This document summarizes an experiment that uses optical pumping to measure the nuclear spins and energy levels of rubidium-85 and rubidium-87 isotopes. Helmholtz coils are used to apply a magnetic field to samples, inducing Zeeman splitting of the energy levels. Resonance frequencies are measured and analyzed using the Breit-Rabi equation to determine the nuclear spins match predicted values of 5/2 for Rb-85 and 3/2 for Rb-87. The earth's magnetic field is also measured and found to be consistent with known values. Sources of error are discussed and results show strong agreement with theoretical predictions.

1. Optical Pumping
Alexander Barriga
Abstract
The resonance frequencies of rubidium isotopes 85 and 87 are used to excite the atoms from the ground state to
an excited state by the method of optical pumping. The nuclear spins and energy levels of both isotopes, and the
earth’s magnetic field is measured. Helmholtz coils are used to pass an applied magnetic field through samples
in order to induce Zeeman splitting of the energy levels.
Keywords
Optical Pumping — Zeemen Splitting — Breit-Rabi
Department of Physics, University of California, Berkeley, United States
*Author: alexander physics@berkeley.edu
Contents
Introduction 1
Theory 1
Apperatus & Proceedure 1
Analysis 2
Error Analysis 3
Conclusion 3
Introduction
Optical pumping is a classic experiment designed to measure
the nuclear spins of atomic nuclei. The solid rubidium sam-
ples are heated into a gas state and then a Helmholtz magnetic
field is generated by the Helmholtz coils passes through the
sample. This system is understood by basic quantum me-
chanics concepts: energy difference between discrete levels,
energy level selection rules, energy level shifts (Zeeman Ef-
fect), and probability of excitation into a higher energy level.
The experiment is conducted by using standard physics lab-
oratory equipment: oscilloscope, voltage and current meter,
signal generator, and basic circuitry.
Theory
This experiment takes measurements of the familiar physi-
cal quantities frequency and voltage. In this lab, we take
these physicals quantities and analyse them the Breit-Rabi
law, which relates frequency to the total external magnetic
field of the sample.
ν
BExt
=
2.799
2I +1
(1)
Equation (1) contains the nuclear spin, however this quantity
cannot be directly measured. So instead, the Breit-Rabi law is
manipulated by substituting the total magnetic field equation
which contains all the different field sources, one of them
being the Helmholtz field whose expression is provided below.
BT otal = BEarth +BHelm (2)
BHlem = 0.9×10−6 NI
a
(3)
Apperatus & Proceedure
The apparatus of this experiment consist of a metal box that
houses the rubidium samples. It has a heating coil inside
and polarized monochromatic light is shined into the box
by a bulb. The Helmholtz coils’s current is controlled by
the users. A photo diode inside the box is used to detect
the energy transition of the rubidium samples. The photo
diode is connect to the oscilloscope which is where the users
observe the sample’s emitted light. The detection of the energy
transitions is done by scanning across a range of frequencies.
Once a figure eight is detected on the oscilloscope, we know
that the resonance frequency has been reached. The current
is then changed and the frequency scanned; the search for
the figure eight on the scope is repeated. This is done for
current values between 0.0 amps and 2.0 amps in intervals of
0.2 amps. First the current scan is done upward from 0.0 amps
to 2.0 amps then, to check for hysteresis effects, a downward
scan is performed: from 2.0 amps back down to 0.0 amps.
Once consecutive upward and downward scans have been
been completed, the polarity of the applied magnetic field is
reversed and the procedure is repeated. Once this procedure

2. Optical Pumping — 2/4
has been completed for the first rubidium isotope, the whole
procedure is repeated for the other isotope.
There are two know sources of error in this experiment. First,
the optical signal from the isotopes is damped by high tem-
peratures. The optimum temperature for the Helmholtz coils
is around 38◦. Whenever the temperature exceeds this value,
the applied magnetic field is switched off and allowed to cool.
Second, the isotope samples produce a well understood hys-
teresis effect when it’s internal magnetization is influenced by
an external magnetic field. Hence the need to complete a full
upward frequency scan followed by a complete downward
frequency scan.
Analysis
The goal is to experimentally verify the nuclear spin of Ru-
bidium isotopes 85 and 87 as 5/2 and 3/2, respectively. Data
for Rb87 is collected using bulb M1, which contains the Rb87
sample. Data for Rb87 is collected using bulb M10, which
contains the Rb85 sample.
The Breit-Rabi equation is used to find a ratio between the
resonance frequencies of both isotopes at the same current.
Selecting resonance frequency and voltage values from the
data, we see that the ratio is
ν85
ν87
=
2I87 +1
2I85 +1
(4)
ν85
ν87
=
3
2
(5)
This expression however is not enough information to identify
the nuclear spins of the isotopes. For instance, although I85 =
5
2 and I87 = 3
2 satisfy the ratio, so do I85 = 7
2 and I87 = 11
2 .
Since the each nuclear spin is expected to be a half-integral
value, equation 1 does not uniquely determine the spin of
Rb85 and RB87.
Using equations (1),(2), and (3), we can derive the following
equation which is the primary equation used with the data.
Notice that equation (6) is linear and that resonance frequency
is a function of the current. This enables us to equate the
slopes of the plots to the expression that are to the left of i,
which contain the factor N
a , and equate the intercepts of the
plots to the expression that contains the earth’s magnetic field,
BE.
ν = ν(i) = [(0.9·10−6 N
a )(2.799
2I+1 )]i+(2.799
2I+1 )BE (6)
Table 2 shows that the experimental isotope atomic spins are
in agreement with the predicted theoretical values. While
a.
b.
c.
d.
Figure 1. Plot a displays the upward scan for isotope 85 and
plot b display the downward scan for isotope 85. Plot c
displays the upward scan for isotope 87 and plot d display the
downward scan for isotope 87.

3. Optical Pumping — 3/4
Isotope Slope(MHz
amp ) Atomic Number
Rb85 up scan 2.0251±0.0136 2.5532±0.0206
Rb85 down scan 2.0340±0.0203 2.5399±0.0303
Rb87 up scan 3.0238±0.0034 1.5448±0.0023
Rb87 down scan 3.0264±0.0035 1.5431±0.0024
Table 1. Experimental values of Rubidium isotope atomic
spins calculated by using the slope term in equation (6).
table 3 shows that the experimentally determined value of the
earth’s magnetic field is in agreement with the known value at
the equator, 3.10 G.
Isotope Intercept(MHz) Earth’s Field(G)
Rb85 up scan −0.1403±0.0164 0.3062±0.0349
Rb85 down scan −0.1511±0.0250 0.3283±0.0528
Rb87 up scan −0.2288±0.0041 0.3343±0.0058
Rb87 down scan −0.2301±0.0042 0.3360±0.0060
Table 2. Experimental values of the earth’s magnetic field
calculated by using the intercept term in equation (6).
Error Analysis
A simple diagnostic plotting (in the R Studio statistical soft-
ware program) of the data shows that it is very accurately
described by a least squared regression model. An unweighed
least squares approach is chosen because the error in res-
onance frequency measurements are all given by the same
standard deviation: σf req = 0.01043631MHz. Several mea-
surements were taken for the same resonance frequency value–
σf req is the same for both isotopes. Controls on the experi-
ment ensured that the only accountable uncertainty in mea-
surement was the hysteresis effect from the magnetization of
the rubidium samples and the temperature fluctuations. Fur-
thermore, vertical error bars were not included in the linear
plots of the data because σf req is small and not visual on the
same scale as the data.
The method that was used to locate the standard deviations of
the isotopes and of the earth’s magnetic field was the propa-
gation of error. First, the σ of the slope is given by R Studio
in the regression summary. Then slope±σ are individually
plugged into the Breit-Rabit equation and the Iright and Ile f t
atomic spins are solved for, the absolute value of their differ-
ence divided by 2 gives σI for each isotope. The same method
is used to find the σB values for the earth’s magnetic field
values.
The results from the fit summary of the upward frequency
scan for Rb85 give a goodness of fit value of R2 = 0.999,
which is unsurprising and expected in a physics experiment
like Optical Pumping in which controls for error are made
and well thought out in advance. The P-value for the linear
fit is reported as 2.2·10−6. Such a low P-value makes sense
for several reasons. First, since it’s inception, Quantum Me-
chanics has proven to be a highly reliable scientific theory that
produces results that strongly agree with theoretical predic-
tions. Second, highly precise instruments were used to take
measurements and, three, factors that tend to produce errors–
like temperature fluctuations and hysteresis were taken into
account.These results are characteristic of all isotope data.
Conclusion
The goal of this experiment is to obtain experimental verifica-
tion of two different types of physical parameters: the nuclear
spin of Rb85 and Rb87 isotopes and the earth’s magnetic field.
The isotope’s experimental values are listed in table 1 and are
certain up to the tenths place. The experimental values for
the earth’s magnetic field are also certain up the tenths place.
Although the earth’s magnetic field strength at 50◦ latitude is
about 5.0G, the experimental values are on the same order of
magnitude and nevertheless accurate for this type of measure-
ment. The Breit-Rabi equation and the linear plots seem to
predict that a linear relationship can be extrapolated into even
higher resonance frequencies. Although this is undoubtedly
true for sufficiently short extrapolations, one must be careful
with these kind of inferences. The Breit-Rabit accounts for
the relationship between frequency and current to a 1st order
approximation. Although 1st order approximation are often
surprising accurate (as they are in this experiment) they are
invariable limited to a short range of values.