- Email: [email protected]
Elastic moduli and phase transitions in Sn0.8In0.2(γ-Sn)
J. Phys.
Chem.
Solids
Vol.
53, No.
6, pp.
827430.
1992
0022-3697/92
Printed in Great Britain.
$5.00
+ 0.00
Pergamoa Press Ltd
ELASTIC
MODULI
AND PHASE TRANSITIONS
IN
Sno.&h2 (Y-W A. N. VAsIL’Ev,t A. S. IVANOV,$ E. A. POPOVA? and A. M. SHORTAMBAEV~ GLOWTemperature
Department, Moscow State University, Russia $Institute of Atomic Energy, 123182 Moscow, Russia
and Superconductivity
(Received 9 April 1991; accepted 25 October
119899 Moscow,
1991)
Abstract-The complete set of elastic moduli for the simple hexagonal alloy SnJn,-,2 (y-Sn) is determined above liquid nitrogen temperature, using a non-contact electromagnetic technique for the excitation and detection of ultrasound. The values obtained are compared with the data for the body-centered tetragonal structure of p-Sn on the basis of the orientational relationship between these two modifications of tin. The temperature dependencies of the longitudinal and transverse sound velocities show pronounced softening below liquid nitrogen temperatures, which can be attributed to successive phase transitions first to the unstable y’-Sn structure and then to the p-Sn structure. Keywords:
Elastic moduli, ultrasonic measurements, %1,,~1n,*, phase transitions.
[3,4]. The sequence of phase transitions in these elements appears to be essentially the same with increasing pressure, beginning with a diamond structure (like a-Sn), then transforming into bet (like B-Sn), sh (like y -Sn), and finally into a close-packed structure (hcp in Si and double hcp in Ge). In spite of the considerable differences between the f3-Sn and y-Sn structures a simple relationship for them can be established. It was found [l] that if one of the bet sublattices forming the structure of /?-Sn is shifted by a distance c/4 along the tetragonal axis (see Fig. l), the result is a base-centered orthorhombic structure with one atom per primitive cell. A fully hexagonal lattice of y-Sn can be constructed by altering just a few per cent of the /?-Sn lattice. It is understood that in a narrow composition range between the y-Sn and fi-Sn structures the intermediate unstable, y’-Sn phase exists, which possesses an orthorhombic structure. This occurs when one of fl-Sn sublattices is shifted less than c/4, i.e. the y-fi transition begins, but is not complete. Since the stable sh structure of y-Sn exists under normal conditions, one can study its properties using a variety of physical approaches. In neutron scattering measurements conducted at room temperature [5] the elastic moduli of Sn,,$,,, alloy were estimated by extrapolation of the slopes of the phonon dispersion curve to the center of the Brillouin zone (q = 0). It should be noted, however, that the precision of these estimates is somewhat limited, because of the marked change of the dispersion curve slopes in the wavevector range investigated, and because of the restriction of this range to q > 0.1 A-’ in the neutron scattering experiments.
INTRODUCTION majority of metal elements are known to have rather densely-packed crystal modifications, i.e. body-centered cubic (bee), face centered cubic (fee) or hexagonal close-packed (hcp) structures, which allow the lowest possible values of the interionic interaction energy. The very existence of metals that do not possess highly packed structures, as well as the presence of a marked crystal anisotropy, reveals evidence of the strong influence of electron subsystems on interionic interactions. Considering the non-transition metals, one can state that tin possesses the most unusual properties. Its metallic modification with two atoms per primitive cell, known as white tin or b-Sn, is characterized by an anisotropic body-centered tetragonal (bet) structure with the lattice parameter ratio c/a = 0.54. On cooling, /3-Sn transforms at T = 286 K into a zero-gap semiconducting phase cr-Sn (or grey tin) having the diamond structure. This transition is accompanied by a change of volume AVIV 30%, which is among the largest for any solid-solid transition. Additional non-trivial crystal modifications can be found in tin-based substitutional alloys with In and Hg [l, 21. These alloys, which are usually referred to as y-Sn, crystallize in monatomic simple hexagonal (sh) structure with the lattice parameter ratio c/u = 0.93. While the two lattices are completely different, the change of volume for the y-/l transition is relatively small (A V/ V < 1X). The same unusual structure has been discovered recently in high pressure experiments on chemical relatives of Sn, i.e. in Si at 160 kbar and in Ge at 750 kbar The
827
828
A. N. VASIL’EVet
Bsn (a)
7Sn
(b) Fig. 1. Crystal lattices of /?-Sn and y-Sn. Arrows show the direction of atomic displacements for /?-Sn and y-Sn. Full and open circles represent the atoms belonging to the two different bet sublattices of j?-Sn.
al.
normal to the surface, transverse ultrasonic waves were excited in the metal. In the latter case, the polarization of the ultrasound thus excited was determined by the alternating magnetic field direction with respect to the orientation of the crystal axes. A secondary coil wound parallel to the primary one was used to detect the resonant singularities of the sample surface impedance, which occurs when a half wavelength of the ultrasonic wave matches the plate thickness. The ultrasonic wave velocities, proportional to the resonant frequencies, were measured for low heating rates of about 20 K h-r. A detailed description of the technique used is given elsewhere [7].
EXPERIMENTAL
RESULTS AND DISCUSSION
In the present work we have made ultrasonic velocity measurements on Sn,,In,,, in the temperature range 4-1.50 K, using the non-contact electromagnetic technique for the generation and detection of ultrasound [6]. A monocrystalline boule of the Sn,,,In,., alloy (non-regular solid solution) was grown by the Stokburger method. An X-ray study made at room temperature showed that it had a hexagonal structure with a mosaic spread of not more than 30’. The boule was spark-cut and optically polished into plane-parallel plates 0.1 cm in thickness and l-2 cm in diameter. The normals to the plate surfaces were chosen to be aligned with the [OOl], [loo] or [loll crystal axes. The experimental arrangement for non-contact electro-magnetic excitation and detection of standing acoustic waves in a metal plate under the combined action of alternating and constant magnetic fields is shown in Fig. 2. The set-up includes a MHz-range variable frequency oscillator (l), the sample, which is put into the primary, as well as into the secondary inductive coils (2), a low-noise amplifier (3), a phase detector (4) and an X-Y recorder (5). A constant magnetic field H = 50 kOe was provided by a superconducting solenoid. For the case when H was applied parallel to the surface of the plate and perpendicular to the direction of the alternating current in the skin layer, a Lorentz force directed into the metal resulted in the excitation of longitudinal ultrasonic waves. In a magnetic field H directed
The complete set of y -Sn elastic moduli consists of five independent moduli, that is C,, , Cnr C,,, C,, and C,, while the modulus C, = (C,, - C,,)/2. The measurements of longitudinal and transverse sound velocities in the [loo] and [OOl] directions give the values of C,, , Cjj, C, and C,, directly. The value of Cn was obtained as C,, - 2C,,, while the value of C,, was calculated from the data for the longitudinal sound velocity I’, in the [ 1011direction. The temperature dependencies of the C, thus obtained are shown in Fig. 3a-e. A specific feature of the temperature dependencies measured is that there is some softening of the elastic moduli below liquid nitrogen temperature, which is most pronounced for the C,] and Cn moduli. In fact, these dependencies are highly indicative that successive phase transitions from the high temperature sh structure occur. We assume that these can be attributed first to a transition to the unstable y’-Sn phase at T = 72.4 K, and then to the /l-Sn structure at T = 31.7 K. Additional evidence for the low temperature phase transitions in y-Sn was obtained in a study of the temperature dependencies of the phonon frequencies, where the appearance of superstructure lines in the neutron scattering spectrum was found on cooling [8]. The values of the elastic moduli of y-Sn at the arbitrarily chosen temperature T = 120 K were used to calculate the Debye characteristic temperature, Tn, of this tin modification by a procedure suitable for hexagonal crystals [9]. The value for Tn of y-Sn was found to be 128 f 1 K, significantly less than that of /I-Sn, T, = 165 f 1 K, estimated at the same temperature [lo]. It is interesting to note that the softening of the phonon spectrum of y-Sn is accompanied by an increase of the superconducting transition temperature, Tc. Tc = 4.5 K in Sn,,Jn,,,
Fig. 2. Experimental set-up for electromagnetic excitation and detection of standing ultrasonic waves in metal plates.
Elastic moduli and phase transitions in Sn,,,In,,, (y-Sn)
829
6.1
6.0
cz‘g7.9
t
0”
7.8
q 0 =
77
u 76
3.65 0
750--1
Temperature
I 20
I 4060
I
I
Temperatze
(K)
I
I Ix)
I] I40
( I??
5.6 CT ‘g
5.7
e 6
5.6
=” 0
5.5
q 0
“a
00 “?
54 i
I
I
530
20
I 40
I 60
I 60
Temperature
I 100
I I20
I 140
(K)
Temperature
(K)
2.65
2.45
Temperature
(K)
Fig. 3. Temperature dependencies of the elastic moduli of SuJn,,, phase transitions are denoted by arrows. a-C,, , b--C,,,
[ll, 121, which
may
be compared
in Sn and Tc = 3.4 K in In. Estimates of the compressional
with
Tc = 3.7 K
K and
shear
G
moduli for y-Sn, as well as for /3&t at T = 120 K in the Voigt (V) and Reuss (R) approximations are as follows (in 10” dyne cm-*) K+
= (K” + KR)/2 = 5.903,
Gy_Sn= (G’ + CR)/2 = 2.249, KB_Sn= (K” + KR)/2 = 5.861, GB+, = (G” + GR)/2 = 2.200.
(y-Sn). Temperatures of successive c-C,,, d-C,,, e-C,.
of the shear moduli, G, as well as of the Debye temperatures, T,, depend on the complete set of elastic constants, while the values of the compressional moduli K are independent of C, and C,,. The y-8 transition takes place due to a finite displacement of the atoms in the primitive cell. The orientational relations for these lattices reflect, in fact, the existence of their subgroup connection. That is, the symmetry groups of these lattices have subgroups in common and symmetry elements that are retained partly at the y-/I transition. To establish the orientational relations for the elastic moduli of y-Sn and /?-Sn, we use the standard, as well as the following The values
830
A. N. VASIL’EVei al. LY -
c,
The relationship between the elastic moduli of y -Sn and /I-%-i at T = 120 K is shown in Fig. 4. The values measured show that while an overall softening of y -Sn takes place, the individual elastic moduli may increase or decrease as compared with p-Sn.
IO >
C+
C” -6
-6
CONCLUSION P
-4
0
Fig. 4. Orientational relations for the elastic moduli of b-Sn and y-Sn at 120 K.
non-standard
notations
In the present work we have determined the complete set of elastic moduli of y-Sn in the temperature range 4-l 50 K. Using the appropriate orientational relations, the values obtained are compared with those for jl-Sn, and the bulk and shear moduli and the Debye temperature of the sh structure are determined. The temperature dependencies measured indicate that y-Sn undergoes phase transitions to y’-Sn at T = 72.4 K, and from y ‘-Sn to p-Sn at T = 31.7 K.
for the elastic moduli (see
PI).
Acknowledgements-The
authors are grateful to A. Yu. Rumiantsev, E. V. Melnikov and A. V. Granato for helpful discussions.
In y-Sn c+ = [(A + B)/2] + {[(A- B)/212 + D2}"2,
REFERENCES
c- = [(A+ B)/2]- ([(A- B)/2]2+D2}"2, C, = C, sin’ ci + C,, cos2 ti,
1. Raynor G. V. and Lee J. A., Acta Met. 2, 616 (1954). 2. Kane R. H., Giessen B. C. and Grant N. J., Acta Met. 14, 605 (1966).
3. Olijnyk H., Sikka S. K. and Holzapfel W., Phys. Lett. A 1984, 103, 137 (1984). 4. Vohra Y. K., Brister K. E., Desgreniers S. et al.,
where
Phys. Rev. L&t. 56, 1944 (1986).
A=C,sin2cc+C,,cos2tl, B = C,, sin2 c( + C, cos2 a, D = (CIj+ C,)cos2u, and tl = arctg (a3/2c) 43". In j?-Sn C’ = (C,, C”
=
C66 +
c,,n
(C,,
+
C&/2.
5. Ivanov A. S., Rumiantsev A. Yu., Dorner B. et al., J. Phys. F.: Metal Phys. 17, 1925 (1987). 6. Vasil’ev A. N. and Gaidukov Yu. P., Sov. Phys. Usp. 26, 952 (1983).
7. Vasil’ev A. N., Gaidukov Yu. P., Kaganov M. I. et al., Sov. J. Low Temp. Phys. 15, 91 (1989). 8. Ivanov A. S., Mitrofanov N. L., Rumiantsev A. Yu. and Alba M., to be published in Physica B (1991). 9. Betts D. D., Bhatia A. B. and Horton G. K., Phys. Rev. 104, 43 (1956). 10. Rayne J. A. and Chandrasekhar B. S., Phys. Rev. 120, 1658 (1960). 11. Merriam M. F. and von Herzen M., Phys. Rev. 131, 637 (1963). 12. Ivanov A. S., Mitrofanov, N. L., Rumiantsev A. Yu. et al.. Sov. Phys. Solid State 29, 977 (1987).
Chem.
Solids
Vol.
53, No.
6, pp.
827430.
1992
0022-3697/92
Printed in Great Britain.
$5.00
+ 0.00
Pergamoa Press Ltd
ELASTIC
MODULI
AND PHASE TRANSITIONS
IN
Sno.&h2 (Y-W A. N. VAsIL’Ev,t A. S. IVANOV,$ E. A. POPOVA? and A. M. SHORTAMBAEV~ GLOWTemperature
Department, Moscow State University, Russia $Institute of Atomic Energy, 123182 Moscow, Russia
and Superconductivity
(Received 9 April 1991; accepted 25 October
119899 Moscow,
1991)
Abstract-The complete set of elastic moduli for the simple hexagonal alloy SnJn,-,2 (y-Sn) is determined above liquid nitrogen temperature, using a non-contact electromagnetic technique for the excitation and detection of ultrasound. The values obtained are compared with the data for the body-centered tetragonal structure of p-Sn on the basis of the orientational relationship between these two modifications of tin. The temperature dependencies of the longitudinal and transverse sound velocities show pronounced softening below liquid nitrogen temperatures, which can be attributed to successive phase transitions first to the unstable y’-Sn structure and then to the p-Sn structure. Keywords:
Elastic moduli, ultrasonic measurements, %1,,~1n,*, phase transitions.
[3,4]. The sequence of phase transitions in these elements appears to be essentially the same with increasing pressure, beginning with a diamond structure (like a-Sn), then transforming into bet (like B-Sn), sh (like y -Sn), and finally into a close-packed structure (hcp in Si and double hcp in Ge). In spite of the considerable differences between the f3-Sn and y-Sn structures a simple relationship for them can be established. It was found [l] that if one of the bet sublattices forming the structure of /?-Sn is shifted by a distance c/4 along the tetragonal axis (see Fig. l), the result is a base-centered orthorhombic structure with one atom per primitive cell. A fully hexagonal lattice of y-Sn can be constructed by altering just a few per cent of the /?-Sn lattice. It is understood that in a narrow composition range between the y-Sn and fi-Sn structures the intermediate unstable, y’-Sn phase exists, which possesses an orthorhombic structure. This occurs when one of fl-Sn sublattices is shifted less than c/4, i.e. the y-fi transition begins, but is not complete. Since the stable sh structure of y-Sn exists under normal conditions, one can study its properties using a variety of physical approaches. In neutron scattering measurements conducted at room temperature [5] the elastic moduli of Sn,,$,,, alloy were estimated by extrapolation of the slopes of the phonon dispersion curve to the center of the Brillouin zone (q = 0). It should be noted, however, that the precision of these estimates is somewhat limited, because of the marked change of the dispersion curve slopes in the wavevector range investigated, and because of the restriction of this range to q > 0.1 A-’ in the neutron scattering experiments.
INTRODUCTION majority of metal elements are known to have rather densely-packed crystal modifications, i.e. body-centered cubic (bee), face centered cubic (fee) or hexagonal close-packed (hcp) structures, which allow the lowest possible values of the interionic interaction energy. The very existence of metals that do not possess highly packed structures, as well as the presence of a marked crystal anisotropy, reveals evidence of the strong influence of electron subsystems on interionic interactions. Considering the non-transition metals, one can state that tin possesses the most unusual properties. Its metallic modification with two atoms per primitive cell, known as white tin or b-Sn, is characterized by an anisotropic body-centered tetragonal (bet) structure with the lattice parameter ratio c/a = 0.54. On cooling, /3-Sn transforms at T = 286 K into a zero-gap semiconducting phase cr-Sn (or grey tin) having the diamond structure. This transition is accompanied by a change of volume AVIV 30%, which is among the largest for any solid-solid transition. Additional non-trivial crystal modifications can be found in tin-based substitutional alloys with In and Hg [l, 21. These alloys, which are usually referred to as y-Sn, crystallize in monatomic simple hexagonal (sh) structure with the lattice parameter ratio c/u = 0.93. While the two lattices are completely different, the change of volume for the y-/l transition is relatively small (A V/ V < 1X). The same unusual structure has been discovered recently in high pressure experiments on chemical relatives of Sn, i.e. in Si at 160 kbar and in Ge at 750 kbar The
827
828
A. N. VASIL’EVet
Bsn (a)
7Sn
(b) Fig. 1. Crystal lattices of /?-Sn and y-Sn. Arrows show the direction of atomic displacements for /?-Sn and y-Sn. Full and open circles represent the atoms belonging to the two different bet sublattices of j?-Sn.
al.
normal to the surface, transverse ultrasonic waves were excited in the metal. In the latter case, the polarization of the ultrasound thus excited was determined by the alternating magnetic field direction with respect to the orientation of the crystal axes. A secondary coil wound parallel to the primary one was used to detect the resonant singularities of the sample surface impedance, which occurs when a half wavelength of the ultrasonic wave matches the plate thickness. The ultrasonic wave velocities, proportional to the resonant frequencies, were measured for low heating rates of about 20 K h-r. A detailed description of the technique used is given elsewhere [7].
EXPERIMENTAL
RESULTS AND DISCUSSION
In the present work we have made ultrasonic velocity measurements on Sn,,In,,, in the temperature range 4-1.50 K, using the non-contact electromagnetic technique for the generation and detection of ultrasound [6]. A monocrystalline boule of the Sn,,,In,., alloy (non-regular solid solution) was grown by the Stokburger method. An X-ray study made at room temperature showed that it had a hexagonal structure with a mosaic spread of not more than 30’. The boule was spark-cut and optically polished into plane-parallel plates 0.1 cm in thickness and l-2 cm in diameter. The normals to the plate surfaces were chosen to be aligned with the [OOl], [loo] or [loll crystal axes. The experimental arrangement for non-contact electro-magnetic excitation and detection of standing acoustic waves in a metal plate under the combined action of alternating and constant magnetic fields is shown in Fig. 2. The set-up includes a MHz-range variable frequency oscillator (l), the sample, which is put into the primary, as well as into the secondary inductive coils (2), a low-noise amplifier (3), a phase detector (4) and an X-Y recorder (5). A constant magnetic field H = 50 kOe was provided by a superconducting solenoid. For the case when H was applied parallel to the surface of the plate and perpendicular to the direction of the alternating current in the skin layer, a Lorentz force directed into the metal resulted in the excitation of longitudinal ultrasonic waves. In a magnetic field H directed
The complete set of y -Sn elastic moduli consists of five independent moduli, that is C,, , Cnr C,,, C,, and C,, while the modulus C, = (C,, - C,,)/2. The measurements of longitudinal and transverse sound velocities in the [loo] and [OOl] directions give the values of C,, , Cjj, C, and C,, directly. The value of Cn was obtained as C,, - 2C,,, while the value of C,, was calculated from the data for the longitudinal sound velocity I’, in the [ 1011direction. The temperature dependencies of the C, thus obtained are shown in Fig. 3a-e. A specific feature of the temperature dependencies measured is that there is some softening of the elastic moduli below liquid nitrogen temperature, which is most pronounced for the C,] and Cn moduli. In fact, these dependencies are highly indicative that successive phase transitions from the high temperature sh structure occur. We assume that these can be attributed first to a transition to the unstable y’-Sn phase at T = 72.4 K, and then to the /l-Sn structure at T = 31.7 K. Additional evidence for the low temperature phase transitions in y-Sn was obtained in a study of the temperature dependencies of the phonon frequencies, where the appearance of superstructure lines in the neutron scattering spectrum was found on cooling [8]. The values of the elastic moduli of y-Sn at the arbitrarily chosen temperature T = 120 K were used to calculate the Debye characteristic temperature, Tn, of this tin modification by a procedure suitable for hexagonal crystals [9]. The value for Tn of y-Sn was found to be 128 f 1 K, significantly less than that of /I-Sn, T, = 165 f 1 K, estimated at the same temperature [lo]. It is interesting to note that the softening of the phonon spectrum of y-Sn is accompanied by an increase of the superconducting transition temperature, Tc. Tc = 4.5 K in Sn,,Jn,,,
Fig. 2. Experimental set-up for electromagnetic excitation and detection of standing ultrasonic waves in metal plates.
Elastic moduli and phase transitions in Sn,,,In,,, (y-Sn)
829
6.1
6.0
cz‘g7.9
t
0”
7.8
q 0 =
77
u 76
3.65 0
750--1
Temperature
I 20
I 4060
I
I
Temperatze
(K)
I
I Ix)
I] I40
( I??
5.6 CT ‘g
5.7
e 6
5.6
=” 0
5.5
q 0
“a
00 “?
54 i
I
I
530
20
I 40
I 60
I 60
Temperature
I 100
I I20
I 140
(K)
Temperature
(K)
2.65
2.45
Temperature
(K)
Fig. 3. Temperature dependencies of the elastic moduli of SuJn,,, phase transitions are denoted by arrows. a-C,, , b--C,,,
[ll, 121, which
may
be compared
in Sn and Tc = 3.4 K in In. Estimates of the compressional
with
Tc = 3.7 K
K and
shear
G
moduli for y-Sn, as well as for /3&t at T = 120 K in the Voigt (V) and Reuss (R) approximations are as follows (in 10” dyne cm-*) K+
= (K” + KR)/2 = 5.903,
Gy_Sn= (G’ + CR)/2 = 2.249, KB_Sn= (K” + KR)/2 = 5.861, GB+, = (G” + GR)/2 = 2.200.
(y-Sn). Temperatures of successive c-C,,, d-C,,, e-C,.
of the shear moduli, G, as well as of the Debye temperatures, T,, depend on the complete set of elastic constants, while the values of the compressional moduli K are independent of C, and C,,. The y-8 transition takes place due to a finite displacement of the atoms in the primitive cell. The orientational relations for these lattices reflect, in fact, the existence of their subgroup connection. That is, the symmetry groups of these lattices have subgroups in common and symmetry elements that are retained partly at the y-/I transition. To establish the orientational relations for the elastic moduli of y-Sn and /?-Sn, we use the standard, as well as the following The values
830
A. N. VASIL’EVei al. LY -
c,
The relationship between the elastic moduli of y -Sn and /I-%-i at T = 120 K is shown in Fig. 4. The values measured show that while an overall softening of y -Sn takes place, the individual elastic moduli may increase or decrease as compared with p-Sn.
IO >
C+
C” -6
-6
CONCLUSION P
-4
0
Fig. 4. Orientational relations for the elastic moduli of b-Sn and y-Sn at 120 K.
non-standard
notations
In the present work we have determined the complete set of elastic moduli of y-Sn in the temperature range 4-l 50 K. Using the appropriate orientational relations, the values obtained are compared with those for jl-Sn, and the bulk and shear moduli and the Debye temperature of the sh structure are determined. The temperature dependencies measured indicate that y-Sn undergoes phase transitions to y’-Sn at T = 72.4 K, and from y ‘-Sn to p-Sn at T = 31.7 K.
for the elastic moduli (see
PI).
Acknowledgements-The
authors are grateful to A. Yu. Rumiantsev, E. V. Melnikov and A. V. Granato for helpful discussions.
In y-Sn c+ = [(A + B)/2] + {[(A- B)/212 + D2}"2,
REFERENCES
c- = [(A+ B)/2]- ([(A- B)/2]2+D2}"2, C, = C, sin’ ci + C,, cos2 ti,
1. Raynor G. V. and Lee J. A., Acta Met. 2, 616 (1954). 2. Kane R. H., Giessen B. C. and Grant N. J., Acta Met. 14, 605 (1966).
3. Olijnyk H., Sikka S. K. and Holzapfel W., Phys. Lett. A 1984, 103, 137 (1984). 4. Vohra Y. K., Brister K. E., Desgreniers S. et al.,
where
Phys. Rev. L&t. 56, 1944 (1986).
A=C,sin2cc+C,,cos2tl, B = C,, sin2 c( + C, cos2 a, D = (CIj+ C,)cos2u, and tl = arctg (a3/2c) 43". In j?-Sn C’ = (C,, C”
=
C66 +
c,,n
(C,,
+
C&/2.
5. Ivanov A. S., Rumiantsev A. Yu., Dorner B. et al., J. Phys. F.: Metal Phys. 17, 1925 (1987). 6. Vasil’ev A. N. and Gaidukov Yu. P., Sov. Phys. Usp. 26, 952 (1983).
7. Vasil’ev A. N., Gaidukov Yu. P., Kaganov M. I. et al., Sov. J. Low Temp. Phys. 15, 91 (1989). 8. Ivanov A. S., Mitrofanov N. L., Rumiantsev A. Yu. and Alba M., to be published in Physica B (1991). 9. Betts D. D., Bhatia A. B. and Horton G. K., Phys. Rev. 104, 43 (1956). 10. Rayne J. A. and Chandrasekhar B. S., Phys. Rev. 120, 1658 (1960). 11. Merriam M. F. and von Herzen M., Phys. Rev. 131, 637 (1963). 12. Ivanov A. S., Mitrofanov, N. L., Rumiantsev A. Yu. et al.. Sov. Phys. Solid State 29, 977 (1987).