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Atomic and Molecular Spectroscopy
High Resolution Laser
Spectroscopy
2016
High resolution laser spectroscopy
Reading instructions: Svanberg, Sections 2.6, 6.2.3, 6.4.5 (in particular Fig 6.50) 8.5.1
(especially understand figures), 9.7.2 (or alternatively Sections 5.6, 11.2.2b, 8.3, 8.4.3 and
11.5.9 in Demtröder, Atoms, molecules and photons). Page references in the text below
refer to Ref. 1, Svanberg, if nothing else is stated.
The following points are especially important in this lab:
1. By discussing both Doppler limited and Doppler free spectra you will see that Doppler
free spectra contain a large amount of information beyond that which Doppler limited
spectra contain. We believe that when you compare these two, you will realize that it is
necessary in certain cases to use Doppler free techniques in order to obtain maximum
information from the studied substance.
2. You will have the opportunity to familiarize yourself with modern optical equipment,
as in many of the course’s other lab exercises.
3. We want you to get an understanding of how to realize absolute or relative
measurements of fundamental quantities, for example length or frequency, with very high
precision (10 - 12 digit accuracy).
Moreover, we hope that while you are learning all this you will have fun.
ASSIGNMENT
In this lab you will study the spectra of the iodine molecule (I2) and the sodium atom.
From quantum mechanics and the fundamental symmetry of the atomic particles and the
molecular wave functions it follows that each of the iodine’s absorption lines are made up
of either 15 or 21 hyperfine components (see the theoretical chapter in the Appendix).
Your assignment is to verify this experimentally for some of the lines and then adjust the
laser to a certain component in a transition, which is valid as the secondary standard for
absolute measurements of length and frequency, according to a decision in Paris 1983.
Measurements on sodium will consist of recording the hyperfine structure for different
adjustments of the laser light’s intensity and the pressure of the sodium vapour. The theory
of sodium’s hyperfine structure is assumed to be known from earlier courses.
SET-UP AND EQUIPMENT
The splitting of spectral lines that the hyperfine interaction gives rise to is very small. If
we want to use emitted or absorbed light we see that the splitting of the energy levels gives
a smaller frequency shift than the Doppler width (see Chapter 6.1.1). There are a couple of
alternative ways to reveal structures hidden under the Doppler profile. In this lab exercise
you will use saturation spectroscopy. The experimental setup is depicted in Fig. 1.
The pump laser is a 𝑁𝑁: 𝑌𝑌𝑂4 diode-pumped green laser delivering CW light at 532 nm
with a power of around 6 W. The dye laser is of the commercial type described in Ref 1.
1
on page 251-255 (see especially the figures 8.24-27). A wave-meter for both pulsed and
continuous lasers based on measurement of ring patterns from Fabry-Pérot etalons is used.
A Fabry-Pérot interferometers (FPI) with N=4 (see fig. 6.31, Ref 1) and a free spectral
range, FSR (FSR = Free Spectral Range), of 50 MHz is used for getting a precise
frequency scale for measuring the separation of the Doppler free structures.
Figure 1. Experimental setup for measurement spectroscopy
The path of the beam in the experimental setup is similar to that which is depicted on page
457, fig. 11.60. The signal is detected using a lock-in technique, where an acousto-optic
modulator (AOM) is used to modulate the pump beam intensity and a lock-in amplifier
detects any signal that varies with the same frequency as the modulation. The main
elements of the AOM are an ultrasound transmitter and a crystal. A pulse generator steers
the ultrasound transmitter so that the sound wave turns on and off. The sound wave creates
a travelling wave in the crystal. The laser beam strikes the crystal perpendicular to the
sound wave, see fig. 2.
The expansions and compressions that the sound wave creates in the crystal change the
refractive index and build a lattice for the incoming light. We see the incoming light and
how it is diffracted in different orders. The frequency of the light shifts by the amount
k⋅N, where k is the order in which the light spreads and N is the frequency of the
ultrasound (not to be confused with the frequency of the pulse generator, which
determines how quickly the ultrasound transmitter is turned on and off).
2
ν+3N
ν+2N
ν +N
ν
ν -N
ν-2N
ν-3N
Optical
Wave
Acoustic
Wave
Figure 2. The principles of an acousto-optic light modulator.
The pulse frequency can be varied up to a few tens of MHz and the formed transmission
grating has a conversion efficiency to the first order of about 50%. The pulse generator
also gives a reference frequency to the lock-in amplifier. The lock-in amplifier collects
from the signal the Fourier component which has the same frequency as the reference
signal (see also Ref. 4 and the lab exercise: “Optical pumping” in atomic physics for F).
Photodiodes are used as detectors for the saturation spectroscopy signal and the Doppler
broadened fluorescence signal. The cells with iodine or sodium are temperature controlled
in order to achieve a suitable vapour pressure. If the pressure is too high, you will get
collision broadening of the signal and if it is too low you will get a too weak signal. The
suitable start value is ~90° C for sodium and 5° C for iodine. The iodine cell therefore
needs to be cooled, which is done with a Peltier element. In the beam path there is an
adjustable intensity attenuator. Unnecessarily high intensity on the pump or the probe
beam gives saturation broadening (compare with the lab exercises: “Diode laser
spectroscopy” and “Optical pumping” in atomic physics for F). The signals from the lockin amplifier and the 50 MHz reference cavity is recorded with a data acquisition device in
a LabView program (SIGAV-LV) that also control the laser scan. The same program can
then evaluate the spectra by fitting Lorentzian or Gaussian peaks to the spectrum. A minimanual is available at the lab.
3
EVALUATING SPECTRA
Sodium
For the sodium calculations you need to know which transition we will use, which will be
the D-line, that is to say, 2S1/2-2P1/2 at approximately 590 nm. The nuclear spin for
sodium is 3/2. What will the saturation spectroscopy signal look like if the a-factor for the
lower state is assumed to be approximately 10 times larger than the a-factor for the upper
state?
Iodine
As was mentioned earlier, and explained in more detail in the Appendix, a rotational line
in iodine consists of 15 or 21 hyperfine components and these can be described with the
help of the quantum numbers (M1, M2). When you are trying to decide the a and b-factors
for the lines you are measuring (especially the P(62)17-1 line) you will need some
guidance. In fig. 3 there are two typical iodine spectra. These are calculated for a J
quantum number of approximately one hundred and with realistic a and b-factors. You
can note that both the upper (15-lines) and the lower (21-lines) spectra consist of 6 groups
of lines. The groups at the points: 0.50, -0.10 and -0.40 consist of one peak in the top
figure and three peaks in the bottom figure. The other three groups always consist of four
peaks (note - this is valid for lines with a high J quantum number). The values, (M1, M2)
for the different peaks are marked in the figures. According to the theoretical chapter in
the Appendix, there are transitions between one (M1, M2)-level in the lower state to the
same (M1, M2)-level in the upper, this means that unfortunately we cannot determine the
a- and b-factors for the state individually, but only the differences ∆a=alow-aupp and
∆b=blow-bupp. This is enough to calculate the hyperfine structure in for example the upper
state if we assume that the a- and b-factors in the lower level are zero. If we use the Ehfsformula from the theory chapter with (F1, F2) = (M1+J, M2+J) and the (M1, M2) values
you can find in the figure you will see that the b-factor (∆b-factor) will be multiplied by
approximately: 0.50, 0.20, 0.05, -0.10, -0.25 and -0.40 for the different groups in the
spectra. The terms at the bottom of the figure come from this.
For iodine the b-factor is approximately -1 GHz and it is the electrical quadruple
interaction that provides the greatest contribution to the hyperfine structure. The magnetic
dipole interaction gives only small corrections to the positions of the energy levels. The afactor for the lines you will be measuring is only a few tens of kilohertz. It is important to
remember that the formula for the hyperfine structure is only an approximation.
The deviation between the calculated and the measured line positions are approximately 5
MHz (the entire structure is approximately 1 GHz). The lines which give the worst
agreement are those with (M1, M2) = (-5/2,-3/2), (5/2,3/2), (-3/2,-1/2) and (3/2,1/2). You
should therefore avoid using these when you calculate a and b, there are still enough
positions so that you will be able to solve the unknown from the system of equations. If
you make an exception for these lines, the deviations between the calculated and the
measured line positions should be less than 1 MHz. Also, if you select the three peaks at
(-5/2,5/2), (-3/2,3/2) and (-1/2,1/2) you will get a system of equations that are hard to solve
(the lines are almost parallel), select at least one position from the other transitions or use
least-squares technique to use all lines when calculating the a and b-factor.
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(1/2,-1/2)
(-3/2,-1/2)
(-3/2,-1/2)
(3/2,1/2)
(3/2,-3/2)
(-5/2,-1/2)
(-5/2,-1/2)
(-3/2,-3/2)
(3/2,-3/2)
(3/2,3/2)
(5/2,3/2)
(5/2,3/2)
(5/2,1/2)
(-5/2,3/2)
(-5/2,3/2)
(5/2,1/2)
(-5/2,-3/2)
(5/2,-3/2)
(-5/2,-3/2)
(5/2,-5/2)
(-5/2,-5/2)
(5/2,-5/2)
(5/2,5/2)
(5/2,-3/2)
(M1,M2)
(M1,M2)
(-3/2,1/2)
(3/2,-1/2)
(-5/2,1/2)
(5/2,-1/2)
I2
Hyperfine Structure
(1/2,1/2)
(1/2,-1/2)
(-1/2,-1/2)
(3/2,1/2)
(-3/2,1/2)
(3/2,-1/2)
(-5/2,1/2)
(5/2,-1/2)
Even, Jlow
Odd, Jlow
0.50
0.20
0.05
-0.10
-0.25
-0.40
Figure 3. Calculated 15 and 21 lines iodine spectra.
More accurate calculations of the hyperfine structure require a computer. The reason for
the deviations in the positions of the lines is that the states are not pure (M1, M2)-states.
Through a more precise energy level calculation you express the states with the help of the
quantum numbers I, J and F in the usual way and calculate the energy matrix:
<α',I',J',F'|Hhfs|α",I",J",F">.
The energy matrix is not diagonal but by diagonalizing it, that is to say, calculating the
eigenvalues and eigenvectors, you get the energy levels (eigenvalues) and the wave
functions (eigenvectors) for the levels.
One of the reasons that iodine is used as a wavelength reference (secondary standard) is
that there are tens of thousands of sharp lines in the wavelength range 500-700 nm. This
is of course an advantage when iodine is used as a reference but it creates problems when
you are trying to identify which line you are measuring. In the lab you will search for line
P(62)17-1 and you have fig. 4 to help you.
The upper spectrum is from a table of the iodine spectrum. The spectrum is recorded
(Doppler broadened) with a Fourier spectrometer (Ref. 5). Thus, each absorption line in
the spectrum consists of 15 or 21 hyperfine components. The laser can scan continuously
5
for 0.03nm and the interesting range is marked in the figure. The enlarged spectra in the
figure is a recording of the fluorescence light from the iodine cell when the laser scans
0.03 nm. The different rotational transitions are identified in the figure.
Available in the lab is a program called (I2WL) which calculates the wavelengths for all
the strong rotational transitions within a 10 cm-1 interval and identifies these. The table
below is a sample from a calculation for the P(62)17-1 line. The lines can be identified
with the help of the wavelength list (Ref. 5), the wavelength table from the program and
the wave meter for wavelength determination of the dye laser.
Table 3.
Transition
R( 67)17-1
R(191)27-3
P( 74)15-0
P(139)19-1
P(118)16-0
P( 43)19-2
R( 59)24-4
R(113)18-1
P(148)17-0
R(195)19-0
P( 55)24-4
P( 62)17-1
R( 47)19-2
P( 96)20-2
R(143)19-1
Wave number
cm-1
17351.6320
17351.7925
17351.7936
17351.8074
17351.8435
17351.9672
17351.9798
17352.0273
17352.0903
17352.0922
17352.1470
17352.2511
17352.3874
17352.4902
17352.5465
Vacuum wavelength
nm
576.3147
576.3093
576.3093
576.3088
576.3076
576.3035
576.3031
576.3015
576.2994
576.2994
576.2976
576.2941
576.2896
576.2862
576.2843
6
Air wavelength
nm
576.1549
576.1496
576.1495
576.1491
576.1479
576.1438
576.1433
576.1418
576.1397
576.1396
576.1378
576.1343
576.1298
576.1264
576.1245
2200
2195
2190
2185
2180
17350
17352
17354
I2
P(139)19-1
P(74)15-0
P(62)17-1
Fluorescence Spectra
R(113)18-1
R(67)17-1
P(43)19-2
R(59)24-4
R(47)19-2
P(118)16-0
P(96)20-2
R(143)19-1
P(148)17-0
R(191)27-3
R(195)19-0
P(55)24-4
0
FREQUENCY/GHz
30
Figure 4. The Fourier spectrum and fluorescence spectrum for the wavelength interval
around the P(62)17-1 transition (Doppler broadened).
7
A CONNECTION TO FUNDAMENTAL
MEASUREMENT TECHNIQUES
If you want to do accurate measurements then frequency stabilized lasers are especially
powerful tools. If you lock a stabilized laser with a frequency bandwidth, ∆v, of for
example 100 Hz to a narrow absorption line with a known frequency then you have a
reference source with an accuracy of the frequency of 12 digits. The frequency of an
unknown light source can then in principle be determined by mixing its light with the light
from the stabilized laser and measuring the beat frequency. However, the beat frequency
is often so high that you are forced to mix light from yet one or more stabilized light
sources in order to come down to a range where the beat frequency can be measured with
good accuracy. In this lab you will tune the laser to the fifteenth hyperfine component in
the transition (v'=17,J'=61) → (v"=1,J"=62) in 127I2 (v is the vibration quantum number,
" denotes the lower and ' the upper level) which is considered a secondary standard for the
length unit, ever since the new meter definition was accepted in Paris the 20th of October
1983 at 1 pm at the “Conference Generale des Poids et Mesures”. This transition
corresponds to a wavelength of 576294760.27·10-15 m.
APPENDIX: THE HYPERFINE STRUCTURE OF THE IODINE MOLECULE
The structure of molecules is described broadly in Ref. 1, chapter 3. Iodine atoms build
diatomic molecules, I2. The term for the electron state for diatomic molecules, where both
the atomic nuclei are of the same type (homonuclear molecules), has index g or u (g stands
for the German “gerade”, u stands for “ungerade”) depending on if the wave function for
the state is symmetrical or asymmetrical, with regards to reflection in the origin (that is to
say if Ψ(x,y,z)= Ψ(-x,-y,-z) or Ψ(x,y,z)=- Ψ(-x,-y,-z)). A plane, which contains the joining
axis between the two atomic nuclei, is a symmetrical plane for all diatomic molecules.
The electron states whose wave function is symmetrical (anti-symmetrical) with regards to
reflection in this plane are indicated with +(-). When one now speaks of the basic state for
iodine molecules as being 1Σ𝑔+ this provides a lot of information on the state of this
molecule.
The nuclear spin for 127I is 5/2. When you calculate the hyperfine interaction in the iodine
molecule, if you first calculate the individual contributions for each of the nuclei and then
add these two contributions you get a good approximation of the correct value.
2 
3 C C +1 − I I +1 J J +1 
) n( n ) ( )
4 n( n
Ehfs = ∑ a 2 Cn + b

2 In (2 In − 1) J ( 2 J − 1)
n =1 

Cn = Fn ( Fn + 1) − In ( In + 1) − J ( J + 1)
where Fn=J+In, In is the nuclear spin (with direction) for nucleus n and Fn is the total
angular momentum including the nuclear spin for nucleus n. Everything, even the
definitions of the constants a and b are completely analogous with what is valid for atoms
(see slides atom repetition lecture especially the slides on electric hyperfine structure).
For a fast rotating molecule (high value for J) the individual nuclear spins (I1,I2) are
quantized along J. The projections along J are denoted as M1 and M2 respectively (for
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levels with a large J, F can be approximated with Fn=Mn+J since F and J are then almost
parallel). With I1=I2=I=5/2 you expect (2·5/2+1)(2·5/2+1)=36 different combinations of
(M1,M2) for each J. We will be studying transitions from the ground state to the state
3 +
Π0𝑢 . As a result we will be looking at a Ω=0 → Ω=0 transition (Ω= Λ+Σ  where Λ and
Σ are the projections of the molecules’ angular momentum and spin along the internuclear
axis respectively.) In this case, ∆J=0 is forbidden (Ref. 2 page 240), thus we have ∆J=±1.
During absorption of a photon the nuclei do not change their positions. This means that
the nuclear spin’s projection on J is unchanged and we then get the selection rule
∆M1=∆M2=0. There are other selection rules but these suffice in order to move the
discussion further. The last of the selection rules above implies that, for a certain
combination J,M1,M2,Ju (here  stands for the lower and u for the upper level) there is
for each level (J1, M1,M2) only one level (Ju,M1u,M2u) for which the transition can
happen. One should then expect that each absorption line should consist of 36 hyperfine
components; one for each combination of (M1,M2). Degeneration between the different
directions of the nuclear spin has been cancelled by the hyperfine interaction. It so
happens that one observes either 15 or 21 components. This can be explained in the
following way.
All particles with a half-integral spin obey the Fermi-Dirac statistics (e.g. iodine atoms,
which have I=5/2). Particles with integral spin obey the Bose-Einstein statistics (see for
example Ref. 3 chapter 9). A fundamental characteristic of these particles, which obey the
Fermi-Dirac statistics, is that their total wave function is anti-symmetrical during the
exchange of two particles. This is the reason why two electrons in an atom can not have
all their quantum numbers equal, that is to say, identical wave functions (Pauli’s exclusion
principle). If that were the case, the total wave function for the atom would of course not
change if you were to exchange the electrons with each other. Let us now study the iodine
molecule. It is not easy to directly see how the wave function would change if you
exchange the two nuclei. We see however that the exchange of the nuclei is equivalent to
performing the following three operations.
1. Rotation of the molecule (the nucleus as well as the electrons) 180 degrees around an
axis that is perpendicular to the nuclei’s binding axis and goes through the inversion
centre.
2. Reflection of the electron wave function in a plane that contains the molecule’s binding
axis.
3. Inversion of the electron wave function at the origin.
These three operations performed after one another shall thus change the sign of the total
wave function. This (Ψtot) can, with a good approximation, be written as (the following
description is similar to Ref. 2 page 128-129)
Ψtot=Ψe(1/r) ΨvΨRβ
Ψe
r
Ψv
is the electron wave function
is the distance between the atomic nuclei
describes the molecule’s vibration movement and depends only upon the distance
between the two nuclei
9
ΨR
β
describes the molecule’s rotation movement and depends upon the molecule’s
position in space with respect to the co-ordinate system with the origin in the
inversion centre
is the nuclear spin wave function.
The nuclear spin of the iodine atom is 5/2. With the ordinary rules for addition of angular
momentum, the total nuclear spin (Itot) for the iodine molecule must take one of the values
0,1,2,3,4,5. Intuitively it is possible that if I=5 then I1 and I2 have the same direction and
β should therefore be symmetric during the exchange of the nuclei. If on the other hand
Itot is zero then the individual nuclear spins must be in the opposite direction to each other
and β is expected to be asymmetric. A more strict mathematical treatment (for example in
Ref. 2, page 137) gives that β is symmetrical if Itot is odd and asymmetrical if Itot is even;
which also corresponds to what is written above.
Let’s now see how Ψe(1/r)ΨvΨR behaves during the last three operations described above.
(1/r)Ψv which depends only on the distance between the nuclei is unchanged during all
three operations. One can show that rotation of the molecule by 180 degrees gives the
new ΨR of ΨR(new)=(-1)JΨR(old). The other operations do not influence the rotational
wave function. Ψe does not change when you rotate the molecule. Besides, both of the
+
electron states we are observing ( 1Σ𝑔+ and 3Π0𝑢
) are symmetric during operation two (is
given by “+” -sign). However, operation three changes the sign of the wave function for
+
the 3Π0𝑢
state.
We can now make a table over the wave functions’ symmetries in the lower state during
exchange of the nuclei.
Table 1
J odd, I even
J odd, I odd
J even, I odd
J even, I even
(1/r)ΨvΨRΨe
asymmetric
asymmetric
symmetric
symmetric
β
asymmetric
symmetric
symmetric
asymmetric
Ψtot
symmetric
asymmetric
symmetric
asymmetric
allowed
no
yes
no
yes
From the table we see that for the iodine molecule’s lower state J and Itot must either both
be odd or both even. For the upper state the following table is valid.
Table 2
J odd, I even
J odd, I odd
J even, I odd
J even, I even
(1/r)ΨvΨRΨe
symmetric
symmetric
asymmetric
asymmetric
β
asymmetric
symmetric
symmetric
asymmetric
Ψtot
asymmetric
symmetric
asymmetric
symmetric
allowed
yes
no
yes
no
We see here that one and only one of the quantum numbers J and Itot must be odd. For the
lower state and an even J, Itot can be 0, 2 or 4.
This corresponds to
(2⋅0+1)+(2⋅2+1)+(2⋅4+1)=15 different combinations of (M1,M2). For odd J we get in the
corresponding way (2⋅1+1)+(2⋅3+1)+(2⋅5+1)=21 levels. 15+21=36 which was the total
number of different combinations of (M1,M2). The selection rule ∆J=±1 implies that, for
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example, an odd J in the lower state with 21 components can only make a transition to an
even J in the upper state during photon absorption, which according to the above also has
exactly 21 components corresponding to exactly the same combination of (M1,M2).
According to the earlier argument we had at the transition a “one-to-one mapping” that is
to say, each hyperfine level in the lower state combines with exactly one level in the upper
state and every absorption line consists of 15 or 21 components depending upon if J in the
lower state is even or odd. It is hard to find a direct analogy for this in the macroscopic
world. That makes it hard to explain just why this occurs. Those who are philosophically
inclined can for example contemplate over how iodine molecules know that they are
Fermi-Dirac particles.
REFERENCES
1. Atomic and Molecular Spectroscopy, S. Svanberg.
2. Spectra of Diatomic Molecules 2:nd ed, G. Herzberg, Van Nostrand Reinhold
Company Inc. 1950.
3. Thermal Physics, C. Kittel, John Wiley & Sons Inc, 1969.
4. http://en.wikipedia.org/wiki/Lock-in_amplifier.
5. Atlas du Spectre D’absorption de la Molecule D’iode, S. Gerstenkorn et P. Luc, 1978.
6. Atoms, Molecules and Photons, W. Demtröder, Springer
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