From c066f70fdfee3b8f0f26def3f29acfbfed4ef63e Mon Sep 17 00:00:00 2001 From: Eugeniy Mikhailov Date: Fri, 13 Sep 2013 21:15:50 -0400 Subject: chapters now compiles separately as well via subfiles package unused chapters moved to separate folder --- unused_chapters/HeNelaser.tex | 291 ++++++++++++++++++++++++ unused_chapters/appendices.tex | 154 +++++++++++++ unused_chapters/fabry-perot.tex | 298 ++++++++++++++++++++++++ unused_chapters/hspect.tex | 436 ++++++++++++++++++++++++++++++++++++ unused_chapters/intro.tex | 145 ++++++++++++ unused_chapters/michelson.tex | 311 +++++++++++++++++++++++++ unused_chapters/millikan.tex | 254 +++++++++++++++++++++ unused_chapters/mo.tex | 156 +++++++++++++ unused_chapters/naspec.tex | 229 +++++++++++++++++++ unused_chapters/report_template.tex | 138 ++++++++++++ unused_chapters/sched.tex | 58 +++++ unused_chapters/spol.tex | 141 ++++++++++++ 12 files changed, 2611 insertions(+) create mode 100644 unused_chapters/HeNelaser.tex create mode 100644 unused_chapters/appendices.tex create mode 100644 unused_chapters/fabry-perot.tex create mode 100644 unused_chapters/hspect.tex create mode 100644 unused_chapters/intro.tex create mode 100644 unused_chapters/michelson.tex create mode 100644 unused_chapters/millikan.tex create mode 100644 unused_chapters/mo.tex create mode 100644 unused_chapters/naspec.tex create mode 100644 unused_chapters/report_template.tex create mode 100644 unused_chapters/sched.tex create mode 100644 unused_chapters/spol.tex (limited to 'unused_chapters') diff --git a/unused_chapters/HeNelaser.tex b/unused_chapters/HeNelaser.tex new file mode 100644 index 0000000..9918f80 --- /dev/null +++ b/unused_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/unused_chapters/appendices.tex b/unused_chapters/appendices.tex new file mode 100644 index 0000000..d310873 --- /dev/null +++ b/unused_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}&=&\\ +\overline{n_1}&=&=\overline{n}\\ +&=&0\\ +\overline{n_2}&=&\\ +\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/unused_chapters/fabry-perot.tex b/unused_chapters/fabry-perot.tex new file mode 100644 index 0000000..d74745f --- /dev/null +++ b/unused_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/unused_chapters/hspect.tex b/unused_chapters/hspect.tex new file mode 100644 index 0000000..565c64f --- /dev/null +++ b/unused_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/unused_chapters/intro.tex b/unused_chapters/intro.tex new file mode 100644 index 0000000..a3ca846 --- /dev/null +++ b/unused_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, “” +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/unused_chapters/michelson.tex b/unused_chapters/michelson.tex new file mode 100644 index 0000000..101fa64 --- /dev/null +++ b/unused_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/unused_chapters/millikan.tex b/unused_chapters/millikan.tex new file mode 100644 index 0000000..085c22f --- /dev/null +++ b/unused_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 youd 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 1020 +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/unused_chapters/mo.tex b/unused_chapters/mo.tex new file mode 100644 index 0000000..2a52bea --- /dev/null +++ b/unused_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/unused_chapters/naspec.tex b/unused_chapters/naspec.tex new file mode 100644 index 0000000..8f88b83 --- /dev/null +++ b/unused_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=$5.9\AA . Compare this approach to the use of the +Fabry-Perot interferometer. + +\end{document} +\newpage diff --git a/unused_chapters/report_template.tex b/unused_chapters/report_template.tex new file mode 100644 index 0000000..c4f2436 --- /dev/null +++ b/unused_chapters/report_template.tex @@ -0,0 +1,138 @@ +\documentclass[aps,prb,preprint]{revtex4} +%\documentclass[aps,twocolumn,prl]{revtex4} +\usepackage{epsfig} +\begin{document} + +\title{Compton Scattering Lab} +\author{Y. O. Urname, P. A. Rtner1, and P. B. Rtner2} +\affiliation{Physics 251 College of William and Mary} +\date{today} +\begin{abstract} +This lab demonstrates Compton scattering. A Na$^{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}{$$} & +\multicolumn{1}{c}{$$} & +\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/unused_chapters/sched.tex b/unused_chapters/sched.tex new file mode 100644 index 0000000..8227a18 --- /dev/null +++ b/unused_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/unused_chapters/spol.tex b/unused_chapters/spol.tex new file mode 100644 index 0000000..6c0446e --- /dev/null +++ b/unused_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} -- cgit v1.2.3