nuclear magnetic resonance rabi bloch in condensed matter · spin echo ‐> fourier transform...
TRANSCRIPT
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2015 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
MgB2, HighTcnanotubes
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
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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)
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.
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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
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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
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‐echoesE. Hahn, Phys. Rev. 80, 5801 (1950)
A spin echo seen in the rotating frame
the race track analogy
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Pulse NMR Electronics
~ 1kV
~ 1uV
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
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
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Spectrum for a nucleus with I = 3/2NQR
1st – OrderQuadrupole
| ±3/2⟩
| ±1/2⟩
Zeeman +1st Ord Quadrupole
|‐3/2⟩
|‐1/2⟩|+1/2⟩|+3/2⟩
PERT NMRE
I = 3/2
Quadrupole Spectra
Zeeman
|‐3/2⟩
|‐1/2⟩|+1/2⟩
|+3/2⟩
PURE NMR
Quadrupole 1st ‐ Ord. Quadrupole
FERROMAGNETIC NMR (zero‐field NMR)Due to internal fields generated by ordered electronic moments
νQνQ νoνo
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
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.
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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 µ 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 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
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)
8
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)
Zero‐field Ferromagnetic NMR at atomic sites with 3 different valences55Mn NMR I=5/2
Examples
[Mn12O12(CH3COO)16(H2O)4].2CH3COOH.4H20
Mn4+ Mn3+
Mn3+
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
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
NMR Lineshape transition to superconducting state17O central transition YBCO
Examples
Idealized NMR lineshape due to vortex
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)
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NMR determination of hyperfine field from ordered moments122 pnictides
ExamplesHyperfine field from ordered moments: 122 pnictides
THE END
NOTES
(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.
∞
∞
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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
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