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Measurement of quadrupolar spin population by solid-state NMR
JOURNAL
OF MAGNETIC
77, 14% 154 (1988)
RESONANCE
NOTES Measurement of QuadrupoIar Spin Population by Solid-State NMR P. P. MAN* Laboratoire des Dispositifs Infra-Rouge et Physique Thermique, C.N.R.S., UA836, ESPCI, IO Rue Vauquelin 75231, Paris Cedex 5, France Received
May
5, 1987; revised
July
3 1, 1987
NMR has recently become a routine method for structural investigations in solidstate chemistry (I). Useful information like chemical shifts of nuclei in different crystallographic environments and relative populations is easy to obtain. For nuclei with spin 1 > f, the strength of the RF pulse WRFand its length tr are very important and affect the lineshapes as well as relative intensities (2-4), because wRF is not always strong enough to permit neglect of the quadrupolar interaction during the irradiation of the spin system. The value of wo/2r varies from zero to several megahertz, depending on the nuclei and the structure of the compound, whereas ww/27r is about 50 kilohertz. In the last few years, it has therefore become customary to specify the irradiation conditions (5). In the two extreme cases, ho 9 ww and 4 ORF, the pulse length which maximizes the FID is related to WRFby (6, 7) WQ
‘dRFt, P = - , 2
and T o&T
=-)
121
2
respectively. It is worth noting that tl* = (1+ &)tr . In the former case, only the central transition is irradiated, and in the latter all the transitions. When the pulse length is short enough, signal intensity is independent of (2, 4). For I= ;, analytical results are available for any value of , and wRF. They have been obtained either via the matrix formalism (8) or using the fictitious spin-4 operator formalism (9). I give some experimental results which show the effects of the pulse length on the line intensity measurement for spin I = i. Two systems are studied: (a) 23Na nuclei in a mixture of powdered NaCl and NaN02. For the sodium nuclei in NaN02, e2qQ/ h = 1.1 MHz and 17= 0.1, whereas in NaCl, e2qQ/h = 0. The NaCl/NaN02 molar ratio R’ in the sample was 6.5. (b) ‘*B in powdered ferroelectric colemanite CaB304(OH)3H20; the unit cell contains four polyanions [B304(OH),12-, each conuQ
tI
* Present address: CB2 IEP, England. 0022-2364/t%
Department
uQ,
of Physical
$3.00
Copyright 0 1988 by Academic Press, Inc. All rights of reproduction in any form reserved.
Chemistry,
University
148
of Cambridge,
Lensfield
Road,
Cambridge
149
NOTES
sisting of a trigonal B03 (e2qQ/h = 2.4 MHz) and two tetrahedral B02(OH)2 and B03(0H) units (e2qQ/h = 1.4 MHz) @a, ZZ). The details of the computation of the line intensity of a spin Z = 5 system have been given earlier (9). Only the main points are repeated here. The initial value of the FID is equal to the area of the line. This initial point F(t, , t2 = 0) is lost in the deadtime of the receiver. On the other hand, it is also related to the density matrix p(tl , t2 = 0) at the end of the pulse by F(t,, t2 = 0) = Tr[p(tl,
t2 = O)Z,].
[31
The normalization factor is chosen to be equal to unity instead of (21 + 1). In this particular case, we have z, = &(Z$' + zy> + 21$3, [41
t51
z, = 3(Z$2 + z;,4) + 4z$3,
where Z’s2 Z$*, Zy , and Z;” are fictitious spin-i operators associated with the two satellite &sitions, whereas Z$’ and Zf,’ are associated with the central transition (Z0). The numerical factors, 3 and 4 in Eq. [Sj, are related respectively to the intensity of the satellite lines and of the central line of a spectrum obtained with a cw spectrometer. The central line corresponds to 4/10 of the total intensity. From Eq. [3], we deduce for the central transition Fc(tl, t2 = 0) = 2 Tr[p(tl,
t2 = O)Z$3],
[61
t2 = O)Z$2].
[71
and for a satellite transition Fs(tl, t2 = 0) = & Tr[p(t,,
Thus FC(tl,
12
= 0) = - k (A2,jsin
cd2,3tl
+ A1,,sin 0 1,4tl +
Al,2sin
J3 Fs(tl, t2 = 0) = - 4 {Z32,3sinw2,3tl + Bj,4sin w l,dl + Bl,in
+ A#in
q2tl
qdl
03,4tl},
+ &,4sin w,dl).
The expressions of the A,, B,, and oii are reported in the Appendix where some minor mistakes of the previous paper are rectified (9). Figure 2 in Ref. (9) and Fig. 1 in this note give respectively F”(t, , t2 = 0) and Fs(tl , t2 = 0) versus the pulse length for several values of with uRF/2r = 50 kHz. Both figures are useful for single crystal study because depends on the orientations of the efg tensor. For the central transition, the maximum of the signal as well as the associated pulse length decrease when increases, but both reach limiting values. The result is a loss of line intensity, and the magnetization seems to precess faster. For this special case (I = i), Eqs. [l] and [2] are verified. There appears also a linear region, defined by < 0.5 PS, where the line intensity = 0) is proportional to and independent of This linear region is therefore valid for a distribution of uQ
uQ
uQ
tl
tl
uQ.
F”(t,,
wQ.
This occurs in a powdered sample.
t2
150
NOTES
L
I
I
1
I
I
0
1
2
3
4
5
tl (y=c) FIG. 1. The intensity of a satellite line of a Z = 4 spin system versus the pulse length t, for several values of wQ. w&~?T = 50 kHz.
For the satellite transition, the maximum of the signal as well as the associated pulse length decrease toward zero, when uQ increases, without reaching some finite limit as in the case of the central transition. The line intensity ratio 3/10:4/10:3/10 for I = f spins is correct only when uQ < wRF. Furthermore, there is no linear region because the magnetization rotates faster and faster when uQ increases. This is due to the increase of the deviation of the satellite lines from the Larmor frequency. The 23Na MAS experiments were performed on a Bruker CXP300 and the sample was spun at about 3.5 kilohertz with an Andrew-Beams rotor. The value of t, was increased from 0.25 to 4.5 ps in steps of 0.25 ps. For each t, ,20 scans were accumulated with a recycle time of 60 s due to the long spin-lattice relaxation time of 23Na in NaCl. The “B experiment was performed on a Bruker CXPlOO without magic-angle spinning because the linewidths are very broad. The pulse length was 0.25 ps with a recycle time of 10 s and 5000 scans were accumulated. The a/2 pulse duration of 2.2 puswas determined in trimethyl borate solution. 23Na spectra are given in Fig. 2 for increasing tr . The symmetrical and sharp lineshape is that of NaCl. The other shape, which has a shoulder, is that of NaN02. It is typical of the powder spectrum of the central line of a compound rotating at the magic angle and characterized by an asymmetry parameter r - 0. This figure illustrates well Fig. 2 in Ref. (9). The magnetization of the 23Na nucleus in NaN02 seems to precess twice as fast as that of the 23Na nucleus in NaCl. The ratio R of the two line intensities changes with tr. If tr were to be disregarded, any intensity ratio could be obtained. For the shortest pulse length (tr < 0.25 ps), R is equal to 2.5 1 (Fig. 3). It is quite different from the
151
NOTES
I
I
0
-1.5 (kHz)
FIG. 2. Stacked plot of 23Na nuclei spectra in a powdered mixture of NaCl and NaN02 versus the RF pulse length t, .
molar ratio R’. But the central line of NaN02 represents only 4/10 of the total signal whereas all the line intensity of NaCl is present. So we must multiply R by 10/4 to obtain the ratio ;
R = 6.28.
For the other quadrupolar spins, in a similar situation, we must multiply R by 35/9 for I = 3, 84/16 for I = $, or 165/25 for I = g. This experiment with a model compound in which 23Na nuclei are surrounded by an efg only in NaNO* shows the effect of the ti on the line intensities. In the case of colemanite, ’ ‘B nuclei are in two efg’s associated with two kinds of sites. As the sample is a powder, we expect two central lines in the spectrum. But the spectrum (Fig. 4) seems to have three peaks. This means that the two small outer peaks correspond to 2.51
I
I 1
I
I
I
0
I
I
I
-2
I
c
-3
(Kt$ FIG.
= 0.25
3. /a).
23Na spectrum of powdered mixture of NaCl and NaN02 obtained with a short RF pulse (tI
152
NOTES
c a
I
I
50
I
0
-50
(kHz) FIG. 4. (a) “B spectrum of powdered ferroelectric colemanite CaB304(OH)3H20 obtained with a short RF pulse (t, = 0.25 ps). (b) Simulated spectrum of tetrahedrally coordinated “B in colemanite.
the central line of the trigonal sites, whereas the sharp one corresponds to the central line of the tetrahedral sites. I have checked that at a higher Larmor frequency (CXP300), there is only one signal in the spectrum. I have simulated the spectrum of “B nuclei in tetrahedral sites with a Gaussian lineshape (Fig. 4b). Incidentally, a peak with a symmetrical lineshape can represent either a central line (“B nuclei in colemanite) or all the spectrum (23Na nuclei in NaCl). We know that the tetrahedral sites represent 2/3 of the sites. Experimentally, the ratio of the two line intensities plotted in Figs. 4a and 4b gives 3.59 5.34
2 3.
1101
In summary, pulse length is a critical parameter for quantitative determination in a quadrupolar spin system. Although analytical calculations were only carried out with the first-order term of the quadrupolar interaction, the results show that the irradiation condition remains valid even when the second-order effect is present. But we must check if all the absorption lines in the spectrum are central lines or not. The ideal way to distinguish these two situations is to perform the experiments at several static magnetic fields. Since the linewidth of the central transition is inversely proportional to the Larmor frequency, its shape changes and becomes sharper when wg increases. If we do not have access to several static magnetic fields, we can perform some quadrupolar nutation experiments (2, 9, 12, 13). An anisotropic lineshape along the F, dimension gives a clear indication of the presence of the quadrupolar interaction,
153 APPENDIX
We have chosen to define the Hamiltonian netic field by ZRF
associated with the radiofrequency mag-
= %FIx,
with t = f 1, instead of rRF = -oRFiT,. In this case, the Ai,j, Bi,j and wi,j are defined by the following expressions: A*,3 = (cm o+ - 2 cos o-)(cos o+ + cos e_>, A1,4 = (cm O+ + 2 cos Be)(cos O+- cos k), A1,2 = -(sin O+ - 2 sin k)(sin 19++ sin fl-), A3,4 = (sin 19++ 2 sin k)(sin 8+ - sin k), B2,3 = -(cos d+ - 2 cos O-)sin O-, B1,4 = (cos O++ 2 cos B-)sin O-, B1,2 = -(sin 8+ - 2 sin &)cos B-, B3,4 = -(sin 0, + 2 sin O-)cos O-, @2,3
=
E‘%F
+
W2,4/2
+
w,3/2,
W1,4
=
EwRF
-
W2,4/2
-
w1,3/2,
W1,2
=
-(eWRF + W&4/2 - WI,@),
03,4
=
~WRF
W2,4
=
-E
WI,3
= d(2WQ f EW& + 3&F,
-
W2,4/2
+
q3/2
(~WQ
-
~WRF)~
+
3W&,
0-t = 8, + 02,
ACKNOWLEDGMENTS The author thanks Dr. J. Klinowski for critical reading of the manuscript, and Dr. A. Trokiner and Dr. H. Zanni for helpful discussions. REFERENCES 1. 2.
C. A. FYFE, “Solid State NMR for Chemists,” C.F.C. Press, Guelph, Ontario, Canada, 1983 A. SAMOSONAND E.LIPPMAA, Phys.Rev. B 28, 6567 (1983).
154
NOTES
3. D. FREUDE, Adv. Colloid Interface Sci. 23, 21 (1985). 4. D. FENZKE, D. FREUDE, T. FROHLICH, AND J. HAASE, Chem. Phys. Lett. 111, 17 1 (1984). 5. (a) G. L. TURNER, K. A. SMITH, R. J. KIRKPATRICK, AND E. OLDF’IELD, J. Magn. Reson. 67, 544 (1986); (b) L. B. ALEMANYAND G. W. KIRKER, J. Am. Chem. Sot. 108,6158 (1986). 6. A. ABRAGAM, “Principles of Nuclear Magnetism,” Oxford Univ. Press, London, 196 1. 7. V. H. SCHMIDT, Proc. Ampere Int. Summer School Znd, 75 ( 197 I). 8. LAKSHMAN PANDEY, S. TOWTA, AND D. G. HUGHES, J. Chem. Phys. 85,6923 (1986). 9. P. P. MAN, J. Magn. Reson. 67, 78 (1986). 10. R. R. ERNST, G. BODENHALJSEN, AND A. WOKAUN, “Principles of NMR in One and Two Dimensions,” Oxford Univ. Press (Clarendon), London/New York, 1987. II. H. THEVENEAU AND P. PAPON, Phys. Rev. B 14,381O (1976). 12. A. P. M. KENTGENS, J. J. M. LEMMENS, F. M. M. GEURTS, AND W. S. VEEMAN, J. Magn. Reson. 71,
62 (1987). 13. (a) A. TROKINER,
P. P. MAN, H. THEVENEAU, AND P. PAPON, Solid State Commun. 55, 929 (1985); (b) P. P. MAN, H. THEVENEAU, AND P. PAPON, J. Magn. Resort. 64, 271 (1985); (c) P. P. MAN, “Proceedings, XXIII Congress Ampere on Magnetic Resonance, Rome, 1986,” p. 574.
OF MAGNETIC
77, 14% 154 (1988)
RESONANCE
NOTES Measurement of QuadrupoIar Spin Population by Solid-State NMR P. P. MAN* Laboratoire des Dispositifs Infra-Rouge et Physique Thermique, C.N.R.S., UA836, ESPCI, IO Rue Vauquelin 75231, Paris Cedex 5, France Received
May
5, 1987; revised
July
3 1, 1987
NMR has recently become a routine method for structural investigations in solidstate chemistry (I). Useful information like chemical shifts of nuclei in different crystallographic environments and relative populations is easy to obtain. For nuclei with spin 1 > f, the strength of the RF pulse WRFand its length tr are very important and affect the lineshapes as well as relative intensities (2-4), because wRF is not always strong enough to permit neglect of the quadrupolar interaction during the irradiation of the spin system. The value of wo/2r varies from zero to several megahertz, depending on the nuclei and the structure of the compound, whereas ww/27r is about 50 kilohertz. In the last few years, it has therefore become customary to specify the irradiation conditions (5). In the two extreme cases, ho 9 ww and 4 ORF, the pulse length which maximizes the FID is related to WRFby (6, 7) WQ
‘dRFt, P = - , 2
and T o&T
=-)
121
2
respectively. It is worth noting that tl* = (1+ &)tr . In the former case, only the central transition is irradiated, and in the latter all the transitions. When the pulse length is short enough, signal intensity is independent of (2, 4). For I= ;, analytical results are available for any value of , and wRF. They have been obtained either via the matrix formalism (8) or using the fictitious spin-4 operator formalism (9). I give some experimental results which show the effects of the pulse length on the line intensity measurement for spin I = i. Two systems are studied: (a) 23Na nuclei in a mixture of powdered NaCl and NaN02. For the sodium nuclei in NaN02, e2qQ/ h = 1.1 MHz and 17= 0.1, whereas in NaCl, e2qQ/h = 0. The NaCl/NaN02 molar ratio R’ in the sample was 6.5. (b) ‘*B in powdered ferroelectric colemanite CaB304(OH)3H20; the unit cell contains four polyanions [B304(OH),12-, each conuQ
tI
* Present address: CB2 IEP, England. 0022-2364/t%
Department
uQ,
of Physical
$3.00
Copyright 0 1988 by Academic Press, Inc. All rights of reproduction in any form reserved.
Chemistry,
University
148
of Cambridge,
Lensfield
Road,
Cambridge
149
NOTES
sisting of a trigonal B03 (e2qQ/h = 2.4 MHz) and two tetrahedral B02(OH)2 and B03(0H) units (e2qQ/h = 1.4 MHz) @a, ZZ). The details of the computation of the line intensity of a spin Z = 5 system have been given earlier (9). Only the main points are repeated here. The initial value of the FID is equal to the area of the line. This initial point F(t, , t2 = 0) is lost in the deadtime of the receiver. On the other hand, it is also related to the density matrix p(tl , t2 = 0) at the end of the pulse by F(t,, t2 = 0) = Tr[p(tl,
t2 = O)Z,].
[31
The normalization factor is chosen to be equal to unity instead of (21 + 1). In this particular case, we have z, = &(Z$' + zy> + 21$3, [41
t51
z, = 3(Z$2 + z;,4) + 4z$3,
where Z’s2 Z$*, Zy , and Z;” are fictitious spin-i operators associated with the two satellite &sitions, whereas Z$’ and Zf,’ are associated with the central transition (Z0). The numerical factors, 3 and 4 in Eq. [Sj, are related respectively to the intensity of the satellite lines and of the central line of a spectrum obtained with a cw spectrometer. The central line corresponds to 4/10 of the total intensity. From Eq. [3], we deduce for the central transition Fc(tl, t2 = 0) = 2 Tr[p(tl,
t2 = O)Z$3],
[61
t2 = O)Z$2].
[71
and for a satellite transition Fs(tl, t2 = 0) = & Tr[p(t,,
Thus FC(tl,
12
= 0) = - k (A2,jsin
cd2,3tl
+ A1,,sin 0 1,4tl +
Al,2sin
J3 Fs(tl, t2 = 0) = - 4 {Z32,3sinw2,3tl + Bj,4sin w l,dl + Bl,in
+ A#in
q2tl
qdl
03,4tl},
+ &,4sin w,dl).
The expressions of the A,, B,, and oii are reported in the Appendix where some minor mistakes of the previous paper are rectified (9). Figure 2 in Ref. (9) and Fig. 1 in this note give respectively F”(t, , t2 = 0) and Fs(tl , t2 = 0) versus the pulse length for several values of with uRF/2r = 50 kHz. Both figures are useful for single crystal study because depends on the orientations of the efg tensor. For the central transition, the maximum of the signal as well as the associated pulse length decrease when increases, but both reach limiting values. The result is a loss of line intensity, and the magnetization seems to precess faster. For this special case (I = i), Eqs. [l] and [2] are verified. There appears also a linear region, defined by < 0.5 PS, where the line intensity = 0) is proportional to and independent of This linear region is therefore valid for a distribution of uQ
uQ
uQ
tl
tl
uQ.
F”(t,,
wQ.
This occurs in a powdered sample.
t2
150
NOTES
L
I
I
1
I
I
0
1
2
3
4
5
tl (y=c) FIG. 1. The intensity of a satellite line of a Z = 4 spin system versus the pulse length t, for several values of wQ. w&~?T = 50 kHz.
For the satellite transition, the maximum of the signal as well as the associated pulse length decrease toward zero, when uQ increases, without reaching some finite limit as in the case of the central transition. The line intensity ratio 3/10:4/10:3/10 for I = f spins is correct only when uQ < wRF. Furthermore, there is no linear region because the magnetization rotates faster and faster when uQ increases. This is due to the increase of the deviation of the satellite lines from the Larmor frequency. The 23Na MAS experiments were performed on a Bruker CXP300 and the sample was spun at about 3.5 kilohertz with an Andrew-Beams rotor. The value of t, was increased from 0.25 to 4.5 ps in steps of 0.25 ps. For each t, ,20 scans were accumulated with a recycle time of 60 s due to the long spin-lattice relaxation time of 23Na in NaCl. The “B experiment was performed on a Bruker CXPlOO without magic-angle spinning because the linewidths are very broad. The pulse length was 0.25 ps with a recycle time of 10 s and 5000 scans were accumulated. The a/2 pulse duration of 2.2 puswas determined in trimethyl borate solution. 23Na spectra are given in Fig. 2 for increasing tr . The symmetrical and sharp lineshape is that of NaCl. The other shape, which has a shoulder, is that of NaN02. It is typical of the powder spectrum of the central line of a compound rotating at the magic angle and characterized by an asymmetry parameter r - 0. This figure illustrates well Fig. 2 in Ref. (9). The magnetization of the 23Na nucleus in NaN02 seems to precess twice as fast as that of the 23Na nucleus in NaCl. The ratio R of the two line intensities changes with tr. If tr were to be disregarded, any intensity ratio could be obtained. For the shortest pulse length (tr < 0.25 ps), R is equal to 2.5 1 (Fig. 3). It is quite different from the
151
NOTES
I
I
0
-1.5 (kHz)
FIG. 2. Stacked plot of 23Na nuclei spectra in a powdered mixture of NaCl and NaN02 versus the RF pulse length t, .
molar ratio R’. But the central line of NaN02 represents only 4/10 of the total signal whereas all the line intensity of NaCl is present. So we must multiply R by 10/4 to obtain the ratio ;
R = 6.28.
For the other quadrupolar spins, in a similar situation, we must multiply R by 35/9 for I = 3, 84/16 for I = $, or 165/25 for I = g. This experiment with a model compound in which 23Na nuclei are surrounded by an efg only in NaNO* shows the effect of the ti on the line intensities. In the case of colemanite, ’ ‘B nuclei are in two efg’s associated with two kinds of sites. As the sample is a powder, we expect two central lines in the spectrum. But the spectrum (Fig. 4) seems to have three peaks. This means that the two small outer peaks correspond to 2.51
I
I 1
I
I
I
0
I
I
I
-2
I
c
-3
(Kt$ FIG.
= 0.25
3. /a).
23Na spectrum of powdered mixture of NaCl and NaN02 obtained with a short RF pulse (tI
152
NOTES
c a
I
I
50
I
0
-50
(kHz) FIG. 4. (a) “B spectrum of powdered ferroelectric colemanite CaB304(OH)3H20 obtained with a short RF pulse (t, = 0.25 ps). (b) Simulated spectrum of tetrahedrally coordinated “B in colemanite.
the central line of the trigonal sites, whereas the sharp one corresponds to the central line of the tetrahedral sites. I have checked that at a higher Larmor frequency (CXP300), there is only one signal in the spectrum. I have simulated the spectrum of “B nuclei in tetrahedral sites with a Gaussian lineshape (Fig. 4b). Incidentally, a peak with a symmetrical lineshape can represent either a central line (“B nuclei in colemanite) or all the spectrum (23Na nuclei in NaCl). We know that the tetrahedral sites represent 2/3 of the sites. Experimentally, the ratio of the two line intensities plotted in Figs. 4a and 4b gives 3.59 5.34
2 3.
1101
In summary, pulse length is a critical parameter for quantitative determination in a quadrupolar spin system. Although analytical calculations were only carried out with the first-order term of the quadrupolar interaction, the results show that the irradiation condition remains valid even when the second-order effect is present. But we must check if all the absorption lines in the spectrum are central lines or not. The ideal way to distinguish these two situations is to perform the experiments at several static magnetic fields. Since the linewidth of the central transition is inversely proportional to the Larmor frequency, its shape changes and becomes sharper when wg increases. If we do not have access to several static magnetic fields, we can perform some quadrupolar nutation experiments (2, 9, 12, 13). An anisotropic lineshape along the F, dimension gives a clear indication of the presence of the quadrupolar interaction,
153 APPENDIX
We have chosen to define the Hamiltonian netic field by ZRF
associated with the radiofrequency mag-
= %FIx,
with t = f 1, instead of rRF = -oRFiT,. In this case, the Ai,j, Bi,j and wi,j are defined by the following expressions: A*,3 = (cm o+ - 2 cos o-)(cos o+ + cos e_>, A1,4 = (cm O+ + 2 cos Be)(cos O+- cos k), A1,2 = -(sin O+ - 2 sin k)(sin 19++ sin fl-), A3,4 = (sin 19++ 2 sin k)(sin 8+ - sin k), B2,3 = -(cos d+ - 2 cos O-)sin O-, B1,4 = (cos O++ 2 cos B-)sin O-, B1,2 = -(sin 8+ - 2 sin &)cos B-, B3,4 = -(sin 0, + 2 sin O-)cos O-, @2,3
=
E‘%F
+
W2,4/2
+
w,3/2,
W1,4
=
EwRF
-
W2,4/2
-
w1,3/2,
W1,2
=
-(eWRF + W&4/2 - WI,@),
03,4
=
~WRF
W2,4
=
-E
WI,3
= d(2WQ f EW& + 3&F,
-
W2,4/2
+
q3/2
(~WQ
-
~WRF)~
+
3W&,
0-t = 8, + 02,
ACKNOWLEDGMENTS The author thanks Dr. J. Klinowski for critical reading of the manuscript, and Dr. A. Trokiner and Dr. H. Zanni for helpful discussions. REFERENCES 1. 2.
C. A. FYFE, “Solid State NMR for Chemists,” C.F.C. Press, Guelph, Ontario, Canada, 1983 A. SAMOSONAND E.LIPPMAA, Phys.Rev. B 28, 6567 (1983).
154
NOTES
3. D. FREUDE, Adv. Colloid Interface Sci. 23, 21 (1985). 4. D. FENZKE, D. FREUDE, T. FROHLICH, AND J. HAASE, Chem. Phys. Lett. 111, 17 1 (1984). 5. (a) G. L. TURNER, K. A. SMITH, R. J. KIRKPATRICK, AND E. OLDF’IELD, J. Magn. Reson. 67, 544 (1986); (b) L. B. ALEMANYAND G. W. KIRKER, J. Am. Chem. Sot. 108,6158 (1986). 6. A. ABRAGAM, “Principles of Nuclear Magnetism,” Oxford Univ. Press, London, 196 1. 7. V. H. SCHMIDT, Proc. Ampere Int. Summer School Znd, 75 ( 197 I). 8. LAKSHMAN PANDEY, S. TOWTA, AND D. G. HUGHES, J. Chem. Phys. 85,6923 (1986). 9. P. P. MAN, J. Magn. Reson. 67, 78 (1986). 10. R. R. ERNST, G. BODENHALJSEN, AND A. WOKAUN, “Principles of NMR in One and Two Dimensions,” Oxford Univ. Press (Clarendon), London/New York, 1987. II. H. THEVENEAU AND P. PAPON, Phys. Rev. B 14,381O (1976). 12. A. P. M. KENTGENS, J. J. M. LEMMENS, F. M. M. GEURTS, AND W. S. VEEMAN, J. Magn. Reson. 71,
62 (1987). 13. (a) A. TROKINER,
P. P. MAN, H. THEVENEAU, AND P. PAPON, Solid State Commun. 55, 929 (1985); (b) P. P. MAN, H. THEVENEAU, AND P. PAPON, J. Magn. Resort. 64, 271 (1985); (c) P. P. MAN, “Proceedings, XXIII Congress Ampere on Magnetic Resonance, Rome, 1986,” p. 574.