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Chapter 8
Magnetic Resonance
9.1 Electron paramagnetic resonance
9.2 Ferromagnetic resonance
9.3 Nuclear magnetic resonance
9.4 Other resonance methods
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A resonance experiment involves a specimen placed in a uniform magnetic field B0 B0
and applying an AC magnetic 2b1cos!t field in the perpendicular direction
2b1cos!t
A magnetic resonance experiment
B0
2b1cos!t
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Larmor frequency
m = "l
! = m x B0
! = dl/dt
dm/dt = -" m x B0
B
µ
Bµ!= "
Torque ! cause µ to precess about B with the Larmor frequencye
eB
m# =
m
! = m x B0
Solution is m(t) = m ( sin# cos!Lt, sin# sin!Lt, cos# ) where !L = "B0
Magnetic moment precesses at the Larmor precession frequency fL = "B0/2"
The Larmor precession is half the cyclotron frequency for orbital moment, but " =
-e/2me equal to it for spin moment. " = -e/me
NB. The electron precessescounterclockwise becauseof the negative charge, " is
negative.
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x
y
b = 2b1cos !t
b = b1[exp!t + exp-!t]
!t
-!t
An alternating field along the x-axis can be decomposed into two counter-rotating fields.
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m = "hS
H Z = - "!B0Sz
Ei = - "!B0MS
MS = S, S-1, …
S = 1/2
MS
1
0
-1
Zeeman-split enegy levels for anelectronic system with S = 1
Splitting is "!B0; ! = "B0
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Why does the AC field have to be applied perpendicular to B0 ?
H = -"!(B0Sz + 2b1Sx)
If the field is applied in the z-direction, the Hamiltonian is diagonal so there is no mixing of different Ms states
However, Sx has nonzero off-diagonal elements (n, n±1). The second term mixes states with $MS = ±1.
Electronic energy levels; Electronic Paramagnetic Resonance (EPR) GHz range
Nuclear energy levels; Nuclear Magnetic Levels (NMR) MHz range
Ferromagnetic moment precession Ferromagnetic Resonance (FMR) GHz range
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9.1 Electron paramagnetic resonance (EPR)
Larmor precession frequency for electron spin is 2% fL = !L = (ge/2m)B0
fL = 28.02 GHz T-1.
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Microwave cavity delivers b1 in a TM100 mode.
X-band radiation, ! 9 GHz, B0 ! 300 mT.
Energy splitting of ±1/2 levels is 0.2 K.
Polarization of the spin system is
P = (n& - n')/ (n& + n')
= [1 - exp(-gµBB0/kT)]/ [1 + exp(-gµBB0/kT])]
! gµBB0/2kT
At RT in 300 mT this is only 7 10-4.
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Derivative lineshape
Integrated lorentzian lineshape
EPR lineshape. Fix frequency ! and amplitude b1, scan magnetic field at a constant rate.
Absorption line is measured by modulating the field B0 with a small ac field and using lockin detection
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MS
1/2
-1/2
E = h(
Microwave power w Switch off power; relaxation time is T1
spin-lattice relaxation
n
t
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EPR works best for S-state ions with half-filled shells.
Free radicals 2S1/2
Mn2+ Fe3+ 6S5/2
Gd3+ 8S7/2
Ions should be dilute in a crystal lattice to diminish dipole-dipole interactions.
The outer electrons in these shells interact strongly with surroundings.
Crystal-field interactions may mix different MS states.
Second order $MJ ± 2
Fourth order $MJ ± 4
Sixth order $MJ ± 6
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Spin hamiltonian
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Zero-field splitting DSz2
H spin = DSz2 - "!B0Sz
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Hyperfine interactions in epr
These interactions are ! 0.1 K. They represent coupling of the spin of the nucleus to the magnetic field produced bythe atomic electrons.
Nuclear spin I. MI = I, I-1 ……… -1.
mn = gnµN MI
Hyperfine Hamiltonian Hhf = A I.S
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Hyperfine interactions in epr
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9.2 Ferromagnetic resonance (FMR)
Resonance frequencies are similar to those for EPR. The coupled moments are due to electrons.
# = -(e/m)
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Kittel equation
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Ferromagnetic resonance can give values of Ms and K as well as ", without the need to know the dimensions or
mass of the sample.
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9.2.1 Spin-wave resonance
Spin-wave dispersion. !! = Dk2
K = n%/t
t
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9.2.2 Antiferromagnetic resonance
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9.2.2 Damping
Two forms of the damping; Landau-Lifschitz and Gilbert
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9.2.3 Domain wall resonance
)w = %(A/K1)1/2
d#/dx = sin#/ )w Apply a field B along Oz. Pressure on the wall is 2BMs
The
z
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9.3 Nuclear magnetic resonance (NMR)
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NMR experiment
MI
-1/2
1/2
E = h(
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Chemical shift
Proton resonance spectrum of an organic compound
Knight shift
Shift in resonance due to shielding of the applied field by the conduction electrons. ! 1 %
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9.3.1 Hyperfine interactions
eQ nuclear quadrupole moment
eq = Vzz electric field gradient at thenucleus
Vxx 0 0
0 Vyy 0
0 0 Vzz
efgVxx + Vyy + Vzz = 0
* = (Vxx - Vyy)/Vzz
Hyperfine field has contact, orbital and dipolar contributions
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9.3.2 Relaxation
T1 Spin lattice relaxation
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T2 Spin-spin relaxation
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Bloch’s Equations
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9.3.2 Rotating frame
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Bloch’s equations in the rotating frame
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9.3.3 Pulsed nmr
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Spin echo
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A typical free induction decay, and its spectrum
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9.4.1 Mossbauer effect
9.4 Other resonance methods
Recoilless fraction f = exp -k"2<x2>
F is the probability of a zero-phononemission or absorption event in a solidsource. E "= hk"
2
<x2> is rms displacement of the nucleus
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5/2
3/2
1/2
3/2
1/2
Source Absorber
57Co (t1/2 250d)
57Fe
14.4 keV "-ray
14.4 keV "-ray
7.3 keVconversionelectron
substrate
interface
surface
t
"-rayEmittedelectron
Electrondetector
Conversion electron Mossbauer spectroscopy
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9.4.2 Muon spin rotationA muon is an unstable particle withspin 1/2Charge ± eMass 250 me
Half-life +µ = 2.2 microseconds.
Pions are produced in collisions ofhigh-energy protons with a target. Theydecay in 26 ns to give muons
%+ , µ+ + (µNeutrino, muon have their spinantiparallel to their momentum, S%= 0
The MeV muons are rapidlythermalized in a solid specimen. Aftertime t, probability of muon decay is 1 -exp(-t/ +µ)
µ+ , e+ + (e + (’e
The direction of emission of thepositron is related to the spin directionof the muon. The muon precessesaround the local field at 135 GHz T-1
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8.5 Superparamagnetism
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8.6 Bulk nanostructures
Recrystallization of amorphous Fe-Cu-Nb-Si-B to obtain a two-phase crystalline/amorphous soft nanocomposite
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The hysteresis loop shows the irreversible, nonlinear response of a ferromagnet to amagnetic field . It reflects the arrangement of the magnetization in ferromagnetic domains.The magnet cannot be in thermodynamic equilibrium anywhere around the open part ofthe curve! M and H have the same units (A m-1).
coercivity
spontaneous magnetization
remanence
major loop
virgin curveinitial susceptibility
The hysteresis loop
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Magnetostatics
Poisson’s equarion
Volume charge
Boundary condition
1. solid
2. air
M( r) , H( r) BUT H( r) , M( r)
Experimental information about the domain structure comes from observations at the surface.The interior is inscruatble.
en
M
+
++