An effective analytical potential including plasma effects

Journal of Quantitative Spectroscopy &Radiative Transfer 75 (2002) 539–557

www.elsevier.com/locate/jqsrt

An e'ective analytical potential including plasma e'ectsJ.M. Gila ; ∗, P. Martela, E. M./nguezb, J.G. Rubianoa, R. Rodr./guezb, F.H. Ruanoc

aDepartamento de F� sica, Universidad de Las Palmas de Gran Canaria, 35017 Las Palmas de Gran Canaria,Spain

bInstituto de Fusi�on Nuclear, Universidad Polit�ecnica de Madrid, 28006 Madrid, SpaincDepartamento de Senales y Comunicaciones, Universidad de Las Palmas de Gran Canria,

35017 Las Palmas de Gran Canaria, Spain

Received 19 October 2001; accepted 7 March 2002

Abstract

In this work, we have developed a method to build an e'ective analytical potential for ions in slightlynonideal plasmas. This proposed potential is obtained from an analytical isolated potential with one or twoparameters depending on the total number of electrons of the ion. The plasma e'ects are included by meansof the linearized Debye–H;uckel approximation taking into account the reaction of the plasma-charge densityto the optical electron. Due to the in>uence of the plasma over the atomic potential, this permits to obtainlevel energies and wave functions as a function of the inverse of Debye radius, the quantum numbers, thenuclear charge, the bound electron number and the ionization state of the ion. Also, we compare the analyticale'ective potential proposed in this paper with other ones very well known in the available literature. ? 2002Elsevier Science Ltd. All rights reserved.

1. Introduction

The availability of atomic properties is essential in the study of hot plasmas. In a plasma ofan element with atomic number Z , many of the ionic states of that element can be present in theplasma, and for each one of these states, a great number of con@gurations corresponding both to theground state and excited states can be found. In order to determine the properties of these ions, someself-consistent models including temperature and density e'ects have been developed [1–10]. Theself-consistent models which handle each con@guration into the plasma involve a high computationalcost [1–5]. To reduce this e'ect, self-consistent models solving an average ion which includes anaverage behavior of ions in the plasma have been developed [6–10].Another alternative to reduce the computational cost is to use analytical potentials. The

linearized Debye–H;uckel model [11] has been extensively used to calculate atomic properties of

∗ Corresponding author. Fax: +7-34-28-45-29-22.E-mail address: jmgil@@sica.ulpgc.es (J.M. Gil).

540 J.M. Gil et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 75 (2002) 539–557

hydrogenic ions inside plasmas at high temperatures and low and intermediate densities. From thismodel, an analytical expression of the e'ective potential (Debye potential) for hydrogenic ions isgiven by

U (r) =−Zre−r=D; (1)

Z being the nuclear charge and D the Debye radius. This potential has been used to provide eigen-values and eigenfunctions [12–30], bound–bound, bound–free and free–free cross section [31–34],and scattering cross section [35].The linearized Debye–H;uckel model has also been used to obtain e'ective potentials for mul-

tielectronic ions [36–38], in the approximation of independent particle in the central @eld. In ageneral context, considering the electronic-screening problem and taking into account the screeningdue to the bound and the free electrons and the rest of ions of the plasma, Rouse [36] proposed thefollowing potential:

U (r) =−Zr{(N − 1)e−�r + Z − N + 1}e−r=D; (2)

N being the number of bound electrons and � a positive parameter which is determined assumingthe isolated ion.Rogers [37] obtained a particular potential only for helium-like ions

U (r) =−(Z − 1)e−r=Dr

− e−r

r; (3)

where was @tted to 1:067Z=a0, a0 being the Bohr radius. This potential was generalized for ionswith N electrons [38] in the following expression:

U (r) =−1r

{n∗∑n=1

Nne−nr + (Z − N + 1) e−r=D}; (4)

Nn being the number of electrons in the shell with principal number n; n∗ the maximum value of nfor the bound levels occupied by the electrons of the ion except one (optical electron) and n thescreening parameter for electrons in shell n. This n was determined for isolated ions by iterativelysolving the Dirac equation and comparing the eigenvalues to experimental energies values. Thispotential is valid for plasmas which inverse Debye radius is less than the n parameter of each shell.The above potential that uses the linearized Debye–H;uckel model produced shifts in the energy

levels with respect to those experimentally observed [17,18]. This disagreement is attributed at leasttwo reasons [39]. First, the linearization is only valid at large distances from the ions, and secondly,the electrostatic potential obtained from this model ignores completely the reaction of the plasmacharge density due to the presence of bound electrons. This last behavior is reported in severalpapers [40–42], and all of them reported that at large distances the optical electron is moved in aCoulomb potential instead of in a potential given by Eq. (1). In this way Schl;uter [42] proposed an

J.M. Gil et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 75 (2002) 539–557 541

analytical potential for hydrogenic ions given by

U (r) =−1r

(Z − doZ2

n− 1

)e−ar + e−2Zr=n

2n−1∑k=0

dk(n; l)(Zrn

)k]; (5)

where n and l are the principal and the orbital quantum numbers, respectively, and the coeMcientsdk are obtained from a recurrent relation which can be seen in Ref. [42].In Section 2 we introduce a method to generate analytical potentials including temperature and

density e'ects and in Section 3 we propose a new potential obtained from an isolated analyticalpotential previously developed [43]. This new method involved the linearized Debye–H;uckel ap-proximation and the reaction of the plasma-charge density to the optical electron. In Section 4 wecompare this e'ective analytical potential with those proposed by other authors, and moreover, theenergy levels and the wave functions are analyzed. Finally, in the last section principal remarksrelated to the new analytical potential and conclusions are presented.

2. Methodology to include plasma e�ects in analytical potentials

For an ion having a nuclear charge Z , and N bound electrons into a plasma of density �(r), thetotal potential Ut (r) caused by the ion and the plasma is given as

Ut (r) = UN (r) + Up(r); (6)

UN (r) being the potential generated by all bound electrons and the nuclear charge and Up(r) thepotential due to the rest of ions and free electrons of the plasma.Taking into account that �p(r) and Up(r) can be related by the Poisson equation, we obtain

∇2Ut (r) =∇2UN (r)− 4��p(r): (7)

Assuming for each potential U (r) the charge function Z(r) de@ned by

Z(r) =−rU (r) (8)

Eq. (7) may be transformed to

d2Ztd r2

=d2ZNd r2

+ 4�r�p(r); (9)

which allows to calculate the total charge function for each pair of �p(r) and ZN (r). In this paper,we have used the linearized Debye–H;uckel approximation which proposes a plasma density givenby

�p(r) =a2

4�rZt(r); (10)

a being the inverse of Debye radius:

a=1D=(�ionZ∗(Z∗ + 1)

)1=2; (11)

where k is the Boltzmann’s constant, T the plasma temperature, �ion the density of the ions, andZ∗ the average ionicity of the plasma. Taking into account this approximation for �p(r), Eq. (9)becomes

d2Ztdr2

− a2Zt =d2ZNdr2

: (12)

The e'ective potential of an electron in the level k di'ers from the total potential due to thecontribution of the electron self-charge distribution, i.e. Ue' (r)=Ut(r)−Ue(r) or Ze' (r)=Zt(r)−Ze(r),Ze(r), being the optical electron charge function. Considering this relation, the charge function of theN − 1 bound electrons and the nuclear charge given by ZN−1(r) = ZN (r)− Ze(r), Eq. (12) becomes

d2Ze'dr2

− a2Ze' =d2ZN−1dr2

+ a2Ze: (13)

In this di'erential equation the optical electron charge function which represents the reaction of theplasma charge density to the optical electron is included. The solution of this di'erential equation is

Ze' (r) = ZN−1(r) + C e−ar − 12a∫ ∞

0e−a|s−r|(ZN−1(s) + Ze(s)) ds; (14)

C =12a∫ ∞

0e−as(ZN−1(s) + Ze(s)) ds; (15)

where the constant C has been determined from the following conditions:

limr→0

ZN−1(r) = Z; limr→∞ZN−1(r) = Z − N + 1; (16a)

limr→0

Ze(r) = 0; limr→∞Ze(r) =−1; (16b)

limr→0

Ze' (r) = Z: (16c)

and by considering a @nite limit for the e'ective charge function at large distances.Eq. (14) allows us to obtain Ze' (r) by a self-consistent procedure. This procedure starts from a

ZN−1(r) and Ze(r) which corresponds to an isolated ion. Then, the e'ective potential obtained wouldbe used to determinate the eigenfunctions of the single-electron levels which permits to calculate thenew ZN−1(r) and Ze(r). The procedure is repeated until self-consistency is achieved. In a similarway, Schl;uter [42] obtained an e'ective potential by a self-consistent solution of a rather di'erentequation to (14), only for hydrogen. From Eq. (14), we have determinated the following analyticalexpression for the e'ective potential:

U 0Ae' (r) =−1

{Z0N−1(r) + 1 + C

0 e−ar − 12a∫ ∞

0e−a|s−r|Z0N−1(r) ds

}; (17)

C0 =−1 + 12a∫ ∞

0e−asZ0N−1(r) ds: (18)

Here, we have approximated ZN−1(r) and Ze(r) by the corresponding isolated functions, Z0N−1(r)and Z0e (r), and moreover, we have approximated the behavior of optical electron charge function byZe(r) ≈ −1. We have called this approximation as order zero approximation with asymptotic opticalelectron (U 0A

e' ). Indeed, this late expression is analytical if an analytical expression for Z0N−1(r) is

assumed.Using the isolated e'ective potential given by

U 0N−1(r) =−1

r{(N − 1)�(r) + Z − N + 1}; (19)

where �(r) is called the screening function, we obtain from Eq. (17) the following e'ective analyticalpotential including plasma e'ects:

U 0Ae' (r) =−1

r{(N − 1)(�(r)− �(r)) + [Z − N + (N − 1)�(0)] e−ar + 1}; (20)

where the new screening function, �(r), is given by

�(r) =12a∫ ∞

0e−a|s−r|�(s) ds: (21)

This is called plasma structural screening function, because this function depends on the ion structurethrough the screening function of the analytical isolated potential, and also the environment plasma,through the inverse of the Debye radius.

3. A new analytical potential with plasma e�ects

In this work the isolated potential used [43] has a screening function given by

�(r) =

e−a1r

a3 if N¿ 12;

(1− a2r) e−a1r if 86N6 11 or N = 2; 3;

e−a1r if 46N6 7;

where the parameters a1, a2 and a3 were @tted to

ak = c1kZ4 + c2kZ3 + c3kZ2 + c4kZ + c5k (k = 1; 2; 3) (23)

and the coeMcients, cik , were obtained for the ground state of ions from He- to U-like [43,44]. Bysubstituting Eq. (21) into (20) we obtain the following plasma screening functions:For ions with N ¡ 12:

�(r) = a2(

2a1a2(a21 − a2)2

− 1a21 − a2

a21 − a2r)e−a1r : (24)

For N¿ 12:

�(r) =12a∫ ∞

0e−a|s−r| e−a1s

a3 ds: (25)

The above integral can be solved numerically although it could be interesting to obtain a moreeasy analytical expression to be used in analytical calculations involving this e'ective potential. Toconsider this objective, an approximation of this integral has been done in Appendix A. In Ap-pendix B we apply Eqs. (17) and (18) for an isolated potential given by Eq. (19) but with amore general screening function including several analytical isolated potentials as those reported inRefs. [45,46].In the following we study the e'ective potential proposed by us, given by Eqs. (20)–(25). We

analyze the in>uence of the plasma over the potential, and also energy levels and wave functionsobtained for each nlj-subshell by solving the Dirac equation using the proposed potential. In orderto compare the e'ects of density and temperature introduced by our potential, we have determinedtwo additionals non-isolated analytical potentials from the same isolated analytical potential [43]by using Rouse [36] and Roger [37] methods. Using the Rouse method, the following potential isobtained:

U Ie' (r) =−1

r{(N − 1)�(r) + Z − N + 1} e−ar (26)

and using the Roger method the following potential is obtained:

U IIe' (r) =−1

r{(N − 1)�(r) + (Z − N + 1) e−ar} (27)

where in both potentials �(r) is the screening function given by (22). Then we have writtenU Ie' and U II

e' in the same formulation as our non-isolated potential, to arrive at the plasmae'ects.

4. Results and discussion

4.1. E8ective potential

There are several di'erences between the e'ective potential proposed in this work and the po-tentials U I

e' and UIIe' . The @rst di'erence is that the potential U

0Ae' has an asymptotic behavior as

the coulomb potential (−1=r), while the potentials U Ie' and U

IIe' tend to in@nity as −e−ar=r. We

show this behavior in Fig. 1 for iron and silver ions in several plasma conditions given for di'erentvalues of the inverse of Debye radius. As a consequence of the asymptotic behavior, the potentialU 0Ae' has in@nite bound levels and the levels with high-principal quantum numbers have atomic prop-

erties similar to the hydrogen atom, while potentials U Ie' and U

IIe' have a @nite number of bound

levels.Other important di'erence is that UA0

e' involves a new screening function, �(r), which dependson plasma conditions and also the ion structure by means of the parameters of analytical isolatedpotential, a1, a2 and a3. In order to analyze the in>uence of this new screening function, we considerthe potential produced by a particle of plasma which is given by

Up(r; a) = Ue' (r; a)− U 0N−1(r) (28)

Fig. 1. Plot of the function rU (r) (in a.u.) versus r for the isolated potential U 0N−1 and the non-isolated e'ective potentials

U 0Ae' , U

Ie' and U

IIe' .

and then, for each e'ective potential we can write

U 0Ap (r) =−1

r{(N − 1)(�(0) e−ar − �(r)) + (Z − N )(e−ar − 1)}; (29)

U Ip(r) =−1

r{(N − 1)�(r) + Z − N + 1}(e−ar − 1); (30)

U IIp (r) =−1

r{(Z − N + 1)(e−ar − 1)}: (31)

In the Eqs. (29)–(31) we can observe that the potential U IIp does not depend on the ion structure

but depends only on the ionization of the ion. Therefore for a given inverse of Debye radius, theplasma potential will be the same for all ions of the di'erent elements with the same ionization. Ingeneral, potentials U 0A

p and U Ip depend on the ion structure by means of the functions �(r) and �(r),

respectively. We have found at large distances potential UA0p is less than U I

p and UIIp , while at short

distances we have U IIp ¡U 0A

p ¡U Ip for low ionizations and U 0A

p ¡U IIp ¡U I

p for high ionizations(Fig. 2). Also, we have observed that the plasma structural screening function is more importantwhen the parameter a is close to the parameters a1, a2 or a3, and for an a given, this situation ismore favorable when the ionization is low. On the other hand, if a is much less than a1, a2 or a3 thefunction �(r) can be neglected and then the plasma potential does not depend on the ion structure(Fig. 3). In this situation the plasma potential is given by

U 0Ap (r) =−1

r{(Z − N )(e−ar − 1)} (32)

and the e'ective potential by

U 0Ae' (r) =−1

r{(N − 1)�(r) + (Z − N ) e−ar + 1}: (33)

The last expression has been obtained from Eq. (13) but neglecting the function �(r). PotentialsU Ie' and U

IIe' can been obtained from the e'ective potential given in (13) by neglecting the plasma

structural screening function, and also, the optical electron charge function. Then, these potentialsare particular cases of the proposed potential when the above condition is satis@ed.In Table 1 we show the average relative shift, 〈�Urel〉, for the three e'ective potentials. This is

given by

〈�Urel(a)〉= 1nrd

nrd∑i=1

|Ue' (ri; a)− U 0N−1(ri)|

U 0N−1(ri)

; (34)

where the sum is over the radial mesh, nrd is the number of points considered, and we haveconsidered the isolated potential as referenced potential. We can see that 〈�Urel〉 increases whenthe inverse of Debye radius rises. The di'erences between U 0A

p and the potentials U Ie' and U

are observed for low ionized, i.e., when it cannot neglect the plasma structural screening function.Taking into account that the variation of 〈�Urel〉 with respect of ionization degree is light for ourpotential, we can say that the three e'ective potentials provide relative shifts of the order of the 4%at 0:01 a:u:, 11% at 0:05 a:u:, 15% at 0:1 a:u: and 22% at 0:5 a:u:

Fig. 2. Plot of the function rU (r) (in a.u.) versus r for the plasma potentials U 0Ap , U

Ip and U

Fig. 3. Plot of the function rU (r) (in a.u.) versus r for the plasma potential U 0Ap for Fe and Ag ions.

Table 1Average relative shifts, 〈�Urel〉, for U 0A

e' , UIe' and U

IIe' potentials

(� RU )rel

Z N Z-N a= 0:01 a= 0:05 a= 0:1 a= 0:5

10 5 5 A0 3.55 9.72 12.77 19.73I 4.27 11.69 15.39 23.97II 4.26 11.67 15.33 23.68

20 10 10 3.87 10.61 13.95 21.594.27 11.69 15.39 23.974.26 11.67 15.34 23.75

30 10 20 4.06 11.13 14.64 22.754.27 11.69 15.39 23.974.27 11.69 15.37 23.90

40 10 30 4.13 11.32 14.88 23.164.27 11.69 15.39 23.974.27 11.69 15.39 23.93

50 10 40 4.16 11.41 15.01 23.364.27 11.69 15.39 23.974.27 11.69 15.38 23.95

70 20 50 4.18 11.46 15.07 23.494.27 11.69 15.39 23.974.27 11.69 15.37 23.91

90 30 60 4.20 11.50 15.12 23.524.27 11.69 15.39 23.974.27 11.69 15.37 23.91

4.2. Energy levels

To analyze the plasma e'ect on level energies of the subshell k(=nlj), we consider the shift levelenergies given by

�Ek = Ek − E0k ; (35)

E0k and Ek being the energies obtained when the isolated potential and non-isolated potential are,respectively, used.As may be expected, for a given ion the energy levels increase as the inverse Debye radius rises,

i.e. are shifted toward the continuous. In Figs. 4 we plot Ek in absolute value versus inverse ofDebye radius for iron at several ionization states, and in Table 2 we show �Ek for the levels 1s1=2,3s1=2, 5s1=2 and 7s1=2.

Fig. 4. Plot of the absolute value of energy levels (in a.u.) versus inverse Debye radius calculated by using the potentialsU 0Ae' , U

Ie' and U

IIe' .

For inner levels, as 1s1=2 (Fig. 4(a) and (b)), we can see that the potentials U Ie' and U

IIe' provide

energy levels that have a linear dependence with respect to the inverse of Debye radius for allionization states, while our potential has a non-linear dependence for low ionization states due to thescreening function �(r). Also we observe that potential U I

e' produces an energy shift larger than theshift of potentials U 0A

e' and U IIe' , mainly when the ionization is low and the nuclear charge arises.

Table 2Energy level shifts (in a.u.) of the levels 1s1=2, 3s1=2 and 7s1=2 for several ions, calculated by using the potentialsUA0e' (�EA0), U I

e' (�EI) y U II

e' (�EII)

1s1=2 3s1=2 7s1=2

Z N Z − N a �EA0 �EI �EII �EA0 �EI �EII �EA0 �EI �EII

0.01 0.0499 0.0816 0.0599 0.0495 0.0602 0.0594 0.0475 0.0568 0.056710 5 5 0.1 0.4961 0.8102 0.5953 0.4564 0.5541 0.5477 0.2941 0.3493 0.3491

0.5 2.4052 3.9312 2.8862 1.6151 1.9462 1.9261 0.3767 0.3959 0.39520.01 0.0511 0.4307 0.0612 0.0507 0.1951 0.0603 0.0489 0.0124 0.0582

50 45 5 0.1 0.6134 4.3003 0.5991 0.5661 1.9203 0.5900 0.4101 0.4213 0.40990.5 4.9700 21.3764 2.9779 3.8910 8.9335 2.7614 0.6992 0.7102 0.70830.01 0.1500 0.1808 0.1600 0.1494 0.1602 0.1594 0.1467 0.1566 0.1565

20 5 15 0.1 1.4943 1.8011 1.5940 1.4435 1.5477 1.5398 1.2114 1.2923 1.29190.5 7.3592 8.8721 7.8498 6.2208 6.6700 6.6342 2.6422 2.8422 2.83140.01 0.1516 0.5326 0.1603 0.1503 0.2849 0.1599 0.1480 0.1604 0.1571

60 45 15 0.1 1.5790 5.3116 1.5984 1.5302 2.8150 1.5790 1.3129 1.4180 1.39270.5 9.2828 26.4359 7.9524 8.0985 13.3377 7.4955 3.8647 4.0835 4.04360.01 0.1999 0.2652 0.2099 0.1994 0.2118 0.2094 0.1968 0.2067 0.2066

30 10 20 0.1 1.9939 2.6470 2.0947 1.9463 2.0673 2.0437 1.7062 1.7927 1.79130.5 9.8508 13.1068 10.3695 8.7524 9.3009 9.1904 4.5830 4.7566 4.75530.01 0.2004 0.6284 0.2031 0.2003 0.3625 0.2100 0.1979 0.2116 0.2073

70 50 20 0.1 2.0781 6.2820 2.9690 2.0304 3.5904 2.0775 1.8042 1.9225 1.88540.5 11.7996 31.2876 10.4485 10.6196 17.1900 9.9561 6.0087 6.3064 6.20930.01 0.2499 0.2806 0.2599 0.2494 0.2602 0.2594 0.2466 0.2565 0.2565

30 5 25 0.1 2.4938 2.7981 2.5936 2.4402 2.5462 2.5378 2.1863 2.2744 2.27380.5 12.3452 13.8528 12.8390 11.0990 11.5827 11.5426 6.4879 6.7137 6.71270.01 0.2503 0.6311 0.2687 0.2501 0.3802 0.2603 0.2476 0.2609 0.2633

70 45 25 0.1 2.5598 6.3054 2.5969 2.511 2 3.7658 2.5722 2.2774 2.3950 2.36170.5 13.8887 31.4126 12.9360 12.6959 18.0396 12.3273 7.9120 8.2428 8.1424

For outer levels, we observe (Fig. 4(c) and (d)) that our potential gives more bound levels thanU Ie' and U

IIe' potentials for large inverse Debye radius. Indeed, our potential has an in@nite number

of bound levels, and we observe in the plot that the energy levels tend to a constant value for eachlevel; this value is the energy level of the hydrogen atom for each level. This is consistent withthe asymptotic behavior of the potential, as it was mentioned above. With this behavior observedin our potential, which has been obtained by taking into account the reaction of the charge plasmato the optical electron, the function partition calculated by using the occupation number formalismwill be @nite and will not present discontinuities in thermodynamic and optical properties [47,48].Our potential is obtained from plasma static charge distribution and the e'ect on bound level is anenergy shift toward the continuous but the ionization pressure is not produced. We consider that otherphysics process, as electronic collisions or >uctuations of the electromagnetic @eld, are responsiblefor the ionization pressure.It has been observed that �Ek decreases both for highest principal and total angular momentum

quantum numbers. A more diMcult evolution is observed for the angular quantum number because

Fig. 5. Plot of the energy level shifts (in a.u.) versus bound electron number calculated by using the potential U 0Ae' .

�Ek increases for higher angular quantum number of inner shell levels and �Ek decreases for principalquantum numbers or when the number of the bound electrons is higher, that means when theionization state is lower. Also we have observed that the energy levels shifts increase when the

ionization state arises. Moreover, the energy level shifts has a linear behavior with the number ofbound electrons or nuclear charge when the inverse Debye radius is small. This behavior disappearswhen the inverse Debye radius arises (Fig. 5).

4.3. Wave functions

We can observe that the maximum and minimum absolute values of the wave function are smallerfor the three potentials in relation to the wave functions from isolated potentials. These e'ects aremore important as the inverse Debye radius increases and for outer shell levels. Also, these wavefunctions have a displacement of the nodes to farthest positions respect to the origin. These facts arerelevant in calculations of expected values, i.e., oscillator strength of the bound–bound, bound–freeand free–free transitions. In Fig. 6 the comparison between the wave functions using the non-isolatedpotentials U 0A

e' and the isolated potential is shown.

5. Conclusions

In this work we have proposed a new general analytical expression for the e'ective potential,and we have built an analytical e'ective potential from an isolated analytical potential. We haveanalyzed the in>uence of the plasma over the potential, energy levels and wave functions, and wehave compared the results obtained with our analytical e'ective potential with those obtained frome'ective potentials built by other methods.The main di'erence with other e'ective potentials is that the analytical potential proposed in this

work includes a new plasma screening function which is di'erent from the typical Debye exponentialscreening. Also, our potential presents a columbian asymptotic behavior which permits an in@nitenumber of bound states.From the analysis of results we can also conclude that our e'ective potential produces for in-

termediate and highly ionized atoms an average relative shift in the potential similar to the othere'ective potentials, although for lower ionization degrees this shift is less than the observer for otherpotentials.Finally, the use of this potential provides level energies and wave functions for a large number

of con@gurations without large time of computation.

Acknowledgements

This work has been partially supported by the program Keep-in Touch of the European Union.

Appendix A

In this appendix we present an approximation of the integral given by Eq. (25). For obtainingthis approximation we write this integral as

�(r) =12a[e−ar

0eas−a1s

a3 ds+ ear∫ ∞

re−as−a1s

a3 ds]: (A.1)

Fig. 6. Plot of (a) the large components of the radial wave functions calculated by using the isolated potential U 0N−1 and

non-isolated potential U 0Ae' and (b) the di'erence between isolated a non-isolated wave functions.

In the @rst integral on the right-hand side of above expression we make the change t = a1sa3 − asand then we @nd that s can be approximated by tp1=(a1 − a)p1 , as p1 = (a1 − a)=(a1a3 − a). Thisleads to∫ r

0eas−a1s

a3 ds=p1

(a1 − a)p1!(p1; a1ra3 − ar): (A.2)

In a similar way, we obtain for the second integral∫ ∞

re−as−a1s

a3 ds=p2

(a1 − a)p2"(p2; a1ra3 + ar) (A.3)

substituting t = a1sa3 + as, and the approximated expression is now s = tp2=(a1 + a)p2 asp2 = (a1 + a)=(a1a3 + a).In the above approximation if t=a1sa3 +as and a3 is close to the unity then, s is close to t=(a1+a)

and to (t=(a1 + a))p. For this reason we propose a solution like s=(t=(a1 + a))p with the parameterp depending on a; a1 and a3, giving as result

a1sa3 + as= (a1 + a)s1=p: (A.4)

p=ln s

ln (a1sa3 + as=a1 + a)(A.5)

where p=1=a3, and 1 for s=0 and ∞, respectively. In a direct numerical test, the best results havebeen found for s=1. Then we obtained s=(t=(a1 + a))p. The solution tp1=(a1− a)p1 is obtained bysubstituting for parameter a by −a.

Appendix B

In this appendix we obtained the analytical e'ective potential from Eqs. (17) and (18) given inthis work, using the isolated potential given by (19) and the following general expression for thescreening function:

�(r) =1

N − 1n∗∑n=1

NnPmn (r)e−anr ; (B.1)

where Nn is the number of electrons in the shell having principle quantum number n; n∗ the max-imum value of n for the bound levels occupied by electrons of the ion except the optical electron,an positive parameters, N − 1 =∑n∗

n=1 Nn, and Pmn (r) is the following m degree polynomial

Pmn (r) =m∑k=0

bnkrk ; (B.2)

where bnk is another set of the parameters with the condition bn0 = 1. Then, the isolated potential isgiven by

U 0N−1(r) =−1

{n∗∑n=1

NnPmn (r)e−ar + Z − N + 1

}: (B.3)

By substituting Eqs. (B.1) and (B.2) into Eq. (21) we obtain

�(r) =a

2(N − 1)n∗∑n=1

m∑k=0

∫ ∞

0e−a|s−r|ske−ans ds

2(N − 1)n∗∑n=1

m∑k=0

[e−as

0easske−ans ds+ eas

∫ ∞

re−asske−ans ds

](B.4)

and taking into account the following integral∫ r2

e±asske−ans ds=e−(an∓a)r

(an ∓ a)

k∑i=0

k!(an ∓ a)i(k − i)!

rk−i∣∣∣∣∣r2

after some straightforward manipulations, we can @nally write the analytical e'ective potential as

U 0Ae' (r) =−1

r{(N − 1)(�(r)− �(r)) + [Z − N + (N − 1)�(0)]e−ar + 1} (B.6)

�(r) =1

(N − 1)n∗∑n=0

NnQmn (r)e

−anr ; (B.7)

Qmn (r) =

k∑i=0

(k − i)!(an − a)i+1 − (an + a)i+1

(a2n − a2)i+1rk−i: (B.8)

References

[1] Salzmann D, Wendin G. Phys Rev A 1978;18:2695.[2] Skupsky S. Phys Rev A 1980;21:1316.[3] M./nguez E, Falquina R. Laser Part Beams 1992;10:651.[4] Abdallah Jr. J, Clark REH. J Appl Phys 1991;69:1.[5] Massacrier G. JQSRT 1994;51:221.[6] Liberman DA. Phys Rev A 1979;10:4981.[7] Davis J, Blaha M. JQSRT 1982;27:307.[8] Cauble R, Blaha M, Davis. J Phys Rev A 1984;29:3280.[9] M./nguez E, G.amez ML. Laser Part Beams 1990;1:103.[10] Crowley BJB. Phys Rev A 1990;41:2179.[11] Debye P, H;uckel E. Phys Rev 1923;24:185.[12] Ecker G, Weizel W. Ann Phys 1956;17:126.[13] Harris GM. Phys Rev 1962;125:1131.[14] Smith CR. Phys Rev A 1964;134:1235.[15] Rouse CA. Phys Rev 1967;159:41.[16] Rouse CA. Phys Rev 1969;188:525.[17] Iafrate GJ, Mendelsohn LB. Phys Rev 1969;182:244.[18] Rogers FJ, Graboske Jr. HC, Harwood D. J Phys Rev A 1970;1:1577.[19] Nauenberg M. Phys Rev A 1973;8:2217.[20] Rousel KM, O’Connell RF. Phys Rev A 1974;9:52.

[21] Gazeau JP, Maquet A. Phys Rev A 1979;20:727.[22] Barcza S. Astron Astrtophys 1979;72:26.[23] Dutt R, Ray A, Ray PP. Phys Lett 1981;83A:65.[24] Green AES. Phys Rev A 1982;26:1759.[25] Gerry CC, Lamb. J Phys Rev A 1984;30:1229.[26] Patil SH. J Phys A 1984;33:1433.[27] Dutt R, Chowdhury K, Varshni YP. J Phys A 1985;18:1379.[28] Vrscay ER. Phys Rev A 1986;33:1433.[29] Sever R, Tezcan C. JQSRT 1991;45:245.[30] Nunez MA. Phys Rev A 1993;47:3620.[31] Roussel KH, O’Connell RF. Phys Rev A 1974;9:52.[32] Shore BW. J Phys B 1975;12:2023.[33] H;ohue FE, Zimmermann R. J Phys B 1982;15:2551.[34] Lange R, Schl;uletr D. JQSRT 1985;33:237.[35] Grandjouan N, Deutsch C. Phys Rev A 1975;11:522.[36] Rouse CA. Phys Rev A 1971;4:90.[37] Rogers FJ. Phys Rev A 1974;10:2441.[38] Rogers FJ. Phys Rev A 1981;23:1008.[39] Benredjem D, Caby M, Couland G. Phys Rev A 1986;33:1279.[40] Ecker G, Weizel W. Ann Phys 1956;17:126.[41] Theimer O, Kepple P. Phys Rev A 1970;1:957.[42] Schl;uter D. JQSRT 1990;43:407.[43] Martel P, Doreste L, M./nguez E, Gil JM. JQSRT 1995;54:621.[44] Martel P, Rubiano JG, Gil JM, Doreste L, M./nguez E. JQSRT 1998;4:623.[45] Klapisch M. Comput Phys Commun 1971;2:39.[46] Rogers FJ, Wilson BG, Iglesias CA. Phys Rev A 1988;38:5007.[47] D;appen W, Mihalas D. Astrophys J 1987;319:195.[48] Hummer DG, Mihalas D. Astrophys J 1988;331:794.

An effective analytical potential including plasma effects

Documents

Transcript of An effective analytical potential including plasma effects

Analytical Hierarcy Process (AHP)

Road Safety Procedure including Journey Management ...

Introduction to Analytical Chemistry

ANALYTICAL ELECTROCHEMISTRY Second Edition

Including RNAV and GNSS

Influence of nebulizer design and aerosol impact bead on analytical sensitivities of inductively coupled plasma mass spectrometry

FARS Analytical User's Manual

Analytical Thesis

Experimental and Analytical - CORE

GRAVITY AND ARCH DAMS INCLUDING HYDRODYNAMIC ...

Analytical Reference Standards

TIMED GENERAL AUCTION INCLUDING TOOLS, CLOTHING ...

Analytical Methods - RSC Publishing

SI 01120.200 Trusts – General, Including

product monograph including patient medication information

Including over 120 pages - Moss Europe

Treating spondyloarthritis, including ankylosing spondylitis ...

Cost Calculation of Construction Projects Including ...

Pre-analytical and analytical variables affecting the measurement of plasma-derived microparticle tissue factor activity

THE MAXIMUM CAPTURE PROBLEM INCLUDING RELOCATION