User:Boleszek/Notebook/Physics 307l, Junior Lab, Boleszek/2008/11/24

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Equipment

 * type w5mt3 variac autotransformer
 * fluke 111 true rms multimeter
 * homemade transformer
 * homemade phase shifter
 * soar model ps-3630 dc power supply
 * 2x's leybold didactic gmbh helmholtz coils n=320 diam=.15m
 * 514 55 esr-grundgerat esr basic unit
 * tektronix tds 1012 oscilloscope
 * 514 56 esr adapter
 * HP 721A power supply x2
 * dpph electron source

Purpose
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.

Theory
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 ($$ h\nu = g_s \mu_B B $$). This photon, when absorbed by an electron with the lower energy ($$ E_o + g_s \mu_B B / 2 $$ 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. In our setup we use a DPPH (Diphenyl-Picryl-Hydrazyl) sample because it has only 1 unpaired electron and exhibits the simplifying properties of 0 net orbital angular momentum and only one resonance point.

Initial Proceedings
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.

Day2
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.
 * Medium


 * When we switched the coil from the medium to small coil, the current readings initially were extremely high, and the voltages were greater than 20 volts from the DC power source. Doing so, we blew two capacitors and the fuse in the multimeter, requiring for us to get a different fuse and capacitor to put in the circuit.


 * Small


 * The signals with the small coil are extremely noisy, and the measured frequency does not stabilize enough to get an accurate reading for all the data points. Because of this we stopped at 5 data points for the small coil, as with values greater than 90kHz the current is greater than 2 Amps, which exceeds the maximum value on the fuse for the multimeter, and values below 65kHz do not settle.


 * Large


 * For the large coil we decided to take 4kHz steps as opposed to 5kHz steps so that we could get 5 distinct data points. Otherwise we either ran into the frequency limit of the RF generator or the current limit of the multimeter.

Calculations
I want to To do this I recall the resonance condition
 * 1) Calculate $$ g_s $$ for each data point
 * 2) Calculate $$ \sigma $$ and errors
 * 3) Calculate $$ g_s $$ from a linear least squares fit
 * $$ h\nu = g_s \mu_B B \,$$
 * Where:
 * $$\nu\;$$ = the RF frequency
 * $$B\;$$ = the field at resonance
 * $$\mu_B\;$$ = the Bohr magneton = $$e \hbar /2m_e$$ = 5.788 × 10^−9eV /G = 9.274 009 15(23) × 10-24 J•T-1

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


 * $$ B = \frac{\mu_0 n I R^2}{2(R^2+x^2)^{3/2}}=\frac{2\mu_0 n I R^2}{2(R^2+(R/2)^2)^{3/2}}$$


 * Where:


 * $$\mu_0\;= 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}$$
 * $$I\;$$ = coil current, in A
 * $$R\;$$ = coil radius, in m

This is then simplified to
 * $$ B = {\left ( \frac{4}{5} \right )}^{3/2} \frac{\mu_0 n I}{R}$$

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 $$ g_s$$ 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 $$g_s ≈ 2$$ my results are noticeably off. I provide here the MATLAB calculations for each data set:


 * SMALL
 * 1) 0.8203
 * 2) 0.8067
 * 3) 0.7958
 * 4) 0.8022
 * 5) 0.7943
 * mean=0.8039
 * sdm=0.0047


 * MEDIUM
 * 1) 0.8788
 * 2) 0.8695
 * 3) 0.8588
 * 4) 0.8607
 * 5) 0.8622
 * 6) 0.8399
 * 7) 0.8486
 * 8) 0.8502
 * 9) 0.8508
 * 10) 0.8517
 * mean=0.8571
 * sdm=0.0036


 * LARGE
 * 1) 0.8654
 * 2) 0.9233
 * 3) 0.8636
 * 4) 0.8498
 * 5) 0.8855
 * mean=0.8775
 * sdm=0.0128

$$g_s=0.84(70)\,$$
 * Average g

The sdm's were calculated using the by now familiar formula:
 * $$\sigma_m= \sqrt{\frac{1}{N*(N-1)} \sum_{i=1}^N (x_i - \overline{x})^2}$$



Linear Least Squares Fits
I performed linear least square fits on each of the three data sets using the formula:
 * $$ \frac{h \nu}{\mu_B}=g_s B$$

so that the g factor is immediately obtained from the slope. The slopes of the three graphs are Small: g_s = 0.9014 Medium: g_s = 0.8886 Large: g_s = 0.8821
 * Average: 0.8907

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).

Retrospective Thoughts
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.
 * One unavoidable source of systematic error

I think that the signal from the small coil is very noisy for a few reasons:
 * What's with the small coil signal?
 * 1) Fewer coils do not cover the sample completely, thus allowing more edge effects
 * 2) The manufacturers tried to stretch the smaller coil so that it would fully cover the sample (though it didn't really), but in doing so the coil became less of a tightly wound wire approximating consecutive circles. The more a coil is stretched, the less it inducts until finally, as a straight line, no induction occurs.
 * 3) After rereading the procedure in Dr. Gould's manual I found that we were supposed to set the AC Variax current source to 2 Volts, but we set it to 20. This large AC component may have introduced interference that couldn't be handled by the capacitor.

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.
 * OOOPS!!!

It has been my fortune to encounter a use of ESR in actual scientific practice during the course of this two week lab. In my thermodynamics class we read a paper written by Carl E. Wieman of UC Boulder who received the Nobel prize in 1995 for achieving a Bose-Einstein Condensate of super-cooled rubidium-87 atoms. The process of laser cooling (μK), though quite effective in reaching low temperatures fairly quickly by the trapping of atoms with low enough velocity to be stopped by the radiation pressure, could not cool the atoms below the condensation temperature of bosons with rubidium atom mass (100nK). Wieman had to rely on evaporation, much like the cooling of a cup of coffee, to get rid of the high energy electrons. Since the laser cooled atoms were already freezing, they would not effectively evaporate on their own so they had to be influenced in some way such that the atoms at the highest energies would be energetic enough to leap out of the trap, lowering the average energy of the trapped atoms. They gave the extra "kick" to the highest energy electrons but not the lower energy ones by selectively setting an applied magnetic field to the right magnitude such that an applied RF radiation would literally flip the high energy atoms out of the trap. Then the B-field was lowered to target the next highest, and so on until the atoms were cooled to the point of condensation. They used ESR as an effective "puncture hole" in the potential holding the atoms in place, allowing selected electrons to escape, thus becoming one of the first scientists to achieve a Bose Einstein Condensate with a material other than helium...and got the Nobel prize for it!
 * Real ESR


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