*Fax: #91-581-822-330.E-mail address: munish

}dixit@yahoo.com (M. Kumar).

Physica B 292 (2000) 173}175

Author's reply on the remark of Prieto and Reneroon Kumar equation of state

Munish Kumar*

Department of Physics, SRMS College of Engineering & Technology, Bareilly-243001, India

Received 2 November 1999

Abstract

It is found that the comparison presented by Prieto and Renero (PR) between the theories of high-pressure equation ofstate due to Kumar and PR is useful for readers. But several ideas in the study of PR are not adequate. A detailed analysisis presented along with the comments on the several concepts presented by PR. ( 2000 Elsevier Science B.V. All rightsreserved.

PACS: 62.50#P; 64.10#h

Keywords: Equation of state; High pressure; High temperature

It is the purpose of the present communication todiscuss in detail the remark of Prieto and Renero(PR) [1] presented on the theory of equation ofstate (EOS) recently reported by Kumar [2].Though the remark is useful for readers. However,many statements are #awed. Thus, it is pertinentand may be useful to discuss the statements of PRso that the readers are not confused. PR havepresented a comparative study of two methods forthe evaluation of EOS, reported by Kumar [3,4]and PR [1]. The following statement is quotedfrom PR [1]:`Although this equation is incomplete in the

sense that it does not contain a term in temperatureand is in consequence valid only at room temper-ature (293 K)a, it seems to the author that PR [1]have not consulted the literature which emerged in

the form of the above statement. Kumar andcoworkers [2,5] have given the complete EOS be-cause it contains a term in the temperature and is inconsequence valid for room temperature upto themelting temperature, and at the pressures varyingfrom atmospheric pressure upto the structuraltransition pressure [2,5]. A detailed analysis isavailable elsewhere [2,5] and the mathematicalform reads as follows:

<

<0

"1!1

AlnC1#

A

B0

MP!a0B

0(¹!¹

0)ND. (1)

Here the terms have their usual meaning [2,5]. InEq. (1) the last term represents the temperaturecontribution. At ¹"¹

0"293 K, Eq. (1) reduces

to the following form [2]:

<

<0

"1!1

AlnC1#

AP

B0D (2)

0921-4526/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.PII: S 0 9 2 1 - 4 5 2 6 ( 0 0 ) 0 0 4 5 3 - 1

or

P"

B0

A CexpAA1!<

<0B!1D. (3)

Eq. (3) is the relation of </<0

and P at constanttemperature and therefore is called isothermalequation of state. Eq. (3) has been quoted by PR [1]and Eq. (1) is ignored. This is the reason by whichPR have given the above statement frequently intheir paper [1], which is wrong. Eq. (1) has beenexposed in the literature [2,5] and has a well-established origin. The form of Eq. (3) which isa special case of Eq. (1), is very simple and useful asacknowledged by PR [1].

PR [1] stated `It should be remarked thatKumar's Eq. (1) is essentially based on the sameassumptions than the Murnaghan equationa. Hereit is pertinent to disclose that both the EOS aredi!erent from each other because these are basedon di!erent assumptions. Murnaghan EOS [6] isbased on the assumption that bulk modulus de-pends linearly on pressure which reads as follows[6,7]:

B

B0

"C1#B@

0B

0

PD. (4)

The integration of Eq. (4) gives the well-knownMurnaghan EOS [6,7] which has been found to bethe best EOS for low-pressure ranges [7,8], as far asthe simplicity and applicability is concerned.Murnaghan EOS can also be derived in an equiva-lent way viz. by using the concept that aB is a con-stant [3,4,7]. Thus, Eq. (4) may be written as [3,4,7]

B

B0

"A<

<0B~dT

, (5)

where </<0

is the relative change in volume due topressure and d

Tthe Anderson}Gruneisen para-

meter. Eq. (5) is based on the fact that dT

is inde-pendent of P which is a crude approximation [7].The variation of d

Twith pressure has been taken

into account properly by Kumar [2}4]. This givesthe following relation for bulk modulus [7]:

B

B0

"C1#AP

B0DC1!

1

AlnA1#

AP

B0BD (6)

or

B

B0

"

<

<0

expAA1!<

<0B. (7)

Eqs. (6) and (7) are entirely di!erent from Eqs. (4)and (5). Therefore, Eq. (3) derived by Kumar [2,5] isentirely di!erent from Murnaghan EOS [6]. Sincethe approximation d

Tindependent of P is valid in

low-pressure range [3,4,7], Murnaghan EOS isfound to work well only in low-pressure range[7,8]. On the other hand, Eq. (3) has been found towork well for the entire range of pressure [3,4],because it includes the pressure dependence of d

T.

Using an adequate set of scaling factors, PR [9]presented an EOS for solids, and claimed the sameto be universal. The EOS as presented by PR [9]seems to be di$cult to handle because it is hard tomanage the terms appearing in the EOS. Due tothis di$culty the EOS of PR has not generally beenreferred to in the literature by the authors, otherthan PR. Now, let us come to the comparison of thetwo EOS as presented by PR [1]. These authorshave selected Cu, Mo, K and Pd to present a com-parison in favour of their EOS. Thus, it is legitimateto discuss the comparison of these authors [1]. Tojudge on the superiority of the results as claimed byPR [1], the present author has calculated the per-centage deviations corresponding to each value ofthe pressure reported by PR [1]. The results ofthree crystals, K, Mo and Pd are reported in Table1. PR [1] pointed out, `For K, two equations are ingood agreementa. The statement may be true but inany case it does not re#ect the better performanceof the EOS of PR. As far as the results are con-cerned, the percentage deviation at each point isless in the results reported by Eq. (3) (Table 1) ascompared with PR [1] EOS. Therefore, it may beconcluded that Eq. (3) is in better agreement withthe experimental data as compared with the EOS ofPR [1]. Thus, the remark of PR on K is unjusti"edin respect of the comparison of two EOS. For Moand Pd the following sentence is quoted from PR[1]. `For high pressure, the PR equation givesbetter agreement than Kumar's equation. For Moand Pd similar conclusions are obtaineda. If werefer to Table 1, the above conclusion does notseem to be correct. In the case of Mo the percentage

174 M. Kumar / Physica B 292 (2000) 173}175

Table 1Percentage deviations in the results reported by PR [1]!

</<0

P%91

Pk

PPR

K0.8809 5 4 6.20.8073 10 3 5.60.7550 15 1.3 4.70.7132 20 1.0 4.90.6047 40 1.75 5.75

Mo0.9741 74 0.67 0.310.9574 126 0.16 1.50.9426 177 0.11 1.100.9178 270 0.29 0.440.8555 563 0.03 0.130.8211 769 0.06 0.350.7950 950 0.04 0.32

Pd0.968 65 1.0 0.060.933 155 2.5 1.30.894 278 1.6 0.190.813 659 1.0 0.710.796 774 1.8 0.16

!Values of pressure are in kbar, P%91

is the experimental valueof pressure, P

,is the percentage deviation in the values of

pressure, calculated using Kumar EOS [3,4], and PPR

due toPrieto and Renero, EOS [1].

Table 2Values of B@

0for Cu compiled from literature

B@0

Ref.

1 4.29 [10]2 4.96 [1]3 5.14 [10]4 5.48 [11]5 5.65 [12]

deviations are less in the results obtained by Eq. (3)as compared with the equation of PR [1]. Certain-ly, the above statement of PR [1] for Mo is #awed.For Pd both equations seem to be equally good,comparable to each other.

In the case of Cu, it has been shown by PR [1]that their EOS gives a better agreement with theexperimental data at high pressure (P'600 kbar)when compared with Eq. (3). It is pertinent tomention here, that in the above discussion for Mo,we realised that Eq. (3) gives a better agreementwith the experimental data as compared with theEOS of PR upto the pressure range, P"950 kbar.Thus, it is not reasonable to doubt on the suitabil-

ity of Eq. (3) for Cu in the same pressure range.Actually, Eq. (3) is very sensitive to the parameterA"(B@

0#1) for which di!erent values have been

reported in the literature [10}12] as compiled inTable 2. For example, PR [1] have pointed out thatB@0"(4b!1). This gives the value of B@

0"4.96,

which di!ers largely from the value of B@0"5.65

quoted by the same authors [12] for Cu. However,this di!erence is very small for other solids con-sidered by PR [1]. It seems that the deviation of Eq.(3) for Cu at highest pressure (P"950 kbar) (max.value 5%) occurred due to the high value ofB@0

[12]. If we take B@0"5.36 which is about an

average of two values by PR [1,12], Eq. (3) givesgood agreement with the experimental data at highpressures.

References

[1] F.E. Prieto, C. Renero, Physica B 253 (1998) 138.[2] M. Kumar, S.S. Bedi, Phys. Stat. Sol. B 196 (1996) 303.[3] M. Kumar, Physica B 212 (1995) 391.[4] M. Kumar, Physica B 217 (1996) 143.[5] S.S. Bedi, K. Ahuja, M. Kumar, Proceedings of the DAE

SSP Symposium, Cochin, India 40 (1997) 350.[6] F.D. Murnaghan, Proc. Nat. Acad. Sci. USA 30 (1944) 244.[7] M. Kumar, Indian J. Pure Appl. Phys. 36 (1998) 119.[8] M. Kumar, S.S. Bedi, J. Phys. Chem. Solids 57 (1996) 133.[9] F.E. Prieto, C. Renero, J. Phys. Chem. Solids 43 (1982) 142.

[10] Z.H. Fang, L.R. Chen, Phys. Stat. Sol. B 180 (1993) K5.[11] U. Walzer, High Temp. } High Press. 19 (1987) 161.[12] F.E. Prieto, C. Renero, J. Phys. Chem. Solids 53 (1992) 485.

M. Kumar / Physica B 292 (2000) 173}175 175