Baria, J K

Gajjar, P N

Jam, A R

Trivedi, H J

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Phys. 78 (9). 943-945 (2004)A'<.K\10N.4^\%U PThermodynamics of d  a n d / -  shell liquid metals -  a variational approachJ K Baria, H J Trivcdi, P N Gajjar and A R Jam’*D epartm ent of Physics, Sardhi I’atcl U n ive is ity . V allabh V id yanai!a i-3$8  120, IndiaR *m ail . |ay„baria@ yah oo  eomR e ce ived  2S  Mciv 2003, a a e p ie d  29 M arch 20(N\bstracl . Keccntly proposed potential is used to calculate internal energy (enthalpy), entropy and H c lm h o lt/ free eneigy of d  and / - shell liquid  „ciaK with the variational approach The param eter o f the potential is determined w ith the standard zero pressuie condition along w ith w t II established l.uitM screening function fo r exchange and correlation effect The structure factor derived by Percus-Yevick solution for haid sphere nuids. which is jur.uten/ed by hard sphere d iam eter, is used. A goixl agreement between theoretical investigations and experim ental findings has confirmed the ability  >! the model potential to the liqu id  d  and J  -  shell metalskt'V'^nrds ■ P se u d o p o te n tia l, th e rm o d y n a m ic s , en tropyPU S Nos. . 65.40 (Jr, 71.15 Dx1. IntroductionIn the Iasi two decades., the considerable etforts have been made to the understanding o f  structure and thermodynamic piof^rlies o f several liquid noble, transition and rarc-carth metals ! 1 -131 A large number o f experimenial measurements on various properties of these metals arc now available in literature 114-15|. The interatomic potentials o f simple metals 116-18| have been fully investigated and their thermodynamic properties could be derived with sufficient accuracy; but in the ca.se o f transition metals, the hybridization o f  r/electron with .y electron makes the tilings complex. Despite the success o f the theory in the solid state, results for the structure factor o f liquid 3^ transitic^n rnctals using molecular dynamics and other complicated liquid .state theories, have not been that reliable. |3 , 19].Wills and Harrison [20] have derived the interatomic P^ itentials for transition metals and obtained thermodynamic properties reasonably good. The potentials o f Wills and Harrison also been used for the determination o f  the surface properties o f noble metals |21] and thermodynamic properties transition metals in the liquid state [3 ,11]. The variational ^^ t'hnique with a hard sphere fluid as a reference system have used by Bretonnet and Derouiche [11] for the calculation of Helmholtz free energy c^ f a series o f liquid transition metals.orresponding AuthorThe potentials of Wills and Harrison |20] and Bretonnet and Derouichc (M l arc a lso  su ccessfu lly  exploited  tor the computation o f several equilibrium proterties. A number of theoretical calculations based on the atomistic model and the use ot interatomic potentials have been used for the study of liquid metals [22-241. Computer experiments are always intended to propose a plausible interpretation of experimental results in some cases to give the solution to an experimentally inaccessible problem. The reliability of the predicted values, however, entirely depends on the validity o f a given interatomic potential and the model used.Keccntly, wc have proposed [25, 2 6 j a model potential depending on an effective core radius but otherwise parameter free. We have reported large number o f physical properties ot transition metals successfully. In the present paper, we have calculated structure factor, radial distribution function, longitudinal and transverse phonon frequencies, elastic  constants, bulk modulus, longitudinal and transverse velocities, Debye temperature, autocorrelation function, characteristic frequencies. Viscosity coefficient and surface tension of liquid Cu, Ag, Au, Ni. Pd, Pt, Rh and Ir at and above melting point temperatures.2. The temperature dependence of thermodynamic propertiesFor the investigation o f  thermodynamic properties oi liquid metals, the Helmholtz free energy F  lies at the heart o f the020041A C S944 J K Baria, H J Trivedi, P N Gajjar and A R Janip scu d o p o te n tia l p ertu rb ation  sc h e m e . T he standard  thermodynamic relation for the free energy is [27,28],F ^ E - - T S . (1)where E  is the internal energy and S  is the entropy o f the system  at a temperature T.Under the usual perturbation theory, the internal energy E  can be expres.sed as |2 7 ,28]E ^ E  - ^ E ,ion elc< clcf -ton (2)H ere, the first term is com p osed  o f  k inetic energy  {(3 /2) o f  ions plus the contribution due to ion-ion  interactions, usually known as Madelung contribution. Thus, we write [27,28](3)where Z is the valency and a ( q )  is the structure factor. For the structure-dependent contribution, the structure factor a {q )  for liquid metals is calculated from the Pcrcus-Ycvick solution for HS fluids which is characterized by the ILS diameter ( a )  or, equivalently, by the packing fraction r] = ( n a ^ / b Q yThe term in eq. (2) is the energy o f the homogeneous electron gas, which is the sum o f  kinetic energy o f  electrons, exchange energy, correlation energy, and the low temperature sp ecific  heat contribution for the electron gas. H ence, the expression for becom es [27 ,28]- 0 .0 4 7 4 - 0 .0 1 5 5  In k .is the Fermi w ave vector and N  is the total nuinhei atoms.In the framework of pseudopotential, second otdckd perturbation theory Eele, has been obtained by 127. ?S|1,,, , , . ^ 7 t Z \ (  Z-el..-,on -  I ' H  «'„(<■/)+ —q Q16;rj j { W „ U l ) } - o ( q )  ' - l \ q * c h i ,  ^  \ q )where the first term on right hand side represents the lust mdci energy and the second is the band structure energy. Heie, H is bare ion pscudopotential |25,26| and C ( q )  is the modific.i Hartree dielectric function.In this expression, 6 , ,  { q )  is the Hartree dielectric lunuitMi and G ( q )  is the correction factor for the exchange and coi u l.ui.l motion o f  the conduction electrons. In the present stikh wc consider the local field conection  G { q )  due to Tayloi |29|7'he second most essential part to investigate the free em i:. is the entropy S  o f  the HS fluids. Using the information of Us diameter ( cr) or packing fraction q  , one can evaluate the eniiv.!t. 5 as [27,28]let »£2\with V . I n5^ = ^ •8 r J (3 r /-4 )( l-r ;^ ^M V -4;r(SiTable 1. Enthalpy, entropy and Helmholtz free energy of liquid d  and /-.shell metals ut melting temperatureVarious contributions lo the enthalpy X 10"^  (au)Total enthalpy (H) X 10“’ (au)Various contributions to the entropy TotalentropyS/k,Helmholl/ tin' energy (F) x 10 (au)Metal H., H dec BOi -5 9 .2 2 3 -3 7 9  169 60 458 -3 7 7 .9 3 3 14.256 -4 .1 3 3 0 .085 10 209 -14528 202Ag -7 4 .5 9 0 -3 2 7 .0 9 2 72.891 -3 2 8 .7 9 2 5.386 -5  000 0 .106 10.493 -13558  15''Au -7 4 .4 5 3 -3 2 8 .2 8 4 81 392 -321  345 16.440 -4  133 0 1 18 12.425 -17081 000Ni -4 9 .7 5 9 -8 8 7 .2 7 5 125.128 -8 1 1 .9 0 5 14.398 -3 .2 6 2 0 .077 11.213 -19882  052Pd -7 7 .2 8 5 -8 4 1 .0 6 7 166.101 -7 5 2 .2 5 0 15.660 -4 .3 3 3 0 .099 11.425 -21 172 084Pt -7 9 .4 4 2 -8 3 4 .7 6 9 169.309 -7 4 4 .9 0 2 16.742 -4 .3 3 3 0.111 12.520 -25704 522Rh -7 2 .4 1 2 -8 3 5 .6 6 9 157.270 -750 .811 15.817 -3 .9 4 2 0 .114 11.989 -26810  240Ir -7 6 .1 5 5 -7 7 6  987 161.044 -6 9 2 .0 9 8 17.079 -3 .9 4 2 0.141 13.270 -36120  375La -1 4 7 .6 3 7 -1 081  456 433.211 -7 9 5 .8 8 2 16.389 -3 .5 8 6 0 .102 12.904 -16041 350Yb -141  314 -1 1 4 5 .8 2 1 434 .9 7 2 -8 5 2 .1 6 6 16,443 -3 .5 8 6 0 .084 12.941 -14534.442Cc -1 4 3 .4 6 7 -1 0 8 4 .2 9 7 4 1 5 .0 3 0 -8 1 2 .7 3 4 16.194 -2 .9 6 6 0 .088 13.316 -15221 0I4Th -1 4 3 .7 9 2 -1 1 1 3 .2 8 5 391 .447 -8 6 5 .6 3 0 17.766 -3 .5 8 6 0 .153 14.333 -29111 02<^Thermodynamics o f d a n d f -  shell liquid metals -  a variational approach 945andn ^ k o T(9)\Vc have calculated the various contributions to the internal -^nergy, entropy and Helm holtz free energy o f  some d  and/- shel 1 metals at melting temperature. In expressions (3) and (3), the integration has been carried out up to 4 0 k to avoid any artificial cuK'ff in the calculation and to achieve proper convergence. The  input parameters used in present calculations are shown in Table I •The presently calculated values o f  internal energy, entropy ,,nd Helmholtz free energy are comparedwith experimental | .301 js well as others theoretical data 19, 10| in Table 2.lahk' 2. Entropy of liquid d  and /-shell metals at mcltinj; temperatureEntropyMcliilExpt 130] O th e rs  [3 0 ] Oiheis 111('llAuNiVdPiRhIrLaYb(VTh10 28 10.06 1 2  no1 I 69 12 11 12 931 3 0014.49.87 10.84 12 04 1 0 02 1 1.50 12 4612.8714 6712 76ReferencesThus, it is confirmed from Tables 1 and 2 that the presently calculated values o f  internal energy and entropy are in good agreement with the experimental data and are comparable to the »»ther reported data in the literature [9, 10, 30]. This confirms the ^ipplicability o f  our m odel potential in the investigation ot thermodynamic properties o f  liquid d  and /-sh ell metals.A ckno wledgmente^he work is supported under the special assistance program at*he level o f DRS by the U niversity Grants Com m ission (UGC), New Delhi. One o f  the authors (J K B) is thankful to the UGC for providing Teacher Fellow ship.(6[7IH19(10(M[1^I 13114115 (16117118(19120(21122(23(24(2 5(26127(28[29[30r  HausIcMtnei ami J Hafnct ./ /Vjv\ F18 1025 ( 0)8X1R N Jourdci ami R V (Jopal Ran /Vns Suu Sot ih) 109 137 (1982)C' Regnaiill Z  /Viv\ B76 179 (l«)89)S Natio, T Aral, 1 Yokoyama and Y Waseda Ivmp Maicr Plot ((!B) 8 125 (1989)() Akmladc, A M Urnar and 1. A Hussain Pnunanu J Phw 47 ^71 (1990)I Yokoyama ami T Aiai Matter lran\ JIM 37 527 (1990)H vSasajima. T Aiai and I Yokoyama Maft i hmis JIM 39 551 (1998)I Yokoyama. 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Thermodynamics of d and f - Shell Liquid Metals

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Thermodynamics of d and f - Shell Liquid Metals

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