Index of Refraction of Glass—Dispersion
1
Theory
A prism is a device that is used to controllably refract light. It typically is made of high
quality glass, with an index of refraction near 1.5. The index changes with wavelength and
so different wavelengths are refracted by different angles. This is called dispersion, and
the theory of dispersion is typically covered in a course in electricity and magnetism or
physical optics. The Cauchy Theory of dispersion says that to second order in wavelength
the index can be written as
B
n≈N+ 2
(1)
λ
Prism dispersion is useful for spectroscopy, splitting a beam of light by wavelength1 .
Figure 1: Dispersion angle δ for a ray passing through a prism of apex angle A.
Most prisms used in spectroscopy have apex angles of A = 60◦ . A ray that hits the prism
is deviated by a total deviation angle δ. If we rotate the prism, we will find that the angle
1
Another method of producing spectra is with a diffraction grating as you will discuss in a physical
optics course. Spectra formed using a prism are generally brighter than those formed using a grating.
Spectral lines that are too dim to be seen with a grating can often be seen using a prism. Unfortunately,
the increased brightness of the spectral lines is offset by a decreased resolution (angular spread), since the
prism doesn’t separate the different lines as effectively as the grating.
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of deviation has a minimum, and for that minimum we can express the index of refraction
as
sin A+δ
2
n=
(2)
sin A2
You can see this in the animation at http://www.mtholyoke.edu/~mpeterso/classes/
phys103/geomopti/MinDev.html.
2
Apparatus
You will use a high precision spectrometer illustrated in Figure 2. Light from a lamp
passes through a slit into a collimator that makes the incoming light emerge as a plane
wave beam. The light hits the prism and is deflected through an angle of deviation that
can be measured on the Angular Scale. The telecope focuses the light onto the image plane
of the eyepiece. An image of crosshairs is also focussed onto the image plane.
The telescope can also be locked or moved very slowly by a fine adjustment screw and the
instrument is provided with a heavy base for stability. To obtain sharp spectral lines the
slit width should be quite small, about 0.1-0.3 mm.
Figure 2: Prism Spectrometer
You will always adjust the prism table for the angle of minimum deviation for the particular
wavelength being observed, and then read the angle using the vernier scale. You will be
able to measure to the nearest minute of arc (1◦ = 600 ). Figure 3 shows the vernier and
main scales.
The top scale (0 to 300 ) is the vernier, and the bottom scale is the main scale (0 to 360◦ ).
First read the main scale where the 0 mark of the vernier is. Here it is between 40.5◦ and
41◦ , or in terms of minutes, between 40◦ 300 and 41◦ 000 . Now look to see which mark on
the vernier matches a main scale line exactly—this is a judgment call, especially with this
2
Figure 3: Angular vernier scale. Bottom main scale in degrees, top vernier scale in minutes.
The reading shown is 40.5◦ + 120 = 40◦ 420 . = 40.70◦ = 0.7103 rad
image, but I would say about 120 . The angle is then the sum of the main scale and the
vernier, 40◦ 420 = 40.70◦ .
There are a number of adjustments that can be made on the spectrometer, but only a
few concern you. To rotate the telescope, make sure that the knob at the center base is
loose, then just rotate. If the center screw is locked, the other screw on the base makes
fine adjustments.
You will have a prism on the prism table, held in place by a spring clamp. Once you are
happy with the prism placement, be careful to avoid changing the prism location. There
are 3 leveling screws for the prism stage, but these should be properly set and not need
adjustment. The prism table can be rotated and raised or lowered. A locking ring fixes
the vertical height of the prism table, and should be left in place. There is a screw that
locks the prism table from movement. If this is loose the prism table can be rotated and
raised or lowered.
At the source end is the collimator, with a screw that adjusts the slit width. Springs work
to pull the knife edges apart, and these may be a bit balky due to corrosion. The telescope
eyepiece can be moved in or out to focus the crosshairs on the slit image.
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Procedure
a. Familiarize yourself with the apparatus.
b. Center the prism on the table. Check the height and level of the prism table and adjust
only if needed. Once these adjustments are made be careful not to nudge the prism, or
change the leveling screws. Only rotate the prism table.
c. Measure the prism angle A. Figure 4 should be self-explanatory. Use white light so
that you have a bright white reflection on each side. Repeat the measurements so that
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you can get an uncertainty in the prism angle. Each member of the group should do
the measurements.
Figure 4: Finding the prism apex angle, A, by measuring the angle of reflection of a white
light source on either side. Be certain you are seeing reflections (white) and not refractions
(colored).
d. Dealing with the uncertainties. Your measurements are of angles. The vernier allows
you to measure to the nearest minute of arc, at least in principle, but the experimental
uncertainty is certainly larger than that. There are several sources of uncertainty.
One is rotating the prism table to get the minimum deviation. Another is setting the
crosshairs on the line (left side of line, center, right side?), can you get the same result
each time? A third is slop in the fine adjustment screw—will you get the same angle
measurement if you approach a line from the left as from the right?
Consider all these uncertainties and determine a way of finding the overall uncertainty
in an angle measurement. Clearly summarize your procedure in your labbook.
e. Begin with a helium source (if available) or a mercury source and measure angles of
minimum deviation for each spectral line (wavelengths are in Table 1 at the end.)
Figure 2 and Figure 5 suggest the orientation needed to find the angle of minimum
deviation for the telescope on the left of center. There is also an angle of minimum
deviation on the right side. The prism table must be rotated to find these two angles.
Locate the refracted image of the slit (in multiple colors!) in the telescope and rotate
the prism table to get the angle of minimum deviation. This is slightly different for
each line of the spectrum, so the process must be repeated for each wavelength.
Measure the angle of minimum deviation on the left and right. By taking half the
difference you have the angle of minimum deviation. Each lab partner should make
measurements, and the measurements should be repeated to allow determination of
uncertainty.
f. Use a mercury tube to add data for other wavelengths. You must have data for at least
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Figure 5: Angles of minimum deviation on left and right. The prism table must be rotated
to get these.
5 different wavelengths ranging from red to violet, but use data from two tubes if time
permits
g. Calculate the index of refraction for each wavelength using Equation 2.
h. Getting uncertainties. If A and δ are independent measurements, and the errors are
normally distributed, the uncertainty in the index is
σn2
=
∂n
∂A
2
2
σA
+
∂n
∂δ
2
σδ2
(3)
Evaluate this numerically for one of your measurements.
Hint: The partial derivatives are somewhat messy. You should be able to do them
by hand, but can check with online Mathematica. Although a very small amount of
simplification is possible using trig identities, you can also just put numbers in. This
may be easiest in something like Excel. Assume that the errors in the rest of the
measurements are similar to this one calculation.
i. Make a plot of index n versus wavelength, λ. Include error bars.
j. Check the Cauchy expression, Eqn 1, by plotting appropriate quantities that should
produce a straight line graph. Be sure to include error bars on your graph. Get values
of N and B with estimates of uncertainty.
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Report
By next lab period you should have calculations done in your labbook to submit to me. I
will check for any problems and get the book back to you quickly. Feel free to collaborate
with your lab partner in doing the calculations.
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Once you have good lab book results, I want you to produce a more formal lab report (the
only one this quarter).
Each person should write a report in LATEX. It should include an introduction laying out
what the objective is, along with the theory, a section that briefly describes important
parts of the data collection (much briefer than this document!) especially noting any
difficulties that needed to be overcome and how they were solved, a section outlining
the error propagation to get uncertainty in the index (this is mostly symbols, with one
calculated result), a section on data, with a good data table, the graphs (scan in a hand
drawn graph or use gnuplot) and discussion of the graphs, the values of B and N from the
graph, and a conclusion.
No abstract or bibliography is required.
Table 1: Wavelengths (in nm) of prominent lines (f = faint)
Hydrogen
Helium
Mercury
Red (f)
706.5
Red
656.3
Red
667.8
Orange
587.6 Yellow 579.1
Yellow 577.0
Blue-green
501.5 Green 546.1
Blue-green 486.1 Blue-Green (f) 492.2
Blue (f)
471.3
Violet
434.0
Violet
447.1 Violet 435.8
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