Anharmonic properties of potassium halide crystals

Turkish Journal of Physics Turkish Journal of Physics

Volume 35 Number 3 Article 10

1-1-2011

Anharmonic properties of potassium halide crystals Anharmonic properties of potassium halide crystals

KRISHNA MURTI RAJU

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Recommended Citation Recommended Citation RAJU, KRISHNA MURTI (2011) "Anharmonic properties of potassium halide crystals," Turkish Journal of Physics: Vol. 35: No. 3, Article 10. https://doi.org/10.3906/fiz-0906-19 Available at: https://journals.tubitak.gov.tr/physics/vol35/iss3/10

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Turk J Phys

35 (2011) , 323 – 340.

c© TUBITAK

doi:10.3906/fiz-0906-19

Anharmonic properties of potassium halide crystals

Krishna Murti RAJUDepartment of Physics, Brahmanand Post-Graduate College

Rath, Hamirpur, U.P., 210 431, INDIAe-mail: [email protected]

Received: 29.07.2009

Abstract

An effort has been made to obtain the anharmonic properties of potassium halides starting from primary

physical parameters viz. nearest neighbor distance and hardness parameters assuming long- and short- range

potentials at elevated temperatures. The elastic energy density for a deformed crystal can be expanded as

power series of strains for obtaining coefficients of quadratic, cubic and quartic terms which are known as

the second, third and fourth order elastic constants respectively. When the values of the higher order elastic

constants are known for a crystal, many of the anharmonic properties of the crystal can be treated within the

limit of the continuum approximation in a quantitative manner. In this study, higher order elastic constants

are computed up to their melting temperature for potassium halides. The first order pressure derivatives

of second and third order elastic constants, the second order pressure derivatives of second order elastic

constants and partial contractions are also evaluated at different temperatures for these substances. The

results thus obtained are compared with experimental data and found in well agreement with present values.

Key Words: Elastic energy density, elastic constants, pressure derivatives

PACS Nos.: 61.50.Ah, 62.20.Dc, 43.25.Dc

1. Introduction

In the present decade, considerable interest has been taken in investigation of anharmonic properties ofmaterials of various kinds [1–4]. Many workers have contributed to this field through their experimental andtheoretical work. Several efforts have been made in the study of physical and anharmonic properties of solidsof different types [5–10] utilizing different physical conditions and using several techniques. Some interestingresults have been presented by several investigators while studying the anharmonic properties of the substancespossessing various crystal structures. Some have studied temperature variation of anharmonic properties ofmixed alkali halides and cyanides [11], of a few alkali cyanides, of rare gas materials [12], of alkali halides [10,

11, 13] using ultrasonic [14–15], theoretical [6] and Brillouin scattering [16] methods. No complete experimentalor theoretical effort has been made so for in obtaining the temperature variation of anharmonic propertiessuch as higher order elastic constants and their pressure derivatives of materials possessing different crystalstructures.

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In this work, a theory for obtaining anharmonic properties such as higher order elastic constants ofmaterials which possess face centered cubic crystal structure has been developed starting from primary physicalparameters viz. nearest neighbor distance and hardness parameter using long- and short- range potentials. Theelastic energy density for a deformed crystal can be expanded as a power series of strains using Taylor’s seriesexpansion. The coefficients of quadratic, cubic and quartic terms are known as the second, third and fourthorder elastic constants (SOECs, TOECs and FOECs) respectively. When the values of these elastic constants ofcrystals are known, many of the anharmonic properties of the substances can be treated within the limit of thecontinuum approximation in a quantitative manner. Several physical properties and crystal anharmonicitiessuch as thermal expansion, specific heat at higher temperature, temperature variation of acoustic velocityand attenuation, the first order pressure derivatives (FOPDs) of SOECs, Gruneisen numbers and temperaturederivatives of SOECs are directly related to SOECs and TOECs. While discussing higher order anharmonicitiessuch as the FOPDs of TOECs, the second order pressure derivatives (SOPDs) of SOECs, partial contractionand deformation of crystals under large forces, the FOECs are to be considered extensively.

The present work is concerned with the formulation to evaluate the second, third and fourth order elasticconstants, the FOPDs of the SOECs and TOECs and the SOPDs of SOECs and the partial contractions; usinglong-and short- range potentials starting from the nearest neighbor distance and hardness parameter. Section2 deals with the derivation of the theory. In Section 3, the theory is tested for potassium halides. The resultsthus obtained are widely discussed in Section 4.

2. Formulation

The elastic energy density for a crystal [17, 18] of a cubic symmetry can be expanded up to quartic termsas shown below:

U0 = U2 + U3 + U4[1/2!]Cijklxijxkl + [1/3!]Cijklmnxijxklxmn + [1/4!]Cijklmnpqxijxklxmnxpq

=12C11(x2

11 + x222 + x2

33) + C12(x11x22 + x22x33 + x33x11) + 2C44(x212 + x2

23 + x231)

+16C111(x3

11 + x322 + x3

33) +12C2

112

[x2

11(x22 + x33) + x222(x33 + x11) + x2

33(x11 + x22)]

+C123x11x22x33+2C144(x11x223+x22x

231+x33x

212)

+2C166

[x2

12(x11+x22)+x223(x22+x33)+x2

31(x33+x11)]+8C456x12x23x31

+124

C1111(x411+x4

22+x433)+

16C1112

[x3

11(x22+x33) + (x322(x33+x11)+x3

33(x11+x22)]

+14C1122(x2

11x222+x2

22x233+x2

33x211)+

12C1123x11x22x33(x11+x22+x33)

+C1144(x211x

223+x2

22x231+x2

33x212) + C1155

[x2

11(x231+x2

12)+x222(x

212+x2

23)+x233(x

223+x2

31)]

+2C1255

[x11x22(x2

23+x231)+x22x33(x2

31+x212)+x33x11(x2

12+x223)

]+2C1266(x11x22x

212+x22x33x

223+x33x11x

231) + 8C1456x12x23x31(x11+x22+x33)

+23C4444(x4

12+x423+x4

31) + 4C4455(x212x

223+x2

23x231+x2

31x212). (1)

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Here, Cijkl , Cijklmn and Cijklmnpq are the SOECs, TOECs and FOECs in tensorial form; xij are theLagrangian strain components; CIJ , CIJK and CIJKL are the SOECs, TOECs and FOECs in Brugger’sdefinition and Voigt notations.

The SOECs, TOECs and FOECs are

Cijkl = CIJ = (∂2U/∂xij∂xkl)x=0Cijklmn = CIJK = (∂3U/∂xij∂xkl∂xmn)x=0,

andCijklmnpq = CIJKL = (∂4U/∂xij∂xkl∂xmn∂xpq)x=0. (2)

The free energy density [19, 20] of a crystal at a finite temperature T is

UTotal = U◦ + Uvib

Uvib =KT

NVC

3sN∑i=1

ln 2 sinh(�ωi/KBT ), (3)

where U◦ is the internal energy per unit volume of the crystal when all ions are at rest on their lattice points,

Uvib is the vibrational free energy, VC is the volume of the primitive cell, N is the number of the primitivecells in the crystal and s is the number of ions in the elementary cell. Other notations used in this equationhave their usual meanings.

The elastic constants each have two terms as follows:

CIJ = C0IJ + Cvib

IJ , CIJK = C0IJK + Cvib

IJK, and CIJKL = C0IJKL + Cvib

IJKL. (4)

The first part is the strain derivative of the internal energy Uo and is known as the “static” elastic constant.

The second part is the strain derivative of the vibrational free energy Uvib and is called the “vibrational” elasticconstant. The superscript “0” has been introduced to emphasize that the static elastic constants correspond totemperature T = 0 K.

The energy density of the non- deformed crystal is expressed as:

U0 =[12VC

] s∑v=1

∑|m�=o

u �=v |Quv (Rmo

uv ) =∑′ Quv(R)

2VC. (5)

Here, Rmouv is the distance between the vth ion in the oth cell and the uth ion in the mth cell and Quv is

the interaction potential between the ions. The indices (v , o) and (u , m) are sometimes dropped when noconfusion occurs. One assumes that Quv is the sum of the long-range Coulomb and the short-range Born-Mayer[21] potentials:

Quv(r0) = ±(

e2

r0

)+ A exp

(−r0

q

). (6)

Here, e is the electric charge, the sign ± applies to like and unlike ions, respectively, r0 is the nearest-neighbordistance, q is hardness parameter and A is

A =−0.29126q e2

r20

exp(−r0

q

)+ 2

√2 exp

(−r0

√2

q

) (7)

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It is assumed that the crystal is deformed homogeneously. When the crystal is deformed homogeneously, thedistance between ions (v , o) and (u , m) in the deformed and non- deformed states, Rmo

uv and rmouv , are related

to the Lagrangian strains xij via

(Rmouv )2 − (rmo

uv )2 = 2Y mouvi Y

mouvj xij = 2Zmo

uv , (8)

where Y mouvi is the ith Cartesian component of the vector rmo

uv The definition of the quantity Zmouv is as expressed

in equation (8). The internal energy Uo given by equation (5) can be expanded in terms of Zmouv , which will

yield quadratic, cubic and quartic terms as given below:

U2 =12Vc

∑′[

12!

Z2D2Q(R)]

R=r

=14Vc

[xijxkl

∑′YiYjYkYlD

2Q(R)]

R=r

U3 =12Vc

∑′[

13!

Z3D3Q(R)]

R=r

=112

Vc

[xijxklxmn

∑′YiYjYkYlYmYnD3Q(R)

]R=r

U4 =12Vc

∑′[

14!

Z4D4Q(R)]

R=r

=148

Vc

[xijxklxmnxpq

∑′YiYjYkYlYmYnYpYqD

4Q(R)]R=r

. (9)

Here, is defined the operator D ≡ dRdR

.

With reference to equations (3) and (4), and comparison of equations (1) and (9), one may obtain thestatic elastic constants presented in Table 1. For a central force model, there are only two independent SOECs,three independent TOECs and four independent FOECs at absolute zero temperature. As in the case of theinternal energy U0 , the vibrational free energy is also expanded in terms of strains, the quadratic, cubic andquartic terms are as below:

U2 = [1/Vc2!]∑′∑′′

[Z′Z(D′D)Uvib]Z=0 = [1/2Vc]xijxklfijkl

U3 = [1/Vc3!]∑′∑′′∑′′′

[Z′Z′′Z(D′D′′D)Uvib]Z=0 = [1/6Vc]xijxklxmnfijklmn

U4 = [1/Vc4!]′∑ ∑′′∑′′′∑′′′

[Z′Z′′Z′′′Z(D′D′′D′′′D)Uvib]Z=0 = [1/24Vc]xijxklxmnxpqfijklmnpq (10)

where,

fijkl =∑′∑′′

[YiYjY′kY ′

l (D′D)Uvib]R=r

fijklmn =′∑ ∑′′∑′′′

[YiYjY′kY ′

l Y ′′mY ′′

n (D′′D′D)Uvib]R=r

and

fijklmnpq =∑′∑′′∑′′′∑′′

”[YiYjY′

kY ′l Y ′′

mY ′′n Y ′′′

p Y ′′′q (D′′′D′′D′D)Uvib]R=r.

Here, the abbreviationsZm′ou′v′ → Z′

[d

Rm′ou′v′dRm′o

u′v′

]→ D′ , etc., are used. On comparison of equations (1) and (10),

one determines the vibrational elastic constants. The prime marks in the summations∑′ ,

∑′′ etc., denotesummation over all lattice points except m = 0, u = v .

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Table 1. Expression for the SOECs, TOECs and FOECs at 0 K for potassium halides.

C011 = -1.56933G + G1 + 2G2

C012 = C0

44 = 0.347775G + G2

C0111 = 10.2639G – G3 – 2G4

C0112 = C0

166 = 1.208625G – G4

C0123 = C0

144 = C0456 = 0.678375G

C01111 = -80.71455G+ G5 + 2G6

C01112 = C0

1155 = 4.43205G + G6

C01122 = C0

1266 = C04444 = 5.615925G + G6

C01123 = C0

1144 = C01255 = C0

1456 = C04455 = -1.584975G

where, G = e2/r40, G1 = (1/r0 + 1/q)Q (r0)/ qr0

G2 = (√

g/2r0 + 1/q)Q (r0√

g)/qr0

G3 = (3/r20 + 3/qr0 + 1/q2)Q (r0)/q

G4 = (3√

g/r20 + 6/qr0 + 2

√g/q2)Q (r0

√g)/4q

G5 = (15/r30 + 15/qr2

0 + 6/q2r0 + 1/q3)r0Q(r0)/q

G6 = (15√

g/4r30 + 15/2qr2

0 + 3√

g2/q2r0 + 1/q3)r0Q(r0√

2)/2q

Vibrational contributions to SOECs, TOECs and FOECs are shown in Table 2. These are shown as acombination of gn ’S and Fn ’S which are evaluated conveniently by taking crystals symmetry [22] into accountand the expressions for gn and Fn are tabulated in Tables 3 and 4. By adding the vibrational elastic constantsto the static elastic constants, one may get SOECs, TOECs and FOECs at any temperature for monovalent fcccrystals.

Table 2. Expressions for Vibrational Contribution to the SOECs, TOECs and FOECs for potassium halides.

Cvib11 = g1F

21 + g1F2

Cvib12 = g2F

21 + g1F5

Cvib44 = g1F5

Cvib111 = g3F

31 + g2F2F1 + g1F3

Cvib112 = g1F

31 + g2F1(2F5 + F2) + g1F6

Cvib123 = g3F

31 + 3 g2F1F5

Cvib144 = g2F1F5

Cvib166 = g2F1F5 + g1F6

Cvib456 =0

Cvib1111 = g4F

41 + 6g3F

21 F2 + 3g2F

22 + 4g2F1F3 + g1F4

Cvib1456 =0

Cvib1112 = g4F

41 + 3g3F

21 (F5 + F2) + 3g2F5F2 + g2F1(3F6 + F3) + g1F7

Cvib1122 = g4F

41 + 2g3F

21 (2F5 + F2) + g2 (2F 2

5 + F 22 ) + 4g2F1F2 + g1F7

Cvib1123 = g4F

41 + g3F

21 (5F5 + F2) + g2F1 (2F5 + F2 ) + 2g2F1F6

Cvib1144 = g3F

21 F5 + g2F5F2

Cvib4444 = 3 g2F

25 + g2F7

Cvib1155 = g3F

21 F5 + g2F5F2 + 2g2F1F6 + g1F7

Cvib4455 = g3F

25

Cvib1255 = g3F

21 F5 + g2F

25 + g2F1F6

Cvib1266 = g3F

21 F5 + g2F

25 + 2g2F1F6 + g1F7

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Table 3. Expression for gn ’S for potassium halides.

g1 = g0S;g2 = g0[(X/S1) + S]/2;g0 = ω0/8r3

0;g3 = g0[(2X2S/3S1) + (X/S1) + S]/48;X = gω0 / 2KT;g4 = -g0 [(X3S2/3S1) + (X3/6S2

1) + (X2S/S1) + (5X/4S1) + (5S/4)]/144;ω0 = (1/M+ + 1/M−)/qr0F0;S = Coth X; S1 = Sinh2X.

Table 4. Expression for Fn ’s for potassium halides.

F0 = 1/[(q0 – 2)(Q(r0) + 2(q0 –√

g)Q(r0√

g)q0 = r0/q;F1 = 2[(2 + 2q0 – q2

0)Q(r0) + 2 (√

g + 2q0 –√

gq20)Q(r0

√g)]F0;

F2 = 2( - 6 – 6q0 – q20 + q3

0)Q (r0)F0 + 2F5;F3 = 2( - 30 – 30q0 – 9q2

0 + q30 – q4

0)Q (r0)F0 + 2F6;F4 = 2( - 210 – 210q0 – 75q2

0 - 5q30 + 4q4

0 + q50)Q (r0)F0 + 2F7;

F5 = ( - 3√

g – 6q0 –√

gq20 + 2q3

0)Q(r0√

g)F0;F6 = [(15/

√g) + 15q0 – (9/

√g)q2

0 – q30 –

√gq4

0] Q (r0√

g)F0;F7 = [-(105/ 2

√g) – (105/2)q0 – (75/ 2

√2)q2

0 – (5/2)q30 + 2

√2 q4

0 + q50 ]Q (r0

√g)F0;

The FOPDs of SOECs are concerned with SOECs and TOECs. The FOPDs of TOECs and SOPDs ofSOECs are directly related to the SOECs, TOECs and FOECs. The Partial contractions are mere combinationof FOECs. The expressions for the FOPDs and SOPDs of SOECs and the FOPDs of TOECs [23, 24], partial

contractions for monovalent fcc solids [25, 26] are given in Tables 5 and 6.

Table 5. Expression for the FOPDs of the SOECs and TOECs for potassium halides.

dC11/dP = (C11 + QQ + C111 + C112)C0;CQ = C11 + 2C12

dC12/dP = −( - C11 + C12 + C123 + 2C112)C0; C0 = 1/CQ;dC44/dP = −(CQ + C44 + C144 + 2C166)C0;dC111/dP = −(−3CQ + 3C111 + C1111 + 2C1112)C0;dC112/dP = −(CQ + 3C112 + C1112 + C1122 + C1123)C0;dC113/dP = −(−CQ + 3C113 + 3C1123)C0;dC144/dP = −(CQ + 3C144 + C1144 + 2C1244)C0;dC166/dP = −(−CQ + 3C166 + C1166 + 2C1244)C0;dC456/dP = −(−CQ + 3C456 + 3C1456)C0;

Table 6. Expression for the SOPDs of the SOECs and for Partial Contraction of the FOECs.

d2C11/dP 2 = [(1 + 3CP )C11 + (4 + 3CP )(C111 + 2C112) + C1111 + 4C1112 + 2C1122 + 2C1123] C02;d2C12/dP 2 = [( 1 + 3CP )C12 + (4 + 3CP )(2C112 + C123) + 2C1122 + 5C1123]C02;d2C44/dP 2 = [(1 + 3CP )C44 + (4 + 3CP )(C144 + 2C166) + C1144 + 2C1166 + 4C1244 + 2C1266] C02;CP = (4C11 + C111 + 6C112 + 2C123)C0;Y11 = C1111 + 4C1112 + 2C1122 + 2C1123;Y12 = 2C1112 + 2C1122 + 5C1123;Y44 = C1144 + 2C1166 + 4C1244 + 2C1266.

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3. Evaluation

Extensive efforts have been committed by the theoretical workers to study the attention-grabbing featuresof materials [27–31]. The detailed study of formulation is given in the preceding Section 2. The expressions fordifferent elastic constants and pressure derivatives of the potassium halides are shown in Tables 1–6. Using theconcept of nearest-neighbour distance and hardness parameter [6], the elastic constants and pressure derivatives

for potassium halides are evaluated at different temperatures (from 100 K to up to their melting points [32];

given in Tables 7) using the expression of Tables 1–6 and shown in Figures 1–15. The values of SOECs, TOECs,FOECs, FOPDs and SOPDs of SOECs and FOPDs of TOECs at 0 K and at room temperature for thesecrystals are given in Table 7–11. The experimental and theoretical data are also given, wherever possible, forcomparison. The whole evaluation is based on the assumption that the fcc crystal structure of the material doesnot change when temperature varies up to their melting point. The values of the nearest neighbor distance (r0)

and hardness parameter (q) [6, 32, 33] are given in Table 7. Thermal expansion coefficients (α) [15, 31–33] for

different solids are taken into account as r = r0 (1 + αT), where α = A1 + A2 T + A3T2 + A4T

3 + A5T4

(for KI, KCl, KBr). Equations for α , are computed using curve fitting and A1 , A2 , A3 ,A4 , A5 are shown inTable 12.

Table 7. The nearest neighbors distance (r0) , hardness parameter (q) (10−10 m), melting points and the SOECs and

TOECs in 1010 Newton/m2 at 0 K. Comparison data taken from A. Cox et al, J. Phys C, 15 (1982) 4473. (Experimental

values are given in bold.)

Crystal MeltingPoint, K

r0 q C011 C0

44 C0111 C0

112 C0123

KF 1153 2.6568 0.278 6.74 1.861.281.34

−113.77 −7.55 3.14

KCl 1063 3.1150 0.296 4.28 0.950.810.66

−76.61 −3.79 1.66

KBr 1003 3.2580 0.305 3.68 0.790.690.52

−66.37 −3.15 1.38

KI 996 3.4840 0.319 2.92 0.600.580.37

−53.51 −2.38 1.06

4. Results and discussions

A literature survey shows that, at present time, several efforts have been made by the experimental andtheoretical workers to study the motivating features of these solids, such as anharmonic effects, higher orderelastic constants, pressure derivatives and phonon-induced phase transition etc. in elevated temperature re-

gion. The SOECs and TOECs in 1010 N/m2 at 0 K for halides of potassium are shown in Table 7 along withthe experimental values reported by other workers. For a cubic crystal there are three independent second or-

der elastic constants at absolute zero namely C011 , C0

12 and C044 . In this investigation Brugger’s definition [21] of

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Table 8. The SOECs and TOECs in 1010 N/m2 at room temperature. (Experimental values are given in bold numerals.)

Crys. C11 C12 C44 C111 C112 C123 C144 C166 C456 Ref.KF 7.615

6.1857.71

1.5551.4401.59

1.8851.2501.29

-119.28 -6.327 1.266 3.207 3.207 -7.6301334

KCl 3.8693.838a

4.940c

1.1610.683a

0.664

0.9220.633a

0.662

-81.19-72.6b

-70.1

-2.669-2.4b

-2.24

0.0351.1b

1.33

1.6222.3b

1.27

-3.7012.6b

-2.45

1.5861.6b

1.1813a,54b

2556c,34

KBr 3.3043.2634.250d

0.5500.5640.510d

0.7650.5040.583

-70.37 -2.124 -0.098 1.354 -3.065 1.3241335d,34

KI 2.6012.5773.499

0.7630.4560.299

0.5790.3700.389

-56.88 -1.492 -0.238 1.034 -2.312 1.0101334

Table 9. FOECs in 1010 N/m2 at room temperature.

Crys. C1111 C1112 C1122 C1123 C1144 C1155 C1255 C1266 C1456 C4444 C4455 Ref

KF 184

1716

1865

2.79

26

31.117

3.22

31.4

41.19

−1.09

−7.49

−6.29

−0.929

−7.45

−5.53

−0.564 −0.738 0.230

31.0

30.99

−0.623

−4.64

−0.214

31.0

32.18

−1.30

47

48

KCl 124

1141

1220

0.978

17

13.85

0.789

20.1

−0.08

−0.530

−2.95

−1.18

−0.460

−3.56

−0.97

−0.486 −0.372 −0.404

26.8

14.22

−0.338

2.59

−0.568

27.7

17.31

−0.607

47

48

KBr 108

991

1085

0.490

20

13.13

0.528

22.7

−5.90

−0.453

−2.22

−0.14

−0.385

−2.88

−0.11

−0.871 −.309 −0.551

30.0

12.89

−0.276

−1.99

−0.695

30.9

16.68

−0.522

47

48

KI 88.3

792

957

0.478

25

13.2

0.425

27.5

−1.44

−0.377

−1.48

−0.10b

−0.305

−2.11

−0.25

−0.869 −0.238 −0.603

34.6

12.19

−0.202

−1.31

−0.738

35.5

15.25

−0.441

47

48

second order elastic constants have been used. In the central force model for the elastic constants; the Cauchy’s

relations are as C012 = C0

44 . Hence only two independent second order elastic constants at absolute zero have

been used here. The Cauchy’s relation C012 = C0

44 , which is a consequence of any central force law, is of coursesatisfied in our study. The abnormal behaviour of the temperature dependence of the elastic constant C12 isrelated to the existence of many body potential and non-central potentials in solids, which are responsible for

the breakdown of the Cauchy relation C12 = C44 [27]. The Cauchy relation C012 = C0

44 is valid only when allinteratomic forces are central under static lattice conditions. The following Cauchy relations are satisfied by

these solids: C0166 = C0

112 and C0144 = C0

456 = C0123 . Since these studies were based on two-body potentials and

could explain Cauchy relation, which are significant in all the monovalent crystals. The semi-empirical studies[28, 29] on lattice dynamics and statics have shown that non additive three-body interactions are importantin these types of materials as there occurs appreciable decrease in their nearest neighbor separations at highpressures. The need for inclusion of three-body interaction forces was also emphasized by Sims et al. [30]for better matching of results. However, the recent experimental data on elastic constants measured at lowtemperatures, show that the Cauchy relations are strongly violated by many ionic crystals and these violations

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cannot be ignored. It is possible that the discrepancy in respect of the elastic behaviour of solids inherent in thecurrent theories is responsible for physically unrealistic values of the parameters obtained in the models whenthey are fitted with the neutron data. Obviously, the search for a model that gives the correct description ofthe dielectric behaviour as well as the elastic behaviour of these solids is quite pertinent. A possible explanationof this behaviour can be sought from the fact that the many-body and/or, thermal effects might be morepronounced in SOECs than TOECs.

Table 10. The FOPDs and SOPDs (in 10−11 N/ m2) of the SOECs and partial contractions (in 1012 N/m2) .

(Experimental values are given in bold numerals.)

Crystal dC11

dPdC12

dPdC44

dPdsdP

dkdP

d2C11

dPd2C12

dPd2C44

dPY11 Y12 Y44 Ref

KF 10.59 1.63 −0.05

-0.43a

4.34 4.81

5.26a

−5.78

−5.26

−5.78

−1.07

−0.47

−1.07

−0.84

−0.76

−0.838

1868 77.4 76

5a,47

37

KCl 11.81

12.93a

1.52

1.58a

−0.21

-0.39a

5.03

5.61a

5.10

5.34a

−14.6

−8.87

−14.6

−1.50

−0.59

−1.50

−1.18

−1.53

−1.18

−12.84 −24.13 −5.15

55a,47

37

KBr 12.01 1.51 −0.23

-0.33a

5.14 5.16

5.38a

−12.5

−10.0

−12.5

−1.99

−0.61

−1.99

−1.64

−2.10

−1.46

−3.063 −12.845 −2.699

55a,47

37

KI 12.31 1.49 −0.26

-0.24a

5.31 5.23

5.47a

−15.8

−16.9

-12.7

−15.8

−2.61

−1.40

-1.6

−2.61

−2.00

−4.42

-1.08

2.0 0

−.03 −7.92 −1.66

55a,47

26

37

Table 11. The FOPDs of the TOECs at room temperature.

Crystal dC111dP

dC112dP

dC123dP

dC144dP

dC166dP

dC456dP

Ref.

KF 10.1−132

211−4.21

2182.25

−1.190.24

72.31.59

2.172.24 47

KCl 0.42−155

130−5.23

1371.76

−1.160.08

46.70.18

2.052.25 47

KBr −52.1−153

81.6−6.86

87.71.60

−1.160.01

29.8−0.75

2.032.20 47

KI −73.9−174

63.9−11.7

69.81.50

−1.160.04

24.2−2.97

2.012.31 47

Table 12. Numerical Coefficients for different compounds.

Coefficients A1 A2 A3 A4 A5

KI 4.3749 0.3395 −1.1212× 10−3 1.5876×10−6 −7.4395× 10−10

KCl −1.7935 0.3031 −8.0212× 10−4 9.2505×10−7 −3.6340× 10−10

KBr −0.7222 0.3828 −1.2936× 10−3 1.8220×10−6 −8.5402× 10−10

Cauchy (1822) has derived the general mathematical theory of elasticity. That hypothesis suggeststhe strains in terms of differential displacements of neighboring points in the material and the stresses in

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terms of attractive forces on infinitesimal areas in the similar position. The theory does not utilize theserepresentations completely, but following Cauchy, implements them in modified forms on the foundation ofarguments proposed by him and considered as reliable ever since. However, a crucial check of those argumentsmakes them indefensible [31]. Cauchy’s assumptions are limited to the homogeneous strains only; and the moregeneral case of heterogeneous strains, including especially all cases of wave-propagation and static deformationsin the nature of torsion and flexure, lie outside its range. On the other hand, the mathematics of elasticity hasbeen applied to these cases and the constants appearing in the formulae have been evaluated experimentally.For instance, the results of experimental work on cubic crystals have been expressed in terms of three constantsgenerally selected as C11 , C12 and C44 , respectively. Hence, by an examination of the experimental data forthose cubic crystals which have been investigated with adequate precision by different methods, it should bepossible to decide whether those data are expressible in terms of three constants only, or whether four constantsare actually needed [31].

The SOECs and TOECs in units of ×1010 N/m2 at room temperature for halides of potassium are shownin Table 8. The experimental values reported by other workers are also given in this Table. The experimentalvalues of potassium halides [13, 25, 34, 35] are of the same order and are in well agreement with present results,which shows the validity of the present theory. The Temperature variation of SOECs for potassium halides areshown in Figures 1(a–c) along with available experimental data [34, 35]. The agreement is satisfactory in viewof the large experimental uncertainties. The elastic constants of solids in general decrease with temperatureand such a decrease has been explained by many available theories. But in the NaCl- like structure theelastic constant C12 of some alkali halides (for example KCl, KBr etc.) is increasing with temperature. Thisphenomenon is known as the anomalous temperature dependence of C12 in these solids. In the present work,the temperature dependence of C12 is found to explain the observed anomalous temperature dependence of C12

in alkali halides with NaCl-Structure. Results are presented for potassium halides. We see that an anomaloustemperature dependence of C12 does not occur in sodium halides, but does in potassium halides. These resultsare firmly supported by the available experimental data. The variation of C11 with temperature is found to belarge as compared with C12 and C44 . The constant C11 represents elasticity in length. A longitudinal strainproduces a change in volume without change in shape. The volume change is closely related to the temperatureand thus produces a large change in C11 . On the other hand, the constant C12 and C44 are related to theelasticity in shape which is a shear constant. A transverse strain or shearing causes a change in shape, without achange in volume. Therefore, C12 and C44 are less sensitive to the temperature. Thus, study of the temperaturedependence of C11 may provide a more critical test of the theory.

The higher order elastic constants are strongly related to other anharmonic properties; such as thermalexpansion, thermo elastic constants and thermal conductivity. The knowledge of TOECs may provide furthercritical data for testing the machines for non-destructive-testing. Furthermore, we expected to obtain additionaldata for the discussion of the influence of asymmetric ions on non-linear elastic properties, in particular forcrystals of rock salt type. Third order elastic constants play an important role in solid-state physics. Theyallow an evaluation of first order anharmonic terms of the inter-atomic potential or of generalized Gruneisenparameters, which enter the theories of all anharmonic phenomena, such as the interaction of acoustic andthermal phonons and the equation of state. The present study of the temperature variation of TOECs couldprove useful in studies of various anharmonic properties of ionic solids in general. The TOECs play an importantrole when it comes to explain anharmonic phenomena in solids (interactions of ultrasonic vibration with thermal

phonons, harmonic generators, equation of state etc.). As a result of the anharmonicity of the crystal lattice

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vibrations, the elastic constants vary with temperature. The present results of TOECs in 1010 N/m2 at roomtemperature for halides of potassium are shown in Table 8. The experimental values obtained by other workersare also given in this Table. The experimental values of potassium halides [25, 34] are of the same order andare in well agreement with present results, which shows the validity of the present theory. The Temperaturevariation of TOECs potassium halides are shown in Figures 2 and 3. Expressions obtained in the present workare more general than those derived previously [36, 37]. This is in view of the fact that we have taken the

thermal expansion coefficient [15, 31–33] into account. Among the calculated third order elastic constants ofthese materials, C111 ’s are the largest in their absolute values and an order of magnitude larger than the SOEC.Magnitude of other Cijk ’s are markedly smaller than those of C111 .

(a) (b)1

2

3

4

5

6

7

100 300 500 700 900

C11

Temperature (K)

KCl

KBr

KI

Expt KBr

KF

-2

3

8

13

18

23

28

33

38

100 300 500 700 900

C12

Temperature (K)

KCl

KBr

KI

Expt KBr

KF

(c)

5

6

7

8

9

10

11

12

13

14

100 300 500 700 900

C44

Temperature (K)

KCl

KBr

KI

Expt KBr

KF

Figure 1. Temperature variation of SOECs for potassium halides. (a) Temperature variation of elastic constant

C11(×1010 Newton/m2) . Experimental data is from [34]. (b) Temperature variation of C12(×109 Newton/m2) .

Experimental data is from [35]. (c) Temperature variation of C44(×109 Newton/m2) . Experimental data is from [34].

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-16

-14

-12

-10

-8

-6

-4100 300 500 700 900

C11

1

Temperature (K)

KCl

KBr

KI

KF0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

100 300 500 700 900

C14

4

Temperature (K)

KCl

KBr

KI

KF

Figure 2. Temperature variation of C111 (in ×1011

Newton/m2) .

Figure 3. Temperature variation of C144 .

Third and fourth order elastic constants are required to study many anharmonic properties of crystalsand therefore their accurate evaluation is essential. Recent attempts have been made to calculate anharmonicproperties of ionic crystals [38–44]. Only a few of them [45, 46] have taken account the temperature dependenceof these properties. The thermal contribution to elastic constants is very significant. The experimental datareveal that in going from 100 K to higher temperatures, the values of second order elastic constants (SOECs) arechanged considerably even for highly ionic solids like alkali halides. We have already discussed the temperaturevariation of second and third order elastic constants of potassium halides. Since the contribution from thirdand fourth order coupling parameters to many anharmonic properties are of the same order of magnitude, theknowledge of FOECs is equally important as that of TOECs. The FOECs for halides of potassium are given inTable 9. Some theoretical results [47, 48] are also presented. Due to non-availability of experimental data, the

comparison is not made. The Partial Contractions in 1012 N/m2 for potassium halides are given in Table 10.Calculated results of fourth order elastic constants at different temperatures are reported in Figures 4–10. ThePartial Contractions are shown in Figure 11.

-18

-13

-8

-3

2

7

12

100 300 500 700 900

C11

11

KCl KBr

KI KF

-350

-300

-250

-200

-150

-100

-50

0100 300 500 700 900

Temperature (K)

C1123

Figure 4. Temperature variation of C1111(×1012

Newton/m2) .

Figure 5. Temperature variation of FOEC for KCl

(×1011 Newton/m2) .

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-0.33

-0.32

-0.31

-0.3

-0.29

-0.28

-0.27

-0.26100 300 500 700 900

Temperature (K)

C1144

C1255

(b)(a)

1.21

1.215

1.22

1.225

1.23

1.235

1.24

1.245

100 300 500 700 900Temperature (K)

C1155

Figure 6. (a) Variation of C1144 and C1255 with temperature for KBr (×1012 N/m2) . (b) Variation with temperature

of C1155 for KBr (×1012 N/m2) .

16

16.2

16.4

16.6

16.8

17

17.2

17.4

17.6

17.8

18

100 300 500 700 900Temperature (K)

C1266

C4444

-60

-50

-40

-30

-20

-10

0

10

100 300 500 700 900

Temperature (K)

C1122

Figure 7. Variation of FOECs with temperature for KBr

(×1011 N/m2) .

Figure 8. Temperature variation of FOEC for KI (×1011

N/m2) .

(a)

-2.5

-2.4

-2.3

-2.2

-2.1

-2

-1.9100 300 500 700 900

Temperature (K)

C1144

C1255

C4455

(b)

8.95

9

9.05

9.1

9.15

9.2

100 300 500 700 900

Temperature (K)

C1155

Figure 9. (a) Temperature variation of FOECs for KI (×1010 N/m2) . (b) Temperature variation of C1155 for KI

(×1010 N/m2) .

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3

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

100 300 500 700

(a)

900Temperature (K)

C1155

C1266

C4444

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2100 300 500 700 900

Temperature (K)

(b)

C1144

C1255

C4455

Figure 10. (a) Temperature variation of the selected FOECs for KF (×1011 N/m2) . (b) Temperature variation of the

selected FOECs for KF (×1011 N/m2) .

-10

-8

-6

-4

-2

0

2

100 300 500 700 900

Temperature (K)

Y11

Y12

Y44

Figure 11. Temperature variation of Partial Contractions (in ×1012) for KI.

Recent extension of ultrasonic techniques to high pressure and high frequencies renewed interest in thehigher order coefficients of non- linear elasticity. Much theoretical work has been done on the temperaturedependence of the elastic constants of ionic crystals [49–53]. An investigation into the higher order elasticconstants and their pressure derivatives provides useful information on the inter-atomic forces, inter-ionicpotentials and on anharmonic properties of crystalline solids. This is why recently [38–41, 45, 46, 49–54]there have been several attempts to determine the elastic constants of higher order, particularly for alkali halidecrystals, using theoretical [7, 8] as well as experimental techniques. The FOPDs and SOPDs of the SOECs of

potassium halides are presented in Table 10 along with experimental [26, 55, 56] and theoretical [47, 48] data.On comparison, one may state that the present results are in well agreement at a great extent. Calculatedresults of first and second order pressure derivatives of second order elastic constants at different temperaturesare reported in Figures 12, 13 and 14. An important aspect of the present investigation is the calculationof second order pressure derivatives of SOECs at different temperatures. Experimental values of SOPD areavailable corresponding to 300 K. The good agreement between theoretical and experimental values supportsthe validity of the present work.

The FOPDs of the TOECs of potassium halides are presented in Table 11 along with theoretical [47]

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data. On comparison, one may state that the present results are in well agreement at a great extent. Calculatedresults at different temperatures are reported in Figure 15.

(b)

(a)

1.45

1.5

1.55

1.6

1.65

1.7

100 300 500 700 900Temperature (K)

KCl

KBr

KIKF

-40

-35

-30

-25

-20

-15

-10

-5

0100 300 500 700 900

Temperature (K)

KClKBrKIKF

Figure 12. (a) Temperature variation of the first order pressure derivative of C12 . (b) Temperature variation of the

first order pressure derivative of C44(×10−2) .

4.2

4.4

4.6

4.8

5

5.2

5.4

5.6

5.8

100 300 500 700 900

ds/d

p

Temperature (K)

KClKBrKIKF

-100

-80

-60

-40

-20

0

20

40

100 300 500 700 900

Temperature (K)

KCl

KBr

KIKF

Figure 13. Temperature variation of ds/dp. Figure 14. Temperature variation of d2 C11 /dp2

(10−10) .

-200

0

200

400

600

800

1000

1200

1400

100 300 500 700 900

dC11

1/dp

Temperature (K)

KCl

KBr

KI

KF

Figure 15. Temperature variation of FOPD of a TOEC.

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5. Conclusion

The extensive investigations of second-, third- and fourth- order elastic constants and their pressurederivatives carried out in present study appear to be important in revealing the anharmonic elastic propertiesof solids. It may thus be concluded that the “deformation-mechanism” used in present model provides muchbetter interpretation of the crystal properties in general. The cases discussed in present study are overall ingood agreement with theoretical and experimental results, which shows the validity of present theory. Thesedata are also useful for the interpretation of the anomalous elastic behavior of cyanides, halides and similarsystems. The new data may provide a further chance to improve the theoretical models developed recently forthe interpretation of the behavior of elastic constants in higher temperature region. But as the non-availabilityof experimental data, a detailed discussion of these properties may be left for a later investigation when themain effects are better understood. We have thus presented a simple method to study the elastic properties ofsolids under varying conditions of temperatures. The results obtained are encouraging. Due to the simplicityof the method, it can be applied to the more complicated solids, like minerals of geophysical importance andapplications. The results on different types of solids at deferent temperatures and composition are in progress.

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