The purpose of the present paper is to derive a new empirical  relationship for the volume dependence of Grüneisen ratio (ϒ) by using  simple and straightforward approach. The results thus obtained for  Forsterite (Mg2SiO4) from the two different  methods are identical to each other. Consistency of calculated values  with those values compiled by Cynn H, Carnes J D, Anderson O L, J Phys  Chem Sol, 57 (1996) 1593 reveals the validity of the formulation. It is  also found that the heat capacity does not influence the change in (ϒ)  with the volume ratios in the studied range

Panwar, M 

Panwar, S 

Sharma, S  K

English

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Indian Journal of Pure & Applied Physics Vol. 55, March 2017, pp. 203-206       Analysis of volume dependence of Grüneisen ratio of Forsterite M Panwara,b*, S K Sharmab & S Panwara aDepartment of Physics, MMEC, Maharishi Markandeshwar University, Ambala 133 207, India bDepartment of Physics, Shivalik Institute of Engineering & Technology, Aliyaspur, Ambala 133 206, India Received 29 February 2016; revised 25 December 2016; accepted 29 December 2016 The purpose of the present paper is to derive a new empirical relationship for the volume dependence of Grüneisen ratio (  by using simple and straightforward approach. The results thus obtained for Forsterite (Mg2SiO4) from the two different methods are identical to each other. Consistency of calculated values with those values compiled by Cynn H, Carnes J D, Anderson O L, J Phys Chem Sol, 57 (1996) 1593 reveals the validity of the formulation. It is also found that the heat capacity does not influence the change in (  with the volume ratios in the studied range. Keywords: Grüneisen ratio, Anderson-Grüneisen parameter, Thermoelastic properties, Forsterite 1 Introduction  Grüneisen ratio (  is a very important parameter used to quantify the relationship between thermal and elastic properties of solids. The Grüneisen ratio (  can be considered as a measure of the change of pressure resulting from the increase energy density at constant volume1. Grüneisen ratio (  is useful to investigate the anharmonic property of materials. There is a long standing interest in the behaviour of the Grüneisen ratio (  at high pressure or compression because of its importance in geophysics, thermodynamics and condensed matter physics2. The Grüneisen ratio (  has both a microscopic and macroscopic definitions. Vibrational Grüneisen ratio (3 may be defined as the logarithmic volume derivative of phonon frequency, , i.e.:   lnlniiV    ... (1)  and the thermodynamic Grüneisen ratio (4:  TthVK VC    ... (2)  where  is volume thermal expansivity,  is the isothermal bulk modulus,  is volume and  is the heat capacity at constant volume. So many researchers5-18 have reported the relationships for Grüneisen ratio (  by using different approaches.  In the present study we have extended the work of Kumar et al.14 by using the concept that  changes with increase in compression or pressure. We have tested the validity of present formulation to Mg2SiO4. It is known that Mg2SiO4 is an important material as well as geophysical mineral1. It is one of the few materials for which sufficient data of its properties are available. The wide range of stability in temperature-pressure space and the fact that it is regarded as a major component of the earth layer mantle make Mg2SiO4 attractive for the study. Forsterite-rich olivine (Mg2SiO4) is the most abundant mineral in the Earth’s mantle above depth of about 410 km, where  GPa19. Also, laboratory- synthesized nano-crystalline forsterite has been considered as a possible successor to calcium phosphate bioceramics, due to its exceptionally high fracture toughness20. The geophysical importance of forsterite as well as its possible application in medicine justify, in general, a work on the volume dependence of its Grüneisen ratio ( , since  is an important parameter in thermodynamics, geophysics, and solid state physics.   2 Method of Analysis Stacey and Davis21 have given the following identity:   ' ln1lnVT TTCq KV         ... (3)  where  are respectively the isothermal Anderson-Grüneisen parameter, first order —————— *Corresponding author (E-mail: monikaverma05@gmail.com) INDIAN J PURE & APPL PHYS, VOL 55, MARCH 2017   204 pressure derivative of isothermal bulk modulus and heat capacity at constant volume and second Grüneisen parameter. All these parameters are  defined as:  PTTTTKK1   ... (4)  ' TTTKKT      ... (5)  in which is the isothermal bulk modulus,  defined as:   TTTKK VP        ... (6)  and  lnln TqV       ... (7)  Sharma and Sharma22 have generalised the isothermal Anderson-Grüneisen parameter in the following manner:   00mT T T TVV             ... (8)  where  and are respectively the values of isothermal Anderson-Grüneisen parameter at zero and infinite pressure, is a dimensionless adjustable parameter. Srivastava and Sinha23 have reported the expression:   '0' ' ' '00KTVK K K KV          ... (9)  where  and are the values of first order pressure derivative of isothermal bulk modulus at zero and at infinite pressure. Using Eqs (3, 7–9) we get:   00lnlnmT T TTVV V                 '0' ' '00ln1lnKVTV CK K KV V                    ... (10)  On integration of the above equation, we can get the following equation:   11exp'00'00100KmVVKVVBVVACCVVT   ... (11)  where ,  are respectively the values of specific heat , and Grüneisen parameter at zero pressure and  and are temperature dependent parameter:  0T TAm        ... (12)    '0''0KKKB  ... (13)  3 Results and Discussion At infinite pressure, i.e., P  ∞ or V  0, Eq. (3) becomes:  ' '1T TK q C        ... (14)  Since at infinite pressure, i.e., P  or V  0 ,  tends to zero21 and  tends to zero24, now Eq. (14) takes the following form:  ' 1T K     ... (15)  Following Thomas-Fermi theory21, 25-28, i.e., 3 Eq. (11) results 3/2T. The values of  for both models21,25-28 satisfy the constraint29 '0   KT . We have proposed a simple method to investigate the volume dependence of the Grüneisen ratio (  at high temperatures of Mg2SiO4 down to a range of volume ratio 0.90.  Using Eq. (15) in Eq.(11) we get:  '000 0 0exp 1 1m KVVC V VA BC V V                            ... (16)  where all the parameters are having their as usual meaning.  Recently, Kumar et al.14 reported the following relation for the volume dependence of Grüneisen ratio (  by using the concept that  remains constant1:  '00' '0'0 0 0 0exp 1 1m KT T V K K Vm V K V                                            ... (17)   where all the parameters are having their as usual meaning. PANWAR et al.: VOLUME DEPENDENCE OF GRÜNEISEN RATIO OF FORSTERITE   205 The values of input parameters used in present study are cited in Table 1. The values of  are taken from reference30. We have investigated the values of volume dependence of the Grüneisen ratio (  through Eqs. (16) and (17) for Forsterite. The results obtained through Eqs (16) and (17) are compared with those values calculated by Cynn et al.30 of  in Table 2. It is found that the results obtained through Eqs (16) and (17) are almost identical to each other and are compatible with those values of  compiled by Cynn et al.30. For direct vision we have also plotted the graph for the dependence of Grüneisen ratio (  on  at different values of  in Fig. 1. Figure 1 reflects that as the temperature increases the values of  Grüneisen ratio (  decrease and show good agreement with those values of  compiled by Cynn et al.30 which supports the validity of the present model. It has also been seen that  changes monotonically above  K temperature. It is pertinent that the present paper proposes only a small correction to Eq. (16) from the paper of Kumar et al.14. This correction is reduced to the  multiplier on the right side of Eq. (16) from the present study. It is readily seen from Table 2 of Cynn et al.30 that  practically does not vary with pressure. Therefore, the multiplier  and does not influence the change of  with  in the studied range of compression ratios. Also,  influences the results very slightly because of the very   Table 1 – Values of input parameters for Mg2SiO4 used in calculations  1  1 1  (J/gK)30 31 300 5.940 5.370 1.290 0.8324 2.380 400 5.580 5.400 1.210 0.9760 2.240 500 5.490 5.440 1.180 1.0482 2.210 600 5.480 5.470 1.170 1.0929 2.190 700 5.490 5.500 1.160 1.1244 2.140 800 5.470 5.540 1.150 1.1480 2.100 900 5.460 5.570 1.150 1.1669 1.980 1100 5.460 5.630 1.140 1.1972 1.660 1200 5.490 5.670 1.150 1.2095 1.270 1300 5.440 5.700 1.150 1.2205 1.240 1600 5.400 5.800 1.140 1.2489 1.240    Fig. 1 – Values of Grüneisen ratio ( ) of Mg2SiO4 as a function of temperature at different volume ratios calculated here with Cynn et al.30 INDIAN J PURE & APPL PHYS, VOL 55, MARCH 2017   206  Table 2 – Grüneisen ratio ( ) of Mg2SiO4 as a function of volume ratio and temperature calculated through (a) Eq. (16),  (b) Eq. (17) and (c) Cynn et al.30              (a) (b) (c) (a) (b) (c) (a) (b) (c) (a) (b) (c) (a) (b) (c) (a) (b) (c) 300 1.29 1.29 1.29 1.27 1.25 1.27 1.24 1.20 1.24 1.20 1.16 1.20 1.17 1.11 1.17 1.13 1.06 1.13 400 1.21 1.21 1.21 1.19 1.18 1.20 1.19 1.15 1.20 1.13 1.11 1.17 1.10 1.07 1.15 1.06 1.03 1.13 500 1.18 1.18 1.19 1.16 1.15 1.18 1.16 1.12 1.18 1.10 1.09 1.13 1.07 1.05 1.10 1.03 1.01 1.08 600 1.17 1.17 1.17 1.15 1.14 1.15 1.15 1.11 1.15 1.09 1.08 1.09 1.06 1.05 1.08 1.02 1.01 1.06 700 1.16 1.16 1.15 1.14 1.13 1.13 1.14 1.11 1.13 1.08 1.07 1.09 1.05 1.04 1.06 1.01 1.00 1.04 800 1.15 1.15 1.14 1.13 1.13 1.12 1.13 1.10 1.12 1.07 1.07 1.07 1.04 1.03 1.05 1.00 0.99 1.03 900 1.15 1.15 1.15 1.13 1.13 1.11 1.13 1.10 1.11 1.07 1.07 1.07 1.04 1.03 1.04 1.00 0.99 1.02 1100 1.14 1.14 1.02 1.12 1.12 1.11 1.12 1.09 1.11 1.06 1.06 1.07 1.03 1.02 1.04 0.99 0.98 1.02 1200 1.15 1.15 1.15 1.13 1.13 1.10 1.13 1.10 1.10 1.07 1.06 1.06 1.03 1.02 1.03 0.99 0.98 1.02 1300 1.14 1.14 1.11 1.13 1.13 1.10 1.13 1.10 1.10 1.07 1.07 1.05 1.03 1.03 1.03 0.99 0.99 1.01 1600 1.14 1.14 1.14 1.12 1.12 1.12 1.10 1.10 1.10 1.06 1.07 1.04 1.03 1.03 1.02 0.99 0.99 0.99  narrow range, it varies in. Thus the Eq. (16) is an asymptotic approximation of Eq. (17) in the limit of P  ∞ or V  0.  4 Conclusions  We have proposed a simple and straight forward empirical relationship to estimate the values of volume dependence of Grüneisen ratio (  for Mg2SiO4 down to a range of volume ratio 0.90. It is found that the results obtained through Eq. (16) are in good agreement with those values of  compiled with Cynn et al.30. Compatibly of results obtained in the present study with values of  compiled by Cynn et al.30 shows the validity of the present model. Results thus obtained through Eqs (16) and (17) are identical to each other. Henceforth, the Eq. (16) is an asymptotic approximation of Eq. (17). There is no significant effect of heat capacity on Grüneisen ratio ( . However, this requires further investigations that heat capacity influences the values of volume dependence of Grüneisen ratio ( . It may be studied in future for those materials that have data on  and at high temperatures and high pressures.   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Analysis of volume dependence of Grüneisen ratio of Forsterite

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Analysis of volume dependence of Grüneisen ratio of Forsterite

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