Second year lab 2, Winter, 2008
Physics 390, Lab 5:
Diffraction and Optical Spectroscopy
(modified from materials by KJ Park & Stephen Gregory)
Reality provides us with facts so romantic that imagination itself could add nothing to them.
Jules Verne.
What is the meaning of it all, Mr. Holmes? Ah. I have no data. I cannot tell, he said.
Conan Doyle— The Adventures of the Copper Beeches.
Hi there! what’s your spectrum (overheard as a pickup line by…
Arthur
Physics Lab Instructor)
Introduction
Because light behaves macroscopically as a wave (when averaged over millions and millions of
photons), it diffracts when encountering regularly spaced scatterers such as grooves or lines in a
diffraction grating. By diffract, we mean that light waves appear to “bend” when encountering
obstacles whose size is on the order of the wavelength of the light under consideration. It is
perhaps better to think of light from a source as forcing the atoms or molecules of the diffraction
centers (grooves, lines, etc.) into vibration, whereupon they re-radiate the original light in many
directions. If the path length from successive diffraction centers to an observation point is an
integer multiple of the wavelength, constructive and destructive interference will occur and
bright and dark diffraction patterns will be observed. The spacing between bright parts of the
pattern will depend upon the wavelength of the light and so, for example, longer wavelength
light will appear to “bend” more when encountering a diffraction grating. By measuring how
light from a particular source is bent upon passing through a diffraction grating, one can learn
interesting things about the source. This branch of science is called spectrometry.
The optical spectrometer enables us to study the emission and absorption of light by atoms
(among many other things). Measurements of optical spectra at the end of the 19th and beginning
of the 20th Century lead to the creation of atomic physics and the understanding of atomic
behavior in terms of quantum theory. In more automated versions the grating spectrometer is
still one of the most important laboratory instruments.
We shall use a grating spectrometer to measure the positions of “lines” in the emission spectrum
of mercury, hydrogen and sodium. These lines correspond to the discrete energy levels of the
electrons in the atoms, and measurements of these lines directly test our understanding of the
atomic model. Hydrogen, a two body system, is the only atomic spectrum which can be
analytically solved in quantum mechanics. Even before the “invention” of quantum mechanics,
Bohr had discovered a certain pattern in the spectrum and had proposed a set of discrete energy
levels with a certain geometric relationship (the Bohr model). Other elements, like Sodium,
exhibit some characteristics of this single-electron spectrum owing to their single valence
electron outside fully-populated inner shells.
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Second year lab 2, Winter, 2008
Experiment Goals
• Establish spacing of optical spectrometer diffraction grating.
• Using a mercury gas discharge source, measure the angle of diffraction of the zero-th
order for the green line.
• Use the measured angle for the green line and the known wavelength of the green line
to determine the spacing between lines of the diffraction grating.
• Measure diffraction angles for the Balmer series of hydrogen
• Use your setup and a hydrogen gas source to determine the wavelengths of the
Balmer series (of lines) for hydrogen. Use the diffraction grating spacing determined
above.
• Compare your values for the Balmer series to those stated in textbooks.
• Use your Balmer series wavelengths to determine the Rydberg constant for hydrogen.
• Fine structure in sodium
• Using a sodium lamp, observe the three primary wavelengths of the sodium spectrum.
• Measure the fine structure splitting of these primary lines
• Relate this structure to the model of the sodium atom.
Background
Light Sources
Our light sources are gas-discharge tubes. These consist of a glass tube with a metal electrode at
each end. The tubes contain low-pressure gases of various types. When a high voltage is applied
between the electrodes the gas ionizes (the atoms split into electrons and positive ions). These
charged particles are accelerated in the electric field and collide with atoms, causing these to
enter a higher energy “excited state”. (Another way of looking at this process is that the ions are
carrying a current which heats up the atoms, creating more ions and also exciting the atoms.)
When the atoms return to their original states, they emit photons of particular wavelengths. The
set of wavelengths which are emitted— the “spectrum” of the atoms— is characteristic of the
atoms and can be used to identify them. In fact, a standard way of identifying materials is to burn
them in a flame and observe the emission spectrum.
The Spectrometer
The spectrometer consists of three basic elements, the collimator, the diffraction grating and the
telescope.
First, some of the light from the tubes is collected by the collimator:
Collimator
The light to be analyzed enters the collimator through a narrow slit whose position can be
adjusted to put it at the focal point of the collimator lens. The light leaving the collimator should
therefore be a parallel beam, which ensures that all the light from the slit strikes the diffraction
grating at the same angle of incidence. This is necessary if a sharp slit image is to be formed.
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Second year lab 2, Winter, 2008
Think about the best method you can use to ensure that the light leaving the collimator really IS a
parallel beam.
The parallel light beam now should enter the diffraction grating perpendicularly:
Diffraction Grating
Diffraction gratings are made by scribing closely spaced grooves on glass or some other
substrate. The phenomenon of diffraction is one of many manifestations of interference in
physics. The incoming beam hits the grooves and scatters in all directions. However, because
the initial, unscattered, wavefront had a common phase, the scattered light will interfere with
itself in such a way as to produce maxima and minima as a function of angle (usually measured
from the perpendicular to the exit face of the grating .
θ
d
Fig. 1. Diffraction Geometry
Many text books contain the theory of the diffraction grating. The basic result is that maxima in
the diffraction pattern are found for angles satisfying the relationship
d sin θ = n λ
where d is the spacing between grooves, θ is the angle between the perpendicular and the
diffracted beam, λ is the wavelength and n is the “order” of the diffraction where n = 0, 1, 2,
…etc.
There is considerably more to the general theory of the diffraction grating. In fact, it is
nowadays presented in terms of Fourier Transforms. However, the above expression is quite
correct and useful for present purposes.
The light from the gas-discharge tube is not monochromatic. There will therefore be an angle
which gives a maximum, i.e. a bright image of the slit which we will call a “line”, for each
wavelength present in the spectrum of the gas. This means that there will be a characteristic set
of lines. Further, this set will repeat as the angle, θ, increases because the phase relationship for
a maximum repeats for each increment of 2π. Each repeat is called an order.
Telescope
The telescope can be rotated to collect the diffracted light at very precisely measured angles.
With the telescope focused at infinity and positioned at an angle to collect the light of a
particular color, a precise image of the collimator slit can be seen. For example, when the
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Second year lab 2, Winter, 2008
telescope is at one angle of rotation, the viewer might see a red image of the slit, at another angle
a green image, and so on. By rotating the telescope, the slit images corresponding to each
constituent color can be viewed and the angle of diffraction for each image can be measured.
This measurement is aided by cross-hairs in the telescope.
Operating the Spectrometer
In order to save you some time, the main alignment of the spectrometer will be performed at the
start of the week before you come to lab. You should only need to adjust the focus of the
collimator and telescope to obtain reasonable measurements. To gain a better appreciation of
what is involved, the full alignment procedure is described in the appendix.
The attached material from the manufacturer will be referred to as the MANUAL.
1) Read the Equipment page of the MANUAL.
2) Identify the lock–screws for the telescope and the spectrometer table. Practice rotating the
two bases coarsely by hand (release lock–screws) and using the fine-adjustment screws
when the lock screws are engaged.
3) NOTE: The scales on the spectrometer are in degrees, minutes and seconds. This is
ridiculous (in the opinion of Steve Gregory), but gives you some of the feel of what it would
have been like to be some guy in a starched shirt and suit sitting at the lab bench at the end of
the 19th Century. Well, anyway, we can convert to decimal values in order to do the
calculations.
Reading the Vernier Scales(with the magnifying glass): To read the angle, first find where
the zero point of the vernier scale aligns with the degree plate and record the value. If the
zero point is between two lines, use the smaller value. In Figure 8 the zero point on the
vernier scale is
between the 172o 20'
mark and the 172o
40' mark on the
degree plate, so the
recorded value is
172o 20'. Now use
the magnifying glass
to find the line on
the vernier scale that
aligns most closely
with any line on the
degree plate. In the
figure, this is the line
corresponding to a
measurement of
12'30'' of arc. Add
Figure 2. Reading the Vernier Scales
this value to the
reading recorded
above to get the correct measure ment to within 30 seconds of arc: that is, 172o 20' + 12'30" =
172o 32'30".
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Second year lab 2, Winter, 2008
4) When analyzing a light source, angles of diffraction are measured using the telescope
vernier. In general, we will measure the difference between angles of diffraction on either
side of the undeflected position and divide by two, but it is useful to establish a vernier
reading for the undeflected beam (see Figure 8 of the MANUAL).
To obtain a vernier reading for
the undeflected beam, first you
would align the vertical
cross-hair of the telescope with
the fixed edge of the slit image
for the undeflected beam. Then
read the vernier scale. This is
the zero point reading,
Figure 3. Measuring an Angle of Diffraction.
The rotational position of the spectrometer table can be measured with the same accuracy
using the spectrometer table vernier.
Note: The telescope and the spectrometer table each have two vernier scales, which are
exactly 180o apart. Unless you use the same vernier scale for both the initial and final
readings, you will need to add (or subtract) 180o from your result.
6) Take turns to practice reading the vernier for arbitrary angular settings in darkened room with
a flashlight or table light. You are to read verniers to 0.5 minutes. (1 minute = 1' = 60
seconds = 60'')
Checking the Focusing
You only have to make minor adjustments for your eyes. Remember not to change the focus of
the collimator or of the telescope by large amounts. To get the cross-hairs into focus (this is
independent of the actual focussing of the telescope) move the eyepiece in and out.
Do not make any adjustment to the leveling screws under the telescope and the collimator. If
you suspect that you need to realign, please see your instructor.
1) If the grating is present, remove it. Illuminate the slit with a mercury discharge source,
placed about 1cm in front of the slit. Line up the collimator and telescope and look in the
telescope. The slit image and the cross–hairs should show no parallax. (i.e. if you move
your head side to side, the slit image should not move with respect to the cross–hairs.) If
you notice some parallax, the following sequence of fine adjustments should accomplish the
final alignment: Remember! These are minute adjustments!
2) Try a small clock-wise adjustment of the collimator focus. See if the parallax is decreased or
increased. If decreased, go to 3; otherwise, make a counter-clock-wise adjustment.
3) When the parallax has been decreased, the slit image should be slightly fuzzy. Adjust the
telescope focus, followed by cross–hair focus.
4) Recheck the parallax. Repeat 2 – 3 until the parallax is not there.
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Second year lab 2, Winter, 2008
Install the diffraction grating:
1) Study the MANUAL section, `Using the Diffraction Grating.'
2) Steps 1–3 are done for you already. In step 4, insert the grating in an orientation in which it
is wider than it is tall. Make sure that the tag tape faces the source slit. Can you think of a
reason why this is important. What would be the consequences of installing the grating with
the tag tape facing the telescope?
3) Plug in the power plug for the Gaussian Eyepiece.
4) Follow steps 1–16 of the MANUAL section, `Appendix.' Again, complete this task without
leveling adjustments for the collimator and the telescope. When using the Gaussian
eyepiece note that the reflected image of the crosshairs is actually dark, because it is a
shadow.
5. Make sure that the Mercury vapor discharge tube is approximately 1-4 cm in front of the slit
and in line with the collimator axis. Follow steps 6–9 of the MANUAL section, `Using the
Diffraction Grating.' Make sure that the Mercury green is symmetrically located with
respect to the central maximum. If not, make a tiny rotation of the spectrometer table to
achieve this. (See the last paragraph of the MANUAL section, 'Appendix.' [You will have to
take three sets of measurements and average the results for the three angular positions. You
will want to repeat with the table angular position changed by increments of approximately
1 degree. You should converge toward the symmetric situation and should be satisfied if you
are within several minutes.]
Accuracy of angular position measurements:
The spectrometer verniers have a precision of 30 seconds. You should determine your
measurement errors by setting the telescope vertical cross–hair to the fixed edge of the
illuminated slit. [Some people prefer to line up the cross–hair to the middle of slit, whose width
has been adjusted slightly wider than the cross–hair width. But this tends to make lines very
dim.] Make five repeated settings of the cross hairs on the slit and measure the angles. Calculate
the standard deviation.
Measurements and Analysis
1) Determination of the grating characteristics using a standard source:
The mercury discharge tube will be used as a standard reference source. Determine the
diffraction angle for the green line of mercury by sighting the line on both sides of the
undeflected position and dividing by two. You should make 5 measurements of the zero
diffraction and five of the first order diffraction positions. (This means resighting each time!)
2) Measure the Positions of the Lines in the Hydrogen Spectrum:
Now switch to the hydrogen discharge tube in order to measure the position of as many of the
lines as you can see in the first order and in the second order. Use the method of taking data on
both sides of the zero point and dividing by two. You only need one measurement of each line.
3) Measure the Splitting of the Lines in the Sodium Spectrum:
Switch the discharge tube for a sodium lamp. You should be able to see three spectral lines (red,
yellow and green) which are each split into closely-spaced pairs. Measure the wavelengths of
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Second year lab 2, Winter, 2008
these lines in both the first and second order if possible. We actually care more about the
splitting than the actual position of the lines. Since there are many lines here to measure, you
probably don’t need to take a full five readings for each line, but try to estimate how accurately
you can determine the splitting by comparing results left vs. right, or first order vs. second order.
Take the grating and put it away in the grating box when you are done.
Mercury/Hydrogen Analysis:
1. Calculate the mean and the standard deviation of the multiple readings taken for the mercury
green line.
2. Calculate the slit spacing of the grating from the mercury line measurement. The Handbook
of Physics lists the wavelength of the mercury green as 546.0735 nm.
3. The visible part of the Hydrogen spectrum is called the Balmer series. The Bohr model
predicts that each electron orbital has an energy proportional to –n2. The visible lines are
transitions between these energy levels. Use your data for the Balmer series of Hydrogen to
obtain a value for the Rydberg constant, R, in the formula
" 1 1%
1
= RZ 2 $ 2 _ 2 ' ; R = 1.0973732x 10 7 m_ 1
# n 1 n2 &
!
where n1 = 2, n2 = 3,4,5, which refer to the red, blue and violet lines (do you know what Z
is?) Discuss your uncertainties on measuring R.
Sodium Analysis:
The Sodium atom (Z=11) has a single valence electron outside 10 electrons which completely fill
the first and second inner shells. This makes the Sodium spectrum somewhat analogous to the
single-electron Hydrogen spectrum. The “fine structure” splitting can be understook by looking
a the energy level diagram shown in Figure 4, where the states have been separated by their
angular momentum. Photons can only be emitted between states where angular momentum
changes by at least one unit. This leads to the transitions shown, which are all in the visible.
Other allowed transitions are in the ultraviolet or infrared Figure 4: Selected Sodium atomic levels
The splitting in the optical spectra can be
explained if the 2P energy levels are split by
the spin-orbit coupling between the valence
electron and the magnetic field produced by
the rest of the atom. The 2P levels actually
consist of J=1/2 and J=3/2 states with
different total angular momenta, and the
energy levels of these two states in the
presence of a magnetic field are slightly
different. The ration of this energy splitting
to the gross energy of the 2P (n=4) state was
originally measured to be α2 = (1/137)2 or
the “fine structure constant.” A similar
splitting is not seen in Hydrogen due to
higher-order QED corrections called the
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Second year lab 2, Winter, 2008
“Lamb Shift” but it is much too small for us to resolve.
1. Measure the wavelength of the splitting observed in the three sets of visible Sodium lines.
Include uncertainties (statistical and systematic) which you have estimated.
2. Calculate the energy difference associated with each set of lines. Recall that the energy
of a photon is given by E=hν=hc/λ.
3. Is your data (with uncertainties) consistent with the hypothesis described above? Why or
why not?
Questions & Discussion:
Here are some items for you to discuss in your report:
• The mercury green is often used as a reference, because its wavelength is extremely stable
even when the discharge is not very good. The fact that we referenced the Balmer lines to
this mercury green should make the result quite reliable. How might this process take care of
systematic errors of the spectrometer?
• Where is your major source of error?
• All these wavelengths are vacuum values. How does the refractive index of air affect the
diffraction pattern? How about the refractive index of the glass substrate in which the
grating is sandwiched?
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Second year lab 2, Winter, 2008
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Second year lab 2, Winter, 2008
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