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Lattice dynamical calculations for the bcc cerium
Materials Letters 59 (2005) 2166 – 2169 www.elsevier.com/locate/matlet
Lattice dynamical calculations for the bcc cerium Engin DeligfzT, Yasemin C ¸ iftci, Kemal C ¸ olakog˘lu Gazi University, Faculty of Art and Science, Physics Department, 06500, Ankara, Turkey Received 22 September 2004; accepted 19 February 2005 Available online 19 March 2005
Abstract Lattice dynamical calculations are performed on cerium with bcc structure using the improved third-neighbor Clark–Gazis–Wallis (CGW) model. The theory has been applied to compute the dispersion curves, frequency spectra, lattice specific heat, and Debye temperature of the bcc cerium. In general, the obtained results agree reasonably well with the experimental data at the high temperature of the bcc cerium which has been very recently measured by K. Nicolaus et al. D 2005 Elsevier B.V. All rights reserved. Keywords: Bcc cerium; Lattice dynamics; Phonon dispersion; Ion–ion interactions; Specific heat
1. Introduction It is known that the most of the transition elements change their phase with increasing temperature. These metals with 1 to 4 d-electrons in outer shell transform to the bcc structure at elevated temperature. The phonon dispersion in these bcc phase strongly reflects increasing delectron occupancy [1]. The rare earth metal cerium exhibits [2] the following four phases sequence at ambient pressure: CeðmeltÞ1068KYd CeðbccÞ999KYc CeðfccÞ326K Yb CeðdhcpÞ96KYa CeðfccÞ All of these transitions at ambient pressure are expected to be martensitic, i.e., diffusionless, of first order and often produce metastable phases. Only yYg transition temperature is well defined, whereas for examples the gYy transition strongly depends on minor impurity contents. Similar to the group 3 metals cerium is trivalent in its two high-temperature phases y- and g-Ce [2].
T Corresponding author. Tel.: +90 312 2126030/3005; fax: +90 312 2122279. E-mail address: [email protected] (E. Deligfz). 0167-577X/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.matlet.2005.02.056
The phonon dispersion of the high temperature (at 1036 K) bcc or y-phase of Cerium has been measured by using inelastic neutron scattering, growing the single crystals in situ on the spectrometer by K. Nicolaus et al. [3]. The extensive measurements of phonon frequencies of several metals and alloys by means of neutron scattering have provided a point-to-point check of the wave vector vs. frequency relations for solids. The theoretical aspect of the lattice dynamical calculation of solids has seen a progress of force constants models, such as central pair potential (CPP), general tensor force(GTF) [4–6], De Launay angular force (DAF) [7], and Clark–Gazis–Wallis (CGW) [8] models. Several authors [11–13] have discussed the superiority of one model over another by comparing fits of the phonon frequencies with or without the ion-electron interactions (or three-body contribution). Therefore many theoretical lattice dynamical models have received considerable attention to reproduce the complete phonon dispersion curves along the principal symmetry directions for metals and alloys. In the last three decades, the DAF and CGW models have been extensively applied in the lattice dynamical calculations and they are still in use [9,10,14–22]. In this work, an angular force model, originally proposed by Clark, Gazis, and Wallis (CGW) [8] and later improved by [9,10] is used to reproduce the phonon frequencies for bcc cerium. Although we have not included the electron–ion
E. Deligo¨z et al. / Materials Letters 59 (2005) 2166–2169
interactions, which is significant in the case of metals [23], the obtained results are almost, excellent, except T 2 branch, by using only the third-neighbor CGW model.
2. Theory 2.1. Dynamical matrix for bcc structure (third-nearest neighbor) In the harmonic and adiabatic approximations, the phonon frequencies of cubic systems are determined by solving the usual secular equation, given by jDðqÞ x2 M Ij ¼ 0
ð1Þ
where w, D( q), M and I are in their usual meanings. By following Moore and Upadhyaya [10] and Upadhyaya et al. [12] one can write the dynamical matrix elements for ion– ion interactions, applying the CGW model up to thirdnearest neighbor, as follows: D11 ¼
8 ða1 þ 2a1VÞð1 C1 C2 C3 Þ 3 þ 4a2 S12 þ 4a2V S22 þ S32 þ 2ða3 þ a3VÞ ð2 C21 C22 C21 C23 Þ þ 4 a3V þ aN3 ð1 C22 C23 Þ
D12 ¼
8 ða1 a1VÞS1 S2 C3 þ 2ða3 a3VÞS21 S22 3
ð2Þ
where S i = sin q i (a / 2), C i = cos q i (a / 2), S 2i = sin q i a, C 2i = cos q i a, i = 1,2,3 and a is the lattice constant. The parameters of this model are as below: a1 ¼ b1 þ
16 8 c2 þ c4 ; 3 3
a2 ¼ b2 þ
32 c þ 4c6 ; 9 1
8 4 16 8 c þ c þ c3 þ c4 ; 3 1 3 2 3 4
a2V ¼
4 8 c c þ 4c5 þ 2c6 ; 3 2 9 1
a3V ¼ c5 þ aN3 ¼
Table 1 Input data and calculated force constants for bcc cerium Input data
c7 ; 2
16 2 c c3 c4 þ c5 þ 7 : 9 3 2
wavelength limits the following relations between elastic constants are obtained: 2 16 aC11 ¼ b1 þ 2b2 þ 4b3 þ ð2c1 þ c2 þ c3 Þ; 3 3 2 8 aC12 ¼ b1 þ 2b3 ð2c1 þ c2 þ 2c3 Þ; 3 3 2 8 ð4Þ aC44 ¼ b1 þ 2b3 þ ð2c1 þ 9c2 þ 2c3 þ 9c5 Þ: 3 9 The bulk modulus of electron gas in CGW model is also given for the bcc structure as 2 4 Ke ¼ ða1 þ a2 Þ ða1V þ a2V þ a3VÞ: ð5Þ 3 3 The model parameters (b 1, b 2, b 3 , c 1, c 2, c 3, and c 5) are calculated by using Eqs. (3)–(5) and phonon frequency relations for m L[100], m L[110], m T1[110], m L[111]. The input data and calculated force constants for this third-neighbor CGW model are given in Table 1. In order to make an analytical calculation of density of states in three dimensions, it would be necessary to integrate (|jk x|)1 over a constant frequency surface in k-space. Since this is not normally feasible, the usual approach is to compute the density of states numerically by the method firstly described by Walker [25]. This is usually referred to as the broot sampling methodQ since it builds up the density of states by finding the roots of the secular equation at large number of points in the Brillouin Zone. The usual expression of specific heat (C v) in the Debye model in terms of the Debye temperature h D = fx D/k B, the temperature T and the related frequency x D, are given as 3 Z ðhD =T Þ T x4 ex hD Cv ¼ 9nN kB dx ¼ f ð6Þ 2 x hD T ðe 1Þ 0 where 18nN p2 f : ð7Þ h3D ¼ 1 2 kB V 3 þ 3 ct c1 and x = (fx / k BT). The quantity that reflects the properties of material is h D, which depends on n, V, c l, and c t. Where c l and c t elastic wave velocity for longitudinal and trans-
8 3 a3 ¼ b3 þ c3 þ c5 þ c7 ; 9 2 a1V ¼
2167
ð3Þ
where b 1, b 2, b 3 (central), c 1, c 2, c 3, and c 5 (angular) are the force constants to be determined in this model. By expanding the secular determinant, Eq. (1), in the long-
a (A)a M (a.m.u)b K (1011 N/m2)b q (kg/m3)b C 11 (1010 N/m2)a C 12 (1010 N/m2)a C 44 (1010 N/m2)a m L(100) (f = 1.0) (THz)a m L(110) (f = 0.5) (THz)a m T1(110) (f = 0.5) (THz)a m L(111) (f = 0.5) (THz)a a b
[3]. [24].
Calculated force constant (N/m) 4.10 140.12 0.239 13,490 2.20 1.71 1.47 2.28 2.68 0.82 1.62
b1 b2 b3 c1 c2 c3 c5
16.62856 2.536106 0.640738 2.072465 0.814094 1.144205 0.137949
2168
E. Deligo¨z et al. / Materials Letters 59 (2005) 2166–2169
Fig. 1. Phonon dispersion and density of states (DOS) curves along the principal symmetry directions of bcc Ce. The solid curves represent the present calculations, the symbols denote the experimental points from Ref. [3].
verse mode, respectively, n is the number of atom, k B is Boltzmann constants and V is molar volume. The molar volume is simple to calculate, being the molecular weight divided by density. The longitudinal and transverse sound velocities, for the isotropic solid, can be related to the bulk modulus K, the shear modulus G, and density q ,by simple elasticity and wave theory, which gives sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffi G K þ 4=3G ct ¼ and cl ¼ : ð8Þ q q
G is the arithmetic mean of G V and G R, which is generally considered to be a good approximation. The integral in Eq. (6) must be evaluated separately for each temperature intervals by using SimpsonTs rule calculation. The Debye temperature calculated by Eq. (7) is 111.1 K for bcc Ce. The program code [26] uses a small number of macroscopic parameters, namely, density, molecular weight, and bulk moduli. Input data are given in Table 1.
If the material is elastically anisotropic, as crystalline materials normally are; there are only upper and lower bounds on G, G V, and G R, which are given by the usual Voight notation GbGV and GNGR
3. Results and discussions
1 ðC11 C12 þ 3C44 Þ and 5 5 4 3 ¼ þ : GR C11 C12 C44
GV ¼
The obtained results for the phonon frequencies and density of states (DOS) of bcc Ce in principal symmetry directions are plotted in Fig. 1 with the experimental points of [3]. Relevant force constants are listed in Table 1. The shapes of dispersion curves are remarkably similar to the results for other bcc metals. The present force constants give a satisfactory description of the phonon dispersion for the studied bcc Ce. Unexpectedly, the present model disagrees with experimental data for T 2 branch in the [110] direction. 400
Debye Temperature (K)
Specific Heat (/3R)
1 0.8 0.6
Bcc Ce 0.4 0.2 0
Bcc Ce 380 360 340 320 300
0
100
200
300
400
Temperature, T (K) Fig. 2. Temperature dependence of specific heat for bcc Ce.
500
0
20
40
60
80
100
T(K) Fig. 3. Temperature dependence of Debye temperature for bcc Ce.
E. Deligo¨z et al. / Materials Letters 59 (2005) 2166–2169
The deviation is order of 25% at the zone boundary for this branch and no satisfactory explanation is given for this disagreement. We believe that the further work is needed to clarify the behaviour of this branch. The specific heat at different temperatures are calculated and the obtained results are given in Fig. 2. Several thermodynamic quantities may readily be calculated from the frequency–distribution function if it is assumed that g(v) is independent of temperature (i.e. harmonic model). As an example, we have calculated the temperature dependence of the Debye temperature, h D for the bcc phase of Cerium, and the obtained results are given in Fig. 3. Although the expected trend can be seen from graphs, there are no experimental data for C v –T and h D–T curves available for the bcc Ce with which the result in Figs. 2 and 3 can be compared. To our best knowledge, this is the first paper based on the phenomenological lattice dynamics for bcc Ce, and the agreement between the theory and experiment is very good except for T 2 branch in [110] direction. It may be concluded that the present model represents the actual interactions responsible for the lattice vibrations in studied bcc Cerium.
References [1] W. Petry, J. Phys., IV, France 5 (1995) C2 – C15. [2] J.M. Wills, O. Eriksson, A.M. Boring, Phys. Rev. Lett. 67 (1991) 2215.
2169
[3] K. Nicolaus, J. Neuhaus, W. Petry, J. Bossy, Eur. Phys. J., B Cond. Matter Phys. 21 (2001) 357. [4] G.H. Begbie, M. Born, Proc. R. Soc. London, Ser. A 188 (1947) 179. [5] G.H. Begbie, Proc. R. Soc. London, Ser. A 188 (1947) 189. [6] E.H. Josepsen, Phys. Rev. 97 (1955) 654. [7] J. De Launay, Solid State Phys. 2 (1956) 219. [8] B.C. Clark, D.C. Gazis, R.F. Wallis, Phys. Rev., A 134 (1964) 1486. [9] J.C. Upadhyaya, S. Lehri, D.K. Sharma, S.C. Upadhyaya, Phys. Status Solidi, B Basic Res. 159 (1990) 659. [10] R.A. Moore, J.C. Upadhyaya, Can. J. Phys. 57 (1979) 2053. [11] Y.P. Varshni, P.S. Yuen, Phys. Rev. 174 (1968) 766. [12] M.M. Shukla, H. Closs, J. Phys. F. Met. Phys. 3 (1973). [13] J. Behari, B.B. Tripathi, J. Phys. C 3 (1970) 659. [14] V.P. Sing, H.L. Kharoo, M. Kumar, M.P. Hemkar, Nouvo Cimento B 32 (1976) 40. [15] S. Garg, H.C. Gupta, Phys. Status Solidi, B Basic Res. 136 (1986) 453. [16] V.B. Gupta, H.C. Gupta, Physica. B 50 (1988) 297. [17] S. Garg, H.C. Gupta, T.K. Bansal, B.B. Tripathi, J. Phys. F. Met. Phys. 15 (1985) 1895. [18] A. Sharma, R.P.S. Rathore, Ind. J. Phys. A 66 (1992) 315. [19] K. C ¸ olakog˘lu, G. Ug˘ur, M. C ¸ akmak, H. Tqtqncq, Tr. J. Phys. 23 (1999) 479. [20] K. C ¸ olakog˘lu, Y. C ¸ iftci, Tr. J. Phys. 24 (2000) 123. ¨ ztekin, G. Ug˘ur, X. AltVndal, Proc. Suppl. BPL 5 [21] K. C ¸ olakog˘lu, Y. O (1997) 470. ¨ ztekin, Il Nuova Cimento D 20 [22] K. C ¸ olakog˘lu, H. C ¸ olak, G. Ug˘ur, Y. O (12) (1998) 1863. [23] E.G. Brovman, Y. Kagan, Sov. Phys., Usp. 17 (1974) 125. [24] C. Kittel, Introduction to Solid State Physics, 5th ed., Wiley, Newyork, 1976. [25] C.B. Walker, Phys. Rev. 103 (1956) 547. [26] I. Johnston, G. Keeler, R. Rollins, S. Spicklemire, Solid State Physics Simulations the Consortium for Upper-Level Physics Software, John Wiley & Sons, Inc., 1996.
Lattice dynamical calculations for the bcc cerium Engin DeligfzT, Yasemin C ¸ iftci, Kemal C ¸ olakog˘lu Gazi University, Faculty of Art and Science, Physics Department, 06500, Ankara, Turkey Received 22 September 2004; accepted 19 February 2005 Available online 19 March 2005
Abstract Lattice dynamical calculations are performed on cerium with bcc structure using the improved third-neighbor Clark–Gazis–Wallis (CGW) model. The theory has been applied to compute the dispersion curves, frequency spectra, lattice specific heat, and Debye temperature of the bcc cerium. In general, the obtained results agree reasonably well with the experimental data at the high temperature of the bcc cerium which has been very recently measured by K. Nicolaus et al. D 2005 Elsevier B.V. All rights reserved. Keywords: Bcc cerium; Lattice dynamics; Phonon dispersion; Ion–ion interactions; Specific heat
1. Introduction It is known that the most of the transition elements change their phase with increasing temperature. These metals with 1 to 4 d-electrons in outer shell transform to the bcc structure at elevated temperature. The phonon dispersion in these bcc phase strongly reflects increasing delectron occupancy [1]. The rare earth metal cerium exhibits [2] the following four phases sequence at ambient pressure: CeðmeltÞ1068KYd CeðbccÞ999KYc CeðfccÞ326K Yb CeðdhcpÞ96KYa CeðfccÞ All of these transitions at ambient pressure are expected to be martensitic, i.e., diffusionless, of first order and often produce metastable phases. Only yYg transition temperature is well defined, whereas for examples the gYy transition strongly depends on minor impurity contents. Similar to the group 3 metals cerium is trivalent in its two high-temperature phases y- and g-Ce [2].
T Corresponding author. Tel.: +90 312 2126030/3005; fax: +90 312 2122279. E-mail address: [email protected] (E. Deligfz). 0167-577X/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.matlet.2005.02.056
The phonon dispersion of the high temperature (at 1036 K) bcc or y-phase of Cerium has been measured by using inelastic neutron scattering, growing the single crystals in situ on the spectrometer by K. Nicolaus et al. [3]. The extensive measurements of phonon frequencies of several metals and alloys by means of neutron scattering have provided a point-to-point check of the wave vector vs. frequency relations for solids. The theoretical aspect of the lattice dynamical calculation of solids has seen a progress of force constants models, such as central pair potential (CPP), general tensor force(GTF) [4–6], De Launay angular force (DAF) [7], and Clark–Gazis–Wallis (CGW) [8] models. Several authors [11–13] have discussed the superiority of one model over another by comparing fits of the phonon frequencies with or without the ion-electron interactions (or three-body contribution). Therefore many theoretical lattice dynamical models have received considerable attention to reproduce the complete phonon dispersion curves along the principal symmetry directions for metals and alloys. In the last three decades, the DAF and CGW models have been extensively applied in the lattice dynamical calculations and they are still in use [9,10,14–22]. In this work, an angular force model, originally proposed by Clark, Gazis, and Wallis (CGW) [8] and later improved by [9,10] is used to reproduce the phonon frequencies for bcc cerium. Although we have not included the electron–ion
E. Deligo¨z et al. / Materials Letters 59 (2005) 2166–2169
interactions, which is significant in the case of metals [23], the obtained results are almost, excellent, except T 2 branch, by using only the third-neighbor CGW model.
2. Theory 2.1. Dynamical matrix for bcc structure (third-nearest neighbor) In the harmonic and adiabatic approximations, the phonon frequencies of cubic systems are determined by solving the usual secular equation, given by jDðqÞ x2 M Ij ¼ 0
ð1Þ
where w, D( q), M and I are in their usual meanings. By following Moore and Upadhyaya [10] and Upadhyaya et al. [12] one can write the dynamical matrix elements for ion– ion interactions, applying the CGW model up to thirdnearest neighbor, as follows: D11 ¼
8 ða1 þ 2a1VÞð1 C1 C2 C3 Þ 3 þ 4a2 S12 þ 4a2V S22 þ S32 þ 2ða3 þ a3VÞ ð2 C21 C22 C21 C23 Þ þ 4 a3V þ aN3 ð1 C22 C23 Þ
D12 ¼
8 ða1 a1VÞS1 S2 C3 þ 2ða3 a3VÞS21 S22 3
ð2Þ
where S i = sin q i (a / 2), C i = cos q i (a / 2), S 2i = sin q i a, C 2i = cos q i a, i = 1,2,3 and a is the lattice constant. The parameters of this model are as below: a1 ¼ b1 þ
16 8 c2 þ c4 ; 3 3
a2 ¼ b2 þ
32 c þ 4c6 ; 9 1
8 4 16 8 c þ c þ c3 þ c4 ; 3 1 3 2 3 4
a2V ¼
4 8 c c þ 4c5 þ 2c6 ; 3 2 9 1
a3V ¼ c5 þ aN3 ¼
Table 1 Input data and calculated force constants for bcc cerium Input data
c7 ; 2
16 2 c c3 c4 þ c5 þ 7 : 9 3 2
wavelength limits the following relations between elastic constants are obtained: 2 16 aC11 ¼ b1 þ 2b2 þ 4b3 þ ð2c1 þ c2 þ c3 Þ; 3 3 2 8 aC12 ¼ b1 þ 2b3 ð2c1 þ c2 þ 2c3 Þ; 3 3 2 8 ð4Þ aC44 ¼ b1 þ 2b3 þ ð2c1 þ 9c2 þ 2c3 þ 9c5 Þ: 3 9 The bulk modulus of electron gas in CGW model is also given for the bcc structure as 2 4 Ke ¼ ða1 þ a2 Þ ða1V þ a2V þ a3VÞ: ð5Þ 3 3 The model parameters (b 1, b 2, b 3 , c 1, c 2, c 3, and c 5) are calculated by using Eqs. (3)–(5) and phonon frequency relations for m L[100], m L[110], m T1[110], m L[111]. The input data and calculated force constants for this third-neighbor CGW model are given in Table 1. In order to make an analytical calculation of density of states in three dimensions, it would be necessary to integrate (|jk x|)1 over a constant frequency surface in k-space. Since this is not normally feasible, the usual approach is to compute the density of states numerically by the method firstly described by Walker [25]. This is usually referred to as the broot sampling methodQ since it builds up the density of states by finding the roots of the secular equation at large number of points in the Brillouin Zone. The usual expression of specific heat (C v) in the Debye model in terms of the Debye temperature h D = fx D/k B, the temperature T and the related frequency x D, are given as 3 Z ðhD =T Þ T x4 ex hD Cv ¼ 9nN kB dx ¼ f ð6Þ 2 x hD T ðe 1Þ 0 where 18nN p2 f : ð7Þ h3D ¼ 1 2 kB V 3 þ 3 ct c1 and x = (fx / k BT). The quantity that reflects the properties of material is h D, which depends on n, V, c l, and c t. Where c l and c t elastic wave velocity for longitudinal and trans-
8 3 a3 ¼ b3 þ c3 þ c5 þ c7 ; 9 2 a1V ¼
2167
ð3Þ
where b 1, b 2, b 3 (central), c 1, c 2, c 3, and c 5 (angular) are the force constants to be determined in this model. By expanding the secular determinant, Eq. (1), in the long-
a (A)a M (a.m.u)b K (1011 N/m2)b q (kg/m3)b C 11 (1010 N/m2)a C 12 (1010 N/m2)a C 44 (1010 N/m2)a m L(100) (f = 1.0) (THz)a m L(110) (f = 0.5) (THz)a m T1(110) (f = 0.5) (THz)a m L(111) (f = 0.5) (THz)a a b
[3]. [24].
Calculated force constant (N/m) 4.10 140.12 0.239 13,490 2.20 1.71 1.47 2.28 2.68 0.82 1.62
b1 b2 b3 c1 c2 c3 c5
16.62856 2.536106 0.640738 2.072465 0.814094 1.144205 0.137949
2168
E. Deligo¨z et al. / Materials Letters 59 (2005) 2166–2169
Fig. 1. Phonon dispersion and density of states (DOS) curves along the principal symmetry directions of bcc Ce. The solid curves represent the present calculations, the symbols denote the experimental points from Ref. [3].
verse mode, respectively, n is the number of atom, k B is Boltzmann constants and V is molar volume. The molar volume is simple to calculate, being the molecular weight divided by density. The longitudinal and transverse sound velocities, for the isotropic solid, can be related to the bulk modulus K, the shear modulus G, and density q ,by simple elasticity and wave theory, which gives sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffi G K þ 4=3G ct ¼ and cl ¼ : ð8Þ q q
G is the arithmetic mean of G V and G R, which is generally considered to be a good approximation. The integral in Eq. (6) must be evaluated separately for each temperature intervals by using SimpsonTs rule calculation. The Debye temperature calculated by Eq. (7) is 111.1 K for bcc Ce. The program code [26] uses a small number of macroscopic parameters, namely, density, molecular weight, and bulk moduli. Input data are given in Table 1.
If the material is elastically anisotropic, as crystalline materials normally are; there are only upper and lower bounds on G, G V, and G R, which are given by the usual Voight notation GbGV and GNGR
3. Results and discussions
1 ðC11 C12 þ 3C44 Þ and 5 5 4 3 ¼ þ : GR C11 C12 C44
GV ¼
The obtained results for the phonon frequencies and density of states (DOS) of bcc Ce in principal symmetry directions are plotted in Fig. 1 with the experimental points of [3]. Relevant force constants are listed in Table 1. The shapes of dispersion curves are remarkably similar to the results for other bcc metals. The present force constants give a satisfactory description of the phonon dispersion for the studied bcc Ce. Unexpectedly, the present model disagrees with experimental data for T 2 branch in the [110] direction. 400
Debye Temperature (K)
Specific Heat (/3R)
1 0.8 0.6
Bcc Ce 0.4 0.2 0
Bcc Ce 380 360 340 320 300
0
100
200
300
400
Temperature, T (K) Fig. 2. Temperature dependence of specific heat for bcc Ce.
500
0
20
40
60
80
100
T(K) Fig. 3. Temperature dependence of Debye temperature for bcc Ce.
E. Deligo¨z et al. / Materials Letters 59 (2005) 2166–2169
The deviation is order of 25% at the zone boundary for this branch and no satisfactory explanation is given for this disagreement. We believe that the further work is needed to clarify the behaviour of this branch. The specific heat at different temperatures are calculated and the obtained results are given in Fig. 2. Several thermodynamic quantities may readily be calculated from the frequency–distribution function if it is assumed that g(v) is independent of temperature (i.e. harmonic model). As an example, we have calculated the temperature dependence of the Debye temperature, h D for the bcc phase of Cerium, and the obtained results are given in Fig. 3. Although the expected trend can be seen from graphs, there are no experimental data for C v –T and h D–T curves available for the bcc Ce with which the result in Figs. 2 and 3 can be compared. To our best knowledge, this is the first paper based on the phenomenological lattice dynamics for bcc Ce, and the agreement between the theory and experiment is very good except for T 2 branch in [110] direction. It may be concluded that the present model represents the actual interactions responsible for the lattice vibrations in studied bcc Cerium.
References [1] W. Petry, J. Phys., IV, France 5 (1995) C2 – C15. [2] J.M. Wills, O. Eriksson, A.M. Boring, Phys. Rev. Lett. 67 (1991) 2215.
2169
[3] K. Nicolaus, J. Neuhaus, W. Petry, J. Bossy, Eur. Phys. J., B Cond. Matter Phys. 21 (2001) 357. [4] G.H. Begbie, M. Born, Proc. R. Soc. London, Ser. A 188 (1947) 179. [5] G.H. Begbie, Proc. R. Soc. London, Ser. A 188 (1947) 189. [6] E.H. Josepsen, Phys. Rev. 97 (1955) 654. [7] J. De Launay, Solid State Phys. 2 (1956) 219. [8] B.C. Clark, D.C. Gazis, R.F. Wallis, Phys. Rev., A 134 (1964) 1486. [9] J.C. Upadhyaya, S. Lehri, D.K. Sharma, S.C. Upadhyaya, Phys. Status Solidi, B Basic Res. 159 (1990) 659. [10] R.A. Moore, J.C. Upadhyaya, Can. J. Phys. 57 (1979) 2053. [11] Y.P. Varshni, P.S. Yuen, Phys. Rev. 174 (1968) 766. [12] M.M. Shukla, H. Closs, J. Phys. F. Met. Phys. 3 (1973). [13] J. Behari, B.B. Tripathi, J. Phys. C 3 (1970) 659. [14] V.P. Sing, H.L. Kharoo, M. Kumar, M.P. Hemkar, Nouvo Cimento B 32 (1976) 40. [15] S. Garg, H.C. Gupta, Phys. Status Solidi, B Basic Res. 136 (1986) 453. [16] V.B. Gupta, H.C. Gupta, Physica. B 50 (1988) 297. [17] S. Garg, H.C. Gupta, T.K. Bansal, B.B. Tripathi, J. Phys. F. Met. Phys. 15 (1985) 1895. [18] A. Sharma, R.P.S. Rathore, Ind. J. Phys. A 66 (1992) 315. [19] K. C ¸ olakog˘lu, G. Ug˘ur, M. C ¸ akmak, H. Tqtqncq, Tr. J. Phys. 23 (1999) 479. [20] K. C ¸ olakog˘lu, Y. C ¸ iftci, Tr. J. Phys. 24 (2000) 123. ¨ ztekin, G. Ug˘ur, X. AltVndal, Proc. Suppl. BPL 5 [21] K. C ¸ olakog˘lu, Y. O (1997) 470. ¨ ztekin, Il Nuova Cimento D 20 [22] K. C ¸ olakog˘lu, H. C ¸ olak, G. Ug˘ur, Y. O (12) (1998) 1863. [23] E.G. Brovman, Y. Kagan, Sov. Phys., Usp. 17 (1974) 125. [24] C. Kittel, Introduction to Solid State Physics, 5th ed., Wiley, Newyork, 1976. [25] C.B. Walker, Phys. Rev. 103 (1956) 547. [26] I. Johnston, G. Keeler, R. Rollins, S. Spicklemire, Solid State Physics Simulations the Consortium for Upper-Level Physics Software, John Wiley & Sons, Inc., 1996.