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13. Magnetic Resonance

13. Magnetic Resonance. Nuclear Magnetic Resonance Equations of Motion Line Width Motional Narrowing Hyperfine Splitting Examples: Paramagnetic Point Defects F Centers in Alkali Halides Donor Atoms in Silicon Knight Shift Nuclear Quadrupole Resonance Ferromagnetic Resonance

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13. Magnetic Resonance

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  1. 13. Magnetic Resonance • Nuclear Magnetic Resonance • Equations of Motion • Line Width • Motional Narrowing • Hyperfine Splitting • Examples: Paramagnetic Point Defects • F Centers in Alkali Halides • Donor Atoms in Silicon • Knight Shift • Nuclear Quadrupole Resonance • Ferromagnetic Resonance • Shape Effects in FMR • Spin Wave Resonance • Antiferromagnetic Resonance • Electron Paramagnetic Resonance • Exchange Narrowing • Zero-Field Splitting • Principle of Maser Action • Three-Level Maser • Ruby Laser

  2. Notable Resonance Phenomena / Instruments: NMR: Nuclear Magnetic Resonance NQR: Nuclear Quadrupole Resonance EP(S)R: Electron Paramagnetic (Spin) Resonance FMR: Ferromagnetic Resonance SWR: Spin Wave Resonance AFMR: Anti-Ferromagnetic Resonance CESR: Conduction Electron Spin Resonance • Information gained: • Fine struction of absorption: Electronic structure of individual defects. • Change in line width: Motion of the spin or its surroundings. • Chemical or Knight shift: Internal magnetic field felt by the spin. • Collective spin excitations. Prototype of all resonance phenomena is NMR. • Main applications of NMR: • Identification & structure determination for organic / biochemical componds. • Medical (MRI).

  3. Nuclear Magnetic Resonance Consider nucleus with magnetic moment μand angular momentum  I. γ = gyromagnetic ratio In an applied field, → Resonance at I = ½ , mI= ½ Nuclear magneton

  4. Equations of Motion Gyroscopic equation: → → In thermal equilibrium, M // Ba. with → For I = ½ , where N1 is the density of population of the lower level →  If system is slightly out of equilibrium, the relaxation towards equilibrium can be described by T1 = spin-lattice (longitudinal) relaxation time

  5. → Let an unmagnetized specimen be placed at t = 0 in field Then so that →

  6. Dominant relaxation mechanism: phonon absorption + re-emission phonon emission phonon inelastic scattering

  7. For and in the presence of relaxation: T2 = transverse relaxation time → relaxation of Mx & My doesn’t affect U. With initial conditions we have where

  8. With an additional transverse rf field and setting d Mz / d t = 0, we have The particular solutions are obtained by setting →  Half-width =

  9. Line Width Magnetic field seen by a magnetic dipole μ1 due to another dipole μ2 is →  nearest neighbor interaction dominant: For protons 2A apart, H2 O

  10. Motional Narrowing Li7 NMR in metallic Li rigid lattice τ = diffusion hopping time Motion narrowed low diffusion rate high Effect is more prominent in liquid, e.g., proton line in water is 10–5 the width of that in ice. (rotational motion)

  11. T2 ~ time for spin to dephase by 1 radian due local perturbation Bi . After t = nτ(random walk)  Average number of steps for a spin to dephase by 1 radian is → whereas for a rigid lattice since H2O: τ ~10 ̶ 10 s

  12. Hyperfine Splitting Hyperfine interaction : between μnucl = μI & μe Contact hyperfine interaction: when e is in L = 0 state L = 0 → Bohr magneton current loop Compton wavelength Dirac: μB ~ circulation of e with v = c, Current in loop = → field at loop center = Probability of e overlaping the nucleus : Average field seen by nucleus: Contact hyperfine interaction:

  13. intersellar H High field: μB B >> a S = I = ½ Selection rule for e: ΔmS = 1 ΔmI = 0 Selection rule for nucl: ΔmS = 0 Number of hf splittings = (2I + 1) (2S+1)

  14. Examples: Paramagnetic Point Defects F centers in Alkali Halides ( negative ion vacancy with 1 excess e ) K39 , I = 3/2 Vacancy surrounded by 6 K39 nuclei → Number of hf components: 2Imax +1 = 19 Number of ways to arrange the 6 spins: ( 2I + 1)6 = 46 = 4096

  15. Donor Atoms in Silicon P in Si (outer shell 3s23p3): 4e’s go diamagnetically into covalent bonding; 1e acts as paramagnetic center of S = 1/2. motion narrowing due to rapid hopping

  16. Knight Shift Knight (metallic) shift: B0 required to achieve the same nuclear resonance ω for a given spin depends on whether it is embedded in a metal or an insulator. For conduction electrons: → Knight shift : aatom ≠ ametal → for Li, | ψmetal(0)|2 ~ 0.44|ψatom(0)|2

  17. Nuclear Quadrupole Resonance Q in field gradient Q > 0 Nuclei of spin I 1 have electric quadrupole moment. Ref: C.P.Slichter, “Principles of Magnetic Resonance”, 2nd ed., Chap 9. Q > 0 for convex (egg shaped) charge distribution. Wigner –Eckart Theorem: Axial symmetry: Number of levels =  I  + 1 Built-in field gradient (no need for H0 ) . App. : Mine detection.

  18. Ferromagnetic Resonance Similar to NMR with S = total spin of ferromagnet. Magnetic selection rule: ΔmS = 1. • Special features: • Transverse χ & χ very large ( M large). • Shape effect prominent (demagnetization field large). • Exchange narrowing • (dipolar contribution suppressed by strong exchange coupling). • Easily saturated (Spin waves excited before rotation of S ).

  19. Shape Effects in FMR Consider an ellipsoid sample of cubic ferromagnetic insulator with principal axes aligned with the Cartesian axes. Bi = internal field . B0 = external field. N = demagnetization tensor Lorenz field = (4 π / 3)M. Exchange field = λM. ( don’t contribute to torque) Bloch equations: → → FMR frequency: uniformmode

  20. For a spherical sample, → For a plate  B0, → For a plate // B0, → Shape-effect experiments determine γ & hence g. Polished sphere of YIG at 3.33GHz & 300K for B0 // [111]

  21. Spin Wave Resonance Spin waves of odd number of half-wavelenths can be excited in thin film by uniform Brf Condition for long wavelength SWR: D = exchange constant For wave of n half-lengths: Permalloy (80Ni20Fe) at 9GHz

  22. Antiferromagnetic Resonance Consider a uniaxial antiferromagnet with spins on 2 sublattices 1 & 2. Let BA = anistropy field derived from θ1 = angle between M1 & z-axis. → Exchange fields: For

  23. With the linearized Bloch equations become: → exchange field  AFMR frequency

  24. MnF2 : TN = 67K

  25. Electron Paramagnetic Resonance Consider paramagnet with exchange J between n.n. e spins at T >> TC . Observed line widths << those due to Udipole-dipole . → Exchange narrowing Treating ωex J /  as a hopping frequency 1/τ , the exchange induced motion-narrowing effect gives (Δω)0 = dipolar half-width For the paramagnetic organic crystal DPPH (DiPhenyl Picryl Hydrazyl), also known as the g marker (used for H calibration), Δω ~ 1.35G is only a few percent of (Δω)0 Zero-Field Splitting Some paramagnetic ions has ground state crystal field splittings of 1010 - 1011 Hz (~MW). E.g. Mn2+ as impurities gives splittings of 107 - 109 Hz.

  26. Principle of Maser Action Maser = Microwave Amplification by Stimulated Emission of Radiation Laser = Light Amplification by Stimulated Emission of Radiation Transition rate per atom (Fermi’s golden rule): Net radiated power: Ambient: Brf at ω thermal equilibrium → nu << nl . stimulated emission → nu >> nl . (inversion) In an EM cavity of volume V and Q factor Q, power loss is Maser condition: → line-width

  27. Three-Level Maser Population inversion is attained by pumping n3 > n2 saturation: n3 n1 n2 > n1 Steady state at saturation: Er3+ are often used in fibre optics amplifiers ( n2 > n1 mode). Signal: λ ~ 1.55 m, bandwidth ~ 41012 Hz.

  28. Ruby Laser Cr3+ in Ruby Optical pumping by xenon flash lamps For 1020 Cr3+ cm−3 , stored U ~ 108 erg cm−3 . → High power pulse laser. Efficiency ~ 1%. Continuous lasing: no need to empty G.S. 4 level Nd glass laser

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