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ISSN 1063-7761, Journal of Experimental and Theoretical Physics, 2017, Vol. 125, No. 4, pp. 663–678. © Pleiades Publishing, Inc., 2017. Original Russian Text © A.V. Demura, D.S. Leont’iev, V.S. Lisitsa, V.A. Shurygin, 2017, published in Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2017, Vol. 152, No. 4, pp. 781–798.

STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS

Statistical Dielectronic Recombination Rates for Multielectron Ions in Plasma A. V. Demuraa,*, D. S. Leont’ieva, V. S. Lisitsaa,b, and V. A. Shurygina a National b National

Research Center “Kurchatov Institute,” Moscow, 123182 Russia Research Nuclear University “MEPhI,” Moscow, 115409 Russia *e-mail: [email protected] Received December 29, 2016

Abstract—We describe the general analytic derivation of the dielectronic recombination (DR) rate coefficient for multielectron ions in a plasma based on the statistical theory of an atom in terms of the spatial distribution of the atomic electron density. The dielectronic recombination rates for complex multielectron tungsten ions are calculated numerically in a wide range of variation of the plasma temperature, which is important for modern nuclear fusion studies. The results of statistical theory are compared with the data obtained using level-by-level codes ADPAK, FAC, HULLAC, and experimental results. We consider different statistical DR models based on the Thomas–Fermi distribution, viz., integral and differential with respect to the orbital angular momenta of the ion core and the trapped electron, as well as the Rost model, which is an analog of the Frank–Condon model as applied to atomic structures. In view of its universality and relative simplicity, the statistical approach can be used for obtaining express estimates of the dielectronic recombination rate coefficients in complex calculations of the parameters of the thermonuclear plasmas. The application of statistical methods also provides information for the dielectronic recombination rates with much smaller computer time expenditures as compared to available level-by-level codes. DOI: 10.1134/S1063776117090138

1. INTRODUCTION The dielectronic recombination (DR) process plays an important role in formation of ionization equilibrium in the astrophysical and laboratory plasmas [1–34]. In recent years, DR of complex multielectron ions, which is interesting for the application of metals like tungsten in structural elements of thermonuclear fusion devices with magnetic confinement [13, 18, 21, 24–33] as well as for astrophysical studies [25, 34], has been analyzed especially actively. It is well known [1–34] that the dielectronic recombination rate can be expressed as the sum of different realizations of branching coefficients, which are associated with the competition of radiative decay and autoionization of doubly excited states of atoms or ions. Such level-by-level calculations for multielectron atoms or ions are quite complicated and laborious [1– 15, 17–21, 23–34]. At the same time, complex calculations of the transport and confinement of particles, which are accompanied by simultaneous calculation of ionization equilibrium, the kinetics of population of atomic energy levels, and radiation transport, as well as energy equilibrium and balance [21, 24], are required as applied to the thermonuclear plasma. In this connection, the calculations of ionization equilibrium of the plasma and the kinetics of atomic level population naturally require simplification and accel-

eration. One of the ways for such a simplification is the application of statistical methods [35] that make it possible to substantially facilitate the procedure for obtaining the required rate coefficients and to reveal universal scaling of these processes. In our recent publications, we succeeded in solving this problem as applied to ionization, excitation, and radiative decay processes using the formulated statistical methods [36–40] for describing radiative losses in the thermonuclear plasmas in a wide range of temperatures and concentrations. For closing the system of equations of ionization equilibrium and atomic kinetics in the statistical approach, statistical methods of DR analysis must be developed; this forms the subject of the present research. 2. STRUCTURAL FORMULA FOR THE DIELECTRONIC RECOMBINATION RATE COEFFICIENT The dielectronic recombination is a two-stage process in which the capture of an electron by an ion with charge Zi leads to the formation of a doubly excited state of an ion with a lower degree of ionization Zi – 1 [7, 9]. The energy of this state lies above the ion single ionization threshold Zi – 1 [7, 9]. This doubly excited state of the ion with Zi – 1 is the excited state i of the

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ion core with Zi and the excited state of the trapped electron, which is external relative to the core with the principal quantum number n and orbital angular momentum l [1–9]. At the second stage of the process, doubly excited state i, nl experiences the radiative decay in which the ion core with Zi passes to unexcited state f, while the outer electron remains in state nl [7, 9]. This stage of the process is known as radiative stabilization and completes the dielectronic recombination process [7, 9]. The doubly excited state can also experience nonradiative decay as a result of autoionization, in which ion core with Zi passes from initial excited state i to unexcited state f, transferring the excess energy to the outer electron in the nl state, which passes to a state in the continuum, leading as a result to the ionization of the Zi – 1 ion and the formation of initial Zi ion [1–9]. If final state f of an ion core with Zi coincides with the initial state α of the ion core prior to the electron capture, we speak of elastic scattering; if, however, the final state differs from α, we have different inelastic channels which become open if certain relations between the energy of incident electron, the binding energy of the outer excited trapped electron, and the energy difference between the final and initial states of the ion core are satisfied [1—14]. Proceeding from the most general considerations concerning the determination of the dielectric recombination, we can express its rate coefficient as the sum of the branching factors connected with the competing channels of radiative and autoionization decays of doubly excited states of atoms or ions [7]. The doubly excited state is traditionally marked simultaneously by excited state i of the atomic core and quantum numbers nl as i, nl. Denoting the quantum-mechanical probability of radiative decay of the doubly excited state of the ion by WR(i, nl; f, nl), the rate of the autoionization (nonradiative) decay of the doubly excited atomic energy level by Wa(i, nl; f), and using the principle of detailed balance and the Saha equation, we can write the sought structural expression for the rate of dielectric recombination in the form [7]

QDR (T ) = ×

∑g

inl , f

g inl f

(

a0 4πRy 2 T 3

)

3/2

⎛ E ⎞ W (i, nl; f )W R (i ) exp ⎜ − inl, f ⎟ a , ⎝ T ⎠ W R (i ) + W a (i, nl )

W R (i ) ≡



(1)

W R (i, nl; f ', nl ),

f'

W a (i, nl ) ≡

∑W (i, nl; f ''), a

f ''

where a0 is the Bohr radius, 2Ry is the atomic energy unit, Zi is the ion charge, ginl and g f are the statistical weights of doubly excited state i, nl of the ion with charge Zi – 1 and state f of the ion core with charge

Zi, Einl, f is the energy difference of the doubly excited state i, nl with the charge Zi – 1 and the state of ion core f with the charge Zi and T is the temperature of electrons of the plasma. In many classical works [1–9] devoted to DR, the hydrogen-like approximation is also used for describing the energy of the outer excited electron to simplify calculations. Then it is assumed [3, 6–9] that the main contribution to the DR rate comes from doubly excited states in which the outer electron is weakly coupled with the excited atomic core and, hence, is at a large distance from the nucleus. This approximation makes it possible to write the ratio of statistical weights and the exponential factor in relation (1) in the form [1–9] ⎛ ω 2(2l + 1)g i Z 2Ry ⎞ exp ⎜ − if + i 2 ⎟ , gf nT ⎠ ⎝ T

where gi is the statistical weight of the excited state of the ion core and 2(2l + 1) is the statistical weight of state nl of the outer excited electron [6, 9] with binding energy – Z i2Ry/n2, and ωif is the frequency of radiative transition i → f in the ion core. The natural condition for such an approximation is n ≫ l ≫ 1. Obviously, we assume that the set of quantum numbers f describes the stationary unexcited states of the ion core and i describes the excited states. It should be noted that even such a traditional representation of the DR rate coefficient contains the assumption that the radiative i → f transition in the ion core is weakly affected by the outer excited electron and can be neglected. Therefore, the probability of radiative transition i, nl → f, nl does not contain information on further evolution of the outer excited electron [7], which allows us to write it as only the probability WR(i, nl; f, nl) ≈ WR(i; f) of the radiative transition in the ion core. A certain asymmetry appears in analysis even at this stage because the autoionization rate coefficient preserves the compound two-electron structure and the corresponding representation [6, 7]. At the same time, modern calculations of dielectronic recombination using level-by-level quantummechanical codes are performed taking into account the extremely diversified realization of cascades of radiative and autoionization-induced decays for which states f ' may also belong to the continuum and their relaxation must also be described with allowance for additional branching factors (see [13–15, 18, 25, 27– 34]). In addition, we must take into account the considerable effect of subbarrier resonances in the vicinity of the ionization thresholds (see [26]). The number of included states can be very large. For example, in the so-called iron project devoted to the calculation of the cross sections of photoabsorption, photoionization, and DR, as well as opacity of the spectra of various ions of iron whose atomic structure is much simpler than that of tungsten, the number of atomic states

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taken into account in calculations attained thousands (see [25]). However, such a detailed level-by-level description requires the knowledge of fine details of the energy structure of various electron configurations and considerable computer time and unfortunately cannot be obtained in a universal and unified manner. At the same time, in the statistical approach to DR (its development forms the subject of this study), a universal and unified description of DR can be obtained. In view of its universality, the statistical approach is found to be less resource consuming and more convenient for clarification of the laws of similarity of the elementary process being described [36–40]. This forms an important advantage of this method in generating atomic data and in calculating the atomic kinetics together with complex plasma codes.

2

if σ ph (nl, E )

ω W a (i, nl; f ) = 3 ωa  2c f fi if2 12 2 σ ph (nl, E ) 4π e ω a Z i a0 Z 2Ry for E =  ωif − i 2 , n 2

(2)

where σph(nl, E) is the photoionization cross section of the electron (external relative to the core) excited upon the absorption of a virtual photon of frequency ωif; ωa = me4/  3 = 2Ry/  is the atomic frequency unit; e and m are the electron mass and charge, and c is the speed of light. For describing the photoionization cross section σph(nl, E), we apply the results of the semiclassical method [41], which is admissible in view of the application of the statistical theory proposed below. The semi-classical photoionization cross section at frequency ωif has the form (cf. [41])

4 2 ⎛ l ωif ω ⎞ 2 = 4 a0 e ⎜ a ⎟ 13 3  c ⎝ ωif ⎠ n 3ωa

3 ⎡ 2 ⎛ ωif l 3 ⎞ 2 ⎛ ωif l ⎞⎤ × ⎢K 2/3 ⎜⎜ 2⎟ ⎟ + K 1/3 ⎜⎜ 3ω Z 2 ⎟⎟⎥ ⎣ ⎝ 3ωa Z i ⎠ ⎝ a i ⎠⎦

(3)

for E = ωif – Z i2Ry/n2 and under obvious assumptions concerning large values of n ≫ l ≫ 1. In expression (3) and below, K2/3(z) and K1/3(z) are the Macdonald functions with fractional indices [41]. Finally, we arrive at the traditional result for the autoionization decay rate using the semi-classical approach (cf. [3–9, 22]):

f l ⎛ ω l3 ⎞ W a (i, nl; f ) = 1 ωa fi3 G ⎜⎜ if 2 ⎟⎟ , π n ⎝ 3ωa Z i ⎠ 2 2 G (u) = u[K 2/3(u) + K 1/3 (u)].

3. STRUCTURAL FORMULA FOR THE AUTOIONIZATION DECAY RATE Considering the dielectronic transition corresponding to the autoionization process under the action of the Coulomb electron–electron interaction in the dipole approximation, we can easily find that the rate of the autoionization decay is proportional to the product of squared matrix elements of the dipole moments of the transitions of the ion core and the outer electron, which describes radiative relaxation of the ion core and the ionization of the outer excited electron to which the energy released during the transition in the ion core was transferred (see [6–9]). This allows us to express the probability Wa(i, nl; f) of nonradiative autoionization in terms of the product of the oscillator strength ffi of the radiative transition in the ion core and the photoionization cross section for the outer electron that has absorbed the energy of a virtual photon from radiative relaxation of the ion core in the form (cf. [6–9])

665

(4)

4. DIELECTRONIC RECOMBINATION RATE IN THE STATISTICAL APPROACH The starting point of the statistical approach for describing elementary atomic processes is the introduction for transitions the effective oscillator strengths, associated with the atomic electron density distribution [36–40] and satisfying the Kuhn–Reiche sum rule [42, 43] f ij ∝ n(r )d 3r.

(5)

In this case, the atomic electron density distribution can be described, for example, by the Thomas–Fermi statistical model nTF(r) [35, 42]. To take advantage of this circumstance in deriving the expression for DR in the statistical approach, we express the radiative relaxation rate coefficient for WR(i, f '), as well as the autoionization rate coefficient for Wa(i, nl; f), in terms of the oscillator strength (cf. [39]): 2 2 gi g W R (i, f ) = − 2 e 3 ωif2 i f if = 2 e 3 ωif2 f fi . gf gf mc mc

(6)

To carry out summation over the initial and final states in statistical models, we must introduce additional assumptions connecting transition frequency ωif, which we naturally equate to current frequency ω, with spatial variable r, thus closing the system of equations. If the corresponding dependence ωif = ω(r) has been chosen on the basis of certain consideration, it is reasonable to associate the integral with respect to d3r with the summation over final states i and the integral with respect to dω with the summation over initial states f in complete agreement with the construction of expression (1) [7, 9]. Let us now express the emission and autoionization probabilities in terms of oscillator strength and current

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frequency ω (cf. [39]) taking into account the aforementioned closure condition

45]), we can write the partial density in the form of integrals with respect to energy E of bound states:

3 ⎞ ωl ⎛ W a (i, nl; f ) = a3 G ⎜ ω l 2 ⎟ π n ⎝ 3ωa Z i ⎠ 3 ×n(r )δ(ω − ω(r ))d rd ω.

(7)

Then the sought total rate coefficients of the radiative and autoionization decay of the final state, which are required for calculating the branching factor in the general case [7–9], are defined as 2 W R (i ) = 2 e 3 ω2(r )n(r )d 3r, mc

W a (i, nl ) =

(9)

ωal ⎛ ω(r )l 3 ⎞ 3 G⎜ n(r )d r. 3 2⎟ π n ⎝ 3ωa Z i ⎠

(10)

As a result, after cancellation of identical factors in the numerator and denominator, the DR rate in the statistical approach in the general case [7] is transformed to st QDR (T )

=

4π a03ωa

(

4π Ry T

)

3/2

∑ ∫ drr

2



(13)

− 1/2

2 ⎛ (L + 1/2) 2 ⎞ × ⎜ 2(E − U (r )) −  , ⎟ m r2 ⎝ ⎠ where U(r) < 0 is the self-consistent statistical potential and C is the corresponding normalization constant. Partial (in L) atomic electron density nL(r, E) is defined in expression (13) for a positive radicand between the turning points. Such a distribution imitates the atomic density distribution over the electron shells analogously to the familiar shell model of atomic structure [42, 44, 45]. It is worth noting that the semi-classical form of the centrifugal potential is taken into account explicitly in formula (13) (see [41– 45]). Using expressions (12) and (13), we can generalize result (11). In view of splitting of the total density in L, the total dielectronic recombination rate in the statistical approach is obviously equal to the sum of partial st(L) DR rates QDR corresponding to fixed values of L:

QDR = st

n(r )

∑Q

st(L) DR .

(14)

L

nl rmin

In the general case, transition frequency ω(L)(r) also st depends on L. As a result, expression (11) for QDR is transformed to

⎛  ω(r ) × exp ⎜ − + + 1) nT ⎠ ⎝ T ⎡ ⎛ e 2 ⎞ 3 ω2(r ) l ⎛ ω(r )l 3 ⎞⎤ G × ⎢2 ⎜ ⎟ ⎜ 2 3 2 ⎟⎥ ⎢⎣ ⎝  c ⎠ ωa πn ⎝ 3ωa Z i ⎠⎥⎦

(11) st (T ) = 4πa03ωa QDR

−1

⎡ ⎛ 2 ⎞ 3 ω2(r ) ⎛ ω(r )l 3 ⎞⎤ × ⎢2 ⎜ e ⎟ + l 3G ⎜ ⎟⎥ . 2 πn ⎝ 3ωa Z i2 ⎠⎥⎦ ⎢⎣ ⎝  c ⎠ ωa

(

4πRy T

)

3/2

⎛ Z i2Ry ⎞ ⎟ n 2T ⎠

∑ exp ⎜⎝ L,nl

rmax (L)

×

Result (11) also permits the generalization to the case when the atomic electron density distribution is equal to the total density from different values of the orbital angular momentum L of the ion core, (12)

where nL(r) is the partial atomic electron density. Following general considerations (see, for example, [44,

⎛  ω(L)(r ) ⎞ drr 2nL (r ) exp ⎜ − ⎟ (2l + 1) T ⎝ ⎠ (L)



rmin

In this expression and below, the values of rmax, min are chosen on the basis of the properties of function n(r) and the relation between the values of ω(r) and the ionization potentials for ions with charges Zi and Zi – 1.



⎛ ⎞  2 L + 1/2 1 dE ⎜ 13 ⎟ 3/2 2 ⎝ a0 (2Ry) ⎠ m r U (r )

rmax

2 Z i Ry ⎞ ⎟ (2l 2

n(r ) = dLnL (r ),

0

=C (8)



nL (r ) = dEnL (r, E )

2

2 3 W R (i, f ) = 2 e 3 ω n(r )δ(ω − ω(r ))d rd ω, mc

⎡ ⎛ e 2 ⎞ 3 ⎛ ω(L)(r ) ⎞ 2 l ⎛ ω(L)(r )l 3 ⎞⎤ G × ⎢2 ⎜ ⎟ ⎜ ⎜ ⎟ 3 2 ⎟⎥ ⎢⎣ ⎝  c ⎠ ⎝ ωa ⎠ π n ⎝ 3ωa Z i ⎠⎥⎦

(15)

−1 (L) 3 ⎡ ⎛ 2 ⎞ 3 ⎛ ω(L)(r ) ⎞ 2 l G ⎛ ω (r )l ⎞⎤ , × ⎢2 ⎜ e ⎟ ⎜ + ⎜ ⎟⎥ ⎟ πn3 ⎝ 3ωa Z i2 ⎠⎦ ⎢⎣ ⎝  c ⎠ ⎝ ωa ⎠ where partial density nL(r) with orbital angular momentum L is defined as

⎛ ⎞  2 L + 1/2 1 nL (r ) = C ⎜ 13 3/2 ⎟ 2 ⎝ a0 (2Ry) ⎠ m r 2 ⎛ (L + 1/2) 2 ⎞ × ⎜ 2| U (r )| −  ⎟ 2 m r ⎝ ⎠

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Let us now carry out traditional transformation (15), st as a result of which the DR rate QDR (T) in the statistical approach assumes the form st QDR (T ) = 4π a03ωa

(

4π Ry T

)

3/2

QDR (T ) = 2 st

drr 2nL (r )

L rmin



L,nl rmin (L)

×

⎡  ω(L)(r ) ⎛ 1 ⎞⎤ dt exp ⎢− ⎜1 − 2 ⎟⎥ T ⎝ t ⎠⎦ ⎣

B(r, L, l ) 21/2 ⎛  c ⎞ = ⎜ ⎟ π ⎝ e2 ⎠ n13

(20)

3

1/2

(18)

n ≥ n1. Nevertheless, we can also indicate other lower physical limits for the value of n. It is clear from simple considerations that n cannot be smaller than the effective quantum number nv of the valence shell, which is determined by the first ionization potential I Z i of the ion core [6, 9], (19)

It should be recalled that the actual principle quantum number of the ground state for the outer shell of the multielectron atom or ion differs from the effective value of nv determined in the hydrogen-like model. Since function G(u) decreases exponentially for large values of l, we can replace the sum over l by an improper integral with an admissible accuracy, extending the upper integration limit to infinity. Analogous considerations also allow us to pass from summation to integration with respect to n with the subsequent replacement of integration variable by t. expression

1 Z i2

⎛ ω(L)(r ) ⎞ dr r 2nL (r ) ⎜ ⎟ ⎝ ωa ⎠ (L)

A(r, L, l ) =

as well as variable t = n/n1. Vanishing of the exponent in expression (17) indicates the lower boundary of the continuous spectrum. Then the excitation quantum of the core coincides with the binding energy of the highly excited state of the trapped electron, and the natural limitation imposed on admissible values of n is

After such transformations, assumes the form

3/2



(17)

⎛ ω(L)(r )l 3 ⎞ , lG ⎜ 2 ⎟ ⎝ 3ωa Z i ⎠ ωa = tZ i − 1. (L) 2ω (r )

⎛ ω ⎞ × 13 ⎜ (L)a ⎟ Z i ⎝ ω (r ) ⎠

Analogously to [7], we introduce the value of quantum number n = n1 for the outer electron, for which the exponent in expression (17) vanishes:

nv ~ (Z i + 1)(Ry/ I Z i )1/2.

( ) 2Ry T

⎛ tmax (l + 1/2)lG (ω(L)(r )l 3 /3ω Z 2 ) ⎞ a i ⎟ × ⎜ dl , 3 ⎜ ⎟ t A r L l + ( , , ) ⎝ 0 ⎠

(17)

2

ωa 2ω(L)(r )



t min

3 ⎛ ω(L)(r )l 3 ⎞ ⎛ ω ⎞  c 1 ⎛ ⎞ B(r, L, l ) = ⎜ 2 ⎟ ⎜ (L)a ⎟ lG ⎜ . 2 ⎟ 2π ⎝ e ⎠ ⎝ ω (r ) ⎠ ⎝ 3ωa Z i ⎠

n1 = Z i

3/2 3 a0 ω a

∑ ∫

×

⎛  ω(L)(r ) Z i2Ry ⎞ × exp ⎜ − + 2 ⎟ (2l + 1) l T π nT ⎠ ⎝ (L) 3 −1 ⎛ ω (r )l ⎞ 3 ⎡n + B(r, L, l )⎤ , ×G ⎜ 2 ⎟⎣ ⎦ ⎝ 3ωa Z i ⎠

π

rmax (L)

rmax (L)

∑ ∫

11/2

667

l max

Pay attention to the fact that operating with the atomic electron density partial in the orbital angular momentum of the ion core, we find that the total rates of radiative decay and partial and total autoionization rates in the statistical models under investigation are functions of orbital angular momentum L.

5. AVERAGING OF AUTOIONIZATION RATE OVER OPBITAL ANGULAR MOMENTUM L OF THE OUTER EXCITED ELECTRON As a rule, level-by-level quantum-mechanical calculation of branching factor and their subsequent summation involves considerable difficulties; to simplify these calculations, the complete summation is traditionally replaced by substitution of autoionization rates averaged over quantum numbers L, S, J [7] into the denominator of general expression (1) for the DR rate. In our case, an analog of this approximation can be the replacement of the total summation over l by the substitution of autoionization rates averaged over quantum numbers l, which has already been used in the literature [3, 6–9]. For this, Wa(i, nl) must be multiplied by the number of projections in magnetic quantum number m for the given orbital angular momentum l of the outer electron (which is known to be equal to (2l + 1)), summed over all values of l, and divided by the total number of states n2 for the given value of the principal quantum number n [42–45]. In view of the properties of the asymptotic forms of the Macdonald functions with fractional indices, the summation can be replaced by integration, and the upper limit can be set at infinity. In addition, since the expressions for the photoionization cross section used here was obtained under the assumption that n ≫ l ≫ 1, we can

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neglect 1/2 as compared to l in the expression (2l + 1)/n2 = 2(l + 1/2)/n2 ≈ 2l/n2. Then we can immediately single out the total differential of the argument of the Macdonald functions in the integrand and reduce the integrals of the squares of these functions to the following analytically integrable form [46]: ∞

∫ du[K

2 2/3(u)

+ K 1/3(u)] = 2

0

π =1 π = π. sin π/3 2 3/2 3

As a result, the autoionization rate averaged over the orbital quantum number of the electron external relative to the core assumes the form (cf. [3, 9, 22])

〈W a (i, L, n)〉 l Z = 2 15 (L) i nL (r )d 3r. ωa 3 n ω (r )/ ωa 2

(21)

This result is general for any definition of n(r) and ω(r). At the same time, this is insufficient for calculating the DR rate because factor (2l + 1) already appears in the numerator. If, however, we substitute 〈Wa(i, L, n)〉l into the denominator, it is sufficient to perform simple summation or integration over l in the numerator. Precisely this recommendation is given in monograph [7]. It can easily be seen, however, that in the given case, this is equivalent to the substitution of averaged quantity 〈Wa(i, L, n)〉l into the numerator. Indeed, the substitution of 〈Wa(i, L, n)〉l into the numerator and denominator of expression (19) makes it possible to perform summation over l of factor (2l + 1) in the numerator of expression (1) upon a variation of l from 0 to n – 1, which is known to give n2. In this case, expression (20) can be reduced to the form st QDR (T )

=

3 a0 ω a

11/2

2

π 3

5/2

( ) 2Ry T

∑ ∫

6. BRANDT–LUNDQUIST MODEL OF LOCAL PLASMA FREQUENCY In this model, it is assumed [46] that the current frequency is equal to the local plasma frequency determined by the density distribution in the ion in accordance with the Thomas–Fermi (TF) statistical model [35, 42, 44, 45]:

×



t min

QDR,BLd (T ) = 2 st

11/2

rmax

×

∫ drr

rmin ∞

×

(2Ry T )

3/2

1 Z i2

⎛ ω (r ) ⎞ nTF (r ) ⎜ p ⎟ ⎝ ωa ⎠

⎡  ω p (r ) ⎛ 1 ⎞⎤ ⎜1 − 2 ⎟⎥ T ⎝ t ⎠⎦

∫ dt exp ⎢⎣−

A(r, l ) = (22)

t , t 5 + A(ω(L)(r )/ ωa ) 3

3/2 3 a0 ω a



drr 2nL (r )

2

5/2

π

⎛ lmax (l + 1/2)lG (ω (r )l 3 /3ω Z 2 ) ⎞ p a i ⎟, × ⎜ dl 3 ⎜ ⎟ t A r l + ( , ) ⎝ 0 ⎠ B(r, l ) 21/2 ⎛  c ⎞ = ⎜ ⎟ π ⎝ e2 ⎠ n13

3

1/2

⎛ ω p (r )l 3 ⎞ lG ⎜⎜ 2 ⎟ ⎟, ⎝ 3ωa Z i ⎠ ωa = tZ i − 1. 2ω p (r )

⎛ ω ⎞ × 13 ⎜ a ⎟ Z ⎝ ω p (r ) ⎠ l max

1/2

⎛ ω (r ) ⎞ 2 ⎛  c ⎞ 1 ⎛ ωa ⎞ A⎜ ⎜ ⎟ ⎟= ⎜ ⎟ . 3 ⎝ e 2 ⎠ Z i3 ⎝ ω(L)(r ) ⎠ ⎝ ωa ⎠ Further, we derive the expressions for three statistical models in which the choice and the form of dependences n(r), ω(r) or nL(r), ω(L)(r) will be specified for the case of direct summation over l in expression (19) or after the above approximate procedure of the substitution of autoionization rates (22) averaged over l, which were obtained in this section. This also allows us to choose more exactly the upper and lower limits of (L)

2

t min

⎡ 2Ry ω(L)(r ) ⎛ 1 ⎞⎤ dt exp ⎢− ⎜1 − 2 ⎟⎥ ω a ⎝ t ⎠⎦ ⎣ T ×

(23)

where nTF(r, Zi) is the Thomas–Fermi statistical distribution of the atomic electron density in the ion with charge Zi and nuclear charge Z. Using expressions (11), (18), and (20), we can define the dielectronic recombination rate in the case of direct summation over l by the formula

L rmin (L)



4πe 2nTF (r, Z i ) , m

ω(r ) = ω p (r ) =

3/2

rmax (L)

×

integration with respect to r in formulas (17), (20), (22).

(24)

Index “BLd” indicates the Brandt—Lundquist model for the frequency with the Thomas–Fermi density distribution and with summation over orbital angular momenta of the outer electron. In expressions (24), the total atomic electron density and frequency are naturally independent of the orbital angular momentum L of the ion core and can be written in terms of the universal Thomas–Fermi function χ(q, x) [35, 42, 44] for an ion with charge Zi and nuclear charge Z:

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( (

) )

2 χ(q, x) nTF (r ) = 13 Z 1282 π x 4 a0 9π 1/2 ω p ( x) χ(q, x) = Z ⎛⎜1282 ⎞⎟ ωa x ⎝ 9π ⎠

3/2

, (25)

3/4

,

where q = Zi/Z and x = r/rTF, rTF = a0Z–1/3(9π2/128)1/3 being the Thomas–Fermi radius. In the Thomas– Fermi model, the ion has a finite size r0(q) = x0(q)rTF, for which the atomic electron density vanishes [35, 42, 44]. Since we are dealing with doubly excited states of the ion with Zi – 1, the energy of these states must lie in the continuum above the ionization threshold for the ions with Zi – 1. In the TF model, this threshold is defined by the expression

I Z i −1 ⎛128 ⎞1/3 Z 4/3(q − Z −1) . =⎜ ⎟ 2Ry ⎝ 9π 2 ⎠ x 0(q − Z −1, Z )

(26)

As a rule, an ion with a lower degree of ionization has a larger size; therefore, the core excitation energy is higher than ionization energy (26) and readily satisfies this condition on account of the additional energy of the (n, l) state. If the summation over l is replaced by the substitution of averaged quantities 〈Wa(i, L, n)〉l, which are independent of L in this case, we obtain st 3 QDR ,BLa (T ) = a0 ω a



×

( )

211/2 π 5/2 2Ry T 3

3/2

rmax

∫ drr

2

nTF (r )

rmin

⎡ 2Ry ω p (r ) ⎛ 1 ⎞⎤ ⎜1 − 2 ⎟⎥ T ω a ⎝ t ⎠⎦

∫ dt exp ⎢⎣−

t min

×

669

while the upper integration limit rmax is determined by the distance to the nucleus, for which plasmon energy ωp(rmax) becomes equal to the difference between the ionizations potentials of ions with charges Zi and Zi – 1, which are defined by expressions (26) and (28),

 ω p (rmax ) = I Z i − I Z i −1. It is interesting to note that the result turns out to be not very sensitive to the value of rmax. For example, it remains almost unchanged if we extend the upper integration limit in expressions (24) and (27) to the ion core size r0 in the Thomas–Fermi model [35, 42]. The lower limit of the inner integral over n in expressions (24) and (27) (i.e., after the change of variables to t) is naturally limited by unity in this model (tmin = 1), which is in conformity with the choice of (28). 7. BRANDT–LUNDQUIST MODEL FOR PARTIAL ELECTRON DENSITY OF THE ION CORE In the generalization of the Brandt–Lundquist model [47] to the electron density distribution nTF, L(r, q) of the ion core, which partial in orbital momentum L [44, 45], the resonance condition determining the frequency of collective excitations of the ion core now has the form

(27)

ω(r ) =

ω(pL)(r )

4π e 2nTF ,L (r, Z i ) , = m

(29)

where nTF, L(r, Zi) is defined by Eq. (16) with C = 1/π2, and the potential energy in Eq. (16) satisfiers the relation (cf. [33, 42, 44])

2

t , 5 t + A(ω p (r )/ ωa ) 1/2

3 5/2 ⎛ ω ⎞ ⎛ ω (r ) ⎞ A ⎜ p ⎟ = 2 ⎛⎜  2c ⎞⎟ 13 ⎜ a ⎟ 3 ⎝ e ⎠ Z i ⎝ ω p (r ) ⎠ ⎝ ωa ⎠

1/3

.

Here, subscript “BLa” indicates the Brandt–Lundquist model for the frequency with the Thomas– Fermi distribution and with the substitution of the autoionization rates averaged over the orbital angular momenta of the outer electron. Therefore, in the model described here, the processes of radiative emission, absorption, and excitation of ions as a result of collisions with electrons (and, hence, dielectronic recombination) are described in terms of collective excitations corresponding to local plasma frequencies. The lower limit rmin of integration over r in expressions (24) and (27) corresponds to the attainment of the value of frequency ωp(r) corresponding to the ionization potential of the ion core with charge Zi [35], 1/3 4/3 I Z q  ω p (rmin ) = Z i = ⎛⎜1282 ⎞⎟ , 2Ry ⎝ 9π ⎠ x 0(q, Z )

| U (r )| = | U TF (r )| = 2Ry Z 4/3 ⎛⎜1282 ⎞⎟ ⎝ 9π ⎠

χ( x, q) . x

(30)

Using further the general result (20) for this case, we get the expression

(28)

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS

QDR,BLLd (T ) = 2 st

11/2

rmax (L)

∑ ∫

×

L rmin (L)

π

3/2 3 a0 ω a

(2Ry T )

3/2

1 Z i2

ω(pL)(r ) drr nTF ,L (r ) ωa 2



⎡  ω(L)(r ) ⎛ 1 ⎞⎤ dt exp ⎢− p ⎜1 − 2 ⎟⎥ T ⎝ t ⎠⎦ ⎣ t min

×



⎛ lmax (l + 1/2)lG (ω(L)(r )l 3 /3ω Z 2 ) ⎞ p a i ⎟, × ⎜ dl 3 ⎜ ⎟ t A r L l + ( , , ) ⎝ 0 ⎠



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A(r, L, l ) =

B(r, L, l ) 21/2 ⎛  c ⎞ = ⎜ ⎟ π ⎝ e2 ⎠ n13

⎛ ω ⎞ × 13 ⎜⎜ (L)a ⎟⎟ Z i ⎝ ω p (r ) ⎠

8. ROST MODEL AND APPLICATION OF KRAMERS ELECTRODYNAMICS

3

1/2

⎛ ω(L)(r )l 3 ⎞ lG ⎜⎜ p 2 ⎟ ⎟, ⎝ 3ωa Z i ⎠ ωa = tZ i − 1. (L) 2ω (r )

l max

In the case of the replacement of the summation over l by substitution of averaged values 〈Wa(i, L, n)〉l, which depend on L in the given model, we obtain 11/2

QDR,BLa (T ) = a0 ωa 2 st

3

π 3

5/2

( ) 2Ry T

3/2

rmax (L)

∑ ∫

×

drr 2nTF ,L (r )

L rmin (L)



⎡ 2Ry ω(pL)(r ) ⎛ 1 ⎞⎤ dt exp ⎢− ⎜1 − 2 ⎟⎥ ω a ⎝ t ⎠⎦ ⎣ T t min

×



×

(32)

2

t , t + A(ω(pL)(r )/ ωa ) 5

1/2

⎛ ω(L)(r ) ⎞ 2 5/2 ⎛  c ⎞ 3 1 ⎛ ωa ⎞ A ⎜⎜ p ⎟⎟ = ⎜ ⎟ ⎜ ⎟ 3 ⎝ e 2 ⎠ Z i3 ⎜⎝ ω(pL)(r ) ⎟⎠ ⎝ ωa ⎠

.

It should be noted that the upper and lower limits of integration with respect to r in formulas (31), (32) are now also determined by the turning points in expression (16). It can easily be seen that the use of the upper turning point as the upper integration limit is a more stringent limitation for distribution (16) as compared to size r0 of the ion core. As regards the limitation imposed on the possible value of the lower integration limit, we must consider (apart from the lower turning point) the constraint imposed by the following inequality analogous to that used in expression (27):

 ω(pL)(r ) ≤ I Z i ,

(33)

where I Z i is defined in accordance with expression (28). Analysis shows that condition (33) is more stringent than the lower turning point for all L ≥ 0 up to large values of L after which the lower turning point comes into play, because the value of r corresponding to condition (33) does not fall in the interval of r values corresponding to positive definiteness of the radicand in distribution (16). At the same time, the lower limit in internal integration over t is defined by unity as in the previous section.

In this approach, the autoionization rate is expressed in terms of the photoionization cross section of the ion core. For calculating such cross sections as applied to photoionization processes, Rost [48] proposed a simple model that made it possible to express these cross sections also in terms of the atomic electron density distribution with transition frequency ω

equal to the change in the centrifugal energy ω(KL) (r) in dipole transitions, which corresponds to a change in the orbital angular momentum by unity. The Rost model was tested for hydrogen and helium atoms; comparison with the exact results for hydrogen, experimental data, and numerical calculations for helium has demonstrated a high accuracy with a discrepancy smaller than 5% [48]. This approximation is an analog to the Frank–Condon principle as applied to atomic structures, for which the main contribution to the transition comes from the turning points of electron trajectories (see [48] and also [49]). Here, we will demonstrate the conformity between this approximation and the principles of the Kramers electrodynamics [16, 50], which were widely used in analysis of bremsstrahlung of electrons in complex atoms. According to these principles, the main contribution to a radiative transition comes from the turning point of the electron trajectory, at which the electron kinetic energy considerably exceeds the initial energy due to acceleration in the field of the target, and orbital moment M remains the single integral of motion. Therefore, a radiative dipole transition is determined by the change in the centrifugal potential upon a change in the squared orbital moment M2. The corresponding transition frequency turns out to be equal to the angular velocity of rotation near the turning point,  ω = ΔM2/2mr2, ΔM2 = 2  2 (L + 1/2) (cf. [48]). It is this frequency that will be used below in general expressions (11) and (15) for the DR rate. As before, like everywhere in this study, the partial atomic electron density distribution is calculated using the Thomas–Fermi model.

We will denote the above model with direct summation over orbital angular momenta l of the outer st electron by subscripts “KLd.” Total DR rate QDR ,KLd in this model is also determined by the sum of partial st(L) rate coefficients QDR ,KLd :

st 11/2 3/2 3 QDR π a0 ω a ,KLd = 2

( ) 2Ry T

rmax (L)

∑ ∫

×

drr 2nTF ,L (r )

L rmin (L)

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3/2

1 2 Zi

ω(KL)(r ) ωa

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×



t min

⎡  ω( L ) ⎤ dt exp ⎢− K ⎛⎜1 − 12 ⎞⎟⎥ ⎣ T ⎝ t ⎠⎦

⎛ lmax (l + 1/2)lG (ω(L)(r )l 3 /3ω Z 2 ) ⎞ K a i ⎟ × ⎜ dl , 3 ⎜ ⎟ t A r L l + ( , , ) ⎝ 0 ⎠ 3 1/2 B(r, L, l ) 2 ⎛  c ⎞ A(r, L, l ) = = ⎜ ⎟ π ⎝ e2 ⎠ n13



(34)

1/2

⎛ ω(L)(r )l 3 ⎞ lG ⎜ K , 2 ⎟ ⎝ 3ωa Z i ⎠ ωa = tZ i − 1. 2ω(KL)(r )

⎛ ω ⎞ × 13 ⎜ (L)a ⎟ Z i ⎝ ω K (r ) ⎠ l max

It should be noted that frequency ω(KL) (r) in expression (34), which is defined by the ratio

ω(KL)(r ) L + 1/2 (35) , = ωa (r / a0 )2 is often referred to in the literature as the Rost frequency [48]. If we replace the summation over l in expression (34) by the substitution of averaged quantities 〈Wa(i, L, n)〉l, we obtain 11/2

st 3 2 QDR ,KLa (T ) = a0 ω a

( )

π 5/2 2Ry T 3

3/2

rmax (L)

∑ ∫

×

drr 2nTF ,L (r )

L rmin (L)



×



t min

⎡ 2Ry ω(KL)(r ) ⎛ 1 ⎞⎤ dt exp ⎢− ⎜1 − 2 ⎟⎥ ω a ⎝ t ⎠⎦ ⎣ T ×

(36)

t2 , t 5 + A(ω(KL)(r )/ ωa ) 1/2

⎛ ω(L)(r ) ⎞ 2 5/2 ⎛  c ⎞ 3 1 ⎛ ωa ⎞ A⎜ K ⎜ ⎟ ⎟= ⎜ ⎟ 3 ⎝ e 2 ⎠ Z i3 ⎝ ω(KL)(r ) ⎠ ⎝ ωa ⎠

.

The admissible domain of integration over the volume in expressions (34) and (36) is determined analogously to the previous model by the turning points of distribution (16), while the frequencies correspond to their effective values following from the Kramers electrodynamics [16, 50] and the Rost model analogous to it in this respect [48]. Like in the previous section, we choose as the lower limit of integration with respect to r for a given L its minimal value from the lower turning point in expression (16) and following from the condition

 ω(KL)(r* ) ≤ I Z i ,

r*(L) ≥

, (L + 12) 2Ry I Zi

(37)

671

with turning points from expression (16) and with the Rost frequency ω(KL) (r)/ωa = (L + 1/2)/(r/a0)2 from expression (35). Analysis shows that in this model, condition (37) for any L is more stringent in determining the lower limit of integration over r than the lower turning point in expression (16). In further analysis, we will refer to this statistical model as the Rost model for brevity.

9. BEHAVIOR OF DR RATE COEFFICIENTS FOR DIFFERENT TUNGSTEN IONS We have calculated numerically the DR rate coefficients for seven tungsten ions W18+, W20+, W29+, W37+, W41+, W43+, and W56+ as functions of temperature using the formulas of six statistical models described above as well as functions of charge of the ions for two characteristic values of temperature. We have also compared the statistical rate coefficients with experimental data [27, 31] and with the results of calculations with level-by-level codes ADPAK [21], FAC [23, 28, 30, 33], and HULLAC [13, 18]. In our calculations aimed at obtaining the description of the electron density distribution according to the Thomas–Fermi model, we used the tabulated values for function χ(x) corresponding to the universal density distribution in neutral atoms with the relevant Sommerfeld approximation for describing the atomic electron density in ions [35]. Given ion sizes x0(q) were calculated using approximations [51] and [52]. All calculations were performed with simultaneous control of normalization of the ion density distribution following from the Kuhn–Reiche sum rule for oscillator strengths [42, 43], as well as the convergence of the results depending on the step size on each of three sequential integrations and on the number of terms in the sum over L. In determining the turning points, the discrimination procedure was used, which made it possible to take into account only inner turning points in the range of positive values of the radicand in expression (16). In calculations, NumPy [53] and SciPy [54] libraries for the Python programming language were used. Comparison of the results of calculation of the DR rate coefficients as functions of the plasma temperature in accordance with the above six statistical models is illustrated in Figs. 1 and 2 for the W18+ and W37+ tungsten ions. The comparison shows that for the W18+ ion (Fig. 1), the highest values of the DR rate in the exact summation are obtained using the model with the density distribution partial in L in accordance with the Thomas–Fermi model with the Brandt–Lundquist frequency. The approximate averaging operation in l leads to higher values of DR rate in all statistical models. Comparison of statistical DR rates in the case of exact summation based on three of the above models for W18+ and W20+ ions depending on the plasma tem-

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Q, cm3/s 10−8

4 3

Q, cm3/s 10−8 6

3

W18+

5 6

5 2 10−9

10−9

1

10−10 102

1

W37+ 4

2

10−10 102

103

103

104 T, eV

T, eV Fig. 1. (Color online) Temperature dependences of the DR rate for W18+ ion in different statistical models: (1) Thomas–Fermi model and Brandt–Lundquist model; (2) Thomas–Fermi model averaged over l and Brandt– Lundquist model in accordance with (27); (3) Thomas– Fermi model with electron shells of the ion core with orbital angular momentum L and Brandt–Lundquist model with the electron density partial in L; (4) Thomas– Fermi model averaged over l with electron shells of the ion core with orbital angular momentum L and Brandt– Lundquist model with electron density partial in L in accordance with (32); (5) Thomas–Fermi model with electron density partial in L and Kramers electrodynamics for frequency (Rost model); (6) Thomas–Fermi model with electron density partial in L and Kramers electrodynamics for frequency (Rost model) averaged over l in accordance with (36).

perature with experimental data [27, 31] and the results of calculations using the ADPAK level code [21] is illustrated in Figs. 3 and 4. As can be seen in these figures, the best agreement with experiments for the W18+ and W20+ ions is attained using the Thomas– Fermi model with the Brandt–Lundquist local plasma frequency and the model with the Rost frequency. It is important to note that the experimental data are described by a nearly straight line with a much larger angle with the ordinate axis as compared to the calculated curves. For this reason, the position of the theoretical curves relative to the experimental line in the temperature interval 0.1–1.3 keV changes significantly. All theoretical curves demonstrate a certain similarity upon a change in temperature. The largest difference in calculations is observed in the range of low temperatures. All statistical models give values exceeding the experimental data (by 2–3 times in the Brandt–Lundquist model) and the results of calculations using the ADPAK code [21] (by 1.5–2 times for the Brandt–Lundquist model). Such a relation between the results is not surprising, because it follows from the general physical considerations that the statistical approach gives, in fact, the upper estimate for

Fig. 2. (Color online) Temperature dependences of the DR rate for W37+ ion for different statistical models. The notation of curves 1–6 is the same as in Fig. 1.

the oscillator strength of the transitions. Therefore, the results of the statistical approach generally also give the upper estimate of the DR rates at the modern state-of-the art of this problem. It should be noted that the experimental data were obtained on the storage rings for which the conditions differ substantially from the plasma medium (in particular, in the existence of metastable states of ions) (see [32, 55, 56]). Figures 5–9 illustrate the comparison of the results of three statistical models with exact summation over l for the W29+, W37+, W41+, W43+, and W56+ ions, depending on the plasma temperature with one another and with the data obtained using the HULLAC [13, 18], FAC [23, 28, 30, 33], and ADPAK [21] codes. It can be seen that the temperature dependences, the form, and the differences between the results of statistical models and the level-by-level quantum-mechanical codes are different for different ions. The results of the Brandt–Lundquist model for the W29+ ion in Fig. 5 coincide well with the results obtained using the Rost model, and the FAC data [28] correspond with the ADPAK data from [21]. For example, the results corresponding to the Brandt– Lundquist model in Fig. 5 exceed the ADPAK data by 1.5–2 times, and the maximum difference from the FAC data at temperatures near 1 keV is by three times. The curves describing the results obtained using other statistical models are approximately 3–4 times higher than the results of level codes. The data for the W37+ ion in Fig. 6 obtained using the Brandt–Lundquist model are in good agreement with the data of the Rost model and with the FAC data [30, 33] (the difference constitutes approximately less than by a factor of 1.25), while the ADPAK data from [21] are noticeably lower. The data obtained using the Brandt–Lundquist model

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Q, cm3/s

Q, cm3/s

10−8

10−8 W18+

W20+ 2

2 3

3

10−9

10−9

1

10

−10

102

673

4

4

1

5

5 10−10 103

102

103

T, eV

T, eV

Fig. 3. (Color online) Comparison of statistical temperature dependences of the DR rate for W18+ ion with experiment [19] and with calculations using level-by-level ADPAK code [21]: (1) Thomas–Fermi model and Brandt–Lundquist model; (2) Thomas–Fermi model with electron shells of the ion core with orbital angular momentum L and Brandt–Lundquist model with the electron density partial in L; (3) Thomas–Fermi model with electron density partial in L and Kramers electrodynamics for frequency (Rost model); (4) experiment [31]; (5) ADPAK [21].

Fig. 4. (Color online) Comparison of the statistical temperature dependences of the DR rate for the W20+ ion with experiment [27] and with calculations based of level-bylevel code ADPAK [21]. The notation of curves 1–5 is the same as in Fig. 3.

for W41+ ion in Fig. 7 are in good agreement with the data of the Rost model, while the ADPAK data from [21] coincide with the FAC data [30, 33]. The results of the above statistical models exceed the data from codes in the temperature range under investigation by 3–4 times. The data obtained using the Brandt– Lundquist and Rost models for the W43+ ion (Fig. 8) are in satisfactory agreement with each other, while the FAC [30, 33] data are close to ADPAK data from [21]. The maximal difference between the results obtained using Brandt–Lundquist model and the FAC data (by a factor of three) is observed at a temperature of 1 keV and becomes smaller at lower temperatures. Conversely, the difference between the Brandt–Lundquist model and ADPAK increases in this region. Finally, the HULLAC data for the W56+ ion [18] (Fig. 9) differ from the ADPAK data from [21] only slightly, but have another slope of the curve describing the temperature dependence; the curves describing the results of statistical models with plasma frequencies lie above the curves obtained in the Rost model, which correspond with the HULLAC data [18] in the range of low temperatures. The maximal difference between the results of the Brandt–Lundquist model and the HULLAC data (by about 4 times) is observed again in the region of 1 keV, while the results of the Rost model differ by 2.5 times. The results of the Brandt–Lundquist model with the atomic electron density distribution partial in the

orbital angular momentum exceed the data of the codes by approximately 4 times. In the whole, the difference in the results obtained using different models of the ions under investigation lie in the limits from 2 to 4 times upon a change in temperature by two orders of magnitude. Figure 9 also shows the results of calculations based on the semi-empirical Burgess–Mertz (BM) formula [1, 2, 4, 5] from [18]. In the range of low temperatures, these results are many orders of magnitude smaller than the results of other calculations [18]. However, at high temperatures (beginning with T > 200–300 eV), the BM formula is in good agreement with the HULLAC data [18] (with an error ≤20–30%) and ADPAK data [21], although the corresponding curve has a smaller slope. In this temperature range, the calculations performed using other codes (FAC [28] and HULLAC [57]) exhibit analogous agreement with the BM data. In accordance with Fig. 9, the results of statistical approach in the entire temperature range noticeably exceed the BM data. In the range of relatively low temperatures, the statistical approach gives more reliable estimates of the DR rate as compared to the BM formula. In Figs. 10 and 11, the dependences of DR rates on ion charge Zi obtained for two characteristic values of temperature (1.1 keV and 2.5 keV) using the above statistical models as well as using level-by-level quantummechanical codes ADPAK [21, 24] and FAC [23, 28, 30, 33] are compared. Our analysis shows that the data obtained with quantum-mechanical codes demonstrates sharp jumps of the DR rates even for ions of adjacent degrees of ionization (see also [14, 15, 19, 20]). The behavior of such jumps can in principle be traced in level-by-level calculations from the change

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Q, cm3/s

Q, cm3/s 10−8

10−8 2 3

10

2

W29+

3

1

−9

W37+

1 10−9

4 5

10

−10

5 4

10−11 2 10

103

10−10 104 T, eV

102

103

104 T, eV

Fig. 5. (Color online) Comparison of the statistical temperature dependences of the DR rate for the W29+ ion with calculations based of level-by-level codes FAC [23, 28] and ADPAK [21]. The notation of curves 1–3 and 5 is the same as in Figs. 3, 4; (4) results of application of the FAC code from [28].

Fig. 6. (Color online) Comparison of the statistical temperature dependences of the DR rate for the W37+ ion with calculations based of level-by-level codes FAC [23, 30, 33] and ADPAK [21]. The notation of curves 1–3 and 5 is the same as in Figs. 1–5; (4) results of application of the FAC code from [30, 33].

Q, cm3/s 10−8

Q, cm3/s 10−8 2

2

W41+ 3

3 1 10

−9

W43+

1

4

10

−9

4 5

5 10−10 102

10−10 103

104 T, eV

102

103

104 T, eV

Fig. 7. (Color online) Comparison of the statistical temperature dependences of the DR rate for the W41+ ion with calculations based of level-by-level codes FAC [23, 30, 33] and ADPAK [21]. The notation of curves 1–3 and 5 is the same as in Figs. 3–6; (4) results of application of the FAC code from [30, 33].

Fig. 8. (Color online) Comparison of the statistical temperature dependences of the DR rate for the W43+ ion with calculations based of level-by-level codes FAC [23, 30, 33] and ADPAK [21]. The notation of curves 1–3 and 5 is the same as in Figs. 3–7; (4) results of application of the FAC code from [30, 33].

in the contributions of different open channels upon a change in temperature [14, 15, 19–21, 23, 24, 28, 30, 33]; ions with different degrees of ionization exhibit different localizations of such extrema on the temperature scale [14, 15, 19–21, 23, 24, 28, 30, 33]. However, the reported results obtained using different

codes noticeably differ in the prediction of jumps in the DR rates for ions of the same degree of ionization. Moreover, the noticeable difference in the results is also revealed from comparison of different versions of the same code (e.g., ADPAK [21, 24]; see Fig. 10). The first version [21] is the result of developers of this

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Q, cm3/s

Q, cm3/s W56+ 1 10−9

–1 –2 –3 –4 –5 –6

2

4 3

10−9

5 10−10

6 10−10

10−11 1 10

102

103

104 T, eV

Fig. 9. (Color online) Comparison of statistical temperature dependences of the DR rate for the W56+ ion with calculations based on level-by-level HULLAC [18] and ADPAK [21] codes and with semi-empirical Burgess– Mertz formula [18]. The notation of curves 1–3 and 5 is the same as in Figs. 3–8; (4) results of application of the HULLAC code from [18]; (6) calculation based on the Burgess–Mertz formula [1, 2, 4, 5] from [18].

code, while the second version [24] is the code in which fitting parameters for the rate coefficients are introduced for the rate coefficients of different elementary processes to satisfy the results of complex large-scale experiments on nuclear fusion devices with magnetic confinement. The problem of determining such coefficients is an incorrect multiparametric inverse problem which cannot always be solved with a preset accuracy. This is clearly manifested in a significant difference in the DR rates of these versions of the ADPAK code, which is mainly observed in the region of DR rate jumps in [24]. The form of jumps in the DR rates in [24] can probably be due, among other things, to the fitting of the calculated data to the experimental values observed in specific conditions (see discussion in [58]). The dependence on Zi in the given statistical models is quite smooth and predictable on the basis of the analytic expressions derived above for these models. The results of statistical models in Figs. 10 and 11 lie noticeably higher than the values of quantummechanical codes averaged over jumps. At the same time, it can be seen that the scale of this excess is of the same order of magnitude as the characteristic spread in the values of DR rates obtained from level-by-level codes in the regions of jumps [21, 24, 28, 30, 33]. It should also be noted that the statistical method makes it possible to obtain the required data much faster and with smaller computation consumption as compared to level-by-level quantum-mechanical codes. Therefore, the above results and their comparison demonstrate at least the reasonableness of the application of the statistic approach for estimating DR rates.

10

20

30

40

50

60 Zi

Fig. 10. (Color online) Comparison of the dependences of the DR rate on the charge of the tungsten ion at a temperature of 1.1 keV. The notation of curves 1–3 is the same as in Figs. 3–9; (4) ADPAK [21]; (5) ADPAK version from [24]; (6) results of application of the FAC code from [23, 28, 30, 33].

The DR rate coefficients slowly decrease with increasing ion charge Zi, which can clearly be traced in the data obtained in [21, 24, 28, 30, 33], where the temperature dependences are given for a number of Zi values. The temperature dependences are characterized by different degrees of nonmonotonicity and similarity for different fixed values of Zi. The lower values of DR rates obtained in [28, 30] are mainly due to the known suppression of the Auger decay and autoionQ, cm3/s 10−9 –1 –2 –3 –4 –5 –6 10−10

10

20

30

40

50

60 Zi

Fig. 11. (Color online) Comparison of the dependences of the DR rate on the tungsten charge at a temperature of 2.5 keV. The notation is the same as in Fig. 10.

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Q, cm3/s 10−9 Ar5+ 10−10

10−11 1 10

−12

2 10

100

1000

10000 T, eV

Fig. 12. (Color online) Comparison of the temperature dependences of the DR rate for the argon Ar5+ ion: (1) statistical Thomas–Fermi and Brandt–Lundquist model with averaging over l; (2) calculation according to [17].

ization rates upon an increase in Zi [38]. In the statistical models, a sharp decrease in the effective frequency with increasing Zi can also be traced as long as the energy spectrum of the ion core differs significantly from that of a hydrogen-like atom. Something of this kind apparently happens in level-by-level calculations [14, 15, 28, 30]. It can be seen from comparison illustrated in Figs. 10 and 11 that the DR rate coefficients strongly decrease in the limiting cases of very small and very large stripping in level-by-level calculations as well as in accordance with the statistical models. Such a behavior completely corresponds to qualitative physical considerations. 10. DISCUSSION It was shown in our previous publications [36–40] devoted to the description of excitation, ionization, and radiation losses in a plasma with multielectron ions of heavy elements that the statistical approach operates with the concepts of collective excitations of the electron density. This allows us to obtain reliable universal characteristics of elementary processes on the average without claiming at the description of resonance effects. The dielectronic recombination in this context differs from the processes that have been considered earlier in that it begins with resonant nonradiative capture of a plasma electron by an ion [1–9]. Therefore, the process is naturally sensitive to the structure of the electronic configuration of the ion under investigation. This explains the jumps in the DR rates in the results of the level-by-level codes HULLAC and FAC, when the main contribution (at least for temperatures in the range of 100–500 eV) comes from the transitions between different values of L with Δn = 0,

which have larger oscillator strengths. In this connection, analysis of the possible supplement of the statistical result by the contribution from such transitions is physically justified. At the same time, the statistical approach demonstrates the importance of the inclusion of the collective nature of excitations of atomic oscillators (see, for example, [59]), which is reflected, for example, in the description of the oscillator strengths and their frequency in terms of the atomic electron density functional based on this approach; this leads to slight exaggeration of the results as compared to the level-by-level code data even in the region in which their results experience jumps. The same situation is observed for the results obtained in [3], in which, according to [9], the approximation of averaging in l was used for the first time, and comparison carried out in [9] demonstrated a noticeable excess of the results obtained in [3] over the data obtained using a more detailed level-by-level description of branching factors in terms of angular momenta in contrast to level-by-level calculations for n in [3]. The statistical model is applicable for multielectron systems. For testing its possible application for lighter ions, we calculated the DR rates for the Ar5+ ion. Comparison of the results for DR rates of the Ar5+ ion obtained in the Thomas–Fermi–Brandt–Lundquist model (27) with averaging over l with analytic approximation of compilation of different numerical calculations [17] is demonstrated in Fig. 12. As expected, the deviation of the results of the statistical model upon a decrease of the number of bound electrons and nuclear charge in the temperature range to 100 eV from the level-by-level calculations increases and amounts to 2–4 times and up to 6 times near 1 keV in the given example. It should be noted that the data reported in [17] also include the Burgess–Mertz approximation [1, 2, 4, 5]. Thus, we have demonstrated that the statistical approach provides a universal description of the DR process for complex multielectron ions in the range of temperature variation over more than two orders of magnitude. Comparison with the known level-bylevel codes ADPAK [21, 24], FAC [23, 28, 30, 33], and HULLAC [13, 18] reveals the excess of the statistical model data by 1.5–4 times for different type of tungsten ions; the difference between the data obtained using different codes can also be significant for some ions and temperature intervals. Such values of DR rates in the statistical approach are apparently due to the overestimation of the number of doubly excited states. In view of substantially smaller computation consumption as compared to the level-by-level quantummechanical codes and its universality, the statistical approach can be used for estimating the DR rates in cumbersome complex calculations aimed at determining the parameters of hot thermonuclear and astrophysical plasmas. Our analysis has demonstrated that

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among the statistical models considered here, the Rost model [48] based on the principles of the Kramers electrodynamics [50] gives the most optimal results. Our results are also valid for an arbitrary atomic electron density distribution and therefore permit, apart from the Thomas–Fermi statistical distribution [35, 42, 44, 45] used in this work, a generalization to the Hartree–Fock electron density distribution [35, 42] and other distributions. ACKNOWLEDGMENTS One of the authors (A.V.D.) is sincerely grateful to Dragan Nikolic for numerous fruitful discussions. REFERENCES 1. A. Burgess, Astrophys. J. 139, 776 (1964). 2. A. Burgess, Astrophys. J. 141, 1588 (1965). 3. W. H. Tucker and R. J. Gould, Astrophys. J. 144, 244 (1966). 4. A. Burgess and H. Summers, Astrophys. J. 157, 1007 (1969). 5. A. L. Mertz, R. D. Cowan, and N. H. Magee, LAMS6220 (1976). 6. V. P. Zhdanov, Sov. Phys. JETP 48, 611 (1978). 7. L. A. Vainshtein, I. I. Sobel’man, and E. A. Yukov, Excitation of Atoms and Broadening of Spectral Lines (Nauka, Moscow, 1979; Springer, New York, 2002). 8. I. L. Beigman, L. A. Vainshtein, and B. N. Chichkov, Sov. Phys. JETP 53, 490 (1981). 9. V. P. Zhdanov, in Plasma Theory Problems, Ed. by M. A. Leontovich (Energoatomizdat, Moscow, 1982), No. 12, p. 79 [in Russian]. 10. M. Arnaud and R. Rothenflug, Astron. Astrophys. Suppl. Ser. 60, 425 (1985). 11. R. M. More, G. B. Zimmerman, and Z. Zinamon, AIP Conf. Proc. 168, 33 (1988). 12. M. Arnaud and J. Raymond, Astrophys. J. 398, 394 (1992). 13. E. Behar, P. Mandelbaum, J. L. Schwob, A. Bar-Shalom, J. Oreg, and W. H. Goldstein, Phys. Rev. A 52, 3770 (1995). 14. K. B. Fournier, M. Cohen, W. H. Goldstein, A. L. Osterheld, M. Finkenthal, M. J. May, J. L. Terry, M. A. Graf, and J. Rice, Phys. Rev. A 54, 3870 (1996). 15. K. B. Fournier, M. Cohen, and W. H. Goldstein, Phys. Rev. A 56, 4715 (1997). 16. L. A. Bureeva and V. S. Lisitsa, An Excited Atom (Izdat, Moscow, 1997) [in Russian]. 17. P. Mazzotta, G. Mazzitelli, S. Colafrancesco, and N. Vittorio, Astron. Astrophys. Suppl. Ser. 133, 403 (1998). 18. A. Peleg, E. Behar, P. Mandelbaum, and J. L. Schwob, Phys. Rev. A 57, 3493 (1998). 19. K. B. Fournier, At. Data Nucl. Data Tables 68, 1 (1998). 20. K. B. Fournier, M. Cohen, W. M. J. May, and H. Goldstein, At. Data Nucl. Data Tables 70, 231 (1998).

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