A tentative model for estimating the compressibility of

Transcription

1 A tentative moel for estimating the compressibility of rock-salt AgCl x Br 1-x alloys VASSILIKI KATSIKA-TSIGOURAKOU * an EFTHIMIOS S. SKORDAS Department of Soli State Physics, Faculty of Physics, University of Athens, Panepistimiopolis, Zografos, Greece Abstract Ab initio etaile calculations of the elastic properties of AgCl x Br 1-x alloys recently appeare using ensity-functional perturbation theory an employing the virtual crystal approximation or by means of the full potential linearize augmente plane wave metho. Here, we suggest a simple theoretical moel that enables the estimation of the isothermal compressibility of these alloys in terms of the elastic ata of en members alone. The calculate values are in satisfactory agreement with the experimental ones. The present moel makes use of an early suggestion that interconnects the Gibbs energy for the formation an/or migration of efects in solis with bulk properties. Keywors: compressibility; point efects; mixe crystals; Pacs 61.7.Bb; 61.7.J-; 6.0.D-; Fq; h *vkatsik@phys.uoa.gr 1

2

3 1. Introuction The silver halies exhibit interesting properties compare to the alkali halies, such as lower melting point an higher ionic conuctivity. Silver halies are of great importance as photographic materials, as soli electrolytes an as liqui semiconuctors (e.g., see Refs [1-5]). Although they all have the same NaCl structure, as the alkali halies, we emphasize that the elastic properties of the silver halies cannot be explaine with the simple theories that successfully escribe the elastic properties of the alkali halies [6]. Many experimental [6 1] an theoretical [13] stuies have been carrie out to unerstan the structural an the elastic properties, the phase transformation at high pressure, an the lattice ynamics of the AgBr, the AgCl an the AgBr 1-x Cl x ternary alloys. For example, recently, Shigeki Enou et al. [1] have measure the temperature epenence of the elastic constants in the silver halie crystals, above room temperature, by using the Resonant Ultrasoun Spectroscopy metho [14]. As a secon example, we refer to Ref. [15], in which the elastic properties an the lattice ynamics of AgBr 1-x Cl x have been stuie as a function of the composition (x) in the NaCl (B1) phase, by using the ensity-functional perturbation theory an employing the virtual-crystal approximation. Thirly, Amrani et al. [], in orer to help unerstan an control the alloy system between AgCl an AgBr an behavior of bowing an relate properties, have investigate the effect of the Cl concentration on the structural an electronic properties of the AgCl x Br 1-x alloys, with Cl contents between 0 an 1, 3

4 using the Full Potential-Linearize Augmente Plane Wave metho. The effect of composition on bulk moulus was investigate. This property was foun to epen nonlinearly on alloy composition x. The question arises whether one can etermine the values of bulk moulus of a AgCl x Br 1-x mixe system, solely in terms of the elastic ata of the en members AgBr an AgCl. This paper aims to answer this question. We employ here a simple moel, that has been also recently [16] use for the calculation of the compressibility of multiphase mixe alkali halies crystals grown by the melt metho [17] using the miscible alkali halies, i.e., NaBr an KCl, which have a simple cubic space lattice of the NaCl-type an measure in a etaile experimental stuy by Pama an Mahaevan [17]. This moel has been also successfully applie [18] to the mixe crystal NH 4 Cl 1-x Br x consiering that NH 4 Cl an NH 4 Br have a simple cubic space lattice structure of the CsCl-type. In this paper we report the remarkable fining that this simple moel prouces in the case of AgCl x Br 1-x alloys equally successful results as in the mixe alkali halies an mixe ammonium halies espite the aforementione significant ifferences in their physical properties an especially the lack [6] of a unifie explanation with simple theories of the elastic properties of silver halies an alkali halies, as mentione above. We emphasize, however, that the proceure through which this simple moel is applie here to AgCl x Br 1-x iffers essentially from the one followe for its application to mixe alkali an ammonium halies as it is explaine in the last paragraph of the next section. 4

5 . The metho We first recapitulate the moel that explains how the compressibility ( 1 B) of a mixe system AB x 1 x can be etermine in terms of the compressibilities of the two en members A an B. Let us call the two en members A an B as pure components (1) an (), respectively an label 1 the volume per molecule of the pure component (1) (assume to be the major component in the aforementione mixe system AB x 1 x), the volume per molecule of the pure component (). Furthermore, let enote V 1 an V the corresponing molar volumes, i.e. V1 N1 an V N (where N stans for Avogaro s number) an assume that 1. Defining a efect volume [19] as the increase of the volume V 1, if one molecule of type (1) is replace by one molecule of type (), it is evient that the aition of one molecule of type () to a crystal containing N molecules of type (1) will increase its volume by 1 (see Chapter 1 of Ref. [19] as well as Ref. [0]). Assuming that is inepenent of composition, the volume VN nof a crystal containing N molecules of type (1) an n molecules of type () can be written as: VNn N1 n( 1) or VNn [1 ( n N)] V1 n (1) The compressibility of the mixe crystal can be foun by ifferentiating Eq.(1) with respect to pressure which gives: 5

6 V [1 ( n N)] V n Nn 1 1 or V V n N N V () Nn where enotes the compressibility of the volume, efine as (1 ) ( P). T Within the approximation of the har-spheres moel, the efect volume can be estimate from: ( V V1 ) N or 1 (3) Thus, since VN n can be etermine from Eq.(1) (upon consiering Eq.(3) ), the compressibility can be foun from Eq.() if a proceure for the estimation of will be employe. In this irection, we aopt a thermoynamical moel for the formation an migration of the efects in solis escribe below which has been of value in various categories of solis incluing [1-6] metals, ionic crystals, rare gas solis etc as well as in high T c superconuctors [7] an in complex ionic materials uner uniaxial stress [8] that emit electric signals before fracture, in a similar fashion with the signals observe [9, 30] before the occurrence of major earthquakes. Accoring to the latter thermoynamical moel, the efect Gibbs energy i g is interconnecte with the bulk properties of the soli through the relation g i i c B (usually calle cb moel) where B stans for the isothermal bulk moulus (=1/κ ), the mean volume 6

7 per atom an i c is imensionless quantity. (The superscript i refers to the efect process uner consieration, e.g. efect formation, efect migration an self-iffusion activation). By ifferentiating this relation i i in respect to pressure P, we fin that efect volume [ ( g P) ]. T i, The compressibility efine by i, [ ( i ) ] n P T, is given by [, 3]: i, (1 B) ( B P ) [( B P) T 1] (4) i, This relation states that the compressibility oes not epen on the type i of the efect process. Thus, it is reasonable to assume now that the valiity of Eq. (4) hols also for the compressibility involve in Eq. (), i.e., ( B P ) [( B P) 1] (5) T where the subscript 1 in the quantities at the right sie enotes that they refer to the pure component (1). The quantities B1 P an B1 P, when they are not experimentally accessible, can be estimate from the moifie Born moel accoring to [19, 0]: B B1 P ( n 7) 3 an B B ( B P ) (4 9)( n 3) (6) 1 1 where B n is the usual Born exponent. This is the proceure that has been successfully applie in Ref. [16] for the multiphase mixe alkali crystals, as well as in mixe ammonium halies [18]. Attention is rawn, however, to cases like AgCl x Br 1-x where the Born moel oes not provie an aequate escription [6], as oes for alkali halies. 7

8 Thus, here, for the case of AgCl x Br 1-x we shall solely rely on Eq. (4), but not on Eq. (6). In other wors in our former publications [16, 18] ealt either with mixe alkali halies or with ammonium halies, we calculate the first an secon pressure erivatives of the bulk moulus on the basis of Eq. (6) obtaine from the moifie Born moel- an then inserte them into Eq. (4). On the other han in the present case of AgCl x Br 1-x we o not use at all the moifie Born moel, but we insert into Eq. (4) the first an secon pressure erivative of the bulk moulus euce from the elastic ata of AgBr uner pressure using a least squares fit to a secon orer Murnaghan equation as it will be escribe in the next section. 3. Results Let us apply this proceure to the mixe system: AgBr-AgCl. In this application we shall intentionally take as starting material AgBr (1) ( V 1 =8.996 cm 3 /mole) an by consiering that for the pure AgCl () the volume is V =5.731 cm 3 /mole, one gets N V V1 3.65cm 3. We now consier the aiabatic values measure for various compositions in Ref. [6] an transform them to the isothermal ones with the stanar thermoynamical proceure escribe in Ref. [19]. Using these isothermal -values, for various compositions x, we actually fin that VN nversus nn is a straight line the slope of which, accoring to the Eq. (), is ( ) ( AgBr) V ( AgBr) 63.99x10 - cm 3 GPa -1 By inserting N 1 1 8

9 the -value we fin GPa -1. Note that, the - value is appreciably higher than the compressibility of AgBr ( 1 = GPa -1 ) an AgCl ( = GPa -1 ), as expecte from thermoynamic arguments forware in Ref. [19] We now procee to the calculation of on the basis of Eq. (5), by using the elastic ata uner pressure [31], which are well escribe if the expansion of the isothermal bulk moulus is carrie out to secon orer, i.e., P B 1 B B( P) B P P lnv P P T T the investigation of which yiels a secon orer Murnaghan equation (the subscript 0 correspons to values close to zero pressure). The resulting expression for the bulk moulus of AgBr was foun to be [31]: 1 B( P) P (0.087) P where B an P are in B kilobars thus 7.49 P an B T P T By inserting these values into Eq. (5) we fin GPa -1, with 1 = GPa -1 the isothermal compressibility for AgBr [18]. From Eq. (1) we calculate the volume VN n for each concentration of the mixe crystals an from Eq. () the values of the isothermal bulk moulus. All the calculate values for the isothermal 9

10 bulk moulus are epicte with asterisks in Fig. 1 where they are plotte versus the composition (x). In the same figure, we also insert with crosses the experimental values euce from the aiabatic values measure in Ref. [6] an transforme to the isothermal ones by the stanar thermoynamical proceure [19], as alreay mentione. We now turn to the values of the aiabatic bulk moulus. The theoretical values calculate in Refs. [] an [15] are plotte in Fig. 1 with soli circles an open reverse triangles, respectively. We also insert with open squares the values calculate by the aforementione simple thermoynamical moel, where we followe the same proceure as above, but by consiering the aiabatic values instea of the isothermal ones. In the same plot, we also show with soli triangles the experimental aiabatic values of the bulk moulus as reporte in Ref. [6]. An inspection of these values reveals that there exists a isparity between the values calculate in Refs. [] an [15]. Furthermore, we see that the values resulte from the simple moel iscusse here lie between those calculate in Refs. [] an [15]. 4. Conclusions Here, we mae use of the key-concept that the volume variation prouce by the aition of a foreign molecule to a host crystal can be consiere as a efect volume. Then the compressibility of this efect volume was calculate on the basis of an early thermoynamical moel which interconnects the efects Gibbs energy with bulk properties. This way enables the estimation of the isothermal 10

11 compressibility of the rock-salt AgCl x Br 1-x alloys in terms of the elastic ata of the pure constituents (i.e., AgBr an AgCl) alone. In all the composition range for which experimental ata are available, the calculate values of the isothermal compressibility of these alloys are in reasonable agreement with the experimental ones. If we consier the aiabatic compressibility of these alloys, instea of the isothermal one, the values obtaine by the present moel lie between those resulte from the microscopic calculations carrie out by other authors [, 15]. 11

12 References [1] S. Enou, Y. Michihiro, K. Itsuki, K. Nakamura an T. Ohno, Soli State Ionics 180, (009) [] B. Amrani, F. El Haj Hassan an M. Zoaeter, Physica B 396, (007) [3] J. Kapecki an J. Roberts, in: M. Howe-Grant (E.), Kirk-Othmer Encyclopeia of Chemica Technology, fourth e., vol. 6 (Wiley, New York, 1993) [4] I. Ebbsjo, P. Vashishta, R. Dejus an K. Skol, J. Phys. C 0, L441-L447 (1987) [5] J.E. Enerby an E.C. Barnes, Rep. Prog. Phys. 53, (1990) [6] L. S. Cain, J. Phys. Chem. Solis 38, (1977) R. Barber, M. Downing, J.B. Shaw an L.S. Cain,. J. Phys. Chem. Solis 50, (1989) [7] S. Hull an D.A. Keen, Phys. Rev. B 59, (1999) D.S. Tannhauser, L.T. Bruner an A.W. Lawson, Phys. Rev. 10, (1956) [8] L.S. Cain an G. Hu, Phys. Rev. B 64, (001) [9] W.C. Hughes an L.S. Cain, Phys. Rev. B 53, (1996) [10] A. Fujii, H. Stolz an W. von er Osten, J. Phys. C: Soli State Phys. 16, (1983) B. Bootz, W. von er Osten an N. Uhle, Phys.Status Solii (b) 66, (1974) [11] J. Winscheif an W. Von er Osten, Soli State Phys. 13, (1980) K. Nakamura, J. Winscheif an W. Von er Osten, Soli State Commun. 39, (1981) N. Koumvakalis an W. Von er Osten, Phys. Status Solii (b) 9, (1979) [1] P. R. Vijayaraghavan R. M. Nicklow, H. G. Smith an M. K. Wilkinson., Phys. Rev. B 1, (1970) J. Weber, Phys. Status Solii (b) 78, (1976) [13] T. Benmessabih, et al., Physica B 39, (007) L.A. Palomino-Rojas et al., Soli State Sci. 10, (008) P.T.Jochym an K. Parlinski, Phys. Rev. B 65, (001) 1

13 Yan Li, et al., Phys. Rev. B 74, (006) [14] T. Goto an O.L. Anerson, Rev. Sci. Instrum. 59, (1988) W.M.Visscher, A. Migliori, T. M. Bell an R. A. Reinert, J. Acoust. Soc. Am. 90, (1991) A. Migliori an J. H. Sarro, Resonant Ultrasoun Spectroscopy (John Willy an Sons, New York, 1997) [15] Kh. Bouamama, P. Djemia, K. Daou an S.M. Chérif, Comp.Mater. Sci. 47, (009) [16] V. Katsika-Tsigourakou an A. Vassilikou-Dova, J. Appl. Phys. 103, (008) [17] C. M. Pama an C. K. Mahaevan, Mater. Manuf. Processes, (007) C. M. Pama an C. K. Mahaevan, Physica B 403, (008) [18] V. Katsika-Tsigourakou, Pramana J. Phys. 77, (011) [19] P. Varotsos an K. Alexopoulos, Thermoynamics of Point Defects an Their Relation With the Bulk Properties (North-Hollan, Amsteram, 1986) [0] P. Varotsos, Phys. Status Solii B 99, K93-K96 (1980) P. Varotsos, J. Phys. Chem. Solis 4, (1981) P. Varotsos, K. Alexopoulos, an K. Nomicos, Phys. Status Solii B 111, (198) [1] P. Varotsos an K. Alexopoulos, Phys.Rev. B 15, (1977) P. Varotsos an K. Alexopoulos, Phys.Rev. B 15, (1977) P. Varotsos an K. Alexopoulos, Phys.Rev. B 1, (1980) P. Varotsos an K. Alexopoulos, Phys.Rev. B 4, (1981) P. Varotsos an K. Alexopoulos, Phys.Rev. B 30, (1984) P. Varotsos an K. Alexopoulos, Phys. Status Solii A 47, K133- K136 (1978). [] P. Varotsos, W. Luwig an C. Falter, J. Phys. C: Soli State 11, L311-L315 (1978) P. Varotsos an W. Luwig, J. Phys. C: Soli State 11, L305-L309 (1978) 13

14 [3] P. Varotsos, W. Luwig an K. Alexopoulos, Phys. Rev. B 18, (1978) P. Varotsos, Phys.Rev. B 75, (007) P. Varotsos, Soli State Ionics 179, (008) [4] P. Varotsos an K. Alexopoulos, J. Phys. Chem. Solis 38, (1977) P. Varotsos an K. Alexopoulos, J. Phys. C: Soli State 1, L761- L764 (1979) [5] P. Varotsos an K. Alexopoulos, Phys. Status Solii B 110, 9-31 (198) K. Alexopoulos an P. Varotsos, Phys.Rev. B 4, (1981) P. Varotsos, Phys. Rev. B 13, 938 (1976) P. Varotsos, J. Appl. Phys. 101, (007) [6] P. A. Varotsos, J. Physique (France) Lettr. 38, L455-L458 (1977) P. A. Varotsos, Phys. Status Solii B 90, (1978) P. A. Varotsos, Phys. Status Solii B 100, K133-K138 (1980) [7] H. Su, D. O. Welch an W. Wong-Ng, Phys. Rev. B 70, (004) [8] P. Varotsos, N. Sarlis an M. Lazariou, Phys. Rev. B 59, 4-7 (1999) P. Varotsos, N. Sarlis, M. Lazariou an P. Kapiris, J. Appl. Phys. 83, (1998) P. Varotsos, K. Alexopoulos an M. Lazariou, Tectonophysics 4, 1-37 (1993) [9] P. Varotsos, K. Alexopoulos, K. Nomicos an M. Lazariou, Nature (Lonon) 3, 10 (1986) P. Varotsos an K. Alexopoulos, Tectonophysics 110, (1984) P. Varotsos an K. Alexopoulos, Tectonophysics 110, (1984) P. Varotsos, K. Alexopoulos, K. Nomicos an M. Lazariou, Tectonophysics 15, (1988) P. Varotsos an M. Lazariou, Tectonophysics 188, (1991) [30] S. Abe et al., Phys. Rev. Lett. 94, (005) 14

15 P. A. Varotsos, N.Sarlis an E. S. Skoras, Phys. Rev. E 66, (00); 67, (003); 68, (003). P.A. Varotsos, N.V. Sarlis, E.S. Skoras, S. Uyea an M. Kamogawa, Proc Natl Aca Sci USA 108, (011) P.A. Varotsos, N.V. Sarlis, E.S. Skoras, H.K. Tanaka an M.S. Lazariou, Phys. Rev. E 73, (006) P.A. Varotsos, N.V. Sarlis, E.S. Skoras, H.K. Tanaka an M.S. Lazariou, Phys. Rev. E 74, 0113 (006) [31] K. F. Loje an D. E. Schuele, J. Phys. Chem. Solis 31, (1970) 15

16 FIG.1 The asterisks an the crosses mark the theoretical an the experimental values of the isothermal bulk moulus (broken lines). The latter come from the aiabatic values measure in Ref. [6] after transforming them to the isothermal ones by means of the stanar thermoynamical manner (see Ref. [19]). We also plot for the aiabatic bulk moulus the theoretical (soli circles from Ref. [], open reverse triangles form Ref. [15] an open squares from the simple moel presente here) along with the experimental values [6] (soli triangles). 16