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Determination of Hyper fine Splitting in Potassium through Spin Noise Spectroscopy
Christopher McDonough
Methods of Experimental Physics II
Introduction
The hyper fine structure of alkali atoms such as 41K was seen by Albert Abraham Michelson in 1881, and first explained by Wolfgang Pauli in 1924 by assigning a nuclear magnetic moment also known as spin to the atom. In a collection of 39K or 41K atoms the most populated state is the 4S1/2 state. In this state the magnetic moment associated with the electron traveling around the nucleus is zero so the only magnetic moments present are the spin of the nucleus and the spin of the electron. This leads to two different states one state associated with the two spins being aligned and the other being associated with the two spins being anti aligned. Applying a constant magnetic field causes new energy levels to arise from these different states and by measuring these energy levels relative to each other at different magnetic fields the magnetic dipole of the atom can be determined.
Traditionally absorption spectroscopy is used to measure the energy levels inside of atoms, instead this experiment uses a different optical technique Faraday rotation. The main benefit to using Faraday rotation instead of absorption is that you are perturbing the system less since the atoms aren't absorbing all the energy from the photons. Faraday rotation is magneto-optical effect discovered by Micheal Faraday in 1845. This effect causes the polarization of light to rotate when it's exposed to a magnetic field along its direction of propagation.
The magnetic field inside a sample of Potassium is determined from the net spin of that sample. In thermal equilibrium each spin state should be equally populated so the net spin spin will be zero however fluctuations will be observed resulting in a net spin and therefore a Faraday rotation proportional to that net spin. When applying a constant magnetic field the spins in the sample will precess with a frequency given by the transitions between hyper fine energy levels. The Faraday rotation signal will therefore also depend on these frequencies so taking the Fourier transform of this signal will give the frequencies.
Theory
Spin noise spectroscopy is a technique which observers transitions in the hyper fine structure by observing the net spin of a collection of atoms through Faraday rotation (angle of polarization of light proportional to net spin) and then applying a magnetic field to the sample causing those spins to percess with a frequency given by the transitions in the hyper fine structure. This is superior to absorption spectroscopy because it allows measurements to be made at frequencies farther away from resonance meaning you perturbing the system less.
The frequencies can be determined using the Breit Rabi Equation and the frequencies are transitions with Δmf=1
ahf is the hyper fine constant, μB is Bohr magneton, gs and gi are the electron and nuclear g factors.
Apparatus
Faraday Rotation induced by sample on laser detected using optical bridge which creates electric signal proportional to rotation.
The hyper fine structure of the 39K and 41K was probed by observing the precession of their spins about a constant magnetic field using Faraday rotation. The hyper fine magnetic dipole constant was determined by preforming a non linear fit using the Breit Rabi equation and the data for the transition frequency vs magnetic field. These values obtained are all within $4\sigma$ of the literature value. The calculated values can be seen in the Table above and are accurate up to 3 significant figures to the literature however this experiment could give up to five significant figures. The main cause of error is from the drift in magnetic field as the run is taking place. The power supply attached to the Helmholtz coil responsible for the constant magnetic field would change it’s current output by $1mA$ over the course of a run ~1min. Also as can be seen on the example of alternating magnetic field data, the alternating field is having no effect on the peaks. This would imply there is something wrong with the coil which should be looked into.
Transition frequency fitted using Breit Rabi Equation for 41K
Conclusion
Transition frequency fitted using Breit Rabi Equation for 39K
Two runs one with the alternating magnetic field on (in the red) and one without the alternating magnetic field on (in the black). No stark effects were observed.
An example of a typical run with 39K
Runs were then done with a sample of pure 41K at different magnetic fields. Then trials were done with an alternating magnetic field and an example is shown below.
Oven designed to reach 200C to make K into a vapor.
Vertical Helmholtz designed to have an impedance of 50Ω at 10MHz to ensure proper power.
Data
Initially trails were performed with natural potassium which is 93.3% 39K and 6.7% 41K so it was treated as a sample of 39K. Trails were preformed from 8G all the way to 90G. A sampling frequency of 500MHz was used. An example of a typical run is shown below.
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