\section{L-5: Spectrometer and the H Balmer Series }\label{l05_spect}

\noindent NOTE: Consult your instructor for assignment of 
parts suitable for a single lab, e.g.~do 
Part I, II (A or B), Part IV or Part V).

\noindent {OBJECTIVES}: \begin{enumerate}
\item[] \latex{\vspace*{-.07in}}  
To become familiar with a precision spectrometer and some of its 
uses. 
\end{enumerate}

\noindent {APPARATUS}:  \begin{enumerate}
\item[] \latex{\vspace*{-.07in}} 
Precision spectrometer; mercury and hydrogen discharge tubes;
prism; diffraction grating; light source for Gaussian eyepiece 
\& its 12V power supply (L-2); ring stand \& achromatic lens.
\end{enumerate}

\begin{figure}[h] \centering 
\includegraphics[height=3.3in]{figs/l104/l05-4.eps}
\latex{\vspace*{-.1in}}
\caption{The spectrometer (side view). }
\latex{\vspace*{-.1in}}
\end{figure} 

\begin{center}  {INTRODUCTION~and~ADJUSTMENTS} \end{center}
\noindent {PART I - A}  
\begin{enumerate}
\item \latex{\vspace*{-.07in}}Become familiar with the clamping and fine adjustment controls for
telescope and prism table angles.  {\bf Never force a motion - you 
may damage the instrument.  {\bf If at the end of a fine adjustment, 
loosen and reclamp}!} The prism table clamp sets table 
elevation: no fine adjustment is necessary.
\item \latex{\vspace*{-.07in}}Note that 180$^\circ$ apart are two {\bf angle
scale reading ports and verniers, I and J}.  
The verniers have 30 divisions per half degree. Hence each division is 
(1/30)$\times $(1/2)~=~(1/60)$^\circ$ or one minute ($'$) of arc.
\item \latex{\vspace*{-.07in}}The {\bf knurled ring about the collimator, B,} controls the {\bf slit opening}.
\item \latex{\vspace*{-.08in}}Staff have already focussed the collimator for parallel light and 
have set collimator and telescope $\perp $ to spectrometer's rotation axis.
%
\end{enumerate}

\noindent REMINDER ON READING A VERNIER\\  
\html{\begin{tabular}{p{4.5in}l} }
\latex{\parbox{0.65\linewidth}}
{\begin{enumerate}  
\item[1)] \latex{\vspace*{-.07in}}Position and tighten the telescope. 
\item[2)] \latex{\vspace*{-.07in}}Move the magnifying glass to see the scale marking 
\item[3)] \latex{\vspace*{-.07in}}Look where the zero mark on the upper scale is located.
The degree reading is the firstu {\em full} degree mark to the {\em left} of the zero
($210^\circ$ in the top figure; $229^\circ$ in the bottom figure).
\item[4)] \latex{\vspace*{-.07in}}Now make the {\em minute} reading:  if the zero mark on the
upper scale comes {\em after} a $\frac{1}{2}$ degree mark (the short lines on the lower scale)
then start at $30'$, otherwise at $0'$.
\item[5)] \latex{\vspace*{-.07in}} Next look for the two lines which match up {\em best}
between the two scales.  Read the number of the appropriate line on the upper scale and add
it to the zero mark is just past a $\frac{1}{2}$ degree mark so we begin with $30'$. The $15'$
mark is the one that lines up best so we get $210^\circ15'$ for the top reading and 
$229^\circ146'$ for the bottom one.
\item[6)] \latex{\vspace*{-.07in}}Now convert to decimal degrees, in this case $15/60 = 0.25$,
so $\theta = 210.25 ^\circ$. latex{\vspace*{-.07in}}
\end{enumerate} 
\hrule
}
\html{ & }
\latex{\parbox{0.35\linewidth}}
{\hspace*{.10in}
\latex{\includegraphics[height=3.5in]{figs/l104/l05-6.eps}}
\begin{center} ~~{Figure~2:} The verniers  \end{center}
\html{\includegraphics[height=4.5in]{figs/l104/l05-6.eps}}
}
\html{ \cr \end{tabular}  }



\noindent {PART I - B}:  {FOCUS THE TELESCOPE FOR PARALLEL RAYS}
\begin{enumerate}
% 
\item[1.]  
\begin{enumerate}
\item[i)] \latex{\vspace*{-.08in}}Slide telescope eyepiece in or out until crosshairs are in sharp 
\item[ii)] \latex{\vspace*{-.04in}}Sight 
telescope at a distant object (thru an open window); then focus 
telescope by rotating its focus ring 
(Fig.~1) until the object's clear image falls on the cross 
hairs.  The test for proper focus is absence of parallax 
between the image and cross hairs (appendix 4).  The telescope, now 
focused for parallel rays, will stay so as long as the focus ring is 
unmoved; but one may still adjust the eyepiece to suit the observer.
\end{enumerate}
%
% 
\item[2.] \latex{\vspace*{-.08in}}An alternative method to focus for parallel light is to use the
Gaussian eyepiece + light source as described below in PART II B. 
\item[3.] \latex{\vspace*{-.08in}}With collimator and telescope both properly focused one 
should get a sharp image of the slit and no parallax between slit image 
and cross hairs.  If you still get parallax, recheck your 
telescope focus and/or consult instructor.  
\item[4.] \latex{\vspace*{-.08in}}{\em Optimizing light thru the collimator}:  A properly located short focal 
length lens can gather a large fraction of the light from a source and 
redirect it \latex{\vspace*{-.08in}}
%

\begin{figure}[h] \centering 
\includegraphics[height=1.in]{figs/l5-02.eps}\latex{\vspace*{-.08in}}
%\caption{Various schematics for the eye}
\end{figure} 


thru the collimator thereby facilitating detection of weak 
spectral lines.
For strong 
lines it permits narrowing the slit width and thus improving resolution.
Use such a lens (mounted on a ring stand) to find an arrangement which 
fills the collimator with light.\latex{\vspace*{-.08in}}
\end{enumerate}

\setcounter{figure}{2}
\begin{figure}[h] \centering 
\includegraphics[height=2.05in]{figs/l5-03.eps}
\latex{\vspace*{-.1in}}
\caption{Proper, a), and improper, b) prism locations. }
\latex{\vspace*{-.2in}}\end{figure} 

\noindent
{PART~II-A:  MEASUREMENT OF PRISM ANGLE}
\begin{enumerate}
\item \latex{\vspace*{-.08in}}Mount the prism (in holder) on the dowel pins (in the prism 
table) so that the prism is far enough from the collimator that beams B$_1$ 
and B$_2$ are centered with respect to the spectrometer axis; otherwise much 
of the reflected light may miss the telescope. Turn the table so that 
the apex angle A (Fig.~3) splits the 
beam into nearly equal parts.  
\item \latex{\vspace*{-.09in}}Using the {\em prism table levelling screws} adjust the plane of the 
prism table so it is close to horizontal.
\item \latex{\vspace*{-.09in}}With the prism table clamped at the proper elevation, set 
the telescope to receive beam B$_1$ 
and form an image of the slit on the intersection of the telescope cross hairs.  
{\bf Record both VERNIER readings (in minutes)}.  
Average the two vernier readings
(to eliminate any systematic error from misalignment of the circle 
scale with respect to bearing axis), 
and add the result to {\em one} of the angle scale readings.  {\em Note 
that the angle
scale reads in half-degree units and the vernier in minutes ({\bf not} in decimal 
degrees}!). The zero position on the vernier determines 
the angle to the nearest 
half-degree (30 minutes) and the vernier reads the number of minutes past the 
half-degree mark.
%
%
\item \latex{\vspace*{-.09in}}Then with telescope in the 2nd position,
set on the slit image. The change in reading of the 
{\em same} angle scale should be the angle D.  (See Fig.~3.)  {\em Be sure you don't 
get the angle scales mixed up and subtract the first reading of angle scale 
1 from a reading of angle scale 2}.  Also some students incorrectly handle 
the subtraction when the scale passes through $360^\circ $ or $0^\circ $.
\item \latex{\vspace*{-.09in}}OPTIONAL: Try to prove that angle D is twice the prism angle A.\latex{\vspace*{-.09in}} 
%
\end{enumerate}

\noindent{PART~II-B:~}{ALTERNATE~method~for~prism~angle~measurement using~Gaussian~eyepiece}
\latex{\vspace*{-.15in}}
\begin{figure}[h] \centering 
\includegraphics[height=1.5in]{figs/l5-04.eps}
\latex{\vspace*{-.1in}}
\caption{Spectrometer telescope}
\latex{\vspace*{-.1in}}
\end{figure} 

\begin{enumerate}

\item[] \latex{\vspace*{-.08in}}{\em Introduction}:  
The {\em Gaussian eyepiece}  (Fig.~4) 
has a partially reflecting 
glass plate G set at 45$^\circ $ to telescope axis so that 
light from the lamp reflects down the telescope tube, past 
the cross hairs and out the objective.  If in front of the objective
you place a reflecting surface {\em perpendicular} ($\perp $) to the telescope axis,
the light will reflect back into the objective and form 
a real image of the 
cross hairs. If the cross hairs are in the focal 
plane of the objective, their reflected image will form in the 
same plane.  Thus both the 
cross hairs and the image will be in focus through the eyepiece.  
When one orients the 
reflecting surface so that the image coincides with the cross 
hairs, the reflector is accurately $\bot $ to telescope axis. \medskip

Thus the Gaussian eyepiece permits both focusing the telescope for 
parallel rays and setting a reflecting surface $\bot $ to the 
telescope axis.
\item \latex{\vspace*{-.03in}}To {\em focus the telescope for parallel light}:
\begin{enumerate}
\item[a)] \latex{\vspace*{-.05in}}Rotate the Gaussian eyepiece to open the hole between 
reflecting plate G and lamp. Adjust eyepiece to 
give a sharp image of the cross hairs, but {\bf don't turn the 
eyepiece to block the light hole}. Next, with prism on the table, adjust 
the prism table screws to make
the table nearly horizontal.
\item[b)] \latex{\vspace*{-.05in}}Now rotate the prism table until the prism
face is approximately $\perp$ 
to telescope axis.  Clamp the telescope and mount a 
light on the Gaussian eyepiece. ({\bf Check that the light hole is 
still open}).  With no illumination on the prism 
except that from the telescope, next rotate the prism table back and forth 
a few degrees until maximum 
reflected light appears in the eyepiece.  
Clamp the table in this position, and 
then focus the telescope (by turning the focus ring) until the reflected 
image of the cross hairs appears and shows no parallax with respect to the 
cross hairs.  The telescope is now focused for parallel rays.  Fine 
adjustments of the prism table leveling screws may help
the images coincide and thus set telescope axis accurately
$\perp $ to the reflecting surface.  See Part VI, Sec.~3. 
\end{enumerate}
\item \latex{\vspace*{-.08in}}To find the prism angle set the telescope $\bot $ to first one 
prism face and then the other.  The angle between these two positions is the 
supplement of the prism angle.  See Part IIA for detail about angle 
readings.

\end{enumerate}

\noindent{PART~III:  INDEX OF REFRACTION}\begin{enumerate}
\item[] \latex{\vspace*{-.07in}} 
{\em Introduction}:  When the path thru a prism is symmetric, the deviation is 
a minimum. At this angle of minimum deviation the index of 
refraction, $n$, is

\html{ \begin{tabular}{p{3.5in}l} }
\latex{\parbox{0.5\linewidth}}
{\[n = \frac{\sin \left[ ( A + d)/\delta\right]}{\sin \left( A/2\right) }\]
where $A$~=~angle of prism\\
\hspace*{.43in} $\delta$~=~angle of minimum deviation\\
\hspace*{.41in} $n$~=~refractive index of the prism.
}
\html{ & }
\latex{\parbox{0.50\linewidth}}
{\hspace*{.55in}
\latex{\includegraphics[height=1.2in]{figs/l5-04a.eps}
%\begin{center} {Figure~2:} Cylindrical lens. \end{center}
}
%\html{\begin{center} {\bf Figure~2:} Cylindrical lens.\end{center}}
\html{\includegraphics[height=1.8in]{figs/l5-04a.eps}}
}
\html{ \cr \end{tabular}  }



\noindent Devise your own methodology and  
determine the prism's refractive index for one or more lines
of the Hg spectrum. \\

\end{enumerate}

\newpage
\noindent PART~IV:  CALIBRATION OF PRISM SPECTROSCOPE

\begin{figure}[h] \centering 
\includegraphics[height=1.5in]{figs/l5-05.eps}
%\caption{}
\latex{\vspace*{-.15in}}
\end{figure} 

\begin{enumerate}

\item \latex{\vspace*{-.05in}}Set the prism for minimum deviation for a green line in the Hg 
spectrum.  
\item \latex{\vspace*{-.05in}}Determine the angle readings for the yellow, green, 
blue-violet, and deep-violet Hg lines. Repeat this for the red line from a 
hydrogen discharge.  
\item \latex{\vspace*{-.05in}}Plot telescope angle vs accepted $\lambda $'s for these lines.
\item \latex{\vspace*{-.05in}}For the same setting of the prism table, find the angles
for the blue-green and two 
violet lines of the hydrogen spectrum.
\item \latex{\vspace*{-.05in}}Use your calibration curve to determine $\lambda $ of the 
blue-green and violet lines in the atomic hydrogen spectrum.  Compare 
results to accepted values.

\end{enumerate}
\noindent {OPTIONAL}:  The calibration curve is very non-linear.  Since a more linear 
plot facilitates interpolation of an unknown $\lambda $, try plotting deviation 
{\it vs}~1/$\lambda ^2$. \\

\noindent {TABLE~OF~WAVELENGTHS}:  \\ \smallskip 

\begin{tabular}{|lllc|lcc|} \hline 
\multicolumn{4}{|c|}{Mercury lines} & \multicolumn{3}{c|}{Hydrogen lines}\\ \hline
&\multicolumn{1}{l}{$\lambda$ (in nm)} & ~ &\multicolumn{1}{c|}{(l/$\lambda ^2$)}& ~&
\multicolumn{1}{c}{$\lambda$  (in nm)} & (1/$\lambda ^2$)\\
Yellow$_1$ & 579.0$\left\rceil      \right. $  (unresolved)
     & ~ & $\times 10^6$(nm)$^{-2}$ & ~ & ~ & $\times 10^6$(nm)$^{-2}$ \\  
Yellow$_2$   & 577.0$\left\rfloor      \right. $ Avg.~= 578 &~
& 2.993 & Red & 656.3 & 2.322\\
Green &  546.1&& 3.353 & Blue-green & 486.1 &   4.232\\
Blue &  496.0$^{\ast}$ & &   4.065 & ~&~ &~\\      
& 49l.6$^{\ast}$&~ & 4.138 & Violet & 434.0 &   5.309\\
Blue &435.8 &~&   5.265 & ~&~ &~  \\   
~~-violet& 434.7$^{\ast}$ &~&~&  Deep Violet & 410.2 & 5.943 \\
Deep- &  433.9$^{\ast}$ & ~&~ &~ & ~ & \\             
~~~violet & 407.8$^{\ast}$&~& 6.013 & \multicolumn{2}{l}{NOTE: $^{\ast}$ implies} & ~\\
& 404.7$^{\ast}$&~&  6.106 & \multicolumn{2}{l}{~~~~~~usually faint}  &~ \\ \hline
\end{tabular}

\newpage
\noindent {PART~V: DIFFRACTION GRATING}

\begin{center}
\begin{tabular}{|c|} \hline
CAUTION:  Do not touch grating surface under any circumstance! \\
\hline
\end{tabular}
\end{center}

\begin{enumerate} 

\item {\em Grating~constant (number~of~lines~per~cm)}:  The lines/cm marked on these replica gratings is only approximate because the plastic often changes dimensions when it is stripped from the master grating. To achieve quantitative results calibration of the grating constant is desireable and can
be performed using the known $\lambda$, 546.1 nm, of the Hg green line. \\

CALIBRATION PROCEDURE:
\begin{enumerate}
\item[a)] Align telescope with collimator so the slit image falls exactly on the 
vertical cross hair.  
\item[b)] Mount grating on the dowel pins in the prism table. Adjust and 
clamp the table so the grating is approximately $\perp$ to telescope-collimator axis. 
\item[c)] Then adjust the table so that the front grating face is accurately 
$\perp $ to telescope.  Use the Gaussian eyepiece (see Part II - B)
for the final adjustment:  the reflected image of the cross hairs should fall on the cross hairs.  (Two reflected images can result if the sides of the grating glass are not parallel.  If so, test which image comes from the grating 
side by wetting the other side.)  Clamp prism table in this position and record telescope direction.  
\item[d)] Next set 
telescope on the first and second orders of the Hg green line.  Use both
$\pm $~angles.  The $\pm $~angles should of course agree.  Use them and 
$\lambda _{\mbox{\small green}} = 546.1~\mbox{nm}$ to calculate the grating spacing d from
\[n\lambda = d \sin \theta  .\]
\end{enumerate}
\item {\em Measurement~of~the~wavelengths~in~the~Balmer~series~in~hydrogen}:  
\begin{enumerate}
\item[a)] Use the calibrated grating spectroscope to measure $\lambda $ for 
three or four atomic hydrogen spectral lines.
\item[b)] Compare these $\lambda $'s to those from the Balmer formula
\[\frac{1}{\lambda }~=~R\left[ \frac{1}{2^2}~-~\frac{1}{n^2}\right] 
\; \; \mbox{where} \; R  = 10,967,758~\mbox{m}^{-1}\]
or calculate the frequencies of these lines and compare to 
\[f = cR \left[ \frac{1}{2^2} - \frac{1}{n^2}\right]  \; \mbox{where} \; R =
10,967,758~\mbox{m} ^{-1 } .\]
\item[c)] Use Planck's relation, $\Delta E = h f$, to calculate the energies in 
eV of the photons for each frequency and, therefore, the energy change in the hydrogen atom associated with each frequency.
\item[d)] Assuming that each observed $\Delta E$ leaves hydrogen in the $n=2$ 
state, show the observed transitions on a hydrogen energy level diagram (drawn to scale; consult textbook).
\end{enumerate}

\end{enumerate}

%\begin{center}
{PART~VI:  ADJUSTMENT OF A SPECTROMETER}
%\end{center}
\begin{rawhtml}  
<IMG SRC="../gif/caution.gif" ALIGN=center>  <br> 
\end{rawhtml}
\begin{center}
\begin{tabular}{|c|} \hline
 NOTE:  Not to be performed without permission of the instructor \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}
\begin{rawhtml}  
<IMG SRC="../gif/caution.gif" ALIGN=center>  <br> 
\end{rawhtml}

\item Use Gaussian eyepiece and prism face method to focus telescope for parallel 
rays. If necessary adjust prism table to make reflected image of the cross
hairs coincide with the cross hairs.
\item Remove prism from the table and align telescope and collimator approximately.  Focus the collimator until there is no parallax between the image of the slit and the cross hairs.  Adjust the ring on the collimator focusing sleeve so that the collimator is in focus with the ``V'' in the slot.  Clamp the telescope in line with the collimator so that the image of the slit (vertical) falls on the vertical cross hairs.  Now rotate the slit through 90$^\circ$ as permitted by the ``V'' projection and slot.  Adjust the level of the telescope so that the image of the slit falls on the horizontal cross hair.  The axes of telescope and collimator are now in line but not necessarily perpendicular to the axis of rotation of the instrument.
\item
\item[]
\latex{\vspace*{-.18in}}
\html{ \begin{tabular}{p{3.5in}l} }
\latex{\parbox{0.40\linewidth}}
{Replace the prism on the prism table and clamp it relative to the 
adjusting screws as shown.  (Line AB~$\perp $~face 1.  Line AC~$\perp $~face 2.  
Thus screw B will adjust face 1 without disturbing face 2, and screw C will 
adjust face 2 without disturbing face 1.
}
\html{ & }
\latex{\parbox{0.60\linewidth}}
{\hspace*{.55in}
\latex{\includegraphics[height=1.4in]{figs/l5-06.eps}
%\begin{center} {Figure~6:} Cylindrical lens. \end{center}
}
%\html{\begin{center} {\bf Figure~6:} Cylindrical lens.\end{center}}
\html{\includegraphics[height=1.8in]{figs/l5-06.eps}}
}
\html{ \cr \end{tabular}  }

\noindent Bring the telescope to within $<90^\circ$ of the collimator and use a 
face of the prism to reflect light from the collimator down the telescope 
tube.  Adjust the prism face so that the center of the slit image falls on 
the intersection of the cross hairs.  The prism face is now parallel to the 
instrument axis. 
\item[4.] Set the telescope $\perp$ to the adjusted prism face (use Gaussian eyepiece method).  Return to part 3 setup and adjust the collimator so that the center of the slit image falls on the cross hair's intersection.  Telescope and collimator are now properly focused and are $perp$ to the instrument axis.
\item[5.] To adjust the second prism face $\perp$ to the axis of the telescope (again use the Gaussian eyepiece method). Turn only the proper screw on the prism table: otherwise one disturbs the adjustment of the first face.  Recheck the first face and, if necessary, readjust it.  Continue this process until both faces are parallel to the axis of the instrument.  Turn collimator slit to vertical position.  The spectrometer should now be in adjustment.

\end{enumerate}
