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+%\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