Evaluation of pressure and bulk modulus for alkali halides under high pressure and temperature using different EOS

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1 Journal of the Association of Arab Uniersities for Basic and Applied Sciences (3) 4, Uniersity of Bahrain Journal of the Association of Arab Uniersities for Basic and Applied Sciences ORIGINAL ARTICLE Ealuation of pressure and bulk modulus for alkali halides under high pressure and temperature using different EOS Ibtisam Y. Alkammash Department of Physics, College of Science, Uniersity of Mosul, Iraq Receied 3 August ; reised Noember ; accepted 5 January 3 Aailable online February 3 KEYWORDS Equation of state; Kumar General EOS; Interionic potential; Compressibility; Bulk modulus; Alkali halides Abstract In the present work, the non-inerted (isothermal) type equations of state (EOS) which are based on interionic potential such as Vinet Rydberg EOS,, as well as Kumar isothermal EOS are used to study the ariation of relatie compression olumes in range and the isothermal bulk modulus with high pressure at temperature T = 3 K for four alkali halides (NaF, NaCl, NaBr and NaI). The results for relatie compression olume obtained from the four EOS hae been compared with experimental data which indicated that up to 5 GPa they are in good agreement. But beyond this, the results which hae been obtained from the Vinet Rydberg EOS and Bardeen EOS as well as from the experimental data showed that they are also in good agreement in the wide range of high pressure. While results obtained from Kumar isothermal EOS showed that beyond 5 GPa they deiated from the other EOS. Moreoer the results of isothermal compressibility hae shown that at the alue, the calculated high pressures are equal to (7, 9, 78 and 6) GPa for NaF, NaCl, NaBr and NaI respectiely. The results for the isothermal bulk modulus (K T ) hae been compared with four EOS and showed that the bulk modulus increases continuously with increase in pressure. The results indicated that up to GPa are found to be in good conergence, but beyond this range, the results do dierge from each other for the four EOS. Finally, the ariations of non-isothermal compression (inerted) and non-isothermal bulk modulus with high pressure hae been inestigated by using a non-isothermal (Kumar General EOS). ª 3 Uniersity of Bahrain. Production and hosting by Elseier B.V. All rights resered.. Introduction address: ibtisamyahya@yahoo.com Peer reiew under responsibility of Uniersity of Bahrain. Production and hosting by Elseier The understanding of the ionic solids at nonambient conditions is an integral part of the work of material scientists, solid state physicists, chemists, and solid earth geoscientists that are routinely confronted with problems inoling ionic solids at high pressure and high temperature (Anderson, 995). The EOS play a central role in the study of the material. These equations describe the relationships between the ª 3 Uniersity of Bahrain. Production and hosting by Elseier B.V. All rights resered.

2 Ealuation of pressure and bulk modulus for alkali halides under high pressure 39 pressure, temperature and olume (Kushwah et al., 7) and (Kushwah and Bhardwaj, 9). Using the EOS, some important properties such as compressibility of solid can be calculated. In addition, the EOS enable one to determine the depth dependence of important thermoelastic parameters, such as thermal expansiity and temperature sensitiity of bulk modulus. The theoretical attempt for obtaining EOS may be diided mainly into two categories (Anderson, 995) (i) interionic potential based model (ii) interionic potential independent models. Howeer, it has been obsered that the methods based on the theory of interionic potential are difficult to deal with (Singh et al., 99) and not een be possible to realize in some complicated solids like minerals. On the other hand the second one is free from these difficulties. A theoretical study based on potential model was performed by Sun (Sun, 7). He obtained bulk modulus results that are in good agreement with the Vinet Rydberg EOS. A potential independent model was deeloped by Sharma (Sharma et al., ) for C 6 solid under arying conditions of pressure and temperature. The aim of this work is to study some thermodynamical properties of some alkali halides (NaF, NaCl, NaBr and NaI) under high pressure using the non-inerted (isothermal) equation of state based on interionic potential such as Vinet Rydberg EOS,, Bardeen and. In order to achiee this study, the theoretical calculation for compressibility and bulk modulus must be done at different high pressures and at 3 K. The calculation results will be compared with experimental data obtained from Sorensen (Sorensen, 983). Moreoer, the ariation of compressibility and bulk modulus ersus pressure at different temperatures will be inestigated using the inerted (non-isothermal) Kumar General EOS for alkali halides.. Theory.. Preamble The pressure olume (P V) relationship for a solid at a gien temperature can be expressed as non-inerted type as gien in Eq. (), or as inerted type as gien in Eq. () (Digpratap et al., 7) P ¼ f ; K ; K ; K ðþ ¼ fðp; K ; K ; K Þ ðþ where V o is the olume at pressure P =.K ; K and K are the alues of bulk modulus, first pressure deriaties and second pressure deriaties respectiely, all at P =. Equation of state for inerted and non-inerted types for solids is deried from the relation (Tosi, 964): P ¼ de ð3þ dv where E is the interionic potential energy and V is the olume The bulk modulus (K T ) for inerted and non-inerted types can be deried from the following equation: K T ¼ V dp ð4þ dv.. Non-Inerted Type EOS During the last years, many researchers hae used Birch- Murnghan EOS as the non-inerted type (isothermal) in the field of solid state physics and geophysics ((Digpratap et al., 7). This EOS is based on the Eulerien finite strain theory (Birch, 95). But recent studies (Hama and Suito, 996), (Poirier and Tarantola, 998), (Shanker et al., 999) and (Sushil et al., 4) reeal that Birch-Murnghan EOS does not describe adequately the compression of the solid at high pressures. Howeer, the following EOS hae been found to be the best way to describe adequately the compression of solids at high pressure.... Vinet Rydberg (V R) EOS One of the most successful isothermal EOS is that proposed by Vinet et al. (Vinet et al., 986) and (Vinet et al., 986) which is alid for all classes of solids in compression and in the absence of phase transition. The basis of this EOS is a uniersal relation for the binding energy in terms of the interionic distance. In the deriation of this EOS contribution of the thermal pressure is neglected, and the olume deriatie of the binding energy is used to approximate the internal energy. The Vinet EOS based on the Rydberg potential function has been widely used in spite of its main shortcoming regarding the extreme compression behaior of solids. Vinet Rydberg hae obtained an EOS which is based on the Rydberg potential energy E(r) expressed as a function of the interionic distance (r) (Rydberg,93). EðrÞ ¼EðaÞ½ b r exp b r ð5þ a a where a and b are the potential parameters. The (V R) EOS is written as (Vinet et al., 989): P ¼ 3K x =3 ð x =3 Þ exp½gð x =3 ÞŠ ð6þ where x = V/V o and g ¼ 3 ðk Þ The isothermal bulk modulus can be written as: K T ¼ K x =3 ½ þfgx =3 þ gð x =3 ÞŠ exp½gð x =3 ÞŠ... Owing to the fact that the pressure and the isothermal bulk modulus may be expressed as a function of the lattice potential energy, and also the deriaties of potential energy with respect to olume may be expressed in terms of the deriaties of (E) with respect to the interionic distance (r), Shanker (Shanker et al., 997) and (Shanker et al., 999) introduced a force constant (F) in terms of Laplacian operator (Born and Huang, 954), and he found that F may be expressed as a function of olume mentioned in references Shanker (Shanker et al., 997) and (Shanker et al., 999) then with some mathematical manipulation he obtained a result known as Shanker EOS. Shanker et al. hae obtained an EOS using the olume dependence of the short-range force constant for interionic potentials. The short range force constant (F) is defined as follows: F ¼ d E 3 dr þ de ð8þ r dr ð7þ

3 4 I.Y. Alkammash Table Values of input parameters used in the present work. NaF NaCl NaBr NaI K o (GPa) K a 4(K-) d T The expression based on the is gien below (Shanker et al., 997 and (Shanker et al.,999). x 4=3 P ¼ K þ t t þ fexpðtyþ gþy þ y expðtyþ t t where x = V/V o,t ¼ K 8 3 and y = -(V/V o). The isothermal bulk modulus can be written as: K T ¼ K x =3 ð þ y þ y Þ expðtyþþ 4 3 P where x = V/V o,t ¼ K 8 3 and y = -(V/V o). ð9þ ðþ..3. The Bardeen equation of state is deried from the potential function E(r) (Bardeen, 938): EðrÞ ¼ a r þ b 3 r c ðþ r where a, b and c are constants. The is written as (Butler and Anderson, 978): P ¼ 3K ½x 5=3 x 4=3 Š þ 3 ðk 3Þðx =3 Þ ðþ where x = V/V The isothermal bulk modulus (K T ) has been deried by (author), using Eq. (4). K T ¼ 5 3 x 5=3 þ 4 3 x 4=3 þ 3 ðk 3Þðx =3 Þ þ K ðx 5=3 x 4=3 Þ 3 ðk 3Þx =3 ð3þ Exp.data Exp.data.8.8 V/Vo.75 V/Vo Exp.data Exp.data.8.8 V/Vo.75 V/Vo Figure Comparison between the calculated isothermal compression cure at T = 3 K using four EOS with experimental data: (a) NaF (b) NaCl (c) NaBr (d) NaI.

4 Ealuation of pressure and bulk modulus for alkali halides under high pressure 4 x 7 x x x Figure Isothermal bulk modulus ersus pressure at T = 3 K using four isothermal EOS: (a) NaF (b) NaCl (c) NaBr (d) NaI..3. Inerted Type EOS There hae been arious attempts to propose inerted type EOS (non-isothermal) as reiewed by Freund and Ingalls (Freund and Ingalls, 989). These EOS are based on different physical properties such as Kumari and Dass (Kumari and Dass, 99), Kumar (Kumar, 995), Bose-Roy and Bose-Roy, (Roy et al., 999) and (Bose Roy and Bose Roy, 4). Howeer, the following EOS hae been found to be the best way to describe the compression of solids at high pressure and temperature..3.. Kumar General EOS Kumar (Kumar, 995) and (Kumar, 996) hae obtained an inerted type EOS (non-isothermal) containing a term of temperature and in consequence alid for room temperature up to melting temperature, and at the pressure arying from atmospheric pressure up to the structural transition pressure. Kumar General EOS is gien below: ¼ A ln þ A fp a K ðt T Þg ð4þ K Or P ¼ K A expa þ a K ðt T Þ ð5þ where A =(d T + ), d T is the Anderson Gruneisen parameter, a the coefficient of olume thermal expansion at room temperature (T o ) and d T is gien below (Singh and Kumar,4): s T d T ¼ d T ffi K ð6þ T d T is the alue of Anderson Gruneisen parameter at T o and s is a dimensionless thermoelastic parameter whose alue is about (.4.5), as gien by (Anderson et al., 99), (Anderson and Isaak, 993) for some alkali halides (from room temperature to melting temperature). In the present study we take s = Special Case. By assuming the temperature T = T o = 3 K, Eq. (4) is reduced to the following form: ¼ A ln þ A P ð7þ K or

5 4 I.Y. Alkammash..9 T=3 T=45 T=6 T=75 T=9..9 T=3 T=45 T=6 T=75 T=9 T=5 V/V o.8 V/V o T=3 T=45 T=6 T=75 T=9 T=..9 T=3 T=45 T=6 T=75 T=9 V/V o.8 V/V o Figure 3 P ¼ K A exp A ð8þ Eq. (8) is the relation of ð Þ and (P) at constant temperature and therefore is called the isothermal equation of state. The inerted type (non-isothermal) bulk modulus deried from Kumar General EOS Eq. (4) and from Eq. (4) is mentioned below: K T ¼ K AP ln þ Aa ðt T Þ A K þ AP K Aa ðt T Þ ð9þ The isothermal bulk modulus deried from special Kumar General EOS Eq. (8) is gien below: K T ¼ K exp A 3. Results and discussion 3.. Isothermal Compression Non-isothermal cure at different temperatures using : (a) NaF (b) NaCl (c) NaBr (d) NaI. ðþ The pressures hae been calculated at different relatie isothermal compression olumes ranging from to at T = 3 K for solid alkali halides (NaF, NaCl, NaBr and NaI) using (V R) EOS,, and Kumar isothermal EOS, Eqs. (,6,9,8) respectiely. The input parameters are listed in Table, where the alues of K and K are taken form (Roberts and Smith, 97), a is obtained from (Dhoble and Verma,986) and d T from (Shanker and Singh, 98). The results are shown in Fig. and compared with experimental data obtained from (Sorensen, 983). From this figure one can see that the relatie compression olume decreases continually with the increase in pressure. It is found that the results obtained from (V R) EOS, Shanker EOS and are similar to each other and coincide with experimental data for four alkali halides. Although, the approach of the three equations is different, it indicates the alidity of these models at low and high pressures. One can see from this figure that the alues of applied high pressure for NaCl, NaBr, NaF and NaI at the compressibility () are equal to (9, 78, 7 and 6) GPa respectiely. This is due to the pressure dependence on the bulk modulus (K o ) which has a lower alue for NaI and a higher alue for NaF. Howeer, Kumar isothermal EOS results for four alkali halide solids at the compression range (.75) are quit coincident with three EOS and experimental data. But at a range

6 Ealuation of pressure and bulk modulus for alkali halides under high pressure 43.5 x.5 T=3 T=45 T=6 T=75 T=9 3.5 x T=3 T=45 T=6 T=75 T=9 T= x.5.5 T=3 T=45 T=6 T=75 T=9 T= 8 x T=3 T=45 T=6 T=75 T= Figure 4 Non-isothermal bulk modulus ersus pressure at different temperatures using : (a) NaF (b) NaCl (c) NaBr (d) NaI. (.75 ), the results deiated from the other results. This indicates that is not based on the interionic potential. 3.. Isothermal bulk modulus The isothermal bulk modulus (K T ) has been calculated at different relatie compression olumes from to for four alkali halides using Eqs. (7,,3,). The input parameter alues used are listed in Table. Fig. shows the isothermal bulk modulus (K T ) ersus pressure (P) for (V R) EOS,, and Kumar isothermal EOS at T = 3 k. This figure exhibits that the isothermal bulk modulus (K T ) increases continuously with increase in pressure. In Fig. one can see that, at low pressure the equations mentioned aboe coincide with each other. At high pressure, Fig. a shows that all equations dierge from each other. But in Fig. b and d, and coincide with each other, Fig. a shows that (V R) EOS, Shanker EOS and are conerge to each other but Bardeen EOS dierge from them. Fig. b,c and d, show that at the pressure range ( 3GPa), the (V R) EOS, and Kumar EOS conerge with each other, while dierge from those equations. At pressure range aboe 3GPa we shows four equations dierge to each other s Non-Isothermal Compression The pressure has been calculated at different relatie non-isothermal compression olume ranges ( ) at different temperatures from T = 3 to temperature close to the melting temperature for solid alkali halides (NaF, NaCl, NaBr and NaI) using the Kumar General EOS Eq. (4). The input parameters are listed in Table. The relationships between compression and pressure (P) at different temperatures are plotted in Fig. 3. It has been seen from this figure that the compression (V/V o ) decreases with increasing pressure at a gien temperature. Howeer, the compression cure for all alkali halides increases as the temperature increases. This can be attributed to the olume expansion.

7 44 I.Y. Alkammash 3.4. Non-Isothermal bulk modulus The non-isothermal bulk modulus (K T ) has been worked out at different relatie compression olumes ( ) at different temperatures from T = 3 K to temperature close to the melting temperature for alkali halides (NaCl, NaBr, NaF and NaI) using the Kumar General EOS (Eq. (9)). The input parameters alues used are listed in Table. The relationships between bulk modulus (K T ) and pressure (P) at different temperatures are plotted in Fig. 4. In Fig. 4, it is clear that the bulk modulus (K T ) increases with pressure dramatically at a gien temperature. (K T ) increases with (T) at a gien pressure. When increasing the temperature the olume of solid increases and hence increases the bulk modulus. This increase is due to a coefficient of Anderson Gruneisen parameter, that contains the formula (T/T o ) s where its alue increases with increasing the temperature. 4. Conclusions In the present paper, the non-inerted (isothermal) type equations of state (EOS) which are based on interionic potential such as Vinet Rydberg EOS,, Bardeen EOS as well as Kumar isothermal EOS are used to study the ariation of relatie compression olumes in range ( ) and the isothermal bulk modulus with high pressure at temperature T= 3 K for four alkali halides (NaF, NaCl, NaBr and NaI). In this work, we hae found that the relatie compression olume obtained from the four EOS gie a good agreement with experimental data up to 5 GPa. But beyond this point, the compression results stay in coincidence for the Vinet Rydberg EOS, and, while the results obtained from the Kumar isotherm EOS deiated from others EOS and from experimental data. Moreoer, at the isothermal compression alue, the calculated pressures for NaF, NaCl, NaBr and NaI were found to be 7, 9, 78 and 6 GPa respectiely. Moreoer, at the isothermal compression alue, the calculated pressures for NaF, NaCl, NaBr and NaI were found to be 7, 9, 78 and 6 GPa respectiely. The results of isothermal bulk modulus for four EOS were found to be superimposed on each other at pressure from ( GPa). But beyond this alue, the results of Vinet Rydberg EOS and are rather in good conergence. But the results of and do dierge largely to other results. Howeer, the result of non-isothermal olume compression by using Kumar General EOS had shown that the compression cure for NaCl, NaBr, NaF and NaI increases as the temperature increases. This behaior also applied to the non-isothermal bulk modulus. References Anderson, O.L., 995. Equations of State of Solids for Geophysics and Ceramic Science. Oxford Uniersity Press, New York. Anderson, O.L., Isaak, D.G., 993. The dependence of the Anderson Gru neisen parameter d T upon compression at extreme conditions. J. Phys. and Chem. Solids 54, 7. 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