ELEMENTARY PARTICLE PHYSICS AND FIELD THEORY

THOMAS-FERMI MODEL OF A MANY-PARTICLE OSCILLATOR

A. V. Glushkov UDC 539.19+539.142

An equation of the Thomas--Fermi type has been obtained and solved numerically for a many-particle oscillator - a system of N electrons in a potential field of a harmonic oscillator. The electron density distribution in the system has been found. An analogous model has been formulated in the theory of atom- ic nucleus and used to calculate the charge density distribution in 4°Ca in good agreement with the available experimental data.

I. Let us consider a system of N electrons (electron gas) in a potential field of har- monic oscillator of the form

U (r) = i ~r ~, (1) 2

where k is the coefficient of quantum "hardness" [1]. The study of this type of system is stimulated, first of all, by their possible unique properties, particularly magnetic ones. The case of N = i corresponds to a well-known classical problem in quantum mechanics, that of a harmonic oscillator [2]. Calculation of the electron density distribution, n(r), in the system being considered using traditional self-consistent field methods for large N is, naturally, lengthy, and, therefore, in this case, it makes sense to use a more universal statistical Thomas--Fermi method [2, 3]. In the statistical approach below, the ground state of the many-particle oscillator is considered, and, therefore, the electron gas should be assumed to exist at zero temperature and should be considered semiclassically because the majority of electrons in the system occupy states with relatively high quantum numbers. Quasiclassical approximations are applicable under these conditions [2].

Within the Thomas--Fermi method we shall obtain an equation for the general potential % in which electrons of the system are assumed to be moving. Using the well-known relations (see [2, 3]) for ~(r) we can write the following expression (in atomic units):

a (~ - ,%) = 8 K 2 ( ~ _ ~ov~,,,, , ( 2 ) 3r=

where ¢0 is the chemical potential. For the electron density of the system

n ( r ) = 2 Vf[~ .a,=, , (r) - s]3/2, ~ >- %, (3)

n ( r ) = o, ,~ < %.

Starting from the natural assumption about a spherically symmetric distribution of the electron density, performing a substitution of the form

(4)

where ~ = k 116, i n t o Eq. ( 3 ) , and making a t r a n s f o r m a t i o n t o a new d i m e n s i o n l e s s v a r i a b l e x = ~r, we will arrive at the following equation of the Thomas-Fermi type for the function x ( x ) :

6X (x) + 6xx' (x) + x~x " (x) = A~3X ~12 (x), (5)

where A = 8 / ] / 3 ~ . The bounda ry c o n d i t i o n a t z e r o , t a k i n g Eq. (4) i n t o a c c o u n t , i s o b v i o u s l y g i v e n by

Odessa Hydrometeorological Institute. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 66-72, July, 1992. Original article submitted January 13, 1991.

1064-8887/92/3507-0641512.50 © 1993 Plenum Publishing Corporation 641

x(O) ~--I. (6)

The boundary of the spherically symmetric oscillator in the statistical model is deter-

mined by the condition ~ (r 0) = ~0. be determined through the expression

On the boundary itself

1 Kr~ ,~ (to) = 7

Outside of the charge distribution, the potential may

,~(r) = l ~ r ~ - - N (7) 2 r

In particular, according to Eq. (8), ~0 = 0 for an oscillator with k = 2N/r03 In or- der to determine constants ~P0 and r0, one should consider in addition to Eq. (8) one other relation (which is a consequence of the Gauss electrostatic theorem)

Q = - - - - ~ - - / 4~.r o = - - r~- ( 9 ) 4~ \ c)r].=.o ~, Or , .=.o

F o r a d i m e n s i o n l e s s v a r i a b l e , t h e b o u n d a r y c o n d i t i o n t a k e s t h e f o l l o w i n g f o r m :

Z(xo) = 0 , Xo=~ro. (10)

We note a universal character of the condition (6) for various oscillators with differ- ent k, N, whereas condition (i0) is, generally speaking, nonuniversal because quantity x0 may be different for different oscillators. Relation (9), expressed in terms of new vari- ables, takes the form

Q = - V E x ~ [z' ( x ) l x : x o - K f i x ~ z ( X o ) 2

or, taking Eq. (i0) into account

Q - xg ['Z.' (x)]x=~o. (11) 2

For an oscillator with k = 2N/r03 it follows from Eq. (ii) that

z'(Xo)=O. (12)

It is easy to verify that Eq. (5) has an exact solution in a power form [i]

X ( x ) _ 144 (13) A 2 x 6

We n o t e a l s o t h a t f o r a T h o m a s - - F e r m i a t o m , an a n a l o g o u s s o l u t i o n h a s t h e f o r m [2]

144 z (x) -

X 3

One can show that solution X(X), which goes to 0 as i/x 6 for x ÷ ~, can be written as

the following series:

144 { 1 - F F ~ F :~ '~ Z(x) =A2x6~ ~ / , ( T ) -F x2 Tn-f.(T). -- x3--~, 5 ( 7 ) + . . . ) , (14)

where "/ = (~/73 - 7 ) / 2 , f : ( ¥ ) = 1; n(n+ I ) n t n + i )

r~=O Tn=O

F is some parameter, which determines the slope of the curve ×(x) at a point x = 0; Pnm and qnm are some coefficients calculated by substituting expansion (14) into Eq. (5). It is curious to note that for a Thomas--Fermi atom, the coefficient analogous to 7 in an expansion of the type (14) is also equal to (v~73 - 7)/2 (see [3]). Substitution of Eq. (13) into rela-

tionship (4), and then into formula (3) gives

I Iim n (r) Go -- r ~ o o r 6

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As a result, the electron density in the model for a many-particle oscillator being considered, decreases on the boundary as i/r 6 (a quantum mechanical calculation would ob- viously give an exponential decrease). As should have been expected, this result points to the known incorrect behavior of the statistical model when describing boundary regions of a given many-electron system, where the electron density becomes too low. One should remember that the local relationship used in the Thomas--Fermi model, according to which the kinetic energy density is proportional to nS/3(r), is a serious deficiency of the theory. The situ- ation in this case is again analogous to the one in the theory of the Thomas-Fermi atom. As is shown by the analysis, in point x = 0, Eq. (5) has a discontinuous solution. If one follows an analogy with the Thomas--Fermitheory of nuclear matter [5], this discontinuity may be explained by the fact that exchange interaction is not taken into account.

Solution of Eq. (5) with boundary conditions (6), (12) was found numerically with the help of the Adams method [4]. The graph of function X(X) is given in Fig. i. The electron density distribution n(r) [the normalization condition is fn(t)dV = N] is easily found tak- ing (3) and (4) into account.

/¢3/2

n (r) = 3= 5 r ' z 3t2 (~r), ( 1 5 )

Using the fact that the chemical potential is constant within the charge cloud and the virial theorem, we will find the following value for energy of the oscillator:

Of x0

E = * ' to 7/¢, I = [ xTx a/2 ( x ) d x "~ 10 -9. ( 1 6 ) 3= o J

3 In the specific case of the condition k = 2N/r 0 we will have from the above expression

213fG. I NTlO E= - - (17)

3= r~12

We note that generalization of the model to the case when exchange, correlation, quan- tum, and shell effects are taken into account may be performed in a standard way, by analogy with the Thomas--Fermi model of a many-electron atom.

2. A natural application of this Thomas--Fermi model to a many particle oscillator is a calculation of the charge density distribution in atomic nuclei. Formulation of the model for nucleons in a nucleus is analogous to the electron model. Unperturbed potential energy, acting onto nucleons in a nucleus, may be written in the form [5]

1 A4m2r2 ( 1 8 ) u ( r ) = - c + 2

w h e r e c o n s t a n t C . i s a p p r o x i m a t e l y e q u a l t o t h e p o t e n t i a l e n e r g y i n t h e n o r m a l n u c l e a r m a t t e r (C = - 8 0 MeV) . I n o r d e r t o d e t e r m i n e ~ and t h e r e f o r e k = M~ 2 one can use a s t a n d a r d f o r m u l a [5]

h~o=40.A-1/3, MeV. (19)

Furthermore, if one takes Eq. (18) into account one can derive the Thomas--Fermi equa- tion of type (5) for a nuclear function Xn, and also can write down an expression for density n n and energy E of a nucleus. The equation for a general potential ~, in which nucleons are assumed to move, takes the form

(? - - ~o) 8 ~ 2 ,~3,2 (~ ,~,2 . . . . ?o) ' • 3= z

We have for the nucleon density distribution the following expression

n n ( r ) - 2 V 2 A43~2 (? _ ?oy/2, ? (r) >~ ,~o, 3~ 2

n~ (r) = O, ? (r) < %.

Performing now a substitution of the form

~ - - ?o = ( l t c r ~ - - C) Zn(ar)

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~z,~ 4

°,e i ~ o, o6 J,2 _~ o, o2 o o, oo~ o, oo3 qoo,0 x o

~.,,3. ~ \

~r

t i L - -

2 4 r ,~

Fig. 1 Fig. 2

Fig. 2. Charge density distribution in 4°Ca. Curve 1 is from experimental data on electron scattering; curve 2 is a Hartree-Fock calculation based on forces independent of the density; curve 3 is a Hartree-Fock calculation but based on the forces which depend on the density; curve 4 a Hartree- Fock calculation with 3 and 4 particle correlations taken into account; curve 5 is from the Lagrange quasiparticle method; curve 6 is from the present calculation.

and making a transformation to a new dimensionless variable x = ~r, we arrive at the follow- ing equation of the Thomas--Fermi type for the function Xn(X):

X~ ~. (20)

Taking i n to account the d e f i n i t i o n ~ = C/~ 4, Eq. (20) w i l l be r e w r i t t e n in the fo l low- ing form:

~×~(x) + (6x - ~8/x) zR(x) + (x ~ - 28) zR(x) - 161/~Maj2 (x~ \B'2 8': 3~ 7 - a , J z ~ (x). (21)

For density n n we obtain

n n ( r ) = _ _ 2 ~r~ iV/a12 Kr 2 - - C Zn • ( 2 2 ) 8~ 2 /

Figure 2 shows the charge density distribution in a 4°Ca nucleus, obtained with the help of the relationship (22) and using a preliminary numerical solution for function Xn [we emphasize that ~(k) was determined according to (19)], with an additional numerical co- efficient g(x) introduced into Eq. (22) [see (23)]. In this case the charge density distri- bution that we have calculated agrees well with experimental data, especially in comparison with the results of other calculations also given in Fig. 2, such as Hartree-Fock calcula- tions with the forces independent of the density, analogous calculations but with the forces dependent on the density, similar calculations with addition of 3 and 4 particle correla- tions, and calculations using the Lagrange quasiparticle method [5-14]. Unfortunately, without taking into account the additional numerical coefficient, the charge density dis- tribution calculated in the present model agrees rather poorly with experiment. This should be attributed to the known simplicity of the Thomas--Fermi model, noninclusion of such impor- tant effects as exchange, correlation, shell, etc. However, one should remember that up to now attempts to apply the Thomas--Fermi model to atomic nucleus turned out not to be very successful, and, unlike atoms, a complete Thomas--Fermi model does not exist for a nucleus. It is sufficient to indicate first consistent attempts to construct such a theory, under- taken in [7, 8], where equations for the charge density and for the general potential were obtained based on a realistic effective density functional using the variational principle but their numerical solution was not successful. Calculations by Gombas, et al. [15] within the framework of the standard Thomas-Fermi technique lead to too smooth change in the den- sity, whose dependence on the radius turned out to be similar to the Gauss curve, which con- tradicts experiment. In a number of papers attempts were made to derive the so-called im-

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proved Thomas-Fermi equation in an integral form; however, these attempts were also unsuc- cessful. A number of papers on the use of the Thomas-Fermi method in the theory of nuclear matter is described in [13, 14]. A more detailed description of various attempts to con- struct the Thomas-Fermi theory for a nucleus will be presented in a separate paper, where we will also give a more detailed comparison between our model with some improvements and various versions of the Thomas-Fermi model of the nucleus. The result similar to the one given for a 4°Ca nucleus (we are referring here to the charge density distribution) was also obtained for a 2°sPb nucleus within the framework of this model. Agreement with experiment is also achieved by introducing into Eq. (19) an additional multiplier g(x). For small x < x 0 0.003, the calculated density, without g(x) taken into account, exceeds experimental value by more than a factor of 2, and for x > x °, the curve for the density distribution is lower than the experimental curve. This fact is, obviously, related to the deficiencies of the model discussed above. In order to correct these deficiencies the multiplier g(x) was chosen in the form

g ( x ) = B exp[610 ,71x -- 15054,24x ~ + 142378,56x3], (23)

where constant B is found from the normalization condition for the new nucleon density n(x) = g(x)n(x) (B = 0.35).

3. In conclusion, we consider a relativistic generalization of the Thomas-Fermi model for a many particle oscillator, using the method analogous to the Wallart-Rosen method in the relativistic Thomas-Fermi theory for an atom (see, for example, [3]). Following the nonrelativistic approach but subtracting from the beginning the rest energy mc 2, we will write down the quasiclassical equation of motion for the fastest electron. If PF(ri) is the momentum of the fastest electron in point ri, and ~(r), as usual, is the self-consistent potential energy, in which electron moves, then we will have, using the standard relativis- tic expression for kinetic energy (a reminder - atomic units are used)

- % = [ d p ~ - + c,],r'- - c~ - ~ ( r ) . (24)

Momentum PF is related to the electron density n by a standard relation

n (r) = p~ (r),/3~'. ( 25 )

We also take into account the self-consistency condition, contained in the Poisson equa- tion

A (~0 - - ~) ------4~n (r). ( 2 6 )

A p p l y i n g now s c a l e t r a n s f o r m a t i o n s in t h e same way as in a n o n r e l a t i v i s t i c c a s e , we u s e d i m e n s i o n l e s s q u a n t i t i e s ( x ) and x d e t e r m i n e d e a r l i e r . T a k i n g s p h e r i c a l s y m m e t r y o f t h e p r o b l e m and Eq. ( 4 ) i n t o a c c o u n t , we w r i t e down e q u a t i o n (16 ) i n t h e f o r m

1 d" r d r 2 [r (% - - ~)] = - - ~- [6X (x) + 6x X' (x) + x~z " (x)] = - - 4~n (r). ( 27 )

Interchanging the terms in Eq. (24) and taking a square we will have

(~ _ % ) 2 + 2c2 (~ _ % ) = d p ~ . (28)

I n t h i s c a s e , we w i l l h a v e in t e r m s o f v a r i a b l e s ~ , x , and n

p ~ _ I (3~)~!8nv 3 (~--~0) 2 t~2/3 2 2 2C 2 + ('~ - - %) = u41a x~"2 + x2z" ( 29 ) 8c ~ ~ T

Using Eq. (27) for density n, and performing certain interchanges of the terms, we will arrive at the final result

l o Z + 6 x z ' ( x ) + x ' z " ( x ) ] 2 ~ 3 - - x ~ X ( X ) l + - ~ c , X ~ X ( x } . (30 )

It is easy to see that Eq. (30) in the nonrelativistic limit, i.e., c + ~ [with the fact that the left-hand side of (30) is independent of the speed of light taken into account] is reduced to the correct dimensionless nonrelativistic equation (5). Numerical solution of Eq. (30) may be obtained in the same as the above solution of Eq. (5) by using the Adams method. It is obvious that the total energy of a many-particle oscillator in a relativistic

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approximation will depend not only on parameter k, number of electrons N, but also on the quantity i/c 2 (in atomic units or (e2/hc) 2 in normal units, i.e., the fine structure con- stant.

The author expresses his deep gratitude to Yu. E. Lozovik for useful suggestions, which stimulated the present work, as well as to A. V. Tarchenko for his help in performing numer- ical calculations.

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