Electron Spin Resonance Study Using Microwave Techniques
University of California, Santa Cruz
September 12, 2006
Contents
0.1
0.2
0.3
0.4
0.5
Introduction . . . . . . . . . . . . . .
Kramers-Krönig relations . . . . . .
Microwave Equipment & Techniques
The Microwave Spectrometer . . . .
Suggested Guide to Experiments . .
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. 1
. 3
. 4
. 7
. 10
Magnetic Resonance References
1. Melissinos - Experiments in Modern Physics
2. Pake - Paramagnetic Resonance
3. Andrew - Nuclear Magnetic Resonance
4. Slichter - Magnetic Resonance
5. Poole - Theory of Magnetic Resonance
Microwave References
1. Moreno - Microwave Transmission Design Data.
2. MIT Radiation Laboratories Series Vol. 8, 9, 10. (Somewhat difficult to read in places.)
3. Jackson - Classical Electrodynamics Chapter 8
4. Reitz & Milford - Classical Electrodynamics Chapter 16 (discussion of waveguide and cavity
modes only)
0.1
Introduction
Many particles (neutrons, protons, many nuclei, and electrons) have a spin angular momentum and
associated with it, a magnetic dipole moment µ. If one calculates classically the magnetic moment
produced by a rotating charged sphere, with a one electron mass, one obtains a moment of order
one Bohr magneton µB .
i.e.,
eh̄
µ ≈ µB =
2Me c
1
whereMe is the electron mass and c is the speed of light.
The observed moment of the electron is 2µB . For heavier particles, such as protons (Mp /M e ≈
1800), one might expect the moment to be approximately 2000 times smaller. The observed nuclear
moments are in fact within an order of magnitude of the nuclear magnetron
eh̄
µn =
where Mp = mass of proton.
2Mp c
−
→
When a system of spins is placed in a magnetic field (Say H = H0 ẑ) the moments tend to align
−
→
−
→
along the field and the energy of the moment →
µ is classically −−
µ · H.
However as a result of the quantization of angular momentum the quantum mechanical descrip→
−
tion of the moments gives a finite number of discrete energy levels. For an electron spin S , the
→
−
−
associated magnetic moment is →
µ = gµB S , where µB is the Bohr magneton defined earlier, and g
is a numerical factor which describes the fact that the actual moment is not µB . (For free electrons
g is very close to 2 both theoretically and experimentally. In solids, the observed g value varies
−
→
around 2.) The component of S along the magnetic field (Sz in our case) is quantized and for a
spin of 12 , SZ has the values ± 12 . Therefore the allowed energies are ± gµB2 H . Magnetic resonance
may occur when the frequency ν of the applied electromagnetic fields is equal to the energy level
splitting divided by Planck’s constant or
hν = h̄ω = gµB H.
Whether or not a resonance is observed depends on several other considerations such as the matrix
element for the transition, total number of spins, the number of spins in each state, the relaxation
rate, etc.
Now consider only solids. Most of the electrons on each atom are in closed shells in which the
electron spins are paired—one spin up and one spin down. Consequently closed shells have no net
moment. In insulators (pure) such as Na+ Cl− or Ca+2 O−2 all the electrons are in closed shells
(that’s why they are insulators) and no net moment is present. In metals, the valence electrons
are free to move in the solid and in a magnetic field these valence electron magnetic moments do
have a net moment, but the resonance in metals is difficult to observe. Alternating fields do not
penetrate very far into a metal, and therefore only a few spins experience the applied fields. This
means that the signal is small and thus difficult to measure. Next consider different impurities in
insulators (impurities in metals are difficult to see for the same reasons given above for metals).
An atom such as Mn has an unfilled d shell of electrons and two S electrons. If it is introduced into
CaO, there is a net moment arising from the electrons in the d shell. Such impurities are called
paramagnetic impurities and many have been studied using electron spin resonance. Finally a forth
source of unpaired or “free electrons” occurs in the formation of a complex ion called a free radical
(often an organic molecule). A well known example which we will use in the laboratory is DPPH
(diphenyl - picryl - hydrazil).
Earlier in discussing the energy levels of free electrons we introduced the g value, which for
free electrons is 2. In a solid the electron spin interacts with the lattice and the electron motion.
−
→
One such interaction is the spin-orbit interaction between the electron spin S and its angular
→
−
momentum L . It is of the form
−
→ −
→
Hspin-orbit = λ L · S .
Another interaction called the contact interaction occurs when the nuclear spin I is not zero and
usually has the form
−
→ −
→
H =AI · S.
2
Other interactions include dipole-dipole interactions between the electron spin and other spins
(both electrons and nuclear) and electrostatic interactions between the electron and the electric
fields in the crystal etc. The effects of such interactions are:
1. A shift of the g value from 2
2. A broadening of the absorption line
3. In some systems, there may be many energy levels and then there can be more than one
absorption line.
Thus the careful study (accuracies of 1 part in 106 are often achieved) of electron spin resonance
gives information about the interactions within the host crystal.
The student should study the basic theory of magnetic resonance. The easiest sources to
understand are
• Melissinos - Beginning of Chapter 8
• Andrew - Chapter 2
• Pake - Introduction
In studying the basic general theory the following points should be understood:
• The energy levels of the system.
• The meaning of absorption and dispersion.
• How the spins absorb power and the meaning of the saturation parameter.
• The Bloch equations and the r-f susceptibilities χ0 & χ00 .
• The relationship between χ0 & χ00 given by the Kramers-Krönig relations.
0.2
Kramers-Krönig relations
For any linear system the response to an applied sinusoidal excitation is at the same frequency but
has a net phase shift which depends on the excitation frequency ω and the natural frequency of the
system ωo (see for example the response of an LRC circuit to a sine wave excitation at frequency ω).
This response can be decomposed into an in-phase component and a 90◦ -out-of-phase component.
For magnetic systems the susceptibility χ is given by:
χ = χ0 − iχ00
M
= χH
(1)
(2)
where χ0 is called the dispersion and χ00 is the absorption. These two quantities are related by the
Kramers-Krönig relations. For the special case of a narrow resonance, as is the case for ESR, these
equations can be written as:
3
00
χ (ω) =
P
π
χ0 (ω) − χ∞ = −
Z ∞
(χ0 (ω 0 ) − χ0∞ )dω 0
−∞
Z
P ∞
π
−∞
(ω 0 − ω)
χ00 (ω 0 )dω
(ω 0 − ω)
(3)
(4)
where P means the principle part of the integral - i.e.
Z ∞
−∞
dω =
Z ω−²
−∞
dω +
Z ∞
ω+²
dω,
(5)
in the limit that ² goes to 0. These equations can be checked experimentally in ESR since both χ0
and χ00 are independently measured. In doing the integrals set them up to miss the point at which
the denominator goes to zero.
When the width of the resonance is determined by the lifetime, χ0 and χ00 are given by:
χ0 (ω) =
χ00 (ω) =
χo ωo T2 (ωo − ω)T2
2
1 + (ω − ωo )2 T22
1
χo ωo T2
2
1 + (ω − ωo )2 T22
(6)
(7)
where ωo is the resonance frequency, T2 is the dephasing time which determines the width of the
resonance, and χo is the dc susceptability. Also since ω = gµB H, we can replace the variable ω in
these equations by H.
0.3
Microwave Equipment & Techniques
The magnitude of the electron magnetic moment is such that at magnetic fields of a few kilogauss,
the resonant frequency will be of order 10 GHz. (1 GHz = 109 Hz. Also note 1 T = 10 kG). This is
in the microwave region where the microwave energy is carried down hollow pipes instead of along
wires and the concepts of capacitance, inductance and impedance are somewhat hazy. Therefore
to make measurements using the microwave equipment, some knowledge of microwave equipment
is necessary.
It is a straightforward and useful exercise for the student to solve Maxwell’s equations in a
rectangular waveguide (and in a rectangular cavity) for sinusoidally varying fields (Use the dimensions 0.9” × 0.4” for the xy cross-section). One finds that energy will propagate along the guide in
many different modes or field configurations. For each mode, there is a frequency called the cut-off
frequency below which, that mode will no longer propagate. In practice waveguides are used in the
lowest order mode at frequencies above cut-off (for the lowest mode) but below the cut-off for the
next higher mode. In rectangular waveguides this is called the TE10 mode.
The student is referred to the microwave references given at the beginning. The field configurations of modes are given on pages 59 and 63 of MIT Series Vol. 10. The nomenclature has changed
since this series was written.
TEnm means transverse electric fields—no electric fields in the direction of propagation - zaxis. These modes are sometimes called Hnm modes (n half-wavelengths along the x-axis (wider
dimension) and m half-wavelengths along the y-axis.
4
TMnm means transverse magnetic fields—no magnetic field in the direction of propagation.
These modes are sometimes called Enm modes. Thus the TE10 and H10 modes are the same mode.
There are a number of devices which are used in the microwave spectrometer, some of which are
discussed in Melissinos. More information is available in the book by Moreno (or the MIT series).
These devices are
Klystron: A stable microwave generator. The unit we have is tunable both electrically and mechanically over the range 8.4-9.7 GHz. The mechanical tuning is achieved by turning a screw
on the top of the klystron. The electrical tuning is achieved by varying the “reflector” voltage.
Isolator: A device which passes power in only one direction. It protects the klystron from reflected
power in the system.
If the device at A reflects a large amount of power back toward the klystron it is possible,
without the isolator, for the entire system to resonate and the microwave fields may become
large enough to damage the klystron. The isolator attenuates power traveling towards the
klystron and this isolates it from the load at A.
Magic Tee: A four port device with the following properties when the arms are suitably matched.
(Matched means loosely that the device attached to a port does not reflect power back toward
the tee.)
5
The first property is approximately true for any 4-port tee made of rectangular waveguide;
it is a result of the orthogonality of the fields in the guides 1 & 2. The second property is
not true for a tee which consists only of pieces of guide soldered together as in the figure.
Power would propagate from arm 3 to arm 4. This is where the“magic” tee gets its name in
that it is magic that one can see through arms 3 & 4 but microwaves do not pass for certain
frequencies. This property is achieved by placing a post and an iris (a short plate in the guide
- look at our magic tee and diagram in Moreno) near the junction. Can you suggest what is
happening physically?
Variable Attenuator: A precision device which absorbs microwave energy and produces little or
no reflections or phase shifts.
Variable Phase Shifter: A precision device which shifts the phase of the microwaves (i.e., changes
the effective length of the line) but produces little or no attenuation or reflections.
Directional Coupler: A device that couples out a fraction of the power in a wave traveling in
one direction in the guide only.
6
The coupler we have couples 10 db (10%) of the power out.
Crystal Detector: A device for observing the magnitude of the microwaves in a guide. It consists
of a crystal diode rectifier and is similar in operation to rectifiers used at lower frequencies.
The maximum signal occurs roughly when the maximum microwave E field is at the diode
junction; one of our detectors has two tuning adjustments to achieve this condition.
Microwave Cavity: A resonant device analogous to LC circuits at low frequencies. It consists of
a rectangular hollow metal box and in the present apparatus is soldered onto the side of a
waveguide. A small hole couples power from the waveguide into the cavity. A moveable short
in the guide can be used to adjust the amount of coupling into the cavity.
You should calculate the modes of such a cavity and determine which mode the cavity operates
in. What is the advantage of using a cavity? What is the meaning of the quality factor Q?
See Moreno Chapter 13 and Melissinos.
Wave Meter: A precisive high Q cavity which absorbs a small amount of power at its resonant
frequency. It is used to measure the frequency of the microwaves and the unit in our laboratory
is variable between 8.2 and 10 GHz.
0.4
The Microwave Spectrometer
A bare spectrometer for detecting both χ0 and χ00 is sketched below.
7
Directional couplers are used to couple out a small amount of power to look at reflected waves
and to measure the frequency.
Consider how you would put the components together. Where would you measure the frequency? How would you obtain a dummy load with a variable phase shift and attenuation? Why
is such a dummy load necessary? Follow the power through all arms of the bridge and determine
what is detected at the detector. How do you measure χ0 and χ00 with this spectrometer? (A
spectrometer is already assembled in the lab for your use.)
In setting up the spectrometer you must first adjust the Klystron to be at the cavity frequency.
It is easiest to sweep the reflector voltage of the Klystron at some convenient frequency and observe
the Klystron mode on the scope. (The “mode” is one of the resonances of the Klystron oscillator it has a low Q, and hence a fairly broad range of frequencies.)
The frequency of the Klystron is adjusted mechanically until the cavity resonance appears as a
dip on the mode. It is perhaps best to experiment with the cavity of the wavemeter first since it is
a sharp resonance. Then go on to find the sample cavity resonance.
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Discontinue the sweep procedure and adjust the Klystron frequency to that of the cavity:
How would you do this?
Next the bridge must be balanced and then adjusted to observe the absorption or
dispersion—how do you do this? (Talk to the instructor or TA about this.)
For many samples the signal will be small and will be below the level of the noise observed
at the crystal when observing only the dc level of the signal. For most systems the noise in a
bandwidth around zero frequency (dc) is high compared to the noise in the same small bandwidth
at some audio frequency, say 1 KHz. i.e.,
Therefore a large signal-to-noise improvement is made by detecting at some finite frequency
such as 1 KHz instead of observing changes in the dc level. The usual method to do this is to
modulate the magnetic field at some frequency (say 1 KHz) and then detect only in narrow band of
frequencies around 1 KHz (Shaded portion of figure). A device which does this is a lock-in detector
which “locks-in” to the modulation frequency. Consider the absorption signal, which if observed
by dc techniques would look like
and assume the amplitude Hm of the modulation field is less than ∆H. What will the modulation
signal look like? What will it look like for the dispersion signal? Show that for small Hm , the
detected signal is proportional to the first derivative (of either χ0 or χ00 ).
9
0.5
Suggested Guide to Experiments
1. Place the DPPH sample in the cavity. Adjust the klystron frequency until you can see the
cavity dip on the Klystron mode and adjust the coupling to maximize the dip. At this step
the attenuator in the reference arm of the microwave bridge should be at maximum (∼ 50db)
and the phase shifter near zero. If however the dip actually goes completely to zero, there
can be a problem as there is no reflected signal to detect. In that case adjust the coupling
at the back of the cavity until the voltage at the dip is ∼ 5% of the mode height. Now
slowly increase the magnetic field until you observe some change in the dip (You should see
a change). Do you understand what is happening?
2. Set up the microwave bridge for absorption (later repeat for dispersion.) First reduce the
modulation amplitude to the klystron to zero and adjust the dc level of the reflector voltage
until you are at the bottom of the dip. Next adust the attenuator and the phase shifter
iteratively (one after the other starting first with the attenuator), to reduce the detected
voltage to zero. This will require several iterations. To observe absorption, unbalance the
bridge approximately 3 db using the attenuator; to observe dispersion, unbalance the bridge
using the phase shifter. Since there is a small amount of absorption in the phase shifter, you
may need to slightly adjust the attenuator to get a “clean” anti-symmetric dispersion curve.
3. With the DPPH sample in the cavity, set up the spectrometer to observe the absorption
signal. Use the sweep generator connected to the magnet current regulator to sweep the
magnetic field across the signal; at the same time use this signal to sweep the oscilloscope
horizontal axis. Does the signal have the expected shape? Repeat with the spectrometer set
up for dispersion.
4. Use the Labview virtual instrument ESR.vi with the computer A to D to collect data. The
inputs are the signal from the output of the oscilloscope plug-in, the voltage across the magnet,
and the magnetic field output from the TEL-Atomic Tesla Meter 2000. First, calibrate the
magnet voltage by using a fairly wide sweep to find ∆B/∆V . Then take data on DPPH using
a narrow sweep. The magnet voltage readings will allow you to interpolate between the Tesla
Meter readings. Check the Kramers-Kronig relations with one set of measurements. For this
you need a good measure of χ0 and χ00 over the same range of magnetic field.
5. Another way to calibrate the magnet voltage is to take readings on DPPH with two different
frequencies from the klystron. Since the g-factor for DPPH is known accurately, this will give
you very accurate measurements of the magnetic field, which can be used for interpolation.
6. Set up the system to look at weaker signals with the lock-in. Run the lock-in output through
the oscilloscope plug-in before it goes into the the A to D. Take data for P-Si (obtain from
instructor), teflon tape, CuSO4 , and anything else you might want to look at. Measure the g
value of the various signals. One might measure the g values of the other materials relative
to the g value of DPPH by placing a small amount of DPPH in along with the sample.
For CuSO4 the g-factor is anisotropic. If you use the powder sample, you observe a very
broad line. There are also some single crystals (they are quite easy to grow from a water
solution.) These will give narrower linewidths and the resonance position will change with
the orientation of the sample in the magnetic field.
10
7. Set up the system to look at weak signals and observe the signals for the various doped
insulator samples. Explain your results. Can you say anything about the impurities in these
samples? Compare your results with accepted values in the literature.
There are a number of concepts and devices which have not been discussed or only briefly
described. Among these are the concepts of impedance and coupling. If you are interested in
pursuing these ideas, please see Frank Bridges.
11