User:Boleszek/Notebook/Physics 307l, Junior Lab, Boleszek/2008/11/24
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ESRSJK Incomplete Feedback Notice
The purpose of this experiment is to measure the periodic spin flips of the electrons in a sample of DPPH that is exposed to an RF radiation source and an oscillating magnetic field in order to calculate the electrons' intrinsic g factor.
A magnetic field applied to an electron orbiting an atom will lift the energy degeneracy that the electron usually has in any given orbit. Spin resonance only occurs when the difference between an electron's spin up and spin down magnetic orientational energies equals the energy of an incident photon (hν = gsμBB). This photon, when absorbed by an electron with the lower energy (Eo + gsμBB / 2 is the lower because E is negative), will provide just enough energy for the electron to flip its spin and exist at the higher energy. In practice, though, it is very difficult to tune the photon to just the right frequency (which happens to be in the range of MHz because of the numerical values associated with electron energies and dipole magnitudes) so instead we set the desired frequency f such that hf is reasonably close to the energy difference and then let the energy difference "tune into" the set frequency by modulating the B-field. If we do this periodically we will flip the spins at twice the frequency of the applied B-field (once on the way up, once on the way down). This change in spin corresponds to a change in energy, which corresponds to a change in the permeability of the sample. Since the sample basically fills the inside of the coils this change in permeability corresponds directly to a change in the magnetic inductance of of the coil. It took me a while to realize this, but it is not the change in spin that induces a current in the coil, for this change is very brief and is not the constant change in B-field required for magnetic inductance to occur. Rather, the RF frequency, which excites the electrons, also constantly induces a current in the coil proportional to the number of coils. We notice a short change in the current signal when the coil inductance changes for the near instant that all the lower energy electrons are resonantly flipped into the higher energy. The RF frequency, therefore plays two important roles 1) exciting electrons into resonance, 2) allowing us to see the blips by producing a constant AC signal in the coil.
Upon our approach to ESR table we encounter a fully setup experiment, but nonetheless read the manual to see if the circuitry is correct. The Helmholtz coils are connected in parallel and require a small AC current superimposed on a larger DC current so that the magnetic field oscillates without reversing its direction. This is supplied by the Variac (set at 20 V)/small transformer and the Soar DC power supply (set at 1 A), respectively, which are connected in parallel, with the 1000 μF capacitor isolating the AC from the DC to prevent wave distortion. Our ESR apparatus is connected to the ESR adaptor which is supplied with 12V and is connected. We began fumbling with the oscilloscope controls and the RF and found that the resonance signal is very noisy and only on the order of a few hundred Hz. It should be clear and on the order of a few kHz. After discussing with Aram we realized that the oscilloscope display does not read the correct frequency, which is why there was a multimeter connected to the ESR adaptor. After switching the multimeter to frequency display we saw the correct kHz range. After this development we began making measurements.
We find that upon turning on the instruments the resonnance signal is very noisy. We change the DC current, modulate the phase and turn down the current output knob on the ESR apparatus to arrive at our desired signal. We make measurements on small, medium, and large field-inducing coils and decide to proceed in 5 kHz steps for higher resolution data.
I want to
To do this I recall the resonance condition
B is found from the Biot-Savart Law. Since our point of interest is halfway between the coils (x = R/2 if they are a distance R apart) it follows that
This is then simplified to
This derivation can be found on Wikipedia's page for Helmholtz coils and any basic physics textbook.
It is quite obvious that the calculation of gs is nothing but a matter of division.
Note: Since the measurements for the medium coil done on Day1 are close to those done on Day2 I just use the Day 2 data and therefore my errors we be calculated directly from a statistical analysis on the g factors from each data pair instead of propagated from the current measurement errors. If I were to redo this lab I would, however, take multiple measurement runs for each coil.
Individual Data Point Analysis
After rechecking my work a few times I confidently say that, although our accuracy is far off, we had good consistency in our results and thus a large systematic error must have influenced the measurements. As compared to the accepted gs2 my results are noticeably off. I provide here the MATLAB calculations for each data set:
SJK 17:20, 18 December 2008 (EST)
The sdm's were calculated using the by now familiar formula:
Linear Least Squares Fits
I performed linear least square fits on each of the three data sets using the formula:
so that the g factor is immediately obtained from the slope.
I did not constrain the graphs to go through the origin since the RF frequency and the B-field magnitude are not coupled (a change in one does not automatically create a change in the other like charge and speed do in the Millikan, for instance).
Since the probe unit is made of metal and part of it must protrude into the B-field region so that the sample can be just in the middle of the Helmholtz coils the electrons in the probe unit will oscillate to the B-field, radiating a different frequency of EM radiation than RF. This also induces a current in the probe coil resulting in AC interference and possibly a higher magnitude current. If this is the case, then I would expect that the g-factor should be a little smaller than the accepted value since it is inversely proportional to current.
I think that the signal from the small coil is very noisy for a few reasons:
After rereading the procedure I also found out that we were supposed to make sure that the Helmholtz coils were separated by a distance equal to their radius. We did not make sure that this was the case and now I'm pretty sure we had them further apart because the manual says this distance results in the coils flush against the probe unit. The importance of correct positioning is so that we can apply the equation derived for this specific case (namely the x=R/2). I believe this is the gravest error we made in our proceedings (other than exploding two capacitors, of course) and is the main reason why my values are all around 0.85 instead of 2.