Nuclear Electric Quadrupolar Interactions in NMR Spectroscopy- An Introductionin NMR Spectroscopy An Introduction

Hellmut EckertI i fü Ph ik li h Ch iInstitut für Physikalische Chemie

WWU Münster

• Electric Interactions - General• Nuclear electric quadrupole momentsq p• Electric field gradients• The Quadrupolar Hamiltonian – NQR SpectroscopyThe Quadrupolar Hamiltonian NQR Spectroscopy• Influence on the NMR Spectra

– Static Solid NMR Lineshapesp– MAS-NMR Spectra– Nutation Behavior– Quantification Aspects

• Applications in Solid State/Materials Chemistry

Nuclear electric quadrupole moment:

A B C D

Nuclear electric quadrupole moment:non-spherical distribution of nuclear charge A B C D

I = 0 I = 1/2 >=I 1 ; eQ > 0 >=I 1 ; eQ < 0

eQ ~ 10-25 to 10-30 m2

The physical picture

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Prof. Dr. Hellmut Eckert

This quadrupole moment interacts with local electric field di t t d b th b di i t f th l igradients created by the bonding environment of the nuclei.

-> probe of local symmetry

Nuclear spin valuesza

hlA

nz

10B 138La 50V 176Lu

Spin

Solid State NMR Periodic Table

2H 3H

Solid State NMR Periodic Table

7Li 9Be

H

11B 13C 14N 17O 19F Ne

3He

21Ne

39K 51V 53Cr 55Mn 57Fe 59Co 61Ni 63Cu 67Zn 71Ga73Ge 75As 77Se 79Br 87Kr

23Na25Mg

43Ca

27Al 29Si 31P 33S 35Cl Ar

45Sc 49Ti39K 51V 53Cr 55Mn 57Fe 59Co 61Ni 63Cu 67Zn 71Ga73Ge 75As 77Se 79Br 87Kr

89Y 93Nb95Mo 99Tc 99Ru103Rh105Pd109Ag113Cd 115In 119Sn121Sb125Te 131Xe

43Ca

87Sr87Rb

45Sc 49Ti

91Zr 127I

133Cs 179Hf 181Ta 183W 185Re187Os 191Ir 195Pt 197Au201Hg 205Tl 207Pb 209Bi Po At Rn

Fr Ra Ac

137Ba 139La

Fr Ra AcStandard

Mössbauer isotope availableNMR restricted b q adr polar interactionsNMR restricted by quadrupolar interactions

Dominant quadrupolar interactionVery small magnetic moment

The electrostatic interaction

E r V r del el

2

,0

10 ...2! ß

ßr

V VV r V x x xx x x

,0 ßr r o

,10 ........2!elE V d V x d V x x d ,

,2!

Coulomb term dipole term quadrupole term

V Vx

r

0

Cartesian electric field components

V Vx x

r

2

0

, Cartesian electric field gradient components

Electric field gradient

V V Vxx xy xz V ’ ’V ’ ’ V ’ ’

Symmetric second-rank tensor diagonal in the principal axis system

E V V VV V V

y

yz yy yz

zx zy zz

,Laplace equation Vx’x’ + Vy’y’ + Vz’z’ = 0

Vz’z’Vy’y’ Vx’x’

V V

Vy y x x' ' ' '

2 parameters: Vz‘z‘ = eq and Vz z' 'Absolute size deviation from cylin-

drical symmetry

The Quadrupolar Hamiltonian

10 ........2!elE V d V x d V x x d ,

,2!el

Coulomb term dipole term quadrupole term

Q x x r d 3 2

E V QQ 16

,

( ) ( )

QI I I I

I eQI I

32 2 1

2 Expressed in spin coordinatesWigner-Eckart-Theorem

( )

' ' 'H e qQ

I II I I IQ z y x

22 2 2 2

4 2 13 ( )I IQ z y x4 2 1

Orientational quantization of spins along the EFG axis results in energy splittingsaxis results in energy splittings

Spin-3/2 Spin-5/2 Spin-7/2

|±7/2>

|±5/2>

|±3/2>|±3/2>

|±3/2>

|±5/2>

|±1/2> |±1/2> |±1/2>|±3/2>

( f “)Nuclear quadrupole resonance („Zero-field NMR“)

Effect of the Quadrupolar Interaction on the NMR Spectraon the NMR Spectra

For HQ ~ 0.05 H :

z H H H

For HQ 0.05 Hz: first-order perturbation theory sufficient.

zz'

H H H Q 0

E (1) = E (0) + <m|HQ|m>z'

z

c o sÎ

Î

Î

ÎzEm

( ) Em( ) + <m|HQ|m>

x

z

xÎ

Case of axial symmetry ( = 0)

x sinÎ x'

( ) cos sinH B I e qQ

I II Io z z x

22 2 2 2

4 2 13 3

cos sin'I I Iz z x sin cos I I I I Iz x x z 23

G t

m H m e qQ m I I m I IQ cos ( ) sin sin ( )

22 2 3

22 3

22 23 1 1

For axially symmetric EFG, the 1st order correction is:

m H mI I

m I I m I IQ ( )cos ( ) sin sin ( )

2 24 2 1

3 1 1

e qQ( ) ( ) cos12

22

3 1 3 1 E m B e qQI I

m I Im o( )

( )( ) cos1 2

4 2 13 1 3 1

2

E l l di f I 3/2

= 90° = 0°

Energy level diagram for I = 3/2

Powder samples: orientational averagingcase I = 3/2case I = 3/2

Central transition

Satellite transitions

Powder pattern for spin-7/2Powder pattern for spin-7/2

Energy-7/2

Central transition-5/2

Central transition

-3/2

-1/2

1/2

3/2

5/27/2 Satellite transitions

Stronger Quadrupole Coupling:g p p g

Second order perturbation theorySecond-order perturbation theory

anisotropic

More complicated geometry dependence of the second-order quadrupolar perturbation

Em(2) = m‘ (<m|HQ|m‘><m‘|HQ|m>)/(Em‘

o- Em°)

q p p

inverse dependence on the Larmor frequency

For axial symmetry:

E (2) = - hQ2/12 × m { 3/2 cos2(1-cos2)(8m2-4I(I+1) + 1] +Em hQ /12o × m { 3/2 cos (1 cos )(8m 4I(I+1) + 1] +

3/8(1-cos2)2 (-2m2 + 2I(I+1) – 1}

F th t l |1/2> < > | 1/2> hFor the central |1/2> <-> |-1/2> coherence:

= - Q2/16 o × {I(I+1) – ¾(1-cos2)(9cos2-1)}Q o { ( ) ( )( )}

where Q = 3CQ/2I(2I-1)

second-order quadrupolar shift

Anisotropic lineshape broadening caused byelectric quadrupolar interactionselectric quadrupolar interactions

1st order 2nd order1st order = 0

= 0.5 = 1

Effect of MAS on lineshape

0.11

0.64axially symm.EFG

0.94

A. AA. Samoson, E. Kundla, E. Lippmaa, J. Magn. Reson. 49, 350 (1982)

23Na MAS-NMR of T – Na3PO4

17.6 T 11.7 T

-50 -40 -30 -20 -10 0 10 20 30 40 ppm -15-10-50510 15 20 253035 ppm ppm

Linewidth decreases with increasing field strengthResonance frequency increases with increasing BResonance frequency increases with increasing BoComposite shapes are field dependent

11B MAS in borate glasses

Trigonal planar D3h

Three-coord. C2v

Tetrahedral Td

23Na MQMAS @ 30kHz of crystalline Na2PO3F P t l l t d f th t f itParameters calculated from the center of gravity

Na(I): Pq = 2.3 MHziso = 2.3 ppm

ppm

-2 Na(III): Pq = 3.2 MHz 5 9

Na(II): Pq = 2.5 MHziso = 0.4 ppm

2

0

2Na(IV) IV

iso = -5.9 ppm

Na(IV): Pq = 2.7 MHziso = -4.8 ppm

2

4

6 Na(II)

Na(III)

II

III

6

8

10

Na(I)( )

I

10

12

ppm-40-30-20-100102030

14

Spectral simulation of each 23Na site resolved in the F1 dimensionin the F1 dimension

Cq = 1.98 MHziso = 3.1 ppm = 0.64

Cq = 3.0 MHziso = -6.1 ppm = 0.50Na(I) Na(III)

(ppm)-20-15-10-50510

(ppm)-55-50-45-40-35-30-25-20-15-10-5051015

Cq = 2.38 MHziso = 0.65 ppm = 0.47

Cq = 2.37 MHziso = -5.1 ppm = 0.72

Na(II)Na(IV)

( )-30-25-20-15-10-505

(ppm)-35-30-25-20-15-10-50510

(ppm) (ppm)

L. Zhang, C: Fehse, J. Vannahme, H. Eckert, to be published

Deconvolution of the MAS-NMR lineshape

Impurity NaF

(ppm)-45-40-35-30-25-20-15-10-5051015

L. Zhang, C: Fehse, J. Vannahme, H. Eckert, to be published

The intensity distribution of 1st order quadrupolar ssb-s issensitive to the EFG asymmetry parameter ( I = 7/2)

Higher Resolution in the satellite transitions

2

2 QL QS 22

L

C3 6I(I 1) 34m(m 1) 13m m g , ,128 I 2I I)

)isotropic part anisotropic part

2Q 2

QS 22 2L

C I I 1 9m m 1 33 1m 140 3I 2I 1

L I 2I 1

I = 3/2 I = 5/2 STCT CTST

2nd order effects greatly diminished for |±1/2> <-> |±3/2> coherences (I = 5/2)

A. Samoson, Chem. Phys. Lett. 119, 29 (1985)

and for |±3/2> <-> |±5/2> coherences (I = 9/2)

27Al NMR of aluminoborate glass (7.0 T)

MAS ssb

Central transition

C Jä W Müll W th C M dC. Jäger. W. Müller-Warmuth, C. Mundus, L. Van Wüllen, J. Non-cryst. Solids 149 (1992), 209

Excitation Behavior of Q-nuclei

The effective nutation frequency depends on the ratio Q/1Th t di ti t iThere are two distinct regimes:

Non selective excitation limit: << :Non-selective excitation limit: Q << 1: - Effective rf amplitude is idential to liq measured in liquids- All the Zeeman transitions are being observed.

Selective excitation limit: Q >> 1: Effective rf nutation frequency is given by (I+ ½)

g

- Effective rf nutation frequency is given by (I+ ½) liq- only the central |1/2> <-> |-1/2> coherence is detected

t1

t2 2-Dt2

detection

2-D FourierTransform

2-D Nutation NMR Spectroscopy

Kentgens et al., J. Magn. Reson. 71 (1987), 62

Spin Echo measurements of homodipole couplings between Q nucleicouplings between Q-nuclei

z y y

xx x

tD

y

x

yy dephasingyy

tD

dephasing

xx 180y (or x)

tD

refocusing

O BB O BB O BB

23Na Spin Echo Decay Spectroscopy

221(2 ) ( )IIdMI t

90° 180° t1 t1Echo

2211

0

(2 ) exp (2 )2

dMI t tI

90° 180° Echo90 180 Echot1 t1

222 cos313

90° t1 t1

Echo180°

j ij

IId r

IFM 3240

2cos31

23

4)(

Regime of validity: HQ(2) ~ HD << H1 << HQ

(1)

(selective excitation)l t t ( 100 K)low temperatures (~100 K)(no dynamic contribution)

Example of Spin Echo Curvep p

)2()2( 2 MI

2)2(

exp)0()2( 2

2dM

II

1,0

1,1

2)0(I

0 7

0,8

0,9 Rb3

ms202 for:

0,5

0,6

0,7

/I(0)

ms2,02 for:

0 2

0,3

0,4

I/

0 0 0 5 1 0 1 5 2 00,0

0,1

0,2

0,0 0,5 1,0 1,5 2,0

2 [ms]

22

23Na Spin Echo Decay: Model Compounds

18

20

22

B0=7,04T B0=9,40T

12

14

16

ad2 /s

2 ]

8

10

12

2(exp

)[106 ra

2

4

6M

2 4 6 8 10 12 14 16 18 20

2

M2(calc)[106rad2/s2]2

M = 0 9562 (µ /4)4h2r -6M2 = 0.9562 (µo/4) h rij

B.Gee, H.Eckert, Solid State Nucl. Magn. Reson. 5,113 (1995)

Spatial sodium distribution in oxide glasses

M2 (23Na -23Na) vs. number density

5.5

6.0

6.5

4.0

4.5

5.0

2.5

3.0

3.5

M2 [

106 s-2

]

1 0

1.5

2.0

sodium silicate glassessodium borate glasses

M

0.00E+000 5.00E+027 1.00E+028 1.50E+028 2.00E+028 2.50E+0280.0

0.5

1.0 sodium borate glasses

NNa [Na/m3]

Spectra of spin-1/2 nuclei coupled to quadrupolar nuclei

31P MAS-NMR of (CuI) P S

quadrupolar nuclei

Cu2Cu2S3

P MAS-NMR of (CuI)3P4S4

P3

P3 P2P2P2

S1P1

S2S2P1

P2

exp fit

P1

180 160 140 120 100 80 60

[ppm]Cu1

G. Brunklaus, J. C.C. Chan, H. Eckert, S. Reiser, T. Nilges, A. Pfitzner, Phys. Chem. Chem. Phys. 5, 3678 (2003)

Practical AspectsPractical Aspects

Dipolar multiplets become asymmetric in case of strong Q-interactions

Indirect determination of CQ even if direct observationfails owing to too strong interaction

Indirect determination of CQ also possible via S{I} TRAPDOR experiments, stepping the irradiationTRAPDOR experiments, stepping the irradiationfrequency of the recouple nucleus I

S ti i 1/2 l i b d/ b blSometimes spin-1/2 nuclei are broad/unobservable, when strongly coupled with quadrupolar nuclei:certain C-N C-Cl C-Br groups in 13C CPMAScertain C-N, C-Cl, C-Br groups in C CPMAS

Further important topics• Resolution Enhancement via MQ-excitation:

A. Medek, J. S. Harwood, L. Frydman, J. Am. Chem. Soc. 1995, 117, 12779.S i l ki d CPMAS f d l l i• Spin-locking and CPMAS of quadrupolar nuclei: A. J. Vega, Solid State Nucl. Magn. Reson. 1, 16 (1992).

• Re-coupling dipolar interactions with quadrupolar nucleiT. Gullion, Chem. Phys. Lett. 1995, 246, 325. W. Strojek, M. K l i H E k t J Ph Ch B 198 7061 (2004)Kalwei, H. Eckert, J. Phys. Chem B 198, 7061 (2004)

• Signal Enhancement via Population TransferJ H M C d Ch Ph L tt 209 (1993) 287J. Haase, M. Conrady, Chem. Phys. Lett. 209 (1993), 287

• Dynamic Information via Modulation of Quadrupolar InteractionsInteractions

LiteratureLiterature• A. Abragam (1961), Principles of nuclear

magnetism, Oxford University Press.• M.H. Cohen, F. Reif, Solid State Physics 5

(1957), 321.( )• D. Freude, J. Haase, NMR-Basic Principles and

Progress 29 (1993), 3Progress 29 (1993), 3• J. Autschbach, S. Zheng, R. W. Schurko,

Concepts Magn Reson A 36 (2010) 84Concepts Magn. Reson. A 36 (2010), 84.

Signal Enhancement via Population Transfer from SatellitesTransfer from Satellites

J. Haase, M. Conrady, Chem. Phys. Lett. 209 (1993), 287

Comparison of experimental and calculated (WIEN2k) 45Sc quadrupole CQ-values in intermetallic compounds

20

Cexp.Q / MHz

16

18

20

10

12

14

6

8

10

0

2

4 tatsächliche Regression Einheitsgerade SATRAS-Messung MQMAS-BestimmungLinienformanalyse

0 2 4 6 8 10 12 14 16 18 20 22

0 Linienformanalyse

Ctheo.Q / MHz

C. Fehse, T. Harmening, R: Pöttgen, H. Eckert, to be published

Experimental Protocol

Results on Al2O3