Several methods to derive thermodynamic sum rules for a system including charged particle are proposed and applied to charged mixtures as well as one-component systems. The validity of the statements is examined carefully with respect to the ordering in the powers of the wave number. As for the mixture of electrons and ions, it is shown how the aspect of the one-component plasma or the ionic mixture appears when electrons become strongly degenerate

Totsuji, Hiroo

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Okayama University Scientific Achievement Repository

Memoirs of the Faculty of Engineering, Okayama University, Vol. 22, March 1988Thermodynamic Sum Rules for Mixturesof Charged Particles*Hiroo Totsuji(Received March 16, 1988)AbstractSeveral methods to derive thermodynamic sum rules for asystem including charged particle are proposed and applied tocharged mixtures as well as one-component systems. Thevalidity of the statements is examined carefully with respect tothe ordering in the powers of the wave number. As for themixture of electrons and ions, it is shown how the aspect ofthe one-component plasma or the ionic mixture appears whenelectrons become strongly degenerate.1. In troductionThe behavior of response functions in the limit of low frequency and longwavelength are related to thermodynamic functions of the system. These relationsha ve been used to construct approximation schemes for response functions or tocheck their thermodynamic consistency.In the case of charged particles, these sum rules take different forms fromthose for systems of neutral particles reflecting the long range nature of theCoulomb interaction. In this paper, we analyse the long wavelength limit of densityresponse functions in mixtures of charged particles by taking the long range naturein several different ways and clarify how sum rules in various idealized models,such as one-component plasma, are related to the properties of the original systemswhich the models try to describe.2. Sum Rules2.1 Long Wavelength Behavior of Density Response FunctionsWe consider a mixture of charged particles in thermal equilibrium at' thetemperature T in a volume V. We denote the quantities related to electrons by* Department of Electrical and Electronic Engineering3536 Hiroo TOTSUJIthe subscript 0 and use the subscripts a=l, 2,... to denote ions of species a:Roman letters will be used as running parameters including electrons, e.g., i=O, 1,2,.... In order to include the cases of one-component plasma and ionic mixtures,the existence of uniform background is assumed when the charged particles underconsideration are not charge neutral. We denote the char.ge, the total number,and the number density of charges of species i=O, 1, 2,... bye., N., and n.,I I Irespectively.We now introduce into the system an infinitesimal fictitious test charge density{j (J • which interacts only with the charges of the species i. We assume thateX,1the wavelength of perturbation is sufficiently large and all quantities related tothe perturbation have the spatial and temporal dependence exp(ik· r-iwt) with w=O.The condition for the chemical potential of the species j, tJ /.1.=0, is writtenJaswhere2v(k)=4'IT. /kis the Fourier transform of the Coulomb interaction and(2.1)(2.2)(2.3)The first term on the left hand side of (2.1) is the change of the chemical potentialdue to induced densities, {j nk, the second term is the interaction with the testcharge, and the third term is the potential energy due to induced mean field. Theinduced chat'ge density is given by the density response function X ..(k,w=O) asIJ(2.4)whereand (J .(k,t) is the density fluctuation of speciesJN.J~ exp[-ik· r.(t)].i=1 IWe thus have(2.5)(2.6)Thermodynamic Sum Rules for Mixtures of Charged Particles-1or, in terms of the inverse of the density response function X ij37(2.7)It is to be noted that we have not assumed that the system is classical nordegenera teeThe above relations can also be obtained by introducing an imaginary backgroundcharge density {j P b to maintain the charge neutrality asIn this case, both the test charge density {j PeX,i and the introduced background{j Pb equivalently work as the external charge density for our system. The induceddensity {j nk is thus given by(2.9)Since the charge neutrality is maintained, the condition for the chemical potentialof the species j is now written as(2.10)the term due to mean filed being absent. From Eqs. (2.8), (2.9), and (2.10),we havethe same results (2.7).The static form factor for species and is defined by(2.11)where < > denotes the statistical average. For classical systems, the static densityresponse function is related to the static form factor by the fluctuation-dissipationtheorem as(2.12)where kB is the Boltzmann constant.2.2 compressibility Sum Rule38 Hiroo TOTSUJIenergy F) is given by P = -(aF/aV)TN N, 0' 1'··the isothermal compressibility .t T is calculated asThe relation between the compressibility of a system and the long wavelengthbehavior of the response function or the static form factor is called compressibilitysum rUles.1 The compressibility sum rule is included in the relations derived in 2.1.Since the Helmholtz free energy F is written as F(T, V, NO' Nl'•••) = Vf(T, nO'n1 ,•••), the pressure of charged mixture (defined by the volume derivative of theHelmhol tz free-v(aPI aV)T,N ,N = n(aPI an)T = :Eijnl·njllloJo.o 1'··· ,xO,x1'···(2.13)Here n is the total (unperturbed) number density and x j is the concentration ofspecies j:n = (2.14)(2.15)From (2.7) and (2.13) we have(2.16)Here < > denotes the average as(2.17)The above equation relates the long wavelength limit of the inverse density responsefunction and the compressibility of the charged mixture.The compressibility sum rule for charge mixtures (2.16) can also be obtainedby applying the test charge densities lJ p ., i=O, 1, 2, •••, adjusted so as to induceeX,lthe number densities lJ nj in proportion to the unperturbed number densities nj ,simulating the volume change of the system:(j = 0, 1, 2, •••), (2.18)where A is an arbitrary infinitesimal constant. From(2.19)the test charge densities are determined asThermodynamic Sum Rules for Mixtures of Charged Particles 39(2.20)The force on charges (per unit volume)is balanced by the pressure gradient-ik(8P/8n)T :E .6n.,xi J JasFrom (2.20) and (2.21), the relation (2.16) is again derived.3. Exampleso. (2.21)3.1 One-Component PlasmaThe one component plasma (OCP) is a system of charged particles of one speciesin a uniform neutralizing backgrourid of opposite charges. We denote the chargeand the number density by e and n, respectively. In this case with only onespecies, the relations (2.7) and (2.16) are identical:(3.1)Here we note that 11 itT is the compressibility of charges in uniform background andmay become negative without causing any instability of the system.2Especially when these charges are classical, we have the long wavelengthbehavior of the static structure factor to the order of k2 as(3.2)where kD is the Debye wave number defined by(3.3)oand Ie T is the compressibility of the classical ideal gas40 Hiroo TOTSU]Io" T = linkST. (3.4)In terms of the Fourier transform of the direct correlation function c(k) defined by(3.2) is expressed asS(k) - 1 c(k) + c(k)[S(k) - ll, (3.5)(3.6)3.2 Classical Ionic MixturesEqs.(2.7) and (2.16) hold also for mixtures of charged particles in a uniformbackground. As in the case of one-component plasma, we note that <e>*O and "Tis the compressibility of mixtures in a uniform background which may become negativewithout causing the instability of the whole system. As a model of real systems,it is natural to regard electrons as a background and assume that the mixture iscomposed of ions. We may therefore expect that the mixture is classical. Eq.(2.7)is thus written as(3.7)The Fourier transform of the direct correIation function c ()( p(k) is related to thestatic form factor by(3.8)The long wavelength limit of the direct correlation function is thus given to theorder of k0 as(3.9)This generalizes Eq.(3.6) to ionic mixtures. From (2.16) and (3.9), we haveFor the charge density fluctuation spectrum, (3.7) gives the long-wavelengthlimit as(3.11)Thermodynamic Sum Rules for Mixtures of Chorged ParliclesFor classical binary ionic mixtures (81M), the structure factors are calculatedas41where(3.l2a)(3.l2b)(3.l2c)A (3.l2d)We note that the terms of the order k2 and k4 cannot be determined from (3.7):Terms of these orders neglected in (2.7) or (3.7) should be taken into account.To the total charge density fluctuation, however, careful calculations showtha t these neglected k2 and k4 terms do not contribute and we havewherekBT[l-C/Av(k}]lAv(k) + O(k6 ), (3.13)(3.14)When we use the temperature T, the concentration of l-st species xl =nl/n, andthe total charge number density n:e le}n (e being the unit charge) as independentavariables, (3.l3) reduces to the expression obtained earlier by the method ofthermodynamic fluctuations.33.3 Two Component PlasmaWe now consider the mixture of electrons and one species of ions, sometimescalled two-component plasma (TCP). Electrons are denoted by the subscript 0 andions are denoted by the subscript 1.Taking the inverse of (2.7), we have for long wavelenghtswhere K T is the compressibility of the total system.the density response function have the same longsystem becomes thermodynamically unstable when the(3.1S)Thus all the components ofwavelength behavior and thetotal compressibility vanishes.42 Hiroo TOTSUJIThis behavior of the response function has already been derived as a result ofanalyses based on temperature Green's functions.4 The simple derivation shown above,however, seems to have not 'appeared in the literatures. We note that the relation(2.7) is accurate to the order of kO and the next order term has the order of k2•Therefore the exact statement on the response function is limited to the order ofkO in (3.15).3.4. Ion. System in Highly Incompressible ElectronsWe here consider the mixtures of ions and electrons where electrons are highlydegenerate and almost incompressible, simulating the role of the background charges.We thus assume that the Fermi energy of electrons EF is much larger than thethermal energy of ions:(3.16)In this case, the element /100 may be estimated by the value for the ideal Fermigas as/I 00 (3.17)and we may assume thatLet us first consider the case where the ions are of one species.calculations lead to the density response functions as-(kBT/nlh:n(k,(d=O) = Sn(k)2 2{= nl kBT K T + (k/kn) + O[(k/kn) (kBTIEF)] 2for 1» kBT/EF)(k/kn) ,2 '4 2(k/kn) + nlkBTKT - (k/kn) (nl/ln/nlkBT) + O[(k/kn) (kBT/EF)](3.18)Careful(3.l9a)(3.l9b)(3.l9c)Thermodynamic Sum Rules for Mixtures of Charged Particles 43(3.l9d)Here It T is the compressibility of the whole system(3.20)which can be approximated by the compressibility of electrons in Eqs.(3.l9c) and(3.l9d).Eq.(3.l9a) is consistent with the result (3.15) and gives the next order term forions. In Eq.(3.l9b), we note that nlPn/nlkaT is the inverse compressibility of ionsnormalized by the ideal gas value nlkaT. We thus see how the aspect of theone-component plasma appears with the decrease of the electronic compressibility orthe increase of P 00.When ions are composed of several species, we haveand112-[kaT/(nan~) ha~(k,(j)=O) = Sap(k)-[kaT/(nan~F2h/a~(k,(j)=0)+ O[(k/ko)2, (kaT/Ea )) (3.21)(3.22)Here X I D is the density response function of the ionic mixture of the same, a fJcomposition in the uniform background. An example of such response functions isgiven in 3.2 for binary mixtures.3.5 Screened One-Component Plasma and Ionic MixturesAs a model of ions in highly degenerate electrons, there has been used theion system interacting via the Coulomb interaction screened by the dielectric functionof electrons e (k,(j) =0). The results in the foregoing sections hold also for theseesystems when the COUlomb potential is replaced by the screened Coulomb potentialasv(k) -) v(k)1 e e(k,O).In the case of one species of ions, the screened OCP, we thus have5(3.23)S(k) (3.24)44 Hiroo TOTSUJIAs has been pointed out by Leger and Deutsch,6 (3.24) is consistent with the results(3.15), (3.19a) and (3.19b), when the dielectric function of electrons satisfies thecompressibility sum rule (of electron system).AcknowledgementsThe author thanks Professor C. Deutsch .and Dr. D. L~ger for discussions atOrsay supported by Japan Society for Promotion of Science and Centre National de1a Recherche Scientique of France.References1. D. Pines and P. Nozieres, The Theory of Quantum Liquids I (Benjamin, New York,1966).2. M. Baus and J. -Po Hansen, Phys. Rep. 59, 1(1980):S. Ichimaru, Rev. Mod. Phys. 54, 1017(1982).3. J. -Po Hansen, G. M. Torrie, and P. Vieillefosse, Phys. Rev. A16, 2153(1977).4. M. Watabe and M. Hasegawa, in The properties of Liquid Metals (Proceedings ofthe Second International Conference, Tokyo, 1972), edited by S. Takeuchi (Taylorand Francis, London, 1973), p.1l3.5. H. Totsuji and K. Tokami, Phys. Rev. A30, 3175(1984).6. D. L~ger and C. Deutsch, J. Physique Lett. 46, L475(1985).

Thermodynamic Sum Rules for Mixtures of Charged Particles

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