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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