An introduction to Solid State NMR and its Interactions From tensor to NMR spectra CECAM Tutorial September 9 Calculation of Solid-State NMR Parameters Using the GIPAW Method Thibault Charpentier - CEA Saclay thibault.charpentier@cea.fr
Nuclear Magnetic Resonance Spin Magnetic Field The Rabi oscillation N = N ℏ I Rabi oscillations in quantum dots Source: http://iramis.cea.fr/drecam/spec/ Pres/Quantro/Qsite/projects/qip.htm
The Zeeman effect 3 I= B E= N. B m= 3/ = N ℏ I. B =ℏ I Z ℏ H B = m= 1/ The Larmor frequency m= 1/ = N B N = B m= 3/ NMR spectrum of an isolated nucleus m= 1
The spin operators 3/ 1/ I Z= 1/ 3/ 1 I Y = I i I 3 I Z m =m m I m = I I 1 mm 1 m 1 I m = I I 1 mm 1 m 1 IX I = I Y IZ 1 I X = I I I= I = 3/ 3/ I = 3/ 3/
The Larmor precession z B The classical approach to Larmor precession N y d N dt = N N t B x The quantum approach to Larmor precession iℏ d I dt =i ℏ d dt, I ] t =i ℏ I B t I t = t [ H N
The nuclear magnetization In NMR, the only direct observable is the macroscopic magnetization i M= i But NMR spectroscopist know how to play with more complex objects... At equilibrium, N up z B y x N down exp E y kbt the magnetization is along the magnetic field x M z Boltzman populations M x =M y=
Sensitivity of NMR The Curie Law (high temperature limit) M = B =N I ℏ I I 1 3kT B The quantum approach = N I N =N I N ℏ Tr I eq M =ℏ I Z H H eq exp kb T In the high temperature limit eq IZ M =N ℏ Tr I eq I N Z kb T kt Description of NMR experiments in term of I operators
The Resonance Phenomenum B Nutation z M t = I H t H Z RF y T x T Pulse Angle = 1 w w Application of a tranverse RF magnetic field at the Larmor frequency produces the precession of the nuclear magnetization around the RF field (nutation) in the socalled rotating frame t = B I cos t = I cos t H 1 x 1 x RF N RF RF d I dt LAB = I z N B1 cos RF t x Time Dependent Rotating frame at RF around B : x, y, z LAB x, y, z T d I dt T = I { RF } z 1 x T Time Independent
Pulsed Fourier Transform NMR Typical Timing of a Solid State NMR experiment RF Pulse Thermal Equilibrium & Relaxation p ms-hours s Free Precession Decay s-ms Macroscopic Magnetization Motion p =M x M = M z t p =M x cos t y sin t M M The complex NMR signal s t M exp i t Fourier Transform
Pulsed Fourier Transform NMR z Bloch equations dm x dt dm y dt dm z dt Mx B = i M y T = = My T i M x y M z M eq x T1 Density operator (quantum state) t =a x t I x a y t I y a z t I z M t Tr I. t y Coherent state M x M y x I z : magnetization ; I, I single quantum coherence The only directly observable entity
Pulsed Fourier Transform NMR Free Precession Decay Fourier Transform (phase correction) Signal Processing (Apodization) NMR Spectrum
The NMR signal in theory t I exp i H t s t=tr I exp i H x Initial state: I x System evolves under: H One quantum transitions are observed: I s t m m 1 I m exp i m, m 1 t The NMR frequencies m, m 1= m 1 H m 1 m H m The contribution of each transition to the signal amplitude is (isotropic!!) m 1 I m Hamiltonian of NMR interactions are needed H DFT calculations provide H!
NMR Interactions B B t, RF = ℏ I B H H B = H N ext local Z inter. External fields to manipulate the system: EPR Electrons NMR A complex situation... Nuclear Spins I B loc Nuclear Spins S B B t, RF Internal fields: B loc NMR inter. Phonons MicroWaves,... =H H H H H... H Z CS J D Q Chemistry: inter.
NMR Interactions are tensors A xx =AX = A B loc yx A zx A xy A xz Xx A yy A yz. X y A zy A zz Xz = ℏ I. A. X H N Second-rank Tensor = A=1, X B, B RF Zeeman Interaction =B A=, X Magnetic shielding = A=D, X S Dipolar Magnetic couplings (through space) = A= J, X S Indirect Magnetic couplings (through bond) = I A=Q, X Quadrupolar Interaction (electric couplings)
NMR Interactions: PAS Diagonalization of A yields the Principal Axis System (PAS) A xx A= A yx A zx A xy A xz A XX A yy 1 = X. A yz A AYY A zy A zz A ZZ. X A =X 1. APAS. X A A Principal Axes labeling: 1 Aiso= Tr A 3 AZZ A ZZ Aiso A XX Aiso AYY Aiso AYY Aiso A XX
NMR Interactions: PAS Introducing a convenient representation for encoding this orientational dependence 1/ 1 APAS = Aiso 1 A 1/ 1 1 Isotropic shift: Aiso=1/3Tr A Strength of the anisotropy: A= A ZZ Aiso Symmetry of the anisotropy A = A XX AYY / A (asymmetry parameter):
NMR Interactions: PAS Relative orientation of the PAS with respect to a reference frame (crystallographic axes, laboratory frame,...) Factorization of XA provides the three Euler angles A, A, A X A=R A, A, A =exp i A I z exp i A I y exp i A I z (x,y,z) is the PAS of the reference frame B A ZZ In NMR the position of a line (single crystal) is dependent upon the six parameters: Aiso, A, A, A, A, A A XX AYY
The secular approximation High Field NMR: B B loc Perturbation Expansion of the NMR interactions 1 1 1 1 H Z H inter. H inter. H CS H Q H Q H D H J... Keeping terms invariant under rotation around B B A zz A XX A ZZ AYY
A general representation of the NMR interactions Powerful Tensorial approach to derive all formula! m m =C H 1 R, m T m m= Euler angles =,, Spatial dependence R = D, m n,m n, n,± NMR parameters = Spin Operator dependence T, m [ I z,t, m ]=m T, m,,±1 II, =, = T =, 1 3I 6 (first order ) secular approximation easy : m=! 1 H =C R, T Z 3 I I 1
The Chemical Shift Tensor Absolute chemical shielding (GIPAW output) Isotropic chemical shift = REF 1 iso = iso REF exp iso ppm=1 6 Chemical Shift Anisotropy (CSA), exp iso REF Hz REF Hz z PAS H CSA= CS, I Z = R, I Z 3 CS, = XX sin cos YY sin sin ZZ cos CS, = iso {3 cos 1 B sin cos } x PAS y PAS
NMR powder spectrum S powder t = sin d d exp i, t TF S powder
The quadrupole Interaction I 1/ Electric coupling between the nuclear quadrupole moment Q and local electric field gradient V(rnuc) at the nucleus Quadrupolar Coupling Constant ( ~ MHz ) C Q= Quadrupolar asymmetry parameter First order H = 1 Q CQ 6I I 1 Q= Second order (complex...) h V XX V YY Q R T eq T,= V ZZ 1 6 V ZZ V iso= 3 I Z I I 1 CQ 1 l l k H = l=,,4 A k= l, l B k Dl, 6I I 1 Q l 3 Z A = f I, I Z Second Order Quadrupolar Induced Shift! = A B Q Isotropic shift
Quadrupolar nuclei Powder Quadrupolar Static spectra
z LAB B The Dipolar Interaction = H D r S r IS I ℏ I S y LAB 3 IS { = D PAS ℏ I S = 3 r IS Homonuclear H = Heteronuclear H = 1 D ℏ I S 3 IS r ℏ I S 1 r 3 IS D R T D 1 x LAB 1 D } 3 I. S I r IS S r IS = I D S r IS R IS 3 I Z SZ 1
Indirect J coupling Small interaction (~Hz), anisotropic effects (so far) neglected H J = I J iso 1 J ani S J iso I S I iso J iso Unlike spins J iso I iso S iso H J =J iso I S H J =J iso I Z S Z IS 3 IS IS I S iso Like spins I I
Multiple interactions Lineshape is also dependant upon the relative orientation of CSA PAS with respect to quadrupolar PAS (or vice-versa) In the crystallographic axes frame (GIPAW calculation) X, c =R, c,, c,, c X Q, c =R Q, c, Q, c, Q, c X,c X 1 Q,c =R, Q,, Q,, Q B V ZZ ZZ XX V V YY XX YY
High Resolution NMR Brownian Motion in Liquids t = H H t H iso ani t = H ani Only isotropic contributions! iso, J iso In rigid lattices, broad lines iso, J iso + CSA, CSA, C Q, Q, CSA,Q, CSA, Q, CSA, Q
Magic Angle Spinning Sample B D M R 3cos 1 3 cos M 1= Second Rank A ZZ A ZZ A XX AYY B AYY Aiso M A XX A ZZ M = t = A ZZ A ZZ t A ZZ A ZZ t
MAS at work I =1/ +1/ <-> -1/ Static spectrum Low speed MAS spectrum High speed MAS spectrum CSA + Dipolar ( 13C-13C ) Spinning sidebands ROT ROT Proton decoupling is necessary (sample rotation + spin rotation! )
MAS at work Na I =3/ 3 NaAlH4 +1/ <-> -1/ +3/ <-> +1/ -3/ <-> -1/ B = 7.5 T vrot=1 khz
NaAlH4 MAS at work Al I =5/ 7 +1/ <-> -1/ Central transition Satellite transitions B = 7.5 T vrot=1 khz
The MAS NMR signal in theory t s t m m 1 I m exp i m, m 1 udu Time-dependent transitions frequencies: t m 1 m H t m m, m 1= m 1 H Euler angles of the PAS in the rotor fixed frame m, m 1,, =, m m, exp { i m ROT t } d exp { i m,m 1 u du }=e t i t,, : MAS lineshape I k, : spinning sidebands pattern i k ROT t k I k, e
The NMR Laboratory torturing probe... MAS probe MAS Rotor Superconducting magnet
NMR parameters: DFT vs Experiment Quadrupolar parameters (I>1/) (MHz) C Q, Q Isotropic Chemical shift (ppm) iso Chemical shift anisotropy (ppm) CSA, CSA Relative orientation of CSA PAS in Quad. PAS (Three Euler angles) CSA, Q Isotropic J couplings (1-3 bonds) 1, CSA, Q, CSA,Q J Si O J Si O Si NMR provides methods for measurements of Dipolar interactions (only the structure is needed)