Semiclassical calculation of oscillator-strengths

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Semiclassical calculation of oscillator-strengths

R.M. More, K.H. Warren

To cite this version:

R.M. More, K.H. Warren. Semiclassical calculation of oscillator-strengths. Journal de Physique, 1989,

50 (1), pp.35-44. �10.1051/jphys:0198900500103500�. �jpa-00210898�

35

Semiclassical calculation of oscillator-strengths ^*

R. M. More and K. H. Warren (1)

Lawrence Livermore National Laboratory ^{and Ecole} Polytechnique

(1) Lawrence Livermore National Laboratory, University ^of California, ^{PO Box} 808, Livermore, U.S.A.

(Reçu ^{le 17} juin 1988, accepté le 18 août 1988)

Résumé.

Parce que les sections efficaces dans les ions fortement chargés jouent ^{un rôle}

fondamental dans le calcul des puissances d’arrêt d’électrons liés, des libres parcours moyens des

photons ^et des sections efficaces d’excitation par impact électronique, ^{on a} besoin d’une méthode

pratique de calcul de ces quantités. ^{Dans cet article, nous décrivons une} approximation ^semi- classique simple qui permet de calculer les éléments de matrice électronique ^{et les forces}

d’oscillateur. La méthode est testée grâce à ^une comparaison ^{détaillée avec} des résultats exacts connus pour le potentiel coulombien ; elle s’avère numériquement ^{fiable et} raisonnablement

précise pour ^toutes les transitions examinées.

Abstract.

Because they play a fundamental role in the calculation of bound electron stopping-

powers, photon ^{mean free} paths and electron impact-excitation cross-sections, ^{there is a need for}

a convenient method to calculate oscillator strengths ^for highly charged ions. In this paper ^we describe a simple semiclassical approximation for electron matrix-elements and oscillator

strengths. The method is tested by ^detailed comparison with known exact results for the Coulomb

potential and proves ^to be numerically ^{robust and} reasonably ^accurate for all transitions examined.

J. Phys. ^{France 50} (1989) ^{35-44 1°r JANVIER 1989,}

Classification

Physics ^Abstracts

32.70C

1. Stopping-power ^and oscillator-strengths.

The electronic stopping-power ^for high-velocity ions, dE/dx, ^can be calculated from the radiative opacity ^or photon absorption cross-section of the target ^medium [1]. ^{This is because}

the electromagnetic ^{field of the fast ion is} approximately equivalent ^{to a} group of virtual

photons ^carried along by ^{the fast ion. When the ion} penetrates ^{a dense} plasma target, ^{some of} the virtual photons ^can be absorbed by ^{electrons of the} target material. After being ^absorbed,

the virtual photons ^are regenerated by the electric current of the fast ion, but the energy

required ^{for this} regeneration is taken from the center-of-mass motion of the ion and therefore the absorption ^{of virtual} photons ^is equivalent ^to electronic energy-loss. ^{From this}

* Work performed ^{under the} auspices of the U.S. Department ^of Energy by the Lawrence Livermore National Laboratory ^{under contract} number W-7405-ENG-48.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500103500

point ^of view, dE/dx ^{can be regarded as an average or mean} opacity ^for absorption ^{of virtual} photons.

A quantitative expression ^{of this intuitive} picture is obtained by calculating ^the absorption length for virtual photons. ^{We use the} frequency spectrum of virtual photons,

Here Zl ^{is the ion} charge, v ^{is its} velocity, ^{c is the} speed ^of light, ^«

e 21hc is the fine- structure constant and the coefficient a is a dimensionless number of order unity [2]. ^{We have}

used a minimum impact parameter bmin

hlMV, ^as is normal for high-velocity ^ion

interactions.

The virtual photons ^{are absorbed} by line transitions of bound electrons of the target plasma, ^as determined by the bound-bound absorption opacity, ^{which we can write in the} language ^{of the} screened-hydrogenic average-atom model,

There are also bound-free and free-free absorption processes ; equation (2) gives ^the opacity

for photon ^line absorption n - n’. Pn is the number of electrons in the initial state of principal quantum number n and (1- P n’/ Dn’) ^{is a} correction for partial filling ^{of states} of the final shell n’ (Dn,

² n’2). I (hv ) ^{is the line} absorption profile, ^{which for this} purpose ^can be

replaced by ^a delta-function, I(hv) == 8 (hv - (E,,, - En». Finally, f n, n,

^{is the oscillator}

strength, ^{summed over states in the final shell n’and} averaged ^{over states} of the initial shell ^n.

The energy lost by absorption ^{of virtual} photons ^{is then} [1]

This integral gives ^the following expression ^{for the} energy-loss :

This is now recognizably ^the bound-bound part ^of the Bethe formula for high-velocity energy- loss [2].

There are several interesting physical questions which arise at this point : ^{does it matter that}

the photons considered are virtual photons rather than real photons ? ^For example, ^should

this slightly modify ^the energy-conservation relation ? When we apply ^these equations ^to plasmas, ^{should we include} the downward transitions, ^{in which an excited atom} apparently gives energy ^to the fast ion ? Can there be stimulated emission by ^excited target ions caused

by ^virtual photons ^{of the fast ion ?} Is it clear these transitions have no effect on the _energy- loss ? While we are inclined to answer these questions ^{in the} affirmative, perhaps ^further study ^is required.

At any rate, ^{in the} Bethe theory ^one writes the stopping power due ^to bound electrons as

37

where NB ^{is the} number of bound electrons per ^atom (ion) ^and 1», ^{the mean} excitation- ionization potential, is defined by :

At this point ^the photoelectric (bound-free) contribution is merely ^{indicated in a schematic}

fashion. Because of the f-sum rule, ^the denominator in equation (6) ^is simply NB

^{Z -} Q,

the number of bound electrons.

The evaluation of Ï requires ^{some atomic} data, ^the energy-levels ^and absorption oscillator-

strengths for transitions between these eigenstates [3]. ^To proceed ^{further we} would need a

tractable formula for these quantities.

There is a simple approximation ^for hydrogen-like ^ions, given by ^{Bethe and} Salpeter [4]

and Menzel and Pekeris [5] :

Equation (7) ^{is accurate to within a factor of two, for} example, ^{the ls -} 2p matrix-element is

(ao

h2 Ime2= ^Bohr radius)

which gives ^{an exact} oscillator-strength ^of /(ls-2p)

4 (2/3 r

0.4162, ^while equation (7) gives f (1, 2 )

0.5808, which is 40 % high. However, equation (7) ^{is more accurate for states}

with higher quantum ^numbers.

Equation (7) ^{can be derived} by extrapolation of the Kramer’s absorption cross-section for

Bremsstrahlung ^to negative energies ^{of both initial and final states for the} absorbing ^electron.

The details of this extrapolation technique ^are given by Zel’dovich and Raizer [6] ^and

More [1].

Reference [1] ^also generalizes equation (7) ^to non-hydrogenic ^atoms, within the context of the screened hydrogenic model. The electron energy-levels ^{are taken to be}

where Qn ^{is the effective} charge ^for electrons of shell n

En° ^{is an} outer-screening correction (also ^a simple ^{function of} Pn, Qn). Repeating ^the

extrapolation ^to negative energies ^{with this} representation ^{of the} energies, ^{one finds :}

For the hydrogenic ^case E n (0) ^{is zero} and the effective charges Qn ^are equal ^to Z ; ^{in this case}

the Qn’S ^{cancel and} equation (11) ^{reduces to} equation (7). ^For Is-Atp transitions of partially

ionized heavy ^atoms, equation (11) ^{can differ from} equation (7) by large ^factors (5 ^to 20) ^and

is systematically ^{more accurate} [1]. ^However equation (11) still differs from the best theoretical calculations by factors of two due to quantum ^and relativistic effects.

For energy-loss calculation, equation (11) ^{is not} sufficient because low-energy transitions

are dominant and subshell splitting greatly ^{affects the answer.} One would like a more accurate and more detailed oscillator-strength formula which describes transitions between states of

specified angular ^momentum, f (nf, n’f’).

The oscillator-strength ^is required for many calculations of atomic processes ^{in hot}

plasmas : calculation of stopping-power, calculation’of emission and absorption of radiation and in addition the oscillator strength is used in many approximate ^{formulas for electron-} impact excitation and ionization. For this reason we have studied the possibility ^of developing

a formula more detailed and ^accurate than equation (11) ^which would retain its simplicity ^and generality.

Although ^this problem ^{has attracted a} great deal of attention [7-18], ^the existing ^{results are}

not very satisfactory. ^{There are} many direct numerical calculations of fn,;", ^{but these give}

numerical answers without enough insight ^{into the} physics.

There are interesting semiclassical results based on Fourier analysis ^{of the classical motion}

on an average trajectory corresponding ^to average quantum numbers nc, Î,, of the electron

[11-13]. These methods assume n - ni 1

An « n. for the hydrogenic system the result is an

expression involving Bessel functions related to Kramers’ original result ; ^{with an} optimal

choice of nc this method gives very good results. The results apply ^to the Coulomb potential,

and the generalization ^{is not} simple.

We are interested in two types ^of non-Coulomb potentials : ^{those with atomic} screening by

core electrons, and those with external plasma screening by free electrons and ions of a dense

plasma environment.

For these problems ^we have studied the use of WKB methods. However the published

WKB methods (e.g., ^Ref. [14]) ^{fail for} simple problems ^{such as} the Coulomb potential ^{or the}

linear harmonic oscillator. Something ^{better is,needed.}

2. VVKB approximation for oscillator strengths.

It is well-known that the semiclassical WKB approximation gives very good (exact) results for the energy-levels ^{of a} nonrelativistic hydrogenic ion, ^and reasonably ^{accurate values for} expectation ^{values or} averages of powers of the radius (rk) , ^{the velocity} (vj), ^{etc. Here we}

extend the WKB method to calculate matrix elements. Our results are within a few percent ^of

the exact answers for transitions in the Coulomb potential, ^{even for low} quantum ^numbers such as the 1s --+ 2p transition, ^{and this is a} surprise.

The method developed ^here applies ^{to other} matrix-elements and other potentials ^without

much modification. It has clear advantages ^for transitions to/from states with high quantum numbers or for sums over many transitions. Because the method is numerically ^{robust we}

believe it will succeed for many other applications.

In the WKB theory ^for spherically-symmetric systems, the radial wave-function ’Fnt ^{is a}

sum of incoming ^and outgoing ^waves,

39

where Ent ^{is the} energy eigenvalue, V (r) ^{is the} electrostatic potential, and ri

ri (n, Q ) ^{is an}

inner turning point, ^defined by qnt(r1)

^{0. The} quantity qnt(r) is the wave-vector, pro-

portional ^{to the radial} velocity, ^{and ço is the} phase-integral, ^{related to} the action function of classical mechanics.

The normalization Cnt of the wave-function ^can be determined by ^{the WKB} normalization rule

For the hydrogenic ^{case we have}

and ri

(ao nz/Z)(1 - s ) ^where

is the eccentricity ^{of the} elliptic orbit. In this case the phase integral ^cp (r) ^{can be evaluated in}

closed from ; this is useful but not essential to what follows.

In our work we allow the radius r to be a complex variable. Then qnt (r) ^{has a branch-cut}

which may be taken along the allowed region (ri ^{-- r «} r2) ; q (r) is defined to be positive ^real just above the real axis and is negative real below it. The two signs correspond physically ^to

the possibility ^of incoming ^and outgoing motion of the electron. Because of the centrifugal potential, q (r) ^{has a} pole ^{at the} origin,

as

and if V (r) ^{--+ 0 at} large ^r, q(r) ^{becomes constant there.}

The square-root ^of q (r) ^has additional branch-points which would make 1JI’ Jf) ^a

complicated ^function except for the remarkable fact that the WKB quantization ^condition (with ^the fractional quantum number)

implies ^{that the} exponential phase-factor exp(::t

J } will ^exactly compensate ^the branch-cut of

q(r). ^To compensate ^the branch-point ^of q(r) ^{at r}

^0, it is necessary ^to employ

(f ⁺ 1/2) ^{in the} centrifugal potential ^{even for f}

0. With thèse conditions, W§i )(r) ^and

Pn(j )(r) ^are analytic functions ^{of r with} possible singularities ^{at r}

^{0, r}

^{00, and a branch-}

cut in the allowed region (rl r r2) of the real axis. With the sign-choices ^{we have} described,

This choice of branch-cut seems most natural but no physical quantity ^can ultimately depend

on the placement of branch-cuts. In fact, complex arithmetic on the CRAY-1 computer places

the branch-cuts differently ^unless special modifications are made to the square-root ^functions.

When we calculate the matrix-element of r with the WKB wave-functions, ^we first write

This is a good approximation ^for large ^n, n’ because the omitted terms, integrals ^of

W§1 )W§7/, ^and 1/1’ JI )1/1’ Jtl’, ^{oscillate rapidly and} symmetrically ^{about zero.} Physically, ^this

can be understood as follows : the integral ^we keep,

corresponds ^{to the} transition of an incoming ^{n’ f’ electron onto an} incoming ^nf orbit. This is the transition involving the smaller change ^{of momentum} for the electron. The transition to an outgoing ^{nf orbit,} corresponding ^{to the} integral

has a much smaller probability because it requires ^{a reversal of} the electron velocity. ^The photon ^{momentum is} negligible compared ^to the electron momenta, ^{and so} this reversal is

extremely unlikely. (The probability ^{is not} exactly ^{zero because momentum is not conserved}

in the electrostatic field of the nucleus.)

We use equation (21) for all values of n, ^{n’ and are} very happy with the results.

The integrand ⁱⁿ equation (21) ^{has a saddle} point ^{at a} complex radius rd defined by

where G (r)

G (r ; nf ; n’ f’) ^{is defined} by

For the Coulomb potential ^{the saddle} point ^can easily ^{be found} by ^{a search} (using ^Newton’s

rule) starting ^at

41

In many ^cases equation (26) ^{is an accurate} approximation ^{to the} saddle-point, and this is not

accidental, because ro is determined by ^the condition that the two orbits should intersect with second-order contact. In any event, equation (26) ^{is a} reasonably good ^first guess of the solution of equation (24).

We note that the quantities G (r), G’(r) ^and r sad ^are calculated directly ^{from the} potential V (r), ^{which must be known for} complex values of radius r.

The saddle-point integration gives

Evaluation of this expression requires integrating equation (14) ^{to find the} phase ^{cp at the} saddle-point. However the integration ^{is not} especially delicate and even coarse zoning produces reasonable results.

The oscillator strength ^{is now}

for dipole-allowed transitions nf - n’ f ± 1. The factor in brackets arises from a sum over

final states and average ^over initial states of given ^n, f, n’, Q’.

The complete shell-range oscillator-strength ^is

For f = 0, ^{the term} fn nit - 1does ^{not exist.}

In the next section we will give examples, ^{tests and} applications of these formulas. Aside from simplicity ^{and intuitive} appeal, ^{the main} advantage ^of equation (27) ^{is the} possibility ^of extending ^{it to} non-hydrogenic systems ^where one can now hope ^for comparable ^results

without a significant increase in the complexity ^of calculation.

3. Results and applications.

First we illustrate the success of equation (27) in calculation of dipole matrix-elements for the

hydrogen ^atom (atomic ^{units are} used, i.e., R (n, f, n’, f’ ) ^is given in units of the Bohr radius ao = 0.529 10- 8 cm).

For the 1s --+ 2p transition, ^{the WKB} integral ^{has a} saddle-point ^at rs

(1.576-1.649i ) ao/Z

and equation (27) gives

compared ^{to exact} values of R = 1.2903 ao/Z, f

^{0.4162 and a} Kramers’ value of

f

^0.5808.

For matrix-elements of the series ls --+ np the ^error grows but does ^not exceed 10 %. As will be seen from table I, ^{other allowed} dipole transitions have comparable accuracy. ^{We note the}

enormous range of the matrix elements, ^{some R’s} being 1 000 times larger ^than others, ^while

equation (27) is almost always within 10 %. Evidently ^the method works.

Table 1

43

An important practical characteristic of equation (27) ^is that it is simple enough ^{that one}

can contemplate analytic summation of large ^{numbers of} transitions. We have tentatively attempted ^{this for} evaluation of the mean excitation potential ^{in the Bethe formula for the} stopping-power. ^While conclusive results await the (straightforward) extension of our method

to photoelectric transitions, ^{we have} observed that the saddle points ^become independent ^{of n}

for transitions of the series ls --+ np ; this means that most of the factors in the summation over n can be pulled ^out.

Next, ^{we consider} quadrupole integrals, ^defined by

Again ^{we are} considering the Coulomb potential, ^{and the exact} answers are readily ^calculated

for small quantum ^{numbers. Some} examples (we give the absolute value of Q in ^{units of}

(ao/ Z)2).

The exact values in..the last three rows are taken from Oumarou, ^{Picart et al.} [19]. ^{We also} compared ^{other cases} given by these authors and found excellent agreement.

The one difficult case is the quadrupole transition ls-3d where the error is fifty percent. ^For this case the classical orbits do not overlap ^at any angle, i.e., ^the apogee of the ls orbit is inside the perigee ^of the 3d orbit.

Finally ^we consider non-Coulomb potentials. There have been a number of studies of the effect of plasma screening ^on the oscillator strengths ^for bound-bound and bound-free transitions [20, 21]. ^{It is} generally found that the strengths ^of individual transitions

n, f , n’, ^{Q’ are} strongly ^{reduced as the state} n’, f’ ^{comes close to the} (lowered) ^continuum.

For the Debye-screened ^Coulomb potential,

Hohne and Zimmerman give ^some representative calculations. We compare with ^one of their

cases :

The agreement ^is again very impressive, considering ^the simplicity ^and convenience of the

WKB calculation. In this _case, while the oscillator strength /(ls ^--+ 4p ) ^{is reduced} by ^{a factor}

of two the saddle point ^is only changed ^{a few} percent by plasma screening. ^{The main} plasma

effect is on the normalization of the WKB wave-function ; in the screened Coulomb potential

the 4p ^{state relaxes outward} and has reduced overlap with the inner ls wave-function.

In this paper ^we have simply presented ^{results of} calculations based on equation (27).

However the success of these calculations points ^{to a} surprising aspect ^{of the} semiclassical limit, ^{which we} propose ^to analyze ^{in a future} publication. Results from equation (27) ^are

better than those which are obtained if we normalize the wave-functions and compute ^the integrals directly according ^{to the} quantum-mechanical rules. That is to say, both

equations (16) ^and (21) differ somewhat from the normal quantum rules, through ^the

omission of terms like that in equation (23), and also work better.

We do not advocate changing the very successful rules of quantum mechanics, ^{of course,}

but rather interpret the result as indicating ^{that the omitted terms contain the most inaccurate} portion of the WKB approximation, ^and physically correspond ^to processes which have very low probabilities.

Acknowledgment.

R. M. More wishes to acknowledge ^{the kind} hospitality of Drs. E. Fabre of the Ecole

Polytechnique and Dr. N. Hoe of the University ^of Paris, ^the majority of this work was

performed during ^a sabbatical visit to the GRECO-ILM Laboratory of the C.N.R.S.

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