The improved Thomas-Fermi model : chemical and

ionization potentials in atoms

I.K. Dmitrieva, G.I. Plindov

To cite this version:

I.K. Dmitrieva, G.I. Plindov. The improved Thomas-Fermi model : chemicaland ionization potentials in atoms. Journal de Physique, 1984, 45 (1), pp.85-95.<10.1051/jphys:0198400450108500>. <jpa-00209743>

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Submitted on 1 Jan 1984

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85

Résumé. 2014 Les expressions analytiques des valeurs non-relativistes du potentiel chimique 03BC(N, Z) et du potentield’ionisation I(N, Z) sont obtenues dans le cadre du modèle de Thomas-Fermi, tenant compte des corrections duesà l’échange et à l’inhomogénéité de la densité électronique. Ces expressions reproduisent bien l’évolution moyennedu potentiel d’ionisation. On montre que, pour de faibles ionisations, I(N, Z) et 03BC(N, Z) sont déterminés princi-palement par les contributions quantiques dues à l’échange et à la corrélation et qu’ils peuvent être mis sous la formeA + B (Z - N). La relation entre 03BC et I s’écrit, pour les atomes de faible degré d’ionisation (m = Z - N ) :

03BCm+1~ - (2 m + 1) (2 m + 2)-1Im+1

qui doit être valable pour les atomes avec des électrons de valence de type s et p. On démontre que le modèle simplede Thomas-Fermi donne la valeur asymptotique exacte dans la limite N/Z ~ 0, la contribution d’oscillation ayantl’ordre relatif N-1/3.

Abstract. 2014 Analytical expressions for non-relativistic values of the chemical 03BC(N, Z) and ionization I(N, Z)potentials are derived from the TF model including corrections for the exchange and inhomogeneity of electrondensity. These expressions reproduce well the smoothed ionization potential of an atom. It is shown that I(N, Z)and 03BC(N, Z) in a weak ionization limit are determined mainly by quantum contributions due to the exchange andcorrelation and may be written as A + B (Z - N). The relationship between 03BC and I for atoms with a small ioni-zation degree, m = Z - N, is found

03BCm+1 ~ - (2 m + 1) (2 m + 2)-1 Im+1 which should be valid for atoms with s- and p-valence electrons. It is proved that the simple TF model gives anasymptotically exact value I(N, Z) within N/Z ~ 0 and the oscillation correction has a relative order N-1/3.

The improved Thomas-Fermi model :chemical and ionization potentials in atoms

I. K. Dmitrieva and G. I. Plindov

A. V. Luikov Heat and Mass Transfer Institute, ByelorussianAcademy of Sciences, Minsk 220728, U.S.S.R.

(Reçu le 7 juin 1983, accepté le 5 septembre 1983)

J. Physique 45 (1984) 85-95 JANVIER 1984,

Classification

Physics Abstracts31.10 - 31.20L

1. Introduction.

Chemical and ionization potentials are importantcharacteristics of an atom. The ionization potential Iof an atom with N electrons and nucleus charge Zis uniquely defined both in theory and in experiment.Determination of the chemical potential = aE/aNfor isolated microscopic quantum systems (such as anatom or a molecule) is difficult since there is no exactcontinuous energy function E(N). Recently, an interestto p has increased essentially [1-4, 25-27] due toestablishment of relationship between chemical poten-tial and electronegativity [1].The present paper pursues a double aim, namely,

to study systematic trends in p and I variation with Nand Z and to obtain useful relations between them.

As a tool for such investigation the Thomas-Fermi(TF) model including quantum corrections is used.This model has enabled us to obtain an accurate

analytical expression for the non-relativistic atomicbinding energy E(N, Z) [5]. We shall demonstratethat this model is able to give asymptotic estimateseven of such oscillating quantities as I(N, Z) andp(N, Z).

2. Basic equations.In a simple TF model, the potential of a sen-consistentatomic field is written as [6]

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198400450108500

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where I/J(x) is the screening function solution of theTF equation

x is the dimensionless coordinate defined by theexpression

The quantity PTF in (1) is the chemical potential in theTF model

xo is the boundary radius beyond which the electrondensity is zero. For a neutral atom PTF = 0 since

1 - N - 0 and xo oo. Here and below atomicZ o

units are used.As is seen from (2) and (3) the solution of the TF

equation and xo value are fully determined by theratio N/Z. It follows from (4) that JlTF (as well as otheratomic properties in the TF model) can be presentedin universal form

A numerical estimate of f(N/Z ) can readily bederived from the known xo values [6-8].The analytical expression for I1TF(N, Z) has been

obtained for the first time in [9] on the basis of theexplicit function ETF(N, Z). Using the expression forETF(N, Z) [5]

JlTF can easily be obtained as an N/Z expansion

Equation 7 reproduces well the numerical values ofYTF [8] up to N/Z = 0.5 (error is within 0.5 %). ForN/Z &#x3E; 0.5, the error of (7) increases quickly withN/Z. For a correct description of PTF with N/Z - 1,one should know an asymptotic behaviour of JJTFor xo in the weak ionization limit.The function FTF(N, Z) at N/Z ---+ 1 has been

obtained in [10] by solving analytically TF equation 2for a weakly ionized atom

Here

Recently [3], the first r.h.s. term of (8) has been obtained by using the same procedure but with a slightlydifferent value of the coefficient. The difference is due to the use [3] of only the first-order correction to thescreening function of a neutral atom (see Appendix).

Equation 8 reproduces well the numerical values of JlTF [7] (the error being less than 2 %) only at smallvalues of 1 - N/Z 0.01 due to slow convergence of series (8). To illustrate this we should say that the useof the first term of (8) alone gives an error of 30 % even at 1 - N/Z = 0.01.

In order to obtain a good analytical estimate of JlTF(N, Z) in the entire range of 0 NIZ , 1, we rearrangeseries (8) and (7). First of all, note that the expansion coefficients in (8) are very close to the coefficients of geome-tric progression. Bearing this in mind we write JlTF for N/Z - 1 in the form

Introduction of this simple geometric approximation drastically improves the estimates (see Table I).Then, we rearrange series (7) using the weak ionization limit

One can see that summation (10) for N/Z = 1 does not give the correct value of the coefficient (- 0.110 3) of(Z - N)4/3. This results in essential inaccuracy of (10) for N/Z - 0.8. The error can be eliminated by using the

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Table I. - JlTF.Z -4/3 as a function of N/Z.

(") JlTF.Z -4/3, found from (11) are compared with JlTF.Z -4/3 from [8].(b) JlTF.Z -4/3 derived from (9) are compared with UTF’Z -4/3 from [7].

[1, 3] Pad6 approximant of series (10)

The coefficient in the denominator is chosen from the condition of matching (11) and (9) for N/Z = 0.88. Com-parison shows that analytical approximations (9) and (11) perfectly reproduce the accurate values of Jl for anatom with an arbitrary degree of ionization (Table I).

Consider quantum contributions to p due to the exchange interaction and the inhomogeneity of electrondensity. As has been shown earlier [11], when taking into account the exchange interaction in the first order withrespect to p = ’(6 nZ)-2/3, the ion binding energy, as well as many other atomic properties, can be written as

The exchange contribution, Eex, is written as [11]

When evaluating the chemical potential of an ion, we obtain the expression for p with regard to the exchange

where JlTF is determined by (7) and

We have failed to evaluate the accuracy of (15) for all N/Z because no data on ge. are available. Comparing theratios of expansion coefficients (15) and (7) we may conclude that (15), like (7), gives sufficiently accurate Uexvalues for highly ionized atoms. The error should increase with N/Z. For more precise determination of M,,xfor N/Z &#x3E; 0.5 we rearranged series (16) taking into account the behaviour of uex for a weakly ionized atom.For this purpose we used the relation between /.t and ro within the TF model with regard to the exchange [12]

It has been shown earlier [13] that the boundary radius, ro, of a heavy ion with regard to the exchange inthe first order in P may be presented as

Here, the ai(m) are complex functions of m = Z - N which are determined from the hierarchy of algebraicequations obtained from the boundary conditions at r = ro. Most important for studying an asymptotic formis the equation for ao(m)

The solution of (18) for small m may be written as an expansion in m, for large m, in m-2/3. Taking into accountthat (16) includes ro 1, it is convenient to obtain aõl(m) expansions in m-2/3 and m

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Comparison of ao ’(m) from (19a) and (19b) with the numerical solutions of (18) shows that (19a) reproducesaccurate values within an error 3 % at m &#x3E; 2, while (19b), at m 2.

Combining (19a, b), (16) and (17) yields

Equations 20 and 21 give p(m, Z) for a weakly ionized heavy atom with regard to the exchange.One can easily see that expression 20 has the form of partition (14). The first term in (20) is PTF for a heavy

atom when only the first order in the ionization is taken into account (see Appendix), the second one is the mainpart of fLex at N /Z "-1 1

We suppose that corrections of higher orders in the ionization do not change essentially the coefficient in (20a),like asymptotic form for PTF (8).

The chemical potential of a neutral atom and single (double)-charged ions (21) cannot be written in theform of equation 14, since the exchange contribution is not a small correction. Comparison of (9) and (21) showsthat the exchange interaction leads to a change in a limiting form of Jl(NjZ ---+ 1).

Expression 20a enables us to rearrange series (15) by taking outside the dependence (Z - N)2/3. Usingthe [1, 2] Pade-approximants of the rearranged series we arrive at

The coefficient in the denominator is chosen from the coupling condition of (20a) and (22).

Let us now discuss the contribution to p due to theeffect of the electron density inhomogeneity. Thecorrection to the binding energy due to this effectcoincides fully in its form with (13) differing onlyin the factor 2/9 [5]. Bearing this in mind we write

Expressions 14 or 23 where JlTF and ,uex are deter-mined from (11) and (22) permit evaluation of p in a

/

The weak ionization limit, Z - 2 N Z, withallowance for inhomogeneity preserves the form of (21)with changed coefficients whose accurate values cannotbe obtained because of the absence of a generallyaccepted scheme for taking into account the inho-mogeneity at the boundary of a weakly ionized atom.However, it is known that the effect of the inhomo-

geneity leads to a slight (as compared to the allowancefor the exchange alone) decrease in ro and to a smallincrease in I JlI.

Let us consider the linkage of p with the ionizationpotential

The Taylor expansion of the second part of equality 24at a point N - 1/2 yields an approximate relation

Relation 25 is obtained from the condition of a conti-nuous dependence p(N) which is always fulfilledwithin the TF model. Making use of all the expressionsfor p we can easily evaluate I.

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3. Results and discussion.

Now we show that the obtained functions p(N, Z)and I(N, Z) allow a number of important conclusionsfor real atomic systems.

First of all it follows from the TF model that theionization Im+ 1 (Z) and chemical ttm+ 1 (Z) potentialsof a heavy weakly ionized atom decrease with increas-ing Z and tend to a finite value depending on themultiplicity m = Z - N. This conclusion is validboth with [10] and without introducing the exchange[13].

In the simple TF model (8) and (24) yield

From (8) and (26) we obtain the limiting form ofI]t 1 and p§t i as Z - oo (m Z)

It is easy to see from (8) and (26) that the limiting valuesare attained at Z far outside the periodic table.

Introducing the exchange interaction changes thefunctional form of Im+ 1 (Z ---+ oo) and Jlm+ 1 (Z ---+ oo)for small m, which makes impossible the derivationof the weak ionization limit from the TF model.

Taking the limit Z - oo from (21 ) and (24) at m = const.(0 m , 2) we get

Thus, allowance for the exchange leads to an essen-tial increase of the limiting value Im+ 1 (Z ---+ oo). Thisis physically substantiated by decreasing electron-electron repulsion due to the exchange interactionwhich leads to decreasing mean distance of electronsfrom the nucleus. In particular, for a neutral atomwe have

Taking account of the exchange results in increasingthe non-zero chemical potential of a neutral atom.

It is of interest that Im+ 1 (Z ---+ aJ) for 0 , m 2calculated within the Fermi-Amaldi model [10] ap-proaches the values obtained from (29).

In order to understand how much the obtainedrelations describe the reality, we consider experi-mental ionization potentials of neutral and single-(double)-ionized atoms (Fig. 1). Let us point out themain trends in Im+ 1 (Z) variation :

1) 1m + 1 (Z) is a periodic function ; an oscillationperiod is determined by nm, maximum value of themain quantum number;

Fig. 1. - Experimental I0(Z), I1(Z ) and I2(Z) [23] as

functions of Zli3. 1. lo(Z); 2. Ii(Z);3. 12(Z).

2) inside one column of the periodic table, 1m + 1 (Z)decreases with increasing Z ;

3) an oscillation amplitude falls with increasing Z ;the ratio

for noble gases and isoelectron ions characterizing arelative oscillation amplitude decreases with increa-sing Z.The TF model cannot, naturally, describe periodic

properties (trend 1). Nevertheless this model repro-duces correctly trend 2. Taking into consideration thatwith increasing Z the ionization potential is smoothed(trend 3) and that the TF model reproduces well thebehaviour of the smooth part Im+ 1 (Z) one maypredict that for ions of small multiplicity with suffi-ciently large Z, the value of Im+ 1 is independent of Z.Bearing in mind the relation for I and p (25), it isnatural to suppose. that the chemical potential of aheavy ion does not depend on Z either. Approximatevalues of 1m + 1 (Z ---+ oo ) and Jlm + 1 (Z ---+ oo ) are givenby (29) and (30).These values can be refined by taking into account

the contribution due to correlation effects. Using anasymptotic expression of the correlation energyEcor - 0.05 N [11] we obtain (0 m , 2) :

It can be easily seen that (31) and (32) yield approximate

90

relationships

As (33) and (34) are obtained by including mainquantum effects, it may be expected that they are validfor real atoms. Indeed, as shown in [14], a linearrelation between successive ionization potentials forsmall m (33) is obeyed approximately for real atoms.It is most accurately obeyed for s and p-valenceelectrons. This was explained in [15] based on detailedstudies of accurate quantum energies of valenceelectrons.

Taking into account the relationship between Iand p (25), one may expect that (34) is also fulfilled fors- and p-valence electrons. This is confirmed by theanalysis of sufficiently accurate p values for the ionsof small ionization degree [4].Combining (25), (31) and (32) we can establish the

relationship between the ionization and chemical

potentials of the ions with small ionization degree :

These relationships allow Jlm+ 1 for real atoms withs- and p-valence electrons to be found.From equation 32, one more interesting corollary

follows. By integrating (32) we get

Thus, the TF model, with exchange and correlationeffects taken into account, enables one to obtain a

qualitatively correct dependence of the binding energyon m in the weak ionization limit. The r.h.s. coefficientin (3 7) for real ions is a function of Z [15].

Relationships (29) through (37) follow from (21) andare valid only for the ions of small ionization degree.With increasing m, the dependence of Im+ l’ Pm+ 1and E on m for heavy atoms changes. Using (20), (23)and (26), we derive, for 3 m Z

From (39) and (38) it follows that the linear relationbetween successive Im (or J1m) breaks down when mincreases.

Equations 38 through 40 involve both the electro-static (TF) contribution and quantum corrections due

to exchange interaction and electron density inhomo-geneity (correlation corrections with the order m°are neglected in these equations). It is seen that form &#x3E; 3 the TF contribution prevails even for the

energy difference. In contrast to this, for 0 m 2the quantum contribution is dominant (1).The oscillation contribution, being rather signifi-

cant for the ions of small ionization degree, cannotbe taken into account within the TF model. This is thereason why expressions 31, 32, 38-40, reproducingcorrectly the trends of I, p, AE variation, do not giveaccurate quantitative values for the ions with greatZ and small m.The question is how much correctly the improved

TF model reproduces the dependence of I and p on Nand Z for an atom with an arbitrary degree of ioniza-tion. We shall show now that for a highly ionized atomthe simple TF model gives the main part of p(N, Z)and I(N, Z), while the quantum oscillations define theleading correction of the relative order N-1/3

It is well known [16] that the ionization potentialof the isoelectronic series may be calculated by using aZ-1 expansion

Now represent the chemical ion potential in the formof a Z-1 expansion

Using expressions 7, 15, 23 and 25, we write the coeffi-cients of Z-1 expansions of I(N, Z) and p(N, Z)within the TF model allowing for the contributionsof the exchange interaction and electron density inho-mogeneity

The first coefficients are specified from the equations

Exact values of bo(N) and do(N) may be determinedfrom the non-interacting electrons model.The binding energy of atoms with N electrons in

(1) This behaviour is not considered in reference 3 whoseauthor has incorrectly extended TF dependence (40) to theweak ionization limit. The same error is made in the deri-vation of the function p(Z - oo) - Z -1/3 for neutralatoms [25].

91

this model is

where p is the fractional occupation of the outermostshell (0 p 1).Taking into account the N - nm link [11].

and eliminating nm from (46) and (45) we arrive at

whence follows the exact value

The exact value of bo(N) is given by the ionizationpotential of a hydrogen-like atom bo = 2 nm-2. Using(46), we expand bo(N) in powers of N -1/3

Comparison of (47) and (48) allows the conclusionthat for a highly-ionized atom up to the terms of~ N-1 the following equality is valid

It should be noted that for closed shells (49) is

destroyed since po(N, Z) undergoes a break andbecomes ambiguous. We may choose Jlo(N, Z) for anion with a closed shell in the form

It is seen from (47) and (44) that the first term in theexact do(N) expansion with respect to N-1/3 coincideswith dlF(N), while the second, the oscillation term,has a relative order N-1/3. Thus, the TF model givesasymptotically (at N » 1) exact expressions p(N, Z)and I(N, Z) at N/Z I.

Evaluation of the oscillation contribution to conse-

quent coefficients of Z -1 expansion is very difficultsince inclusion of the interelectronic interaction

requires the summation of multiparticle diagramswhich does not yield explicit dependence bk(N) atk &#x3E; 0. Numerical estimation ofb1(N) may be based onthe summation of hydrogen-like radial integrals [17].An approximate analytical estimate is given by usingthe expression for the oscillation contribution to thesecond term of Z -1 expansion of binding energy [11]

Differentiating this expression, we represent the mainpart of the oscillation contribution to bi(N) as

Analysis of the numerical values shows (Fig. 2) thatlinear dependence (50) fairly describes b*s’(N) assum-ing that A1(N - 1/2) is a slowly varying function

Comparison of (47) and (50) with (44) indicates that,with the TF model giving the main part bk(N) atN &#x3E;&#x3E; 1, the oscillation contribution may be verypronounced at moderate N especially at the beginningof a shell. We think that the main part of the oscilla-tion contribution to subsequent coefficients of Z-1expansion has also relative order N- 1/3 and is propor-tional to (p - 1/2).Combining (47), (50) and (51) we represent the

oscillation contribution to I (N, Z) of a many-electronhighly ionized atom as

Fig. 2. - The plot of bfC(N) = b1(N) - dlF(N - 1/2) +do(N - 1/2) for N 28; bl(N) are calculated by usinghydrogen-like integrals [17].

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The main part of lose(N, Z) has a shell structureand depends linearly on p. Iose(N, Z) is the result ofthe substitution of a discrete dependence on quantumnumbers by a continuous dependence on N.

It follows from (52) and figure 2 that lose for a highlyionized atom is a periodic function of = 3/2N2013 1/2.When nm increases by unity, maximum lose shiftsto minimum, which is consistent with the criticalvalues Nc = 3,11, 29, 61...Taking into account that an oscillation period of a

weakly ionized atom is also determined by nm (trend 1,page 89) - we may present Iosc for an atom withan arbitrary degree of ionization as

Here, k(N l/Z) N f/3 = nm ; F(x) is the periodic func-

tion, A N N 1 is the relative amplitude of oscilla-tions.Now let us discuss a change of A(N, N/Z) and nm

along isoelectronic series. As N/Z increases for N 18,the oscillation period changes slightly. Electron shellsrearrange along isoelectronic series due to electronicinteraction. This is followed by increasing nm [19] and,as a result, by decreasing the oscillation period.The most pronounced change occurs around a neutralatom. At some N/Z, a new quantity, Nc, appears. Fora neutral atom Nc = 19, 37, 55, 87...Now consider the dependence of the oscillation

amplitude on N and N/Z. From (41) and (44) theexpression of I(N, Z) for a highly ionized atomwithin the TF model with corrections for the exchangeand inhomogeneity results

Combining (54) and (52) allows approximate expan-sion of A(N1, N1/Z) for N/Z 1

It follows from (55) that the variation of A(N, N/Z )along isoelectronic series is different for great andsmall N. With N 18, the second term of expansion

(55) is positive and A(N/Z) grows with N/Z (Fig. 3).As a measure of A(N, N/Z), the parameter

is assumed for N = Nr. This is valid since for N = N c - 1the function F reaches a maximum, while for N = N c’it is minimum.For N &#x3E; 20, the coefficient of Z -1 in (55) changes

sign, which results in decreasing A(N, N/Z) along anisoelectronic series. Near a neutral atom, as followsfrom the analysis of experimental I(N, Z) values, therelative amplitude of oscillations abruptly increasesirrespective of N (Table II). Thus, the oscillation con-tribution, like other quantum effects, i.e. exchangeinteraction and electron density inhomogeneity, in-creases for a weakly ionized atom. But, in contrast toexchange and inhomogeneity, the relative oscillationamplitude for many-electron ions is minimum at

N/Z # 0.

Fig. 3. - Relative oscillation amplitude vs. N/Z for the Neisoelectronic series.

Table II. - A(N, Z) for neutral and weakly ionizedatoms.

I I I

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This raises hopes that the TF model with allowancefor exchange and inhomogeneity may give a. satisfac-tory estimate of I(N, Z) and p(N, Z) in a wide rangeof N and Z. We have compared I (N, Z) (11), (22), (25)with experimental data [19] to verify that an error ofthe improved TF model does not exceed 30 % for12 N 29 and Z &#x3E; N + 4. A typical dependenceof the error on Z - N is plotted in figure 4. Takinginto account that A(N, N/Z) falls with increasing None may assert that the improved TF model givesa semi-quantitative estimate of non-relativistic valuesof I(N, Z) and p(N, Z) for all atoms with N &#x3E;, 12atZ &#x3E; N + 4.The existence of quantum oscillations leads to a

complex configuration (hills and ravines) of I(N, Z)and p(N, Z) surfaces for real atoms. There is a phy-sical problem implying a study of heavy-ion pene-tration through matter [20] for which a smoothedionization potential for an atom with arbitrary Zand degree of ionization must be known. From expe-rimental conditions averaging is performed with

respect to angular momentum for many different,but energetically close, states of ionization. The im-proved TF model gives this very potential, I(N, Z).We have calculated I(N, Z) for a wide range of ionswith Z &#x3E; 20, N Z - 2 by using expressions 9, 11,22, 23 and 25. Comparison with realistic I(N, Z)values [18, 19] shows that, with the exception of thesystem having a small electron number (N - 2 -7 4),the improved TF model yields a reliable estimate ofsmoothed ionization potentials. The asymptotic beha-viour of the expressions displays the growing accuracywith increasing Z and N (Fig. 5).

4. Conclusion.

Based on expressions previously obtained for non-relativistic binding energy, analytical expressions ofp(N, Z) and I(N, Z) are derived within the TF model,including corrections for exchange interaction ofelectrons and inhomogeneity of electron density((9), (11), (22), (23), (25)). These expressions fairlyreproduce the smoothed ionization potential for

arbitrary N and Z.It follows from the TF model that for heavy ions

with small ionization degree, m, the values of Im+ 1 (Z)and Jlm+ 1 (Z) decrease with increasing Z and tend tothe limit determined by rn (27), (28). The effect ofexchange interaction (and other quantum effects)changes the weak ionization limit for Jl, I and AE

(equations 29, 30, 37). Based on these equations,approximate relationships between Im+ 1 and Jlm+ 1(36)-(39) are derived which may be useful to estimatethe chemical potential for ions with s-p-valenceelectrons.Within the quantum model of non-interacting

electrons it is proved that the simple TF model givesan asymptotically exact dependence of I(N, Z) forN/Z 1, while the oscillation contribution is a

Fig. 4. - Error of the improved TF model vs. Z - N;I(N, Z) are taken from [19, 24]. 1. Ar isoelectronic series;2. K isoelectronic series.

Fig. 5. - Ionization potentials of zinc and uranium asfunctions of N/Z. 1. our results; 2. accurate data [18, 19].

leading correction of relative order N - 1/3. The studyof isoelectronic series shows that the oscillation perioddecreases with increasing N/Z but the relative ampli-tude has a minimum at an intermediate value0 N/Z 1. Owing to this the improved TF modelgives a reliable semi-quantitative estimate of non-relativistic I(N, Z) values (the error being less than30 %) for all atoms with N &#x3E; 12, Z a N + 4.

94

Appendix.The TF expression for Xo at q L

Solution of TF equation (2, 3) for a weakly ionizedatom may be represented as [10]

where t/J o(X) is the solution of the TF equation for aneutral atom; parameter C depends on N/Z, its

explicit form is given below.Substituting (A. 1) into equation 2 and equating

the expressions with equal exponents C we derive arecurrent system of second-order differential equa-tions

The first equation of system (A. 2) has been derivedby Fermi when investigating the first-order correctionfunction; its solution is written in quadratures [21]

Solution to the n-th equation of system (A. 2) satisfyingthe conditions t/J(O) = 4/0(0) = 1 and C = t/J’ (0) - 0’ 0 (0),is [10]

The relationship between C, Xo and N/Z is deter-mined from boundary conditions 3

Using asymptotic expansions of t/Jo(X) [22], /l(X)and I2(X)atX &#x3E;&#x3E; 1 [10]

proves that all ql,,(X &#x3E;&#x3E; 1) are of the form

In formulae A. 7 and A. 8 Ao = Bo = 1; Ak and Bkare determined from the recurrent relations and are

expressed in terms of A 1 = - 13.270 974 [22],ex1 = [48 O"A1(7 + 2 0")]-1; Bk and Ynl values are

given in [10].Introducing (A. 8) into (A. 6) and substituting

cxri+cr = D, we obtain the system (q= I-NjZ)

Solution of system (A. 9) gives the weak ionizationlimit for Xo and D

Insertion of each consequent term of expansion (1)leads to readtimation of ak and Dk.For the first order we obtain

and introducing the second one we have

More accurate ak values may be obtained when usingan increasing number of terms of the asymptoticseries (A. 1). The author of [3] has not noticed this andhas believed erroneously that the first term in (A. 10)gives the weak ionization limit for X0(q -+ 0). At ourrequest S. K. Pogrebnya has calculated ao by using20 orders to found ao = 10.232 0 that differs slightlyfrom the value given in (A. 12).

95

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