103
Solid State Nuclear Magnetic Resonance, l(1992) 103-109
Elsevier Science Publishers B.V., Amsterdam
Inhomogeneous angle spinning
interactions
in solids under slow off-magic
Shangwu Ding, Jianzhi Hu and Chaohui Ye Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics, The Chinese Academy of Sciences, Wuhan 430071, China
(Received 2 March 1992; accepted 11 March 1992)
Abstract
The central band of an inhomogeneous interaction under slow off-magic angle spinning (off-MAS) is no longer a scaled static spectrum with scaling factor p,(cos 91, that is, the spectrum distortion occurs in comparison with that under the rapid rotation condition. The dependence of this distortion in the chemical shift spectrum, for instance, on anisotropy, the asymmetry parameter, spinning rate and the angle between the spinning axis and the applied magnetic field is studied in this paper. It is shown that the distortion can be partly eliminated by using TOSS in certain circumstances. Experimental evidence is presented in this study. Keywords: off-MA% slow spinning; inhomogeneous
interactions; spinning sidebands
Introduction The magic angle spinning (MAS) technique is widely used in high resolution solid state NMR [1,2] because it is an effective and convenient method to eliminate the anisotropy of inhomogeneous interactions and, therefore, to obtain resolved isotropic spectra. Under the so-called slow spinning condition, when the MAS spinning rate is less than the anisotropy width of an inhomogeneous interaction, a MAS spectrum consists of a central peak and several spinning sidebands [3-51 which contains all the information of the inhomogeneous interaction. Therefore, the principal elements of an interaction tensor can be determined
Correspondence
to; Professor C. Ye, Laboratory of Magnetic Resonance and atomic and Molecular Physics, Wuhan Institute of Physics, The Chinese Academy of Sciences, Wuhan 430071, China.
09262040/92/$05.00
by the simulation of the intensities of the sidebands [5,6]. The properties of inhomogeneous interactions in solids under MAS have been thoroughly studied [3-91. In the off-MAS case, however, less attention has been paid. Menger et al. [lo] noticed that quite dramatic lineshapes may arise when the deviation from the magic angle (MA) is large and the spinning rate is low. The investigation of off-MAS is important in two aspects. First, the analysis of the asymptotic behaviour of inhomogeneous interaction, when the angle is between the spinning axis and the applied magnetic field towards MA, can be used to accurately determine the direction of the spinning axis [9]. This analysis has provided a quick and accurate MA setting approach. Second, it is generally accepted that the central band of an off-MAS spectrum is the scaled static spectrum with scaling factor P,(cos S>, where 6 is the angle between the spinning axis and the applied magnetic field and P,(cos -9) = 0 when 6 = I?“, =
0 1992 - Elsevier Science Publishers B.V. All rights reserved
104
S. Ding et al. /Solid
State Nucl. Magn. Reson. 1 (1992)_7) 103-109
54.74”. Therefore, it turns out that if 6 is known in advance, the anisotropy of the interaction can be extracted. Combined with the isotropic components read from the MAS spectrum, the principal values of the interaction tensors at chemically nonequivalent sites can be determined, e.g., the off-MAS measurement of chemical shift tensors [ll]. This measurement is more straightforward than sideband intensity analysis [6] in the case when the off-MAS spectrum is well resolved. We will theoretically demonstrate in the next section, that the statement that the central band of an off-MAS spectrum is the scaled static one is exact only when the rapid spinning condition is met. At slow spinning rate, however, the central band will be distorted with respect to the scaled static spectrum. For a system with large anisotropy an experiment at high magnetic field, the rapid condition is difficult to realize, as this distortion will be appreciable. In this paper, the inhomogeneous interaction under slow off-MAS condition is systematically studied in the time domain as is usual for MAS. The quantitative relationships between the distortion and the relative spinning rate, the asymmetry parameter of the interaction tensor and the direction of spinning axis are deduced in this study.
where C” is the coupling constant of the interaction, p,, the tensor elements of the interaction in the principal axis system (PAS), D the Wigner matrix which transforms the interaction from PAS to the laboratory frame, (a, p, y) are the Euler angles between PAS and the rotor system, 13 is the angle between the spinning axis and the applied magnetic field and o, is the spinning rate. It can be easily seen that the eigenvalues of H”’ and HQ are similar. Therefore, we concentrate on the consideration of Hcs in the following. According to eqn. (21, with first order approximation, the eigenvalue wcs of Hcs consists of two parts: time independent w” and time dependent w*, with o(’ = wo[ (To+ 6(3 cos26 - 1) X
(3 cos”p - 1 + n sir@ cos 2~u)/4]
(3)
where oa is the isotropic chemical shift, the second term in brackets is the scaled anisotropy with scaling factor P,(cos 8) = (3~0~~19- 1)/2, which corresponds experimentally to the scaled static powder spectrum, and w* = c, cos( 0,lf
y) + c, cos 2( w,c + y)
+ S, sin( o,t + r) + S, sin 2( w,t + r)
(4)
where Theory
The Hamiltonian of a spin system in an applied magnetic field can be written as H=H,+H”
(1)
where Hz = -woZ, is the Zeeman interaction, H” is the internal interaction of the spin system, and A stands for the type of internal interaction. The most important inhomogeneous interactions are the chemical shift interaction (A = cs) and the quadrupole interaction (A = Q). For spinning solids, H” can be written as [ll I H”=C”
c
c
I c
I c
(-1)‘”
C, = w,6 sin 2w sin 2p( 77cos (Y- 3)/4
(5)
C2=wo6[2sin’~+~(1+cos2~)
(6)
cos2a]/4
S, = -w,&q
sin 26 sin 2a/2
(7)
S, = -wo6n
sin26 cos p sin 2a/2
(8)
If w,6 is replaced by CQGQ = e2QV,,/6Z(21 llh, i.e., the anisotropy of quadrupole interaction, then eqns. (3)-(8) can also be used to describe Ho, Obviously, those time dependent terms modulated by W, and 2w, are due to sample spinning. Under the rapid spinning condition, i.e., w, B woS, the time dependent terms vanish. Therefore, in this case the anisotropy of the interaction can readily be obtained from the off-MAS spectrum if 6 is given. This method is of course much simpler than simulation of sideband intensities [61.
State Nucl. Magn. Reson. 1 (19921 103-109
105
Actually, however, due to the mechanical restrication of the rotor, the rapid spinning condition often cannot be realized when the experiment is carried out at high magnetic field (for the pursuit of higher resolution and sensitivity) or when the anisotropy of the interaction is large. It is expected that the central band of an off-MAS spectrum is no longer a scaled static powder spectrum and large errors would emerge when interaction tensor elements are evaluated from the central band. In fact, the time dependent part w* can cause the distortion of the central band with respect to the scaled static powder spectrum. Therefore, the effect of w* should be taken into consideration in the calculation of the spectrum pattern under the slow spinning condition. The FID signal for an inhomogeneous interaction, can be written as [2,5]
will simultaneously appear as noticed by Raleigh et al. [8]. Notice that the difference in eqn. (11) between MAS and off-MAS is that for off-MAS, the integration should be made not only for I,. but also for o0 since w” becomes anisotropic in the offMAS case. It is this significant difference that makes the central band of a slow off-MAS spectrum not only decreases in intensity, but also changes in pattern, i.e., the distortion with respect to the rapid spinning spectrum. The sidebands of an inhomogeneous interaction can be eliminated by means of TOSS [12]. For a MAS/TOSS spectrum, the intensity of the central peak can be restored to some extent. These two conclusions are valid for the offMAS/TOSS spectrum. When TOSS is used here, only the central band remains and eqn. (11) becomes
S. Ding et al. /Solid
g(t) = (I/SrTT2) ldo
exp( -@(a,
P, Y, f)} g(r) = (l/477)1
da F, exp( --io’t)
(9) where 4(t)
=
coot +j-“’dt
w*
From eqns. (3)-(lo),
we have
g(r) =(1/4~)C/
dfl IN exp[-i(wO+Nw,)t]
(IO)
N
(11) where I, = I FN I ’ FN = (l/2+,*=
(12)
d x exp[-i(Nx+4(x))] (13)
43(x) = (2C, sin x - 2S, cos x + C, sin 2x -s,
cos 2x)/2w,
( 14)
It is well known that when w, B w,S (rapid spinning condition), only I, (N = 0) in eqn. (11) is left, but when w, < wo6, the sidebands appear. In the MAS case, the slow spinning brings about the intensity loss o f the central peak, which has been studied by many authors [3-8,121. For slow offMAS, the intensity loss and the lineshape change
(15)
It can be seen from the above expression that the off-MAS/TOSS spectrum is still different from the scaled static powder spectrum since F, is anisotropic. It means that the distortion of the central band exists even though TOSS is applied. To confirm the above analysis, we have made some numerical calculations and simulations on off-MAS and off-MAS/TOSS spectra at various spinning rates, taking the chemical shift interaction as an example. Here we show two cases, one is the spectrum of an asymmetric chemical shift interaction (Fig. l), the other is that of an axially symmetric one (Fig. 2). Obviously, the distortions of both off-MAS and off-MAS/TOSS spectra decrease with the increase in the relative spinning rate w,/o& For a quantitative description of the spectrum distortion under the slow spinning condition, we define the following distortion factor: D=/
dw[f(w)
-f,(o)]‘,‘Aw
(16)
where do is the spectrum width, f(w) and f,,(w) are the normalized intensities of the central band under slow and rapid spinning conditions, respectively.
S. Ding et al./Solid
106
State Nucl. Magn. Reson. 1 (1992) 103-109
The relationship between D and R = wr/w,6 is calculated and diagramatically shown in Fig. 3 for different asymmetry parameters. As expected, the distortion becomes less and less as R increases. It can be seen that when R > 1, D vanishes. When R > 0.5, D is negligibly small for any case of q. Therefore, the condition R > 0.5 can be approximately referred to as the distortionless condition. We can also see that as R > 0.3, TOSS can reduce distortion, but when R < 0.3, the distortion of an off-MAS/TOSS spectrum may be larger than that of an off-MAS spectrum without
165
90
241I
PPm
Fig. 2. Same as Fig. 1 but with axially symmetric chemical shift interaction. The parameters used are: g,, = 90 ppm, eI = 240 ppm and P,kos 6) = -0.5.
R=0.2
TOSS. When R is very small, the distortion of an off-MAS/TOSS spectrum becomes very large. Here, we give an explanation of this characteristic. The ratio of intensity differences between the scaled static spectrum and the off-MAS spectra with and without TOSS is
:--.
dt exp( -iwt)
ppm Fig. 1. The off-MAS spectra of the asymmetric chemical shift interaction under various relative spinning rates, R = wr /w,S. The solid lines stand for rapid spinning cases or the scaleddown static lineshape, dashed lines for the slow spinning with TOSS and dotted lines for slow spinning without TOSS. The parameters used in this figure are: or, = 60 ppm, q2 = 120 ppm, ms3= 240 ppm and P,(cos 6) = - 0.5.
X
/
d0( 1 - Fa) exp( --hot)
I
dt exp( --icoOt)
X / dL!(l - I F, I’) exp( --iw’t) I
(17)
S. Ding et al. /Solid
107
State Nucl. Magn. Reson. 1 (1992) 103-109
From eqns. (S)-(8), (13) and (14) we can see that when the relative spinning rate R = wJw,6 is large, both F,, and I F, I 2 are positive and the change of F, with (a, p) is smaller than that of I F, I ‘, hence r < 1, meaning that the distortion of an off-MAS spectrum can be partly eliminated by TOSS. In the limit, when R is so small that the change of -F, is greater than that of I F. I 2, therefore r > 1, i.e., at a very slow spinning rate, the TOSS makes the off-MAS spectrum more distorted. The above calculations show that the TOSS can partly restore the intensity of central band only when the relative spinning rate is not too slow, e.g., R > 0.3. This means that there is a restrication on TOSS which has been widely used in solid state NMR. A similar phenomenum also
_-__._ ,/* ,_I'
12.0
I
2.0 L 12.01
I
b 8.0 6.0 __a-
___.~~ ________.-------_.
4.0 1
2.0
a ..--------
;:JJi-:ll:_I:g -0.4
1.4
0.2 s.
Fig. 4. The relationship between D and S = 2P,(cos -9) = 3 cos24 - 1. Solid line: off-MAS/TOSS spectrum: dashed line; off-MAS spectrum. (r = 200 ppm, 6 = 100 ppm. (a) T= 0, o,= 3 kHz; (b) ~=0.5, w,=5 kHz; and Cc) 7 =l, w,=5 kHz.
0.0’ ‘--I
_I.-~-.-
/+L--.0.4 ._.
0.2
_____
’
R
0.6
’
0.8
I.0
’
Fig. 3. The relationship between the distortion factor D and relative spinning rate R = wr/w06. Solid line corresponds to the off-MAS/TOSS spectrum and the dashed line to the off-MAS spectrum: o0 = 200 ppm, 6 = 100 ppm and P,(cos 6) = 0.5. (a) TJ= 0: (b) 77= 0.5; and Cc) 7 = 1.
occurs in MAS experiments [13] where TOSS is ineffective when R is very small. Also noted from Figs. 1-3 is that the more symmetric the interaction tensor, the less the distortion of its spectrum. We find that D also depends upon 6, the direction of the spinning axis relative to the applied magnetic field. Figure 4 shows the relationship between D and S, here S = 2P, (cos 19) = 3 co?19 - 1. The general behaviour is that no matter how large are the spinning rate, the anisotropy and asymmetry parameters of the interaction, and no matter whether TOSS is used or not, D decreases monotonically with the increase of 6 (0’ through 90’). This fact can be understood from eqns. (5)--(8): as 6 increases,
108
S. Ding et al. /Solid
the decrement in S, and S, is greater than that of C, and C,, so both F, and IF, 1’ decrease with the increase of 6. Therefore, the distortion of a slow MAS spectrum is not at a minimum, although it has highest resolution. We can further see from Fig. 4 that TOSS can partly eliminate the distortion as long as R is not very small.
rate was 4 kHz. For many L-amino acids, in which the anisotropy of the 13C chemical shift interaction in the carboxyl group is as large as 100 ppm, the relative spinning rate R is - 0.4. According to the theory, TOSS can partly restore the spectrum pattern. The experimental and theoretical results for the chemical shift powder spectrum of the carboxyl carbon in L-histine, for instance, are shown in Fig. 5. From this computer simulation, the chemical shift tensor components of this carbon are 116, 168 and 235 ppm, respectively, in good agreement with the results for the sideband intensity simulation of the MAS spectrum. For other samples, similar results were obtained.
Experimental For slow spinning solids, the measurement of an inhomogeneous interaction tensor is no longer as simple as that under rapid spinning condition. The distortion due to an insufficient spinning rate should be taken into consideration. But the interaction tensor can still be elucidated from the experimental spectrum with computer simulation based on the above theory. The experimental measurement of the chemical shift tensor components of 13C in some Lamino acids agree with the above theoretical analysis. The spectrometer used in our experiments was a Bruker MSL-400, with a working frequency for 13C of 100.63 MHz. The regular cross-polarization sequence is used. The contact time was taken as - 1 ms. The distortion of spectra caused by the inhomogeneous crosspolarization was neglected. The sample spinning
.I._..-,
180
.__..
160
140
State Nucl. Magn. Reson. 1 (19921 103-109
Conclusions For an inhomogeneous interaction in a spin system under off-MA& we conclude that when the rapid spinning condition is reached, an offMAS spectrum is exactly as the scaled static powder spectrum with a scaling factor P,(cos 8). When the rapid spinning condition cannot be realized, the distortion of the central band is inevitable. Approximately, as the relative spinning rate R is greater than 0.5, the distortion can be neglected; if R is greater than about 0.3, TOSS can be used to reduce the distortion. Moreover, the greater the angle between the spinning axis and the applied magnetic field, the less the distortion is. We further make a remark about the off-MAS experiment for measuring anisotropies: it has no preference over the sideband analysis approach [6] since the calculation of a distortive spectrum under the slow off-MAS condition is not time saving as compared to the simulation of sideband intensities. Nevertheless, the present work can be regarded as a general description of the off-MAS inhomogeneous interaction.
ppm
Fig. 5. The chemical shift powder spectrum of 13C of the carboxyl carbon of L-histine. The solid line is the theoretical result (with TOSS) and the dashed line is the experimental result. The sample spinning rate is 4 kHz and P,(cos a)= -0.06. The 13C chemical shift tensor components of the group were obtained from computer simulation as: oI1 = 116 ppm. gZ2 = 168 ppm and 033 = 235 ppm.
Acknowledgment This work was supported by the National Natural Science Foundation of China.
S. Ding et al. /Solid State Nucl. Magn. Reson. 1 (1992) 103-109
References 1 U. Haeberlen, High Resolution NMR in Solids: Selectke Ar.eraging, Academic Press, New York, 1976. 2 M. Mehring. Principles of High Resolution NMR in Solids, Springer-Verlag, Berlin, 1983. 3 J. Waugh, M. Maricq and R. Cantor, J. Magn. Reson., 29 (1978) 183. 4 M. Maricq and J. Waugh, Chem. Phys. Lett., 47 (1977) 327. 5 M. Maricq and J. Waugh, J. Chem. Phys., 70 (1979) 3300. 6 J. Herzfield and A. Berger. J. Chern. Phys.. 73 (1980) 6027.
1OY 7 E.T. Olejniczak, S. Vega and R.G. Griffin. J. Clzem. Phys., 81 (1984) 4804. 8 D.P. Raleigh, E.T. Olejniczak and R.G. Griffin, .L Chem. Phys., 89 (1988) 1333. 9 C. Ye, B. Sun. G. Maciel, J. Magn. Reson.. 70 (1986) 241. IO E.M. Menger. D.P. Raleigh and R.G. Griffin. J. Magn. Resort. , 63 (1985) 579. II N. Sethi, D.M. Grant and R.J. Pugmire, J. Mugn. Reson., 71 (1987) 476. 12 W.T. Dixon, J. Chem. Phys., 77 (1982) 1800. 13 B. Sun and C. Ye. Chin. J. Magn. Reson., 2 (19X5) 364.
Solid State Nuclear Magnetic Resonance, l(1992) 103-109
Elsevier Science Publishers B.V., Amsterdam
Inhomogeneous angle spinning
interactions
in solids under slow off-magic
Shangwu Ding, Jianzhi Hu and Chaohui Ye Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics, The Chinese Academy of Sciences, Wuhan 430071, China
(Received 2 March 1992; accepted 11 March 1992)
Abstract
The central band of an inhomogeneous interaction under slow off-magic angle spinning (off-MAS) is no longer a scaled static spectrum with scaling factor p,(cos 91, that is, the spectrum distortion occurs in comparison with that under the rapid rotation condition. The dependence of this distortion in the chemical shift spectrum, for instance, on anisotropy, the asymmetry parameter, spinning rate and the angle between the spinning axis and the applied magnetic field is studied in this paper. It is shown that the distortion can be partly eliminated by using TOSS in certain circumstances. Experimental evidence is presented in this study. Keywords: off-MA% slow spinning; inhomogeneous
interactions; spinning sidebands
Introduction The magic angle spinning (MAS) technique is widely used in high resolution solid state NMR [1,2] because it is an effective and convenient method to eliminate the anisotropy of inhomogeneous interactions and, therefore, to obtain resolved isotropic spectra. Under the so-called slow spinning condition, when the MAS spinning rate is less than the anisotropy width of an inhomogeneous interaction, a MAS spectrum consists of a central peak and several spinning sidebands [3-51 which contains all the information of the inhomogeneous interaction. Therefore, the principal elements of an interaction tensor can be determined
Correspondence
to; Professor C. Ye, Laboratory of Magnetic Resonance and atomic and Molecular Physics, Wuhan Institute of Physics, The Chinese Academy of Sciences, Wuhan 430071, China.
09262040/92/$05.00
by the simulation of the intensities of the sidebands [5,6]. The properties of inhomogeneous interactions in solids under MAS have been thoroughly studied [3-91. In the off-MAS case, however, less attention has been paid. Menger et al. [lo] noticed that quite dramatic lineshapes may arise when the deviation from the magic angle (MA) is large and the spinning rate is low. The investigation of off-MAS is important in two aspects. First, the analysis of the asymptotic behaviour of inhomogeneous interaction, when the angle is between the spinning axis and the applied magnetic field towards MA, can be used to accurately determine the direction of the spinning axis [9]. This analysis has provided a quick and accurate MA setting approach. Second, it is generally accepted that the central band of an off-MAS spectrum is the scaled static spectrum with scaling factor P,(cos S>, where 6 is the angle between the spinning axis and the applied magnetic field and P,(cos -9) = 0 when 6 = I?“, =
0 1992 - Elsevier Science Publishers B.V. All rights reserved
104
S. Ding et al. /Solid
State Nucl. Magn. Reson. 1 (1992)_7) 103-109
54.74”. Therefore, it turns out that if 6 is known in advance, the anisotropy of the interaction can be extracted. Combined with the isotropic components read from the MAS spectrum, the principal values of the interaction tensors at chemically nonequivalent sites can be determined, e.g., the off-MAS measurement of chemical shift tensors [ll]. This measurement is more straightforward than sideband intensity analysis [6] in the case when the off-MAS spectrum is well resolved. We will theoretically demonstrate in the next section, that the statement that the central band of an off-MAS spectrum is the scaled static one is exact only when the rapid spinning condition is met. At slow spinning rate, however, the central band will be distorted with respect to the scaled static spectrum. For a system with large anisotropy an experiment at high magnetic field, the rapid condition is difficult to realize, as this distortion will be appreciable. In this paper, the inhomogeneous interaction under slow off-MAS condition is systematically studied in the time domain as is usual for MAS. The quantitative relationships between the distortion and the relative spinning rate, the asymmetry parameter of the interaction tensor and the direction of spinning axis are deduced in this study.
where C” is the coupling constant of the interaction, p,, the tensor elements of the interaction in the principal axis system (PAS), D the Wigner matrix which transforms the interaction from PAS to the laboratory frame, (a, p, y) are the Euler angles between PAS and the rotor system, 13 is the angle between the spinning axis and the applied magnetic field and o, is the spinning rate. It can be easily seen that the eigenvalues of H”’ and HQ are similar. Therefore, we concentrate on the consideration of Hcs in the following. According to eqn. (21, with first order approximation, the eigenvalue wcs of Hcs consists of two parts: time independent w” and time dependent w*, with o(’ = wo[ (To+ 6(3 cos26 - 1) X
(3 cos”p - 1 + n sir@ cos 2~u)/4]
(3)
where oa is the isotropic chemical shift, the second term in brackets is the scaled anisotropy with scaling factor P,(cos 8) = (3~0~~19- 1)/2, which corresponds experimentally to the scaled static powder spectrum, and w* = c, cos( 0,lf
y) + c, cos 2( w,c + y)
+ S, sin( o,t + r) + S, sin 2( w,t + r)
(4)
where Theory
The Hamiltonian of a spin system in an applied magnetic field can be written as H=H,+H”
(1)
where Hz = -woZ, is the Zeeman interaction, H” is the internal interaction of the spin system, and A stands for the type of internal interaction. The most important inhomogeneous interactions are the chemical shift interaction (A = cs) and the quadrupole interaction (A = Q). For spinning solids, H” can be written as [ll I H”=C”
c
c
I c
I c
(-1)‘”
C, = w,6 sin 2w sin 2p( 77cos (Y- 3)/4
(5)
C2=wo6[2sin’~+~(1+cos2~)
(6)
cos2a]/4
S, = -w,&q
sin 26 sin 2a/2
(7)
S, = -wo6n
sin26 cos p sin 2a/2
(8)
If w,6 is replaced by CQGQ = e2QV,,/6Z(21 llh, i.e., the anisotropy of quadrupole interaction, then eqns. (3)-(8) can also be used to describe Ho, Obviously, those time dependent terms modulated by W, and 2w, are due to sample spinning. Under the rapid spinning condition, i.e., w, B woS, the time dependent terms vanish. Therefore, in this case the anisotropy of the interaction can readily be obtained from the off-MAS spectrum if 6 is given. This method is of course much simpler than simulation of sideband intensities [61.
State Nucl. Magn. Reson. 1 (19921 103-109
105
Actually, however, due to the mechanical restrication of the rotor, the rapid spinning condition often cannot be realized when the experiment is carried out at high magnetic field (for the pursuit of higher resolution and sensitivity) or when the anisotropy of the interaction is large. It is expected that the central band of an off-MAS spectrum is no longer a scaled static powder spectrum and large errors would emerge when interaction tensor elements are evaluated from the central band. In fact, the time dependent part w* can cause the distortion of the central band with respect to the scaled static powder spectrum. Therefore, the effect of w* should be taken into consideration in the calculation of the spectrum pattern under the slow spinning condition. The FID signal for an inhomogeneous interaction, can be written as [2,5]
will simultaneously appear as noticed by Raleigh et al. [8]. Notice that the difference in eqn. (11) between MAS and off-MAS is that for off-MAS, the integration should be made not only for I,. but also for o0 since w” becomes anisotropic in the offMAS case. It is this significant difference that makes the central band of a slow off-MAS spectrum not only decreases in intensity, but also changes in pattern, i.e., the distortion with respect to the rapid spinning spectrum. The sidebands of an inhomogeneous interaction can be eliminated by means of TOSS [12]. For a MAS/TOSS spectrum, the intensity of the central peak can be restored to some extent. These two conclusions are valid for the offMAS/TOSS spectrum. When TOSS is used here, only the central band remains and eqn. (11) becomes
S. Ding et al. /Solid
g(t) = (I/SrTT2) ldo
exp( -@(a,
P, Y, f)} g(r) = (l/477)1
da F, exp( --io’t)
(9) where 4(t)
=
coot +j-“’dt
w*
From eqns. (3)-(lo),
we have
g(r) =(1/4~)C/
dfl IN exp[-i(wO+Nw,)t]
(IO)
N
(11) where I, = I FN I ’ FN = (l/2+,*=
(12)
d x exp[-i(Nx+4(x))] (13)
43(x) = (2C, sin x - 2S, cos x + C, sin 2x -s,
cos 2x)/2w,
( 14)
It is well known that when w, B w,S (rapid spinning condition), only I, (N = 0) in eqn. (11) is left, but when w, < wo6, the sidebands appear. In the MAS case, the slow spinning brings about the intensity loss o f the central peak, which has been studied by many authors [3-8,121. For slow offMAS, the intensity loss and the lineshape change
(15)
It can be seen from the above expression that the off-MAS/TOSS spectrum is still different from the scaled static powder spectrum since F, is anisotropic. It means that the distortion of the central band exists even though TOSS is applied. To confirm the above analysis, we have made some numerical calculations and simulations on off-MAS and off-MAS/TOSS spectra at various spinning rates, taking the chemical shift interaction as an example. Here we show two cases, one is the spectrum of an asymmetric chemical shift interaction (Fig. l), the other is that of an axially symmetric one (Fig. 2). Obviously, the distortions of both off-MAS and off-MAS/TOSS spectra decrease with the increase in the relative spinning rate w,/o& For a quantitative description of the spectrum distortion under the slow spinning condition, we define the following distortion factor: D=/
dw[f(w)
-f,(o)]‘,‘Aw
(16)
where do is the spectrum width, f(w) and f,,(w) are the normalized intensities of the central band under slow and rapid spinning conditions, respectively.
S. Ding et al./Solid
106
State Nucl. Magn. Reson. 1 (1992) 103-109
The relationship between D and R = wr/w,6 is calculated and diagramatically shown in Fig. 3 for different asymmetry parameters. As expected, the distortion becomes less and less as R increases. It can be seen that when R > 1, D vanishes. When R > 0.5, D is negligibly small for any case of q. Therefore, the condition R > 0.5 can be approximately referred to as the distortionless condition. We can also see that as R > 0.3, TOSS can reduce distortion, but when R < 0.3, the distortion of an off-MAS/TOSS spectrum may be larger than that of an off-MAS spectrum without
165
90
241I
PPm
Fig. 2. Same as Fig. 1 but with axially symmetric chemical shift interaction. The parameters used are: g,, = 90 ppm, eI = 240 ppm and P,kos 6) = -0.5.
R=0.2
TOSS. When R is very small, the distortion of an off-MAS/TOSS spectrum becomes very large. Here, we give an explanation of this characteristic. The ratio of intensity differences between the scaled static spectrum and the off-MAS spectra with and without TOSS is
:--.
dt exp( -iwt)
ppm Fig. 1. The off-MAS spectra of the asymmetric chemical shift interaction under various relative spinning rates, R = wr /w,S. The solid lines stand for rapid spinning cases or the scaleddown static lineshape, dashed lines for the slow spinning with TOSS and dotted lines for slow spinning without TOSS. The parameters used in this figure are: or, = 60 ppm, q2 = 120 ppm, ms3= 240 ppm and P,(cos 6) = - 0.5.
X
/
d0( 1 - Fa) exp( --hot)
I
dt exp( --icoOt)
X / dL!(l - I F, I’) exp( --iw’t) I
(17)
S. Ding et al. /Solid
107
State Nucl. Magn. Reson. 1 (1992) 103-109
From eqns. (S)-(8), (13) and (14) we can see that when the relative spinning rate R = wJw,6 is large, both F,, and I F, I 2 are positive and the change of F, with (a, p) is smaller than that of I F, I ‘, hence r < 1, meaning that the distortion of an off-MAS spectrum can be partly eliminated by TOSS. In the limit, when R is so small that the change of -F, is greater than that of I F. I 2, therefore r > 1, i.e., at a very slow spinning rate, the TOSS makes the off-MAS spectrum more distorted. The above calculations show that the TOSS can partly restore the intensity of central band only when the relative spinning rate is not too slow, e.g., R > 0.3. This means that there is a restrication on TOSS which has been widely used in solid state NMR. A similar phenomenum also
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12.0
I
2.0 L 12.01
I
b 8.0 6.0 __a-
___.~~ ________.-------_.
4.0 1
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a ..--------
;:JJi-:ll:_I:g -0.4
1.4
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Fig. 4. The relationship between D and S = 2P,(cos -9) = 3 cos24 - 1. Solid line: off-MAS/TOSS spectrum: dashed line; off-MAS spectrum. (r = 200 ppm, 6 = 100 ppm. (a) T= 0, o,= 3 kHz; (b) ~=0.5, w,=5 kHz; and Cc) 7 =l, w,=5 kHz.
0.0’ ‘--I
_I.-~-.-
/+L--.0.4 ._.
0.2
_____
’
R
0.6
’
0.8
I.0
’
Fig. 3. The relationship between the distortion factor D and relative spinning rate R = wr/w06. Solid line corresponds to the off-MAS/TOSS spectrum and the dashed line to the off-MAS spectrum: o0 = 200 ppm, 6 = 100 ppm and P,(cos 6) = 0.5. (a) TJ= 0: (b) 77= 0.5; and Cc) 7 = 1.
occurs in MAS experiments [13] where TOSS is ineffective when R is very small. Also noted from Figs. 1-3 is that the more symmetric the interaction tensor, the less the distortion of its spectrum. We find that D also depends upon 6, the direction of the spinning axis relative to the applied magnetic field. Figure 4 shows the relationship between D and S, here S = 2P, (cos 19) = 3 co?19 - 1. The general behaviour is that no matter how large are the spinning rate, the anisotropy and asymmetry parameters of the interaction, and no matter whether TOSS is used or not, D decreases monotonically with the increase of 6 (0’ through 90’). This fact can be understood from eqns. (5)--(8): as 6 increases,
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S. Ding et al. /Solid
the decrement in S, and S, is greater than that of C, and C,, so both F, and IF, 1’ decrease with the increase of 6. Therefore, the distortion of a slow MAS spectrum is not at a minimum, although it has highest resolution. We can further see from Fig. 4 that TOSS can partly eliminate the distortion as long as R is not very small.
rate was 4 kHz. For many L-amino acids, in which the anisotropy of the 13C chemical shift interaction in the carboxyl group is as large as 100 ppm, the relative spinning rate R is - 0.4. According to the theory, TOSS can partly restore the spectrum pattern. The experimental and theoretical results for the chemical shift powder spectrum of the carboxyl carbon in L-histine, for instance, are shown in Fig. 5. From this computer simulation, the chemical shift tensor components of this carbon are 116, 168 and 235 ppm, respectively, in good agreement with the results for the sideband intensity simulation of the MAS spectrum. For other samples, similar results were obtained.
Experimental For slow spinning solids, the measurement of an inhomogeneous interaction tensor is no longer as simple as that under rapid spinning condition. The distortion due to an insufficient spinning rate should be taken into consideration. But the interaction tensor can still be elucidated from the experimental spectrum with computer simulation based on the above theory. The experimental measurement of the chemical shift tensor components of 13C in some Lamino acids agree with the above theoretical analysis. The spectrometer used in our experiments was a Bruker MSL-400, with a working frequency for 13C of 100.63 MHz. The regular cross-polarization sequence is used. The contact time was taken as - 1 ms. The distortion of spectra caused by the inhomogeneous crosspolarization was neglected. The sample spinning
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180
.__..
160
140
State Nucl. Magn. Reson. 1 (19921 103-109
Conclusions For an inhomogeneous interaction in a spin system under off-MA& we conclude that when the rapid spinning condition is reached, an offMAS spectrum is exactly as the scaled static powder spectrum with a scaling factor P,(cos 8). When the rapid spinning condition cannot be realized, the distortion of the central band is inevitable. Approximately, as the relative spinning rate R is greater than 0.5, the distortion can be neglected; if R is greater than about 0.3, TOSS can be used to reduce the distortion. Moreover, the greater the angle between the spinning axis and the applied magnetic field, the less the distortion is. We further make a remark about the off-MAS experiment for measuring anisotropies: it has no preference over the sideband analysis approach [6] since the calculation of a distortive spectrum under the slow off-MAS condition is not time saving as compared to the simulation of sideband intensities. Nevertheless, the present work can be regarded as a general description of the off-MAS inhomogeneous interaction.
ppm
Fig. 5. The chemical shift powder spectrum of 13C of the carboxyl carbon of L-histine. The solid line is the theoretical result (with TOSS) and the dashed line is the experimental result. The sample spinning rate is 4 kHz and P,(cos a)= -0.06. The 13C chemical shift tensor components of the group were obtained from computer simulation as: oI1 = 116 ppm. gZ2 = 168 ppm and 033 = 235 ppm.
Acknowledgment This work was supported by the National Natural Science Foundation of China.
S. Ding et al. /Solid State Nucl. Magn. Reson. 1 (1992) 103-109
References 1 U. Haeberlen, High Resolution NMR in Solids: Selectke Ar.eraging, Academic Press, New York, 1976. 2 M. Mehring. Principles of High Resolution NMR in Solids, Springer-Verlag, Berlin, 1983. 3 J. Waugh, M. Maricq and R. Cantor, J. Magn. Reson., 29 (1978) 183. 4 M. Maricq and J. Waugh, Chem. Phys. Lett., 47 (1977) 327. 5 M. Maricq and J. Waugh, J. Chem. Phys., 70 (1979) 3300. 6 J. Herzfield and A. Berger. J. Chern. Phys.. 73 (1980) 6027.
1OY 7 E.T. Olejniczak, S. Vega and R.G. Griffin. J. Clzem. Phys., 81 (1984) 4804. 8 D.P. Raleigh, E.T. Olejniczak and R.G. Griffin, .L Chem. Phys., 89 (1988) 1333. 9 C. Ye, B. Sun. G. Maciel, J. Magn. Reson.. 70 (1986) 241. IO E.M. Menger. D.P. Raleigh and R.G. Griffin. J. Magn. Resort. , 63 (1985) 579. II N. Sethi, D.M. Grant and R.J. Pugmire, J. Mugn. Reson., 71 (1987) 476. 12 W.T. Dixon, J. Chem. Phys., 77 (1982) 1800. 13 B. Sun and C. Ye. Chin. J. Magn. Reson., 2 (19X5) 364.
