Platinum and Aluminium are widely used as a pressure calibration standard. The present proposal which intends to compare the efficiency of the four equations under close examination reports the V/VO versus P values derived from the new modified forms of Murnaghan equation NMME1, NMME2, Birch equation (BE) and Freund-Ingalls (FIE) obtained for the best agreement with the experimental data of Mc Queen et.al

Saud Khan, Mahendra Mehra, Rajendra Kumar and Salil Dubey, Tanveer Ahmad Wani*, Pankaja Singh, A.

English

Update Publishing (E-Journals)

Journal of Experimental Sciences 2011, 2(7): 11-14 ISSN: 2218-1768 www.scholarjournals.org  www.jexpsciences.com 11  JES-Physical Sciences Pressure-volume Relationship for Platinum and Aluminium Tanveer Ahmad Wani*, Pankaja Singh, A.Saud Khan, Mahendra Mehra, Rajendra Kumar and Salil Dubey Department of Physics, MVM Bhopal M.P.-462008, India Article Info  Abstract Article History  Platinum and Aluminium are widely used as a pressure calibration standard. The present proposal which intends to compare the efficiency of the four equations under close examination reports the V/VO versus P values derived from the new modified forms of Murnaghan equation NMME1, NMME2, Birch equation (BE) and Freund-Ingalls (FIE) obtained for the best agreement with the experimental data of Mc Queen et.al.  Received : 11-03-2011  Revised : 16-05-2011  Accepted : 17-05-2011  *Corresponding Author  Tel : +91-9981670354  Fax :  +91-7552764273   Email:  wani_mvm@yahoo.com  ©ScholarJournals, SSR  Key Words: Equation of state, NMME1, NMME2, BE, FIE  PACS numbers: 64.30.-t; 64.10.+h; 62.20.-x   Introduction The materials included in this study are Platinum and Aluminium and have been investigated up to extremely high pressures via the shock wave technique [1] and retain their phase over the respective pressure range investigated. Platinum is among the densest elements (ρ~20gm/cc). The metal does not exhibit any anomalies in the shock velocity-particle velocity relation up to pressures as high as 300GPa. The high pressure change their electronic states from normal metals below about 200 GPa to hot dense plasma state above 5 TPa with intermediate states of condensed liquid and cold plasma. A systematic pressure volume relationship through these states can be extremely useful for the measurement of pressure through the entire range. We have investigated these materials mainly for the purpose.  Theory and Calculations Some physical properties of Platinum and Aluminium are listed below. The properties included are (i) the structure with lattice constants a and c (ii) the zero pressure density ρ0 and (iii) the max. pressure Pmax reached in the particular study and the original reference.  Material a,c(Ao) ρ0      Pmax Pt F.C.C 21.42 58.6 [2]  a=3.92  270 [1]    430 [3]    660 [4] Al F.C.C 2.78 50.5[2]    100 [5]  a=4.05  120 [5]    150 [6]  The values of Bo, Bo` and Bo`` as determined from the polynomial fit with the p-v data are listed below, Pt: BO (GPa) = 277.1; BO`= 4.438; BO`` (GPa-1) = 0.0352 Al:  BO (GPa) = 78.34; BO`= 3.147; BO`` (GPa-1) = 0.109 The values have been numerically determined from equations as,      (    )                                                           (1)        ∑         ∑                                                               (2)  In the limit P=0, we get                                                               (3) Successive differentiation of eqn. 1 with respect to P and then on taking the limit as P=0 yield                                                             (4)                                                          (5) and so on. The suffix o indicates value at zero pressure. It is observed that contrary to expectations Bo`` is positive in this case. However, numerical values of Bo`` evaluated at high pressures show that this derivative takes negative values for the entire pressure range starting from P=20 or 30 GPa that is Tanveer Ahmad Wani et al./J Exp Sci 2 (2011) 11-14    12 nearly the zero pressure. The positive values may therefore be due to small undulations caused by the numerical programme. The shock wave data published by McQueen et al [1] covers a pressure range for the normal metal phase of these materials. The p-v data tabulated by these authors were used to obtain the best fit parameters for the four equations under examination namely the NMME1 [7], NMME2 [7], FIE [8] and BE [9]. The two equations which have been designated as NMME1 and NMME2 indicate that they are new modified forms of Murnaghan equation. The Freund-Ingalls three parameter equation FIE [8] and the Birch equation BE [9] have also been used for comparing the derived equations [7] with these two equations which are known to be giving very good results in similar studies. The equations used for fitting the p-v data are explicitly given here for ready reference.   (    )                                                                                                                                                                                                                                   (6) (Birch equation)                                                              (7) (Freund and Ingalls equation)                                                                         }]                               (8) (New modified form of Murnaghan equation NMME2)  This equation easily goes into NMME1 for b=1  Results The present study which intends to compare the efficiency of the four equations under close examination reports the V/VO versus P values derived from the NMME1, NMME2, BE and FIE obtained for the best agreement with the experimental data of Mc Queen et.al. The shock wave data published by McQueen et.al covers a pressure range for the normal metal phase of these materials. These values are reported in (Tables 1&2). Table 1. Pressure versus volume of Platinum P (GPa) Expt. NMME1 NMME2 FIE  0 1 - - - 10 0.967 - - - 20 0.9395 - - - 30 0.9162 - - - 40 0.8955 - - - 50 0.8775 - - - 60 0.8609 - - - 70 0.8463 - - - 80 0.8328 - - - 90 0.8204 - - - 120 0.7878 0.7878 0.7878 0.7878 150 0.7612 0.7614 0.7616 0.7612 180 0.7389 0.739 0.7393 0.739 210 0.7193 0.7196 0.7199 0.7198 240 0.7023 0.7025 0.7028 0.7031 270 0.6874 0.6871 0.6875 0.6883   Table 2. Pressure versus volume of Aluminium                 The graphs obtained by plotting these values against P together are shown in (Figures 1 & 2).    P(GPa) Expt. NMME1 NMME2 BE  0 1 1 1    1 10 0.8997 0.9001 0.9001 0.9013 20 0.8348 0.835 0.835 0.8358 30 0.7875 0.7875 0.7875 0.7877 40 0.7514 0.7505 0.7504 0.7501 50 0.7202 0.7203 0.7203 0.7196 60 0.695 0.6949 0.6949 0.6941 70 0.6731 0.6731 0.6731 0.6722 80 0.6541 0.6539 0.6539 0.6532 90 0.6362 0.6368 0.6368 0.6364 100 0.6205 0.6214 0.6214 0.6214 110 0.607 0.6073 0.6073 0.608 120 0.5953 0.5943 0.5943 0.5658 Tanveer Ahmad Wani et al./J Exp Sci 2 (2011) 11-14    13                Figure 1. Pressure versus relative volume of Pt.                 Figure 2. Pressure versus relative volume of Al.  The best fit values for BO, BO`, BO`` and BO``` are listed in (Tables-3&4). Table 3. The zero pressure bulk modulus and its pressure derivatives for Pt BO BO` BO`` BO``` Source 279.8 4.7 - - BE 279.53 4.57 -7.018*10-4 - NMME1 278.76 4.6 -7.34*10-4 2.5*10-5 NMME2 282.043 4.46 -7.018*10-4 - FIE 283.8 - - - Ref.[10] 282.7 - - - Ref.[11]  Table 4. The zero pressure bulk modulus and its pressure derivatives for Al  BO BO` BO`` BO``` Source 78.2 4 - - BE 77.3 3.804 -1.759*10-3 - NMME1 77.273 3.805 -1.759*10-3 -1.735*10-4 NMME2 79.4005 3.596 -1.759*10-4 - FIE 75.7 - - - Ref.[5] 76.9 - - - Ref.[12] 73.58 - - - Ref.[9] - 4.85 - - Ref.[13]   Conclusion The numerical results clearly show that the p-v data, of the materials included in this study, upto moderately high pressures of the order of a few hundred GPa are explained very well with several equations of state. Equation NMME1 further shows that the modification in the simple two parameter Murnaghan equation is rather small. To have an exact estimate of the change that is produced by the NMME1 in the ME one has only to evaluate the term              ⁄  which adds to one and then multiplies the ME expression                      to yield the volume ratio V/V0. The values of λu at the highest pressures investigated have been calculated and listed below for the solids under examination. Pt= -0.0054 Al= -0.0122 It is obvious from the values that the NMME modification is merely 2 to 4 percent. In the Birch Scheme of equations it appears that even the parameter Bo` is quite often not contributing to the p-v relationship as the term involving Bo` occurs as multiplying factor (Bo`-4)and the Bo` values for many solids are close to four. The NMME2 is the only equation which yields a value of the third order pressure derivative of the bulk modulus. This value turns out to be extremely small as could be anticipated but the parameter b to which Bo``` is related invariably obtains a significant value suggesting that Bo`` may not be neglected in an accurate description of the equation of state. It is therefore hoped that the values of Bo``` obtained in the present study may prove useful in studies in suitable functional forms.  Acknowledgements The authors acknowledge thanks due to all departmental teachers and researchers. Tanveer Ahmad Wani et al./J Exp Sci 2 (2011) 11-14    14 References [1] R.G. McQueen, S.P. Marsh, J.W. Taylor, J.N. Fritz and W.J. Carter, High velocity impact phenomena, ed. R. Kirslow, Vol. 1 (Academic Press, Orland FL. 1970) p. 165. [2] J.M. Walsh, M.H. Rich, R.G. McQueen and F.L. Yarger, Phys. Rev., 108, (1957), 196. [3] A.C. Mitchell and W.J. Nellis, J. Appl. Phys., 52, (1981), 3363. [4] N.C. Holmas, J.A. Moriarty, G.R. Gathens and W.J. Nellis, J. Appl. Phys., 66, (1989), 7. [5] R.J. Naumann, J. Appl. Phys., 42, (1971), 4945. [6] J.A. Moriarty, D.A. Young and M. Ross, Phys. Rev. B, 30, (1984), 578. [7] P. Singh and M.P. Verma, Proc. Nat. Acad. Sci. India, 64, (1994), 1. [8] J. Freund and R. Ingalls, J. Phys. Chem. Solids, 50, (1989), 263. [9] F.J. Birch, J. Geo. Phys. Res., 57, (1952), 227. [10] F. Seitz and D. Turnbull, Solid State Phys., 16, (1964), 308. [11] N. Singh, S.N. Banger and S.P. Singh, Phys. Rev. B, 28, (1988), 7415. [12] T.R. Long and Charles S. Smith, E. Goens. Ann.  Physik, 38, (1953), 456. [13] N. Trappeniers, P. Van`t Klooster and S. Biswas, High Pressure Science and Technology, eds. K. Timmerhaus and M. Barker, Vol. 1 (Plenum, New York. 1979) p. 108.   

Pressure-volume Relationship for Platinum and Aluminium

https://updatepublishing.com/journal/index.php/jes/article/download/1857/1839

Pressure-volume Relationship for Platinum and Aluminium

Abstract

Similar works

Full text

Available Versions

Update Publishing (E-Journals)