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Dephasing of spin echoes by multiple heteronuclear dipolar interactions in rotational echo double resonance NMR experiments
Solid State Nuclear Magnetic Resonance 15 Ž1999. 139–152 www.elsevier.nlrlocatersolmag
Dephasing of spin echoes by multiple heteronuclear dipolar interactions in rotational echo double resonance NMR experiments Marko Bertmer, Hellmut Eckert
)
Institut fur Wilhelms-UniÕersitat Schlossplatz 7, Munster D-48149, Germany ¨ Physikalische Chemie, Westfalische ¨ ¨ Munster, ¨ ¨
Abstract The application of rotational echo double resonance ŽREDOR. nuclear magnetic resonance ŽNMR. for accurate distance measurements has thus far been largely restricted to isolated heteronuclear two-spin systems. In the present paper, the informational content of REDOR curves is explored for systems characterized by multi-spin interactions. To this end, numerical REDOR simulations are presented for cases in which single observe spins S are dipolarly coupled to groups of spins I in distinct geometries. To develop the utility of REDOR for characterizing dipolar couplings in unknown andror ill-defined geometries, the validity ranges and systematic errors of certain analytical approximations are studied. In the limit of short dipolar evolution times where 0 F DS0S F 0.2 to 0.3, the REDOR difference signal intensity increases approximately proportional to the square of the dipolar evolution time. Here, the curvature depends simply on the second moment M2 characterizing the overall strength of the heterodipolar coupling, irrespective of specific molecular geometries. Fitting experimental REDOR data in this manner produces slight systematic underestimates of M2 . However, these errors tend to be counterbalanced by additional systematic errors made by neglecting weak couplings to more remote spins and distribution effects caused by disorder. Based on these findings, the results suggest a convenient method of obtaining site-resolved second moment information in disordered materials. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Dephasing; Multiple dipolar interaction; NMR
1. Introduction Ever since its invention by Gullion and Schaefer w1x in 1988, rotational echo double resonance ŽREDOR. nuclear magnetic resonance ŽNMR. spectroscopy has enjoyed widespread popularity for a multitude of applications in chemistry, biology, and materials science. This technique allows the quantita)
Corresponding author. Tel.: q49-251-832-9161; fax: q49251-832-9159; e-mail: [email protected]
tive determination of dipole–dipole coupling constants in high-resolution solid-state NMR experiments carried out under magic-angle spinning conditions. In essence, the pulse sequence constitutes a rotor-synchronized spin echo experiment carried out on the observe-spin species S, while p-pulses applied to the I nuclei during the rotor period reintroduce the heteronuclear dipolar coupling. The additional dephasing thereby produced, i.e., the normalized REDOR difference signal D SrS0 , is measured as a function of the dipolar evolution time NTr ,
0926-2040r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 6 - 2 0 4 0 Ž 9 9 . 0 0 0 5 0 - 8
M. Bertmer, H. Eckert r Solid State Nuclear Magnetic Resonance 15 (1999) 139–152
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the number of rotor cycles times the rotor period. For isolated I–S pairs Žspin quantum number 1r2., such ‘‘REDOR curves’’ possess universal shapes, from which the dipole–dipole coupling constants can be calculated. The relation:
D IS s
g Ig S " m0 2p r 3IS 4p
,
Ž 1.
then provides a direct connection to internuclear distances r IS . Alternatively to numerical calculations of this curve, analytic solutions w2x and REDOR transform algorithms have been presented for the two-spin case w3,4x. To date, the most successful applications of REDOR have included 13 C– 15 N distance measurements in singly or doubly labeled biological solids w5–7x. Extensions to inorganic solids Žincluding quadrupolar nuclei. have been less common and non-quantitative in general w8–15x, presumably because the condition of two truly isolated spins is not a very realistic assumption in most systems. The first theoretical approach to calculate REDOR dephasing by more than one spin was presented by Naito et al. w16x who considered additional coupling of S to a second, more remote spin I 2 . Multi-spin heterodipolar interactions among spin-1r2 nuclei in special geometries were also treated recently by Goetz and Schaefer w17x and Fyfe et al. w18x. The goal of the present study is to develop a general strategy for extracting dipolar coupling information in disordered systems such as crystalline solid solutions or glasses. Such systems are generally characterized by multiple-spin interactions, and the precise bonding geometries are frequently unknown andror ill-defined. To achieve this objective, we first extend the approach of Naito et al. w16x to the general case of SI n systems, under specific consideration of respective geometries and multiplicities n. On the basis of the results obtained, we discuss the general informational content of such curves by assessing the systematic errors in dipolar second moments extracted from them. Additional simulations are presented for studying the effects of dipolar couplings to more remote spins and of the influence of distributions in spin geometries on REDOR data.
2. Theory 2.1. Numerical simulations Fig. 1 illustrates the REDOR experiment most commonly used by experimentalists. This is a spinecho sequence on the S-channel, the total evolution time is the product of rotor period Tr and the number of rotor cycles N, N being 2, 4, 6 . . . On the I-channel, 1808 pulses are applied at half and full rotor periods — except at that time when the 1808 pulse is applied to the S spins. As assumed by others w11–13x, our calculations neglect the effect of the pulse sequence on the magnitude of the homodipolar I–I interactions. This assumption is generally admissible if there is no chemical shift difference between the I spins. Furthermore, calculations by Goetz and Schaefer have shown that even for the case of nonequivalent I spins, the effect of I–I dipolar couplings on REDOR behavior can be neglected if a sequence is used that minimizes the number of I pulses during the rotor period and if rotational resonance conditions are carefully avoided. Even if the pulse sequence of Fig. 1 is used, systematic errors are found mostly in the oscillatory region of long evolution times, whereas the short-time decay behavior is affected very little w12x. Finally, the effects of homodipolar S–S interactions can be neglected, because these are generally averaged out by fast MAS.
Fig. 1. REDOR pulse sequence with N s 4 rotor cycles. Two separate experiments are conducted. In the first experiment, an S spin echo is recorded in the absence of I spin irradiation ŽIq.. In the second experiment, the S spin echo is recorded in the presence of I spin irradiation ŽIy..
M. Bertmer, H. Eckert r Solid State Nuclear Magnetic Resonance 15 (1999) 139–152
141
Fig. 2. Molecular axis system with three dipolar vectors, showing the definitions of the angles z , d and ´ .
The REDOR calculation procedure outlined below begins with the definition of a molecular axis system. For a three-spin system SI 1 I 2 , the x-axis of this coordinate system is chosen parallel to vector r SI 1 whereas vector r SI 2 lies in the xy plane, the angle between both denoted as z . For higher-spin systems, the orientations of all additional dipolar vectors need to be specified by two additional angles Ž d , ´ ., since they are oriented in three-dimensional space. Here, ´ is the angle between the z-axis and the dipolar vector and d is the angle between the x-axis and the projection of the dipolar vector onto the xy plane, as can be seen in Fig. 2. Thus, the coordinates are as follows:
™r1 s <™r1 < ™rn s <™ri <
cos z 1 ™ ™ < < 0 ; r 2 s r 2 sin z ; 0 0
0
ž/
cos d i sin ´ i sin d i sin ´ i . cos ´ i
0
With the use of Euler rotation matrices, the molecular axis system is transformed to the spinner axis using RŽ a , b ,g . and to the laboratory coordinate system using RŽ vr t, um , 0.. The dipolar Hamiltonian for an I–S spin pair is: HD i s " v D iŽ u . Iˆz Sˆz , with
v D iŽ u . s 2p Di Ž 3 cos 2 u i y 1 . , Di s
g Ig S " m0 2p r i3 4p
.
Ž 3.
Given the assumptions specified above, the total I–S dipolar Hamiltonian of an SI n multi-spin system can be written as the sum of the individual two-spin Hamiltonians:
Ž 2. HD s HD 1Ž u 1 . q HD 2Ž u 2 . q . . . qHD nŽ un . .
Ž 4.
M. Bertmer, H. Eckert r Solid State Nuclear Magnetic Resonance 15 (1999) 139–152
142
Additionally, with every new I spin, the number of dipolar frequencies doubles:
2.2. Analytical approximations
SI: v i s "v D 1 ;
As shown below, in many cases, the REDOR dephasing behavior at short evolution times is of special interest. Previous calculations have shown that the initial part of the REDOR curve can be approximated analytically, thereby avoiding the full numeric integration procedure. This analytical approximation is based on the usual expansion:
i s 1,2,
SI 2 : v i s " Ž v D 1 " v D 2 . ;
i s 1 to 4,
SI n : v i s " Ž v D 1 " v D 2 " . . . " v D n . ;
i s 1 to 2 n .
After coordinate transformation, the z-component of each dipolar vector r i is given by cos u i in the Hamiltonian of Eq. Ž3.. With this, the term 3 cos 2 u i y 1 has to be calculated. Following the procedure of Pan et al. w19x, the average dipolar transition frequency can then be calculated according to:
vis
1 Tr
Trr2
H0
vi d t y
Tr
HT r2v d t i
.
Ž 5.
cos x s 1 y
DS
As an example, for an SI 2 system, the result is:
S0
=sin 2 b sin g q sin 2 Ž a y z . sin b cos g . .
Ž 6. Using spin density matrix formalism w20x, the REDOR difference signal can be calculated for the sequence of Fig. 1. After one rotor period following the 908 ŽS. pulse, the spin density matrix has evolved to r ŽTr . s Sˆx cos v iTr q 2 Sˆy Iˆz sin v iTr , where x, y and z are the rotating frame coordinates. Due to the cyclic nature of the REDOR pulse sequence w21x, the result after N rotor cycles is given by:
Ž 7.
Observation in x-direction leads to: SŽ a , b ,g . ; cosŽ v iNTr .. This result is only valid for one specific crystallite orientation characterized by a set of Euler angles a , b and g . To obtain the REDOR curve of a powdered crystal or an amorphous system, one has to integrate over all Euler angles: 2n
1 8p
2
2p
ÝH
is1 0
p
2p
H0 H0
cos Ž v iNTr . sin b d a d b dg ,
Ž 8. n being the number of I spins coupled to S.
2
24
y...
Ž 9.
15
2 Ž NTr . D12 q D 22 q . . . qDn2 .
Ž 10 .
Clearly, Ý D 12 is proportional to the heterodipolar second moment. In quantitative terms, we find:
qsin 2 a sin b cos g . q D 2 Ž cos 2 Ž a y z .
Sf s
16 s
r Ž NTr . s Sˆx cos v iNTr q 2 Sˆy Iˆz sin v iNTr .
x4 q
Truncation after the second term in the limit x s v iNTr < 1, results in a first-order approximation:
r
v i s 2'2 NTr D 1 Ž cos 2 a sin 2 b sin g
x2
DS
4 s 3p
S0
2
2 Ž NTr . M2 .
Ž 11 .
Thus, within this short-term limit, the REDOR curve becomes geometry-independent, and the strength of the internuclear dipole coupling can be expressed easily in terms of a second moment. Such an analysis is particularly favorable for structural studies of systems in which there is no ad hoc information on spin geometry. Likewise, a geometry-independent assessment of dipolar coupling strength is advantageous for the structural analysis of disordered systems such as glasses, where the spin geometries may be ill-defined and subject to local distribution effects. Unfortunately, the range of experimental data points available is often too small for reliable analysis of the initial REDOR decay curves. Extension of the data range to larger dipolar evolution times, however, requires the next higher approximation, keeping the term in fourth power of NTr . For a two-spin system, this second-order approximation results in the expression: DS
16 s
S0
15
2 Ž NTr . D 2 y
128 315
4 Ž NTr . D 4 .
Ž 12 .
Fig. 3 compares the two approximations with the universal REDOR curve for a two-spin system. It is
M. Bertmer, H. Eckert r Solid State Nuclear Magnetic Resonance 15 (1999) 139–152
143
Fig. 3. Comparison of the two-spin REDOR curve Ž — — . with the first-order Ž- - -. and second-order approximation ŽP P P ..
shown that Eq. Ž10. is acceptable up to about D SrS0 s 30%, whereas Eq. Ž12. remains valid up to about D SrS0 s 60%. Fig. 4 plots the error in M2 and D that is obtained by fitting the calculated REDOR curve to Eqs. Ž11. and Ž12., respectively, as a function of the data range used in D SrS0 . Clearly, the true second moment is underestimated in the first-
order, and overestimated in the second-order approximation. Taking into account that, in general, M2 values are acceptable within an experimental range of 10% error, it is seen that the second order approximation can give satisfactory results for NDTr values smaller than unity. As discussed in more detail below, however, second-order approximations for
Fig. 4. Relative errors of M2 Ž — — . and D Ž- - -. by fitting the two-spin REDOR curve to the first-order and second-order approximation vs. D SrS0 , respectively.
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M. Bertmer, H. Eckert r Solid State Nuclear Magnetic Resonance 15 (1999) 139–152
higher-spin systems contain angle-geometry dependent terms and are therefore generally less useful in systems with unknown geometries.
3. Experimental The simulation program was written in Pascal language running on an IBM PC, Pentium 100 MHz. Numerical simulations were done using the Simpson rule with integration steps of pr32, whereas pr16 were sufficient for the general curve behavior but leading to less oscillation steps in the long evolution time limit. From these simulated curves, approximate M2 values were extracted by fitting the dephasing curves calculated for short evolution times to Eq. Ž11.. Similar procedures were used to calculate approximate D values for various types of higher spin geometries in the limit of the second-order approximation.
4. Results In the following, selected simulated REDOR curves and analytical approximations are presented
for various multi-spin systems. As for the SI 2 system, there is only one variable angle, we present the differences of the REDOR curve as a function of this angle. Dephasing of single spins S by dipolar couplings to more than two I spins is considered only for selected point symmetries that commonly occur in inorganic compounds.
4.1. SI2 systems Fig. 5 illustrates the results of REDOR calculations in an SI 2 system with equal distances r SI 1 and r SI 2 as a function of angle z . Since the angle dependence is symmetrical to 908, the same curves are obtained for complementary angles, i.e., for z s 608 and z s 1208. With z equal to 1808, the maximum dephasing is only 50%. This result is physically plausible because in a symmetric linear spin system, there is a 50% probability that the I spins have opposing z-components, causing cancellation of the local fields at the S site. The most important result is that the curves are more or less geometry-independent in the limit of short evolution times, as predicted by Eq. Ž10..
Fig. 5. REDOR curves of the SI 2 system with varying angle z and same dipolar coupling.
M. Bertmer, H. Eckert r Solid State Nuclear Magnetic Resonance 15 (1999) 139–152
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Fig. 6. Error plot of M2 and D vs. D SrS0 for the SI 2 system with an angle z of 109.58 Žtetrahedral angle., in the limits of first-order and second-order approximations.
For the SI 2 system, the corresponding secondorder approximation becomes: DS
16 s
S0
15
2 Ž NTr . D12 q D 22
128 y 315
range, 0 F Ž D SrS0 . F 0.4; however, knowledge of the order and geometry of the spin system is required in this case.
4 Ž NTr . D14 q D 24
3 q D 12 D 22 Ž 9 q 4 cos 2 z q 3 cos 4z . , 8
Ž 13 .
including, unfortunately, also a dependence on spin geometry. Fig. 6 plots the relative systematic error in the second moment Ž D M 2 rM 2 . s w M 2 Žsim. y M 2 Žexact.rw M2 Žexact.x extracted from Eq. Ž11. by extending the quadratic approximation to increasing values of D SrS0 . Again, applying the first-order approximation to the data results in underestimates of M2 . Concerning an error assessment of the second-order approximation ŽEq. Ž13.., special geometries have to be considered. Assuming z s 109.58 Žtetrahedral angle. and D 1 s D 2 Žequal distances. results in: DS S0
2
4
s 2.13 Ž NTr . D 2 y 1.81 Ž NTr . D 4 .
Ž 14 .
The second curve in Fig. 6 plots the systematic error, Ž D DrD . s w DŽsim. y DŽexact.xrw DŽexact.x for this special geometry in the limit of the second-order approximation. Note that this approximation can give excellent estimates of D within a fairly wide data
Fig. 7. SI 3 system with trigonal-plane and trigonal-pyramid symmetry. Ža. Simulated REDOR curves; trigonal plane Ž — — ., trigonal-pyramid Ž- - -.. Žb. Error plots for these symmetries. Error in M2 : trigonal plane Ž — — ., trigonal pyramid Ž – P – . Žfirst-order approx..; error in D: trigonal plane ŽP P P., trigonal pyramid Ž- - -. Žsecond-order approx...
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M. Bertmer, H. Eckert r Solid State Nuclear Magnetic Resonance 15 (1999) 139–152
Fig. 8. REDOR curves of higher spin systems. SI 4 : tetrahedron Ž — — ., square plane Ž- - -.; SI 6 : octahedron ŽP P P ..
4.2. Higher-spin systems High-symmetry configurations for the SI 3 spin system are trigonal-plane symmetry and trigonal pyramid. Corresponding angles for simulation are:
trigonal-plane symmetry: z s 1208; ´ s 908; d s y1208; trigonal-pyramid: z s 988, ´ s 128, d s 1318. In Fig. 7a, the simulated REDOR curves are presented. While the oscillatory parts of the two curves are quite different, the behavior in the limit of
Fig. 9. Corresponding error plots for the higher spin systems. Error in M2 : tetrahedron Ž — — ., square plane Ž – P – ., octahedron Ž- - -. Žfirst-order approx... Error in D: tetrahedron ŽP P P ., square plane Ž- - -., octahedron Ž – P – . Žsecond-order approx...
M. Bertmer, H. Eckert r Solid State Nuclear Magnetic Resonance 15 (1999) 139–152
147
Fig. 10. Simulated REDOR curves for a distorted SI 4 tetrahedron with one large coupling Ž D1 s 1 kHz. and three smaller couplings D 2 , D 3 , D4 , the inset showing the initial part of the REDOR curve magnified. Two-spin system with D s 1 kHz Ž — — ., tetrahedron with D 2 , D 3 , D4 s 100 Hz ŽP P P ., 200 Hz Ž- - -. and 300 Hz Ž – P – ..
short evolution times is very similar as expected. Fig. 7b shows the corresponding error plot in M2 and D in the limit of both approximations. Finally, Figs. 8 and 9 provide the analogous simulations and error plots for tetrahedral, square planar, and octahedral environments. The angles for these geometries are, respectively, tetrahedron: z s 109.58, ´ 1 s 368, d 1 s y1258,
´ 2 s 1448, d 2 s y1258, square plane: z s 908, ´ 1 s 908, d 1 s 1808,
´ 2 s 908, d 2 s y908, octahedron: z s 908, ´ 1 s 908, d 1 s 1808,
´ 2 s 908, d 2 s y908, ´ 3 s 08, d 3 s 08, ´4 s 1808, d4 s 08. For all these geometries, the first-order approximation systematically underestimates M2 and D; however, it can give reasonable second moment estimates at sufficiently short evolution times, where Ž D SrS0 . F 0.2 to 0.3. 4.3. Effect of remote spins on REDOR dephasing To assess the effect of weaker dipole–dipole couplings to more remote spins, the simulations carried
Fig. 11. Relative error in D, by fitting the tetrahedra with the first-order Ža. and second-order approximation Žb. of the two-spin system. Pure two spin system Ž — — ., tetrahedron with D 2 , D 3 , D4 s100 Hz ŽP P P., 200 Hz Ž- - -. and 300 Hz Ž – P – ..
148
M. Bertmer, H. Eckert r Solid State Nuclear Magnetic Resonance 15 (1999) 139–152
M. Bertmer, H. Eckert r Solid State Nuclear Magnetic Resonance 15 (1999) 139–152
149
Fig. 13. Relative error in D using the first-order approximation for the simulated REDOR curves shown in Fig. 12c using the same symbols.
out for tetrahedral geometry were altered by assuming a dominant two-spin coupling constant D 1 and three identical smaller coupling constants D 2 –D4 , the magnitude of which was systematically altered. The resulting REDOR curves are summarized in Fig. 10. As expected, the effect of the more remote spins is to increase M2 as measurable from the REDOR dephasing curve in the limit of short evolution times. Furthermore, the oscillatory behavior in the long evolution time limit is damped significantly. Fig. 11 plots the error in D Žand hence internuclear distance. that would be obtained if the weaker dipole–dipole couplings D 2 –D4 to the three more remote spins were neglected, i.e., if the REDOR curve would be analyzed under the assumption of an isolated two-spin system. Note that the error arising from this approximation tends to compensate the negative error produced by the parabola fit; this fortuitous benefit is lost in the second-order approximation. 4.4. Effect of distance distributions in systems with multi-spin interactions In disordered systems, the internuclear distances usually do not possess singular values but rather are
subject to distribution effects. As a consequence, the experimental REDOR curve is often a weighted superposition of individual curves according to a continuous function. To simulate the effect of distance distributions on REDOR curves, calculations were carried out for a tetrahedral geometry. Three Gaussian distributions of distances with different widths were considered and transposed into corresponding distributions of D values as shown in Fig. 12a. Due to the non-linear relationship between r and D, symmetrical distribution functions in r produce asymmetric distribution functions in D. Simulating the resulting REDOR curves then implies a corresponding compression or elongation of the abscissa in the presentation of the universal curve D SrS0 vs. NTr Žsee Fig. 12b.. The resulting REDOR curves obtained by superposition are shown in Fig. 12c. In particular, the initial slope is slightly increased with increasing range of D values compared to that obtained for the uniform distance. As a result, the average dipolar coupling strength is slightly overestimated, while the distances are underestimated. Fig. 13 shows the error in D obtained using the various distance distributions assumed under the first-order approximation; note again the compensatory effect on the error.
˚ ŽP P P ., s s 0.089 A˚ Ž — — . and s s 0.16 A˚ Ž — — .. Ža. Fig. 12. Simulated REDOR curves for Gaussian distributions in r of s s 0.032 A Distribution in D resulting from a Gaussian distribution of distance Žinset.. Žb. Selected REDOR curves showing the compression or elongation of the abscissa. Žc. REDOR curves obtained by superposition of individual curves with different dipolar couplings according to ˚ ŽP P P ., s s 0.089 A˚ Ž- - -. and s s 0.16 A˚ Ž — — .. their ratios in the distribution: Exact tetrahedron Ž – P – ., distribution with s s 0.032 A
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M. Bertmer, H. Eckert r Solid State Nuclear Magnetic Resonance 15 (1999) 139–152
5. Discussion and conclusions 5.1. Dipolar oscillations as probes for spin geometries In the discussion of the features of the REDOR simulations obtained within this study, it is most practical to differentiate between two time regimes: the oscillatory part obtained in the limit of long evolution times and the initial decay measured in the limit NDTr - 1. Our results indicate that the longterm oscillatory part may be a valuable tool for identifying specific spin geometries. This is illustrated by Fig. 14, summarizing the effects of coupling partners and specific geometry considered. This feature may be particularly useful for studies of truly isolated spin clusters as present in some molecular compounds, coordination complexes or dilute species within matrices. In contrast, for infinite solid lattices or inorganic networks, it is generally not possible to simply neglect the additional dipolar coupling to more distant spins. As illustrated in Fig. 10, these additional dipolar couplings lead to very effective damping of the oscillatory behavior so that only a smooth curve is observed. Details depend on the specific spin geometry considered, but the case shown in Fig. 10 represents still one of the most favorable
situations where dipolar structure will be observable. Even in this favorable case, the dipolar oscillation in the REDOR curve vanishes if the ratio D 1rD 2 drops below a value of about three. Denoting D 1 as the dipolar coupling constant with the closest neighbors and D 2 as the one with the next nearest heteronuclear neighbors, D 1rD 2 ratios of 5.2, 7.0 and 15.9 are calculated for the typical three-dimensional lattices, NaCl, ZnS Žzinc blend. and SiO 2 Žcristobalite., respectively. Since multiple primary and secondary interactions are present in these lattices, one can expect that the oscillations will be damped at least in the first two cases. Observation of oscillatory behavior may further be hindered by low signal-to-noise ratio at long evolution times due to spin–spin relaxation of the observed spin coherence. Indeed, published REDOR data on aluminum phosphate networks show no indications of dipolar oscillations w22x. Finally, in glasses, the dipolar oscillations are further damped owing to the presence of distance distribution effects. 5.2. REDOR dephasing in the limit of short eÕolution times The results of the present study illustrate that the REDOR dephasing curve in the limit of short evolu-
Fig. 14. Dephasing behavior in REDOR experiments for SI n multi-spin systems: Ž — — .: n s 1, ŽP P P .: n s 2, Ž — — .: n s 3, trigonal plane, Ž – P – .: n s 4, tetrahedron, Ž – ( – .: n s 6, octahedron.
M. Bertmer, H. Eckert r Solid State Nuclear Magnetic Resonance 15 (1999) 139–152
tion times is independent of the specific spin geometry involved and can be approximated by a parabola, provided the effects of distance distributions are not overly severe. The curvature of this parabola is described by a single parameter related to the dipolar second moment powder average, which is computed from the lattice sum over the corresponding ry6 ij terms of all spin pairs involved. As such, a mean square average of the dipole–dipole coupling constant can be characterized without explicit knowledge of the spin geometry present. This feature is particularly useful when dealing with disordered systems and glasses. In such systems, an approximate dipolar coupling constant Žor second moment. can be obtained by applying the first-order approximation to the initial REDOR dephasing curve. Such experimental values can then be compared with van Vleck M2 calculations for specific bonding scenarios, where the range of distances is usually limited to the nearest neighbors. The whole procedure derives an additional benefit from the fortuitous error compensation effect discussed above. Regardless of the specific geometry considered, our simulations show that fitting the initial part of the REDOR curve to a parabola Žfirst-order approximation. tends to underestimate the dipole–dipole coupling constant in question compared to the accurate simulation. On the other hand, the experimental curves include the additional second moment contributions from magnetic dipole–dipole interactions with more remote magnetic neighbors, whereas the van Vleck calculations for specific bonding scenarios in glasses do not if only the closest neighbours are considered. Thus, if we compare an experimental second moment from a REDOR parabola fit with a van Vleck value for a simple nearest-neighbor bonding scenario Žneglecting longer-range dipolar coupling., the systematic errors have opposite sign and thus partially cancel each other. Based on this favorable situation, REDOR dephasing curves measured in the short evolution time limit are expected to produce valid experimental criteria for hypothetical scenarios involving the short- and intermediate-range order in glasses. We note that Gullion and Pennington have recently presented an elegant alternative approach for disentangling the multi-spin problem in REDOR experiments. In their method, pulses corresponding to small flip angles are applied to the non-observed
151
spins in the middle of each rotor period Žtheta REDOR. w23x. Since in this procedure only a small fraction of spins creating the dipolar field is flipped, the multi-spin problem is reduced to a two-spin problem by mere probability. The disadvantage of theta REDOR, however, is an attenuation of the overall effect, producing larger errors in systems with weak couplings. Furthermore, a quantitative analysis of the experimental results requires knowledge about the spin multiplicity. The method delineated in the present contribution does not suffer from these drawbacks. Validation of this technique on crystalline model compounds and applications to various glass systems are currently under intense study in our laboratory. Acknowledgements This research was funded by the Wissenschaftsministerium Nordrhein-Westfalen. Valuable discussions with Dr. Leo van Wullen are grate¨ fully acknowledged. References w1x T. Gullion, J. Schaefer, J. Magn. Reson. 81 Ž1989. 196–200. w2x K.T. Mueller, J. Magn. Reson. A 113 Ž1995. 81–93. w3x K.T. Mueller, T.P. Jarvie, D.J. Aurentz, B.W. Roberts, Chem. Phys. Lett. 242 Ž1995. 535–542. w4x K.T. Mueller, T.P. Jarvie, D.J. Aurentz, B.W. Roberts, Chem. Phys. Lett. 254 Ž1996. 281–282. w5x A.M. Christensen, J. Schaefer, Biochemistry 32 Ž1993. 2868–2873. w6x W. Hing, N. Tjandra, P.F. Cottam, J. Schaefer, Biochemistry 33 Ž1994. 8651–8661. w7x Y. Pan, N.S. Shenouda, G.E. Wilson, J. Schaefer, J. Biol. Chem. 268 Ž1993. 18692–18695. w8x L.v. Wullen, L. Zuchner, W. Muller-Warmuth, H. Eckert, ¨ ¨ ¨ Solid State Nucl. Magn. Reson. 6 Ž1996. 203–212. w9x C.A. Fyfe, K.T. Mueller, H. Grondey, K.C. Wong-Moon, J. Phys. Chem. 97 Ž1993. 13484–13495. w10x S.M. Holl, T. Kowalewski, J. Schaefer, Solid State Nucl. Magn. Reson. 6 Ž1996. 39–46. w11x L.V. Wullen, B. Gee, L. Zuchner, M. Bertmer, H. Eckert, ¨ ¨ Ber. Bunsenges. Phys. Chem. 100 Ž1996. 1539–1549. w12x C. Rong, K.C. Wong-Moon, H. Li, P. Hrma, H. Cho, J. Noncryst. Solids 223 Ž1998. 32. w13x K. Herzog, B. Thomas, D. Sprenger, C. Jager, J. Noncryst. ¨ Solids 190 Ž1995. 296. w14x C.P. Grey, B.S. Arun Kumar, J. Am. Chem. Soc. 117 Ž1995. 9071.
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w15x A.L. Blumenfeld, D.J. Coster, J.J. Fripiat, Chem. Phys. Lett. 231 Ž1994. 491. w16x A. Naito, K. Nishimura, S. Tuzi, H. Saito, ˆ Chem. Phys. Lett. 229 Ž1994. 506–511. w17x J.M. Goetz, J. Schaefer, J. Magn. Reson. 127 Ž1997. 147– 154. w18x C.A. Fyfe, A.R. Lewis, J.M. Chezeau, H. Grondey, J. Am. Chem. Soc. 119 Ž1997. 12210–12222.
w19x Y. Pan, T. Gullion, J. Schaefer, J. Magn. Reson. 90 Ž1990. 330–340. w20x J. Shriver, Concepts, Magn. Reson. 4 Ž1992. 1–33. w21x U. Haeberlen, J.S. Waugh, Phys. Rev. 175 Ž1968. 453–467. w22x C.A. Fyfe, K.T. Mueller, H. Grondey, K.C. Wong-Moon, J. Phys. Chem. 97 Ž1993. 13484–13495. w23x T. Gullion, C.H. Pennington, Chem. Phys. Lett. 290 Ž1998. 88–93.
Dephasing of spin echoes by multiple heteronuclear dipolar interactions in rotational echo double resonance NMR experiments Marko Bertmer, Hellmut Eckert
)
Institut fur Wilhelms-UniÕersitat Schlossplatz 7, Munster D-48149, Germany ¨ Physikalische Chemie, Westfalische ¨ ¨ Munster, ¨ ¨
Abstract The application of rotational echo double resonance ŽREDOR. nuclear magnetic resonance ŽNMR. for accurate distance measurements has thus far been largely restricted to isolated heteronuclear two-spin systems. In the present paper, the informational content of REDOR curves is explored for systems characterized by multi-spin interactions. To this end, numerical REDOR simulations are presented for cases in which single observe spins S are dipolarly coupled to groups of spins I in distinct geometries. To develop the utility of REDOR for characterizing dipolar couplings in unknown andror ill-defined geometries, the validity ranges and systematic errors of certain analytical approximations are studied. In the limit of short dipolar evolution times where 0 F DS0S F 0.2 to 0.3, the REDOR difference signal intensity increases approximately proportional to the square of the dipolar evolution time. Here, the curvature depends simply on the second moment M2 characterizing the overall strength of the heterodipolar coupling, irrespective of specific molecular geometries. Fitting experimental REDOR data in this manner produces slight systematic underestimates of M2 . However, these errors tend to be counterbalanced by additional systematic errors made by neglecting weak couplings to more remote spins and distribution effects caused by disorder. Based on these findings, the results suggest a convenient method of obtaining site-resolved second moment information in disordered materials. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Dephasing; Multiple dipolar interaction; NMR
1. Introduction Ever since its invention by Gullion and Schaefer w1x in 1988, rotational echo double resonance ŽREDOR. nuclear magnetic resonance ŽNMR. spectroscopy has enjoyed widespread popularity for a multitude of applications in chemistry, biology, and materials science. This technique allows the quantita)
Corresponding author. Tel.: q49-251-832-9161; fax: q49251-832-9159; e-mail: [email protected]
tive determination of dipole–dipole coupling constants in high-resolution solid-state NMR experiments carried out under magic-angle spinning conditions. In essence, the pulse sequence constitutes a rotor-synchronized spin echo experiment carried out on the observe-spin species S, while p-pulses applied to the I nuclei during the rotor period reintroduce the heteronuclear dipolar coupling. The additional dephasing thereby produced, i.e., the normalized REDOR difference signal D SrS0 , is measured as a function of the dipolar evolution time NTr ,
0926-2040r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 6 - 2 0 4 0 Ž 9 9 . 0 0 0 5 0 - 8
M. Bertmer, H. Eckert r Solid State Nuclear Magnetic Resonance 15 (1999) 139–152
140
the number of rotor cycles times the rotor period. For isolated I–S pairs Žspin quantum number 1r2., such ‘‘REDOR curves’’ possess universal shapes, from which the dipole–dipole coupling constants can be calculated. The relation:
D IS s
g Ig S " m0 2p r 3IS 4p
,
Ž 1.
then provides a direct connection to internuclear distances r IS . Alternatively to numerical calculations of this curve, analytic solutions w2x and REDOR transform algorithms have been presented for the two-spin case w3,4x. To date, the most successful applications of REDOR have included 13 C– 15 N distance measurements in singly or doubly labeled biological solids w5–7x. Extensions to inorganic solids Žincluding quadrupolar nuclei. have been less common and non-quantitative in general w8–15x, presumably because the condition of two truly isolated spins is not a very realistic assumption in most systems. The first theoretical approach to calculate REDOR dephasing by more than one spin was presented by Naito et al. w16x who considered additional coupling of S to a second, more remote spin I 2 . Multi-spin heterodipolar interactions among spin-1r2 nuclei in special geometries were also treated recently by Goetz and Schaefer w17x and Fyfe et al. w18x. The goal of the present study is to develop a general strategy for extracting dipolar coupling information in disordered systems such as crystalline solid solutions or glasses. Such systems are generally characterized by multiple-spin interactions, and the precise bonding geometries are frequently unknown andror ill-defined. To achieve this objective, we first extend the approach of Naito et al. w16x to the general case of SI n systems, under specific consideration of respective geometries and multiplicities n. On the basis of the results obtained, we discuss the general informational content of such curves by assessing the systematic errors in dipolar second moments extracted from them. Additional simulations are presented for studying the effects of dipolar couplings to more remote spins and of the influence of distributions in spin geometries on REDOR data.
2. Theory 2.1. Numerical simulations Fig. 1 illustrates the REDOR experiment most commonly used by experimentalists. This is a spinecho sequence on the S-channel, the total evolution time is the product of rotor period Tr and the number of rotor cycles N, N being 2, 4, 6 . . . On the I-channel, 1808 pulses are applied at half and full rotor periods — except at that time when the 1808 pulse is applied to the S spins. As assumed by others w11–13x, our calculations neglect the effect of the pulse sequence on the magnitude of the homodipolar I–I interactions. This assumption is generally admissible if there is no chemical shift difference between the I spins. Furthermore, calculations by Goetz and Schaefer have shown that even for the case of nonequivalent I spins, the effect of I–I dipolar couplings on REDOR behavior can be neglected if a sequence is used that minimizes the number of I pulses during the rotor period and if rotational resonance conditions are carefully avoided. Even if the pulse sequence of Fig. 1 is used, systematic errors are found mostly in the oscillatory region of long evolution times, whereas the short-time decay behavior is affected very little w12x. Finally, the effects of homodipolar S–S interactions can be neglected, because these are generally averaged out by fast MAS.
Fig. 1. REDOR pulse sequence with N s 4 rotor cycles. Two separate experiments are conducted. In the first experiment, an S spin echo is recorded in the absence of I spin irradiation ŽIq.. In the second experiment, the S spin echo is recorded in the presence of I spin irradiation ŽIy..
M. Bertmer, H. Eckert r Solid State Nuclear Magnetic Resonance 15 (1999) 139–152
141
Fig. 2. Molecular axis system with three dipolar vectors, showing the definitions of the angles z , d and ´ .
The REDOR calculation procedure outlined below begins with the definition of a molecular axis system. For a three-spin system SI 1 I 2 , the x-axis of this coordinate system is chosen parallel to vector r SI 1 whereas vector r SI 2 lies in the xy plane, the angle between both denoted as z . For higher-spin systems, the orientations of all additional dipolar vectors need to be specified by two additional angles Ž d , ´ ., since they are oriented in three-dimensional space. Here, ´ is the angle between the z-axis and the dipolar vector and d is the angle between the x-axis and the projection of the dipolar vector onto the xy plane, as can be seen in Fig. 2. Thus, the coordinates are as follows:
™r1 s <™r1 < ™rn s <™ri <
cos z 1 ™ ™ < < 0 ; r 2 s r 2 sin z ; 0 0
0
ž/
cos d i sin ´ i sin d i sin ´ i . cos ´ i
0
With the use of Euler rotation matrices, the molecular axis system is transformed to the spinner axis using RŽ a , b ,g . and to the laboratory coordinate system using RŽ vr t, um , 0.. The dipolar Hamiltonian for an I–S spin pair is: HD i s " v D iŽ u . Iˆz Sˆz , with
v D iŽ u . s 2p Di Ž 3 cos 2 u i y 1 . , Di s
g Ig S " m0 2p r i3 4p
.
Ž 3.
Given the assumptions specified above, the total I–S dipolar Hamiltonian of an SI n multi-spin system can be written as the sum of the individual two-spin Hamiltonians:
Ž 2. HD s HD 1Ž u 1 . q HD 2Ž u 2 . q . . . qHD nŽ un . .
Ž 4.
M. Bertmer, H. Eckert r Solid State Nuclear Magnetic Resonance 15 (1999) 139–152
142
Additionally, with every new I spin, the number of dipolar frequencies doubles:
2.2. Analytical approximations
SI: v i s "v D 1 ;
As shown below, in many cases, the REDOR dephasing behavior at short evolution times is of special interest. Previous calculations have shown that the initial part of the REDOR curve can be approximated analytically, thereby avoiding the full numeric integration procedure. This analytical approximation is based on the usual expansion:
i s 1,2,
SI 2 : v i s " Ž v D 1 " v D 2 . ;
i s 1 to 4,
SI n : v i s " Ž v D 1 " v D 2 " . . . " v D n . ;
i s 1 to 2 n .
After coordinate transformation, the z-component of each dipolar vector r i is given by cos u i in the Hamiltonian of Eq. Ž3.. With this, the term 3 cos 2 u i y 1 has to be calculated. Following the procedure of Pan et al. w19x, the average dipolar transition frequency can then be calculated according to:
vis
1 Tr
Trr2
H0
vi d t y
Tr
HT r2v d t i
.
Ž 5.
cos x s 1 y
DS
As an example, for an SI 2 system, the result is:
S0
=sin 2 b sin g q sin 2 Ž a y z . sin b cos g . .
Ž 6. Using spin density matrix formalism w20x, the REDOR difference signal can be calculated for the sequence of Fig. 1. After one rotor period following the 908 ŽS. pulse, the spin density matrix has evolved to r ŽTr . s Sˆx cos v iTr q 2 Sˆy Iˆz sin v iTr , where x, y and z are the rotating frame coordinates. Due to the cyclic nature of the REDOR pulse sequence w21x, the result after N rotor cycles is given by:
Ž 7.
Observation in x-direction leads to: SŽ a , b ,g . ; cosŽ v iNTr .. This result is only valid for one specific crystallite orientation characterized by a set of Euler angles a , b and g . To obtain the REDOR curve of a powdered crystal or an amorphous system, one has to integrate over all Euler angles: 2n
1 8p
2
2p
ÝH
is1 0
p
2p
H0 H0
cos Ž v iNTr . sin b d a d b dg ,
Ž 8. n being the number of I spins coupled to S.
2
24
y...
Ž 9.
15
2 Ž NTr . D12 q D 22 q . . . qDn2 .
Ž 10 .
Clearly, Ý D 12 is proportional to the heterodipolar second moment. In quantitative terms, we find:
qsin 2 a sin b cos g . q D 2 Ž cos 2 Ž a y z .
Sf s
16 s
r Ž NTr . s Sˆx cos v iNTr q 2 Sˆy Iˆz sin v iNTr .
x4 q
Truncation after the second term in the limit x s v iNTr < 1, results in a first-order approximation:
r
v i s 2'2 NTr D 1 Ž cos 2 a sin 2 b sin g
x2
DS
4 s 3p
S0
2
2 Ž NTr . M2 .
Ž 11 .
Thus, within this short-term limit, the REDOR curve becomes geometry-independent, and the strength of the internuclear dipole coupling can be expressed easily in terms of a second moment. Such an analysis is particularly favorable for structural studies of systems in which there is no ad hoc information on spin geometry. Likewise, a geometry-independent assessment of dipolar coupling strength is advantageous for the structural analysis of disordered systems such as glasses, where the spin geometries may be ill-defined and subject to local distribution effects. Unfortunately, the range of experimental data points available is often too small for reliable analysis of the initial REDOR decay curves. Extension of the data range to larger dipolar evolution times, however, requires the next higher approximation, keeping the term in fourth power of NTr . For a two-spin system, this second-order approximation results in the expression: DS
16 s
S0
15
2 Ž NTr . D 2 y
128 315
4 Ž NTr . D 4 .
Ž 12 .
Fig. 3 compares the two approximations with the universal REDOR curve for a two-spin system. It is
M. Bertmer, H. Eckert r Solid State Nuclear Magnetic Resonance 15 (1999) 139–152
143
Fig. 3. Comparison of the two-spin REDOR curve Ž — — . with the first-order Ž- - -. and second-order approximation ŽP P P ..
shown that Eq. Ž10. is acceptable up to about D SrS0 s 30%, whereas Eq. Ž12. remains valid up to about D SrS0 s 60%. Fig. 4 plots the error in M2 and D that is obtained by fitting the calculated REDOR curve to Eqs. Ž11. and Ž12., respectively, as a function of the data range used in D SrS0 . Clearly, the true second moment is underestimated in the first-
order, and overestimated in the second-order approximation. Taking into account that, in general, M2 values are acceptable within an experimental range of 10% error, it is seen that the second order approximation can give satisfactory results for NDTr values smaller than unity. As discussed in more detail below, however, second-order approximations for
Fig. 4. Relative errors of M2 Ž — — . and D Ž- - -. by fitting the two-spin REDOR curve to the first-order and second-order approximation vs. D SrS0 , respectively.
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M. Bertmer, H. Eckert r Solid State Nuclear Magnetic Resonance 15 (1999) 139–152
higher-spin systems contain angle-geometry dependent terms and are therefore generally less useful in systems with unknown geometries.
3. Experimental The simulation program was written in Pascal language running on an IBM PC, Pentium 100 MHz. Numerical simulations were done using the Simpson rule with integration steps of pr32, whereas pr16 were sufficient for the general curve behavior but leading to less oscillation steps in the long evolution time limit. From these simulated curves, approximate M2 values were extracted by fitting the dephasing curves calculated for short evolution times to Eq. Ž11.. Similar procedures were used to calculate approximate D values for various types of higher spin geometries in the limit of the second-order approximation.
4. Results In the following, selected simulated REDOR curves and analytical approximations are presented
for various multi-spin systems. As for the SI 2 system, there is only one variable angle, we present the differences of the REDOR curve as a function of this angle. Dephasing of single spins S by dipolar couplings to more than two I spins is considered only for selected point symmetries that commonly occur in inorganic compounds.
4.1. SI2 systems Fig. 5 illustrates the results of REDOR calculations in an SI 2 system with equal distances r SI 1 and r SI 2 as a function of angle z . Since the angle dependence is symmetrical to 908, the same curves are obtained for complementary angles, i.e., for z s 608 and z s 1208. With z equal to 1808, the maximum dephasing is only 50%. This result is physically plausible because in a symmetric linear spin system, there is a 50% probability that the I spins have opposing z-components, causing cancellation of the local fields at the S site. The most important result is that the curves are more or less geometry-independent in the limit of short evolution times, as predicted by Eq. Ž10..
Fig. 5. REDOR curves of the SI 2 system with varying angle z and same dipolar coupling.
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145
Fig. 6. Error plot of M2 and D vs. D SrS0 for the SI 2 system with an angle z of 109.58 Žtetrahedral angle., in the limits of first-order and second-order approximations.
For the SI 2 system, the corresponding secondorder approximation becomes: DS
16 s
S0
15
2 Ž NTr . D12 q D 22
128 y 315
range, 0 F Ž D SrS0 . F 0.4; however, knowledge of the order and geometry of the spin system is required in this case.
4 Ž NTr . D14 q D 24
3 q D 12 D 22 Ž 9 q 4 cos 2 z q 3 cos 4z . , 8
Ž 13 .
including, unfortunately, also a dependence on spin geometry. Fig. 6 plots the relative systematic error in the second moment Ž D M 2 rM 2 . s w M 2 Žsim. y M 2 Žexact.rw M2 Žexact.x extracted from Eq. Ž11. by extending the quadratic approximation to increasing values of D SrS0 . Again, applying the first-order approximation to the data results in underestimates of M2 . Concerning an error assessment of the second-order approximation ŽEq. Ž13.., special geometries have to be considered. Assuming z s 109.58 Žtetrahedral angle. and D 1 s D 2 Žequal distances. results in: DS S0
2
4
s 2.13 Ž NTr . D 2 y 1.81 Ž NTr . D 4 .
Ž 14 .
The second curve in Fig. 6 plots the systematic error, Ž D DrD . s w DŽsim. y DŽexact.xrw DŽexact.x for this special geometry in the limit of the second-order approximation. Note that this approximation can give excellent estimates of D within a fairly wide data
Fig. 7. SI 3 system with trigonal-plane and trigonal-pyramid symmetry. Ža. Simulated REDOR curves; trigonal plane Ž — — ., trigonal-pyramid Ž- - -.. Žb. Error plots for these symmetries. Error in M2 : trigonal plane Ž — — ., trigonal pyramid Ž – P – . Žfirst-order approx..; error in D: trigonal plane ŽP P P., trigonal pyramid Ž- - -. Žsecond-order approx...
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Fig. 8. REDOR curves of higher spin systems. SI 4 : tetrahedron Ž — — ., square plane Ž- - -.; SI 6 : octahedron ŽP P P ..
4.2. Higher-spin systems High-symmetry configurations for the SI 3 spin system are trigonal-plane symmetry and trigonal pyramid. Corresponding angles for simulation are:
trigonal-plane symmetry: z s 1208; ´ s 908; d s y1208; trigonal-pyramid: z s 988, ´ s 128, d s 1318. In Fig. 7a, the simulated REDOR curves are presented. While the oscillatory parts of the two curves are quite different, the behavior in the limit of
Fig. 9. Corresponding error plots for the higher spin systems. Error in M2 : tetrahedron Ž — — ., square plane Ž – P – ., octahedron Ž- - -. Žfirst-order approx... Error in D: tetrahedron ŽP P P ., square plane Ž- - -., octahedron Ž – P – . Žsecond-order approx...
M. Bertmer, H. Eckert r Solid State Nuclear Magnetic Resonance 15 (1999) 139–152
147
Fig. 10. Simulated REDOR curves for a distorted SI 4 tetrahedron with one large coupling Ž D1 s 1 kHz. and three smaller couplings D 2 , D 3 , D4 , the inset showing the initial part of the REDOR curve magnified. Two-spin system with D s 1 kHz Ž — — ., tetrahedron with D 2 , D 3 , D4 s 100 Hz ŽP P P ., 200 Hz Ž- - -. and 300 Hz Ž – P – ..
short evolution times is very similar as expected. Fig. 7b shows the corresponding error plot in M2 and D in the limit of both approximations. Finally, Figs. 8 and 9 provide the analogous simulations and error plots for tetrahedral, square planar, and octahedral environments. The angles for these geometries are, respectively, tetrahedron: z s 109.58, ´ 1 s 368, d 1 s y1258,
´ 2 s 1448, d 2 s y1258, square plane: z s 908, ´ 1 s 908, d 1 s 1808,
´ 2 s 908, d 2 s y908, octahedron: z s 908, ´ 1 s 908, d 1 s 1808,
´ 2 s 908, d 2 s y908, ´ 3 s 08, d 3 s 08, ´4 s 1808, d4 s 08. For all these geometries, the first-order approximation systematically underestimates M2 and D; however, it can give reasonable second moment estimates at sufficiently short evolution times, where Ž D SrS0 . F 0.2 to 0.3. 4.3. Effect of remote spins on REDOR dephasing To assess the effect of weaker dipole–dipole couplings to more remote spins, the simulations carried
Fig. 11. Relative error in D, by fitting the tetrahedra with the first-order Ža. and second-order approximation Žb. of the two-spin system. Pure two spin system Ž — — ., tetrahedron with D 2 , D 3 , D4 s100 Hz ŽP P P., 200 Hz Ž- - -. and 300 Hz Ž – P – ..
148
M. Bertmer, H. Eckert r Solid State Nuclear Magnetic Resonance 15 (1999) 139–152
M. Bertmer, H. Eckert r Solid State Nuclear Magnetic Resonance 15 (1999) 139–152
149
Fig. 13. Relative error in D using the first-order approximation for the simulated REDOR curves shown in Fig. 12c using the same symbols.
out for tetrahedral geometry were altered by assuming a dominant two-spin coupling constant D 1 and three identical smaller coupling constants D 2 –D4 , the magnitude of which was systematically altered. The resulting REDOR curves are summarized in Fig. 10. As expected, the effect of the more remote spins is to increase M2 as measurable from the REDOR dephasing curve in the limit of short evolution times. Furthermore, the oscillatory behavior in the long evolution time limit is damped significantly. Fig. 11 plots the error in D Žand hence internuclear distance. that would be obtained if the weaker dipole–dipole couplings D 2 –D4 to the three more remote spins were neglected, i.e., if the REDOR curve would be analyzed under the assumption of an isolated two-spin system. Note that the error arising from this approximation tends to compensate the negative error produced by the parabola fit; this fortuitous benefit is lost in the second-order approximation. 4.4. Effect of distance distributions in systems with multi-spin interactions In disordered systems, the internuclear distances usually do not possess singular values but rather are
subject to distribution effects. As a consequence, the experimental REDOR curve is often a weighted superposition of individual curves according to a continuous function. To simulate the effect of distance distributions on REDOR curves, calculations were carried out for a tetrahedral geometry. Three Gaussian distributions of distances with different widths were considered and transposed into corresponding distributions of D values as shown in Fig. 12a. Due to the non-linear relationship between r and D, symmetrical distribution functions in r produce asymmetric distribution functions in D. Simulating the resulting REDOR curves then implies a corresponding compression or elongation of the abscissa in the presentation of the universal curve D SrS0 vs. NTr Žsee Fig. 12b.. The resulting REDOR curves obtained by superposition are shown in Fig. 12c. In particular, the initial slope is slightly increased with increasing range of D values compared to that obtained for the uniform distance. As a result, the average dipolar coupling strength is slightly overestimated, while the distances are underestimated. Fig. 13 shows the error in D obtained using the various distance distributions assumed under the first-order approximation; note again the compensatory effect on the error.
˚ ŽP P P ., s s 0.089 A˚ Ž — — . and s s 0.16 A˚ Ž — — .. Ža. Fig. 12. Simulated REDOR curves for Gaussian distributions in r of s s 0.032 A Distribution in D resulting from a Gaussian distribution of distance Žinset.. Žb. Selected REDOR curves showing the compression or elongation of the abscissa. Žc. REDOR curves obtained by superposition of individual curves with different dipolar couplings according to ˚ ŽP P P ., s s 0.089 A˚ Ž- - -. and s s 0.16 A˚ Ž — — .. their ratios in the distribution: Exact tetrahedron Ž – P – ., distribution with s s 0.032 A
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5. Discussion and conclusions 5.1. Dipolar oscillations as probes for spin geometries In the discussion of the features of the REDOR simulations obtained within this study, it is most practical to differentiate between two time regimes: the oscillatory part obtained in the limit of long evolution times and the initial decay measured in the limit NDTr - 1. Our results indicate that the longterm oscillatory part may be a valuable tool for identifying specific spin geometries. This is illustrated by Fig. 14, summarizing the effects of coupling partners and specific geometry considered. This feature may be particularly useful for studies of truly isolated spin clusters as present in some molecular compounds, coordination complexes or dilute species within matrices. In contrast, for infinite solid lattices or inorganic networks, it is generally not possible to simply neglect the additional dipolar coupling to more distant spins. As illustrated in Fig. 10, these additional dipolar couplings lead to very effective damping of the oscillatory behavior so that only a smooth curve is observed. Details depend on the specific spin geometry considered, but the case shown in Fig. 10 represents still one of the most favorable
situations where dipolar structure will be observable. Even in this favorable case, the dipolar oscillation in the REDOR curve vanishes if the ratio D 1rD 2 drops below a value of about three. Denoting D 1 as the dipolar coupling constant with the closest neighbors and D 2 as the one with the next nearest heteronuclear neighbors, D 1rD 2 ratios of 5.2, 7.0 and 15.9 are calculated for the typical three-dimensional lattices, NaCl, ZnS Žzinc blend. and SiO 2 Žcristobalite., respectively. Since multiple primary and secondary interactions are present in these lattices, one can expect that the oscillations will be damped at least in the first two cases. Observation of oscillatory behavior may further be hindered by low signal-to-noise ratio at long evolution times due to spin–spin relaxation of the observed spin coherence. Indeed, published REDOR data on aluminum phosphate networks show no indications of dipolar oscillations w22x. Finally, in glasses, the dipolar oscillations are further damped owing to the presence of distance distribution effects. 5.2. REDOR dephasing in the limit of short eÕolution times The results of the present study illustrate that the REDOR dephasing curve in the limit of short evolu-
Fig. 14. Dephasing behavior in REDOR experiments for SI n multi-spin systems: Ž — — .: n s 1, ŽP P P .: n s 2, Ž — — .: n s 3, trigonal plane, Ž – P – .: n s 4, tetrahedron, Ž – ( – .: n s 6, octahedron.
M. Bertmer, H. Eckert r Solid State Nuclear Magnetic Resonance 15 (1999) 139–152
tion times is independent of the specific spin geometry involved and can be approximated by a parabola, provided the effects of distance distributions are not overly severe. The curvature of this parabola is described by a single parameter related to the dipolar second moment powder average, which is computed from the lattice sum over the corresponding ry6 ij terms of all spin pairs involved. As such, a mean square average of the dipole–dipole coupling constant can be characterized without explicit knowledge of the spin geometry present. This feature is particularly useful when dealing with disordered systems and glasses. In such systems, an approximate dipolar coupling constant Žor second moment. can be obtained by applying the first-order approximation to the initial REDOR dephasing curve. Such experimental values can then be compared with van Vleck M2 calculations for specific bonding scenarios, where the range of distances is usually limited to the nearest neighbors. The whole procedure derives an additional benefit from the fortuitous error compensation effect discussed above. Regardless of the specific geometry considered, our simulations show that fitting the initial part of the REDOR curve to a parabola Žfirst-order approximation. tends to underestimate the dipole–dipole coupling constant in question compared to the accurate simulation. On the other hand, the experimental curves include the additional second moment contributions from magnetic dipole–dipole interactions with more remote magnetic neighbors, whereas the van Vleck calculations for specific bonding scenarios in glasses do not if only the closest neighbours are considered. Thus, if we compare an experimental second moment from a REDOR parabola fit with a van Vleck value for a simple nearest-neighbor bonding scenario Žneglecting longer-range dipolar coupling., the systematic errors have opposite sign and thus partially cancel each other. Based on this favorable situation, REDOR dephasing curves measured in the short evolution time limit are expected to produce valid experimental criteria for hypothetical scenarios involving the short- and intermediate-range order in glasses. We note that Gullion and Pennington have recently presented an elegant alternative approach for disentangling the multi-spin problem in REDOR experiments. In their method, pulses corresponding to small flip angles are applied to the non-observed
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spins in the middle of each rotor period Žtheta REDOR. w23x. Since in this procedure only a small fraction of spins creating the dipolar field is flipped, the multi-spin problem is reduced to a two-spin problem by mere probability. The disadvantage of theta REDOR, however, is an attenuation of the overall effect, producing larger errors in systems with weak couplings. Furthermore, a quantitative analysis of the experimental results requires knowledge about the spin multiplicity. The method delineated in the present contribution does not suffer from these drawbacks. Validation of this technique on crystalline model compounds and applications to various glass systems are currently under intense study in our laboratory. Acknowledgements This research was funded by the Wissenschaftsministerium Nordrhein-Westfalen. Valuable discussions with Dr. Leo van Wullen are grate¨ fully acknowledged. References w1x T. Gullion, J. Schaefer, J. Magn. Reson. 81 Ž1989. 196–200. w2x K.T. Mueller, J. Magn. Reson. A 113 Ž1995. 81–93. w3x K.T. Mueller, T.P. Jarvie, D.J. Aurentz, B.W. Roberts, Chem. Phys. Lett. 242 Ž1995. 535–542. w4x K.T. Mueller, T.P. Jarvie, D.J. Aurentz, B.W. Roberts, Chem. Phys. Lett. 254 Ž1996. 281–282. w5x A.M. Christensen, J. Schaefer, Biochemistry 32 Ž1993. 2868–2873. w6x W. Hing, N. Tjandra, P.F. Cottam, J. Schaefer, Biochemistry 33 Ž1994. 8651–8661. w7x Y. Pan, N.S. Shenouda, G.E. Wilson, J. Schaefer, J. Biol. Chem. 268 Ž1993. 18692–18695. w8x L.v. Wullen, L. Zuchner, W. Muller-Warmuth, H. Eckert, ¨ ¨ ¨ Solid State Nucl. Magn. Reson. 6 Ž1996. 203–212. w9x C.A. Fyfe, K.T. Mueller, H. Grondey, K.C. Wong-Moon, J. Phys. Chem. 97 Ž1993. 13484–13495. w10x S.M. Holl, T. Kowalewski, J. Schaefer, Solid State Nucl. Magn. Reson. 6 Ž1996. 39–46. w11x L.V. Wullen, B. Gee, L. Zuchner, M. Bertmer, H. Eckert, ¨ ¨ Ber. Bunsenges. Phys. Chem. 100 Ž1996. 1539–1549. w12x C. Rong, K.C. Wong-Moon, H. Li, P. Hrma, H. Cho, J. Noncryst. Solids 223 Ž1998. 32. w13x K. Herzog, B. Thomas, D. Sprenger, C. Jager, J. Noncryst. ¨ Solids 190 Ž1995. 296. w14x C.P. Grey, B.S. Arun Kumar, J. Am. Chem. Soc. 117 Ž1995. 9071.
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M. Bertmer, H. Eckert r Solid State Nuclear Magnetic Resonance 15 (1999) 139–152
w15x A.L. Blumenfeld, D.J. Coster, J.J. Fripiat, Chem. Phys. Lett. 231 Ž1994. 491. w16x A. Naito, K. Nishimura, S. Tuzi, H. Saito, ˆ Chem. Phys. Lett. 229 Ž1994. 506–511. w17x J.M. Goetz, J. Schaefer, J. Magn. Reson. 127 Ž1997. 147– 154. w18x C.A. Fyfe, A.R. Lewis, J.M. Chezeau, H. Grondey, J. Am. Chem. Soc. 119 Ž1997. 12210–12222.
w19x Y. Pan, T. Gullion, J. Schaefer, J. Magn. Reson. 90 Ž1990. 330–340. w20x J. Shriver, Concepts, Magn. Reson. 4 Ž1992. 1–33. w21x U. Haeberlen, J.S. Waugh, Phys. Rev. 175 Ž1968. 453–467. w22x C.A. Fyfe, K.T. Mueller, H. Grondey, K.C. Wong-Moon, J. Phys. Chem. 97 Ž1993. 13484–13495. w23x T. Gullion, C.H. Pennington, Chem. Phys. Lett. 290 Ž1998. 88–93.