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 - ω NMR 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 Nuclear spin I in a magnetic field H 0 H = γ I H 0-3/2 H 0 = 0-1/2 ω 0 E = ω 0 = γ H loc H loc 1/2 ω 0 3/2 ω 0 Zeeman 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 H j I j e iω 0t H h +, (ω) -1/2 H +,- ω 0-1/2 ω 0 +1/2 +1/2 n n = e ω0 k B T & ω 0 k B T n ω 0 k B T
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. M(T), pulse, signal π/2 Dead time > 2 µs t = 0 time 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 ω 0 H loc ω 0 Frequency
NMR Principles Spin Echo (Hahn, Phys. Rev. 80, 5801 (1950)) H 0
How is it done? Probe Sample
Static NMR Measurements Static NMR Spectrum Measurements Local Magnetic Field Probability Distribution (a) α (b) Cs(A) ω n = γ n H loc = γ n (H 0 + H hf ) Cs(B) Cl Cu α α c c H hf = n A n,k S k b a α a b Hyperfine tensor e - spin operator H 0 averaged over ~ 10 μs Contributions to hyperfine coupling constant (A): - on-site: A = strong & ~ known - transferred: A can be anything - dipole: A can be calculated & weak Magnitude [A. U.] 1.0 0.8 0.6 0.4 0.2 0.0 69.4 Cs(B) offset by 4.8 MHz 69.6 H 0 b = 11.5 T 69.8 70.0 70.2 Frequency [MHz] T = 1.7 K Cs(A) 70.4 H hf = n A n,k S k = n A n,k 1 g k µ B χ k (T ) magnetic susceptibility per site ω n ω 0 ω 0 = K(T ) χ(t )=χ(q =0, ω 0) Magnetic hyperfine shift
Static NMR Measurements Static NMR Spectrum Measurements Local Magnetic Field Probability Distribution ω n = γ n H loc = γ n (H 0 + H hf ) H hf = n A n,k S k Width of an NMR spectrum Distribution of S z (r) K(T ) χ (q =0, ω 0) Shift of an NMR spectrum Magnetic susceptibility In metals: K(T ) N(E F )
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 = 2 nd rank tensor with components along its principal axes (i= X,Y,Z) : V i,j = 2 V/ x i x j & V ZZ V YY V XX Quadrupolar Hamiltonian: H Q = hν Q 2 ν Q = [ I 2 Z I(I + 1) 3 + η ] 6 (I2 + + I ) 2 3eQ 2I(2I 1)h V ZZ & η = V XX V YY V ZZ H 0 0 NMR line spilts into 2I lines H 0 =0 NQR lines with ω Q ν Q
Quadrupolar Interactions - NQR Quadrupolar Hamiltonian: H Q = hν Q 2 ν Q = I = 3/2 [ I 2 Z I(I + 1) 3 + η ] 6 (I2 + + I ) 2 3eQ 2I(2I 1)h V ZZ & η = V XX V YY V ZZ m -3/2 H 0 0 NMR line spilts into 2I lines H 0 =0 NQR lines with ω Q ν Q! 1 H 0 = 0-1/2! 0! 1 1/2! 0! 0! 3 3/2! 0! 2! 2 Zeeman 1 st 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 1 2 h (t) (no energy loss for nuclear system) Spin lattice relaxation rate: T 1 1 h (t) =h +, (t) The nuclear spin-lattice relaxation time measures the time that it takes for excited nuclear spins to return to thermal equilibrium with the lattice (electrons). The nuclei relax to equilibrium (the state in which the population of the nuclear Zeeman levels is described by the Boltzmann population function) by exchanging energy with the electronic system. T 1 1 χ (q, ω 0) T 1 2 χ (q, ω 0)
Spin-spin relaxation rate, T2-1 : How to measure rates? M(t), pulse, signal π/2 π M (t=2τ) - Record M (2 ) as a function of t = t = 0 t = τ t = 2τ time Spin-lattice relaxation rate, T1-1 : - Record M (2 ) as a function of t = t d ω 0 ω 0 t d τ M(t d ) x 10 saturation π/2 π detection
Spin-Spin Relaxation T 1 2 h (t) 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. 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 : M (t) exp t2 /(2(T 2G ) 2 ) & (T2G ) -2 = 2 nd moment of the homogeneous lineshape (excluding the broadening due to the finite lifetime of a spin in an eigenstate).
Spin-Spin Relaxation T 1 2 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/(t 2R) Can be removed from the raw experimental data after T1 is measuremed.
Spin-Spin Relaxation T 1 2 h (t) 3. Indirect nuclear interaction: (C. Kittel, Quantum Theory of Solids) 1 T 2 χ(q) χ(r r )= the real part of the e - spin susceptibility A(r r) = describes the strength of the contact interaction between a nucleaus and e - s Indirect nuclear interaction = 2 step process 1 2 1. H ne r I(r i )A(r i r)s(r) I(r i ) S(r) I(r j ) 2. H en r S(r )A(r r j )I(r j ) 1. + 2. => H nn rr I(r i )A(r i r)s(r)s(r )A(r r j )I(r j ) 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 species at position R : I = nuclear spin operator S = e - spin operator A = hyperfine matrix element
From the fluctuation-dissipation theorem => Spin Lattice Relaxation
Spin Lattice Relaxation T. Moriya, J. Phys. Soc. Jpn 18, 516 (1963).
Spin Lattice Relaxation A(q) and/or F(q) = form factors => partial q dependence In general case - for non-diagonal hyperfine tensor => Shastry-Mila-Rice form factors for HTS, Physica C 157, 561 (1989). 1 ν T 1z T = C q β=x,y,z ( A 2 xβ (q)+a 2 yβ(q) ) Im χ ββ (q, ω nn ) ω nn
Spin Lattice Relaxation 1 T 1 N 2 (E F ) Korringa relation
To Remember Static NMR Spectrum Measurements Local Magnetic Field Probability Distribution ω n = γ n H loc = γ n (H 0 + H hf ) H hf = n A n,k S k Width of an NMR spectrum Distribution of S z (r) K(T ) χ (q =0, ω 0) Shift of an NMR spectrum Magnetic susceptibility In metals: K(T ) N(E F ) T 1 1 χ (q, ω 0) T 1 2 χ (q, ω 0)