JournalofAcousticalSocietyofIndia(JASI)A quarterly publication of the Acoustical Society of IndiaVolume 39, Number 1,January 2012EDITORIALChoosing Concrete Blocks for Sound TransmissionControlMahavir Singh ............................................................................... 2ARTICLESUnified Dispersion Characteristics of Structural AcousticWaveguidesAbhijit Sarkar, M.V. Kunte and Venkata R. Sonti........................ 3Improving the Sound Insulation Performance of Double-leaf PanelsMahavir Singh and Dharam Pal Singh........................................ 8Spectral Characteristics of Traffic Noise Attenuation byTree Belts in and Around Kolkata HighwaysMadhumita Banerjee Lahiri, Ranjan Sengupta,Tarit Guhathakurta, Argha Deb, Dipak Ghosh andAshis Majumder ............................................................................ 15Capturing the Essence of Raga fromHindusthani Singing : an Objective ApproachA.K. Datta, R. Sengupta and N. Dey ............................................ 20Speech Rhythm in Kannada Speaking ChildrenS.R. Savithri................................................................................... 25A Statistical Analysis of Raga BhairaviSwarima Tewari and Soubhik Chakraborty .................................. 34Thermo-Acoustic Properties of Binary Mixtures ofDi-(2-Ethylhexyl) Phosphoric Acid with Dioxane,Cyclohexane and n-Pentane by Ultrasonic MethodS.K. Dash, B. Dalai, S.K. Singh, N. Swain, M.D. Swain andB.B. Swain ..................................................................................... 38INFORMATIONExecutive Council of Acoustical Society of India 48InsidebackcoverJASIMAHAVIR SINGHChief EditorOMKAR SHARMAManaging EditorTRINATH KARAssociate Scientific EditorYudhishter KumarAnil Kumar NainNaveen GargAssistant EditorsEDITORIAL BOARDM L MunjalIISc Banglore, IndiaS NarayananIIT Chennai, IndiaV RajendranKSRCT Erode, IndiaR J M CraikHWU Edinburg, UKTrevor R T NightingleNRC Ottawa, CanadaB V A RaoVIT Vellore, IndiaN TandonIIT Delhi, IndiaP NarangNMI Lindfield, AustraliaE S R RajagopalIISc Banglore, IndiaA L VyasIIT Delhi, IndiaV Bhujanga RaoNSTL Vizag, IndiaYukio KagawaNU Chiba, JapanS DattaLU Loughborough, UKSonoko KuwanoOU Osaka, JapanK K PujaraIIT Delhi (Ex.), IndiaA R MohantyIIT Kharagpur, IndiaAshok KumarNPL New Delhi, IndiaV MohananNPL New Delhi, India Information for AuthorsEDITORSSPACEChoosing Concrete Blocks for Sound Transmission ControlIs it better to use lightweight or normal weight concrete blocks to control sound transmission through demisingwalls between hotel rooms?Lighter blocks reduce the weight and increase the thermal insulating properties, but they are not the bestchoice for lowering sound transmission. Reduction of sound transmission is related to the density of thewall: the higher the density of the units the better the sound transmission reduction. For this reason normalweight units are better than lightweight blocks for reducing sound transmission.The best way to determine the STC rating of an assembly is to perform laboratory tests and review theresults. ASTM E 90-00, Standard Test Method for Laboratory Measurement of Airborne-Sound TransmissionLoss of Building Partitions, is used to determine the STC rating of building partitions.If test data is not available, Document 0302, "Standard Method for Determining the Sound TransmissionClass Rating for Masonry Walls", published by The Masonry Society, can be used. Data in this document ison the conservative side and typically underestimates the actual STC rating.The rating is determined by the equation STC = 0.47 Block Weight + 39.4 where Block Weight is equal tothe weight of the wall expressed in terms of kg. The equation is used only for uncoated fine or mediumtextured concrete masonry units. Coarse texture concrete masonry units may allow airborne sound to enterthe wall system. Surface treatments may be needed to increase the STC rating when coarse textured units areused.The equation cannot be used if the wall system is thinner than 3 inches. The units must be laid in fullmortar joints, and all holes, voids, or cracks in the masonry must be solidly filled with mortar.Itisimportanttorecognizethatverysmallopeningsinamasonrywallsignificantlyaffectsoundtransmission through it. Openings through the wall for a pipe or conduit greatly increase sound transmissionif the penetration is not properly detailed. Back-to-back receptacle boxes in walls or receptacle boxes notcoated on the backside to reduce sound transmission greatly increase the sound transmission through themasonry walls.Any joints in the wall and the joint along the top of the wall significantly impact the sound transmissionif not properly detailed. These joints must be covered with elastomeric sealant on the exterior face. Spacebetween the top of the wall and the underside of the structure above must be filled with mortar or grout if it isnot designed as a movement joint, or filled with foam or fiber material where movement must be accommodated.Doors in demising walls are also a common cause of sound transmission problems. Using doors withhigh sound transmission ratings, as well as proper detailing, are essential to prevent sound transmissionthrough demising walls in hotels.MahavirSinghUnified Dispersion Characteristics of Structural Acoustic Waveguides3Journal of Acoustical Society of IndiaUnified Dispersion Characteristics of StructuralAcoustic WaveguidesAbhijit Sarkar1, M.V. Kunte2 and Venkata R. Sonti21Department of Mechanical EngineeringIndian Institute of Technology Madras, Chennai-600 036 (tamil Nadu)2Vibro-Acoustics Lab, Facility for Research In Technical Acoustics (FRITA)Department of Mechanical Engineering, Indian Institute of Science, Bangalore-560 012 (Karnataka)ABSTRACTIn this article, we show with some formalism that infinite flexible structural acoustic waveguideshave a general form for the dispersion equation. This allows us to bring out the common features inthe dispersion characteristics of structural acoustic waveguides. We take three examples todemonstrate this fact, namely, the rectangular, the circular cylindrical and the elliptic geometries.The individual coupled dispersion equations are cast into the generic form. It is then shown that thecoupled wavenumber solutions of all these systems can be represented on a single schematic.1. INTRODUCTIONRecently, several studies were presented on the dispersion characteristics of waveguides (with rectangular,circular cylindrical and elliptic cylindrical geometries) by Sarkar and Sonti [3-6], [7, 8] and Kunte, Sarkar, andSonti (2010) using the asymptotic methods. Despite the geometrical differences between the systems, thereseems to be an underlying feature common to these waveguides in the manner in which the uncoupledstructural and acoustic waves form coupled waves. In this article, we intend to bring out this feature bypresenting the coupled dispersion equation for the different geometries in a generic form. And along with thisgeneric equation is presented a generic solution of the coupled waves in a schematic onto which all thecoupled wavenumbers for the different geometries can be superposed. The structural acoustic systems thatare considered are as follows :1. The two-dimensional fluid-filled rectangular waveguide (Fig. 1a).2. The fluid-filled circular cylindrical shell (Fig. 1b).3. The fluid-filled elliptic cylindrical shell (Fig. 1c)2. A GENERAL FLUID-FILLED WAVEGUIDEFor a general fluid-filled waveguide of constant cross-section, the coupled dispersion relation has the followinggeneric form,Equation 1 : Generic form of the coupled dispersion relationJournal of Acoustical Society of India : Vol. 39, No. 1, 2012 (pp. 3-7)2012 Acoustical Society of Indiaf f pf f1 2 3 4 += 0, [Received: 12.11.2011; Revised: 28.12.2011; Accepted: 10.01.2012]4Journal of Acoustical Society of IndiaAbhijit Sarkar, M.V. Kunte and Venkata R. SontiFig. 1. Structural acoustic waveguide systems studied in this article : (a). Two dimensional rectangularwaveguide, (b). Fluid-filled circular cylindrical shell, (c). Fluid-filled elliptic cylindrical shellwhere is the fluid-structure coupling parameter and physically represents the ratio of the mass of the fluidto the mass of the structure per unit length of the waveguide.The functions f1, f2, f3, f4and p represent the uncoupled wave solutions (Fig. 2). With = 0, the followingroots are obtained: the roots of f1 = 0 that represent the in vacuo structural wavenumbers and the roots of f2 =0 representing the cut-on wavenumbers of a rigid-acoustic duct (referred to as R = 0 in figure 2). In otherwords, this latter wavenumber originates when the structure is completely rigid. The solution of p = 0 is theacoustic plane wave. In some cases, the waveguide configuration is such that the only plane wave possible isthat of zero amplitude. With = , the roots of f3 = 0 represent the cut-on wavenumbers of the pressure-releaseacoustic duct where the structural boundary is perceived as a free surface by the acoustic fluid (referred to asPR = 0 in Fig. 2). Lastly, the roots of f4 = 0 give the structural wavenumbers corresponding to = . Thesesolutions turn out to be the longitudinal and torsional wavenumbers for cylindrical shell systems. Thesewaves, unlike the transverse flexural waves couple with the acoustic fluid through the Poisson's ratio. Thefunctions f1, f2, f3, f4 and p are different for different geometric configurations and will be presented later. Thus,for intermediate values of (0Fructose>Citric acid. The viscosity values for all three systems have beenpresented in Tables 1-3. It is evident from theseTables, thatviscosity () values increase with increase of concentration of solute, showing the presence ofstrong solute- solvent interactions. Also suggest more association.Table 1.Variation in Density, viscosity and Velocity of (aqueous NaCl +Fructose) at differentTable 2.Variation in Density, viscosity and Velocity of (aqueous NaCl +Glycine) at differentconcentration and temperature at 2MHzTable 3.Variation in Density, viscosity and Velocity of (aqueous NaCl +Citric acid) at differentconcentration and temperature at 2MHzswt% Density() kg/m3Viscosity() Nsm-2wt% Density() kg/m3Viscosity() Nsm-2wt% Density() kg/m3Viscosity() Nsm-2Velocity(u) m/sCon. 298.15303.15308.15K298.15 303.15 308.15K298.15 303.15 308.15K0 0996.3 0995.7 0994.0 08.2100 07.9800 07.1800 1500.24 1583.60 1583.901 1070.0 1069.0 1068.0 11.9754 12.9250 11.0385 1620.01 1613.03 1612.092 1078.0 1076.0 1075.0 12.4979 12.0432 11.0649 1612.04 1612.00 1624.033 1099.0 1097.0 1096.0 11.2120 12.9958 13.3273 1580.10 1609.36 1617.464 1101.0 1099.0 1098.0 10.6265 13.2514 13.5570 1537.74 1602.66 1602.265 1106.0 1105.5 1104.0 10.8479 13.3860 13.8760 1496.00 1546.00 1553.20Velocity(u) m/sCon. 298.15303.15308.15K298.15 303.15 308.15K298.15 303.15 308.15K0 1058.0 1060.4 1060.0 09.158 10.964 09.915 1583.90 1500.24 1583.601 1076.7 1067.0 1068.4 11.321 11.221 11.160 1575.08 1580.04 1584.082 1081.3 1081.1 1079.5 11.400 11.332 11.275 1599.40 1600.08 1664.643 1091.3 1090.5 1087.2 11.475 11.447 11.709 1503.20 1565.44 1565.444 1109.4 1108.1 1104.1 11.502 11.551 11.806 1612.76 1624.08 1634.445 1123.9 1121.4 1195.0 11.652 11.780 11.915 1632.68 1635.52 1637.20Velocity(u) m/sCon. 298.15303.15308.15K298.15 303.15 308.15K298.15 303.15 308.15K0 1060.4 1060.0 1058.0 10.9637 09.9150 09.1575 1500.24 1583.60 1583.901 1062.4 1051.6 1051.1 10.3060 09.6309 09.0717 1561.53 1572.20 1576.202 1067.0 1065.0 1064.0 11.1288 10.1002 10.4020 1563.67 1581.21 1576.903 1073.2 1064.7 1061.3 12.1288 10.9401 10.1402 1564.28 1589.70 1597.804 1074.0 1072.2 1070.2 15.2598 13.1559 10.4132 1568.14 1601.80 1607.005 1077.4 1076.7 1075.6 15.6058 14.2682 10.6920 1613.80 1610.80 1612.00212Journal of Acoustical Society of IndiaAlka Tadkalkar and Govind K. BichileTable4. Variation in Adiabatic compressibility, Change in compressibility and Relative change incompressibility of (aqueous NaCl +Fructose) at different concentration and temperature at 2MHz.Wt% Adiabatic compressibility()x10-10 m2N-1Change in compressibilityx10-12 m2N-1Relative change in compressibility/(0x10-12) m2N-1Con. 298.15 303.15 308.15K 298.15303.15308.15K 298.15303.15308.15K0 3.7675 3.7684 3.7617 - - - - - -1 3.7435 3.7428 3.7300 +24.0 +25.6 +31.7 +6.37 +6.79 +8.432 3.6152 3.6128 3.3430 +15.23 +15.56 +41.87 +4.04 +1.07 +11.133 4.0553 3.7419 3.7533 -28.78 +2.65 +2.65 -7.63 +0.70 +0.704 3.9076 3.4214 3.3904 -14.01 +34.7 +37.13 -3.72 +9.21 +9.875 3.3378 3.3337 3.1219 +42.97 +43.47 +63.98 +11.41 +56.57 +17.00Wt% Adiabatic compressibility()x10-10 m2N-1Change in compressibilityx10-12 m2N-1Relative change in compressibility/(0x10-12) m2N-1Con. 298.15 303.15 308.15K 298.15303.15308.15K 298.15303.15308.15K0 3.7684 3.7617 3.7675 - - - - - -1 3.8602 3.8471 3.8294 -9.18 -8.54 -6.19 -2.44 -2.27 -1.642 3.8330 3.7555 3.7844 -6.46 -6.21 -1.69 -1.71 -1.65 -0.493 3.8079 3.7165 3.6907 -3.95 -4.52 +7.68 -1.05 -1.20 -2.044 3.7863 3.6350 3.5987 -1.79 +12.67 +16.88 -0.48 +3.37 +4.485 3.5638 3.5794 3.5778 +20.46 +18.23 +18.97 +5.43 +4.85 +5.04Table 5. Variation in Adiabatic compressibility, Change in compressibility and Relative change incompressibility of (aqueous NaCl +Glycine) at different concentration and temperature at 2MHz.Wt% Adiabatic compressibility()x10-10 m2N-1Change in compressibilityx10-12 m2N-1Relative change in compressibility/(0x10-12) m2N-1Con. 298.15 303.15 308.15K 298.15303.15308.15K 298.15303.15308.15K0 4.4500 4.4000 4.3400 - - - - - -1 3.5600 3.5900 3.6000 +89.0 +81 +74 +20.0 +18.40 +17.052 3.5800 3.5700 3.5200 +87.0 +43.0 +82 +19.55 +9.77 +18.893 3.6900 3.5100 3.4800 +76.0 +89 +86 +17.8 +20.22 +19.824 3.8400 3.5300 3.5200 +115.88 +87 +82 +26.04 +19.77 +18.895 4.0400 3.7800 3.7300 +41 +116.4 +61 +9.21 +26.45 +2.12Table 5. Variation in Adiabatic compressibility, Change in compressibility and Relative change incompressibility of (aqueous NaCl +Glycine) at different concentration and temperature at 2MHz.The values of in system 1 and 2 show a decreasing trend. The values of are larger in system 2 thansystem 1.this shows that the molecular interaction is more in system 2 than system 1. Glycine molecules in thepolar solution have interaction with surrounding water molecules. The electrostrictive compression of wateraround the molecules results in larger. Table 3-6 show the values of change in and relative change in of the three systems there is non linearlychange in these parameters , depicted in Figures 1-3 ,which may is attributed to solute solvent interactions.Acoustical Study of Glycine, Fructose and Citric Acid in Aqueous NaCl at Different Temperatures213Journal of Acoustical Society of IndiaWt% Adiabatic compressibility()x10-10 m2N-1Change in compressibilityx10-12 m2N-1Relative change in compressibility/(0x10-12) m2N-1Con. 298.15 303.15 308.15K 298.15303.15308.15K 298.15303.15308.15K0 4.4500 4.4000 4.3400 - - - - - -1 3.5600 3.5900 3.6000 +89.0 +81 +74 +20.0 +18.40 +17.052 3.5800 3.5700 3.5200 +87.0 +43.0 +82 +19.55 +9.77 +18.893 3.6900 3.5100 3.4800 +76.0 +89 +86 +17.8 +20.22 +19.824 3.8400 3.5300 3.5200 +115.88 +87 +82 +26.04 +19.77 +18.895 4.0400 3.7800 3.7300 +41 +116.4 +61 +9.21 +26.45 +2.12Table 6. Variation in Adiabatic compressibility, Change in compressibility and Relative change incompressibility of (aqueous NaCl +Citric acid) at different concentration and temperature at 2MHzFig. 1. Variation of relative change in adiabatic compressibility with different concentration andtemperature for the system 1 (aqueous NaCl +Fructose) at 2 MHz214Journal of Acoustical Society of IndiaAlka Tadkalkar and Govind K. BichileFig. 3. Variation of relative change in adiabatic compressibility with different concentration andtemperature for the system 1 (aqueous NaCl +Citric acid) at 2 MHzFig. 2. Variation of relative change in adiabatic compressibility with different concentration andtemperature for the system 1 (aqueous NaCl +Glycine) at 2 MHzAcoustical Study of Glycine, Fructose and Citric Acid in Aqueous NaCl at Different Temperatures215Journal of Acoustical Society of India5. REFERENCES[1] M.K. RAWAT andSHARMA GEETA, 2007. J. Indian Chem. Soc., 81, p. 46.[2] S.U. PATIL, P.B. RAGHUWANSHI and D.T. TAYADE, 2007. J. Indian Chem. Soc., 84, p. 389.[3] SHESHAGIRI and M.G.RAO, 1977. Ind. J. Pure Appl. Phys., 9, p. 169.[4] R.P. VARMA and SURENDRAKUMAR, 2006.Ind. J. Pure Appl. Phys., 38(2), p. 96.[5] SHESHAGIRI and K.C. REDDY, 1973. Acoustica, 29, p. 09.[6] S.K. UPADHYAY, 2000. Ind. J. Chem., 39, p. 537.[7] S. THIRUMARAN and S. SUDHA, 2010. J. Chem, Pharm. Res., 1, p. 327.[8] S. GHANANBA B.R. RAO, 1969. Ind. J. Pure Appl. Phys., 7.[9] M.L. PARMAR, R.K. AWASTHi and M.K. GULERIA, 2000. J. Chem. Sci., 116, p. 33.[10] G.V. RAMA RAO,A. VISWANATHA SARMA,D. RAMCHANDRAN and C. RAMBABU, 2005. Ind. J.Pure & Appl. Phys., 43, p. 602.[11] S. THIRUMARAN and K. JOB SABU, 2009. Ind.J. Pure & Appl. Phys., 47, p. 87.[12] S. SHAKEEL, AHMADKHALIL A. NASIR and RUBINA SALEEM, 2004. Pak. J. Sci. Ind. Res., 47(5),349-355.[13] CHERKUPALLY, SANJEEVA REDDY, 2007. J. Ind. Chem. Soc., 84, 165-170.[14] V.A. TABHANE, O.P. CHIMANKAR, S. MANJA and T.K. NAMINARAYANAN, 1999. Pure. Appl. Ultraso.,21, p. 67.[15] S. THIRUMARAN and A.N.KANNAPAN, 2009. Global J.Mole. Sci., 4(2), p. 160.[16] R. PALANI, G. SRINIVASAN and B. GEETA LAKSHMI, Ind. J.Chem. Tech. Res. , 3 (1),264-89.[17] S.R. KANHEKAR, PRAVINA PAWAR and G.K. BICHILE, 2010. Ind. J. Pure & Appl. Phys., 48, 95-99.[18] KANNAPPAN and PALANI, 2007. Ind. J. Pure and Appl. Phy., 45, 575-579.[19] S. THIRUMARAN and K. JOB SABU, 2009. Ind. J. Pure &Appl. Phys., 47, 87-96.216Journal of Acoustical Society of IndiaKailash, S.K. Shrivastava, Jitendra Kumar and Virendra Kumar2012 Acoustical Society of IndiaHigh Temperature Study of Anharmonic Properties forHalides of Na and RbKailash1*, S.K. Shrivastava2, Jitendra Kumar3 and Virendra Kumar11Department of Physics, BN PG College, Rath, Hamirpur-210 431 (UP)2Department of Physics, Bundelkhand University, Jhansi-284 128 (UP)3Department of Physics, Govt. Girls P.G. College, Banda-210 001 (UP)*e-mail: kailashrath@gmail.comABSTRACTA method for evaluating the temperature variation of second, third and fourth order elastic constantsfor FCC crystal structure solids is developed on the basis of long range Coulomb and short rangeBorn-Mayer potentials using nearest-neighbour distance and hardness parameter. The theory istested for higher order elastic constants and related physical properties of halides of Sodium andRubidium from low temperature to near their melting points. The temperature variation of higherorder elastic constants and related physical properties for these crystals has been discussed properlyto provide valuable information. The values obtained are compared with other theoretical andexperimental data available which shows that the general behaviour of the constants confirms thevalidity of simple theoretical model.Keywords:Higher order elastic constants, pressure derivatives, partial contractions, sodium halides,rubidium halides, mono - valent crystals, FCC crystals.1. INTRODUCTIONThe alkali halide crystals have always been at the centre stage of solid-state physics. These have been used asmodel crystals for testing many solid-state theories. In recent decade, these have also been proved useful inseveral applications.The elastic properties of ionic solids provide valuable information about cohesive energies,interatomic forces and anharmonic properties such as thermal expansion and attenuation of high frequencysound waves. For this reason, there have been numerous experimental and theoretical studies of the elasticproperties of the alkali halides. The alkali halides are the simplest ionic solids and the computation of theirelastic properties have been very useful in providing information on short range interatomic forces, anharmonicproperties and equations of state.A theory for obtaining anharmonic properties such as higher order elastic constants of materials whichpossess face centered cubic crystal structure has been developed starting from primary physical parametersviz., nearest neighbour distance and hardness parameter using long- and short- range potentials. The elasticenergy density for a deformed crystal is expanded as a power series of strains. The coefficients of quadratic,cubic and quartic terms are known as the second, third and fourth order elastic constants (SOECs, TOECs andFOECs) respectively. When the values of elastic constants of any crystalline solid are known, many of theJournal of Acoustical Society of India : Vol. 39, No. 4, 2012 (pp. 216-229) [Received: 17.04.2012 ; Revised: 18.07.2012 ; Accepted: 21.09.2012]High Temperature Study of Anharmonic Properties for Halides of Na and Rb217Journal of Acoustical Society of Indiaanharmonic properties of the substance can be treated within the limit of the continuum approximation in aquantitative manner. These elastic constants are used to compute ultrasonic parameters [1, 2] such as ultrasonicvelocities, thermal relaxation time, pressure dependence of anharmonic properties [3]etc. The variation ofelastic constants [4-5] with respect to pressure can reveal many important features of the short range forces athigh pressure. The ultrasonic studies [6-8] can provide interesting information on the specificities of ion-solvent interaction related to the structure of the solute and the reciprocal effects which arise in the solvent.Several physical properties and crystal anharmonicities such as thermal expansion, specific heat at highertemperature, temperature variation of acoustic velocity and attenuation [9-11] and the first order pressurederivatives (FOPDs) of SOECs, Grneisen numbers and temperature derivatives of SOECs are directly relatedto SOECs and TOECs. While discussing higher order anharmonicities such as the FOPDs of TOECs, thesecond order pressure derivatives (SOPDs) of SOECs, partial contractions and deformation of crystals underlarge forces, the FOECs are to be considered extensively.In these days, keen interest has been taken in investigation of anharmonic properties of materials ofvarious kinds [12-19]. Many workers have contributed in this field through their experimental and theoreticalapproach.Several efforts have been made in the study of physical and anharmonic properties of solids ofdifferent types [20-22] 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 [23-26] of a few alkali cyanides, of mixed alkali halides and cyanides, of rare gas materials, of alkali halides usingultrasonic, theoretical and Brillouin scattering methods and others the temperature variation of latticeanharmonicities [27] and the role of collinear and noncolinear phonons in anharmonic scattering processesand in ultrasonic attenuation for different structured solids [28].No complete experimental or theoretical efforts have been made so for obtaining the temperature variationof anharmonic properties such as the SOECs, TOECs and FOECs, the FOPDs of SOECs and TOECs, SOPDs ofSOECs and partial contractions of materials possessing different crystal structures up to near melting point.The present work is concerned with the formulation to evaluate the SOECs, TOECs and FOECs, the FOPDs ofthe SOECs and TOECs and the SOPDs of SOECs and the partial contractions at an elevated temperature usinglong-and short- potentials starting from the nearest neighbour distance and hardness parameter. Section 2deals with the derivation of the theory. In Section 3, the theory is tested for halides of Sodium and Rubidium.The results thus obtained are widely discussed in Section 4.2 THEORYThe theory consists of two parts. In first part we discuss the temperature variation of elastic constants and insecond part the pressure derivatives of these constants.2.1Elastic ConstantsThe elastic energy density for a crystal of a cubic symmetry can be expanded up to quartic terms as shownbelow [29];0 1 2 3ijkl ab cd ijklmn ab cd mn ijklmnpq ab cd mn pqU U U U{[1/ 2!]C e e } {[1/ 3!] C e e e } {[1/ 4!]C e e e e }= + += + +(1)where Cijkl, Cijklmn and Cijklmnpq are the SOECs, TOECs and FOECs in tensorial form; eij are the Lagrangian straincomponents. The SOECs, TOECs and FOECs are as given below:2ijkl IJ ab cd 0C C ( U/ e e ) ,== = 3ijklmn IJK ab cd mn 0C C ( U/ e e e )== = 4ijklmnpq IJKL ab cd mn pq 0and C C ( U/ e e e e )== = (2)218Journal of Acoustical Society of IndiaKailash, S.K. Shrivastava, Jitendra Kumar and Virendra Kumarwhere CIJ, CIJK and CIJKL are the SOECs, TOECs and FOECs in Brgger's definitionand Voigt notations30.The free energy density of a crystal at a finite temperature T isvibTotal 0U U U , = +3sNvibi ci 1U [KT/NV ] ln2Sinh( /KT)= =where U0 is the internal energy per unit volume of the crystal when all ions are at rest on their lattice points, Uvibis the vibrational free energy, Vc is the volume of the primitive cell, N is the number of the primitive cells in thecrystal and s is the number of ions in the elementary cell. Other notations used in this equation have their usualmeanings.An elastic constant consists of two parts as follows:The first part is the strain derivative of the internal energy Uoand is known as static elastic constant and thesecond part is the strain derivative of the vibrational free energy Uvib and is called vibrational elastic constant. Thesuperscript 0 has been introduced to emphasize that the static elastic constants correspond to 0 K.The energy density of the non- deformed crystal is expressed as:wheremouvRis the distance between the v-th ion in the o-th cell and the u-th ion in the m-th cell and Quvis theinteraction potential between the ions. The indices (v, o) and (u, m) are sometimes dropped when no confusionoccurs. One assumes that Quv is the sum of the long-range Coulomb and the short-range Brn-Mayer potentials. where e is the electric charge,sign apply to like and unlike ions respectively,r0 is the nearest-neighbourdistance,is hardness parameter and A isIt is assumed that the crystal is deformed homogeneously. When the crystal is deformed homogeneously,the distance between (v, o) and (u, m) ion in the deformed and non- deformed states,and, are relatedto the Lagrangian strains eab as follows:whereis the i-th Cartesian component of the vector . The definition of the quantity is alsoexpressed in Eq.(8). The internal energy U0 given by Eq.(5) can be expanded in terms of, which will yieldquadratic, cubic and quartic terms as given below:0 vibIJK IJK IJKC C C = +0 vibIJKL IJKL IJKLandC C C = +(3)(4)smo0 c uv uv uv cm o v 1u vU [1/2V ] Q (R ) 'Q (R)/2V = = = (5)2uv 0 0 0Q (r ) (e /r ) Aexp( r / ) = + (6)(7)(8)U2= [1/2Vc] [Z2D2Q(R)/2!]R=r = [1/4Vc]eabecdYiYjYkYlD2Q(R)]R=rU3= [1/2Vc] [Z3D3Q(R)/3!]R=r = [1/12Vc]eabecdemnYiYjYkYlYmYnD3Q(R)]R=rU4 = [1/2Vc] [Z4D4Q(R)/4!]R=r = [1/48Vc]eabecdemnepqYiYjYkYlYmYnYpYqD4Q(R)]R=r(9)0 vibIJ IJ IJC C C , = +2 2 40 0 0A 0.29126 (z e /r )/[exp( r / ) 2 2 exp( r 2 / )] = + mouvRmouvrmo 2 mo 2 mo mo mouv uv uvi uvj ab uv(R ) (r ) 2Y Y e 2Z = =mouviYmouvrmouvZmouvZHigh Temperature Study of Anharmonic Properties for Halides of Na and Rb219Journal of Acoustical Society of Indiavib0 Z c 2'U [1/V 2!] ' '[Z' Z(D' D)U ] == c ab cd ijkl[1/2V ]e e f =vib0 Z c 3' ''U [1/V 3!] ' ' '[Z' Z'' Z(D' D'' D)U ] == c ab cd mn ijklmn[1/6V ]e e e f =' '' '''c ab cd mn pq ijklmnpq[1/24V ]e e e e f =' ' vibijkl i j k l R r'f ' '[YYY Y(D' D)U ] == ' ' '' '' vibijklmn i j r R n m l k' ''f ' ' '[YYY YY Y (D'' D' D)U ] == b i v ' ' ' ' ' ' ' ' ' ' ' 'ijklmnpq i j k l m n p q R r' '' '''f ' ' ' '[YYY YY Y Y Y (D''' D'' D' D)U ] == With reference to Eqs. (3) and (4) and comparison of Eqs. (1) and (9), one may obtain the static elasticconstants which are 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:whereandOn comparison Eqs. (1) and (10); one determines the vibrational elastic constants. Vibrational contributionsto SOECs, TOECs and FOECs are given in Table 2. These are shown as a combination ofn's and n's which areevaluated by taking crystal's symmetry into account and the expressions forn and n are presented in Tables3 and 4. By adding the vibrational elastic constants to the static elastic constants, one may get SOECs, TOECsand FOECs at any temperature for monovalent FCC crystals.The FOPDs of SOECs are concerned with SOECs and TOECs. The FOPDs of TOECs and SOPDs of SOECsare directly related to the SOECs, TOECs and FOECs. The Partial contractions are mere combination of FOECs.TheexpressionsfortheFOPDsandSOPDsofSOECsandtheFOPDsofTOECs,partialcontractionsformonovalent FCC solids are given in Tables 5 and 6.0 0 02 4 4 2 1 2 1 1 10 0 0 0 0 06 5 4 4 4 1 3 2 1 4 6 6 1 2 1 1 4 3 1 1 10 0 0 06 5 4 1 5 5 2 1 4 4 1 1 3 2 1 1 6 5 1 1 1 1C1.56933 2 ,C C 0.347775C 10.2639 -2 , C C 1.208625 ,CC C 0.678375C -80.714552 ,CC CC= + + = = + = = = = = = = + + = = =0 044550 0 0 0 06 4 4 4 4 6 6 2 1 2 2 1 1 6 5 5 1 1 2 1 1 12 2 40 1 0 0 0 2 0 0 03 C -1.584975 CC 4.43205 ,CCC5.615925Where; = z e /r ,= (1/r+ 1/) Q (r )/ r , = ( 2/2r+ 1/) Q (r 2)/r= (3= = = = + = = = + 2 2 2 20 0 0 4 0 0 03 2 2 35 0 0 0 0 03 2 2 36 0 0 0 0 0/r+ 3/ r+ 1/ ) Q (r )/ , = (32/r+ 6/ r+ 2 2/) Q (r 2)/4= (15/r+ 15/ r + 6/r+ 1/ )rQ(r )/ = (152 /4r +15/2 r+ 3 2 / r+ 1/) r Q(r 2)/2 Table 1. Expression for the SOECs, TOECs and FOECs at 0K forfcc Crystalsvib4 c Z 0' '' '''U [1/V 4!] ' ' ' '[Z' Z'' Z''' Z(D' D'' D''' D)U ] == 220Journal of Acoustical Society of IndiaKailash, S.K. Shrivastava, Jitendra Kumar and Virendra Kumarb i v 2 b i v 2 b i v3 4 4 3 1 2 2 1 2 1 1 13 b i v 3 b i v111 3 1 2 2 1 5 112 1 1 2 1 3 2 1 6vib 3123 3 1 2 C, C, C = C = ++, C = + (2+ ) +C = + 31 1 1 11 + = + = vib1 3 144 2 1 3b i v b i v166 2 1 3 1 6 456b i v 2 2 4 b i v1111 4 1 3 1 2 2 2 2 1 5 4 1456vib 4 21112 4 1 3 1, C =C = + , C= 0C= + 6+ 3 + 4+, C = 0C=+ 31 3 2 2 3 2 2 6 3 72 2 2 4 b i v1122 4 1 3 1 3 2 2 3 2 2 1 2 7vib 4 21123 4 1 3 1 3 2 2 3 2 2 1 6vi1144 (+) + 3+ (3+)+ C=+ 2(2+)+ (2+)+ 4 + C=+ (5+)+ (2+)+ 2C1 111 2 b i v 2 b3 1 3 2 3 2 4444 2 3 2 72 b i v 2 b i v1155 3 1 3 2 3 2 2 1 6 7 4455 3 3vib 2 2 vib1255 3 1 3 2 3 2 1 6 1266 +, C = 3+ C= ++ 2+,C = C= ++, C = 1= 2 23 1 3 2 3 2 1 6 7++ 2+ 1 Table 2. Expressions for Vibrational Contribution to the SOECs, TOECs and FOECs for fcc CrystalsTable 3. Expression for n's for fcc CrystalsTable 4. Expression for n'S for fcc Crystals0 0 0 0 0 0 02 20 0 0 0 0 0 0 12 32 0 0 0 0 0 331/ [( -2) (Q (r ) 2( - 2) Q (r 2)) , r / ;2[(2 2 - )Q(r )2 ( 2 2 -2)Q(r 2)];2( - 6 - 6 -) Q (r )2 ;=(- = + = = + + + = + + 2 30 0 0 002 3 4 57 0 0 0 0 0 0 0 42 3 45 0 0 0 0 0 0 62 30 0 0 63 2 - 6 - 2 2) Q (r 2);=2(- 210 - 210- 75- 5+ 4 +) Q (r ) + 2;=2(- 30 - 30- 9+ -) Q (r )+ 2 ;=[(15/ 2 ) + 15- (9/2)-- + 40 0 02 3 4 50 0 0 0 0 0 0 7 2] Q (r 2) ;=[-(105/ 22) - (105/2)- (75/ 22) - (5/2) + 22 +]Q (r 2) 21 0 2 0 1 3 0 1 13 3 2 24 0 2 1 1 1 13 20 0 0 0 0; [(X/ )]/2; [(2X /3 ) (X/ )]/48;- [(X /2 )(X /6 )(X / ) (5X/4 ) (5 /4)]/144;/8r ;X / 2KT;(1/ = = + = + + = + + + + = = = 1 2 0 021m1/m )/ r ; Coth X; Sinh X+ = =High Temperature Study of Anharmonic Properties for Halides of Na and Rb221Journal of Acoustical Society of India2.2 Pressure Derivatives of Elastic ConstantsUsing the second, third and fourth order elastic constants, the FOPDs of SOECs and TOECs can be calculatedusing following expressions:Table 5. Expression for the FOPDs of the SOECs and TOEC for fcc CrystalsAlso the second order pressure derivatives of SOECs and Partial Contractions can be calculated usingfollowing expressions.Table 6. Expression for the SOPDs of the SOECs and for Partial Contraction of the FOECs for fcc Crystals3. EVALUATIONThe detailed study of formulation is given in preceding Section 2. The expressions for various anharmonicproperties of solids possessing FCC crystal structure are shown in Tables 1- 6. Starting from basic parameterssuch as nearest-neighbour distance and hardness parameter, the SOECs, TOECs and FOECs for NaX and RbX(where X= F, Cl, Br and I) are evaluated up to their nearest melting point. The values of SOECs, TOECs and11 11 12 111112 111 11 12 111 1111 111212 11 12 123 112 44 11 12d dC(2C 2 CCC ) S;C - (-3C-6 C3CC2C ) S;dp dpd dC(- C- CC2C) S;C - (C2 C Cdp dp= + + + = + + += + + = + +44 144 166112 11 12 112 1112 1122 1123 11 12113 11 12 113 1123 456 11 12 4 C2C S;dC - (C 2 C3CCCC)S;S 1/C2Cdpd dC - (-C -2 C3C3C)S;C-(-C - 2 C3Cdp dp+ += + + + + + = += + + = +56 1456144 11 12 144 1144 1244 166 11 12 166 1166 1244 3C)S;d dC - (C 2 C3CC 2C)S; C-(-C -2 C3CC2C)S;dp dp+= + + + + = + + +[[2211 P 11 P 111 112 1111 1112 1122 112322212 P 12 P 112 123 1122 11232244 P 44 P 144 166 1144 1166 1244 12dC (1 3C )C (4 3C )(C 2C ) C 4C 2C 2C ] ;SdpdC[(13C )C(4 3C )(2CC ) 2C 5C ] ;SdpdC (1 3C )C (4 3C )(C 2C ) C 2C 4C 2Cdp= + + + + + + + += + + + + + += + + + + + + + +2266P 11 111 112 12311 11111112 1122 112312 1112 1122 112344 1144 1122 1144 1266] ; S C (4CC6C2C ) SYC4C 2C 2C;Y2C 2C 5C ;Y C2C 4C 2C;= + + += + + += + += + + +222Journal of Acoustical Society of IndiaKailash, S.K. Shrivastava, Jitendra Kumar and Virendra KumarFOECs at 300K for these crystals are given in Tables 7 - 9. The FOPDs of SOECs and TOECs, the SOPDs ofSOECs and partial contractions for these materials are evaluated utilizing room temperature data ofTables 7 - 9 and the results are shown in Tables 10 and 11. The experimental and theoretical data are also givenfor comparison.4. RESULTS AND DISCUSSIONThe SOECs and TOECs in 1011dyne/ cm2 at 300K for NaX and RbX are presented in Tables 7 and 8 along withtheoretical and experimental values [31, 32]. The experimental values of SOECs and TOECs for these crystalsare of the same order and are in well agreement with present results. The values of FOECs in 1011dyne/ cm2 atroom temperature are also shown in Table 9. The FOPDs of the SOECs and TOECs are presented in Table 10along with theoretical data [31, 32] which are in well agreement at a great extent.The SOPDs of the SOECs in10-10 cm2/ dyne and partial contractions in 1013 dyne/ cm2 are shown in Table 11 along with theoretical values [33].The temperature variation of various anharmonic properties such as SOECs, TOECs and FOECs, FOPDsof SOECs and TOECs, the SOPDs of SOECs and partial contractions are presented graphically for NaX andRbX (Figs. 1 - 22). There are three second order elastic constants for monovalent crystals. One may state that allSOECs are positive in nature. It is clear from Figs. 1 and 2; the values of C11 and C44 are increasing and the valueof C12 is decreasing as temperature increases for NaF and RbF. There are six third order elastic constants forHalides. For all Halides; the values of C111, C112 and C166 are negative in nature while C123, C144 and C456 arepositive in nature. The values of C111, C123, C144 and C166 are increasing; the value of C112 is decreasing astemperature increases and C456 remains constant. Figs. 3 and 4 show that the value of C111 and C123 is increasingbut C112 is decreasing as temperature increases for NaF and RbF. The same results are found for other halides.It is seen that C144 and C166 are increasing but C456 remains constant for all halides and presented graphically forNaF and RbF in Figs 5 and 6. There are eleven fourth order elastic constants for monovalent crystals. For allhalides, the values of C1111, C1112, C1122, C1144, C1155, C1255, C1266, C4444 and C4455 is decreasing as temperature increasesand the value of C1123 is increasing as temperature increases. The value of C1456 remains constant. In Figs. 7 and8; one can see that the values of C1111, C1112 and C1122 are decreasing as temperature increases for NaCl and RbCl.The same is true for remaining halides. It is also seen that the value of C1123 is increasing and the value of C1144and C1155 is decreasing as temperature increases and the value of C1255, C1266, C4444 and C4455 are decreasing astemperature increases for all halides and presented graphically for NaCl and RbCl in Figs. 9 - 12. There arethree FOPDs of SOECs for FCC crystals. For all halide single crystals; the values of dC11/ dp and dC12/ dp aredecreasing and the value of dC44/ dp first increases and after some temperature it decreases as temperatureincreases. It is remarkable that the FOPDs are dimensionless quantities. From Figs. 13 and 14; one can see thatthe temperature variation of FOPDs of SOECs for NaBr and RbBr, the values of dC11/ dp and dC12/ dp aredecreasing and the value of dC44/ dp first increases and after some temperature it decreases. The same resultsare obtained for other halides. There are six FOPDs of TOECs for monovalent FCC crystals. The values of dC111/dp, dC112/ dp, dc144/ dp, dC166/ dp and dC456/ dp are increasing and the value of dc123/ dp is decreasing. It is seenthat the values of dC111/ dp, dC112/ dp are increasing and dc123/ dp is decreasing as temperature increases andthe values of dc144/ dp, dC166/ dp and dC456/ dp are increasing in nature for all halides and presented graphicallyfor NaBr and RbBr in Figs. 15 - 18. There are three SOPDs of SOECs for these monovalent crystals. It may bestated that all the SOPDs are positive in nature. Figs. 19 and 20 present that the values of d2C11/ dp2 and d2C44/dp2 are decreasing but d2C12/ dp2 is increasing as temperature increases for NaI and RbI. It is seen for remaininghalides also. There are three partial contractions for monovalent crystals. The values of partial contractions forall halide Crystals are decreasing. Figs. 21 and 22 shows that the values of Y11, Y12 and Y44 are decreasing forNaI and RbI.The cases discussed in present study are overall in good agreement with theoretical and experimentalresults which shows the validity of present theory. The data obtained in present investigation will be helpfulto those workers who are engaged in such studies.High Temperature Study of Anharmonic Properties for Halides of Na and Rb223Journal of Acoustical Society of IndiaFig.1. Temperature Variation of SOECsfor NaFFig. 2. Temperature Variation of SOECsfor RbFFig. 3. Temperature Variation of TOECsfor NaFFig. 4. Temperature Variation of TOECsfor RbFFig. 5. Temperature Variation ofTOECsfor NaFFig. 6. Temperature Variation of TOECsfor RbF224Journal of Acoustical Society of IndiaKailash, S.K. Shrivastava, Jitendra Kumar and Virendra KumarFig. 7. Temperature Variation of FOECsfor NaClFig. 8. Temperature Variation of FOECsfor RbClFig. 9. Temperature Variation of FOECsfor NaClFig.10. Temperature Variation of FOECsfor RbClFig. 11. Temperature Variation of FOECsfor NaClFig. 12. Temperature Variation of FOECsfor RbClHigh Temperature Study of Anharmonic Properties for Halides of Na and Rb225Journal of Acoustical Society of IndiaFig. 13. Temperature Variation of FOPDsfor NaBrFig. 14. Temperature Variation of FOPDsfor RbBrFig. 15. Temperature Variation of FOPDsfor NaBrFig.16. Temperature Variation of FOPDsfor RbBrFig. 17. Temperature Variation of FOPDsfor NaBrFig. 18. Temperature Variation of FOPDsfor RbBr226Journal of Acoustical Society of IndiaKailash, S.K. Shrivastava, Jitendra Kumar and Virendra KumarFig. 19. Temperature Variation of SOPDsfor NaIFig.20. Temperature Variation of SOPDsfor RbIFig. 21. Temperature Variation ofPartial Contractions for NaIFig. 22. Temperature Variation ofPartial Contractions for RbITable 7. The nearest-neighbour distance (r0) in 10-8cm and hardness parameter ( ) in 10-8cm and SOECsin 1011 dyne/cm2 at room temperature for Sodium and Rubidium Halide Crystals.Crystal MP(K) r0 C11C12C44RefNaF 1265 2.3032 0.256 15.1011.803.102.963.403.41Present [31]NaCl 1073 2.7865 0.296 8.076.001.381.271.561.57Present [31]NaBr 1028 2.9567 0.310 6.634.901.071.061.221.23Present [31]NaI 924 3.1967 0.326 5.253.950.760.690.890.89Present [31]RbF 795 2.7914 0.357 5.29 1.54 1.67 PresentRbCl 718 3.2564 0.39 3.494.290.780.640.880.49Present [32]RbBr 693 3.4263 0.36 3.723.860.590.470.680.40Present [32]RbI 647 3.6220 0.41 2.603.210.490.360.560.29Present [32]High Temperature Study of Anharmonic Properties for Halides of Na and Rb227Journal of Acoustical Society of IndiaTable 8. TOECs in 1011dyne/cm2 at room temperature for Sodium and Rubidium Halide CrystalsCrystal C111C112C123C144C166C456RefNaF -163.34-179.66-40.79-27.006.803.226.185.66-13.44 -13.515.555.56Present [31]NaCl -84.47-86.36-21.8-5.713.202.842.932.60-6.09 -6.082.592.59Present [31]NaBr -68.48-77.22-18.01-4.402.530.692.232.09-4.76 -4.982.040.04Present [31]NaI -53.34-61.62-14.63-2.691.830.321.711.53-3.43 -3.451.491.49Present [31]RbF -102.32-83.83-25.70-5.262.991.662.892.56-5.82 -5.742.572.47Present [32]RbCl -67.14-67.87-1.58 -2.211.470.381.561.36-2.98 -2.791.391.32Present [32]RbBr -55.32-50.72-15.76-1.831.190.301.281.13-2.42 -2.381.131.10Present [32]RbI -46.35 -50.72-13.95-1.230.910.0021.030.88-1.91 -1.740.900.85Present [32]Table 9. FOECs in 1011 dyne/cm2 at room temperature for Sodium and Rubidium Halide CrystalsCrystal C1111C1112C1122C1123C1144C1155C1255C1266C1456C4444C4455RefNaF 772.2 -234.3 -68.60 -11.81 -13.5 53.17 -18.09 68.70 -12.9 74.31 -13.12 PresentNaCl 185.2 -169.2 -66.60 -4.76 -6.39 22.73 -8.92 30.61 -6.05 33.84 -6.12 PresentNaBr 83.46 -151.3 -63.07 -3.55 -5.05 17.38 -7.16 23.76 -4.77 26.48 -4.83 PresentNaI -20.25 -136.1 -61.90 -2.35 -3.71 12.07 -5.37 16.92 -3.49 19.07 - 3.53 PresentRbF 106.01 -69.39 -2.680 -6.151 -6.31 25.11 -8.82 32.52 -6.02 35.85 -6.145 PresentRbCl 39.19 -63.88 -15.49 -3.204 -3.44 12.41 -5.04 16.80 -3.25 18.80 -3.318 PresentRbBr 10.45 -58.98 -16.27 -2.399 -2.82 9.85 -4.19 1.35 -2.65 1.53 -2.708 PresentRbI -302.3 -180.8 -1.062 -0.868 -2.26 6.42 -3.27 9.20 -2.12 10.56 2.318 Present228Journal of Acoustical Society of IndiaKailash, S.K. Shrivastava, Jitendra Kumar and Virendra KumarTable 10. The First Order Pressure Derivatives of the Second and Third Order ElasticConstants for Sodium and Rubidium Halide CrystalsCrystal dC11/ -dpdC12/ -dpdC44/ -dpdC111/ -dpdC112/ -dpdC123/ -dpdC144/ -dpdC166/ -dpdC456/ -dpRef.NaF -12.89-9.84-5.471.690.0450.0614.27243.0025.43341.01.912.281.2110.402.41115.02.342.26Present[33]NaCl -14.2310.40-6.151.65-0.02-0.0254.5846.6037.83153.01.591.931.330.262.7052.82.312.20Present[33]NaBr -14.69-10.56-6.401.630.042-0.0471.06-7.6442.60101.01.491.841.390.172.8035.62 .312.18Present[33]NaI -15.62-10.9-6.961.61-0.08-0.09100.30-30.0051.5283.41.841.331.430.082.9230.02.292.14Present[33]RbF -10.729.66-5.181.900.34-0.0321.49 18.49 2.27 1.78 2.93 2.62 Present[34]RbCl -12.323.10-5.881.700.21-0.2354.02 29.89 1.93 1.88 3.21 2.54 Present[34]RbBr -14.922.52-6.851.70-0.01-0.22100.83 48.45 1.40 1.69 3.16 2.37 Present[34]RbI -20.453.66-9.491.38-0.30-0.34249.87 100.38 0.95 1.16 2.98 2.12 Present[34]5. ACKNOWLEDGEMENTThe authors are thankful to the University Grants Commission, New Delhi for financial support.Table 11. The SOPDs of the SOECs in 10-10 (dyne/cm2)-1 and Partial Contractions in 1013 dyne/cm2for Sodium and Rubidium Halide CrystalsCrystal dC11/ -dpdC12/ -dpdC44/ -dpY11Y12Y44Ref.NaF 2.09-2.850.53-4.331.95-1.01-3.25-150.3-6.64-182.51.52-38.32Present [33]NaCl 4.04-1.221.07-3.790.34-0.64-6.34-25.41-4.95-40.320.61-7.98Present [33]NaBr 4.91-3.961.33-3.010.40-0.32-6.55-11.01-4.46-21.390.46-4.00Present [33]NaI 6.46-6.781.86-3.240.50-0.24-6.93-4.45-4.07-13.430.30-2.46Present [33]RbF 3.7 1.7 0.54 -1.26 -1.75 0.17 PresentRbCl 6.4 -11.31.9-3.07.88-1.10-2.5311.20-0.20-2.720.330.16Present [35]RbBr 7.9 -15.62.4-3.50.71-1.20-4.849.10-2.72-0.300.230.18Present [35]RbI 10.2-18.43.5-2.30.53-1.30-12.407.70-5.78-0.410.140.21Present [35]High Temperature Study of Anharmonic Properties for Halides of Na and Rb229Journal of Acoustical Society of India6. REFERENCES[1] D. SINGH, 2009. Mat. Chem. & Phys., 115, p. 65.[2] D. SINGH and D.K. PANDEY, 2009. Pramana., 32, p. 389.[3] A. GRZECHNIK, 2010. et al, J. Phys.: Condens. Matter., 22, p. 095402.[4] A. RUKMANGAD, M. AYNYAS and S.P. SANYAL, 2009. Ind. J. Pure& Appl. Phys., 47, p. 114.[5] B. S. ARYA, AYNYAS and S.P. SANYAL, 2008. Ind. J. Pure& Appl. Phys., 46, p. 722.[6] M.K. RAWAT and SANGEETA, 2008. Ind. J. Pure& Appl. Phys., 46, p. 187.[7] C.H. FARNY and G.T. CLEMATE, 2008. J. Acoust. Soc. Am., 123, p. 1773.[8] P.F. YUAN and Z.J. DING, 2008. Physica. B, 403, p. 1996.[9] Z.H. FANG, 2004. Phys Stat Sol (b), 241, 2886.[10] O.N. AWASTHI and V.K. PUNDHIR, 2007. Ind. J. Pure& Appl. Phys., 45, p. 434.[11]V. KANNAPPAN and N.I. GANDHI, 2007. Ind. J. Pure& Appl. Phys., 45, p. 221.[12] R.K. SINGH, 2006. Ind J Pure& Appl Ultras, 28, 59.[13] E. GUILRBORIT, C. BACON and B. HOSTEN, 2002. J. Acoust. Soc. Am., 122, p. 65.[14] J.J. FINNERAN, C.E. SCHLUNDT, R. DEAR, D.A. CARDER and S.H. RIDGWAY, 2002.J. Acoust. Soc.Am., 112, p. 322.[15] R.R. REDDY, Y.N. AHAMMED, K.R. GOPAL, P.A. AZEEM, B.S. DEVI and T.V. R. RAO, 2003. J. Pure&Appl.Ultras., 25, p. 19.[16] M. ZHAO and Q. JIANG, 2004. Mat. Chem. Phys., 87, p. 1.[17] A. KUMAR, M.R. KHAN and N.K. CHAUDHARI, 2002. Ind. J. Pure& Appl. Ultras., 24, p. 6.[18] S.E. DOSSOand N.E. COLRBSION, 2002. J. Acoust. Soc. Am., 111, p. 2166.[19] R. RAIDUTIS and L. MAPEIKA, 2001. Ultrag, 38, 43.[20] E.S. R. GOPAL and B.J. THANGARAJU, 2000. Ind. J. Pure& Appl. Ultras., 22, p. 29.[21] P.S. NAIDU and K.R. PRASAD, 2002. Ind. J. Pure& Appl. Ultras., 24, p. 18.[22] K. KIM, W.R.L. LAMBRECHT and B. SEGALL, 1996. Phys. Rev. B, 53, p. 16310.[23] W. KRASSER, U. BUCHENON and S. HASSUHL, 1976. Solid StateCommun., 18, p. 287.[24] S. HASSUHL, 1973. Solid StateCommun, 13, 147.[25] S. HASSUHL, J. EEKSTEIN, K. ROCKER and F.W. SOFEN, 1977. Acta. Crys., A33, p. 847.[26] S.K. SATEZA and C.H. WANG, 1979.J. Chem. Phys., 70, p. 4437.[27] R.P. SINGH and R.K. SINGH, 2010. Current Applies Phys.[28] I.G. KULEYEV ET AL., 2010. J. Phys: Condens. Matter., 22, p. 095403.[29] F. BIRCH, 1947. Phys. Rev., 71, p. 809[30] K. BRGGER, 1964. Phys. Rev., A133, p. 1611.[31] KAILASH, K.M. RAJU, S.K. SHRIVASTAVA and K.S. KUSHWAHA, 2007. Physica B, 390, 270.[32] V.K. GARG, D.S. PURI and M.P. VERMA,1977. Phys. Stat. Sol. (b), 82, p. 481.[33] KAILASH, 1996.Acta. Physica. Pol., 89, p. 75.[34] B. GHATE P, 1965. Phys. Rev., 139, p. 1666.[35] Z.P. CHANG and G.R. BARSCH, 1971. J. Phys. Chem. Solids, 32, p. 27.AcousticalSocietyofIndia(Regn.No.65-1971)ExecutiveCouncil(2010-2014)President : DrVRajendran[KSRCT,Tiruchengode;veerajendran@gmail.com;+91-9994130303]VicePresident : NSNaidu[NSTL,Vizag;nsnaidu04@yahoo.com;+91-9490750582]GeneralSecretary : PVSGaneshKumar[NSTL,Vizag;gkpakki@rediffmail.com;+91-9866400894]Jt.Secretary : DrKTrinadh[NSTL,Vizag;hello_trinath@yahoo.co.in;+91-9704719500]Treasurer : ProfAVSharma[AU,Vizag;sarmavakella@yahoo.co.in;+91-9490431726]ChiefEditor : DrMahavirSingh[NPL,NewDelhi;mahavir@nplindia.org;+91-9871693346]CouncilMembers : DrSVRangaNayakulu[VITAE,Hyderabad;nayakulu@rediffmail.com;+91-9866532613]DrIJohnson[SJCollege,Trichy;jnaadarsh@hotmail.com;+91-9442904820]DrRajivKUpadhayay[GovtPGCollege,Rishikesh;rku8@rediffmail.com;+91-9412972890]DrSShekhar[OxfordCollege,Trichy;acousticssekar@yahoo.co.in;+91-9994920030]DrVBhujangaRao[PastPresident;NSTL,Vizag;vepcrew1@rediffmail.com;+91-9866441074]Co-optedMembers : RajshekharUchil[Josts,Bangalore;ruchil@josts.in;+91-9880170895]DrNKNarayanan[CIT,Kozhikode;csirc@rediffmail.com;+91-9446955830]