1. JOURNAL OF MAGNETIC RESONANCE 71,62-14 (1987)
Two-DimensionalSolid-StateNutation NMR of Half-Integer
Quadrupolar Nuclei
A. P. M. KENTGENS, J. J. M. LEMMENS, F. M. M. GEURTS, AND W. S. VEEMAN
Department of Molecular Spectroscopy, Faculty of Science, University of Nijmegen, Toernooiveld,
6525 ED Nijmegen, The Netherlands
Received May 13, 1986; revised July 11, 1986
The two-dimensional solid-state nut&ion NMR experiment for the determination of
quadrupole parameters, asintroduced by Samoson and Lippmaa is evaluated. A complete
series of spectra (for spin Z = f, i, f, and p) resulting from density-matrix calculations is
presented, and some experimental aspectsof the method am discussed.Finally applications
of the method to *‘Al (I = 3) in spodumene and to 45Sc(Z = i) in Sc2(SO& are shown.
Q 1987 Academic Pres, Inc.
INTRODUCTION
The majority of elements’ in the periodic table have nuclei with half-integer quad-
rupole spins. It is therefore not surprising that there is interest in high-resolution NMR
spectra of quadrupole nuclei in solids. Especially the recent studies of zeolites, clays,
and ceramics have focused attention on obtaining structural information from NMR
spectra of quadrupolar spins.
In comparison to nuclear spins with spin quantum number I = 1, the NMR spectra
of quadrupolar spins contain two new features: ( 1)several transitions are possible and
(2) the transition frequencies are determined not only by the interaction between the
magnetic moment of the nucleus and the static external field but also by the interaction
between the nuclear electric quadrupole moment and the electric field gradient of the
surroundings of the particular nucleus. The chemical information which an NMR
spectrum of a quadrupolar spin offers is not limited to the chemical-shift data, but
such a spectrum can also provide the parameters that describe the quadrupolar inter-
action. These parameters depend on the local symmetry around the nucleus in con-
sideration and thus give direct structural information.
Traditionally, quadrupole interaction parameters can be determined by NQR. There
is usually a small magnetic field or none, and the nuclear spin levels are split mainly
by the quadrupole interaction. The frequencies of the transitions between these levels
provide the quadrupole parameters. The disadvantages of NQR are its low sensitivity
when the quadrupole interaction is rather small (0- 10 MHz) and the wide frequency
range one has to search for possible resonances. Both of these disadvantages can be
overcome in principle, at least for small quadrupole interactions, in an NMR ex-
periment.
0022-2364187 $3.00
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AU rights of reprcduction in any form -ed.
62

2. NMR OF QUADRUF’OLAR NUCLEI 63
The description of the NMR spectra of quadrupolar spins depends strongly on the
spin quantum number Zand the relative magnitude of the Zeeman and the quadrupole
interactions. Here we limit our discussion to half-integer spins (I = 5,I, 5,:) and assume
the quadrupole interaction to be small with respect to the Zeeman interaction, The
(anisotropic) contribution of the quadrupole term to the NMR transition frequencies
can then be calculated by perturbation theory.
In high magnetic fields all Zeeman transitions WZ,m’ shift in first order because of
the quadrupole interaction except the f, - 1 transition which experiences a (much
smaller) second-order shift. As a result of this, typical spectra of polycrystalline samples
containing quadrupolar spins give characteristic powder patterns for the 1, - 1 tran-
sition whereas all other transitions are usually broadened beyond detection (I). Because
the magic angle has no “magic” properties for the second-order quadrupolar inter-
action, MAS will not average this interaction, but yields powder patterns for the 1,
- f transition which are approximately four times narrower than for static
samples (2).
Although, in principle, it is possible to extract information about the electric field
gradient from the spectra of static or spinning polycrystalline samples, in practice the
powder patterns are often blurred by a spread in chemical shift and/or by the presence
of more than one quadrupole interaction. To overcome these problems Samoson and
Lippmaa (3) introduced a simple two-dimensional experiment which allows one to
separate the quadrupole interaction from the chemical-shift interaction. This technique
is based on a nutation experiment (4) where the evolution of the spin system in the
presence of a radiofrequency field Br is studied in the rotating frame. This evolution
yields a low-field (B, - O.OOl&) NMR spectrum, the nutation spectrum, with the
sensitivity of the high-field spectrometer. In this respect it can directly be compared
to the zero-field NMR technique developed by Pines and co-workers (5) where the
sample is pneumatically shifted in and out of the magnet. The advantage of the nutation
experiment is that, in contrast to the zero-field technique, there is no limitation on
the T, spin-lattice relaxation time. A nutation spectrum, however, is more difficult
to analyse than a zero-field spectrum.
In this paper we discuss the calculation of these nutation spectra using the density
matrix formalism, and present a number of simulated spectra for all half-integer quad-
rupole spins (I = $to Z = 5) which can be used as fingerprints. In a previous paper on
27A1nutation NMR of zeolites (6) it was shown how nutation NMR can be used to
separate the signals from 27A1with a large quadrupole interaction from the Al signals
with a small quadrupole interaction. Here we want to show that by comparing an
experimental spectrum to a set of simulated spectra one can determine the quadrupole
parameters. This will be demonstrated for 27Al in spodumene and for 45Scin S@SO4)3.
Further, some experimental aspects of the method will be discussed.
THEORY
The pulse scheme of the experiment is outlined in Fig. 1. During the evolution
period tl , an rf field is present and the system evolves under the secular Hamiltonian
&“, in the rotating frame, assuming that the sample is static (i.e., no MAS):
9, =~o~+&“~+‘zQ=(n,-a)zz-n,zx+n~(3z~-z(z+ 1)) 111

3. 64 KENTGENS ET AL.
b F, -projection
FIG. 1. (a) Pulse scheme of the experiment; a free induction decay is acquired during tzas a function of
the pulse length tI . Subsequent Fourier transformation of the signals obtained yields a typical 2D powder
pattern as shown in (d). This spectrum was calculated for Z = f with a ratio Q&l,+ = 0.45 and 9 = 0. The
F2 projection of this pattern (c) gives the second-order quadrupole powder pattern, and the F, projection
(b) is a characteristic nutation spectrum for the ratio Q,$&.
with
% = ;;;!;)(3 cos28- 1+ 7jsin28 cos 2@) = Qo(3 cos2B- 1+ 7)sin28cos 2@)
where Q,+= y& and B and @are the polar angles orienting the magnetic field &, in
the principal axis system of the field gradient, and 1 represents the asymmetry param-
eter. Here dipolar interactions and the nonsecular part of the quadrupole Hamiltonian
have been neglected. During t2 there is no rf field present and the FID of the f, - f
transition is acquired (assuming that all other transitions are too broad to be detected).
The system is now governed by the Hamiltonian S2:
z2=&“z+2Yf3+‘zQ PI
where SF’=and Sacs are the Zeeman and the chemical-shift interaction and So is the
quadrupolar interaction which only contributes in second order to the lineshape of

4. NMR OF QUADRUPOLAR NUCLEI 65
the central 4, - 4 transition (7). With the knowledge of these Hamiltonians we can
calculate the signal s(t,, fz) using the density-matrix formalism. At time tl = 0 we
start with the equilibrium density matrix u(O)in the high-temperature approximation.
As one can seethe dominating Zeeman interaction X’z is not present in 3?‘, . So if we
irradiate close enough to resonance (zOf small) then %‘o and 9’ti are the most im-
portant terms of 2,. This means that the eigenfunctions of ~3?rdepend on the ratio
of Qo and Q,. From the Liouville-von Neumann equation we get at time tl
u(tl) = exp(-i~ltl/~)u(0)ex~i~,~,/~). [31
Starting in a basis of eigenfunctions of Z,, we get after (numerical) diagonalization of
z,, using the orthogonal transformations T
u(tr) = Texp(-iE~l/ti)~+~(0)Texp(iE~I/~)~+. [41
Here Z?represents the diagonalized matrix of X1 giving the eigenvalues
Ej= C Z TpjzlpqTqj 151
P 4
and eigenfunctions
lj) = C Tij14 m). [61
We assume that we only detect the coherence between the f and the - f states
during t2, so only the element ~r(t~)~~~,-~,~has to be evaluated. The signal of the central
transition then can be calculated as a function of tl and t2:
- a(t1)1/2,-1,2exp(-i~2t2)
= C(R-*,2,,,2)i,jeXp(i~2iitl)eXp(-i~2t~)
i,j
171
where Q2 represents the Larmor precession of the spins during t2 and Q2,
= (Ei - Ej)/h is the transition frequency in the rotating frame between the states Ii)
and /j). The coefficients (R-1/2,1/&
(R-1/2,1/& = Tl/2,iT-l/2,j C Tk,iTk,jdO)k,k 181
k
represent the contributions of the coherence between Ii) and lj) in the rotating frame
during tl to the 4, - 1 coherence detected during t2.
To summarize, we have during tl a system with eigenfunctions and eigenvalues
which depend on the ratio of no and C&r(Fig. 2). The coherences between all these
levels, with frequency a,,, develop during tl and give a certain contribution to the 1,
-1 coherence detected during tz. Therefore, a two-dimensional Fourier transformation
of the acquired signal will give a characteristic powder pattern (Fig. Id), whose pro-
jection on the F2 axis (Fig. lc) yields the normal powder lineshape due to the combined
effect of chemical-shift anisotropy and quadrupole interaction. Projection onto the F,
axis (Fig. 1b) gives the nutation spectrum which depends on the quadrupole parameters
e2qQ and 1, the spin quantum number Zand the rf field strength B, , and is independent

5. 66 KENTGENS ET AL.
energy (MHz)
.04
.03
-.os
0 25 5 1.5
I
Ok-----
:;‘::I:-
-%?ii-xiy6 (kHz) -
FIG. 2. Energy diagram of an isolated spin I = $ in the presence of a magnetic field B with quadrupole
parameters e2qQ/h = 1 MHz and t) = 0. On the left is the low-field situation Zz < &“o and on the right is
the situation &“= $ Zo, where X’o appears to be negligible.
of the chemical shift. As we shall seelater, and aswe reported before (6), the nutation
spectra show more detail than MAS NMR spectra and therefore we concentrate on
the nutation spectra instead of on the whole 2D powder pattern.
In the extreme cases l.%‘o[6 (%‘A and l%‘o[ + I%‘,.rjthe nutation spectrum consists
of a single line. In the first situation X’o may be neglected with respect to Xti and the
nutation frequency is simply f&r. In the second situation straightforward perturbation
theory shows that the nutation frequency becomes (I + $)fiti (6-84b).
In intermediate cases, I~??‘ol- IsA, the spectra are complicated and several peaks
can occur because many transition frequencies t&in the rotating frame exist. In addition
the nutation spectra will be powder patterns because of the anisotropic nature of the
quadrupole interaction. Therefore for intermediate casesthe experimentally obtained
nutation spectra have to be compared to simulated spectra, calculated as de-
scribed above.
Figures 3-6 each show a series of calculated nutation spectra for different ratio
t&&.r, for the different spins Z = 5 to Z = 3. In all spectra the asymmetry parameter
71and the resonance offset are assumed to be 0. These spectra were calculated for 7500
crystallite orientations, taking up to 5 minutes computer time on a mainframe (NAS
9060). The spectra appear to be very characteristic powder patterns so for these in-
termediate cases with !& known, one can determine no (no = e2gQ/h8Z(2Z - 1)).
Because no depends on the nuclear quadrupole moment Q and the spin quantum
number Z, and with Q,+limited for experimental reasons, what range of electric field
gradients eq can be determined without getting in the extreme situation with only one
nutation frequency left will depend on the nucleus in question.
Figure 7 shows the effect of the change of the asymmetry parameter 17for a spin
Z = $ with G-J& = 0.6. The calculations are for 100 X 100 crystallite orientations
taking up to 10 minutes computer time. The overall appearance of the powder pattern

6. NMR OF QUADRUPOLAR NUCLEI 67
I = 3/2
i, Orf 2 Orf
FIG. 3. Calculated nutation spectra for an isolated spin I = 3 as function of the ratio &/&, with 9 = 0.
No resonance offset is taken into account. The scale factors of the spectra are 1, 4.7, 15.2, 14.9, 10.2, 7.3,
5.7, and 3.0 with respect to the G@, = 0 spectrum. In all spectra a Lorentzian line broadening of 2.5 kHz
was applied.
remains the same but the intensity of some of the lines changes drastically. So it is
also possible to get a good estimate of T)from the nutation spectra.’
So far it has been assumed that the sample is static, i.e., no magic-angle spinning
applied. Magic-angle spinning would be desirable in view of the resolution along F2.
However, numerical computer calculations of the simulated nutation spectra under
MAS conditions becomes very time-consuming because the change of the orientation
of the sample during tl and thus of &“r cannot be neglected. Experimentally it has
been observed that the nutation spectra change with MAS. The discussion of these
spectra and their interpretation will be postponed to a later publication.
EXPERIMENTAL ASPECTS
Resonance ofiet. When the system is irradiated on resonance (Q = s2,) then the
matrix of &“r in Eq. [2] becomes symmetrical with respect to both diagonals of the
matrix. As a result of this
’ A complete set of simulated spectra with asymmetry parameter variation for all spins(I = $to I = $)
can be obtained on request to the authors.

7. I = 5/2
c,
r
Qf 34f
FIG 4. As in Fig. 3, but for spin I = $. Scale factors: 1, 7.4, 8.2, 25.0, 25.1, 19.4, 15.7, and 6.9.
0.6
0.05
0
0 Qf Wf
FIG. 5. As in Fig. 3, but for spin I = $. Scale factors: 1, 6.7, 16.0, 22.0, 39.6, 32.6, 26.2, and 12.7.
68

8. -0
0 Orf %f
FIG. 6. As in Fig. 3, but for Z = f. Scale factors: 1, 12.1, 25.2, 19.8, 72.0, 61.6, 52.3, and 25.7.
FIG. 7. Nutation spectra for a spin Z = f as lknction of the asymmetry parameter t (&/Q, = 0.6). The
overall appearance of the powder pattern remains the same, but the intensity of some lines decreaseswith
increasing 9.
69

9. 70 KENTGENS ET AL.
and
W1/2,1,2h,j = -v-1/2,1/2)j,i i#j
(R-1/2,I&j = 0 i=j.
Substitution in Eq. [7] shows that the signal then becomes
[91
WI, f2) = 2 U- I,z,l,z)i,jsin QijtlexP(-ifizt2).
iJ
[lOI
The amplitude of the signal detected during t2 is sine modulated which allows us to
obtain pure absorption spectra (9). The presence of an off-resonance term in z1 lowers
the symmetry of its matrix and the modulation now becomes a phase-modulation,
which cannot be changed to amplitude-modulation by phase cycling. Thus there will
be dispersive contributions to the lineshapes which can easily distort powder patterns.
Another effect of off-resonance irradiation is the appearance of a dispersive line at
fir = 0. This is due to the magnetization component along the BeEfield in the rotating
frame which does not evolve during tl .
Both effects of off-resonance irradiation can even be observed for spins with Z = 5
for which the signal in such a case can be written as
S(tl, t2) - [A( 1- cos QeEtl)+ iB sin !&&l]exp(-iQ2t2) [Ill
where A = Q,AQ/2(Q$ + AQ2) and B = Qd2w. When A0 = Q - Q0# 0 then
A # 0 and thus a constant and a cos fled1 term is introduced causing, respectively,
the Q, = 0 signal and the phase modulation. To avoid complications it is clearly
recommendable to irradiate on resonance, i.e., with the excitation frequency Q equal
to the Larmor frequency yB0. That “on resonance” in this case does not mean at the
frequency of the 1, - 4 transition is shown by the following magic-angle spinning
experiment. Here the Larmor frequency lies outside of the quadrupole lineshape (2)
and Fig. 8 shows the nutation spectra of NaN02 (e2qQ/h = 1.1 MHz, 9 = 0.1) as a
function of the excitation frequency. In this case we are in the extreme situation
no b Qti and MAS does not influence the nutation spectrum. It is clear that the line
at Qi = 0 increases relative to the line at 2Qfi. The line at Bi = 0 is minimal when we
irradiate outside the powder pattern near the Larmor frequency.
Line broadening. An important cause for line broadening in the F, dimension is
the inhomogeneity of the rf magnetic field. The effect of rf inhomogeneity cannot
simply be described with a Lorentzian broadening exp(-tr/rJ because a spread in !&
will also cause a spread in the ratio Qo/s2ti, and will thus change the whole nutation
spectrum. Furthermore the lineshapes due to rfinhomogeneity will not be Lorentzian
but asymmetric (see Ref. (ZO), Fig. 3). Only in the extreme case Q, $ no, do we get
a line at nutation frequency Qti with a linewidth directly determined by the rf inho-
mogeneity. In the other extreme case Qd 4 no there will be a line at (I + f)&, so the
linewidth will be (I + 4)~ times the rf inhomogeneity.
Recycle delay. Another important experimental aspect in the nutation experiment
is that one has to ensure that the recycle delay or relaxation delay is long enough for
the system to return to equilibrium before the next pulse arrives. If the recycle delay
is short with respect to T, then the build up of magnetization along the z axis is
incomplete. This will distort the pure sine amplitude modulation of the FID

10. NMR OF QUADRUPOLAR NUCLEI 71
NaN02
a b
1
F1 %f 0
C
,
5 *%f 0
FIG. 8. MA!3 spectra of NaN02 together with their nut&ion spectra as a function of the excitation frequency
fi. (a) Off-resonance irradiation; in the nutation spectrum we seea large line at R, = 0. (b) Irradiation close
to the Larmor frequency (outside the powder pattern) gives the best nutation spectrum. (c) With the carrier
frequency equal to the average 4, - f transition frequency (in the middle of the powder pattern) the line at
ti, = 0 in the nutation spectrum increases again.
(Eq. [ lo]). Fourier transformation of a distorted sin QIt wave will give aline at frequency
Q, plus a number of harmonics at 2%) 3!4, . . . . The number and amplitude of
these harmonics depends on the distortion of the sine wave, and they can easily be
mistaken for components with a large quadrupole frequency. Figure 9 shows this effect
- F2
FIG. 9. 2D nutation spectrum of ‘Li in Lick where the recycle delay (0.25 s) is short with respect to T, .
Li has a small no in Lick so only one line at t& is expected. Because of the short recycle delay, however,
there are several harmonics of this frequency present.

11. 72 KENTGENS ET AL,
for the ‘Li nutation spectrum of LiCl. Here Li has a very small quadrupole interaction,
but due to the short repetition rate we see not only a line at 52, = Qfi but also at
Q, = 2&, 3&, and 4Q+ In fact this experiment is proposed asa method to determine
Q,.rin solution, for nuclei with long T, and low natural abundance (II).
EXPERIMENTAL
NaN02, LiCl, and SCZ(SO& were all commercially available chemicals. NaN02
and spodumene spectra were recorded on a Bruker CXP-300 with, respectively, 64
and 128 tl increments of 2 ~.ls.A standard Bruker probe with an rf field of 36 kHz
was employed. LiCl and Sc#O.& spectra were recorded on a Bruker WM-500 (with
256 t, increments of 2 and 1.5 ps, respectively). Here a specially constructed probe
equipped with a 6 X 12 mm solenoid perpendicular to BO, operating with an rf field
strength up to 70 kHz, was used.
SPODUMENE
J‘_~-_--
/ L-...cd.--L.-,.,--r--r- L+------- .~_
, I I0 Yf 150
- KHz
FIG. 10. (a) 27AIspectrum of the central transition of spodumene recorded at 78.2 MHz. (b) F, projection
of the 2D nutation spectrum of spodumene recorded on a Bruker CXP-300 (128 t, increments of 2 ps and
Q, = 36 kHz). (c) Simulated spodumene spectrum using the quadrupole parameters e*qQlh = 2.95 MHz
and 7 = 0.94 (12) and a Lorentzian broadening of 2.5 kHz.

12. NMR OF QUADRUPOLAR NUCLEI 73
Sc*(SO&
SC 1=7/2
200 khz
-F 2
FIG. 11. 2D nutation spectrum of 45Scin SC&SO& recorded on a Bruker WM 500, together with its F,
projection (256 tl increments of I .5 ps with a 70 kHz rf field).
RESULTS
To demonstrate the effectiveness of the nutation experiment, Fig. 10a displays the
NMR spectrum of 27Al in powdered spodumene recorded on a Bruker CXP-300 at
78.2 MHz. The spectrum consists of one featureless line 5 kHz wide. It is clear that
no accurate quadrupole interaction parameters can be extracted from this spectrum.
In addition, MAS does not solve the problem; again we see a rather featureless line,
1.5 kHz wide. In Ref. (2~) it is shown that this MAS spectrum can be reproduced
theoretically (using a large line broadening) with the known quadrupole parameters
(e*qQ/h = 2.95 MHz, 1 = 0.94 (Z2)), but it will be 1c ear that it is almost impossible
to do so without any preknowledge of the quadrupole parameters. The result of the
2D nutation experiment, however, appears to be a well structured pattern. From the
F1 projection of this pattern (Fig. lob) we can estimate the magnitude of the quadrupole
parameters e*qQ and rl using our set of calculated spectra. It appears that flo/Q,.r - 1
with an rf field strength of 36 kHz; this means that e*qQ/h - 3 + 0.5 MHz, and that
I) must be between 0.8 and 1.
Another example is the 45Sc(I = $) nutation spectrum of Scz(SO& (Fig. 11). The
spectrum of the central transition measured at 121.5 MHz is 2.6 kHz wide and shows
no structure, MAS narrows the spectrum to 600 Hz. The MAS spectrum recorded at
43.8 MHz does show some structure from which it becomes clear that there must be
siteswith different quadrupole parameters but the same chemical shift. The 2D nutation
spectrum is well structured (Fig. 11). Comparing its projection to our set of calculated
spectra reveals that the major constituent has a e*qQ/h - 2 f. 0.5 MHz with a low

13. 74 KENTGENS ET AL.
asymmetry parameter (7 between 0.1 and 0.4). P. P. Man recently also showed an
intermediate case for Mn in KMn04 (8~).
CONCLUSIONS
The 2D nutation method appears to be very useful for the determination of quad-
rupole interaction parameters of half-integer quadrupolar nuclei in a field gradient,
especially when these parameters cannot be determined from MAS experiments. The
method combines the sensitivity of high-field measurements with the information one
gets from experiments at low or zero field. The spectra can easily be simulated with
a straightforward density matrix calculation. To get high sensitivity and maximum
resolution in the F2 dimension, which is advantageous if nuclei with different chemical
shifts are present, one preferably performs the experiment in the highest available
magnetic field. Because there is no obvious need for MAS the method will also be
very suited for high-temperature studies (e.g., of zeolites).
ACKNOWLEDGEMENTS
Mr. P. van Dael and Dr. C. Haasnoot, of the SON hf-NMR facility at the University of Nijmegen, are
thanked for their support with the experiments on the 500 MHz spectrometer. We thank Mr. J. W. M. van
OS and Mr. P. van Dijk for their technical assistance and Prof. dr. E. de Boer for critically reading the
manuscript. Dr. B. de Jong kindly supplied the spodumene sample. This work was carried out under the
auspices of the Netherlands Foundation of Chemical Research (SON) and with the aid of the Netherlands
Organization for the Advancement of Pure Research (ZWO).
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