, 1 _.------_.r'-.

Tlieoretical and Mathematical Physics, Vol. 126, No. 2, pp. 212-223, 2001

MICROSCOPIC QUANTUM HYDRODYNAMICS OF SYSTEMS OFFERMIONS: 11

L. S. Kus'menkov;' S. G. Maksimov;! and V. V. Fedoseev'

A closed system oi nonstationary equations [01' a multielectron atom is constructed using a metlwd prc-viously developed [or investigating multipartic1e fornuon systems. Tlie well-lmoll'lI Tlwmas--FermiDi¡-acmethod is a particular case oE the system. TI¡e exchange spín-spin correlatíon» are c:alculated lISillg animproved spín-spín Interacrion Hamiltonian. TIIC dependence of the electron COlWCIJtl'atioll0.n tlu: self-consistent electric field potential and a discrete set of parameters detel'lui/led by t,JJ(' t()/;al euergy ilml tJlCangular momentum of tJ¡e atom is derived for statiollitr,}' atom states.

The general method for constructing the self-consistent microscopio quantum hydrodynamics devel-oped in [1] relies on calculating the correlation functions. These functions can he calculated through aniteration procedure. We use the orthogonal one-particle wave functions that are the solutions of one-particleSchródínger equation (1-45) (hereinafter we refer to formulas in [1] as (I-N), where N is the formula numberin Part 1 of this work) as the zeroth-order iteration. The only interaction we ret.ain in this equation is theinteraction between the spin and the external magnetic field. It is necessary t.o take this interaction int.oaccount to derive the dependence of the exchange correlation functions 011 the dcnsity of the propor mag-netic moment of the electrons. Neglecting this term in the one-partícle Hamiltoninu leads to significant lossof information about the spin in the quantum hydrodynamic equations. Hence, the zeroth-order itorationfor the one-particle Schródinger equation has the form

A 2

'fJ)<P¡ fp (" B )t,~-= -<PI .- Jts (T o <PI'8t 2m (1)

It is clear that the coordinate and spinor parts of the wave function <PI are separated, and we obtain t.1\('

equation for the spinor w:

i1i~~ = -JLn(uBo)w,

where the spinor w is connected with <PI through the relation

(2)

p2e --

p - 2111.'

We also assume that the spins of all electrons are involved in a precession but remain parallel or antiparallclat all moments of time, i.e., the precession is uniform for the entire spin system. In the further aualysis, wcuse the polarization density matrix at an arbitrary instant t,

(3)

1Moscow State University, Moscow, Russia, e-mail: Isk@lIi\bla.phys.msu.su.serguey@t.eor.:l·12.phys.msll.sll.wladdy@teor242.phys.msu.su.

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 126, No. 2, )JJl. 258--270, February, 2001.Original artic1esubmitted June 22, 2000.

212 0040-5779/01/1262-0212$25.00 © 2001 Plenum Publishíng Corporation

where

(4)

is the unit vector determining the direction of the electron magnetíc moment at an arbitrary instant,WL = eBo/2mc is the Larmor electron precession frequency, and n is the unit vector determining themagnetic moment direction at t = O.

1. Calculation of the exchange spin+spin correlation function

We now calculate the exchange functions g(r,r', t), maP(r,r', t), and ~a"YP(r,r', t). Taking into accountthat the partic1es are identical, we obtain

MaP(r,r',t) = N(N -1) L L !dRN-2IJtcp*(r,lI;r',II';RN-2,SN-2;t) xv.v' SN-2

(AaAP)( .:» '.R S . )_~l(Ia!3( ') ¡aP( '»)x CTICT2CPr,lI,r ,11, N-2, N-2,t - ~ 2- vv' r,r ,t + v.v' r,r ,t, II.V'

(5)

from formula (1-29), where the bar over the tensor ¡:~indicates complex conjugation, RN-2 = dr3··· drN,

and LSN_2 = L.3 •••••• N• We obtain

¡:~(r,r',t) == N(N -1) L!dRN-2 IJ! {cp*(r, 11;r', 11';RV-2,SN-2; t) x. SN-2

x (ufa{cp}(r,lI¡r',II'¡RN-2,SN-2¡t) + ..

+ (ü~cp)*(r, 11;r', II'¡ RN-2, SN-2¡ t)(üfcp}(r, 1/; r', 11';RN-2, SN-2; t) } (6)

for the rank-two tensor introduced in Eq. (5). Substituting wave function (1-39) in Eq. (6), where the one-partic1e functions cp/ are the solutions of Eq. (1), and using the double «xpanslon of the Slater determinant

(r,w¡r',W'¡RN-2,SN-2Inl,n2' ... ¡t) = L L ~J Nn~ 1(-I)E!'slI<! n, x/ r-a

x ((r,wIJ;t)(r',w'I1';t) - (r',w'IJ¡t)(r,wl!,¡t») x\

x (RN-2,SN-2Inl', ... ,'(Il/, -1), ... ,(n/-l), ... ;t), (7)

we obtain

I:~= ~IJt L ((J¡ tlr, I,)(J'¡ tlr', 1I')(r, IIluaIJ;t)(r', 1I'lü!3I!,¡t) +s.r#1'

+ (J¡ tlr, 11)(J'ICTPlr' ,II')(r, IIlua IJ¡ t)(r', 11'11.':t) -

- (J¡ tlr',II') (1'; tlr, lI)(r, IIlua IJ; t)(r', 1I'lü.dI!'¡ t) -

- (J¡ tlü!3lr', II'){!'; tlr, lI)(r, IIlüa IJ; t)(r/, 11'11'¡ t) ).

213

The expression in the parentheses vanishes for 1= /'; therefore, the sum also applies to the case wher«f = [':

I:t. (r, r', t) = ~ (I~(r, t)I~ (r', t) + I~(r, t)I~ (r', t») -

- ~ Ln¡J.l.B(r,vluCl'lf; t}(J; tlr', v'} Lnf'ltn(r','/lai3If';t}(J';tlr,v}-¡ l' .

- ~E nJJL~(r, VluOIf¡t}(J; tlai3lr', v') E nI' (r', I/I/'¡ t)(J'; tlr, v),¡ r

(8)

where

I~(r,t) = LJ.l.B7l¡(J; tlr, v}(r, vW'lf; t},¡

M~(r,t) = ~(I~(r,t) + I:?(r,t»).

Substituting Eq. (8) in Eq. (5), we obtain the two-particle magnetic mornent density t.ensor..

M°{J(r, r', t) = L M:t. (r, r', t),V,II'

M:t.(r,r',t) = M~(r,t)M~(r',t) + m~~(r,r',t),

m~~,(r,r',t) = - ~J.I.~(r,vluo¡3(t)lr',v'}(r',v'lu{J¡3(t)lr,v) +

+ (r,vl¡3(t)aJ~lr'II/)(r',I/I¡3(t)atl'lr,I/) +

+ (r,vlutl'¡3(t)ui3lr',1/)(r',v'I¡3(t)lr,v) +

+ (r, vl¡3(t) Ir' ,v'}(r', v'lu{J ¡3(t)uCt Ir IV»),

(9)

where p(t) is the one-particle density matrix of the fonn

p(t) = Ln¡lf¡ t}(J¡ ti·¡

Therefore, we have the expression for the exchange spin-spin correlation funct.ion Ill~~, (r, r', t) in (9) forarbitrary one-particle wave functions 11¡ t}. Choosing specífic wave functions as the zeroth-order iteration,we can derive the quantum hydrodynamic equations for specific physical systems, such as crystals, atoms,etc. For example, using the one-particle Bloch functions, we obtain the equations for a system with aspatially periodic structure, in particular, for crystals. However, every crystal is characterized by thecorresponding symmetry group. Therefore, if we choose the wave functions that are invariant with respectto a certain transformation group, we can obtain the model for a specific crystal.

Because the one-particle wave vector ís

(10)

214

we obtain

mO'P(r, r', t) = L m~~,(r, r', t) = - ~lt~ ';2 L I>·p,/Tnp' ,/T' XII,V' ptp' (1,0'

x (exp[~(p - p')(r - r')] Tr(an{JR(t)aIJpu,(t») -1-

+ exp [- ~(p - p')(r - r')] Tr(af3 P/T(t,)a<l Pu' (t)) ) (11)

from (9) through summation with respect to the indices u and v'. Using the commutation properties ofPauli matrices and formula (3), we obtain

Tr(UapCT(t)aP PCT'(t») = ~óap + ~fP(t) Tr(a(a<lal'aJJ) + o' (aoa{3 a"») +

+ ~ao' fP(t)¡V(t) Tr(a<la"aI3a") ==

= ~Óap(l - 0'0") + ~iJl'(t)énll{3(a - a') + 0'0" r'(t)f(J(t).

Substituting the resulting equation in formula (11) and passing from t.he sumrnation with respect to themomenta to the integration, we obtain t.he exchange energy density for the spin-spin interaction:

€spin(r, t) = - ~f dr' GoP(r - r')maP(r, r', t) =

= (2:ñ)6~JL~ f dp f dp' :Enp,/Tnp"CT,GaJJ(p-p')Tr(a~fi<T(t)aJJpCT') =(T .a'

= (2:ñ)6 ~JL~ f dp f dp' :E np,/Tnp',U' (1 - oo')Gu",(p - p') + 20'0" r'(t)ftJ(t)Go,J(p - p'j].tT ,(T'

In this equation, the symmetry of the tensor GOItJwith respectoto index permutation is takeu into account.We now use formula (1-7) for the spín-spín interaction in the Fourier n-presentation and obtaín

- 2 '2 2 1 f f ' (p - p')f(t»)éspin(r,t) = 21fJLBn -21fJLB(21fñ)6 dp dp (np.+l-np,_¡)(np'.ll-np',_¡) Ip-p'12 . (12)

Because the integration in Eq. (12) is performed with the time fixed, W(' choose the coordinate fraine z axisaccording to the condition f = ez, and because the integration domain is spherically syrnmetric, we obt.ainthe final form

€spin(r, t) = 21fJL~n2 - 21fJL~(2:ñ)6 f dp f dp' ~ (np.+1 - np.-¡)(np'.+1 - 1tp' ,-¡) =

2 ) 21f 2= 21fJLBn(r, t - 3"M (r, t). (13)

Expression (13) is derived from an improved spin-spín interaction Hamiltonian and is always positive, whilethe exchange energy calculated from conventional Hamiltonian (1-1) is

215

and, as can be easily seen, is negative definite. In the latter equation, n± is the density of the el~ctronswhose spins are oriented along (or against) the vector f.

The expression for the exchange energy density of the spín-spin interactron allows calculating theexchange spin interaction force density involved in momentum balance equation (1-42):

.r~in(r, t) = f dr' VQGt3'Y(r - r')mt3'Y(r, r', t) =

= va (f dr' Gt3'Y(r - r')mt3"Y(r, r', t)) - f dr' Gt3"Y(r- r')v"'m{3"Y(r, r', t) =

= 2Va€(r, t) - !dr' Gt3"Y(r- r')J.l~ (2:ñ)6 !dp!dp' L np,.,.np',.,.' X

tI,a'

I~ the system is not bounded, the second term vanishes, because Gat3(r - r') is an even function, i.e.,Ga{3(r - r') = Ga{3(r' - r), and is integrated over an unbounded domain together with the odd functionsin [(p - p'){r - r')/ñ]. As a result, we obtain

.r~in (r, t) =!dr' VaGt3"Y(r - r')mt3"Y(r, r', t) = -2Va€spin(r, t).

2. Exchange contribution to the Bloch equation

(14)

The express ion for the exchange spin-spin correlation function obtained aboye allows calculating theexchange contribution to the Bloch equation. We obtain

BO!(r, t) = ..!:....ea{3"y f dr' G"Y6(r - r')mt36(r. r', t) =me

= - ~eJ.l~ (2:ñ)6 !dp!dp' ~ np,.,.np".,.,uu'ealh ft3(t)G'YÓ(p - p')f6(t). t7,tS

from formula (1-43). As a result, we obtain the formula

B( ) e 2 1 !d !d ' ~ ,[f x (p - p')] '))r,t =mcJLB(21l'ñ)6 P p f.:,np,.,.np"",'41l'UU Ip-p'12 (f·(p-p. (15)

Because np,.,.is independent of the direction of the vector p,

!dP!dp' ~ n n (p"y - p,"y)(¡f' - p'P) '" 8'YPf::. p,'" p',.,.' [p - p'12 •

The integral in Eq. (15) is therefore equal to zero,

(16)

Therefore, the exchange spín-spin interaction does not contribute to magnetic moment density balanceequation (1-42) in the approximation considered aboye, where one-particJe wave functions (10) are chosenas the zeroth-order iteration.

216

3. Coulomb exchange correlatíons

We consider the two-particle probability density

1t2(r,r',t) = L1Ll/v,(r,r',t),fI,,~'

where

nllV,(r, r', t) = N(N - 1) L J dRN-2 (nI, 1/.2,···; t[r, 11;1", v'; RN-2, SN-2) XSN-2

Using expansion (7), we obtain

•.It is then easy to see that

g(r,r',t) = - L[(r,v[p(t)[r',v'}[2 =,.I,V'

= - :2 LL np,<Tnp',<T' exp{ ~(p - p')(r - 1")} Tl'(p<T(t)pu' (t»),p,p' tT ,u' .

(17)

where

Tr(P<T(t)P<T,(t») = ~ Tr(l -1-(O' -1-a')(uf(t» + a/7'f2(O) = ~ (1 + 0'0").

The Coulomb interaction exchange energy was calculated by Dirar.. We recall the final.result for theenergy density of the Coulomb exchange interaction:

.( 3 )4/3{ (2 )4/3 (2 )4/:i}ec(r,t)=-211"e2

811" n+[;[M(r,t)[ + n-[:~~[M(r,t)[ . (18)

Formula (18) for M(r, t) = O was derived by Kirzhnits [2].(see also [3]·[5]).The force density of the Coulomb exchange interact.ion is

:F~(r, t) = -1ds: e2'\7°G(r - r')g2(r, r', t) =

= _'\70 (1dr' e2G(r - r')92(r, r', t») +1dr' e2G(r - r')'\7°92(r, r', t) =

= -2'\7°ec - (2:ñ)61 dp 1 dp' ~ np,<Tnp',IT' ~(1J" -¡r)e2G(p - p') X

a,a .

(19)

because the integration with respect to the momenta is carried out ovcr the entire space and the Laplaceoperator Green's function is an even functíon,

217

4. Explicit form of the pressure tensor

For the closure of the balance equations, an explicit form of the pressure tensor must be obtained. Thekinetic pressure tensor pOlfJis determined by formula (1-26), where the wave function q,s(R, t) is changedto <ps(R, t) from relation (1-32). Because the particles are identical, we obtain

pOlfJ(r,t) = LN L I dRN-l 4~ {DfJ*(nl,n2, ... ;tlr,v;RN_l,SN_l) xv SN-l

(20)

from Eq. (1-26) and Eq. (1-39). We now use the Slater determinant expansion with respect to one of itsUnes,

pOl{J(r,t) = ~ L L nl-41

(DfJ* (J; tlr, v) DOl(r, Vl!i t) + (Ji tlr, v)D{J DQ(r, Vl!i t) + c.c.). (21)v I m

Expression (21) for the pressure tensor is obtained for arbitrary one-particle wave functíons (r, Vl!i t).Therefore, choosing specific wave functions, we can derive the equations of state applicable to correspondingspecific systems, such as a crystal, an atom nucleus, etc. In particular, wave funct.ions in the form of planewaves can be chosen as the zeroth-order iteration for systems with high electron density (10). Substitutingone-particle wave functions (10) in Eq. (21), we obtain the equation for the kinetic pressure:

0l{J( ) _ 1 Id"" 1 0l-'3 "0l{J (.)p r, t - (21T/i)3 P ~ np,v;;,p l' = 6 PF r, t ,v

(22)

where

(21T/i)2 ( 3 ) 2/3 { ( 2rnc ) 5/3 ( 2mc' ) 5/3}PF(r, t) = lOm 871" n + lel/i IMI + n - lelli IMI (23)

is the Fermi pressure. It is easy to see that for M = 0, the pressure tensor derived from the Schródingerequation yields the Fermi pressure at zero temperatura.

Equation (1-43) for the magnetic moment balance contaíns the magnetic moment flux density j~f(r, t).Following a procedure similar to that in the previous cases, we obtain

j~(r, t) = ~ L L nI;/-{ (Ji tlr, v)DIt (r, VI&OlI!it) + D{J* (Ji tlr, v)(r. VI&OlI!it)} = O (24)v I m

because the momentum and the proper magnetic moment of an electron are not rorrelated in the approxi-mation we consider.

Summíng the results forrnulated in Eqs. (13), (14), (16), (18), (19), (23), and (24), we finally obtain

mn(r,t)(%t +v(r,t)v)vOl(r,t) + \1Olp(r,t) =

= en(r,t)( EOl(r,t) + ~éO{J"Yv{J(r,t)B'(r,t») + MJ3(r,t)\1OlBIJ(r,t), (25)

aM~~r,t) + \1IJ(vIJ(r,t)MOl(r,t») = ~céOl¡hMIJ(r,t)B"Y(r,t). (26)

218

In this equation, the term p(r, t) accounts for the total exchangc inter.otíon, and we thus obtaiu

p = ¡JF + 2tc + 2t.pin (2i)

for the quantum pressure, where tspin and te are determined by Eqs. (13) and (18) derived nbov«. Equa-tions (25) and (26) should be used together with field equations (1-44).

5. Multielectron atom

We now consider the problem of a multielectron at.om with a nucleus charge Ze (e > O). The magnetk:field is set to be zero. If all lower energy levels are occupied, t.he propor magnetíc moment deusity of theelectrons is nonzero only in the outermost shell of the atom and can lx- neglected (pr multíelectron atorns.In the nonstationary case, a multielectron atom is described by the system of Eqs. (25) and (26) and thecontinuity equation. These equations in the spherical coordinat.es havo the forms

ñn 1 8 2 1 8 . 1 a _-8 +"2-8 (r nvr) +-. -()8()(nvesmO) +-. -()-8 (nv'I') - O,t r r r sin r S1l1 <p

(8Vr 8vr Ve 8vr .s«.8vr _ v~ + 1)~ ) 81J ~ Q~~

mn 8 +vr a + aB + . B a + o - en.... ,t r r r sin ip r r (Ir

(8ve 8vo Ve ave v'l' oVn VrVe 1J~ ) 1(JI) 1al)

mn -8 +vr-a +- 8B +-,-() -a +-- - - cot.B+ --;-)-0 = (~n-8B't r r r sin ip r r l' ( r

(28)

(8V", 8v", Ve OV'" v'" 01J", V"V'I' 1JOV'I' o) 1 o¡) 1 é)<I!mn -+vr-+--+-,--+--+--cot. +-,--=en--.----;:-··.

8t 8r r a() r sin B o<p r r _ r SlI1 B 8<p r SU} O d'f

The potential l)(r, B, <p) of the atom electric 'field satisfies the equation

o ( 2 ol)) . ( 1 a (. 8l) ) 1 82l)) _ . .r() ,2ar r 8r + sin B 8B S1l1 () 8B + sin:!()O<p2 - --eZu r +41rem , (29)

We seek the stationary and axially symmetric solutions. Choosing the roordinate frame whose z axis is thosymmetry axis and setting Vo = O, we obtain thc continuity equatiou

1 o 2 1 {) ."2-a (r nvr) + -'-B o-(nv'l') = ()r r rsm <p

from system (28), Because the system has a symmctry axis, derivativos of the field functions with respectto <pvanísh, Jt immediately follows that

fr(B)vr(r,()= 2 ( B)'r n r, (30)

and Eqs. (28) become

o (mv;' ) _ l1W~- -- + tu - e<J! - --,ar 2 r

(31)

1 a rnv2

;: 8B (w - e<J!) = 7cotO,

(a1.'", v"') _ ov,, + -,

81' r

(32)

(33)

219

where the variable w is determined by the relation

and has the form

w = an2/3 - bn1/3 + en, (34)

= (211"1i)2 (~) 2/3

a 2m 811" 'e = 811"JI.~. (35)

Equation (33) has two solutions, including the solution Vr = O. Integrating this equation for Vr t- O,we find that the 'P component of the hydrodynamic velocity is

v!p(r,O) = f!p(O).r

(36)

Substituting Eqs. (36) and (30) in Eqs. (31) and (32), we obtain the system

mf;(O). mf;,(O)2r4n2 + w(r, O) - e<I>(r, O) + 2r2 = C(O),

D(O)w(r, O) - e<I>(r, O) = -2-'

r

D1(O) = mf;,(O) cot é.

The functions fr(O), f!p(O), C(O), and D(O) are determined from system (3i). It is easy to obtain theexpression for the electron density from system (37):

(37)

n(r,O) = /m~ Ifr(O)1 .y 2" r JC(O)r"J - D(O) - mf~(O)/2

(38)

We show that this expression leads to an incorrect result for the number of electrons. Indeed, theintegral of expression. (38) over the entire volume occupied by the system must he finite and equal to thenumber of electrons in the atom. However, the number of electrons in the atom calculated from formula (38)is

N = 211" /m r sinOdO Ifr(O)1 reo ---r===r=d=r====V 2" Jo Jo vC(O)r2 - D(O) - mf~(H)/2

It is easy to see that this integral diverges; therefore, Vr = O. Taking this result into account, we obtain

(r!... - tan o!...) (w - e<I» = OBr 00

from Eqs. (31)-(33). The variables r and O in this equation are separated, Le.,

w(r, O) - e<I>(r, O) = R(r)T(O) - e<Po,

where the functions R(r) and T(O) satisfy the equations

_r_dR(r) _ kR(r) dr - ,

tan OdT(O) = kT(O) dO '

220

~o is the value of the potential ~(r, O) at infinity, and k is a constant determined from t;!J(' boundarvconditions. Integratin!? these equations, we obtain R(r) and T(O):

T(O) = A2 sill~: O.

We then obtain the integral of Eqs. (31)-(33),

(30)

The electron density and the pot.ential ~(r, O) rapidly decrease as l' ~ oo; therefore, k: < O iu Eq, (39). 011

the other hand, substituting Eq. (39) in Eq, (31), we obtain

Because k < O and the right-hand side of the equation is positive, A > O. In the right-hand side, sin O cantake either positive or negativc values; therefore, the express ion in t.he right-hand side takes positivo valuesonly for even k. Hence.•k = -2/, where 1= 0,1, .... Setting the potential to be zero atoinfinity, we obtuiuthe final expressions for the velocity v", and integral (39) of Eqs. (31)--(33):

mv2 .--'" = 2A.fr-12/+1) sin-21 O,r

(40)

A .'an2/3 - bn}/3 + en - ec!l(1', O) + ,.

21sin--.2f fl = O. (41)

Quantity (40) has the sense of the specific electron angular momentum. It can be seen from Eq. (40) tha.tthe conditions at the system boundary lead to a discrete spectrum of t.his quantity. The expression in t.IH'ríght-hand side of Eq. (40) is infinite for O = O,7r, but. this should notoresult in any inconsistency, hecauscthe expressions for the integral system characteristics (i.e., the total enorgy, the t.ot.a.Iangular 1I10lllcntUII1.

etc.) contain the hydrodynamic velocity v",(-r, O) only in the product with the electron density 11(1', O). Forexample, the atom angular momentum is determined by the formula

171' 100. 11lV2 171' . 100.L",(A,f) = 27r sinOdiJ ,,2 dr --'" = 47rAf (sinfl)-2f+1 dO r-2/+1 d7'1t(r,O).

o o r o o(42)

The requirement for no singularities in nv", can be interpreted as one of t.he conditions we irnpose on thedensity n(r, O) when solving the problem.

Although the left-hand side of Eq. (41) is the integral of the quautum hydrodynamic equations aIHI

its form and dimension are similar to those of the specific energy of HII electron in an atom, it is 1I0t theelectron specific energy. In fact., the energy c1ensityof the systern has tite form

(43)

(where the translational energy is taken into account.). On thc other haud, comparing relatíous (13), (18),and (23) with Eq. (34), it is easy to obtain the relat.ion

3 15-PF = _an5/32 2 '

- lb 4/3é = -- ne 8 '_ 1 2

éS11ill = ¡C1L (44)

221

- ------------------

It then follows that the electron specific energy is

15 2/3 1 1/3 1 Al. 2/ 1 ) 1 eZe(r (}) = -an - -n + -en + - sin" (}- -e~(r (} - --o, 2 8 2 r2/ 2' 2 r (45)

Using the expression for integral (39), we reduce the power of the concentration n(r, (}) in formula (45):

e(r, (})= 59 bn1/3 _ 7en _ 13 Al sin-2/ (}+ 13 e~(r, (}) _ !eZ.8 2 r2/ 2 2 r

Then the total energy of the system is

(46)

{~ {OO (~E(A,I) = 211' Jo sin (}dOJo r2 dr n(r, O) Sbn1/3(r, O) - Tctiir, O) -

13 Al . -2/' ( 1 eZ)- - - sm O + 7e~ r O) - --2 r2/ '2 r . (47)

Equation (41) is considered together with Eq. (29) for the electric field potencial ~(r, O). It is convenientto introduce the auxiliary function ~ connected with ~ through the relation

eZ Aje. -2/ -~(r, (}) = - + -r/ sin O + ~(r, O).r r

Equations (41) and (29) become

an2/3 _ bn1/3 + en _ e~(r, O) _ e2Z = O,

r (48)

8 ( 28~) 1 8 (. 8~) .)Aje 2(/+1) .)- r - + -.-- smO- = -(21)-- sin- (}+ 47l'er-n.8r 8r sm 080 80 r2/(49)

The system of Eqs. (48) and (49) includes the Thomas-Ferrni-Dirac model for the atom ground stat.e.Indeed, because of the boundary conditions for the ground state (where I = O), we must set A = O,and we thus obtain the Thomas-Fermi-Dirac model for a multielectron atom. The only difference is thatthe equations we derived here account for the exchange spin-spin interaction energy, ca1culated using animproved spin-spin interaction Hamiltonian (1-9).

Finally, we estimate the contribution from the exchange spin-spin interaction to the energy of a multi-electron atom (compared with the Coulomb interaction energy). We find the elertron concentration valuecorresponding to the situation where the exchange energy density of the spin-spin interact.ion is equal tothat of the Coulomb interaction, i.e., I¿el = I¿spinl. In this case, we irnmediately obtain

n = (~)3/221:

from Eq. (44). Substituting the expressions for b and e (see Eq. (35)) in this equation, we obtain

1 ( 3) 2 (me) 3 . (me) 3 28 _:n = 81/2;;: T ~0.332 T ~14.4 . 10 cm .

Therefore, for the electron concentration ofthe order of 1028 cm-3, the exchange energy densities of the spin-spin and the Coulomb interactions practically coincide. If the electron concentration is above this value,

222

--------~-

the exchange energy density of the spin-spin interaction is higher thau that of the Coulomb interaction,because the latter energy increases more slow ly as the concentratiou incroases, as can be seen from Eq. (4.1).

The electrou density can be estimat.ed as

Zn~-4 3/3'71Ta •

where Ta is the atom radius and Z is the number of elect.rons in the atom. Because the atom radius is ofthe order of 1Q-8cm, the electroll density is of the order of 11. '" Z· 1(fHcm-3. 1"01'example, in an ironatorn, this conceutration is of the order of 1026 cm-3•

Usiug the quantum hydrodynamic equations, it is easy 1.0 describo t.he linear kinetic properties 01' aquantum collisionless plasma with a self-consistent field [6J.

REFERENCES

1. L. S. Kuz'menkov, S. G. Maksimov, and V. V. Fedoseev, Tlieor. Mat;!I. l'iI,l''<;., 126, 110 (2001).2. D. A. Kirzhnits, JETP, 5, 64 (1957).3. M. Gell-Mann and K. A. Breuckner, Pbys. Rev., 106,364 (1957).4. M. Gell-Mann, P!lys. Rev., 106, 369 (1957).5. A. S. Kompaneets, JETP, 4, 730 (1957).6. M. V. Kuzelev and A. A. Rukhadze, Phys. Usp., 169,687 (1999).

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