Chapter 8 Magnetic Resonance

9.1 Electron paramagnetic resonance 9.2 Ferromagnetic resonance 9.3 Nuclear magnetic resonance 9.4 Other resonance methods

TCD March 2007 1 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

TCD March 2007 2 Larmor frequency

m = "l m = m x B µ ! 0 ! = dl/dt

m m x B d /dt = -" 0

= m x B ! != µ"B0 NB. The electron precesses counterclockwise because of the negative charge, " is Solution is m(t) = m ( sin# cos! t, sin# sin! t, cos# ) where ! = "B L L L 0 negative. eB Torque ! cause µ to precess about B with the Larmor frequency# = me

B /2 Magnetic moment precesses at the Larmor precession frequency fL = " 0 "

The Larmor precession is half the cyclotron frequency for orbital moment, but " = -e/2me equal to it for spin moment. " = -e/me

TCD March 2007 3 An alternating ﬁeld along the x-axis can be decomposed into two counter-rotating ﬁelds.

b = 2b1cos !t y b = b1[exp!t + exp-!t]

-!t x !t

TCD March 2007 4 m = "hS

H Z = - "!B0Sz

Ei = - "!B0MS

MS = S, S-1, … S = 1/2

MS 1

-1

Zeeman-split enegy levels for an electronic system with S = 1

Splitting is "!B0; ! = "B0

TCD March 2007 5 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

TCD March 2007 6 TCD March 2007 7 9.1 Electron paramagnetic resonance (EPR)

Larmor precession frequency for electron spin is 2% fL = !L = (ge/2m)B0 -1 fL = 28.02 GHz T .

TCD March 2007 8 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.

TCD March 2007 9 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

Integrated lorentzian lineshape

Derivative lineshape

TCD March 2007 10 MS E = h( 1/2

-1/2

Microwave power w Switch off power; relaxation time is T1 spin-lattice relaxation n

TCD March 2007 11 EPR works best for S-state ions with half-ﬁlled shells.

2 Free radicals S1/2 2+ 3+ 6 Mn Fe S5/2 3+ 8 Gd S7/2

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

TCD March 2007 12 TCD March 2007 13 Spin hamiltonian

TCD March 2007 14 2 Zero-ﬁeld splitting DSz

2 H spin = DSz - "!B0Sz

TCD March 2007 15 Hyperﬁne interactions in epr

These interactions are ! 0.1 K. They represent coupling of the spin of the nucleus to the magnetic ﬁeld produced by the atomic electrons.

Nuclear spin I. MI = I, I-1 ……… -1.

mn = gnµN MI

Hyperﬁne Hamiltonian Hhf = A I.S

TCD March 2007 16 Hyperﬁne interactions in epr

TCD March 2007 17 9.2 Ferromagnetic resonance (FMR)

Resonance frequencies are similar to those for EPR. The coupled moments are due to electrons. # = -(e/m)

TCD March 2007 18 Kittel equation

TCD March 2007 19 Ferromagnetic resonance can give values of Ms and K as well as ", without the need to know the dimensions or mass of the sample.

TCD March 2007 20 TCD March 2007 21 9.2.1 Spin-wave resonance

Spin-wave dispersion. !! = Dk2 K = n%/t

TCD March 2007 22 9.2.2 Antiferromagnetic resonance

TCD March 2007 23 9.2.2 Damping

Two forms of the damping; Landau-Lifschitz and Gilbert

TCD March 2007 24 TCD March 2007 25 TCD March 2007 26 TCD March 2007 27 9.2.3 Domain wall resonance

1/2 )w = %(A/K 1)

d#/dx = sin #/ )w Apply a ﬁeld B along Oz. Pressure on the wall is 2BMs

The

TCD March 2007 28 TCD March 2007 29 9.3 Nuclear magnetic resonance (NMR)

TCD March 2007 30 NMR experiment

MI E = h( -1/2

1/2

TCD March 2007 31 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 %

TCD March 2007 32 9.3.1 Hyperfine interactions

Hyperﬁne ﬁeld has contact, orbital and dipolar contributions

eQ nuclear quadrupole moment

eq = Vzz electric ﬁeld gradient at the nucleus

Vxx 0 0 V + V + V = 0 efg xx yy zz 0 Vyy 0 * = (Vxx - Vyy)/Vzz 0 0 Vzz

TCD March 2007 33 TCD March 2007 34 9.3.2 Relaxation

T1 Spin lattice relaxation

TCD March 2007 35 T2 Spin-spin relaxation

TCD March 2007 36 Bloch’s Equations

TCD March 2007 37 9.3.2 Rotating frame

TCD March 2007 38 Bloch’s equations in the rotating frame

TCD March 2007 39 TCD March 2007 40 9.3.3 Pulsed nmr

TCD March 2007 41 TCD March 2007 42 TCD March 2007 43 Spin echo

TCD March 2007 44 TCD March 2007 45 TCD March 2007 46 TCD March 2007 47 A typical free induction decay, and its spectrum

TCD March 2007 48 9.4 Other resonance methods 9.4.1 Mossbauer effect

2 2 Recoilless fraction f = exp -k" F is the probability of a zero-phonon emission or absorption event in a solid 2 source. E "= hk" is rms displacement of the nucleus

TCD March 2007 49 TCD March 2007 50 Conversion electron Mossbauer spectroscopy

Electron detector 57Co (t 250d) "-ray surface 1/2 Emitted electron t interface

substrate 7.3 keV conversion electron 5/2 14.4 keV "-ray 57Fe

3/2 14.4 keV "-ray 3/2 1/2 1/2

Source Absorber TCD March 2007 51 9.4.2 Muon spin rotation A muon is an unstable particle with spin 1/2 Charge ± e Mass 250 me Half-life +µ = 2.2 microseconds.

Pions are produced in collisions of high-energy protons with a target. They decay in 26 ns to give muons

+ + % , µ + (µ Neutrino, muon have their spin antiparallel to their momentum, S%= 0

The MeV muons are rapidly thermalized in a solid specimen. After time t, probability of muon decay is 1 - exp(-t/ +µ)

+ + µ , e + (e + (’e

The direction of emission of the positron is related to the spin direction of the muon. The muon precesses around the local ﬁeld at 135 GHz T-1

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TCD March 2007 71 8.5 Superparamagnetism

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TCD March 2007 73 8.6 Bulk nanostructures

Recrystallization of amorphous Fe-Cu-Nb- Si-B to obtain a two-phase crystalline/ amorphous soft nanocomposite

TCD March 2007 74 The hysteresis loop

spontaneous magnetization

remanence

coercivity virgin curve initial susceptibility

major loop

The hysteresis loop shows the irreversible, nonlinear response of a ferromagnet to a magnetic ﬁeld . It reﬂects the arrangement of the magnetization in ferromagnetic domains. The magnet cannot be in thermodynamic equilibrium anywhere around the open part of the curve! M and H have the same units (A m-1).

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TCD March 2007 76 Magnetostatics

Poisson’s equarion

Volume charge

Boundary condition

en 2. air + 1. solid + M +

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.

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