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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

+

++