Full Experiments Prism Spectrometer

The Prism Spectrometer

Learning Outcomes (the skills you will acquire)

 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 line spectrum of cadmium
 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 at:-
http://level1.physics.dur.ac.uk/ISE/ISEs.php
Also work through 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 through the prism deviate because of refraction. Simple ray (geometrical)
optics and the use of Snell‟s law [1] 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 cadmium 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, . The deviation
of the incident ray (at angle ) through the prism is shown as angle . It can be shown
using knowledge of simple geometry and Snell‟s law that the refractive index, , is
found by evaluating
Full Experiments Prism Spectrometer

Where 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

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, determine the minimum deviation angle for a prism
having if the refractive index at some wavelength is known to be .
Assume that the prism angle is exactly Furthermore, determine the
maximum and minimum values of this angle (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

2. Preparation
Arrange the cadmium lamp (blue 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
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.
Full Experiments Prism Spectrometer

Collimator

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 . 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. and or perhaps and ) Ideally, all measurements should
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.
Full Experiments Prism Spectrometer

Task 5
Use this precision to calculate the precision to which refractive index can in
principle be measured, using your nominal value for obtained from Task 1
and assuming 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 cadmium lamp into a number of spectral lines (see
the table below). Light travelling through the prism and re-emerging 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 Wavelength (nm)

Cadmium red 643.8
Cadmium green 508.6
Cadmium green/blue 480.0
Cadmium blue 474.0
Cadmium violet 435.8

Table 1: The first five lines in the atomic series of the cadmium 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 five of the lines listed in the above table.

Bring the Cadmium red 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 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 five 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,
Full Experiments Prism Spectrometer

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 , 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

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 against the inverse square of the wavelength at which is
measured yields a straight line of gradient and intercept . Thus, is the value of
at very large wavelengths, and determines the wavelength dependence of .
Task 8
Provide a plot of refractive index versus and obtain the constants and
for the glass of the prism from a least squares fit, using “Excel” and
incorporating your error bars. Specify the dimensions of and .

5. Exploration
Replace the cadmium 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 origin of
energy levels in atoms. 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, measured by the above method and given by
equation 1 is expressed formally by the following equation.
Full Experiments Prism Spectrometer

which, using equation 1, reduces to

where and are the uncertainties in the angles and 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
rather than the nominal value of and recompute the refractive index
and the error in the refractive index, which now incorporates this “systematic” error in
. Comment on your result.