Download Determining g-factors in Rubidium-85 and Rubidium

Determining g-factors in Rubidium-85 and Rubidium-87 using optical
pumping
Will Weigand,1 Adam Egbert,1 and Hannah Sadler1
University of San Diego
(Dated: 25 September 2015)
In this paper we quantify the Zeeman effect in rubidium. We find the g-factor for Rb-85 to be gF =
0.0114 ± 1.0552 ∗ 10−5 and for Rb-87 to be gF = 0.0171 ± 7.0506 ∗ 10−5 . The theoretical values of gF are 13
and 12 respectively. The nuclear magnetic quantum number for Rb-87 is I = 57.98 and for Rb-85 I = 87.22.
The accepted values for I are 23 and 52 respectively.
I.
INTRODUCTION
Optical pumping allows us to probe the quantum
world and understand the Zeeman effect, the splitting of
energy levels when an atom is perturbed with a magnetic
field. Alfred Kastler won the Nobel Prize in 1966 for
using optical pumping to determine nuclear magnetic
moments. The ideas of optical pumping are useful in the
designs of MRI and lasers.
Conceptually, optical pumping is easy to understand.
Suppose we have a two-state system consisting of a
set of particles that are distributed evenly between the
two states. We can define a set of rate equations that
describe the transitions between the two states of this
system. Now suppose that the rate of transition from
state 1 to state 2 is greater than the reverse. Over time,
all the particles in state one will end up in state 2.
This is optical pumping: disturbing the thermodynamic
equilibrium of a collection of atoms.
From a quantum mechanical viewpoint, the way to
force a transition rate to be greater or lesser than another
is to have a selection rule that has an associated ’forbidden’ transition. For our experiments the selection rule of
concern is 4M = +1 where M is the magnetic quantum
number. This rule only applies to absorption of photons
which means that there will exist a state that cannot
absorb photons but it is not forbidden to decay into
this state which is exactly what optical pumping requires.
In this paper we will investigate optical pumping in
a Rubidium gas. Section II will lay down more of the
theory of optical pumping. Section III will explain the
experimental apparatus and measurements. In Section
IV we present and discuss our results.
electrons total angular momentum and the nuclear spin,
there is an additional third energy level splitting from
the application of an external magnetic field. This is
the so-called Zeeman effect and will be the focus of our
theoretical discussion.
The Zeeman effect can be divided into three different
regions: weak, intermediate, and strong, dependening on
the relative strengths of the magnetic field interaction
and the spin-orbit coupling. The perturbing Hamiltonian
including spin-orbit coupling, spin-spin coupling, and the
Zeeman effect is:
H = H0 + AL · S + BSp · Se + Hz .
Our main concern is the Zeeman perturbation which
reads:
µJ
µI
Hz = haI · J −
J ·B−
I · B,
(2)
J
I
where µJ is the electron magnetic dipole moment and µI
is the nuclear magnetic dipole moment. If we ignore the
nuclear magnetic moment, which is a good approximation when B is small, we can define the Lande g-factor
to be gJ = 1 + (j(j + 1) + s(s + 1) − l(l + 1))/2j(j + 1).
Adding in the interaction with the nucleus gives a new g
factor, gF , which is defined as:
gF = gJ
f (f + 1) + j(j + 1) − i(i + 1)
.
2f (f + 1)
THEORY
The concepts behind optical pumping arise from
quantum mechanical perturbation theory.1,2 In addition
to the spin-orbit coupling, the interaction of an electron’s
intrinsic spin and its orbital angular momentum, and
the spin-spin coupling, the interaction between the
(3)
Both of the above g factors along with the following
interaction energies can be determined from either the
vector or matrix models of quantum mechanics.
When taking into account both the electronic and nuclear magnetic dipole moments, and the magnetic field is
small, we can define the interaction energy as:
W = gF µ0 BM,
II.
(1)
(4)
where M is the component of electron spin along the B
field and µ0 is the Bohr magneton. Equation 4 is by
far the most useful equation for quantifying the linear
Zeeman effect. Suppose we take a sample of rubidium
gas and impinge on it a certain frequency of light, as
described in Section III, then by the Planck-Einstein relation the interaction energy is simply proportional to
2
the frequency, ν. Thus, one can determine the g factors
and finally the nuclear spin quantum number.
In the case where the external magnetic field is strong
equation 2 can be diagonalized to retrieve the Breit-Rabi
equation:
4W
µI
4W
4M
− BM ±
[1 +
x + x2 ]1/2 .
2(2I + 1)
I
2
2I + 1
(5)
Here we define 4W to be the interaction energy due to
µ0 B
µI
.
hyperfine splitting, x = (gJ − gI ) 4W
and gI = − Iµ
0
The quadratic Zeeman effect splits the energy levels into
2F+1 sublevels with unequal spacing, where F is the
atoms total angular momentum of the atom. In addition to the energies involved there are various selection
rules that define the different allowed transitions between
energy levels. In optical pumping experiments we exploit
the 4M = ±1 selection rule.
W =−
III.
METHODS
To observe optical pumping we use an optical pumping device supplied by TeachSpin, Inc. A rubidium
discharge cell containing equal amounts of Rb85 and
Rb87 and Xenon buffer gas is heated to 323K. This
gas is simultaneously excited an oscillatory frequency
of 100Mhz. This allows for the gas to ionize enabling
electrons to excite rubidium atoms by collisions. This
produces two resonance lines at 780nm and 795nm.
The resonance light then passes through an interference
filter that removes the 780nm light. An interference
filter is an optical device similar to a low pass or high
pass filter built in such a way that it blocks off all light
above and below the desired frequency. The light is then
right handed circularly polarized through the use of a
linear polarizer and a quarter-wave plate. Right handed
circularly polarized light forces 4M = +1 meaning that
there will exist an energy that that cannot absorb light.
After the light has been circularly polarized it enters
the rubidium absorption cell. The cell is surrounded by
two Helmoltz coils. One coil produces a horizontal magnetic field and the other produces a vertical magnetic
field. These are necessary to cancel out the vertical and
horizontal parts of the Earth’s magnetic field. We orient
the pumping device along the third component of the
Earth’s magnetic field. The magnetic field, in units of
gauss, can be measured by the equation for the magnetic
field in a Helmholtz coil:
8.991 ∗ 10−3 N I
,
(6)
R̄
where N is the number of turns in the coil, I is the current, and R̄ is the mean coil radius. An RF magnetic
field is used to examine both the linear and quadratic
Zeeman effects. After passing through the absorption
cell the transmitted light is collected by a photodiode
which converts the intensity of light to a voltage.
B=
FIG. 1. A schematic of the optical pumping equipment with
all parts labeled.
We conducted two experiments using the optical
pumping apparatus. The first experiment we conducted
was used to determine the nuclear spin quantum number.
To accomplish this, we first determine at which current
the zero field transition occurs. We can calculate the
magnetic field produced by this current using equation
6. This is numerically equivalent to the Earth’s magnetic
field in that direction. The RF frequency is then varied
from 40kHz to 100kHz and the currents where absorption increases are recorded. The g factors, and therefore
the nuclear spin quantum number, are calculated from
Equation 4.
IV.
RESULTS AND DISCUSSION
We find that the ratio of g-factors is 1.49.
The calculated g factor for Rubidium 87 is
gF = 0.0171 ± 7.0506 ∗ 10−5 .
The calculated g
factor for Rubidium 85 is gf = 0.0114 ± 1.0552 ∗ 10−5 .
The g factors were calculated from the slope of the
plot in Figure 2 of the frequency versus the magnetic
field. This slope The magnetic field is defined to be the
magnetic field where resonance is determined to occur
minus the zero field. These values are approximately 30
times less than the expected gf values of 12 for Rubidium
87 and 31 for Rubidium 85. Through the use of equation
3 and considering that J = 12 we find that the nuclear
spin quantum number, I, for Rubidium 87 is I = 57.98.
Likewise for Rubidium 85, I = 87.22. These values are
about a factor of 30 off from the accepted theoretical
values of 23 and 52 respectively.
The large variation from the theoretical values is difficult to pinpoint. As stated in the previous paragraph
the ratio of the g-factors was 1.49. This is within 0.01
of the theoretical value of the ratio. This leads us to believe that a coil was connectedthat should not have been.
If an unwanted coil was on this would mean that the
cancellation of Earth’s magnetic field may not have been
calibrated correctly leading to the incorrect slope values
but a correct ratio.
3
V.
CONCLUSIONS
Optical pumping is a unique and novel way to determine small magnetic fields. Through our understanding
of optical pumping we were able to determine the quantum numbers of Rubidium atoms which teaches us much
about the atom. Our results confirm theoretical predictions on the ratio of g-factors but leaves up to debate the
correct values.
ACKNOWLEDGMENTS
The authors would like to thank the USD Physics and
Biophysics department for providing the optical pumping
equipment.
FIG. 2. Plot of resonant frequency versus the magnetic field
of transition.
1 Optical
Pumping of Rubidium OP1-A. (TeachSpin Inc., Buffalo,2002).
2 D.J. Griffiths. Introduction to Quantum Mechanics. (Pearson Education Inc., Upper Saddle River, 2005). Annalen der Physik,
322(10):891921, 1905.