jerschow NMR elec qpole etc
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A published review (Progress in Nuclear Magnetic Resonance Spectroscopy 46, 2005, pp. 63-78) by A. Jerschow of NYU, apparently saved from the web. It derives the quadrupolar NMR Hamiltonian from nuclear and electronic multipole moments using spherical tensors and SI units, covering electric field gradients, NQR, and first- and second-order effects. It also outlines DOR, DAS, MQMAS and STMAS.
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From nuclear structure to the quadrupolar NMR interaction
and high-resolution spectroscopy
Alexej Jerschow *
Chemistry Department, New York University, New York, NY 10003, USA
Received 7 September 2004
Keywords: Nuclear structure and spin; Quadrupolar coupling; Nuclear quadrupole moment; Electric field gradient; Solid-state NMR; MQMAS; STMAS;
DAS; DOR; NQR
Contents
1. Introduction . . . . ....................................................................... 6 4
2. Nuclear structure and nuclear and electronic multipole moments . . . ................................. 6 4
2.1. Nuclear structure and spin . . ........................................................... 6 4
2.2. The classical nucleus–electron interaction . ................................................ 6 5
2.3. The quantum nucleus–electron interaction . ................................................ 6 5
2.4. The multipole moment representation, spherical tensors . . ..................................... 6 5
2.5. The electric field gradient . . ........................................................... 6 6
2.6. Ionic model for electric field gradients, Sternheimer factor ..................................... 6 7
3. NMR Hamiltonian and spectra . . ........................................................... 6 7
3.1. NMR Hamiltonian for the quadrupolar interaction . . . . . . ..................................... 6 7
3.2. Nuclear quadrupole resonance . . . . . . .................................................... 6 8
3.3. High magnetic fields and truncation . . .................................................... 6 8
3.4. Higher order terms of HQperturbing the Zeeman Hamiltonian . ................................. 6 8
3.5. Rapid sample spinning ............................................................... 7 0
3.6. Nutation spectroscopy . ............................................................... 7 1
3.7. Overtone spectroscopy ............................................................... 7 1
4. High-resolution spectroscopy of quadrupolar nuclei . . ............................................ 7 2
4.1. Double rotation—DOR ............................................................... 7 2
4.2. Dynamic-angle spinning—DAS . . . . . .................................................... 7 2
4.3. Multiple-quantum MAS—MQMAS . . .................................................... 7 3
4.4. Satellite transition MAS—STMAS . . .................................................... 7 4
4.5. Other experiments ................................................................... 7 4
5. Conclusions . . . . ....................................................................... 7 5
0079-6565/$ - see front matter q2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.pnmrs.2004.12.001
Progress in Nuclear Magnetic Resonance Spectroscopy 46 (2005) 63–78
www.elsevier.com/locate/pnmrs
* Tel.: C1 212 998 8451; fax: C1 212 260 7905.
E-mail address: [email protected].
Acknowledgements . . . .................................................................. 7 5
Appendix A . . . . . ...................................................................... 7 5
A.1. Other second-order interactions . . ....................................................... 7 5
A.1.1. Quadrupolar/dipolar coupling . . ................................................ 7 5
A.1.2. Quadrupolar/chemical shift . . . ................................................ 7 5
A.1.3. Additional cross-terms . . . .................................................... 7 5
A.2. Sine and cosine representation of the first- and second-order quadrupolar frequencies ................. 7 6
A.2.1. First order . ............................................................... 7 6
A.2.2. Second order . . . ........................................................... 7 6
References . . . . . . ...................................................................... 7 7
1. Introduction
NMR spectroscopy of isotopes with nuclear spin
quantum numbers greater than 1/2 has received widepopularity recently due to the development of powerful
high-resolution techniques. The theoretical description of
these experiments is well established and many reviewarticles discuss the appearance of the quadrupolar inter-
action in NMR spectra. In this context, however, the
physical origin of the quadrupolar interaction is rarely
discussed, and the quadrupolar NMR Hamiltonian is usually
not derived from more fundamental principles.
The authoritative texts of Abragam [1]and Slichter [2]
remain the only relevant references in this respect. On the
other hand, they use spherical and spatial tensor formalismswhich are not in line with recent literature, which often can
cause confusion over the correct definition of quadrupolar
parameters. This may have contributed to the widespreaduse of different conventions for the quadrupolar interaction
(u
Q,CQ,c, etc.), and often disguises the contributions of the
nuclear and electronic effects to the quadrupolar coupling. Itis the aim of this review to provide a link from nuclear
physics aspects to the actual manifestation in NMR spectra.
The use of modern spherical tensor definitions and SI unitswill help the reader to make this connection more easily and
to make a comparison with recent literature more
transparent.
It is not unusual that significant differences in nuclear
quadrupole moment data are reported and NMR spec-
troscopy may be very helpful in providing more accuratedata. A clear connection between NMR observables and
nuclear and electronic quadrupole moment data is therefore
highly valuable, especially in view of the wide variety ofconventions used in the nuclear physics literature.
To pay tribute to the success of new high-resolution
NMR techniques for the study of quadrupolar nuclei, asection on the most important of these experiments is
included, outlining the equations for their theoretical
description. This part is kept brief to avoid overlap withrecent review articles, and references are given that discuss
these techniques in more detail. A number of topics that are
often left out, such as nuclear quadrupole resonance,overtone spectroscopy, and the ionic model for the
calculation of electric field gradients, are included.
The article is organized as follows. We provide a brief
introduction into nuclear structure and spin, followed by thederivation of the quadrupolar NMR Hamiltonian via themultipole moments. The appearance of simple spectra underthe action of the quadrupolar NMR Hamiltonian is thendiscussed, including first- and second-order effects, as wellas, the nuclear quadrupole resonance case, rapid sample
spinning, nutation and overtone spectroscopy, and cross-
terms with the dipolar and chemical shift anisotropyinteractions are given. The high-resolution techniques arethen discussed, including double rotation, dynamic-anglespinning, multiple-quantum magic-angle spinning, andsatellite transition magic-angle spinning.
2. Nuclear structure and nuclear and electronic
multipole moments
2.1. Nuclear structure and spin
The nuclear spin responsible for NMR phenomena is the
result of the interplay between the individual spins andangular momenta of the protons and neutrons that reside inthe nucleus. While exact calculations of the states of differentnuclides are rather difficult, models based on simplifiedpotentials may describe the nuclear configurations to variousdegrees of accuracy. It is found that nuclides with certainnumbers of protons and neutrons are particularly stable,
namely those with numbers of particles of one kind of 2, 8,
20, 28, 50, 82, and higher [3]. These numbers are often
referred to as magic numbers and can be compared insignificance to the numbers of electrons in closed-shell atomsand ions (noble-gas arrangement: 2,8,18, .,). Nuclides in
which both the numbers of neutrons and protons are ‘magic’are called ‘doubly magic’ nuclides and are known to beexceptionally stable. The sequence of these numbers can be
explained by a so-called shell model, which treats
the nucleons as particles occupying orbits influenced by theaverage strong attractive forces exerted by the othernucleons. As one adds protons or neutrons to the nucleusA. Jerschow / Progress in Nuclear Magnetic Resonance Spectroscopy 46 (2005) 63–78 64
they drop into the lowest-energy shells permitted by the
Pauli-Principle. Filled shells have a total angular momentumIequal to zero, and the next added nucleon (a valence
nucleon) determines the Iof the new ground state. To a very
good approximation, one can fill up the shells first
individually by protons and neutrons, and then couple the
total angular momenta of both types of valence nucleons.
The nuclear spin can be calculated by examining the
coupling of the individual spins of all valence nucleons.Unlike electrons, nucleons tend to pair their spins ratherthan to fill the orbitals according to Hund’s rule. This is dueto relativistic spin–orbit couplings, which also occur forelectrons in heavy atoms. Since pairs couple to spin zero, a
nuclide with an even number of protons and an even number
of neutrons usually has spin zero.
The final spin value of a nuclide is usually
determined by the spins of the unpaired protons andneutrons if any. If there is an odd number of one type ofnucleons and an even number of the other, the spin isdetermined by the last nucleon added of the former typeand is a half-integer value ( lCsby spin–orbit coupling).
If the number of both types of nucleons is odd then
proton–neutron spin coupling occurs. Whether theycouple in a parallel or antiparallel manner is difficultto predict but the total spin value will be an integervalue. Chapter 1 of Ref. [4]and Chapter 26 of Ref. [5]
are recommended for a very instructive introductioninto nuclear structure.
The following subsections are partly based on material
from Abragam [1], Slichter [2], and Cohen and Reif [6],
using a more common spherical tensor formalism and SIunits.
2.2. The classical nucleus–electron interaction
Classically, the collective electrostatic energy from two
charge distributions, r
n(rn) for the nucleus, and re(re) for the
electrons, is
EZ1
4pe0ðð
dredrnreðreÞrnðrnÞ
jreKrnj(1)
This can be expanded in spherical coordinates ( rn,qn,fn
for the nucleus, and re,qe,fefor the electrons) using
1
jreKrnjZ4pXN
lZ0Xl
mZK11
2lC1rl
small
rlC1
largeYm/C3
lðqn;fnÞYm
lðqe;feÞ
(2)
where Ym
lare the spherical harmonics, and rsmall andrlarge
are the smaller and the larger of the numbers reandrn. Since
the nucleus is so much smaller than average electrondistances, we can safely assume that r
smallZrnandrlargeZ
re. For compactness we can rewrite the energy as
EZXN
lZ0Xl
mZK1Am
lBm/C3
l (3)where
Am
lZffiffiffiffiffiffiffiffiffiffiffiffiffi
4p
2lC1r ð
drnrnðrnÞrl
nYm
lðqn;fnÞ (4)
and
Bm
lZ1
4pe0ffiffiffiffiffiffiffiffiffiffiffiffiffi
4p
2lC1r ð
drereðreÞrKlK1
eYm
lðqe;feÞ (5)
It is useful to include the factor 1/(4 pe0) in the electronic
portion as it allows one to incorporate that factor into theexpression for the electrostatic potential.
2.3. The quantum nucleus–electron interaction
In order to perform the transition to quantum mechanics,
we make use of the correspondence principle. r
e,n,Ym
l, and
re,nbecome operators and Am
landBm
lexpectation values.
Forrnwe use, in particular
rnðrnÞZ JnXA
iZ1eidðrnKRiÞ/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12J
n*+
(6)
where jJniis the ground state wave function of the nucleus,
Ais the number of nucleons, and
eiZe;the unit charge ;for proton
0;for neutron(
(7)
A similar expression holds for the electrons, and the
operators producing the expectation values Am
landBm
lcan
then be written as
Al;mZffiffiffiffiffiffiffiffiffiffiffiffiffi
4p
2lC1rXA
iZ1eirl
iYm
lðqi;fiÞ (8)
Bl;mZKe1
4pe0ffiffiffiffiffiffiffiffiffiffiffiffiffi
4p
2lC1rXN
iZ1rKlK1
iYm
lðqi;fiÞ (9)
summing over the Anucleons and Nelectrons, respectively.
Here, riandYm
lare operators. Al,mandBl,mare spherical
tensor operators of rank land order m.
The Hamiltonian becomes a contraction of the two
spherical tensors
HZXN
lZ0Xl
mZKlAl;mB†
l;mZXN
lZ0Xl
mZKlðK1ÞmAl;mBl;Km (10)
2.4. The multipole moment representation, spherical tensors
The first term in the series is A0,0B0,0,w h i c h
represents the electrostatic energy between two concen-trically arranged spherically symmetric charges and is
hence isotropic. Stationary nuclear states are generally
assumed to have defined parity [1,3] , i.e. they are
unchanged under inversion of the coordinate system.This appears to be well established by experiment.A. Jerschow / Progress in Nuclear Magnetic Resonance Spectroscopy 46 (2005) 63–78 65
Since under the inversion P
ðr;q;fÞ /C255/C255/C255/C255/Pðr;pKq;pCfÞ; (11)
and
Ym
lðfKq;pCfÞ /C255/C255/C255/C255/PðK1ÞlYm
lðq;fÞ (12)
all terms Al,mwith lodd vanish.
The first relevant term is therefore the quadrupole
moment ( A2,m). All moments in Eq. (10) carry a factor
proportional to rl
n=rlC1
eand since rn/rethe expansion
converges rapidly. For this reason, the next relevant term,the hexadecapole moment, would be extremely difficult toobserve [7–9] . The extent to which higher multipole
moments occur is further limited by the spin value I.B y
the Wigner–Eckart theorem [10,11] the matrix elements of a
spherical tensor operator can be written as
hI
0M0jAl;mjIMiZhlm;IMjI0M0ihI0kAlkIi (13)
where the first factor on the right side is the Clebsch–Gordan
coefficient, Ithe total angular momentum quantum number
(corresponding to the nuclear spin), Mthe magnetic
quantum number, and hI0kAlkIia constant characteristic of
the particular tensor chosen, also called the ‘reduced matrixelement’. The possible values of MandM
0areKI,.,I. The
Clebsch–Gordan factors are non-zero only for
jIKI0j%l%jICI0j (14)
hence they are zero whenever lO2I.T h eq u a d r u p o l e
moment is therefore only observable for spins IR1 and
the hexadecapole moment would be seen only for IR2.
Using the expansions of the spherical harmonics in
Cartesian coordinates [10,11] one obtains
A2;0Z1
2X
ieið3z2
iKr2
iÞ (15)
A2;G1ZHffiffiffi
3
2rX
ieiziðxiGiyiÞ (16)
A2;G2Zffiffiffi
3
8rX
ieiðxiGiyiÞ2(17)
The quadrupole moment of the nucleus is defined as
eQZ IIX
ieið3z2
iKr2
iÞ/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12II*+
Z2hIIjA
2;0jIIi (18)
which is a number that is usually derived from experiment.
The second entry in the ket means that the magneticquantum number MZI. Only one constant need to be
specified in Eq. (18) since all the components of the A
2,m
tensor are related to this parameter via the Wigner–Eckart
Theorem. As is apparent from Eq. (18), the actual strengthof the interaction depends on the state of the ensemble of thenucleons. This expression may be converted into a moreuseful form by replacing A
2,mby a tensor Q2,mthat involves
the total spin-operators only of the coupled ensemble ofnucleons. Again, making use of the Wigner–EckartTheorem, we obtain
Q
2;0Za1
2ð3I2
zKIðIC1ÞÞ (19)
Q2;G1ZHaffiffiffi
3
8r
ðIzIGCIGIzÞ (20)
Q2;G2Zaffiffiffi
3
8r
I2
G; (21)
where the factor ais to be determined. Here
IGZIxGiIy: (22)
Occasionally, this equation is represented using the same
IGsymbols denoting the spherical tensor operators
TGZHð1=ffiffiffi
2p
ÞðIxGIyÞ, in which case Q2,G1and Q2,G2
acquire additional factors of Hffiffiffi
2p
and 2, respectively. We
will continue to use the definition of Eq. (22).
The factor aof Eq. (19) is to be determined by
introducing a number eQ, defined by
hIIjA2;0jIIiZeQ=2ZahIIjQ2;0jIIi
ZaII1
2½3I2
zKIðIC1Þ/C138/C12/C12/C12/C12/C12/C12/C12/C12II/C28/C29
Zða=2ÞIð2IK1Þ (23)
which gives
aZ
eQ
Ið2IK1Þ: (24)
2.5. The electric field gradient
Likewise, we can derive a spherical tensor operator for
the electronic portion of the form
B2;0Z1
21
4pe0ðN
iZ1ðKeÞð3z2
iKr2
iÞ
r5
i(25)
along with the other components.
IfV(x,y,z) is the electrostatic potential produced by the
electrons at point ( x,y,z)
B2;0Z1
2v2V
vz2/C18/C19
rZ0Z1
2Vzz: (26)
Vzzis an electron operator with a corresponding
expectation value. The other components are
B2;G1ZH1ffiffiffi
6pðVxzGiVyzÞ (27)
B2;G2Z1
2ffiffiffi
6pðVxxKVyyG2iVxyÞ: (28)A. Jerschow / Progress in Nuclear Magnetic Resonance Spectroscopy 46 (2005) 63–78 66
If we choose as coordinate axes, the principal axes of the
symmetric tensor Vij, the cross-terms vanish, i.e.
VxyZVxzZVyzZ0. It is a common convention to label
the remaining components such that
jVzzjRjVyyjRjVxxj (29)
and to define
VzzZeq (30)
hZðVxxKVyyÞ=Vzz: (31)
By Lagrange’s Equation, VxxCVyyCVzzZ0, and hence
jVzzjZjVxxCVyyj, which, combined with Eq. (29) shows
thatVxxandVyymust have the same sign, opposite to the
sign of Vzz. Eq. (31) then shows that 0 %h%1.
Using these definitions, the tensor components for the
electronic portion become
B2;0Z1
2eq (32)
B2;G1Z0 (33)
B2;G2Z1
2ffiffiffi
6peqh: (34)
in the principal axis frame.
2.6. Ionic model for electric field gradients,
Sternheimer factor
The calculation of the electric field gradients Vaaby ab
initio methods can be very time-consuming and anacceptable precision is not always achievable even usinghigh-performance computers. Frequently, it is therefore
convenient to use a point-charge approximation to calculate
the field gradient produced at the site of the nucleus of interestby the surrounding charged atoms [12]. One first calculates
the partial charges of the atoms surrounding the quadrupolarnucleus (these could be Mulliken charges, for example, orother types of charge interpolations). In order to calculateeqZV
zzandhZðVxxKVyyÞ=Vzzone needs to first calculate
VLAB
abin the laboratory frame. By explicit differentiation of
the electrical potential of the point charges (with charge nie)
one obtains
VLAB
aaZ1
4pe0X
inieð3a2
iKr2
iÞ=r5
i (35)
VLAB
abZ1
4pe0X
inie3aibi=r5
i (36)
Diagonalizing the associated matrix produces VPAS
aaZVaa
in the principal axis frame and the corresponding transform-
ation matrix.
In practice one finds, however, that the actual field
gradient at the site of the nucleus of interest is a multiple ofthe one calculated by Eq. (35) and is written as
Vactual
aa Zð1KgÞVcalc
aa: (37)
where gis called the Sternheimer antishielding factor [1,2] .
The factors are often large and negative for heavy atoms (aslarge as several hundred). This effect is caused by the
polarization of the closed-shell electrons in the presence of
external field gradients.
3. NMR Hamiltonian and spectra
3.1. NMR Hamiltonian for the quadrupolar interaction
Combining Eqs. (10), (19)–(21), (24), (32)–(34) one
arrives at the definition of the quadrupolar interaction in theprincipal axis frame of V
abas
HQZe2qQ
4Ið2IK1Þ3I2
zKIðIC1ÞCh
2ðI2
CCI2
KÞhi
(38)
in units of Joules.
In NMR literature, it is common practice to rewrite
Eq. (38) for the quadrupolar interaction as
HQZuQX2
mZK2ðK1ÞmR2;KmT2;m (39)
where
uQZe2qQ
2Ið2IK1ÞZZ2pCQ
2Ið2IK1Þ: (40)
In this notation, HQanduQhave units of radians, and CQZ
e2qQ/h, often referred to as the ‘quadrupole coupling constant’,
has units of Hertz. The spatial tensors R2,mcan be related to the
tensors of the principal axis frame, rl,m0by a Wigner rotation
parameterized by the Euler angles a,b,g[10,11]
Rl;mZX
m0Dl
m0;mða;b;gÞrl;m0; (41)
with
r2;0Zffiffiffiffiffi
3
2;r
r2;G1Z0;r2;G2Z1
2h: (42)
The spin tensors T2,mare defined in analogy to Eq. (19) as
T2;0Z1ffiffiffi
6p/C2
3I2
zKIðIC1Þ/C3
;
T2;G1ZH1
2ðIGIzCIzIGÞ;T2;G2Z1
2I2
G:(43)
If spherical tensors are used for IGrather than the definition
of Eq. (22), additional factors of Hffiffiffi
2p
and 2 apply for T2,G1
andT2,G2, respectively.
The advantages of the choice of the normalizations in the
RandTtensors are not immediately clear but reflect a very
common convention in recent literature (one may point outA. Jerschow / Progress in Nuclear Magnetic Resonance Spectroscopy 46 (2005) 63–78 67
that in this convention the Clebsch–Gordan coefficient
h10;10j20iZffiffiffiffiffiffi ffi
3=2p
Zr2;0, which highlights the fact that
the second rank tensor can be produced by coupling twoone-rank tensors).
In the principal axis system, Eq. (39) takes on the form
H
QZðuQ=2Þ3I2
zKIðIC1ÞCh
2ðI2
CCI2
KÞno
: (44)
Often, another notation is used in which u0QZ(3e2qQ)/
2I(2IK1)Z. This quantity then corresponds to the distance
between resonance lines in a zero-field experiment or in ahigh-field experiment with hZ0. To avoid confusion, we
denote this quantity converted to Hertz as f
Q
fQZ3uQ=2pZ3e2qQ
h2Ið2IK1Þ: (45)
Frequently, also half that value is used ðf0
QZð3e2qQÞ=h
4Ið2IK1ÞÞ[13,14] . For a spin 1, f0
Qthen corresponds to the
distance between the peaks in a Pake doublet when hZ0.
3.2. Nuclear quadrupole resonance
Without the application of an external magnetic field, the
Hamiltonian of Eq. (44) creates a discrete spectrum, a mode ofoperation that is commonly termed Nuclear QuadrupoleResonance (NQR) [15]. In particular, for hZ0 one finds 2 I
resonance lines for a spin I,s p a c e db y3 u
Q/(2p), centered at
zero frequency. Slight differences are found for hs0. This is
observed regardless of the orientation of the system withrespect to the measuring apparatus. The orientation merely hasan influence on the relative intensities of the resonance lines.
The same is true for field-cycling experiments, where the
zero-field quadrupolar frequencies are observed indirectlyin a two-dimensional experiment. The spin-system is firstpolarized in a high magnetic field, then the system evolves
at zero magnetic field and is finally observed again at a high
magnetic field [16–18] . The orientation of the sample with
respect to the polarizing magnetic field solely influences therelative intensities of the discrete resonance lines.
3.3. High magnetic fields and truncation
If the system is placed in a strong magnetic field such that
the Zeeman interaction can be assumed to be much strongerthan the quadrupolar interaction, the truncated version ofEq. (44) can be used, i.e.
H
QZuQR2;0T2;0ZuQR2;01ffiffiffi
6p½3I2
zKIðIC1Þ/C138: (46)
which represents the portion of the Hamiltonian that
commutes with the Zeeman interaction (the same equationis applicable to NQR in the case of axial symmetry). Since
in this case, a preferred axis of quantization is set by the
Zeeman interaction through the orientation of the externalmagnetic field, we have to take into account the orientationof the principal axis frame of the Rtensor with respect tothe magnetic field direction, i.e.
R
2;0Zffiffiffiffiffiffi ffi
3=2p
½P2ðcosbÞCðh=2Þcos 2 asin2b/C138; (47)
where aandbare two of the three Euler angles. If a sample
is investigated that contains crystals with random distri-
bution of orientations, a so-called powder spectrum isobserved, such as those represented in Fig. 1 .
A similar type of truncation may be obtained in NQR if one
attempts to rotate the sample rapidly around an axis in theabsence of an external magnetic field. Since the tensorcomponents R
2,mforms0 oscillate rapidly and average out
over time, it appears as though the rotation axis imposes a
quantization direction onto the spin-system. Eqs. (46) and (47)
then apply as well, with the angles aandbdetermining the
rotation from the principal axis system onto the rotatingcoordinate system. As a result, a powder lineshape is obtained.Frequently, however, the mechanical reorientation is not rapidenough and a series of discrete resonance lines arises.
By the same token, the case of rapid sample spinning in a
high magnetic field can be considered as a ‘double
truncation’, whereby the quadrupolar interaction is first
quantized along the direction of the Zeeman Hamiltonian,resulting in a powder lineshape, and then along the directionof the rotor axis. The spatial part R
2,mforms0 oscillates
rapidly, and if the rotor axis is inclined at the magic anglewith respect to the magnetic field (magic-angle spinning—MAS) the spatial portion R
2,0is exactly zero. The case of
double rotation (DOR) [19] may then be regarded as a triple
truncation. Due to practical limitations, the truncation
imposed by sample spinning is seldom complete and aseries of spinning sidebands occurs.
If the Zeeman interaction is not overwhelmingly stronger
than the quadrupolar interaction, higher terms in aperturbation series need to be considered. If the twointeractions are of comparable strengths then the fullHamiltonian needs to be diagonalized and direct computer
simulations may be the only applicable method for
obtaining representations of NMR spectra [20]. A discus-
sion on the transition from the NQR situation to the high-field NMR case can be found in Ref. [21].
In the following, we restrict ourselves to the case of a
strong Zeeman interaction and we will only consider thefirst few terms in a perturbation expansion.
3.4. Higher order terms of H
Qperturbing the Zeeman
Hamiltonian
The second-order quadrupolar interaction can be calcu-
lated from the second-order perturbation series for the non-degenerate case (the degenerate case would be found inNQR, which is irrelevant here)A. Jerschow / Progress in Nuclear Magnetic Resonance Spectroscopy 46 (2005) 63–78 68
Eð2Þ
jZ1
u0X
jskhjjHQjkihkjHQjji
kKj
ZKu2
Q
u0X
ms0R2;KmR2;mhjjT2;mT2;Kmjji
m(48)
Splitting the sum over minto two equal parts and
regrouping terms of opposite minto commutators, we get
Hð2Þ
QZu2
Q
u0X
mO0R2;KmR2;m½T2;Km;T2;m/C138
m; (49)
When one attempts to derive the second-order term by
Average Hamiltonian Theory, one obtains an additionalterm proportional to [ T
2,m,T2,0], which can be shown to bean artifact of the method. These issues were discussed
extensively in [22–24] .
The third-order contribution, which appears to be
important in some experiments [25,26] can be derived from
Hð3ÞZ1
u2
0X
ks0;ms01
km½HKk;½HkKm;Hm/C138/C138; (50)
where Hmis the portion of the Hamiltonian whose spin part
transforms like an m-rank tensor.
Using
K1Z½T2;K1;T2;1/C138Z1
2Iz½4IðIC1ÞK8I2
zK1/C138;
K2Z½T2;K2;T2;2/C138ZIz½2IðIC1ÞK2I2
zK1/C138(51)
Fig. 1. First-order quadrupolar lineshapes in the high-field limit. The frequencies are plotted in units of 3 uQ/2p. The lineshapes of the individual transitions are
plotted underneath the overall powder spectrum.A. Jerschow / Progress in Nuclear Magnetic Resonance Spectroscopy 46 (2005) 63–78 69
we rewrite Eq. (49) as
Hð2Þ
Q¼u2
Q
u0R2;1R2;K1K1þ1
2R2;2R2;K2K2/C18/C19
(52)
Converting the products R2,mR2,Kmto the coupled
representation Al,mwe get
Hð2Þ
QZu2
Q
u0X
mZ1X
2lZ0;2;41
mAl;0Kmh2;Km;2;mjl;0i (53)
In the principal axis frame we have
A0;0Zr0;0Z3Ch2
2ffiffiffi
5p;A2;0Zr2;0ZK3Ch2
ffiffiffiffiffi
14p;
A4;0Zr4;0Z18Ch2
2ffiffiffiffiffi
70p:(54)
Evaluating the sum over mwe obtain
Hð2Þ
QZu2
Q
u01ffiffiffi
5pA0;0Iz½3I2
zKIðIC1Þ/C138/C26
C1
2ffiffiffiffiffi
14p A2;0Iz½8IðIC1ÞK12I2
zK3/C138
C1
2ffiffiffiffiffi
70p A4;0Iz½18IðIC1ÞK34I2
zK5/C138/C27
ð55Þ
The term containing A0,0transforms as a zero rank tensor,
hence is independent of orientation and produces an isotropicshift, called the quadrupolar isotropic shift u
QISin radians or
in Hertz. Using Eqs. (54) and (55) gives the shift as
C2
Qð3Ch2ÞIz½3I2
zKIðIC1Þ/C138
10v0½2Ið2IK1Þ/C1382: (56)
in units of Hertz. For central transitions (from Kmtom)w e
may substitute IzZp/2, where pis the coherence order, and
considering that a transition frequency will be twice
the energy shift we arrive at a frequently used expressionfor the quadrupolar isotropic shift v
QIS
vQISZC2
Qð3Ch2Þp½ð3=4Þp2KIðIC1Þ/C138
10v0½2Ið2IK1Þ/C1382: (57)
3.5. Rapid sample spinning
The second and fourth rank tensors Al,0display
orientational dependence. In order to obtain the appropriateinteraction in the laboratory reference frame one needs toperform rotations from their principal axis frame. It iscustomary to perform two Wigner rotations: One from the
principal axis frame of the electric field gradient tensor into
the frame of the rotor (including the current rotor angle), andfrom there to the laboratory frame. The sequence ofrotations can be represented byA
l;0ðtÞZX
mX
nDl
m0ðurt;q;0ÞDl
nmða;b;gÞrl;n
ZX
mX
nDl
m0ð0;q;0ÞDl
nmða;b;gCurtÞrl;n
ZX
mX
ndl
m0ðqÞdl
nmðbÞexpðKinaKim½gCurt/C138Þrl;n
(58)
where a,b,gare the three Euler angles to rotate the tensor
from the principal axis frame into the rotor frame, qis the
angle between the static magnetic field and the rotor axis,
and uris the rotor speed. The non-zero tensors in
the coupled representation in the principal axis frame are
r0;0Z3Ch2
2ffiffiffi
5p;r2;0ZK3Ch2
ffiffiffiffiffi
14p;
r2;G2Zhffiffiffi
3
7r
;r4;0Z18Ch2
2ffiffiffiffiffi
70p;
r4;G2Z3h
2ffiffiffi
7p;r4;G4Zh2
4:(59)
For numerical calculations, it is often convenient to cast
Eq. (58) in a form of the type
Al;0ðtÞZcl;0CX
mO0cl;mcosðmg0ÞCsl;msinðmg0Þ (60)
where g0ZgCurt. The coefficients cl,mandsl,mare then
determined as follows
cl;0Zdl
00ðqÞdl
00ðbÞrl;0C2X
nZ2;4dl
n0ðbÞcosðnaÞrl;n"#
(61)
cl;mZ2dl
m0ðqÞX
nZ0;2;4½dl
nmðbÞCð1Kdn0Þdl
KnmðbÞ/C138rl;ncosðnaÞ
(62)
sl;mZK2dl
m0ðqÞX
nZ2;4½dl
nmðbÞKdl
KnmðbÞ/C138rl;nsinðnaÞ(63)
where we have used the Kronecker dn0. The explicit
calculation of these terms is shown in Appendix A.
Under conditions of infinite spinning one can neglect the
cl,0term. If one is not interested in studying the spinning
sidebands in detail, this is a fairly reliable approach.
Eqs. (61)–(63) and their specific forms given in
Appendix A become increasingly complicated if onestudies cases where several interactions are present withdifferent principal axis frames. These can couple in secondor higher orders (e.g. second-order dipolar/quadrupolarcouplings [27], or second-order CSA–quadrupolar terms
[28]). In these situations, it is probably more straightfor-
ward to perform the simulation in the uncoupled represen-tation. For considerations of computational speed, one maystill invoke the infinite spinning assumption if oneA. Jerschow / Progress in Nuclear Magnetic Resonance Spectroscopy 46 (2005) 63–78 70
combines only tensors with opposite order in the rotor axis
frame.
The coupled representation is nevertheless quite useful
for general discussions and for the study of the underlying
symmetries [29–31] .
If the spinning angle is qZ54.74 8then Eq. (61) together
with the infinite spinning assumption predicts that the only
residual term left is c4,0.
Exact expressions for the MAS frequency spectra
obtained from quadrupolar nuclei under both the first- and
second-order quadrupole couplings have been given else-where [32]. The terms c
l,mandsl,mcan be used to derive the
intensities of spinning sidebands in a variety of experiments
[33,34] .Fig. 2 shows the powder lineshapes obtained under
static and spinning conditions at different qangles.
3.6. Nutation spectroscopy
A very useful method for determining quadrupolar
coupling constants is the nutation experiment [35]. In this
two-dimensional experiment, one applies a long rf
irradiation either to the spin system in equilibrium, or tothe spins after preparation by a soft pulse, followed by the
acquisition of a free induction decay. The duration of the rf
irradiation is incremented and represents the time variable t
1for the indirect dimension of the two-dimensional exper-
iment. After Fourier transformation, the spectra showcorrelations between the peaks of the central transitions
(the method is mostly applied to half-integer spins), and
peaks located at the nutation frequencies in the indirectfrequency dimension. Quadrupolar couplings and aniso-
tropy parameters can be determined by comparing these
nutation spectra to computer simulations.
The method draws on the fact that applying rf irradiation
with power u
rfleads to a nutation speed of the magnetiza-
tion components which depends sensitively on the ratiou
rf/uQ. The extreme cases are when urf/uQ, for which the
nutation frequency is equal to urf(IC1/2), and urf/uQ,
when it is urf. In powder samples, the experimental spectra
show sufficient structure to be fitted with appropriate
simulations [35]. Nutation spectroscopy appears most
powerful in cases with very large quadrupolar couplingconstants where methods such as MQMAS may not be
applicable.
3.7. Overtone spectroscopy
If the quadrupolar coupling constant is large such that u
Q
is no longer negligibly small compared to u0, higher orders
of perturbation need to be taken into account. The z-axis is
Fig. 2. Second-order quadrupolar lineshapes for different spinning angles and anisotropies. The frequencies are displayed in units of
ðu2
Q=2pu0Þ½IðIC1ÞK3=4/C138.A. Jerschow / Progress in Nuclear Magnetic Resonance Spectroscopy 46 (2005) 63–78 71
no longer a good quantization axis. The corrections to the
eigenvectors are calculated to first order as [22]
jjð1ÞiZKuQ
u0X
ms0ðK1ÞmR2;KmT2;m
mjji: (64)
To order uQ/u0the transition moments of the operators
IC, and Izare then non-zero for DjZ0,2,3 and DjZ1,2,
respectively (neglecting those cases where negative Larmorfrequencies would occur). Therefore, one may excite anddetect transitions at twice and triple the Larmor frequency
by transverse rf-fields, and at the Larmor frequency and
twice that frequency by rf-fields collinear with the staticmagnetic field. Higher order terms in the perturbation willgive resonances at higher multiples of the Larmor frequencybut with lower signal intensity.
Overtone spectroscopy is especially interesting for study-
ing
14N, where by observing the mZK1t oC1 transition at
u0one obtains spectra devoid of first-order quadrupolar
broadening [36–38] . This was exploited for a number of
applications, including cross-polarization, separated-localfield measurements [36–38] , overtone-nutation spectra [39],
and heteronuclear recoupling [40]. The effect of spinning on
the appearance of overtone spectra has been analyzed byTakegoshi and Hikichi [41]. Recently, a set of overtone
transitions was observed for
35Cl in a single crystal [42].
4. High-resolution spectroscopy of quadrupolar nuclei
4.1. Double rotation—DOR
Even if it were possible to rotate a powder sample at
infinite speeds, one would not succeed in eliminating theadditional linebroadening originating in the second andhigher orders of perturbation. In the limit where the second-order perturbation term is sufficient, the terms A
2,0Zc2,0and
A4,0Zc4,0of Eq. (61) determine the orientational depen-
dence of the resonance frequencies. As there is no spinningangle qat which both the factors d
2
00ðqÞZP2ðcosqÞand
d4
00ðqÞZP4ðcosqÞcan be made zero at the same time a
residual broadening remains, whatever the value of q.
In the Double Rotation experiment (DOR) [19], the
sample is spun at two different angles in a rotor-in-a-rotordesign. One can formally write down the orientational and
spinning dependence in a way similar to Eq. (58) as
A
l;0ðtÞZX
mX
nX
pDl
p0ðuð2Þ
rt;qð2Þ;0ÞDl
mp
ðuð1Þ
rt;qð1Þ;fÞDl
nmða;b;gÞrl;n
ZX
mpndl
m0ðqð2ÞÞdl
mpðqð1ÞÞdl
nmðbÞexp½KinaKimg
where q(1),uð1Þ
r, and q(2),uð2Þ
rdefine the parameters of the
first and the second spinning axes, and fis the relative
spinning angle between the first rotor and the second rotor atthe start of the experiment. Under conditions of infinitespinning speed the only remaining term is
A
t;0ZX
mpndl
00ðqð2ÞÞdl
00ðqð1ÞÞdl
n0ðbÞexp½Kina/C138rl;n: (66)
If the spinning axes are chosen such that both d2
00ðqð1ÞÞZ
0 and d4
00ðqð2ÞÞZ0, then no additional broadening arises
and a high-resolution spectrum may be obtained (e.g. whenq
(1)Z54.74 8andq(2)Z30.56 8).
In the experimental implementations, the spinning
speeds uð1Þ
randuð2Þ
rare much smaller than uQand a large
number of spinning sidebands appears. Nevertheless, theindividual sidebands and the centerband can be obtained insufficiently high resolution such that different crystal-lographic sites may be distinguished. The sideband patternsand intensities were given by Sun et al. [43]
4.2. Dynamic-angle spinning—DAS
A mechanically much less demanding approach is to spin
at two different angles sequentially rather than simul-
taneously [44,45] . One can find two different spinning
angles q
(1)andq(2)and constants aandbsuch that
ad2
00ðqð1ÞÞZKbd2
00ðqð1ÞÞ;
ad4
00ðqð2ÞÞZKbd4
00ðqð2ÞÞ;aCbZ1:(67)
One can then perform a partial echo of the residual
quadrupolar interaction during the t1period of a two-
dimensional experiment as illustrated in Fig. 3 . This is done
by spinning in one period ( a!t1) at an angle q(1)and during
the other, ( b!t1), atq(2). When aZbZ1=2 there is only one
such solution, q(1)Z37.4 8andq(2)Z79.2 8.Fig. 2 shows the
powder spectra due to second-order broadening atboth these spinning angles. The patterns at q
(1)Z37.4 8and
q(2)Z79.2 8are exact mirror images of each other, which
highlights the fact that a partial echo can be performed torefocus the residual broadening.
Fig. 3. Dynamic-angle spinning experiment.(65)A. Jerschow / Progress in Nuclear Magnetic Resonance Spectroscopy 46 (2005) 63–78 72
When asbmany more such combinations can be
found. When the spinning speed is finite, sidebands form[43,46,47] .
In principle, pulses are not required between the two t
1
periods. In practice, the switching time between the two
spinning angles is relatively long (of the order of 50 ms)
and the magnetization is usually stored along the zdirection
and subsequently brought back to the transverse plane. Anyother interactions of rank two or four will also be eliminatedin this way.
While in many areas this experiment has been super-
seded by MQMAS, it should be noted that, it is currently theonly high-resolution method applicable to integer spins.
4.3. Multiple-quantum MAS—MQMAS
When spinning at the magic angle, the only term
responsible for the residual broadening is
A
4;0Iz½18IðIC1ÞK34I2
zK5/C138: (68)
The orientational dependence is determined by the spatial
part A4,0, whereas the spin part Iz½18IðIC1ÞK34I2
zK5/C138
leads to scaling of the powder lineshape. The MQMASexperiment uses a combination of two different spintransitions in order to refocus the evolution based on thissecond-order term. The t
1period is divided into two, of
durations at1andbt1, with aCbZ1, as illustrated in Fig. 4
[48,49] .
Usually, odd multiple-quantum coherence is created first,
and is then converted to single-quantum coherence. Nofirst-order broadenings appear for symmetric transitions.Furthermore, the scaling factor relating the powderlineshape of the multiple-quantum coherence to the one ofthe single-quantum coherence should be negative to insure
the formation of an echo. If it is positive, the coherenceorder with opposite sign has to be used. In a more generalapproach one may use two different multiple-quantumcoherences during the t
1period in order to form the echo
[50].
The timings of the two periods at1andbt1are adjusted
such that
aZC4
0ðp2Þ=½C4
0ðp1ÞCC4
0ðp2Þ/C138;
bZC4
0ðp1Þ=½C4
0ðp1ÞCC4
0ðp2Þ/C138(69)
where C4
0ðpÞrepresents the spin-dependent part of the
fourth-rank term responsible for the broadening, defined in
analogy to Eq. (68) as
C4
0ðpÞZðp=2Þ18IðIC1ÞK17
2p2K5/C20/C21
: (70)
p1and p2are the two different multiple-quantum or
single-quantum coherence orders used during the t1-period.
Table 1 summarizes the factors aandbto be used for a
number of spin values and coherence order combinations.
Fig. 4. Multiple-quantum magic-angle spinning experiment.Table 1
P2
QZC2
Qð1Ch2=3Þ, in MHz2,n0is given in MHz, d1,d2,dCSin ppm
p1 p2 ab dCS P2
Q
SZ3/2
319
167
162ð4d1C5d2Þ
27ð8d1K17d2Þn2
0
675000
SZ5/2
3 K112
311931 31d1C10d2
27ð31d1K17d2Þn2
0
162000
5112
372537 37d1C50d2
135ð37d1K85d2Þn2
0
810000
5319
442544 2ð11d1C25d2Þ
135ð22d1K85d2Þn2
0
810000
SZ7/2
3 K145
146101146 73d1C10d2
2749ð73d1K17d2Þn2
0
3375000
5 K19
201120 10ðd1Cd2Þ
2749ð10d1K17d2Þn2
0
3375000
5 K3101
15655
15639d1C50d2
13549ð39d1K85d2Þn2
0
16875000
7145
206161206 103d1C140d2
3787ð103d1K238d2Þn2
0
6750000
73101
262161262 131d1C140d2
9457ð131d1K595d2Þn2
0
16875000
7555
216161216 2ð54d1C175d2Þ
9457ð108d1K595d2Þn2
0
16875000
SZ9/2
3 K136
12791
127127d1C10d2
27ð127d1K17d2Þn2
0
37500
5 K136
13195
131131d1C50d2
135ð131d1K85d2Þn2
0
187500
5 K391
18695
18693d1C50d2
135ð93d1K85d2Þn2
0
187500
7 K118
257
255ð5d1C14d2Þ
189ð25d1K119d2Þn2
0
262500
7 K313
152
153d1C10d2
27ð3d1K17d2Þn2
0
1312500
7 K595
10914
109109d1C350d2
945ð109d1K595d2Þn2
0
1312500
916
373137 37d1C50d2
135ð37d1K85d2Þn2
0
187500
9391
277186277 277d1C810d2
2187ð277d1K1377 d2Þn2
0
3037500
9595
281186281 281d1C1050 d2
2835ð281d1K1785 d2Þn2
0
3937500
977
10093
10010ð5d1C21d2Þ
567ð50d1K357d2Þn2
0
787500
The two columns to the right show how the chemical shifts and the
quadrupolar coupling constant (in the form of PQ) can be extracted using
the center-of-gravity peak positions in a MQMAS experiment.A. Jerschow / Progress in Nuclear Magnetic Resonance Spectroscopy 46 (2005) 63–78 73
There exist a number of variants of the basic two-pulse
MQMAS experiment [48,49] , including the z-filter [51],t h e
shifted-echo [52], and an experiment that uses several different
coherence pathways simultaneously [53]. Sensitivity
improvements in the coherence transfers have been achieved
by applying fast amplitude-modulated pulses [54–59] ,
double-frequency sweeps [60–63] , or taking advantage of
rotational resonance effects [64–74] . The use of a CPMG-
pulse train can yield a substantial signal enhancement [75–77] .
The MQMAS experiment is now a widely popular
technique for obtaining high-resolution spectra of quad-rupolar nuclei in solids, since it allows one to use routinehardware components without special modifications.
The representations of the spectra are often produced
using different conventions for labeling the ppm axis inthe isotropic dimension. Two frequent procedures are: (1)the frequencies in F1a r ed i v i d e db yt h eL a r m o r
frequency of the observed nucleus—this is probably themost natural convention; (2) the frequencies in F1 are
divided by the Larmor frequency times the chemical shiftscaling factor, so that on a ppm-scale chemical shifts
stay the same whatever the multiple-quantum coherence
order used [78,79] . The latter appears inconvenient since
quadrupolar isotropic shifts lead to different peakpositions at different magnetic fields and need to becorrected for this additional scaling factor. Some otherconventions are based on the use of only the firstmultiple-quantum period as the ‘actual t
1-period’, i.e. at1.
In this case, the axis needs to be multiplied by the factor
ato obtain the original ppm-scale. Other conventions
may require one to change the sign of the F1-axis
labeling [78,79] . A discussion of different conventions
can be found in Refs. [14,78,79] .
We feel that adoption of convention (1), and assuming an
experiment of the type displayed in Fig. 4 , where aCbZ1,
is a convenient choice. This procedure is also therecommended one in a recent review article [14].
One can use the center of gravity peak positions in a
MQMAS experiment to determine the chemical shifts andquadrupole coupling constants for the individual resonancesas follows.
The evolution frequency (in ppm) of p-quantum
coherence is given by
dðpÞZK½d
CSpCnQISðpÞ=n0/C138; (71)
where dCSis the chemical shift in ppm, and nQIS(p) is given
by Eq. (57). The positions of the peaks in F1 and F2 are then
given by
d1Zadðp1ÞCbdðp2Þ;d2ZdðK1Þ: (72)
Given the position of the center of gravity of a resonance
in a two-dimensional MQMAS spectrum one may extract
the chemical shifts and the quadrupolar coupling parameterP
QZCQffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1Ch2=3p
by inverting Eq. (72). These
expressions are shown in Table 1 for a number of relevantcases. In line with common practice in the NMR literature,
the resonance frequencies of positive coherence ordersduring the t
1interval are inverted in sign. The hparameters
can often be extracted by fitting the lineshapes of the
resonances along the F2 dimension.
A number of review articles describe MQMAS and
STMAS experiments and their applications in more detail
[14,80–87] .
4.4. Satellite transition MAS—STMAS
It is also possible to obtain high-resolution spectra by
correlating a satellite transition with the central transition of
a half-integer nucleus [88,89] . The experiment requires very
accurate rotor synchronization and setting of the magicangle [90]. The appearance of the first-order quadrupolar
interaction is avoided by assuring exact rotor synchroniza-tion during the t
1-period, which makes all spinning
sidebands overlap on top of each other. Recently, a pulse
sequence has been described, that allows one to reduce thesensitivity to magic-angle offsets [91,92] . One may also use
double-quantum transitions which involve satellite tran-sitions during the t
1evolution delay [91]. Double-quantum
filtration can further be used to avoid the appearance of a
number of unwanted signals [93] in STMAS experiments.
Proper aandbparameters for these types of experiments
may be derived from Eq. (68).
STMAS experiments are in general subject to odd-order
broadening [25], broadening due to motion, and inaccura-
cies related to magic-angle missetting [94], which are
normally absent or less problematic for central-transitionMQMAS experiments. For further details, we wish to point
the reader to the comprehensive review by Ashbrook and
Wimperis [14].
STMAS may further be applicable to any spins larger
than one, whether it is an integer spin or a half-integer spin.For spin one nuclei such a procedure is not helpful because
it is impossible to find a combination of transitions that
affords a partial echo for the second-order quadrupolarinteraction that does not at the same time refocus theisotropic chemical shifts.
4.5. Other experiments
MQMAS experiments have been combined with cross-
polarization (CP) to single quantum or to multiple-quantum coherences. Heteronuclear correlation exper-iments between two quadrupolar nuclei, or between a
quadrupolar nucleus and a spin 1/2 nucleus have been
performed [95–106] . Recoupling schemes have been
investigated [107–110] . Some homonuclear quadrupolar
correlation experiments have been documented [111–114] ,
as well as exchange experiments [115] , and J-resolved
experiments [116] .A. Jerschow / Progress in Nuclear Magnetic Resonance Spectroscopy 46 (2005) 63–78 74
5. Conclusions
Very advanced techniques for the study of quadrupolar
nuclei are available, and the theoretical description of theseexperiments is well established. The existence of a number
of different definitions for the quadrupolar interactions
frequently leads to confusion in comparing work fromdifferent sources. In this review article, the author hasattempted to establish a link between nuclear structureconstants and the NMR Hamiltonian and the spectraobtained by NMR methods. It is hoped that the article willprovide a useful reference point for describing thequadrupolar interaction, but also to facilitate the interface
between NMR and nuclear physics literature.
Acknowledgements
The author wishes to express his gratitude for very
insightful and stimulating discussions with S. Vega, C.P.Slichter, S. Wimperis, P.K. Madhu, G. Denninger, and S.C.Shekar. This work also benefitted from infrastructureprovided through a National Science Foundation Grant,‘Research Coordination Network for NMR of BiologicalSolids’ (MCB-0233854). The author is a member of theNew York Structural Biology Center supported by NIH
Grant 1P41GM66354 and the Center’s Metagroup Meetings
are acknowledged for helpful discussions.
Appendix A
A.1. Other second-order interactionsA.1.1. Quadrupolar/dipolar coupling
The second-order cross-term between the quadrupolar
coupling and the dipolar coupling can be determined from
equations similar to Eqs. (48) and (49). For the hetero-nuclear case one obtains [27]
H
ð2Þ
Q;DDZKuI
QuD;eff
2uI
0ðRQ;I
2;K1RD;eff
2;1CRQ;I
2;1RD;eff
2;K1Þ
!½3I2
zKIðIC1Þ/C138SzKuS
QuD;eff
2uS
0ðRQ;S
2;K1RD;eff
2;1
CRQ;S
2;1RD;eff
2;K1Þ½3S2
zKSðSC1Þ/C138Iz (A1)
where
uD;effZKm0ZgIgS
4pr3
ISCJIS
aniso; (A2)
andRD;eff
2;0Zffiffiffi
6p
in the principal axis frame. RQ;S
l;mandRQ;I
l;m,a s
well as, uS
Qand uI
Qare the quadrupolar tensors and
constants for spin SandI, respectively, and uS
0anduI
otheir
Larmor frequencies.For the homonuclear case, one obtains [27]
Hð2Þ
Q;DDZKuI
QuD;eff
2uI
0ðRQ;I
2;K1RD;eff
2;1CRQ;I
2;1RD;eff
2;K1Þ
!½3I2
zKIðIC1Þ/C138SzKuS
QuD;eff
2uS
0ðRQ;S
2;K1RD;eff
2;1
CRQ;S
2;1RD;eff
2;K1Þ½3S2
zKSðSC1Þ/C138Iz
CuS
QuD;eff
4uS
0ðRQ;S
2;K1RD;eff
2;1CRQ;S
2;K2RD;eff
2;2Þ
!SCIKð2SzC1ÞCuI
QuD;eff
4uI
0ðRQ;I
2;K1RD;eff
2;1
CRQ;I
2;K2RD;eff
2;2ÞSKICð2IzC1Þ
CuS
QuD;eff
4uS
0ðRQ;S
2;1RD;eff
2;K1CRQ;S
2;2RD;eff
2;K2ÞSKICð2SzK1Þ
CuI
QuD;eff
4uI
0ðRQ;I
2;1RD;eff
2;K1CRQ;I
2;2RD;eff
2;K2ÞSCIKð2IzK1Þ:
(A3)
A.1.2. Quadrupolar/chemical shift
The second-order cross-term between the quadrupolar
coupling and the chemical shift Hamiltonian can determinedfrom Eq. (48) to be [28]
H
ð2Þ
Q;CSZKuQ
2½ðRQ
2;1RCS
2;K1CRQ
2;K1RCS
2;1ÞCðRQ
2;1RCS
1;K1
CRQ
2;K1RCS
1;1Þ/C138½3I2
zKIðIC1Þ/C138; (A4)
where the chemical shift spatial tensors are taken as RCS
lmZP
m0Dm0mða;b;gÞrCS
lmwith the non-zero components in the
principal axis frame of
rCS
10ZKffiffiffi
2p
isa
XY;rCS
1G1Zsa
XZGisa
YZ;
rCS
20Zffiffiffi
3
2r
sCSA;rCS
2G2ZhCSsCSA=2:(A5)
In Eq. (A4) the antisymmetric components involving
RCS
1;G1can be shown to cancel if the average over a full
rotation period is considered. As with all such second-order
contributions, an overall frequency shift results from this
cross-term.
Because the only level-dependent component in the spin
part is I2
z, these effects are not visible when symmetric
transitions are observed.
A.1.3. Additional cross-terms
It has further been shown that for quadrupolar couplings
of the order of the spinning speed, i.e. uQzuradditional
line-broadening mechanisms may occur. They stem fromA. Jerschow / Progress in Nuclear Magnetic Resonance Spectroscopy 46 (2005) 63–78 75
the interplay between the non-commuting dipolar and
quadrupolar coupling Hamiltonians during sample spinning[117] . These effects are similar to nZ0 rotational resonance
effects shown for the interplay between chemical shiftanisotropy and dipolar terms. These mechanisms are only
relevant for small quadrupolar coupling constants ( u
Qz
5–25 kHz). The remarkable feature of these effects is theincrease of the linewidth with an increasing spinningfrequency and may be quite surprising when studying aneveryday setup sample such as KBr.
A.2. Sine and cosine representation of the first-
and second-order quadrupolar frequencies
A.2.1. First order
c
0Zffiffiffi
3
2r
P2ðcosqÞðP2ðcosbÞCðh=2Þcos 2 asin2bÞ;
c1Zffiffiffi
3p
4ffiffiffi
2pðK3Chcos 2 aÞsin 2bsin 2q;
c2Zffiffiffi
3p
8ffiffiffi
2pðhcos 2 að3Ccos 2 bÞC6 sin2bÞsin2q;
s1ZKffiffiffi
3
8r
hsinbsin 2asin 2q;
s2ZKffiffiffi
3
8r
hcosbcos 2 asin2q(A6)
A.2.2. Second order
c2;0Z1ffiffiffiffiffi
14p P2ðcosqÞ½ðK3Ch2ÞP2ðcosbÞ
C3hcos 2 asin2b/C138 (A7)
c4;0Z1
128ffiffiffiffiffi
70p P4ðcosqÞ½162C9h2C35ð18Ch2Þcos 4 b
C1200 hcos 2 asin2bC280h2cos 4 asin4b
C20 cos 2 bð18Ch2C84hcos 2 asin2bÞ/C138
c2;1Z3
4ffiffiffiffiffi
14p ½3Kh2C2hcos 2 a/C138sin 2 bsin 2 q
c2;2Z3
4ffiffiffiffiffi
14p ½hð3Ccos 2 bÞcos 2 aCðK3Ch2Þsin2b/C138sin2qc4;1ZKffiffiffi
5p
1024ffiffiffiffiffi
14p ðK56h2cosbcos 4 asin3b
C2ð18Ch2C12hcos 2 aÞsin 2 bC7ð18Ch2
K12hcos 2 aÞsin 4 b/C138ð2 sin 2 qC7 sin 4 qÞ
c4;2ZKffiffiffi
5p
128ffiffiffiffiffi
14p ð5C7 cos 2 qÞ½3hð5C4 cos 2 b
C7 cos 4 bÞcos 2 aCð90C5h2C21h2cos 4 a
C7 cos 2 bð18Ch2Ch2cos 4 aÞÞsin2b/C138sin2q
c4;3ZKffiffiffiffiffi
35p
128ffiffiffi
2pcosq½96hcos3bcos 2 asinb
C8ð18Ch2Þcosbsin3bKh2cos 4 að14 sin 2 b
Csin 4 bÞ/C138sin3q
c4;4Zffiffiffiffiffi
35p
1024ffiffiffi
2p½h2ð35C28 cos 2 bCcos 4 bÞcos 4 a
C8 sin2bð6hð3Ccos 2 bÞcos 2 a
Cð18Ch2Þsin2bÞ/C138sin4q
s2;1ZK3ffiffiffiffiffi
14p hsinbsin 2 asin 2 q
s2;2ZK3ffiffiffiffiffi
14p hcosbsin 2 asin2q
s4;1ZKffiffiffi
5p
128ffiffiffiffiffi
14p hsinb½3ð5C7 cos 2 bÞsin 2 a
C7hsin2bsin4a/C138ð2 sin 2 qC7 sin 4 qÞ
s4;2ZKffiffiffi
5p
64ffiffiffiffiffi
14p hð5C7 cos 2 qÞ½3ðcosbC7 cos 3 bÞsin 2 a
C14hcosbsin2bsin 4 a/C138sin2q
s4;3ZKffiffiffiffiffi
35p
32ffiffiffi
2ph½K6K18 cos 2 bC3hcosð2bK2aÞ
C10hcos 2 aC3hcosð2bC2aÞ
!cosqsinbsin 2 asin3q/C138
s4;4ZKffiffiffiffiffi
35p
128ffiffiffi
2ph½hcos 3 bsin4aCcosbð24 sin2bsin 2 a
C7hsin 4 aÞ/C138sin4qA. Jerschow / Progress in Nuclear Magnetic Resonance Spectroscopy 46 (2005) 63–78 76
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