B

DD

a

ARRA

KCAL

1

nttbhepAieddwpc(sctEn

(

0d

Materials Chemistry and Physics 115 (2009) 65–68

Contents lists available at ScienceDirect

Materials Chemistry and Physics

journa l homepage: www.e lsev ier .com/ locate /matchemphys

ehaviour of acoustic attenuation in rare-earth chalcogenides

evraj Singh ∗

epartment of Applied Physics, AMITY School of Engineering and Technology, Bijwasan, New Delhi-110061, India1

r t i c l e i n f o

rticle history:eceived 1 April 2008eceived in revised form 16 July 2008ccepted 7 November 2008

a b s t r a c t

The acoustic attenuation due to phonon–phonon interaction and thermoelastic relaxation mechanismshave been studied for longitudinal and shear waves in B1 structured CeS, CeSe, CeTe, NdS, NdSe and NdTealong 〈1 0 0〉, 〈1 1 0〉 and 〈1 1 1〉 crystallographic directions at room temperature. The second- and third-

eywords:halcogenidescoustic propertiesattice dynamics

order elastic constants (SOEC and TOEC) of the chalcogenides are also computed for the evaluation ofacoustic parameters. The acoustic attenuation due to phonon–phonon interaction is predominant overthermoelastic loss in these materials. NdS is more ductile, stable and less imperfected material in com-parison to other chalcogenides systems (CeS, CeSe, CeTe, NdSe, NdTe, LaS, LaSe, LaTe, PrS, PrSe and PrTe)and rock salt-type LiF single crystal due to its lowest value of attenuation. The value of attenuation along〈1 1 1〉 crystallographic direction is smaller than the attenuation along 〈1 0 0〉 and 〈1 1 0〉 directions. Hence

pprop

〈1 1 1〉 direction is most a

. Introduction

The acoustic attenuation study of the materials has gainedew dimensions with progress in material science. The tempera-ure dependent part of acoustic attenuation has been explained inerms of model where the acoustic phonon interacts with num-er of thermal phonons in the lattice. Most of the workers [1–4]ave been assumed three phonons interaction in which the lownergy acoustic phonon interacts with one thermal phonon toroduce another, both thermal phonons having higher energy.ll these studies indicate that major portion of the attenuation

s caused by direct conversion of acoustic energy into thermalnergy via phonon–phonon interaction. The acoustic attenuationue the thermoelastic phenomenon is caused by the thermal con-uction between compressed and rarefied parts of propagatingave. Modified Mason–Bateman approach is used to study thehysical properties of Ce- and Nd-chalcogenides through the pro-ess of computing acoustic attenuation under Akhieser’s regimeω� ≤ 1). The acoustic coupling constant is the measure of conver-ion of acoustic energy into thermal energy. The acoustic couplingonstant is related to Grüneisen number, which in turn is correlatedo the second- and third-order elastic constants (SOEC and TOEC).

lastic constants of the materials provide valuable insight into theature of the atomic binding forces.

Compounds whose main component elements include sulphurS), selenium (Se), and tellurium (Te) constitute a large group among

∗ Tel.: +91 11 28062106; fax: +91 11 28062105.E-mail address: dsingh1@aset.amity.edu.

1 Affiliated to G.G.S.I.P.University, Delhi.

254-0584/$ – see front matter © 2008 Elsevier B.V. All rights reserved.oi:10.1016/j.matchemphys.2008.11.025

riate to study the physical properties of these materials.© 2008 Elsevier B.V. All rights reserved.

semiconductors. In this case of compounds, rare-earth chalco-genides are broadly studied. These compounds belong to the wideclass of binary rare-earth chalcogenides with NaCl-type (B1) struc-ture, which has been intensively studied because of their interestingphysical properties. It was shown that Ce- and Nd-chalcogenidesare compounds having metallic conductivity [5–13]. In presentinvestigation particular attention is given to theories, which cal-culates SOEC, TOEC, thermal relaxation time, acoustic couplingconstants and acoustic attenuation at room temperature.

2. Theory

2.1. Second- and third-order elastic constants by Coulomb andBorn–Mayer potential

The potential is used for evaluation of second- and third-orderelastic constants (SOEC and TOEC) of the form

�(r) = �(C) + �(B) (1)

where �(C) is electrostatic/Coulomb potential and �(B) is therepulsive/Born-Mayer potential, given as

�(C) = ±(e2/r) and �(C) = A exp(−r/b) (2)

where e is electronic charge, r is the nearest neighbour distance, bis the hardness parameter and A is the strength parameter.

According to lattice dynamics developed by Leibfried and Lud-

wig [14], lattice energy changes with temperature, hence addingvibrational energy contribution to static elastic constants, one getsCij and Cijk at required temperature.

Cij = C0ij + CVib

ij and Cijk = C0ijk + CVib

ijk (3)

6 try and Physics 115 (2009) 65–68

w0dttzxoxcfT[

2d

eiippbiiorpewAaEct

pw

A

A

wivs

rtc

�

wvs

Table 1The second and third-order elastic constants in 1010 N m−2 and bulk modulus (B) inGPa of Ce and Nd-chalcogenides at room temperature.

Elastic constants Material

CeS CeSe CeTe NdS NdSe NdTe

C11 5.40 4.72 4.26 5.73 5.00 4.76C12 1.09 0.94 0.68 1.18 1.16 0.71C44 1.35 1.17 0.91 1.45 1.39 0.95C111 −84.30 −73.71 −68.83 −89.31 −76.20 −78.26C112 −4.48 −3.86 −2.72 −4.84 −4.781 −2.77C123 0.80 0.64 0.17 0.90 1.01 0.07C 2.30 2.00 1.58 2.46 2.33 1.68

〈1 0 0〉, 〈1 1 0〉 and 〈1 1 1〉 crystallographic directions are computedusing Grüneisen number by means of Eq. (8). The obtained results ofacoustic coupling constants and thermal relaxation time for Ce- andNd-chalcogenides are presented in Table 2. Finally acoustic attenu-

Table 2Acoustic coupling constants (Dl and DS) along 〈1 0 0〉, 〈1 1 0〉 and 〈1 1 1〉 directionsfor longitudinal and shear waves and relaxation time (�th in 10−11 s) at roomtemperature.

Physical property Direction CeS CeSe CeTe NdS NdSe NdTe

Dl 〈1 0 0〉 11.14 11.17 17.90 16.57 15.47 18.45〈1 1 0〉 15.49 15.45 16.13 15.51 15.33 16.27〈1 1 1〉 13.20 13.17 13.60 13.21 13.01 13.67

DS 〈1 0 0〉a 1.20 1.20 1.17 1.20 1.23 1.17〈1 1 0〉b 2.11 2.06 1.39 2.17 2.93 1.26〈1 1 0〉c 26.42 24.46 29.98 26.33 24.75 30.74〈1 1 1〉d 17.88 17.91 19.85 17.82 16.73 20.85

6 D. Singh / Materials Chemis

here superscript 0 has been used to denote SOEC and TOEC atK (static elastic constants) and superscript Vib has been used toenote SOEC and TOEC at a particular temperature. A stress appliedo a solid in a given direction may be resolved into six components,hree tensile stresses �x, �y and �z along the principal axes x, y andrespectively and three shear stresses �yz, �xz and �yx about axes

, y and z. In the notation for shear, subscripts indicate the actionf the shear—thus yz denotes a shear in the yz-plane acting about-axis. Also it is noted that �yz = �zy and so on. The correspondingomponents of strains are εx, εy, εz, εyz, εxy and εzy. In general, theollowing stress-strain relationship can be generalized �ij = Cijεij.he expressions of CVib

ijand CVib

ijkare given in our previous paper

12].

.2. Acoustic attenuation due to phonon–phonon interaction andue to thermoelastic relaxation mechanisms

The main cause of acoustic attenuation for perfect antiferro-lectric and antiferromagnetic crystals are due to phonon–phononnteraction, the thermoelastic relaxation, and electron–phononnteraction mechanisms. At room temperature, electron mean freeath is not equal to phonon mean free path and no coupling will takelace. Thus the attenuation due to electron–phonon interaction wille absent. Hence the acoustic attenuation due to phonon–phonon

nteraction and thermoelastic relaxation mechanisms are govern-ng processes at room temperature. Both type of attenuation arebserved in these chalcogenides. It has assumed that the equilib-ium distribution phonons in a crystal can be distributed by theropagation of an acoustic phonon and the reestablishment of thequilibrium is the relaxation phenomenon. A theory of this typeas first proposed by Akhieser [15] and this region is often calledkhieser region (ω� ≤ 1). A simplified version was given by Bömmelnd Dransfeld [16] and a more complete version by Woodruff andhrenreich [17]. Finally most simple form of this theory was enun-iated by Mason [3,18]. In present study, modified form of Mason’sheory is used to evaluate acoustic attenuation.

The expressions for acoustic attenuation produced ofhonon–viscosity effect for longitudinal and shear acousticaves, are, respectively

l = E0ω2(Dl/3)�l

2�V3l

(4)

S = E0ω2(DS/3)�S

2�V3S

(5)

here the condition ω� ≤ 1 has already been considered. Here ωs the angular frequency, E0 is the thermal energy density, V is theelocity of ultrasonic wave, � is the density of the crystal. The sub-cripts l and S stand for longitudinal and shear waves respectively.

The relaxation time for longitudinal waves (�1) is about twice theelaxation time (�th), i.e. �1 = 2�th. While �th is equal to relaxationime for shear waves (�S), i.e. �S = �th. The thermal relaxation timean be expressed as

th = 12

�l = �S = 3K

CVV2

(6)

here K is thermal conductivity, CV is the specific heat per unit

olume and V is the average Debye velocity, which for isotropicolid is given by

3

V3

= 1

V3l

+ 2

V3S

(7)

144

C166 −5.49 −4.77 −3.64 −5.88 −5.68 −3.82C456 2.25 1.96 1.54 2.40 2.28 1.64B 25.27 22.00 18.73 26.93 24.40 20.60

The acoustic coupling constant (D) in Eqs. (4) and (5) is evaluatedfrom SOEC and TOEC using relation for Grüneisen parameters.

D = 9⟨

(�ji)2⟩

− 3〈�ji〉2

�CVT

E0(8)

where �ji

are the Grüneisen parameters corresponding to a par-ticular direction of propagation and polarization. The attenuationcaused by thermoelastic mechanism is calculated from the formula

Ath = ω2〈�ji〉2

KT

2�V5l

(9)

3. Results and discussion

The theory is tested for rare-earth chalcogenides, i.e. rock salt-type structured materials. Starting with nearest neighbour distance(r0) [13,19] and hardness parameter (b) [20] for these chalcogenides,the SOEC and TOEC are computed using Coulomb and Born–Mayerpotential. The evaluated values of SOEC and TOEC in these chalco-genides are presented in Table 1. The SOEC and TOEC provideaverage Debye velocity and Grüneisen numbers along 〈1 0 0〉, 〈1 1 0〉and 〈1 1 1〉 directions for longitudinal and shear waves from Mason’stable [3]. The values of CV and E0 evaluated as a function of (D/T)[D and T being Debye and Kelvin temperature respectively] aretaken from the AIP Handbook [21]. The thermal relaxation time(�th) is calculated by Eq. (6) using density (�) and thermal con-ductivity (K) data [13]. The acoustic coupling constant (D) along

�th 1.09 1.24 1.58 1.22 5.37 1.47

a Polarisation along 〈1 0 0〉.b Polarisation along 〈0 0 1〉.c Polarisation along 〈1 1 0〉.d Polarisation along 〈110〉.

D. Singh / Materials Chemistry and Physics 115 (2009) 65–68 67

Ftt

aa(m

pip

atmBZ1Soot

1tTa1e

htfam

Fda

Table 3Comparison of acoustic attenuation coefficient (A) in dB �s−1 of Ce- and Nd-chalcogenides with La- and Pr-chalcogenides and LiF at 900 MHz (at roomtemperature).

Material 〈1 0 0〉l 〈1 0 0〉Sa 〈1 1 0〉l 〈1 1 0〉S

b 〈1 1 0〉Sc 〈1 1 1〉l 〈1 1 1〉S

d

CeS 0.5 0.1 0.6 0.3 1.9 0.5 1.3CeSe 0.5 0.1 0.7 0.2 2.3 0.6 0.6CeTe 1.0 0.2 0.8 0.2 3.6 0.7 2.4NdS 0.2 0.1 0.6 0.2 2.0 0.1 0.1NdSe 3.0 0.4 3.0 1.0 8.6 1.3 5.8NdTe 0.6 0.1 0.6 0.1 0.01 0.1 0.01LaS [12] 1.3 0.2 1.5 0.2 4.5 0.1 0.3LaSe [12] 1.5 0.3 2.0 0.1 6.0 0.1 0.4LaTe [12] 2.6 0.4 3.3 0.1 10.0 0.2 0.8PrS [11] 0.9 0.1 0.9 0.3 2.5 0.7 1.7PrSe [11] 0.9 0.1 0.8 0.3 2.7 0.7 1.8PrTe [11] 1.2 0.2 1.0 0.3 4.4 1.5 3.0LiF [22] 3.5 0.8 1.3 0.8 10.0 0.8 5.0

a Polarisation along 〈1 0 0〉.

ig. 1. Acoustic attenuation due to thermoelastic mechanism along different direc-ions of the materials (1. CeS, 2. CeSe, 3. CeTe, 4. NdS, 5. NdSe, and 6. NdTe) at roomemperature.

tion (A/f 2)th, (A/f 2)l and (A/f 2)S are evaluated using Eqs. (9), (4)nd (5) respectively. The acquired results of (A/f 2)th, (A/f 2)l andA/f 2)S are visualised in Figs. 1 and 2 along three directions in these

aterials.The calculations have been carried out both manually and a com-

uter program in C++ language, which is based on formulae givenn the paper. The program has been checked and verified by knownrevious results for other B1-structured materials.

It is depicted from Table 1 that the evaluated values of SOECnd TOEC are in good agreement with other chalcogenides sys-ems [11,12]. The values of bulk modulus [B = (C11 + 2C12)/3] of these

aterials are lying between 18 GPa and 25 GPa, while the value ofin SmS, SmSe, SmTe, UTe, NpTe, PuTe, AmTe and rock salt type

rC are 47.6 GPa [7], 40 GPa, 40 GPa [7,8], 45 GPa, 65 GPa, 34 GPa,5 GPa [7,9] and 22.23 GPa [10] respectively. The bulk moduli ofmS, SmSe, SmTe, UTe, NpTe, PuTe are found greater than thosef chosen materials, while the B of AmTe and ZrC are very close topted chalcogenides systems. Hence the theoretical approach forhe evaluation of SOEC and TOEC is justified.

The evaluated thermal relaxation time (�th) is of the order of0−11 s in all these semiconductors, which is expected as in lan-hanum and praseodymium chalcogenides [11,12]. It is obvious fromable 2 that the value of Dl/DS are be positioned between 5 and 16long 〈1 0 0〉a and 〈1 1 0〉b directions, while the value DS/Dl is about.5–2.0 along 〈1 1 0〉c and 〈1 1 1〉d orientations in these materials, asxpected in literature [11,12].

It can be observed from Figs. 1 and 2 that NdSe and NdS have

ighest and lowest attenuation respectively. Hence NdS has bet-er performance in comparison to other materials. It is also clearrom the Figs. 1 and 2 that all materials have lowest value ofttenuation along 〈1 1 1〉. It is concluded that the rock salt-typeaterial like chosen materials has a peculiar behaviour along 〈1 1 1〉

ig. 2. Acoustic attenuation due to phonon–phonon interaction along differentirections of the materials (1. CeS, 2. CeSe, 3. CeTe, 4. NdS, 5. NdSe, and 6. NdTe)t room temperature.

b Polarisation along 〈0 0 1〉.c Polarisation along 〈1 1 0〉.d Polarisation along 〈1 1 0〉.

direction. The compared results of acoustic attenuation in La- andPr-chalcogenides and LiF (experimental) [22] with these chalco-genides are presented in Table 3 at 900 MHz. It can be seen fromthe Table 3, that the values of acoustic attenuation of chosen chalco-genides are smaller than those of La- and Pr-chalcogenides. Hencethe decided materials have better prospects in comparison to lan-thanum and praseodymium chalcogenides.

Therefore the attenuation in these materials is mainly dueto lattice part of thermal conductivity. The acoustic attenuationover frequency square (A/f 2)th due to thermoelastic relaxationphenomenon is negligible in comparison to Akhieser loss. Henceacoustic attenuation due phonon–phonon interaction is predom-inant over total attenuation, i.e. (A/f 2)Total = (A/f 2)th + (A/f 2)l +(A/f 2)S. Since the order of acoustic attenuation is same as for La-and Pr-chalcogenides in these materials. It is confirmed by mea-surement that at room temperature the acoustic attenuation isindependent of dislocation count and it has not been consideredhere [23].

On the basis of above compared results, the acoustic attenuationhas been computed in these materials at room temperatures estab-lishing modified Mason–Bateman approach observing the effectof lattice thermal conductivity values. The achieved results in thepresent work can be used for further investigations using othermethods such as polarizing microscope, X-rays scattering, surfacetension, viscosity, NMR and various transport phenomena. Theseresults may expand future prospects for the applications of CeS,CeSe, CeTe, NdS, NdSe and NdTe.

Acknowledgements

Author wishes to express sincere thanks to Dr. Ashok K. Chauhan,Founder president, Prof. B.P. Singh, Senior Director and Prof. PremPrakash, Director, AMITY School of Engineering and Technology,Bijwasan, New Delhi, India for their continuous encouragement andproviding necessary facilities to complete the work. Author is grate-ful to Dr. D.K. Pandey and Dr. P.K. Yadawa for valuable discussionsand their permanent interest in this work.

References

[1] S.K. Kor, P.K. Mishra, U.S. Tandon, Solid Stat. Commun. 15 (1974) 499.[2] R. Nava, M.P. Veechi, J. Romero, B. Fernandez, Phys. Rev. B14 (1976) 800.[3] W.P. Mason, Physical Acoustics, vol. IIIB, Academic Press, New York, 1965.[4] S.D. Lambade, G.G. Sahasrabudhe, S. Rajagopalan, Phys. Rev. B51 (1995) 15861.[5] G. Vaitheeswaran, V. Kanchana, S. Heathman, M. Idiri, T. Le Bihan, A. Svane, A.

Delin, B. Johanansson, Phys. Rev. B75 (2007) 184108.

6 try an

[[[[

8 D. Singh / Materials Chemis

[6] E. Bucher, K. Andres, F.J. di Salvo, J.P. Maita, A.C. Gossard, A.S. Cooper, G.W. HillJr., Phys. Rev. B11 (1975) 500.

[7] U. Benedict, W.B. Holzafel, Handbook on the Physics and Chemistry of Rare-earth Lanthanides/Actinides Physics I, vol. 17, Elsevier Press, Amsterdam, 1993.

[8] A. Chatterjee, A.K. Singh, A. Jayaraman, Phys. Rev. B6 (1972) 2285.

[9] P. Watcher, M. Filmoser, J. Rebizant, Physica B293 (2001) 199.10] W. Weber, Phys. Rev. B8 (1973) 5098.11] R.R. Yadav, D. Singh, Acoust. Phys. 49 (2003) 595.12] R.R. Yadav, D. Singh, J. Phys. Soc. Jpn. 70 (2001) 1825.13] V.P. Zhuze, A.V. Golubkov, E.V. Goncharova, V.M. Sergeeva, Sov. Phys. Solid Stat.

6 (1964) 205.

[[[[

d Physics 115 (2009) 65–68

[14] G. Leibfried, W. Ludwig, Solid State Physics, vol. 12, Academic Press, New York,1961, p. 276.

[15] A. Akhieser, J. Phys. (USSR) 1 (1939) 277.[16] H.E. Bömmel, K. Dransfeld, Phys. Rev. 117 (1960) 1245.[17] T.O. Woodruff, H. Ehrenreich, Phys. Rev. 123 (1961) 1553.

[18] W.P. Mason, J. Acoust. Soc. Am. 42 (1967) 2537.[19] R.W.G. Wychoff, Crystal Structure, Interscience Publishers, New York, 1963.20] M.P. Tosi, Solid State Physics, vol. 16, Academic Press, New York, 1965.21] D.E. Gray, AIP Handbook, IIIrd edition, McGraw Hill, New York, 1972.22] R.C. Hanson, J. Phys. Chem. Solids 28 (1967) 475.23] L. Lamb, M. Redwood, Z. Steirschlifier, Phys. Rev. Lett. 3 (1959) 332.