Sound Attenuation at High Temperatures in Pt

Transcription

1 Vol ) ACTA PHYSICA POLONICA A No. Sound Attenuation at High Temperatures in Pt R.K. Singh and K.K. Pandey H.C.P.G. College, Varanasi-1001, U.P., India Received October 4, 005) Ultrasonic attenuation due to phonon phonon interaction and thermoelastic loss was evaluated in VIII group transition metal Pt in a wide temperature range 100 K to 1500 K) or longitudinal and shear waves along 100, 110 and 111 directions and or shear waves polarised along dierent directions. Electrostatic and Born Mayer potentials were used to obtain second and third order elastic constants, taking nearest-neighbour distance and hardness parameter as input data. Second and third order elastic constants data obtained at dierent temperatures were used to obtain Gruneisen parameters and non-linearity or anisotropy parameters which in turn were used to evaluate and in Akhiezer regime. The results were discussed and it was ound that at lower temperatures increases )l )s rapidly with temperature and then rate o increase becomes very small. Contribution to the total attenuation due to thermoelastic loss is negligible so that due to phonon phonon interaction establishing that major part o energy rom sound wave is removed due to interaction with thermal phonons lattice vibrations). PACS numbers: 6.0.Dc, k ) 1. Introduction Platinum Z = 78) is a silvery white transition metal o VIII group in VI period with an outer electronic coniguration o 5d 9 6s 1 and shows high catalytic activity. It is highly malleable and ductile metal with high melting point K). Platinum is corrosion resistant and its wear and tarnish resistance characteristics are well suited or making ine jewelry. Other distinctive properties include resistance to chemical attack, excellent high temperature characteristics and sta- corresponding author; rksingh 17@redimail.com. Present address: Physics Department, B.H.U., Varanasi-1005, India. 19)

2 0 R.K. Singh, K.K. Pandey ble electrical properties. All these properties have been exploited or industrial applications. In recent past, elastic [1], thermal and mechanical [] properties o Pt have been investigated extensively at room temperature as well as in wide temperature region up to 1500 K). So ar as sound characterisation o platinum is concerned, no result is available or this metal in wide temperature range. Ultrasonic attenuation studies have been made in dierent types o solids viz. dielectric [3], mixed dielectric [4], semiconducting [5, 6] and metallic [7, 8] substances at room temperature as well as at higher temperatures. But results are available up to 500 K. Due to high melting point o platinum and its important high temperature uses, ultrasonic attenuation study has been made or Pt rom 100 K to 1500 K. Dierent causes may be attributed to the attenuation o ultrasonic waves propagating through solids crystalline or amorphous). O these important actors or sound attenuation are electron phonon interaction, phonon phonon p p) interaction, thermoelastic loss and loss due to screw and edge dislocations. Phonon phonon interaction is the principal cause o ultrasonic attenuation at high temperatures viz. 100 K and above in all types o solids i.e. metallic, semiconducting and dielectric [4 8]. In metals a small contribution about to 5%) o total attenuation coeicient occurs due to thermoelastic loss i.e. due to heat transer rom compressed to rareied parts o metals due to strain caused by applied sound waves. Sound attenuation studies oer the possibility to detect and characterize microstructural properties as well as laws in materials, controlling material behaviours, based on physical mechanisms to predict uture perormance o the material. Structural inhomogeneities, elastic parameters, non-linearity parameters are well connected with requency or temperature dependence o ultrasonic attenuation mechanisms. Starting rom electrostatic and Born Mayer repulsive potentials and taking nearest-neighbour distance and Born repulsive parameters as input data and considering interactions up to nd nearest neighbours, second and third order elastic constants have been obtained at dierent temperatures which in turn have been used to obtain non-linearity coupling parameters and hence ultrasonic absorption coeicients. The present communication deals with temperature variation o ultrasonic attenuation due to p p interaction rom 100 K to 1500 K along 100, 110 and 111 crystallographic directions o propagation or longitudinal and shear waves and or shear waves polarized along dierent directions. Ultrasonic attenuation due to thermoelastic loss has been evaluated and it has been ound that its contribution relative to contribution made by p p interaction is about 5%. Temperature variation o ultrasonic attenuation has been ound to increase rapidly at lower temperatures and aterwards rate o increase becomes small which nearly saturates beyond 1000 K along all the directions o propagation or longitudinal and shear waves.

3 Sound Attenuation at High Temperatures in Pt 1. Theory.1. Theory o second and third order elastic constants Second and third order elastic constants SOEC and TOEC), C 0 ij and C0 ijk, at 0 K have been obtained using electrostatic and Born Mayer potentials and ollowing Brugger s [9] deinition o elastic constants. Repulsive parameter and nearest-neighbour distance have been used as input data and interaction up to next nearest-neighbours has been considered. According to Brugger s deinition, n-th order elastic constant is deined as C ijklmn... = u/ ε ij ε kl ε mn...), 1) where u is the crystal ree-energy density and ε ij is strain tensor. In Voigt notation C IJK replaces C ijklmn... in which ij I... etc. For cubic crystals Pt has cc structure) three independent SOEC C 11, C 1, and C 44 ) and six independent TOEC C 111, C 11, C 144, C 166, C 456, and C 13 ) occur. Using the theory discussed in [9], SOEC and TOEC at 0 K viz. Cij 0 and C0 ijk obtained are given as C 0 11 = z e r 4 0 C 0 1 = C 0 44 = z e ) Qr 1 ) ) Qr ), qr 0 r 0 q qr 0 r0 q r ) Qr ), qr 0 r0 q C111 0 = z e 3 r0 4 qr0 + 3 q + 1 ) r 0 q 3 Qr 1 ) 3 r 0 q + 3 ) q + r 0 q 3 Qr ), C11 0 = C166 0 = z e 3 r0 4 + r0 q + 3 ) q + r 0 q 3 Qr ), C13 0 = C144 0 = C456 0 = z e 3 r0 4 r0 q + 3 q + 1 ) Qr ), r 0 q 3 Qr 1 ) = A exp r ) 0, Qr ) = A exp r ) 0. q q According to lattice dynamics developed by Leibried and Ludwig [10] temperature variation o SOEC and TOEC have been obtained by adding vibrational contribution to elastic constants. SOEC and TOEC at any temperature are given by and C ij T ) = C 0 ij + ij )

4 R.K. Singh, K.K. Pandey C ijk T ) = Cijk 0 + Cijk vib, 3) where Cij vib and Cijk vib are vibrational contributions to elastic constants where where 11 = 1,1 G 1 + G, 1 = 1,1 G 1 + G 1,1, 44 = G 1,1, 111 = 1,1,1 G ,1 G G G 3, 11 = 1,1,1 G 3 1 +,1 G 1,1 + G )G G,1, 13 = 1,1,1 G ,1 G 1 G 1,1 + 3 G 1,1,1, 144 =,1 G 1 G 1,1 + 3 G 1,1,1, 456 = 3 G 1,1, 166 =,1 G 1 G 1,1 + 3 G,1, G 3 = Qr 1 ) q 0 + 9q0 q0 3 q0) 4 H + G.1, G,1 = Qr ) q q q 0 3 ) q0 4 H, 0 G = Qr 0 ) 6 6q 0 q0 + q0) 3 H + G1.1, G 1,1 = Qr ) 3 6 q 0 ) q0 + q0 3 H, G 1 = Qr 0 ) + q 0 q 0) + Qr ) + q0 q 0 ) H, where Qr 1 ) = A exp = 3 = ηw 0 8r 3 0 r ) 0 q coth X, and Qr ) = A exp r ) 0, q 1,1 =,1 = 1 ηw 0 r [ ηw0 1,1,1 = ηw 0 384r 3 0 X = ηw 0 kt, H = kt η = h π X sinh + coth X, X ) coth X 6 sinh X + X ] sinh X + coth X, and k is Boltzmann constant, [ ) r0 r0 Qr 1 ) + q q ) 1 Qr )].

5 Sound Attenuation at High Temperatures in Pt 3.. Theory o sound attenuation In the Akhiezer regime a sound wave passing through a solid can be attenuated by two processes [11]. First, i the wave is longitudinal, periodic contractions and dilations in the solid induce a temperature wave via thermal expansion. Energy is dissipated by heat conduction between regions o dierent temperatures. This is called thermoelastic loss. Second, dissipation occurs as the gas o thermal phonons tries to reach an equilibrium characterized by a local sound wave induced) strain. This is an internal riction mechanism. The physical basis or obtaining attenuation coeicient is that the elastic constants contributed by the thermal phonons relax [1]. The phonon contribution to the unrelaxed elastic constants is evaluated by taking into consideration the change in energy o the thermal phonons due to applied instantaneous strain. The requency o each mode ν i is changed by ν i ν i = γ j i S j, where γ j i is generalised Gruneisen parameter and S j is instantaneous strain. It is assumed that all the phonons o a given direction o propagation and polarization have equal change in requency. Then phonons o i-th branch and j-th mode suer a change in temperature T i T 0 = γ j i S j T is the temperature). A relaxed elastic constant is obtained ater there is phonon phonon coupling among various branches and the T i relax to a common temperature change, T, given by T T = γj i S j, where γ j i is the average value o γj i. Thermal relaxation time is τ = τ s = τ l = 3K C V v, 4) where K is thermal conductivity, C V is speciic heat per unit volume and v is Debye average velocity. According to Mason and Batemann [1], SOEC and TOEC which are measure o anharmonicity o the crystal) are related by Gruneisen parameter γ j i and hence by non-linearity parameter, D. Ultrasonic attenuation due to phonon phonon interaction in Akhiezer regime ωτ 1) [1] is given by ) = π DE 0 τ 3v 3, 5) p p where non-linearity coupling constant D = 9 γ j i ) 3 γj i C V T. 6) E 0 γ j i ) and γ j i are square average and average square Gruneisen parameters, v is sound wave velocity or longitudinal waves v l ) and or shear waves v s ) and d is density. Debye average velocity is given by 3 v 3 = 1 vl 3 + vs 3. 7) Propagation o sound wave through crystal produces compression and rareactions as a result heat is transmitted rom compressed region at higher

6 4 R.K. Singh, K.K. Pandey temperature) to rareied region at lower temperature) and hence thermoelastic loss occurs, which is given by [13] ) = 4π γ j i KT dvl 5. 8) th 3. Results and discussions Using the theory discussed above values o SOEC and TOEC obtained at absolute zero temperature are given in Table I. Thermal energy density, E 0, and speciic heat per unit volume, C V are evaluated as unction o the Debye temperature, θ D, using physical constants table [14]. TABLE I C 0 ij and C 0 ijk in [10 11 dyne/cm ] or Pt. C11 0 C1 0 C111 0 C11 0 C Thermal relaxation time has been obtained using Eq. 4), taking thermal conductivity rom physical constants table [15], Gruneisen parameter γ j i, square average Gruneisen parameter γ j i ), and non-linearity constants D l and D s along dierent crystallographic directions are evaluated using Eq. 6) and D l and D s are given in Table II. Sound absorption coeicients or longitudinal wave, and or shear )l wave, are evaluated using Eq. 5). Temperature variation o )s and )l along dierent crystallographic directions o propagation and polarization )s are shown in Figs Temperature variation o along dierent directions )th or longitudinal wave has been presented in Table III. Values o SOEC and TOEC at 0 K are given in Table I. Cagin et al. [] have evaluated SOEC or Pt using computer simulation technique at dierent temperatures. Their values at 300 K are C 11 = dyne/cm, C 1 = dyne/cm, and C 44 = dyne/cm, while our values at the same temperature are C 11 = dyne/cm, C 1 = dyne/cm C 44 = dyne/cm. Thus there is good agreement between values o C 11 and C 44. Deviations in value o C 11 and slight deviation in value o C 44 can be attributed to the approach adopted by them. They have used NVE molecular dynamics simulation, in which they have not considered certain parameters. Also our approach is very simple, which involves only two basic parameters, nearest- -neighbour distance and hardness parameter.

7 Sound Attenuation at High Temperatures in Pt 5 TABLE II Temperature variation o D l and D s along dierent directions or Pt Temp. [K] D l D s D l Ds Ds D l D s + D s Average and square average Gruneisen parameters have the values as expected [16]. Non-linearity constants ratio along all the three directions o propagation is as expected [16]. Fig. 1. Variation o ) 100 direction or platinum. with temperature or longitudinal and shear waves along

8 6 R.K. Singh, K.K. Pandey Fig.. Variation o ) 110 direction or platinum polarisations are shown). with temperature or longitudinal and shear waves along Fig. 3. Variation o ) 111 direction or platinum polarisations are shown). with temperature or longitudinal and shear waves along From Figs. 1 3 it can be seen that temperature variation o along all the directions or longitudinal and shear waves. and )l ) )s is similar increase with temperature. At lower temperatures rate o increase is large, as temperature increases beyond ) 500 K, rate o increase becomes small. Beyond 500 K rate o variation o with temperature is small.

9 Rate o increase in Sound Attenuation at High Temperatures in Pt 7 )l with temperature is little larger or longitudinal waves compared to shear waves. This may be attributed to larger value o thermal relaxation time, τ or longitudinal waves compared to that o shear waves Eqs. 4) and 5)). Parameter in [10 18 db s cm 1 ] at dierent temperature or Pt. Temp. [K] )th TABLE III Direction o propagation From Table III and rom the values o given in Figs. 1 3, it can be )th seen that along all the directions is negligible in comparison to )th or )l )s. Although experimental results are not available in the temperature range studied, yet on the basis o the above discussions it can be concluded that the present approach is correct. Acknowledgments One o us R.K. Singh) grateully acknowledges inancial support F. No. 6-70/SPT/00MRP/NRCB) rom University Grants Commission, Government o India.

10 8 R.K. Singh, K.K. Pandey Reerences [1] D. Verma, M.L. Verma, A. Verma, Ind. J. Phys. A 78, ). [] T. Cagin, G. Dereli, M. Uludogan, M. Tomak, Phys. Rev. B 59, ). [3] S.K. Kor, R.K. Singh, Acta Phys. Pol. A 80, ). [4] R.K. Singh, J.P. Acad. Sci. 5, ). [5] S.K. Kor, R.K. Singh, J. Acoust. Soc. India 1, ). [6] S.K. Kor, R.K. Singh, Acustica 79, ). [7] S.K. Kor, R.K. Singh, Acustica 78, ). [8] R.R. Yadav, O.K. Pandey, Acta Phys. Pol. A 107, ). [9] K. Brugger, Phys. Rev. A 133, ). [10] G. Leibried, W. Ludwig, in: Solid State Physics, Eds. F. Seitz, D. Turnbull, Academic Press Inc., New York [11] J. Fabian, P.B. Allen, Phys. Rev. Lett. 8, ). [1] W.P. Mason, Physical Acoustics, Vol. III B, Academic Press, New York [13] R.R. Yadav, D. Singh, Acoust. Phys. 49, ). [14] AIP Handbook, Academic Press, New York [15] CRC Data Series, CRC Press, Florida [16] S.K. Kor, R.K. Yadav, Kailash, J. Phys. Soc. Japan 55, ).