ALUMINUM AT T ≈ TF : THEORY OF THE DYNAMICALLY COMPRESSED METALLIC STATE

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ALUMINUM AT T ≈ TF : THEORY OF THE

DYNAMICALLY COMPRESSED METALLIC STATE

R. Dandrea, N. Ashcroft

To cite this version:

R. Dandrea, N. Ashcroft. ALUMINUM AT T ≈ TF : THEORY OF THE DYNAMICALLY COM- PRESSED METALLIC STATE. Journal de Physique Colloques, 1984, 45 (C8), pp.C8-13-C8-18.

�10.1051/jphyscol:1984803�. �jpa-00224301�

JOURNAL DE PHYSIQUE

Colloque C8, supplément au n

l l , Tome 45, novembre 1984 page C8-13

ALUMINUM AT T « T

: THEORY OF THE DYNAMICALLY COMPRESSED METALLIC STATE *

R.G. Dandrea and N.W. A s h c r o f t

Laboratory of Atomic and Solid State Physios, Cornell University, Ithaca, NY 148S3, U.S.A.

Résumé

Nous rendons compte ici d'une méthode de calcul de l'équation d ' é t a t et des courbes Hugoniot des métaux simples sous haute compres- sion, a la fois dynamique et statique. La méthode est fondée sur l'observation que les énergies libres de Helmholtz des phases isochoriques solide et liquide différent surtout par leurscomposantes entropiques respectives. Par conséquent, nous utilisons une théorie pseudopotentielle au second ordre pour calculer les fonctions thermodynamiques du métal simple par une méthode variationnelle qui traite la phase solide, à des fins structurelles, comme un liquide supra-refroidi. Dans le cas de l'aluminium, nous avons examiné des compressions de volume de l'ordre de V / V

= l / 3 , correspondant aux valeurs atteintes dans de récentes mesures de choc nucléaire. Les températures associées sont dans les environs de Tp et de telles conditions requièrent que l'on tienne pleine- ment compte de la dépendance en température de l'énergie du gaz d'électrons et de la fonction diélectrique.

Abstract

This work reports a method Tor calculating the equation of state and Hugoniot curves of simple metals under high compression, both dynamic and static. The method is based on the observation that the Helmholtz free energies of isochoric solid and liquid phases differ mainly in their respective cntropic components. Accordingly, we use second order pseu- dopotential theory to calculate the thermodynamic functions of the simple metal by a variational method t h a t treats the solid phase, for structural purposes, as a supercooled liquid. In the case of aluminum, we have con- sidered volume compressions in the range V / V '

= l / 3 which correspond to values reached in recent nuclear shock measurements. Associated tem- peratures arc in the neighborhood of Tp and such conditions require that the temperature dependence of the electron gas energy and dielectric function be fully taken into account.

1. I n t r o d u c t i o n

In the past decade, experimental high pressure probes have undergone both a con- siderable refinement in precision and a considerable extension in their compressive capa- bilities. Diamond anvil experiments have recently achieved static compression pressures of 1.7 Mbar

, whereas dynamic compression pressures in excess of 4000 Mbar have recently been reported in convergent nuclear shock wave experiments

. The challenge t o theoretically model the behavior of matter under such extreme conditions has thus

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984803

C8-14 J O U R N A L DE PHYSIQUE

become increasingly significant. It is the purpose of this paper to present a preliminary scheme to calculate the equation of state of simple metals a t ultrahigh

Specifically, second order pseudopotential theory is used to calculate the room tempera- ture isotherm, bulb: modulus and binding energy, as well as the shock Hugoniot. Alr~mi- num is chosen as the prototype system for these calculations because of its importauce aa a possible high pressure standardg, and because of the wealth of experimental data, both static and dynamic, curreutly available for it.

In the following section we present our calculational scheme, emphasizing the specific approximations made. The results are presented in section 3, and we conclude with commentary in section 4.

2. C a l c u l a t i o n a l S c h e m e

Consider a system of N ions of charge +Ze and ZN electrons of charge -e in a volume V. Because we restrict our attention to simple metals, we may treat the ions (nuclei plus core electrons) as unpolarizable particles, and consider the electron-ion interaction as describable by a local pseudopotential. The Hamiltonian can thus be writ- ten

H = H e + HI + ~ " ~ ( < - r i , ) ,

i , t

(1)

?here He and Hi are the standard electronic and ionic Hamiltonians, and where 3 and R j denote the electronic and ionic coordinates, respectively. For simplicity we have used a modified Heine-Abarenkov form for the pseudopotential, i.e.

The effective ion core radius r, and the constant c present two free parameters, which are obtained by fitting the calculated room temperature isotherm t o experimental data.

The essence of our calculation is to use second order pseudopotential theory to evaluate an approximate energy of the Hamiitonian (1). Thus, upon making the stan- dard adiabatic approximatiou and assuming a linear response of the electronic density to the ionic density as a perturbation, the following "pair potential" expression is obtained for the energy4:

E ( T , v ) = E, + ^E, + ^T, + ~ c w R , , ) . (3)

~ Z J

Here E, is the energy of the fully interacting homogeneous electron gas, E, is a structure-independent term (which contains the overall Hartree energy), T, is the kinetic energy of the ions, and d r ) is the temperature and volume dependent interionic pair potential.

Because we shall be calculating high temperature properties ( T - T F ) along the Hugoniot, the full temperature dependence of the electron gas energy must be included.

We thus write

The kinetic energy of the electron gas, E,', is approximated with the kinetic energy of a

gas of ideal fermions, which is obtained, along with the chemical potential, by numerical

integration. The quantity U, is the temperature dependent interaction energy of the

electron gas. We approximate it with an expression that interpolates between the

exchange-correlation energy of degenerate electrons and the interaction energy of a clas-

sical one-component plasma. In the density range of interest here (where r, *2), such an

expression is

where a =1.03937, b =0.91303, d =0.305, V / N Z = = 4 d r , a, )3/3 and 8= T I T F . U, ( T =0) is the exchange-correlation energy

determined by Rahman and vignale5.

Note that U, extrapolates into the classical (8>> 1) regime t o values determined by Car- ley6 and, in the extreme weak coupling limit, by Debye-Hiickel theory. The term E, in Eq. (3) is given by

( N Z

E " ( T , v ) = & ^{+ &? I"(k)} I2W.

where c(k , T )

1 + ^{4rre2n(k ,T} )/k2 is the temperature dependent dielectric ruuction of the electron gas, n(k , T ) being its irreducible polarization. Finally, the state-dependent pair potential appearing in (3) is given by

We use the following form for the electronic polarization:

where no(k ,T)=IIo(O,T)/ ( k ) is (at T << T F ) the free electron (RPA) polarization, J (k ) being the Lindhard function, i.e.

We have chosen a lledin-Lundqvist form' for the correlation enhancement function G(k),namely

G ( k ) = 1.01434r,O(k /2kF )' , (10)

wbere u=0.08607. T h e prefactor of k2 above has been choeen so t h a t the resulting T =O polarization satisfies the compressibility sum rules. The quantity n,(O,T) is

~ ~ ( 0 , T ) = -kF [n'e'a. J 1 + ( 3 0 / 2 ) ~ 1-I . ₍₁₁₎ Note again t h a t because of the high temperatures reached along the Hugoniot curve, a finite temperature correction to the screening wave vector that interpolates brtween Thomw-Fermi at T << TF and Debye-Hiickel at T >> TF is included in Il,(k = o ) ~ .

T o evaluate E, and b(r ), the k-space integrals are carried out analytically by using a I'adC approximaut for the Lindbard functionlo: / (k ) = N ( k )/D (k ), where

and

D ( k ) = 1

0 . 2 0 8 1 8 8 7 ~ ~ + ^0.01910z4 + 0.0191657z6

with z =k /2kF. Upon making this approximation, b(r ) can be written in closed form as the sum of six complex Yukawa potentials.

Notice that the application of second order pseudopotential theory to the Hamil-

tonian (1) results in a separation of the electronic and ionic degrees of freedom, and t h a t

the electronic degrees of freedom are traced over quantum mechanically in arriving a t

the expression (3) for the total energy. However, within the density and temperature

range of interest, the ionic thermal de Broglie wavelength is much smaller than the inter-

'particle spacing, and thus classical statistical mechanics can be used t o trace over the

ionic degrees of freedom. This classical averaging can be greatly simplified, and a varia-

tional principle admitted, if the long range positiollal order of the solid is neglected and

the structural energy is evaluated using a liquid-like pair correlation function:

JOURNAL DE PHYSIQUE

Such a "supercooled-liquid" approximation has been recently applied to argon by Jones and ~ s h c r o f t " . Fluid variational theory, based on the Gibbs-Bogolubov inequalityI2, can then be used to evaluate the free energy in a fluid reference system:

F = E, + E, + ^{$NkB T}

^TS,

where 9 is the variational parameter. An approximate expression for the electron gas entropy S, ( T , V ) is obtained by a 6rst order finite temperature perturbation theory cal- culation, which gives

TS, = $E,D-PN . (16)

For calculations of the crystalline state we use a hard sphere reference system, where the variational parameter 9 is the hard sphere packing fraction. Because the pair potential can be expressed as the sum of complex Yukawa potentials, the energy integral in (15) can be evaluated in closed form within a Percus-Yevick approximation13. The excess hard sphere entropy is obtained from molecular dynamics as"

St.(q) = [1n(qa3)+ a 2 - 0 . 4 4 0 - 2.1559 NkB I , ⁽¹⁷⁾ where o =( V - VCp )/ V c p , Vcp being the close-packed volume. For calculations of the liquid state ( a t compressions beyond the point where the Ilugoniot crosses the melting curve), we use the soft sphere model of ~ o s s ' ~ , which has been shown to be the preferred reference system for liquid aluminum16. T h e soft sphere system uses the same pair correlation function as the hard sphere system, but the hard sphere Carnahan-Starling excess entropy is modified t o the form

where 9 is again the packing fraction.

Our calculational scheme thus consists of two parts. First, second order pseudopo- tential theory is employed t o express the total energy in the pair potential form of Eq.

(3); second, by treating the solid for certain structural purposes as a supercooled liquid, fluid variational theory can be used to calculate the final thermodynamics. In the fol- lowing section we report the results of our calculations on the equation of state of alumi- num.

3. Results of Calculations

T h e first step in implementing the above calculational scheme is to obtain the two pseudopotential parameters c and r, of Eq. (2) by fitting the calculated room tempera- ture isotherm t o experimental data". The resulting theoretical curve plotted in Fig. ( I ) is seen t o agree quite well with experiment. The values of the pseudopotential parame- ters obtained are c =0.437 and r , =1.424a0.

T h e bulk ~nodulus of aluminum is easily obtained by diEerentiating the p ( V ) rela- tion shown in Fig. (1). .4t V = V o and T=300 K, we obtain a value Bo=0.66 Mbar, which is to be compared to the experimentally observed value 0.794 bar'^.

Because we have used p ~ e u d o p o t e ~ ~ t i a l theory, the results of our total energy calcu- lations give the binding energy of aluminum, which is the total energy relative t o its tri- ply charged ions and valence electrons separated at infinity. At room temperature and zero pressure, we obtain E =-4.12 Ryd/atom, which is indced quite close to the experi- mental value of -4.14 ~ ~ c i / a t o m ' ~ .

T h e major goal of this calculation, however, is the determination of the shock

Hugoniot curve, which is the locus of states in the p - V plane attainable by a shock

compression. The Hugoniot of course lies above the isotherm, by virtue of the heating

0.4 0.6 0.8 1 .O

v / V o

Figure 1. Fitted T=300 K isotherm of aluminum. Theory (solid line) vs. exper-

iment (+'s) from Ref. 17. F l p r e 1. Aluminum Hugoniot. Theory (solid line) vs. experiment (+Is) from Ref.'s 21 and 22. In the insert, T M and EM refer to the theoretical and experi- mental melting regions, respectively.

t h a t occurs during shock loading. Because the compressive stresses attained in the shock front s o exceed t h e yield strength of t h e metallic sample, it is common t o neglect t h e shear stresses produced and to treat the shocked material as a fluid. T h e conservation laws of mass, momentum and energy can then be straightforwardly applied t o describe t h e Hugoniot curve by t h e Rankine-Hugoniot equationz0 for isotropic systems:

E - E o = + ( V o - V ) ( p + ~ o ) . (19)

Here (Eo,Vo,po) is the initial thermodynamic state, while (E , V , p ) describes t h e thermo- dynamic state of t h e shock front.

In Fig. (2) is plotted t h e theoretical Hugoniot curve, obtained by solution of Eq.

(19), along with t h e experimental d a t a of Mitchell and elli is" and RaganZ2. In t h e insert is shown d a t a up t o 2 Mbar obtained by impact loading, along with t h e experi- mental16 and tbeoretical regions in which melting occurs. This region was calculated theoretically by plotting t h e Hugoniot pressures of both the solid and liquid phases against t h e melting curve of Moriarty, Ross and ~ o u n ~ " . In t h e larger plot of Fig. (2), t h e tlugoniot curve is extended up t o 30 Mbar, in order t o compare theory with a d a t a point recently obtained by Ragan using nuclear explosive shock loading. T h e theoretical curve is found t o lie within the 5% error b a n on compression about t h e R a g m point a t 29.33 Mbar, and in general the calculated Hugoniot agrees quite well with experiment.

A t a compression of V / V0=0.3, we obtain a Hugoniot temperature of 331,000 K , which is a b o u t 1.1 TF. T h e slight discrepancy in melting along the IIugoniot is t o be expected because of o u r ueglect of t h e structural aspects of the free energy.

4. Conclusion

We have presented a simplified scheme t o calculate the bulk properties of simple metals a t high static and dynamic compreusion. By applying t h e "supercooled liquid"

approximation of Jones and Ashcroft t o a second order pseudopotential Hamiltonian, we

arc able t o successfully model the iuothcrm, binding energy and shock IIugoniot curve of

C8-18 JOURNAL DE PHYSIQUE

aluminum. Significantly, in calculat~ng the h ~ g h tempcrature properties along the Hugoniot, we have noted that the temperature dependence of the electron gas energy and polarization must be fully included. Future work will aim a t a more systematic interpolatiou of these quantitiev between their degenerate quantum mechanical values and classical high temperature values. Note t h a t this method does not require the intro- duction of

phenomenological Griineisen parameter.

Reference9

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