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Dynamic angle spinning and double rotation quadrupolar NMR
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trends in analytical
chemistry,
vol. 13, no. 8, 1994
Dynamic angle spinning and double rotation quadrupolar NMR D.B. Zax Ithaca, NY, USA The problem of high-resolution solid state NMR of quadrupolar nuclear spins (where I> l/2) has been recently solved. The twin techniques of dynamic angle spinning (DAS) and double rotation (DOR) demonstrate resolution comparable to that observed in highresolution spectra of 13Cor 15N.This presents a substantial extension as most NMR-active spins are quadrupolar. In this review we describe previous techniques, and explain why DAS and DOR yield superior results.
1. Introduction The question of how best to achieve high resolution (and therefore good chemical sensitivity) in NMR has intrigued the field since the first work of Purcell, Torrey and Pound [ 1 ] and of Bloch, Hansen and Packard [2] nearly 50 years ago. In the earliest years, research emphasized applications to solids where the larger internal interactions lead to broad spectral lines. In magnetic fields of low homogeneity, the finer spectral details associated liquid state NMR are with high resolution, obscured and little more than the profile of the magnetic field can be measured. With the development of high resolution magnet coils, the superior chemical sensitivity associated with the routine measurement of chemical shifts made liquid state NMR the more popular alternative. Broad-line spectra in solids are a result of the orientation-dependence of these internal interacthe dipole-dipole and quadrupolar tions, couplings. High resolution, as achieved in isotropic liquids, reflects nature’s efforts to eliminate the direct effects of these anisotropic interactions through ‘motional narrowing’. Where NMR experiments normally monitor the resonance frequency only on time scales of microseconds to millisec016%9986/94/$07,00
onds, in low viscosity liquids collisions between molecules cause substantial molecular reorientation every few picoseconds. Observed transition frequencies correspond to averages over all those orientations accessed during the measurement period. This unrestricted, rapid motion averages all anisotropic interactions precisely to zero. (Manifestations of these interactions may still be found in relaxation measurements.) Only the isotropic chemical shift (w,,,) and the J coupling, (Ji,,) survive this motional averaging process. Spectra of solids reflect all of these unaveraged interactions superposed upon one another. The broad lines characteristic of NMR spectra of solidstate samples therefore contain all of the spectroscopic information available in liquids, and much more as well. Often, the available information so exceeds methods of analysis that it is desirable to mimic nature and sacrifice some spectral parameters so as to more accurately measure others; or, as in multidimensional NMR techniques, to use one experimental parameter to spread out the interesting information in multiple dimensions. If resolution between sites is good in one dimension, overlapping broad lines may be sufficiently separated that they become amenable to analysis. Achieving high resolution in solid-state NMR spectra generally requires that we somehow mimic nature, and remove that orientational broadening. But while nature’s technique is unquestionably effective, it is also hopelessly inefficient as a model for high-resolution NMR in solids. No mechanical process can reasonably expect to reorient crystallites through all possible orientations on sub-microsecond timescales. Key to a practical implementation of high resolution NMR in solids is the realization that the anisotropies have well-behaved directional properties. Even where only a small number of orientations are sampled, the anisotropy may be suppressed if those orientations are carefully chosen. A particularly efficient example of averaging by sample reorientation is provided by the example 0
1994 Elsevier Science B.V. All rights reserved
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trends in analytical chemistry, vol. 13, no. 8, 1994
b
50
40
30
20
10
Chemacal
0
shill
-10
-20
30
40
(p.p.m.1
Fig. 1. MAS and DOR spectra of *‘AI in the dehydrated and rehydrated aluminophosphate sieve VPI-5, at 104.23 MHz. The crystal lattice consists of alternating AIO, and PO, tetrahedra, with two crystallographically inequivalent Al sites in the abundance ratio 2:l. (a) MAS NMR spectrum of the dehydrated (at 1O-5 Torr for 48 h) VPI-5 sample. Only a single asymmetric line is observed. (b) DOR NMR spectrum of the dehydrated (at lop5 Torr for 48 h) VPI-5 sample. Two peaks are resolved with intensity ratios 2:l. (c) DOR NMR spectrum of VPI-5 after rehydration for two days, showing the appearance of octrahedrally coordinated Al (at - 18.4 ppm) and a variety of different tetrahedral sites (labeled a,b, and g in the figure). Intensity is preserved in the line marked d during partial rehydration, indicating that only the doubly populated sites are initially effected by hydration. (d) As in (c), except that rehydration has continued for a total of 23days. (Reprinted with permissionfrom Ref. [32].)
of magic angle hopping in which Bax, Szeverenyi and Maciel [ 31 showed that chemical shift anisotropy could be completely averaged, independent of initial crystallite orientation, by reorienting the container holding the sample through only three
discrete orientations. It is experimentally simpler to work with samples contained in cylindrically rotating containers so that orientation-space is sampled continuously. In nuclei with spin I= l/2, the chemical shift anisotropy is routinely removed via magic angle spinning (MAS) [ 4,5]. Disappointingly, neither magic angle hopping nor magic angle spinning yields comparable high resolution spectra in quadrupolar (12 1) spin systems. Two experiments have recently been introduced which do: dynamic angle spinning (DAS) [ 6,7], in which the sample is spun sequentially around a small number of directions and double rotation (DOR) [7], in which the sample spins inside a rotor which itself is spinning inside a second rotor spinning at the magic angle with respect to the field. The result of each of these experiments is a spectrum where orientational broadening due to the anisotropic quadrupole coupling is removed. The result is a striking improvement in attainable resolution for the vast majority of nuclear spin species where quadrupolar broadening often seemed an insurmountable obstacle to the effective use of solid-state NMR methods. An example of the unparalleled chemical sensitivity now routinely available is shown in Fig. I, where DOR NMR spectroscopy is demonstrated to be capable of following subtle changes in zeolite chemistry as a function of degree of hydration and time. For comparison, the MAS NMR spectrum of the same system is shown at the top of the figure. Where chemical evolution is clearly monitored in the successive 27Al DOR spectra, the MAS spectral linewidth is sufficiently broad that no resolved features are observable. In this article, our plan of attack is as follows: first, we explain how magic angle techniques achieve high resolution in the important case of anisotropic chemical shifts of spin - l/2 nuclei. We briefly consider the differences between spinning and hopping. We then explore the partial success of MAS in narrowing lineshapes observed in quadrupolar spin systems. This is followed by an analysis of how DAS and DOR achieve high resolution. We then conclude with some examples of its use in systems of practical interest, and some further comments on its prospects for the future. 2. Achieving high-resolution in solids: magic angle hopping and spinning Andrew, Bradbury and Eades [4], and separately Lowe [5] provided a simple solution for
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narrowing the broad lines associated with powder patterns due to anisotropic dipole-dipole couplings in solid-state NMR. They realized, and demonstrated experimentally, that powder pattern lineshapes could be narrowed by bulk sample reorientation. Optimum narrowing was achieved when the sample was spun at the ‘magic angle’ 19,=54.74”. In what follows we specialize to the main contemporary application of MAS, the narrowing of powder pattern lineshapes associated with anisotropic chemical shifts. The same technology that is the basis for the drill found in a modern dentist’s office can be used to rapidly rotate solids in an NMR sample holder. The air-bearing can be tilted at any arbitrary angle with respect to the applied field. As the sample container is rotated, the orientation of any particular molecule as referenced to the external magnetic field varies in a simply-describable fashion. This orientation dependence is equivalent to a time-dependent transition frequency. Our goal is to understand how magic angle spinning accomplishes ‘motional averaging’ such that all molecules, independent of initial orientation, have the same time-averaged transition frequency. We can formalize this problem using a simple geometric model [ 81. The orientation of any chemical shift tensor in an external field can be specified by two angles LY,,and &. With little loss of generality we will choose to treat explicitly only molecular systems of axial or higher symmetry, so that there is no explicit dependence of any of what follows on (Y,.. (With somewhat more attention to the details, all of our results can be straightforwardly extended to the more general case.) Where only the chemical shift is observed, the NMR transition frequency Aw can be separated into orien(isotropic) and orientation-independent tation-dependent (anisotropic) components via Am= Awis, + Aaaniso where Awiso = WO(1 - a) and Am)aniso=$3
cos2Pcs - 1)
for o0 the resonance frequency of a suitably defined reference compound, (Tthe conventionally defined isotropic chemical shift, and 6 the magnitude of the anisotropic shift. An example of the solid-state powder pattern associated with the anisotropy of
Static
r-----8 = 45”
e = em
$ = 60”
150
100
50 ppm
0
-50
-100
-150
from q,,
Fig. 2. Axially symmetric chemical shift powder patterns, as are commonly found for spin- l/2 nuclei, where the shift is referenced to CT,,, and the anisotropy is 6= 100 ppm. At top, the signal observed for static samples. Below, the scaled powder pattern observed for rapid rotation about the indicated axes, defined as the tilt angle with respect to the applied magneticfield.
the chemical shift (the CSA) for a spin - l/2 species such as 13C or 15N is shown in Fig. 2a. Characteristic values of PCs can be associated with well-defined positions in the powder pattern; for example, the intense maximum shifted by 6/2 from corresponds to &=90”, while the other 6w extreme at - 6 from ~iso corresponds to PCs= 0”. This lineshape function was first analyzed by Bloembergen and Rowland [9], and as it recurs throughout problems in magnetic resonance we will refer to as the standard powder pattern lineshape. As the sample rotates, /3_(t) varies in time. Choosing (as the angle a given site is initially tilted with respect to the rotor axis, 0 the angle the rotor is tipped with respect to the laboratory z-axis, and qt the (time-dependent) angle swept out by the rotor as it spins, it can be shown [ 81 that cos&,( t) = cos0 co+
sin0 sin
and sin pcs( t) = sin0 cosl+ cos0 sin< cosqt
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In lockstep with PCs(t), Ao,,i,, varies periodically. Our goal is to find the trajectory-averaged anisotropic contribution to the transition frequency. After a goodly dose of straightforward if tedious algebraic manipulations, we find
+ terms periodic in w, and 20, The isotropic contribution, as it is independent of orientation, is unaffected. Equation ( 1) states that, without regard to initial orientation l, the averaged contribution to the anisotropic shift is scaled by (3 cos2 19- 1) /2. Magic angle spinning corresponds to the choice 8= 0,, = 54.74” where the scaling factor is zero and optimum line narrowing occurs. This is illustrated in Fig. 2a-d, where spectra corresponding to the powder lineshape associated with a static sample, as well as rapidly spinning samples including a magic angle spectrum and the spectrum observed at two angles which are not quite magic ( 8= 45” and 8= 60”), are shown. In this limit only the time-averaged transition frequency is observed. For Am,“,= AWiso the specific case of magic angle spinning, each orientation has precisely the same average transition frequency that of the isotropic chemical shift. The only evidence of the anisotropy 6 is preserved in the spin-rate dependent terms. Where w, < 6, these correspond to spinning sidebands observed at frequencies Wisef no,, with little observed in intensity for IIZW,]> 6 [ IO]. Magic angle spectra, where the spin rate varies from slow to fast, are shown in Fig. n
5.
The magic angle hopping experiment [ 31 works on a similar principle. In contrast to MAS, the sample is reoriented in discrete steps so that each chemical shift tensor is found in three discrete orientations only. Each is related to the other two by a 120” rotation of the whole sample container about an angle 8. In analogy to the averaging performed above, we find that AWaniso =
=
cos2P&)
- 11
6(3cos21-l)(3cos2&l) 2 2
As in magic angle spinning, it is sufficient that we choose to hop about the 8= 8, to average away all
““‘* K= 2 =i 150
100
50 wm
0 from
-50
-100
-150
oisO
Fig. 3. Axially symmetric chemical shift powder patterns, where the sample is placed in a rotor aligned at 0, with respect to the applied magnetic field. At top, the signal observed for static samples. Below, the sideband pattern observed as a function of rotation rate K = 6( o.I~cJ,).
anisotropy. There are no sidebands observed in magic angle hopping, which may be a substantial advantage in systems where 6 is large. Nonetheless, it remains largely unappreciated, and suffers from the considerable disadvantage that a practical implementation requires that the experiment be repeated a large number of times while the time spent in each of the three orientations is continuously incremented - i.e. in two-dimensional form. 3. Magic angle spinning: quadrupolar
nuclear spins Based on our long experience of high resolution in systems where spectral features are dominated by large chemical shift differences, it has been a source of some disappointment that MAS fails to achieve similar results for the more numerous halfinteger quadrupolar nuclei [ 1 l-l 31. In these systems fine structure in the NMR spectrum is often dominated by the interaction of the nuclear quadrupole moment (corresponding to a non-spherical charge distribution of the atomic nucleus) with local electric field gradients. There are 21-t 1 dif-
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ferent orientations allowed for a nucleus with spin I. In available magnetic fields, quadrupole couplings are often orders of magnitude larger than chemical shifts. Under these conditions, the transition frequency depends simultaneously upon both the chemical shift and quadrupolar interactions, and Ati= A@,, + Am, Each interaction is directional, and generally not coincident; thus, for any given site, we are required to track (assuming axial symmetry in both tensors) PCsand Pa. (This will turn out to be a bookkeeping nuisance but of little fundamental import.) The chemical shift contribution, Aw,,, has been given above. It is independent of which of the 2Z+ 1 different Zeeman energy levels we begin from. In contrast, while the change in quadrupolar energy does depend upon which transition is observed. We will designate a transition by its initial (mi) and final (mr) values of Z,. The first order quadrupolar energy change is Aob’) = wo[;(3cos2&-
l)(m;
al /
! 1
bJ
n
cl
75
-$)I
50
25 k/-/z
where 3e2qQ/4Z( 2Z- 1) fi. Because the Z, states are defined with respect to the spin orientation in the externally applied field and we observe transitions in the vicinity of o. only, we can make use of the selection rule mi = mr + 1 so as to simplify our expression, and wQ
Am(‘) = Q
=
~Q[~(3cos2pQ-
1)
(mi
-:)I
The superscript indicates that this is a first-order approximation to Aoa, derived by standard methods of perturbation theory. Due to the dependence on mi, where 0 a single chemical environment gives rise to a series of 21 transitions corresponding to absorption from each of the possible initial states. The spacing between any pair of transitions is constant. In a sample where all local environments are similarly aligned, such as a single crystal, each of these transitions appear with comparable intensity, as shown in Fig. 4a. In randomly oriented samples, the spectrum corresponds to 21 overlapping standard powder patterns, as might be discerned in Fig. 4b. As the anisotropy of Ati&‘) depends on in precisely the same way as the anisotropic chemical shift depends on PO, it should be averaged under rapid spinning in precisely the same way. In fact, magic angle spinning studies of 2H do show that aQ
#
PQ
0
-25
from
ITis,
-50
-75
Fig. 4. Simulations of axially symmetric quadrupolar powder patterns for I= 5/2, as a function of $qQ/h. Each simulation assumes a Larmor frequency of 75 MHz. (a) Single orientation spectrum, with e2qQlh =300 kHz. (b) Randomly oriented powder sample, with $qQ/h= 300 kHz. (c) Randomly oriented powder sample, with $qQ/h= 1000 kHz. (d) Randomly oriented powder sample, with &qQ/h= 3000 kHz.
the quadrupolar powder pattern breaks up into a centerband characteristic of a chemical shift and a series of spinning sidebands, as shown in Fig. 5 [ 14-161. But the case of “H is exceptional, due to its small nuclear quadrupole moment. Magic angle spinning of more typical quadrupolar nuclear spins rarely achieves the desired level of resolution enhancement and there is no rotor frequency sufficiently rapid to reproduce the familiar spectrum of a sharp centerband and sidebands. An example of more typical results are shown in Fig. 6, where the 23Na resonance in Na,C,O, is seen to narrow by only a factor of 3 under conditions of MAS. This failing is due to an oversimplification. As shown in Fig. 4, for the central transition /l/2) + I - l/2) of a half-integer spin quadrupolar nucleus, Aw,(I) = 0 independent of either or because mi - l/2 = 0. In a powder Sample mQ
PQ,
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must be dominated by higher order effects. Secondorder perturbation theory suggests that Aw,( 2) depends differently on orientation and, for the central transition I1 /2) -+ I - I /2)
_ o$[41(1+
1) - 31
(9cos’po-
l)sin*&
1600 This corresponds to a new kind of powder pattern lineshape, as shown in Fig. 7a. We can decompose the powder pattern of Fig. 7a into three separate components, as shown in Fig. 7b-d. (This particular decomposition is far from arbitrary, and can be justified by arguments from the theory of the rotation group. The details of that justification are far outside the scope of this article.) These correspond to a new type of anisotropic powder pattern (Fig. 7b), an anisotropic component similar to the standard powder pattern treated above when we described the chemical shift anisotropy (Fig. 7c, elf Fig. 2a), and an isotropic component independent of orientation (Fig. 7d) shifted somewhat from a,,,,. Which, if any, of these patterns dominates the observed spectrum depends upon what mechanical reorientation is imposed on the sample. The goal of high-resolution NMR is to transform the specFig. 5. *H (I= 1) spectra of a physical mixture of deuterated hexamethylbenzene and deuterated ferrocene. At top, static sample spectrum. Two quadrupolar powder patterns are observed; a narrow pattern associated with the methyl groups in hexamethylbenzene, and a broader component associated with the deuterons in ferrocene. Middle spectrum: under conditions of MAS, the quadrupolar powder patterns break up into a series of spinning sidebands. At bottom: closer examination of the center of the MAS spectrum reveals two isotropic peaks corresponding to the chemical shifts of the two different deuterated molecules. (Reprintedwith permission from Ref. [15].)
with randomly oriented sites, the intensity associated with each of the other transitions is spread over a bandwidth of order wo. Intensity associated with the central transition, however, appears in a narrow bandwidth. Experimentally, where probes, transmitters and receivers are narrowbanded so as to increase signal-to-noise ratios, only the central transition is observed (see Fig. 4). Because AU:) =O, structure in the observed lineshape
I
10000
I
I
5000
0 Frequency
-5000
-
10000
(Hertz)
Fig. 6. 23Na(I= 3/2) spectra of Na,C,O,. At top, static sample spectrum. At bottom, magic angle spinning spectrum, revealing a partially narrowed secondorder powder pattern and spinning sidebands. (Reprinted with permission from Ref. [18].)
trends in analytical chemistry, vol. 13, no. 8, 1994
r
to (3-30 cos2 p + 35 cos4 p), that is the source of our difficulty. It can only be partially averaged under MAS, and does not break up into a simple sideband pattern. As such it is the culprit in the rotation-rate independent broadening observed for most quadrupolar spins under MAS, as is demonstrated in Fig. 8. For typical values of and 6, the residual linewidth associated with the second-order quadrupolar powder pattern exceeds 6 and is the source of the low resolution associated with quadrupolar spin systems under conditions of MAS. We might choose to spin about an angle designed to eliminate the residual broadening from the powder pattern of Fig. 8b by choosing to rotate about an angle ‘magic’ for it ( 0,: = 30.56” or 70.12”), but under these conditions, residual broadening is observed due to the incompletely averaged, conventional powder pattern of Fig. SC. Moreover, independent of spin angle or spin rate, the center of gravity of the entire powder pattern is shifted with respect to the conventionally described isotropic chemical shift, by an amount (in frequency units) mQ
Fig.7. Expanded view of the axially symmetric (static)
second order quadrupolar powder pattern (simulation), for eQQ/h=3000 MHz, l-5/2, and a Larmor frequency of 75 MHz. (a) Complete powder pattern, of full width
Compare this to the same plot found in Fig. 3d, where the same spectrum is drawn on a different horizontal scale. (b-d) Subspectra corresponding to the rotationally invariant components of (a); (b) is an anisotropic lineshape which is incompletely averaged by MAS; (c) is an anisotropic lineshape analogousto that of the chemical shift anisotropy, and completely averaged by MAS, and (d) is an isotropic, field- and quadrupole-dependent shift by
trum of Fig. 7a into a spectrum similar to that of Fig. 7d. The isotropic component (Fig. 7d) is unaffected by mechanical motion. The standard powder pattern of Fig. 7c, which results from an angular dependence proportional to (3 cos2 1 ), can be treated in exact analogy to our discussion of the averaging of anisotropic chemical shifts. It is therefore completely averaged away under rapid MAS. It is the additional anisotropic pattern of Fig. 7b, resulting from an angular dependence proportional pQ
40
20
0
-20
-40
ppm from CJiso
-
Fig. 8. As in Fig. 7, except under conditions of rapid MAS and emphasizing how the subspectra individually transform. Subspectrum (b) is scaled by -7/l 8, (c) is completely averaged away, and (d) is unaffected.
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4. Achieving high resolution in quadrupolar spin systems
fJ
SJ
hl
jJ
I
I
-2
I
I
0
1
1
1
2
I
I
-2
I
1
0
I
I
I
2
kHz from Uiso Fig. 9. Variation in axially symmetric second-order powder pattern observed under rapid sample rotation, as a function of rotation angle and for e2qQ/h= 3000 MHz, I= 5/2, and a Larmor frequency of 75 MHz. (aj) Rotation about 8=0”, lo”, 20”, 30”, 40”, 50”, 60”, 70”, 80”, and 90”, respectively.
or in ppm units, as
Both the residual broadening and line shifts are inversely proportional to the applied field strength, and a dramatic increase in resolution is achieved in high magnetic fields. Due to partial averaging of different anisotropies, at rotor angles other than the conventional magic angle narrower lines might be observed in a technique called variable angle spinning, or VAS [ 171, as is illustrated in Fig. 9. At no single angle can both contributions be simultaneously eliminated. (As an additional complication, the chemical shift anisotropy will be only partially averaged at rotation angles other than 13,,.)
Much can be learned from the spin-axis dependence of the VAS spectra. One interesting feature is that at certain pairs of spinning angles the spectra appear nearly as mirror images of one another. We can ask whether there are pairs of angles where the anisotropies are of equal magnitude and opposite sign and find that this condition is met for 8 = 79.19” and 8= 37.38”. Mirror image patterns, where the anisotropies are identical to within a scaling constant, are achievable for a variety of angle pairs [ 181. In analogy to the averaging effected by magic angle hopping between different orientations, Samoson, Lippmaa and Pines [6], and Llor and Virlet [7], realized that hopping between multiple orientations - for example, spinning about 8, = 79.19” and 6)*= 37.38” for one rotor period each - could average all the anisotropic terms to zero! The choice of angle pairs is determined by the further requirement that, at the end of evolution through a number of rotor periods ~1,and ~1~at each of the two angles 8, and & (3cos%,
- l)n, = - (3cos2& - l)n,
and (3 - 30cos28, + 35cos48,)n, = - (3 -
3oCOS262
+ 3%OS48,)TZ2
Under these conditions evolution away from the isotropic frequency during the first ~1~rotor cycles is exactly balanced by the opposite sense of evolution in the next ~1~rotor cycles. For ~1,=rz2 the solution is the pair of angles described above. Of course, other angle pairs (which require unequal evolution times at the two settings) can be easily designed, and implemented [ 201. As in magic angle hopping, DAS achieves high resolution in discontinuous steps and is executed as a two-dimensional experiment, where the highresolution axis is measured only one time-point at a time. As a hybrid of a hopping-like experiment and a spinning experiment, spinning sidebands are still observed associated with the (possibly incomplete) averaging of broad lines where the rotor frequency is small as compared to the second-order broadening. Generalizations to more than 2 sets of angles, or to continuously varying angle settings, are of course imaginable [ 191. Samoson, Lippmaa and Pines [6] further realized that a continuously rotating version of the
308
experiment was possible, with the advantage that the evolving signal could be sampled directly. In the absence of any fundamental mechanical or technical constraint, the averaging about two angles might be carried out simultaneously rather than sequentially. Again, the VAS spectra of Fig. 9 can help guide our choice of angle pairs. We have already seen that spinning about 13,,,leaves only the broadening due to the higher-order quadrupolar powder pattern (see Fig. 7). In Fig. 9, we see that for H=: 30” or 8- 70”, the residual broadening appears to be completely analogous to that found in standard chemical shift powder patterns. Superimposing rotation about 0,,,might therefore average away this contribution to the residual broadening. Double rotation spectroscopy (DOR) does just that, by placing a rotor-in-a-rotor. The outer rotor spins about a rotation axis tilted by 8, and at frequency w, with respect to the applied field, while the sample inside spins about a rotation axis tilted from the first by H2and at frequency Oi. The averaged transition frequency can be found by applying the logic used above in describing the magic angle experiment, but by carrying out the averaging process with respect to two angles, simultaneously. Effective line narrowing is achieved by averaging one anisotropic pattern under rotation about one rotor axis, and further averaging the residual broadening through rotation about the second axis. The required rotor angles are simply the magic angles associated with the two distinct anisotropic contributions to the powder pattern, the first of which is our familiar magic angle of 54.74”. The second angle can be chosen to be either 70.12” or 30.56”. Effective resolution enhancement, and mechanical stability, further requires wj/oo 2 5 [ 2 I]. The two different experimental approaches to quadrupolar line narrowing are illustrated in Fig. 10. In either DAS or DOR a single chemical environment results in a centerband flanked by a series of sidebands. The centerband frequency is, as discussed above, shifted from the traditionally understood chemical shift due to the isotropic second-order shift we identified above so that the observed central transition frequency is Awizo = AW,s,ino+ Am&?<<> or, expressed in the common units of ppm,
Unlike a pure chemical shift, the transition frequency (measured in ppm) varies with the measurement field, and contains information about
trends in analytical chemistry, vol. 13, no. 8, 7994
/
(b)
/ Fig. 10. Two different approaches to quadrupolar linenarrowing. In (a), narrowing is achieved by rotating about two angles 13,and 0, successively, so that the rotor axis is time dependent. Second-order quadrupolar broadening is removed stroboscopically. In (b), narrowing is achieved by rotating about two angles 8, and 0, simultaneously using two independent rotors. Second-order quadrupolar broadening is removed continuously. (Reprinted with permission from Ref.
E’8l.J both the quadrupole coupling parameter (and weakly on the asymmetry parameter 7) and the chemical shift, (7. One obvious advantage is that the applied field can be chosen so as to optimize resolution. At very high fields, chemical shift differences between sites may provide good resolution. At lower fields, differences in quadrupole coupling constants may provide a second spreading parameter that results in improved site-differentiation [20]. This dependence of absorption frequency on and wL means that reported chemical shift values require careful scrutiny to ensure that the quadrupolar contributions to observed shifts have been correctly quantified. Fewer sidebands are found in DAS than in DOR. In DAS, sidebands are observed at scaled multiples of the DAS rotor frequency, [ ~1,/ (~2, + IQ) ] 0,. For wQ
wQ
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experiments where II , = n2, sidebands appear at half the rotor period and with modern high speed rotors the spacings between sidebands can be quite large. A clever combination of rotor-synchronized pulses and dynamic angle spinning can completely suppress these sidebands so as to produce spectra representing only isotropic shifts corresponding to distinguishable environments [ 221. In DOR experiments, spinning sidebands appear at the frequencies wi and w, corresponding to the rotation rates of the inner and outer rotors, respectively. As a practical implementation will spin the inner rotor much faster than the outer rotor, most of the sidebands are associated with multiples of the relatively slow outer rotor frequency, only. A sideband-less, hopping-only experiment, where the entire averaging process is executed by reorienting the sample through a small number of discrete orientations, has been described [ 191 but no examples are yet available in the literature. DAS and DOR have different strengths and weaknesses. DAS yields high resolution spectra with fewer of the complications associated with the multiplicity of sidebands found in DOR [ 231. Successful implementations of DAS probes are substantially less expensive than are implementations of DOR probes. Cross-polarization, where signal enhancement is desirable, is a particular problem for spinning quadrupolar systems [ 24,251 that may be ameliorated by spinning about 0= 0” or 8= 90” in a preparation period though there appears to be progress in solving the problem of successfully cross-polarizing to quadrupolar spins in DOR [ 261. In addition, the anisotropic quadrupole coupling is often a more useful diagnostic parameter than is the isotropic shift alone. The full twodimensional DAS experiment may be analyzed more readily than any one-dimensional spectrum due to the clean separation between isotropic shifts in ol. Thus, the scaled anisotropic quadrupolar powder patterns in o2 are experimentally deconvoluted and can be individually analyzed so as to yield better chemical sensitivity than is available from the isotropic shift alone. DOR would appear to be the superior technique where high sample throughput is desired. Where DAS is inherently two-dimensional, requiring numerous repetitions of the experimental sequence independent of the need for signal averaging, in DOR the entire spectrum can be acquired in a single scan lasting only milliseconds. Many of the quadrupolar spin species have good inherent sensitivity and need little signal averaging to produce quality
Diopside (CaMc&Os)
h
Clmoenstatlte
\
(MgSi03) \
/-
\
-__&f,, , ,, ,>:-,:,; 75
50
25
0 Wollastonlte
‘-7
Larmte
_JL, ,,, \
(Ca2SI0,)
\,
140
120
~-_--_
100
80
Frequency (ppm) Fig. 11. I70 MAS spectra at 54.25 MHz of a series of inorganic solid silicates, as illustrated. MAS spin rate 5.4 kHz. Note the limited resolution achievable by this method. (Reprinted with permission from Ref. [28].)
spectra. Where rapid and substantial chemical evolution occurs, DOR is better suited to yielding ‘instantaneous’ snapshots of the chemical environment and its time-evolution. Unlike DAS, DOR spectra are not adversely influenced by short spinlattice relaxation times T,, and in DOR spectra of high-abundance quadrupolar spins the spectral lines are generally narrower than when observed by DAS. Presumably, unrefocused spin diffusion via dipole-dipole couplings during the rotor transition periods are responsible for this observation. (An implementation of DAS where the spinning axis is continuously varied might, therefore, show narrower lines.) 5. Some applications spectroscopy
of DAS and DOR
Fig. 11 compares the resolution achievable by MAS to that of DAS or DOR in Fig. 12 and com-
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Dynamic-Angle
80
60
-...
40
Spinning
20
Double
80
0
Rotation
SO-~40’20’
0
i__...______I
75
50
25
75
0
50
25
0
Wollastonlte
-__I_
140
120
100
140
80
Frequency
120
100
80
(ppm)
Fig. 12. “0 DAS and DOR spectra at 54.25 MHz for the same series of inorganic compounds as are found in Fig. 10. For DAS experiments, spin rate is 5.4 kHz. For DOR experiments, the inner rotor spins at 5 kHz and the outer rotor at 800 Hz. This leads to more sidebands in DOR than in DAS. Conversely, residual linewidths are typically broader in DAS than in DOR. (Reprinted with permission from Ref. [28].) Table 1 Observed and deconvoluted
RbCl RbCIO, Rb,SO, RbNO,
pares spectra of a “0 at a Larmor frequency of 54.25 MHz in a number of inorganic silicates [ 27291. Vastly superior resolution is achieved in experiments where the second order effects are averaged out. Based on experiments where the isotropic shift is probed at several fields, or combining measurements of the isotropic shift with magic angle spinning powder pattern results, it is possible to obtain the chemical shift contribution to the isotropic shift as well as the contribution of the quadrupolar coupling to the isotropic shift. Based on the latter, and a generalization to non-axial symmetry of Eq. 2, the parameter P,=(l+ 712/3) “* e’qQ/h can be extracted. Separation of q (where 77= 0 corresponds to axial symmetry and q= 1 corresponds to the maximum asymmetry) and e*qQ/h can be accomplished by analyzing the scaled quadrupolar powder patterns observed in o2 in DAS spectroscopy. The analysis is simplified if detection occurs while the sample is spinning at the conventional magic angle, so as to eliminate possible contributions from chemical shift anisotropies. This, of course, requires an additional rotor angle hop at the end of t,. For most common nuclear spins, and even at the highest available magnetic fields, the second-order contributions to 6iso from ~o,iso are large. Tables 1 and 2 compare the observed field-dependent shifts in a number of inorganic salts (Table 1) and inorganic oxides (Table 2) and suggest the magnitudes of the corrections necessary when accurate chemical shift information is desired. The “0 data show that isotropic shift resolution need not be best in the highest available fields. Often, the pure chemical shifts SC, differ little from site-to-site and the additional shifts associated with Sgii, dominate the shift differences. As these are largest where the applied field is smallest, spectral dispersion may be largest in these low applied fields. Where wQ is
“Rb chemical shifts in inorganic salts
127+1 -23_tl -10&l 34&l -29*1 -32&l -34&l
127ki -28kl -25kl 29&l -32kl -3651 -37&l
127+2 -14+2 16k2 42k2 -24+2 -25+2 -29+2
0
0 -9 -26 -8 -5 -7 -5
-14 -41 -13 -8 -11 -8
Measured isotropic shifts 6,,, as a function of applied field strength, and decomposition of the observed shift into a chemical shift S,, and second-order quadrupolar contribution to the shift SE’. All shifts are referenced to 1 M RbNO, solution used as an external standard.
311
trends in analytical chemistry, vol. 13, no. 8, 1994
Table 2 Observed and deconvoluted Compound CaMgSi,O, (diopside) Mg,SiO, (forsterite) MgSiO, (clinoenstatite)
Ca,SiO, (larnite)
?&‘:,”
170 shifts and quadrupole 6947 ,50
6c5
couplings in silicates
s(g),” 7T
l;d”,” 4r
C, (MHz)
v
&(I
+ 772/3)“2(MHz)
75.1
69.2
86
-11
-17
2.83kO.05
0.13,O.lO
2.8kO.2
54.0
48.5
64
-10
-16
2.74kO.05
0.00*0.10
2.7kO.2
43.3
28.6
69
-26
-40
4.39kO.05
0.36_tO.O5
4.5fO.l
57.1
49.0
72
-15
-23
3.3kO.i
54.8
49.0
64
-9
-15
2.8kO.2
37.5
30.8
49
-11
-18
3.OkO.2
45.5
39.3
57
-11
-18
2.950.2
44.1
34.5
61
-17
-26
3.6rtO.i
42.0
32.3
59
-17
-27
3.6kO.i
39.0
26.3
62
-23
-36
4.2,O.l
36.8
18.0
70
-33
-52
5.1 kO.1
34.7
15.0
70
-35
-55
5.2kO.l
123.3
117.3
134
-11
-17
2.9kO.2
118.5
113.3
128
-9
-15
2.7kO.2
113.4
108.8
122
-9
-13
2.5iO.2
112.0
106.3
122
-10
-16
K,Si,O,
64
(glass) KMg, ,Si,O,
24
2.8kO.2 5.7kO.4
0.2+0.1
5.7+0.4
0.2+0.1
5.8
0.0
6 6
SiO, (cristobalite)
-16.6
52
-69
Measured isotropic shifts S,,, as a function of applied field strength, and decomposition of the observed shift into a chemical shift S,, and second-order quadrupolar contribution to the shift 82). Based on an analysis of the multidimensional DAS NMR lineshapes, the quadrupolar parameters C, = (e2qQ)lh and 7, or SC:)and Eq. (2) can be combined so as to extract the product C,(l + 77*/3)“*. All shifts are referenced to Hi’0 used as an external standard.
highly variable from site-to-site, high-to-low-field ordering of the lines may differ at different field strengths. The quadrupole coupling constants themselves are diagnostic of local electronic environment and are often more readily interpreted than are chemical shifts. In silicates, correlations have been established between the magnitude (e*qQ/h) and symmetry (7) of 170 quadrupole coupling constants and Si-0-Si bond angles (cy), so that, for example, e*qQlh a cos cu/(cos (Y- 1) and n= 1 fcos (Y [ 30,3 I]. Approximate values of (Ycan be derived from the field-dependence of the second order isotropic shift in DAS or DOR, because S$,Ti’,,is proportional to Po = ( 1 + q2/3) “2 e*qQlh. These important relationships have been combined with the diagnostic power of two dimensional “0 DAS spectroscopy so as to quantify disorder in network-modified silicate glasses. Using the isotropic shift dimension wI so as to ‘spread out’ differing chemical environments, the otherwise overlapping quadrupolar lineshapes are well enough separated so that the distribution of bond angles cr, and its dependence on network
modifiers can be probed straightforwardly [ 301. While the most probable bond angle is the same ( 143” us. 144”)) bond angle distributions for the bridging oxygens in K2Si409 are narrower (halfwidth 21”) than for pure SiO, glass (half-width 36”). The quadrupolar parameters associated with non-bridging oxygen sites appear unaffected when Mg replaces half of the K cations.
6. Conclusions The promise of DAS and DOR is that they will provide the same wealth of high-resolution data for solid-state inorganic chemistry that CPMAS has yielded in 20 years of applications to solid-state organic and polymer chemistry. Already there are a number of applications to zeolite and zeolite-like samples [ 1,32-381. As improved spectroscopic techniques combine with wider distribution of the specialized probes needed for these experiments, the number of applications are certain to grow rap-
312
idly as entirely new branches of chemistry become amenable to straightforward spectroscopic analyses.
References [ I] E. M. Purcell, H. C. Torrey and R. V. Pound, Phys.
Rev., 69 (1946) 37. 12 I F. Bloch, W. W. Hansen and M. Packard, Phys. Rev., 69 (1946) 127. [ 3) A. Bax, N. M. Szeverenyi and G. E. Maciel, J. Mngn. Reson., 52 (1983) 147. (41 E. R. Andrew, A. Bradbury and R.G. Eades, Nature, 182 ( 1958) 1659. [5 1 I. J. Lowe, Phys. Rev. Lett., 2 ( 1959) 285. [ 61 A. Samoson, E. Lippmaa and A. Pines, Mol. Phys., 65 (1988) 1013. 171 A. Llor and J. Virlet, Chem. Phys. Lett., 152 ( 1988) 248. [ 8 1 E. R. Andrews, Int. Rev. Phys. Chem., 1 ( 1980) 195. [ 91 N. Bloembergen and T. J. Rowland, Acta. Metall., 1 (1953) 731. [ IO] J. Herzfeld and A. E. Berger, J. Chem. Phys., 73 (1980) 6021. [ 111 E. Kundla, A. Samoson and E. Lippmaa, Chem. Phys. Lett., 83 ( 1981) 229. [ 121 A. Samoson, E. Kundla and E. Lippmaa, J. Map. Reson., 49 ( 1982) 350. [ 131 M. M. Meadows, K. A. Smith, R. A. Kinsey, T. M. Rothgeb, R. P. Skarjune and E. Oldfield, Proc. Natl. Acad. Sci. USA, 79 ( 1982) 135 1. [ 141 M. M. Maricq and J. S. Waugh, J. Chem. Phys., 70 ( 1979) 3300. [ 151 J. L. Ackerman, R. Eckman and A. Pines, Chem. Phys. 42 ( 1979) 423. [ 161 R. Eckman, M. Alla and A. Pines, J. Magn. Resort., 41 ( 1980) 440. [ 171 S. Ganapathy, S. Schramm and E. Oldfield, J. Chem. Phys., 77 ( 1982) 4360. [ 181 K. T. Mueller, B. Q. Sun, G. C. Chingas, J. W. Zwanziger, T. Terao and A. Pines, J. Magn. Reson., 86 ( 1990) 470. [ 191 A. Samoson, B. Q. Sun and A. Pines, in D. M. S. Bagguley (Editor), Pulsed Magnetic Resonance: NMR, ESR, and Optics, Clarendon Press, Oxford, 1992, Ch. 3. [ 201 J. H. Baltisberger, S. L. Gann, E. W. Wooten, T. H. Chang, K. T. Mueller and A. Pines, J. Am. Chem. Sot., 114 ( 1992) 7489. [ 211 A. Samoson and A. Pines, Rev. Sci. Instrum., 60 ( 1989) 3239. [ 221 S. L. Gann, J. H. Baltisberger and A. Pines, Chem. Phys. Lett., 210 (1993) 405.
trends in analytical chemistry
vol. 13, no. 8, 1994
[231 B. Q. Sun, J. H. Baltisberger, Y. Wu, A. Samoson and A. Pines, Solid State Nucl. Magn. Reson., 1 ( 1992) 267. [241 A. J. Vega, J. Map. Reson., 96 ( 1992) 50. ~251 A. J. Vega, Solid State Nucl. Magn. Reson., I (1992) 17. L2.61Y. Wu, D. Lewis, J. S. Frye, A. R. Palmer and R. A. Wind, J. Map. Reson., 100 ( 1992) 425. u71 B. F. Chmelka, K. T. Mueller, A. Pines, J. Stebbins, Y. Wu and J. W. Zwanziger, Nature, 339 ( 1989) 42. [281 K. T. Mueller, Y. Wu, B. F. Chmelka, J. Stebbins and A. Pines, J. Am. Chem. Sot., 113 ( 1991) 32. [29] K. T. Mueller, J. H. Baltisberger, E. W. Wooten and A. Pines, J. Phys. Chem., 96 ( 1992) 7001. [30] I. Faman, P. J. Grandinetti. J. H. Baltisberger, J. F. Stebbins, U. Werner, M. A. Eastman and A. Pines, Nature, 358 ( 1992) 3 1. [3ll J. A. Tossell and P. Lazzeretti, Phys. Chem. Miner., I5 ( 1988) 564. c321 Y. Wu, B. F. Chmelka, A. Pines, M. E. Davis, P. J. Grobet and P. A. Jacobs, Nature, 346 ( 1990) 550. [331 P. J. Barrie, M. E. Smith and J. Klinowski, Chem. Phys. Lett., 180 ( 199 1) 6. [341 R. Jelinek, B. F. Chmelka, Y. Wu, P. J. Grandinetti, A. Pines, P. J. Barrie and J. Klinowski, J. Am. Chem. Sot., 113 ( 1991) 4097. [351 R. Jelinek, S. ozkar and G. A. Ozin, J. Am. Chem. Sot., 114 (1992) 4907. [ 361 R. Jelinek, S. Gzkar, and G. A. Ozin, J. Phys. Chem., 96 ( 1992) 5949. [37] R. Jelinek, B. F. Chmelka, A. Stein and G. A. Ozin, J. Phys. Chem., 96 ( 1992) 6744. [ 381 J. Rocha, J. Klinowski, P. J. Barrie, R. Jelinek and A. Pines, Solid State Nucl. Map. Reson., 1 (1992) 217. D.B. Zax is at the Baker Laboratory, Deparfment ofChemistrv, CornellUniversitv. ,I Ithaca, NY 148531301, USA.<
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1
trends in analytical
chemistry,
vol. 13, no. 8, 1994
Dynamic angle spinning and double rotation quadrupolar NMR D.B. Zax Ithaca, NY, USA The problem of high-resolution solid state NMR of quadrupolar nuclear spins (where I> l/2) has been recently solved. The twin techniques of dynamic angle spinning (DAS) and double rotation (DOR) demonstrate resolution comparable to that observed in highresolution spectra of 13Cor 15N.This presents a substantial extension as most NMR-active spins are quadrupolar. In this review we describe previous techniques, and explain why DAS and DOR yield superior results.
1. Introduction The question of how best to achieve high resolution (and therefore good chemical sensitivity) in NMR has intrigued the field since the first work of Purcell, Torrey and Pound [ 1 ] and of Bloch, Hansen and Packard [2] nearly 50 years ago. In the earliest years, research emphasized applications to solids where the larger internal interactions lead to broad spectral lines. In magnetic fields of low homogeneity, the finer spectral details associated liquid state NMR are with high resolution, obscured and little more than the profile of the magnetic field can be measured. With the development of high resolution magnet coils, the superior chemical sensitivity associated with the routine measurement of chemical shifts made liquid state NMR the more popular alternative. Broad-line spectra in solids are a result of the orientation-dependence of these internal interacthe dipole-dipole and quadrupolar tions, couplings. High resolution, as achieved in isotropic liquids, reflects nature’s efforts to eliminate the direct effects of these anisotropic interactions through ‘motional narrowing’. Where NMR experiments normally monitor the resonance frequency only on time scales of microseconds to millisec016%9986/94/$07,00
onds, in low viscosity liquids collisions between molecules cause substantial molecular reorientation every few picoseconds. Observed transition frequencies correspond to averages over all those orientations accessed during the measurement period. This unrestricted, rapid motion averages all anisotropic interactions precisely to zero. (Manifestations of these interactions may still be found in relaxation measurements.) Only the isotropic chemical shift (w,,,) and the J coupling, (Ji,,) survive this motional averaging process. Spectra of solids reflect all of these unaveraged interactions superposed upon one another. The broad lines characteristic of NMR spectra of solidstate samples therefore contain all of the spectroscopic information available in liquids, and much more as well. Often, the available information so exceeds methods of analysis that it is desirable to mimic nature and sacrifice some spectral parameters so as to more accurately measure others; or, as in multidimensional NMR techniques, to use one experimental parameter to spread out the interesting information in multiple dimensions. If resolution between sites is good in one dimension, overlapping broad lines may be sufficiently separated that they become amenable to analysis. Achieving high resolution in solid-state NMR spectra generally requires that we somehow mimic nature, and remove that orientational broadening. But while nature’s technique is unquestionably effective, it is also hopelessly inefficient as a model for high-resolution NMR in solids. No mechanical process can reasonably expect to reorient crystallites through all possible orientations on sub-microsecond timescales. Key to a practical implementation of high resolution NMR in solids is the realization that the anisotropies have well-behaved directional properties. Even where only a small number of orientations are sampled, the anisotropy may be suppressed if those orientations are carefully chosen. A particularly efficient example of averaging by sample reorientation is provided by the example 0
1994 Elsevier Science B.V. All rights reserved
301
trends in analytical chemistry, vol. 13, no. 8, 1994
b
50
40
30
20
10
Chemacal
0
shill
-10
-20
30
40
(p.p.m.1
Fig. 1. MAS and DOR spectra of *‘AI in the dehydrated and rehydrated aluminophosphate sieve VPI-5, at 104.23 MHz. The crystal lattice consists of alternating AIO, and PO, tetrahedra, with two crystallographically inequivalent Al sites in the abundance ratio 2:l. (a) MAS NMR spectrum of the dehydrated (at 1O-5 Torr for 48 h) VPI-5 sample. Only a single asymmetric line is observed. (b) DOR NMR spectrum of the dehydrated (at lop5 Torr for 48 h) VPI-5 sample. Two peaks are resolved with intensity ratios 2:l. (c) DOR NMR spectrum of VPI-5 after rehydration for two days, showing the appearance of octrahedrally coordinated Al (at - 18.4 ppm) and a variety of different tetrahedral sites (labeled a,b, and g in the figure). Intensity is preserved in the line marked d during partial rehydration, indicating that only the doubly populated sites are initially effected by hydration. (d) As in (c), except that rehydration has continued for a total of 23days. (Reprinted with permissionfrom Ref. [32].)
of magic angle hopping in which Bax, Szeverenyi and Maciel [ 31 showed that chemical shift anisotropy could be completely averaged, independent of initial crystallite orientation, by reorienting the container holding the sample through only three
discrete orientations. It is experimentally simpler to work with samples contained in cylindrically rotating containers so that orientation-space is sampled continuously. In nuclei with spin I= l/2, the chemical shift anisotropy is routinely removed via magic angle spinning (MAS) [ 4,5]. Disappointingly, neither magic angle hopping nor magic angle spinning yields comparable high resolution spectra in quadrupolar (12 1) spin systems. Two experiments have recently been introduced which do: dynamic angle spinning (DAS) [ 6,7], in which the sample is spun sequentially around a small number of directions and double rotation (DOR) [7], in which the sample spins inside a rotor which itself is spinning inside a second rotor spinning at the magic angle with respect to the field. The result of each of these experiments is a spectrum where orientational broadening due to the anisotropic quadrupole coupling is removed. The result is a striking improvement in attainable resolution for the vast majority of nuclear spin species where quadrupolar broadening often seemed an insurmountable obstacle to the effective use of solid-state NMR methods. An example of the unparalleled chemical sensitivity now routinely available is shown in Fig. I, where DOR NMR spectroscopy is demonstrated to be capable of following subtle changes in zeolite chemistry as a function of degree of hydration and time. For comparison, the MAS NMR spectrum of the same system is shown at the top of the figure. Where chemical evolution is clearly monitored in the successive 27Al DOR spectra, the MAS spectral linewidth is sufficiently broad that no resolved features are observable. In this article, our plan of attack is as follows: first, we explain how magic angle techniques achieve high resolution in the important case of anisotropic chemical shifts of spin - l/2 nuclei. We briefly consider the differences between spinning and hopping. We then explore the partial success of MAS in narrowing lineshapes observed in quadrupolar spin systems. This is followed by an analysis of how DAS and DOR achieve high resolution. We then conclude with some examples of its use in systems of practical interest, and some further comments on its prospects for the future. 2. Achieving high-resolution in solids: magic angle hopping and spinning Andrew, Bradbury and Eades [4], and separately Lowe [5] provided a simple solution for
302
trends in analytical chemistry, vol. 13, no. 8, 1994
narrowing the broad lines associated with powder patterns due to anisotropic dipole-dipole couplings in solid-state NMR. They realized, and demonstrated experimentally, that powder pattern lineshapes could be narrowed by bulk sample reorientation. Optimum narrowing was achieved when the sample was spun at the ‘magic angle’ 19,=54.74”. In what follows we specialize to the main contemporary application of MAS, the narrowing of powder pattern lineshapes associated with anisotropic chemical shifts. The same technology that is the basis for the drill found in a modern dentist’s office can be used to rapidly rotate solids in an NMR sample holder. The air-bearing can be tilted at any arbitrary angle with respect to the applied field. As the sample container is rotated, the orientation of any particular molecule as referenced to the external magnetic field varies in a simply-describable fashion. This orientation dependence is equivalent to a time-dependent transition frequency. Our goal is to understand how magic angle spinning accomplishes ‘motional averaging’ such that all molecules, independent of initial orientation, have the same time-averaged transition frequency. We can formalize this problem using a simple geometric model [ 81. The orientation of any chemical shift tensor in an external field can be specified by two angles LY,,and &. With little loss of generality we will choose to treat explicitly only molecular systems of axial or higher symmetry, so that there is no explicit dependence of any of what follows on (Y,.. (With somewhat more attention to the details, all of our results can be straightforwardly extended to the more general case.) Where only the chemical shift is observed, the NMR transition frequency Aw can be separated into orien(isotropic) and orientation-independent tation-dependent (anisotropic) components via Am= Awis, + Aaaniso where Awiso = WO(1 - a) and Am)aniso=$3
cos2Pcs - 1)
for o0 the resonance frequency of a suitably defined reference compound, (Tthe conventionally defined isotropic chemical shift, and 6 the magnitude of the anisotropic shift. An example of the solid-state powder pattern associated with the anisotropy of
Static
r-----8 = 45”
e = em
$ = 60”
150
100
50 ppm
0
-50
-100
-150
from q,,
Fig. 2. Axially symmetric chemical shift powder patterns, as are commonly found for spin- l/2 nuclei, where the shift is referenced to CT,,, and the anisotropy is 6= 100 ppm. At top, the signal observed for static samples. Below, the scaled powder pattern observed for rapid rotation about the indicated axes, defined as the tilt angle with respect to the applied magneticfield.
the chemical shift (the CSA) for a spin - l/2 species such as 13C or 15N is shown in Fig. 2a. Characteristic values of PCs can be associated with well-defined positions in the powder pattern; for example, the intense maximum shifted by 6/2 from corresponds to &=90”, while the other 6w extreme at - 6 from ~iso corresponds to PCs= 0”. This lineshape function was first analyzed by Bloembergen and Rowland [9], and as it recurs throughout problems in magnetic resonance we will refer to as the standard powder pattern lineshape. As the sample rotates, /3_(t) varies in time. Choosing (as the angle a given site is initially tilted with respect to the rotor axis, 0 the angle the rotor is tipped with respect to the laboratory z-axis, and qt the (time-dependent) angle swept out by the rotor as it spins, it can be shown [ 81 that cos&,( t) = cos0 co+
sin0 sin
and sin pcs( t) = sin0 cosl+ cos0 sin< cosqt
frendsinanalyticalchemistry, vol.13,no. 8, 1994
303
In lockstep with PCs(t), Ao,,i,, varies periodically. Our goal is to find the trajectory-averaged anisotropic contribution to the transition frequency. After a goodly dose of straightforward if tedious algebraic manipulations, we find
+ terms periodic in w, and 20, The isotropic contribution, as it is independent of orientation, is unaffected. Equation ( 1) states that, without regard to initial orientation l, the averaged contribution to the anisotropic shift is scaled by (3 cos2 19- 1) /2. Magic angle spinning corresponds to the choice 8= 0,, = 54.74” where the scaling factor is zero and optimum line narrowing occurs. This is illustrated in Fig. 2a-d, where spectra corresponding to the powder lineshape associated with a static sample, as well as rapidly spinning samples including a magic angle spectrum and the spectrum observed at two angles which are not quite magic ( 8= 45” and 8= 60”), are shown. In this limit only the time-averaged transition frequency is observed. For Am,“,= AWiso the specific case of magic angle spinning, each orientation has precisely the same average transition frequency that of the isotropic chemical shift. The only evidence of the anisotropy 6 is preserved in the spin-rate dependent terms. Where w, < 6, these correspond to spinning sidebands observed at frequencies Wisef no,, with little observed in intensity for IIZW,]> 6 [ IO]. Magic angle spectra, where the spin rate varies from slow to fast, are shown in Fig. n
5.
The magic angle hopping experiment [ 31 works on a similar principle. In contrast to MAS, the sample is reoriented in discrete steps so that each chemical shift tensor is found in three discrete orientations only. Each is related to the other two by a 120” rotation of the whole sample container about an angle 8. In analogy to the averaging performed above, we find that AWaniso =
=
cos2P&)
- 11
6(3cos21-l)(3cos2&l) 2 2
As in magic angle spinning, it is sufficient that we choose to hop about the 8= 8, to average away all
““‘* K= 2 =i 150
100
50 wm
0 from
-50
-100
-150
oisO
Fig. 3. Axially symmetric chemical shift powder patterns, where the sample is placed in a rotor aligned at 0, with respect to the applied magnetic field. At top, the signal observed for static samples. Below, the sideband pattern observed as a function of rotation rate K = 6( o.I~cJ,).
anisotropy. There are no sidebands observed in magic angle hopping, which may be a substantial advantage in systems where 6 is large. Nonetheless, it remains largely unappreciated, and suffers from the considerable disadvantage that a practical implementation requires that the experiment be repeated a large number of times while the time spent in each of the three orientations is continuously incremented - i.e. in two-dimensional form. 3. Magic angle spinning: quadrupolar
nuclear spins Based on our long experience of high resolution in systems where spectral features are dominated by large chemical shift differences, it has been a source of some disappointment that MAS fails to achieve similar results for the more numerous halfinteger quadrupolar nuclei [ 1 l-l 31. In these systems fine structure in the NMR spectrum is often dominated by the interaction of the nuclear quadrupole moment (corresponding to a non-spherical charge distribution of the atomic nucleus) with local electric field gradients. There are 21-t 1 dif-
304
trends in analytical chemistry, vol. 13, no. 8, 1994
ferent orientations allowed for a nucleus with spin I. In available magnetic fields, quadrupole couplings are often orders of magnitude larger than chemical shifts. Under these conditions, the transition frequency depends simultaneously upon both the chemical shift and quadrupolar interactions, and Ati= A@,, + Am, Each interaction is directional, and generally not coincident; thus, for any given site, we are required to track (assuming axial symmetry in both tensors) PCsand Pa. (This will turn out to be a bookkeeping nuisance but of little fundamental import.) The chemical shift contribution, Aw,,, has been given above. It is independent of which of the 2Z+ 1 different Zeeman energy levels we begin from. In contrast, while the change in quadrupolar energy does depend upon which transition is observed. We will designate a transition by its initial (mi) and final (mr) values of Z,. The first order quadrupolar energy change is Aob’) = wo[;(3cos2&-
l)(m;
al /
! 1
bJ
n
cl
75
-$)I
50
25 k/-/z
where 3e2qQ/4Z( 2Z- 1) fi. Because the Z, states are defined with respect to the spin orientation in the externally applied field and we observe transitions in the vicinity of o. only, we can make use of the selection rule mi = mr + 1 so as to simplify our expression, and wQ
Am(‘) = Q
=
~Q[~(3cos2pQ-
1)
(mi
-:)I
The superscript indicates that this is a first-order approximation to Aoa, derived by standard methods of perturbation theory. Due to the dependence on mi, where 0 a single chemical environment gives rise to a series of 21 transitions corresponding to absorption from each of the possible initial states. The spacing between any pair of transitions is constant. In a sample where all local environments are similarly aligned, such as a single crystal, each of these transitions appear with comparable intensity, as shown in Fig. 4a. In randomly oriented samples, the spectrum corresponds to 21 overlapping standard powder patterns, as might be discerned in Fig. 4b. As the anisotropy of Ati&‘) depends on in precisely the same way as the anisotropic chemical shift depends on PO, it should be averaged under rapid spinning in precisely the same way. In fact, magic angle spinning studies of 2H do show that aQ
#
PQ
0
-25
from
ITis,
-50
-75
Fig. 4. Simulations of axially symmetric quadrupolar powder patterns for I= 5/2, as a function of $qQ/h. Each simulation assumes a Larmor frequency of 75 MHz. (a) Single orientation spectrum, with e2qQlh =300 kHz. (b) Randomly oriented powder sample, with $qQ/h= 300 kHz. (c) Randomly oriented powder sample, with $qQ/h= 1000 kHz. (d) Randomly oriented powder sample, with &qQ/h= 3000 kHz.
the quadrupolar powder pattern breaks up into a centerband characteristic of a chemical shift and a series of spinning sidebands, as shown in Fig. 5 [ 14-161. But the case of “H is exceptional, due to its small nuclear quadrupole moment. Magic angle spinning of more typical quadrupolar nuclear spins rarely achieves the desired level of resolution enhancement and there is no rotor frequency sufficiently rapid to reproduce the familiar spectrum of a sharp centerband and sidebands. An example of more typical results are shown in Fig. 6, where the 23Na resonance in Na,C,O, is seen to narrow by only a factor of 3 under conditions of MAS. This failing is due to an oversimplification. As shown in Fig. 4, for the central transition /l/2) + I - l/2) of a half-integer spin quadrupolar nucleus, Aw,(I) = 0 independent of either or because mi - l/2 = 0. In a powder Sample mQ
PQ,
305
trends in analytical chemistry, vol. 13, no. 8, 1994
must be dominated by higher order effects. Secondorder perturbation theory suggests that Aw,( 2) depends differently on orientation and, for the central transition I1 /2) -+ I - I /2)
_ o$[41(1+
1) - 31
(9cos’po-
l)sin*&
1600 This corresponds to a new kind of powder pattern lineshape, as shown in Fig. 7a. We can decompose the powder pattern of Fig. 7a into three separate components, as shown in Fig. 7b-d. (This particular decomposition is far from arbitrary, and can be justified by arguments from the theory of the rotation group. The details of that justification are far outside the scope of this article.) These correspond to a new type of anisotropic powder pattern (Fig. 7b), an anisotropic component similar to the standard powder pattern treated above when we described the chemical shift anisotropy (Fig. 7c, elf Fig. 2a), and an isotropic component independent of orientation (Fig. 7d) shifted somewhat from a,,,,. Which, if any, of these patterns dominates the observed spectrum depends upon what mechanical reorientation is imposed on the sample. The goal of high-resolution NMR is to transform the specFig. 5. *H (I= 1) spectra of a physical mixture of deuterated hexamethylbenzene and deuterated ferrocene. At top, static sample spectrum. Two quadrupolar powder patterns are observed; a narrow pattern associated with the methyl groups in hexamethylbenzene, and a broader component associated with the deuterons in ferrocene. Middle spectrum: under conditions of MAS, the quadrupolar powder patterns break up into a series of spinning sidebands. At bottom: closer examination of the center of the MAS spectrum reveals two isotropic peaks corresponding to the chemical shifts of the two different deuterated molecules. (Reprintedwith permission from Ref. [15].)
with randomly oriented sites, the intensity associated with each of the other transitions is spread over a bandwidth of order wo. Intensity associated with the central transition, however, appears in a narrow bandwidth. Experimentally, where probes, transmitters and receivers are narrowbanded so as to increase signal-to-noise ratios, only the central transition is observed (see Fig. 4). Because AU:) =O, structure in the observed lineshape
I
10000
I
I
5000
0 Frequency
-5000
-
10000
(Hertz)
Fig. 6. 23Na(I= 3/2) spectra of Na,C,O,. At top, static sample spectrum. At bottom, magic angle spinning spectrum, revealing a partially narrowed secondorder powder pattern and spinning sidebands. (Reprinted with permission from Ref. [18].)
trends in analytical chemistry, vol. 13, no. 8, 1994
r
to (3-30 cos2 p + 35 cos4 p), that is the source of our difficulty. It can only be partially averaged under MAS, and does not break up into a simple sideband pattern. As such it is the culprit in the rotation-rate independent broadening observed for most quadrupolar spins under MAS, as is demonstrated in Fig. 8. For typical values of and 6, the residual linewidth associated with the second-order quadrupolar powder pattern exceeds 6 and is the source of the low resolution associated with quadrupolar spin systems under conditions of MAS. We might choose to spin about an angle designed to eliminate the residual broadening from the powder pattern of Fig. 8b by choosing to rotate about an angle ‘magic’ for it ( 0,: = 30.56” or 70.12”), but under these conditions, residual broadening is observed due to the incompletely averaged, conventional powder pattern of Fig. SC. Moreover, independent of spin angle or spin rate, the center of gravity of the entire powder pattern is shifted with respect to the conventionally described isotropic chemical shift, by an amount (in frequency units) mQ
Fig.7. Expanded view of the axially symmetric (static)
second order quadrupolar powder pattern (simulation), for eQQ/h=3000 MHz, l-5/2, and a Larmor frequency of 75 MHz. (a) Complete powder pattern, of full width
Compare this to the same plot found in Fig. 3d, where the same spectrum is drawn on a different horizontal scale. (b-d) Subspectra corresponding to the rotationally invariant components of (a); (b) is an anisotropic lineshape which is incompletely averaged by MAS; (c) is an anisotropic lineshape analogousto that of the chemical shift anisotropy, and completely averaged by MAS, and (d) is an isotropic, field- and quadrupole-dependent shift by
trum of Fig. 7a into a spectrum similar to that of Fig. 7d. The isotropic component (Fig. 7d) is unaffected by mechanical motion. The standard powder pattern of Fig. 7c, which results from an angular dependence proportional to (3 cos2 1 ), can be treated in exact analogy to our discussion of the averaging of anisotropic chemical shifts. It is therefore completely averaged away under rapid MAS. It is the additional anisotropic pattern of Fig. 7b, resulting from an angular dependence proportional pQ
40
20
0
-20
-40
ppm from CJiso
-
Fig. 8. As in Fig. 7, except under conditions of rapid MAS and emphasizing how the subspectra individually transform. Subspectrum (b) is scaled by -7/l 8, (c) is completely averaged away, and (d) is unaffected.
307
trends in analytical chemistry, vol. 13, no. 8, 1994
4. Achieving high resolution in quadrupolar spin systems
fJ
SJ
hl
jJ
I
I
-2
I
I
0
1
1
1
2
I
I
-2
I
1
0
I
I
I
2
kHz from Uiso Fig. 9. Variation in axially symmetric second-order powder pattern observed under rapid sample rotation, as a function of rotation angle and for e2qQ/h= 3000 MHz, I= 5/2, and a Larmor frequency of 75 MHz. (aj) Rotation about 8=0”, lo”, 20”, 30”, 40”, 50”, 60”, 70”, 80”, and 90”, respectively.
or in ppm units, as
Both the residual broadening and line shifts are inversely proportional to the applied field strength, and a dramatic increase in resolution is achieved in high magnetic fields. Due to partial averaging of different anisotropies, at rotor angles other than the conventional magic angle narrower lines might be observed in a technique called variable angle spinning, or VAS [ 171, as is illustrated in Fig. 9. At no single angle can both contributions be simultaneously eliminated. (As an additional complication, the chemical shift anisotropy will be only partially averaged at rotation angles other than 13,,.)
Much can be learned from the spin-axis dependence of the VAS spectra. One interesting feature is that at certain pairs of spinning angles the spectra appear nearly as mirror images of one another. We can ask whether there are pairs of angles where the anisotropies are of equal magnitude and opposite sign and find that this condition is met for 8 = 79.19” and 8= 37.38”. Mirror image patterns, where the anisotropies are identical to within a scaling constant, are achievable for a variety of angle pairs [ 181. In analogy to the averaging effected by magic angle hopping between different orientations, Samoson, Lippmaa and Pines [6], and Llor and Virlet [7], realized that hopping between multiple orientations - for example, spinning about 8, = 79.19” and 6)*= 37.38” for one rotor period each - could average all the anisotropic terms to zero! The choice of angle pairs is determined by the further requirement that, at the end of evolution through a number of rotor periods ~1,and ~1~at each of the two angles 8, and & (3cos%,
- l)n, = - (3cos2& - l)n,
and (3 - 30cos28, + 35cos48,)n, = - (3 -
3oCOS262
+ 3%OS48,)TZ2
Under these conditions evolution away from the isotropic frequency during the first ~1~rotor cycles is exactly balanced by the opposite sense of evolution in the next ~1~rotor cycles. For ~1,=rz2 the solution is the pair of angles described above. Of course, other angle pairs (which require unequal evolution times at the two settings) can be easily designed, and implemented [ 201. As in magic angle hopping, DAS achieves high resolution in discontinuous steps and is executed as a two-dimensional experiment, where the highresolution axis is measured only one time-point at a time. As a hybrid of a hopping-like experiment and a spinning experiment, spinning sidebands are still observed associated with the (possibly incomplete) averaging of broad lines where the rotor frequency is small as compared to the second-order broadening. Generalizations to more than 2 sets of angles, or to continuously varying angle settings, are of course imaginable [ 191. Samoson, Lippmaa and Pines [6] further realized that a continuously rotating version of the
308
experiment was possible, with the advantage that the evolving signal could be sampled directly. In the absence of any fundamental mechanical or technical constraint, the averaging about two angles might be carried out simultaneously rather than sequentially. Again, the VAS spectra of Fig. 9 can help guide our choice of angle pairs. We have already seen that spinning about 13,,,leaves only the broadening due to the higher-order quadrupolar powder pattern (see Fig. 7). In Fig. 9, we see that for H=: 30” or 8- 70”, the residual broadening appears to be completely analogous to that found in standard chemical shift powder patterns. Superimposing rotation about 0,,,might therefore average away this contribution to the residual broadening. Double rotation spectroscopy (DOR) does just that, by placing a rotor-in-a-rotor. The outer rotor spins about a rotation axis tilted by 8, and at frequency w, with respect to the applied field, while the sample inside spins about a rotation axis tilted from the first by H2and at frequency Oi. The averaged transition frequency can be found by applying the logic used above in describing the magic angle experiment, but by carrying out the averaging process with respect to two angles, simultaneously. Effective line narrowing is achieved by averaging one anisotropic pattern under rotation about one rotor axis, and further averaging the residual broadening through rotation about the second axis. The required rotor angles are simply the magic angles associated with the two distinct anisotropic contributions to the powder pattern, the first of which is our familiar magic angle of 54.74”. The second angle can be chosen to be either 70.12” or 30.56”. Effective resolution enhancement, and mechanical stability, further requires wj/oo 2 5 [ 2 I]. The two different experimental approaches to quadrupolar line narrowing are illustrated in Fig. 10. In either DAS or DOR a single chemical environment results in a centerband flanked by a series of sidebands. The centerband frequency is, as discussed above, shifted from the traditionally understood chemical shift due to the isotropic second-order shift we identified above so that the observed central transition frequency is Awizo = AW,s,ino+ Am&?<<> or, expressed in the common units of ppm,
Unlike a pure chemical shift, the transition frequency (measured in ppm) varies with the measurement field, and contains information about
trends in analytical chemistry, vol. 13, no. 8, 7994
/
(b)
/ Fig. 10. Two different approaches to quadrupolar linenarrowing. In (a), narrowing is achieved by rotating about two angles 13,and 0, successively, so that the rotor axis is time dependent. Second-order quadrupolar broadening is removed stroboscopically. In (b), narrowing is achieved by rotating about two angles 8, and 0, simultaneously using two independent rotors. Second-order quadrupolar broadening is removed continuously. (Reprinted with permission from Ref.
E’8l.J both the quadrupole coupling parameter (and weakly on the asymmetry parameter 7) and the chemical shift, (7. One obvious advantage is that the applied field can be chosen so as to optimize resolution. At very high fields, chemical shift differences between sites may provide good resolution. At lower fields, differences in quadrupole coupling constants may provide a second spreading parameter that results in improved site-differentiation [20]. This dependence of absorption frequency on and wL means that reported chemical shift values require careful scrutiny to ensure that the quadrupolar contributions to observed shifts have been correctly quantified. Fewer sidebands are found in DAS than in DOR. In DAS, sidebands are observed at scaled multiples of the DAS rotor frequency, [ ~1,/ (~2, + IQ) ] 0,. For wQ
wQ
309
trends in analytical chemistry, vol. 13, no. 8, 1994
experiments where II , = n2, sidebands appear at half the rotor period and with modern high speed rotors the spacings between sidebands can be quite large. A clever combination of rotor-synchronized pulses and dynamic angle spinning can completely suppress these sidebands so as to produce spectra representing only isotropic shifts corresponding to distinguishable environments [ 221. In DOR experiments, spinning sidebands appear at the frequencies wi and w, corresponding to the rotation rates of the inner and outer rotors, respectively. As a practical implementation will spin the inner rotor much faster than the outer rotor, most of the sidebands are associated with multiples of the relatively slow outer rotor frequency, only. A sideband-less, hopping-only experiment, where the entire averaging process is executed by reorienting the sample through a small number of discrete orientations, has been described [ 191 but no examples are yet available in the literature. DAS and DOR have different strengths and weaknesses. DAS yields high resolution spectra with fewer of the complications associated with the multiplicity of sidebands found in DOR [ 231. Successful implementations of DAS probes are substantially less expensive than are implementations of DOR probes. Cross-polarization, where signal enhancement is desirable, is a particular problem for spinning quadrupolar systems [ 24,251 that may be ameliorated by spinning about 0= 0” or 8= 90” in a preparation period though there appears to be progress in solving the problem of successfully cross-polarizing to quadrupolar spins in DOR [ 261. In addition, the anisotropic quadrupole coupling is often a more useful diagnostic parameter than is the isotropic shift alone. The full twodimensional DAS experiment may be analyzed more readily than any one-dimensional spectrum due to the clean separation between isotropic shifts in ol. Thus, the scaled anisotropic quadrupolar powder patterns in o2 are experimentally deconvoluted and can be individually analyzed so as to yield better chemical sensitivity than is available from the isotropic shift alone. DOR would appear to be the superior technique where high sample throughput is desired. Where DAS is inherently two-dimensional, requiring numerous repetitions of the experimental sequence independent of the need for signal averaging, in DOR the entire spectrum can be acquired in a single scan lasting only milliseconds. Many of the quadrupolar spin species have good inherent sensitivity and need little signal averaging to produce quality
Diopside (CaMc&Os)
h
Clmoenstatlte
\
(MgSi03) \
/-
\
-__&f,, , ,, ,>:-,:,; 75
50
25
0 Wollastonlte
‘-7
Larmte
_JL, ,,, \
(Ca2SI0,)
\,
140
120
~-_--_
100
80
Frequency (ppm) Fig. 11. I70 MAS spectra at 54.25 MHz of a series of inorganic solid silicates, as illustrated. MAS spin rate 5.4 kHz. Note the limited resolution achievable by this method. (Reprinted with permission from Ref. [28].)
spectra. Where rapid and substantial chemical evolution occurs, DOR is better suited to yielding ‘instantaneous’ snapshots of the chemical environment and its time-evolution. Unlike DAS, DOR spectra are not adversely influenced by short spinlattice relaxation times T,, and in DOR spectra of high-abundance quadrupolar spins the spectral lines are generally narrower than when observed by DAS. Presumably, unrefocused spin diffusion via dipole-dipole couplings during the rotor transition periods are responsible for this observation. (An implementation of DAS where the spinning axis is continuously varied might, therefore, show narrower lines.) 5. Some applications spectroscopy
of DAS and DOR
Fig. 11 compares the resolution achievable by MAS to that of DAS or DOR in Fig. 12 and com-
310
trends in analytical chemistry, vol. 13, no. 8, 1994
Dynamic-Angle
80
60
-...
40
Spinning
20
Double
80
0
Rotation
SO-~40’20’
0
i__...______I
75
50
25
75
0
50
25
0
Wollastonlte
-__I_
140
120
100
140
80
Frequency
120
100
80
(ppm)
Fig. 12. “0 DAS and DOR spectra at 54.25 MHz for the same series of inorganic compounds as are found in Fig. 10. For DAS experiments, spin rate is 5.4 kHz. For DOR experiments, the inner rotor spins at 5 kHz and the outer rotor at 800 Hz. This leads to more sidebands in DOR than in DAS. Conversely, residual linewidths are typically broader in DAS than in DOR. (Reprinted with permission from Ref. [28].) Table 1 Observed and deconvoluted
RbCl RbCIO, Rb,SO, RbNO,
pares spectra of a “0 at a Larmor frequency of 54.25 MHz in a number of inorganic silicates [ 27291. Vastly superior resolution is achieved in experiments where the second order effects are averaged out. Based on experiments where the isotropic shift is probed at several fields, or combining measurements of the isotropic shift with magic angle spinning powder pattern results, it is possible to obtain the chemical shift contribution to the isotropic shift as well as the contribution of the quadrupolar coupling to the isotropic shift. Based on the latter, and a generalization to non-axial symmetry of Eq. 2, the parameter P,=(l+ 712/3) “* e’qQ/h can be extracted. Separation of q (where 77= 0 corresponds to axial symmetry and q= 1 corresponds to the maximum asymmetry) and e*qQ/h can be accomplished by analyzing the scaled quadrupolar powder patterns observed in o2 in DAS spectroscopy. The analysis is simplified if detection occurs while the sample is spinning at the conventional magic angle, so as to eliminate possible contributions from chemical shift anisotropies. This, of course, requires an additional rotor angle hop at the end of t,. For most common nuclear spins, and even at the highest available magnetic fields, the second-order contributions to 6iso from ~o,iso are large. Tables 1 and 2 compare the observed field-dependent shifts in a number of inorganic salts (Table 1) and inorganic oxides (Table 2) and suggest the magnitudes of the corrections necessary when accurate chemical shift information is desired. The “0 data show that isotropic shift resolution need not be best in the highest available fields. Often, the pure chemical shifts SC, differ little from site-to-site and the additional shifts associated with Sgii, dominate the shift differences. As these are largest where the applied field is smallest, spectral dispersion may be largest in these low applied fields. Where wQ is
“Rb chemical shifts in inorganic salts
127+1 -23_tl -10&l 34&l -29*1 -32&l -34&l
127ki -28kl -25kl 29&l -32kl -3651 -37&l
127+2 -14+2 16k2 42k2 -24+2 -25+2 -29+2
0
0 -9 -26 -8 -5 -7 -5
-14 -41 -13 -8 -11 -8
Measured isotropic shifts 6,,, as a function of applied field strength, and decomposition of the observed shift into a chemical shift S,, and second-order quadrupolar contribution to the shift SE’. All shifts are referenced to 1 M RbNO, solution used as an external standard.
311
trends in analytical chemistry, vol. 13, no. 8, 1994
Table 2 Observed and deconvoluted Compound CaMgSi,O, (diopside) Mg,SiO, (forsterite) MgSiO, (clinoenstatite)
Ca,SiO, (larnite)
?&‘:,”
170 shifts and quadrupole 6947 ,50
6c5
couplings in silicates
s(g),” 7T
l;d”,” 4r
C, (MHz)
v
&(I
+ 772/3)“2(MHz)
75.1
69.2
86
-11
-17
2.83kO.05
0.13,O.lO
2.8kO.2
54.0
48.5
64
-10
-16
2.74kO.05
0.00*0.10
2.7kO.2
43.3
28.6
69
-26
-40
4.39kO.05
0.36_tO.O5
4.5fO.l
57.1
49.0
72
-15
-23
3.3kO.i
54.8
49.0
64
-9
-15
2.8kO.2
37.5
30.8
49
-11
-18
3.OkO.2
45.5
39.3
57
-11
-18
2.950.2
44.1
34.5
61
-17
-26
3.6rtO.i
42.0
32.3
59
-17
-27
3.6kO.i
39.0
26.3
62
-23
-36
4.2,O.l
36.8
18.0
70
-33
-52
5.1 kO.1
34.7
15.0
70
-35
-55
5.2kO.l
123.3
117.3
134
-11
-17
2.9kO.2
118.5
113.3
128
-9
-15
2.7kO.2
113.4
108.8
122
-9
-13
2.5iO.2
112.0
106.3
122
-10
-16
K,Si,O,
64
(glass) KMg, ,Si,O,
24
2.8kO.2 5.7kO.4
0.2+0.1
5.7+0.4
0.2+0.1
5.8
0.0
6 6
SiO, (cristobalite)
-16.6
52
-69
Measured isotropic shifts S,,, as a function of applied field strength, and decomposition of the observed shift into a chemical shift S,, and second-order quadrupolar contribution to the shift 82). Based on an analysis of the multidimensional DAS NMR lineshapes, the quadrupolar parameters C, = (e2qQ)lh and 7, or SC:)and Eq. (2) can be combined so as to extract the product C,(l + 77*/3)“*. All shifts are referenced to Hi’0 used as an external standard.
highly variable from site-to-site, high-to-low-field ordering of the lines may differ at different field strengths. The quadrupole coupling constants themselves are diagnostic of local electronic environment and are often more readily interpreted than are chemical shifts. In silicates, correlations have been established between the magnitude (e*qQ/h) and symmetry (7) of 170 quadrupole coupling constants and Si-0-Si bond angles (cy), so that, for example, e*qQlh a cos cu/(cos (Y- 1) and n= 1 fcos (Y [ 30,3 I]. Approximate values of (Ycan be derived from the field-dependence of the second order isotropic shift in DAS or DOR, because S$,Ti’,,is proportional to Po = ( 1 + q2/3) “2 e*qQlh. These important relationships have been combined with the diagnostic power of two dimensional “0 DAS spectroscopy so as to quantify disorder in network-modified silicate glasses. Using the isotropic shift dimension wI so as to ‘spread out’ differing chemical environments, the otherwise overlapping quadrupolar lineshapes are well enough separated so that the distribution of bond angles cr, and its dependence on network
modifiers can be probed straightforwardly [ 301. While the most probable bond angle is the same ( 143” us. 144”)) bond angle distributions for the bridging oxygens in K2Si409 are narrower (halfwidth 21”) than for pure SiO, glass (half-width 36”). The quadrupolar parameters associated with non-bridging oxygen sites appear unaffected when Mg replaces half of the K cations.
6. Conclusions The promise of DAS and DOR is that they will provide the same wealth of high-resolution data for solid-state inorganic chemistry that CPMAS has yielded in 20 years of applications to solid-state organic and polymer chemistry. Already there are a number of applications to zeolite and zeolite-like samples [ 1,32-381. As improved spectroscopic techniques combine with wider distribution of the specialized probes needed for these experiments, the number of applications are certain to grow rap-
312
idly as entirely new branches of chemistry become amenable to straightforward spectroscopic analyses.
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Rev., 69 (1946) 37. 12 I F. Bloch, W. W. Hansen and M. Packard, Phys. Rev., 69 (1946) 127. [ 3) A. Bax, N. M. Szeverenyi and G. E. Maciel, J. Mngn. Reson., 52 (1983) 147. (41 E. R. Andrew, A. Bradbury and R.G. Eades, Nature, 182 ( 1958) 1659. [5 1 I. J. Lowe, Phys. Rev. Lett., 2 ( 1959) 285. [ 61 A. Samoson, E. Lippmaa and A. Pines, Mol. Phys., 65 (1988) 1013. 171 A. Llor and J. Virlet, Chem. Phys. Lett., 152 ( 1988) 248. [ 8 1 E. R. Andrews, Int. Rev. Phys. Chem., 1 ( 1980) 195. [ 91 N. Bloembergen and T. J. Rowland, Acta. Metall., 1 (1953) 731. [ IO] J. Herzfeld and A. E. Berger, J. Chem. Phys., 73 (1980) 6021. [ 111 E. Kundla, A. Samoson and E. Lippmaa, Chem. Phys. Lett., 83 ( 1981) 229. [ 121 A. Samoson, E. Kundla and E. Lippmaa, J. Map. Reson., 49 ( 1982) 350. [ 131 M. M. Meadows, K. A. Smith, R. A. Kinsey, T. M. Rothgeb, R. P. Skarjune and E. Oldfield, Proc. Natl. Acad. Sci. USA, 79 ( 1982) 135 1. [ 141 M. M. Maricq and J. S. Waugh, J. Chem. Phys., 70 ( 1979) 3300. [ 151 J. L. Ackerman, R. Eckman and A. Pines, Chem. Phys. 42 ( 1979) 423. [ 161 R. Eckman, M. Alla and A. Pines, J. Magn. Resort., 41 ( 1980) 440. [ 171 S. Ganapathy, S. Schramm and E. Oldfield, J. Chem. Phys., 77 ( 1982) 4360. [ 181 K. T. Mueller, B. Q. Sun, G. C. Chingas, J. W. Zwanziger, T. Terao and A. Pines, J. Magn. Reson., 86 ( 1990) 470. [ 191 A. Samoson, B. Q. Sun and A. Pines, in D. M. S. Bagguley (Editor), Pulsed Magnetic Resonance: NMR, ESR, and Optics, Clarendon Press, Oxford, 1992, Ch. 3. [ 201 J. H. Baltisberger, S. L. Gann, E. W. Wooten, T. H. Chang, K. T. Mueller and A. Pines, J. Am. Chem. Sot., 114 ( 1992) 7489. [ 211 A. Samoson and A. Pines, Rev. Sci. Instrum., 60 ( 1989) 3239. [ 221 S. L. Gann, J. H. Baltisberger and A. Pines, Chem. Phys. Lett., 210 (1993) 405.
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vol. 13, no. 8, 1994
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