ARTICLE IN PRESS

Physica B 403 (2008) 3672– 3676

Contents lists available at ScienceDirect

Physica B

0921-45

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/physb

Temperature dependence of interatomic separation

Manoj Kumar, M. Kumar �

Department of Physics, G.B. Pant University of Agriculture and Technology, Pantnagar 263145, Uttarakhand, India

a r t i c l e i n f o

Article history:

Received 11 October 2007

Received in revised form

5 May 2008

Accepted 5 June 2008

PACS:

64.30.+t

65.70.+y

91.35.�X

91.60.Ki

Keywords:

Interatomic separation

Thermal expansion

Thermodynamic relation

Equation of state

26/$ - see front matter & 2008 Elsevier B.V. A

016/j.physb.2008.06.010

esponding author. Tel.: +915944 233689; fax

ail address: munish_dixit@yahoo.com (M. Ku

a b s t r a c t

A simple unification of various relations for the temperature dependence of interatomic separation is

presented. It is found that the relations reported by the earlier workers as new are contained in a simple

thermodynamic relation. Some other relations based on different physical origins are also included in

the study and compared with the thermodynamic relation. The results are reported for ten crystals, viz.

KF, KCl, KBr, KI, RbF, RbCl, RbBr, RbI, MgO and CaO. A good agreement between unified relation and

experiment demonstrates the superiority of the simple theory based on the thermodynamic analysis.

& 2008 Elsevier B.V. All rights reserved.

1. Introduction

The properties of solids at high pressures and high tempera-tures are of fundamental interest for the understanding of theEarth’s deep interior. Considerable efforts have been made todetermine the properties of solids under high pressures. More-over, at room pressure, relatively fewer efforts have been made.The investigators have tried to study the temperature dependenceof interatomic distances using different approaches. A detailedreview of the experimental work has been provided by Fiquetet al. [1]. Kumar and Upadhyay [2] as well as Anderson [3]proposed that the coefficient of volume thermal expansion,a, depends on temperature, T, as follows:

aa0¼ ½1� a0dTðT � T0Þ�

�1 (1)

or

a ¼ a0 þ a20dTðT � T0Þ þ a3

0d2TðT � T0Þ

2þ � � �1 (2)

where dT is the Anderson–Gruneisen parameter and 0 refers to theinitial condition, a is the coefficient of volume thermal expansion,

ll rights reserved.

: +915944 233473.

mar).

which is defined as

a ¼ 1

V

dV

dT

� �P

(3)

Using the definition of a, the integration of Eq. (1) gives thefollowing relation, as reported by Kumar and Upadhyay [2]:

r

r0¼

1

1� a0dTðT � T0Þ

� �1=3dT

(4)

Here, the relation V/V0 ¼ (r/r0)3 has been used, which is a merescaling, coinciding with an actual interatomic distance just in someparticular cases, those, for instance, in which atoms’ positions areprimitive cubic lattice points. r is the interatomic distance.

A thermodynamic analysis of the material under hightemperature–high pressure has been performed by Kumar [4,5]by developing the theory of equation of state (EOS). It has beendiscussed that the theory may be used to study the properties ofsolids for a wide range of pressure and temperature, viz. fromroom temperature upto the melting temperature of solids andfrom atmospheric pressure upto the structural transition pressureof solids. The theory may be used to study the temperaturedependence of interatomic separation.

Gruneisen theory of thermal expansion as formulated by Bornand Huang [6] has been used by Shanker et al. [7] and claimed toreport what became known as ‘Shanker formulation’, which hasbeen widely used in the literature [7–15] for the determination of

ARTICLE IN PRESS

M. Kumar, M. Kumar / Physica B 403 (2008) 3672–3676 3673

the temperature dependence of interatomic separation. Moreover,it has been found that the formulation does not work under highpressure and needs modification [16].

He and Yan [17] used the quadratic expansion of a with T, tostudy the temperature dependence of r, and claimed to have got anew relation. Singh and Chauhan [18] used the linear expansion ofa with T and also claimed to have got a new relation for thetemperature dependence of r. Kushwah and Shanker [19] studiedthe thermal expansion of MgO in the temperature range(300–1800 K) using Guillermet and Gustafson [20] relation, whichis based on the linear expansion of a with T.

Thus, there are many formulations with their advocates. It istherefore legitimate and may be useful to discuss that thesemodels are not different and can be unified. The unification mayhelp the researchers to reach on a single platform to study thetemperature–volume EOS for solids, which is the purpose of thepresent paper.

2. Method of analysis

Kushwah et al. [19,21] as well as Chauhan and Singh [22]reported the following relation:

V

V0¼ ½1� a0dTðT � T0Þ�

�1=dT (5)

A comparison of Eqs. (4) and (5) shows that Eq. (5) is exactlysame as Eq. (4).

He and Yan [17] assumed that a depends quadratically on T asgiven below:

a ¼ a0 þ a20dTðT � T0Þ þ a3

0d2TðT � T0Þ

2 (6)

Using the definition of a, the integration of Eq. (6) gives thefollowing relation as presented by He and Yan [17]:

r

r0¼ exp

1

3a0ðT � T0Þ þ

1

2a2

0dTðT � T0Þ2þ

1

3a2

0d3TðT � T0Þ

3

� �� �(7)

It should be mentioned here that Eq. (6) is an approximate formof Eq. (2). The integration of Eq. (6) gives Eq. (7), while theintegration of Eq. (2) gives Eq. (4). Therefore, Eq. (7) may beregarded as the approximate form of Eq. (4).

Singh and Chauhan [18] assumed that a depends linearly on T

as given below:

a ¼ a0 þ a20dTðT � T0Þ (8)

Using the definition of a, the integration of Eq. (8) gives thefollowing relation as presented by Singh and Chauhan [18]:

r

r0¼ exp

1

3a0ðT � T0Þ þ

1

2a2

0dTðT � T0Þ2

� �� �(9)

Eq. (8) is the more approximate form of Eq. (2). Therefore, Eq. (9)may be regarded as the more approximate form of Eq. (4). It is alsopertinent [23] to mention here that Eq. (9) as presented by Singhand Chauhan [18] is exactly same as the relation of Guillermet andGustafson [20] as used by Kushwah and Shanker [19].

Kumar [24] simply assumed that a depends linearly on T asgiven below:

a ¼ ao þ a2odTT (10)

Using the definition of a, the integration of Eq. (10) gives thefollowing relation [24]:

r

r0¼ exp

1

3a0ðT � T0Þ þ

1

2a2

0dTðT2� T2

0Þ

� �� �(11)

Thus, Eq. (11) may also be treated as the approximate form ofEq. (4).

A theoretical analysis of high pressure–high temperature EOSgives the following relation [4,25]:

V

V0� 1 ¼ �

ln½1� aoðdT þ 1ÞðT � T0Þ�

ðdT þ 1Þ(12)

In Eq. (12), thermal pressure, PTh, is defined as

PTh ¼ a0B0ðT � T0Þ (13)

using the approximation, dT ¼ B00, where B00 is the first-orderpressure derivative of B0, Eq. (12) reads as

V

V0� 1 ¼ �

1

ðB00 þ 1Þln 1�

ðB0o þ 1ÞPTh

B0

� �(14)

or

PTh ¼ �Bo

ðB00 þ 1ÞexpðB00 þ 1Þ 1�

V

V0

� �� �� 1

� �(15)

Neglecting higher-order terms, Eq. (15) gives

PTh ¼ �B0 1�V

V0

� �þ

B00 þ 1

2

� �1�

V

V0

� �2" #

(16)

or

V

V0� 1 ¼

1� 1� 2ððB00 þ 1Þ=B0ÞPTh

� 1=2

ðB00 þ 1Þ(17)

or

V

V0� 1 ¼

1� ½1� 2a0ðdT þ 1ÞðT � T0Þ�1=2

ðdT þ 1Þ(18)

Thus, Eq. (12) may be written in the form of Eq. (18).Using the Gruneisen theory of thermal expansivity, Shanker

et al. [7] reinvestigated Eq. (17), and named it ‘Shanker formula-tion’, which has been widely used to study the thermal propertiesof solids [7–15]. Here it is very clear that Eq. (17) is theapproximate form of Eq. (12).

Now, there remain Eqs. (4) and (18) for analysis. The expansionof (1�x)�n reads

ð1� xÞ�n¼ 1þ nxþ

nðnþ 1Þ

2!x2 þ

nðnþ 1Þðnþ 2Þ

3!x3 þ � � �1 (19)

Eq. (4) therefore may be written as

V

V0¼ 1þ a0ðT � T0Þ þ

a20

2ðdT þ 1ÞðT � T0Þ

2

þa3

0

6ðdT þ 1ÞðdT þ 2ÞðT � T0Þ

3þ � � �1 (20)

The expansion of ln(1�x) reads as follows:

lnð1� xÞ ¼ �x�x2

2�

x3

3� � � �1 (21)

Eq. (12), therefore, may be written as

V

V0¼ 1þ a0ðT � T0Þ þ

a20

2ðdT þ 1ÞðT � T0Þ

2

þa3

0

3ðdT þ 1Þ2ðT � T0Þ

3þ . . .1 (22)

Eqs. (20) and (22), which are the series expansion of Eqs. (4) and(12), are same up to the third term. Moreover, the higher-orderterms are slightly different. Thus, Eq. (12) may also be regarded asthe approximate form of Eq. (4). Thus, all these relations may beunified as Eq. (4).

ARTICLE IN PRESS

Table 1Values of input parameters used in the present work [1,3,17]

Crystals a0 (in 10�4 K�1) dT

KF 1.02 6.16

KCl 1.11 6.29

KBr 1.16 5.88

KI 1.23 5.83

RbF 0.94 6.80

RbCl 1.03 6.73

RbBr 1.08 6.64

RbI 1.23 6.53

MgO 0.309 5.26

CaO 0.405 6.19

1

1.005

1.01

1.015

1.02

1.025

1.03

1.035

1.04

300 400 500 600 700 800 900

T (K)

r/r o

eq. (4)

eq. (12)

eq. (18)

Exp. [17]

Fig. 2. KCl: Temperature dependence of interatomic separation, r/r0, using

different equations.

1.02

1.025

1.03

1.035

1.04

r/r o

M. Kumar, M. Kumar / Physica B 403 (2008) 3672–36763674

3. Results and discussion

We have thus presented a critical analysis of various relationsreported in the literature to study the temperature dependence ofr. It is found that the relations reported by He and Yan [17] as wellas Singh and Chauhan [18] are the approximate forms of Eq. (4).The relation of Singh and Chauhan [18] is exactly same [23] as thatof Guillermet and Gustafson [20] relation as used by Kushwah andShanker [19]. Eq. (17) reported as ‘Shanker formulation’ in theliterature [7–15] is the approximate form of Eq. (12). Now thereremain Eqs. (4) and (12) for discussion. The series expansion ofthese equations shows that they are same up to the third term anddifferences occur in the higher-order terms.

To present numerical analysis, we therefore selected Eqs. (4),(12) and (18). The input data [1,3,17] required for the present workare given in Table 1. We used Eqs. (4), (12) and (18) to calculate thetemperature dependence of interatomic separation of ten crystalsviz. KF, KCl, KBr, KI, RbF, RbCl, RbBr, RbI, MgO and CaO. The resultsare reported in Figs. 1–10 along with the available experimentaldata [1,17] for the sake of comparison. It is found that the resultsobtained from Eqs. (4) and (12) are very similar. The reason is veryclear from the above discussion that Eqs. (4) and (12) are same ifhigher-order terms are neglected. The experimental data [1,17] areavailable up to third decimal place; we therefore computedthe results up to third decimal place. The results obtained fromEq. (18) also agree with the experimental data up to a certaintemperature limit and a fast increase is obtained at hightemperatures. In some cases (KI, RbI, MgO and CaO) resultsbecome imaginary at high temperatures. In these cases, at hightemperatures the quantity within the bracket becomes negative.The square root of negative quantity becomes imaginary. Thus,Eq. (18) becomes inadequate at high temperatures. This may bedue to the fact that in the derivation of Eq. (18) higher-order termshave been neglected [7], which makes Eq. (18) an approximaterelation [25]. This approximate nature may be responsible for itsfailure in high-temperature range. To confirm the above discus-sion, we have also calculated the percentage deviations (PD) at thehigh temperatures considered in the present work (Table 2). It hasalso been realized that Eq. (18) does not work under high pressureand needs modification [16].

Thermal expansion data can be used to study various proper-ties of solids under the effect of temperature. Vijay [15] haspresented a comparative study of Eqs. (14) and (17) by calculatingV/V0 as a function of temperature. On the basis of the calculatedvalues, Vijay [15] demonstrated that Eq. (17) is better than

1

1.005

1.01

1.015

1.02

1.025

1.03

1.035

300 400 500 600 700 800 900

T (K)

r/r o

eq. (4)

eq. (12)

eq. (18)

Fig. 1. KF: Temperature dependence of interatomic separation, r/ro using different

equations.

1

1.005

1.01

1.015

300 400 500 600 700 800 900

T (K)

eq. (4)

eq. (12)

eq. (18)

Exp. [17]

Fig. 3. KBr: Temperature dependence of interatomic separation, r/r0, using

different equations.

Eq. (14), and used the results to study the temperaturedependence of second-order elastic constants. Here, it is veryclear that Eq. (17) is an approximate form of Eq. (14). Therefore,Eq. (17) can never be better than Eq. (14). The analysis of Vijay [15]is based on inadequate results. We found that Eq. (17) is not betterthan Eq. (14) and there is computational error in the resultsreported by Vijay [15].

ARTICLE IN PRESS

1

1.005

1.01

1.015

1.02

1.025

1.03

1.035

300 400 500 600 700 800 900

T (K)

r/r o

eq. (4)

eq. (12)

eq. (18)

Exp. [17]

Fig. 4. KI: Temperature dependence of interatomic separation, r/r0, using different

equations.

1

1.005

1.01

1.015

1.02

1.025

1.03

300 400 500 600 700 800 900

T (K)

r/r o

eq. (4)

eq. (12)

eq. (18)

Fig. 5. RbF: Temperature dependence of interatomic separation, r/r0, using

different equations.

1

1.005

1.01

1.015

1.02

1.025

1.03

1.035

1.04

300 400 500 600 700 800 900

T (K)

r/r o

eq. (4)

eq. (12)

eq. (18)

Exp. [17]

Fig. 6. RbCl: Temperature dependence of interatomic separation, r/r0, using

different equations.

1

1.005

1.01

1.015

1.02

1.025

1.03

1.035

1.04

1.045

300 400 500 600 700 800 900

T (K)

r/r o

eq. (4)

eq. (12)

eq. (18)

Exp. [17]

Fig. 7. RbBr: Temperature dependence of interatomic separation, r/r0, using

different equations.

1

1.005

1.01

1.015

1.02

1.025

1.03

1.035

1.04

300 400 500 600 700 800 900

T (K)

r/r o

eq. (4)

eq. (12)

eq. (18)

Exp. [17]

Fig. 8. RbI: Temperature dependence of interatomic separation, r/r0, using

different equations.

1

1.005

1.01

1.015

1.02

1.025

1.03

1.035

1.04

1.045

1.05

505 874 1211 1557 2153 2573 2973

T (K)

r/r o

eq. (4)

eq. (12)

eq. (18)

Exp. [1]

Fig. 9. MgO: Temperature dependence of interatomic separation, r/r0, using

different equations.

M. Kumar, M. Kumar / Physica B 403 (2008) 3672–3676 3675

ARTICLE IN PRESS

1

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

300 664 1022 1427 1733 2114 2573 3073

T (K)

r/r o

eq. (4)

eq. (12)

eq. (18)

Exp. [1]

Fig. 10. CaO: Temperature dependence of interatomic separation, r/r0, using

different equations.

Table 2Percentage deviations (PD) at the highest temperatures

Crystal T (K) PD

Eq. (4) Eq. (12) Eq. (18)

KCl 900 0.09 0.09 0.79

KBr 900 0.09 0.09 0.80

KI 900 0.19 0.19 a

RbCl 900 0 0 0.68

RbBr 900 0.19 0.19 1.17

RbI 900 0.48 0.58 a

MgO 2973 0.86 0.86 a

CaO 3073 2.00 2.10 a

a Means the corresponding value of r/r0 is imaginary.

M. Kumar, M. Kumar / Physica B 403 (2008) 3672–36763676

4. Conclusion

We conclude that various relations that appeared in theliterature are the approximate forms of Eq. (4) and thereforethese relations may be unified as Eq. (4). The unified relation

(Eq. (4)) has been found to work well for simple solids, viz.potassium and rubidium halides, as well as some importantminerals of geophysical importance (MgO and CaO). The unifica-tion presented in the present paper makes the theoreticalformulation as simple as possible. Such a unified analysis hasnot been presented by the earlier workers, and therefore it mayhelp the researchers to bring their ideas on a single platform. Wehave thus presented a simple analysis devoted to compare severalT–V EOSs which demonstrates that several EOSs may be thoughtof as being derivable from Eq. (4).

Acknowledgment

We are thankful to the referee for his valuable comments,which have been used in the revised manuscript.

References

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Science, Oxford University Press, Oxford, 1995.[4] M. Kumar, Physica B 212 (1995) 391.[5] M. Kumar, Physica B 365 (2005) 1.[6] M. Born, K. Huang, Dynamical Theory of Crystal Lattices, Oxford University

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