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

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).

Nuclear Magnetic Resonance Consider nucleus with magnetic moment μ and angular momentum  I. γ = gyromagnetic ratio In an applied field, → I = ½, m I =  ½ Resonance at γ [s -1 G –1 ]ν [MHz] proton  B 0 [kG] electron  B 0 [kG] Nuclear magneton

Equations of Motion Gyroscopic equation:→→ In thermal equilibrium, M // B a. → with For I = ½, where N 1 is the density of population of the lower level →  If system is slightly out of equilibrium, the relaxation towards equilibrium can be described by T 1 = spin-lattice (longitudinal) relaxation time

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

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

and in the presence of relaxation:For T 2 = transverse relaxation time → relaxation of M x & M y doesn’t affect U. With initial conditions where we have

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

Line Width Magnetic field seen by a magnetic dipole μ 1 due to another dipole μ 2 is →  nearest neighbor interaction dominant: For protons 2A apart, H 2 O

Motional Narrowing Li 7 NMR in metallic Li diffusion ratehighlow rigid lattice Motion narrowed τ = diffusion hopping time Effect is more prominent in liquid, e.g., proton line in water is 10 –5 the width of that in ice. (rotational motion)

T 2 ~ time for spin to dephase by 1 radian due local perturbation B i. After t = n τ (random walk)  Average number of steps for a spin to dephase by 1 radian is whereas for a rigid lattice → since H 2 O: τ ~10 ̶ 10 s

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

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

Examples: Paramagnetic Point Defects F centers in Alkali Halides ( negative ion vacancy with 1 excess e ) K 39, I = 3/2 Vacancy surrounded by 6 K 39 nuclei → Number of hf components: 2I max +1 = 19 Number of ways to arrange the 6 spins: ( 2I + 1) 6 = 4 6 = 4096

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

Knight Shift Knight (metallic) shift: B 0 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 : a atom ≠ a metal → for Li, | ψ metal (0)| 2 ~ 0.44|ψ atom (0)| 2

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

Ferromagnetic Resonance Similar to NMR with S = total spin of ferromagnet. Magnetic selection rule: Δ m S =  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 ).

Shape Effects in FMR Consider an ellipsoid sample of cubic ferromagnetic insulator with principal axes aligned with the Cartesian axes. B i = internal field. B 0 = external field. N = demagnetization tensor Lorenz field = (4 π / 3)M. Exchange field = λ M. Bloch equations: → → ( don’t contribute to torque) FMR frequency:uniform mode

For a spherical sample,→ For a plate  B 0, → For a plate // B 0,→ Polished sphere of YIG at 3.33GHz & 300K for B 0 // [111] Shape-effect experiments determine γ & hence g. FeCoNi g

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

Antiferromagnetic Resonance Consider a uniaxial antiferromagnet with spins on 2 sublattices 1 & 2. LetB A = anistropy field derived from θ 1 = angle between M 1 & z-axis. → Exchange fields: For

Withthe linearized Bloch equations become: → exchange field  AFMR frequency

MnF 2 : T N = 67K

Electron Paramagnetic Resonance Consider paramagnet with exchange J between n.n. e spins at T >> T C. Observed line widths << those due to U dipole-dipole. → Exchange narrowing Treating ω ex  J /  as a hopping frequency 1/τ, the exchange induced motion-narrowing effect gives For the paramagnetic organic crystal DPPH (DiPhenyl Picryl Hydrazyl), also known as the g marker (used for H calibration), Δ ω ~ 1.35G is only a few percent of (Δ ω) 0 (Δ ω) 0 = dipolar half-width Zero-Field Splitting Some paramagnetic ions has ground state crystal field splittings of Hz (~MW). E.g. Mn 2+ as impurities gives splittings of Hz.

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

Three-Level Maser Population inversion is attained by pumping saturation: n 3  n 1 Steady state at saturation: Er 3+ are often used in fibre optics amplifiers ( n 2 > n 1 mode). Signal: λ ~ 1.55  m, bandwidth ~ 4  Hz. n 3 > n 2 n 2 > n 1

Ruby Laser Cr 3+ in Ruby Optical pumping by xenon flash lamps For Cr 3+ cm −3, stored U ~ 10 8 erg cm −3. → High power pulse laser. Efficiency ~ 1%. 4 level Nd glass laser Continuous lasing: no need to empty G.S.