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Quantum-number-dependent energy level shifts of ions in dense plasmas: A generalized analytical approach

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a Letters Journal Exploring the: Frontiers or Pnrsics

August 2012

EPL, 99 (2012) 33001 www.epljournal.org

doi: 10.1209/0295-5075/99/33001

Quantum-number-dependent energy level shifts of ions in dense plasmas: A generalized analytical approach

X. Li1(a) and F. B. Rosmej2'3

1 State Key Laboratory of High Field Laser Physics, Shanghai Institute of Optics and Fine Mechanics Shanghai 201800, PRC

2 Sorbonne Universités, Pierre et Marie Curie, UMR 7605, LULI - case 128, 4 Place Jussieu, F-75252 Paris Cedex 05 France, EU

3 Ecole Polytechnique, Laboratoire pour l'Utilisation des Lasers Intenses, Physique Atomique dans les Plasmas Denses - F-91128 Palaiseau, France, EU

received 11 May 2012; accepted in final form 4 July 2012 published online 6 August 2012

PACS 32.70.Jz - Line shapes, widths, and shifts

PACS 32.30.Rj - X-ray spectra

PACS 52.25.Jm - Ionization of plasmas

Abstract - Dense plasma effects on bound energy levels have been studied with relativistic atomic structure simulations that include intermediate coupling and configuration interaction. A general analytical formula is proposed that predicts the energy of LSJ-split levels for wide conditions of density and finite temperature and any atomic element. The achieved precision of the analytical approach allowed even to describe energy level crossings and the breakdown of level degeneration induced by dense plasma effects. Finally, a further simplification of the proposed method is developed that requests only information on energy level energies and quantum numbers of isolated atoms that are readily available.

Copyright © EPLA, 2012

Introduction. — Energy level shifts of ions in dense plasmas have been found experimentally and theoretically for many decades [1-15]. The reason for this phenomenon is related to the fact that the plasma shields the nuclear attraction of the bound electron that in turn shifts the energy level toward the continuum. As the overlap of the bound electron with the free electrons depends on the bound electron configurations, different shifts are encountered for upper and lower levels thereby inducing the so-called plasma polarization shift (PPS) [16].

Although the PPS and some relations to the bound orbital quantum numbers have been indicated, no general rules could be established to predict plasma screening effects in dependence on the bound electron characteristics. To reveal such dependences is an urgent need in order to provide a clear picture about the plasma influence on the energy levels and in order to study the behavior of the spectral fine structure that is also at the origin of the spectral line shift [5,11].

In this work the self-consistent field ion sphere model (SCFISM) [17-20] is used. Generally, two basic screening

(a)E-mail: xiangdong_li8mail.siom.ac.cn

models have been developed at present to estimate the energy level shift for ions embedded in plasma. Debye screening model is valid only for weakly coupled plasma. The ion sphere model is applicable in dense plasma. In the frame of ion sphere model, the ion is represented by a point nucleus Z embedded at the centre of a spherical cavity containing enough electrons to ensure global neutrality. Beyond the ion sphere radius R0, the plasma is assumed to produce an electrically neutral background. The SCFISM uses an iteration to obtain the self-consistent density distribution between the bound and free electrons within the ion sphere. Then a self-consistent plasma screening potential is obtained to calculate the bound wave functions by the relativistic atomic structure calculation theory.

Based on scaling studies of H-like ions we develop a general analytical formula that can be applied to any ion/configuration at wide conditions of density and finite temperature. Detailed numerical results are presented for H- and He-like aluminum up to densities of Ne ^ 5 x 1023 cm-3 and temperatures down to Te ^ 50 eV (corresponding to an ion-ion coupling parameter up to 15) that provide a stringent test of the analytical approach for strongly coupled plasma and configurations where even

ra 2.0-

<D c LU

Plasma electron density (in cm"

Fig. 1: (Colour on-line) Relative energy level structures for the energy levels in the shell of H-like Al of n = 5. The energy for each level in the shell 51 at a given plasma parameters is relative to the energy of the 5s(2S1/2) level.

2P(P„> 3P(P„)

4P(P,„)

5P(P„)

6P(P,„) 2p(P„)

3P(P„> 4P(P,:) 5P(P,,) 6p(P,,)

■ > •A

x- 7- < >

Te= 50eV

Plasma electron density (in cm"3)

Fig. 2: (Colour on-line) Comparison of energy shifts between the present analytical approach (dotted lines) and the numerical SCFISM calculations (symbols) for the orbits p of H-like Al in different shell at plasma temperatures of 50 eV. The energy shifts are normalized to the energy of a given orbit at Ne = 0.

intermediate coupling and level crossings are important. Such analytical formula not only shows the clear physics of how the bound electron responses to the plasma screening, but also provides an easy way to calculate energy level structures and populations that are indispensable in plasma radiation simulations.

Results and discussion. — In order to study the scaling properties of dense plasma effects on LSJ-split atomic structure we consider first H-like Al. The orbits with principal quantum number n ^ 7 (shells K, L, M, N, O, P, and Q) and with orbital angular quantum number l ^ 4 (subshells s, p, d, f and g) are included in the configuration interaction (CI) calculation. Thus, the electron configurations considered in our CI calculation are as follows: 1s, 2(s,p), 3(s,p,d), 4(s,p,d,f), 5(s,p,d,f,g), 6(s,p,d,f,g) and 7(s,p, d, f, g). Figure 1 illustrates the relative energy shifts for the fine-structure levels in the shell of n = 5. The energies of nl at certain plasma conditions are presented relative to the energy of the state ns(2Sx/2) in order to better demonstrate the variation within the fine structure. The numerical results show that the relative energy shifts within the same shell depend strongly on the orbital angular momentum l. With increasing density energy levels with large l shift more strongly toward the high-energy region. Moreover, the energy level structure within the same shell is no longer degenerated. Figure 1 also shows that the energy difference between orbits with different l becomes larger as the plasma density increases. The numerical results show also that the spin-orbit splitting is almost unchanged as the plasma density increases. This indicates that for the bound electrons of a certain quantum shell of H-like ions the different response to the plasma is mainly decided by the orbital angular momentum l.

Figure 2 illustrates the energy shift for p orbits with different principal quantum numbers n. In order to present a large variation of n quantum numbers, the energy shift presented is the energy of a given orbit relative to the energy of this orbit at Ne = 0. The figure shows that for the same orbital angular momentum larger quantum numbers n correspond to a smaller relative energy shift. The relative energy shift within the same shell is strongly dependent on the orbital angular momentum quantum number l as has been shown in fig. 1.

In general, it is difficult to obtain an exact analytical expression for the plasma screening potential for any plasma parameters. However, in the limit of infinite plasma temperature the free-electron distribution is uniform and an exact analytic screening potential Vf of the free electron in the ion sphere can be derived as

(Z - Nb) 2Ro

where Z is the nuclear charge, R0 is the radius of the ion sphere (in atomic units), Nb is the number of bound electrons in ion sphere.

Considering the free-electron screening, the Hamil-tonian for an ion embedded in a plasma can be written as

(Z - Nb)

H = Ho + Vf = Ho + ■

where H0 is the unperturbed Hamiltonian or the Hamil-tonian in vacuum. Apparently, the Hamiltonian is made up of two parts. The first part is the contribution from the nucleus that allows negative energies and corresponding bound electrons. The second part is the contribution from

the plasma free-electron screening that adds a positive potential energy to the bound electrons. Consequently, the free-electron screening increases the energy of the bound levels thereby shifting their positions towards the continuum. Supposing ^ is the wave function of an energy level for certain plasma conditions, the energy can be calculated according to

(■Ro

E — Eq + E

Eq — *qHq ^odr,

fRo (Z - Nb) Iq 2R0

3 R2 J

^Qdr :

3(Z - Nb) (Z - Nb) 2

(r2> = 2Z2 [5n2 + 1- 31(1 + 1)].

the energy for a bound level of H-like ions at infinite temperature is given by (in atomic units)

Z2 ßr.,.

+ + 1)- 1(1 + 1)- s(s +1)]

2n2 2 '3(Z-Nb) (Z-Nb) n2 2

+ Vf )tfdr. (3)

In order to proceed with an analytical description, we apply the first-order perturbation theory to evaluate the perturbed energy according to eq. (3),

+ 1^2+ 1-31(1 + 1)]}, (8)

where ß [j(j + 1) - 1(1 + 1) - s(s + 1)] is the energy of spinorbital splitting and

ß (n,Z) = Z4

4ny B n3a31(Z + 1/2)(1 + 1)

: 2.66566 x 10-5 x

n31(1 + 1/2)(1 +1)

(in atomic units).

The ion sphere radius (in atomic units) is given by

Ro — —

3(Z - Nb)

where is the unperturbed wave function. E0 is the unperturbed energy (or the energy produced by nuclear and the spin-orbital splitting) and Ef is the energy shift induced by the free-electron screening. The range of validity of the first-order perturbation theory can be approximately judged by eq. (34) in ref. [21], which is Ne < 1.6 x 1028(Z4/n6) cm-3. For Z = 6 and n = 5, the free-electron density should be Ne ^ 1.33 x 1027 cm-3 that is consistent with the range Ne ^ 5 x 1023 cm-3 used in this paper. Indeed, the upper limit increases when nuclear charge increases. Apparently, the energy shift is decided by the quantum average value (r2). As (r2) increases, the free-electron screening contribution to the energy level shift Ef decreases. The quantum average value (r2) can be analytically expressed as

Ne is the free-electron density in units of cm-3 and a0 =0.529177 x 10-8 cm. The results from eqs. (8)-(10) are in very good agreement with numerical calculations even in the high-plasma-density region. This good agreement suggests that the first-order perturbation theory as proposed by eqs. (4)-(6) is very useful to study dense plasma effects on atomic structure.

For finite-temperature plasmas an exact analytical expression for free-electron screen is not available albeit highly demanding by the dense plasma physics community. Recent studies [21], however, have proposed an asymptotic expansion to express the free-electron screening potential in finite-temperature plasmas in a closed analytical expression (ne, R0 and kbTe are in atomic units)

Vf (r) — 4nne(Ro)

Ro _ r2 +

2 6 3^/n V kbTe

4 ((Z- Nb)\ 1/2R3/2

8 /(Z - Nb)

15^^ V

Equations (6) and (7) show that the plasma free-electron screening contribution to the energy shift Ef is related to the principal quantum number n and to the orbital angular momentum l. Large n and small l correspond to large (r2), and large (r2) means a small energy shift. For the energy levels in the same shell (with the same n), the relative energy shift is orbital angular momentum quantum number dependent. The energy shift Ef becomes large when the orbital angular momentum quantum number l is large. This general behavior is clearly seen from the numerical simulations depicted in fig. 1. Expressing E0 and Ef by orbital quantum numbers

where ne(R0) is the free-electron density at the surface of the ion sphere. It can be calculated by the boundary condition Vf (Ro) = (Z - Nb)/R0.

In order to investigate analytical expressions for the finite-temperature screening potential, we compare in fig. 3(a) the results from eqs. (11) with the SCFISM calculations for Ne = 1 x 1023 cm-3 and different temperatures. The comparison shows that eqs. (11) provide a reasonable approximation for high temperatures but for low temperatures the precision is limited. Moreover, eqs. (11) do not allow deriving analytical expressions with respect to quantum numbers as integrals over the terms

1. 17-

^ 16 I 15

1312 H 11

Te-20eV o Te=50eV a Te=500eV v Te=infinite -

Te= =20eV =50eV =500eV infinite

:: Sjv

Ov -Te=

Ne=1x10*Jcm"J R =5.78224519153a,

20 19 18 17 16 15 14 13 12 11

r (in a )

Vf (r) = 4nne(Ro)

R2 r2 4

—---h

f(Z-N6) 3^/n V k6Te

8 ( (Z-N6) N 1/2

V k6Te

r +--r

ne(Ro) should be calculated by eq. (12) according to the boundary condition Vf (R0) = (Z — Nb)/R0.

n-e(Ro)

(Z—Nb) 4nRo

R0 R^ 4 /(Z-Nb)^1/2 R3/2

_o__o +

2 6 ^Vn

8 AZ-Nb)

Ro + 777 Ro

As before ne, r, R0 and kbTe are in atomic units. Figure 3(b) provides a comparison of the free-electron potential between the present formula (eq. (12)) and the numerical SCFISM calculations. It is found that the modified formula provides improvements especially for the desired low temperature region (strongly coupled plasma). Moreover, with the help of eq. (12) an entirely analytical expression for the energy levels in dense plasmas can be derived,

E = - 2n2 + 2 [j(j + 1)- 1(1 + 1)-s(s + 1)]

+ 4nne(Ro )

6 + 3Vn

4 AZ-NbA 1/2 r3/2

V kbTe

(Z - Nb)\ 1/V 4 , . 1 ( M 1 n (r> +10 (r

(r> = 2Z [3n2 - Z(Z + 1)]

Fig. 3: (Colour on-line) Absolute values of the free-electron potential within the ion sphere. The same colours indicate the same temperature. (a) Comparison of the free-electron potential between the asymptotic expansion of [19] (the solid lines) and the SCFISM calculations (symbols). (b) Comparison of the free-electron potential between the present analytical formula eq. (12) (solid lines) and the SCFISM calculations (symbols).

r3/2 have no analytical solution. We are therefore seeking for a description of the plasma screening potential that has an analytical solution not only for finite temperatures but with respect to quantum numbers too. An extended study of the SCFISM simulations permitted us to propose an analytical expression for the finite-temperature plasma potential that contains only terms in powers of r2 and r

with ne(R0) given by eq. (13), 0 by eq. (9) and R0 by eq. (10). At infinite Te, eq. (14) goes exactly over into eq. (8). Using eq. (14), we calculate the energy shift of the np(Px/2) levels for different plasma densities and temperatures and compare with numerical calculations in fig. 2. It is obvious that the proposed formula according to eq. (14) provides excellent agreement with the numerical SCFISM calculations for a large interval of densities, temperatures and quantum numbers. Both (r) and (r2) are increasing with n and decrease with 1. Thus, according to eq. (14) the absolute energy shift for a H-like orbit at any given plasma condition become large as the orbital principal quantum number decreases and the orbital angular momentum quantum number increases.

Based on the foregoing analysis for different plasma parameters and quantum numbers we are now seeking to generalize eq. (14) to non-H-like ions. The following expression (in atomic units) is proposed:

E(YLSJ) = Eo(YLSJ)

Nb f + ^ 4nne(Ro)

_' 1 ' +

8 /(Z - Nb)

_±_ ((Z - Nb)\ 1/2 R3/2 3 V^V kbTe

15Vn V kbTe

yn (ri>+10(r-

(r> = 2Z^[3n2 - l(l + 1)], (17)

2Zeff n2

(r2> = [5n2 + 1- 3l(l + 1)], (18)

Zfff = nyj2 |Eo(nl)|, (19)

AEpiasma = E(7LSJ) - Eo(7LSJ), (20)

where ne(R0) is given by eq. (13), R0 by eq. (10). E and E0 are the energies in plasma and vacuum, respectively, and AEPlasma is the energy level shift due to dense plasma effects (energies are in a.u. where 1 a.u. = 27.2114 eV) and kbTe is the electron temperature (also in a.u.). L,S and J are the coupled angular quantum numbers, y indicates all other quantum numbers to define the energy level. The summation is over the energy shifts for all bound electrons with number Nb. E0(7LSJ) is the total energy of the considered level, Zeff is the effective nuclear charge of the single bound electron with energy E0(nl) (in the sum of eq. (16)) that moves in the effective hydrogenic potential with charge Zeff.

Let us illustrate the application of the above equations (16)-(20) with an example: He-like level 1s2p(1 Pi) of aluminum, Z = 13, Nb = 2. According to the GRASP0 calculation: E0(1s2p(1P1)) = -102.58579a.u., E0(1s) = -83.3652a.u. and E0(2p) = -18.1557a.u. This provides Zeff (1s) = 12.9124, Zeff (2p) = 12.0517, (r>(1s)=0.11617 a.u., (r>(2p) = 0.41348 a.u., (r2>(1s) = 0.01799 a.u. and (r2>(2p) = 0.20655 a.u. Let us assume the following plasma parameters: electron density Ne = 1023 cm-3 = 0.014818 a.u., providing R0 = 5.6169 a.u., electron temperature kbTe = 50 eV = 1.8375 a.u. According to eq. (13) ne(R0)= 0.0067a.u. which gives for the energy according to eq. (16) E(1s2p(1P1)) = -102.58579 + 6.6736 = -95.9122. The SCFISM calculations provide -95.9639 a.u. demonstrating excellent agreement (precision of the order of 10-3) with the analytical result from eq. (16).

We are now seeking for an application of eq. (16) that does not request employing atomic structure calculation in order to calculate E(7LSJ), E0(7LSJ) and E0(nl) but rather energies that can conveniently be obtained from standard data bases. Let us illustrate the application of eq. (16) employing the data available in the NIST-data base [22]. This data base indicates the following energies: ENIST(1s2p(1P1)) = 1598.2902 eV, E^Sf^ (1s2 (1S0)) = 2085.9756 eV and ENnSTtion (1s1 (2 S1/2)) = 2304.1399 eV from which it follows E0 (1s2p(1P1)) = -(2085.9756 -1598.2902 + 2304.1399)/27.2114 a.u. = -102.5976 a.u. (being equivalent to the energy needed to ionize all electrons from the configuration 1s2p(1P1)). The bound electron energies E0(nl) where an electron nl moves in an effective hydrogenic potential with charge Zeff can be obtained as follows. The averaged energy of the 1s2p state is obtained from a statistical average

of the states 1s2p1P1, 1s2p3P0, 1s2p3P1, 1s2p3P2: Enist (1s2p) = 1590.9178 eV. This provides E0(2p) = -(2085.9756-1590.9178)/27.2114 a.u. = -18.1930 a.u. and Zeff (2p) = 12.0642 with (r>(2p) = 0.41445 a.u. and (r2>(2p) = 0.20612 a.u.

The consideration of E0(1s) is more complex as the 1s-electron corresponds to an inner-shell electron that moves in the nuclear potential screened by a 2p-electron. The following procedure is proposed to approximate E0(1s). First, we consider the energy ignoring all screening of other electrons: E£(1s) = -EN^f^(1s1 (2S1/2)) = -2304.1399/27.2114 a.u. = -84.6755 a.u. Second, we consider the energy E0full(1s) that is obtained from full screening (the term full screening is selected in order to indicate that "exact" calculations or experimental data for energy are employed rather than simple screening constants). For our example it is the difference of the energies E(1s2p) and E(2p) which is the ionization energy of an 1s-electron from the 1s2p-configuration. In general, the difference of energies that involve inner shells can be obtained calculating the energies necessary to ionize all electrons from certain configurations. For E(1s2p) it is the sum of the ionization energies of the 2p-electron and the 1s-electron, E(1s2p) = (ENSt"^^2^)) -Enist (1s2p)) + Enist (1s1(2^1/2)) = (2085.9756 - 1590. 9178) + 2304.1399 eV = 2799.1977 eV. After ionization of the 1s-electron from the configuration 1s2p the configuration 2p is left. E(2p) is the energy of the 2p-electron and given by the statistical average of the states 2p(2P1/2) and 2p(2P3/2), the Nist-Database provides Enist (2p) = 1728.5551 eV and therefore E(2p) = Enist(1s1(2^1/2)) - Enist(2p) = (2304.1399 -1728.5551) eV = 575.5848 eV. Therefore Eful (1s) = -(E (1s2p)-E (2p)) = -(2799.1977- 575.5848)/27.2114 a.u. = 81.7162a.u. For E0(1s) we propose a simple average for full and neglected screening: E0(1s) = 0.5(E0(1s) + EfuU(1s)) =83.1959a.u. This provides Zeff = 12.8993, (r> (1s) = 0.11629 a.u. and (r2> (1s) = 0.01803 a.u. From these data we obtain for the energy according to eq. (16) E (1s2p(1P1)) = -102.5976 + 6.6737 = -95.9239 being likewise in excellent agreement with the numerical calculations. We therefore conclude that eq. (16) provides an excellent description of the energy level shift AEPlasma.

In order to provide further insight in the precision of eq. (16), fig. 4 shows the comparisons between the SCFISM and the analytical approach for the relative energy level position of the 1s4l levels in aluminum. Excellent agreement is obtained for all fine-structure levels over many orders of magnitude in density even for a rather low electron temperature of Te = 50 eV (where finite-temperature effects are very pronounced). Moreover, as can be seen from fig. 4 the high precision of the proposed analytical approximation even permits to study the energy level crossing due to dense plasma effects (that have been predicted recently [20]), see, e.g., the crossing of the 1s4p1P1 with the 1s4f levels at about Ne = 1.3 x 1022 cm-3.

.225 .200 .175.150.125 .100 .075 .050 .025 .000

■ 1s4s(sS,) • 1s4s(lSl>) 1s4p(3P2) T 1s4p(sP,)

1s4p(1P,) ^pi'Pu) 1s4d(®Ds) • 1s4d(sD,)

* 1s4d(1D2) • 1s4d(3D1) ■» 1s4f(®F4) + 154^)

x l^'F,) x 154^

..■S*

♦..............

p : : : : : : : : : : : : " : : J: • ' • • • • • • ^ • ' íS': " * '

♦ ♦

0.225 0.200 0.175 0.150 0.125 0.100 0.075 0.050 0.025 0.000

Plasma electron density (in cm"3)

Fig. 4: (Colour on-line) 1s41 energy levels of He-like Al. The energy of each fine-structure level in 1s41 at a given electron density is relative to the energy of 1s4s(3Si). The solid symbols indicate the results from the numerical SCFISM calculations. The dotted lines are the results from eq. (16). The same colours correspond to the same fine structure. The plasma electron temperature is 50 eV. Excellent agreement is demonstrated between the numerical calculations and the generalized analytical approximation.

Conclusion. — A powerful quantum-number-dependent analytical approximation has been developed to describe finite-temperature dense plasma effects on the energy level structure of H-like ions. Comparison with numerical self-consistent ion sphere simulations indicate excellent agreement for wide ranges of plasma parameters and all combinations of quantum numbers. A generalization of the analytical approach allowed describing dense plasma effects for arbitrary configurations and ions that are based only on energy level information from free ions (electron energies and quantum numbers). This generalization will allow the application of dense plasma effects by a large scientific community as it does not require numerical calculations or the solution of complex integrals anymore. Detailed numerical simulations carried out for He-like ions indicated also that spectroscopic precision is achieved that permits even to investigate level crossings and the loss of degeneracy for the fine-structure energy eigenvalues.

This work was supported by the NSAF under grant No.

10776036, the "ExtreMe Matter Institute - EMMI" and

the project "Emergence-2010: Metaux transparents crees

sous irradiations intenses emises par un laser XUV/X a

electrons libres" of the University Pierre and Marie Curie.

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