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diff --git a/chapters/hspect.tex b/chapters/hspect.tex deleted file mode 100644 index 565c64f..0000000 --- a/chapters/hspect.tex +++ /dev/null @@ -1,436 +0,0 @@ -%\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 |