Concise History of NMR

1926 ‐ Pauli’s prediction of nuclear spin

Gorter 1932 ‐ Detection of nuclear magnetic moment by Stern using Stern molecular beam (1943 Nobel Prize) 1936 ‐ First theoretical prediction of NMR by Gorter; attempt to

detect the first NMR failed (LiF & K[Al(SO4)2]12H2O) 20K. 1938 ‐ Prof. Rabi, First detection of nuclear spin (1944 Nobel) 2015 Maglab Summer School 1942 ‐ Prof. Gorter, first published use of “NMR” ( 1967, Fritz Rabi Bloch London Prize)

Nuclear Magnetic Resonance 1945 ‐ First NMR, Bloch H2O , Purcell paraffin (shared 1952 Nobel Prize) in Condensed Matter 1949 ‐ W. Knight, discovery of Knight Shift 1950 ‐ Prof. Hahn, discovery of spin echo. Purcell 1961 ‐ First commercial NMR spectrometer Varian A‐60 Arneil P. Reyes Ernst 1964 ‐ FT NMR by Ernst and Anderson (1992 Nobel Prize) NHMFL 1972 ‐ Lauterbur MRI Experiment (2003 Nobel Prize) 1980 ‐ Wuthrich 3D structure of proteins (2002 Nobel Prize) 1995 ‐ NMR at 25T (NHMFL)

Lauterbur 2000 ‐ NMR at NHMFL 45T Hybrid (2 GHz NMR) Wuthrichd 2005 ‐ Pulsed field NMR >60T

Concise History of NMR ‐ Old vs. New Modern Developments of NMR Magnets Technical improvements parallel developments in electronics cryogenics, superconducting magnets, digital computers.

Advances in NMR Magnets 70 100T Superconducting 60 Resistive Hybrid 50 Pulse

40

Nb3Sn 30

NbTi 20 MgB2, HighTc nanotubes 10

0 1950 1960 1970 1980 1990 2000 2010 2020 2030

NMR in medical and industrial applications

¬ MRI, functional MRI ¬ non‐destructive testing ¬ dynamic information ‐ motion of molecules ¬ petroleum ‐ earth's field NMR , pore size distribution in rocks Condensed Matter ChemBio ¬ liquid chromatography, flow probes ¬ process control – petrochemical, mining, polymer production. Samples Materials, Crystals Complex molecules, proteins Magnetometers 63 27 207 139 1 13 15 14 Nuclei Cu, Al, Bi, La, … H, C, P, N, ¬ Pharmacology‐designer drugs Science Focus electronic correlations Molecular structure ¬ Quantum computing, nuclear qubits Spectra Broad lines, large BW –MHz Narrow lines & BW high res –Hz Signal strength Low S/N High S/N

Lifetime Short T1’s , ~us‐ms Long T1’s, 100ms‐10’s s Technique Simple pulse sequences Exotic pulse sequences Environment Cryogenic temperatures Room temperature Instrumentation Homemade systems Commercial spectrometers Peripheral Equip Sweepable magnets Fixed magnets Pressure , transport, Optics MAS, 2D, Multi‐D Cost $104 $106

1 NMR as a TOOL to study condensed matter systems More than 100 naturally occurring nuclei are NMR active!

¬Local, microscopic, site‐specific probe ‐ Virtually all elements are NMR active Transport leads ‐ study electronic spin /lattice structure

¬ Non‐invasive –no current, no contacts on the sample

‐ ωNMR ≈ 0 (neV‐μeV), low energy ¬ can be combined with other techniques: 2mm2 mm ‐ transport, magnetization, dielectric, optical, esr ‐ Maglab: extreme conditions: high field, temperature, pressure Related local techniques: Electron Spin Resonance (ESR) Neutron scattering Mössbauer Effect muon spin rotation (μSR) RDNMR Surface Coil

How is NMR useful in Condensed Matter Research? Nuclear Magnetic Resonance Phenomena

references: Hyperfine interaction C.P. Slichter, Principles of Magnetic Resonance, 3rd Ed. (Springer Verlag, 1989) A. Abragam. The Principles of Nuclear Magnetism (Clarendon Pres, Oxford, 1961). Fukushima and Roeder, Pulsed NMR Nuts and Bolts Approach (Wiley,1987)

Hhyp = I ⋅ A ⋅ S Behavior of the nuclear spin in magnetic field Electron cloud = Aiso I ⋅ S

nucleus μ = γNħI

γN: nuclear gyromagnetic ratio; fingerprint μ : magnetic moment E=hν ~ 10–9 -10–6 eV I : nuclear spin Energy (Hamiltonian) of the nuclear spin

in uniform magnetic field Ho

¬ interactions due to orbital, dipole, contact (electronic overlap) H Z = – μ · Ho = – γNħ I · Ho ¬ Nuclei are invisible spies to the electronic environment

Classical Treatment on nuclear spins Quantum theoretical Treatment

Heisenberg equation: ħ dI/dt = i [H , I ]

Torque acting on a magnetic moment: μ Ho = time derivative of the angular momentum with H = – γNħ I · Ho = – γNħ IZHo Ho z gives ħ dI/dt = μ Ho = γNħ I Ho

dIX/dt = – i γN Ho [ IZ , IX ] = γNIYHo

dIY/ddt = – γNIXHo dI /dt = 0 Larmor Precession Frequency: Z

¬ dI/dt = γN ( I × Ho ) ωL = γNHo ¬ identical to the classical expression. ¬ Radio frequency range! ~ kHz to ~ GHz

2 Typical Values of gyromagnetic ratios NMR Periodic Table

Nucleus Spin γΝ (MHz/T) 1H 1/2 42.5774 Copper 13C 1/2 10.7054

63Cu 3/2 11.285

65Cu 3/2 12.089

27Al 5/2 11.094 Oxygen 17O 5/2 5.7719

33As 3/2 7.2919

139La 7/2 6.0146

195In 9/2 9.3295

Resonance Condition – spin manipulation Spin Precession on a Bloch Sphere (on‐resonance)

¬ When an oscillating field is applied that matches the Larmor frequency, resonance will occur

ωo = ωL = γNHo

Ho

H1 H1

Lab frame Rotating frame M rotates on Oscillating magnetic field, H H1 1 y‐z plane

ω1 = γNH1

Spin Precession on a Bloch Sphere (off‐resonance) Quantum Mechanical Description of the Resonance Condition

Nuclear Zeeman levels in the presence of magnetic field Ho E = – m γ ħ I H m z N z o mZ = I Spin I γNHo transverse field ~ H1 cos ωt mZ = I – 1

causes transitions between mz and mz –1

V ~ – HXIX = – HX (I+ – I–) /2 mZ = – I when ω = γNHo I½I = ½ Fermi Golden Rule –½ γΝHo Selection rule: Δm = ± 1 + ½

Population difference is tiny Resonance condition: ωο = γΝHo

absorption

H1 Ho H

3 Experimental Setup MRI Setup

NMR Probe H0 z Cryostat H0 z

H1 H1

Resistive Magnet Sample

RF Coil

Pulsed NMR, observation of resonance Free Induction Decay and Phase Coherence

¬ 90º (π/2) pulse –tips the magnetization towards x’‐y’ plane ¬ Inhomogeneous magnetic field ‐introduces dephasing ‐ some spins precess faster than others γNH1 tw= π/2 ¬ signal decays (FID!) Precession induces voltage across the coil as a change in susceptibility V(t) ~ dM/dt Ho z Ho z ’ ’ HX HX t t

MZ

t FID t H x’ We want M’ 1 H1 x’ X this signal M X MX Lab frame Rotating frame t Rotating frame

Spin‐echoes

E. Hahn, Phys. Rev. 80, 5801 (1950) kilovolt pulses sub microvolt signals ¬ for broad lines, the FID may not be observable, due to limitation of electronics t

electronic “deadtime”

the race track analogy

A spin echo seen in the rotating frame

4 Pulse NMR Electronics Condensed Matter NMR User Facility at NHMFL Low Temperature Wideline NMR - probes electronic interactions in Condensed Matter Systems via electron- High BW ~ 1kV nuclear hyperfine coupling. Magnets • 25T 52mm bore, 1 ppm/mm resistive (Cell 6) 31T 32mm bore, 3 ppm/mm resistive (Cell 2), Optics (Cell 3) Dual axis Rotator • 45T hybrid, 32 mm bore, 25ppm/mm (Cell 15) • 12T 39mm, 40ppm/cm field-sweepable superconducting Resistively Detected NMR • 15T 40mm, 4ppm/cm field-sweepable superconducting (Simultaneous transport) • 17T 40mm, 10ppm/cm , sweepable superconducting • 18T 25mm, 100ppm, SC dil-fridge equipped (SCM1) milliKelvin Spectrometers and probes Dilution Fridge ~ 1uV • Five MagRes2000 homemade portable homodyne quadrature-detected console 2MHz-2GHz system, Optical pumping Labview interface, 25ns pulse widths, up to 600W OPNMR • 9 High Field Probes – >900 MHz, 20mK-350K vacuum sealed, ~micron to 10mm sample dia , single and dual axis goniometry, optical access, high pressure, stepper Uniaxial stress motor bottom tuning, simultaneous transport and NMR • Q=1 probe, top tuning for ultrawide frequency sweeps Cryogenics Pulse • 4 Adjustable flow VT cryostats- 1.4 to 325K, fast Fields Works like a Cell phone! cooldown, for 31mm bucket dewars • 3He sorption 350mK Janis cryostat High • 20mK-300mK Oxford Dilution Fridge (SCM1) Pressure SCH ready!

Homebuilt NMR Spectrometer Homebuilt NMR Spectrometer

Console 2MHz‐2GHz homodyne and Labview software developed in‐house. Console 2MHz‐2GHz homodyne and Labview software developed in‐house.

Magnet Systems Quadrupole Interaction

¬ For I > ½ nuclei, the nuclear quadrupole moment Q couples with the electric field gradient (EFG) ∇E arising from the surrounding electronic charge distribution with symmetry less than cubic. ∇E =

nd (i,j= X,Y,Z) : Cell 2 High homogeneity 2 rank tensor NMR grade magnet 31T, 45T Hybrid 20ppm/mm <4ppm/mm Quadrupole Hamiltonian:

+ ‐ ‐ + Field sweepable 15/17T superconducting magnet SCM1 dilution fridge ¬ Needs I >1/2 in non‐cubic environment. 4ppm/cm 100ppm/mm ¬ Useful for study of lattice deformations

5 Quadrupole Spectra The Hyperfine Interaction – manifestations in CM NMR Spectrum for a nucleus with I = 3/2 ¬ electron‐nuclear interaction (magnetic) NQR PERT NMR PURE NMR ‐ nuclear spins interact with the surrounding E |‐3/2〉 |‐3/2〉 electronic (spin or orbital) magnetic moments | ±3/2〉 |‐1/2〉 |‐1/2〉 = γ ħI · H ~ I · A · S Ho I = 3/2 |+1/2〉 H N hf hf | ±1/2〉 |+1/2〉 hyperfine field |+3/2〉 |+3/2〉 1st –Order Zeeman + Zeeman Quadrupole 1st ‐ Ord. Quadrupole Effective field acting on the nuclear spin

Heff = Ho + Hhf (r, t)

statistical average for the electronic system spatial and temporal function. νQ νo νQ νo

NUCLEAR QUADRUPOLE RESONANCE (NQR, H0=0) ¬ Electric quadrupole interaction induces magnetic transitions Two major effects: 1) static: shift of resonance frequency: Knight Shift, K FERROMAGNETIC NMR (zero‐field NMR) 2) dynamic: nuclear spin‐lattice relaxation, T ¬ Due to internal fields generated by ordered electronic moments 1

1. Static Measurements: the Knight Shift Knight Shift in Metals

¬ Shift of resonance due to “additional” field coming from within the material 1. Simple metals‐ temperature independent Pauli susceptibility Li Na Al Cu Sn Pb Spectrum: Spin echo ‐> Fourier Transform ‐> energy spectrum in frequency (or Field) domain K(%) 0.026 0.114 0.164 0.24 0.78 1.54 Field sweep or frequency sweep slope = γN(1+K ) “Spin” Knight shift – due to unpaired conduction electrons. ν Only s‐orbitals have finite probability at the nuclear position (r=0).

2 〈Ahf 〉 = (8π/3) gμB|ψ(0)| 〈S〉 slope = γN core s‐orbital 2. Transition metals, rare earths ν Δν o + –strongly T‐dependent Curie paramagnetism Η o χ = χ + χ + χ + χ time average of Hhf(r,t) dia s,p, d,f, orb,, Definition: K = K + K +K +K dia s,p, d,f, orb,, p or d‐orbital

Kd,f (T)= Ad,f χd,f (T) core polarization: Ad,f < 0 K = Δν/νo (usually in units of %) chemical shift σ – solely orbital in nature. In general, include orbitals and ω = 2πν = γN (1 + K ) Ho transferred fields from neighboring atoms, molecules.

2. Dynamic Measurements T1 and T2 Nuclear Spin‐lattice Relaxation ‐ approach to equilibrium ¬ measure of local magnetic field fluctuations

mZ = I N(mZ) = exp (– γNħ mZHo/kBT) I – 1 A. Spin lattice relaxation rate: ~ 1 – γNħ mZHo/kBT

¬ Interaction between nuclear spin system and external “lattice” (electrons, ¬ MZ eq ~ Ho/kBT phonons, etc.) I ‐ relaxation toward the Boltzmann distribution population nuclear relaxation π/2) pulse

MZ eq time: T1 rate: 1/T B. Spin decoherence (spin‐spin relaxation) rate: 1 t ¬ irrecoverable decay of the spin echo due to loss of phase coherence recovery: MZ(t) = Meq (1 – exp(– t /T1)

Application to MRI –contrast between bones and soft tissues, blood flow, Gd contrast.

6 T1 Measurement, FID T2 Measurement, echo

π/2 pulse t t’ t” exponential growth of FID

recovery: MZ(t) = Meq (1 – exp(– t /T1)

π/2 π t t

t’ t’

t” t”

exponential decay: MZ(t) = Meq exp(– t /T2)

NMR Lineshapes NMR in Superconductors

Distribution of local magnetic field ‐magnet inhomogeneity 1/T = |ρ(ε )|2 P(H) 1 F ‐anisotropic chemical shift K µ ρ(ε ) ‐quadrupole interaction s F energy gap ‐internal fields • Hebel‐Slichter peak, classic s‐wave pairing ‐local spin structure DOS ‐superconducting vortices 1/Τ1 Τ ‐nuclear dipolar fields H

After π/2 pulse: M (t) = P(H) cos(γ Ht) dH energy X ∫ N Τc Τ εF

MY(t) = ∫ P(H) sin(γNH t) dH • spin‐pairing, pairing‐symmetry Fourier transform of Κ exponential – isotropic gap power law- anisotropic

MX(t) + iMY(t) = ∫ P(H) exp(iγNHt) dH

gives the distribution function P(H). Τc Τ • pseudo‐gap behavior Τ * material behaves like a Κ superconductor above its transition temperature.

Τc Τ

NMR in Condensed Matter at NHMFL Examples ƒ Field driven new magnetic phases field‐induced states and Quadrupole splitting of NMR line at two identical sites but 90 degree apart phenomena

Materials and Physics 11 B NMR in SmB6 30K 200 MHz

ƒSDW,CDW, organics, oxides, perovskites, spinel B ƒManganites, ruthenates, cobaltites

ƒCarbon nanotubes, buckyballs A ƒRare earth intermetallics A B

ƒhigh Tc, FFLO, pseudo‐gap, Vortex structures, pnictides FT-Sum

F ƒNFL behavior, Heavy fermion superconductors, Kondo insulators ƒspin‐Peierls systems ƒAF multiferroics ƒweak ferromagnets, SDW Spin Fluids 14.85 14.90 14.95 15.00 Field (T) ƒmolecular nanomagnets ƒamorphous glasses ƒMagnetization plateau, BEC in frustrated dimers ƒFQHE, IQHE, Skyrmions in quantum‐well 2DEG systems

7 Examples Quadrupole split I=5/2 spectrum at 4 different crystal sites Examples

HZ + HQ HZeeman 100K O(4) Zero‐field Ferromagnetic NMR at atomic sites with 3 different valences H || c 55 0 -5/2 Mn NMR I=5/2 O(2,3) * -3/2 [Mn12O12(CH3COO)16(H2O)4].2CH3COOH.4H20 -1/2 * T 1 1/2 * 3/2 * *T1 * 5/2 * * I = 5/2 Mn4+ Mn3+

11.6 11.8 12.0 12.2 12.4 Field (T) Mn3+

Examples Topological Kondo Insulator SmB6 Nuclear Quadrupole Resonance of 2 Cu two isotopes and two sites, I=3/2 10 11 B NMR in SmB6 H||c

1 ρ(ε) ρ(ε) i

ρ(ε) f )

-1 0.1 ( s

1 ε = 0 δ/2 Δ/2 T δ /2 Δ /2

1/ o o

0.01 [111] 1.2T 1.16T 13.9T 2.1T 20.9T 6.07T 37T TI band structure Hybridization gap

10 100 Temperature (K)

11B Field dependent relaxation Energy level diagram and model density of states I = 3/2 ∇ E 63 ±3/2 T. Caldwell, A. P. Reyes, W. G. Moulton, P. L. Cu Kuhns, M. J. R. Hoch, P. Schlottmann, and Z. 65Cu Fisk, Phys Rev B 75, 075106 (2007). SmB6:Topological Insulator, Takimoto, JPSJ 80, ±1/2 123710(2011) In-gap states and field suppression of gap

Examples Spin-Nematic Phase in Frustrated AF LiCuVO4 (New state of matter) NMR Lineshape transition to superconducting state 17O central transition YBCO

•Phase transition at 40T •Spin-nematic - new exotic state of matter similar to liquid crystals •Rotational symmetry, no LR spin order •Results of competition between AF and FM interaction •Magnon pairs undergo BEC above Tc ~ 40T. •NMR shows narrowing of line where all magnons line up with field

Idealized NMR lineshape due to vortex

Buettgen et al. (2013)

8 Examples Hyperfine field from ordered moments: 122 pnictides NMR determination of hyperfine field from ordered moments 122 pnictides

THE END

NOTES

Appendix 1: Nuclear Spin-Lattice Relaxation ∞ (general formulation) using δ(En – Em+ ħωo ) = 1/(2πħ) ∫–∞ dt exp {i[(En – Em)/ħ + ωo]t} ¬ dynamical fluctuations of electronic magnetic moments ∞ W = ¼γ 2 ∫ dt Σ |〈 m | H + | n 〉|2 exp {i[(E – E )/ħ + ω ]t} exp(– E /k T) ‐ unpaired electron spin exchanges energy with nucleus, causes transition –½ ↔ ½ N –∞ m,n hf n m o m B ∞ = ¼γ 2 ∫ dt Σ 〈 n | H – | m 〉 〈 m | H +(t) | n 〉 exp (iω t) I = 1/2 N –∞ m,n hf hf o S –½ I S ħωo + – ½ since 〈 m | Hhf | n 〉∗ = 〈 n | Hhf | m 〉 and time dependent (Heisenberg representation)

+ + typical process in metals Hhf (t) = exp (iH t/ħ) Hhf exp (–iH t/ħ) + + 〈 m | Hhf | n 〉 exp [ i(En – Em)t/ħ ]= 〈 m | Hhf (t) | n 〉 H = γNħI · Hhf = γ ħ [ I H z + ½ (I H – + I H + ) ; I = I + iI and I = I – iI ∞ N Z hf + hf – hf + X Y – X Y W = ¼γ 2 ∫ dt 〈 H – H +(t) 〉 exp (iω t) perturbation causing transition –½ ↔ ½ –½ ↔ ½ N –∞ hf hf o time correlation function of the hyperfine field (statistical thermal average) Transition probability (Golden Rule) 55 1/T1 = 2W–½ ↔ ½ ; if Hhf = Ahf Si e.g. Mn in MnO, etc. + 2 W–½ ↔ ½ = 2π/ħ Σm,n |〈 –½,m | –½γ ħI– H | ½, n 〉| exp(– En/kBT) δ(En – Em+ ħωo ) ∞ N hf 2 2 + – 1/T1 = ½ γN A hf ∫–∞ dt 〈 S (t) S (0)〉 exp (iωot) where |n〉, |m〉 are electronic states 2 + 2 relaxation is given by the spin auto‐correlation function. = ½πγN ħ Σm,n |〈 m | Hhf | n 〉| exp(– En/kBT) δ(En – Em+ ħωo )

9 Example: interacting localized moments, 4f, 3d electrons Relation to dynamical susceptibility τ : correlation time • response to space-time varying field J c ħ/τc ~ J >> ωo H(r,t) = Hq exp[ i(q·r – ωt)] S(r,t) = Sq exp[ i(q·r – ωt)]

+ – ⅔S(S+1) 〈 S (t) S (0)〉 linear response spin system

Dynamical susceptibility: χ (q, ω) = Sq/Hq Imaginary part: χ" (q, ω) Æ dissipation of the system t τc Fluctuation-dissipation theorem: ∞ 1/T = ½ γ 2 A2 [⅔S(S+1)] τ + – 1 N hf c χ" (q, ω) = ω/kBT ∫–∞ dt 〈 Sq (t) Sq (0)〉 exp (iωt) ; kBT >> ω ∞ and Σ A(q, ω) = ∫ dt 〈 S +(t) S – (0)〉 exp (iωt) independent of temperature! q –∞ i i S1 A1 2 xx 2 yy 2 1/T1 = ½ γN kBT Σq |Aq | χ"xx(q, ω)/ω +|Aq | χ"yy(q, ω)/ω S3 I where form factor: Aq = Σi Ai exp (i q·r ) and Ho||z A3

isotropic local A case: S2 A2 2 2 1/T1 = ½ γN A kBT Σq χ"(q, ω)/ω

Appendix 2: Korringa Relation Appendix 3: The Hyperfine Interaction Korringa relation ( free electron) k’ S 3 5 3 Raman scattering process: S I H el-n = (8π/3) gμBγNħδ(r) I · S – gμBγNħ I · [ S/r – 3r(S · r)/r ] – gμB γNħ I· l /r

k fermi-contact (s-states) spin dipolar (non-s) orbital (non-s)

Effective field for the nuclear spins + 2 1/T1 = 4π/ħ Σk,k’ |〈 k’↑| –½γNħAS |k↓ 〉| f(ε)[(1– f(ε)] δ(εk – εk’+ ħωo ) 3 5 3 〈Hhf 〉 = (8π/3) gμB〈δ (r)S 〉 – gμB〈 S/r – 3r(S · r)/r 〉 –2μB 〈l /r 〉 expectation value for particular state = πγ 2 ħA2 dε dε’ f(ε)[(1– f(ε)] δ(ε – ε ) ; ω ~ 0 N ∫∫ k k’ o f(ε) first and second term ≠ 0 for unpaired electrons last term ≠ 0 for electrons in open shell k T ρ(ε ) ρ(ε ) B F F ε ε 2 2 2 F Finite 〈Hhf 〉 examples = πγN ħA kBT |ρ(εF)| 1. Ferromagnetic materials ( Fe, Co, Ni ...) La Sr CoO 1-x x 3 0.1 spin Knight shift K ∝ Αχ ∝ Αρ(ε ) ; Fermi gas, non-interacting magnetization: M = gμ 〈S〉 ≠ 0 0.14 s s F B 0.18 4000 spectra as afunction of x 2 0.20 0 external field field 〈H 〉 = (8π/3) gμ |ψ(0)| 〈S〉= H 3500 0.30 hf B int Pulse Width 6.0ps 0.40 2 2 3000 T TK = (ħ/4πk ) (γ / γ ) 0.5 1 s B e N 0.51 2500 -resonance field is observed at zero external field at ωN = γNHint 2000 59 1500 enhancement over Korringa constant for highly correlated electrons. Co 230MHz 22.9 T (AU) amplitude 1000 57 Li Na Rb Cu Al Fe 46.5MHz 33.8 T 500 T (expt, ms.) 150 15.9 2.75 3.0 6.3 61Ni 28.5MHz 7.5 T 0 1 -500 55 50 100 150 200 250 T1(Korringa,ms.) 88 10 2.1 2.3 5.1 Mn 375MHz 35.7 T frequency MHz

2. Paramagnetic materials (linear response to external fields) 3. superconductors 2 Mspin = gμB 〈 S 〉 = χspin Ho conduction electron susceptibility χspin = 2μB Σk ∂fk/ ∂εk Morb = gμB 〈 l 〉 = χorb Ho ~ Σk (εF – εk) = ρ(εF ) (density of states) magnetic susceptibility 2 2 1/2 2 3 quasiparticle energy (superconductor, s‐wave) Ek = (εk + Δ ) gap 〈Hhf 〉 = (8π/3)|ψ(0)| χspin Ho + dipolar + 〈2/r 〉 χorb Ho

spin susceptibility: χspin (T → 0)∝ exp (-Δ/kBT) hyperfine coupling constant, Ahf energy gap Knight shift, definition in metals

K = 〈Hhf 〉 / Ho = Ahf χ Î ω = (1+K)γNH DOS

(chemical shift σ – solely orbital in nature) In general, include orbitals and transferred fields from neighbors εF energy

K = As χs,spin + Ap(d,f...) χp(d,f..)spin + Bp(d,f...) χp(d,f..)orb Κ exponential – isotropic gap Core polarization effect – spin polarization of p, d, f states produces a spin-dependent power law- anisotropic exchange potential for the inner core s-state, resulting in a spin polarization of the inner s- state in the opposite direction.

Τc Τ

10