The dielectric constant of Na 0.4 K 0.6 Br and its large temperature variation

Transcription

1 The ielectric constant of Na.4 K.6 Br an its large temperature variat V. Katsika-Tsigourakou*, P. Bekris Sect of Soli State Physics, Department of Physics, Natal an Kapoistrian University of Athens, Panepistimiopolis, Zografos, Greece PACS 6.7.Ji, 6..Dc, 66..Fq, 66..-h, 6.7.Bb * Corresponing author: vkatsik@phys.uoa.gr, Since polycrystals of alkali halies are highly useful as components in optical evices, a number of mixe crystals of NaBr an KBr have been prepare from melt by other workers. Among these crystals, it was reporte that the polycrystal Na.4 K.6 Br exhibits the strongest temperature variat of the ielectric constant. Here, we show quantitatively that this is ue to the temperature variat of the ic polarizability. Introuct It has been suggeste, long ago, that several physical properties (e.g., the ielectric constant [], the compressibility [], the conuctivity []) of alkali halie mixe crystals can be etermine in terms of the corresponing properties of the pure en members (for a relevant review see chapter of Ref. [4]). Several experimental stuies appeare [5-] on single phase an multiphase mixe crystals of alkali halies as a result of the intensifie interest when it was realize that polycrystals of alkali halies are highly useful as components in optical evices []. As a first attempt towars unerstaning these experimental results, a proceure has been suggeste [4], which enables the estimat of the compressibility of the multiphase mixe crystals in terms of the elastic ata of the en members alone (this will be summarize later in Sect ). The key point of that proceure propose [4] was the consierat of the volume variat prouce by the ait of a foreign molecule to a host crystal as a efect volume whose compressibility was calculate on the basis of a thermoynamic moel (terme cb moel) that has been foun of value for the calculat of the efect (format an migrat) parameters in a large variety of solis [5-]. Here, we eal with the experimental stuy [] that refers to multiphase mixe crystals of NaBr an KBr. In this stuy, Pama an Mahaevan [] hieve the preparat of mixe crystals Na x K -x Br of NaBr an KBr from melt an their X-ray iffrt analysis inicate the existence of two phases in the mixe crystals. The growth of the following five systems was reporte []: Na. K.8 Br, Na.4 K.6 Br, Na.5 K.5 Br, Na.6 K.4 Br, Na.8 K. Br (the composit written is taken for crystallizat, see their Table ). Beyon the stuy of thermal parameters (Debye-Waller ftor, mean square amplitue of vibrat, Debye temperature an Debye frequency), Pama an Mahaevan performe electrical measurements in all these (polycrystalline) mixe systems in the temperature range 8 o to 4 o K. Namely, they measure the c an conuctivity (labele c an ) as well as the (real part of the) ielectric constant ( ) an the ielectric loss ftor ( tan ). They foun that the values of,, tan an increase with increasing temperature. However, this increase is ifferent for ifferent mixe systems. The results obtaine inicate that the bulk composit has nonlinear influences on the electrical parameters. (Such a behavior has been also observe earlier [] for single crystals NaBr-KBr). This nonlinearity is foun to increase with the increase in temperature in the c,, tan an values. In particular, Pama an Mahaevan [] plotte eh of these parameters versus the composit x an a maximum was ientifie at x=.4 (see their figures 5, 6, 7 an 8). This maximum was very pronounce at the highest temperature (4 o K) stuie. In other wors, among the five mixe systems mente above, the secon one, i.e., Na.4 K.6 Br exhibite a maximum in the corresponing plots for eh of the parameters c

2 c,, tan an versus x, which was very pronounce for the temperature of 4 o K. In this paper we solely focus on the explanat of the variat of with temperature for which Pama an Mahaevan [] offere the following qualitative remarks: This is generally attribute to the crystal expans, the electronic an ic polarizats an the presence of impurities an crystal efects. The variat at low temperatures is mainly ue to the expans an electronic an ic polarizats. In the case of alkali halie crystals, the electronic polarizability has no role to play. The increase at higher temperatures is mainly attribute to the thermally generate charge carriers an impurity ipoles. So, the observe increase in ielectric constant with temperature is essentially ue to the temperature variat of ic polarizability. Before proceeing, an concerning the aforemente qualitative arguments of Pama an Mahaevan [], we note the following. Analytical reagent grae samples of NaBr an KBr were the starting materials they use for the growth of crystals. As they notice, the ominant impurities present in NaBr incluing iron (.%) an those present in KBr inclue the ivalent cats (calcium.% an magnesium.%). No specific controls were provie to prevent these impurities from entering the crystals. Actually, early stuies [4] have shown that in alkali halies the presence of the aforemente ivalent cats prouce (for reasons of charge compensat) aital cat vancies; at low temperatures, a port of these vancies which epens on temperature- are boun, i.e., they are attrte by the nearly ivalent cats, thus forming electric ipoles (usually terme complexes [4]) that contribute to the real (an imaginary) part of the ielectric constant. The remaining free vancies (i.e., locate far away from the impurities), contribute to the c conuctivity of the material (this contribut raises the so calle extrinsic reg [5] of the conuctivity plot n( T) vs T ). A theoretical calculat of the extent to which this phenomenon contributes to the observe temperature epenence of the ielectric constant is teious an, at the present stage, not possible, since it emans the knowlege of the associat (an issociat ) parameters that govern the populat of the electric ipoles ivalent impurity cat vancy at eh temperature. Unfortunately, these parameters have not been reporte either in Ref. [4] or in other publicats. In view of this ifficulty, we restrict ourselves here to the calculat of the contributs of the electronic an ic polarizability to the temperature epenence of the ielectric constant. Along these lines, a theoretical moel is presente in the next sect. The moel that counts for the temperature epenence of the electronic an ic polarizability Here, we exten an early moel suggeste in Refs. [6-8] for the pure alkali halies to the case of mixe systems. Szigeti [9], within the frame of classical theory, propose the following two relats that interconnect the low frequency ( ) an high frequency ( ) ielectric constant of (unope) alkali halies with the transverse optical moe frequency : f T T T () 4 ( ) e () 9 where f is the short-range force constant, the volume per pair, the reuce mass an e the so calle Szigeti effective charge. If we enote an the electronic polarizabilities of the cat an an respectively, the classical Lorentz Lorentz relat reas: 4 ( ) while the low frequency (static) ielectric constant can be calculate from: 4 ( ) where stans for the ic polarizability given by e f () (4) (5) It has been foun, however, that Eq. (4) agrees with the experimental ata only when takes the value: e f (6) Consiering that the number of cats (or ans) per unit volume is equal to 4, where enotes the lattice constant in a NaCl structure, an that can be approxi- mately calculate by the relat T const B, where B stans for the bulk moulus, a combinat of the aforemente relats finally leas to: 6 ( ) const B in which epens on the temperature through the relat 4 T (7) T exp T, where is the thermal volume expans coefficient. bviously, the term in the right han sie of Eq. (7) refers to the ic polarizability, while the term 6 ( ) to the electronic polarizability.

3 In orer to overcome the ifficulty of the unknown value of the const involve in the right han sie of Eq. (7), we first apply this equat to a temperature T (e.g., at room temperature, R.T.) an then etermine the value of at any other temperature (after isregaring, to a first approximat, corrects ue to the volume epenence of the Szigeti charge). This leas to: 4 B ( ) AB ( ) ( ) B ( A ) 4 B ( ) AB ( ) ( ) B ( A) where the quantity A, efine as (8) 6 A ( ), is known from the measure value of through Eq. (). Since, however has not been measure in Ref. [] at various temperatures, we have to rely on the approximat that A is temperature inepenent. Furthermore, note that in Eq. (8) the values of the quantities, B an correspon to any esire temperature T, while the subscript o enotes the relevant values at T T. Applicat of Eq. (8) to the mixe system Na.4 K.6 Br Let us now apply Eq. (8) to the system Na.4 K.6 Br which, among the five mixe systems stuie, exhibits the strongest temperature variat, as iscusse above. We take as T the lowest temperature of 8 o K for which the ielectric constant measure by Pama an Mahaevan to be 8.7. At this temperature, they reporte the value of the compressibility 8.89 m / N. The latter gives for the corresponing bulk moulus B ( / ) the ue B =.9 GPa, for the mixe system. In orer to apply Eq. (8), to the highest temperature (4 o K) at which the ielectric constant has been measure in Ref. [], we nee to know the corresponing values of B an, which unfortunately have not been measure by Pama an Mahaevan [] at temperature higher than 8 o K. Hence, we must rely on certain approximats to estimate them: We start with the lattice constant, which is calculate as follows: The molar volume of the mixe crystal is given by V xv ( x) V, where V an V stan for the corresponing volumes of NaBr an KBr. Hence x ( x), where, correspon to the lattice constants of the pure en members KBr an NaBr 6.6 m an 5.98 m for T =8 o K we fin 6.9 m for the mixe system. The same respectively. ( Thereafter, the subscripts an at all the quantities use will refer to KBr an NaBr, respectively). Thus, consiering the values is repeate for the highest temperature T =4 o K, by consiering the corresponing values of the thermal expans coefficient 4.4 K for KBr an 4.46 K for NaBr (obtaine from an interpolat of their corresponing values of at room temperature an their melting temperature []). Thus we fin the value system. 6.4 m at T =4 o K for the mixe We now turn to the value of B. We first consier for KBr the compressibility values 44. an 6.8 m / N reporte by Vai et al [] for the melting temperature (7 o K) an room temperature (R.T.), respectively, an therefrom we fin the corresponing values of the bulk moulus. Then, by consiering that in the high temperature range the B -value ecreases almost linearly upon increasing the temperature [4], we fin by linear interpolat- that the corresponing B -value at 4 o K is aroun 9.74 GPa. We now procee to the calculat of B for the mixe system, by following the proceure evelope in Ref. [4]. This, in general, can be summarize as follows: Let be the volume per molecule of the pure component () (usually assume to be the major component in the aforemente mixe system). Without losing generality, we assume that is smaller than the volume per molecule of the pure component (). bviously V N an V N, where N stans for Avogaro s number. We now efine a efect volume as the increase of the volume V if one molecule of type () is reple by one molecule of type (). Thus, the ait of one molecule of type () to a crystal containing N molecules of type () will increase its volume by, where is a efect volume (see also below). If is inepenent of composit, the volume N n V of a crystal containing N molecules of type () an n molecules of type () shoul be equal to:

4 VNn N n( ) (9) which upon ifferentiating with respect to pressure leas to: V [ ( n / N)] V n Nn () where, an enote the compressibility of the mixe crystal,the en member an the volume, respectively. In the har-spheres moel, the efect volume can approximately etermine from: ( V V ) / N or () an the compressibility of the volume is given by: (/ B) ( B / P ) /[( B / P) T ] () when consiering the so-calle cb moel, which has been foun to be successful for escribing the parameters for the format an migrat of the efects in a variety of solis [5-]. Thus, since VN n can be irectly compute from Eqs. (9) an (), the compressibility of the mixe system is calculate from Eq. () Let us now apply the aforemente proceure to the case of the mixe system, i.e., (NaBr).4 (KBr).6 in which the en member (pure) crystal () with the higher composit is of course KBr. In orer to calculate, from Eq. (), we use the following values for KBr: (B /P) T =5.8 [] an B/P =-.49 GPa - obtaine from the relat [4] B ( B /P ) = - (4/9)(n B +), where n B is the usual Born exponent [, ]- along with the aforemente value of B =9.74GPa at T=4 o K. By inserting these values into Eq. () we fin =.5 GPa -. Furthermore, by consiering the an values of KBr an NaBr respectively, we fin from Eq. () an V from Eq. (9), at T=4 o K an finally obtain from Eq. N n () the value of B( / ) system at that temperature. =.96 GPa for the mixe By inserting into Eq. (8) the values of =6.4 - m an B =.96 GPa erive in the previous paragraphs, we fin that the value of the ielectric constant is =7.45. This agrees nicely with the experimental value 69. reporte in Ref. []. 4 Discuss We first comment on the ft that at 4 o K the calculate value iffers slightly, i.e. only by % (a ifference which is anyhow between the experimental error), from the experimental one. This is remarkable, especially if we take into count that the -value measure at the highest temperature (T=4 o K ) is almost by a ftor of aroun two larger than the corresponing value measure at the lowest temperature T=8 o K. The origin for the success of Eq. (8) to count for such a consierable temperature epenence of, coul be better unerstoo from an inspect of Eq. (7), which reveals the following: In general, since excees at the most by a few percent, what counts for the large temperature of is the temperature ecrease of the bulk moulus upon increasing the temperature. This means that, as the temperature increases, the ic polarizability exhibits a consierable increase, which reflects a large temperature increase of as well. In other wors, Eq. (7) quantifies the merit of the qualitative argument of Pama an Mahaevan mente in the Introuct that the increase in ielectric constant with temperature is essentially ue to the temperature variat of the ic polarizability. We now turn to an alternative usefulness of Eq. (7), which is of prtical importance. Let us now assume that the temperature remains constant, but we vary the (external) pressure P. Then Eq. (7) preicts that shoul also change upon varying the pressure mainly ue to the pressure variat of B (cf. in ic crystals the quantity (B/P) T usually rehes consierable values, i.e. aroun 5 or larger [4]). This seems to explain, in principle, the observat of the so calle coseismic signals [] i.e., the electric signals generate upon the arrival of the seismic waves at measuring site. (cf. These electric signals are entirely ifferent from the precursory electric signals that are etecte ays to weeks before an earthquake occurrence [4, 5]). This is so, because the arrival of seismic waves causes a time variat of pressure aroun the measuring electroes which reflects coring to Eq. (7)- a time variat of the polarizat, thus proucing electric signals (in a manner qualitatively similar to the well known piezoelectric phenomenon). These signals are intensifie when a consierable ensity of charge islocats is also present [].. 5 Conclus Among the five polycrystalline mixe systems of NaBr an KBr, one of them, i.e., Na.4 K.6 Br, exhibits the strongest temperature variat in the ielectric constant. In particular, the -value at 4 o K is about twice that at 8 o K. Here, we showe that this variat can be almost exclusively counte for from the temperature increase of the ic polarizability. References [] P. Varotsos, Phys. Status Solii B, K (98). 4

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