectral +components by a diffraction grating. You will measure the angle at which each +line occurs on the left ($\theta_L$) and ($\theta_R$) right sides for as many +diffraction orders $m$ as possible, and use Eq.(\ref{mlambda}) to calculate +$\lambda$, using the following expression, derived in the Appendix. +\begin{equation}\label{mlambda} +m\lambda = \frac{h}{2}\left(\sin\theta_L+\sin\theta_R\right) +\end{equation} + Then the same +expression will be used to check/calibrate the groove spacing $h$ by making +similar measurements for a sodium spectral lines with known wavelengths. + +We will approach the data in this experiment both with an eye to confirming + Bohr's theory and from Balmer's early perspective of someone + trying to establish an integer power series linking the + wavelength of these four lines. + +\section*{Spectrometer Alignment Procedure} + +Fig. \ref{expspec} gives a top view of the Gaertner-Peck optical spectrometer +used in this lab. +\begin{figure} +\includegraphics[height=4in]{expspec.eps} +\caption{\label{expspec}Gaertner-Peck Spectrometer} +\end{figure} + +\subsubsection*{Telescope Conditions:} Start by adjusting the +telescope eyepiece in + or out to bring the crosshairs into sharp focus. Next aim the + telescope out the window to view a distant object such as + leaves in a tree. If the distant object is not in focus or if + there is parallax motion between the crosshairs and the + object, pop off the side snap-in button to give access to a + set screw. Loosen this screw and move the ocular tube in or + out to bring the distant object into sharp focus. This should + result in the elimination of parallax. Tighten the set screw + to lock in this focussed condition. + +\subsubsection*{Collimator Conditions:} Swing the telescope to view the collimator + which is accepting light from the hydrogen discharge tube + through a vertical slit of variable width. The slit opening + should be set to about 5-10 times the crosshair width to + permit sufficient light to see the faint violet line and to be + able to see the crosshairs. If the bright column of light is + not in sharp focus, you should remove a side snap-in button + allowing the tube holding the slit to move relative to the + collimator objective lens. Adjust this tube for sharp focus + and for elimination of parallax between the slit column and + the crosshairs. Finally, tighten the set screw. + +\subsubsection*{ Diffraction Grating Conditions:} +\textbf{Appendix in this handout describes the operation of a diffraction +grating!} + Mount a diffraction grating which nominally + has 600 lines per mm in a grating baseclamp. + %Put a piece of + % doublesided scotch tape on the top surface of the table plate. + Fix the grating baseclamp to the table such that the grating's + vertical axis will be aligned with the telescope pivot axis. + Since the table plate can be rotated, orient the normal of the + grating surface to be aligned with the collimator axis. Use + the AUTOCOLLIMATION procedure to achieve a fairly accurate + alignment of the grating surface. This will determine how to + adjust the three leveling screws H1, H2, and H3 and the + rotation angle set screw for the grating table. + + \textbf{AUTOCOLLIMATION} is a sensitive way to align an optical + element. First, mount a ``cross slit'' across the objective lens of + the collimator, and direct a strong light source into the + input end of the collimator. Some of the light exiting through + the cross slit will reflect from the grating and return to the + cross slit. The grating can then be manipulated till this + reflected light retraces its path through the cross slit + opening. With this the grating surface is normal to the + collimator light. + Then, with the hydrogen tube ON and in place at + the collimator slit, swing the rotating telescope slowly + through 90 degrees both on the Left \& Right sides of the forward + direction. You should observe diffraction maxima for most + spectral wavelength, $\lambda$, in 1st, 2nd, and 3rd order. If these + lines seem to climb uphill or drop downhill + the grating will have to be rotated in its baseclamp to + bring them all to the same elevation. + +\section*{Data acquisition and analysis} + +Swing the rotating telescope slowly and determine which spectral lines from +Balmer series you observe. + +\emph{Lines to be measured:} +\begin{itemize} +\item \emph{Zero order} (m=0): All spectral lines merge. +\item \emph{First order} (m=1): Violet, Blue, Green, \& Red on both Left \& + Right sides. +\item \emph{Second order} (m=2): Violet, Blue, Green, \& Red on + both Left \& Right sides. +\item \emph{Third order} (m=3): Blue, \& Green. +\end{itemize} + You might not see the Violet line due to its low + intensity. Red will not be seen in 3rd order. + +Read the angle at which each line occurs, measured with the crosshairs centered +on the line as accurately as possible. Each lab partner should record the +positions of the spectral lines at least once. Use the bottom scale to get the +rough angle reading in degrees, and then use the upper scale for more accurate +reading in minutes. The width of lines is controlled by the Collimator Slit +adjustment screw. If set too wide open, then it is hard to judge the center + accurately; if too narrow, then not enough light is available + to see the crosshairs. For Violet the intensity is noticeably + less than for the other three lines. Therefore a little + assistance is required in order to locate the crosshairs at + this line. We suggest that a low intensity flashlight be + aimed toward the Telescope input, and switched ON and OFF + repeatedly to reveal the location of the vertical crosshair + relative to the faint Violet line. + +\subsubsection*{ Calibration of Diffraction Grating:} Replace the hydrogen tube with + a sodium (Na) lamp and take readings for the following two + lines from sodium: $568.27$~nm (green) and $589.90$~nm (yellow). Extract from + these readings the best average value for $h$ the groove + spacing in the diffraction grating. Compare to the statement + that the grating has 600 lines per mm. Try using your measured value + for $h$ versus the stated value $600$ lines per mm in + the diffraction formula when obtaining the measured + wavelengths of hydrogen. Determine which one provide more accurate results, and discuss the conclusion. + +\subsubsection*{ Data analysis} +\textbf{Numerical approach}: Calculate the wavelength $\lambda$ for each line +observed. For lines observed in more than one order, obtain the mean value +$\lambda_ave$ and the standard error of the mean $\Delta \lambda$. Compare to +the accepted values which you should calculate using the Bohr theory. + +\textbf{Graphical approach}: Make a plot of $1/\lambda$ vs $1/n_1^2$ where +$n_1$ = the principal quantum number of the electron's initial state. Put all +$\lambda$ values you measure above on this plot. Should this data form a +straight line? If so, determine both slope and intercept and compare to the +expected values for each. The slope should be the Ryberg constant for +hydrogen, $R_H$. The intercept is $R_H/(n_2)^2$. From this, determine the value +for the principal quantum number $n_2$. Compare to the accepted value in the +Balmer series. + +\textbf{Example data table for writing the results of the measurements}: + +\noindent +\begin{tabular}{|p{1.in}|p{1.in}|p{1.in}|p{1.in}|} +\hline + Line &$\theta_L$&$\theta_R$&Calculated $\lambda$ \\ \hline + m=1 Violet&&&\\ \hline + m=1 Blue&&&\\ \hline + m=1 Green&&&\\ \hline + m=1 Red&&&\\ \hline + m=2 Violet&&&\\ \hline + \dots&&&\\ \hline + m=3 Blue&&&\\ \hline + \dots&&&\\\hline +\end{tabular} + +\section*{Appendix: Operation of a diffraction grating-based optical spectrometer} + +%\subsection*{Fraunhofer Diffraction at a Single Slit} +%Let's consider a plane electromagnetic wave incident on a vertical slit of +%width $D$ as shown in Fig. \ref{frn}. \emph{Fraunhofer} diffraction is +%calculated in the far-field limit, i.e. the screen is assume to be far away +%from the slit; in this case the light beams passed through different parts of +%the slit are nearly parallel, and one needs a lens to bring them together and +%see interference. +%\begin{figure}[h] +%\includegraphics[width=0.7\linewidth]{frnhfr.eps} +%\caption{\label{frn}Single Slit Fraunhofer Diffraction} +%\end{figure} +%To calculate the total intensity on the screen we need to sum the contributions +%from different parts of the slit, taking into account phase difference acquired +%by light waves that traveled different distance to the lens. If this phase +%difference is large enough we will see an interference pattern. Let's break the +%total hight of the slit by very large number of point-like radiators with +%length $dx$, and we denote $x$ the hight of each radiator above the center of +%the slit (see Fig.~\ref{frn}). If we assume that the original incident wave is +%the real part of $E(z,t)=E_0e^{ikz-i2\pi\nu t}$, where $k=2\pi/\lambda$ is the +%wave number. Then the amplitude of each point radiator on a slit is +%$dE(z,t)=E_0e^{ikz-i2\pi\nu t}dx$. If the beam originates at a hight $x$ above +%the center of the slit then the beam must travel an extra distance $x\sin +%\theta$ to reach the plane of the lens. Then we may write a contributions at +%$P$ from a point radiator $dx$ as the real part of: +%\begin{equation} +%dE_P(z,t,x)=E_0e^{ikz-i2\pi\nu t}e^{ikx\sin\theta}dx. +%\end{equation} +%To find the overall amplitude one sums along the slit we need to add up the +%contributions from all point sources: +%\begin{equation} +%E_P=\int_{-D/2}^{D/2}dE(z,t)=E_0e^{ikz-i2\pi\nu +%t}\int_{-D/2}^{D/2}e^{ikx\sin\theta}dx = A_P e^{ikz-i2\pi\nu t}. +%\end{equation} +%Here $A_P$ is the overall amplitude of the electromagnetic field at the point +%$P$. After evaluating the integral we find that +%\begin{equation} +%A_P=\frac{1}{ik\sin\theta}\cdot +%\left(e^{ik\frac{D}{2}\sin\theta}-e^{-ik\frac{D}{2}\sin\theta}\right) +%\end{equation} +%After taking real part and choosing appropriate overall constant multiplying +%factors the amplitude of the electromagnetic field at the point $P$ is: +%\begin{equation} +%A=\frac{\sin (\frac{\pi D}{\lambda}\sin\theta)}{\frac{\pi +%D}{\lambda}\sin\theta} +%\end{equation} +%The intensity is proportional to the square of the amplitude and thus +%\begin{equation} +%I_P=\frac{(\sin (\frac{\pi D}{\lambda}\sin\theta))^2}{(\frac{\pi +%D}{\lambda}\sin\theta)^2} +%\end{equation} +%The minima of the intensity occur at the zeros of the argument of the sin. The +%maxima are near, but not exactly equal to the solution of: +%\begin{equation} +% (\frac{\pi D}{\lambda}\sin\theta)=(m+\frac{1}{2})\pi \end{equation} +%for integer $m$. +% +%The overall pattern looks like that shown in Fig. \ref{sinxox}. +%\begin{figure} +%\includegraphics[width=\linewidth]{sinxox.eps} +%\caption{\label{sinxox}Intensity Pattern for Fraunhofer Diffraction} +%\end{figure} + +%\subsection*{The Diffraction Grating} +A diffraction grating is a common optical element, which consists of a pattern +with many equidistant slits or grooves. Interference of multiple beams passing +through the slits (or reflecting off the grooves) produces sharp intensity +maxima in the output intensity distribution, which can be used to separate +different spectral components on the incoming light. In this sense the name +``diffraction grating'' is somewhat misleading, since we are used to talk about +diffraction with regard to the modification of light intensity distribution to +finite size of a single aperture. +\begin{figure}[h] +\includegraphics[width=\linewidth]{grating.eps} +\caption{\label{grating}Intensity Pattern for Fraunhofer Diffraction} +\end{figure} + +To describe the properties of a light wave after passing through the grating, +let us first consider the case of 2 identical slits separated by the distance +$h$, as shown in Fig.~\ref{grating}a. We will assume that the size of the slits +is much smaller than the distance between them, so that the effect of +Fraunhofer diffraction on each individual slit is negligible. Then the +resulting intensity distribution on the screen is given my familiar Young +formula: +\begin{equation} \label{2slit_noDif} +I(\theta)=\left|E_0 +E_0e^{ikh\sin\theta} \right|^2 = 4I_0\cos^2\left(\frac{\pi +h}{\lambda}\sin\theta \right), +\end{equation} +where $k=2\pi/\lambda$, $I_0$ = $|E_0|^2$, and the angle $\theta$ is measured +with respect to the normal to the plane containing the slits. +%If we now include the Fraunhofer diffraction on each slit +%same way as we did it in the previous section, Eq.(\ref{2slit_noDif}) becomes: +%\begin{equation} \label{2slit_wDif} +%I(\theta)=4I_0\cos^2\left(\frac{\pi h}{\lambda}\sin\theta +%\right)\left[\frac{\sin (\frac{\pi D}{\lambda}\sin\theta)}{\frac{\pi +%D}{\lambda}\sin\theta} \right]^2. +%\end{equation} + +An interference on $N$ equidistant slits illuminated by a plane wave +(Fig.~\ref{grating}b) produces much sharper maxima. To find light intensity on +a screen, the contributions from all N slits must be summarized taking into +account their acquired phase difference, so that the optical field intensity +distribution becomes: +\begin{equation} \label{Nslit_wDif} +I(\theta)=\left|E_0 ++E_0e^{ikh\sin\theta}+E_0e^{2ikh\sin\theta}+\dots+E_0e^{(N-1)ikh\sin\theta} +\right|^2 = I_0\left[\frac{sin\left(N\frac{\pi +h}{\lambda}\sin\theta\right)}{sin\left(\frac{\pi h}{\lambda}\sin\theta\right)} +\right]^2. +\end{equation} + Here we again neglect the diffraction form each individual slit, assuming that the + size of the slit is much smaller than the separation $h$ between the slits. + +The intensity distributions from a diffraction grating with illuminated + $N=2,5$ and $10$ slits are shown in Fig.~\ref{grating}c. The tallest (\emph{principle}) maxima occur when the denominator + of Eq.(~\ref{Nslit_wDif}) becomes zero: $h\sin\theta=\pm m\lambda$ where + $m=1,2,3,\dots$ is the diffraction order. The heights of the principle maxima are + $I_{\mathrm{max}}=N^2I_0$, and their widths are $\Delta \theta = + 2\lambda/(Nh)$. + Notice that the more slits are illuminated, the narrower diffraction peaks + are, and the better the resolution of the system is: + \begin{equation} +\frac{ \Delta\lambda}{\lambda}=\frac{\Delta\theta}{\theta} \simeq \frac{1}{Nm} +\end{equation} +For that reason in any spectroscopic equipment a light beam is usually expanded +to cover the maximum surface of a diffraction grating. + +\subsection*{Diffraction Grating Equation when the Incident Rays are +not Normal} + +Up to now we assumed that the incident optical wavefront is normal to the pane +of a grating. Let's now consider more general case when the angle of incidence +$\theta_i$ of the incoming wave is different from the normal to the grating, as +shown in Fig. \ref{DGnotnormal}a. Rather then calculating the whole intensity +distribution, we will determine the positions of principle maxima. The path +length difference between two rays 1 and 2 passing through the consequential +slits us $a+b$, where: +\begin{equation} +a=h\sin \theta_i;\,\, b=h\sin \theta_R +\end{equation} +Constructive interference occurs for order $m$ when $a+b=m\lambda$, or: +\begin{equation} +h\sin \theta_i + \sin\theta_R=m\lambda +\end{equation} +\begin{figure}[h] +\includegraphics[width=\columnwidth]{pic4i.eps} +%\includegraphics[height=3in]{dn.eps} +\caption{\label{DGnotnormal}Diagram of the light beams diffracted to the Right +(a) and to the Left (b).} +\end{figure} +Now consider the case shown in Fig. \ref{DGnotnormal}. The path length between +two beams is now $b-a$ where $b=h\sin\theta_L$. Combining both cases we have: +\begin{eqnarray} \label{angles} +h\sin\theta_L-\sin\theta_i&=&m\lambda\\ +h\sin\theta_R+\sin\theta_i&=&m\lambda \nonumber +\end{eqnarray} +Adding these equations and dividing the result by 2 yields Eq.(\ref{mlambda}): +\begin{equation}m\lambda=\frac{h}{2}\left(\sin\theta_L+\sin\theta_R\right) +\end{equation} + +\end{document} +\newpage diff --git a/manual_source/chapters/interferometry.tex b/manual_source/chapters/interferometry.tex new file mode 100644 index 0000000..112e493 --- /dev/null +++ b/manual_source/chapters/interferometry.tex @@ -0,0 +1,274 @@ +%\chapter*{Michelson Interferometer} +%\addcontentsline{toc}{chapter}{Michelson Interferometer} +%\documentclass{article} +%\usepackage{tabularx,amsmath,boxedminipage,epsfig} +% \oddsidemargin 0.0in +% \evensidemargin 0.0in +% \textwidth 6.5in +% \headheight 0.0in +% \topmargin 0.0in +% \textheight=9.0in + +%\begin{document} +\chapter{Optical Interferometery} +\setcounter{figure}{1} +\setcounter{table}{1} +\setcounter{equation}{1} + +%\date {} +%\maketitle + +\noindent + \textbf{Experiment objectives}: Assemble and align Michelson and Fabry-Perot +interferometers, calibrate them using a laser of known wavelength, and then use them characterize the bright yellow emission line in sodium (Na). + +\section*{Introduction} + +Optical interferometers are the instruments that rely on interference of two or more superimposed reflections of the input laser beam. These are one of the most common optical tools, and are used for precision measurements, surface diagnostics, astrophysics, seismology, quantum information, etc. There are many configurations of optical interferometers, and in this lab you will become familiar with two of the more common setups. + +The \textbf{Michelson interferometer}, shown in Fig.~\ref{fig1mich.fig}, is based on the interference of two beams: the initial light is split into two arms on a beam splitter, and then these resulting beams are reflected and recombined on the same beamsplitter again. The difference in optical paths in the two arms leads to a changing relative phase of two beams, so when overlapped the two light fields will interfere constructively or destructively. +\begin{figure}[h] +\centering +\includegraphics[width=0.8\linewidth]{./pdf_figs/fig1} \caption{\label{fig1mich.fig}A Michelson Interferometer setup.} +\end{figure} +Such an interferometer was first used by Michelson and Morley in 1887 to determine that electromagnetic waves propagate in vacuum, giving the first strong evidence against the theory of a \textit{luminiferous aether} (a fictious medium for light wave propagation) and providing insight into the true nature of electromagnetic radiation. Michelson interferometers are widely used in many areas of physics and engineering. At the end of this writeup we describe LIGO, the world's largest Michelson interferometer, designed to measure the graviational waves and thus test general relativity. + +Figure~\ref{fig1mich.fig} shows the traditional setting for a Michelson interferometer. A beamsplitter (a glass plate which is partially silver-coated on the front surface and angled at 45 degrees) splits the laser beam into two parts of equal amplitude. One beam (that was initially transmitted by the beamsplitter) travels to a fixed mirror $M_1$ and back again. One-half of this amplitude is then reflected from the partially-silvered surface and directed at 90 degrees toward the observer (you will use a viewing screen). At the same time the second beam (reflected by the beamsplitter) travels at 90 degrees toward mirror $M_2$ and back. Since this beam never travels through the glass beamsplitter plate, its optical path length is shorter than for the first beam. To compensate for that, it passing twice through a clear glass plate called the compensator plate, that has the same thickness. At the beamsplitter one-half of this light is transmitted to an observer, overlapping with the first beam, and the total amplitude of the light at the screen is a combination of amplitude of the two beams: +\begin{equation} +\mathbf{E}_{total} = \mathbf{E}_1 + \mathbf{E}_2 = \frac12 \mathbf{E}_0 + \frac12 \mathbf{E}_0\cos(k\Delta l) +\end{equation} +Here $k\Delta l$ is a phase shift (``optical delay'') between the two wavefronts caused by the difference in +optical pathlengths for the two beams $\Delta l$ , $k=2\pi n/\lambda$ is the wave number, $\lambda$ is the +wavelength of light in vacuum, and $n$ is the refractive index of the optical medium (in our case - air). + +Mirror $M_2$ is mounted on a precision traveling platform. Imagine that we adjust its position (by turning the micrometer screw) such that the distance traversed on both arms is exactly identical. Because the thickness of the compensator plate and the beamsplitter are the same, both wavefronts pass through the same amount of glass and air, so the pathlength of the light beams in both interferometer arms will be exactly the same. Therefore, the two fields will arrive in phase to the observer, and their amplitudes will add up constructively, producing a bright spot on the viewing screen. If now you turn the micrometer to offset the length of one arm by a half of light wavelength, $\Delta l = \lambda/2$, they will acquire a relative phase of $\pi$, and total destructive interference will occur: +\begin{displaymath} +\mathbf{E}_1 +\mathbf{E}_2=0\;\;\mathrm{or} \;\;\mathbf{E}_1 = -\mathbf{E}_2. +%\end{displaymath} +% or +%\begin{displaymath}\mathbf{E}_1(t) = -\mathbf{E}_2(t). +\end{displaymath} +It is easy to see that constructive interference happens when the difference between pathlengths in the two interferometer arms is equal to the integer number of wavelengths $\Delta l = m\lambda$, and destructive interference corresponds to a half-integer number of wavelengths $\Delta l = (m + 1/2) \lambda$ (here $m$ is an integer number). Since the wavelength of light is small ($600-700$~nm for a red laser), Michelson interferometers are able to measure distance variation with very good precision. + + +In \textbf{Fabry-Perot configuration} the input light field bounces between two closely spaced partially reflecting surfaces, creating a large number of reflections. Interference of these multiple beams produces sharp spikes in the transmission for certain light frequencies. Thanks to the large number of interfering rays, this type of interferometer has extremely high resolution, much better than a Michelson interferometer. For that reason Fabry-Perot interferometers are widely used in telecommunications, lasers and spectroscopy to control and measure the wavelengths of light. In this experiment we will take advantage of the high spectral resolution of the Fabry-Perot interferometer to resolve two very closely-spaces emission lines in Na spectra by observing changes in overlapping interference fringes from the two lines. +\begin{figure}[h] +\centering +\includegraphics[width=0.8\linewidth]{./pdf_figs/fpfig1} \caption{\label{fpfig1}Sequence of Reflection and Transmission for a ray arriving at the treated inner surfaces $P_1 \& P_2$.} +\end{figure} + +A Fabry-Perot interferometer consists of two parallel glass plates, flat to better than 1/4 of an optical +wavelength $\lambda$, and coated on the inner surfaces with a partially transmitting metallic layer. Such +two-mirror arrangement is normally called an {\it optical cavity}. The light in a cavity by definition bounces +back and forth many time before escaping; the idea of such a cavity is crucial for the construction of a laser. +Any light transmitted through such cavity is a product of interference between beams transmitted at each bounce +as diagrammed in Figure~\ref{fpfig1}. When the incident ray arrives at interface point $A$, a fraction $t$ is +transmitted and the remaining fraction $r$ is reflected, such that $t + r = 1$ ( this assumes no light is lost +inside the cavity). The same thing happens at each of the points $A,B,C,D,E,F,G,H\ldots$, splitting the initial +ray into parallel rays $AB,CD,EF,GH,$ etc. Between adjacent ray pairs, say $AB$ and $CD$, there is a path +difference of : +\begin{equation} + \delta = BC+CK +\end{equation}%eq1 + where $BK$ is normal to $CD$. In a development +similar to that used for the Michelson interferometer, you can show that: +\begin{equation} + \delta = 2d\cos\theta +\end{equation}%eq.2 + If this path difference produces +constructive interference, then $\delta$ is some integer multiple of $\lambda$, namely, +\begin{equation} + m\lambda = 2d\cos\theta %eq.3 +\end{equation}%eq.3 + +This applies equally to ray pairs $CD$ and $EF, EF$ and $GH$, etc, so that all parallel rays to the right of +$P2$ will constructively interfere with one another when brought together. + +Issues of intensity of fringes \& contrast between fringes and dark background are addressed in Melissinos, {\it +Experiments in Modern Physics}, pp.309-312. + +\subsection*{Laser Safety} +\textbf{Never look directly at the laser beam!} Align the laser so that it is not at eye level. Even a weak +laser beam can be dangerous for your eyes. + + +\section*{Alignment of Michelson interferometer} + +\textbf{Equipment needed}: Pasco precision interferometry kit, a laser, Na lamp, adjustable-height platform (or a few magazines or books). + +To simplify the alignment of a Michelson interferometer, it is convenient to work with diverging optical beams. In this case an interference pattern will look like a set of concentric bright and dark circles, since the components of the diverging beam travel at slightly different angles, and therefore acquire different phase, as illustrated in Figure \ref{fig2mich.fig}. Suppose that the actual pathlength difference between two arms is $d$. Then the path length difference for two off-axis rays arriving at the observer is $\Delta l = a + b$ where $a = +d/\cos \theta$ and $b = a\cos 2\theta$: +\begin{equation} +\Delta l=\frac{d}{\cos \theta}+\frac{d}{\cos \theta}\cos 2\theta +\end{equation} +Recalling that $\cos 2\theta = 2(\cos \theta)^2-1$, we obtain $ \Delta l=2d\cos\theta$. The two rays interfere +constructively for any angle $\theta_c$ for which $ \Delta l=2d\cos\theta = m\lambda$ ($m$=integer); at the same +time, two beams traveling at the angle $\theta_d$ interfere destructively when $ \Delta l=2d\cos\theta = +(m+1/2)\lambda$ ($m$=integer). Because of the symmetry about the normal direction to the mirrors, this will mean +that interference ( bright and dark fringes) appears in a circular shape. If fringes are not circular, it means +simply that the mirrors are not parallel, and additional alignment of the interferometer is required. + +\begin{figure} +\centering +\includegraphics[width=0.8\linewidth]{./pdf_figs/fig2} \caption{\label{fig2mich.fig}Explanation of circular fringes. Notice that to simplify the figure we have ``unfold'' the interferometer by neglecting the reflections on the beamsplitter.} +\end{figure} + +When the path length difference $\Delta l$ is varied by moving one of the mirrors using the micrometer, the fringes appear to ``move''. As the micrometer is turned, the condition for constructive and destructive interference is alternately satisfied at any given angle. If we fix our eyes on one particular spot and count, for example, how many bright fringes pass that spot as we move mirror $M_2$ by a known distance, we can determine the wavelength of light in the media using the condition for constructive interference, $\Delta l = 2d\cos +\theta = m\lambda$. + +For simplicity, we might concentrate on the center of the fringe bull's eye at $\theta = 0$. The equation above +for constructive interference then reduces to $2\Delta l = m\lambda$ (m = integer). If $X_1$ is the initial +position of the mirror $M_2$ (as measured on the micrometer) and $X_2$ is the final position after a number of +fringes $\delta m$ has been counted, we have $2(X_2-X_1) = \lambda\Delta m$. Then the laser wavelength, +$\lambda$, is then given as: +\begin{equation}\label{old3} +\lambda = 2(X_2-X_1)/\Delta m. +\end{equation} + +Set up the interferometer as shown in Figure~\ref{fig1mich.fig} using components from the PASCO interferometry kit. A mirrors $M_{1,2}$ are, correspondingly, a movable and an adjustable mirror from the kit. Align the interferometer with a laser beam. Adjust the beam so that it is impinging on the beamsplitter and on the viewing screen. Try to make the beams to hit near the center of all the optics, including both mirrors, the compensator plate and beam splitter. The interferometer has leveling legs which can be adjusted. Align the beams such that they overlap on the viewing screen, and so that the reflected beam is directed back into the laser. This can be tricky to get right the first time. Be patient, make small changes, think about what you are doing, and get some help from the instructor and TA. + +Once the interferometer is aligned, insert a convex lens ($f=\unit[18]{mm}$ works well) after the laser to spread out the beam (ideally the laser beam should be pass through the center of the lens to preserve alignment). Adjust the adjustable mirror slightly until you see the interference fringes in the screen. Continue make small adjustments until you see a clear bull's eye circular pattern. \emph{A word of caution: sometimes dust on a mirror or imperfections on optical surfaces may produce similar intensity patterns. True interference disappears if you block one arm of the interferometer. Try it!} + +\textbf{Note}: before starting the measurements, make sure you understand how to read the micrometer properly! +\begin{figure}[h] +\centering +\includegraphics[width=0.7\columnwidth]{./pdf_figs/fig3} +\caption{\label{fig3mich.fig}Micrometer readings. The course division equals to 100~$\mu$m, and smallest division on the rotary dial is 1~$\mu$m (same as 1 micron). The final measurements is the sum of two. } +\end{figure} + +\section*{Wavelength measurements using Michelson interferometer} + +\subsection*{ Calibration of the interferometer} + +Record the initial reading on the micrometer. Focus on a the central fringe and begin turning the micrometer. You will see that the fringes move. For example, the central spot will change from bright to dark to bright again, that is counted as one fringe. A good method: pickout a reference line on the screen and then softly count fringes as they pass the point. Count a total of about $\Delta m = 50$ fringes and record the new reading on the micrometer. + +Each lab partner should make at least two independent measurements, starting from different initial positions of the micrometer. For each trial, approximately 50 fringes should be accurately counted and tabulated with the initial $X_1$ and final $X_2$ micrometer settings. Do this at least five times (e.g., $5\times 50$ fringes). Consider moving the mirror both forward and backward. Make sure that the difference $X_2-X_1$ is consistent between all the measurements. Calculate the average value of the micrometer readings $<X_2-X_1>$. + +%When your measurements are done, ask the instructor how to measure the wavelength of the laser using a commercial wavemeter. Using this measurement and Equation~\ref{old3} calculate the true distance traveled by the mirror $\Delta l$, and calibrate the micrometer (i.e. figure out precisely what displacement corresponds to one division of the micrometer screw dial). + +%\noindent\textbf{\emph{Experimental tips}}: +\subsection*{Experimental tips} +\begin{enumerate} +\item Avoid touching the face of the front-surface mirrors, the beamsplitter, and any other optical elements! +\item The person turning the micrometer should also do the counting of fringes. It can be easier to count them in bunches of 5 or 10 (\textit{i.e.} 100 fringes = 10 bunches of 10 fringes). +\item Use a reference point or line and count fringes as they pass. +\item Before the initial position $X_1$ is read make sure that the micrometer has engaged the drive screw (There can be a problem with "backlash"). Just turn it randomly before counting. +\item Avoid hitting the table which can cause a sudden jump in the number of fringes. +\end{enumerate} + +\subsection*{Measurement of the Na lamp wavelength} + +A calibrated Michelson interferometer can be used as a \textbf{wavemeter} to determine the wavelength of different light sources. In this experiment you will use it to measure the wavelength of strong yellow sodium fluorescent light, produced by the discharge lamp.\footnote{Actually, sodium might be a better calibration source than a HeNe laser, since it has well known lines, whereas a HeNe can lase at different wavelenths. Perhaps an even better calibration source might be a line from the Hydrogen Balmer series, which can be calculated from the Standard Model.} + +Without changing the alignment of the interferometer (i.e. without touching any mirrors), remove the focusing lens and carefully place the interferometer assembly on top of an adjustable-hight platform such that it is at the same level as the output of the lamp. Since the light power in this case is much weaker than for a laser, you won't be able to use the viewing screen. You will have to observe the interference looking directly to the output beam - unlike laser radiation, the spontaneous emission of a discharge is not dangerous\footnote{In the ``old days'' beams in high energy physics were aligned using a similar technique. An experimenter would close his eyes and then put his head in a columated particle beam. Cerenkov radiation caused by particles traversing the experimenter's eyeball is visible as a blue glow or flashes. This is dangerous but various people claim to have done it... when a radiation safety officer isn't around.} However, your eyes will get tired quickly! Placing a diffuser plate in front of the lamp will make the observations easier. Since the interferometer is already aligned, you should see the interference picture. Make small adjustments to the adjustable mirror to make sure you see the center of the bull's eye. + +Repeat the same measurements as in the previous part by moving the mirror and counting the number of fringes. Each lab partner should make at least two independent measurements, recording initial and final position of the micrometer, and you should do at least five trials. Calculate the wavelength of the Na light for each trial. Then calculate the average value and its experimental uncertainty. Compare with the expected value of \unit[589]{nm}. + +In reality, the Na discharge lamp produces a doublet - two spectral lines that are very close to each other: \unit[589]{nm} and \unit[589.59]{nm}. Do you think your Michelson interferometer can resolve this small difference? Hint: the answer is no - we will use a Fabry-Perot interferometer for that task. + +\section*{Alignment of the Fabry-Perot interferometer} + +\begin{figure} +\centering +\includegraphics[width=0.8\linewidth]{./pdf_figs/fpfig3} \caption{\label{fpfig3.fig}The Fabry-Perot Interferometer. For initial alignment the laser and the convex lens are used instead of the Na lamp.} +\end{figure} +Disassemble the Michelson Interferometer, and assemble the Fabry-Perot interferometer as shown in +Figure~\ref{fpfig3.fig}. First, place the viewing screen behind the two partially-reflecting mirrors ($P1$ and $P2$), and adjust the mirrors such that the multiple reflections on the screen overlap. Then place a convex lens after the laser to spread out the beam, and make small adjustments until you see the concentric circles. Is there any difference between the thickness of the bright lines for two different interferometers? Why? + +Loosen the screw that mounts the movable mirror and change the distance between the mirrors. Realign the interferometer again, and comment on the difference in the interference picture. Can you explain it? + +Align the interferometer one more time such that the distance between two mirrors is $1.0 - \unit[1.5]{mm}$, but make sure the mirrors do not touch! + +\subsection*{Sodium doublet measurements} + +\begin{enumerate} +\item Turn off the laser, remove the viewing screen and the lens, and place the interferometer on the adjustable-height platform, or alternatively place the Na lamp on it's side and plan to adjust it's height with books or magazines. With the diffuser sheet in front of the lamp, check that you see the interference fringes when you look directly to the lamp through the interferometer. If necessary, adjust the knobs on the adjustable mirror to get the best fringe pattern. + +\item Because the Na emission consists of two light at two close wavelengthes, the interference picture consists of two sets of rings, one corresponding to fringes of $\lambda_1$, the other to those for $\lambda_2$. Move the mirror back and forth (by rotating the micrometer) to identify two sets of ring. Notice that they move at slightly different rate (due to the wavelength difference). + +\item Seek the START condition illustrated in Fig.(\ref{fpfig4.fig}), such that all bright fringes are evenly spaced. Note that alternate fringes may be of somewhat different intensities. Practice going through the fringe conditions as shown in Fig.(\ref{fpfig4.fig}) by turning the micrometer and viewing the relative movement of fringes. Do not be surprised if you have to move the micrometer quite a bit to return to the original condition again. + +\item Turn the micrometer close to zero reading, and then find a place on the micrometer ($d_1$) where you have the ``START'' condition for fringes shown in Fig.(\ref{fpfig4.fig}). Now advance the micrometer rapidly while viewing the fringe pattern ( NO COUNTING OF FRINGES IS REQUIRED ). Note how the fringes of one intensity are moving to overtake those of the other intensity (in the manner of Fig.(\ref{fpfig4.fig})). Keep turning until the ``STOP'' pattern is achieved (the same condition you started with). Record the micrometer reading as $d_2$. + +\item Each lab partner should repeat this measurement at least one time, and each group should have at least three independent measurements. + +\item Make sure to tabulate all the data taken by your group. + +\end{enumerate} + +We chose the ``START'' condition (the equally spaced two sets of rings) such that for the given distance between two mirrors, $d_1$, the bright fringes of $\lambda_1$ occur at the points of destructive interference for $\lambda_2$. Thus, the bull's eye center ($\theta= 0 $) we can write this down as: + +\begin{equation} +2d_1=m_1\lambda_1=\left(m_1+n+\frac{1}{2}\right)\lambda_2 +\end{equation} + +Here the integer $n$ accounts for the fact that $\lambda_1 > \lambda_2$, and the $1/2$ for the condition of destructive interference for $\lambda_2$ at the center. The ``STOP'' condition corresponds to the similar situation, but the net action of advancing by many fringes has been to increment the fringe count of $\lambda_2$ by one more than that of $\lambda_1$: + +\begin{equation} +2d_2=m_2\lambda_1=\left(m_2+n+\frac{3}{2}\right)\lambda_2 +\end{equation} + +Try to estimate your uncertainty in identifying the START and STOP positions by turning the micrometer back and forth, identifying the points at which you can begin to see doublet, rather than equally spaced, lines. The variance from multiple measurements should agree, at least approximately, with this estimate. + +Subtracting the two interference equations, and solving for the distance traveled by the mirror $d_2-d_1$ we obtain: + +\begin{equation} +2(d_2-d_1)=\frac{\lambda_1\lambda_2}{(\lambda_1-\lambda_2)} +\end{equation} + +Solving this for $\Delta \lambda = \lambda_1-\lambda_2$, and accepting as valid the approximation that $\lambda_1\lambda_2\approx \lambda^2$ ( where $\lambda$ is the average of $\lambda_1$ and $\lambda_2 \approx 589.26 nm$ ), we obtain: + +\begin{equation} +\boxed{\Delta\lambda=\frac{\lambda^2}{2(d_2-d_1)}} +\end{equation} + +Use this equation and your experimental measurements to calculate average value of Na doublet splitting and its standard deviation. Compare your result with the established value of $\Delta \lambda_{Na}=0.598$~nm. + +\begin{figure}[h] +\centering +\includegraphics[width=0.7\linewidth]{./pdf_figs/fpfig4} \caption{\label{fpfig4.fig}The Sequence of fringe patterns encountered in the course of the measurements. Note false colors: in your experiment the background is black, and both sets of rings are bright yellow.} +\end{figure} + +\newpage +\section*{Detection of Gravitational Waves} + +\textbf{A Michelson interferometer can help to test the theory of relativity!} +% +Gravity waves, predicted by the theory of relativity, are ripples in the fabric +of space and time produced by violent events in the distant universe, such as +the collision of two black holes. Gravitational waves are emitted by +accelerating masses much as electromagnetic waves are produced by accelerating +charges, and often travel to Earth. The only indirect evidence for these waves +has been in the observation of the rotation of a binary pulsar (for which the +1993 Nobel Prize was awarded). +% +\begin{figure}[h] +\centering +\includegraphics{./pdf_figs/LIGO} \caption{\label{LIGO.fig}For more details see http://www.ligo.caltech.edu/} +\end{figure} +Laser Interferometry Gravitational-wave Observatory (LIGO) sets the ambitious +goal to direct detection of gravitational wave. The measuring tool in this +project is a giant Michelson interferometer. Two mirrors hang $2.5$~mi apart, +forming one "arm" of the interferometer, and two more mirrors make a second arm +perpendicular to the first. Laser light enters the arms through a beam splitter +located at the corner of the L, dividing the light between the arms. The light +is allowed to bounce between the mirrors repeatedly before it returns to the +beam splitter. If the two arms have identical lengths, then interference +between the light beams returning to the beam splitter will direct all of the +light back toward the laser. But if there is any difference between the lengths +of the two arms, some light will travel to where it can be recorded by a +photodetector. + +The space-time ripples cause the distance measured by a light beam to change as the gravitational wave passes by. These changes are minute: just $10^{-16}$ centimeters, or one-hundred-millionth the diameter of a hydrogen atom over the $2.5$ mile length of the arm. Yet, they are enough to change the amount of light falling on the photodetector, which produces a signal defining how the light falling on changes over time. LlGO requires at least two widely separated detectors, operated in unison, to rule out false signals and confirm that a gravitational wave has passed through the earth. Three interferometers were built for LlGO -- two near Richland, Washington, and the other near Baton Rouge, Louisiana. +% +%\begin{figure} +%\centering +%\includegraphics{LISA.eps} \caption{\label{LISA.fig}For more details see http://lisa.nasa.gov/} +%\end{figure} + +%\emph{ +% +%LIGO is the family of the largest existing Michelson interferometers, but just wait for a few years until LISA (Laser Interferometer Space Antenna) - the first space gravitational wave detector - is launched. LISA is essentially a space-based Michelson interferometer: three spacecrafts will be arranged in an approximately equilateral triangle. Light from the central spacecraft will be sent out to the other two spacecraft. Each spacecraft will contain freely floating test masses that will act as mirrors and reflect the light back to the source spacecraft where it will hit a detector causing an interference pattern of alternating bright and dark lines. The spacecrafts will be positioned approximately 5 million kilometers from each other; yet it will be possible to detect any change in the distance between two test masses down to 10 picometers (about 1/10th the size of an atom)! +% +%} + diff --git a/manual_source/chapters/intro.tex b/manual_source/chapters/intro.tex new file mode 100644 index 0000000..a3ca846 --- /dev/null +++ b/manual_source/chapters/intro.tex @@ -0,0 +1,145 @@ +\chapter*{Introduction} +\addcontentsline{toc}{chapter}{Introduction} + Welcome to Experimental Atomic Physics Laboratory! What is + this class all about? In this class you will learn more + details about how experimental physics is done. The + experiments you do here will help you further learn the + concepts you are being introduced to in Physics 201, Modern + Physics. You will learn by doing. You will learn about the + scientific process and be introduced to what it takes to be a + physicist. Hopefully, you will have fun too! You will be doing +. some experiments which are very fundamental, some of which + have won the Nobel Prize! + +The goals of this class are to: + +\begin{enumerate} +\item Gain understanding of physical principles. +\item Become familiar with the setup of experimental equipment, how to + use equipment, and how to make measurements. +\item Learn how to analyze your data, determine the error in your data, +how to graph data and how to fit the data to a curve to extract parameters. +\item Learn how to draw conclusions from your data. +\item Learn how to keep a scientific journal. +\item Learn how to approach a problem +\item Learn to communicate your findings to other people in a way which is +clear and concise. +\end{enumerate} + +Some or most of these principles you were introduced to in Physics +Laboratory 101. How is this class different? The experiments you will +do are taking you a step closer to the work that actual physicsist do +in the laboratory. You will be repeating some very fundamental, +complex experiments through which physical principles were discovered. +You will be carrying out the steps required to do experimental +physics: setup equipment, make measurements, record data, analyze +data, draw conclusions and communicate your findings in a scientific +report. You may say, well, I am only going to be a theorist, or a +banker, why should I care about experimental physics? Well, atleast +you should learn about the scientific process so that when you hear +about some major scientific discovery you can judge its merit. In this +way you will learn what it takes to extract a physical principle from +an experiment, so you understand how scientists make the connections +that they do, and what the limitations are to scientific +experimentation. + +This laboratory manual briefly summarizes the principles of general +laboratory practice, treatment of error and curve fitting, how to +write a laboratory report, and then each of the laboratories you will +be conducting this semester. Please read the chapter on the experiment +you will do before you do the actual experiment. + + +\section*{General Laboratory Practices} + +\begin{enumerate} +\item When conducting your experiments- be safe! You will be using + equipment which poses some hazards, such as lasers and high-voltage + power supplies. Listen to the safety instructions and heed + them. Also, if a piece of equipment isn’t working even after you + have followed all the instructions, be careful what you fiddle + with! Some fiddling is good, but if you are planning to do anything + major (like take a piece of equipment apart), it is best to ask an + instructor first. Also, it is generally good lab practice not to + eat or drink in the lab. It keeps crumbs and liquid out the + equipment, and prevents you from eating or drinking something you + didn’t intend to. + + +\item Keep a good laboratory book and record your data and the steps you + take! It is recommended you buy a separate notebook to keep as a + laboratory journal. Don’t scribble inledigbly on pieces of scrap + paper you will only lose later. When conducting an experiment, + right down what you did, how you setup the equipment and if + anything unusual happened. Write down data in a neat and organized + way. The goal is to read and understand what you did after you + leave the laboratory. Don’t think this is trivial! Major scientific + discoveries were made because of some anomaly in data, which + scientists were able to exactly repeat because they had recorded + exactly what they did! If you are a research scientist in a + corporation and you make a discovery like this and couldn’t repeat + the conditions under which it was found- you would be in big + trouble! + +\item Related to the last point: don’t leave this laboratory for the day +without making sure you understand your data. You won’t be able to go +back and redo the experiment- so it is best to check your data and +make sure it is reasonable. 4. +Do not fudge data! If your data is +off and you admit it and speculate why it is off, you will be given +credit for this. Everyone has a bad day. Forging data not only breaks +the Honor Code but is also a very, very bad habit which can have +serious consequences in the future. Some sceintists have been tempted +under pressure to do this. Those who did were usually found out and +the ramifications are very serious. Not only does it hurt society, but +the scientists found doing this ruined their careers. See the book, Ҡ+for further discussion of this problem. + +\item Treatment of errors/curve +fitting + +\item How to Write a Lab Report + +\framebox{You are not writing a laboratory report just for a grade! It is +important that you learn how to communicate your findings. In reading +your report, someone should be able to understand: your hypothesis or +theory, 1) how you did the experiment, 2) what equipment you used 3) +did anything unusual happen? conditions? data – in table and graph +form, analysis you did, conclusions, If anything strange happened- +mention it! If someone cannot tell what you did or gather useful +information, your paper is worthless!} + +\begin{description} + +\item[TITLE OF EXPERIMENT] +\item[LIST PARTNERS by name] + +\end{description} + + +The components of your laboratory report shall include: + +\begin{description} +\item[INTRODUCTION]: Briefly state objective(s) of experiment +\item[THEORY]: Introduce important equations or at least restate in +your own words; + Specify any "Working Eqs.", defining the variables involved. +\item[PROCEDURES] : + Describe in terms of equipment and layout diagrams, + Describe important steps in producing data, + Describe variables to be measured. +\item[DATA / ANALYSIS TABLES]: + Present data in spreadsheet form (rows \& columns which are +clearly defined). + Analyzed results can be listed in the same spreadsheet. +\item[DATA ANALYSIS]: Carry out explicit "sample calculations" to show how +results are produced +Results can appear in DATA / ANALYSIS TABLES +\item[CONCLUSIONS]: Conclusions should be developed; attempt to put them in +quantitative terms + +\end{description} +\end{enumerate} + +\fbox{this is some text.} +\newpage + diff --git a/manual_source/chapters/michelson.tex b/manual_source/chapters/michelson.tex new file mode 100644 index 0000000..101fa64 --- /dev/null +++ b/manual_source/chapters/michelson.tex @@ -0,0 +1,311 @@ +%\chapter*{Michelson Interferometer} +%\addcontentsline{toc}{chapter}{Michelson Interferometer} +\documentclass{article} +\usepackage{tabularx,amsmath,boxedminipage,epsfig} + \oddsidemargin 0.0in + \evensidemargin 0.0in + \textwidth 6.5in + \headheight 0.0in + \topmargin 0.0in + \textheight=9.0in + +\begin{document} +\title{Michelson Interferometer} +\date {} +\maketitle + +\noindent + \textbf{Experiment objectives}: Assemble and align a Michelson +interferometer, and use it to measure wavelength of unknown laser, and the +refractive index of air. + +\section*{History} + +Michelson interferometer is an extremely important apparatus. It was used by Michelson and Morley in 1887 to +determine that electromagnetic waves propagate in vacuum, giving the first strong evidence against the theory of +a \textit{luminiferous aether} (a fictious medium for light wave propagation) and providing the insight into +the true nature of electromagnetic radiation. Nowedays, Michelson interferometer remains a widely used tool in +many areas of physics and engineering. In this laboratory you will use the interferometer to accurately measure +the wavelength of laser light and the index of refraction of air. +\begin{figure}[h] +\centerline{\epsfig{width=0.8\linewidth,file=fig1.eps}} \caption{\label{fig1mich.fig}A Michelson Interferometer +setup.} +\end{figure} + +\section*{Theory} + + The interferometer works by combining two light waves + traversing two path lengths. A diagram of this type of + interferometer is shown in Figure!\ref{fig1mich.fig} + A beamsplitter (a glass + plate which is partially silver-coated on the front surface + and angled at 45 degrees) splits the laser beam into two parts of equal + amplitude. One beam (reflected by the + beamsplitter) travels at 90 degrees toward mirror $M_2$ and back + again, passing twice through a clear glass plate called the + compensator plate. At the beamsplitter one-half of + this light is transmitted to an observer (you will use a + viewing screen). At the same time the other beam (that was initially transmitted by the beamsplitter) + travels to + a fixed mirror $M_1$ and back again. One-half of this amplitude + is reflected from the partially-silvered surface and directed + at 90 degrees toward the observer. Thus, the total amplitude of the light the observer + records is a combination of amplitude of the two beams: +\begin{equation} +\mathbf{E}_{total} = \mathbf{E}_1 + \mathbf{E}_2 = \frac12 \mathbf{E}_0 + \frac12 \mathbf{E}_0\cos(k\Delta l) +\end{equation} + +Here $k\Delta l$ is a phase shift (``optical delay'') between the two wavefronts caused by the difference in +pathlengths for the two beams $\Delta l$ , $k=2\pi n/\lambda$ is the wave number, $\lambda$ is the wavelength +of light in vacuum, and $n$ is the refractive index of the optical medium (in our case - air). + +Mirror $M_2$ is mounted on a precision traveling platform. Imagine that we adjust its position (by turning the +micrometer screw) such that the distance traversed on both arms is exactly identical. Because the thickness of +the compensator plate and the beamsplitter are the same, both wavefronts pass through the same amount of glass +and air, so the pathlength of the light beams in both interferometer arms will be exactly the same. Therefore, +two fields will arrive in phase to the observer, and their amplitudes will add up constructively, producing a +bright spot on the viewing screen. If now you turn the micrometer to offset the length of one arm by a half of +light wavelength, $\Delta l = \lambda/2$, they will acquire a relative phase of $\pi$, and total destructive +interference will occur: +\begin{displaymath} +\mathbf{E}_1 +\mathbf{E}_2=0\;\;\mathrm{or} \;\;\mathbf{E}_1 = -\mathbf{E}_2. +%\end{displaymath} +% or +%\begin{displaymath}\mathbf{E}_1(t) = -\mathbf{E}_2(t). +\end{displaymath} +It is easy to see that constructive interference happens when the difference between pathlengths in two +interferometer arms is equal to the integer number of wavelengths $\Delta l = m\lambda$, and destructive +interference corresponds to the half-integer number of wavelengths $\Delta l = (m + 1/2) \lambda$ (here $m$ is +an integer number). Since the wavelength of light is small ($600-700$~nm for a red laser), Michelson +interferometers are able to measure distance variation with very good precision. + + + + +%Figure 1. The Michelson Interferometer + +To simplify the alignment of a Michelson interferometer, it is convenient to work with diverging optical beams. +In this case an interference pattern will look like a set of concentric bright and dark circles, since the +components of the diverging beam travel at slightly different angles, and therefore acquire different phase, as +illustrated in Figure \ref{fig2mich.fig}. Suppose that the actual pathlength difference between two arms is $d$. +Then the path length difference for two off-axis rays arriving at the observer is $\Delta l = a + b$ where $a = +d/\cos \theta$ and $b = a\cos 2\theta$: +\begin{equation} +\Delta l=\frac{d}{\cos \theta}+\frac{d}{\cos \theta}\cos 2\theta +\end{equation} +Recalling that $\cos 2\theta = 2(\cos \theta)^2-1$, we obtain $ \Delta l=2d\cos\theta$. The two rays interfere +constructively for any angle $\theta_c$ for which $ \Delta l=2d\cos\theta = m\lambda$ ($m$=integer); at the same +time, two beams traveling at the angle $\theta_d$ interfere destructively when $ \Delta l=2d\cos\theta = +(m+1/2)\lambda$ ($m$=integer). Because of the symmetry about the normal direction to the mirrors, this will mean +that interference ( bright and dark fringes) appears in a circular shape. If fringes are not circular, it means +simply that the mirrors are not parallel, and additional alignment of the interferometer is required. + +\begin{figure} +\centerline{\epsfig{width=0.8\linewidth,file=fig2.eps}} \caption{\label{fig2mich.fig}Explanation of circular +fringes. Notice that to simplify the figure we have ``unfold'' the interferometer by neglecting the reflections +on the beamsplitter.} +\end{figure} + +When the path length difference $\Delta l$ is varied by moving one of the mirrors using the micrometer, the +fringes appear to "move". As the micrometer is turned, the condition for constructive and destructive +interference is alternately satisfied at any given angle. If we fix our eyes on one particular spot and count, +for example, how many bright fringes pass that spot as we move mirror $M_2$ by known distance, we can determine +the wavelength of light in the media using the condition for constructive interference, $\Delta l = 2d\cos +\theta = m\lambda$. + +For simplicity, we might concentrate on the center of the fringe bullseye at $\theta = 0$. The equation above +for constructive interference then reduces to $2\Delta l = m\lambda$ (m = integer). If $X_1$ is the initial +position of the mirror $M_2$ (as measured on the micrometer) and $X_2$ is the final position after a number of +fringes $\delta m$ has been counted, we have $2(X_2-X_1) = \lambda\delta m$. Then the laser wavelength, +$\lambda$, is then given as: +\begin{equation}\label{old3} +\lambda = 2(X_2-X_1)/\delta m. +\end{equation} + +\section*{Procedure} + +\subsection*{Laser Safety} +While this is a weak laser caution should still be used. \textbf{Never look directly at the laser beam!} Align +the laser so that it is not at eye level. + +\subsection*{Set Up} +\textbf{Equipment needed}: Pasco precision interferometry kit, a laser, +adjustable-hight platform. + +Set up the interferometer as shown in Figure~\ref{fig1mich.fig} using the components of Pasco precision +interferometry kit. A mirrors $M_{1,2}$ are correspondingly a movable and an adjustable mirror from the kit. +Make initial alignment of the interferometer with a non-diverging laser beam. Adjust the beams so that it is +impinging on the beamsplitter and on the viewing screen. Make sure the beam is hitting near the center of all +the optics, including both mirrors, the compensator plate and beam splitter. The interferometer has leveling +legs which can be adjusted. + +Then insert a convex lens after the laser to spread out the beam (ideally the laser beam should be pass through +the center of the lens to preserve alignment). After the beams traverse through the system, the image of the +interfering rays will be a circular pattern projected onto a screen. The two beam reflected off the mirrors +should be aligned as parallel as possible to give you a circular pattern. + +\subsection*{ Measurement of laser wavelength} + +Note the reading on the micrometer. Focus on a particular fringe (the center is a good place). Begin turning the +micrometer so that the fringes move (for example, from bright to dark to bright again is the movement of 1 +fringe). Count a total of about 100 fringes and record the new reading on the micrometer. Calculate the +wavelength from Eq. \ref{old3} above, remembering that you may need to convert the distance traveled on the +micrometer to the actual distance traveled by the mirror. + + + Each lab group must make at least four (4) measurements of $\lambda$. Each + partner must do at least one. For each trial, a minimum of 100 + fringes should be accurately counted, and related to an + initial $X_1$ and final $X_2$ micrometer setting. A final mean + value of $\lambda$ and its uncertainty should be + generated. Compare your value with the accepted value (given + by the instructor). + +\textbf{\emph{Experimental tips}}: +\begin{enumerate} +\item Avoid touching the face of the front-surface mirrors, the beamsplitter, and any other optical elements! +\item Engage the micrometer with both hands as you turn, maintaining +positive torque. +\item The person turning the micrometer should also do the counting of +fringes. It can be easier to count them in bunches of 5 or 10 (\textit{i.e.} +100 fringes = 10 bunches of 10 fringes). +\item Before the initial position $X_1$ is read make sure that the micrometer has engaged the +drive screw (There can be a problem with "backlash"). +\item Before starting the measurements make sure you understand how to read a +micrometer! See Fig.\ref{fig3mich.fig}. +\item Move the travel plate to a slightly different location for the +four readings. This can done by loosening the large nut atop the traveling +plate,and then locking again. +\item Avoid hitting the table which can cause a sudden jump in the +number of fringes. + +\end{enumerate} + +\begin{figure}[h] +\centerline{\epsfig{width=0.7\columnwidth,file=fig3.eps}} \caption{\label{fig3mich.fig}Micrometer readings. The +course scale is in mm, and smallest division on the rotary dial is 1~$\mu$m (same as 1 micron). The final +measurements is the sum of two. } +\end{figure} + + +\subsection*{Measurement of the index of refraction of air} + + If you recall from the speed of light experiment, the value +for air's index of refraction $n_{air}$ is very close to unity: +$n_{air}$=1.000293. Amazingly, a Michelson interferometer is precise enough to +be able to make an accurate measurement of this quantity! + +Let's remind ourselves that a Michelson interferometer is sensitive to a phase +difference acquired by the beams travelling in two arms +\begin{equation}\label{phase} +k\Delta l=2\pi n\Delta l/\lambda. +\end{equation} +In previous calculations we assumed that the index of refraction of air $n$ is exactly one, like in vacuum. +However, it is actually slightly varies with air pressure, as shown in Fig.~\ref{fig4mich.fig}. Any changes in +air pressure affect the phase $k\Delta l$. +% +\begin{figure} +\centerline{\epsfig{file=macfig1add.eps}} \caption{\label{fig4mich.fig}Index of refraction as a function of air +gas pressure} +\end{figure} + +To do the measurement, place a cylindrical gas cell which can be evacuated in +the path of light heading to mirror $M_1$ and correct alignment of the +Michelson interferometer, if necessary. Make sure that the gas cell is +initially at the atmospheric pressure. + +Now pump out the cell by using a hand pump at your station and count the number of fringe transitions $\delta +m$ that occur. When you are done, record $\delta m$ and the final reading of the vacuum gauge $p_{fin}$. +\textbf{Note}: most vacuum gauges display the difference between measured and atmospheric pressure . If +absolute pressure is needed, it should be found by subtracting the gauge reading from the atmospheric pressure +($p_0=76$~cm Hg). For example, if the gauge reads $23$~cm Hg, the absolute pressure is $53$~cm Hg. +Alternatively, you can pump out the air first, and then admit air is slowly to the cell while counting the +number of fringes that move past a selected fixed point. + +The shifting fringes indicate a change in relative optical phase difference for the two arms caused by the the +difference in refractive indices of the gas cell at low and atmospheric pressures $\Delta n$. According to +Eq.(\ref{phase}), this difference is +\begin{equation} \label{delta_n} +\Delta n=\delta m \frac{\lambda}{2d_{cell}} +\end{equation} +where $d_{cell}=3$~cm is the length of the gas cell. + +Since the change in the refractive index $\Delta n$ is linearly depends on the +air pressure $\Delta p=p_0-p_{fin}$, it is now easy to find out the +proportionality coefficient $\Delta n/\Delta p$ and calculate the value of the +refractive index at the atmospheric pressure $n_{air}$. + +Each partner should make one measurement of the fringe shift quantity $\delta m$. Use Eq.(\ref{delta_n}) to find +mean values of the relative change of the refractive index $\Delta n$, proportionality coefficient $\Delta +n/\Delta p$ and $n_{air}$ with corresponding uncertainties. Compare your measurements to the following +accepted experimental values: \\ +Index of Refraction of Air(STP) = 1.000293 \\ + + +\subsection*{\emph{Detection of Gravitational Waves}} + +\textbf{\emph{A Michelson interferometer can help to test the theory of +relativity!}} \emph{ +% +Gravity waves, predicted by the theory of relativity, are ripples in the fabric +of space and time produced by violent events in the distant universe, such as +the collision of two black holes. Gravitational waves are emitted by +accelerating masses much as electromagnetic waves are produced by accelerating +charges, and often travel to Earth. The only indirect evidence for these waves +has been in the observation of the rotation of a binary pulsar (for which the +1993 Nobel Prize was awarded).} +% +\begin{figure}[h] +\centerline{\epsfig{file=LIGO.eps}} \caption{\label{LIGO.fig}For more details see http://www.ligo.caltech.edu/} +\end{figure} +\emph{ +% +Laser Interferometry Gravitational-wave Observatory (LIGO) sets the ambitious +goal to direct detection of gravitational wave. The measuring tool in this +project is a giant Michelson interferometer. Two mirrors hang $2.5$~mi apart, +forming one "arm" of the interferometer, and two more mirrors make a second arm +perpendicular to the first. Laser light enters the arms through a beam splitter +located at the corner of the L, dividing the light between the arms. The light +is allowed to bounce between the mirrors repeatedly before it returns to the +beam splitter. If the two arms have identical lengths, then interference +between the light beams returning to the beam splitter will direct all of the +light back toward the laser. But if there is any difference between the lengths +of the two arms, some light will travel to where it can be recorded by a +photodetector.} + +\emph{ +%The space-time ripples cause the distance measured by a light beam to change as +the gravitational wave passes by. These changes are minute: just $10^{-16}$ +centimeters, or one-hundred-millionth the diameter of a hydrogen atom over the +$2.5$ mile length of the arm. Yet, they are enough to change the amount of +light falling on the photodetector, which produces a signal defining how the +light falling on changes over time. LlGO requires at least two widely separated +detectors, operated in unison, to rule out false signals and confirm that a +gravitational wave has passed through the earth. Three interferometers were +built for LlGO -- two near Richland, Washington, and the other near Baton +Rouge, Louisiana.} +% +\begin{figure} + \centerline{\epsfig{file=LISA.eps}} \caption{\label{LISA.fig}For +more details see http://lisa.nasa.gov/} +\end{figure} + +\emph{ +% +LIGO is the family of the largest existing Michelson interferometers, but just +wait for a few years until LISA (Laser Interferometer Space Antenna) - the +first space gravitational wave detector - is launched. LISA is essentially a +space-based Michelson interferometer: three spacecrafts will be arranged in an +approximately equilateral triangle. Light from the central spacecraft will be +sent out to the other two spacecraft. Each spacecraft will contain freely +floating test masses that will act as mirrors and reflect the light back to the +source spacecraft where it will hit a detector causing an interference pattern +of alternating bright and dark lines. The spacecrafts will be positioned +approximately 5 million kilometers from each other; yet it will be possible to +detect any change in the distance between two test masses down to 10 picometers +(about 1/10th the size of an atom)! +% +} + +\end{document} +\newpage diff --git a/manual_source/chapters/millikan.tex b/manual_source/chapters/millikan.tex new file mode 100644 index 0000000..085c22f --- /dev/null +++ b/manual_source/chapters/millikan.tex @@ -0,0 +1,254 @@ +%\chapter*{Millikan Oil Drop Experiment} +%\addcontentsline{toc}{chapter}{Millikan Oil Drop Experiment} +\documentclass{article} +\usepackage{tabularx,amsmath,boxedminipage,epsfig} + \oddsidemargin 0.0in + \evensidemargin 0.0in + \textwidth 6.5in + \headheight 0.0in + \topmargin 0.0in + \textheight=9.0in + +\begin{document} +\title{Millikan Oil Drop Experiment} +\date {} +\maketitle +\noindent + +\textbf{Experiment objectives}: \\ \textbf{Week 1}: explore the experimental +apparatus and data + acquisition procedure; develop the data analysis routing using a mock Millikan experiment. \\ \textbf{Week 2}: extract the value of a unit charge $e$ by observing the motion + of charged oil drops in gravitational and electric field. + + \begin{boxedminipage}{\linewidth} +\textbf{Warning: this is a hard experiment!} \\ +% +You have two class sessions to complete this experiment - for a good reason: +this experiment is very hard! After all, it took R. A. Millikan 10 years to +collect and analyze enough data to make accurate measurement of the electron +charge. It takes some (often considerable) time to learn how to use the +apparatus and get reliable data with it, so make sure you take take good notes +during the first session on what gives you good and bad results. Also prepare +and debug all the data analysis routines (such as calculations of drop +parameters from the velocity measurements). Then you will hopefully have enough +time + to make reliable measurements during the second session. +\end{boxedminipage} + + \section*{Introduction and Theory} +The electric charge carried by a particle may be calculated by measuring the +force experienced by the particle in an electric field of known strength. +Although it is relatively easy to produce a known electric field, the force +exerted by such a field on a particle carrying only one or several excess +electrons is very small. For example, a field of $1000$~Volts per cm would +exert a force of only $1.6\cdot l0^{-14}$~N dyne on a particle bearing one +excess electron. This is a force comparable to the gravitational force on a +particle with a mass of $l0^{-l2}$~gram. + +The success of the Millikan Oil Drop experiment depends on the ability to +measure forces this small. The behavior of small charged droplets of oil, +having masses of only $l0^{-l2}$~gram or less, is observed in a gravitational +and an electric field. Measuring the velocity of fall of the drop in air +enables, with the use of Stokes’ Law, the calculation of the mass of the drop. +The observation of the velocity of the drop rising in an electric field then +permits a calculation of the force on, and hence, the charge carried by the oil +drop. + +Consider the motion of a small drop of oil inside the apparatus shown in Fig. +\ref{moplates}. +\begin{figure}[h] +\centerline{\epsfig{width=3in, file=modexp.eps}} \caption{\label{moplates} +Schematic Millikan Oil Drop System with and without electric field.} +\end{figure} + + +Because of the air drag tiny droplets fall very slowly with some constant +terminal velocity $v_f$: +\begin{equation}\label{fall} +mg=kv_f +\end{equation} +where $q$ is the charge on the droplet, $m$ is the mass of the droplet, $g$ is +the acceleration due to gravity, and $k$ is a drag coefficient which will be +related to the viscosity of air and the radius of the droplet. + +Because of its small mass the motion of the droplets is sensitive to an +external electric field $E$ even if they carry charges of only a few electrons. +A sufficient electric field can cause the oil drop to rise with a constant +velocity $v_r$, such that: +\begin{equation}\label{rise} +Eq=mg+kv_r +\end{equation} +Combining Eqs.~(\ref{rise},\ref{fall}) we can find the charge $q$: +\begin{equation}\label{q} +q=\frac{mg(v_f+v_r)}{Ev_f} +\end{equation} + +Therefor, the charge of the droplet can be found by measuring its terminal +velocity $v_t$ and rising velocity in the external magnetic field $v_r$. +However, we also need to know the mass and the radius of a drop. These data has +to be extracted from the same data. The drag coefficient, $k$, can be +determined from the viscosity, $\eta$, and the radius of the droplet, $a$, +using Stokes law: +\begin{equation} +k=6\pi a\eta +\end{equation} +The mass of a drop can be related to its radius: +\begin{equation}\label{m} +m=\frac{4}{3}\pi a^3 \rho, +\end{equation} +and one may solve for $a$ using Eq.~(\ref{fall}): +\begin{equation}\label{simple} +a=\sqrt{\frac{9\eta v_f}{2g\rho}} +\end{equation} +Here $\rho=.886\cdot 10^3 \mathrm{kg/m}^3$ is the density of the oil. + +The air viscosity at room temperature is $\eta=1.832\cdot 10^{-5}$Ns/m$^2$ for +relatively large drops. However, there is a small correction for this +experiment for a small drops because the oil drop radius is not so different +from the mean free path of air. This leads to an effective viscosity: +\begin{equation}\label{etaeff} +\eta_{eff}=\eta\frac{1}{1+\frac{b}{Pa}} +\end{equation} +where $b\approx 8.20 \times 10^{-3}$ (Pa$\cdot$m) and $P$ is atmospheric +pressure (1.01 $10^5$ Pa). The idea here is that the effect should be related +to the ratio of the mean free path to the drop radius. This is the form here +since the mean free path is inversely proportional to pressure. The particular +numerical constant can be obtained experimentally if the experiment were +performed at several different pressures. A feature Milikan's apparatus had, +but ours does not. + +To take into the account the correction to the air viscosity, one has to +substitute the expression for $\eta_{eff}$ of Eq.~(\ref{etaeff}) into Eq.~( +\ref{simple}) and then solve this more complex equation for $a$: +\begin{equation}\label{complex} +a=\sqrt{\frac{9\eta v_f}{2g\rho}+\left(\frac{b}{2P}\right)^2}-\frac{b}{2P} +\end{equation} + +Therefor, the calculation of a charge carried by an oil drop will consists of +several steps: +\begin{enumerate} +\item Measure the terminal velocities for a particular drop with and without +electric field. +\item Using the falling terminal velocity with no electric field, calculate +the radius of a droplet using Eq.~(\ref{complex}), and then find the mass of +the droplet using Eq.~(\ref{m}). +\item Substitute the calculated parameters of a droplet into Eq.~(\ref{q}) to +find the charge of the droplet $q$. + +\end{enumerate} +%This second approach leads to: +% +% +%Having found $a$ one can then find $m$ using Eq. \ref{m} and then find +%$q$ from Eq. \ref{q}. I would use this approach rather than the Pasco one. + + + + +\section*{Experimental procedure} + +\subsection*{Mock Millikan experiment - practice of the data analysis} +\textit{The original idea of this experiment is described here: +http://phys.csuchico.edu/ayars/300B/handouts/Millikan.pdf} + +The goal of this section is to develop an efficient data analysis routine for +the electron charge measurements. You will be given a number of envelopes with +a random number of Unidentified Small Objects (USOs), and your goal is to find +a mass of a single USO (with its uncertainty!) without knowing how many USOs +each envelope has. This exercise is also designed to put you in Robert Millikan +shoes (minus the pain of data taking). + +Each person working on this experiment will be given a number of envelopes to +weight. Each envelope contain unknown number of USO plus some packing material. +To save time, all the data will be then shared between the lab partners. + +Then analyze these data to extract the mass of a single USO and its uncertainty +in whatever way you’d like. For example, graphs are generally useful for +extracting the data - is there any way to make a meaningful graph for those +measurements? If yes, will you be able to extract the mean value of USO mass +and its uncertainty from the graph? \textit{Feel free to discuss your ideas +with the laboratory instructor!} + +After finding the mass of a USO, work with your data to determine how the size +of the data set affects the accuracy of the measurements. That will give you a +better idea how many successful measurements one needs to make to determing $e$ +in a real Millikan experiment. + +This part of the experiment must be a part of the lab report, including the +results of your measurements and the description of the data and error analysis +routine. + + + +\subsection*{Pasco Millikan oil drop setup} + +Follow the attached pages from Pasco manual to turn on, align and control the +experimental apparatus. Take time to become familiar with the experimental +apparatus and the measurement procedures. Also, it is highly recommended that +you develop an intuition about ``acceptable'' drops to work with (see Pasco +manual, ``Selection of the Drop'' section). + +\subsection*{Data acquisition and analysis} + +\begin{itemize} + +\item Choose a ``good'' drop and make about 10 measurements for its fall and rise +velocities $v_t$ and $v_r$ by turning the high voltage on and off. Try to find +a drop that does not rise too quickly for it will likely have a large number of +electrons and, further, it will be difficult to determine the $v_r$. If you +can't find slow risers, then lower the voltage so as to get better precision. + + +\item Calculate the charge on the droplet. If the result of this first +determination for the charge on the drop is greater than 5 excess electron, you +should use slower moving droplets in subsequent determinations. Accepted value +of the electron charge is $e=1.6\times10^{-19}$~C. + +\item If the drop is still within viewing range, try to change its charge. To +do that bring the droplet to the top of the field of view and move the +ionization lever to the ON position for a few seconds as the droplet falls. If +the rising velocity of the droplet changes, make as many measurements of the +new rising velocity as you can (10 to 20 measurements). If the droplet is still +in view, attempt to change the charge on the droplet by introducing more alpha +particles, as described previously, and measure the new rising velocity 10–20 +times, if possible. Since making measurements with the same drop with changing +charge allows does not require repeating calculations for the drop mass and +radius, try ``recharging'' the same drop as many times as you can. + +\item Be sure to measure the separation $d$ between the electrodes and the voltage potential in order to +determine the field from the voltage. + +\end{itemize} + +Each lab partner should conduct measurements for at least one drop, and the +overall number of measurements should be sufficient to make a reliable +measurement for the unit electron charge. Make a table of all measurements, +identify each drop and its calculated charge(s). Determine the smallest charge +for which all the charges could be multiples of this smallest charge. Estimate +the error in your determination of $e$. + +% Answer these questions somewhere in your report: +% +%\begin{enumerate} +%\item You will notice that some drops travel upward and others downward +% in the applied field. Why is this so? Why do some drops travel +% very fast, and others slow? +%\item Is the particle motion in a straight line? Or, do you notice that +% the particle "dances" around ever so slightly? This is due to +% Brownian motion: the random motion of a small particle in a gas or +% fluid. +% +% +%\item We made three assumptions in determining the charge from Equation 1 +% above. What are they? Hint: They are related to Stoke's Law. +% +% +%\item Would you, like Millikan, spend 10 years on this experiment? +% +%\end{enumerate} +% +%Extra credit: Millikan and his contemporaries were only able to +%measure integer values of electron charge (as you are). Has anyone +%measured free charges of other than integer multiples of e? + +\end{document} diff --git a/manual_source/chapters/mo.tex b/manual_source/chapters/mo.tex new file mode 100644 index 0000000..2a52bea --- /dev/null +++ b/manual_source/chapters/mo.tex @@ -0,0 +1,156 @@ +\documentclass{article} +\usepackage{tabularx,amsmath,boxedminipage,epsfig} +\begin{document} +\title{Millikan ``Oil Drop'' Experiment} +\author{} \date{} +\maketitle + \section*{Introduction and Theory} +Consider Fig. \ref{moplates} +\begin{figure}[h] +\centerline{\epsfig{width=3in, file=moplates.eps}} +\caption{\label{moplates}Very schematic Millikan Oil Drop System} +\end{figure} +It turns out that very small droplets fall very slowly. Clouds, for +example, are very small water droplets, trying to fall, but held aloft +by very slight air currents. + + +An electric field can take the place of the air current and +even cause the oil drop +to rise. Thus, for a rising oil drop: + +\begin{equation}\label{rise} +Eq=mg+kv_r +\end{equation} +where $E$ is the electric field, $q$ is the charge on the droplet, $m$ +is the mass of the droplet, $g$ is the acceleration due to gravity, $v_r$ +is the velocity rising, and $k$ is a drag coefficient which will be +related to the viscosity of air and the radius of the droplet. + +If the field is off and the droplet is just falling, then: +\begin{equation}\label{fall} +mg=kv_f +\end{equation} +Combining Eqs. \ref{rise} and \ref{fall} we can find the charge $q$: +\begin{equation}\label{q} +q=\frac{mg(v_f+v_r)}{Ev_f} +\end{equation} + + +The drag coefficient, $k$, can be determined from the viscosity, $\eta$, and +the radius of the droplet, $a$: + using Stokes law: +\begin{equation} +k=6\pi a\eta +\end{equation} + +Since +\begin{equation}\label{m} +m=\frac{4}{3}\pi a^3 \rho +\end{equation} +one may solve for $a$: +\begin{equation}\label{simple} +a=\sqrt{\frac{9\eta v_f}{2g\rho}} +\end{equation} +Here $rho=.886\cdot 10^3 kg/m^3$ is the density of the oil. (We ignore the +density of air, which is roughly 1/1000 less.) + +There is a +small correction because the oil drop radius is not so different +from the mean free path of air. This leads to an effective viscosity: +\begin{equation}\label{etaeff} +\eta_{eff}=\eta\frac{1}{1+\frac{b}{pa}} +\end{equation} +where $b\approx 8.20 \times 10^{-3}$ (Pa m) and $p$ is atmospheric +pressure (1.01 $10^5$ Pa). The idea here is that the effect should be +related to the +ratio of the mean free path to the drop radius. This is the form +here since the mean free path is inversely proportional to pressure. +The particular numerical constant can be obtained experimentally if +the experiment were performed at several different pressures. A feature +Milikan's apparatus had, but ours does not. + +There are two approaches at this point that one could take. +\begin{enumerate} +\item One could use Eq. \ref{simple} to determine $a$ using an uncorrected +$\eta$, then use this to determine $\eta_{eff}$ then use this viscosity +in Eq. \ref{simple} again to find a somewhat better $a$, and then proceed +around the loop again until convergence is achieved. If the correction +is large, this can get tedious. + +\item Put $\eta_{eff}$ of Eq. \ref{etaeff} into Eq. \ref{simple} and +then solve this more complex equation for $a$. +\end{enumerate} +This second approach leads to: +\begin{equation}\label{complex} +a=\sqrt{\frac{9\eta v_f}{2g\rho}+\left(\frac{b}{2p}\right)^2}-\frac{b}{2p} +\end{equation} + +Having found $a$ one can then find $m$ using Eq. \ref{m} and then find +$q$ from Eq. \ref{q}. I would use this approach rather than the Pasco one. +Try to find drops that do not rise too quickly for they will likely have +a large number of electrons on them and, further, it will be difficult to +determine the $V_r$. If you can't find slow risers, then lower the voltage +so as to get better precision. Be sure to measure the space thickness in +order to determine the field from the voltage. + + + +\newpage\ + + + + +\subsection*{Required for your report}: + +Make a table of your measurements. Identify the drop and charge. +Determine the charge in each case. Make a table of charge differences. +Determine the smallest charge for which all the charges could be multiples +of this smallest charge. Estimate the error in your determination of e. + + Answer these questions somewhere in your report: + +\begin{enumerate} +\item You will notice that some drops travel upward and others downward + in the applied field. Why is this so? Why do some drops travel + very fast, and others slow? +\item Is the particle motion in a straight line? Or, do you notice that + the particle "dances" around ever so slightly? This is due to + Brownian motion: the random motion of a small particle in a gas or + fluid. + +\item Do you notice distinct steps in the terminal velocity in applied + field? That is, do the terminal velocities appear to clump around + similar values? What does this say about the discrete nature of + charge? + +\item We made three assumptions in determining the charge from Equation 1 + above. What are they? Hint: They are related to Stoke's Law. + +\item How does the average particle diameter you extracted from the + terminal velocity without the field on compare to the value given + on the bottle? Try to explain any discrepancies. + +\item Would you, like Millikan, spend 10 years on this experiment? + +\end{enumerate} + +Extra credit: Millikan and his contemporaries were only able to +measure integer values of electron charge (as you are). Has anyone +measured free charges of other than integer multiples of e? + + + + +\end{document} + + + + + + + + + + + diff --git a/manual_source/chapters/naspec.tex b/manual_source/chapters/naspec.tex new file mode 100644 index 0000000..8f88b83 --- /dev/null +++ b/manual_source/chapters/naspec.tex @@ -0,0 +1,229 @@ +%\chapter*{Spectrum of Sodium } +%\addcontentsline{toc}{chapter}{Spectrum of Sodium} + +\documentclass{article} +\usepackage{tabularx,amsmath,boxedminipage,epsfig} + \oddsidemargin 0.0in + \evensidemargin 0.0in + \textwidth 6.5in + \headheight 0.0in + \topmargin 0.0in + \textheight=9.0in + +\begin{document} +\title{Spectrum of Sodium} +\date {} +\maketitle \noindent + \textbf{Experiment objectives}: measure the energy spectrum of sodium (Na), + determine values of quantum defects of low angular momentum states, and measure fine splitting +using Na yellow doublet. + +\section*{Theory} +Sodium (Na) belongs to the chemical group of \emph{alkali metals}, together +with lithium (Li), potassium (K), rubidium (Rb), cesium (Cs) and Francium (Fr). +All elements of this group have a closed electron shell with one extra unbound +electron. This makes energy level structure for this free electron to be very +similar to that of hydrogen, as shown in Fig.~\ref{nae}. + +For example, a Na atom has 11 electrons, and its electronic configuration is +$1s^22s^22p^63s$, as determined from the Pauli exclusion principle. Ten +closed-shell electrons effectively screen the nuclear charge number ($Z=11$) to +an effective charge $Z^*\approx 1$, so that the $3s$ valent electron experience +the electric field potential similar to that of a hydrogen atom. As a result, +the electron spectrum of all alkali metal atoms is quite similar to that of +hydrogen: +\begin{equation}\label{Hlevels_Naexp} +E_n=-hcRy\frac{1}{n^2} +\end{equation} +% +where $n$ is the principle quantum number, and $Ry=\frac{2\pi +m_ee^4}{(4\pi\epsilon_0)^2ch^3}$ is the Rydberg constant ($Ry = 1.0974 +\times 10^5 cm^{-1}$ and $hcRy = 13.605 eV$). For each particular value of +angular momentum $l$ the energy spectrum follows the same scaling as hydrogen +atom. However, the absolute values of energies obey Eq.(\ref{Hlevels_Naexp}) +only for electron energy states with orbits far above closed shell - the ones +with large value of an angular momentum $l$. Electron with smaller $l$ spends +more time closer to the nuclear, and ``feels'' stronger bounding electrostatic +potential. As a result the corresponding energy levels are pulled down compare +to those of hydrogen, and the states with the same principle number $n$ but +different angular momenta $l$ are split (\emph{i.e.} have different energies). +\begin{figure} +\includegraphics[height=\columnwidth]{nae.eps} +\caption{\label{nae}Energy spectrum of Na. The energy states of H are shown in +far right for comparison.} +\end{figure} + +For each particular value of angular momentum $l$ the energy spectrum follows +the same scaling as hydrogen atom, but with an effective charge $Z^*$: +\begin{equation}\label{heq} +E_n=-\frac{1}{2}\frac{Z^{*2}e^4}{(4\pi\epsilon_0)^2}\frac{mc^2}{\hbar^2c^2} +\frac{1}{n^2}=-Z^{*2}\frac{hcRy}{n^2} +\end{equation} +The value of the effective charge $Z^*$ depends on the angular momentum $l$, +and does not vary much between states with different principle quantum numbers +$n$ but same $l$\footnote{The accepted notation for different electron angular +momentum states is historical, and originates from the days when the proper +quantum mechanical description for atomic spectra has not been developed yet. +Back then spectroscopists had categorized atomic spectral lines corresponding +to their appearend: for example any spectral lines from electron transitions +from $s$-orbital ($l=0$) appeared always \textbf{S}harp on a photographic film, +while those with initial electron states of $d$-orbital ( $l=2$) appeared +always \textbf{D}iffuse. Also spectral lines of \textbf{P}rinciple series +(initial state is $p$-orbital, $l=1$) reproduced the hydrogen spectrum most +accurately (even though at shifted frequencies), while the \textbf{F}undamental +(initial electron state is $f$-orbital, $l=3$) series matched the absolute +energies of the hydrogen most precisely. The orbitals with higher value of the +angular momentum are denoted in an alphabetic order ($g$, +$h$, \textit{etc}.) }:\\ +\begin{tabular}{ll} +States&$Z^*$\\ +s~($l=0$)&$\approx$ 11/9.6\\ +p~($l=1$)&$\approx$ 11/10.1\\ +d~($l=2$)&$\approx$ 1\\ +f~($l=3$)&$\approx $ 1\\ +\end{tabular} +\\These numbers mean that two states with the lowest angular momentum ($s$ and +$p$) are noticeably affected by the more complicated electron structure of Na, +while the energy levels of the states with the higher values of angular +momentum ($d$, $f$) are identical to the hydrogen energy spectrum. + +An alternative (but equivalent) procedure is to assign a {\it quantum defect} +to the principle quantum $n$ instead of introducing an effective nuclei +charge. In this case Eq.(\ref{heq}) can be written as: +\begin{equation}\label{qdef} +E_n=-\frac{hcRy}{(n^*)^2}=-\frac{hcRy}{(n-\Delta_l)^2} +\end{equation} +where $n*=n-\Delta_l$, and $\Delta_l$ is the corresponding quantum defect. +Fig. \ref{nadell} shows values of quantum defects which work approximately for +the alkalis. One sees that there is one value for each value of the angular +momentum $l$. This is not exactly true for all alkali metals, but for Na there +is very little variation in $\Delta_l$ with $n$ for a given $l$. + +\begin{figure} +\includegraphics[width=0.5\columnwidth]{nadell.eps} +\caption{\label{nadell}Quantum Defect $\Delta_l$ versus $l$ for different +alkali metals. Taken from Condon and Shortley p. 143} +\end{figure} +%\begin{figure} +%\includegraphics[height=3in]{nadel.eps} +%\caption{\label{nadel}Quantum Defect $\Delta_l$ variation with $n$. The +%difference between the quantum defect of each term and that of the lowest term +%of the series to which it belongs is plotted against the difference between +%the total quantum numbers of these terms. Again from Condon and Shortley p. 144.} +%\end{figure} + +The spectrum of Na is shown in Fig. \ref{nae}. One can immediately see that +there are many more optical transitions because of the lifted degeneracy of +energy states with different angular momenta. However, not all electronic +transition are allowed: since the angular momentum of a photon is $1$, then the +electron angular momentum cannot change by more than one while emitting one +spontaneous photon. Thus, it is important to remember the following +\emph{selection rule} for atomic transitions: +\begin{equation}\label{selrules} +\Delta l = \pm 1. +\end{equation} +According to that rule, only transitions between two ``adjacent'' series are +possible: for example $p \rightarrow s$ or $d \rightarrow p$ are allowed, while +$s \rightarrow s$ or $s \rightarrow d$ are forbidden. + +The strongest allowed optical transitions are shown in Fig. \ref{natrns}. +\begin{figure} +\includegraphics[height=\columnwidth]{natrans.eps} +\caption{\label{natrns}Transitions for Na. The wavelengths of selected +transition are shown in {\AA}. Note, that $p$ state is now shown in two +columns, one referred to as $P_{1/2}$ and the other as $P_{3/2}$. The small +difference between their energy levels is the ``fine structure''.} +\end{figure} +%\begin{figure} +%\includegraphics[height=4in]{series.eps} +%\caption{\label{series}Series for Hydrogen, Alkalis are similar.} +%\end{figure} +Note that each level for given $n$ and $l$ is split into two because of the +\emph{fine structure splitting}. This splitting is due to the effect of +electron \emph{spin} and its coupling with the angular momentum. Proper +treatment of spin requires knowledge of quantum electrodynamics and solving +Dirac equation; for now spin can be treated as an additional quantum number +associated with any particle. The spin of electron is $1/2$, and it may be +oriented either along or against the non-zero electron's angular momentum. +Because of the weak coupling between the angular momentum and spin, these two +possible orientation results in small difference in energy for corresponding +electron states. + +\section*{Procedure and Data Analysis} +Align a diffraction-grating based spectrometer as described in ``Atomic +Spectroscopy of Hydrogen Atoms'' experimental procedure. + +Then determine the left and right angles for as many spectral lines and +diffraction orders as possible. Each lab partner should measure the postilions +of all lines at least once. + +Reduce the data using Eq. \ref{nlambda} to determine wavelengths for each +spectral line (here $m$ is the order number): +\begin{equation}\label{nlambda} +m\lambda=\frac{d}{2}(\sin\theta_r+\sin\theta_l) +\end{equation} +Determine the wavelengths of eight Na spectral lines measured in both first +and second order. Combining first and second order results obtain the mean and +standard deviation (error) of the mean value of the wavelength for each line. +Compare these measured mean wavelengths to the accepted values given in +Fig.~\ref{natrns} and in the table below: + +\begin{tabular}{lll} + Color&Line$_1$(\AA)&Line$_2$(\AA)\\ +Red&6154.3&6160.7\\ +Yellow & 5890.0&5895.9\\ +Green & 5682.7&5688.2\\ +&5149.1&5153.6\\ +& 4978.6&4982.9\\ +Blue&4748.0&4751.9\\ +&4664.9&4668.6\\ +Blue-Violet&4494.3&4497.7\\ +\end{tabular} + +Line$_1$ and Line$_2$ corresponds to transitions to two fine-spitted $3p$ +states $P_{1/2}$ and $P_{3/2}$. These two transition frequencies are very +close to each other, and to resolve them with the spectrometer the width of the +slit should be very narrow. However, you may not be able to see some weaker +lines then. In this case you should open the slit wider to let more light in +when searching for a line. If you can see a spectral line but cannot resolve +the doublet, record the reading for the center of the spectrometer line, and +use the average of two wavelengthes given above. + + Identify at least seven of the lines with a particular transition, e.g. +$\lambda = 4494.3${\AA} corresponds to $8d \rightarrow 3p$ transition. + +\subsection*{Calculation of a quantum defect for $n=3, p$ state} +Identify spectral lines which corresponds to optical transitions from $d$ to +$n=3,p$ states. Since the energy states of $d$ series follows the hydrogen +spectra almost exactly, the wavelength of emitted light $\lambda$ is given by: +\begin{equation} +\frac{hc}{\lambda}=E_{nd}-E_{3p}=-\frac{hcRy}{n^2}+\frac{hcRy}{(3-\Delta_p)^2}, +\end{equation} +or +\begin{equation} +\frac{1}{\lambda}=\frac{Ry}{(3-\Delta_p)^2}-\frac{Ry}{n^2}, +\end{equation} + where $n$ is the principle number of the initial $d$ state. To verify this +expression by plotting $1/\lambda$ versus $1/n^2$ for the $n$= 4,5, and 6. From +the slope of this curve determine the value of the Rydberg constant $Ry$. From +the intercept determine the energy $E_{3p}$ of the $n=3,p$ state, and calculate +its quantum defect $\Delta_p$. +\subsection*{Calculation of a quantum defect for $s$ states} +Now consider the transition from the $s$-states ($n=5,6,7$) to to the $n=3, p$ +state. Using $hc/\lambda=E_{ns}-E_{3p}$ and the results of your previous +calculations, determine the energies $E_{sn}$ for different $s$ states with +$n=5,6,7$ and calculate $\Delta_s$. Does the value of the quantum defect +depends on $n$? + +Compare the results of your calculations for the quantum defects $\Delta_s$ and +$\Delta_p$ with the accepted values given in Fig. \ref{nadell}. + +\subsection*{Calculations of fine structure splitting} +For the Na D doublet measure the splitting between two lines +$\Delta\lambda=\lambda_{3/2}-\lambda_{1/2}$ in the second diffraction order +(why the second order is better than the first one?). Compare to the accepted +value: $\Delta\lambda=$5.9\AA . Compare this approach to the use of the +Fabry-Perot interferometer. + +\end{document} +\newpage diff --git a/manual_source/chapters/pe-effect.tex b/manual_source/chapters/pe-effect.tex new file mode 100644 index 0000000..b28d000 --- /dev/null +++ b/manual_source/chapters/pe-effect.tex @@ -0,0 +1,266 @@ +%\chapter*{Photoelectric Effect} +%\addcontentsline{toc}{chapter}{Photoelectric Effect} +%\documentclass{article} +%\usepackage{tabularx,amsmath,boxedminipage,epsfig} +% \oddsidemargin 0.0in +% \evensidemargin 0.0in +% \textwidth 6.5in +% \headheight 0.0in +% \topmargin 0.0in +% \textheight=9.0in + +%\begin{document} +\chapter{Photoelectric Effect} +\setcounter{figure}{1} +\setcounter{table}{1} +\setcounter{equation}{1} + +%\date {} +%\maketitle \noindent + \textbf{Experiment objectives}: measure the ratio of Planck's constant to the electron charge $h/e$ using the photoelectric effect. + +\section*{History} + + + The photoelectric effect and its understanding was an important step in development + of quantum mechanics. You probably know that Max Planck was the first to postulate that + light was made up of discrete packages of energy. At that time it was a proposed hypothetical light property that + allowed for the proper description of black body radiation. Several years after Planck made this + suggestion, Albert Einstein refined the idea to explain the + strange features of electrons which are emitted from metal + surfaces after absorbing energy from light. Later he received a Nobel prize for this work. + +\section*{Theory} + +\begin{figure}[h] +\centering \includegraphics[height=3in]{./pdf_figs/pefig1} \caption{\label{pefig1} A simplified schematic of the photo-electric apparatus.} +\end{figure} + Consider an apparatus as outlined in Figure \ref{pefig1} (it is the +apparatus which Heinrich Hertz used to inadvertently discover the ``photoelectric effect''). Light of a +frequency $\nu$ strikes a cathode, causing electrons to be emitted with velocity $v$. A positive voltage applied +between the anode and the cathode can accelerate emitted electrons towards the positive anode, producing an +electrical current in the circuit. A reverse bias potential applied between the anode and cathode will slow down +the electrons, and even stop them from reaching the anode by matching their kinetic energy. This is the way to +carefully measure the kinetic energies of the photo electrons. +\begin{equation} +KE_{max} = (\frac{1}{2}mv^2)_{max} = eV_0 +\end{equation} +% +where $KE_{max}$ is the maximum kinetic energy, $m$ and $e$ are the mass and the charge of an electron, and +$V_0$ is a potential required to stop the electrons (known as the stopping potential). + +From the point of view of a wave theory of light, the energy carried by the light wave is proportional to its +intensity, and independent of light frequency. Thus, it was logical to expect that stronger light should +increase the energy of photoelectrons. However, the experiments of Philipp Lenard in 1900 showed something +different: although the maximum current increased with light intensity, the stopping potential $V_0$ was +independent of light intensity. This meant that increasing the rate of energy falling onto the cathode does not +increase the maximum energy of the electrons. This was quite a surprising result from the classical point of +view. + +This ``controversy'' was elegantly resolved five years later by Albert Einstein, who postulated that light +arrived in discrete quanta known as ``photons'', and each photon led to an emission of a single electron. The +energy of each photon is determined by the frequency of light - $h\nu$. Thus the energy of an emitted +photoelectron (and therefore the value of the stopping potential $V_0$) is determined by the frequency of an +individual incident photon: +\begin{equation} +\label{eq:pe} + eV_0= \frac{1}{2}mv^2 = h\nu-\phi = \frac{hc}{\lambda} - \phi +\end{equation} +where $\phi$ is known as the ``work function'' - the amount of energy needed to free the electron from the +cathode. The value of the work energy is a property of the cathode material. Also, it is now clear why the +intensity of light does not affect the stopping potential: more intense light has higher photon flux and thus +increase the number of emitted photoelectrons, but the energy of each electron is determined only on the energy +of a single photon, and that depends only by light frequency $\nu$. + +\begin{boxedminipage}{\linewidth} +\textbf{Where do all those electrons come from?} \\ +% +At Jefferson Lab in nearby Newport News, high-energy electrons (GeV range) are +aimed at targets to probe the fundamental properties of quarks. But where do +all those electrons come from? From the photoelectric effect, of course! A +laser beam is aimed at a cathode consisting of a material like copper or GaAs. +The frequency of the laser is such that electrons will be emitted from the +cathode. In this way, high current electron beams are produced. One benefit of +using a laser and the photoelectric effect is that if the laser is "pulsed" in +time, the electron beam will be also (this allows synchronized timing in the +experiments). Also, the polarization of the laser can be manipulated to allow +for the emission of electrons with particular spin. See: www.jlab.org +\end{boxedminipage} + +\section*{Procedure} + +\textbf{Equipment needed}: Pasco photo-electric apparatus, Hg lamp, digital +voltmeter. The Pasco apparatus contains a circuit which automatically determines the stopping potential, which you measure off of a voltmeter, so there is no need to adjust the stopping potential yourself or measuring the current (lucky you!). Read the brief description of its operation in the appendix. + +\begin{figure} +\centering \includegraphics[width=\linewidth]{./pdf_figs/pefig3} \caption{\label{pefig3} +The Pasco photoelectric effect setup.} +\end{figure} + +Set up the equipment as shown in Fig. \ref{pefig3}. First, place a lens/grating assembly in front of the Mercury +lamp, and observe a dispersed spectrum on a sheet of paper, as shown in Fig.~\ref{pefig4}. Identify spectral +lines in both the first and the second diffraction orders on both sides. Keep in mind that the color +``assignment'' is fairly relative, and make sure you find all lines mentioned in the table in Fig.~\ref{pefig4}. +Often the first/second order lines on one side are brighter than on the other - check your apparatus and +determine what orders you will be using in your experiment. + +After that instal the $h/e$ Apparatus and focus the light from the Mercury Vapor Light Source onto the slot in +the white reflective mask on the $h/e$ Apparatus. Tilt the Light shield of the Apparatus out of the way to +reveal the white photodiode mask inside the Apparatus. Slide the Lens/Grating assembly forward and back on its +support rods until you achieve the sharpest image of the aperture centered on the hole in the photodiode mask. +Secure the Lens/Grating by tightening the thumbscrew. Align the system by rotating the $h/e$ Apparatus on its +support base so that the same color light that falls on the opening of the light screen falls on the widow in +the photodiode mask, with no overlap of color from other spectral + lines. Return the Light Shield to its closed position. + +Check the polarity of the leads from your digital voltmeter (DVM), and connect them to the OUTPUT terminals on +the $h/e$ Apparatus. + + +\section*{Experimental procedure} + + + +\section*{Part A: The dependence of the stopping + potential on the intensity of light} +\begin{enumerate} +\item Adjust the $h/e$ Apparatus so that one of the blue first order spectral lines falls upon the opening of the mask of the photodiode. +\item Press the instrument discharge button, release it, and observe how much time\footnote{Use the stopwatch feature of your cell phone? You don't need a precise measurement.} is required to achieve a stable voltage. +\item Place the variable transmission filter in front of the white reflective mask (and over the colored filter, if one is used) so that the light passes through the section marked 100\% and reaches the photodiode. Record the DVM voltage reading in the table below. +\item Do the exact same thing for another blue spectral line. +\item It's important to check our data early on to make sure we are not off in crazyland. This procedure is technically known as ``sanity checking''. According to equation Eq.~\ref{eq:pe} the frequency (or, better, wavelength) of light can be found from the measured voltage with knowledge of the work function $\phi$. Unfortunately, we don't know the work function. You'll measure it in the second part of the lab. For now, compute $\Delta V_0$ between the two measurements and use it to compute $\Delta \lambda$ or $\Delta \nu$. Does your data agree with the accepted value? For this computation, it might be helpful to realize that $eV_0$ is the electron charge times the voltage. If you measured $V_0=\unit[1]{V}$ then $eV_0 =1$ ``electron-volt'' or ``eV''. This is why electron-volt units are useful. Also, $hc=\unit[1240]{nm~eV}$. +\item Go back to the original spectral line. +\item Move the Variable Transmission Filter so that the next section is directly in front of the incoming light. Record the new DVM reading, and time to recharge after the discharge button has been pressed and released. +\item Move the Variable Transmission Filter so that the next section is +directly in front of the incoming light. Record the new DVM reading and +approximate time to recharge after the discharge button has been pressed and +released. + +Repeat step 3 until you have tested all five sections of the filter. + +Repeat the procedure with a second color from the spectrum. + +\end{enumerate} + +\begin{figure}[h] +\centering \includegraphics[width=\linewidth]{./pdf_figs/pefig4} \caption{\label{pefig4} +The Pasco photoelectric effect setup.} +\end{figure} + +\section*{Part B: The dependence of the stopping potential on the frequency + of light} +\begin{enumerate} +\item You can easily see five brightest colors in the mercury light spectrum. Adjust the $h/e$ Apparatus so that the 1st order yellow colored band falls upon the opening of the mask of the photodiode. Take a quick measurement with the lights on and no yellow filter and record the DVM voltage. Do the same for the green line and one of the blue ones. + +\item Repeat the measurements with the lights out and record them (this will require coordinating with other groups and the instructor). Are the two sets of measurements the same? Form a hypothesis for why or why not. +\item Now, with the lights on, repeat the yellow and green measurements with the yellow and green filters attached to the h/e detector. What do you see now? Can you explain it? (Hint: hold one of the filters close to the diffraction grating and look at the screen). +\item Repeat the process for each color using the second order lines. Be sure to use the the green and yellow filters when you are using the green and yellow spectral lines. + +\end{enumerate} + +\section*{Analysis} +\section*{Classical vs. Quantum model of light} +\begin{enumerate} +\item Describe the effect that passing different amounts of the same +colored light through the Variable Transmission Filter has on the stopping +potential and thus the maximum energy of the photoelectrons, as well as the +charging time after pressing the discharge button. +\item Describe the effect that different colors of light had on the stopping +potential and thus the maximum energy of the photoelectrons. +\item Defend whether this experiment supports a classical wave or a +quantum model of light based on your lab results. +\end{enumerate} +Read the theory of the detector operation in the Appendix, and explain why +there is a slight drop in the measured stopping potential as the light +intensity is reduced. \\{\bf NOTE:} While the impedance of the unity gain +amplifier is very high ($10^{13}~\Omega$), it is not infinite and some charge +leaks off. + +\section*{The relationship between Energy, Wavelength and Frequency} + +\begin{enumerate} +\item +Use the table in Fig.~\ref{pefig4} to find the exact frequencies and wavelengths of the spectral lines you used and plot the measured stopping potential values versus light frequency for of measurements of the first and second order lines (can be on same graph). + +\item Fit the plots according to $eV_0 = h\nu-\phi$, extracting values for slopes and intercepts. Find average value for slope and its uncertainty. From the slope, determine $h$ counting $e=1.6\cdot10^{-19}$~C. Do your measured values agree with the accepted value of $h=2\pi\cdot 10^{-34}$J$\cdot$s within experimental uncertainty? + +\item From the intercepts, find the average value and uncertainty of the work function $\phi$. Look up some values of work functions for typical metals. Is it likely that the detector material is a simple metal? + +\end{enumerate} + +\section*{Appendix: Operation principle of the stopping potential detector} + +The schematics of the apparatus used to measure the stopping potential is shown in Fig.~\ref{pefig5}. Monochromatic light falls on the cathode plate of a vacuum photodiode tube that has a low work function $\phi$. Photoelectrons ejected from the cathode collect on the anode. The photodiode tube and its associated electronics have a small capacitance which becomes charged by the photoelectric current. When the potential on this capacitance reaches the stopping potential of the photoelectrons, the current decreases to zero, and the anode-to-cathode voltage stabilizes. This final voltage between the anode and cathode is therefore the stopping potential of the photoelectrons. +\begin{figure}[h] +\centering \includegraphics[width=0.7\linewidth]{./pdf_figs/pe_det} +\caption{\label{pefig5} The electronic schematic diagram of the $h/e$ +apparatus.} +\end{figure} + +To let you measure the stopping potential, the anode is connected to a built-in +differential amplifier with an ultrahigh input impedance ($> 10^{13}~\Omega$), +and the output from this amplifier is connected to the output jacks on the +front panel of the apparatus. This high impedance, unity gain ($V_{out}/V_{in} += 1$) amplifier lets you measure the stopping potential with a digital +voltmeter. + +Due to the ultra high input impedance, once the capacitor has been charged from +the photodiode current it takes a long time to discharge this potential through +some leakage. Therefore a shorting switch labeled ``PUSH TO Zero'' enables the +user to quickly bleed off the charge. +\newpage +\section*{Sample data tables:} +{\large +%\begin{tabular}{|p{27mm}|p{27mm}|p{27mm}|p{27mm}|} +\begin{tabular}{|c|c|c|c|} +\hline + Color & \%Transmission & Stopping Potential & +Approx. Charge Time \\ +\hline +&100&&\\\hline +&80&&\\ \hline +&60&&\\ \hline +&40&&\\ \hline +&20&&\\ \hline +\hline + Color & \%Transmission & Stopping Potential & +Approx. Charge Time \\ +\hline +&100&&\\\hline +&80&&\\ \hline +&60&&\\ \hline +&40&&\\ \hline +&20&&\\ \hline +\end{tabular} +} +\vskip .1in +% +% +%{\large +%\begin{tabular}{|p{27mm}|p{27mm}|} +%\hline +% Light Color & Stopping Potential \\\hline +%Yellow&\\\hline +%Green&\\ \hline +%Blue&\\ \hline +%Violet&\\ \hline +%Ultraviolet&\\ \hline +%\end{tabular} +%} + +{\large +\begin{tabular}{|c|c|c|c|} +\hline + 1st Order Color&$\lambda$ (nm) &$\nu$ ($10^{14}Hz$) & +Stopping Potential (V) \\ +\hline Yellow&&&\\\hline Green&&&\\ \hline Blue&&&\\ \hline Violet&&&\\ +\hline Ultraviolet&&&\\ \hline \hline +2nd Order Color&$\lambda$ (nm) &$\nu$ ($10^{14}Hz$) & +Stopping Potential (V) \\ +\hline Yellow&&&\\\hline Green&&&\\ \hline Blue&&&\\ \hline Violet&&&\\ +\hline Ultraviolet&&&\\ \hline \hline + +\end{tabular} +} +%\end{document} +%\newpage diff --git a/manual_source/chapters/report_template.tex b/manual_source/chapters/report_template.tex new file mode 100644 index 0000000..c4f2436 --- /dev/null +++ b/manual_source/chapters/report_template.tex @@ -0,0 +1,138 @@ +\documentclass[aps,prb,preprint]{revtex4}
+%\documentclass[aps,twocolumn,prl]{revtex4}
+\usepackage{epsfig}
+\begin{document}
+
+\title{Compton Scattering Lab}
+\author{Y. O. Urname, P. A. Rtner1, and P. B. Rtner2}
+\affiliation{Physics 251 College of William and Mary}
+\date{today}
+\begin{abstract}
+This lab demonstrates Compton scattering. A Na$^{22}$ $\gamma$-ray source is
+used. The decay $\gamma$-ray has an energy of approximately .66 MeV.
+\end{abstract}
+\maketitle
+
+
+
+
+
+
+\section{Introduction}
+\subsection{Overview ...}
+
+Start writing here ..
+
+g
+%++++++++++++++++++++++++++++++++++++++++
+How to cite a reference:
+In 1989, experiments at CERN \cite{EMC} reported ...
+
+%++++++++++++++++++++++++++++++
+How to make a nice equation:
+\begin{equation}
+\label{equ:aperp}
+A_{\perp} = \frac{\sigma^{\downarrow\leftarrow} -
+\sigma^{\uparrow\leftarrow}} {\sigma^{\downarrow\leftarrow} +
+\sigma^{\uparrow\leftarrow}} = f_kE^{\prime}\sin\theta\left(g_1(x,Q^2)
++ \frac{2E}{\nu}g_2(x,Q^2)\right),
+\end{equation}
+
+%++++++++++++++++++++++++++++++
+How to include a figure into the document:
+%%%%%%%%%%%%%%%%%%%%%%%%% Figure - Helicity %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{figure}[hbt]
+\begin{center}
+\epsfysize=2.5in \epsfxsize=1.5in
+\rotatebox{70}{\leavevmode\epsffile{mypicture.eps}}
+\end{center}
+\caption{Every figure MUST have a caption}
+\label{fig:helicity}
+\end{figure}
+%%%%%%%%%%%%%%%%%%%%%%%% End Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%+++++++++++++++++++++++++++++++++++++
+An easier way to include a figure in a document:
+%%%%%%%%%%%%%%%%%%%%%%%%% Figure - Helicity %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{figure}[hbt]
+\begin{center}
+\epsfig{file=mypicture.eps, width=3in, angle=270}
+\end{center}
+\caption{Did you hear that every figure needs a caption?}
+\label{fig:helicity1}
+\end{figure}
+%%%%%%%%%%%%%%%%%%%%%%%% End Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Theory}
+\section{Experiment}
+
+
+
+%++++++++++++++++++++++++++++++++
+How to make a nice table. Note that I had to put the newpage in to get it
+placed right. (Take the newpage out and see what happens.)
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Table bins %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{table}[h]
+\begin{center}
+\caption{Every table needs a caption
+ }
+\label{tbl:bins}
+\begin{tabular}{|ccccccc|} \hline
+\multicolumn{1}{|c}{Polarization} &
+\multicolumn{1}{c}{Target} &
+\multicolumn{1}{c}{Bin} &
+\multicolumn{1}{c}{$<x>$} &
+\multicolumn{1}{c}{$<Q^2>$} &
+\multicolumn{1}{c}{$A_{\perp}^{meas}$} &
+\multicolumn{1}{c|}{$\Delta A_{\perp}$} \\
+\hline
+$-$ & LiD & 1 & 0.0233323 & 0.8429978 & 0.0044151 & 0.0030871 \\
+ & & 2 & 0.0638046 & 1.5017358 & 0.0021633 & 0.0021343 \\
+ & & 3 & 0.1892825 & 3.1877837 & 0.0006640 & 0.0022467 \\
+ & & 4 & 0.4766562 & 7.1827556 & -0.0197585 & 0.0085528 \\
+ & NH$_3$ & 1 & 0.0232572 & 0.8454089 & 0.0003600 & 0.0018642 \\
+ & & 2 & 0.0633156 & 1.4870013 & 0.0023831 & 0.0013287 \\
+ & & 3 & 0.1923955 & 3.1753302 & -0.0024246 & 0.0013771 \\
+ & & 4 & 0.4830315 & 7.3245904 & -0.0284834 & 0.0047061 \\
+$+$ & LiD & 1 & 0.0233503 & 0.8340932 & -0.0086018 & 0.0031121 \\
+ & & 2 & 0.0638688 & 1.4785886 & -0.0018465 & 0.0021452 \\
+ & & 3 & 0.1892192 & 3.1277721 & -0.0017860 & 0.0022525 \\
+ & & 4 & 0.4778486 & 7.0313856 & -0.0041773 & 0.0084659 \\
+ & NH$_3$ & 1 & 0.0232964 & 0.8439092 & -0.0022961 & 0.0018851 \\
+ & & 2 & 0.0633764 & 1.4814540 & 0.0021355 & 0.0013354 \\
+ & & 3 & 0.1924094 & 3.1580557 & -0.0065302 & 0.0013775 \\
+ & & 4 & 0.4825868 & 7.3191291 & -0.0290878 & 0.0047329 \\
+\hline
+\end{tabular}
+\end{center}
+\end{table}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\newpage
+\subsection{The Target} %For example
+\section{Results}
+\section{Conclusions}
+
+%+++++++++++++++++++++++++++++++++++++++
+%Here's how to do references:
+
+%\begin{thebibliography}{99} % Remove leading % if this style is to your taste.
+%\bibitem{EMC}
+%J.~Ashman {\it et al.} [European Muon Collaboration],
+%``An investigation of the spin structure of the proton in deep inelastic scattering of polarized muons on polarized protons,''
+%Nucl.\ Phys.\ {\bf B328}, 1 (1989).
+%\end{thebibliography}
+
+\bibliography{dummy}
+
+\end{document}
+
+
+
+
+
+
+
+
+
+
+
+
diff --git a/manual_source/chapters/sched.tex b/manual_source/chapters/sched.tex new file mode 100644 index 0000000..8227a18 --- /dev/null +++ b/manual_source/chapters/sched.tex @@ -0,0 +1,58 @@ +\noindent +Wednesday/Thursday, 2-5 pm in Rooms 203\& 205 of Small Hall\\ +Instructors:\\ + +\begin{tabular}{llll} +W. J. Kossler & Rm 129 & 221 3519 & kossler@physics.wm.edu\\ +&& home 229 8060&\\ +Assistant: Cara Campbell &Rm 243 &cell 540 850 4606 & cacamp@wm.edu\\ +\end{tabular} +\vskip .2in + + + Reports are due one week after the lab work. Collaboration is +allowed and in fact expected, but the writeup should be individually done. + + +\section*{Schedule} + +The content of the cells is the week(s) during which the experiment +will be carried out. +\vskip .2in + +\begin{tabular}{||l||l|l|l|l|l|l||}\hline\hline +Exp$\setminus$Group& 1 & 2 & 3 & 4 & 5 & 6\\\hline\hline +C&1&2&3&4&5&6\\\hline +Michelson&2&1&1&1&6&5\\\hline +Fabry Perot&3&3&2&2&7&9\\\hline +e/m&4&4&5&5&8&11\\\hline +$\hbar$/e&5&5&4&6&3&3\\\hline +Electron Diffraction& 6&6&6&3&4&4\\\hline +Black Body& 9&9&7&7&1&1\\\hline +Oil Drop& 7,8&7,8&9,10&9,10&9,10&7,8\\\hline +Hydrogen Spectrum & 11 & 11 & 8 & 8 & 2 & 2\\\hline +Na Spectrum &10 & 10 &12 &12 &12 &10\\\hline +Superconductivity &12& 12 &11 &11 &11 &12\\\hline +Tour & 13 & 13 & 13 & 13 & 13 & 13 \\\hline\hline +\end{tabular} + + +\subsection*{Mapping Date $\rightarrow$ Week Number} +\begin{tabular}{||l|l||}\hline\hline +Aug. 30,31 & 0\\\hline +Sept 6, Sept. 7& 1\\\hline +Sept. 13,14& 2 \\\hline +Sept. 20,21 &3 \\\hline +Sept. 27,28 & 4\\\hline +Oct. 4,5 & 5 \\\hline +Oct. 11,12 & 6 \\\hline +Oct. 18,19 8& 7 \\\hline +Oct. 25,26 & 8 \\\hline +Nov. 1,2 & 9 \\\hline +Nov. 8,9 & 10 \\\hline +Nov. 15,16 & 11 \\\hline +Nov. 22,23 &Thanksgiving \\\hline +Nov. 29,30& 12\\\hline +Dec. 6,7 & 13\\\hline\hline +\end{tabular} +\newpage diff --git a/manual_source/chapters/spectr.tex b/manual_source/chapters/spectr.tex new file mode 100644 index 0000000..634bec0 --- /dev/null +++ b/manual_source/chapters/spectr.tex @@ -0,0 +1,640 @@ +%\chapter*{Atomic Spectroscopy of the Hydrogen Atom} +%\addcontentsline{toc}{chapter}{Hydrogen Spectrum} +%\documentclass{article} +%\usepackage{tabularx,amsmath,boxedminipage,epsfig} +% \oddsidemargin 0.0in +% \evensidemargin 0.0in +% \textwidth 6.5in +% \headheight 0.0in +% \topmargin 0.0in +% \textheight=9.0in + +%\begin{document} +\chapter{Atomic Spectroscopy} +\setcounter{figure}{1} +\setcounter{table}{1} +\setcounter{equation}{1} +%\date {} +%\maketitle \noindent + \textbf{Experiment objectives}: test a diffraction grating-based spectrometer, study the energy spectrum of atomic hydrogen (H) and a hydrogen-like atomic sodium (Na), determine values of quantum defects of low angular momentum states of Na, and measure fine splitting using Na yellow doublet. + +\subsection*{History} + + The observation of discrete lines in the emission spectra of + atomic gases gives insight into the quantum nature of + atoms. Classical electrodynamics cannot explain the existence + of these discrete lines, whose energy (or wavelengths) are + given by characteristic values for specific atoms. These + emission lines are so fundamental that they are used to + identify atomic elements in objects, such as in identifying + the constituents of stars in our universe. When Niels Bohr + postulated that electrons can exist only in orbits of discrete + energies, the explanation for the discrete atomic lines became + clear. In this laboratory you will measure the wavelengths of + the discrete emission lines from two elements - hydrogen and sodium - + to determine the energy levels in the hydrogen-like atoms. + +\section*{Hydrogen Spectrum} + + The hydrogen atom is the simplest atom: it consists of a single proton and a single +electron. Since it is so simple, it is possible to calculate the energy spectrum of the electron +bound states - in fact you will do that in your Quantum Mechanics course. From this calculations it +follows that the electron energy is quantized, and these energies are given by the expression: +\begin{equation}\label{Hlevels_inf} +E_n=- \frac{2\pi^2m_ee^4}{(4\pi\epsilon_0)^2h^2n^2} + = -hcRy\frac{1}{n^2} +\end{equation} +% +where $n$ is the {\bf principal quantum number} of the atomic level, and +\begin{equation} \label{Ry} +Ry=\frac{2\pi m_ee^4}{(4\pi\epsilon_0)^2ch^3} +\end{equation} +is a fundamental physical constant called the {\bf Rydberg constant} (here $m_e$ is the electron +mass). Numerically, $Ry = 1.0974 \times 10^5 cm^{-1}$ and $hcRy = 13.605 eV$. + +Because the allowed energies of an electron in a hydrogen atom, the electron can change its state +only by making a transition ("jump") from an one state of energy $E_1$ to another state of lower +energy $E_2$ by emitting a photon of energy $h\nu = E_1 - E_2$ that carries away the excess energy. +Thus, by exciting atoms into high-energy states using a discharge and then measuring the frequencies +of emission one can figure out the energy separation between various energy levels. Since it is +more convenient to use a wavelength of light $\lambda$ rather than its frequency $\nu$, and using the +standard connection between the wavelength and the frequency $h\nu = hc/\lambda$, we can write the +wavelength of a photon emitted by electron jumping between the initial state $n_1$ and the final +state $n_2$ as follows: +\begin{equation} \label{Hlines_inf} +\frac{1}{\lambda_{12}}=\frac{2\pi^2m_ee^4}{(4\pi\epsilon_0)^2ch^3} +\left[\frac{1}{n_2^2}-\frac{1}{n_1^2}\right]= Ry \left[\frac{1}{n_2^2}-\frac{1}{n_1^2}\right] +\end{equation} +%Based on this description it is clear that by measuring the frequencies (or +%wavelengths) of photons emitted by an excited atomic system, we can glean +%important information about allowed electron energies in atoms. + +%To make more accurate calculation of the Hydrogen spectrum, we need to take +%into account that a hydrogen nucleus has a large, but finite mass, M=AMp (mass +%number A=1 and Mp = mass of proton)\footnote{This might give you the notion +%that the mass of any nucleus of mass number $A$ is equal to $AM_p$. This is not +%very accurate, but it is a good first order approximation.} such that the +%electron and the nucleus orbit a common center of mass. For this two-mass +%system the reduced mass is given by $\mu=m_e/(1+m_e/AM_p)$. We can take this +%into account by modifying the above expression (\ref{Hlines_inf}) for +%1/$\lambda$ as follows: +%\begin{equation}\label{Hlines_arb} +%\frac{1}{\lambda_A}=R_AZ^2\left[\frac{1}{n_2^2}-\frac{1}{n_1^2}\right] \mbox{ +%where } R_A=\frac{R_{\infty}}{1+\frac{m_e}{AM_p}} +%\end{equation} +%In particular, for the hydrogen case of ($Z=1$; $M=M_p$) we have: +%\begin{equation}\label{Hlines_H} +%\frac{1}{\lambda_H}=R_H\left[\frac{1}{n_2^2}-\frac{1}{n_1^2}\right] +%\end{equation} +%Notice that the value of the Rydberg constant will change slightly for +%different elements. However, these corrections are small since nucleus is +%typically several orders of magnitude heavier then the electron. + + +Fig.~\ref{Hspec.fig} shows the energy levels of hydrogen, and indicates a large number of observed +transitions, grouped into series with the common electron final state. Emitted photon frequencies +span the spectrum from the UV (UltraViolet) to the IR (InfraRed). Among all the series only the +Balmer series, corresponding to $n_2$ = 2, has its transitions in visible part of the spectrum. +Moreover, from all the possible transition, we will be able to to observe and measure only the +following four lines: $n_1=6 \rightarrow 2$, $5 \rightarrow 2$, $4 \rightarrow 2$, + and $3 \rightarrow 2$. + +\begin{figure} +\includegraphics[width=0.7\linewidth]{./pdf_figs/spec} +\caption{\label{Hspec.fig}Spectrum of Hydrogen. The numbers on the left show the energies of the +hydrogen levels with different principle quantum numbers $n$ in $eV$. The wavelength of emitted +photon in {\AA} are shown next to each electron transition. } +\end{figure} + +%In this lab, the light from the hydrogen gas is broken up into its spectral +%components by a diffraction grating. You will measure the angle at which each +%line occurs on the left ($\theta_L$) and ($\theta_R$) right sides for as many +%diffraction orders $m$ as possible, and use Eq.(\ref{mlambda}) to calculate +%$\lambda$, using the following expression, derived in the Appendix. +%\begin{equation}\label{mlambda} +%m\lambda = \frac{h}{2}\left(\sin\theta_L+\sin\theta_R\right) +%\end{equation} +% Then the same +%expression will be used to check/calibrate the groove spacing $h$ by making +%similar measurements for a sodium spectral lines with known wavelengths. +% +%We will approach the data in this experiment both with an eye to confirming +% Bohr's theory and from Balmer's early perspective of someone +% trying to establish an integer power series linking the +% wavelength of these four lines. + +\section*{Sodium spectrum} +Sodium (Na) belongs to the chemical group of \emph{alkali metals} [together with lithium (Li), +potassium (K), rubidium (Rb), cesium (Cs) and Francium (Fr)]. All these elements consist of a closed +electron shell with one extra unbound electron. Not surprisingly, the energy level structure for this +free electron is very similar to that of hydrogen. For example, a Na atom has 11 electrons, and its +electronic configuration is $1s^22s^22p^63s$. Ten closed-shell electrons effectively screen the +nuclear charge number ($Z=11$) to an effective charge $Z^*\approx 1$, so that the $3s$ valent +electron experiences the electric field potential similar to that of a hydrogen atom, given by the +Eq.~\ref{Hlevels_inf}. + +However, there is an important variation of the energy spectrum of alkali metals, related to the +electron angular momentum $l$. In hydrogen the energy levels with same principle quantum number $n$ +but different electron angular momentum $l=0, 1, \cdots (n-1)$ are degenerate. For Na and others the +levels with different values of $l$ are shifted with respect to each other. This is mainly due to the +interaction of the unpaired electrons with the electrons of the closed shells. For example, the +orbits of the electron with large angular momentum value $l$ is far above closed shell, and thus +their energies are basically the same as for the hydrogen atom. An electron with smaller $l$ spends +more time closer to the nuclear, and ``feels'' stronger bounding electrostatic potential. As a result +the corresponding energy levels are pulled down compare to those of hydrogen, and the states with the +same principle number $n$ but different angular momenta $l$ are split (\emph{i.e.} have different +energies). + + +To take onto account the modification of the atomic spectra while still using the same basic +equations as for the hydrogen, it is convenient to introduce a small correction $\Delta_l$ to the +principle quantum number $n$ to take into account the level shifts. This correction is often called a +{\it quantum defect}, and its value % an effective nuclei charge $Z^*$ keeping the For each particular value +%of angular momentum $l$ the energy spectrum follows the same scaling as hydrogen atom, but with an +%effective charge $Z^*$: +%\begin{equation}\label{heq} +%E_n=-\frac{1}{2}\frac{Z^{*2}e^4}{(4\pi\epsilon_0)^2}\frac{mc^2}{\hbar^2c^2} +%\frac{1}{n^2}=-Z^{*2}\frac{hcRy}{n^2} +%\end{equation} +%The value of the effective charge $Z^*$ +depends on the angular momentum $l$, and does not vary much between states with different principle +quantum numbers $n$ but same $l$\footnote{The accepted notation for different electron angular +momentum states is historical, and originates from the days when the proper quantum mechanical +description for atomic spectra has not been developed yet. Back then spectroscopists had categorized +atomic spectral lines corresponding to their appearend: for example any spectral lines from electron +transitions from $s$-orbital ($l=0$) appeared always \textbf{S}harp on a photographic film, while +those with initial electron states of $d$-orbital ( $l=2$) appeared always \textbf{D}iffuse. Also +spectral lines of \textbf{P}rinciple series (initial state is $p$-orbital, $l=1$) reproduced the +hydrogen spectrum most accurately (even though at shifted frequencies), while the +\textbf{F}undamental (initial electron state is $f$-orbital, $l=3$) series matched the absolute +energies of the hydrogen most precisely. The orbitals with higher value of the angular momentum are +denoted in an alphabetic order ($g$, $h$, \textit{etc}.) }: +\begin{equation}\label{qdef} +E_{nl}=-\frac{hcRy}{(n-\Delta_l)^2}%=-\frac{hcRy}{(n-\Delta_l)^2} +\end{equation} + +%\begin{tabular}{ll} +%States&$Z^*$\\ +%s~($l=0$)&$\approx$ 11/9.6\\ +%p~($l=1$)&$\approx$ 11/10.1\\ +%d~($l=2$)&$\approx$ 1\\ +%f~($l=3$)&$\approx $ 1\\ +%\end{tabular} +In particular, the energies of two states with the lowest angular momentum ($s$ and $p$) are +noticeably affected by the more complicated electron structure of Na, while the energy levels of the +states with the higher values of angular momentum ($d$, $f$) are identical to the hydrogen energy +spectrum. +% +%An alternative (but equivalent) procedure is to assign a {\it quantum defect} to the principle +%quantum $n$ instead of introducing an effective nuclei charge. In this case Eq.(\ref{heq}) can be +%written as: +% +%where $n*=n-\Delta_l$, and $\Delta_l$ is the corresponding quantum defect. Fig. \ref{nadell} shows +%values of quantum defects which work approximately for the alkalis. One sees that there is one value +%for each value of the angular momentum $l$. This is not exactly true for all alkali metals, but for +%Na there is very little variation in $\Delta_l$ with $n$ for a given $l$. +\begin{figure} +\includegraphics[height=\columnwidth]{./pdf_figs/nae} +\caption{\label{nae}Energy spectrum of Na. The energy states of H are shown in far right for +comparison.} +\end{figure} +%\begin{figure} +%\includegraphics[width=0.5\columnwidth]{nadell.eps} +%\caption{\label{nadell}Quantum Defect $\Delta_l$ versus $l$ for different alkali metals. Taken from +%Condon and Shortley p. 143} +%\end{figure} +%\begin{figure} +%\includegraphics[height=3in]{nadel.eps} +%\caption{\label{nadel}Quantum Defect $\Delta_l$ variation with $n$. The +%difference between the quantum defect of each term and that of the lowest term +%of the series to which it belongs is plotted against the difference between +%the total quantum numbers of these terms. Again from Condon and Shortley p. 144.} +%\end{figure} + +The spectrum of Na is shown in Fig. \ref{nae}. One can immediately see that there are many more +optical transitions because of the lifted degeneracy of energy states with different angular momenta. +However, not all electronic transition are allowed: since the angular momentum of a photon is $1$, +then the electron angular momentum cannot change by more than one while emitting one spontaneous +photon. Thus, it is important to remember the following \emph{selection rule} for atomic transitions: +\begin{equation}\label{selrules} +\Delta l = \pm 1. +\end{equation} +According to that rule, only transitions between two ``adjacent'' series are possible: for example $p +\rightarrow s$ or $d \rightarrow p$ are allowed, while $s \rightarrow s$ or $s \rightarrow d$ are +forbidden. The strongest allowed optical transitions are shown in Fig. \ref{natrns}. +\begin{figure} +\includegraphics[height=\columnwidth]{./pdf_figs/natrans} +\caption{\label{natrns}Transitions for Na. The wavelengths of selected transition are shown in {\AA}. +Note, that $p$ state is now shown in two columns, one referred to as $P_{1/2}$ and the other as +$P_{3/2}$. The small difference between their energy levels is the ``fine structure''.} +\end{figure} +%\begin{figure} +%\includegraphics[height=4in]{series.eps} +%\caption{\label{series}Series for Hydrogen, Alkalis are similar.} +%\end{figure} +Note that each level for given $n$ and $l$ is split into two because of the \emph{fine structure +splitting}. This splitting is due to the effect of electron \emph{spin} and its coupling with the +angular momentum. Proper treatment of spin requires knowledge of quantum electrodynamics and solving +Dirac equation; for now spin can be treated as an additional quantum number associated with any +particle. The spin of electron is $1/2$, and it may be oriented either along or against the non-zero +electron's angular momentum. Because of the weak coupling between the angular momentum and spin, +these two possible orientation results in small difference in energy for corresponding electron +states. + +\section*{Experimental setup} +\textbf{Equipment needed}: Gaertner-Peck optical spectrometer, hydrogen discharge lamp, sodium +discharge lamp. + +Fig. \ref{expspec} gives a top view of the Gaertner-Peck optical spectrometer used in this lab. The +spectrometer consists of two tubes. One tube holds a vertical collimator slit of variable width, and +should be directed toward a discharge lamp. The light from the discharge lamp passes through the +collimator slit and then get dispersed on the diffraction grating mounted in the mounting pedestal in +the middle of the apparatus. The other tube holds the telescope that allows you to magnify the image +of the collimator slit to get more accurate reading on the rotating table. This tube can be rotated +around the central point, so that you will be able to align and measure the wavelength of all visible +spectral lines of the lamp in all orders of diffraction grating. +\begin{figure} +\includegraphics[height=4in]{./pdf_figs/expspec} +\caption{\label{expspec}Gaertner-Peck Spectrometer} +\end{figure} + +It is likely that you will find a spectrometer at nearly aligned condition at the beginning on the +lab. Nevertheless, take time making sure that all elements are in order to insure good quality of +your data. + +\textit{Telescope Alignment:} Start by adjusting the telescope eyepiece in or out to bring the +crosshairs into sharp focus. Next aim the telescope out the window to view a distant object such as +leaves in a tree. If the distant object is not in focus, you will have to adjust the position of the +telescope tube - ask your instructor for directions. + +\textit{Collimator Conditions:} Swing the telescope to view the collimator which is accepting light +from the hydrogen discharge tube through a vertical slit of variable width. The slit opening should +be set to about 5-10 times the crosshair width to permit sufficient light to see the faint violet +line and to be able to see the crosshairs. If the bright column of light is not in sharp focus, you +should align the position of the telescope tube again (with the help of the instructor). + +\textit{Diffraction Grating Conditions:} In this experiment you will be using a diffraction grating +that has 600 lines per mm. A brief summary of diffraction grating operation is given in the Appendix +of this manual. If the grating is not already in place, put it back to the baseclamp and fix it +there. The table plate that holds the grating can be rotated, so try to orient the grating surface to +be maximally perpendicular to the collimator axis. However, the accurate measurement of angle does +not require the perfect grating alignment. Instead, for each spectral line in each diffraction order +you will be measuring the angles on the left ($\theta_l$) and on the right ($\theta_r$), and use both +of the measurements to figure out the optical wavelength using the following equation: +\begin{equation}\label{nlambda} +m\lambda=\frac{d}{2}(\sin\theta_r+\sin\theta_l), +\end{equation} +where $m$ is the diffraction order, and $d$ is the distance between the lines in the grating. + + % Use +% the AUTOCOLLIMATION procedure to achieve a fairly accurate +% alignment of the grating surface. This will determine how to +% adjust the three leveling screws H1, H2, and H3 and the +% rotation angle set screw for the grating table. +% +% \textbf{AUTOCOLLIMATION} is a sensitive way to align an optical +% element. First, mount a ``cross slit'' across the objective lens of +% the collimator, and direct a strong light source into the +% input end of the collimator. Some of the light exiting through +% the cross slit will reflect from the grating and return to the +% cross slit. The grating can then be manipulated till this +% reflected light retraces its path through the cross slit +% opening. With this the grating surface is normal to the +% collimator light. + Then, with the hydrogen tube ON and in place at + the collimator slit, swing the rotating telescope slowly + through 90 degrees both on the left \& right sides of the forward + direction. You should observe diffraction maxima for most + spectral wavelength, $\lambda$, in 1st, 2nd, and 3rd order. If these + lines seem to climb uphill or drop downhill + the grating has to be adjusted in its baseclamp to + bring them all to the same elevation. + + Also, turn on the sodium lamp as soon as you arrive, since it requires 10-15 minutes to warm up + and show clear Na spectrum. + +\subsection*{Experimental studies of Hydrogen Balmer line} + +Swing the rotating telescope slowly and determine which spectral lines from Balmer series you +observe. You should be able to see three bright lines - Blue, Green and Red - in the first (m=1) and +second (m=2) diffraction orders on both left \& right sides. In the third order (m=3) only the Blue, +\& Green lines are visible, and you will not see the Red. + +One more line of the Balmer series is in the visible range - Violet, but its intensity is much lower +than for the other three line. However, you will be able to find it in the first order if you look +carefully with the collimator slit open a little wider. +% +%\emph{Lines to be measured:} +%\begin{itemize} +%\item \emph{Zero order} (m=0): All spectral lines merge. +%\item \emph{First order} (m=1): Violet, Blue, Green, \& Red on both left \& +% right sides. +%\item \emph{Second order} (m=2): Violet, Blue, Green, \& Red on +% both left \& right sides. +%\item \emph{Third order} (m=3): Blue, \& Green. +%\end{itemize} +% You might not see the Violet line due to its low +% intensity. Red will not be seen in 3rd order. + +After locating all the lines, measure the angles at which each line occurs. The spectrometer reading for each line should be measured at least \emph{twice} by \textit{different} lab partners to avoid systematic errors. \textbf{Don't forget}: for every line you need to measure the angles to the right and to the left! + +You should be able to determine the angle with accuracy of $1$ minute, but you should know how to +read angles with high precision in the spectrometer: first use the bottom scale to get the rough +angle reading within a half of the degree. Then use the upper scale for more accurate reading in +minutes. To get this reading find a tick mark of the upper scale that aligns perfectly with some tick +mark of the bottom scale - this is your minute reading. Total angle is the sum of two readings. + +To measure the frequency precisely center the crosshairs on the line as accurately as possible. +Choose the width of lines by turning the collimator slit adjustment screw. If the slit is too wide, +it is hard to judge the center of the line accurately; if the slit is too narrow, then not enough +light is available to see the crosshairs. For Violet the intensity is noticeably less than for the +other three lines, so you may not see it. However, even if you find it, a little assistance is +usually required in order to locate the crosshairs at this line. We suggest that a low intensity +flashlight be aimed toward the Telescope input, and switched ON and OFF repeatedly to reveal the +location of the vertical crosshair relative to the faint Violet line. + +%\subsubsection*{ Calibration of Diffraction Grating:} Replace the hydrogen tube with +% a sodium (Na) lamp and take readings for the following two +% lines from sodium: $568.27$~nm (green) and $589.90$~nm (yellow). Extract from +% these readings the best average value for $h$ the groove +% spacing in the diffraction grating. Compare to the statement +% that the grating has 600 lines per mm. Try using your measured value +% for $h$ versus the stated value $600$ lines per mm in +% the diffraction formula when obtaining the measured +% wavelengths of hydrogen. Determine which one provide more accurate results, and discuss the conclusion. + +\subsubsection*{ Data analysis for Hydrogen Data} +Calculate the wavelength $\lambda$ for each line observed in all orders, calculate the average +wavelength value and uncertainty for each line, and then identify the initial and final electron +states principle numbers ($n_1$ and $n_2$) for each line using Fig.~\ref{Hspec.fig}. + +Make a plot of $1/\lambda$ vs $1/n_1^2$ where $n_1$ = the principal quantum number of the electron's +initial state. Put all $\lambda$ values you measure above on this plot. You data point should form a +straight line. From Equation~(\ref{Hlines_inf}) determine the physical meaning of both slope and +intercept, and compare the data from the fit to the expected values for each of them. The slope +should be the Rydberg constant for hydrogen, $Ry$. The intercept is $Ry/(n_2)^2$. From this, +determine the value for the principal quantum number $n_2$. Compare to the accepted value in the +Balmer series. + +\subsection*{Experimental studies of Sodium Spectrum} + +Switch to Sodium lamp and make sure the lamp warms up for approximately 5-10 minutes before starting +the measurements. + +Double check that you see a sharp spectrum in the spectrometer (adjust the width of the collimator +slit if necessary). In the beginning it will be very useful for each lab partner to quickly scan the +spectrometer telescope through all first-order lines, and then discuss which line you see corresponds +to with transition in Table~\ref{tab:sodium} and Fig.~\ref{natrns}. Keep in mind that the color +names are symbolic rather than descriptive! + +After that carefully measure the left and right angles for as many spectral lines in the first and orders +as possible. The spectrometer reading for each line should be measured at least \emph{twice} by +\textit{different} lab partners to avoid systematic errors. + +Determine the wavelengths of all measured Na spectral lines using Eq. \ref{nlambda}. Compare these +measured mean wavelengths to the accepted values given in Fig.~\ref{natrns} and in the table below. +Identify at least seven of the lines with a particular transition, e.g. $\lambda = 4494.3${\AA} +corresponds to $8d \rightarrow 3p$ transition. + +\begin{table} + +\centering +\begin{tabular}{l|l|l} + Color&Line$_1$(\AA)&Line$_2$(\AA)\\ \hline +Red&6154.3&6160.7\\ +Yellow & 5890.0&5895.9\\ +Green & 5682.7&5688.2\\ +&5149.1&5153.6\\ +& 4978.6&4982.9\\ +Blue&4748.0&4751.9\\ +&4664.9&4668.6\\ +Blue-Violet&4494.3&4497.7\\ +\end{tabular} +\caption{\label{tab:sodium}Wavelength of the visible sodium lines.} +\end{table} +Line$_1$ and Line$_2$ correspond to transitions to two fine-spitted $3p$ states $P_{1/2}$ and +$P_{3/2}$. These two transition frequencies are very close to each other, and to resolve them with +the spectrometer the width of the slit should be very narrow. However, you may not be able to see +some weaker lines then. In this case you should open the slit wider to let more light in when +searching for a line. If you can see a spectral line but cannot resolve the doublet, record the +reading for the center of the spectrometer line, and use the average of two wavelengthes given above. + +\textbf{Measurements of the fine structure splitting}. Once you measured all visible spectral lines, +go back to some bright line (yellow should work well), and close the collimator slit such that you +can barely see any light going through. In this case you should be able to see the splitting of the +line because of the \emph{fine structure splitting} of states $P_{1/2}$ and $P_{3/2}$ . For the Na D +doublet the splitting between two lines $\Delta\lambda=\lambda_{3/2}-\lambda_{1/2}$. + +Measure the splitting between two lines in the first and the second order. Which one works better? +Discuss this issue in your lab report. Compare to the accepted value: $\Delta\lambda=5.9$~\AA. +Compare this approach to the use of the Fabry-Perot interferometer. + + +\subsection*{Data analysis for Sodium} + +\textbf{Calculation of a quantum defect for $n=3, p$ state. } + +Once you identified all of your measured spectral lines, choose only those that correspond to optical +transitions from any $d$ to $n=3,p$ states. Since the energy states of $d$ series follows the +hydrogen spectra almost exactly, the wavelength of emitted light $\lambda$ is given by: +\begin{equation} +\frac{hc}{\lambda}=E_{nd}-E_{3p}=-\frac{hcRy}{n_d^2}+\frac{hcRy}{(3-\Delta_p)^2}, +\end{equation} +or +\begin{equation} +\frac{1}{\lambda}=\frac{Ry}{(3-\Delta_p)^2}-\frac{Ry}{n_d^2}, +\end{equation} + where $n_d$ is the principle number of the initial $d$ state. To verify this +expression by plotting $1/\lambda$ versus $1/n_d^2$ for the $n_d$= 4,5, and 6. From the slope of this +curve determine the value of the Rydberg constant $Ry$. The value of the intercept in this case is +$\frac{Ry}{(3-\Delta_p)^2}$, so use it to find the quantum defect $\Delta_p$. + +Compare the results of your calculations for the quantum defect with the accepted value +$\Delta_p=0.86$. + +%\subsection*{Calculation of a quantum defect for $s$ states} +%Now consider the transition from the $s$-states ($n=5,6,7$) to to the $n=3, p$ state. Using +%$hc/\lambda=E_{ns}-E_{3p}$ and the results of your previous calculations, determine the energies +%$E_{sn}$ for different $s$ states with $n=5,6,7$ and calculate $\Delta_s$. Does the value of the +%quantum defect depends on $n$? +% + +%\textbf{Example data table for writing the results of the measurements}: +% +%\noindent +%\begin{tabular}{|p{1.in}|p{1.in}|p{1.in}|p{1.in}|} +%\hline +% Line &$\theta_L$&$\theta_R$&Calculated $\lambda$ \\ \hline +% m=1 Violet&&&\\ \hline +% m=1 Blue&&&\\ \hline +% m=1 Green&&&\\ \hline +% m=1 Red&&&\\ \hline +% m=2 Violet&&&\\ \hline +% \dots&&&\\ \hline +% m=3 Blue&&&\\ \hline +% \dots&&&\\\hline +%\end{tabular} + +\section*{Appendix: Operation of a diffraction grating-based optical spectrometer} + +%\subsection*{Fraunhofer Diffraction at a Single Slit} +%Let's consider a plane electromagnetic wave incident on a vertical slit of +%width $D$ as shown in Fig. \ref{frn}. \emph{Fraunhofer} diffraction is +%calculated in the far-field limit, i.e. the screen is assume to be far away +%from the slit; in this case the light beams passed through different parts of +%the slit are nearly parallel, and one needs a lens to bring them together and +%see interference. +%\begin{figure}[h] +%\includegraphics[width=0.7\linewidth]{frnhfr.eps} +%\caption{\label{frn}Single Slit Fraunhofer Diffraction} +%\end{figure} +%To calculate the total intensity on the screen we need to sum the contributions +%from different parts of the slit, taking into account phase difference acquired +%by light waves that traveled different distance to the lens. If this phase +%difference is large enough we will see an interference pattern. Let's break the +%total hight of the slit by very large number of point-like radiators with +%length $dx$, and we denote $x$ the hight of each radiator above the center of +%the slit (see Fig.~\ref{frn}). If we assume that the original incident wave is +%the real part of $E(z,t)=E_0e^{ikz-i2\pi\nu t}$, where $k=2\pi/\lambda$ is the +%wave number. Then the amplitude of each point radiator on a slit is +%$dE(z,t)=E_0e^{ikz-i2\pi\nu t}dx$. If the beam originates at a hight $x$ above +%the center of the slit then the beam must travel an extra distance $x\sin +%\theta$ to reach the plane of the lens. Then we may write a contributions at +%$P$ from a point radiator $dx$ as the real part of: +%\begin{equation} +%dE_P(z,t,x)=E_0e^{ikz-i2\pi\nu t}e^{ikx\sin\theta}dx. +%\end{equation} +%To find the overall amplitude one sums along the slit we need to add up the +%contributions from all point sources: +%\begin{equation} +%E_P=\int_{-D/2}^{D/2}dE(z,t)=E_0e^{ikz-i2\pi\nu +%t}\int_{-D/2}^{D/2}e^{ikx\sin\theta}dx = A_P e^{ikz-i2\pi\nu t}. +%\end{equation} +%Here $A_P$ is the overall amplitude of the electromagnetic field at the point +%$P$. After evaluating the integral we find that +%\begin{equation} +%A_P=\frac{1}{ik\sin\theta}\cdot +%\left(e^{ik\frac{D}{2}\sin\theta}-e^{-ik\frac{D}{2}\sin\theta}\right) +%\end{equation} +%After taking real part and choosing appropriate overall constant multiplying +%factors the amplitude of the electromagnetic field at the point $P$ is: +%\begin{equation} +%A=\frac{\sin (\frac{\pi D}{\lambda}\sin\theta)}{\frac{\pi +%D}{\lambda}\sin\theta} +%\end{equation} +%The intensity is proportional to the square of the amplitude and thus +%\begin{equation} +%I_P=\frac{(\sin (\frac{\pi D}{\lambda}\sin\theta))^2}{(\frac{\pi +%D}{\lambda}\sin\theta)^2} +%\end{equation} +%The minima of the intensity occur at the zeros of the argument of the sin. The +%maxima are near, but not exactly equal to the solution of: +%\begin{equation} +% (\frac{\pi D}{\lambda}\sin\theta)=(m+\frac{1}{2})\pi \end{equation} +%for integer $m$. +% +%The overall pattern looks like that shown in Fig. \ref{sinxox}. +%\begin{figure} +%\includegraphics[width=\linewidth]{sinxox.eps} +%\caption{\label{sinxox}Intensity Pattern for Fraunhofer Diffraction} +%\end{figure} + +%\subsection*{The Diffraction Grating} +A diffraction grating is a common optical element, which consists of a pattern +with many equidistant slits or grooves. Interference of multiple beams passing +through the slits (or reflecting off the grooves) produces sharp intensity +maxima in the output intensity distribution, which can be used to separate +different spectral components on the incoming light. In this sense the name +``diffraction grating'' is somewhat misleading, since we are used to talk about +diffraction with regard to the modification of light intensity distribution to +finite size of a single aperture. +\begin{figure}[h] +\includegraphics[width=\linewidth]{./pdf_figs/grating} +\caption{\label{grating}Intensity Pattern for Fraunhofer Diffraction} +\end{figure} + +To describe the properties of a light wave after passing through the grating, +let us first consider the case of 2 identical slits separated by the distance +$h$, as shown in Fig.~\ref{grating}a. We will assume that the size of the slits +is much smaller than the distance between them, so that the effect of +Fraunhofer diffraction on each individual slit is negligible. Then the +resulting intensity distribution on the screen is given my familiar Young +formula: +\begin{equation} \label{2slit_noDif} +I(\theta)=\left|E_0 +E_0e^{ikh\sin\theta} \right|^2 = 4I_0\cos^2\left(\frac{\pi +h}{\lambda}\sin\theta \right), +\end{equation} +where $k=2\pi/\lambda$, $I_0$ = $|E_0|^2$, and the angle $\theta$ is measured +with respect to the normal to the plane containing the slits. +%If we now include the Fraunhofer diffraction on each slit +%same way as we did it in the previous section, Eq.(\ref{2slit_noDif}) becomes: +%\begin{equation} \label{2slit_wDif} +%I(\theta)=4I_0\cos^2\left(\frac{\pi h}{\lambda}\sin\theta +%\right)\left[\frac{\sin (\frac{\pi D}{\lambda}\sin\theta)}{\frac{\pi +%D}{\lambda}\sin\theta} \right]^2. +%\end{equation} + +An interference on $N$ equidistant slits illuminated by a plane wave +(Fig.~\ref{grating}b) produces much sharper maxima. To find light intensity on +a screen, the contributions from all N slits must be summarized taking into +account their acquired phase difference, so that the optical field intensity +distribution becomes: +\begin{equation} \label{Nslit_wDif} +I(\theta)=\left|E_0 ++E_0e^{ikh\sin\theta}+E_0e^{2ikh\sin\theta}+\dots+E_0e^{(N-1)ikh\sin\theta} +\right|^2 = I_0\left[\frac{sin\left(N\frac{\pi +h}{\lambda}\sin\theta\right)}{sin\left(\frac{\pi h}{\lambda}\sin\theta\right)} +\right]^2. +\end{equation} + Here we again neglect the diffraction form each individual slit, assuming that the + size of the slit is much smaller than the separation $h$ between the slits. + +The intensity distributions from a diffraction grating with illuminated + $N=2,5$ and $10$ slits are shown in Fig.~\ref{grating}c. The tallest (\emph{principle}) maxima occur when the denominator + of Eq.(~\ref{Nslit_wDif}) becomes zero: $h\sin\theta=\pm m\lambda$ where + $m=1,2,3,\dots$ is the diffraction order. The heights of the principle maxima are + $I_{\mathrm{max}}=N^2I_0$, and their widths are $\Delta \theta = + 2\lambda/(Nh)$. + Notice that the more slits are illuminated, the narrower diffraction peaks + are, and the better the resolution of the system is: + \begin{equation} +\frac{ \Delta\lambda}{\lambda}=\frac{\Delta\theta}{\theta} \simeq \frac{1}{Nm} +\end{equation} +For that reason in any spectroscopic equipment a light beam is usually expanded +to cover the maximum surface of a diffraction grating. + +\subsection*{Diffraction Grating Equation when the Incident Rays are +not Normal} + +Up to now we assumed that the incident optical wavefront is normal to the pane of a grating. Let's +now consider more general case when the angle of incidence $\theta_i$ of the incoming wave is +different from the normal to the grating, as shown in Fig. \ref{DGnotnormal}(a). Rather then +calculating the whole intensity distribution, we will determine the positions of principle maxima. +The path length difference between two rays 1 and 2 passing through the consequential slits us $a+b$, +where: +\begin{equation} +a=h\sin \theta_i;\,\, b=h\sin \theta_R +\end{equation} +Constructive interference occurs for order $m$ when $a+b=m\lambda$, or: +\begin{equation} +h\sin \theta_i + h\sin\theta_R=m\lambda +\end{equation} + +\begin{figure} +\includegraphics[width=\columnwidth]{./pdf_figs/pic4i} +%\includegraphics[height=3in]{dn.eps} +\caption{\label{DGnotnormal}Diagram of the light beams diffracted to the right +(a) and to the left (b).} +\end{figure} + +Now consider the case shown in Fig. \ref{DGnotnormal}(b). The path length between two beams is now +$b-a$ where $b=h\sin\theta_L$. Combining both cases we have: +\begin{eqnarray} \label{angles} +h\sin\theta_L-h\sin\theta_i&=&m\lambda\\ +h\sin\theta_R+h\sin\theta_i&=&m\lambda \nonumber +\end{eqnarray} +Adding these equations and dividing the result by 2 yields the following expression connecting the +right and left diffraction angles: +\begin{equation}m\lambda=\frac{h}{2}\left(\sin\theta_L+\sin\theta_R\right) +\end{equation} + diff --git a/manual_source/chapters/spol.tex b/manual_source/chapters/spol.tex new file mode 100644 index 0000000..6c0446e --- /dev/null +++ b/manual_source/chapters/spol.tex @@ -0,0 +1,141 @@ +%\chapter*{Measuring the Speed of Light} +%\addcontentsline{toc}{chapter}{Measuring the Speed of Light} +\documentclass{article} +\usepackage{tabularx,amsmath,boxedminipage,epsfig} + \oddsidemargin 0.0in + \evensidemargin 0.0in + \textwidth 6.5in + \headheight 0.0in + \topmargin 0.0in + \textheight=9.0in + +\begin{document} +\title{Measuring the Speed of Light} +\date {} +\maketitle + +\noindent + \textbf{Experiment objectives}: Determine the speed of light directly by + measuring time delays of pulses. + +\section*{History} + + The speed of light is a fundamental constant of nature, the value +we now take for granted. In 1983, the internationally adopted value in vacuum became: + +\[ +c = 2.99792458 \times 10^8 m/s\,\, \mbox{exactly} +\] + +But considering that light travels seven and a half times around the world in one second, you can imagine how +challenging a measurement it would be to determine the exact value of the speed of light. In fact, it took +several attempts over many centuries to determine the value (some of the measurements are shown in Table 1). +\begin{figure}[hbt] +\centerline{\epsfig{file=ctable.eps, width=6in, angle=0}} \label{fig:ctable} + +\end{figure} + +The first attempt at a measurement was made by Galileo in 1600 using two lanterns on hills. He had an assistant +on a distant mountain who would signal when he saw a lantern be masked, and then Galileo would measure the +interval between his own signaling and the response of his assistant. He only could find the speed of light to +be ``very fast''. But interestingly enough, the technique you will use is nowhere near the best, but it is +direct and in some ways similar to Galileo's. + +Several other experiments followed over the centuries until Michelson and Morely made a very accurate +measurement in 1887 using a specially design interferometer (which by lucky coincidence you explore during +another lab in our course). The currently accepted value was not determined until the advent of the laser. + +You might wonder why the speed of light is now a defined quantity. The +measurements at the end of the Table are measurements of the wavelength +and frequency of light, both referenced to the wavelength of atomic transitions +and to the frequency of atomic transitions. Distances can be measured to +small fractions of the wavelength of light, and this over distances of +meters. Frequencies are compared by beating one light signal against another +so that the difference frequency can be directly compared to atomic clocks. +You can estimate the accuracy of this by considering a meter to be measured +to $10^{-3}$ of $\lambda$ of some visible lightwave, and $\nu$, the frequency +can be measured to $10^{-5}$ Hz out of the frequency of an atomic transition. + +\section*{Procedure} + +\subsection*{Laser Safety} +While this is a weak laser caution should still be used. \textbf{Never look directly at the laser beam!} Align +the laser so that it is not at eye level. + +\subsection*{Set Up} +\textbf{Equipment needed}: diode laser, photodetector, lens, Pasco magnetic platform, large mirror on a rolling +table, small reference mirror, function generator, oscilloscope. + +In the experiment you modulate the power sent to the laser to produce short pulses of light, and then measure +the time it takes for these pulses to travel from the laser to the mirror and back to the photodetector, as +shown in the layout for the experiment in Fig. \ref{fig:solapp}. This measurement is repeated for several +displacements of the mirror (the more the better) by rolling the table with the mirror along the corridor (if +you like challenges, you can try to see how far you can go). + + +% +\begin{figure}[hbt] +\begin{center} +\epsfig{file=solapp.eps, width=5in, angle=0} +\end{center} +\caption{Speed of light Apparatus} \label{fig:solapp} +\end{figure} +% +\subsection*{Data acquisition} + +\begin{itemize} + +\item Put a rolling table as close as possible to the stationary table with the laser and the photodetector. Make +sure you have enough clearance to push the table along the corridor (you may need to move the tables). Make sure +that the laser beam hits the mirror relatively close to the center, and use fine tuning on the mirror to reflect +the beam to the photodetector - first without the lens, then with the lens in place. + +\item Plug in the output of the photodetector to the oscilloscope, and use a TTL pulse output as a trigger. If +everything works, you will see a train of nearly square pulses. Before starting the measurements, you first need +to think about two issues (\textit{the instructor will ask you about them}!): \\ +1) How will the detected signal change as you start pushing the mirror farther and farther? \\ +2) What is a suitable characteristic feature(s) of the detected signal to trace this change? Also, Make +yourself familiar with the scope features, such as ``measurements'' and ``save traces'' (your instructor or TA +will be able to help you with that). That will make your data acquisition easier. + + +\item Vary the position of the mirror by moving the rolling table from as close as possible to as far as possible +in about $10$ steps (the more measurements the more accurate final result you will have). For each step measure +the position of the table $D$. The floor tiles make a reasonable gauge - each tile is a 9 inch square (remember +to convert to meters!). Count the tile squares and double check. + +For each position each member of the group determine the light pulse time delay $T_{1,2,\cdots}$ by comparing +the time difference between the chosen characteristic features for the light reflected off the large ``distant'' +mirror and small ``reference'' mirror placed near the detector. Calculate average value $T_{ave}$ and the +uncertainty $\Delta T$. Below is the example of a table for data recording. + +\end{itemize} + +\vskip .1in + +\begin{tabular}{|l|l|l|l|l|l|l|}\hline +$D \pm \Delta D$ ($\#$ of tiles)& $D \pm \Delta D$ (m) &$T_1$($\mu$s)& $T_2$($\mu$s) +& $T_3$($\mu$s) & $T_{ave}$ ($\mu$s) & $\Delta T$ ($\mu$s) \\ +\hline &&&&&&\\\hline +&&&&&&\\$\dots$&$\dots$&$\dots$&$\dots$&$\dots$&$\dots$&$\dots$\\\hline +&&&&&&\\\hline &&&&&&\\\hline &&&&&&\\\hline + +\end{tabular} + +\vskip .2in +\noindent + +\subsection*{Analysis} + +Plot the results of the measurements as a distance vs time delay graph $D vs. T$. If the measurements are done +properly, the data will be scattered close to a straight line, and the slope of this line is inversely +proportional to the speed of light $1/v$. Thus, the measured $v$ and its uncertainty can be extracted from +fitting the experimental data. This method should give more accurate results than calculating $v$ from each +measurements, since it allows avoiding systematic errors due to an offset in the distance measurements. From +same fit determine the distance intercept. It the obtained value reasonable? + +In the lab report compare the measured speed of light with the theoretical +value. Is it within experimental uncertainty? If it is not, discuss possible +systematic errors which affected your results. + +\end{document} diff --git a/manual_source/chapters/supcon.tex b/manual_source/chapters/supcon.tex new file mode 100644 index 0000000..c74b856 --- /dev/null +++ b/manual_source/chapters/supcon.tex @@ -0,0 +1,193 @@ +%\chapter*{Properties of Solids: Superconductivity} \author{} \date{} +%\addcontentsline{toc}{chapter}{Superconductivity} + +%\documentclass{article} +%\usepackage{tabularx,amsmath,boxedminipage,epsfig} +% \oddsidemargin 0.0in +% \evensidemargin 0.0in +% \textwidth 6.5in +% \headheight 0.0in +% \topmargin 0.0in +% \textheight=9.0in + + +\chapter{Superconductivity} +\setcounter{figure}{1} +\setcounter{table}{1} +\setcounter{equation}{1} + + \textbf{Experiment objectives}: study behavior of a high temperature superconducting material + Yttrium-Barium-Copper-Oxide (YBCO, $YBa_2Cu_3O_7$) in magnetic field, measure the critical + temperature for a phase transition in a superconductor. + +\subsection*{History} + +Solids can be roughly divided into four classes, according to the way they +conduct electricity. They are: Metals, Semiconductors, Insulators and +Superconductors. The behavior of these types of materials is explained by +quantum mechanics. Basically, when atoms form a solid, the atomic levels of the +electrons combine to form bands. That is over a finite range of energy there +are states available to electrons. Since only one electron can occupy a given +state, the {\bf Pauli Exclusion Principle}, electrons will fill these states up +to some maximum, the Fermi Energy: $E_f$. A solid is a metal if it has an +energy band which is not full; the electrons are then free to move about, +making a metal a good conductor of electricity. If the solid has a band which +is completely full, with an energy gap to the next band, that solid will not +conduct electricity very well, making it an insulator. A semiconductor is +between a metal and insulator: while it has a full band (the valence band), +the next band (the conduction band) is close enough in energy and so that the +electrons can easily reach it. Superconductors are in a class by themselves. +They can be metals or insulators at room temperature. Below a certain +temperature, called the critical temperature, the electrons "pair" together (in +Cooper pairs) and travel through the solid without resistance. Current in a +superconductor below the critical temperature will travel indefinitely without +dissipation. + + Superconductivity was discovered in 1911 by H. Onnes. He + discovered that simple metals (Pb, Nb) superconduct when + placed in liquid helium (4 K). This was an important + discovery, but the real excitement came in 1986 when Swiss + scientists discovered that certain ceramics would superconduct + at 35 K. Several groups later discovered materials that would + superconduct at temperatures up to 125 K. These materials are + called high temperature superconductors (HTS). Their discovery + was a breakthrough, because this means that these + superconductors will work in liquid nitrogen (at 77 K), which + is relatively cheap and abundant. + + Some fascinating facts about superconductors: they will carry + a current nearly indefinitely, without + resistance. Superconductors have a critical temperature, above which they lose their + superconducting properties. + + Another striking demonstrations of superconductivity is the \textbf{Meissner effect}. + Magnetic fields cannot penetrate superconducting surface, instead a + superconductor attempts to expel all magnetic field + lines. It is fairly simple to intuitively understand the Meissner effect, if you imagine a perfect + conductor of electricity. If placed in a magnetic field, + Faraday's Law says an induced current which opposes the field + would be setup. But unlike in an ordinary metal, this induced current does not dissipate in + a perfect conductor. So, this + induced current would always be present to produce a field + which opposes the external field. In addition, microscopic dipole moments + are induced in the superconductor that oppose the applied field. This induced field +repels the source of the applied magnetic field, and will consequently repel +the magnet associated with this field. Thus, a superconductor will levitate a +magnet placed upon it (this is known as magnetic levitation). + +\subsection*{Safety} +\begin{itemize} +\item Wear glasses when pouring liquid nitrogen. Do not get it on your +skin or in your eyes! +\item Do not touch anything that has been immersed in liquid nitrogen until the +item warms up to the room temperature. Use the provided tweezers to remove and +place items in the liquid nitrogen. +\item Do not touch the superconductor, it contains poisonous materials!. +\item Beware of the current leads, they are carrying a lethal current! +\end{itemize} + + +\section*{Experimental procedure} +\textbf{Equipment needed}: YBCO disc, tweezers, styrofoam dish, small magnet. + + + +\subsection*{Magnetic Levitation (the Meissner effect)} + +\begin{enumerate} + +\item Place one of the small magnets (provided) on top of the superconducting +disc at room temperature. Record the behavior of the magnet. + +\item Using the tweezers, place the superconducting disk in the styrofoam + dish. Attach the thermocouple leads (see diagram) to a multimeter + reading on the mV scale. Slowly pour liquid nitrogen over the disk, + filling the dish as much as you can. The nitrogen will boil, and + then settle down. When the multimeter reads about 6.4 mV, you are + at liquid nitrogen temperature (77 K). + + +\item After the disc is completely covered by the liquid nitrogen, use the tweezers +to pick up the provided magnet and attempt to balance it on top of the +superconductor disk. Record what you observe. + +\item Try demonstrating a \emph{frictionless magnetic bearing}: if you carefully set the magnet rotating, +you will observe that the magnet continues to rotate for a long time. Also, try +moving the magnet across the superconductor. Do you feel any resistance? If you + feel resistance, why is this. + +\item Using tweezers, take the disk (with the magnet on it) out of the + nitrogen (just place it on side of disk), allowing it to + warm. Watch the thermocouple reading carefully, and take a reading + when the magnet fails to levitate any longer. This is a rough estimate of the + critical temperature. Make sure you record it! + +\item Repeat the experiment by starting with the magnet on top of the +superconductor disc and observe if the magnet starts levitating when the disk's +temperature falls below critical. + +\end{enumerate} +\begin{figure} +\includegraphics[height=2in]{./pdf_figs/scnut} +\caption{\label{scnut} The superconducting disk with leads.} +\end{figure} + + +\subsection*{Measuring resistance and critical temperature} + + We will measure the resistance by a {\bf four probe method}, as a + function of temperature. Using four probes (two for current + and two for voltage) eliminates the contribution of resistance + due to the contacts, and is good to use for samples with small + resistances. Connect a voltmeter (with 0.01 mV resolution) to + the yellow wires. Connect a current source through an ammeter + to the {\bf black} wires. Place a current of about 0.2 Amps (200 mA) + through black leads. Note: {\bf DO NOT EXCEED 0.5 AMP!!!!} + %On the +% Elinco power supplies, you hardly have to turn the knobs at +% all! +At room temperature, you should be reading a non-zero + voltage reading. + +\begin{enumerate} + \item With the voltage, current and thermocouple leads attached, + carefully place disk in dish. Pour liquid nitrogen into the + dish. Wait until temperature reaches 77 K. +\item With tweezers, take disk out of nitrogen and place on a side of the + dish. {\bf Start quickly recording the current, voltage and thermocouple readings + as the disk warms up.} When superconducting, the disk should have + V=0 (R=0). At a critical temperature, you will see a voltage + (resistance) appear. + +\item Repeat this measurement several time to acquire significant number of data points +near the critical temperature (6.4-4.5 mV). Make a plot of + resistance versus temperature, and make an estimate of the critical + temperature based on this plot. + +\end{enumerate} + +\section*{Resistance of a ceramic resistor} +\begin{enumerate} +\item Attach a ceramic resistor to a multimeter reading resistance ($k\Omega$ range). Record the room temperature resistance. + +\item Dunk the resistor in liquid nitrogen. Wait until it stops boiling. Record the resistance at this low temperature ($\approx$77 K). + +\item Take the resistor out of the nitrogen and carefully set it down. Record the resistance as the temperature increases. Make a plot of the measured resustance vs temperature. Compare the plots for the superconductor and the normal resistor, and explain the differences. +\end{enumerate} + +\hskip-.8in\includegraphics[height=5in]{./pdf_figs/mvtok} +%\end{document} + + +%\begin{tabular}{|p{17mm}|p{17mm}|p{17mm}|p{35mm}|p{35mm}|}\hline +% V (mV)& I (mA)& R ($\Omega$)& Thermocouple (mV)& Temperature (K)\\ +%&&&&\\\hline &&&&\\\hline &&&&\\\hline &&&&\\\hline &&&&\\\hline &&&&\\\hline +%&&&&\\\hline &&&&\\\hline &&&&\\\hline &&&&\\\hline &&&&\\\hline &&&&\\\hline +%&&&&\\\hline &&&&\\\hline &&&&\\\hline &&&&\\\hline &&&&\\\hline &&&&\\\hline +%&&&&\\\hline &&&&\\\hline &&&&\\\hline &&&&\\\hline &&&&\\\hline &&&&\\\hline +%&&&&\\\hline &&&&\\\hline &&&&\\\hline &&&&\\\hline &&&&\\\hline &&&&\\\hline +%&&&&\\\hline &&&&\\\hline &&&&\\\hline &&&&\\\hline &&&&\\\hline &&&&\\\hline +%&&&&\\\hline &&&&\\\hline &&&&\\\hline &&&&\\\hline &&&&\\\hline &&&&\\\hline +%\end{tabular} + +\newpage diff --git a/manual_source/chapters/two-photon-interference.tex b/manual_source/chapters/two-photon-interference.tex new file mode 100644 index 0000000..f1c026d --- /dev/null +++ b/manual_source/chapters/two-photon-interference.tex @@ -0,0 +1,397 @@ +%\documentclass{article} +%\usepackage{tabularx,amsmath,boxedminipage,epsfig} +% \oddsidemargin 0.0in +% \evensidemargin 0.0in +% \textwidth 6.5in +% \headheight 0.0in +% \topmargin 0.0in +% \textheight=9.0in + +%\begin{document} +\chapter{Two-Slit Interference, One Photon at a Time} +\setcounter{figure}{1} +\setcounter{table}{1} +\setcounter{equation}{1} +%\date {} +%\maketitle + + + +\noindent + \textbf{Experiment objectives}: Study wave-particle duality for photons by measuring + interference pattern in the Young double-slit experiment using conventional light source (laser) and a + single-photon source (strongly attenuated lamp). + + + +\section*{History} + +There is a rich historical background behind the experiment you are about to perform. Isaac Newton first +separated white light into its colors, and in the 1680's hypothesized that light was composed of 'corpuscles', +supposed to possess some properties of particles. This view reigned until the 1800's, when Thomas Young first +performed the two-slit experiment now known by his name. In this experiment he discovered a property of +destructive interference, which seemed impossible to explain in terms of corpuscles, but is very naturally +explained in terms of waves. His experiment not only suggested that such 'light waves' existed; it also +provided a result that could be used to determine the wavelength of light, measured in familiar units. Light +waves became even more acceptable with dynamical theories of light, such as Fresnel's and Maxwell's, in the 19th +century, until it seemed that the wave theory of light was incontrovertible. + + +\begin{figure}[h] +\centering +\includegraphics[width=0.4\linewidth]{./pdf_figs/ambigram} \caption{\label{ambigram.fig} ``Light is a Particle / Light is a Wave'' oscillation ambigram (from \emph{For the Love of Line and Pattern}, p. 30).} +\end{figure} + +And yet the discovery of the photoelectric effect, and its explanation in terms of light quanta by Einstein, +threw the matter into dispute again. The explanations of blackbody radiation, of the photoelectric effect, and +of the Compton effect seemed to point to the existence of 'photons', quanta of light that possessed definite and +indivisible amounts of energy and momentum. These are very satisfactory explanations so far as they go, but +they throw into question the destructive-interference explanation of Young's experiment. Does light have a dual +nature, of waves and of particles? And if experiments force us to suppose that it does, how does the light know +when to behave according to each of its natures? + +It is the purpose of this experimental apparatus to make the phenomenon of light interference as concrete as +possible, and to give you the hands-on familiarity which will allow you to confront wave-particle duality in a +precise and definite way. When you have finished, you might not fully understand the mechanism of duality -- +Feynman asserts that nobody really does -- but you will certainly have direct experience of the actual phenomena +that motivate all this discussion. + +\section*{Experimental setup} \textbf{Equipment needed}: Teachspin ``Two-slit interference'' apparatus, +oscilloscope, digital multimeter, counter. + +\begin{figure} +\centering +\includegraphics[width=0.8\linewidth]{./pdf_figs/tsisetup} \caption{\label{tsifig1.fig}The double slit interference apparatus.} +\end{figure} + +\textbf{Important}: before plugging anything in, or turning anything on confirm that the shutter (which +protects the amazingly sensitive single-photon detector) is \textbf{closed}. Locate the detector box at the +right end of the apparatus, and find the rod which projects out of the top of its interface with the long +assembly. Be sure that this rod is pushed all the way down; take this opportunity to try pulling it vertically +upward by about $2$~cm, but then ensure that it's returned to its fully down position. Also take this occasion +to confirm, on the detector box, that the toggle switch in the HIGH-VOLTAGE section is turned off, and that the +10-turn dial near it is set to $0.00$, fully counter-clockwise. + +To inspect the inside of the apparatus open the cover by turning four latches that hold it closed. The details +of the experimental apparatus are shown in Fig.~\ref{tsifig1.fig}. Take time to locate all the important +components of the experiment: +\begin{itemize} +\item Two distinct light sources at the left end: one a red \emph{laser} and the other a green-filtered \emph{light +bulb}. A toggle switch on the front panel of the light source control box switches power from one source to the +other. + +\item Various \emph{slit holders} along the length of the long box: one to hold a two-slit mask, one for slit blocker, +and one for a detector slit. Make sure you locate \emph{slits} (they may be installed already) and two +\emph{micrometer drives}, which allow you to make mechanical adjustments to the two-slit apparatus. \textbf{Make +sure you figure out how to read the micrometer dials!} On the barrel there are two scales with division of +$1$~mm, shifted with respect to each other by 0.5~mm; every fifth mark is labeled with an integer 0, 5, 10 and +so on: these are at 5-mm spacing. The complete revolution of the drum is $0.5$~mm, and the smallest division on +the rotary scale is $0.01$~mm. + +\item Two distinct light detectors at the right-hand end of the apparatus: a \emph{photodiode} +and a \emph{photomultiplier tube} (PMT for short). The photodiode is used with the much brighter laser light; +it's mounted on light shutter in such a way that it's in position to use when the shutter is closed (pushed +down). The photomultiplier tube is extremely sensitive detector able to detect individual photons (with energy +of the order of $10^{-19}$~J, and it is used with the much dimmer light-bulb source. Too much light can easily +damage it, so \textbf{PMT is safe to use only when the cover of the apparatus is in place, and only when the +light bulb is in use}. It is exposed to light only when the shutter is in its up position. +\end{itemize} + + +\section*{Experimental procedure} + +The experiment consists of three steps: +\begin{enumerate} +\item You will first observe two-slit interference directly by observing the intensity +distribution of a laser beam on a viewing screen. +\item Using the photodiode you will accurately measure the intensity distribution after single- and two-slit interference patterns, +which can be compared to predictions of wave theories of light. \\ +These two steps recreate original Young's experiment. +\item Then using a very weak light source you will record the two-slit interference pattern one photon at a time. +While this measurement will introduce you to single-photon detection technology, it will also show you that +however two-slit interference is to be explained, it must be explained in terms that can apply to single +photons. +\end{enumerate} + +\subsection*{Visual observation of a single- and two-slit interference} + +For this mode of operation, you will be working with the cover of the apparatus open. Switch the red diode laser +on using the switch in the light source control panel, and move the laser in the center of its magnetic pedestal +so that the red beam goes all the way to the detector slit. The diode laser manufacturer asserts that its output +wavelength is $670 \pm 5$~nm, and its output power is about 5~mW. \emph{\textbf{As long as you don't allow the +full beam to fall directly into your eye, it presents no safety hazard.}} Place a double slit mask on the holder +in the center of the apparatus, and then put your viewing card just after the mask to observe the two ribbons of +light, just a third of a millimeter apart, which emerge from the two slits. Move you viewing card along the beam +path to see the interference pattern forming. By the time your viewing card reaches the right-hand end of the +apparatus, you'll see that the two overlapping ribbons of light combine to form a pattern of illumination +displaying the celebrated ``fringes'' named after Thomas Young. + +Position a viewing card at the far-right end of the apparatus so you can refer to it for a view of the fringes. +Now it is the time to master the control of the slit-blocker. By adjusting the multi-turn micrometer screw, make +sure you find and record the ranges of micrometer reading where you observe the following five situations: +\begin{enumerate} +\item both slits are blocked; +\item light emerges only from one of the two slits; +\item both slits are open +\item light emerges only from the other slit; +\item the light from both slits is blocked. +\end{enumerate} +It is essential that you are confident enough in your ability to read, and to set, these five positions that +you'll be able to do so even when the box cover is closed. In your lab book describe what you see at the viewing +card at the far-right end of the apparatus for each of the five settings. + +\textbf{One slit is open:} According to the wave theory of light, the intensity distribution of light on the +screen after passing a single slit is described by Fraunhofer diffraction (see Fig.~\ref{interference.fig} and +the derivations in the Appendix): +\begin{equation} \label{1slit} +I(x) = I_0 \frac{(\sin (\frac{\pi a}{\lambda}\frac{x}{\ell}))^2}{(\frac{\pi +a}{\lambda}\frac{x}{\ell})^2}, +\end{equation} +where $I$ is the measured intensity in the point $x$ in the screen, $I_0$ is the intensity in the brightest +maximum, $a$ is the width of the slit, and $\ell$ is the distance between the slit and the screen (\emph{don't +forget to measure and record this distance in the lab journal!}) + +In your apparatus move the slit blocker to let the light go through only one slit and inspect the +light pattern in the viewing screen. Does it looks like the intensity distribution you expect from +the wave theory? Take a minute to discuss how this picture would change if the slit was much wider or +much narrower. + +\textbf{Two slits are open:} Now move the slit blocker to the position that opens both slits to observe Young's +two-slit interference fringes. Again, compare what you see on the screen with the interference picture predicted +by wave theory: +\begin{equation} \label{2slit_wDif} +I(x)= 4 I_0 \cos^2\left(\frac{\pi d}{\lambda}\frac{x}{\ell} \right)\left[\frac{\sin (\frac{\pi +a}{\lambda}\frac{x}{\ell})}{\frac{\pi a}{\lambda}\frac{x}{\ell}} \right]^2, +\end{equation} +where an additional parameter $d$ is the distance between centers of the two slits. Discuss how this picture +would change if you vary the width and the separation of the two slits, and the wavelength of the laser. Make a +note of your predictions in the lab book. + +\subsection*{Quantitative characterization of interference patterns using laser light} + +At this stage you will use a photodiode to measure the intensity distribution of the interference pattern by +varying the position of the detector slit. You will continue using the red laser. While you may conduct these +measurements with the box cover open, room light will inevitably add some varying background to your signals, so +it is a good idea to dim the room lights or (even better!) to close up the cover of the apparatus. For +convenience, have the slit-blocker set to that previously determined setting which allows light from both slits +to emerge and interfere. + +The shutter of the detector box will still be in its closed, or down, position: this blocks any light from +reaching the PMT, and correctly position a 1-cm$^{2}$ photodiode, which acts just like a solar cell in actively +generating electric current when it's illuminated. The output current is proportional to total power +illuminating the detector area, so it is important to use a narrow slit allow only a selected part of the +interference pattern to be measured. Make sure that a detector slit mask (with a single narrow slit) on a +movable slit holder at the right-hand side of the apparatus is in place. By adjusting the micrometer screw of the +detector slit, you can move the slit over the interference pattern, eventually mapping out its intensity +distribution quantitatively. For now, ensure that the detector slit is located somewhere near the middle of the +two-slit interference pattern, and have the slit-blocker set to the setting which allows light from both slits +to emerge and interfere. + +The electric \emph{current} from the photodiode, proportional to the \emph{light intensity}, is conducted by a +thin coaxial cable to the INPUT BNC connector of the photodiode-amplifier section of the detector box, and +converted to \emph{voltage} signal at the OUTPUT BNC connector adjacent to it. Connect to this output a digital +multimeter set to 2 or 20-Volt sensitivity; you should see a stable positive reading. Turn off the laser first +to record the ``zero offset'' - reading of the multimeter with no light. You will need to subtract this reading +from all the other reading you make of this output voltage. + +Turn your laser source back on, and watch the photodiode's voltage-output signal as you vary the setting of the +detector-slit micrometer. If all is well, you will see a systematic variation of the signal as you scan over the +interference pattern. Check that the maximum signal you see is about 3-8 Volts; if it is much less than this, +the apparatus is out of alignment, and insufficient light is reaching the detector. + +\textbf{Initial tests of wave theory of light:} If we assume that the light beam is a stream of particle, we +would naively expect that closing one of two identical slits should reduce the measured intensity of light at +any point on the screen by half, while the wave theory predicts much more dramatic variations in the different +points in the screen. Which theory provide more accurate description of what you see? + +\begin{itemize} +\item Find the highest of the maxima -- this is the ``central fringe'' or the ``zeroth-order fringe'' which theory +predicts, -- and record the photodiode reading. Then adjust the position of the slit-blocker to let the light +to pass through only one of the slits, and measure the change in the photodiode signal. + +\item To see another and even more dramatic manifestation of the wave nature of light, set the slit blocker again +to permit light from both slits to pass along the apparatus, and now place the detector slit at either of the +minima immediately adjacent to the central maximum; take some care to find the very bottom of this minimum. +Record what happens when you use the slit-blocker to block the light from one, or the other, of the two slits? + +\item Check your experimental results against the theoretical predictions using Eqs.~(\ref{1slit}) and (\ref{2slit_wDif}). +Do your observation confirm or contradict wave theory? + +\end{itemize} + +Once you have performed these spot-checks, and have understood the motivation for them and the obtained results, you are ready to conduct systematic measurements of intensity distribution (the photodiode voltage-output signal) as a function of detector slit position. You will make such measurements in two slit-blocker positions: when both slits are open, and when only one slit is open. You will need to take enough data points to reproduce the intensity distribution in each case. Taking points systematically every 0.05 or \unit[0.1]{mm} on the tick lines will produce a very high quality dataset. One person should turn the dial and the other should record readings directly to paper or a spreadsheet (if you do this, print it out and tape into your logbook). Estimate your uncertainties from the dial and the voltmeter. Cycle through multiple maxima and minima on both sides of the central maximum. It is a good idea to plot the data points immediately along with the data taking -- nothing beats an emerging graph for teaching you what is going on, and your graph will be pretty impressive. \emph{Note: due to large number of points you don't need to include the tables with these measurements in the lab report -- the plotted distributions should be sufficient. Be clear on your uncertainties though.} + +\textbf{Slit separation calculations}: Once you have enough data points for each graph to clearly see the +interference pattern, use your data to extract the information about the distance between two slits $d$. To do +that find the positions of consecutive interference maxima or minima, and calculate average $d$ using +Eq.~\ref{2slit_wDif}. Estimate the uncertainty in these parameters due to laser wavelength uncertainty. Check if +your measured values are within experimental uncertainty from the manufacturer's specs: the center-to-center +slit separation is 0.353 mm (or 0.406 or 0.457 mm, depending on what two-slit mask you have installed). + +Use Igor to fit your data with Eqs.~(\ref{1slit}) and (\ref{2slit_wDif}). You will need to add these functions using ``Add new function'' option. Note that in this case you will have to provide a list of initial guesses for all the fitting parameters. A few tips: +\begin{itemize} +\item Make sure that units of all your measured values are self-consistent - the program will go crazy trying to combine measurements in meters and and micrometers together! +\item Try to plot your function for guesstimated values before doing a fit with it. This catches many silly errors. +\item You will have to fit for a term to account for the overall normalization and also for the fact that the maximum is not set at $x=0$. In other words, substitute $x \rightarrow x-x_0$. Estimate both of these (for the normalization, look at what happens when $x-x_0 = 0$) but include them as free parameters in the fit. +\item You need to add a parameter to account for the non-zero background you observed when the laser was off. +\item Include the minimal number of parameters. If I have parameters $a$ and $b$ but they always appear as $ab$ in my function, then I am much better off including a term $c=ab$ when doing the fit. Otherwise the fit is underconstrained. Adjusting $a$ up has the same effect as adjusting $b$ down so it's impossible to converge on unique values for $a$ and $b$. Fitting algorithms really dislike this situation. +\item If the program has problems fitting all the parameters, first hold the values of the parameters you know fairly well (such as the light wavelength, maximum peak intensity, background, etc.). Once you determine the approximate values for all other parameters, you can release the fixed ones, and let the program adjust everything to make the fit better. + +\end{itemize} + +\subsection*{Single-photon interference} + +Before you start the measurements you have to convince yourself that the rate of photons emitted by the weak filtered light bulb is low enough to have in average less than one photon detected in the apparatus at any time. Roughly estimate the number of photons per second arriving to the detector. First, calculate the number of photons emitted by the light bulb in a 10~nm spectral window of the green filter (between $541$ and $551$~nm), if it runs at 6V and 0.2A, only 5\% of its electric energy turns into light, and this optical energy is evenly distributed in the spectral range between 500~nm and 1500~nm. These photons are emitted in all directions, but all of then are absorbed inside the box except for those passing through two slits with area approximately $0.1\times 10~\mathrm{mm}^2$. Next, if we assume that the beam of photons passing through the slits diffract over a $1~\mathrm{cm}^2$ area by the time they reach the detector slit, estimate the rate of photons reaching the detector. Finally, we have to adjust the detected photon rate by taking into account that for PMT only 4\% of photons produce output electric pulse at the output. That's the rate of event you expect. Now estimate the time it takes a photon to travel through the apparatus, and estimate the average number of detectable photons inside at a given moment of time. \emph{You may do this calculations before or after the lab period, but make sure to include them in the lab report.} + +Now you need to change the apparatus to use the light bulb. Open the cover and slide the laser source to the side (do not remove the laser from the stand). Now set the 3-position toggle switch to the BULB position and dial the bulb adjustment up from 0 until you see the bulb light up. (\emph{The flashlight bulb you're using will live longest if you minimize the time you spend with it dialed above 6 on its scale, and if you toggle its power switch only when the dial is set to low values}). If the apparatus has been aligned, the bulb should now be in position to send light through the apparatus. Check that the green filter-holding structure is in place: the light-bulb should look green, since the green filter blocks nearly all the light emerging from the bulb, passing only wavelengths in the range 541 to 551 nm. The filtered light bulb is very dim, and you probably will not be able to see much light at the double slit position even with room light turned off completely. No matter; plenty of green-light photons will still be reaching the double-slit structure -- in fact, you should now dim the bulb even more, by setting its intensity control down to about 3 on its dial. + +Now close and lock the cover - you are ready to start counting photons. But first a WARNING: a photomultiplier tube is so sensitive a device that it should not be exposed even to moderate levels of light when turned off, and must not be exposed to anything but the dimmest of lights when turned on. In this context, ordinary room light is intolerably bright even to a PMT turned off, and light as dim as moonlight is much too bright for a PMT turned on. + +\textbf{Direct observation of photomultiplier pulses} You will use a digital oscilloscope for first examination of the PMT output pulses, and a digital counter for counting the photon events. Set the oscilloscope level to about 50~mV/division vertically, and 250 - 500~ns/division horizontally, and set it to trigger on positive-going pulses or edges of perhaps $>20$~mV height. Now find the PHOTOMULTIPLIER OUTPUT of the detector box, and connect it via a BNC cable to the vertical input of the oscilloscope. Keeping the shutter closed, set the HIGH-VOLTAGE 10-turn dial to 0.00, and turn on the HIGH-VOLTAGE toggle switch. Start to increase the voltage while watching the scope display. \emph{If you see some sinusoidal modulation of a few mV amplitude, and of about 200 kHz frequency, in the baseline of the PMT signal, this is normal. If you see a continuing high rate ($>10$~kHz) of pulses from the PMT, this is not normal, and you should turn down, or off, the bias level and start fresh -- you may have a malfunction, or a light leak.} Somewhere around a setting of 4 or 5 turns of the dial, you should get occasional positive-going pulses on the scope, occurring at a modest rate of $1-10$ per second. If you see this low rate of pulses, you have discovered the ``dark rate'' of the PMT, its output pulse rate even in the total absence of light. You also now have the PMT ready to look at photons from your two-slit apparatus, so finally you may open the shutter. The oscilloscope should now show a much greater rate of pulses, perhaps of order $10^3$ per second, and that rate should vary systematically with the setting of the bulb intensity. \emph{You may find a small device called Cricket in your table. It allows you to "hear" the individual photon arrivals - ask your instructor to show you how it works.} + +To count the pulses using a counter you will use another PMT output -- the OUTPUT TTL -- that generates a single pulse, of fixed height and duration, each time the analog pulse exceeds an adjustable threshold. To adjust the TTL settings display the OUTPUT TTL on the second oscilloscope channel and set it for 2 V/div vertically. By simultaneously watching both analog and TTL-level pulses on the display, you should be able to find a discriminator setting, low on the dial, for which the scope shows one TTL pulse for each of, and for only, those analog pulses which reach (say) a $+50$~-mV level. If your analog pulses are mostly not this high, you can raise the PMT bias by half a turn (50 Volts) to gain more electron multiplication. If your TTL pulses come much more frequently than the analog pulses, set the discriminator dial lower on its scale. + +Now send the TTL pulses to a counter, arranged to display successive readings of the number of TTL pulses that occur in successive 1-second time intervals. To confirm that this is true, record a series of ``dark counts'' obtained with the light bulb dialed all the way down to 0 on its scale. Now choose a setting that gives an adequate photon count rate (about $10^3$/second) and use the slit-blocker, according to your previously obtained settings, to block the light from both slits. This should reduce the count rate to a background rate, probably somewhat higher than the dark rate. Next, open up both slits, and try moving the detector slit to see if you can see interference fringes in the photon count rate. You will need to pick a detector-slit location, wait for a second or more, then read the photon count in one or more 1-second intervals before trying a new detector-slit location. If you can see maxima and minima, you are ready to take data. Finally, park the slit near the central maximum and choose the PMT bias at around $5$ turns of the dial and the bulb intensity setting to yield some +convenient count rate ($10^3 - 10^4$ events/second) at the central maximum. + +%Before you begin the data collection you need to set the PMT bias voltage to a suitable range. \emph{This +%procedure is \textbf{optional}, and necessary if the event rate you observe is too low and/or the dark rate +%count is too high.} To do that you'll need to measure the dependence of dark count rate (PMT shutter closed) and +%the count rate at the central maximum of the interference pattern (both slits open) on the PMT bias voltage over +%the range 300 to 650 V. When you plot the two count rates on a semi-logarithmic graph, you should see the +%``light rate'' reach a plateau, with the interpretation that you have reached a PMT bias which allow each +%photoelectron to trigger the whole chain of electronics all the way to the TTL counter; you should also see the +%(much lower) 'dark rate' also rising with PMT bias. Based on your graph choose the PMT bias setting at which +%you are counting substantially all true photon events, but minimizing the number of ``dark events''. + +\textbf{Single-photon detection of the interference pattern} Most likely the experimental results in the +previous section has demonstrated good agreement with the wave description of light. However, the PMT detects +individual photons, so one can expect that now one has to describe the light beam as a stream of particles, and +the wave theory is not valid anymore. To check this assumption, you will repeat the measurements and take the +same sort of data as in the previous section, except now characterizing the light intensity as photon count +rate. +\begin{itemize} +\item Like previously, slowly change the position of the detection slit and record the average count rate in each +point. Start with the two-slit interference. Plot the data and confirm that you see interference fringes. + +\item Repeat the measurement with one slit blocked and make the plot. + +\item Use the spacing of the interference maxima to check that the light source has a different wavelength than +the red laser light you used previously. Using the previously determined value of the slit separation $d$, +calculate the wavelength of the light, and check that it is consistent with the green filter specs +($541-551$~nm). + +\end{itemize} + +The plots of your experimental data are clear evidence of particle-wave duality for photons. You've made contact with the central question of quantum mechanics: how can light, which so clearly propagates as a wave that we can measure its wavelength, also be detected as individual photon events? Or alternatively, how can individual photons in flight through this apparatus nevertheless 'know' whether one, or both, slits are open, in the sense of giving photon arrival rates which decrease when a second slit is opened? Discuss these issues in your lab report. + + +\section*{\emph{Two-Slit interference with atoms}\footnote{Special thanks to Prof. Seth Aubin for providing the materials for this section}} + +\emph{According to quantum mechanics, the wave-particle duality must be applied not only to light, but to any +``real'' particles as well. That means that under the right circumstance, atoms should behave as waves with +wavelength $\lambda_{\mathrm{atom}}=h/\sqrt{2mE}=h/p$ (often called deBroglie wavelength), where $h$ is Planck's +constant, $m$ is the mass of the particle, and $E$ and $p$ are respectively the kinetic energy and the momentum +of the particle. In general, wave effects with ``massive'' particles are much harder to observe compare to +massless photons, since their wavelengths are much shorter. Nevertheless, it is possible, especially now when +scientists has mastered the tools to produce ultra-cold atomic samples at nanoKelvin temperatures. As the energy +of a cooled atom decreases, its deBroglie wavelength becomes larger, and the atom behaves more and more like +waves. For example, in several experiments, researches used a Bose-Einstein condensate (BEC) -- the atomic +equivalent of a laser -- to demonstrate the atomic equivalent of the Young's double slit experiment. As shown in +Fig.~\ref{BECinterferfometer.fig}(a), an original BEC sits in single-well trapping potential, which is slowly +deformed into a double-well trapping potential thus producing two phase-coherent atom wave sources. When the +trapping potential is turned off, the two BECs expand and interfere where they overlap, just as in the original +Young's double slit experiment.} +% +\begin{figure}[h] +\centering +\includegraphics[width=0.8\linewidth]{./pdf_figs/BECinterferfometer} \caption{\label{BECinterferfometer.fig} +Atom interferometry version of Young's double slit experiment: \emph{(a)} schematic and \emph{(b)} +experimentally measured interference pattern in an ${}^{87}$Rb Bose-Einstein condensate.} +\end{figure} + +\emph{Fig.~\ref{BECinterferfometer.fig}(b), shows the resulting interference pattern for a ${}^{87}$Rb BEC. Atom +interferometry is an area of active research, since atoms hold promise to significantly improve interferometric +resolution due their much shorter deBroglie wavelength compared to optical photons. In fact, the present most +accurate measurements of accelerations, rotations, and gravity gradients are based on atomic interference. } + +\section*{Appendix: Fraunhofer Diffraction at a Single Slit and Two-Slit interference} + +\textit{Diffraction at a Single Slit} We will use a \emph{Fraunhofer} diffraction model to calculate the +intensity distribution resulting from light passing a single slit of width $a$, as shown in +Fig.~\ref{interference.fig}(a). We will assume that the screen is far away from the slit, so that the light +beams passed through different parts of the slit are nearly parallel. +\begin{figure}[h] +\begin{center} +\includegraphics[width=0.9\linewidth]{./pdf_figs/interference} +\caption{\label{interference.fig}(a) Single slit diffraction pattern formation. (b) Two-slit interference +pattern formation. (c) Examples of the intensity distributions on a viewing screen after passing one slit +(black), two infinitely small slits (red), two slits of finite width (blue). } \end{center} +\end{figure} +To calculate the total intensity on the screen we need to sum the contributions from different parts of the +slit, taking into account phase difference acquired by light waves that traveled different distance to the +screen. If this phase difference is large enough we will see an interference pattern. Let's break the total +height of the slit by very large number of point-like radiators with length $dx$ each and positioned at the +height $x$ above the center of the slit (see Fig.~\ref{interference.fig}(a)). Since it is more convenient to +work with complex numbers, we will assume that the original incident wave is a real part of +$E(z,t)=E_0e^{ikz-i2\pi\nu t}$, where $k=2\pi/\lambda$ is the wave number. Then the amplitude of each point +radiator on a slit is [a real part of] $dE(z,t)=E_0e^{ikz-i2\pi\nu t}dx$. A beam emitted by a radiator at the +height $x$ above the center of the slit must travel an extra distance $x\sin \theta$ to reach the plane of the +screen, acquiring an additional phase factor. Then we may write a contributions at the point $P$ from a point +radiator $dx$ as the real part of: +\begin{equation} +dE_P(z,t,x)=E_0e^{ikz-i2\pi\nu t}e^{ikx\sin\theta}dx. +\end{equation} +To find the overall amplitude at that point we need to add up the contributions from all point sources along the +slit: +\begin{equation} +E_P=\int_{-a/2}^{a/2}dE(z,t)=E_0e^{ikz-i2\pi\nu t}\int_{-a/2}^{a/2}e^{ikx\sin\theta}dx = A_P \times +E_0e^{ikz-i2\pi\nu t}. +\end{equation} +Here $A_P$ is the relative amplitude of the electromagnetic field at the point $P$: +\begin{equation} +A_P= \frac{1}{ik\sin\theta}\cdot \left(e^{ik\frac{a}{2}\sin\theta}-e^{-ik\frac{a}{2}\sin\theta}\right) \propto +\frac{\sin (\frac{\pi D}{\lambda}\sin\theta)}{\frac{\pi D}{\lambda}\sin\theta} +\end{equation} +%After taking real part and choosing appropriate overall constant the amplitude of the electromagnetic field at +%the point $P$ is: +%\begin{equation} +%A_P \propto \frac{\sin (\frac{\pi D}{\lambda}\sin\theta)}{\frac{\pi D}{\lambda}\sin\theta} +%\end{equation} +The intensity is proportional to the square of the amplitude and thus +\begin{equation} +I_P \propto \frac{(\sin (\frac{\pi a}{\lambda}\sin\theta))^2}{(\frac{\pi a}{\lambda}\sin\theta)^2} +\end{equation} +The minima of the intensity (``dark fringes'') occur at the zeros of the argument of the sin function: +$\frac{\pi D}{\lambda}\sin\theta=m\pi$, while the maxima (``bright fringes'') are almost exactly match +$\frac{\pi D}{\lambda}\sin\theta=(m+\frac{1}{2})\pi$ for $m = 0, \pm1, \pm2, \cdots$. + +Let us now consider the case of interference pattern from two identical slits separated by the distance $d$, as +shown in Fig.~\ref{interference.fig}(b). We will assume that the size of the slits is much smaller than the +distance between them, so that the effect of Fraunhofer diffraction on each individual slit is negligible. Then +going through the similar steps the resulting intensity distribution on the screen is given my familiar Young +formula: +\begin{equation} %\label{2slit_noDif} +I(\theta)=\left|E_0e^{ikd/2\sin\theta} +E_0e^{-ikd/2\sin\theta} \right|^2 = 4I_0\cos^2\left(\frac{\pi +h}{\lambda}\sin\theta \right), +\end{equation} +where $k=2\pi/\lambda$, $I_0$ = $|E_0|^2$, and the angle $\theta$ is measured with respect to the normal to the +plane containing the slits. + +If we now include the Fraunhofer diffraction on each slit as we did before, we arrive to the total intensity +distribution for two-slit interference pattern: +\begin{equation} %\label{2slit_wDif} +I(\theta)\propto \cos^2\left(\frac{\pi d}{\lambda}\sin\theta \right)\left[\frac{\sin (\frac{\pi +a}{\lambda}\sin\theta)}{\frac{\pi a}{\lambda}\sin\theta} \right]^2. +\end{equation} + +The examples of the light intensity distributions for all three situations are shown in +Fig.~\ref{interference.fig}(c). Note that the intensity distributions derived here are functions of the angle +$\theta$ between the normal to the plane containing the slits and the direction to the point on the screen. To +connect these equations to Eqs.~(\ref{1slit}) and(\ref{2slit_wDif}) we assume that $\sin\theta \simeq \tan\theta += x/\ell$ where $x$ is the distance to the point $P$ on the screen, and $\ell$ is the distance from the two slit +plane to the screen. + + + +%\end{document} diff --git a/manual_source/fig_sources/angle.fig b/manual_source/fig_sources/angle.fig new file mode 100644 index 0000000..6076fde --- /dev/null +++ b/manual_source/fig_sources/angle.fig @@ -0,0 +1,52 @@ +#FIG 3.2 +Landscape +Center +Inches +Letter +100.00 +Single +-2 +1200 2 +1 3 0 1 0 7 50 -1 -1 0.000 1 0.0000 5550 2700 1802 1802 5550 2700 5475 4500 +2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 2 + 3300 2700 3300 2850 +2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 2 + 3300 2550 3300 2700 +2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 2 + 3300 2700 6900 1500 +2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 2 + 3300 2700 7350 2700 +2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 2 + 6900 1500 6900 3900 +2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 2 + 3450 4500 3450 4950 +2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 2 + 7350 4500 7350 4950 +2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 + 1 1 1.00 60.00 120.00 + 4650 4725 7350 4725 +2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 + 1 1 1.00 60.00 120.00 + 4050 4725 3450 4725 +2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 2 + 5550 2700 6900 1500 +2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 2 + 3450 5100 3450 5475 +2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 2 + 6900 5100 6900 5475 +2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 4800 5250 3450 5250 +2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 + 1 1 1.00 60.00 120.00 + 4800 5250 3450 5250 +2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 + 1 1 1.00 60.00 120.00 + 5400 5250 6900 5250 +4 0 0 50 -1 32 12 0.0000 4 165 120 6075 2550 f\001 +4 0 0 50 -1 32 12 0.0000 4 135 105 4200 2625 q\001 +4 0 0 50 -1 0 12 0.0000 4 90 90 4350 2700 s\001 +4 0 0 50 -1 0 12 0.0000 4 135 120 6300 2175 R\001 +4 0 0 50 -1 0 12 0.0000 4 135 105 4350 4800 L\001 +4 0 0 50 -1 0 12 0.0000 4 90 90 5025 5325 x\001 +4 0 0 50 -1 0 12 0.0000 4 90 60 6750 2100 r\001 diff --git a/manual_source/fig_sources/tester.tex b/manual_source/fig_sources/tester.tex new file mode 100644 index 0000000..fe2ae4f --- /dev/null +++ b/manual_source/fig_sources/tester.tex @@ -0,0 +1,9 @@ +\documentclass{article} +\usepackage{epsfig} +\begin{document} +\begin{figure}\label{fig1mich.fig} +\centerline{\epsfig{file=fig1.eps}} +\vskip .25in +\caption{The Michelson Interferometer} +\end{figure} +\end{document} diff --git a/manual_source/manual.tex b/manual_source/manual.tex new file mode 100644 index 0000000..7800da0 --- /dev/null +++ b/manual_source/manual.tex @@ -0,0 +1,53 @@ +\documentclass[12pt,openany]{book} +\usepackage{tabularx,boxedminipage,amsmath,fullpage,units} +\usepackage{graphicx} +\usepackage[pdftex,final]{hyperref} +\hypersetup{ + colorlinks=true, % false: boxed links; true: colored links + linkcolor=blue, % color of internal links + citecolor=blue, % color of links to bibliography + filecolor=magenta, % color of file links + urlcolor=blue +} + +\newcommand{\vect}[1]{\boldsymbol{#1}} + +\begin{document} +\title{Physics 251 Atomic Physics Lab Manual} +\author{ + W. J. Kossler \and A. Reilly \and J. Kane (2006 edition) \and + I. Novikova (2009 edition) + \and M. Kordosky (2011-2012 edition) + \and E. E. Mikhailov (2013 edition) +} +\date{Fall 2013} +\maketitle + +\tableofcontents + +%\newpage + +%\input{chapters/sched.tex} +%\include{chapters/intro} +\include{chapters/interferometry} +\include{chapters/emratio} +\include{chapters/ediffract} +\include{chapters/blackbody} +\include{chapters/pe-effect} +\include{chapters/two-photon-interference} +\include{chapters/faraday_rotation} +\include{chapters/spectr} +\include{chapters/supcon} + +%\input{chapters/hspect.tex} +%\input{chapters/naspec.tex} + + +%\include{chapters/spol} +%\input{chapters/millikan.tex} + +%\input{chapters/tour.tex} +%\input{chapters/appendices.tex} + + +\end{document} diff --git a/manual_source/pdf_figs/BECinterferfometer.pdf b/manual_source/pdf_figs/BECinterferfometer.pdf Binary files differnew file mode 100644 index 0000000..d431d61 --- /dev/null +++ b/manual_source/pdf_figs/BECinterferfometer.pdf diff --git a/manual_source/pdf_figs/HeNelevels.pdf b/manual_source/pdf_figs/HeNelevels.pdf Binary files differnew file 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