Chapter 8 Magnetic Resonance

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1 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

2 A resonance experiment involves a specimen placed in a uniform magnetic field B 0 B 0 and applying an AC magnetic 2b 1 cos t field in the perpendicular direction 2b 1 cos t B 0 2b 1 cos t A magnetic resonance experiment TCD March

3 Larmor frequency B m = l = m x B 0 = dl/dt mµ dm/dt = - m x B 0 = m = µ x B 0 Solution is m(t) = m ( sin cos L t, sin sin L t, cos ) where L = B 0 eb Torque cause µ to precess about B with the Larmor frequency = me Magnetic moment precesses at the Larmor precession frequency f L = B 0 /2 NB. The electron precesses counterclockwise because of the negative charge, is negative. The Larmor precession is half the cyclotron frequency for orbital moment, but = -e/2m e equal to it for spin moment. = -e/m e TCD March

4 An alternating field along the x-axis can be decomposed into two counter-rotating fields. y b = 2b 1 cos t b = b 1 [exp t + exp- t] x - t t TCD March

5 m = hs H Z = - B 0 S z E i = - B 0 M S S = 1/2 M S = S, S-1, M S Zeeman-split enegy levels for an electronic system with S = 1 Splitting is B 0; = B 0 TCD March

6 Why does the AC field have to be applied perpendicular to B 0? H = - (B 0 S z + 2b 1 S x ) If the field is applied in the z-direction, the Hamiltonian is diagonal so there is no mixing of different M s states However, S x has nonzero off-diagonal elements (n, n±1). The second term mixes states with M S = ±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

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8 9.1 Electron paramagnetic resonance (EPR) Larmor precession frequency for electron spin is 2 f L = L = (ge/2m)b 0 f L = GHz T -1. TCD March

9 Microwave cavity delivers b 1 in a TM 100 mode. X-band radiation, 9 GHz, B mt. Energy splitting of ±1/2 levels is 0.2 K. Polarization of the spin system is P = (n - n )/ (n + n ) = [1 - exp(-gµ B B 0 /kt)]/ [1 + exp(-gµ B B 0 /kt])] gµ B B 0 /2kT At RT in 300 mt this is only TCD March

10 EPR lineshape. Fix frequency and amplitude b 1, scan magnetic field at a constant rate. Absorption line is measured by modulating the field B 0 with a small ac field and using lockin detection Integrated lorentzian lineshape Derivative lineshape TCD March

11 E = h M S 1/2-1/2 Microwave power w Switch off power; relaxation time is T 1 spin-lattice relaxation n t TCD March

12 EPR works best for S-state ions with half-filled shells. Free radicals 2 S 1/2 Mn 2+ Fe 3+ 6 S 5/2 Gd 3+ 8 S 7/2 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 M S states. Second order M J ± 2 Fourth order M J ± 4 Sixth order M J ± 6 TCD March

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14 Spin hamiltonian TCD March

15 Zero-field splitting DS z 2 H spin = DS z 2 - B 0 S z TCD March

16 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. M I = I, I-1-1. m n = g n µ N M I Hyperfine Hamiltonian H hf = A I.S TCD March

17 Hyperfine interactions in epr TCD March

18 9.2 Ferromagnetic resonance (FMR) Resonance frequencies are similar to those for EPR. The coupled moments are due to electrons. = -(e/m) TCD March

19 Kittel equation TCD March

20 Ferromagnetic resonance can give values of M s and K as well as, without the need to know the dimensions or mass of the sample. TCD March

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22 9.2.1 Spin-wave resonance t Spin-wave dispersion. = Dk 2 K = n /t TCD March

23 9.2.2 Antiferromagnetic resonance TCD March

24 9.2.2 Damping Two forms of the damping; Landau-Lifschitz and Gilbert TCD March

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28 9.2.3 Domain wall resonance z w = (A/K 1 ) 1/2 d /dx = sin / w Apply a field B along Oz. Pressure on the wall is 2BM s The TCD March

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30 9.3 Nuclear magnetic resonance (NMR) TCD March

31 NMR experiment E = h M I -1/2 1/2 TCD March

32 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

33 9.3.1 Hyperfine interactions Hyperfine field has contact, orbital and dipolar contributions eq eq = V zz nuclear quadrupole moment electric field gradient at the nucleus efg V xx V yy V zz V xx + V yy + V zz = 0 = (V xx - V yy )/V zz TCD March

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35 9.3.2 Relaxation T 1 Spin lattice relaxation TCD March

36 T 2 Spin-spin relaxation TCD March

37 Bloch s Equations TCD March

38 9.3.2 Rotating frame TCD March

39 Bloch s equations in the rotating frame TCD March

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41 9.3.3 Pulsed nmr TCD March

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44 Spin echo TCD March

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48 A typical free induction decay, and its spectrum TCD March

49 9.4 Other resonance methods Mossbauer effect Recoilless fraction f = exp -k 2 <x 2 > F is the probability of a zero-phonon emission or absorption event in a solid source. E = hk 2 <x 2 > is rms displacement of the nucleus TCD March

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51 Conversion electron Mossbauer spectroscopy 57 Co (t 1/2 250d) Electron detector Emitted electron -ray surface t interface 57 Fe 5/2 7.3 kev conversion electron substrate 14.4 kev -ray 3/2 1/ kev -ray 3/2 1/2 Source Absorber TCD March

52 9.4.2 Muon spin rotation A muon is an unstable particle with spin 1/2 Charge ± e Mass 250 m e 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 TCD March

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72 8.5 Superparamagnetism TCD March

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74 8.6 Bulk nanostructures Recrystallization of amorphous Fe-Cu-Nb- Si-B to obtain a two-phase crystalline/ amorphous soft nanocomposite TCD March

75 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 ). TCD March

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77 Magnetostatics Volume charge Poisson s equarion Boundary condition e n 1. solid 2. air + + 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. TCD March