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High-pressure behavior of some binary 1:1 actinide-arsenide compounds
ARTICLE IN PRESS
Physica B 387 (2007) 271–275 www.elsevier.com/locate/physb
High-pressure behavior of some binary 1:1 actinide-arsenide compounds Kuldeep Kholiya, B.R.K. Gupta Department of Physics, G.B. Pant University of Agriculture and Technology, Pantnagar, Uttaranchal 263145, India Received 22 December 2005; received in revised form 24 March 2006; accepted 15 April 2006
Abstract In the present investigation the B1–B2 phase transition pressure in four binary 1:1 actinide-arsenide compounds has been predicted. The calculated compression curves and values of different high pressure properties for these compounds viz: ThAs, UAs, NpAs and PuAs are reported and compared with the experimental values. The calculated values of elastic properties show predominantly ionic nature of these compounds. The results achieved in the present study are found in good agreement with the available experimental data. r 2006 Elsevier B.V. All rights reserved. PACS: 61.50.Ks; 61.50.Lt; 62.20.Dc Keywords: B1–B2 phase transition; Elastic constants; Binary 1:1 actinide- arsenide compounds
1. Introduction At ambient pressure and temperature, practically allbinary actinides (monocarbides, monopnictides and the monochalcogenides) compounds exhibit NaCl (B1) type structure. However, under pressure a first-order structural phase transition occurs from the sixfold-coordinated rocksalt (B1) to the eightfold-coordinated cesium chloride (B2) structure [1]. In past years, the phase transition pressure for ThAs, UAs, NpAs and PuAs have been reported by using the high-pressure X-ray diffraction technique [2–5]. It has been noted by these workers that these compounds undergo a sluggish phase transition and the values for the pressure range of phase transition are 18–25, 18–30, 26–36 and 35–38 GPa, respectively for ThAs, UAs, NpAs and PuAs. Therefore, the B2 phase for these compounds becomes more stable above the pressure 25, 30, 36 and 38 GPa, respectively. A phenomenon of interest in these compounds is the hybridization of the 5f electrons with conduction electrons which is an important parameter and leads to complex properties such as Kondo-like, heavy fermions or intermediate valance behavior and also responsible for their unusual properties. Experimentally, Corresponding author.
E-mail address: [email protected] (B.R.K. Gupta). 0921-4526/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2006.04.015
several photo-emission investigations favor a picture of a narrow f bands, hybridized with d states [6–8]. In recent years, the extensive studies [9–12] of induced phase transition properties of actinide solids have been performed by using the different theoretical methods. Rajagopalan [9] has used the method based on tightbinding linear muffin tin orbital (TBLMTO) method. The value of phase transition pressure (PT) for ThAs obtained by him is reasonably good as compared to the observed value. Aynyas et al. [10] and Jha et al. [11,12] have analyzed the phase transition and elastic properties of actinide arsenides within the framework of two-body potential model. Though, the model used by them is too simple to explain these properties but there are some weaknesses in their model. (a) The equilibrium condition used by them is not satisfied. (b) They have assumed the unit cellvolume pffiffiffiof CsCl phase equal to (8/3)r3 while it should be 8=3 3 r3 [13]. They have also assumed in their model that the interionic separation in B2 phase is greater than in B1 i.e. r(B2)4r(B1), which means that the volume of unit cell increases at transition pressure. It is noted that the unit cell volume always collapses at transition pressure [13]. (c) The values of ionic radii used by them do not match with the values of Ref. [14], given by them. Moreover, they have considered the same value of anion and cation radii in both phases while the NaCl type structure becomes stable only if
ARTICLE IN PRESS K. Kholiya, B.R.K. Gupta / Physica B 387 (2007) 271–275
272
the ratio of anion radius to the cation radius is between 0.41 and 0.73 and the CsCl type structure becomes stable if the radius ratio is above 0.73 [1,15]. To overcome these difficulties, we have therefore employed the potential model defined elsewhere [15] in which we have not assumed any assumption and considered the realistic approach to analyze the phase transition and elastic properties of actinide-arsenides. The model potential used in the present study is Born–Mayer type and the methodology is quite different than those of the previous workers [10–12]. The method of analysis is given in Section 2 and the results obtained in the present work are discussed in Section 3. 2. Method of analysis The stability of the particular crystal structure is given by the minima of the Gibbs free energy, i.e. G ¼ U þ PV TS,
b¼
G B1 ðrÞ ¼ U B1 ðrÞ þ 2Pr3
(2)
and 8ðr1 Þ3 P pffiffiffi . 3 3
(3)
Within the framework of Born–Mayer type potential which is considered up to nearest and the next-nearest neighbor interactions, the cohesive energies for B1 and B2 phases may be given as follows: aM Z 2 e 2 U B1 ðrÞ ¼ þ 6b expðr=rÞ 4pe0 r pffiffiffi þ 12b exp 2r=r , ð4Þ 1 2 2 aM Z e þ 8b1 expðr1 =r1 Þ 4pe0 r1 pffiffiffi þ 6b1 exp 2r1 = 3r1 ,
U B2 ðr1 Þ ¼
ð5Þ
where aM and a1M be the Modelung constants for the NaCl and CsCl structures respectively. r, b and r1, b1 are, respectively, the hardness and range parameters in B1and B2 phases. To determine the phase transition pressure one has to first determine the hardness parameter and the range parameter in B1 and B2 phases. For B1 phase r and b are determined from Bulk modulus and the equilibrium condition as given below " ( pffiffiffi )# expðr0 =rÞ þ 4 exp 2r0 =r aM e2 Z 2 r0 pffiffiffi B0 ¼ 1 pffiffiffi 2r expðr0 =rÞ þ 2 2 exp 2r0 =r 4pe0 9r40
(6)
aM e 2 Z 2 r pffiffiffi pffiffiffi . 4pe0 6r0 expðr0 =rÞ þ 2 2 exp 2r0 =r 2
(7)
However, for B2 phase the value of the hardness parameter and range parameter cannot be determined by this method because for B2 phase the values of bulk modulus and the equilibrium separation are not known experimentally. The previous workers [10–12] have taken the same values of these parameters in both phases, which is not physically realistic. To remove this difficulty we have followed the work of Mario P. Tosi who concluded that the value of hardness parameter decreases for the more compact CsCl phase up to 10%, whereas the ratio of the range parameters increases by the ratio 8/6 in two different structures [15] such as b1 ¼
(1)
where, U is the internal energy, and S is the vibrational entropy at pressure P, volume V and temperature T. The Gibbs free energy for NaCl (B1) and CsCl (B2) phases is expressed as [16]
G B2 ðr1 Þ ¼ U B2 ðr1 Þ þ
and
8 b, 6
(8)
where 8 and 6 be the coordination numbers of B2 and B1 phase, respectively. In view of Eq. (8) the value of hardness parameter (r1) in B2 phase may thus be calculated from minima of the Gibbs free energy whereas the inter atomic separation r1 can be calculated with the help of volume collapse at the phase transition pressure. The phase transition pressure is the pressure at which the difference of Gibbs free energy of two phases becomes zero (i.e. DG ¼ G B2 G B1 ¼ 0). Further to calculate the elastic constants for B1 phase, we have considered both the Coulomb and the short-range interactions as given by Cowley [17] i.e. C Coul ¼ 11
2:55604e2 Z2 . 4pe0 2r40
(9)
¼ C Coul 12
0:11298e2 Z2 . 4pe0 2r40
(10)
C Coul ¼ 44
1:27802e2 Z2 . 4pe0 2r40
(11)
The short-range (SR) contributions considered up to first and second-nearest neighbor interactions are pffiffiffi ! 1 b r0 1 2b 2r 0 SR exp exp C 11 ¼ þ 2 2 r0 r r0 r r r pffiffiffi ! 2r 0 1 b 2 exp . ð12Þ r r0 r
C SR 12
pffiffiffi ! 1 b r0 1 b 2r 0 ¼ 2 exp exp þ 2 r0 r r r r0 r pffiffiffi ! 2r 0 5 b þ 2 exp . r 2r0 r
ð13Þ
ARTICLE IN PRESS K. Kholiya, B.R.K. Gupta / Physica B 387 (2007) 271–275
C SR 44
pffiffiffi ! 1 b r0 1 b 2r 0 ¼ 2 exp exp þ r0 r2 r r r0 r pffiffiffi ! 3 b 2r 0 2 exp . r r 2r0
3. Results and discussion
ð14Þ
The Bulk modulus for B1 phase is calculated from the relation 1 B ¼ ðC 11 þ 2C 12 Þ 3
273
(15)
and for B2 phase it is calculated from the thermodynamic condition B ¼ V ðd2 U=dV 2 Þ which gives 1 a1 e 2 Z 2 B1 ¼ pffiffiffi M 1 4 4 3 4pe0 ðr0 Þ " ( pffiffiffi )# expðr10 =r1 Þ þ exp 2r10 = 3r1 r1 pffiffiffi pffiffiffi . 1 1 r 2 expðr10 =r1 Þ þ 3 exp 2r10 = 3r1
The input data and calculated model parameters for the compounds under study are given in Tables 1a and b, respectively. It is clear from Table 1b that in binary 1:1 actinide-arsenide compounds for B2 phase the value of the hardness parameter decreases up to 5.5% while in alkali halides it decreases up to 10% [15]. This may be because of the fact that binary 1:1 actinide-arsenides are harder than alkali halides, as there bulk modulus is higher. The calculated values of induced phase transition pressure for ThAs, UAs, NpAs and PuAs along with their cohesive
Table 1a Input parameters used to calculate model parameters
ð16Þ The pressure derivative of Bulk modulus for both the phases may be calculated by fitting P–V data to the Birch [18] and Vinet [19] equations.
Crystal
r0 (A˚)
B0 (GPa)
% volume collapse at transition
ThAs Uas NpAs PuAs
2.989 [5] 2.8835 [5] 2.9183 [4] 2.92825 [5]
118 [5] 98 [5] 70 [4] 68.5 [5]
11 [5] 10 [5] 9 [4] 9 [5]
Table 1b Calculated model parameters Crystal
b (1019 J)
r (A˚)
b1 (1019 J)
r1 (A˚)
% decrease in hardness parameter for B2 phase
ThAs UAs NpAs PuAs
180149.6984 11418.39025 1655.298741 1576.562401
0.2402 0.3069 0.3992 0.4033
240199.5979 15224.52033 2207.064988 2102.083201
0.2365 0.296 0.3773 0.3815
1.54 3.55 5.49 5.41
Table 2 Cohesive and phase transition properties of ThAs, UAs, NpAs and PuAs Crystal
Equilibrium separation (A˚)
Cohesive energy (1019 J)
Transition pressure (GPa)
rB1
rB2
GB1
GB2
ThAs Present Exp. Other
2.989034 2.989 [2] 2.9951 [10]
3.122741
49.5724
48.13437
3.11 [10]
49.454 [10]
48.3546 [10]
UAs Present Exp. Other
2.888346 2.88835 [3] 2.8895 [11]
3.023342
49.92965
48.57729
3.0647 [11]
49.4592 [11]
47.8725 [11]
NpAs Present Exp. Other
2.918302 2.9183 [4] 2.93 [12]
3.065055
47.905
46.6486
3.07 [12]
48.5567 [12]
47.2744 [12]
PuAs Present Exp.
2.928251 2.92825 [5]
3.080841
47.69746
46.37301
26 25 [2] 16.4 [10] 26.2 [9] 30.4 30 [3] 19.75 [11] 34 35 [4] 28.2 [12] 38 38 [5]
ARTICLE IN PRESS K. Kholiya, B.R.K. Gupta / Physica B 387 (2007) 271–275
present study are not available for the comparison, but the values of bulk modulus predicted from these elastic constants are better than those of the previous workers as compared to the experimental values. For UAs 1.5 1
ΔG(10-19Joule)
energies predicted at zero pressure are reported in Table 2. The experimental values of the phase transition pressure are also shown in the table for the sake of comparison. It may be noted from Table 2 that the values of transition pressure (Pt) predicted for these compounds are in excellent agreement with the available experimental values. On the other hand the values obtained by other workers [9–12] deviate largely in comparison to the experimental values. The large deviations of their results with the observed values may be ascribed to the fact that these workers have not predicted the correct values of the model parameters and adopted the same values of both range and hardness parameters in B1 and B2 phases. In phase B2 these parameters have significant different values as compared to those in phase B1. Moreover, their concept that the unit cell volume in B2 phase i.e. (8/3)(r1)3 is greater than B1 phase i.e. 2r3 is not correct, otherwise, the unit cell volume in phase B2 should increase and thus will never collapse. On the other hand the volume in B2 phase always collapses at phase transition pressure and the structural transition occurs. The variations of DG with pressure for these compounds are shown in Figs. 1–4. The experimental values are also represented by filled triangles. It is clear from these figures that our calculated results are quite consistent with the available experimental values. An inspection of Tables 3 and 4 reveals that although the experimental values of the second order elastic constants for the compounds under
-0.5
(a)
-1
0.5 0
0
10
20
30 40 Pressure(GPa)
1.1
50
60
Present work
1
Experimental
0.9 V/ V0
274
0.8 0.7 0.6 0.5 0.4 0
10
20
30
40
50
60
Pressure(GPa)
(b)
Fig. 2. (a) Represents the variation of the difference for Gibbs free energies (DG) in B1 and B2 phase while (b) shows the volume compression with pressure for UAs.
ThAs 2
NpAs 1.5
1 0.5 0 -0.5
0
10
20
30
40
50
60
Pressure (GPa)
-1
ΔG(10-19Joule)
ΔG(10-19 Joule)
1.5
1 0.5 0
0
10
20
(a) -1.5
(a)
1.1 Experimental
0.8
V/ V0
V/ V0
0.9
0.7
50
60
70
-1
1
Present work
0.9
Experimental
0.8 0.7
0.6
0.6
0.5
0.5
0.4
0.4 0
(b)
40
1.1
Present work
1
30
Pressure(GPa)
-0.5
10
20
30
40
0
50
Pressure (GPa)
Fig. 1. (a) Represents the variation of the difference for Gibbs free energies (DG) in B1 and B2 phase while (b) shows the volume compression with pressure for ThAs.
(b)
10
20
30
40
50
60
Pressure(GPa)
Fig. 3. (a) Represents the variation of the difference for Gibbs free energies (DG) in B1 and B2 phase while (b) shows the volume compression with pressure for NpAs.
ARTICLE IN PRESS K. Kholiya, B.R.K. Gupta / Physica B 387 (2007) 271–275
PuAs
ΔG(10-19Joule)
1.5
Table 4 Calculated bulk modulus and its pressure derivative of ThAs, UAs, NpAs and PuAs for NaCl-B1 and CsCl-B2 structures
1
Bulk modulus (GPa)
B10
In B1 phase
In B2 phase
In B1 phase
In B2 phase
ThAs Present Exp. Other
117.892 118 [5] 109 [10]
140.674
5.887 3.4 [5]
6.238
UAs Present Exp.
97.59 98 [5]
118.613
4.934 4.9 [5]
5.15
NpAs Present Exp. Other
69.138 70 [5] 70.12 [12]
89.155
4.349 6 [5]
3.974
PuAs Present Exp.
67.634 68.5 [5]
79.758
4.314 3.3
4.613
Crystal
0.5 0 0
10
20
-0.5
30
40
50
60
70
80
Pressure(GPa)
(a) -1 1.1
V/ V0
275
1
Present work
0.9
Experimental
0.8 0.7 0.6 0.5 0.4 0
(b)
10
20
30
40
50
60
Pressure(GPa)
Fig. 4. (a) Represents the variation of the difference for Gibbs free energies (DG) in B1 and B2 phase while (b) shows the volume compression with pressure for PuAs.
properties of all the binary 1:1 actinide-arsenide compounds.
Table 3 Calculated elastic constants Cij of ThAs, UAs, NpAs and PuAs for B1 phase
References
Crystal
C11 (GPa)
C12 (GPa)
C44 (GPa)
ThAs Present Other
268.747 246.24 [10]
42.465 41.46 [10]
42.781 40.46 [10]
UAs Present
187.239
52.766
53.991
NpAs Present Other
97.037 115.85 [12]
55.188 47.93 [12]
57.8 47.93 [12]
PuAs Present
93.778
54.562
57.184
The pressure derivatives of bulk modulus ðB10 Þ are also found in reasonably good agreement with experimental values in B1 phase. However, B10 could not be compared in phase B2 as the experimental data in B2 phase are not available so far. On the basis of overall description it may thus be concluded that the potential model and its application in the present study has satisfactorily explained the structural stability, cohesive and phase transition
[1] U. Benedict, J. Less-Common Met. 128 (1987) 7. [2] L. Gerward, J. Staun Olsen, U. Benedict, S. Dabos, H. Luo, J.P. Itie, High Temp. High Pressure 20 (1988) 545. [3] J. Staun Olsen, L. Gerward, U. Benedict, S. Dabos, J.P. Itie, High Pressure Res. 1 (1989) 253. [4] S. Dabos, C. Dufour, U. Benedict, J.C. Spirlet, M. Pages, Physica B 144 (1986) 79. [5] S. Dabos, U. Benedict, J.C. Spirlet, J. Less-Common Met. 153 (1989) 133. [6] Y. Baer, Physica B 102 (1980) 104. [7] R. Baptist, M. Belakhovsky, M.S.S. Brooks, R. Pinchaux, Y. Baer, O. Vogt, Physica B 102 (1980) 63. [8] M. Erbudak, F. Meier, Physica B 102 (1980) 134. [9] M. Rajagopalan, Advance in High Pressure Science and Technology, University Press, Nancy, 1997 (p. 264). [10] M. Aynyas, S.P. Sanyal, P.K. Jha, Phys. Stat. Sol. (b) 229 (3) (2002) 1459. [11] P.K. Jha, S.P. Sanyal, Phys. Stat. Sol. (b) 200 (1997) 13. [12] P.K. Jha, M. Aynyas, S.P. Sanyal, Indian J. Pure Appl. Phys. 42 (2004) 112. [13] Alan J. Cohen, Roy G. Gordon, Phys. Rev. B 12 (1975) 3228. [14] C. Kittle, Introduction to Solid State Physics, Wiley Eastern, New Delhi, 1999. [15] M.P. Tosi, Solid State Phys. 16 (1964) 1. [16] K.N. Jog, R.K. Singh, S.P. Sanyal, Phys. Rev. B 31 (1985) 6047. [17] R.A. Cowley, Proc. R. Soc. London A 268 (1962) 121. [18] F. Birch, Phys. Rev. 71 (1947) 809. [19] P. Vinet, J. Ferrante, J.H. Rose, J.R. Smith, J. Geophys. Res. 92 (1987) 9319.
Physica B 387 (2007) 271–275 www.elsevier.com/locate/physb
High-pressure behavior of some binary 1:1 actinide-arsenide compounds Kuldeep Kholiya, B.R.K. Gupta Department of Physics, G.B. Pant University of Agriculture and Technology, Pantnagar, Uttaranchal 263145, India Received 22 December 2005; received in revised form 24 March 2006; accepted 15 April 2006
Abstract In the present investigation the B1–B2 phase transition pressure in four binary 1:1 actinide-arsenide compounds has been predicted. The calculated compression curves and values of different high pressure properties for these compounds viz: ThAs, UAs, NpAs and PuAs are reported and compared with the experimental values. The calculated values of elastic properties show predominantly ionic nature of these compounds. The results achieved in the present study are found in good agreement with the available experimental data. r 2006 Elsevier B.V. All rights reserved. PACS: 61.50.Ks; 61.50.Lt; 62.20.Dc Keywords: B1–B2 phase transition; Elastic constants; Binary 1:1 actinide- arsenide compounds
1. Introduction At ambient pressure and temperature, practically allbinary actinides (monocarbides, monopnictides and the monochalcogenides) compounds exhibit NaCl (B1) type structure. However, under pressure a first-order structural phase transition occurs from the sixfold-coordinated rocksalt (B1) to the eightfold-coordinated cesium chloride (B2) structure [1]. In past years, the phase transition pressure for ThAs, UAs, NpAs and PuAs have been reported by using the high-pressure X-ray diffraction technique [2–5]. It has been noted by these workers that these compounds undergo a sluggish phase transition and the values for the pressure range of phase transition are 18–25, 18–30, 26–36 and 35–38 GPa, respectively for ThAs, UAs, NpAs and PuAs. Therefore, the B2 phase for these compounds becomes more stable above the pressure 25, 30, 36 and 38 GPa, respectively. A phenomenon of interest in these compounds is the hybridization of the 5f electrons with conduction electrons which is an important parameter and leads to complex properties such as Kondo-like, heavy fermions or intermediate valance behavior and also responsible for their unusual properties. Experimentally, Corresponding author.
E-mail address: [email protected] (B.R.K. Gupta). 0921-4526/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2006.04.015
several photo-emission investigations favor a picture of a narrow f bands, hybridized with d states [6–8]. In recent years, the extensive studies [9–12] of induced phase transition properties of actinide solids have been performed by using the different theoretical methods. Rajagopalan [9] has used the method based on tightbinding linear muffin tin orbital (TBLMTO) method. The value of phase transition pressure (PT) for ThAs obtained by him is reasonably good as compared to the observed value. Aynyas et al. [10] and Jha et al. [11,12] have analyzed the phase transition and elastic properties of actinide arsenides within the framework of two-body potential model. Though, the model used by them is too simple to explain these properties but there are some weaknesses in their model. (a) The equilibrium condition used by them is not satisfied. (b) They have assumed the unit cellvolume pffiffiffiof CsCl phase equal to (8/3)r3 while it should be 8=3 3 r3 [13]. They have also assumed in their model that the interionic separation in B2 phase is greater than in B1 i.e. r(B2)4r(B1), which means that the volume of unit cell increases at transition pressure. It is noted that the unit cell volume always collapses at transition pressure [13]. (c) The values of ionic radii used by them do not match with the values of Ref. [14], given by them. Moreover, they have considered the same value of anion and cation radii in both phases while the NaCl type structure becomes stable only if
ARTICLE IN PRESS K. Kholiya, B.R.K. Gupta / Physica B 387 (2007) 271–275
272
the ratio of anion radius to the cation radius is between 0.41 and 0.73 and the CsCl type structure becomes stable if the radius ratio is above 0.73 [1,15]. To overcome these difficulties, we have therefore employed the potential model defined elsewhere [15] in which we have not assumed any assumption and considered the realistic approach to analyze the phase transition and elastic properties of actinide-arsenides. The model potential used in the present study is Born–Mayer type and the methodology is quite different than those of the previous workers [10–12]. The method of analysis is given in Section 2 and the results obtained in the present work are discussed in Section 3. 2. Method of analysis The stability of the particular crystal structure is given by the minima of the Gibbs free energy, i.e. G ¼ U þ PV TS,
b¼
G B1 ðrÞ ¼ U B1 ðrÞ þ 2Pr3
(2)
and 8ðr1 Þ3 P pffiffiffi . 3 3
(3)
Within the framework of Born–Mayer type potential which is considered up to nearest and the next-nearest neighbor interactions, the cohesive energies for B1 and B2 phases may be given as follows: aM Z 2 e 2 U B1 ðrÞ ¼ þ 6b expðr=rÞ 4pe0 r pffiffiffi þ 12b exp 2r=r , ð4Þ 1 2 2 aM Z e þ 8b1 expðr1 =r1 Þ 4pe0 r1 pffiffiffi þ 6b1 exp 2r1 = 3r1 ,
U B2 ðr1 Þ ¼
ð5Þ
where aM and a1M be the Modelung constants for the NaCl and CsCl structures respectively. r, b and r1, b1 are, respectively, the hardness and range parameters in B1and B2 phases. To determine the phase transition pressure one has to first determine the hardness parameter and the range parameter in B1 and B2 phases. For B1 phase r and b are determined from Bulk modulus and the equilibrium condition as given below " ( pffiffiffi )# expðr0 =rÞ þ 4 exp 2r0 =r aM e2 Z 2 r0 pffiffiffi B0 ¼ 1 pffiffiffi 2r expðr0 =rÞ þ 2 2 exp 2r0 =r 4pe0 9r40
(6)
aM e 2 Z 2 r pffiffiffi pffiffiffi . 4pe0 6r0 expðr0 =rÞ þ 2 2 exp 2r0 =r 2
(7)
However, for B2 phase the value of the hardness parameter and range parameter cannot be determined by this method because for B2 phase the values of bulk modulus and the equilibrium separation are not known experimentally. The previous workers [10–12] have taken the same values of these parameters in both phases, which is not physically realistic. To remove this difficulty we have followed the work of Mario P. Tosi who concluded that the value of hardness parameter decreases for the more compact CsCl phase up to 10%, whereas the ratio of the range parameters increases by the ratio 8/6 in two different structures [15] such as b1 ¼
(1)
where, U is the internal energy, and S is the vibrational entropy at pressure P, volume V and temperature T. The Gibbs free energy for NaCl (B1) and CsCl (B2) phases is expressed as [16]
G B2 ðr1 Þ ¼ U B2 ðr1 Þ þ
and
8 b, 6
(8)
where 8 and 6 be the coordination numbers of B2 and B1 phase, respectively. In view of Eq. (8) the value of hardness parameter (r1) in B2 phase may thus be calculated from minima of the Gibbs free energy whereas the inter atomic separation r1 can be calculated with the help of volume collapse at the phase transition pressure. The phase transition pressure is the pressure at which the difference of Gibbs free energy of two phases becomes zero (i.e. DG ¼ G B2 G B1 ¼ 0). Further to calculate the elastic constants for B1 phase, we have considered both the Coulomb and the short-range interactions as given by Cowley [17] i.e. C Coul ¼ 11
2:55604e2 Z2 . 4pe0 2r40
(9)
¼ C Coul 12
0:11298e2 Z2 . 4pe0 2r40
(10)
C Coul ¼ 44
1:27802e2 Z2 . 4pe0 2r40
(11)
The short-range (SR) contributions considered up to first and second-nearest neighbor interactions are pffiffiffi ! 1 b r0 1 2b 2r 0 SR exp exp C 11 ¼ þ 2 2 r0 r r0 r r r pffiffiffi ! 2r 0 1 b 2 exp . ð12Þ r r0 r
C SR 12
pffiffiffi ! 1 b r0 1 b 2r 0 ¼ 2 exp exp þ 2 r0 r r r r0 r pffiffiffi ! 2r 0 5 b þ 2 exp . r 2r0 r
ð13Þ
ARTICLE IN PRESS K. Kholiya, B.R.K. Gupta / Physica B 387 (2007) 271–275
C SR 44
pffiffiffi ! 1 b r0 1 b 2r 0 ¼ 2 exp exp þ r0 r2 r r r0 r pffiffiffi ! 3 b 2r 0 2 exp . r r 2r0
3. Results and discussion
ð14Þ
The Bulk modulus for B1 phase is calculated from the relation 1 B ¼ ðC 11 þ 2C 12 Þ 3
273
(15)
and for B2 phase it is calculated from the thermodynamic condition B ¼ V ðd2 U=dV 2 Þ which gives 1 a1 e 2 Z 2 B1 ¼ pffiffiffi M 1 4 4 3 4pe0 ðr0 Þ " ( pffiffiffi )# expðr10 =r1 Þ þ exp 2r10 = 3r1 r1 pffiffiffi pffiffiffi . 1 1 r 2 expðr10 =r1 Þ þ 3 exp 2r10 = 3r1
The input data and calculated model parameters for the compounds under study are given in Tables 1a and b, respectively. It is clear from Table 1b that in binary 1:1 actinide-arsenide compounds for B2 phase the value of the hardness parameter decreases up to 5.5% while in alkali halides it decreases up to 10% [15]. This may be because of the fact that binary 1:1 actinide-arsenides are harder than alkali halides, as there bulk modulus is higher. The calculated values of induced phase transition pressure for ThAs, UAs, NpAs and PuAs along with their cohesive
Table 1a Input parameters used to calculate model parameters
ð16Þ The pressure derivative of Bulk modulus for both the phases may be calculated by fitting P–V data to the Birch [18] and Vinet [19] equations.
Crystal
r0 (A˚)
B0 (GPa)
% volume collapse at transition
ThAs Uas NpAs PuAs
2.989 [5] 2.8835 [5] 2.9183 [4] 2.92825 [5]
118 [5] 98 [5] 70 [4] 68.5 [5]
11 [5] 10 [5] 9 [4] 9 [5]
Table 1b Calculated model parameters Crystal
b (1019 J)
r (A˚)
b1 (1019 J)
r1 (A˚)
% decrease in hardness parameter for B2 phase
ThAs UAs NpAs PuAs
180149.6984 11418.39025 1655.298741 1576.562401
0.2402 0.3069 0.3992 0.4033
240199.5979 15224.52033 2207.064988 2102.083201
0.2365 0.296 0.3773 0.3815
1.54 3.55 5.49 5.41
Table 2 Cohesive and phase transition properties of ThAs, UAs, NpAs and PuAs Crystal
Equilibrium separation (A˚)
Cohesive energy (1019 J)
Transition pressure (GPa)
rB1
rB2
GB1
GB2
ThAs Present Exp. Other
2.989034 2.989 [2] 2.9951 [10]
3.122741
49.5724
48.13437
3.11 [10]
49.454 [10]
48.3546 [10]
UAs Present Exp. Other
2.888346 2.88835 [3] 2.8895 [11]
3.023342
49.92965
48.57729
3.0647 [11]
49.4592 [11]
47.8725 [11]
NpAs Present Exp. Other
2.918302 2.9183 [4] 2.93 [12]
3.065055
47.905
46.6486
3.07 [12]
48.5567 [12]
47.2744 [12]
PuAs Present Exp.
2.928251 2.92825 [5]
3.080841
47.69746
46.37301
26 25 [2] 16.4 [10] 26.2 [9] 30.4 30 [3] 19.75 [11] 34 35 [4] 28.2 [12] 38 38 [5]
ARTICLE IN PRESS K. Kholiya, B.R.K. Gupta / Physica B 387 (2007) 271–275
present study are not available for the comparison, but the values of bulk modulus predicted from these elastic constants are better than those of the previous workers as compared to the experimental values. For UAs 1.5 1
ΔG(10-19Joule)
energies predicted at zero pressure are reported in Table 2. The experimental values of the phase transition pressure are also shown in the table for the sake of comparison. It may be noted from Table 2 that the values of transition pressure (Pt) predicted for these compounds are in excellent agreement with the available experimental values. On the other hand the values obtained by other workers [9–12] deviate largely in comparison to the experimental values. The large deviations of their results with the observed values may be ascribed to the fact that these workers have not predicted the correct values of the model parameters and adopted the same values of both range and hardness parameters in B1 and B2 phases. In phase B2 these parameters have significant different values as compared to those in phase B1. Moreover, their concept that the unit cell volume in B2 phase i.e. (8/3)(r1)3 is greater than B1 phase i.e. 2r3 is not correct, otherwise, the unit cell volume in phase B2 should increase and thus will never collapse. On the other hand the volume in B2 phase always collapses at phase transition pressure and the structural transition occurs. The variations of DG with pressure for these compounds are shown in Figs. 1–4. The experimental values are also represented by filled triangles. It is clear from these figures that our calculated results are quite consistent with the available experimental values. An inspection of Tables 3 and 4 reveals that although the experimental values of the second order elastic constants for the compounds under
-0.5
(a)
-1
0.5 0
0
10
20
30 40 Pressure(GPa)
1.1
50
60
Present work
1
Experimental
0.9 V/ V0
274
0.8 0.7 0.6 0.5 0.4 0
10
20
30
40
50
60
Pressure(GPa)
(b)
Fig. 2. (a) Represents the variation of the difference for Gibbs free energies (DG) in B1 and B2 phase while (b) shows the volume compression with pressure for UAs.
ThAs 2
NpAs 1.5
1 0.5 0 -0.5
0
10
20
30
40
50
60
Pressure (GPa)
-1
ΔG(10-19Joule)
ΔG(10-19 Joule)
1.5
1 0.5 0
0
10
20
(a) -1.5
(a)
1.1 Experimental
0.8
V/ V0
V/ V0
0.9
0.7
50
60
70
-1
1
Present work
0.9
Experimental
0.8 0.7
0.6
0.6
0.5
0.5
0.4
0.4 0
(b)
40
1.1
Present work
1
30
Pressure(GPa)
-0.5
10
20
30
40
0
50
Pressure (GPa)
Fig. 1. (a) Represents the variation of the difference for Gibbs free energies (DG) in B1 and B2 phase while (b) shows the volume compression with pressure for ThAs.
(b)
10
20
30
40
50
60
Pressure(GPa)
Fig. 3. (a) Represents the variation of the difference for Gibbs free energies (DG) in B1 and B2 phase while (b) shows the volume compression with pressure for NpAs.
ARTICLE IN PRESS K. Kholiya, B.R.K. Gupta / Physica B 387 (2007) 271–275
PuAs
ΔG(10-19Joule)
1.5
Table 4 Calculated bulk modulus and its pressure derivative of ThAs, UAs, NpAs and PuAs for NaCl-B1 and CsCl-B2 structures
1
Bulk modulus (GPa)
B10
In B1 phase
In B2 phase
In B1 phase
In B2 phase
ThAs Present Exp. Other
117.892 118 [5] 109 [10]
140.674
5.887 3.4 [5]
6.238
UAs Present Exp.
97.59 98 [5]
118.613
4.934 4.9 [5]
5.15
NpAs Present Exp. Other
69.138 70 [5] 70.12 [12]
89.155
4.349 6 [5]
3.974
PuAs Present Exp.
67.634 68.5 [5]
79.758
4.314 3.3
4.613
Crystal
0.5 0 0
10
20
-0.5
30
40
50
60
70
80
Pressure(GPa)
(a) -1 1.1
V/ V0
275
1
Present work
0.9
Experimental
0.8 0.7 0.6 0.5 0.4 0
(b)
10
20
30
40
50
60
Pressure(GPa)
Fig. 4. (a) Represents the variation of the difference for Gibbs free energies (DG) in B1 and B2 phase while (b) shows the volume compression with pressure for PuAs.
properties of all the binary 1:1 actinide-arsenide compounds.
Table 3 Calculated elastic constants Cij of ThAs, UAs, NpAs and PuAs for B1 phase
References
Crystal
C11 (GPa)
C12 (GPa)
C44 (GPa)
ThAs Present Other
268.747 246.24 [10]
42.465 41.46 [10]
42.781 40.46 [10]
UAs Present
187.239
52.766
53.991
NpAs Present Other
97.037 115.85 [12]
55.188 47.93 [12]
57.8 47.93 [12]
PuAs Present
93.778
54.562
57.184
The pressure derivatives of bulk modulus ðB10 Þ are also found in reasonably good agreement with experimental values in B1 phase. However, B10 could not be compared in phase B2 as the experimental data in B2 phase are not available so far. On the basis of overall description it may thus be concluded that the potential model and its application in the present study has satisfactorily explained the structural stability, cohesive and phase transition
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