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1 2016 Maglab Summer School Nuclear Magnetic Resonance in Condensed Matter Arneil P. Reyes NHMFL Concise History of NMR 1926 Pauli’s prediction of nuclear spin 1932 Detection of nuclear magnetic moment by Stern using molecular beam (1943 Nobel Prize) 1936 First theoretical prediction of NMR by Gorter; attempt to detect the first NMR failed (LiF & K[Al(SO 4 )2]12H 2 O) 20K. 1938 Prof. Rabi, First detection of nuclear spin (1944 Nobel) 1942 Prof. Gorter, first published use of “NMR” ( 1967, Fritz London Prize) 1945 First NMR, Bloch H O , Purcell paraffin (shared 1952 Gorter Stern Rabi Bloch 1945 First NMR, Bloch H 2 O , Purcell paraffin (shared 1952 Nobel Prize) 1949 W. Knight, discovery of Knight Shift 1950 Prof. Hahn, discovery of spin echo. 1961 First commercial NMR spectrometer Varian A60 1964 FT NMR by Ernst and Anderson (1992 Nobel Prize) 1972 Lauterbur MRI Experiment (2003 Nobel Prize) 1980 Wuthrich 3D structure of proteins (2002 Nobel Prize) 1995 NMR at 25T (NHMFL) 2000 NMR at NHMFL 45T Hybrid (2 GHz NMR) 2005 Pulsed field NMR >60T Purcell Ernst Lauterbur Wuthrichd Concise History of NMR Old vs. New Technical improvements parallel developments in electronics cryogenics, superconducting magnets, digital computers. Modern Developments of NMR Magnets Advances in NMR Magnets 50 60 70 Superconducting Resistive Hybrid Pulse 100T 0 10 20 30 40 1950 1960 1970 1980 1990 2000 2010 2020 2030 32T High Tc NbTi Nb3Sn ChemBio Complex molecules, proteins 1 H, 13 C, 15 P, 14 N, Molecular structure Condensed Matter Materials, Crystals 63 Cu, 27 Al, 207 Bi, 139 La, … electronic correlations Samples Nuclei Science Focus Molecular structure Narrow lines & BW high res – Hz High S/N Long T 1 ’s, 100ms10’s s Exotic pulse sequences Room temperature Commercial spectrometers Fixed magnets MAS, 2D, MultiD $10 6 electronic correlations Broad lines, large BW – MHz Low S/N Short T 1 ’s , ~usms Simple pulse sequences Cryogenic temperatures Homemade systems Sweepable magnets Pressure , transport, Optics $10 4 Science Focus Spectra Signal strength Lifetime Technique Environment Instrumentation Peripheral Equip Cost NMR in medical and industrial applications MRI, functional MRI nondestructive testing dynamic information motion of molecules petroleum earth's field NMR , pore size distribution in rocks liquid chromatography, flow probes process control – petrochemical, mining, polymer production. Magnetometers Pharmacologydesigner drugs Quantum computing nuclear qubits Quantum computing, nuclear qubits
Transcript
Page 1: Nuclear Magnetic Resonance Rabi Bloch in Condensed Matter · NMR Periodic Table Resonance Condition – spin manipulation ¬When an oscillating field is applied that matches the Larmor

1

2016 Maglab Summer SchoolNuclear Magnetic Resonanceg

in Condensed Matter

Arneil P. ReyesNHMFL

Concise History of NMR

1926 ‐ Pauli’s prediction of nuclear spin 1932 ‐ Detection of nuclear magnetic moment by Stern using 

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)1942 ‐ Prof. Gorter, first published use of “NMR” ( 1967, Fritz 

London Prize)   1945 ‐ First NMR, Bloch H2O , Purcell paraffin (shared 1952

GorterStern

Rabi Bloch

1945 First NMR, Bloch H2O , Purcell paraffin (shared 1952 Nobel Prize)

1949 ‐W. Knight, discovery of Knight Shift1950 ‐ Prof. Hahn, discovery of spin echo.1961 ‐ First commercial NMR spectrometer Varian A‐60 1964 ‐ FT NMR by Ernst and Anderson (1992 Nobel Prize)1972 ‐ Lauterbur MRI Experiment (2003 Nobel Prize)1980 ‐Wuthrich 3D structure of proteins (2002 Nobel Prize)1995 ‐ NMR at 25T (NHMFL)2000 ‐ NMR at NHMFL  45T Hybrid (2 GHz NMR)2005 ‐ Pulsed field NMR >60T

Purcell Ernst

LauterburWuthrichd

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

Modern Developments of NMR Magnets

Advances in NMR Magnets

50

60

70

SuperconductingResistiveHybridPulse

100T

0

10

20

30

40

1950 1960 1970 1980 1990 2000 2010 2020 2030

32THigh Tc

NbTi

Nb3Sn

ChemBio

Complex  molecules, proteins1H, 13C, 15P, 14N, Molecular structure

Condensed Matter  

Materials, Crystals63Cu, 27Al, 207Bi, 139La, … electronic correlations

SamplesNuclei

Science Focus Molecular structureNarrow lines & BW high res – HzHigh S/NLong T1’s, 100ms‐10’s sExotic pulse sequencesRoom temperatureCommercial spectrometersFixed magnetsMAS, 2D, Multi‐D$106

electronic correlationsBroad lines, large BW – MHzLow S/NShort T1’s , ~us‐msSimple pulse sequencesCryogenic temperaturesHomemade systemsSweepable magnetsPressure , transport, Optics$104

Science FocusSpectra

Signal strengthLifetime

TechniqueEnvironment

InstrumentationPeripheral  Equip

Cost

NMR in medical and industrial applications

MRI, functional MRInon‐destructive testingdynamic information ‐motion of moleculespetroleum ‐ earth's field NMR , pore size 

distribution in rocksliquid chromatography, flow probesprocess control – petrochemical, mining, polymer 

production.Magnetometers

Pharmacology‐designer drugsQuantum computing nuclear qubitsQuantum computing, nuclear qubits

Page 2: Nuclear Magnetic Resonance Rabi Bloch in Condensed Matter · NMR Periodic Table Resonance Condition – spin manipulation ¬When an oscillating field is applied that matches the Larmor

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NMR as a TOOL  to study condensed matter systems

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

Non‐invasive – no current, no contacts on the sample‐ ωNMR ≈ 0 (neV‐μeV), low energycan be combined with other techniques: 2 mm

Transport leads

can be combined with other techniques: ‐ transport, magnetization, dielectric, optical, esr‐Maglab:  extreme conditions: high field, temperature, pressure

Related  local techniques: Electron Spin Resonance (ESR)Neutron scatteringMössbauer Effectmuon spin rotation (μSR), β‐NMR

2 mm

RDNMR Surface Coil

More than 100 naturally occurring nuclei are NMR active!

How is NMR useful in Condensed Matter Research?

Hyperfine interaction

Hhyp = I ⋅ A ⋅ S= Aiso I ⋅ S Electron cloud

nucleus

interactions due to orbital, dipole, contact (electronic overlap) Nuclei are invisible spies to the electronic environment

E = hν ~ 10–9 - 10–6 eV

references: 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)

Behavior of the nuclear spin in magnetic field

μ = γNħI

Nuclear Magnetic Resonance Phenomena

Energy  (Hamiltonian) of the nuclear spin in uniform magnetic field Ho

H Z =  – μ · Ho = – γNħ I · Ho

γN: nuclear gyromagnetic ratio; fingerprintμ : magnetic momentI : nuclear spin

Torque acting on a magnetic moment: μ × Ho= time derivative of the angular momentum

ħ dI/dt = μ × Ho = γNħ I × Ho

Classical Treatment on nuclear spins

Ho z

Larmor Precession Frequency:

Radio frequency range!    ~ kHz to ~ GHz

ωL = γNHo

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

with  H = – γNħ I · Ho = – γNħ IZHogives

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

dI /d I H

Quantum theoretical Treatment

dIY/dt = – γNIXHo

dIZ/dt = 0

dI/dt = γN ( I × Ho )

identical to the classical expression.

Page 3: Nuclear Magnetic Resonance Rabi Bloch in Condensed Matter · NMR Periodic Table Resonance Condition – spin manipulation ¬When an oscillating field is applied that matches the Larmor

3

Typical Values of gyromagnetic ratios

Copper

Nucleus1H

13C

63Cu

65Cu

γΝ (MHz/T)42.5774

10.7054

11.285

12.089

Spin1/2

1/2

3/2

3/2

Oxygen27Al

17O

33As

139La

195In

11.094

5.7719

7.2919

6.0146

9.3295

5/2

5/2

3/2

7/2

9/2

NMR Periodic Table

Resonance Condition – spin manipulation

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

ωo = ωL = γNHo

Ho

Lab frame Rotating frameM rotates  on   y‐z planeω1 = γNH1

H1

Oscillating magnetic field, H1

H1

Spin Precession on a Bloch Sphere (on‐resonance)

H1

Spin Precession on a Bloch Sphere (off‐resonance)

H1

Nuclear Zeeman levels in the presence of magnetic field Ho

Em = – mzγNħ IzHo

transverse field ~ H1 cos ωt

causes transitions between mz and mz – 1

V ~ – HXIX = – HX (I+ – I–) /2

when ω = γNHo

mZ = I

mZ = I – 1γNHo

mZ = – I

Quantum Mechanical Description of the Resonance Condition

I ½

Spin I

Fermi Golden RuleSelection rule: Δm = ± 1

Population difference is tiny

I = ½

Resonance condition: ωο = γΝHo

absorption

HHo

– ½

+ ½γΝHo

Page 4: Nuclear Magnetic Resonance Rabi Bloch in Condensed Matter · NMR Periodic Table Resonance Condition – spin manipulation ¬When an oscillating field is applied that matches the Larmor

4

H0 z

H1

Experimental Setup

Sample

Cryostat

NMR Probe

Resistive Magnet

RF Coil

Sample

H0 z

H1

MRI Setup

90º (π/2) pulse –tips the magnetization towards x’‐y’ planeγNH1 tw= π/2

Precession induces voltage across the coil as a change in susceptibility

V(t)  ~ dM/dt

HX

MZ

t

Pulsed NMR, observation of resonance

Ho z

Z

M’X

M X

t

H1 x’

Lab frame Rotating frame

Large amount of power (~1kW) for short time (~ us)

Inhomogeneous magnetic field ‐introduces dephasing‐ some spins precess faster than otherssignal decays (FID!)

HX

t

Free Induction Decay and Phase Coherence

Ho z

t

MXt

H1 x’

Rotating frame

FIDWe want this signal

for broad lines, the FID may not be observable, due to limitation of electronics t

kilovolt pulsessub microvolt signals

electronic “deadtime”

Spin‐echoes (Hahn echoes)E. Hahn, Phys. Rev. 80, 5801 (1950)

A spin echo seen in the rotating frame

the race track analogy

Page 5: Nuclear Magnetic Resonance Rabi Bloch in Condensed Matter · NMR Periodic Table Resonance Condition – spin manipulation ¬When an oscillating field is applied that matches the Larmor

5

Pulse NMR Electronics

~ 1kV

Receive

Transmit

~ 1uV

Pulse NMR Electronics

Works like a Cell phone!

Low Temperature Wideline NMR - probes electronic interactions in Condensed Matter Systems via electron-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)

• 45T hybrid, 32 mm bore, 25ppm/mm (Cell 15)• 12T 39mm, 40ppm/cm field-sweepable superconducting • 15T 40mm, 4ppm/cm field-sweepable superconducting • 17T 40mm, 10ppm/cm , sweepable superconducting• 18T 25mm, 100ppm, SC dil-fridge equipped (SCM1)

Condensed Matter NMR User Facility at NHMFL

High BW

Dual axis Rotator

Resistively Detected NMR(Simultaneous transport)

milliKelvinSpectrometers and probes• Five MagRes2000 homemade portable homodyne quadrature-detected console 2MHz-2GHz system, Labview interface, 25ns pulse widths, up to 600W

• 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 motor bottom tuning, simultaneous transport and NMR

• Q=1 probe, top tuning for ultrawide frequency sweeps

Cryogenics• 4 Adjustable flow VT cryostats- 1.4 to 325K, fast cooldown, for 31mm bucket dewars

• 3He sorption 350mK Janis cryostat• 20mK-300mK Oxford Dilution Fridge (SCM1)SCH ready!

Uniaxial stress

Optical pumpingOPNMR

HighPressure

Pulse Fields

milliKelvinDilution Fridge

Homebuilt  NMR  Spectrometer

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

Homebuilt  NMR  Spectrometer

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

45T Hybrid 20ppm/mm

Magnet Systems

Cell 2 High homogeneity NMR grade magnet 31T, 

Field sweepable 15/17T superconducting magnet  4ppm/cm

<4ppm/mm

SCM1 dilution fridge 100ppm/mm

Page 6: Nuclear Magnetic Resonance Rabi Bloch in Condensed Matter · NMR Periodic Table Resonance Condition – spin manipulation ¬When an oscillating field is applied that matches the Larmor

6

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.

2nd rank tensor  (i,j= X,Y,Z) :

Quadrupole Interaction

∇E =

Quadrupole Hamiltonian:

+

+

‐‐

Needs I >1/2 in non‐cubic environment. Useful for study of lattice deformations

Spectrum for a nucleus with I = 3/2NQRNQR

1st – OrderQuadrupole

| ±3/2⟩

| ±1/2⟩

Zeeman +1st Ord Quadrupole

|‐3/2⟩

|‐1/2⟩|+1/2⟩|+3/2⟩

PERT NMRPERT NMREE

I = 3/2I = 3/2

Quadrupole Spectra

Zeeman

|‐3/2⟩

|‐1/2⟩|+1/2⟩

|+3/2⟩

PURE NMRPURE NMR

Quadrupole 1st ‐ Ord. Quadrupole

FERROMAGNETIC  NMR  FERROMAGNETIC  NMR  (zero‐field NMR)Due to internal fields generated by ordered  electronic moments

νQνQ νoνo

NUCLEAR QUADRUPOLE RESONANCE NUCLEAR QUADRUPOLE RESONANCE (NQR, H0=0)Electric quadrupole interaction induces magnetic transitions

electron‐nuclear interaction (magnetic)‐ nuclear spins interact with the surrounding electronic (spin or orbital) magnetic moments

H = γNħI · Hhf ~ I · Ahf · S

hyperfine field

The Hyperfine Interaction – manifestations in CM NMR

Ho

Effective field acting on the nuclear spinHeff = Ho + Hhf (r, t)

statistical average for the electronic systemspatial and temporal function.

Two major effects:1) static: shift of resonance frequency: Knight Shift, K2) dynamic: nuclear spin‐lattice relaxation, T1

1. Static Measurements: the Knight Shift

ν

slope = γN

slope = γN(1+K )

Shift of resonance due to “additional” field coming from within the material

Spectrum:Spin echo ‐> Fourier Transform ‐> energy  spectrum in frequency (or Field) domain 

Field sweep or frequency sweep

time average of Hhf(r,t)

νo + Δν

Definition:

K = Δν/νo (usually in units of %)

ω = 2πν = γN (1 + K ) Ho

Ηo

1. Simple metals‐ temperature independent Pauli susceptibilityLi          Na          Al        Cu       Sn PbK(%) 0.026    0.114    0.164    0.24    0.78   1.54

“Spin” Knight shift – due to unpaired conduction electrons. Only s‐orbitals have finite probability at the nuclear position (r=0).

⟨Ahf ⟩ = (8π/3) gμB|ψ(0)|2⟨S⟩

Knight Shift in Metals

core s‐orbital2. Transition metals, rare earths – strongly T‐dependent Curie paramagnetismχ = χdia + χs,p, + χd,f, + χorb,,

K = Kdia + Ks,p, + Kd,f, + Korb,,

Kd,f (T)= Ad,f χd,f (T) core polarization: Ad,f < 0

chemical shift σ – solely orbital in nature.   In general, include orbitals and transferred fields from neighboring atoms, molecules.

core s orbital

p or d‐orbital

Interaction between nuclear spin system and external “lattice” (electrons, phonons, etc.)‐ relaxation toward the Boltzmann distribution

2. Dynamic Measurements T1 and T2

A. Spin lattice relaxation rate:

measure of local magnetic field fluctuations

B. Spin decoherence (spin‐spin relaxation) rate:

irrecoverable decay of the spin echo due to loss of phase coherence

Page 7: Nuclear Magnetic Resonance Rabi Bloch in Condensed Matter · NMR Periodic Table Resonance Condition – spin manipulation ¬When an oscillating field is applied that matches the Larmor

7

mZ = I

I – 1

I

population

N(mZ) = exp (– γNħ mZHo/kBT)

~ 1 – γNħ mZHo/kBT

MZ eq ~ Ho/kBT

π/2) pulsenuclear relaxation

Nuclear Spin‐lattice Relaxation‐ approach to equilibrium

t

MZ eq

π/2) pulse

time: T1

rate: 1/T1

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

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

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

π/2 pulset t’ t” exponential growth 

of FID

T1 Measurement, FID

T2 Measurement, echo

π/2t t

π

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

t’ t’

t” t”

Distribution of local magnetic field‐magnet inhomogeneity‐anisotropic chemical shift‐quadrupole interaction‐internal fields‐local spin structure‐superconducting vortices‐nuclear dipolar fields

After π/2 pulse: MX(t) = ∫ P(H) cos(γNH t) dH

M (t) = ∫ P(H) sin(γ H t) dH

H

P(H)

NMR Lineshapes

MY(t) = ∫ P(H) sin(γNH t) dH

Fourier transform of 

MX(t) + iMY(t) = ∫ P(H) exp(iγNH t) dH

gives the distribution function P(H).

1/T1 ~ |ρ(εF)|2Ks ~ ρ(εF)

• Hebel‐Slichter peak, classic s‐wave pairing

1/Τ1 Τ

ΤΤc

DOS

energyεF

energy gap

NMR in Superconductors

• spin‐pairing, pairing‐symmetry

• pseudo‐gap behavior

Κ

ΤΤc

exponential – isotropic gappower law- anisotropic

Κ

ΤΤc

Τ * material behaves like a superconductor above its transition temperature.

NMR in ordered magnets, example LiCuVO4

Nuclear probe ( 7Li, 51V)Local moments ( 63Cu)

NMR lineshape gives distributionof local magnetic fields at the nuclear probe site

Hint = Hdip + Hhyp

Ho- Hint Ho Ho + Hint

SDW

uniform

Page 8: Nuclear Magnetic Resonance Rabi Bloch in Condensed Matter · NMR Periodic Table Resonance Condition – spin manipulation ¬When an oscillating field is applied that matches the Larmor

8

Zero‐field Ferromagnetic NMR at atomic sites with 3 different valences55Mn NMR I=5/2

Examples

[Mn[Mn1212OO1212(CH(CH33COO)COO)1616(H(H22O)O)44].2CH].2CH33COOH.4HCOOH.4H2200

Mn4+ Mn3+

Mn3+

Hyperfine field from ordered moments: 122 pnictides

YBa2Cu3O7 17O(2,3) NMR 8.4T

Tc

80

90

100

110

YBa2Cu3O7 17O(2,3) NMR 8.4T

Tc

80

90

100

110

NMR Lineshape transition to superconducting state17O central transition YBCO

Examples

Idealized NMR lineshape due to vortex

Quadrupole splitting of NMR line at two identical sites but 90 degree apart 

B

A

B

A

11B NMR in SmB6 30K 200 MHz

FFT-

Sum

Examples

14.85 14.90 14.95 15.00

F

Field (T)

100KH0 || c

*

**

*

*T1

* T1

*

*O(2,3)

O(4)HZeeman

HZ + HQ

I = 5/2

-5/2

-3/2-1/21/23/25/2

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

11.6 11.8 12.0 12.2 12.4

Field (T)

Nuclear Quadrupole Resonance of 2 Cu two isotopes and two sites, I=3/2

Examples

±3/2

±1/2

I = 3/263Cu 65Cu

∇ E

Energy level diagram

Page 9: Nuclear Magnetic Resonance Rabi Bloch in Condensed Matter · NMR Periodic Table Resonance Condition – spin manipulation ¬When an oscillating field is applied that matches the Larmor

9

0.01

0.1

1

10

ρi(ε)

ρf(ε)

Δo/2

δ/2

ρ(ε)

δo/2

Δ/2ε = 0

11B NMR in SmB6H||c

[111] 1.2T1.16T 13.9T

1/T 1 (

s-1 )

Topological Kondo Insulator SmB6

T. Caldwell, A. P. Reyes, W. G. Moulton, P. L. Kuhns, M. J. R. Hoch, P. Schlottmann, and Z. Fisk, Phys Rev B 75, 075106 (2007).

SmB6:Topological Insulator, Takimoto, JPSJ 80, 123710(2011)

11B Field dependent relaxation and model density of states

10 100

1.16T 13.9T 2.1T 20.9T 6.07T 37T

Temperature (K)

In-gap states and field suppression of gap

Hybridization gapTI band structure

Spin-Nematic Phase in Frustrated AF LiCuVO4 (New state of matter)

5 K

4 K

3 K

2 K

380 mK

spin

-ech

o in

tens

ity

1 K

Hc3

Hsat

paramagnetic

spin nematic?

spin saturated

39,0 39,5 40applied field µ0H (T)

SDW2

planar spiral

H||c

Hc1

Hc2

THE END

NOTES

Spin-Nematic Phase in Frustrated AF LiCuVO4 (New state of matter)

•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

Buettgen et al. (2013)

NMR determination of hyperfine field from ordered moments122 pnictides

Examples

Page 10: Nuclear Magnetic Resonance Rabi Bloch in Condensed Matter · NMR Periodic Table Resonance Condition – spin manipulation ¬When an oscillating field is applied that matches the Larmor

10

NMR in Condensed Matter at NHMFL Field driven new magnetic phases field‐induced states and 

phenomena 

Materials and Physics 

SDW,CDW, organics, oxides, perovskites, spinel

Manganites, ruthenates, cobaltites

Carbon nanotubes, buckyballs

Rare earth intermetallics 

high Tc, FFLO, pseudo‐gap, Vortex structures, pnictides

NFL behavior, Heavy fermion superconductors, Kondo insulators

spin‐Peierls systems 

AF multiferroics

weak ferromagnets, SDW Spin Fluids

molecular nanomagnets 

amorphous glasses

Magnetization plateau, BEC in frustrated dimers

FQHE, IQHE, Skyrmions in quantum‐well 2DEG systems

(general formulation)dynamical fluctuations of electronic magnetic moments

‐ unpaired electron spin exchanges energy with nucleus, causes transition

typical process in metals

SS

I ħωo

I = 1/2– ½

½

Appendix 1: Nuclear Spin-Lattice Relaxation

H = γNħI · Hhf = γNħ [ IZ Hhf

z + ½ (I+ Hhf– + I– Hhf

+ ) ; I+ = IX + iIY and I– = IX – iIYperturbation causing transition – ½ ↔ ½

Transition probability (Golden Rule)

W– ½ ↔ ½ = 2π/ħ Σm,n |⟨ – ½,m | – ½γNħI– Hhf+ | ½, n ⟩|2 exp(– En/kBT) δ(En – Em+ ħωo )

where | n ⟩, | m ⟩ are electronic states= ½πγN

2ħ Σm,n |⟨ m | Hhf+ | n ⟩|2 exp(– En/kBT) δ(En – Em+ ħωo )

using δ(En – Em+ ħωo ) = 1/(2πħ) ∫–∞ dt exp {i[(En – Em)/ħ + ωo]t}

W– ½ ↔ ½ = ¼γN2 ∫–∞ dt Σm,n |⟨ m | Hhf

+ | n ⟩|2 exp {i[(En – Em)/ħ + ωo]t} exp(– Em/kBT)

= ¼γN2 ∫–∞ dt Σm,n ⟨ n | Hhf

– | m ⟩ ⟨ m | Hhf+(t) | n ⟩ exp (iωot)

since ⟨ m | Hhf+ | n ⟩∗ = ⟨ n | Hhf

– | m ⟩and time dependent (Heisenberg representation)

Hhf+(t) = exp (iH t/ħ) Hhf

+ exp (–iH t/ħ)

⟨ m | Hhf+ | n ⟩ exp [ i(En – Em)t/ħ ] = ⟨ m | Hhf

+(t) | n ⟩

W– ½ ↔ ½ = ¼γN2 ∫–∞ dt ⟨ Hhf

– Hhf+(t) ⟩ exp (iωot)

time correlation function of the hyperfine field (statistical thermal average)

1/T1 = 2W– ½ ↔ ½ ; if Hhf = Ahf Si e.g. 55Mn in MnO, etc.

1/T1 = ½ γN2 A2

hf ∫–∞ dt ⟨ S+(t) S– (0)⟩ exp (iωot)

relaxation is given by the spin auto‐correlation function. 

Example: interacting localized moments, 4f, 3d electrons

J

⟨ S+(t) S– (0)⟩ ⅔S(S+1)

τc : correlation timeħ/τc ~ J >> ωo

1/T1 = ½ γN2 A2

hf [⅔S(S+1)] τc

independent of temperature!

τc t

Relation to dynamical susceptibility• response to space-time varying field

H(r,t) = Hq exp[ i(q · r – ωt)] S(r,t) = Sq exp[ i(q · r – ωt)]

spin system

Dynamical susceptibility: χ (q, ω) = Sq/HqImaginary part: χ" (q, ω) dissipation of the system

linear response

Fluctuation-dissipation theorem:χ" (q, ω) = ω/kBT ∫–∞ dt ⟨ Sq

+(t) Sq– (0)⟩ exp (iωt) ; kBT >> ω

and Σq A(q, ω) = ∫–∞ dt ⟨ Si+(t) Si

– (0)⟩ exp (iωt)

1/T1 = ½ γN2 kBT Σq |Aq

xx|2 χ"xx(q, ω)/ω + |Aqyy|2 χ"yy(q, ω)/ω

where form factor: Aq = Σi Ai exp (i q · r ) and Ho||z

isotropic local A case:1/T1 = ½ γN

2 A2 kBT Σq χ"(q, ω)/ω

S2

I

S1

A1

A3

A2

S3

Korringa relation ( free electron)

Raman scattering process:

1/T1 = 4π/ħ Σk,k’ |⟨ k’↑| – ½γNħAS+ |k↓ ⟩|2 f(ε)[(1– f(ε)] δ(εk – εk’+ ħωo )

= πγN2 ħA2 ∫∫ dε dε’ f(ε)[(1– f(ε)] δ(εk – εk’) ; ωo ~ 0

kBT ρ(εF) ρ(εF)

SS

I

k

k’

f(ε)

Appendix 2: Korringa Relation

kBT ρ(εF) ρ(εF)= πγN

2 ħA2 kBT |ρ(εF)|2

spin Knight shift Ks ∝ Αχs ∝ Αρ(εF) ; Fermi gas, non-interacting

T1TKs2 = (ħ/4πkB) (γe / γN)2

enhancement over Korringa constant for highly correlated electrons.Li Na Rb Cu Al

T1(expt, ms.) 150 15.9 2.75 3.0 6.3T1(Korringa,ms.) 88 10 2.1 2.3 5.1

εF ε

Page 11: Nuclear Magnetic Resonance Rabi Bloch in Condensed Matter · NMR Periodic Table Resonance Condition – spin manipulation ¬When an oscillating field is applied that matches the Larmor

11

H el-n = (8π/3) gμBγNħδ(r) I · S – gμBγNħ I · [ S/r3 – 3r(S · r)/r5] – gμB γNħ I · l /r3

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

Effective field for the nuclear spins

⟨Hhf ⟩ = (8π/3) gμB⟨δ (r)S ⟩ – gμB⟨ S/r3 – 3r(S · r)/r5 ⟩ – 2μB ⟨l /r3 ⟩

first and second term ≠ 0 for unpaired electronslast term ≠ 0 for electrons in open shell

expectation value for particular state

Appendix 3: The Hyperfine Interaction

Finite ⟨Hhf ⟩ examples1. Ferromagnetic materials ( Fe, Co, Ni ...)

magnetization: M = gμB ⟨S⟩ ≠ 0⟨Hhf ⟩ = (8π/3) gμB|ψ(0)|2⟨S⟩ = Hint

-resonance field is observed at zero external field at ωN = γNHint

59Co 230MHz 22.9 T57Fe 46.5MHz 33.8 T61Ni 28.5MHz 7.5 T55Mn 375MHz 35.7 T 50 100 150 200 250

-500

0

500

1000

1500

2000

2500

3000

3500

4000

ampl

itude

(AU

)

frequency MHz

0.1 0.14 0.18 0.20 0.30 0.40 0.50.51

spectra as afunction of x 0 external field fieldPulse Width 6.0ps

La1-xSrxCoO3

2. Paramagnetic materials (linear response to external fields)Mspin = gμB ⟨ S ⟩ = χspin HoMorb = gμB ⟨ l ⟩ = χorb Ho

magnetic susceptibility

⟨Hhf ⟩ = (8π/3)|ψ(0)|2 χspin Ho + dipolar + ⟨2/r3 ⟩ χorb Ho

hyperfine coupling constant, Ahf

Knight shift, definition in metals

K = ⟨Hhf ⟩ / Ho = Ahf χ ω = (1+K)γNH

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

K = As χs,spin + Ap(d,f...) χp(d,f..)spin + Bp(d,f...) χp(d,f..)orb

Core polarization effect – spin polarization of p, d, f states produces a spin-dependent exchange potential for the inner core s-state, resulting in a spin polarization of the inner s-state in the opposite direction.

3. superconductorsconduction electron susceptibility χspin = 2μB

2 Σk ∂fk/ ∂εk ~ Σk (εF – εk) = ρ(εF ) (density of states)

quasiparticle energy (superconductor, s‐wave)   Ek = (εk2 + Δ2)1/2 gap

spin susceptibility: χspin (T → 0)∝ exp (-Δ/kBT)

DOS

energy gap

energyεF

Κ

ΤΤc

exponential – isotropic gappower law- anisotropic


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