Extended average-atom model with semiclassical electrons allowing for ion correlations

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1 Extended average-atom model with semiclassical electrons allowing for ion correlations A. L. Falkov 1,2, A. A. Ovechkin 1, P. A. Loboda 1,2 1. Russian Federal Nuclear Center Zababakhin All-Russian research institute of technical physics (P.O.Box 245, Snezhinsk, Chelyabinsk region, Russia 45677) 2. National Research Nuclear University "MEPhI" (Kashirskoe sh.,31, Moscow, Russia 11549) Moscow, NPP-215

2 Table of contents Ion correlation treatment in various models of dense plasmas Self-consistent dense plasma model by C. E. Starrett and D. Saumon Calculation of ion-ion radial distribution functions (RDFs) Comparision with the results of QMD (DFT-MD) modeling Comparision with the results of TFMD (OFMD) modeling Comparision with the experimental RDF data for melted metals under the athmospheric pressure (H. F. Y. Waseda) Deviations in average ionization due to ionic nonideality X-ray Thomson scattering experiments with Al in WDM state Elastic X-ray scattering. Some notes for WDM state ¾Omega-6 experiment with compressed Al (T = 1 ev; ρ = 3ρ ) LCLS-MEC experiment with compressed Al (T = 1.75 ev; ρ = 2.32ρ ) Results of wide-range EOS data calculations Principal Hugoniot (σ, P ) for Al Conclusions Main results of research work Possible advances in further work 2

3 Correlative treatment of ions in various models TFD, VAAQP g II (r) = Θ ( r r I ) ; gii from Ornstein- Zernike (OZ) equations; RESEOS, CP-SC Phenomenology charged hard spheres: ion's intrinstic volume eects + OCP of ions; SCAALP Free arguments for g II (r,...); F. Perrot, Y. Rosenfeld et al. TF + V eff II [r, g II, V tot [g II ]] + + Ornstein-Zernike equations (OZ-HNC) for g II (r); V el [g II ] n e [V el ] V II [n e, c Ie, c ee,...] g II (r) QHNC Average atom TCP (e-i) model "jellium"; B. F. Rozsnyai c Ie, c II "pure Coulomb- without LFC + g II from OZ equations; TFSC, QMSC (C. E. Starrett and D. Saumon) c Ie, c II with LFC + g II from OZ set of equations with hypernetted chain (HNC) closure;... 3

4 Self-consistent description for dense {e, I} plasmas General scheme of modicated C. E. Starrett's and D. Saumon's model [1, 2, 3] Input data 1. Temperature 2. Density 3. Nuclear charge 4. Atomic weight I Complete electron subtask Approximations 1. Electron response function 2. Exchange energy 3. Closure for OZ set of equations «Out» iterations II «External» electron subtask «Out» iterations Output data 1. Ion-ion RDF 2. Average ionization 3. Helmholtz free energy III Ionic subtask (OZ + closure) 4

5 r s Ornstein-Zernike framework for {r, s} systems h rr (k) = c rr (k) + n rc rr (k)h rr (k) + n sc rs (k)h rs (k), h ss (k) = c ss (k) + n sc ss (k)h ss (k) + n rc rs (k)h rs (k), h rs (k) = c rs (k) + n sc ss (k)h rs (k) + n rc rs (k)h rr (k). c(r,s) h(r,r), c(s,s) h(s,s), c(r,r) c(r,s) h(r,s) c(s,s) h(r,s), c(r,s) s r c(r,s) h(r,s) h(s,s) h(r,r) c(r,r) h(r,r) c(s,s) s h(r,s) 3 r r 3 r 1 c(1,3) 1 h(3,2) h(1,2), c(1,2) h II (k) = c II (k) + n I c II(k)h II (k) + n e c Ie (k)h Ie k), h Ie (k) = χ ee(k) [ cie n (k) + n Ic Ie (k)h II (k) + n e c ee (k)h Ie (k) ], e β h ee (k) = c ee (k) + n e c ee (k)h ee (k) + n Ic Ie (k)h Ie (k). r 2 2 QTCP, J. Chihara, [4]. 5

6 QTCP reduction to an eective ionic OCP g II (r) g(r), n I = invar { h(k) = c(k) + n I c(k)h(k), 1 + h(r) = exp ( βv (r) + h(r) c(r) + E(r)) ; χ ee(k), βv (k) = 4πβ ( c ee (k), k 2 Z 2 c Ie k, n e ) n SCR e (k); ( n SCR e (k) c Ie k, n e ) = β nscr e (k) ; χ e (k) χ χ e (k) ee(k) 1 + χ ee(k)c ee (k)/β. n SCR e χ ee(k), c ee (k) χ e (k), n SCR e (k) c Ie (k) V (k) h(r), c(r) c ee (k) = 4πβ k 2 (1 G ee(k)) ("jellium approximation"+ LFC); (r) n P e A (r) n ion e, n P e A (r) n e (r) n ext e Z = drn SCR e V χ ee(k) Lindhard function, n ion e (r) from [5], LFC G ee (k) from [6]. 6 (r).

7 Relationships among n e, n ext e, n P e A, n ion e, and n SCR e TFSC for W at ρ = 4 g/cm 3 and T = 1 ev 4πr 2 n e (r) n e ext n e PA n e ion n e SCR n e n e PA = ne - n e ext n e SCR = ne PA - ne ion (r/r ),5 7

8 I. III. II. Self-consistent TFSC model { h(k) = c(k) + n I c(k)h(k), g(r) 1 + h(r) = exp ( βv (r) + h(r) c(r) + E(r)) ; ( )) n e (r) = C T F I 1/2 (β µ id e V eff Ne (r), β = 1/T, V eff Ne (r) = Z r + dr n e (r ) n eg(r ) V r r [ ] +Vee xc [n e (r)] Vee xc n e ; ( ( n ext e (r) = C T F I 1/2 β µ id e Ve eff,ext (r) )), Ve eff,ext (r) = dr V next e (r ) n eg(r ) r r [ ] +Vee xc [n ext e (r)] Vee xc n e ; ( ) V e,c Z n Ie [n I(r)] = I Z β V + V e,c Ie [n I (r)]+ + V e,c Ie [n I (r)]+ dr ( c Ie r r, n e ) (g(r ) 1), n () I n IΘ ( r r I ), TFIS: ni (r) = n (1) I (r), TFCS: n I (r) n Ig(r) 8

9 Ion-ion RDFs for Al: comparision with the QMD ) RDF for Al at ρ = 2,7 g/cm 3 and various T Al 1 ev 2,7 g/cm 3 4) Al 1 ev 2,7 g/cm QMD HNC-Y TFSC TFSC-MS QMD HNC-Y TFSC 2) Al 2 ev 5) Al 15 ev 2,7 g/cm 3 2,7 g/cm 3 g II (r) ) Al 6 ev 2,7 g/cm 3 6) Al 3 ev 2,7 g/cm r/r I r/r I QMD [3, 7, 8]. 9

10 RDF for shock compressed Fe plasmas Program TFSC (OZ-HNC-AA) for Fe ρ(1 ev) = 22,5 g/cm 3 ; T = 1 ev, TFMD, CEA, 26 ρ(1 ev) = 39,65 g/cm 3 T = 1 ev, TFSC, LANL, 214 T = 1 ev, TFSC, VNIITF, 214 T = 1 ev, TFMD, CEA, 26 T = 1 ev, TFSC, LANL, 214 T = 1 ev, TFSC, VNIITF, 214 g II (r) 1 E F (1 ev) = 6,43 ev; E F (1 ev) = 16,61 ev; Γ(1 ev) = 19,8;.5 Γ(1 ev) = 8,1736; <Z>(1 ev) = 8,786; <Z>(1 ev) = 21,62; r (1 ev) = 1,8796 a B ; r (1 ev) = 1,5561 a B N 1 = 8192 r(a B ) r end = 65 / 97,5 r TFMD, CEA [9]; TFSC, LANL [2]. 1

11 Isochoric Γ-plateau eect for W (ρ = 2ρ ) 2 W, ρ = 4 g/cm ev 2 ev 3 ev g II (r) ev 12 ev 4 ev TFSC OFMD HNC-Y MHNC-Y.5 8 ev ev r (a B ) 5 ev OFMD [1]. 11

12 RDFs for melted Mg and Al (ρ ρ ) g II (r) RDF for liquid metalls (Mg and Al) Mg 953K 1,546 g/cm 3 TFSC experiment Mg 163K 1,433 g/cm 3 Mg 1153K 1,3 g/cm 3 Al 943K 2,366 g/cm 3 Al 123K 2,348 g/cm 3 Al 1323K 2,272 g/cm r (angstrom) r (angstrom) Experimental data [11]. 12

13 Isochores of average ionic charge for C plasmas 5.5 C, ρ =,2 g/cm <Z> Experiment TFIS TFSC T (ev) Experimental data [12]. 13

14 Deviations in average ionization due to ionic nonideality ɛ = 2 ( Z Z ) / ( Z + Z ) 14

15 Theoretical base for calculation of S el (k) elastic static sctructure factor of photonic scatternig S tot (k, ω) = S el (k)δ (ω) + S ne (k, ω) + S ee (k, ω) = = f I (k) + q(k) 2 S II (k) δ (ω) + Z S }{{} ee (k, ω) + S }{{} bf (k, ω), }{{} S el (k) free free bound free S II (k) = 1 + n F I s [g(r) 1], q(k) = F [ s n SCR e (r) ], lim q(k) = Z drn SCR e (r), k V f I (k) = F s [ n ion e (r) ], f I (k) q I (k). k = k 1 k, k = 2k sin (Θ s /2), Recording of S el (k) data: X-ray Thomson scattering experiments with pre-compressed laser plasmas in WDM state. 15

16 ¾Omega-6 experiment with Al WDM (213214) Discrepancies δ (4 45) % from [13, 14] data near absolute maximum of S el (k) TFSC / QMIS for Al at ρ = 8,1 g/cm 3 and T = 1 ev k (a B -1 ) 16 (exp) S el (k) (TFSC, HNC) 2S II (k) (QMIS, HNC) 2S II (k) (TFSC, HNC) S el (k) (QMIS, HNC) S el (k)

17 Renunciation of HNC? Alternative OZ closures PY Percus-Yevick [15], MS Martynov-Sarkisov [16] OZ closure: HNC or Percus-Yevik? QMIS for Al at ρ = 8,1 g/cm 3 and T = 1 ev k (a B -1 ) 17 (exp) S el (k) (HNC) 2S II (k) (PY) 2S II (k) (MS) 2S II (k) (HNC) S el (k) (PY) S el (k) (MS) S el (k)

18 TFSC LCLS-MEC experiment (214215) TFSC for Al at ρ = 6,3 g/cm 3 and T = 1,75 ev LCLS-MEC experiment VASP TFSC (HNC) TFIS (MHNC) S el (k) k (Α -1 ) Experimental data, VASP [17]; MHNC closure [18]. 18

19 QMIS LCLS-MEC experiment (214215) QMIS for Al at ρ = 6,3 g/cm 3 and T = 1,75 ev LCLS-MEC experiment VASP QMIS (HNC) QMIS (MHNC) S el (k) k (Α -1 ) Experimantal data, VASP [17]; MHNC closure [18]. 19

20 Principal Hugoniot (σ, P ) for Al 1 6 ρ = 2,712 g/cm ρ stfd = ρ TFSC = 2,818 g/cm 3 P, GPa L. V. Al tshuler et al., 196 L. V. Al tshuler et al., 1977 L. P. Volkov et al., 1981 A. S. Vladimirov et al., 1984 C. E. Ragan, 1982 C. E. Ragan, 1984 V. A. Simonenko et al., 1985 E. N. Avrorin et al., 1986 M. D. Knudson et al., 23 VNIITF, stfd VNIITF, TFSC, σ = ρ / ρ Experimental data from [19]. 2

21 Main results of the work A close agreement of ion-ion RDFs (H, Be, C, Al, Fe, and W) calculated according to the TFSC model with the results of ab initio TFMD (OFMD), QMD (DFT-MD), QLMD modeling, and PIMC calculations. TFSC model application for calculating RDFs in liquid metals (Mg, Al, and Ti) with near-normal density. Agreement with the X-ray scattering experiments. 5 15% deviations (primarily decreasing) in average ionization Z = Z (T, ρ, Z, g(r)) due to ionic nonideality. The results of recent experimental studies of warm (1.75 ev and 1 ev) compressed (ρ = 2.32ρ and ρ = 3ρ ) aluminium are explained. Using some alternative closures (Percus-Yevick, Martynov- Sarcisov, Ietomy-Ogata-Ichimary (MHNC)) for OZ set of equations considerably improves elastic static structure factor S el (k) calculated from TFSC/QMIS models. 21

22 Possible advances in the future work Improvement for the semiclassical TCP plasmas models (TFIS and TFSC): Vee xc (T = ) Vee xc (T ),... Improvement for the methods of Helmholtz free energy calculation in models with ionic correlations: F = ( ) ( FI id + Fe id + F el + F el) + + (F xc + F xc Ie + F xc II + F xc ee ). TFIS/TFSC nodels generalization in case of dense multicomponent mixtures. Thermodynamic properties of dense mixture plasmas. Hybrid PAMD method i.e. TFSC + (classical) pseudo-atom molecular dynamics. Independent calcultaion of ionic transport coecients 22

23 Pseudoatom molecular dynamics (PAMD) For ion-ion RDFs, EOS, and viscosity calculations [2] N = 64 N = 343 N = 512 TFSC g (r) r / r 23

24 1. Starrett C. E. and Saumon D. Phys. Rev. E, 85:2643(1)2643(1), Starrett C. E. and Saumon D. Phys. Rev. E, 87:1314(1)1314(14), Saumon D., Starrett C. E., Anta J. A. et al. arxiv: v1 [physics.plasm-ph], Chihara J. J. Phys.: Condens. Matter, 3: , Ofer D., Nardi E., and Rosenfeld Y. Phys. Rev. A, 38: , Ichimaru S. and Utsumi K. Phys. Rev. B, 24(12): , Starrett C. E. and Saumon D. HEDP, 8:11 14, Surh M. P., Barbee III T. W., and Yang L. H. Phys. Rev. Lett., 86(26): , Recoules V., Lambert F., Decoster A. et al. Phys. Rev. Lett., 12:752(1)752(4), Cl erouin J., Robert G., Arnault Ph. et al. Phys. Rev. E, 87:6111(R)(1)6111(R)(5), Waseda H. F. Y. The structure of non-cristalline materials Gregori G., Cumpbell K. M., Dewald E. L. et al. Phys. Rev. Lett., 11:453, Ma T., D oppner T., Falcone R. W. et al. Phys. Rev. Lett., 11:651(1)651(5), Ma T., Fletcher L., Pak A. et al. Phys. Plasmas, 21: , Percus J. K. and Yevick G. J. Phys. Rev., 1(11):113, Ã. À. Ìàðòûíîâ. Êëàññè åñêàÿ ñòàòèñòè åñêàÿ ìåõàíèêà. Òåîðèÿ æèäêîñòåé, Fletcher L. B., Lee H. J., D oppner T. et al. Nat. Phot.,DOI: 1.138/NPHOTON S 18. Iyetomy H., Ogata Sh., and Ichimary S. Phys. Rev. A, 46(2):151158, À. Â. Áóøìàí, È. Â. Ëîìîíîñîâ, Ê. Â. Õèùåíêî Ä. Ê. Ðàïàïîðò. Èñêóññòâî ìîëåêóëÿðíîé äèíàìèêè,

Structure Factor and Electronic Structure. of Compressed Liquid Rubidium arxiv:cond-mat/ v1 [cond-mat.mtrl-sci] 13 Jan 1998

Structure Factor and Electronic Structure of Compressed Liquid Rubidium arxiv:cond-mat/9801116v1 [cond-mat.mtrl-sci] 13 Jan 1998 Junzo Chihara Advanced Photon Research Center, Japan Atomic Energy Research