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NMR: Formalism & Techniques
Vesna Mitrović, Brown University
Boulder Summer School, 2008
Why NMR?
- Local microscopic & bulk probe
- Can be performed on relatively small samples (~1 mg +) & no contacts on the sample
- ωNMR ≈ 0 (µeV), partial q info, r ≈ constant
- Extreme conditions: high field, dilution refrigerator, pressure cell
NMR Principles The basic steps of NMR are as follows:
1. A sample is placed in a high magnetic field environment. This breaks the degeneracy of the nuclear Zeeman spin state.
I = 3/2 m
-3/2
1/2
-1/2
3/2
H0 = 0
ω0
ω0
ω0
Zeeman
Nuclear spin I in a magnetic field H0 ⇒ H = −γ! I ·H0
∆E = ω0! = γ!Hloc ∝ Hloc
H⊥ ∝ h+,−(ω)
2. A coil around the sample generates a low amplitude, high frequency oscillating magnetic field transverse to the main
field. This excites nuclei from one state to another.
H⊥ = γ!HjIjeiω0t
+1/2
-1/2
ω0
+1/2
-1/2
ω0
H+,-
n↓n↑
= e!ω0kBT & !ω0 ! kBT ⇒ ∆n ≈ !ω0
kBT
NMR Principles The basic steps of NMR cnt:
3. The oscillating field is removed, and the nuclei begin to relax back to their original state.
time
M(T
) , p
ulse
, sig
nal π/2
Dead time > 2 µs
t = 0
4. As the nuclei relax, a small current is induced in the coil that generated the original oscillating field.
5. This current is amplified and analyzed, yielding information about the energy eigenstates of the sample.
FID FT
ω0Frequency
ω0 ∝ Hloc
NMR Principles Spin Echo (Hahn, Phys. Rev. 80, 5801 (1950))
H0
How is it done?
Probe
Sample
Static NMR Measurements
Static NMR Spectrum Measurements ⇒ Local Magnetic Field Probability Distribution
Cs(A)
Cs(B)
Cl
Cu
ba
c
c
(a) (b)
ba
α
α
α
α
1.0
0.8
0.6
0.4
0.2
0.0
Mag
nitu
de [A
. U.]
70.470.270.069.869.669.4
Frequency [MHz]
Cs(B) offset by 4.8 MHz
Cs(A)
T = 1.7 K H0 || b = 11.5 T
〈Hhf 〉 =∑
n
−An,k〈Sk〉
Contributions to hyperfine coupling constant (A):- on-site: A = strong & ~ known- transferred: A can be anything- dipole: A can be calculated & weak
Hyperfine tensore- spin operator || H0
averaged over ~ 10 μs
〈Hhf 〉 =∑
n
−An,k〈Sk〉 =∑
n
−An,k1
gkµBχk(T )
ωn = γnHloc = γn (H0 + 〈Hhf 〉)
⇒ Magnetic hyperfine shiftωn − ω0
ω0= K(T ) ∝ χ(T ) = χ(q = 0,ω → 0)
χ= magnetic susceptibility per site
Static NMR Measurements
Static NMR Spectrum Measurements ⇒ Local Magnetic Field Probability Distribution
K(T ) ∝ χ′(q = 0,ω → 0)
In metals:
K(T ) ∝ N(EF )
〈Hhf 〉 =∑
n
−An,k〈Sk〉
ωn = γnHloc = γn (H0 + 〈Hhf 〉)
Width of an NMR spectrum ⇒ Distribution of 〈!Sz(r)〉
Shift of an NMR spectrum ⇒ Magnetic susceptibility
Quadrupolar Interactions - NQR
For I > 1/2 ⇒ nuclei have nuclear quadrupole moment Q
For I > 1/2 & non-cubic local symmetry ⇒ Q interacts with the electric field gradient (EFG) arising
from the surrounding electronic charge distribution.
The EFG = 2nd rank tensor with components along its principal axes (i= X,Y,Z) :
HQ =hνQ
2
[I2Z −
I(I + 1)3
+η
6(I2
+ + I2−)
]
Vi,j = ∂2V/∂xi∂xj & |VZZ | ≥ |VY Y | ≥ |VXX |
Quadrupolar Hamiltonian:
νQ =3eQ
2I(2I − 1)hVZZ & η =
|VXX − VY Y |VZZ
H0 != 0⇒
NMR line spilts into 2I lines
H0 = 0NQR lines with ωQ ∝ νQ
Quadrupolar Interactions - NQR
HQ =hνQ
2
[I2Z −
I(I + 1)3
+η
6(I2
+ + I2−)
]Quadrupolar Hamiltonian:
νQ =3eQ
2I(2I − 1)hVZZ & η =
|VXX − VY Y |VZZ
H0 != 0⇒
NMR line spilts into 2I lines
H0 = 0NQR lines with ωQ ∝ νQI = 3/2 m
-3/2
1/2
-1/2
3/2
H0 = 0
!0
!0
!0
!2
!0
!1
!2
!3
!1
Zeeman 1st Order 2 nd Order
Good for study of lattice deformations....
Dynamic NMR Measurements
Dynamic NMR Spectrum Measurements ⇒Measure of Fluctuations of Local Magnetic Field
Spin decoherence (spin-spin relaxation) rate: T−12 ∝ h||(t)
(no energy loss for nuclear system)
Spin lattice relaxation rate: T−11 ∝ h⊥(t) = h+,−(t)
The nuclear spin-lattice relaxation time measures the time that it takes for excited nuclear spins to returnto thermal equilibrium with the lattice (electrons). The nuclei relax to equilibrium (the state in which thepopulation of the nuclear Zeeman levels is described by the Boltzmann population function) by exchanging energy with the electronic system.
T−11 ∝ χ′′(q, ω → 0)
T−12 ∝ χ′(q, ω → 0)
How to measure rates?
Spin-spin relaxation rate, T2-1:
time
M(t)
, pu
lse,
sig
nal π/2
t = 0
π
t = τ t = 2τ
M (t=2τ)
Spin-lattice relaxation rate, T1-1:
- Record M⊥(2τ) as a function of t = τ
π/2 π
τ
td
detectionsaturation
x 10
ω0 ω0
- Record M⊥(2τ) as a function of t = td
M(t d
)
Spin-Spin Relaxation
The nuclear spin–spin relaxation time T2 is the characteristic time for the decay of the M⊥ component of the nuclear magnetisation M. Can be a poverfull tool for probing e.g. vortex dynamics.
T−12 ∝ h||(t)
In correlated electron systems 3 main sources of the decay of the M⊥ :
1. Nuclear-nuclear interaction - the spin exchange between two nuclear spins (nuclear dipole-dipole interaction):
In most solids T2 arises from nuclear dipole-dipole interaction to give :
& (T2G )-2 = 2nd moment of the homogeneous lineshape (excluding the broadening due to the finite lifetime of a spin in an eigenstate).
M⊥(t) ∝ exp−t2/(2(T2G)2)
Spin-Spin Relaxation
T−12 ∝ h||(t)
2. ``T1’’ or the Redfield processes - the fluctuations of the nearby e- spin cause T1 relaxation & provide a decay of M⊥ :
M⊥(t) ∝ exp−t/(T2R)
Can be removed from the raw experimental data after T1 is measuremed.
Spin-Spin Relaxation T−12 ∝ h||(t)
3. Indirect nuclear interaction: (C. Kittel, Quantum Theory of Solids)
1T2∝ χ(q)
χ(r − r′) =A(r′ − r) =
the real part of the e- spin susceptibility
describes the strength of the contact interaction between a nucleaus and e-s
Indirect nuclear interaction = 2 step process
I(ri) I(rj) S(r)
1 2
2. Hen ∝∑
r′
S(r′)A(r′ − rj)I(rj)
1. Hne ∝∑
r
I(ri)A(ri − r)S(r)
1. + 2. => Hnn ∝∑
rr′
I(ri)A(ri − r)S(r)S(r′)A(r − rj)I(rj)
〈S(r)S(r′)〉 = e- spin density correlation function = the real part of the retarded susceptibility χ′(r − r′)
Spin Lattice Relaxation
Hyperfine Hamiltonian - interaction between conduction electrons and nucleus of speciesνat position Rν:
I = nuclear spin operatorS = e- spin operatorA = hyperfine matrix element
Spin Lattice Relaxation
From the fluctuation-dissipation theorem =>
Spin Lattice Relaxation
T. Moriya, J. Phys. Soc. Jpn 18, 516 (1963).
Spin Lattice Relaxation
In general case - for non-diagonal hyperfine tensor =>
1νT1zT
= C∑
q
∑
β=x,y,z
(A2
xβ(q) + A2yβ(q)
) !m χββ(q, ωnn′)!ωnn′
A(q) and/or F(q) = form factors =>
partial q dependence
Shastry-Mila-Rice form factors for HTS,Physica C 157, 561 (1989).
Spin Lattice Relaxation
Korringa relation
1T1∝ N2(EF )
To Remember Static NMR Spectrum Measurements ⇒ Local Magnetic Field Probability Distribution
K(T ) ∝ χ′(q = 0,ω → 0)
In metals:
K(T ) ∝ N(EF )
〈Hhf 〉 =∑
n
−An,k〈Sk〉
ωn = γnHloc = γn (H0 + 〈Hhf 〉)
Width of an NMR spectrum ⇒ Distribution of 〈!Sz(r)〉
Shift of an NMR spectrum ⇒ Magnetic susceptibility
T−11 ∝ χ′′(q, ω → 0)
T−12 ∝ χ′(q, ω → 0)
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