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125Te NMR studies of indirect and direct dipole-dipole coupling in polycrystalline CdTe, HgTe, and PbTe
JOURNAL
OF
MAGNETIC
RESONANCE
40,9-16
(1980)
12’TeNMR Studies of Indirect and Direct Dipole-Dipole Coupling in Polycrystalline CdTe, HgTe, and PbTe R. BALZ, M. HALLER, W. E. HERTLER, 0. LUTZ, A. NOLLE, AND R. SCHAFITEL Physikalisches
Institut
der Universitiit
Tiibingen,
Morgenstelle,
D-7400
Tiibingen,
Germany
Received October 15, 1979; revised December 12, 1979 The NMR signals of ‘*‘Te in CdTe, HgTe, and PbTe powders show a fine structure due to direct and indirect dipole-dipole interactions. A theoretical powder pattern lineshape is given, with which the isotropic and anisotropic contributions of the direct and indirect dipoledipole interaction can be separated. Experimental results for Jll and JL in CdTe, HgTe, and PbTe are presented. A linear dependence of the reduced coupling on the atomic number predicts a vanishingly small coupling for ZnTe and this has been confirmed.
INTRODUCTION
For high-resolution experiments in liquid samples the indirect spin-spin coupling is a very important phenomenon which, together with chemical shift and in some cases with relaxation rate data, makes possible the solutions of structural problems. Until now the indirect spin-spin coupling has played only a minor role in solid state NMR because the influence of this interaction can be detected in only a few cases. This is due to the fact that in solids the direct dipole-dipole interaction usually is not averaged out and for nuclei like ‘H or 19F the direct dipole-dipole coupling is nearly always stronger than the indirect spin-spin coupling. There are only a few solid samples for which the indirect spin-spin coupling has been observed without using special techniques (I-11). Some further data on indirect spin-spin coupling have been reported by Andrew et al. (12-14, who used the magic angle spinning technique, by which the direct dipole-dipole interaction is reduced. This method has the disadvantage, however, that any anisotropic contributions of the indirect spinspin coupling are also reduced. Therefore, it seemed to us that apart from our recent work on lz5Te and II3 Cd NMR in a single crystal and polycrystalline CdTe, no data on anisotropic indirect spin-spin coupling in solid samples have been published (9, 10). Samples like CdTe, HgTe, or PbTe are very favorable for such investigations because the chemical bonding is partly covalent (15, I@, and therefore, an indirect spin-spin coupling is possible. The direct dipoledipole coupling is rather weak due to the small gyromagnetic ratios of the nuclei involved and their small natural abundances, resulting in narrow lines for these solids. Further, one expects from their 9
0022-2364/60/100009-08$02.00/O Copyright @ 1980 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in Great Britain
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behavior in the liquid state that elements with high atomic number will have rather strong indirect spin-spin coupling. rz5Te NMR has not been widely used in solids (9, IO, 17-21) and for the few known examples apart from (9,10, 17) no high-resolution spectra in solids have been observed. But as we showed in (9, IO), with good homogeneity of the magnetic field and with high purity of the samples, high-resolution spectra of lz5Te in solid samples of CdTe, HgTe, and PbTe are possible and from these spectra information on the isotropic and anisotropic indirect spin-spin coupling can be obtained. EXPERIMENTAL
The NMR signal of “‘Te was observed in a magnetic field of 2.114 T at a Larmor frequency of 28.4 MHz with a multinuclei pulsed Bruker SXP 4-100 NMR spectrometer. The temperature was (298 f 3) K. The free induction decays following a 90” pulse were accumulated and Fourier transformed by a B-NC 12 computer. The magnetic field was externally stabilized by a ‘H NMR stabilization unit. Of the stable tellurium isotopes, only lz3Te and lz5Te have a nuclear spin I = i and a nuclear magnetic moment. The natural abundances of lz3Te and “‘Te are 0.9 and 7%) respectively. The NMR receptivity of 12’Te is about 2.2 x 10p3, compared with 1 for the proton for an equal number of hydrogen and tellurium atoms at natural abundance. The receptivity of *23Te is about one order of magnitude lower. The elements cadmium, mercury, and lead also consist mainly of such isotopes, which have either no nuclear spins and moments or which have spin I = i: lliCd, ‘i3Cd, 199Hg, and 207Pb. Only the 201Hg isotope has spin I = $, but the gyromagnetic ratio of this nucleus is rather small. Therefore, mainly interactions between spin-i nuclei will be observed in the i2’Te spectra of CdTe, HgTe, and PbTe. The NMR signals of 125Te were observed in CdTe, HgTe, PbTe, and ZnTe powder samples, which were purchased from Zinsser, Frankfurt. The purity of the samples was 99.999%, with the exception of ZnTe, which was 99.998% pure. The chemical -shift data of these samples have been reported elsewhere (9). THEORY
Imagine a spin system with two spin-4 nuclei 1 and 2, for which besides the Zeeman interaction a direct and indirect dipole-dipole coupling must be taken into account. If the principal axis system of the indirect spin-spin coupling tensor is the same as that of the direct dipole-dipole interaction and if the indirect spin-spin coupling tensor is also axially symmetric with the two principal values .J and JL, two signals are observed. For example, for nucleus 1 at the frequencies (JO), v=v,+J/2+(0’/2)*(1-3cos26),
v’=vo-J/2-(D/2).(1-3cos28),
[II
where v. = (1/27r) * yBo is the Larmor frequency, J = f (2J1 +J&
PI
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the isotropic indirect spin-spin coupling, and D’= D-:
(.T,-Jl)
[31
is the anisotropic part of the indirect spin-spin coupling plus the direct dipole-dipole coupling, which always is anisotropic (22). PO YlY2fi r D=GT
[41
determines the direct dipole-dipole coupling, y1 and y2 are the gyromagnetic ratios of the respective nuclei 1 and 2, r is their internuclear distance, and 8 is the angle between the internuclear axis and the static magnetic field. From Eq. [l] the lineshape function of a powder pattern can be calculated easily according to, for example (23): f(v)-(
v-1+-J/2 D,,2
l-
)
-II2
[51
+( l+V-;;2J’2)-1’2.
This formula means that the two parts of the well-known “Pake-doublet” are shifted by J/2 and -J/2, as shown in Fig. 1. Further, the splitting is not only determined by the direct dipole-dipole coupling but also by the anisotropic part of the indirect spin-spin coupling. From the frequencies, for which in Fig. 1 divergences or shoulders occur, Jiband JI can be evaluated if values of D are known. For example, the frequency separation of the divergences of the powder pattern is equal to D’+J=D+J,
[61
and the frequency separation of the shoulders is 2D’-J=2D-4,.
[71
To calculate the theoretical lineshapes of 12’Te powder spectra in CdTe, HgTe, and PbTe, one must consider the crystallographic data of these materials, which are given in Table 1, and also the natural abundances of the isotopes rlrCd, ‘r3Cd, 199Hg, and 207Pb. 201 Hg has been omitted in this list because of its small gyromagnetic ratio
I’ -
!
’ v,-J/2-D/2
FIG.
coupling spectrum
v,+J/Z-D’
v;J/2.0’
I
Y
v,,+J/Z+D’/2
1. Theoretical powder spectrum for a two spin-$ system with direct as calculated from Eq. [5]. The meanings of J and D’ are explained is plotted for a ratio of D’ : J = 7 : 10.
and indirect dipole-dipole in Eqs. [2] through [4]. The
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Sample CdTe HgTe PbTe
ET AL.
TABLE CdTe,HgTe,
1 AND
PbTe ASTAKENFROMREF.(Z~)
Structure
Space group
Length of the cubic unit cell (lo-‘” m)
ZnS ZnS NaCl
T:F43m TiF43m OiFm3m
6.481 6.429 6.454
Distance between nearest neighbors il O-‘” m) 2.806 2.784 3.227
and its quadrupole moment. It follows from the crystallographic data that a tellurium nucleus in CdTe or HgTe has four nearest cadmium or mercury neighbors and six lead neighbors in PbTe. With natural abundances of “‘Cd and ‘13Cd together of 25%, lg9Hg of 16.8%, and ‘07Pb of 22.6%, the probabilities that 0, 1, 2, . . . of the nearest neighbors of a tellurium atom are 111*113Cd, 199Hg, or *“Pb atoms can be evaluated. If no neighboring atoms have a nuclear magnetic moment, dipole-dipole interaction cannot occur, and only a single signal at the frequency v. is observed for these 125Te nuclei. If of the four (six) neighbors one is an isotope with a nuclear magnetic moment, dipole-dipole interaction takes place and a lineshape as given, for example, in Eq. [5] is expected. If two of the neighbors are such isotopes with a nuclear magnetic moment, a triplet signal with intensities 1: 2 : 1 can be predicted. However, the probability of more than one isotope with a nuclear magnetic moment being the neighbor of a tellurium atom is small and the intensities of the corresponding satellites can be neglected. The strong central component of the triplet coincides with the unsplit 125Te signal and makes a significant contribution, as long as isotope effects are small. From these considerations one gets the following ratios of the integrated intensities of each of the satellites due to a spin-spin coupling with one nearest neighbor to the intensity of the central signal: for CdTe, 0.50 : 1; for HgTe, 0.36 : 1; and for PbTe, 0.53 : 1. Using these ratios and the appropriate Gaussian and Lorentzian lineshape functions for the convolution with the theoretical lineshape given in Eq. [5] one can calculate theoretical spectra of ‘*‘Te in CdTe, HgTe, and PbTe. The parameters of the dipole-dipole coupling can be evaluated by comparison of the theoretical and experimental spectra by a fit procedure. RESULTS
AND
DISCUSSION
The powder patterns of ‘*‘Te in CdTe, HgTe, and PbTe and of r13Cd in CdTe have been measured carefully with high signal-to-noise ratios. In Figs. 2 and 3 examples of powder spectra of 125Te in HgTe and PbTe are given. Signals of ‘25Te and of ‘13Cd in CdTe powder have been reported in (9, 10). All these spectra show satellites whose intensity is proportional to the probability that just one of the four (six) neighbors of the observed nuclei is a ‘llCd, ‘13Cd, ‘*‘Te, 199Hg, or *07Pb nucleus. The splitting was the same for the ‘13Cd and I*’ Te signals in CdTe. These facts indicate a spin-spin interaction between ‘25Te and ‘13Cd, 199Hg, or *07Pb. From a single-crystal
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7 kHz FIG. 2. Uppertrace: Experimental ‘*‘Te NMR absorption signal in HgTe powder. The satellites are due to direct and indirect spin-spin coupling between ‘*‘Te and 199Hg. Resonance frequency, 28.371 MHz; experimental spectrum width, 50 kHz; trigger frequency, 0.1 Hz; measuring time, 7 hr; 200 data points were accumulated followed by 7992 points of zero-filling before the Fourier transformation of 8 K data points. Lower truce: Theoretical lineshape with the data of Table 2.
measurement of *“Te in CdTe the parameters of the direct and indirect spin-spin splitting were evaluated (10). From the satellite signals of a powder spectrum the value of Jl using Eq. [6] and the values of D quoted, for example, in Table 2 can be roughly determined, whereas the values of Jll cannot be given by a simple inspection of the spectrum because of the strong central signal. Therefore, the parameters Jil and JL were evaluated from D’ and J, which were determined by the three-parameter least-squares fit procedure mentioned above. By this computer program the linewidth of the Gaussian or Lorentzian function used for the convolution of the lineshape function and the parameters J and D’ in Eq. [5] were changed. The assumption of an axial symmetry of the indirect spin-spin coupling tensor, which was used for the derivation of Eq. [5], seems reasonable for chemical bonding (30) and, in addition, the principal vector belonging to Jll is assumed to be parallel to the internuclear axis. For the CdTe powder spectrum a Lorentzian lineshape with a halfwidth of 220 Hz and the values of D’ and J given in Table 2 yielded the best result of the fit procedure. The values of JL are mainly determined by the frequencies of the satellite maxima
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3 kHz FIG. 3. Upper trace: Experimental ‘25Te NMR absorption signal in PbTe powder. The satellite signals are due to a direct and indirect spin-spin coupling between ‘*‘Te and “’ Pb. Resonance frequency, 28.409 MHz; measuring time, 35 hr; allother parameters are identical to those in Fig. 2. Lower trace: Theoretical lineshape with the data of Table 2.
and are not very sensitive to whether a Gaussian or a Lorentzian function is used for the convolution. Because the shoulders of Fig. 1, which according to Eq. [7] determine the values of .Q, are hidden by the central signal, the values of .Q depend on the chosen broadening function. In the case of CdTe the standard deviation for the Lorentzian function is about a factor of 2 smaller than for the Gaussian function, so the fitted data using the Lorentzian function for the convolution are more reliable TABLE 2 ISOTROPICANDANISOTROPICINDIRECTSPIN-SPINCOUPLINGCONSTANTSINC~T~,H~T~,AND PbTe” Sample CdTe b CdTe HgTe PbTe
Interacting nuclei
(D’-D) U-W 380(l) 380(l) 308(l) 236(l)
llO(50) 60(100) 1640(600) 1340(400)
41 (Hz) 655(60) 680(100) 5080(600) 2150(400)
765(80) 740(150) 6720(850) 3480(560)
435(120) 560(180) 1800(1000) -520(700)
a The values of the direct dipole-dipole coupling D have been evaluated from Eq. [4] using the values in Table 1 and the gyromagnetic ratios of Refs. (9,25-29). b Results of a single-crystal measurement taken from (10).
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and they are in agreement with the more precise values of the single-crystal measurement (10). This agreement between the single-crystal results and those obtained by the procedure described above enables us to evaluate the spectra of lz5Te in HgTe and PbTe in the same manner. Again, for both samples the fit using a Lorentzian broadening function yielded a smaller standard deviation of the experimental signal from the theoretical spectrum. The results for the spin-spin coupling data are given in Table 2. The values of Jil and Jl for CdTe and HgTe can be compared, because both samples have a zincblende arrangement. As expected, the indirect spin-spin coupling increases from CdTe to HgTe and is anisotropic for both compounds. Assuming a linear dependence of the reduced coupling constant (J/Y~~~)~‘* in the series ZnTe, CdTe, and HgTe on the atomic number of Zn, Cd, and Hg (see, e.g., (5)), from two measured values the indirect spin-spin coupling constant J between 67Zn and **‘Te in ZnTe, which also has a zincblende arrangement, can be estimated. It is nearly zero and D is about 90 Hz. For “‘Te in ZnTe, however, only a single signal with a signal-to-noise ratio of about 100 and a linewidth of 260 Hz was observed. Why is no splitting observed in ZnTe? This is due to the low natural abundance of 67Zn (only 4%) and further due to the spin I = 2. Therefore, instead of a doublet 21+ 1= 6 signals must occur for those ‘25Te nuclei that have only one neighboring 67Zn nucleus, together with an unsplit central signal. The intensity of the satellites is 36 times smaller than the intensity of the central signal. Such small satellites could not be detected. The data for the indirect spin-spin coupling between 207Pb and *25Te in PbTe are somewhat different from the data of CdTe and HgTe. .J seems to be much smaller, but since the error is rather large, it is not quite clear whether the sign of Jil is really different from the sign of JL or not. CONCLUSIONS
In solid compounds of heavy elements an indirect spin-spin coupling is observable in favorable cases due to the fact that it is large compared with the direct dipoledipole coupling. From powder samples the anisotropy of the indirect coupling can be determined with the procedure given. The result for CdTe powder compares well with that from the single-crystal measurement. Indirect spin-spin couplings in the solid are found to be large for the heavy atoms, as in the case of liquids, and the coupling constants increase with increasing atomic number in the IIb-tellurides. ACKNOWLEDGMENTS We would like to thank Professor Dr. H. Kriiger for his continuous support of this work and the Deutsche Forschungsgemeinschaft for financial support. REFERENCES 1. M. A.RUDERMANNAND C.K~-rr~~,f%ys. Rev.96,99 (1954). 2. N.BLOEMBERGENANDT.J.ROWLAND, Phys.Rev.97,1675(1955). 3. K.YOSIDA, Phys. Rev. 106,893 (1957).
16 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.
25. 26. 27.
28. 29. 30.
BALZ E. E. A. A. A. W. A. E.
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FUKUSHIMA AND S. H. MASTIN, J. Magn. Reson. 1,648 (1969). FLJKUSHIMA, J. Chem. Phys. 55,2463 (1971). KAWAMORI, R. TAKAGI, AND G. SODA, Mol. Phys. 25,489 (1973). KAWAMORI AND G. SODA, Mol. Phys. 29, 1085 (1975). KAWAMORI, J. Magn. Reson. 31, 423 (1978). KOCH, 0. LUTZ, AND A. NOLLE, 2. Phys. A 289,17 (1978). NOLLE, 2. Phys. B 34, 175 (1979). R. ANDREW, W. S. HINSHAW, M. G. HUTCHINS, AND A. JASINSKI, Proc. Roy. Sot. London Ser. A 364, 553 (1978). E. R. ANDREW, L. F. FARNELL, AND T. D. GLEDHILL, Phys. Rev. Left. 19, 6 (1967). E. R. ANDREW, M. FIRTH, A. JASINSKI, AND P. J. RANDALL, Phys. Lett. 31A, 446 (1970). E. R. ANDREW, in “Progress in NMR Spectroscopy” (J. W. EMSLEY, J. FEENEY, AND L. H. SUTCLIFFE, Eds.), Vol. 8, p. 1, Pergamon Press, Oxford, 1971. J. C. PHILIPS, Rev. Mod. Phys. 42, 317 (1970). P. J. STILES, Solid State Commun. 11, 1063 (1972). M. J. BENSOUSSAN, J. Phys. Chem. Solids 28, 1533 (1967). A. WILLIG, B. SAPOVAL, K. LEIBLER, AND C. V~~RIX?, J. Phys. C 9, 1981 (1976). S. D. SENTURIA, A. C. SMITH, C. R. HEWES, J. A. HOFFMANN, AND P. L. SAGALYN, Phys. Rev. B 1,4045 (1970). A. WILLIG AND B. SAPOVAL, J. Phys. (Paris) Lett. 38, L 57 (1977). M. BENSOLJSSAN AND M. LANNOO, J. Phys. (Paris) 38,921 (1977). A. ABRAGAM, “The Principles of Nuclear Magnetism,” Oxford, London, 1961. H. SILLESCU, “Kernmagnetische Resonanz,” Springer-Verlag, Berlin, 1966. R. W. G. WYCKOFF, “Crystal Structures,” Vol. 1 p, 85, Wiley, New York, 1965. A. NOLLE, Z. Naturforsch. 33a, 666 (1978). H. KRUGER, 0. LUTZ, A. NOLLE, AND A. SCHWENK, 2. Phys. A 273,325 (1975). 0. LUTZ AND G. STRICKER, Phys. Lett. 35A, 397 (1971). H.KRCGER,O.LUTZ, A.~CHWENK,AND G.STRICKER,Z. Phys.266,233 (1974). K.U.BUCKLER,J.KRONENBI~ER,O.LUTZ,ANDA.NOLLE, Z.Naturforsch. 32a,1263(1977). A. D. BUCKINGHAM AND I. LOVE, J. Magn. Reson. 2,338 (1970).
OF
MAGNETIC
RESONANCE
40,9-16
(1980)
12’TeNMR Studies of Indirect and Direct Dipole-Dipole Coupling in Polycrystalline CdTe, HgTe, and PbTe R. BALZ, M. HALLER, W. E. HERTLER, 0. LUTZ, A. NOLLE, AND R. SCHAFITEL Physikalisches
Institut
der Universitiit
Tiibingen,
Morgenstelle,
D-7400
Tiibingen,
Germany
Received October 15, 1979; revised December 12, 1979 The NMR signals of ‘*‘Te in CdTe, HgTe, and PbTe powders show a fine structure due to direct and indirect dipole-dipole interactions. A theoretical powder pattern lineshape is given, with which the isotropic and anisotropic contributions of the direct and indirect dipoledipole interaction can be separated. Experimental results for Jll and JL in CdTe, HgTe, and PbTe are presented. A linear dependence of the reduced coupling on the atomic number predicts a vanishingly small coupling for ZnTe and this has been confirmed.
INTRODUCTION
For high-resolution experiments in liquid samples the indirect spin-spin coupling is a very important phenomenon which, together with chemical shift and in some cases with relaxation rate data, makes possible the solutions of structural problems. Until now the indirect spin-spin coupling has played only a minor role in solid state NMR because the influence of this interaction can be detected in only a few cases. This is due to the fact that in solids the direct dipole-dipole interaction usually is not averaged out and for nuclei like ‘H or 19F the direct dipole-dipole coupling is nearly always stronger than the indirect spin-spin coupling. There are only a few solid samples for which the indirect spin-spin coupling has been observed without using special techniques (I-11). Some further data on indirect spin-spin coupling have been reported by Andrew et al. (12-14, who used the magic angle spinning technique, by which the direct dipole-dipole interaction is reduced. This method has the disadvantage, however, that any anisotropic contributions of the indirect spinspin coupling are also reduced. Therefore, it seemed to us that apart from our recent work on lz5Te and II3 Cd NMR in a single crystal and polycrystalline CdTe, no data on anisotropic indirect spin-spin coupling in solid samples have been published (9, 10). Samples like CdTe, HgTe, or PbTe are very favorable for such investigations because the chemical bonding is partly covalent (15, I@, and therefore, an indirect spin-spin coupling is possible. The direct dipoledipole coupling is rather weak due to the small gyromagnetic ratios of the nuclei involved and their small natural abundances, resulting in narrow lines for these solids. Further, one expects from their 9
0022-2364/60/100009-08$02.00/O Copyright @ 1980 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in Great Britain
10
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AL.
behavior in the liquid state that elements with high atomic number will have rather strong indirect spin-spin coupling. rz5Te NMR has not been widely used in solids (9, IO, 17-21) and for the few known examples apart from (9,10, 17) no high-resolution spectra in solids have been observed. But as we showed in (9, IO), with good homogeneity of the magnetic field and with high purity of the samples, high-resolution spectra of lz5Te in solid samples of CdTe, HgTe, and PbTe are possible and from these spectra information on the isotropic and anisotropic indirect spin-spin coupling can be obtained. EXPERIMENTAL
The NMR signal of “‘Te was observed in a magnetic field of 2.114 T at a Larmor frequency of 28.4 MHz with a multinuclei pulsed Bruker SXP 4-100 NMR spectrometer. The temperature was (298 f 3) K. The free induction decays following a 90” pulse were accumulated and Fourier transformed by a B-NC 12 computer. The magnetic field was externally stabilized by a ‘H NMR stabilization unit. Of the stable tellurium isotopes, only lz3Te and lz5Te have a nuclear spin I = i and a nuclear magnetic moment. The natural abundances of lz3Te and “‘Te are 0.9 and 7%) respectively. The NMR receptivity of 12’Te is about 2.2 x 10p3, compared with 1 for the proton for an equal number of hydrogen and tellurium atoms at natural abundance. The receptivity of *23Te is about one order of magnitude lower. The elements cadmium, mercury, and lead also consist mainly of such isotopes, which have either no nuclear spins and moments or which have spin I = i: lliCd, ‘i3Cd, 199Hg, and 207Pb. Only the 201Hg isotope has spin I = $, but the gyromagnetic ratio of this nucleus is rather small. Therefore, mainly interactions between spin-i nuclei will be observed in the i2’Te spectra of CdTe, HgTe, and PbTe. The NMR signals of 125Te were observed in CdTe, HgTe, PbTe, and ZnTe powder samples, which were purchased from Zinsser, Frankfurt. The purity of the samples was 99.999%, with the exception of ZnTe, which was 99.998% pure. The chemical -shift data of these samples have been reported elsewhere (9). THEORY
Imagine a spin system with two spin-4 nuclei 1 and 2, for which besides the Zeeman interaction a direct and indirect dipole-dipole coupling must be taken into account. If the principal axis system of the indirect spin-spin coupling tensor is the same as that of the direct dipole-dipole interaction and if the indirect spin-spin coupling tensor is also axially symmetric with the two principal values .J and JL, two signals are observed. For example, for nucleus 1 at the frequencies (JO), v=v,+J/2+(0’/2)*(1-3cos26),
v’=vo-J/2-(D/2).(1-3cos28),
[II
where v. = (1/27r) * yBo is the Larmor frequency, J = f (2J1 +J&
PI
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the isotropic indirect spin-spin coupling, and D’= D-:
(.T,-Jl)
[31
is the anisotropic part of the indirect spin-spin coupling plus the direct dipole-dipole coupling, which always is anisotropic (22). PO YlY2fi r D=GT
[41
determines the direct dipole-dipole coupling, y1 and y2 are the gyromagnetic ratios of the respective nuclei 1 and 2, r is their internuclear distance, and 8 is the angle between the internuclear axis and the static magnetic field. From Eq. [l] the lineshape function of a powder pattern can be calculated easily according to, for example (23): f(v)-(
v-1+-J/2 D,,2
l-
)
-II2
[51
+( l+V-;;2J’2)-1’2.
This formula means that the two parts of the well-known “Pake-doublet” are shifted by J/2 and -J/2, as shown in Fig. 1. Further, the splitting is not only determined by the direct dipole-dipole coupling but also by the anisotropic part of the indirect spin-spin coupling. From the frequencies, for which in Fig. 1 divergences or shoulders occur, Jiband JI can be evaluated if values of D are known. For example, the frequency separation of the divergences of the powder pattern is equal to D’+J=D+J,
[61
and the frequency separation of the shoulders is 2D’-J=2D-4,.
[71
To calculate the theoretical lineshapes of 12’Te powder spectra in CdTe, HgTe, and PbTe, one must consider the crystallographic data of these materials, which are given in Table 1, and also the natural abundances of the isotopes rlrCd, ‘r3Cd, 199Hg, and 207Pb. 201 Hg has been omitted in this list because of its small gyromagnetic ratio
I’ -
!
’ v,-J/2-D/2
FIG.
coupling spectrum
v,+J/Z-D’
v;J/2.0’
I
Y
v,,+J/Z+D’/2
1. Theoretical powder spectrum for a two spin-$ system with direct as calculated from Eq. [5]. The meanings of J and D’ are explained is plotted for a ratio of D’ : J = 7 : 10.
and indirect dipole-dipole in Eqs. [2] through [4]. The
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CRYSTALLOGRAPHICDATAOF
Sample CdTe HgTe PbTe
ET AL.
TABLE CdTe,HgTe,
1 AND
PbTe ASTAKENFROMREF.(Z~)
Structure
Space group
Length of the cubic unit cell (lo-‘” m)
ZnS ZnS NaCl
T:F43m TiF43m OiFm3m
6.481 6.429 6.454
Distance between nearest neighbors il O-‘” m) 2.806 2.784 3.227
and its quadrupole moment. It follows from the crystallographic data that a tellurium nucleus in CdTe or HgTe has four nearest cadmium or mercury neighbors and six lead neighbors in PbTe. With natural abundances of “‘Cd and ‘13Cd together of 25%, lg9Hg of 16.8%, and ‘07Pb of 22.6%, the probabilities that 0, 1, 2, . . . of the nearest neighbors of a tellurium atom are 111*113Cd, 199Hg, or *“Pb atoms can be evaluated. If no neighboring atoms have a nuclear magnetic moment, dipole-dipole interaction cannot occur, and only a single signal at the frequency v. is observed for these 125Te nuclei. If of the four (six) neighbors one is an isotope with a nuclear magnetic moment, dipole-dipole interaction takes place and a lineshape as given, for example, in Eq. [5] is expected. If two of the neighbors are such isotopes with a nuclear magnetic moment, a triplet signal with intensities 1: 2 : 1 can be predicted. However, the probability of more than one isotope with a nuclear magnetic moment being the neighbor of a tellurium atom is small and the intensities of the corresponding satellites can be neglected. The strong central component of the triplet coincides with the unsplit 125Te signal and makes a significant contribution, as long as isotope effects are small. From these considerations one gets the following ratios of the integrated intensities of each of the satellites due to a spin-spin coupling with one nearest neighbor to the intensity of the central signal: for CdTe, 0.50 : 1; for HgTe, 0.36 : 1; and for PbTe, 0.53 : 1. Using these ratios and the appropriate Gaussian and Lorentzian lineshape functions for the convolution with the theoretical lineshape given in Eq. [5] one can calculate theoretical spectra of ‘*‘Te in CdTe, HgTe, and PbTe. The parameters of the dipole-dipole coupling can be evaluated by comparison of the theoretical and experimental spectra by a fit procedure. RESULTS
AND
DISCUSSION
The powder patterns of ‘*‘Te in CdTe, HgTe, and PbTe and of r13Cd in CdTe have been measured carefully with high signal-to-noise ratios. In Figs. 2 and 3 examples of powder spectra of 125Te in HgTe and PbTe are given. Signals of ‘25Te and of ‘13Cd in CdTe powder have been reported in (9, 10). All these spectra show satellites whose intensity is proportional to the probability that just one of the four (six) neighbors of the observed nuclei is a ‘llCd, ‘13Cd, ‘*‘Te, 199Hg, or *07Pb nucleus. The splitting was the same for the ‘13Cd and I*’ Te signals in CdTe. These facts indicate a spin-spin interaction between ‘25Te and ‘13Cd, 199Hg, or *07Pb. From a single-crystal
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13
-I
7 kHz FIG. 2. Uppertrace: Experimental ‘*‘Te NMR absorption signal in HgTe powder. The satellites are due to direct and indirect spin-spin coupling between ‘*‘Te and 199Hg. Resonance frequency, 28.371 MHz; experimental spectrum width, 50 kHz; trigger frequency, 0.1 Hz; measuring time, 7 hr; 200 data points were accumulated followed by 7992 points of zero-filling before the Fourier transformation of 8 K data points. Lower truce: Theoretical lineshape with the data of Table 2.
measurement of *“Te in CdTe the parameters of the direct and indirect spin-spin splitting were evaluated (10). From the satellite signals of a powder spectrum the value of Jl using Eq. [6] and the values of D quoted, for example, in Table 2 can be roughly determined, whereas the values of Jll cannot be given by a simple inspection of the spectrum because of the strong central signal. Therefore, the parameters Jil and JL were evaluated from D’ and J, which were determined by the three-parameter least-squares fit procedure mentioned above. By this computer program the linewidth of the Gaussian or Lorentzian function used for the convolution of the lineshape function and the parameters J and D’ in Eq. [5] were changed. The assumption of an axial symmetry of the indirect spin-spin coupling tensor, which was used for the derivation of Eq. [5], seems reasonable for chemical bonding (30) and, in addition, the principal vector belonging to Jll is assumed to be parallel to the internuclear axis. For the CdTe powder spectrum a Lorentzian lineshape with a halfwidth of 220 Hz and the values of D’ and J given in Table 2 yielded the best result of the fit procedure. The values of JL are mainly determined by the frequencies of the satellite maxima
14
BALZ
ET AL.
3 kHz FIG. 3. Upper trace: Experimental ‘25Te NMR absorption signal in PbTe powder. The satellite signals are due to a direct and indirect spin-spin coupling between ‘*‘Te and “’ Pb. Resonance frequency, 28.409 MHz; measuring time, 35 hr; allother parameters are identical to those in Fig. 2. Lower trace: Theoretical lineshape with the data of Table 2.
and are not very sensitive to whether a Gaussian or a Lorentzian function is used for the convolution. Because the shoulders of Fig. 1, which according to Eq. [7] determine the values of .Q, are hidden by the central signal, the values of .Q depend on the chosen broadening function. In the case of CdTe the standard deviation for the Lorentzian function is about a factor of 2 smaller than for the Gaussian function, so the fitted data using the Lorentzian function for the convolution are more reliable TABLE 2 ISOTROPICANDANISOTROPICINDIRECTSPIN-SPINCOUPLINGCONSTANTSINC~T~,H~T~,AND PbTe” Sample CdTe b CdTe HgTe PbTe
Interacting nuclei
(D’-D) U-W 380(l) 380(l) 308(l) 236(l)
llO(50) 60(100) 1640(600) 1340(400)
41 (Hz) 655(60) 680(100) 5080(600) 2150(400)
765(80) 740(150) 6720(850) 3480(560)
435(120) 560(180) 1800(1000) -520(700)
a The values of the direct dipole-dipole coupling D have been evaluated from Eq. [4] using the values in Table 1 and the gyromagnetic ratios of Refs. (9,25-29). b Results of a single-crystal measurement taken from (10).
DIPOLE-DIPOLE
COUPLING
IN TELLURIDES
15
and they are in agreement with the more precise values of the single-crystal measurement (10). This agreement between the single-crystal results and those obtained by the procedure described above enables us to evaluate the spectra of lz5Te in HgTe and PbTe in the same manner. Again, for both samples the fit using a Lorentzian broadening function yielded a smaller standard deviation of the experimental signal from the theoretical spectrum. The results for the spin-spin coupling data are given in Table 2. The values of Jil and Jl for CdTe and HgTe can be compared, because both samples have a zincblende arrangement. As expected, the indirect spin-spin coupling increases from CdTe to HgTe and is anisotropic for both compounds. Assuming a linear dependence of the reduced coupling constant (J/Y~~~)~‘* in the series ZnTe, CdTe, and HgTe on the atomic number of Zn, Cd, and Hg (see, e.g., (5)), from two measured values the indirect spin-spin coupling constant J between 67Zn and **‘Te in ZnTe, which also has a zincblende arrangement, can be estimated. It is nearly zero and D is about 90 Hz. For “‘Te in ZnTe, however, only a single signal with a signal-to-noise ratio of about 100 and a linewidth of 260 Hz was observed. Why is no splitting observed in ZnTe? This is due to the low natural abundance of 67Zn (only 4%) and further due to the spin I = 2. Therefore, instead of a doublet 21+ 1= 6 signals must occur for those ‘25Te nuclei that have only one neighboring 67Zn nucleus, together with an unsplit central signal. The intensity of the satellites is 36 times smaller than the intensity of the central signal. Such small satellites could not be detected. The data for the indirect spin-spin coupling between 207Pb and *25Te in PbTe are somewhat different from the data of CdTe and HgTe. .J seems to be much smaller, but since the error is rather large, it is not quite clear whether the sign of Jil is really different from the sign of JL or not. CONCLUSIONS
In solid compounds of heavy elements an indirect spin-spin coupling is observable in favorable cases due to the fact that it is large compared with the direct dipoledipole coupling. From powder samples the anisotropy of the indirect coupling can be determined with the procedure given. The result for CdTe powder compares well with that from the single-crystal measurement. Indirect spin-spin couplings in the solid are found to be large for the heavy atoms, as in the case of liquids, and the coupling constants increase with increasing atomic number in the IIb-tellurides. ACKNOWLEDGMENTS We would like to thank Professor Dr. H. Kriiger for his continuous support of this work and the Deutsche Forschungsgemeinschaft for financial support. REFERENCES 1. M. A.RUDERMANNAND C.K~-rr~~,f%ys. Rev.96,99 (1954). 2. N.BLOEMBERGENANDT.J.ROWLAND, Phys.Rev.97,1675(1955). 3. K.YOSIDA, Phys. Rev. 106,893 (1957).
16 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.
25. 26. 27.
28. 29. 30.
BALZ E. E. A. A. A. W. A. E.
ET
AL.
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