TIFR/RAC, Jan. 2010 Quadrupolar Nuclei I to III
Solid-state NMR of spin > 1/2
Anisotropic Interactions
• Anisotropy of the chemical shift 1H 0 ... 40 ppm (max. 16 kHz @ 9.4 T) 13C 0 ... 250 ppm (max. 25 kHz @ 9.4 T) 19F 0 ... 300 ppm (max. 113 kHz @ 9.4 T)
• Dipolar Interaction 1H-1H, 1H-13C: typically ≤ 50 kHz
• Quadrupolar Interaction Only for spin I ≥ 1
2H (I=1) 0 ... 250 kHz
14N (I=1) 0 ... 2 MHz 23Na (I=3/2) 0 ... 10 MHz 27Al (I=5/2) 0 ... 10 MHz 35,37Cl (I=3/2) 0 ... 40 MHz
Nuclear spins with I > 1/2possess an “electricalquadrupole moment”.
TIFR/RAC, Jan. 2010 Quadrupolar Nuclei I to III
Properties of Selected Quadrupolar Nuclei
Q values in millibarn
TIFR/RAC, Jan. 2010 Quadrupolar Nuclei I to III
Quadrupolar Interaction for Spin-1
Energy level diagram Single crystal spectrum
Approximation:
TIFR/RAC, Jan. 2010 Quadrupolar Nuclei I to III
Relative Magnitude of Quadrupolar Interaction
Different situations because of relative magnitude of quadrupolar couplingconstant Cq (or QCC, or χ) and Larmor frequency ω0:
B0 = 0: Pure quadrupolar interaction (NQR, NNuclear QQuadrupole RResonance): Transitions between the quadrupole levels
Cq « ω0: „Quadrupolar interaction of first order“(i.e., Cq of the order of tens to hundreds of kHz)
Cq ≤ ω0: „Quadrupolar interaction of 2nd (and higher) order“ (i.e., Cq of the order of MHz)
Tetrahedral and higher symmetry: eq = 0 ⇒ No quadrupolar coupling!
TIFR/RAC, Jan. 2010 Quadrupolar Nuclei I to III
Quadrupolar Hamiltonian and Eigenfunctions for Spin-1
Quadrupolar Hamiltonian:
First order Hamiltonian:
Eigenvalues andEigenfunctions:
-
TIFR/RAC, Jan. 2010 Quadrupolar Nuclei I to III
The Quadrupolar Splitting (I=1)
Quadrupolar CouplingConstant (QCC, Cq):
Largest component of EFG tensor:
Quadrupole moment:
!
", # Euler angles relating the PAS of the EFG to B0 (lab frame)
!
" asymmetry parameter of quadrupolar coupling tensor Q
TIFR/RAC, Jan. 2010 Quadrupolar Nuclei I to III
The Quadrupolar Coupling Tensor
Keep a direct relation between doublet splitting andquadrupolar interaction by defining a quadrupolarcoupling tensor Q:
asymmetry of Q:
with:
Q in its own PAS:
with:
Obtain doublet splitting simply bythe tensor products:
b - unit vector along the magnetic field B0
TIFR/RAC, Jan. 2010 Quadrupolar Nuclei I to III
From Rotation Pattern to Q-Tensor
Have z-axis of the Q-tensor PAS originally aligned alongdirection of B0, rotate step-wise around y-axis of PAS.
The tensor at 0o rotation is diagonal:
TIFR/RAC, Jan. 2010 Quadrupolar Nuclei I to III
Determine Full Q-Tensor from Rotation Pattern
T. Voosegard et al.
To obtain all elements of the Q-tensor:
⇒ Need more complex transformations for general orientation of Q-tensor.
⇒ Use more than one rotation axis.
⇒ Utilize crystal symmetries to extract all information from one rotation pattern. („Single rotation method“, Tesche et al., JMR 1993)
TIFR/RAC, Jan. 2010 Quadrupolar Nuclei I to III
Relation of the Q-Tensor to Molecular Structure
Haeberlen & co-workers, J. Magn. Reson., (2001), 151, 65-77. For a static (i.e. not motionallyaveraged) Q-tensor of a chemically bound deuteron, the following 3 rules can be formulated:
TIFR/RAC, Jan. 2010 Quadrupolar Nuclei I to III
Dynamic Information by 2H-NMR: Relaxation Time Analysis
Reorientational correlation timesaccessible by NMR methods
T1 relaxation time curve
methyl groupdynamics
phenyl ringdynamics
TIFR/RAC, Jan. 2010 Quadrupolar Nuclei I to III
The Quadrupolar Echo Sequence
Echo experiment for I=1 only!
Systematic derivation:M. H. Levitt, Spin Dynamics,2nd edition
Alternative derivation:
TIFR/RAC, Jan. 2010 Quadrupolar Nuclei I to III
Solid-state 14N-NMR
S.P. Marburger, B.M. Fung, A. K. Khitrin,J. Magn. Reson. (2002) 154, 205--209
TiN(underMAS)
low symmetry:⇒ wide-line 14N-NMR spectra
TIFR/RAC, Jan. 2010 Quadrupolar Nuclei I to III
14N-NMR: Overtone Spectroscopy
14N-Overtone NMR spectra of N-acetyl-D,L-valine
The frequency of a m ⇔ -m transition is unaffected by the 1st order quadrupolarsplitting. Therefore, overtone spectra of integral spin nuclei (e.g., 14N) can havemuch smaller total spectral ranges than the fundamental single-quantum spectra.
TIFR/RAC, Jan. 2010 Quadrupolar Nuclei I to III
Quadrupolar Nuclei in Inorganic Materials: Mostly I > 1
Numerous materials of technologicalinterest, which are functional only in thesolid state:
• Glasses, Ceramics• Minerals, Cements• Catalysts (Zeolites)• Polymers, Biopolymers
Frequently occuringNMR-observable nuclei:
• 1H (I=1/2)• 13C (I=1/2)• 29Si (I=1/2)• 11B (I=3/2)• 17O (I=5/2)• 23Na (I=3/2)• 25Mg (I=5/2)• 27Al (I=5/2)
(2002)
TIFR/RAC, Jan. 2010 Quadrupolar Nuclei I to III
Energy Levels of Spin-5/2
5(2si n22θ + si n4q)
3(si n4θ - 2si n22θ)
- 3(si n4θ - 2si n22θ)
- 5(2si n22θ + si n4θ)
2(si n4θ - 2si n22θ)
- 2(si n4θ - 2si n22θ)CT
ST
ST
ST
ST
HNMR = HZ + HQ (+ HDD + HCS) Zeeman 1st order HQ 2nd order HQ
- hν0m ( hCq/40) (3cos2θ- 1) (9hCq2/6400ν0)
TIFR/RAC, Jan. 2010 Quadrupolar Nuclei I to III
Spin-5/2 Static Spectra
1st order
central transition
Spectrum with 2I components, shifted by
νm = 3Cq(3cos2θ –1)(mz –1/2 )/(4I(2I-1)), with Cq = eQVzz/h
→ quadrupolar interaction parameters can be determined from full 1st orderpattern, or from characteristic 2nd order line shape
satellite transitions
2nd order
A = (I(I +1)-3/4) νq2/ν0
TIFR/RAC, Jan. 2010 Quadrupolar Nuclei I to III
Spectra of Spin-3/2 under MAS
ν r
M. E. Smith, E.R.H. van Eck, Prog. NMR Spec., 34, 159 (1999)
23Na-NMR spectra of Amelia albite (NaAlSi3O8)
2nd order line shape
2nd order quadrupolar lineshapes of the central transition
→ Extraction of quadrupolarparameters from shape ofcentral-transition line.
TIFR/RAC, Jan. 2010 Quadrupolar Nuclei I to III
Signal Enhancement by Spin Population Transfer (SPT)
central transition (CT)
satellite transitions (ST)
Saturation of satellite transitions firstdemonstrated by R. V. Pound in 1950.
Experimental realisation:
• adiabatic passage (Haase et al.)
•Double Frequency Sweeps (DFS) (Kentgens et al.)
• Fast Amplitude Modulated (FAM) pulse trains (S. Vega, P. K. Madhu, A. Goldbourt; P. Grandinetti)
• hyperbolic secant pulses (HS) (Wasylishen et al.)
Signal enhancement of CT:
saturation of ST ⇒ I + 1/2
inversion of ST ⇒ 2I
TIFR/RAC, Jan. 2010 Quadrupolar Nuclei I to III
Signal Enhancement using the QCPMG Sequence
87Rb (I=3/2) static
TIFR/RAC, Jan. 2010 Quadrupolar Nuclei I to III
Excitation Regimes for Quadrupolar Nuclei with I > 1
Non-selective excitation:
!
"RF
>>"Q!
"Q#"
Q
(1)max( ) =
3$
2I 2I %1( )
Selective excitation:
!
"RF
<<"Q
Intermediate case:
!
"RF#"
Q
TIFR/RAC, Jan. 2010 Quadrupolar Nuclei I to III
Selective Excitation of Central Transition
Pulse Nutation Response of CT:
Spin 3/2:
!
"nut
C= 2"
RF
Spin 5/2:
!
"nut
C= 3"
RF!
"nut
C= I +
1
2
#
$ %
&
' ( "RF
!
"nutC
= 2#$ nut
C % & p = 2# I +1
2
'
( )
*
+ , $RF % & p
Correct selective (“solid”) pulse:
“This effect often catches outinexperienced spectroscopists.”
(M. H. Levitt, Spin Dynamics, 2nd edition)
!
"RF
<<"Qfor
TIFR/RAC, Jan. 2010 Quadrupolar Nuclei I to III
How to suppress 2nd order broadening
Simulation of 27Al-NMR MAS spectra ofthe centre band of kyanite (Al2SiO5)
B 0
M. E. Smith, E.R.H. van Eck, Prog. NMR Spec., 34, 159(1999)
field dependenceLegendre poly-nomial P4(cosq)
magi c angl e
P4(cosq)P2(cosq)
→ no angle for averaging P2and P4 simultaneously byrotating around a single axis
four sites resolved!
!"#1/ 2 , +1 / 2
(2nd order) = #Cq
2
6"0
I I +1( ) #3
4
$
% &
'
( )
* Acos4+ + Bcos
2 + + C[ ]
TIFR/RAC, Jan. 2010 Quadrupolar Nuclei I to III
Averaging P4(cosθ) by Double Rotation (DOR)
“High resolution solid-state N.M.R. Averaging ofsecond-order effects by means of a double-rotor”;A. Samoson, E. Lippmaa, & A. Pines,Mol. Phys., 65, 1013 (1988)
→ mechanically demanding: limited rotation speeds, therefore limited resolution→ still impossible to extract both chemical shift and Cq from one DOR spectrum
TIFR/RAC, Jan. 2010 Quadrupolar Nuclei I to III
Dynamic Angle Spinning (DAS)
B. F. Chmel ka et al ., Nature, 339, 42 (1989)
L. M. Bul l et al ., J. Am. Chem. Soc., 120, 3510 (1998)
→ less mechanically demanding than DOR→ because of long switching time (30 ms), the signal vanishes if sample has: * short T1 * strong dipolar interactions
TIFR/RAC, Jan. 2010 Quadrupolar Nuclei I to III
Multi-Quantum MAS (MQMAS) Spectroscopy
MQMAS DAS
coherence order
“Isotropic Spectra of Half-Integer QuadrupoleSpins from Bidimensional MAS NMR”,L. Frydman, and J. S. Harwood,J. Am. Chem. Soc., 117, 5367 (1995)
number of citations (Jan. 2010): 731
General principle: refocus anisotropicparts of interactions, so that at kt1 anecho will form. The amplitude of thisecho evolves only under the isotropicparts of the interactions.
!"#1/ 2 ,+1 / 2
(2nd order)= !" iso
(2nd order)+ Cl
pAl $ ,%( )
l=2,4
& Pl cos'( )
k =P2cos!
1( )P2cos!
2( )=P4cos!
1( )P4cos!
2( )k =
C4
p
C4
1
TIFR/RAC, Jan. 2010 Quadrupolar Nuclei I to III
Resolving Structural Sites with MQMAS
87Rb MQMAS spectra of RbNO3
D. Massiot et al., Solid State NMR, 6,73, (1996)
• anisotropic dimension F2 (2nd order quadrupolar)
• isotropic dimension F1 (chemical shift)→ different sites resolved
• obtain 2nd order line shapes for sites from 1D slices
• simulate line shapes to extract Cq values