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I. Optical Pumping

III. Error Analysis Notes

IV. Data Analysis (OPT)

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Contents 1 Overview 2 Nuclear Spins 3 Earth’s Magnetic Field 4 Errors in the Field

Overview

The purposes of the experiment are

1. to determine the nuclear spins of the two isotopes 85 and 87. It is known that both spins are odd half-integral values, like 1/2, 3/2, 5/2, etc., and
2. to measure Be, the earth’s magnetic field

We have two isotopes of rubidium as gas in a bulb placed in a magnetic field and subject to electromagnetic radiation. The earth’s magnetic field also affects the rubidium energy levels and their populations. We set the current in the coil producing the magnetic field, and adjust the radio frequency to produce a resonance condition. The parameters that we either want to change, vary, or to measure are the current, the field produced by the current, and the frequency of the applied RF radiation at resonance. The relevant equations are:

B_H, magnetic field from the Helmholtz coils

i, current in the coils

a, radius of the coils

The value 0.9 is approximate because the radius a is not exactly the same for each of the N windings and we have no simpler way of incorporating this fact.

ν, frequency of applied em radiation

I, nuclear spin of Rb

B = B_H + B_e, the total magnetic field from the Helmholtz coil and the earth

This equation is called the “Breit-Rabi” [bright-robby] equation.

Nuclear Spins

How and what data should you take? To determine the nuclear spins, rearrange the equation to get an equation that looks like y = mx + b. Then set a value of the current i, adjust the RF to the resonance value, reset i, etc., until you have a table of pairs of values. In fact, you will end up with four tables, because you should take data for both positive and negative currents for each of two isotopes. A plot of resonance frequency vs. current will give a straight line with a slope dependent on 2I + 1, and an intercept dependent only on the earth’s field. [Sketch 4 straight lines]

Another way of determining the spins is to set the current and take the ratio of the resonance frequencies for the two isotopes, at the same current.

In both methods there will be some errors, but it does not matter because we know that the spins have exact half-integral values. This is a case in which the use of error analysis is unwarranted because the results are unambiguous.

Earth’s Magnetic Field

To determine the value of the earth’s magnetic field, we need to find a way to determine B_H exactly, or to eliminate it from the equations. B_H can be eliminated by setting i = 0 and using the frequency intercepts and the Breit-Rabi equation. Recall that the equation for B_H as a function of i is not exact.

There is yet another way to determine the earth’s field. There is a reversing switch on the coil current, to reverse the direction of the field. For the same absolute value of the current, the field must have the same strength, except for possible hysteresis effects. Check for hysteresis by measuring the resonant frequency at a particular current, when the value of current is reached from above (run the current to a max, and then come down to the desired value), and from below (run the current to zero, and then come up to the desired value).

So, at a particular current, measure the resonance frequencies for both positive and negative values of current. Add and subtract the two equations, and get exact expressions for B_e and for the more exact parameter for the relation between current and B_H.

Errors in the Field

How do we treat the errors in the value of the earth’s field? Step back a little, and see what data you have and how you can arrange it to make error computation the simplest. Here are several approaches.

1. Look at plots of your data. You should have 4 lines, two for each isotope, one with positive current and one with negative current. You can do a least-square fit of each line, calculate the position of the zero-current intercepts, and obtain values for the field. Then you can calculate “errors of adjusted coefficients” using the methods given in Lyons and other references.
2. But, by changing negative current points to negative frequencies as well, each isotope has only one plot line and presumably the zero crossing is more accurate. Then there are only two lines to fit, one for each isotope, and the errors can be calculated as described above for adjusted coefficients.
3. A still easier method is to use pairs of points with the same absolute values of current, and add the two resulting equations. We then have values of the earth’s field without adjusting any coefficients, and the errors can be calculated as an error of a single parameter, rather than as an error of an adjusted coefficient. We have

,

,

The standard deviation is calculated as described above in Error of a Single Parameter: Measurement Statistics. Compute the average; compute the differences, square them, add, divide by N(N - 1) and take the square root. Then step back, look at the results, and see if they look reasonable. If not, you’ve goofed somewhere, and must try again.