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\title{Optical Pumping of Rubidium Vapor}
\author{}
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\maketitle
\section{Introduction}
When irradiated with right circularly polarized light, rubidium vapor is forced into higher magnetic substates because the polarization of the light causes transitions upward within the magnetic substates and natural transitions between levels follow a evenly spread Boltzmann distribution, yielding an overall bias into the higher magnetic substates. Spontaneous transitions between magnetic substates occur very infrequently because they are such low energy transitions. By subjecting the vapor to RF waves with energies in the range of the Zeeman splitting, one can attain resonance of RF induced transitions. By measuring this resonance frequency, a value for the energy differences between magnetic substates can be determined.
\section{Procedures}
We measure the variation in resonance frequency as a function of variation in magnetic fields that the vapor is subjected to. Given the value for which the resonance is at a minimum, we can take that magnetic field to be the magnetic field that cancels out the earth's magnetic field.
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Given this, we can than take detailed measurements of the value of the resonance frequency at some arbitrary magnetic field and then from there determine the energy of Zeeman splitting (it will equal the energy of the resonance RF photons) and given the resonances at different fields you can find the level of Zeeman splitting as a funciton of magnetic field.
\section{Analysis}
By determining the coil currents that yielded minimum values for the resonance (Zeeman) energy this yeilds the magnetic field which cancels the Earth's field. Our values for these were as follows. In the $z$ direction (downstream), the field was .154 gauss and in the $x$ direction, the field was .475 gauss. The field in the $y$ direction was extraordinarily small ($< .04$ gauss) and treated as 0.
Then measuring the values of the resonance frequency at a non-zero magnetic field, the energy difference between the magnetic substates can be determined as a function of the magnetic field. With a magnetic field in the Z direction of .554 gauss (in the opposite direction of the earth's field), the energies of the resonances were $6.46 x 10^{-22}$ ergs and $9.59 x 10^{-22}$ ergs.
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With a magnetic field in the Z direction of .511 gauss, the energies of the resonances were $6.01 x 10^{-22}$ ergs $8.81 x 10^{-22}$.
This indicates a Zeeman splitting energy of $10^-21 \frac{ergs}{gauss}$ for the first isotope ($Rb_85$) and $1.8 x 10^{-21} \frac{ergs}{gauss}$ for the second ($Rb_{87}$).
\section{Conclusion}
It slopes in the right direction. As magnetic field increases, the energy difference between magnetic substates increases. Additionally, near zero magnetic field the Zeeman splitting virtually disappears.
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