Download Chapter 4 Dispersion of Glass

Chapter 4
Dispersion of Glass
4.1
Introduction
This experiment will develop skills in choosing a suitable fit for data and
plotting the resulting curve. Curve fitting will count for a big chunk of the
marks in this lab, so don’t try to avoid it!
4.2
Apparatus
For this experiment you will need the following pieces of equipment:
• Spectrometer,
• Bubble Level,
• 60◦ Prism,
• Front Surface Mirror,
• small Allen wrench, screwdriver,
• Sodium Lamp,
• Hydrogen, Helium and Mercury Geissler Tubes with Power Supply,
• Variable Intensity White–Light Source,
• assorted stands and clamps.
4-2
4.3
Dispersion of Glass
Theory
In this experiment we shall use the angle of minimum deviation to measure
the index of refraction of glass as a function of wavelength.
When a ray of light is obliquely incident on the interface separating two
transparent media of different indices of refraction, its direction changes. The
amount of deviation is described by Snell’s Law:
n1 sin θi = n2 sin θr
(4.1)
where n1 is the index of refraction of medium 1, θi is the angle of incidence
(measured from the normal to the interface), n2 is the index of refraction of
medium 2, and θr is the angle of refraction.
For most transparent media, the angle of refraction increases with increasing frequency (for a fixed angle of incidence) — thus, blue light is refracted
through a much larger angle than red light. Since the definition of the index
of refraction is:
c
n=
(4.2)
v
where c is the speed of light in vacuum and v is the speed of light in a
medium, and since the index of refraction is always greater than or equal to
1 for any medium, it is clear that v is less than or equal to c.
The speed of light is related to frequency and wavelength by:
c = νλ
(4.3)
and the frequency does not depend on the medium through which the light
travels. Thus it follows that the wavelength of light must change when it
enters a material medium.
Furthermore, the index of refraction of a material is not a constant, but
depends upon the frequency of the incident light. From measurements of the
index of refraction of a medium as a function of wavelength, we can construct
a dispersion curve by plotting a graph of n(λvac ) vs λvac , where λvac refers
to the vacuum wavelength of the incident light. (For the purposes of this
experiment, we shall assume that the wavelengths in air are identical to the
wavelengths in vacuum). The dispersion of the medium at a particular
wavelength λ0 is defined to be the slope of the dispersion curve at that
wavelength:
!
dn
D=
(4.4)
dλ λ0
4.4 Procedure
4-3
Figure 4.1: Path of Light Through a Prism in Air
Referring to Figure 4.1, when a ray of monochromatic light passes through
the prism, it is deviated through some angle δ. The value of this angle
depends on the angle of incidence at the first surface of the prism. When
the angle of incidence φi is equal to the angle of emergence φe , the angle of
deviation δ is a minimum, called the angle of minimum deviation δmin .
The index of refraction of the glass, n, can then be found from:
n = n1
sin[(α + δmin )/2]
sin(α/2)
(4.5)
where α is the prism angle.
4.4
4.4.1
Procedure
Aligning the Spectrometer
See the appropriate section on the course web site.
4.4.2
Measurement of Prism Angle
Place the prism on the prism table with the prism angle pointing toward
the collimator as in Figure 4.2. Place the white light source in front of the
collimator. Use a narrow collimator slit in the following procedure. Note that
the slit image can be seen reflected in the prism faces. Adjust the telescope
4-4
Dispersion of Glass
Figure 4.2: Orientation of Prism to Find Prism Angle
so that the slit image is aligned with the cross–hairs, read the vernier scale
(30 part vernier every half–degree; thus readings are in degrees and minutes),
and record this reading (the telescope angle).
Be as careful as possible when taking measurements, and be sure to record
values to the nearest minute of angle. Sloppy measurements will give you
meaningless results.
Repeat this for the other prism face and enter your readings into Table 4.1. Repeat these measurements 3 times. The prism angle is one half the
difference between the two vernier readings.
From the three series of measurements recorded in Table 4.1, determine
the prism angle α for each measurement, and calculate the average value of
the prism angle. Include measurement uncertainties in your final answer.
4.4.3
Measuring the Angle of Minimum Deviation
Place the sodium vapour lamp in front of the collimator slit. Rotate the
prism 90◦ so that now light will be refracted through it on the way to the
telescope. Observe the spectral lines and reduce the slit width until they
are sharp and narrow. Rotate the prism table as shown in Figure 4.3 until
the angle of minimum deviation is achieved for one of the Sodium–D lines
4.4 Procedure
Trial #
4-5
Left Reading Right Reading Prism Angle α
(◦ )
(◦ )
(◦ )
1
2
3
Table 4.1: Calculation of Prism Angle
Figure 4.3: Orientation of Prism to Find Prism Angle
4-6
Dispersion of Glass
(2 lines separated by 6 Å; these may be too close together to be clearly
resolved — if so, set the cross–hairs to the center of the doublet and use
the mean wavelength of 5893Å). Record the vernier reading in Table 4.2.
Repeat this measurement three times. If you have any difficulty, see the
laboratory demonstrator. Now rotate the prism through 180◦ , repeat the
above measurements, and average the results.
Q1:How do you determine the uncertainty in the average of a set of numbers?
The difference between the vernier readings from each side is twice the
angle of minimum deviation.
Repeat using the Hydrogen Geissler tube and the four visible lines of the
Hydrogen spectrum (red, blue and two violet).
Repeat using the Mercury Geissler tube and as many as possible of the
following lines of the Hg spectrum: two close yellow lines separated by 20Å;
two greens; a blue–green; and a blue (this is a triplet — use the center of the
image).
Repeat using the Helium Geissler tube and any 4 visible lines.
4.5
4.5.1
Analysis
Plotting the Dispersion Data
Using the angles of minimum deviation for each wavelength measured, along
with their uncertainties, and the average value of the prism angle α, along
with its uncertainty calculated above, determine the index of refraction at
each wavelength, along with the uncertainties, and place them in Table 4.2.
Plot a graph of n vs λ.
If your graph does not suggest a smooth, monotonically decreasing function,
go back and check your data. Bad data will make your results meaningless.
4.5.2
Curve Fitting
This is the most important part of this experiment. Don’t try and take
shortcuts in this section.
4.5 Analysis
Colour
Wavelength
Å
H (r)
6563
Na (y)
5893
Hg (y)
5790
Hg (y)
5770
Hg (g)
5461
Hg (g)
5015
H (b-g)
4861
Hg (b-v)
4358
H (v)
4300
H (v)
4101
Hg (v)
4077
Hg (v)
4047
He (y)
?
He (g)
?
He (b-g)
?
He (b-v)
?
4-7
Left Dev.
(◦ )
Right Dev. δmin
(◦ )
(◦ )
n
Table 4.2: n vs Wavelength
b–blue; g–green; r–red; v–violet; y–yellow (colours are approximate.)
4-8
Dispersion of Glass
For each of the following fits, do the fit and determine A, B, and C (if
applicable). Make a table of these results including the SSE for each fit.
Determine which of the following functional forms best approximates your
dispersion curve:
n
n
n
= A + Bλ
= A + λB2
1
= A + λ−B
n = A + Bλ + λC2
n = A + λB2 + λC4
B
n
= A + λ−C
Note that two of the above fits are non-linear, and will force you to use the
optimizer. You’ll also need to come up with reasonable starting values for
the optimizer. This will take time, but that’s why this part of the lab will
carry a lot of marks.
4.5.3
Extracting Information from the Graph
After finding the best fit to the data, draw the best fitting smooth curve
through the data points.
From your dispersion curve, determine the index of refraction of the glass
for the Fraunhofer F, C and D lines whose wavelengths are 4861Å, 6563Åand
5893Å, respectively. Using this information, calculate the dispersive power
W of the prism from
nF − nC
W =
(4.6)
nD − 1
Determine the unknown wavelengths of the 4 Helium lines from your dispersion curve, using the solve for spreadsheet function and determine uncertainties appropriately. Compare your results with the accepted values found in
the C.R.C. Handbook.