1 1 Solid State NMR workshop 2011 Quadrupolar coupling theory Quadrupolar couplings = electric interaction of spin >1/2 nuclei with the surrounding electric fields interaction between a quadrupole moment (Q) with an electric field gradient (EFG) Very often as strong as Zeeman interaction! Electric field gradient (EFG): ??? Electric quadrudpole moment: ??? 2 Solid State NMR workshop 2011 Quadrupolar coupling Theory: the electric quadrupole moment Q - Feature of all nuclei with spin>1/2: - The non-spherical charge distribution can be described mathematically by a multipole expansion: non-spherical distribution of the nuclear charge =0 (75% of all NMR active nulcei) The quadrupole moment is fixed within the nucleus: alignment along B0 field through the spin
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SS NMR workshop2011 day4 - Stellenbosch University · interaction of spin >1/2 nuclei with the surrounding electric fields ... one pulse MAS MQMAS MQMAS : 2D experiment to obtain
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1Solid State NMR workshop 2011
Quadrupolar couplingtheory
Quadrupolar couplings= electric interaction of spin >1/2 nuclei with the surrounding electric fields
interaction between a quadrupole moment (Q) with an electric field gradient (EFG)
Very often as strong asZeeman interaction!
� Electric field gradient (EFG): ???
� Electric quadrudpole moment: ???
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Quadrupolar couplingTheory: the electric quadrupole moment Q
- Feature of all nuclei with spin>1/2:
- The non-spherical charge distribution can be describedmathematically by a multipole expansion:
non-spherical distribution of the nuclear charge
=0
(75% of all NMR active nulcei)
The quadrupole moment is fixedwithin the nucleus:�alignment along B0 field through the spin
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Quadrupolar couplingTheory: the electric quadrupole moment Q
!!! The strength of quadrupolar coupling depends on Q AND the EFG!
M. Levitt, spin dynamics, 2nd edition, Wiley & sons,(2008)
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Quadrupolar couplingTheory: the electric field gradient (EFG)
- arises if electron density is non-spherical distributed= feature of the electronic environment (depends on compound, morphology…)
- Mathematically described by an tensor (like the chemical shift tensor):
=
zzzyzx
yzyyyx
xzxyxx
VVV
VVV
VVV
V
=PAF
ZZ
PAFYY
PAFXX
PAF
V
V
V
00
00
00
V
0=++=
++PAF
ZZPAF
YYPAF
XX
ZZYYXX
VVV
VVV
The trace of the EFG tensor is zero:
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Quadrupolar couplingTheory: parameters of the EFG tensor
- isotropic component:
Analog calculation like for the chemical shift tensor:
( ) 03
1 =++= ZZYYXXiso VVVV
- anisotropy: PAFzz
PAFzz
isoPAFzz
aniso VVVVV =−=−= 0 eqV PAFzz =with:
� anisotropy of the EFG tensor is the largest component of the vector
- asymmetry: aniso
PAFxx
PAFyy
Q V
VV −=η 10 ≤≤ Qη
(because the trace of theEFG tensor is zero)
� no isotropic component of the EFG tensor
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Quadrupolar couplingTheory: the quadrupolar Hamiltonian
In the PAF: ( )
−+−−
⋅⋅= 222 ˆˆ2
1ˆˆ3)12(4
1ˆPAFPAFPAF yxzQ III
II
eQeqH I
h
Quadrupolar coupling constant [Hz]:
Quadrupolarcoupling constant [rad/s]:
h
eQeqCQ
⋅=
� Measure of the strength for the quadrupolar interaction
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Quadrupolar couplingTheory: impact on the NMR energy level
So far:- all interactions were considered as perturbations of the Zeeman interaction:
XZ HHH ˆˆˆ += this perturbation theory approach is valid , if the interactionX is small compared to the Zeeman interaction
� for quadrupolar coupling very often not the case!
...ˆˆˆˆ )2()1( +++= QQZ HHHH
Perturbation theory must be expanded:
)1(QE )2(
QE “first order/ second order energy correction termof the quadrupolar coupling interaction”
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Quadrupolar couplingTheory: impact on the NMR energy level
Energy correction terms of
- First order:
Same energy shifts for +m and –m!�Central transition (m=1/2↔m=-1/2)
is not affected!
Can be averagedout by MAS !
= CQh
( ) ( )[ ]θφηθ 222)1( sin2cos1cos32
1)1(3
)12(4 QQ IImII
eQeqE
m+−+−
−⋅=
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Quadrupolar couplingSpin-1 nuclei: energy diagramms and lineshapes
1=m
0=m
mE 1−=m
ω0
ω0 + 1/2ω(1)Q
ω0 - 1/2ω(1)Qω0
ω0
+ 1/2ω(1)Q
ω0
- 1/2ω(1)Q
ω0
¾ CQ
3/2 CQ
- Only natural I=1 spins: 2H, 6Li and 14N
-6Li: - nuclei with smallest Q- usually Pake pattern can’t be observed due to distribution of CQ
� Gaussian lineshapes
-14N: - small Q, but often strong EFGs � quadrupolar coupling in order of several MHz� hard to impossible to measure
Pake pattern
Zeeman
Zeeman + 1st orderquadrupolar coupling ηηηηQ=0
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Quadrupolar couplingSpin-1 nuclei: 2H NMR
�Low Q, CQs typically about 130 kHz� correspond to an FID decay within 5 µs (dead time about 4 µs)� 2H NMR spectra can be measured using the Solid Echo pulse sequence
The Solid Echo (quadrupole echo) pulse sequence:
echo
90°x 90°yt t-designed for spin-1 nuclei-can refocus 1st order quadrupolar coupling and homonuclear dipolarcoupling
-Variation for spin-3/2 nulei:90°x-t-64°y-t-aq
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Quadrupolar coupling2H NMR: study of dynamics in solids
MQMAS :2D experiment to obtainhigh-resolution spectra of half-integerquadrupolar nuclei
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Some wise, closing words… ☺☺☺☺
Before you start any Solid State NMR experiments, p lease … :
-Be aware that solution state and solid state NMR are very different techniques(different strengths and weaknesses)
-Think about: What is your research question?� What experiments could be useful to answer this question?
Accept that you won’t find a recipe to interpret Solid State NMR results!�Interpretation requires understanding of
- solid state NMR (interactions, their impact on the spectra, methods)
- your material (what nuclei, nuclei environment,morphology,…)
� Practice makes perfect! ☺
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I hope this workshop was a first step in I hope this workshop was a first step in I hope this workshop was a first step in I hope this workshop was a first step in understanding Solid State NMR! understanding Solid State NMR! understanding Solid State NMR! understanding Solid State NMR!
Thank you for your attention!
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-Proportional to 1/B0
� 2nd order quadrupolar coupling is reduced at strong magnetic fields!!
-Increases with (CQ/h)2
- no orientation dependence� isotropic component
- Energy shifts in opposite directions for +m and –m!
-Complicated orientation dependence� can’t be averaged out by MAS!
( )( ) +−+−×
−⋅−= 22
0
2
)2( 33)1(5
1
)12(4 QmmII
m
II
eQeqEQ η
ω
( )( )( )[ ]
( ) ( )( )
( )
+−+
+−
+−−++
+−−−−++
φθηφθθη
θθη
φθηθη
4cossin4
12cos1cos7sin
7
3
3cos30cos3518140
1534)1(18
8
1
2cossin61cos33312)1(828
1
4222
2422
2222
QQ
Q
QQ
mII
mII
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Quadrupolar couplingTheory: impact on the NMR energy level
Energy correction terms of
- Second order:
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R. Böhmer, K.R. Jeffrey, M.Vogel; J. Prog. Nucl. Magn. Reson. Spectrosc., 50, 87-174, (2007)
slow motion limit: Gaussian line
fast motion limit:Lorentzian line
motional narrowing:superposition of a Gaussian anda Lorentzian line
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The REDOR pulse sequence:
I (23Na)
S (1H)
I (23Na)
S (1H)
ISDipH
ISDipH
S0
S
0 1 2 3 4 NTr
( )1cos3ˆˆ
ˆ 23
−∝ θr
SIH ZZIS
Dip
REDOR= Rotational Echo Double Resonance
100 80 60 40 20 0 -20 -40 -60 -80 -100 -120 -140
0
5
10
15
20
25
30
I(N
TR)
X=0.79NT
R= 2.0 ms
δ(23Na)/ppm
S S
0
∆S
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20dB
nutation curves:
pl1=30dB
no selective excitation of thecentral transition !
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In this shear mapping the red arrow changes direction but the blue arrow does not. Therefore the blue arrow is an eigenvector, with eigenvalue 1 as its length is unchanged.