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diff --git a/chapters/hspect.tex b/chapters/hspect.tex new file mode 100644 index 0000000..565c64f --- /dev/null +++ b/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 |