Penetrating collisions in hydrogen spectral line broadening by plasmas

High Energy Density Physics 35 (2020) 100743

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Penetrating collisions in hydrogen spectral line broadening by plasmas Carlos A. Iglesias

T

Lawrence Livermore National Laboratories, P.O. Box 808, Livermore, CA 94550, USA

ARTICLE INFO

ABSTRACT

Keywords: Spectral line shapes Stark broadening

Spectral line Stark broadening calculations in the “standard theory” typically retain only the long-ranged dipole term in the interaction between the perturbing plasma and emitting or absorbing atom. Thus, penetrating collisions as well as higher multipoles are neglected. The full Coulomb interaction is applied to hydrogen lines using a quantum mechanical treatment. It is found that the line widths from the quantum mechanical approach moderately disagree with earlier semi-classical treatment of penetrating collisions.

1. Introduction Spectral line shapes are an important diagnostic tool for astrophysical and laboratory plasmas. The “standard theory” of Stark spectral line broadening [1], however, approximates the interaction between the perturbing plasma and the emitting or absorbing atomic system by the long-ranged multipole expansion often keeping only the dipole term. This approximation was tested for highly charged ions in laser implosion experiments using the full Coulomb atom-perturber interaction [2]. The resulting unexpected modest corrections were due to a fortuitous overestimate of the dipole interaction simulating the contributions from higher order multipoles in the full Coulomb interaction. More recently the impact of penetrating collisions on hydrogen lines (i.e.; plasma electrons significantly overlapping with the atomic electron wavefunctions during a collision) attracted attention in high density plasmas [3]. In such cases, the atomic states have a spatial size comparable to the average interparticle distance and atom-electron collisions may not be adequately described by the long-ranged multipole expansion. The penetrating collisions were found to soften the atom-electron interaction relative to the long-ranged multipoles [4,5]. This earlier work, however, used a semi-classical description for the perturbing electrons [4–6]. The purpose here is to examine penetrating collisions with a quantum mechanical treatment for neutral hydrogen lines. Following previous work [4,6], the collision operator is treated in second-order perturbation theory but replaces the straight-line classical path trajectories of the perturbing electrons with plane waves. In addition, plasma screening effects are formally introduced through the dynamic structure factor for the perturbing electron gas [7–10] rather than with an ad hoc long-ranged cutoff when averaging over impact parameters [1,3,4,6,7]. For hydrogen lines the quantum mechanical line width expression

reduces to a semi-analytical formula in the weakly-coupled plasma limit. It yields analytic results at line center when only levels with the same principal quantum number are allowed to interact. As in the earlier work, the line widths at line center using the dipole term in the spherical harmonic expansion of the full Coulomb interaction are emphasized. The results are compared with earlier line broadening models. 2. Line broadening by electrons The line shape function at photon energy ℏω, neglecting ion and Doppler broadening, is written as [8–10] 1Re

I( ) =

dte i

t

d · d (t )

o 1Im Tr a

=

d ·{

B

M ( )} 1f (a) d

(2.1)

where Tra is a trace over the hydrogen atom internal states, 〈⋅⋅⋅〉 de-

notes an ensemble average over a perturbing electron gas, and d is the atom dipole operator. For the atom and s perturbing electrons, the reduced distributions are defined as a trace over the remaining plasma electrons,

nes f (a1 s ) =

N! (N

s )!

Trs + 1,

N

(2.2)

with ne the plasma electron number density and ρ the density matrix for the atom-electron gas system. In addition,

=

(2.3)

L (a)

L (a) = [H (a),

]

(2.4)

with [H(a), ⋅⋅⋅] a commutator with the Hamiltonian for the internal

E-mail address: [email protected]. https://doi.org/10.1016/j.hedp.2020.100743 Received 5 November 2019; Received in revised form 9 January 2020; Accepted 9 January 2020 Available online 11 January 2020 1574-1818/ © 2020 Elsevier B.V. All rights reserved.

High Energy Density Physics 35 (2020) 100743

C.A. Iglesias

degrees of freedom of the isolated hydrogen atom. The “width and shift” operator, B + M ( ) , was expressed as a generalized binary collision with many-body aspects contained in reduced distribution functions, screened interactions, mean field terms, and multiple collision effects appearing in its definition. This expression for the line shape assumed a single atom immersed in an electron gas with independent electron, ion, and Doppler broadening. Thus, it is independent of the atom center-of-mass coordinates. It is then convenient to place the coordinate system origin at the atomic nucleus. In addition, the reduced distribution functions in Eq. (2.2) satisfy the equilibrium hierarchy equations and exchange between the bound and perturbing electrons was neglected. A tetradic matrix representation leads to

µM( )

=

Mµ

=

iq · r

dr e 16 q2

2

V (ra , r ) ^

(2.2.2)

with Yℓm a spherical harmonic, x^ a unit vector in the direction of arbitrary vector x , and jℓ(z) a spherical Bessel function. Taking matrix elements,

V˜µ ( q ) =

µ V˜ ( ra , q )

= ( 1)mµ

16 2 (2 q2

µ

+ 1)(2

( ) ( i ) V˜ µ (q)

+ 1)

2 +1 Y *m (q^) 4

µ

×

(2.1.1)

µ

^

o] Y m (ra ) Y *m (q )

( i) [j (qra) m

mµ m m

m

( ) µ

,µ

V˜ (ra , q ) =

(2.2.3)

and the 3-j symbol restricts the sums. The dimensionless atomic radial integrals are defined by

where Θ is an operator in the internal atom subspace. It is convenient to use eigenstates of H(a) defined by |µ = |nµ µ mµ with nμ, ℓμ, and mμ the principal, orbital angular momentum, and magnetic quantum numbers, respectively, describing a non-relativistic hydrogen state. For the assumed atom-electron interaction (see Section 2.2) and absence of exchange, collisions conserved spin; thus, spin quantum numbers are suppressed.

( ) V˜ µ (q) =

µ

drPµ (r ){j (qr )

0 0 0

o} P

(r )

o

(2.2.4)

where Pμ(r) is the reduced radial wavefunction for atomic state |μ〉. For hydrogen, the integrations in Eq. (2.2.4) yield analytical results and examples are given in Appendix A.

2.1. Second-order theory

2.3. Dynamic structure factor

To second order in the atom-electron interaction and using plane waves for the perturbing electron states, get [8,9]

The expression in Eq. (2.1.2) is second order in the atom-electron interaction. Hence, it neglects initial correlations [1] and the density matrix is approximated by a product of independent density matrices for each subsystem. Thus, for a neutral hydrogen radiator the resulting dynamic structure factor in Eq. (2.1.2) is for a homogeneous electron gas, which can be written in terms of the dielectric function [11]

dq 8 3

Mµ(2), µ ( ) =

+

µµ

V˜µ ( q ) V˜ µ ( q ) f f µ 1

d 2

V˜

+i

( q ) V˜ ( q ) +i µ

V˜µµ ( q ) V˜ ( q ) +i

µ

(1 V˜µµ ( q ) V˜ µ

( q ) fµ f µ 1 +i

S (q ,

µ

µ

So (q, ) =

Lim 0

i

( )

q2 Im (q, ) 4 2 (q , ) 2

(2.3.2)

ne q

2 me c 2 exp Te

S (q ,

)=e

/ Te S (q ,

satisfies detailed balance.

The atom-electron interaction for hydrogen is given by

r

ra

2 2

2Te q2

(2.3.3)

is not satisfied by the classical expression and often neglected [13,14]. The replacement

(2.14)

2.2. Atom-electron interaction

V ( ra , r ) =

ec

(2.3.4)

So (q, )

1 r

2m

)

with P denoting the Cauchy principal part yields the real and imaginary parts.

1

(2.3.1)

with Te and me respectively the electron temperature in energy units and mass, α the fine structure constant, and c the speed of light. The detailed balance property for an electron gas in thermodynamic equilibrium [11]

with ωμ and fμ respectively the energy and occupation of the atom internal state |μ〉, the sum σ is over all isolated atomic internal states, η is a positive infinitesimal and using the identity

1 1 =P ±i

)=

where the ideal classical gas result is given by

(2.1.3)

)

) S (q,

So (q , ) (q , ) 2

S (q , )

This, and following expressions, assume atomic energy units (hartree) and lengths in Bohr radius. Here, q and ω′ are respectively the momentum and energy transferred during an atom-electron collision, the detuning is given by

(

/ Te

For weakly coupled, classical systems the expression simplifies to [12]

)

(2.1.2)

=

e

So (q , )

1 e

/ Te

, ,

>0 <0

(2.3.5)

3. Line width The semi-classical line formulation [4,6] emphasized the = 1 contribution at line center. Although not required, their sample calculations only coupled atomic states with the same principal quantum number [4,6]. With those constraints, the line width for level nμℓμ may be written in the form

(2.2.1)

where ra and r are the positions of the bound and perturber electrons, respectively. The Fourier transform and consequent expansion in spherical harmonics is given by 2

High Energy Density Physics 35 (2020) 100743

C.A. Iglesias

el) wn(µmod = µ

2 me c 2 Te

4 ne 3

+ 1) Rn2µ

(2

µ, nµ

( mod el) nµ µ, nµ

momentum transfer (or small impact parameter) cutoff, qmax = nµ 2 , to avoid a logarithmic divergence.

(y mod el ) (3.1)

3.3. Semi-classical with penetration (STP)

for the various models discussed below with

Rnµ

=

µ, n

1 0 0 0 µ

dra Pnµ µ (ra ) ra Pn

(ra)

The semi-classical description for electrons penetrating the atom [4] can be approximated by a semi-analytic formula [6]

(3.2)

o

(STP ) µ

explicitly displaying the quantum number dependence (w is independent of magnetic quantum numbers), but henceforth suppressed with obvious notation. The various models discussed below make different approximations for the dimensionless thermal average ϕ and argument y.

µ

ImMµ

( )

,µ

2 me c 2 Te

Gµ(

)

[Hartree]

(2

+ 1) o

dq q3

(ST ) µ

(3.1.2)

( ) V˜ µ (q)

(q ,

(3.1.3)

=

9 e4 Rµ2

( V˜ µ

dqq

= 1)

2 2 e q

o

(q )

(3.1.4)

where the screening of the atom-electron interaction assumes a weaklycoupled electron gas [12]

(q ,

= 0)

1+

1 (3.1.5)

2 2 e q

with electron Debye length e

=

Te 4 ne

Iµ(QM ) (q) =

3

1

+ O (q 2 )

1 = ln(1 + 2

2 2 e qmax )

(3.2.1)

1

2 2 e qmax 2 + e2 qmax

(3.4.1)

1 q3

2

(1) V˜ µ (q)

(q ,

= 0)

(3.5.1)

1 9q

2

Rµ (q ,

(3.5.2)

= 0)

It is readily shown that where the latter has a cutoff at q > qmax = I(QMP)(q) ≤ I(QM)(q) as well as I(QMP)(q) ≥ I(QM)(qmax) for all q. Thus, if the small impact parameter cutoff removes strong collisions (large momentum transfer) that are not suitably for second-order theory in the QM model, then it follows that all collisions, regardless of momentum transfer size, in the QMP model can be treated in second-order theory.

which substituting into Eq (3.1.4) yields (QM ) µ

rmax rmin

1/ n2 .

The long-ranged multipole expansion assumes 1/q is much larger than the radiator spatial size and the radial integrals in Eq. (2.2.4) reduce to

qRµ

= ln

and

3.2. Quantum mechanical excluding penetration (QM)

0) =

].

= Min [ µ ,

Iµ(QMP ) (q) =

(3.1.6)

The integral over momentum transfer yields analytic results (see Appendix B). It is emphasized that the QMP model yields finite widths without introducing cutoffs.

( ) V˜ µ (q

(3.3.2)

The term strong collisions refers to interactions where the perturber appreciably modifies the atomic wavefunction and second-order perturbation theory is no longer valid. The models without penetrating collisions, ST and QM, include a small impact parameter cutoff. The cutoff is necessary to avoid divergences but also helps enforce unitarity [1,4–6]. On the other hand, neither the semi-classical or the quantum mechanical models including penetrating collisions (STP and QMP models above) exclude possible strong collisions and may violate unitarity. To test this possibility, consider the integrand of Eq. (3.1.4) in the definition of the QMP and QM models,

2

+1

+ 2)]

3.5. Strong collisions

and different multipoles in the atom-electron interaction expansion do not interfere [2]. For = 1, Eqs. (3.1.2) and (3.1.3) yield for the QMP model, (QMP ) µ

<

<( <

Both the STP and ST models require a maximum impact parameter cutoff, rmax = 0.68 e , to acount for plasma screening [3–6] (however, see Appendix C). For the ST model rmin = nµ2 while for STP there is no minimum impact parameter cutoff.

2

= 0)

(3.3.1)

Neglecting penetrating collisions in the semi-classical description leads to [1,6]

with

Gµ( ) = (2 + 1)

y2 y + ln 2 2

E1 (y 2 )] + E1

3.4. Semi-classical without penetration (ST)

(3.1.1)

The expression in Eq. (3.1) only requires the first term in Eq. (2.1.2). Combining results in Section 2, yields at line center

wµ(QMP ) = 4 ne

E

2 [5nµ2 8nµ

=

with

The line width in the tetradic representation is given by

( )=

1 [ 2

where γE is the Euler constant, E1(x) is the exponential integral [16], y = 2rmax / nµ µ , and

3.1. Quantum mechanical including penetration (QMP)

µ ,µ

=

4. Numerical results The various approximations to ϕμν are compared for a few levels. These comparisons exclude corrections for neglected strong collisions ( mod el) or coupling between atomic states [15,17]. Ratios of µv are plotted 2 in Figs. 1 and 2 as a function of λe/n . The ST model produces negative line widths for 0.68λe/n2 < 1; hence, the figures exclude those conditions. It follows from the results above that both the QM and ST models

(3.2.2)

This expression reproduces the second-order line width from Lee [15] at line center. The plasma screening of the atom-electron interaction is included in this model via the weakly-coupled static dielectric function in Eq. (3.1.5). The expression, however, requires a maximum 3

High Energy Density Physics 35 (2020) 100743

C.A. Iglesias

penetrating models have greater sensitivity to the choice of small impact parameter cutoff as the plasma coupling increases. Also plotted in Figs. 1 and 2 is µ(QMP ) normalized to the largest Debye length value. The figures show the expected monotonic decreasing function with increasing electron density with just a slightly flattened slope at the highest densities. Some caution is warranted here since for λe ≈ n2 the states in the n shell may be affected by the plasma environment [18] and describing those levels with isolated atomic wavefunctions is uncertain. The dimensionless plasma parameter, which estimates the ratio of the particles potential to kinetic energy, is often defined by [19]

=

1 Te ae

(4.1)

= 1 gives the average inter electron distance. In prinwhere 4 ciple, the weakly-coupled approximation assumed in all the comparisons above only applies over the portion of the figure where Γ < < 1 [19]. Nevertheless, calculations using analytic fits to the static structure factor, S(q), for an isotropic classical system [20] to account for stronger plasma coupling in the dielectric function [19] ne ae3 /3

(q , Fig. 1.. Ratio of the thermal average for various models (left axis) as a function of λe/n2 for broadening of the hydrogen 2s by the 2p level. Also displayed is (QMP ) (right axis) normalized to the largest Debye length value. 2s 2p

= 0) =

2 2 e q 2 2 e q

S (q )

(4.3)

only yield a few percent correction in the line width for the largest plasma coupling assuming Te = 1 eV in Fig. 1. This is expected since corrections to the Debye-Hückel dynamic structure factor in intermediate coupling plasma conditions had small impact on the Ly-α line [21]. 5. Conclusion An existing quantum mechanical expression for the spectral line width [8,9] was applied to penetrating collisions of a hydrogen radiator by plasma electrons using the full Coulomb atom-electron interaction. Similar to earlier work on penetrating collisions [4,6], the expression is second order in the atom-electron interaction. Contrary to that work, the perturbing electrons are treated quantum mechanically rather than by classical straight-line trajectories. Furthermore, plasma screening of the atom-electron interaction is treated through the dynamic structure factor for the electron gas instead of an ad hoc cutoff over impact parameters at large distance. The quantum mechanical treatment of penetrating collisions yields moderately larger line widths than the semi-classical approach. Furthermore, the quantum mechanical treatment does not support a decrease in line width compared to the long-ranged dipole approximation to the atom-electron interaction. They do support the enhanced line width relative to the standard theory excluding penetrating collisions at higher electron densities. The enhancement highlights the uncertainty of the necessary but ad hoc minimum impact parameter cutoff using the long-ranged dipole approximation. The present quantum mechanical calculations assume weak-coupling plasma screening, however, corrections for intermediate coupling plasma conditions yield only small corrections to the line width. It is emphasized that these conclusions apply strictly to neutral hydrogen and do not address results for hydrogenic ions [5]. The present quantum mechanical calculations remove approximations in the semi-classical treatment [4]; nevertheless, some caution is warranted. All the considered models neglect exchange between the bound and perturbing electrons. It is expected that exchange introduces an effective repulsion further softening penetrating atom-electron collisions.

Fig. 2.. Same as Fig. 1 for broadening of 8d by the 8f level.

scale with λe/n2. The two figures show that the QMP and STP models also approximately scale with λe/n2. The figures show the reproduction of the semi-classical results [6] where penetrating collisions soften the dipole interaction except at extreme high density (smallest Debye length) where they lead to larger widths. On the other hand, the quantum models display a different trend. In the quantum mechanical treatment, the penetrating collisions do not impact the line width except at the higher densities. In addition, the penetrating collisions increase the line width starting at lower densities than the semi-classical model. Also note the larger widths from QMP relative to STP except at low densities where all the models reasonably agree. The enhancement of the line width due to penetrating collisions at higher density may reflect an inadequacy of the strong collision cutoff in the QM model rather than physical phenomena. That is, the non-

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to 4

High Energy Density Physics 35 (2020) 100743

C.A. Iglesias

influence the work reported in this paper.

manuscript. Also, thanks to S. Alexiou for suggesting a discussion on strong collisions. The work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

Acknowledgments Thanks are due to T. Gomez and J.-C. Pain for comments on the Appendix A. Atomic radial integrals

Analytical results for the atomic radial integrals with hydrogen wavefunctions are given for examples used in the figures. The matrix elements in Eq. (2.2.4) for n = 2 levels are (0) V˜2s,2s (q) =

q2 (7 + q2 (2 + q2) 2) (1 + q2) 4

(A.1)

q2 (5 + 6q2 + 4q 4 + q6) (0) V˜2p,2p(q) = 3 (1 + q2 ) 4

(A.2)

q (1 q2 ) (1) V˜2s,2p(q) = (1 + q2) 4

(A.3)

q2 2 15 (1 + q2) 4

(2) V˜2p,2p(q) = 2

(A.4)

The matrix elements for Fig. 2, (1) V˜8d,8f (q) =

4q

33 7

16

(1 + 16q2)

(16q2

16q2)(1

1)(1 + 8q

16q2)

8q

× (1 + 16q (1 2q (3 + 8q (1 q))))(1 16q (1 + 2q (3 8q (1 + q)))) × (1 64q2 (5 24q2 (15 64q2 (5 28q2))))

(A.5)

Obviously, the results become cumbersome with increasing principal quantum number. It follows from Eq. (2.2.4) that these radial integrals scale for hydrogenic ions, ( ) ( ) V˜ µ (q; ZN ) = V˜ µ (q

(A.6)

ZN q; ZN = 1)

with ZN the nuclear charge. Appendix B. Momentum transfer integrals Substituting the dynamic structure factor in Eq. (2.3.2) into the line width expression leads to the momentum transfer integral,

Fµ(

) ,µ

( )= o

2m

dq exp q3

ec

( ) ( ) V˜ µ (q) V˜ µ (q)

2 2

2Te q2

(q , )

2

(B.1)

Two frequency limits are considered for n = 2 levels for

= 1.

B.1. Zero frequency limit The thermal average in Eq. (B.1) for

F2(1) s,2p;2p,2s (0) = {(

2 e

1)(5 8 e (1

+ 420

44 +

2 e

+ 187

2 e )(5

= 0 together with the weakly-coupled static screening in Eq. (3.1.5) yields analytical results. For example, 4 e

+ 10

688 2 e

+

6 e

4013

8 e

4 e )ln

In spite of appearances, this expression does not diverge at

F2(1) s,2p;2p,2s (0;

e

1)

17( e 1) 1 + + O (( 252 1260

e

7268

10 e

1619

12 e )

4 e e} 8 840( e2 1)

(B.1.1) e

= 1,

1)2)

(B.1.2)

For large λe, the thermal integral at ω = 0 yields

F2(1) s,2p;2p,2s (0)

1619 + ln( e ) + O 840

1 2 e

(B.1.3)

B.2. High frequency limit The integration in Eq. (B.1) for hydrogenic radiators yields analytical results in the limit that (q, frequency. As an example,

5

> > e) = 1 with ωe the electron plasma

High Energy Density Physics 35 (2020) 100743

C.A. Iglesias

F2(1) s,2p;2p,2s ( ;

1 7194 + 5040

= 1) =

2

2 [25380 +

2)[2520

e E1 (

2 (22581

2 (22680

+

+

2 (7800

+

2 (42840

+

2 (1201

+

2 (29400 +

+2

2 (41

2 (8925+

+

2))))

2 (1281

+2

2 (42

+

2))))]}

(B.2.1)

with 2

=

2m

ec

2

2

2Te

(B.2.2)

In the large Ω limit,

F2(1) s,2p;2p,2s (

2

)=

60

3600

12

14

+ O(

16)

(B.2.3)

The result involves cancellation of many terms. For example, to leading order in Ω,

F2(1) s,2p;2p,2s (

2

)=

14 e 2E ( 1

1 {2 1680

12

840

{1

2)

1} + O (

2

12 [1

2

42e E1 (

2)]

+

}

10)

(B.2.4)

and eventually leading to Eq. (B.2.3). Also note the different asymptotic form in Eq. (B.2.3) compared to the exponential decay in the Lee [15] model. Appendix C. Large impact parameter cutoff The plasma screening is approximated in the semi-classical method by introducing a maximum impact parameter. For the dipole interaction, equate the QM and ST results in Eqs. (3.2.2) and (3.4.1) to write [7] 2 1 ln 1 + 2 e 2 rmin

2 e 2 rmin

+

2 e

= ln

e

rmin

defining a long-ranged impact parameter cutoff, rmax =

=

1+

2 rmin 2 e

exp

(C.1) e.

This leads to

2 e 2 2(rmin +

2 e)

(C.2)

For λe/rmin > > 1, get

=

1 e

1+

2 rmin 2 e

4 rmin r6 + O min 4 6 4 e e

(C.3)

or β ≈ 0.6065 in agreement with [7] but not exactly with [3,5,6]. In the high electron density limit when the screening length is comparable to the atom size, λe/rmin ≈ 1, then β ≈ 1.1. The larger b value is recommended by Griem [1] and Hussey et al. [22] (the latter was not second order but a unified or all order semi-classical calculation). After some experimentation, however, the present calculations show best agreement between the quantum and semi-classical second-order theories for β ≈ 0.68 at low densities.

[12] [13] [14] [15] [16]

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

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[17] [18] [19] [20] [21] [22]

6

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