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The Thomas-Fermi model: momentum expectation
values
I.K. Dmitrieva, G.I. Plindov
To cite this version:
I.K. Dmitrieva, G.I. Plindov. The Thomas-Fermi model: momentum expectation values.
Journal de Physique, 1983, 44 (3), pp.333-342. <10.1051/jphys:01983004403033300>. <jpa00209602>
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Submitted on 1 Jan 1983
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J.
Physique 44 (1983)333-342
MARS
1983,
333
Classification
Physics Abstracts
31.10
-
31.20L
The Thomas-Fermi model: momentum
expectation
values
I. K. Dmitrieva and G. I. Plindov
Heat and Mass Transfer
Institute, BSSR Academy of Sciences, Minsk, USSR
(Reçu le 12 juillet 1982, accepté le 30 novembre 1982)
Résumé. 2014 Les expressions analytiques de toutes les valeurs moyennes des impulsions pb&#x3E; et de quelques
de la densité électronique 03C1m&#x3E; pour les atomes dans un degré d’ionisation arbitraire sont obtenues
dans le cadre du modèle de Thomas-Fermi compte tenu des corrections d des à l’échange et à la contribution
des électrons fortement liés. On montre que le traitement correct de celle-ci permet d’obtenir une estimation
quantitative de pb &#x3E; et 03C1m&#x3E; lorsque 3 ~ b 5 et 1 ~ m 5/3. La dépendance des coefficients du développement de pb &#x3E; et 03C1m&#x3E; en Z-1 est donnée explicitement en fonction du nombre d’électrons.
puissances
Abstract
Within the Thomas-Fermi model including the exchange interaction and contributions of strongly
bound electrons, analytical expressions are obtained for all momentum expectation values pb&#x3E; and for some
of the expectation values of powers of the electron density pm&#x3E; for an atom with an arbitrary degree of ionization.
It is shown that a correct treatment of strongly bound electrons gives a quantitative estimate of pb&#x3E; and 03C1m&#x3E;
within 3 ~ b
5 and 1 ~ m
5/3. The Z-1 expansion coefficients for pb&#x3E; and 03C1m&#x3E; are given as an explicit
function of the electron number.
-
1. Introduction.
Recently [ 1 ], asymptotic estimates of the expectation values of electron positions
( r a &#x3E; and of momentum pb) have been obtained
for a neutral atom and for an atom without electronelectron interaction within the Thomas-Fermi model.
In the previous work [2], study was made of rO )
for atoms with an arbitrary degree of ionization on
the basis of the improved TF model. Here we shall
study ( p’ &#x3E; and related expectation values of powers
of the electron density ( p‘" ).
The quantum determination of ( pm )
-
in the momentum space. In (2) Io is the
electron momentum density. The range of the validity
of (2) is restricted by the behaviour of lo(p) at p -+ 0,
p - oo namely, lo(p -+ 0) const. [3] and lo(p -+ oo) =
8 Z. p(o) p- 6 [4], and is given as - 3 b 5.
Alternative determination of pb ), relating pb &#x3E;
with the isotropic Compton profile, Jo(q), [4] :i
equation
is a rather tedious problem requiring the solution of
the N-particle Schrodinger equation in the coordinate
space.
Still more difficult is the search for ( pb &#x3E; (p =I p I)
( pb ) to be found from the experimental
Compton profiles. Both these methods cannot give an
analytical dependence of p’ &#x3E; on the electron number
N and nucleus charge Z.
The present work is aimed at obtaining analytical
estimates of expectation values ( pb ) and ( p’" ) by
using the Thomas-Fermi model with account for the
exchange interaction and contributions of strongly
bound electrons. Systematic trends in ( pb ) and
( p"‘ ) will also be analysed.
requires either the Fourier transformation of
spatial wave function or solution of the Schrodinger
2. Statistical model.
In the frame work of the
Thomas-Fermi (TF) and Thomas-Fermi-Dirac (TFD)
allows
which
a
-
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01983004403033300
334
models, the
equal to :
state
density
in the
phase
space
[5] is
equal
The value of
( pb &#x3E; (b
&#x3E; -
3), based
on
(4), is :
beyond
to zero.
atom we
where pF(r) is the Fermi momentum at a distance r
from the nucleus and 0 the Heaviside theta-function.
Integrating (4) over momentum, it is easy to establish the relationship between the particle number
density and the Fermi momentum :
which the electron density is
in
Therefore, the TF model for a neutral
have :
finite radius, ro,
This range includes 4 moments of the momentum
1, 0, 1, 2. The zero moment is reduced to the normalization integral. The
values ( p2 &#x3E; and ( p &#x3E; determine kinetic and exchange
energy (in a local approximation) and ( p-1 ) is proportional to Jo(0), (2). The virial theorem links ( p2
with the binding energy which has been earlier investigated in detail within the TF model [6]. Therefore
p2 &#x3E; values will be considered here only for the
completeness of the analysis.
For an ion, the range of the validity of (6) is extended
distribution, namely, b
=
-
to :
Here and below, atomic units are used.
With regard to (5), the value of pb &#x3E; may be
expressed in terms of pm)
which allows an estimate of p- 2 ).
When the exchange is taken into account, the neutral
atom is bounded and the range of validity of (6)
coincides with (9) within this model.
Let us use the TF equation
or the TF equation with regard for the exchange in
simple relationship between the first order with respect to fl [2, 8] :
momentum and electron density expectation values;
it is exact if the electrostatic and exchange interaction
is taken into account. This relationship is broken by
allowing for contributions of strongly bound electrons, or for the inhomogeneity and the oscillation of
the electron density.
where
the dimensionless radius x
Expression (6) is the basis to study pb &#x3E; as a func- and thef3 = 2 (6 nZ)-2/3,
function
screening
03C8(x) are related to r and
tion of N and Z within the statistical model. The applicability of (6) is specified by the behaviour of the sta- p(r) by :
0 and r
tistical density at r
oo. In the simple TF
Equation (7) gives
a
-
-
model for
p(r - oo)
Let
us
-
neutral atom p(r -+ 0) - r- 3/2 and
r- 6. The ion in the TF model has a
a
present (6) in the following form :
or
without and with account for the exchange interaction, respectively. In (12)-(13), xo is the boundary ion radius
in the TF model (for a neutral atom xo - oo) and xeX is the boundary radius when including the electron exchange
interaction.
335
Within -
b
3/2
3, ( pb &#x3E; for
a
neutral atom may be
given
as a sum :
where
Here
( p6 &#x3E;TF has
a
universal form
[1, 7] :
9.175 8 ;
B(1) 0.693 75 ;
B(2) 1.537 5 .
Special consideration must be made of the exchange contribution.
Using the expansion of qlo(x) and 1 (x) at x &#x3E; 1 [8] :
B(- 1)
6
=
=
=
( 73 - 7)/2, it is easy to
Numerical
=
see
that for -
1/2
b
The situation is
for the
negative
Expression
estimating ( p-1 ). However,
more
complex
may be used for
since the integrals in the P-’ &#x3E;ex estimate substantially depend on xeX, their analytical estimation is
impossible. Because of xex - Z 1/3 [8], expression (15)
gives only
(Z &#x3E; 1) :
a
p6 )eX may be given as :
integration in (15a) yields :
moments of the momentum distribution.
(14)
2, the main part of
qualitative asymptotic dependence
being filled. The SCF-data obtained from HF Compton profiles [ 15] exhibit an explicit periodic dependence
of A p-’ &#x3E; = P-’ /HF B p 1 /TF and C p 2 )HF
on Z 1/3 (Fig. 1). The major maxima of the curves
Ap-1 &#x3E; (Z 113 ) and p-2 )HF(ZI/3) correspond to
alkali-earth atoms. The positions of major minima
correspond to noble gases. As is seen from figure 1,
the oscillation amplitude for heavy atoms is independent
on
Z. Thus, the oscillation contribution of
has a relative order Z - 1/3 , being a leading
(p-l &#x3E;
for (p-2 ).
term
When estimating p - I &#x3E; for a neutral atom, (14) is
not valid since the integrals in (12) and (15) diverge;
when passing to simultaneous consideration of the
integrals in (13) and taking into account that the
integral on the RHS is mainly determined by x - xeX,
we find a qualitative asymptotic dependence at
Z&#x3E; 1 :
which confirms the result given in [ 1].
The expectation values p - ’ 1 &#x3E; ex and p - ’ &#x3E; are
determined by the external regions of atoms, and for
real atoms must oscillate as the last electron shell is
.
pass to the estimate of (ph) for ions.
b
3.
is valid for all b within - 3
In order to obtain ( p’ &#x3E; as an explicit function of N
and Z, we use the expansions of the screening function,
qlo(x), in the TF model and exchange correction,
Now
we
Expression (14)
t/J1 (x),
into series in the parameter ,
336
The closed form of the functions cpk(x/xo) and
xk(x/xo) enables one to obtain the exact values of
Bk(b) and Bk X (b). Here Box(b) is given for arbitrary
b(-3b4) :
Bl (b) is found in a closed form for integer b. This
value being very bulky, we present only numerical
values together with B2(b) and BlX(b) (Table I).
Comparison of Bk(b) and Bke’(b) leads to the conclusion that the exchange interaction increases ( pb &#x3E;
for b &#x3E; 0 and decreases it for b
0. From table I,
and
it
is
(19)
(16)
expressions
easy to see that (19)
with three expansion terms well reproduces ( p2 )TF
and p &#x3E;TF, including the neutral atom (error does
not exceed 0.6 % and 0.9 %, respectively). Based on
(19), the values of p-2 &#x3E;TF and P-’ &#x3E;TF may be
calculated with good accuracy only for small N/Z.
The value of ( p6 &#x3E;TF for a slightly ionized atom
must be studied to improve (19) for ions with NIZ - 1.
At NIZ - 1, #o(x) may be given as :
Fig.
1.
-
The values of p-’&#x3E;,, and Ap-’&#x3E; =
obtained from HF Compton proa function of Z 1/3 : 1, p -2&#x3E;HF; 2, A p-’ &#x3E;.
P-’&#x3E;HF - P-’1 /TF
files
[15]
as
The functions qJi(Y) and Xi(y) are presented in [2, 9].
Substitution of (17) and (18) into (12) and (13) and
regard to J, and xo as a function of N and Z (cf. 2)
give ( pb )TF and p6 &#x3E; ex as an N/Z series expansion :
is the function for a neutral atom and
is
function. Using the asymptothe
correction
t/lOl (x)
tic expressions for t/loo(x) and t/lOl(X) [2], #oo(x) 144 x-3(1 + O(x-U)); t/lo 1 (x)
Ax4+u(1 + O(x-U))
and taking into account
where
=
from
The value of
was
Bo(b) equal
obtained in
Table I.
-
to
[1].
Values
of Bk(b)
and B;X(b).
t/loo(x)
(12),
we
obtain the
asymptotic expressions :
337
Expressions (23a) and (23b) determine an approximate
type of singularity at N/Z - 1. We think it expedient
to present p-2 )TF at NIZ - I as :
series
f 1 (N/Z) is a function having no singularities
N/Z 0 and N/Z 1.
Equations (19), (20) and (23) give systematic trends
of ( p’ &#x3E; to be studied at a large electron number.
It is easy to see that ( pb &#x3E; obtained from (19) and
(20) may be presented as the Z -expansion :
HF data
Table II.
-
Values
of Ne and Ar ;
of p’ /3 &#x3E; for
TF model with
the isoelectronic
exchange (26a).
where
at
=
=
[11]
are
given
in brackets.
--&#x3E;
the asymptotic expressions for the coefficients Dk(N,
within - 3 b
3 being of the form :
b)
For ( p2 ), Dk(N, b) as a function of N and k is well
studied in [6]. It is shown that the TF model gives a
reliable estimate of the Z - expansion coefficients. In
the present work, the values of the three first Z -11
expansion coefficients for p’ &#x3E; for b =1= 2 are obtained
for the first time. The coefficient of the higher power
of N in (25) is exact (see, next section), thus the quality
of Dk(N, b) increases with growing N. The deviation
at moderate N is related to the fact that (25) does not
The value of Go(N,
Bk(b).
We studied pb) in detail. Expression (7) shows
that all results obtained in this section are, to the same
extent, related to expectation values p"’ ). For
example, (25) and (24) are used to estimate Z -1
expansion coefficients for ( p’ &#x3E; :
coincides with the
m)
Expressions (25)
for many-electron
involve the contributions of strongly bound electrons,
the inhomogeneity of the electron density and oscillations. The first of them may be very substantial and
will be considered in section 4. The correction for the
electron density inhomogeneity has the same relative
order as the exchange contribution but with a smaller
factor; this correction being neglected, a small error
will be made in the estimate of ( pb ) at any b and
N/Z. The oscillation contribution will be briefly discussed in the next section.
Equation (25) gives an important property of the
Z -11 expansion coefficients for ( p’ &#x3E; : the ratios
Dk + 1 (N, b)/NDk(N, b) quickly tend to a constant
determined by the TF model and equal to Bk, l(b)l
and
(26)
are
atoms at k &#x3E; 0
the
one
obtained in
[10].
only estimates of the Z -1 coefficients for ( pb ) and pm ) expansions
(except ( p2 &#x3E;). Therefore,
other data. To illustrate the quality of
we could not perform a direct comparison with
(25) and (26), we made a systematic comparison of pl /3 &#x3E; calculated by :
54 and N Z 20 + N. The maximum
with Hartree-Fock data [11] for isoelectronic series 10 N
error of (26a) does not exceed 8 % (isoelectronic series of Ar). The main error of (26a) is due to the absence
of oscillation effects being essential for open shell isoelectronic series as in studying binding energy (or p2 »
[6]. A typical behaviour of ( pl/3 &#x3E; for closed shell ions and open shell ones is demonstrated in table II. The data
of table II show that (25) and (26) may be used to reliably estimate ( pb &#x3E; and pm &#x3E; for an atom with an arbi3 (0 m
1).
trary degree of ionization for 0 b
3.
then
Non-interacting electron model. - If an atom is considered to be with no electron-electron interaction,
are found by summing over all occupied hydrogen-like orbitals :
( p6 ) and pm &#x3E;
338
Here 0,,,(r) are the orthonormalized radial wave functions and q,,, are the occupancy number for the orbitals
with quantum numbers n and l. The values (.pb )H and ( pm &#x3E;H determine exact quantum values of Do(N, b)
and Go(N, b). The last quantity may be calculated only numerically. The analysis of Do(N, b) allows exact analytical expressions to be obtained for closed shells.
The estimate Of pb )nl is given by the expression [3] :
where
Fnl( p) is the normalized radial function of the momentum distribution
Cm(x) are Gegenbauer’s polynomials [12].
Replacement of (n2 p2 - 1 ) (n2 p2 + 1 ) -1
=
u
in
:
(29) gives :
and use of the symmetry property of Gegenbauer’s polynomials C"(x) _ ( - 1 )"’ Cm( - x) results in a relationship between expectation values pb )nl for different b :
Thus, for integer b the problem is reduced
trivial.
To
to calculation of
only four moments,
one
of which, pl
), is
obtain pb )n’, a calculation must be made of the integrals J(n, I, b) equal to :
The
use
The exact
of the
explicit expression for Gegenbauer’s polynomials [12] yields :
expressions for ( p2 &#x3E;n’ and p4 &#x3E;n’ were derived in [3] :
Using (30),
from
(32a)
we
have :
Summing (32a) and (32b) over I for closed shells, ( pb &#x3E;H may be presented for even b as :
Numerical summation of (27) using (29),
odd b
1, 1, 3, DO(N, b) is also described
=
-
(31 a) and (31 b) for
by (33).
the four first electron shells shows
that, for
339
To find Do(N, b) as a function of N, summation is performed over n in (33) with regard
between the maximum main quantum number nm and N for closed electron shell atoms :
When
limiting
Here C
=
to
the terms of the relative order of
N - 2/3,
we
to the
relationship
have
0.557 216 is Euler’s constant.
It is easy to see that for - 1
b 2 the coefficients
with the leading power of N in (34c)434e) coincide
with the values of Bo(b) found by the TF theory,
equation (21). Comparing (34a, b) with the estimates
of ( p4 &#x3E; and ( p3 ) obtained by using the Kompaneets-Pavlovskii (KP) model [1], one may be convinced
that the KP model gives a qualitatively correct
description of ( pb ) within 3 b 5. This is due
to cutting off the. electron density within KP model
at small distances from nucleus and due to the dependence of the internal boundary radius on Z,
XJ(Z) _ z - 2/3
.
The asymptotic expressions (34a-f) perfectly describe Do(N, Z) for closed electron shells (33). For
open shells account must be taken of the effect of
oscillations, whose amplitude is of relative order of
N - 2/3 for positive and of N - 1/3 for negative b. The
oscillation effect is most substantial for expectation
value p-2 &#x3E; (Fig. 2).
The oscillation effects appear in ( p’ &#x3E; due to the
fact that the discrete quantum state electron distribution differs from the continuous one defined by
(34a-f). The analytical estimate of these effects may
be made using simple algebra as it was done for the
energy [6].
4.
Strongly
bound electrons.
of pb &#x3E; for 3
b
-
A correct estimate
5 and of ( p"’ ) for I K m K 5/3
may be made
only if the quantum effects near the
nucleus are allowed for. With these effects taken into
b
3 and of
account, the estimates of ( pb &#x3E; for 1
p"’ ) for 1/3 m 1 may be essentially improved.
A clear physical picture of strongly bound electrons
has been recently elucidated by Schwinger [13].
Based on this method, the expectation value ( pb )
is given as a sum of two contributions :
The first contribution is caused by strongly bound
electrons (with binding energy
B,8 1 - Z’) and is
calculated by the summation over the states of pb &#x3E;nl
for non-interacting electrons (27) :
Fig. 2. 2013 pb &#x3E;H/DS(N, b) as a function of N 1/3 : 1, b = 1 ;
2,b=-2.
340
Here n’ is the main quantum number of strongly
bound electrons ; n’ - (Z 2/2 E) is not obligatorily integer while [n’] is the integer part of n’.
The second contribution is calculated by the TF
model; the strongly bound electron contribution
incorrectly described by the TF model must be
eliminated from (12) :
2 n 2(Z p)- 1.
Here xm is the region of localization of strongly bound electrons, xm Z( - E,u)-1
Let us show that a similar result may be obtained if the TF contribution of N-non-interacting electrons
is eliminated from (12) and replaced by a quantum-mechanical quantity. This approach as applied to pb &#x3E;
=
yields :
Here in the first integral, 1
x/xoo is the TF screenfunction
of
ing
non-interacting electrons and
-
is the dimensionless radius of the
From (15) it follows that for b &#x3E; 4 the first integral
the RHS diverges, which shows that a strongly
bound electron contribution to ( pb &#x3E;ex (b &#x3E; 4) must
be taken into account.
This approach is especially convenient when
combined with the Z -1 perturbation theory. To
determine the expectation value of the local operator,
for example, ( pb ) it is sufficient to replace Do(N, b)
in (25) by the exact quantum quantity conserving
Dk(N, b) at k &#x3E; 0 from (25), i.e.
on
as a
=
ion with rio
electron-electron interaction. Intein
and
(36) and allowance for the relationgration (35)
between
and
n’ demonstrate that both apship
xm
coincide
correctly to the terms of the relative
proaches
order of N -2/3 . Approximately the same method was
used by Scott [14] to estimate the binding energy of a
neutral atom.
The approach based on (35) allows combination
of the advantages of the quantum-mechanical model
for non-interacting electrons and the TF model. The
first model correctly takes into account a contribution
of strongly bound electrons and partially another
quantum contributions (oscillations, inhomogeneity
of the electron density, etc.). The second model gives
an exact asymptotic value of the contribution due
to the electron-electron interaction.
Equation (36) may be supplemented with the
corrections for electron exchange interaction based
on (13). Bearing this in mind, pb &#x3E; (I K b
3)
may be given as :
TF
are calculated to determine
of N for b
function
3, 4 at k a 1.
Dk(N, b)
Partial integration of (36) gives :
p4 &#x3E;TF and p3 &#x3E;’TF
Substitution of (17) into (36a)-(36b), with regard to
the dependence of xo and )B on Z and N [2], yields
Dk(N, 3) and Dk(N, 4) at k &#x3E; 1 in the form of (25).
The values of Bk(b) at b = 3, 4 and of Bkex- 1 (3) at
k
1, 2 are listed in table I.
Expressions (38), (24) and table I give asymptotically exact (at N &#x3E; 1) values of the Z -1 expansion
coefficients for all moments of the momentum distribution.
For a neutral atom, the calculation of the integrals
in ( 15a) and (36) and using the values Of pb &#x3E;H from
(34) result in :
=
In (39d), the term - Z incorporates the contributions both of the exchange interaction and of
strongly bound electrons. Note that the amplitude of
341
Table III.
-
Values
of pb &#x3E; for
a
neutral
atom
(39).
Table IV.
-
Values of p &#x3E;.10- 2 for the electronic
; TF model with account for exchange
bound electron contributions (42).
series ofNe and Ar
and
the oscillation, not taken into account in (39d), is
also proportional to Z.
Expression (39a) was for the first time obtained in
[ 1], and the second term of this expression was found
in [13] when the leading relativistic correction to the
binding energy of a neutral atom was calculated. p3 &#x3E;
and ( p &#x3E; as functions of Z are first obtained here.
Comparison with HF data (Table III) shows that the
error of (39) does not exceed 10 % at Z &#x3E; 10 and falls
with increasing Z. We have calculated pb )HF
(Table III) on the basis of (3) using the isotropic HF
Compton profiles [15]. We think that the accuracy of
the HF data is about 0.5 % since for their calculation
the interpolation procedures involving the exact
asymptotic expression
were
used; p(O) is the electron density
strongly
Substitution of (17) into (40)
electronic series :
Expression (41)
gives ( pm &#x3E;
may be used to
for
an
iso-
obtain pm &#x3E; for
atom with an
arbitrary degree of ionization. To
check the validity of (41), we have calculated p &#x3E; for
the isoelectronic series of Ne and Ar :
an
at the nucleus
[11].
Since data on ( pb &#x3E; for ions are absent in literature,
it is impossible to qualitatively estimate an error of
(38). However, the fact that the limit of a highly ionized
atom is described by this expression exactly and the
limit of a neutral atom, quite accurately, (38) may be
recommended to estimate ( pb &#x3E; for b &#x3E; 0 in an atom
with an arbitrary degree of ionization.
Now we discuss the expectation values p"’ ) with
regard to the contributions of strongly bound electrons.
Similar to
(36), ( pm &#x3E;
may be
presented
as :
Comparison shows that (42) well reproduces the HF
data for isoelectronic series (Table IV). It is easily seen
from table IV that the open shell isoelectronic series
are described worse than the closed shell ones. This
is because of the absence of oscillation contributions
to the coefficients D 1 ( 18, 3) and D2(18, 3).
The above consideration has shown that the inclusion of a strongly bound electron contribution allows
not only investigation of systematic trends in the
expectation values pb &#x3E; and pm &#x3E; but also reliable
quantitative estimates for 0 b 5 and 0 m 5/3.
For negative exponents - 3
b
0 and - 1
m
a
is
bound
electron
contribution
0,
strongly
small
while
the
oscillation
contribution
to
negligibly
pb &#x3E; and pm &#x3E; is very important and its inclusion
requires special considerations.
5. Conclusions.
On the RHS of (40), the first integral diverges for
m &#x3E; 5/3, which limits the range of the validity of (40)
to - 1 K m
5/3.
In quantum-mechanical consideration, ( pm &#x3E; exists
at all m within - 1
m
oo. The limit of the range
of m in (40) is due to the incorrect TF contribution
of strongly bound electrons not completely eliminated
for m &#x3E; 5/3.
-
The main results obtained
are :
( 1 ) the expectation values pb ) and pm &#x3E; are
found as functions of b, m and degree of ionization
within the TF model with account for exchange interaction ;
(2) ( p6 ) is obtained as a function of N within the
non-interacting electron model; it is proved that for
a great electron number the TF model gives the leading
term in
pb &#x3E; which is identical to an exact quantum
quantity
within - 2
b
2;
342
is shown that a correct treatment of strongly
bound electrons gives a reliable estimate of pb &#x3E;
5 and of ( p"’ ) within 1
within 3 b
m
5/3
b
3 and
and essentially improves them within 1
within 1 /3 m
1;
(3) it
,
(4) the expectation values ( p - ’ &#x3E; and ( p - ’ &#x3E; for
neutral atoms are found to obey the periodic law;
1
(5) the three first Z -1 expansion coefficients for
pb &#x3E;, pm &#x3E; are defined as functions of N.
These results
with the data of
[2] and [6]
provides a quantitative estimate of such atomic properties, whose
values are basically determined by bulky and strongly
prove that the
together
improved
TF model
bound electron contributions.
Acknowledgment - The authors would like to
thank S. K. Pogrebnya for his programming assistance
in calculations.
References
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[11] BARTOLOTTI,
[12] WHITTAKER,
L. J. (unpublished).
E. T. and WATSON, G.
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[15] BIGGS, F., MENDELSOHN, L. B. and MANN, J. B.,
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