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

{| width="800"  Anastasia's Main project page
 * style="background-color: #EEE"|[[Image:owwnotebook_icon.png|128px]] Electron Spin Resonance
 * style="background-color: #F2F2F2" align="center"|  Alex's Main project page
 * style="background-color: #F2F2F2" align="center"|  Alex's Main project page


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 * '''Please note that this lab notebook page is the combined efforts of Alex Andrego and Anastasia A. Ierides

Purpose and Brief Description of Electron Spin Resonance

 * The purpose of this lab is to measure the g-factor, $$g_s\,\!$$, 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 $$\mu_s\,\!$$ 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:
 * $$\mu_s = −g_s \mu_B \frac{S}{\hbar}\,\!$$
 * where:
 * $$g_s\,\!$$ is a constant characteristic of the electron, its intrinsic g-factor
 * $$\mu_B=\frac{e \hbar}{2 m_e} \simeq 5.788\times10^{-5} eV/T\,\!$$ is the Bohr magneton
 * $$S\,\!$$ is the spin of the electron
 * $$ \hbar = 6.582 \times 10^{−16} eV \cdot sec\,\!$$ 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:
 * $$ E =- \mu_s \times B\,\!$$
 * The two energies associated with the two different orientations of the electron are given by:
 * $$E = E_0 \pm g_s \mu_B B\,\!$$
 * where $$E_0\,\!$$ 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



 * 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:
 * First and foremost your safety comes first and then the equipments'
 * 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
 * Be careful to ground all power supplies properly before use
 * Make sure the areas containing and around the experiment are clear of obstacles
 * Keep in mind that we do not want to blow the capacitors in this lab, which requires careful monitoring of the passing current
 * 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 in| Professor 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.

Calculations and Analysis

 * To calculate the g-factor of the electron we have
 * $$ E =g_s \mu_B B\,\!$$


 * $$\mu_B=\frac{e \hbar}{2 m_e} \simeq 5.788\times10^{-5} eV/T\,\!$$


 * $$B=\frac{\mu R^2NI}{(R^2+x^2)^{3/2}}\,\!$$


 * $$E = h \nu \rightarrow g_s=\frac{h \nu}{\mu_B B}\,\!$$


 * In the Helmholtz configuration, from Professor Gold's Manual we are given:


 * $$x=R/2\,\!$$, $$N=320\,\!$$ (according to Chad McCoy's lab notebook, after a few calculations), and $$R=0.0675 m\,\!$$


 * The permeability of free space is given as


 * $$\mu=4\pi\times10^{-7}\frac{Wb}{A \cdot m}\,\!$$


 * From these values we can calculate:


 * $$B=\frac{\mu R^2NI}{(R^2+x^2)^{3/2}}\,\!$$


 * $$B=\frac{\mu R^2NI}{(\frac{5}{4}R^2)^{3/2}}\,\!$$


 * $$B=\frac{\mu NI}{(\frac{5}{4})^{3/2}R}\,\!$$


 * $$B=(4.26\times10^{-3}\frac{Wb}{A \cdot m})\times I\,\!$$


 * $$g_s=\frac{4.136 \times 10^{-15} eV \cdot s \times \nu}{(5.788\times10^{-5} eV/T) \times (4.26\times10^{-3}\frac{Wb}{A \cdot m})\times I}\,\!$$


 * $$g_s=1.677\times10^{-8} A \cdot s \times \frac{\nu}{I}\,\!$$


 * An example calculation is as follows:


 * $$\nu=22.49 \times 10^3 Hz \,\!$$


 * $$ I = 0.398 A \,\!$$


 * $$g_s=1.677\times10^{-8} A \cdot s \times \frac{22.49 \times 10^6 Hz}{0.398 A}\,\!$$


 * $$\simeq 0.9479\,\!$$


 * The rest of our values follow in this table:


 * The mean was calculated using excel codes and they are as follows for each coil size:


 * $$ g_{s,mean,small} \simeq 0.9002\pm.0007\,\!$$


 * $$ g_{s,mean,medium} \simeq 0.9226\pm.0025\,\!$$


 * $$ g_{s,mean,large} \simeq 0.8795\pm.0009\,\!$$

Large Coil Analysis






Medium Coil Analysis






Small Coil Analysis







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


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


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


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


 * The accepted g-factor value is given as:
 * $$g_{s,accepted}=2.0023 \,\!$$


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


 * $$g_{s,slope,mean}=0.924633333\,\!$$
 * $$% error_{s,slope}= \frac{|g_{s,accepted}-g_{s,slope}|}{g_{s,accepted}} \times100 $$
 * $$=\frac{|2.0023-0.924633333|}{2.0023}\times100$$
 * $$\simeq 53.8% \,\!$$

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!


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