ELECTRON SPIN RESONANCE
KEN CHENEY
June 6, 2006
PICTURES
http://www.paccd.cc.ca.us/instadmn/physcidv/physics/teachers/cheney/lab%
20manuals/Category.htm
ABSTRACT
Electron spin resonance is observed by observing the adsorption of radio
frequency photons by unpaired electrons flipping in an external field.
HISTORY
The need for electron spin was forced on the world of physics by the need
for a fourth quantum number for the Pauli Exclusion Principle to work on,
by the Stern Gerlach experiment (1922) and by the splitting of spectral lines
in a magnetic field, the anomalous Zeeman Effect.
First postulated by the students Goudsmit and Uhlenbeck in 1925.
How this could be was very obscure until Dirac in 1928 produced a
relativistic quantum theory that automatically gave electrons spin, not to
mention producing anti matter electrons.
WHY OF INTEREST
Electron spin resonance is very sensitive to the magnetic fields due to
surrounding atoms. Therefore the resonance can be used to identify atoms
or molecules.
The necessary unpaired electron occurs in free radicals, which are very
active. One would think that this activity would insure that they would not
be available long enough to be useful but ingenious experimentations have
found uses for ESR in Biological Systems, Chemical Systems, Conduction
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Electrons, Free Radicals, Irradiated Substances, naturally Occurring
Substances, and Spin Labels.
The process is quite analogues to “Magnetic Nuclear Resonance” (or
Magnetic Imaging in the medical profession who don’t want to scare patents
with the word nuclear). As the name implies nuclear resonance senses the
response of nuclei to ambient and driving magnetic fields.
THEORY
Electrons have charge and spin (or more precisely angular momentum). The
combination would lead you to expect that the electrons would also have a
magnetic moment produced by the rotating charge.
The electrons do indeed have a magnetic moment but it cannot be derived
from a rotating charge, in fact the electron’s angular momentum doesn’t
follow from any reasonable (classical) rate of rotation.
In practice one generally just accepts that electrons have a certain angular
momentum and a certain magnetic moment.
In an atom electrons may or may not have orbital angular momentum. We
will only consider the case of atoms with one electron in a zero orbital
angular momentum sub shell (s). Further we will only consider that this
electron is outside otherwise closed shells. The point is to avoid worries
about how to handle orbital angular momentum! This restriction is not
generally necessary but it immensely simplifies our analysis.
Torque and Energy
Of course a magnetic moment  means a torque  is produced by an
external magnetic field B0 making an angle  with the direction of the
magnetic moment:
   B0 sin( )
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(1)
If the electron is rotated by 180 degrees (the only choice since the spin is
quantized with only two directions!) then we can calculate the work U done
by using dU  Td , with Eq. (1):

U    B0 sin( )d   B0  sin( )d   B0 cos( )0  2  B0
(2)
0
Of course whether you put energy in or get energy out depends on how the
electron’s magnetic moment is aligned with the external magnetic field.
Two values for energy
Beware, some values for the energy are for the electron flip, other values
you see are half as much because they are for the energy difference between
zero magnetic field and with a magnetic field B0 .
Interesting complications we will “ignore”
g:
g is the “Lande g factor”
The effective magnetic moment is not, in general, just the magnetic moment
of the electron but also includes contributions from the orbital angular
momentum.
(3)
effective  g(eh / 2m)  g B
Where:
B  eh / 2m
(4)
is the “Bohr Magneton”
For our special case g is approximately 2, actually 2.002 319 304 376 8(86),
perhaps the best known of all constants!
g is called the “gyromagnetic ratio, the ratio of the magnetic dipole moment
to the angular momentum of the electron.
“Parametric Resonance”
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Parametric refers to driving an oscillator at some integer multiple or fraction
of the resonant frequency.
Electron Spin Resonance is also called “Electron Parametric Resonance”.
You might ask “What is resonating?”.
Consider the mechanics of adding (or subtracting) energy from or to the
electron through its magnetic moment:
First recall that spinning objects precesses instead of simply rotating when a
torque is applied.
We are inputting a high frequency magnetic field. It the phase difference
between the electron and magnetic field is random than no net energy will be
transferred. However if the magnetic field is in phase with the precession of
the electron then energy can be added or subtracted.
Check whether the resonance frequency matches the frequency given by
Plank’s E=hv discussed below.
Does the precession frequency for a “gyroscope”:
  Torque/Angular Momentum
Match the frequency from the “Larmor Equation”:
  gB
(5)
(6)
Energy and photons
We are going to put the energy in to flip the electron using another rapidly
varying magnetic field. We can consider that this second field will produce
photons, which in turn will flip the electron.
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To find the frequency needed to flip the electron we can set the energy
needed from Eq. (2) equal to the energy of a photon of frequency  :
(7)
2 B0  h
If most of the electrons in a sample are in the lower energy state (expected!)
then photons with the frequency given by equation (7) will be adsorbed.
MAGNETIC FIELDS
There are two magnetic fields involved here: B0 that is produced by
external Helmholtz Coils and aligns the electrons, and Brf (rf for the radio
frequency driving the coil) which adds or removes energy from the
electrons.
The only component of Brf that will rotate the electrons are components
perpendicular to the magnetic motion, hence perpendicular to B0 .
Magnetic Field
From
Helmholtz
Coils B0
Large Cylinder to
hold Helmholtz
Coils
Magnetic
Field
from Coil
Brf
Sample
and Coil
Helmholtz Coils
Open End
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CONNECTIONS
Power
Supply
and
Frequency
Meter
Oscillator
Head
Power
Frequency
Supply
Power
Frequency
Helmholtz
Coils: HC
Oscilloscope Use
Banana
Plugs
Sense
Video
Source arm
Power HC
x channel
y channel
HC Current
Sample coil volts
Support and
power source for
sample and
sample coil
The Oscillator Head clamps onto a vertical rod in order to reach the correct
height to put the source at the center of the Helmholtz Coils.
WHAT IS GOING TO HAPPEN
Follow the Setup and Operation instructions on page 8 of the manual.
If all goes well you will have set the sample coil circuit to oscillate at around
30Mhz.
The Helmholtz coils will automatically sweep from negative to positive
current (and hence reverse the direction of B0 ).
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For two B0 (same magnitude just plus and minus) here will be a resonance
between the rf magnetic field of the sample coil and the frequency or energy
to flip the electron’s spin. Since there are more electrons in the lower energy
state energy will be adsorbed from the source coil.
For reasons I can’t follow instead of seeing a dip in the “sense” voltage this
lose of energy shows up as a peak in the voltage!? Perhaps just displayed
upside down?
To get the current (I) through the Helmholtz coils use the conversion:
 1amp 
I V 

 1volt 
(8)
Where V is the voltage measured by the x channel of the oscilloscope.
You measure the x (Helmholtz current I) difference between the peaks and
divide by two to get the I 0 that gives resonance. One peak is for - B0 , the
other peak is for B0 so the magnitude of I 0 is half the difference between the
peaks.
To get B0 from I 0 use the equation (10) on page 5:
B0  0.48I 0 millaTesla
(9)
Finally use Eq. (7)
2 B0  h
to calculate the electron’s magnetic moment.
SETTINGS AND CONTROLS
Oscilloscope Setting: xy
Controls:
1. Coil Current: The maximum amplitude of the current to the
Helmholtz Coils.
2. Phase: If you get four peaks instead of two play with the phase
knob!
3. Tuning: Adjusts the resonant frequency of the circuit including the
sample, adjust for maximum size of the peaks.
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4. Feedback: Adjusts how sensitive the sensing circuit is to
resonance, adjust for best results!
WHAT TO DO
1. Check the electron’s magnetic moment as accurately as possible.
2. If you have time try the suggestions on page 12 of Daedalon’s
manual.
3. Do Eq.(5) and Eq. (6) give the same frequency? Does this make
sense?
BIBLIOGRAPHY
Instruction Manual for EN-35 Electron Spin Resonance Apparatus,
Daedalon Corporation
Overview of Electron Spin Resonance and its Applications, Farach and
Poole, www.uottawa.ca/publications/interscientia/inter.2/spin.html
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