Solid State NMR of GlassModelling NMR spectra of glass from first-principles

Thibault Charpentier

CEA / IRAMIS / SIS2MLaboratoire de Structure et Dynamique par Resonance Magnetique

GDR Verre, 9-13 April 2011

Introduction

Basics of NMR: Interactions, MAS, MQMAS

NMR of disordered Materials

NMR from first-principles

Molecular Dynamics versus NMRContent of the tutorial

I Basics of NMR interactions and spectra of solids

I 1D-2D NMR experiments: Static, MAS and MQMAS spectra

I NMR of disordered Materials

I Structure: Molecular Dynamics Simulations (TD)

I NMR parameters: DFT GIPAW calculations (TD)

I NMR parameters & Simulations: specific NMR softwarepackages (fpNMR) (TD)

I Principles of pulsed NMR

I NMR Hamiltonian Art . . .

The Zeeman Interaction and Larmor FrequencyThe NMR spectrum of an isolated nucleus . . .

m 1

0

0

I=32

B00

B0=0

m=3/2

m=1/2

m=1/2

m=3/2 0

The Zeeman effect

0

E=NB0

0=N2

B0

The Larmor frequency and its NMRspectrum.

No information on the chemical surrounding

(~) H = ~N~I ~B0

NMR and the Periodic TableOne-half and quadrupolar nuclei

I Isotope, Nuclear Spin

I Natural Abundance

I Gyromagnetic ratio (rad/s/T)0 = 20 = B0

I Quadrupolar MomentQ (see Pyykko)

Pulsed NMRThe Basic NMR Experiment . . . One pulse !

M0: Nuclear Magnetization at Equilibrium

s s-msms-h

p

B0 M0 M0

BRF

M0

FFT

S t

S

~M(0) = M0~z~M(p) = M0~x

S(t) = M0ei0te

tT2

S() =

0

dt S(t)ei2t N1k=0

S(tk)ei2tk = L ( 0)

Lineshape L(): Gaussian, Lorentzian . . .

Pulsed NMRThe Basic NMR Experiment: Fourier Transform

0 5000 10000 15000Time (s)

-50-40-30-20-1001020304050Chemical shift ( ppm )

23Na in Na2MoO4

S(t) S()

NMR Interactions for NMR spectroscopistsEffects of the local magnetic fields . . .

Electrons

Phonons

Nuclear Spins I Nuclear Spins SNMRNMR

EPR

MicroWaves, ....

Bloc

B0 BRF

0

E=N{B0Bloc }

loc

The NMR spectrum

Information on the chemical surrounding

(~) H = ~N~I {

~B0 + ~Bloc}

= HZ + Hinter.

(~) H = HZ + HCS + HQ + HJ + HD + . . .

CS: Chemical Shift, Q: Quadrupolar, J: J couplings, D: Dipolar

NMR interactions for Quantum ChemistsThe effective Hamiltonian

HS(NMR) = ~

i

i~Ii (1 ) ~B0 +

i ,|Ii |1

~IiQii~Ii

+~2

2

i

j 6=i

ij~Ii (Dij + Jij)~Ij

~B0 External static magnetic field

~BRF (t) External radiofrequency magnetic field (NMR signal)

Internal Interactions

~Ii Nuclear spin operators ~i = i~~Ii Nuclear magnetic shielding tensor (chemical shift)

Dij Nuclear magnetic dipolar coupling tensor (structure)

Jij Indirect nuclear spin-spin coupling tensor

Qii Nuclear quadrupolar coupling tensor ( |Ii | 1 )

The nature of NMR interactionsSecond-rank tensors (i.e. matrices) define NMR interactions.

~B(A)loc = A

~X =

Axx Axy Axz

Ayx Ayy Ayz

Azx Azy Azz

Xx

Xy

Xz

HA = N~~I A ~X

A ~X Interaction

1 ~B0,~BRF Zeeman interaction

~B0 Magnetic Shielding DFT

Q ~I Quadrupolar interaction DFT

D ~S Dipolar interaction (spatial) structure

J ~S Indirect J couplings (through bond) DFT

DFT calculations output A in its tensorial (matrix) form relative to a

reference frame(x , y , z) (crystal axes).

NMR Interactions: The Principal Axes System (PAS) IThe principal values

AZZ

AYYAXX

B0B0

z

x

y

A =

AXX 0 0

0 AYY 0

0 0 AZZ

Isotropic Aiso =

13Tr [A]

Anisotropy A = AZZ AisoAsymmetry A = (AXX AYY ) /A

with 0 A 1

Calculations output the tensor in a reference frame (unit-cell definition).Its diagonalization (3D rotation) provides the principal values (A,) and

its orientation with respect to the reference frame.Convention: |AZZ Aiso | |AXX Aiso | |AYY Aiso |

AisoAZZ AYY AXX

NMR Interactions are Anisotropic . . .. . . NMR spectra of a single molecule (crystal)

O

C

O

C

O

C

zz yy xx

B0 B0 B0

The NMR frequency depends on the orientation with respect to ~B0 0 = int()

Single crystal NMR exists . . . but amorphous materials ?

NMR Interactions: The Principal Axes System (PAS) IIThe relative orientation of the PAS

APAS = Aiso1

+ A

12 (1 + A) 0 0

0 12 (1 A) 0

0 0 1

A = X1A APASXA

XA = R(A, A, A)

= exp(iAIz) exp(iAIy ) exp(iAIz)

(x , y , z): reference frame axis(A, A, A): Euler angles

A NMR interaction is characterized by 6 parameters:Aiso , A, A, A, A, A.

For a single interaction, (A, A, A) does not affect the powderlineshape but can be determined for single crystal NMR.

NMR Interactions are Anisotropic . . .. . . NMR spectrum (CSA) of a powder

B0

x

y

z

,

=90 =0

The powder average

S(t) =

sin dd

exp {i(, )t}

NMR Interactions are Anisotropic . . .. . . but in liquids only the isotropic part in effective

Motional averaging

I Motions affect the anisotropiclineshape.

I In the case of fast motionalaveraging (vs Larmorfrequency), a narrow line at theisotropic frequency.

I Brownian Motion in liquid

I Motions (fluctuations) induce

Relaxation

H ((t)) = Hiso + Hani ((t)) withHani ((t)) = 0 at the Larmor time scale ( 1/0)

The Zeeman Truncation in High Field NMROnly the part along B0 is NMR active to first order.

AZZ

AYYAXX

B0B0

z

x

y

Azz

In case of 1-spin interaction,only Azz (in the referenceframe) contributes to firstorder to the NMR frequency.

Because B0 Bloc (or HZ Hint), wehave the secular approximation

Hint H(1)CS + H(1)Q + H

(2)Q + H

(1)J + H

(1)D . . .

Typical strength of NMR interactionsI Z: 10-1000 MHz

I CS: kHz

I Q: MHz (up to second order)

I D: kHz

I J: Hz

NMR Interactions in generalAn overview . . .

H () = C () R () () T()

C () Constant Obtained from DFT (,Q)

R() () Spatial Dependence Manipulated through (sample) motions

R() () = 1 Isotropic Brownian motions in liquids

T() Spin operators Manipulated through RF (spin rotation)

T() Iz Chemical shift

T(Q) I 2z First order quadrupolar interaction

T(Q) Iz , I 3z Second order quadrupolar interaction

T(D,J) IzSz Dipolar (heteronuclear) or J interactions

T(D)II

IAz IBz

Dipolar (homonuclear) interaction+ 14

(IA+ I

B + I

AI

B+

)

The Shielding / Chemical Shift TensorNMR frequency

The secular contribution of the magnetic shielding is:

HCS = ~I~B0 B0zz()Iz

zz() = XX sin2 cos2 + YY sin

2 sin2 + ZZ cos2

= iso +CS2

((3 cos2 1) + CS sin2 cos 2

)= iso + CSR20()

NMR measure the chemical shift tensor (ppm):

expiso (ppm) = 106

( ref

ref

)ref is a frequency reference value obtained from one tabulated

reference compound, generally in liquid or in a solid (narrow peak,e.g. NaCl for 23Na.

The shielding / Chemical Shift TensorNMR static lineshape

-200 -100 0 100 200ppm

-200 -100 0 100 200ppm

CSA > 0 CSA < 0

= 0

= 0.5

= 1

The magnetic shieldingRelationshield with the chemical environment

-120-110-100-90-80-70-6029Si chemical shift iso (ppm)

O-O-SiO- O-

O-SiOSiO- O-

O-SiOSiOSi O-

Si OSiOSiOSi O-

Si OSiOSiOSi O Si

Q(0) Q(1) Q(2) Q(3) Q(4)

Q(1) SiOSi=180oQ(0)

Q(1)

Q(2)

Q(3)

Q(4)

Tetrahedral silicon Q(n) units in silicates

The quadrupolar interactionfrom electric field gradient. Only for I 1.

Because of its large strength (kHz-MHz), quadrupolar Hamiltonianare often accounted for up to second order (23Na,17O,27Al,11B. . . )

depending on the local environment of the nucleus.

The electric field gradient V is a symmetric traceless(VXX + VYY + VZZ = 0) tensor.

CQ =eQ

hVZZ , Q =

VXX VYYVZZ

,Viso = 0

CQ : quadrupolar couping constant, Q : quadrupolar asymmetryparameter.

The quadrupolar interactionTo first order. Only for I 1.

The firt-order secular quadrupolar Hamiltonian is

H(1)Q =

CQ6I (2I 1)

R20 () T20 , T20 =

1

6

(3I 2z I (I + 1)

)This I 2z yields the NMR frequency as

m,m1 0 =1 2m

2QR20 () =

(1)Q ()

The central transition 12 12 is not affected by H

(1)Q .

0Q1

20

Q1

2

I=1

0

0Q

10Q1

I=32

The quadrupolar interactionTo second order.

The second-order secular quadrupolar Hamiltonian is (morecomplex)

H(2)Q =

1

0

(CQ

6I (2I 1)

)2

l=0,2,4

AlRl0()

Rl0() =lk=l

Bkl ()Dkl0 () ,D

kl0 : Wigner rotation matrix elements

Al = f (I 3z , Iz) and isotropic shift (SOQS) (2)Q = A

0B00 ()

The NMR frequency of the central transition can be written as

1/2,+1/2() 0 = a0 + a2R20() + a4R40()

The quadrupolar interactionfrom electric field gradient

-2000-1000010002000kHz

-200-