Users: Alexandra S. Andrego and Anastasia A. Ierides/Notebook/Physics 307L/2009/11/23

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Please note that this lab notebook page is the combined efforts of Alex Andrego and Anastasia A. Ierides
SJK 17:11, 18 December 2009 (EST)
17:11, 18 December 2009 (EST)
This is a spectacular primary lab notebook!!! Seriously, I love it and am amazed!
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My feedback is incomplete on this page for two reasons. First, the value of the feedback to the students is low, given that the course is over. Second, I'm running out of time to finish grading!

Purpose and Brief Description of Electron Spin Resonance

The purpose of this lab is to measure the g-factor, $\displaystyle g_s\,\!$ , of the electron using Electron Spin Resonance (a.k.a. Electron Paramagnetic Resonance). To do this we look for the spin-flip transition of an unpaired, free electron under the influence of a magnetic field by placing a sample of DPPH (Diphenyl-Picryl-Hydrazyl) whose total angular momentum is zero and has only one unpaired electron in this magnetic field. The electron itself has a magnetic dipole moment $\displaystyle \mu_s\,\!$ due to it's angular momentum/spin, which interacts with the uniform magnetic field, and orients itself in one of two ways, spin up or spin down. The equation of this relation is:
$\displaystyle \mu_s = −g_s \mu_B \frac{S}{\hbar}\,\!$
where:
$\displaystyle g_s\,\!$ is a constant characteristic of the electron, its intrinsic g-factor
$\displaystyle \mu_B=\frac{e \hbar}{2 m_e} \simeq 5.788\times10^{-5} eV/T\,\!$ is the Bohr magneton
$\displaystyle S\,\!$ is the spin of the electron
$\displaystyle \hbar = 6.582 \times 10^{−16} eV \cdot sec\,\!$ is Planck’s constant
The magnetic field interacts with the magnetic dipole moment of the electron to create an electric field given by the equation:
$\displaystyle E =- \mu_s \times B\,\!$
The two energies associated with the two different orientations of the electron are given by:
$\displaystyle E = E_0 \pm g_s \mu_B B\,\!$
where $\displaystyle E_0\,\!$ is the rest energy of the electron.

You can see a more detailed purpose and description in Professor Gold's Electron Spin Resonance Lab Manual.

Equipment

OUR COMPLETE SETUP
FLUKE MULTIMETER AND SPECIAL DUAL CONNECTOR
THE TRIPLE OUTPUT POWER SUPPLY
ESR ADAPTER FUNCTIONING CORRECTLY WITH THE PROPER CONNECTIONS
FLUKE MULTIMETER DISPLAY
THE THREE SIZES OF COILS WE USED
SAMPLE OF DPPH
THE PICTURE OUTPUT OF OUR OSCILLOSCOPE
THE CONNECTIONS TO THE DIGITAL VOLTMETER
VARIAC AUTOTRANSFORMER
SECONDARY TRANSFORMER
PHASE SHIFTER
TWO CAPACITORS IN PARALLEL
THE APPROPRIATE CONNECTIONS ON THE PHASE SHIFTER
OUR HELMHOLTZ COILS IN PARALLEL WITH THE ESR UNIT
OUR DC POWER SUPPLY AND CONNECTIONS

• Tektronix Oscilloscope (Model TDS 1002)
• ESR Adapter(Model Leybold Didactic GMBH 51456)
• Dual Display Multimeter (Model: Fluke 45)
• ESR Basic Unit (Model Leybold Didactic GMBH 51455)
• Digital Volt Meter (Model Wavetek 85XT)
• Hewlett-Packard Triple Output Supply (Model 6236B)
• BCN Cables
• Special dual adapter for FLUKE Dual Display Multimeter
• Phase Shifter
• 2 Helmholtz Coils(Model Leybold Didactic GMBH 55506)
• 2 Capacitors (Model Kippon Chemi-Con CEO2W 25v470 micro Farads)
• Variac Autotransformer (Model W5MT3)
• SOAR DC Power Supply(Model PS-3630)
• Caltronics Transformer (6.3V-117V)

Safety

Before we begin, some points of safety must be noted:
1. First and foremost your safety comes first and then the equipments'
2. Check the cords, cables, and machinery in use for any damage or possible electrocution points on fuses of machinery by making sure the power cords' protective grounding conductor must be connected to ground
3. Be careful to ground all power supplies properly before use
4. Make sure the areas containing and around the experiment are clear of obstacles
5. Keep in mind that we do not want to blow the capacitors in this lab, which requires careful monitoring of the passing current
6. Electrocution and/or being shocked are the two largest risks in this lab, this can be minimized by care in handling and making sure that all equipment is turned off before adjustments are made

Set Up

The procedure we followed was based on the descriptions given in Professor Gold's manual

The following connections were made to make the experiment functional:

• Variac Autotransformer to the secondary side of the Caltronics transformer
• Secondary positive side of the Caltronics transformer to the positive side of the capacitors (2 capacitors connected in parallel)
• Negative side of the capacitors to the negative output port of the DC power Supply
• Negative side of Secondary Caltronics transformer to the COM port of the Digital Voltmeter
• Helmholtz Rings (placed appropriately parallel to one another) were connected in parallel (Z to Z port and A to A port)
• Port A on the front helmholts coil to the negative (black) port of the Phase Shifter
• Port Z on the front helmholts coil to positive (red) port of Phase Shifter
• Negative (black) port of the Phase Shifter to the negative output port of the DC Power Supply
• Positive (red) port of the Phase Shifter to the mA port on the Digital Voltmeter
• ESR Basic Unit to ESR Adapter input
• CH 1 of the Oscilloscope to Phase Shifter
• +12 V port on the ESR Adapter to the +20 V on the Triple Output Power Supply
• 0 port on ESR Adapter to COM port on on the Triple Output Power Supply
• -12 V port on the ESR Adapter to the -20 V on the Triple Output Power Supply
• Y port on the ESR Adapter to the CH 2 port of the Oscilloscope
• f/1000 port to the FLUKE Dual Display Multimeter using a special dual adapter for both left-most ports

Here are two circuit diagrams we used to help make all of our connections. They can be found inProfessor Gold's manual

To actually perform the lab we turned our entire set up on (making sure that the Variac Autotransformer was never left on by itself, to insure that we did not blow our capacitors). Once everything had the chance to warm up we were able to modify the frequency using the ESR Basic Unit, and adjust the current so that our two wave patterns on the oscilloscope met up to have the same frequency (the resonance frequency). We then made a table, and repeated this process fifteen times for each coil size.

Measurements and Data

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Calculations and Analysis

To calculate the g-factor of the electron we have
$\displaystyle E =g_s \mu_B B\,\!$
$\displaystyle \mu_B=\frac{e \hbar}{2 m_e} \simeq 5.788\times10^{-5} eV/T\,\!$
$\displaystyle B=\frac{\mu R^2NI}{(R^2+x^2)^{3/2}}\,\!$
$\displaystyle E = h \nu \rightarrow g_s=\frac{h \nu}{\mu_B B}\,\!$
In the Helmholtz configuration, from Professor Gold's Manual we are given:
$\displaystyle x=R/2\,\!$ , $\displaystyle N=320\,\!$ (according to Chad McCoy's lab notebook [1], after a few calculations), and $\displaystyle R=0.0675 m\,\!$
The permeability of free space is given as
$\displaystyle \mu=4\pi\times10^{-7}\frac{Wb}{A \cdot m}\,\!$
From these values we can calculate:
$\displaystyle B=\frac{\mu R^2NI}{(R^2+x^2)^{3/2}}\,\!$
$\displaystyle B=\frac{\mu R^2NI}{(\frac{5}{4}R^2)^{3/2}}\,\!$
$\displaystyle B=\frac{\mu NI}{(\frac{5}{4})^{3/2}R}\,\!$
$\displaystyle B=(4.26\times10^{-3}\frac{Wb}{A \cdot m})\times I\,\!$
$\displaystyle g_s=\frac{4.136 \times 10^{-15} eV \cdot s \times \nu}{(5.788\times10^{-5} eV/T) \times (4.26\times10^{-3}\frac{Wb}{A \cdot m})\times I}\,\!$
$\displaystyle g_s=1.677\times10^{-8} A \cdot s \times \frac{\nu}{I}\,\!$
An example calculation is as follows:
$\displaystyle \nu=22.49 \times 10^3 Hz \,\!$
$\displaystyle I = 0.398 A \,\!$
$\displaystyle g_s=1.677\times10^{-8} A \cdot s \times \frac{22.49 \times 10^6 Hz}{0.398 A}\,\!$
$\displaystyle \simeq 0.9479\,\!$
The rest of our values follow in this table:

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The mean was calculated using excel codes and they are as follows for each coil size:
$\displaystyle g_{s,mean,small} \simeq 0.9002\pm.0007\,\!$
$\displaystyle g_{s,mean,medium} \simeq 0.9226\pm.0025\,\!$
$\displaystyle g_{s,mean,large} \simeq 0.8795\pm.0009\,\!$

Large Coil Analysis

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Medium Coil Analysis

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Small Coil Analysis

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To calculate the g-factor using the slope we have:
$\displaystyle g_s=1.677\times10^{-8} A \cdot s \times \frac{1}{slope}\,\!$

For the small coil:
$\displaystyle slope_{small}=0.018320354 A/MHz \,\!$
$\displaystyle g_s=1.677\times10^{-8} A \cdot s \times \frac{10^6 Hz}{0.018320354 A }\,\!$
$\displaystyle \simeq 0.9201\,\!$
With a range calculated as:
$\displaystyle g_s=1.677\times10^{-8} A \cdot s \times \frac{10^6 Hz}{0.018320354+0.000389674 A }\,\!$
$\displaystyle \simeq 0.8965 \,\!$
$\displaystyle g_s=1.677\times10^{-8} A \cdot s \times \frac{10^6 Hz}{0.018320354-0.000389674 A }\,\!$
$\displaystyle \simeq 0.9355 \,\!$
$\displaystyle 0.8965 \leq g_{s,small} \leq 0.9355 \,\!$

For the medium coil:
$\displaystyle slope_{medium}=0.01719745 A/MHz \,\!$
$\displaystyle g_s=1.677\times10^{-8} A \cdot s \times \frac{10^6 Hz}{0.01719745 A }\,\!$
$\displaystyle \simeq 0.9754\,\!$
With a range calculated as:
$\displaystyle g_s=1.677\times10^{-8} A \cdot s \times \frac{10^6 Hz}{0.01719745 + 0.000546372 A }\,\!$
$\displaystyle \simeq 0.9454 \,\!$
$\displaystyle g_s=1.677\times10^{-8} A \cdot s \times \frac{10^6 Hz}{0.01719745 - 0.000546372 A }\,\!$
$\displaystyle \simeq 1.0074 \,\!$
$\displaystyle 0.9454 \leq g_{s,medium} \leq 1.0074 \,\!$

For the large coil:
$\displaystyle slope_{large}=0.019096418 A/MHz \,\!$
$\displaystyle g_s=1.677\times10^{-8} A \cdot s \times \frac{10^6 Hz}{0.019096418 A }\,\!$
$\displaystyle \simeq 0.8784\,\!$
With a range calculated as:
$\displaystyle g_s=1.677\times10^{-8} A \cdot s \times \frac{10^6 Hz}{0.019096418 + 0.000359099 A }\,\!$
$\displaystyle \simeq 0.8622\,\!$
$\displaystyle g_s=1.677\times10^{-8} A \cdot s \times \frac{10^6 Hz}{0.019096418 - 0.000359099 A }\,\!$
$\displaystyle \simeq 0.8952 \,\!$
$\displaystyle 0.8622 \leq g_{s,large} \leq 0.8952 \,\!$

The accepted g-factor value is given as:
$\displaystyle g_{s,accepted}=2.0023 \,\!$
The percentage error of our mean calculated and slope values relative to the accepted value of the g-factor can be given as:
$\displaystyle % error_{calculated, mean}=\frac{|g_{s,accepted}-g_{s,calculated}|}{g_{s,accepted}}\times100$
$\displaystyle =\frac{|2.0023-0.904473421|}{2.0023}\times100$
$\displaystyle \simeq 54.8% \,\!$
$\displaystyle g_{s,slope,mean}=0.924633333\,\!$
$\displaystyle % error_{s,slope}= \frac{|g_{s,accepted}-g_{s,slope}|}{g_{s,accepted}} \times100$
$\displaystyle =\frac{|2.0023-0.924633333|}{2.0023}\times100$
$\displaystyle \simeq 53.8% \,\!$

According to our results, any one of our calculated values is less than one half of the accepted value. We believe that this may be due to the fact that there are two coils as the current is split between them. Also we believe that there might have been a high amount of systematic error.

Notes about Our Uncertainty

• The oscilloscope lacks a zoom function, and our data was taken based on the human visual alignment of the two waves on the screen.
• We saw a lot of "noise" in our wave pattern for the large coil trails, though the noise was located at the peaks and not the troughs this could still have caused some error due to the lack of precision that the oscilloscope provides.

Systematic Error:

• The connecting wire between the ESR Basic unit and the ESR adapter seemed to be extremely finicky, if it was jarred or even touched slightly the wave pattern on the oscilloscope would go crazy and take at least a full minute to stabilize again. This caused some problems in taking data because it was hard to not disturb the wire.
• The setup for this lab was very complex and called for a lot of attention to detail. There could have been any amount of systematic error that was overlooked. For example, our connections to our Digital Voltmeter required a tricky clamp to BCN cable connection that was very loose and a few times during our experiment, had to be readjusted.

Summary

If you wish to see Alex Andrego's informal summary of this lab follow this link
If you wish to see Anastasia Ierides's informal summary of this lab follow this link

Acknowledgments

Prof. Gold's Lab Manual served as a loose guideline for our lab procedure and our "Brief Description of Electron Diffraction" above as well as the source of the accepted values of the separation of the carbon atoms corresponding to the inner and outer ring diameters
Professor Koch and Pranav for always being of great help to us!