Download Optical Pumping of Natural Rubidium

Optical Pumping of Natural Rubidium
Erick Arabia
University of San Diego
(Jonathon Garamilla and Enrique Lance)
(Dated: March 26, 2011)
A gas of natural Rubidium, which consists of Rubidium-87 and Rubidium-85 was optically pumped
in order to determine the nuclear spin number (I) for each isotope experimentally, as well as to
confirm the Breit-Rabi equation for large magnetic fields. The g-factor was determined to be 0.47
for Rb-87 and 0.33 for Rb-85, and the results of the large field calculations can be found in Table
(II).
I.
INTRODUCTION
In 1950, French physicist Alfred Kastler introduced the
concept of modern optical pumping, for which he was
awarded the Nobel Prize in 1966. The process of optical
pumping involves using photons of a desired wavelength
and polarization to change the distribution of a gas of
atoms from thermal equilibrium. Today, optical pumping is a critical tool in modern Physics experiments, and
is the basis of lasers. This process utilizes quantum mechanical ideas, and can therefore be used to gain a better
understanding of quantum and atomic theory.
In this experiment optical pumping processes were performed on natural Rubidium gas, which is a mixture of
Rb-87 and Rb-85. Because Rubidium only has a single valence electron, it can be approximated as a singleelectron atom (i.e. Hydrogen). Optically pumping the
Rubidium sample achieved two main goals, which was
the determination of the nuclear spin angular momenta
(I) for each isotope, as well as the confirmation of Zeeman splitting due to a magnetic field according to the
Breit-Rabi equation.
In Section (II), the theory behind the optical pumping is discussed. Section (III) explores the experimental
setup and procedure. The results and their analyzations
are in Section (IV), and the conclusions can be found in
Section (V).
II.
A.
THEORY
Atomic Structure
In order to understand the concept of optical pumping,
it is first necessary to understand a few basic principles
about atomic structure. Because Rubidium is an alkali
atom, it only has a single valence electron. That is to
say that all the electrons are paired except for one. It
follows that it can be approximated as a hydrogen-like
atom, with a single free electron.
This electron has three important quantities associated
with it: the orbital angular momentum L, the spin angular momentum S, and the total electron angular momentum J. These numbers are quantized, are all in values of
h̄, and are related by
J = L + S.
(1)
Each electron is described in spectroscopic notation by
2S+1
LJ ; the valence electron in Rubidium is therefore
designated by 2 S1/2 . This corresponds to J = 1/2, L = 0,
and S = 1/2.
It is also important to consider the angular momentum of the nucleus. The nuclear spin I combines with
J to form the total angular momentum number, F. This
coupling has a few important properties. First of all,
because of the interaction of the nucleus and the electron, the quantized energy levels of the atom are split
into 2S + 1 sublevels for each energy state of the electron
(known as Hyperfine Splitting). Second, this splitting is
degenerate, meaning that atoms with slightly different
quantum numbers will occupy the same Hyperfine split
states, which are designated by F. It should be noted that
the difference in energy between the F = 1 and F = 2
states is insignificant compared to the difference between
the energies of the ground state and first excited state.
B.
Zeeman Splitting
The degeneracy in the Hyperfine splitting described
earlier becomes important when a magnetic field is introduced to the atom in question. When this occurs,
the F energy states are split further into 2F + 1 sublevels, designated by the new quantum number M. At
very small magnetic field levels, the difference between
the Zeeman states is approximately equal for each value
of F, and as the magnetic field is increased the difference
between the states also increases linearly. As is the case
when comparing the energy levels of the Hyperfine splitting and the electron energy states, the Zeeman splitting
is significantly smaller than the Hyperfine splitting.
However, as the magnetic field is further increased beyond the low-field levels, the Zeeman split energy states
begin to change in a non-linear fashion. These changes
are designated by the Breit-Rabi equation,
1/2
∆W
µI
∆W
4M
2
W (F, M ) = −
− BM ±
1+
x+x
,
2(2I + 1) I
2
2I + 1
(2)
2
FIG. 2: Allowed transitions due to photon absorption
during the experiment. Note that while electrons in the
M=2 state cannot absorb photons, excited electrons
can relax back into this state.
D.
FIG. 1: Plot of the Breit-Rabi equation. The horizontal
axis shows the magnetic field strength as the
dimensionless unit x, and the vertical axis shows the
energy as the dimensionless unit W/∆W. Individual
lines are designated by their quantum numbers as
|F M >
where W is the interaction energy and ∆W is the Hyperfine energy splitting. Furthermore,
µ0 B
,
∆W
µI
gI = −
.
Iµ0
x = (gJ − gI )
(3)
(4)
A plot of the Breit-Rabi equation, using the dimensionless axis W/∆W (energy) versus x (magnetic field
strength) can be seen in Figure (1).
C.
Photon Absorption
As photons are absorbed by the atoms, the electrons
are excited to higher energy states designated by the energy level of the particular photon absorbed. When the
electrons relax, they re-emit a photon corresponding to
the drop in energy they undergo. It is important to understand that the re-emitted photon does not have to
be of the same energy of the absorbed photon. Furthermore, because the possible energy levels of the electron
are quantized, and because of the interaction of the quantum numbers, there are certain selection rules that designate how an atom can absorb and emit a photon. The
rules for an atom in a magnetic field are ∆S=0, ∆J=0,
±1, ∆L=0, ±1 (but not L=0 to L=0), ∆F=0, ±1, and
∆M=0, ±1.
Optical Pumping
In this experiment the light introduced is of a very
particular wavelength and therefore energy. The light
introduced to the sample of natural Rubidium will only
induce changes between the ground state and the first
excited state. Because this energy is so much larger than
the Hyperfine and magnetic splitting energies, the selection rules for F and M still apply (and in fact are the
only important selection rules as other transitions are not
possible). The light introduced will also be circularly polarized, which means that it will have a very particular
angular momentum. It follows that for the atoms absorbing these photons, only transitions of ∆M=+1 are
possible (however the normal selection rules for M still
apply when the electrons relax and re-emit photons).
The allowed transitions are shown in Figure (2). Because the incident photons are circularly polarized, only
transitions of ∆M = +1 are possible, and therefore any
electron in the M=2 state will be unable to absorb a photon. Any electron in this state will be unable to leave.
However, the excited electrons do not have this constraint
when they re-emit a photon during their relaxation process, which means that transitions into the M=2 state
are allowed. It follows that eventually all the electrons
will occupy the M=2 state, and the incident light will
not be absorbed. In a sense, the sample of natural Rubidium gas becomes “invisible” to the incident light! This
is process is referred to as optical pumping.
The optical pumping will break down in certain situations. First of all, if the Rubidium atoms collide with the
walls of the container, it is possible to force electrons from
the M=2 state (more on this later). Second, if the magnetic field applied to the sample is just strong enough to
cancel out the magnetic field of the Earth, then there will
be no net magnetic field at the sample and the pumping
will also break down (if there is no magnetic field then
there is no Zeeman splitting). Because the strength of
the magnetic field applied in this experiment is actually
swept over a wide range, this breakdown is observed and
is referred to as the “zero-field transition.”
3
Transmitted Light Intensity
bidium gas, two perpendicular pairs of Helmholtz coils,
a set of RF pulse coils, a Silicon photodiode detector,
and several lenses and optical filters. These components
were all arranged along a straight track. Additionally,
there was a base unit for controlling the apparatus (the
strength of the magnetic fields, the temperature of the
Rubidium lamp, etc.), an oscilloscope for reading the
photodetector output, a computed connected to the oscilloscope capable of printing the oscilloscope data, and
a wave function generator. The experimental setup is
shown in Figure (4).
Magnetic Field
1.
FIG. 3: Observed resonances during the low field
optical pumping trials. The large dip to the left
represents the zero-field transition, and the two dips to
the right represent the Rb-87 and Rb-85 Zeeman
resonances, respectively. Note that the magnetic field
strength increases to the right and the transmitted light
intensity increases upward
There is another way in which the optical pumping can
be broken down, and that is by introducing an RF signal
perpendicular to the path of the incident light. By doing
this at the resonant frequency for the M=2 to M=1 transition, it is possible to induce transitions from the M=2
state and thus to temporarily break the optical pumping.
This has a few important implications. Because the frequency needed to cause this transition is proportional to
the applied magnetic field with relation to the particular
atomic structure of the atom in question, it is possible to
experimentally determine the quantum numbers of the
atom. And because the atomic structure of Rb-87 and
Rb-85 are different, the energy gap corresponding to the
M=2 to M=1 transition is different. It is therefore possible to observe both dips corresponding to each isotope’s
resonant breakdown of the optical pumping, as shown in
Figure (3).
Furthermore, when a large magnetic field is applied to
the sample, it is possible to break down the pumping even
further and elicit transitions between all the split Zeeman
states. In this way the full nature of the Zeeman splitting can be observed (however it should be noted that
the transitions that occur from ∆F = ±1 take place at
several GHz and will not be observed in this experiment).
The lamp was filled with natural Rubidium gas and
a small amount of gaseous Xenon as a buffer. This was
done so that the collisions between the Rubidium atoms
and the walls of the container were minimized. The gases
in the lamp were heated and thus excited, and as they
relaxed photons relating to the relaxation energy were
released. It should be noted that this results in a broad
spectrum of emitted light, but this was filtered by the
optical setup described later.
The natural Rubidium in the sample also contained a
buffer gas, which in this case was Neon. Again, the buffer
gas was used to minimize the collisions with the walls of
the container. If this were not done, collisions with the
container would result in photon emission, which would
destroy the optical pumping.
2.
Optics
The optical setup consisted of two plano-convex lenses,
an interference filter, two linear polarizers, and a quarter
wavelength plate. As the Rubidium lamp was heated,
the gas emitted two main lines at 780nm and 795 nm
(due to the two isotopes found in natural Rubidium).
The interference filter removed the 780nm line, and the
remaining light was linearly polarized. Next, the quarter
wavelength plate turned the linearly polarized light into
circularly polarized light before it was introduced to the
natural Rubidium sample in the cell. This was all done
to ensure that the electrons were excited by photons of a
precise energy and angular momentum. The two planoconvex lenses were used to focus the light.
3.
III.
Natural Rubidium
Helmholtz Coils
EXPERIMENT AND APPARATUS
A.
Apparatus
The TeachSpin optical pumping apparatus was used
for all experiments performed. The apparatus consisted
of a Rubidium discharge lamp, a sample of natural Ru-
There were three main sets of magnetic fields applied
to the Rubidium sample. These were calibrated to counteract the effect of the Earth’s magnetic field along the
x, y, and z axis of the sample. The entire apparatus
(i.e. the track with the sample, lamp, etc.) was alined
along one of the Earth’s magnetic axis. The vertical
4
FIG. 4: Overhead view of the experimental setup.
Helmholtz coil was adjusted until it produced a magnetic field just strong enough to counteract the vertical
field of the Earth. Both of these processes were done by
making adjustments until the observed zero-field dip was
as deep and narrow as possible. It should be noted that
these adjustments did not have any effect on the numerical values of the data, but rather they simply made the
dips easier to find and read.
The other fields generated were the main horizontal
field and the horizontal sweep field. Both fields were
perpendicular to the vertical field axis and parallel to
the track (and thus both the main and sweep fields were
coaxial). The sweep field started from a predetermined
value and increased linearly for a predetermined range
(the maximum current possible through the coils was
1A). The time the sweep coil took to complete a full
sweep was also adjustable. Therefore, by controlling the
value of the start field, the range over which the field was
swept, and the time it took to complete a full sweep, it
was possible to observe many of the characteristics of the
optically pumped Rubidium.
The main field was simply a more powerful horizontal
field coaxial with the sweep field. This was applied in
order to observe the quadratic Zeeman splitting at higher
fields, which the sweep field was not able to find on its
own. Both horizontal fields had a sense resistor on the
base unit, across which it was possible to read the voltage
difference and thus current through the coils by use of
V = IR. The sense resistor for the main field was 0.5 Ω,
and the voltage in the sweep coils was read across a 1 Ω
resistor.
4.
B.
1.
Low Field Resonance Determination of I
First, the value of the residual magnetic field was determined. Because the orientation of the apparatus and the
strength of the vertical field only counteracted two axial
components of the Earth’s magnetic field, the zero-field
transition was actually not observed at B= 0. The voltage across the sense resistor of the sweep coil (and thus
the current through the sweep coils) was measured as the
field was slowly swept through the zero field transition.
The strength of the sweep field was roughly determined
by
B = 8.991 × 10−3 IN/R̄,
(5)
where B is the magnetic field strength in gauss, I is the
current through the coils in Amperes, N is the number of
turns of coil on each side (11 for this setup), and R̄ is the
mean radius of the coils (which was 0.1639 m). The value
determined to be the strength of the residual magnetic
field was then subtracted from all future magnetic field
values determined by Equation (5).
Next, an RF signal was introduced in order to generate
transitions from the M= 2 to M= 1 state for each isotope. Because the two isotopes have different g-factors
(and thus different I values), this resulted in two separate observed dips. The RF signal was applied at six
separate frequencies ranging from 40 KHz to 65 KHz;
and for each frequency the current through the sweep
coils was measured at the two observed dips. From this
data, plots were created of transition frequency versus
the magnetic field strength, and the data was fit to two
separate straight lines.
ν = gF µ0 B/h
RF generator
The RF generator was applied perpendicular to the
horizontal and vertical magnetic fields. It was also connected to the wave function generator. Therefore it was
possible to apply an oscillating RF signal of a known
frequency and amplitude. This was done to allow transitions from the M= 2 state, which made it possible to
observe the transitions between the Zeeman states.
Procedure
(6)
was used to determine the Lande g-factor for each isotope, where ν is the frequency in MHz, µ0 is the Bohr
Magneton, h is Plank’s constant, and B is the magnetic
field in gauss. Once the g-factors were determined,
gF = gJ
F (F + 1) + J(J + 1) − I(I + 1)
2F (F + 1)
(7)
was used to determine I, as gJ is known for both isotopes.
Transmitted Light Intensity
Transmitted Light Intensity
5
Magnetic Field
Magnetic Field
(a) The first 5 dips
(b) The second 5 dips
FIG. 5: Observed resonance at high magnetic fields. The magnetic field strength increases to the right, and the
transmitted light intensity increases upward. Note that there are only 6 total dips, and therefore a 4 dip overlap
between figures.
2.
Quadratic Zeeman effect
For the next experiment the accepted g-factor values
were used, along with the data already taken, to generate
a more precise equation for the magnetic fields with respect to current. The resonance equation (Equation (6)
from above) was used on all the low field data to create
a plot of the magnetic field strength versus the current
through the sweep coils. Next, the main horizontal coils
were turned on in a direction opposite the sweep coils.
The main field was then increased slightly, and the sweep
coil was also increased in the opposite direction so as to
center on the zero-field transition. Because the strength
of the magnetic sweep field with respect to the current
through the sweep coils was known, the main coils were
therefore generating a magnetic field of equal (though opposite) strength. By repeating this procedure over a few
points, it was possible to generate another plot for the
main magnetic field strength versus the current through
the main coils. By doing this, two equations were created (one for the sweep field and one for the main field)
that gave a precise measurement of the magnetic field
strength.
Next, the sweep field was centered on the Rubidium87 dip, and the main field was turned to zero. By slowly
increasing both the main field and RF pulse frequency,
it was possible to “follow” the dip out to a high field
region. In this region transitions between the 2F+1 sublevels were observed as six separate dips. The strength
of the combined magnetic fields was determined at each
dip, and the results compared to the Breit-Rabi equation.
The resonant dips can be observed in Figure (5).
IV.
RESULTS AND ANALYSIS
A.
Low Field Determination of I
The zero field dip was observed to occur at 314 mV,
which corresponds to a current through the coils of 314
mA and a magnetic field of .189 gauss. This value was
then subtracted from the values of the magnetic field
calculated from Equation (5); the final calculated values
can be found in Table (I). The data in the fourth and fifth
column were plotted against the frequency at which they
were observed to form the straight lines described earlier.
From this, the Lande g-factor of Rb-87 was determined
to be 0.48, and the g-factor of Rb-85 was determined to
be 0.33. The accepted values are 1/2 and 1/3, which is
in good agreement with the data. Equation (7) yielded
nuclear spin values of 3/2 for Rb-87 and 5/2 for Rb-85.
B.
RF Spectroscopy
The sweep field was calibrated as described in Section (III), which resulted in the calibrated equation
B = −0.187 + 0.595I. Note that the residual magnetic
field from the Earth is taken into account when I = 0.
Likewise, the calibrated equation for the main field was
B = 6.829I.
The quadratic splitting was observed at a frequency
of 4.064 MHZ, and a main field voltage of 506.8 mV.
This corresponded to a 6.921 gauss magnetic field. The
data from the spectroscopy measurements can be found
in Table (II). The calculated values are actually not in
good agreement with those predicted by the Breit-Rabi
equation, which were all around 5.8 gauss.
6
TABLE I: Low field data used to determine the g-factor for each isotope. There is an uncertainty of ±0.5 mA for
each current measurement, which corresponds to a ±3 × 10−4 uncertainty in the magnetic field calculations.
Frequency (KHz)
40
45
50
55
60
65
Current at Rb-87
dip (mA)
408
422
434
446
459
469
Current at Rb-85
dip (mA)
455
476
496
511
529
546
TABLE II: High field resonant dips. All data taken at a
main field strength of 6.921 gauss. The Breit-Rabi
equation was used for the values in the last column.
Sweep Coil
Current (A)
Sweep Field
(gauss)
Total Field
(gauss)
0.319
0.358
0.397
0.435
0.472
0.509
0.003
0.027
0.050
0.072
0.094
0.116
6.918
6.895
6.871
6.849
6.827
6.805
Predicted
Value
(gauss)
5.805
5.813
5.820
5.813
5.805
5.798
[1] Melissinos, A, & Napolitano, J. (2003). Experiments in
modern physics. San Diego: Academic press.
[2] Optical pumping of Rubidium: Instructor’s manual. Buffalo: TeachSpin
Magnetic field at
Rb-87 dip (gauss)
0.057
0.066
0.073
0.080
0.088
0.094
V.
Magnetic Field at
Rb-85 dip (gauss)
0.086
0.098
0.110
0.119
0.130
0.140
CONCLUSIONS
It appears that there may be something wrong with
the values predicted by the Briet-Rabi equation. For instance, the fact that there is repetition in four of the
values is unsettling, as six dips were clearly observed.
That is not to say that the measurements were perfect,
as the calibration equations could be wrong. If this experiment were to be performed again, more data would
be taken for the calibration measurements to ensure a
more accurate line fitting.
However, it should be noted that the theory behind
the experiment was shown to be sound. The sample of
Rubidium was observed to have been Optically Pumped,
and the values obtained for the nuclear spin numbers
were very close to the accepted values. Furthermore, the
dips observed confirm that Zeeman splitting of the nuclear energy states did in fact take place, and that the
optical pumping was disrupted by the applied RF signal.
The process of optical pumping remains a valuable tool
for any Physicist. Quantum mechanical theories were
explored and confirmed throughout the experiment.
[3] Wolff-Reichert, B. (2009). Conceptual tour of optical pumping. TeachSpin Newsletters,
Retrieved
from
http://teachspin.com/newsletters/OP%
20ConceptualIntroduction.pdf.