The Physical Methods in Inorganic Chemistr

y

(Fall Term, 2004)Department of Chemistry

National Sun Yat-sen University

無機物理方法（核磁共振部分）

Chapter 7

Introduction to Solid State NMR

• 7.0 Summary of internal interactions in solid state NMR

• 7.1 Typical lineshapes for static samples• 7.2 Magic-angle-spinning (MAS)• 7.3 Cross polarization (CP) and CPMAS• 7.4 Homonuclear decoupling pulse sequence

s• 7.5 Multi-quantum MAS (MQMAS) of qua

drupole spins

Single Crystal or Polycrystalline (Powder) Samples

Spin 1 creates a tiny magnetic field at spin 2 and vise versa, introducing direct magnetic coupling between them.

The magnetic field produced by spin 1 at the position of spin 2 is

]2,1)2,1(3[ 111

42,1 312

0 mm rrB r

Which causes an energy of amount

])2,1)(2,1(3[ 21211

42,12 312

0 mmmmmE rrB r

This is the same energy that the spin 1 gains from the magnetic field produced by the spin 2.

||12

12

12ˆr

rr

12r̂

1ˆˆˆˆˆˆ ,12,12,12,12,12,12 zzyyxx rrrrrr

(unit vector)

r1,2

1

2

1|ˆ| 12 r

Expressing the energy in quantum mechanics, we have the direct dipolar interaction Hamiltonian as

]))((3[ ,,3

20

4,, jiji

rijiiijD IIIIIH jiji rrB

ij

ji

ij

ij

]))((3[ 212142,12212, 2,12,1312

212

0 IIrIrIBIHrD

[with ]

2,1, iIm iii which can be written in compact form

i

j

X

Y

Z

zjziyjyixjxiji

zjizjyjiyjxjixjj

zjiziyjiyixjixii

IIIIIIII

rIrIrII

rIrIrII

ji

ji

r

r

,,,,,,

,,,,,,,,,

,,,,,,,,,

,

,

]))((3[]))((3[ ,,,,3

20

4,, jijiDijjijirijiiijD IIIIIIIIIH jirjirjirjirBij

ji

ij

ij

i

j

X

Y

Zzjizjixjziyjizjixjzixjizjixjzi

zjiyjizjyiyjiyjiyjyixjiyjixjyi

zjixjizjxiyjixjiyjxixjixjixjxi

zjizjyjiyjxjixj

zjiziyjiyixjixijijjii

rrIIrrIIrrII

rrIIrrIIrrII

rrIIrrIIrrII

rIrIrI

rIrIrIrIrI

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,

,,,,,,,,,,,

)(

)()ˆ()ˆ(

zjziyjyixjxiji IIIIIIII ,,,,,,(

jijiD IDIH

zjzzijziyjzyijzixjzxijzi

zjyxijyiyjyyijyixjyxijyi

zjxzijxiyjxyijxixjxxijxiD

IDIIDIIDI

IDIIDIIDI

IDIIDIIDIH

,,,,,,,,,

,,,,,,,,,

,,,,,,,,,

)1ˆˆ3(ˆˆ3ˆˆ3

ˆˆ3)1ˆˆ3(ˆˆ3

ˆˆ3ˆˆ3)1ˆˆ3(

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

zjizjiDijyjizjiDijxjizjiDij

zjiyjiDijyjiyjiDijxjiyjiDij

zjixjiDijyjixjiDijxjixjiDij

ij

rrrrrr

rrrrrr

rrrrrr

D

jijiijD IDIH ,

)1ˆˆ3( ,,,,, xjixjiDijxxij rrD yjixjiDijxyij rrD ,,,,, ˆˆ3

)1ˆˆ3( ,,,,, yjiyjiDijyyij rrD

zjixjiDijxzij rrD ,,,,, ˆˆ3

xjiyjiDijyxij rrD ,,,,, ˆˆ3zjiyjiDijyzij rrD ,,,,, ˆˆ3

xjizjiDijzxij rrD ,,,,, ˆˆ3yjizjiDijzyij rrD ,,,,, ˆˆ3 )1ˆˆ3( ,,,,, zjizjiDijzzij rrD

It is symmetric It is traceless (see the reason?)

1ˆˆˆˆˆˆ ,,,,,, zijzijyijyijxijxij rrrrrr

where D is called dipolar coupling tensor.

Principal-Axis System (PAS)

0

)2( ,2,1,2,1,2,112,

,2,1,2,1,2,112,

zzyyxx

zzyyxxD

zzzzyyyyxxxxD

DDD

IIIIII

IIDIIDIIDH

)1ˆˆ3(ˆˆ3ˆˆ3

ˆˆ3)1ˆˆ3(ˆˆ3

ˆˆ3ˆˆ3)1ˆˆ3(

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

zjizjiDijyjizjiDijxjizjiDij

zjiyjiDijyjiyjiDijxjiyjiDij

zjixjiDijyjixjiDijxjixjiDij

ij

rrrrrr

rrrrrr

rrrrrr

D

In the principal-axis system (PAS), only the diagonal elements of D are non-zero and

i

j

Dij

Dij

Dij

ijD

200

00

00

Spherical Coordinates

cos

sinsin

cossin

rz

ry

rx

(x,y,z)

r

HD in Spherical coordinates

)}()(2sin

)(2sin)]()[1cos3{(

,,2

,,2

4

sin3,,,,4

3

,,,,43

,,,,41

,,2

4

2

3

20

jii

jii

zjijzii

ij

zjiji

jzii

ijjijizjziijr

jijiij

D

IIeIIeIIIIe

IIIIeIIIIII

IDIH

ijijijij

ij

ij

ji

ijijijyji

ijijijxji

ijijzji

rr

rr

rr

sinsinˆ

cossinˆ

cosˆ

,,

,,

,,

Zero-quantum terms Single-quantum terms

Double-quantum terms

Principal-Axis System (PAS)

jijijizjzi

r

jijiij

D

IIIIII

IDIH

ij

ji )]([ ,,,,41

,,2

4 3

20

jijizjziijD

jijijizjziijDD

IIII

IIIIIIH

](3[

)]([2

,,,

,,,,41

,,,

2

2

2

2',2',2,'

)2( )0,,()1(m m

mD

mD

ijijmmm

jiD TDH

(dipolar tensor in PAS)

DD

rD

ij

ji

2,21,2

40,2

0

63

20

and spin part (operator tensor)

,,21

2,2

,,,,21

1,2

,,61

0,2

)(

)3(

jiD

jzizjiD

jizjziD

IIT

IIIIT

IIIIT

The most important terms are those commuting with i ziI ,

:

ji

jijizjziijr

D IIIIIIHij

ji )]()[1cos3( ,,,,41

,,2

4sec

3

20

2

2,20,2,0

)2( )0,,()1(m

mDD

ijijmm

jiD TDH

Expressed with irreducible tensors

Why irreducible tensors?

• Rotation is treated most conveniently by means of irreducible tensors

• No matter how many rotations you have, the calculation is straightforward if the Hamiltonian is expressed in terms of irreducible tensors.

2

2',,20,2,0

)2(',

)2( )0,,()0,,()1(mm

mDD

ijijmMrmmm

jiD TDtDH

PASRotorLAB

Electric quadrupolar interaction

For a quadrupolar nucleus (spin>=1), the electric field gradient (EFG) at the nucleus may cause energy shift for the nucleus.

The general form for EFG is a tensor (like dipolar coupling tensor).

r

EV

The quadrupolar Hamiltonian can be derived as

IVIIQIHIIQe

EQ )12(2

2

0

)]()1(3[ 22212

)12(4

2

zzyyxx

V

VVQ

QzIIQVe

Q

VVV

IIIIIH

zz

yyxx

zz

In the principal axis-system (PAS), it is given by

In arbitrary coordinate systems, electric quadrupolar interaction is given by

2

2

2

2',2',2,'

)2( ),,()1(m m

mQ

mQ

mmm

Q TDH

with spatial part (quadrupolar tensor in PAS):

QQQ

Q

QIIQVeQ

2,2

1,2

)12(40,2

0

66 332

and spin part (operator tensor)

IIT

IIIIT

IIIT

Q

zzQ

zQ

21

2,2

21

1,2

2

61

0,2

)(

)]1(3[

β

α

γ

Secular term (First order)

)]1(3)[2cossin1cos3({ 22221

)12(4sec 33

2

IIIH zQIIQVe

Q

For many quadrupolar nuclei, higher orders may becomeappreciable and need to be removed.

Chemical shift interaction

2

0 ',,',,'

)( ),,()1(k

k

kmmmk

CSmk

CSmm

kmCS TDH

)(

0

)(23

33221131

0

21

2,21,223

0,2

12131,1120,100,0

1122

033

1122

CSCS

CSCSCSCS

CSCS

CSCSCS ii

.0,,

,0,

2,2021

1,2032

0,2

021

1,10,1031

0,0

CSCSz

CS

CSCSz

CS

TBITBIT

BITTBIT

zCSCSCS IH )2cossin1cos3({ 2221

The most significant term is

J-coupling interaction

jijiij

J IJIH

2,0

2

2',,',,'

)( ),,()1(k mm

mkJ

mkJ

ijijijmmkm

jiJ TDH

The expression of J tensor is complicated and is not discussed here. Unlike direct dipolar interaction, J-coupling tensor has non-zero isotropic component and in most cases, it is the only term to be considered.

The most important internal interactions in NMR spectroscopy are

• Chemical shift interaction

• J-coupling interaction

• Dipolar coupling interaction

• Quadrupolar interaction

• Spin-rotation interaction (for rotating molecules, not studied here)

All of them can be written in the form of where R is a rank-2 tensor (matrix), varying with the type of interactions.

Coordinate Systems

Lab Frame(XYZ)

),,(

JQDCTRHk

k

kmmkmk ,,,

2

0,',

k

kmm

mkmmk

rMmmkm

mk DtDR",'

',,')(

',")(

, ),,(),,0()1(

How to calculate a solid NMR spectrum

JQDCTRHk

k

kmmkmk ,,,

2

0,',

),,,,,(

dteedddS titi ,,,

),,,,,(sin)(

More generally,

dteeeIdddS titiHtiH

,,,

),,,,,(),,,,,( ])0([sin)(

Chemical shit anisotropy interaction

Direct Dipole-Dipole Coupling

Spin Pair

~80 kHz

Many coupled spins

Decoupling Sequences

• Hetronuclear decoupling:

CW

TPPM

• Homonuclear decoupling

WAHUHA

MREV

HR

CORY etc

CRAMPS (combination of rotation and multi-pulse spectroscopy

Indirect Spin-Spin Coupling

• In contrast to the direct, through space dipole-dipole coupling of two nuclear magnetic moments, the indirect spin-spin coupling interaction is mediated by the electrons of the intervening bonds.

• The isotropic J coupling constant is familiar from solution NMR.  We are also interested in anisotropies in the indirect spin-spin coupling tensor, denoted as J.  This anisotropy can be measured by a few different techniques; solid-state NMR is especially useful in certain cases.

• Wasylishen J. Am. Chem. Soc. 2000, 122, 3197. • "Anisotropy in the 199Hg-31P Indirect Spin-Spin Couplin

g Tensor of a 1:2 Mercury-Phosphine Complex. A Phosphorus Single-Crystal NMR Study", Michael D. Lumsden, Roderick E. Wasylishen, and James. F. Britten J. Phys. Chem. 1995, 99, 16602.

Dipolar-Chemical Shift NMR (1D)• The interplay of

chemical shift anisotropy and spin-spin coupling interactions results in complex line shapes.

• The dipolar-chemical shift method is useful in the case of isolated spin pairs.

Many other cases where more than one interaction are involved.

Cross polarization• CP condition: The nutation frequen

cy must be the same for the two coupled spins:

• CP incorporated with MASCPMAS—one of the most important solid state NMR techniques.

• CP contact time: several hundred microseconds to tens of milliseconds.

• Purpose: To enhance the sensitivity of the lower γ spins such as carbon-13. maximal enhancement factor: γI/γS

• Other advantages: Shorter recycle delay time

• Distinguish the interconnectivity of nuclear spins such as the protonation of a certain carbon nucleus.

SI ,1,1

H B11H XB1 X

Separation of Local Fields

Chemical shift correlation

Chemical shift -dipolar correlation

Chemical shift-quadrupolar correlation

t1 tm t2I

S

Interaction A Interactions B,AMixing

Chemical Shift Correlation Spectrum

3D CSA-D Correlation (with One Quadrupolar Spin)

MQMAS

rfQJDCSAB HHHHHHH

ZB SH SH rf 1

)1()0(QQQ HHH

QQQQ

IIqQe

QQ

QnnmMRmj

nm

Qj

QQZ

QQZZQ

ZQ

Q

jDtDV

VVSS

VVSSSH

SSSVH

L

2222)12(820

222

2

2,2

222222

2121222)1(

2203

2)0(

,66

2,1,0,),,()0,,(

])122

)184[(

)]1(3[

2

Under rapid magic angle spinning (MAS):

)](cos)(),(

)(cos)(),()([

444

22200

2

MS

MSS

II

PICA

PICAICAL

Q

Dig EFGs From This Spectrum!

Energy Levels of a Spin-3/2 Nucleus in a Static Magnetic Filed

m3/2

1/2

-1/2

-3/2

Zeeman Quadrupolar (first-order)

Quadrupolar (second-order)

Quadrupolar Coupling May Be Very Strong! Multiple SitesMultiple Sites

In A PowderIn A Powder

)](cos)(),(

)(cos)(),()([

444

22200

2

PICA

PICAICAS

SSII L

Q

)](cos)(),(

)(cos)(),()([

444

22200

2

PICA

PICAICAS

SSII L

Q

Both The EFG Information And High Resolution Can Be Achieved.

Second Order Quadrupolar FrequencySecond Order Quadrupolar Frequency Second Order Quadrupolar FrequencySecond Order Quadrupolar Frequency

2D Solution:Keep AND Remove2D Solution:Keep AND Remove2D Solution:Keep AND Remove2D Solution:Keep AND Remove0)74.54(cos2 oP 0)74.54(cos2 oP

]0,0[])(cos)()(cos)(

,)(cos)()(cos)([

24241414

22221212

tPICtPIC

tPICtPIC

MS

MS

MS

MS

)()(/ 142421 ICICtt SS )()(/ 142421 ICICtt SS

Excitation Evolution Conversion Acquisition

P1 t1 P2 t2

 

 

 

 

 

 

 

 

 

 

θM θM

MQC SQC Magic AngleMagic Angle

(54.7 ) Spinning(54.7 ) Spinning

oo

Multi-Quantum Magic-Angle Spinning (MQMAS)

L.Frydman, J.S.Harwood, JACS, 1995.L.Frydman, J.S.Harwood, JACS, 1995.

2D-17O-DAS spectrum of the silicate coesite

MQMAS Signal Enhancement

S.Ding,C.A.McDowell, Chem. Phys. Lett. 1997, 270, 81-86.S.Ding,C.A.McDowell, Chem. Phys. Lett. 1997, 270, 81-86.

Other Topics

• Multiple pulse for homonuclear decoupling (WAHUHA, MREV, HR, CORY etc)

• Combination of rotation and multiple pulses (CRAMP)

• Recoupling (Rotational Resonance, REDOR, RFDR etc)

• Other multi-dimensional solid state NMR (HETCOR, CSA/Q correlation, D/Q correlation, 3D correlation spectra)

• Single-Crystal NMR

Effect of MAS on dipolar coupling

Proton Decoupling

Pulsed decoupling (WAHUHA, MREV-8)

Correlation experiment

Homonuclear correlation

Homonuclear correlation : establishing connectivities

• Let us have a tour of solid state NMR following Professor Malcolm H. Levitt.