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Thermodynamic properties of cuprous chloride
Solid State Communications,
Vol. 9, pp. 2115—2118, 1971. Pergamon Press.
Printed in Great Britain
THERMODYNAMIC PROPERTIES OF CIJPROUS CHLORIDEi R. Banerjee and Y.P. Varshni Department of Physics, University of Ottawa, Ottawa, Canada KiN 6N5 (Received 23 August 1971 by R.h’. Silsbee)
The phonon dispersion curves and thermodynamic properties of CuC1 have been obtained on the basis of a rigid ion model. The phonon frequency distribution function shows an interesting feature, it has four sharp peaks.
CUPROUS Chloride (CuC1) is an interesting compound which crystallizes in the zinc-blende structure. As compared to other compounds of Il—VI and Ill—V groups which have this structure, one expects that in CuC1 the binding will be more ionic.1 Martin2 has drawn attention to the peculiar behaviour of CuCI relative to other crystals with the same symmetry. Recently, Carabatos et al.3 have measured the frequencywave-vector dispersion relation for the normal vibrations of CuC1 in the
Table 1. Values of parameters in the SN! model for CuCI. All parameters are given in units of 10~ dyn/cm. a.
= =
12.7218 12.9760
A A 2
= =
0.8815 —0.4911 3.5432
=
—2.8093
x
=
2.9858
=
[1001,[1101 and [iii]
directions using inelastic neutron scattering. In the present note we present results for CuC1 on 4aInrigid modelcalled investigated the thision model, the SN! by model, authors. the interactions consist of Coulomb forces with an effective charge and a general force constant field up toandincluding second neighbours. It has seven parameters and their determination is reasonably straight forward. The details of the model are given in reference 4 and in the notation of that paper, the values of the parameters are shown in Table 1. In Fig. 1 are shown the calculated phonon dispersion curves together with the experimental points. Except for the two acoustic branches in the [110] direction, the agreement is satisfactory. In the [110] direction, the disagreement of the two calculated acoustic branches with the experimental points is in opposite directions, and it is not 2115
expected to affect seriously, except at very low temperatures, the results for those properties which depend on some type of average over phonon frequency distribution. The latter is the shown in Fig. 2. We notice a very interesting feature. There are four sharp peaks, two each in the acoustic and optical branches. It indicates that for those properties in which it is very complicated to take into account the phonon dispersion (e.g. thermal conductivity), it may be a good approximation, in the case of CuC1, to represent the frequency distribution function by a sum of four Einstein terms. A note of caution, however, should be added here as regards the positions of the first two peaks. The major contribution to the first peak (at 1 x 1012 c.p.s.) comes from the TA branches and that to the second one (at 3 x 1012 c.p.s.) from the LA branches. In our model, in the [1101 P’-’
2116
THERMODYNAMIC PROPERTIES OF CUPROUS CHLORIDE
aT 0
—
Vol.9, No.23
01 L
OL
Th~
~-
200 >-
U
~ 0
z
0
a
~Ioo~i
0.0
0.2
0.4 0.6 [100]—
06
101.0
o
0,4 0.2 —[110] REDUCED WAVE VECTOR 06
06
0.0 0
Oi
0.2 0.3 0.4 0.5
1”]—
FIG. 1. Phonon dispersion curves for cuixous chloride along the three major symmetry directions. The
experimental points are taken from Carabatos et al. ~
__________________________________
be considered as somewhat uncertain. The associated calorimetric Debye temperature is shown in Fig. 3. There is very little experimental data to compare with the theoretical 5 results. For the range 298°Kto 703°K, Kelley gives
8
6
—4 235,
I
) J
C,
~
I
2 3 12cps 4 Frequency (IO
unitS) 5
=
5.87
+
19.2 x 10~ T cal/deg. mole
(1)
To compare it with the theoretical results we need the isothermal bulk modulus (K) and the
temperature at There constant iset an pressure coefficient oldCi,, value (,B) for of for ~,~2 volume obtaining ~3O xexpansion 106, the due to(2) difference — sets C~, = KVT Klemm al.6 Two of elastic constant measurements at room temperature have been recently reported. Inoguchi et al:7 c 11 = 2.72 6
7
FIG. 2. Calculated frequency distribution histogram.
c,2 1.87, and c~ = 1.57 (all in i0~dynes/ cm2) and Hallberg and Hanson:8 c 11 = 4.82, c12 = 3.89, c~ = 1.47 (all in 10” dynes/cmz). ~.t 300°K, using Inoguchi’s data, we find C~, (theor.)
direction, the LA branch and one of the TA branches are poorly reproduced and probably a similar situation will exist in other directions which are in the proximity of this direction. This
will reflect in the g(u) function, consequently the positions and widths of the first two peaks should
=
11.47 and with Hallberg and Hanson
data, Ci,, (theor.) = 11.51. These values are to be compared with the value from equation (1), Ci,, = 11.63. It appears likely that the difference is due to contributions from the anharmonic terms. This remark is, of course, subject to the uncertainty in the value of
f3.
Vol. 9, No. 23
THERMODYNAMIC PROPERTIES OF CUPROUS CHLORIDE
2117
270
F230 w )-19O
ISO
00
0
200
300
TEMPERATURE (K)
FIG.
3. Calculated calorimetric Debye temperatures as a function of temperature.
I
0
00
200
300
I
I
I
400
500
600
TEMPERATURE (1()
FIG.
4. Calculated X-ray Debye temperatures as a function of temperature.
We have also calculated the Debye—Waller
factor, and the equivalent X-ray Debye tempera-
ture is shown as a function of temperature in
Fig. 4. There are no experimental data available for this quantity.
REFERENCES 1.
For quantitative results, see VAN VECHTEN J.A., Phys. Rev. 187, 1007 (1969) and PHILLIPS J.C., Rev, mod. Phys. 42, 317 (1970).
2. 3.
MARTIN R.M., Phys. Rev. Bi, 4005 (1970). CARABATOS C., HENNION B., KUNC K., MOUSSA F. and SCHWAB C., Phys. Rev. Lett. 26, 770 (1971).
2118 4.
THERMODYNAMIC PROPERTIES OF CUPROUS CHLORIDE
5.
BANERJEE R. and VARSHNI Y.P., Can. J. Phys. 47, 451 (1969). KELLEY K.K., Bur. Mines Bull. 584 (1960).
6.
KLEMM W., TILK W. and v. MULLENHEIM S., Z. anorg. ailgem. Chem. 176, 1(1928).
7.
INOGUCHI T., OKAMOTO T. and KOBA M., Sharp Tech. J. 12, 59 (1969).
8.
HALLBERG
J.
Vol. 9, No. 23
and HANSON R.C., private communication. We are grateful to Dr. R.C. Hanson for
supplying the elastic constant data prior to publication.
Les courbes de dispersion des phonons et les propriétés thermo-
dynamiques du CuC1 ont été obtenues a partir d’un modéle d’ions rigides. La fonction de distribution des phonons révéle une
structure intéressante: elle contient quatre pics aigus.
Vol. 9, pp. 2115—2118, 1971. Pergamon Press.
Printed in Great Britain
THERMODYNAMIC PROPERTIES OF CIJPROUS CHLORIDEi R. Banerjee and Y.P. Varshni Department of Physics, University of Ottawa, Ottawa, Canada KiN 6N5 (Received 23 August 1971 by R.h’. Silsbee)
The phonon dispersion curves and thermodynamic properties of CuC1 have been obtained on the basis of a rigid ion model. The phonon frequency distribution function shows an interesting feature, it has four sharp peaks.
CUPROUS Chloride (CuC1) is an interesting compound which crystallizes in the zinc-blende structure. As compared to other compounds of Il—VI and Ill—V groups which have this structure, one expects that in CuC1 the binding will be more ionic.1 Martin2 has drawn attention to the peculiar behaviour of CuCI relative to other crystals with the same symmetry. Recently, Carabatos et al.3 have measured the frequencywave-vector dispersion relation for the normal vibrations of CuC1 in the
Table 1. Values of parameters in the SN! model for CuCI. All parameters are given in units of 10~ dyn/cm. a.
= =
12.7218 12.9760
A A 2
= =
0.8815 —0.4911 3.5432
=
—2.8093
x
=
2.9858
=
[1001,[1101 and [iii]
directions using inelastic neutron scattering. In the present note we present results for CuC1 on 4aInrigid modelcalled investigated the thision model, the SN! by model, authors. the interactions consist of Coulomb forces with an effective charge and a general force constant field up toandincluding second neighbours. It has seven parameters and their determination is reasonably straight forward. The details of the model are given in reference 4 and in the notation of that paper, the values of the parameters are shown in Table 1. In Fig. 1 are shown the calculated phonon dispersion curves together with the experimental points. Except for the two acoustic branches in the [110] direction, the agreement is satisfactory. In the [110] direction, the disagreement of the two calculated acoustic branches with the experimental points is in opposite directions, and it is not 2115
expected to affect seriously, except at very low temperatures, the results for those properties which depend on some type of average over phonon frequency distribution. The latter is the shown in Fig. 2. We notice a very interesting feature. There are four sharp peaks, two each in the acoustic and optical branches. It indicates that for those properties in which it is very complicated to take into account the phonon dispersion (e.g. thermal conductivity), it may be a good approximation, in the case of CuC1, to represent the frequency distribution function by a sum of four Einstein terms. A note of caution, however, should be added here as regards the positions of the first two peaks. The major contribution to the first peak (at 1 x 1012 c.p.s.) comes from the TA branches and that to the second one (at 3 x 1012 c.p.s.) from the LA branches. In our model, in the [1101 P’-’
2116
THERMODYNAMIC PROPERTIES OF CUPROUS CHLORIDE
aT 0
—
Vol.9, No.23
01 L
OL
Th~
~-
200 >-
U
~ 0
z
0
a
~Ioo~i
0.0
0.2
0.4 0.6 [100]—
06
101.0
o
0,4 0.2 —[110] REDUCED WAVE VECTOR 06
06
0.0 0
Oi
0.2 0.3 0.4 0.5
1”]—
FIG. 1. Phonon dispersion curves for cuixous chloride along the three major symmetry directions. The
experimental points are taken from Carabatos et al. ~
__________________________________
be considered as somewhat uncertain. The associated calorimetric Debye temperature is shown in Fig. 3. There is very little experimental data to compare with the theoretical 5 results. For the range 298°Kto 703°K, Kelley gives
8
6
—4 235,
I
) J
C,
~
I
2 3 12cps 4 Frequency (IO
unitS) 5
=
5.87
+
19.2 x 10~ T cal/deg. mole
(1)
To compare it with the theoretical results we need the isothermal bulk modulus (K) and the
temperature at There constant iset an pressure coefficient oldCi,, value (,B) for of for ~,~2 volume obtaining ~3O xexpansion 106, the due to(2) difference — sets C~, = KVT Klemm al.6 Two of elastic constant measurements at room temperature have been recently reported. Inoguchi et al:7 c 11 = 2.72 6
7
FIG. 2. Calculated frequency distribution histogram.
c,2 1.87, and c~ = 1.57 (all in i0~dynes/ cm2) and Hallberg and Hanson:8 c 11 = 4.82, c12 = 3.89, c~ = 1.47 (all in 10” dynes/cmz). ~.t 300°K, using Inoguchi’s data, we find C~, (theor.)
direction, the LA branch and one of the TA branches are poorly reproduced and probably a similar situation will exist in other directions which are in the proximity of this direction. This
will reflect in the g(u) function, consequently the positions and widths of the first two peaks should
=
11.47 and with Hallberg and Hanson
data, Ci,, (theor.) = 11.51. These values are to be compared with the value from equation (1), Ci,, = 11.63. It appears likely that the difference is due to contributions from the anharmonic terms. This remark is, of course, subject to the uncertainty in the value of
f3.
Vol. 9, No. 23
THERMODYNAMIC PROPERTIES OF CUPROUS CHLORIDE
2117
270
F230 w )-19O
ISO
00
0
200
300
TEMPERATURE (K)
FIG.
3. Calculated calorimetric Debye temperatures as a function of temperature.
I
0
00
200
300
I
I
I
400
500
600
TEMPERATURE (1()
FIG.
4. Calculated X-ray Debye temperatures as a function of temperature.
We have also calculated the Debye—Waller
factor, and the equivalent X-ray Debye tempera-
ture is shown as a function of temperature in
Fig. 4. There are no experimental data available for this quantity.
REFERENCES 1.
For quantitative results, see VAN VECHTEN J.A., Phys. Rev. 187, 1007 (1969) and PHILLIPS J.C., Rev, mod. Phys. 42, 317 (1970).
2. 3.
MARTIN R.M., Phys. Rev. Bi, 4005 (1970). CARABATOS C., HENNION B., KUNC K., MOUSSA F. and SCHWAB C., Phys. Rev. Lett. 26, 770 (1971).
2118 4.
THERMODYNAMIC PROPERTIES OF CUPROUS CHLORIDE
5.
BANERJEE R. and VARSHNI Y.P., Can. J. Phys. 47, 451 (1969). KELLEY K.K., Bur. Mines Bull. 584 (1960).
6.
KLEMM W., TILK W. and v. MULLENHEIM S., Z. anorg. ailgem. Chem. 176, 1(1928).
7.
INOGUCHI T., OKAMOTO T. and KOBA M., Sharp Tech. J. 12, 59 (1969).
8.
HALLBERG
J.
Vol. 9, No. 23
and HANSON R.C., private communication. We are grateful to Dr. R.C. Hanson for
supplying the elastic constant data prior to publication.
Les courbes de dispersion des phonons et les propriétés thermo-
dynamiques du CuC1 ont été obtenues a partir d’un modéle d’ions rigides. La fonction de distribution des phonons révéle une
structure intéressante: elle contient quatre pics aigus.