Full Experiments
Prism Spectrometer
The Prism Spectrometer
Learning Outcomes (the skills you will acquire)
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Proficiency in the reading of vernier scales
Understand how to use a spectrometer
Obtain a practical understanding of the refraction of light by a prism
Observe the Balmer line spectrum of atomic hydrogen
Learn a method for the measurement of the refractive index of glass
Understand the accuracy and precision limits of your measurement of the
refractive index
Practice in taking clear and intelligible laboratory notes
Preparatory Task:
Work through the prism spectrometer interactive screen experiment on the
laboratory website http://level1.physics.dur.ac.uk/ISE/ISEs.php
Also the angular vernier ISE describing the reading of vernier scales. Read
the entire script and write a short paragraph in your lab book summarizing the
experiment and its aims. Write a brief outline of the theory behind the
experiment and perform Tasks 1 and 2 below.
1. Introduction
The refractive index is probably the most basic descriptor of materials which are
useful for their optical properties. The refractive index governs the design of lenses
and other optical components and is important for the design of photonic devices used
in fibre optic communication systems. Accurate measurement of refractive index of
materials can be made using a prism made out of the material of interest. Rays
passing though the prism deviate because of refraction. Simple ray (geometrical)
optics and the use of Snell’s law1 show that the index is related to the minimum
angular deviation of a ray that passes through the prism at different angles of
incidence. You can measure this minimum deviation directly for any ray that
comprises a single wavelength of light (a “monochromatic” beam). A hydrogen lamp
provides a few discrete wavelengths, and for each wavelength the minimum deviation
angle will be different. This is because the prism refractive index varies with
wavelength (the property known as “dispersion”), and so you will be able to measure
this property as well.
For the isosceles prism you will use (figure 1) there is an apex angle, A. The
deviation of the incident ray (at angle I) through the prism is shown as angle D. It can
be shown using knowledge of simple geometry and Snell’s law that the refractive
index, n, is found by evaluating
1
The properties described by Snell’s law can be found in your physics textbook, H. D. Young and R.
A. Freedman, University Physics, 13th Ed., Pearson Addison-Wesley, San Francisco (2012).
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Prism Spectrometer
 A + Dmin 
 A
n = sin 
sin  

2
2


Equation 1
where Dmin is the minimum observed angular deviation. At the minimum deviation
position, the light passes symmetrically through the prism, and the ray within the
prism is parallel to the base of the prism.
A
I
D
Figure 1: The isosceles prism with rays and measurement angles shown. The
broken lines show the normal to the face of incidence and the incident direction,
respectively. The incident beam is deviated through the angle D when it passes
through the prism. The angle of minimum deviation occurs when the light passes
symmetrically through the prism, and the ray within the prism is parallel to the prism
base.
Tasks 1 and 2: Using equation 1 above to determine the
minimum deviation angle. Assume the prism angle is exactly 60 °
and the refractive index of the prism at some wavelength is known to be
1.755 ± 0.001. Calculate the corresponding minimum deviation angle
and its maximum and minimum values (use the simple common sense
method, substituting the upper and lower limits on the refractive index
values into equation 1). Use this result to show in your report to what
precision you might have to measure angles in order to determine
refractive indices to the third decimal place. Express these angles in
degrees, arcminutes and arcseconds; i.e. as x° y’ z’’
2. Preparation
Arrange the hydrogen lamp (pink light source when switched on) so that it shines
though the slit at the end of the collimator (a device which produces a non-diverging,
parallel light beam and which is attached to the rotation table) directly towards you.
This is the direction of incidence from which deviations will be measured. Take
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Prism Spectrometer
off the prism if it is mounted on its stage and rotate the telescope, also mounted on the
rotation table, so that it is roughly in line with the collimator and light source.
Note that the rotation stages (upper and lower) and the prism stage may be rotated
freely when the locking screws are released. Only tighten them gently to lock again.
Also note that fine angular adjustment on the rotation stages is possible, when the
screws are locked, by using the adjustment wheels. Try this out to see how fine this
adjustment can be.
Collimato
Vernier scales
Telescope
The adjustable slit on the end of the collimator nearest the lamp allows a vertical
line image of the light source to be fed to the telescope. Look though the telescope
and find the pink line image, which can be brought into focus using the knurled knob
on the telescope. The telescope cross hairs (visible if you illuminate the far end of the
telescope with a desk lamp) can be brought into focus using the eyepiece (pull
out/push in gradually). Align the centre of the cross hairs with the centre of the line
image. You can now adjust the image width and orientation by means of the
adjustable collimator slit; it should be quite narrow for accurate measurements of
angular position, but remember that the amount of light coming through decreases as
the slit size is reduced.
Task 3: Read off the angles recorded by the two vernier scales to the left
and right of the telescope arm. Use the eyepiece provided to magnify the
scales. Calculate the difference in angle between these two readings and
state by how much this deviates from exactly 180°. Make a comment in
your report on the systematic uncertainty (affecting the accuracy of the
measurement of angles) that this error might introduce.
You now have two values of angle relative to which you will measure the deviation
of the light from the source when it passes through the prism. Subsequent deviations
will be calculated relative to one of these two values: choose which one. (Give them
names, e.g. θ 0, R and θ 0, L or perhaps D0,R and D0, L .) Ideally, all measurements should
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Prism Spectrometer
be made on both of the vernier scales and the mean value of the displacement
determined from these two readings. This procedure minimizes the effects of the
systematic error; but it is also time-consuming, which is why you should choose and
subsequently use only one of the two scales.
Task 4: Quote the precision to which you can measure angles from the
vernier scales.
Task 5: Use this precision to calculate the precision to which refractive
index can in principle be measured, using your nominal value for Dmin
obtained from Task 1 and assuming A = 60° exactly.
Task 6: Repeat the procedures as for Task 3 but for the scales on the
upper rotation stage.
3. Measurements
The prism splits the light from the hydrogen lamp into a number of spectral lines
(the “Balmer series”: see the table below). Light travelling through the prism and reemerging will consist of some spectacularly coloured line images. If you adopt the
arrangement of prism and collimator shown in the diagram above, you will find these
lines in the region indicated by the arrow for transmitted rays. The intensity of the
lines decreases moving to shorter wavelengths (i.e. to higher frequencies and hence
higher energies).
Line
Hα
Hβ
Hγ
Hδ
Wavelength (nm)
656.2 (red)
486.1 (green/blue)
434.0 (blue)
410.1 (violet/blue)
Table 1: The first four lines in the Balmer series of the hydrogen spectrum.
The lines are best located by first using your naked eye, at the level of the
prism. You should be able to see the image of the end of the collimator, refracted
through the prism, with the coloured lines at differing angles of transmission. Look
for the brightest line – the red line – first. Then move the telescope into position and
identify all four of the Balmer lines listed in the above table. You may see a relatively
weak and almost continuous emission underlying the Balmer lines. This continuum is
emitted by molecular hydrogen. The discharge tube produces atomic hydrogen by
dissociating molecular hydrogen, but the dissociation is not complete. If the
background emission from molecular hydrogen makes it difficult to identify the
Balmer lines, the discharge tube probably needs to be replaced: ask the demonstrator
to take a look.
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Prism Spectrometer
Bring the Hα line into the field of view of the telescope, and then follow its
movement with the telescope while you gradually increase the incidence angle by
rotating the upper stage, on which the prism sits. You will find that the image of the
red line in the telescope eventually reverses its direction of travel. The angle of
incidence at which this “turning point” occurs corresponds to the the minimum
deviation of the red light passing through the prism. Try to locate the exact turning
point by going back and forth in the incidence angle to locate the minimum position.
Fix the incidence angle by locking the upper stage and then measure the angles of
deviation for all four spectral lines (the incidence angle at which minimum deviation
occurs is approximately the same for all four lines, and so it is OK to fix the angle of
incidence using the red line). For each line, make at least four independent
measurements of the minimum deviation angle (two measurements by each student).
Task 7: Using Excel, provide a spreadsheet showing the wavelengths and the
measured values of the angle of minimum deviation. Use Excel to calculate the mean,
standard deviation and standard error in the minimum deviation angle for each of the
four wavelengths. Compare the standard errors with the precision of measurement of
angles by the spectrometer, given in Task 4.
Assuming a prism angle A = 60°, calculate the refractive index and its uncertainty
for each wavelength by means of Equation 1 above.
Transparent optical materials have a refractive index, n, which varies
with wavelength according to the power series
n(λ ) = K + B λ2 + C λ4 + ....
which is known as the Cauchy formula. The first two terms in this series provide an
adequate approximation to the wavelength dependence of the refractive index of
glass, and so a plot of n against the inverse square of the wavelength at which n is
measured yields a straight line of gradient B and intercept K. Thus, K is the value of n
at very large wavelengths, and B determines the wavelength dependence of n.
Task 8: Provide a plot of refractive index versus 1 λ2 and obtain the
constants K and B for the glass of the prism from a least squares fit,
using “Excel” and incorporating your error bars. Specify the dimensions
of K and B.
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Prism Spectrometer
5. Exploration
Replace the hydrogen lamp with the orange sodium lamp. Carry out an
experiment similar to the one that you have just finished to find the angle
of minimum deviation for the sodium ‘D’ lines. (There are two lines, but
the spectral resolution of the glass prism is insufficient to separate them.)
Calculate the wavelength of the sodium ‘D’ lines and compare with the
known value from the literature.
6. To conclude
Make sure your lab note book contains enough guidance for the marker to be able to
follow the presentation of your results. State in words what each numerical result
corresponds to and say how it was obtained.
For your extended report – Give some background to this experiment. Give some
background to the Balmer series in hydrogen. Explain why the measurement of
refractive index might be important technologically. For the historical context, recall
Newton’s discovery of the nature of white light, using a prism.
In the extended report, you should compare the uncertainties in the refractive index,
derived from the mean and the standard error in the minimum deviation angle, and
using Equation 1 (i.e. adopting the “common sense” approach), with those derived by
adopting the following, more rigorous error analysis.
The error in the refractive index, αn measured by the above method and given by
equation 1 is expressed formally by the following equation
2
 ∂n
 ∂n 
α n =   α A2 + 
 ∂A 
 ∂Dmin
which, using equation 1, reduces to
αn =
{ sin (D
2
min
2
 2
 α Dmin

}
/2)α A2 + sin 2 (A /2)cos 2 [(A + Dmin ) /2]α D2 /{2sin2 (A /2)}
where α A and α Dmin are the uncertainties in the angles A and Dmin, respectively.
Neglect the uncertainty in A. Your values for the refractive index and the
corresponding points on the plot should incorporate the error calculated in this way.
In practice, you have no means of measuring accurately the prism angle. Suppose
that A = 59.5°, rather than the nominal value of 60°, and recompute the refractive
index and the error in the refractive index, which now incorporates this “systematic”
error in A. Comment on your result.