An introduction to Solid State NMR and its Interactions

Transcription

1 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

2 Nuclear Magnetic Resonance Spin Magnetic Field The Rabi oscillation N = N ℏ I Rabi oscillations in quantum dots Source: Pres/Quantro/Qsite/projects/qip.htm

3 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

4 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/

5 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

6 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=

7 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

8 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

9 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

10 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

11 Pulsed Fourier Transform NMR Free Precession Decay Fourier Transform (phase correction) Signal Processing (Apodization) NMR Spectrum

12 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!

13 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.

14 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)

15 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

16 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):

17 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

18 The secular approximation High Field NMR: B B loc Perturbation Expansion of the NMR interactions 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

19 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

20 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

21 NMR powder spectrum S powder t = sin d d exp i, t TF S powder

22 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

23 Quadrupolar nuclei Powder Quadrupolar Static spectra

24 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

25 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

26 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

27 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

28 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

29 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! )

30 MAS at work Na I =3/ 3 NaAlH4 +1/ <-> -1/ +3/ <-> +1/ -3/ <-> -1/ B = 7.5 T vrot=1 khz

31 NaAlH4 MAS at work Al I =5/ 7 +1/ <-> -1/ Central transition Satellite transitions B = 7.5 T vrot=1 khz

32 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

33 The NMR Laboratory torturing probe... MAS probe MAS Rotor Superconducting magnet

34 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)