# Physics307L F09:People/McCoy/ESR

## Background

For the lab covering the dates November 24-December 6, 2008, I chose to work with Boleszek on the electron spin resonance experiment. The objective of the lab was to measure the g-factor for the intrinsic spin of the electron. In the intrinsic spin of the electron, there is a slight energy difference between the spin-up and spin down states. Because the difference is related only to the magnetic moment of the electron and the strength of the magnetic field, the g-factor is the proportionality constant for the direct relationship. In the lab, we used Helmholtz coils to create the magnetic field and had a copper coil so that we could measure the energy through the transmission of free electrons interacting with photons emitted by the Diphenyl-Picryl-Hydrazyl (DPPH) sample.

## Data

To take the data we used the medium coil first, followed by the small coil then the large coil and went from the highest value of the frequency in each set to the smallest value that could be distinguished, going by steps of 5MHz on multiples of 5, with the exception of the large coil which we went by 4 on because going by 5 did not allow for us to take 5 data points with that coil. The data that we took can be found in Boleszek's lab notebook that can be accessed here

Having taken the data, I did the numerical analysis using Microsoft Excel 2007, and calculated the value of the g-factor for electron spin, coming out with a value ${\displaystyle g_{s}=.8498(72)}$. This value is extremely low when compared to the accepted value of 2.0023, so the assumption that I have made is that there was a large systematic error. My analysis and calculations can be found in my lab notebook, with that entry located here

## Results

My calculated value of g_s for the electron is ${\displaystyle g_{s}=.8498(72)}$.

As I previously stated, the calculated value of the g-factor for electron spin was significantly far off the calculated value and error. To be more specific, the calculated value lies approximately 160 times the standard error below the accepted value. Having a result this far off confused me greatly, causing me to go back and re-evaluate all my calculations, making sure all my formulas and numbers were correct, and to the best of my knowledge there were no errors in my calculations.

This extreme difference is very noteworthy, because as the g-factor is a natural constant that classically should only be 1, rather than it's value of 2 for the electron, or 2.8 for the proton, generating a value that is within 2 standard deviations of the classically expected value could mean that the formulae were incorrect for use in quantum mechanical objects, but having checked my modern physics text, the formulae were the same as listed in the book.

For the error in my final answer, I used the standard error in the calculated g for all 20 data points. I used this margin for the error, because there was no statistical error in the apparatus, nor was there any given error in any of the instrumentation. By using the standard error of the mean, my error is affected most by the values of g as taken using the small coil, as in that coil, the values of the calculated g were significantly lower than any with the medium or large coils. On the other extreme, the large coil had larger values for two of the five data points and had the least gaussian distribution.

## Conclusions

### Results

The most likely conclusion that can be drawn from my result is that there was a large source of systematic error that resulted in my calculated value being focused around the value of .85 instead of the actual value of approximately 2. This systematic error will be discussed in the next section, but because the calculated values for each individual portion of the experiment were similar, with only the final value being significantly far from the accepted value, it is evident that there was not a random error that resulted in my values being so low. The other possibly conclusion from my results is that either my calculations were incorrect or that there is another leading constant that multiplies with the energy of the photon released in the spin flip process that would increase the value of the calculated g-factor so that it is more accurate.

### Errors

SJK 17:43, 18 December 2008 (EST)
17:43, 18 December 2008 (EST)
This is a really good discussion. Something occurred to me while reading it: Perhaps one of the helmholtz coils is broken, so the field is only half of the expected? I can't remember how their wired...if in series, then this would not be able to explain it (the field would be zero)...but if in parallel, then this could explain things.

Actually, I'm looking at the diagram, and something definitely seems fishy about wiring them in parallel and measuring the current that way...definitely this is something to think about.

The primary error that would occur with this experiment is the apparatus. The experiment had a very complicated and confusing apparatus, such that it took us the first 2+ hours on the first day to figure out what everything was doing. Along with the complicated apparatus is the first possible systematic error, being that there was a connection that was incorrect that caused the current to bypass the variable resistor, or some other component, such that the measured value of the current/frequency was not accurate.

The second error that could have resulted with the apparatus is the spacing of the Helmholtz coils. As the coils need to be spaced such that the distance between them is equal to that of their radius, and as they were free to move, because they did not have a fixed stand, any slight rotation of one of the coils, or of the ESR apparatus would put it in a portion of the magnetic field that was not exactly uniform and that has a different value than what we calculated.

A third error could be that as the plastic tube containing the DPPH sample was broken on one end, air could have came in and contaminated or partially oxidized the sample, such that it would have slightly different characteristics that may result in there being free electrons that are more apt to absorb energy and then release it as a photon, or there could have been deflection of the photons off free electrons that decreased the photon energy, causing the frequencies to be lower than the correct value for each value of the current.

A final error that may have affected the experiment could have been the alignment of the signals using the oscilloscope, as the large amount of noise in the signal from the ESR apparatus had indistinct peaks that changed at a rapid rate, that would have randomly altered the data points and increased the spread between each measurement. This I feel is quite unlikely as the measured results were relatively constant, meaning that there was not a major source of random error.

The first three stated errors would be quite easy to fix though, because as the first two are related to the set-up, more detailed instructions for the connections would allow for us to ensure that the apparatus is set-up correctly. For the ESR apparatus, the simplest correction would be to have bracers that will lock the apparatus into the proper positioning, along with keeping the Helmholtz coils stable so that they remain parallel with the sample properly centered. To correct any possible error related to the sample, the simplest remedy would be to purchase a new sample so as to see if there was any error that came into being because of the broken casing for the sample.