Free-free matrix-elements for two-photon opacity

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Free-free matrix-elements for two-photon opacity R. More , J.-C. Pain , S.B. Hansen , T. Nagayama , J.E. Bailey PII: DOI: Reference:

S1574-1818(19)30052-7 https://doi.org/10.1016/j.hedp.2019.100717 HEDP 100717

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High Energy Density Physics

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14 May 2019 18 September 2019 26 September 2019

Please cite this article as: R. More , J.-C. Pain , S.B. Hansen , T. Nagayama , J.E. Bailey , Free-free matrix-elements for two-photon opacity, High Energy Density Physics (2019), doi: https://doi.org/10.1016/j.hedp.2019.100717

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Free-free matrix-elements for two-photon opacity

R. More(1), J.-C. Pain(2), S. B. Hansen(3), T. Nagayama(3) and J. E. Bailey(3) (1) Retired from the National Institute for Fusion Science, Toki, Gifu, Japan; currently RMorePhysics, Pleasanton, California USA (2) CEA, DAM, DIF, F-91297 Arpajon, France (3) Sandia National Laboratories, Albuquerque, New Mexico, USA

ABSTRACT Recent opacity measurements have inspired a close study of the two-photon contributions to the opacity of hot plasmas. The absorption and emission of radiation is controlled by dipole matrix-elements of electrons in an atom or ion. This paper describes two independent methods to calculate matrix-elements needed for the two-photon opacity and tests the results by the f-sum rule. The usual f-sum rule is extended to a matrix f-sum that offers a rigorous test for bound-bound、 bound-free and free-free transitions. An additional higher-order sum-rule for the two-photon transition amplitudes is described. We obtain a simple parametric representation of a key plasma density effect on the matrix-elements. The perturbation theory calculation of two-photon cross-sections is compared to an independent method based on the solution of the time-dependent Schroedinger equation for an atom or ion in a high-frequency electromagnetic field. This is described as a high frequency Stark effect or AC Stark effect. Two-photon cross sections calculated with the AC Stark code agree with perturbation theory to within about 5%. In addition to this cross check, the AC Stark code is well suited to evaluating important questions such as the variation of two-photon opacity for different elements.

Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-NA-0003525. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U. S. Department of Energy or the United States Government.

1.) Introduction: opacity and radiative matrix-elements The coupling of radiation to an atom or ion is measured in quantitative terms by dipole

matrix-elements. However free-free dipole matrix-elements R(E, l --> E', l±1) for high-charge ions have special (singular) behavior when the energies E, E' are equal. This paper gives a practical way to handle the singular behavior, and solves a central problem for the second-order perturbation theory calculation of two-photon opacity.(1) In that method of calculation, the two-photon cross-section is the square of a sum/integral over intermediate states linked by dipole matrix-elements from the atomic initial state to the final state. The integral over intermediate states includes the singular cases where R  . (See Eq. (2) in section 2 below.) A key tool for this work is an extension of the well-known f-sum rule to "off diagonal" transitions and to free-free matrix elements. The Thomas-Reiche-Kuhn sum rule(2) is derived from a double commutator of the electron position operator z with the one-electron Hamiltonian H,

[ [H, z], z]

= k1

(1)

Here z is the z-component of the electron position inside an atom/ion, k is a constant (= - 2/m), and 1 is the quantum unit operator. The expectation value of Eq. (1) for a bound state gives an expression that should be unity (times k) and that equation is the usual f-sum rule. We can also examine off-diagonal matrix-elements of Eq. (1) between states of different energies. Those matrix-elements should be zero because the states are orthogonal and that condition gives many more tests. The states can be bound or free states. We can describe Eq. (1) as a matrix f-sum. The one-photon sum rule is useful in matrix form because it provides a check for certain necessary matrix elements and it enables a novel calibration of FF matrix elements that is difficult to obtain by other means. The results of the tests are shown in Figs. 1-3 and discussed in detail in section 7. The singular behavior of the matrix-elements for equal-energy free states is a continuation of similar behavior of the bound-bound matrix-elements described in section 4 below. The matrix sum-rule test based on Eq. (1) is only satisfied if the matrix-elements are calculated with sufficient numerical precision. The comparison in section 4 (see Tables II, III) shows the surprising effect of even a small error in the well-known textbook of Bethe and Salpeter.(2) Free-free transitions (bremsstrahlung or inverse-bremsstrahlung) are not necessarily important to opacity but the free-free matrix-element is associated with a "photoelectric" absorption process among the 2-photon contributions. Free-free dipole matrix-elements are calculated in section 5 in two ways: by analytic continuation from the bound-bound matrix-elements and by numerical integration of solutions of the Schroedinger equation for positive energy (free) states. These two methods are compared with published data obtained by a third method.(3) The three results agree but our main concern is with the precision of that agreement. Sections 6 and 7 use the matrix sum-rule to determine singular integrals over products of matrix-elements and to obtain a quantitative measure of the accuracy of the matrix-elements. Details of Eq. (1) and an extension are given in section 8 below. Two-photon absorption can be calculated by an independent method based on the time-dependent Schroedinger equation for an atom in a high-frequency electromagnetic field. This second method treats two-photon absorption as an extension of the high-frequency Stark effect. Two-photon cross-sections calculated this way agree closely with 2nd order perturbation theory when the same atomic data is used for the two methods. This comparison is sketched in sections

9-10 below. This research is motivated by the benchmark opacity experiment reported in 2015 by Bailey et al.(4) The experiment found that Fe foils heated to 182-195 eV temperatures using strong radiation from a z-pinch x-ray source show a surprising extra opacity, a significant disagreement with theoretical opacity codes that are often used to calculate opacities for modeling stellar internal structure. The Sandia experiment has now been repeated many times. An ongoing review of the experimental data finds a persistent discrepancy for high-temperature Fe foils (Te ~ 180 eV).(5) Experiments on other materials (Ni, Cr) show smaller discrepancies but may also disagree depending on the opacity model and spectral range considered.(5) Aside from X-ray elastic scattering, the usual opacity codes omit two-photon processes. If the two-photon cross-sections are large enough, they may explain the Sandia experimental results and may lead to improved opacity for high-temperature stellar interior conditions. To jump ahead of the discussion, the scaling laws suggest that the two-photon processes will be more important for higher temperature plasmas and for ions with more bound electrons. The theory of two-photon processes began in the early days of quantum theory with the research of Maria Goeppert-Mayer.(6) In 1939, Breit and Teller used second-order perturbation theory to calculate two-photon emission from the metastable 2s state of hydrogen. (7) Their formula for the two-photon emission rate has been verified to ~ 1% in beam-foil spectroscopy experiments performed on large accelerators (Marrus et al.).(8) Two-photon emission was distinguished from other emission by a coincidence-counting technique. It was found that other processes also contribute to decay from metastable excited states. Multiphoton absorption and Raman effects are easily observed with intense visible light from optical lasers, and are central to a branch of laser science now called Nonlinear Optics. The nonlinear optical effects have many applications described in text-books by Shen(9) and New(10). There is every reason to expect similar nonlinear phenomena for high-intensity X-rays. X-ray two-photon absorption and Raman scattering processes are likely to be observed in experiments using the newly developed X-ray Free Electron Lasers. This paper will review the difficulties of theory for two-photon opacity. It is probably the complexity of the theory that has impeded researchers from including two-photon processes in opacity computer codes; the calculations represent a task comparable to the challenge of including inner-shell and dielectronic processes in opacity codes.(11) The singular behavior of the dipole matrix-elements(12) is most difficult, but we also emphasize that it is necessary to include many absorption processes to obtain even an estimate of the effect on opacity. Two-photon processes include the well-known Raman effect in which one photon is absorbed and another (of different frequency) is emitted. X-Ray Raman opacity is also generally omitted from hot plasma or astrophysical opacity codes. There is also a difficulty with the energy spectrum of excited states; the simplest average-atom model gives excited-state energy-differences that overlap in an unrealistic way, leading to an exaggerated frequency dependence of the two-photon cross-sections. This paper will concentrate on dealing with the singularities of the dipole matrix-elements.

2.) Perturbation theory for two-photon opacity The relation between the two-photon cross-sections and opacity is a straightforward application of known equations from kinetic theory, radiative transfer and quantum electrodynamics.(1, 13, 14, 15) From the start it is important to distinguish the MGM cross-section (2)(hv1, hv2) for two photons of energies hv1, hv2 to be absorbed in one step from the attenuation cross-section (hv1) which is the integral of (2)(hv1, hv2) weighted by the flux (photons/cm2sec-eV) of the second photon hv2. The two-photon cross-section (2)(hv1, hv2) has units cm4sec (sometimes called "MGM units" in honor of Mrs. Mayer) while the attenuation cross-section (hv1) has units cm2. The orders of magnitude are very different: (2)(hv1, hv2) is in the range 10-50 to 10-60 cm4sec and (hv1) in a range 10-18 to 10-25 cm2. We begin with a basic formula for the MGM absorption cross-section for a transition i -> j -> k, given by second-order perturbation theory:(10, 11, 14)

æ d E eˆ × R d E eˆ × R ö ˆ ˆ 8p 3a 2 ij 1 ij jk 2 jk d Eij e2 × Rij d E jk e1 × R jk ÷ MGM ç s = + å ÷ g hn1hn 2 j ç Ei + hn1 - E j Ei + hn 2 - E j è ø

2 (2)

In Eq. (2),  = e2/c is the fine-structure constant, gi is the initial-state statistical weight, the two R's are dipole matrix-elements (i --> j) and (j --> k), and j is a schematic index for the intermediate state. The E's in the numerator are energy-differences between the atomic states involved in the R's (see Eq. (3) below). The symbols e1, e2 indicate the photon polarization vectors (unit vectors). When this cross-section is used, it is almost always multiplied by population factors n(h1), n(h2) for the two photons involved. In an isotropic thermal equilibrium plasma, the Bose-Einstein population factors are simply n(h) = 1 / [exp(h /kT) -1]. Equation (2) is described in greater detail in the following discussion.(See refs. 1, 10, 11, 14) The relation between the attenuation cross-section (hv1) and the MGM two-color cross-section (2)(hv1, hv2) can be symbolically written

s (hn1) = ò s MGM (hn1, hn 2 ) F(hn 2 ) dhn 2 where (hv2) is the flux (photons/cm2sec-eV) of the second photon. This formula needs to be adapted to the actual geometry which ranges from isotropic LTE plasma to beam-target interaction. As described below the most important angle-dependence of the cross-section runs through the polarization vectors of the two photons. The cross-section of Eq. (2) has a resonance peak whenever one of the denominators equals zero and that peak overlaps an allowed spectral line so that the two-photon process looks like an enhanced line-width. For this paper the more important, and more difficult, question comes from the divergence of the matrix-elements Rij and Rjk in the numerator whenever the states i, j or j, k have equal energies. Rij  1/(Eij)2 so the factors E in the numerator do not solve the problem. The perturbation theory method has the advantage that it is relatively easy to intervene in the

calculation to compare the various contributions or to add physical corrections for experimental conditions (density effects, anisotropic radiation spectra or electron collisions). The calculations are not too demanding for a workstation computer. The perturbation theory has the disadvantage that one must explicitly work with the free-free matrix-elements Rij. As we will see below, the free-free matrix-elements are a special challenge because of their singular behavior. The Breit-Teller emission calculation(8) required bound-free, but not free-free, matrix-elements. Two-photon opacity can be calculated in a series of steps. A first step is to decide the initial (lower) atomic states. For Fe16+, Fe17+ ions, the most important lower states are states of the configurations 2s2 2p6 or 2s2 2p5. In an average-atom or independent-electron calculation the initial states are 2s and 2p states. For a dense plasma at 200 eV temperature, absorption from excited states can also make contributions to the opacity. The second step is the sum over final states. The final states can be bound states or continuum ("free") states reached by two-photon photoionization. The final state is connected to the initial state by selection rules l = 0, ±2. In both cases, bound --> bound and bound --> free, energy conservation requires that the sum of energies of the two photons must equal the difference of energies of the atomic system. For the Raman processes it is the difference of photon energies that is constrained. Unlike the situation for one-photon processes, two-photon absorption is a continuum absorption in all cases. This already has an important practical consequence: the two-photon processes can fill gaps between spectral lines where they only compete with low-opacity inverse bremsstrahlung or scattering. Several researchers have tried to estimate the two-photon process by assuming that two identical photons are absorbed, the "one-color" picture. If the two photons must be identical, then they must have exactly half the excitation energy and this is a sort of line absorption. Our calculations show that this case leads to a small cross-section, as much as six orders of magnitude smaller than nearby "two-color" cases. The one-color case, if we take it literally, also faces a subtlety about detailed balance (see Appendix A). The one-color calculations might be useful for future X-ray laser experiments with highly collimated monochromatic X-rays -- if the laser beam-spread can be neglected. Chains of intermediate states After initial and final states are decided, the next step for the calculation is a sum over intermediate states connected by dipole selection rules l = + 1. The intermediate state, denoted j in Eq. (2), can be bound (B) or free (F) and both possibilities must be included. This sum generates the absorption amplitude for processes which can be indicated Bound to bound

=

BBB + BFB

Bound to free

=

BBF + BFF

and

The amplitude is summed over intermediate states before it is squared to form the cross-section. That means there is interference, which can enhance or reduce the cross-section, depending on the

energies. In many cases there are two chains of intermediate states, for example, absorption p --> s --> p and p --> d --> p. In those cases the squared amplitude has four times as many terms. Photon many-body effect The photon coupling perturbation operator is

H (1) = -

e e p × A = -i d E R × A mc c

(3)

The vector potential field operator A is a sum over all photon normal modes. If the initial and final states of the photon field differ by absorption (or emission) of two photons, the two photons can be absorbed in either order. The "order" of absorption, as indicated in Eq. (2), is decided by which photon remains in the intermediate state. This possibility doubles the number of amplitudes. If the two photons are identical, there are two repeated photon annihilation operators; this raises a technical complication for the detailed balance relation. (See Appendix A) Electron many-body effect In the same way, electrons changing state during the absorption can "move" in two orders, in general. The order is decided by the intermediate state. However, in many cases only one electron moves to and through an unpopulated excited state and only one order occurs. For the case

2s22p5

--> intermediate

--> 2s2p53d

there are two sets of intermediate states: 2s2p5np and 2s22p43d. In the first case we say the 2s electron moves first, 2s --> np, and in the second case the 2p electron moves first 2p --> 3d. There is no reason to think that the physical time-order can be meaningfully resolved; "first" and "second" refers to the order of the mathematical operators that create photons and move electrons. In some cases the sum over orders can produce surprising results. (allowed) transition

2s22p6

--> 2s22p53d

A nice example is the

--> 2s2p63d

This looks like a transition 2s --> 2p --> 3d. Naively such a transition would be forbidden because the 2p shell is fully occupied. This apparent "violation of the exclusion principle" is well-known in other contexts.(16,17, 18) A related surprising result is the exact cancellation of amplitudes when the two electrons that change states do not have even one state in common. This cancellation is related to the "linked cluster theorem", a general rule of many-body perturbation theory.(17) Angular factor Angular factors implied by the dot products e . R in Eq. (2) can be calculated by straightforward (if tedious) angular momentum algebra. Different answers are appropriate to different physical conditions: the angular factors are somewhat different for atoms irradiated by a mono-directional X-ray laser, by a back-lighter beam or by the nearly-isotropic radiation flux in a stellar interior.(1) Angular factors for these different physical conditions differ by as much as a factor ~ 3.

Raman effect Raman processes - based on the photon energies, we can distinguish eight processes - are extensions of two-photon absorption, described by the same formula. The Raman cross-section is essentially the same Eq. (2), evaluated with one photon formally having negative energy. In certain frequency ranges, Raman absorption appears to be the dominant process. The reader might notice that the frequency h2 is a prefactor for the cross-section in Eq. (2). If h2 < 0, does Eq. (2) give a negative cross-section? Actually, that negative frequency, inserted into the Bose-Einstein population factor n(h2), changes it to - (n(|h2 |) + 1), which is a stimulated emission factor. (The minus sign is removed.) This expresses crossing symmetry. It is natural to begin with an "astrophysical" two-photon opacity assuming the atomic populations are LTE (equilibrium) populations and the radiation environment is a black-body distribution at the same temperature. This will be appropriate for astrophysical applications but may not optimally match Laboratory experiments. Inserting information about the actual radiation environment in the experiments should improve the comparison to experiments. This paper mainly considers absorption cross-sections but as always in radiative transfer theory, in a near-LTE plasma the opacity should also include stimulated emission which reduces the absorption cross-section by a factor [1 - exp(-h/kT)] for each absorbed photon. For the two-photon opacity there are two such factors. Atomic data Two-photon opacity calculations require atomic data. The formulas show we need energy levels and dipole matrix-elements for all relevant ions in order to evaluate the cross-sections. At least for low densities, the atomic energy states are classified as bound (= B) and free (= F) and the matrix elements which connect two states are classified as BB, BF and FF. We also need information about the plasma radiation environment in order to average the cross-section over the energy and direction of the "second" photon. In the one-electron self-consistent field (SCF), or average-atom description, the energy eigenvalues are a small set of numbers (20 or 30) that can easily be transferred between codes. We have used energies from several sources. The dipole matrix-elements for radiative transitions are a larger and more difficult data-set, many thousands of numbers. For the matrix-elements, there are important questions about how the states are defined and labeled, and how the free state wave-functions are normalized. These questions can make transferred matrix-elements a source of confusion, so it seems safest to generate them close to the place where they will be used. At present it appears that scaled-hydrogenic matrix-elements provide an appropriate starting point. Difficult points Now we describe some difficulties. A first difficulty is pressure-ionization or related density effects. For the moment this is handled by leaving out bound states whose orbit radii are likely to be too large for the plasma considered. More elaborate treatments may be necessary. For Z* = 16 (Ne-like Fe) the Bohr radius for n = 7 is rn

= ao n2/ Z* ~ 1.6 10-8 cm

At electron density ne ~ 1022/cm3, 2 rn exceeds the ion-sphere radius (~ 2.9 10-8 cm).

(4) If we simply

omit states with n  7, there is a gap between the last bound state and the continuum; a continuum-lowering level shift is introduced to close this gap. A second difficulty is the "retardation" effect, which can occur if the x-ray wavelength is comparable to the orbit-radius of excited states. The original form radiative matrix-element is

 1(r) r exp(iq . r) 2(r) d3r

(5)

The "dipole approximation" replaces the exponential by unity. In Eq. (5), q is the wave-vector of the absorbed or emitted x-ray. For visible light the wave-length is thousands of Ångstroms and the dipole approximation is justified. For X-rays with a short wavelength  ~ 7 Å and for states with n = 7, the exponent is 2rn /  ~ 1.4 and this "retardation" effect could be important. From Eq. (5) it appears that the oscillatory exponential can cause phase-mixing which would reduce the generalized matrix-element. This is an additional reason to question the contribution of excited states with large quantum numbers. This retardation effect afflicts the excited states much like a density effect. (See Appendix B, which gives an approximate curve-fit to the modified matrix-elements.) Another challenge for theory is that preliminary average-atom calculations reported in reference (1) gave a cross-section with oscillatory wavelength-dependence over the experimental range 7 - 10 Å. This is apparently due to the oversimplified atomic excited state spectra built into the average-atom model. When the excitation energies are calculated as differences of eigenvalues, that calculation ignores the term-splitting of the levels of an excited configuration. The term splitting is strong enough (~ 30 eV) to have a significant effect on the two-photon absorption spectrum. In this paper we do not include the effect of term-splitting but can easily see (from Figures 5, 6) that it will be important. A more fundamental mathematical difficulty comes from the singularity of the dipole matrix elements when the energies of two free states become equal.(12) Although we have recent results for the other questions, this paper will concentrate on the singularity of the matrix-elements. The following sections describe a practical solution for that difficulty.

3.) Bound-bound Matrix Elements The bound-bound matrix-elements give a preview of the free-free difficulty and show a practical solution for the difficulty. The discussion here centers on the surprising behavior of the equal-energy matrix-elements, which are important for the Stark effect. A first point is that the matrix elements increase with energy, proportional to the square of the principal quantum number, i.e., R ~ n2. From equation (2), this implies an amazing power law ~ n8 for at least part of the two-photon absorption cross-section.(1) Such a strong dependence may be reduced by density effects, which limit the high-n contributions, but still suggests that two-photon processes can be important for many-electron atoms or ions. Bound-bound (BB) matrix-elements can be calculated directly from the known bound-state wave functions. Table I gives some examples (atomic units ao/Z are used):

TABLE I

Sample hydrogenic bound-bound dipole matrix-elements [a. u.]

R(4s --> 2p) R(4s --> 3p) R(4s --> 4p) R(4s --> 5p) R(4s --> 6p)

= 0.382301 = 2.443534 = - 23.237900 = 8.517827 = 3.454502

R(5s --> 3p) R(5s --> 4p) R(5s --> 5p) R(5s --> 6p) R(5s --> 7p)

= 0.969610 = 4.600278 = - 36.742346 = 12.213855 = 4.869884

The n--> n' matrix-elements are positive, grow with n and are largest when n and n' are nearly equal. However, the n = n' matrix element is negative. The matrix-element R(ns --> np) is approximately - 3n2/2 (see Eq. (20) below). The variation around the maximum at n = n' is not symmetric. Dipole matrix-elements involve two states whose angular momentum differs by one unit. We consider hydrogenic one-electron states which can be labeled Rnl(r) Ylm(), or many-electron states that are made from determinants of such states. After being integrated over , the angular functions Ylm() disappear into an angular factor ang. The matrix-elements are integrals over the radial wave-functions Rn, l (r). R, V, A formulas for matrix-elements There are three well-known formulas for the dipole matrix-elements.(2) If the wave-functions obey the Schroedinger equation, the three forms are equal. The 3D wave-function is  = Rnl(r) Ylm(). If we write R' for dR/dr, the radial wave-equation is:

-

2æ 2 ( +1) 2 ö R = En, R ç R"+ R'÷ + V(r)R + 2m è r ø 2m r 2

(6)

In the hydrogenic approximation the potential V(r) = - Ze2/r. The bound-state wave-functions are normalized by:

¥ 2 2 ò 0 R (r) r dr = 1 The radius form ("R-form") of the dipole matrix-element is

n', ' = ¥ R (r) r R 2 Rn, ò 0 n, n', ' r dr

(7)

For bound states n,l and n',l' the integral converges because the wave-functions decrease exponentially at large radius. The velocity form ("V-form") reads:

æ çE ç è n,

é ö n', ' = 2 ¥ ê R dRn', ' - R dRn, -En', ' ÷÷ Rn, ò 2m 0 êê n, dr n', ' dr ø ë

Here, L = éë ( +1)- '( '+1)ùû

ù ú ú ú û

2 ¥ r2 dr + 2m L ò Rn, Rn', ' rdr 0 (8)

. The acceleration form ("A-form") of the matrix-element is:

(

2 2 n', ' ¶V Ze2 2 ¥ 2 ¥ En, - En', ' Rn, = R r dr = ò 0 Rn, (r) ò 0 Rn, (r) Rn', ' dr m ¶r n', ' m (9)

)

Eq. (9) assumes l' = l ± 1, and the third version assumes V(r) = - Ze2/r. The acceleration form is useful because the simplest model potential for an atom in a dense plasmas is a Coulomb potential truncated to a constant at the ion-sphere radius set by the density. The integral in Eq. (9) can be evaluated for that model, even if the (positive-energy) wave-function does not provide good convergence at large radii. It is often convenient to use a modified radial wave-function (r) = r R(r). For this wave-function, Eq. (1) becomes

-

2 2m

f '' -

The normalization integral is

2 ( +1) Ze2 f + f = En, f r 2m r 2

¥ 2 ò 0 f (r) dr = 1

(10)

(11)

The other formulas are easily translated. BB matrix-elements Hydrogenic wave-functions(2) for levels up to n = 6 were entered into a computer code and checked by testing their orthonormality properties. With these wave-functions it is easy to make a preliminary calculation of the effect of "retardation" (i.e., failure of the dipole approximation) on the matrix-elements. We found a 50% reduction of the n = 6 bound-bound matrix-elements for conditions of the Fe experiments for X-ray wavelengths ~ 7 Ångstroms, consistent with an estimate given after Eq. (5) above. These calculations are described in Appendix B. BB matrix-elements are also given by the Gordon formula, Eqs. (12-19) below, and Bethe-Salpeter give explicit rational functions for the matrix-elements for the first few series (1s --> np, 2s --> np, etc.). There is also a well-known formula for BF or photoelectric matrix-elements, sometimes referred to as the Stobbe formula.(2, 19) Today our best way to correct the Gordon formula for density effects will use Eq. (9). Gordon formula This formula gives dipole BB radial matrix-elements for hydrogen-like ions.(2) The formulas use a dimensionless principal quantum number "n" which is related to the energy by the Bohr formula, (ao = .529 10-8 cm is the Bohr radius, and Z = nuclear charge).

Z 2 e2 E = n, l 2aon 2

(12)

The dipole matrix-element is given by

n', l-1 R = A éêF(a, b, g , z) - q2 F(a ', b ', g ', z)ùú n, l ë û

(13)

F(, z) is the hypergeometric function and the variables are

z = -

4nn'

(14)

(n - n')2 æ n - n' ö q = ç ÷ è n + n' ø

(15)

=l+1-n

 = l - n'

 = 2l

(16)

' = l - 1 - n

' = 

' = 

(17)

The normalization coefficient is  A =

(-1)n'-l (n + l)! (n'+ l -1)! æ ao ö (-z)l+1q(n+n') ç ÷ 4(2l -1)! (n - l -1)! (n'- l)! è Z ø

(18)

The Gordon formula only gives Rn,ln',l-1 but Rn',l-1n,l is the same number:

n, l n', l-1 Rn', l-1 = Rn, l

(19)

The equal-energy case n = n' is special:

n,l-1 = - 3n ao Rn,l 2 Z

n2 - l 2

(20)

The minus sign in Eq. (20) disagrees with Eq. (63.5) of the Bethe-Salpeter textbook.(2) This is a long-known error or misprint in that book.(19, 20) The correct sign in Eq. (20) can be verified in several ways. An easy way uses the wave-functions from Eq. (3.18) of the Bethe-Salpeter textbook. For example:

R2s (r) =

1 æ r ö -r/2 ç1- ÷ e 2 è 2ø

1 R2 p (r) = r e-r/2 2 6

(21)

The matrix-element integral will be found to be - 3 3. Testing other cases confirms Eq. (20). The sign in Eq. (20) is not important for 1-photon dipole emission because the cross-section

uses the absolute square of the matrix-element. However the sign is important for the Stark effect (20) or for the two-photon processes, which involve interference between several contributions. Tests of the matrix f-sum shown in the following Tables II, III also confirm the sign in Eq. (20). The large negative value prefigures the singularity of the free-free matrix-elements at equal energies. That also gives a strong (negative) contribution that partly cancels the nearby large positive matrix-elements. Tables II, III test the matrix sum-rule of Eq. (1) using matrix-elements obtained from the Gordon formula. The matrix-sum should be unity on the diagonal and zero off-diagonal. In Table II, the matrix f-sum is calculated using the Bethe-Salpeter version of Eq. (20). The sum-rule test is very sensitive: in making Table II only 24 out of 24,579,200 matrix elements have the sign error and the result is nothing like the expected 4X4 unit matrix. (Most of the 24 million matrix-elements connect to the finely resolved continuum states.) In Table III the same matrix f-sum is calculated using the correct sign of Eq. (20). The improvement is dramatic. The situation for the free-free f-sum is quite similar because the singular matrix elements are large and negative. The diagonal terms, usually the only thing considered in testing the f-sum rule, are the same in both tables. Tables II, III are for the case s-p-s but the results are very similar for other cases p-s-p, p-d-p, d-p-d, etc. In order to check the tables, the reader will need the (Stobbe) formulas for BF matrix-elements; some cases are given in the references.(2, 19) Table III is strong evidence for the sign in Eq. (20). The comparison shows that the matrix f-sum is a powerful test of the consistency of matrix-elements.

TABLE II.

Test of matrix f-sum using Bethe-Salpeter matrix-elements 1s

2s

3s

4s

1s

0.999357

1.675714

1.948111

2.211244

2s

1.675714

0.999690

1.579838

1.737683

3s

1.948111

1.579838

0.999572

1.554941

4s

2.211244

1.737683

1.554941

0.999388

TABLE III.

Test of matrix f-sum with the correct sign of Eq. (20) 1s

2s

3s

4s

1s

.999357

-.000391

-.000446

-.000596

2s

-.000391

.999690

-.000376

-.000500

3s

-.000446

-.000376

.999572

-.000530

4s

-.000596

-.000500

-.000530

.999388

4.) Free-free matrix-elements Next we consider the free-free matrix elements. There are several questions: How to calculate them? What to do about the singularity at equal energies? How to verify that the calculations are sufficiently accurate? In this section we describe and compare three methods to calculate the free-free matrix-elements. Gao-Starace matrix-elements In 1989, B. Gao and A. Starace published calculations of Coulomb free-free matrix-elements, using an original technique of complex contour rotation. (3) Their results (shown in Table IV) are accurate to five digits. The calculations are limited to the Coulomb potential (H-like ions), but can be scaled using effective charges. Density effects are not included. The original paper does not give information about the singular behavior at E = E', but their eight cases give a solid check on any other method. Analytic continuation The Gordon formula can be extended to positive energies by analytic continuation.(21, 22) The analytic continuation has four main steps: 1.) For positive energies, the "quantum number" n becomes a pure-imaginary quantity ñ.

2 2 n =±i Z e 2aoE

(22)

The sign of the square-root of energy can be chosen two ways and some experimentation is necessary to find good choices. (The "good choices" give matrix-elements that agree with Gao and Starace.) Note that different computer compilers may assign the branch-cut of the complex square-root in different ways. 2.) Factorials in the normalization of Eq. (18) are extended to complex ñ by use of the Gamma function.  3.) The hypergeometric functions F are evaluated for complex arguments which can be large; various well-known transformation formulas help for this.(22, 23) 4.) Bound and free wave-functions are normalized differently.

The bound-state normalization

N

n, l

µ

1 3/2 n

is replaced by the free-state normalization:

N E,l µ

1 1/2

é1- e-2p n ù ë û

(23)

In this change, the units of the wave-function change. Where it works, analytic continuation from the Gordon formula gives high accuracy. It agrees with the Gao-Starace calculations to five digits for their 8 cases (see Table IV below). Both methods are limited to H-like ions. Density effects are not included, and it is not clear how to incorporate them. The comparison with Gao-Starace is especially important because the off-diagonal matrix f-sum test described below verifies the ratios of the matrix-elements while the comparison in Table IV tests the actual values. We have studied the matrix-elements for nearly-equal energies (E near E') to understand the singularity (Table V). The analytic continuation results are good near equal energies but the matrix-element goes to infinity at equal energies and the analytic continuation formula doesn't give an answer. Examples are given in section 5 below. Analytic continuation is not just a simple rough extrapolation. It uses the original Gordon formula for positive energies. If we only had access to the positive-energy version, it would be possible to reverse the continuation and exactly reproduce the bound-bound matrix-elements.(19) Numerical wave-functions It is also possible to directly calculate the free-free matrix-elements from Eqs. (7-9). Radial wave-functions for free states in the Coulomb potential are given in textbooks as a closed-form expression in terms of the confluent hypergeometric function. The series expansion for that function is a convergent series, but is not quite satisfactory on a 64-bit computer. The series ultimately converges, but if it is evaluated at large radii, the partial summations become very large and the series is inaccurate. The series works well at small radii. The series expansion of the confluent hypergeometric wave-functions can be used to begin the numerical calculation of the wave-functions at small radius. After that, a second-order integration method is able to form the continuum wave-function. With numerical wave functions generated in this way, the acceleration formula for the matrix-elements agrees with Gao-Starace and the analytic continuation method to a fraction of a percent (examples are given below). This is good but not perfect. This method can be extended to include density effects. The method could be used in an SCF code. There may be some subtlety about normalizing continuum wave-functions when there is continuum lowering, but that is a long-standing problem with SCF calculations. Examples

In Table IV we give some examples of matrix elements calculated by the three methods described above. R(E,s  E",p) is the free-free dipole matrix element. Atomic units are used: the energies are given in Ry ~ 27.2 eV, and lengths in ao ~ .529 10-8 cm (for hydrogen). For Fe 16+ ions, the energy unit would be 162 Ry = 6.96 keV and length unit is .033 Angstrom. The labels "s" and "p" mean angular momentum l = 0, 1. Several questions motivate the comparisons. Gao and Starace use atomic units and the free-free matrix-elements are "densities" in energy-space, so it is necessary to be careful about the units. The free wave-functions can be normalized in energy-space or in momentum-space. (In our work they are normalized in energy-space.) The matrix-elements become infinite for equal energies of the two free states involved, and when the divergent part is removed, by multiplying by (E - E")2, what's left has some singular behavior too. It seems important to find several ways to check the results.

TABLE IV

Gao-Starace, analytic continuation and numerical wave-functions. Gao-Starace

Analytic continuation

Numerical integral

E = .016 l = 0 E' = .059

l' = 1

RGS = 111.060

RAC = 111.064

RNI = 111.0906

E = .016 l = 1 E' = .059

l' = 0

RGS = 57.943

RAC = 57.9434

RNI = 58.0109

E = .016 l = 1 E' = .059

l' = 2

RGS = 114.24

RAC = 114.243

RNI = 114.22901

E = .016 l = 2 E' = .059

l' = 1

RGS = 33.414

RAC = 33.4126

RNI = 33.479428

E = .100 l = 0 E' = .700

l' = 1

RGS = 1.2993

RAC = 1.29927

RNI = 1.299249

E = .100 l = 1 E' = .700

l' = 0

RGS = .35695

RAC = .356944

RNI = .356911

E = .100 l = 1 E' = .700

l' = 2

RGS = .72339

RAC = .723395

RNI = .723416

E = .100 l = 2 E' = .700

l' = 1

RGS = .08783

RAC = .087818

RNI = .087823

It seems clear that the analytic continuation agrees accurately with Gao-Starace. The numerical integral calculation produces a result that oscillates slightly with radius. The code uses 124,000 radial grid-points with a spacing of .001 Ångstrom. The numerical integral agrees with

the other methods to 3 or 4 digits. Where it works, analytic continuation seems very satisfactory. Numerical integration works even for the equal-energy case and can be extended to include density effects represented by a screened-Coulomb potential. Unfortunately it does not give the singular (delta-function) contribution at equal energies. The numerical integral has a larger numerical error. Why insist on the precision of the calculations? Do the small differences matter? High precision is important because the f-sum must survive cancellation of large terms to produce a small answer. Any inaccuracy in the numbers is a problem.

5.) Free-free matrix elements at equal energies The difficult point for free-free matrix-elements is the singular behavior at equal energies. At equal energies, the matrix element R becomes infinite - this is clear from Eq. (9) above. The analytic continuation method works near equal energies but does not work at that case. We can multiply the matrix element by (E - E")2 and calculate A(E,s  E",p) = (E - E")2 R(E,s  E",p)

(24)

The multiplied matrix-element A stays finite and is not too large. But it is not analytic near the equal-energy curve; there is a discontinuity in its energy-derivative. Table V gives the limiting values of A for equal energies.

TABLE V Limiting value of A(E,s --> E',p) at E = E' Energy [a.u.]

Analytic continuation

E = 0.1 E = 0.2 E = 0.3 E = 0.4 E = 0.5 E = 0.6 E = 0.7 E = 0.8 E = 0.9

0.12995 0.17014 0.19492 0.21221 0.22508 0.23509 0.24311 0.24970 0.25522

Numerical integral 0.133 0.172 0.196 0.213 0.226 0.236 0.244 0.250 0.256

The quantity A(E, E') = (E-E')2 R(E, E') remains finite at E = E' as shown in Table V. This gives a quantitative limit on the singularity. However even if we accurately know A(E, E') we can only reconstruct R(E, E') up to a delta-function contribution at E = E'. In fact, according to Veniard and Piraux(12), there is also a contribution proportional to the derivative '(E - E'). The positive-energy f-sum rule confirms that these contributions exists and are important. In the following section 6, the coefficient U(E) in Eq. (29) gives the integral over the delta function and V(E) gives the integral over the derivative '(E - E').

6.) Matrix f-sum for free-free transitions The f-sum rule is a mathematical theorem involving the dipole matrix elements for one-photon absorption/emission. The term "f-sum rule" refers to several related mathematical statements. Within their limitations, they are rigorous mathematical results from basic quantum theory, but different versions of the sum-rule impose different restrictions on atomic optical properties. A many-electron version of the f-sum is given in Eq. (61.1) of ref. 2. The one-electron version is also described in detail in section 61 of that reference. Elementary algebra connects the sum-rule for operators "z", as in Eq. (1), to similar statements for the dipole matrix-elements Rnln'l'. (See Eqs. (60-7) and (60-11) of reference 2.) These mathematical theorems were extended by Dyson and Bernstein(24), Liberman(25) and Armstrong(26, 27) to provide inequalities for the opacity of hot plasmas. These inequalities are a widely recognized constraint on opacities obtained from theoretical calculations and one author (28) has proposed to use them as a way to test the Sandia experiments. The test is not conclusive because it would require opacity data at all frequencies, which is not available. For that purpose it is also important to keep in mind the limitation that the f-sum rule considers only dipole transitions, omits quadrupole transitions, omits the wavelength effect of Appendix B and, of course, omits the multi-photon and Raman effects. However we have seen that the sum-rule is a powerful test for dipole matrix-elements used in perturbation theory. The matrix sum-rule tests the mutual consistency of dipole matrix-elements. The off-diagonal BB sum-rule immediately reveals the sign error in the Bethe-Salpeter Eq. (63.5). The free-free matrix-elements are more difficult, as described above, and the powerful overall test given by the sum-rule can become a way to accurately determine integrals over the singularities of the matrix-elements. Both the two-photon perturbation theory of Eq. (2) and the matrix f-sum of Eq. (1) require integration over intermediate states in the continuum and in the range of integration the equal-energy singularity is encountered. If we evaluate the matrix f-sum for two free s-states of energies E1 and E2, there is an integral over free p-states (E", p), which is singular when E" = E1 or E" = E2. We need a special integration technique to deal with this situation. A similar singularity occurs with the absorption cross-section of Eq. (2) for BFF transitions. Matrix f-sum is an integral over Rij Rjk For positive energies, the diagonal part of the matrix f-sum in Eq. (1) is a delta-function, something very inconvenient for computer evaluation. If we only consider the off-diagonal part of

the matrix, we can omit the numerical factor denoted k (= - 2/m) and the factors relating matrix elements of z to matrix elements of R (those factors are just a factor 1/3 for the case s-p-s which we consider here). The f-sum test for E1, s --> E", p --> E2, s is a sum plus an integral:

ì

¥

ü

î n"

0

þ

0 = í å + ò dE" ý

( 2E"- E1 - E2 ) R(E1, s ® E", p) R(E", p ® E2, s)

(25)

The sum includes (l = 1) bound states starting with n" = 2 and the integral runs over all E" > 0. This sum + integral should be zero, or, what is essentially the same request, 2/3m times zero. The difficult points in the integration are where E" = E1 and where E" = E2. We assume E1 does not equal E2, so the two difficulties do not occur together. As was seen with Tables II, III, the off-diagonal matrix f-sum is a very exigent test. Singular points in the integral The two singular cases are treated in the same way, so we only describe the integration over the singularity at E" = E1. The sum over bound states is non-singular. The non-singular part of the integral is treated by Simpson's rule with a fixed zone-spacing E". The delicate part is the range E" = E1 - E" to E" = E1 + E". In this range, we write the integrand of Eq. (25) as

f(E") R(E , s ® E", p)

1

with

f(E") = (2E" - E1 - E2) R(E", p --> E2, s)

(26)

At the ends of the range, we have

f- = f(E" = E1 - E") = ( E1 - 2E" - E2) R(E1 - E", p --> E2, s)

(27)

f+ = f(E" = E1 + E") = ( E1+ 2E" - E2) R(E1+ E", p --> E2, s)

(28)

The integral over this range can be written approximately as E1 +DE"

ò

E1 -DE"

æf +f ö æf -f ö = U(E1 ) ç + - ÷ +V (E1 ) ç + - ÷ è 2DE" ø è 2DE" ø

(29)

A similar expression with coefficients U(E2) and V(E2) is introduced for the singularity at E" = E2. The quantity U corrects the average integral over the singularity and V corrects the derivative. These corrections are related to the delta-function and derivative of the delta-function discussed in reference 12. The corrections become large when E" is small, but they cancel large contributions from the neighbor zones of the integration. As written in Eq. (29), the coefficients U(E), V(E) are approximately independent of zone spacing in the continuum. The functions U, V are shown in Figure 4 for 3 cases, sps, pdp and dfd. The calculated f-sum is very sensitive to the functions U, V. Even a part-per-million change in these functions has an effect on the sum-rule. This is unfortunate because we can predict U and V to about 1% accuracy by extrapolating from the bound-bound matrix-elements but those

predictions are not accurate enough for the f-sum test. An example of the sensitivity is shown by the detailed numbers for the case E1 = 0.3, E2 = 0.7 (atomic units). For this case the bound state f-sum (s --> p --> s) is 

=

-.030426

The non-singular part of the continuum integral is much larger, 23,287.6793. This integral is large because with fine energy zones E" = 1.5625 10-5, the integration runs quite close to the singularities at E" = E1, E2. The coefficients in Eq. (29) are found to be

f+ + f= -25, 025.1203 2DE"

f+ - f= -0.362, 509,347 2DE"

at

E" = 0.3

f+ + f= +66, 232.2557 2DE"

f+ - f= +0.593,815, 900 2DE"

at

E" = 0.7

With correction factors U(.3) = - 0.404,613

V(.3) = .472,112,7

U(.7) = - 0.504,489,9

V(.7) = .897,351,7

the final result for the f-sum for this case is -.004,465. The numbers show the severe cancellation that is required to satisfy the f-sum rule. The original numbers ~ 105 must be cancelled to a part-per-million level to obey the f-sum rule. It seems significant when one set of values U(E), V(E) performs this cancellation for the entire matrix. Figures 1, 2 and 3 show the cancellation of the FF matrix f-sum achieved with the functions U(E), V(E) shown in Fig. 4. The numerical calculations are not absolutely perfect, even when they are correct to the ppm level. Numerical issues concern (i.) the number of bound states included in the FBF summation, (ii.) the range and zoning of the continuum integration and (iii.) the possibility of including higher-order corrections in the Taylor series expansion of f(E") around the singular points E 1, E2. The complex-number subroutines used for evaluation of the Gordon formula probably have limited precision, especially for the gamma function. We invite other researchers to use larger computers or more complicated mathematical methods to improve the accuracy of the free-free f-sum. With the corrector functions U(E), V(E) determined by the f-sum rule in this way, we are able to evaluate other integrals over free-free matrix elements.

7.) Singular terms and f-sum test As described above, we approximately determined singularity corrector functions U(E), V(E). An approximate curve-fit matches U(E), V(E) to about 1% accuracy, but for the f-sum tests, this curve-fit is not accurate enough: the functions U, V need part-per million accuracy to satisfy the f-sum test. We can measure the sum-rule test by a figure of merit, the RMS values of the f-sum. If the sum-rule were perfectly satisfied this would be zero.

The RMS figure of merit should be clearly defined. With 20 energies, there are 190 independent off-diagonal f-sum cases (the f-sum is symmetric between the two energies). There are 20 U's and 20 V's, a total of 40 numbers. The figure of merit reported is the RMS f-sum over all 190 off-diagonal cases. The diagonal f-sums are infinite and not included. So we are asking 190 f-sums to be ~ zero and the quantity fRMS is the root-mean-square figure of merit for that test. We are selecting 40 U, V values to pass 190 tests. That is a ratio of about 5 to 1. With the best U, V the RMS off-diagonal part of the matrix f-sum is s-p-s p-d-p d-f-d

fRMS = .000,35 fRMS = .001,1 fRMS = .004,6

The off-diagonal sum-rule is nicely satisfied (answers are small ~ 0.000,35 for E1 ≠ E2) for the transitions s --> p. The same method of extracting the "divergent" integral from the off diagonal f-sum succeeds for other cases (p-d-p, d-f-d), although the numerical errors are larger. Graphical comparisons are shown in Figures (1-3). The "reverse" f-sums for p-s-p and d-p-d have somewhat larger numerical errors. Most of the numerical error looks like a finite width of the central delta-function (E1 - E2). The matrix element of 1 between two p states of different energies should be zero. The p-states couple to both s-states and d-states. Analysis of the m-dependence gives independent sum-rules for the two "channels" of p-s-p and p-d-p. The original matrix f-sum of Eq. (1) can also be tested for a matrix-element < n, l | k 1 | E, l > where E > 0. This is a bound-free test. The numerical evaluation again encounters the singular case where the intermediate energy equals E. Our code actually uses this test to generate preliminary approximate values for the functions U(E), V(E) and the test is satisfied to about the same accuracy as for the FF cases.

8.) New sum-rule for two-photon transitions We can summarize some important features of the f-sum rule: The f-sum is obtained from commutation relations for quantum operators. The formula is true for quantum motion in any one-electron potential V(r), i.e., for pure-Coulomb ("hydrogenic") potentials, for screened Coulomb potentials calculated as a self-consistent field for a many-electron ion, and even for atomic potentials including plasma environment screening effects. The f-sum equation is true for any choice of the potential V(r), because the quantum operators for position z and potential V(r) commute: [ z, V(r) ] = z V - V z = 0

(31)

Here "z" is taken as a generic position variable; it can be thought of as the z-component of a vector

r in 3-dimensional space. The f-sum rule is a rigorous mathematical theorem that involves quantities relevant to the opacity, but the f-sum must be processed to connect it to the opacity. (24,25,26,27) The sum-rule does not include the temperature-dependent stimulated emission correction factor [1 - exp(-/kT)] that appears in the opacity. The sum-rule contains an angular factor , which is simple for one-photon transitions, but for two-photon processes the angular factor is more complicated and depends on the radiation environment. Two-photon sum rule It is natural to ask whether there is an extended or generalized sum-rule that includes or applies to the two-photon opacity. If there is such a generalized sum-rule, could it be derived in the same way as the one-photon sum-rule? We want an equation that would have similar properties involving the quantities of the two-photon absorption cross-section, and an obvious way to find such a formula is to copy/extend the derivation of the one-photon f-sum rule. For two-photon transitions we expect something more complicated than Eq. (1), now involving a sum over two intermediate states involving two energy denominators and four dipole matrix-elements. We can build up such a sum by examining the equation of motion for the composite operator

X(e ) = z G0 (e ) z

(32)

where

1 G0 (e ) = e-H

(33) is the one-electron Green's function. Here  is a number with units of energy, which we may select for convenience. The z is any one component of the electron position vector. It is easy to form the commutator of the Hamiltonian with ,

[ H, X(e ) ] =

-i pz G0 z + z G0 pz m

(

)

(34)

Now the desired generalization of the f-sum rule comes from expanding the 3-operator commutator [ [ H,  )], 2]. The algebra for the expansion is straightforward. The new result can be written, inserting explicit sums over the final state "b" and the two intermediate states "j" and "k",

ååå b jk

(2Eb - E j - Ek ) zaj z jbzbk zka (e1 - E j ) (e2 - Ek )

=

2

zab zba å m b e -E e -E 1 b 2 b

(

)(

)

(35)

Is Eq. (35) the desired sum-rule? The triple sum on the left side can be reorganized to look like the absolute square of a 2nd-order perturbation theory expression, | z z / ( - Ej)|2 and when it is written this way it is clearly related to the two-photon cross-section. It looks like a sum of two-photon cross-sections (summed over final state b) but without the photon population factors and without careful treatment of the angular factors. The right-hand side was simplified by use of

the Heisenberg formula [ pz , z ] = -i. The right side resembles a polarizability evaluated in Eq. (61.21A) of the book by Bethe and Salpeter. (2) It is important to underline that the sums over states refer to a complete set of states, so they require bound-bound, bound-free and in some cases free-free matrix elements. This is also the case for the one-photon sum rule. To make an opacity bound, Eq. (35) must be extended. The important additions are inclusion of the photon density of states, the generalization to matrix-elements of the vector position and momentum, and inclusion of the photon polarization vectors. The improved equations should have indices for the x, y, z components of the position vector. H in Eq. (34) could be replaced by any polynomial function f(H), which would still commute with the Green's function G0(). The Green's functions in the two 's can depend on different variables 1, 2. So we can examine [ [ f(H), ij1) ], 

(2) ]

Here f(H) is a function of H, chosen to give the desired photon frequency factors. The subscripts i,j and l,m on the 's refer to Cartesian coordinate indexes x,y,z. The energy variables  1 and  2 allow the expression to have the desired energy denominators including photon energies. None of these changes necessarily alter the commutation properties of the quantum operators. Is there a physical meaning to the sum-rule formulas? It seems likely that there is. The commutator of any operator A with the Hamiltonian, [ H, A ] is essentially the operator for the time-derivative of A. The commutator of 3 operators [ [ H, A ], B ] can easily be transformed, using the Jacobi identity, to be the operator for the time derivative of [ A, B ]. These time derivatives are restricted by general conservation laws, related to probability conservation and/or energy conservation. The importance of the f-sum rule is due to the fact that it is true for any atomic potential: pure Coulomb or Coulomb potential screened by bound electrons or by the external plasma. When a property is so general, it is no surprise if it comes from conservation laws.

9.) Time-dependent Schroedinger equation for high-frequency Stark effect An alternative approach to calculating the two-photon opacity is based on the idea that two-photon absorption is just an extreme version of the high-frequency ("AC") Stark effect. This approach gives a powerful check of the perturbation theory and can provide estimates for other multiphoton processes such as the 3-photon absorption. This section will describe solution of the time-dependent Schroedinger Eq. for an ion in a high-frequency electromagnetic field, exploiting the close relation between multiphoton absorption and the Stark effect. Improved numerical accuracy is obtained with Gaussian wave-packets for the X-rays, because the wave-packet has precisely known Fourier spectrum & spectral width. When the code is compared to known one-photon cross-section, it agrees to high accuracy. In section 10, we compare the AC Stark code to a hand calculation (paper-pencil calculation using a pocket calculator) for 2-photon absorption and it agrees. The code also gives first tests for

the calculated Raman cross-sections. AC Stark code Microwave heating in a Tokamak is a strong AC electric field that almost resonates with the Zeeman splitting due to the main plasma magnetic field. The AC Stark code was written to test whether plasma microwave heating might change the Stark-Zeeman line profiles of emission lines for hydrogen. The code uses the usual quantum theory of hydrogen atomic structure(2) together with numerical solution of the time-dependent Schroedinger equation for the effect of the applied fields. The atomic states are a fixed basis-set (energies and wave-functions) and the code calculates the effect of the applied electric and magnetic fields by solving the time-dependent Schroedinger equation. The correct matrix-elements of Eq. (20) were used. Originally the code calculated line profiles using the density-matrix method of Baranger, Kubo and Anderson.(20) The code has been modified to treat hard x-rays on Fe16+. It gives estimates for one-, twoand three-photon absorption processes. The code now describes the X-ray pulses by Gaussian wave-packets and that change has greatly improved the extraction of definite cross-sections. The high-frequency Stark calculation can be compared to second-order perturbation theory for two-photon absorption. When the same atomic data is used for both calculations, the agreement is quite remarkable. This AC Stark code does not include free states or photoelectric ionization. Because the electromagnetic fields are classical, there is no spontaneous emission. Other researchers have also used a time-dependent Schroedinger method for nonlinear optical phenomena. Pindzola and Colgan(29) describe a similar calculation for ions Fe15+-17+. Their computations include ionization. Their calculations produce generally similar (but somewhat larger) cross-sections for two-photon absorption, but they only studied the "one-color" case. We calculate the "MGM" cross-section MGM(1, 2), which depends on two photon energies. The results would need to be integrated over 2 to get the attenuation opacity for photon 1. Code structure One-photon and even two-photon transitions are one-electron processes and the total cross-section is additive for the different electrons in a many-electron ion. (This is related to the so-called linked cluster theorem quoted in reference [17].) For ground-state Ne-like Fe, we calculate a cross-section per 2s or 2p electron and should multiply by 2 or 6 to get the total atomic cross-section. The code solves the time-dependent Schroedinger equation for the evolution operator U(s, s'; t), where s, s' are one-electron states and t is the time. Hb contains the static E, B fields, not used for this work. The coupling to AC fields is through a dynamic Hamiltonian Hd(t) = -e X.E(t), where E(t) is the total electric field of the applied photon beams. The photon beams arrive as one or more Gaussian pulses having specified intensity and polarization. The Schroedinger equation

i

¶U = H o + H b + H d (t) U(t) ¶t

(

)

(36)

is solved for a fixed time-interval = 5 fsec for most of the calculations described here. Eq. (36) is solved by a second-order differencing method. The time-step 0.5x10-20 sec gives good

time-resolution for keV X-rays. Tests show that this time-step is small enough. Because one million time-steps are needed for a 5 fsec calculation, a typical code run requires more than an hour. Today the AC Stark code has 54 one-electron states n, l, m - all l's and m's for n = 2 to 5 (L, M, N and O shells). For X-ray interactions the electron spin is inactive. The Hamiltonian Ho has one-electron energy eigenvalues taken from an SCF calculation for Fe16+. The code has matrices for the unperturbed Hamiltonian, for static and dynamic electric/magnetic fields, for the 3-D vector position operator X and for orbital angular momentum L and spin angular momentum S. For example, the position operators Xnlmn'l'm' are three 54 x 54 matrices. In the original hydrogen-like version of the code, the angular momentum matrices are important for spin-orbit coupling and for the magnetic coupling, but for high-frequency radiation the magnetic effects play no role. Static electric and magnetic fields could be contained in Hb but are zero for the calculations described here. The AC electric field for one X-ray pulse is taken to be

æ (t - t )2 ö o ÷ E(t) = Eo cos(wot) exp çç 4To2 ÷ø è

(37)

The X-ray intensity is (c/4 E2 in cgs units. I(t) is the intensity  E2(t) expressed in photons/cm2sec. When there are two X-ray pulses they can have different intensities, polarizations, pulse center t0 or pulse-duration parameter T0. Spectral width of the X-ray pulse. If the pulse is narrow in time the spectral width will be large and the results could co-mingle several transitions. If the pulse is too long, it extends outside the 5 fsec time window. We can see how the excitation rate changes with de-tuning by varying the central X-ray energy o. Here, we Fourier-analyze the intensity I(t) to obtain the spectral width. These two ways to determine the spectral width agree accurately. For one-photon transitions, the frequency-spectrum of the intensity is

I()

=

Io A exp[- ( )2]

(38)

where o and  = / (To2). The one-pulse spectral-width correction is

Wp(1) = A =

1 W p

(39)

For 2-photon transitions with two beams (pulse widths = T1, T2), the frequency spectrum is the product of the one-beam spectra, but absorption is only possible when

1 + 2 = Ei - Ef Integrating over the distributions and enforcing energy-conservation condition with a delta-function, we have

I10 I 20 ò d (Ei - E f - w1 - w 2 )I1( w1 ) I 2 ( w 2 ) d w 2 = p (W12 + W22 ) If the two beams have the same pulse-width T1 = T2 = T0, as in most of our calculations, the two-pulse spectral width correction simplifies to

Wp(2) =

T0

(40)

p

1-photon transitions The AC Stark code can be tested by comparison with known cross-sections for optically allowed one-photon transitions. The one-photon cross-section for absorption ni, l --> nf, l±1 is:

2 é max(l,l ') ù R d (E f - Ei - w ) ú ë 3(2l +1) û if

s abs ( w ) = 4p 2a w ê

(41)

Here  = e2/c. The initial state i has quantum numbers n, l and the final state f is n', l' where l' = l ± 1 according to the dipole selection rules. With the energy-levels that we use, the 2s-->3p transition has energy  = 912.85 eV. This means  = 2.207 1017/sec. The time-step dt = 0.5 10-20 sec gives about 900 steps per cycle of the electric field. Calculations were performed with different X-ray pulse-lengths (To = 0.1 to 0.5 fsec). A pulse-length To = 0.2 fsec is about 125 cycles per pulse. Because the Gaussian pulse is spread over the energy range Wp(1), the effective on-resonance part of the beam energy is I(t)/Wp(1). The cross-section for one-photon absorption is calculated as the population of the excited state (1) divided by the beam exposure (=  I(t) dt / Wp ).

æ è m

s 2s®3p = ç å U(2s,3pm)

2 ö (1) ÷ Wp / ò I(t)dt ø

(42)

where m = -1, 0, 1 identifies the magnetic sublevels of the 3p state. For the transition 2p --> 3d, the populations from the matrix U must be (i) summed over final 3d states and (ii) averaged over initial 2p states, corresponding to the usual definition of cross-section. This gives the cross-section per electron. Cross-section is independent of photon beam intensity Otherwise identical calculations with higher and lower X-ray intensities found the same cross-section to 6 digits. The average intensity was 1019 photons/cm2-sec and the range considered was two orders of magnitude up and down. At still higher intensities there might be a Rabi oscillation or other nonlinear effect. One-photon absorption The one-photon 2s-->3p absorption cross-section from the AC Stark code is

2s-3p = 9.00702 10-17 cm2eV

(43)

This code result is the same to 6 digits for To = 0.1, 0.2, 0.3 fsec. At To = 0.4 fsec the inferred cross-section is smaller by 20 parts in 106; we attribute this to wave-packet tails slightly spilling out of the time-integration window (the pulse duration is several times the parameter To). The excited state population is proportional to To and without the spectral width correction Wp(1) the answers would differ by factors of ~ 3. We compare with paper-pencil calculation of the cross-section from Eq. (41): Transition energy = 912.85 eV. Dipole matrix-element (atomic units) R2s3p = 3.064815. For Z* ~ 16 (Neon-like Fe) this scales by a factor ao/Z* to R2s3p = 8.536 10-10 cm. Eq. (41) then gives

2s->3p = 9.007013 10-17 cm2-eV

(44)

This is the cross-section integrated across the line profile. Agreement with the AC Stark code calculation is exemplary. Of course, both calculations use the same atomic data.

10.) Two-photon transitions The MGM cross-section (2) describes absorption from two beams of X-rays hitting the target atom (ion) at the same time. This is not the cross-section to compare to the Sandia experiment; in the Z-machine experiment there is a radiation field from radiation in the heated foil and from X-rays arriving from the pinch source. The opacity is the integral of the MGM cross-section over the flux of the "second" photon 2. That flux is large, as much as ~ 1028 photons/cm2sec-eV, so the attenuation cross-section is much larger than the MGM cross-section. In this section we describe calculations of the two-photon MGM by the AC Stark method. X-ray pulses and 2-photon absorption We expect there can be no two-photon absorption if the two X-ray pulses do not overlap in time. We normalize the excitation by dividing the excited-state population by the pulse overlap integral

 I1(t)I2(t) dt As long as the X-ray pulses do not escape the time-window of our calculation, we expect the answer to be independent of the center times, or rather, that dependence should be removed when we normalize the excitation by the overlap integral. Several tests verified this expectation. An important input to the calculations is the time-width T1, T2 of the two Gaussian pulses. The answers depend on these widths in at least two ways. One obvious dependence is that the two-photon process occurs only during beam overlap, so the excitation should be proportional to the pulse intensity overlap integral =  I1(t) I2(t) dt. This strong dependence is easily seen in the results from the AC Stark code. The frequency spectra of the two pulses also depend on the pulse widths. If the frequency spectrum is broad, a large part of the photon pair-energy cannot cause absorption because it does not conserve energy. This effect is described by the spectral width factor W(2) of Eq. (40) above. An additional effect also occurs for the two-photon transitions: with broad pulses, the intrinsic energy-dependence of the cross-sections is averaged over.

Pulse-width variation We performed calculations to evaluate these pulse-width effects. One set used the same pulse width for both beams. The transition studied is 2s --> 4d with 1 = 962.85 eV, 2 = 209.43 eV; this is 50 eV detuned from the one-photon chain (2s --> 3p --> 4d) condition. The cross-section expected from 2-photon perturbation theory is  = 4.3726 10-52 cm4sec-eV. For T1 = T2, the spectral-width correction is Wp(2) = T1 / ( . Here, the quantity P4d is the 4d population at the end of the calculation. T1 = T2 = 0.1 fsec ==>

P4d x Wp(2)/ I1(t)I2(t) dt = 4.710607 10-52 cm4sec-eV

T1 = T2 = 0.2 fsec ==>

P4d x Wp(2)/ I1(t)I2(t) dt = 4.682227 10-52 cm4sec-eV

T1 = T2 = 0.3 fsec ==>

P4d x Wp(2)/ I1(t)I2(t) dt = 4.677042 10-52 cm4sec-eV

T1 = T2 = 0.4 fsec ==>

P4d x Wp(2)/ I1(t)I2(t) dt = 4.675233 10-52 cm4sec-eV

With increasing pulse length the inferred cross-section settles to a limit closer to the analytic formula but remains about 7% high. Some of the change between cases is likely to be caused by the intrinsic energy-dependence of the real cross-section (for 2s-4d,  is proportional to E-2, where E is the "detuning" from the one-photon resonance). To summarize: with Wp(2) the predicted cross-section is independent of pulse-width to 0.8 % and agrees with perturbation theory to ~ 7%. Energy-dependence and 1/E2 law The energy-dependence of the 2-photon cross-sections is simple in certain parts of the spectrum. For the 2s-->4d transition with detuned energies 1 = 912.85 eV + E, 2 = 209.43 eV - E, we have the following results from the AC Stark code. These are calculated with To = 0.2 fs for both beams and corrected for that.

E 20 eV 40 eV 60 eV 80 eV 100 eV

 2.978 10-51 7.351 10-52 3.2268 10-52 1.7526 10-52 1.0146 10-52

E2 Column 4 1.1937 10-48 1.1762 10-48 1.1616 10-48 1.1217 10-48 1.0146 10-48

1.147 10-48 1.113 10-48 1.069 10-48 0.997 10-48 0.862 10-48

Over this range, this cross-section is strongly dominated by 2s-3p-4d chains. A second conclusion: the 2-photon bound-bound transitions appear as a line-broadening ~ 1/()2 around the allowed 1-photon line. Experimental spectra have shown "broader lines" than most opacity codes(4,5), and this might be the reason. Column 4 above is the same quantity E2 MGM calculated by the "hand calculation" described in Appendix C. That analytic calculation uses the 2nd order perturbation theory formula but only includes the L, M, N, O-shell states with the same atomic data (energy levels and matrix-elements) as the AC Stark code. It clearly gets almost the same answer with the same energy dependence as the AC Stark calculation. The AC Stark answer is consistently a little higher, probably due to some subtle aspect of the spectral width of the wave-packets; the hand calculation has specified frequencies with no spectral width. The 1/ E2 power-law becomes inaccurate at E ~ 100 eV for this case. For other cases the simple 1/E2 power-law is less accurate. Even in the case shown above it fails by ~ 25% at E > 100 eV. Figure 6 shows the AC Stark calculations in comparison with the "Bridge" code based on the formulas in the Appendix. Angular factors It is easy to use the AC Stark code to test the angular factors for two-photon cross-sections. We compared absorption 2s --> np --> 4d from two pulses of X-rays, one at 962.85 eV, the other at 209.43 eV (these are detuned by E = 50 eV from the one-photon chain energies). For case (1), the two photons have the same polarization vectors (e(1) = e(2) = y). For case (2) the vectors are perpendicular. The results are case (1)

 = 4.68223 10-52 cm4sec-eV

case (2)  = 3.51167 10-52 cm4sec-eV

The ratio of these cross-sections is 1.33333. That ratio agrees to within a part per million with the ratio calculated by several pages of algebra involving spherical harmonic functions. 2p --> 4f also agrees We tried numerical and pocket-calculator calculations for the transition 2p --> nd --> 4f. In this case the intermediate quantum number n can be 3, 4 or 5. The result is a cross-section generally similar to the 2s --> np --> 4d cross-section, with peaks at different energies, and again the AC Stark code agrees with the hand calculation to a few percent. We also show AC Stark calculations for 2s --> nd --> 3d, 5d in Figure 6.

Raman cross-section It is possible to extract a cross-section for the Raman effect from the AC Stark code. This is useful because we have no other check for the Raman effects, which are likely to be among the easiest things to measure at an X-ray FEL. For the transition 2s --> 4d, when the X-ray absorbed 1 has "too much" energy for the 2s-4d energy splitting, excitation occurs if the second photon 2 carries away the extra energy. The AC Stark code cannot calculate spontaneous emission of the second photon but it can calculate stimulated emission. The cross-section for that is obtained from code runs in a straightforward way, dividing the excitation (population) by the integral product of the photon fluxes. The result is a small, frequency-dependent Raman cross-section. This will be a useful check on the formula

calculations. We can imagine an experiment where photon 1 comes from an X-FEL. Photon 2 is an output that can be detected. The energy of the output photon 2 varies as the input photon 1 changes energy. The atom is left in an excited state and spontaneous emission from that excited state would be a second indicator of the Raman excitation.

Comparison of elements The Sandia experiments observe an extra opacity for Fe which does not appear for Ni and Cr at similar temperatures. It was not difficult to compare Ni and Fe for one process, the two-photon absorption 2s-->4d. The 2-photon cross-section is found to be smaller for Ni than for Fe. This cross-section should be weighted by the population of photon 2 and the density of states for that photon. The weighting is important because photon 2 has higher energy for Ni than for Fe, for the same 1. The comparison of cross-sections indicates the two-photon cross-section for Fe is much larger than Ni. Even including the correction (2)2 n(2), the Ni opacity contribution is 4 or 5 times less than for Fe. That is consistent with the experiments but many more comparisons will be necessary before we draw conclusions.

Three-photon transitions At this writing the 3-photon processes do not seem important except to answer the natural question "If 2-photon processes, why not 3?" It is useful to have an estimate of the three-photon cross-sections to answer the question. Calculating the 3-photon processes by perturbation theory would be difficult. The best answer to the question is that the continuous 2-photon absorption competes with the small scattering opacity between lines in the one-photon opacity. The 3-photon opacity would be competing with 2-photon opacity. The 3-photon process might be directly measured in a high-intensity X-FEL experiment. In the past, we had used the AC Stark code with a constant X-ray flux and had observed 3-photon absorption with an effective cross-section of roughly 10-90 cm6sec2eV. New calculations done with Gaussian wave-packets produce a similar number, with much nicer precision. However we have not calculated the spectral width correction for these calculations and that could change the answers by a factor between 2 and 5.

11.) Summary and Conclusions This paper addresses the need for accurate free-free dipole matrix-elements for the perturbation theory treatment of two-photon processes. It has shown how to obtain free-free matrix-elements that obey several criteria or tests. Our best results come from analytic continuation of the Gordon formula for bound-bound matrix-elements. We summarize the tests: The results from analytic continuation agree to five digits with the eight cases calculated by Gao and Starace. Those cases were not near the difficult equal-energy condition.

For many test-cases the new results agree to about three digits with matrix elements calculated from direct numerical solution of the Schroedinger equation. The precision is limited because the numerically integrated matrix-elements oscillate slightly with the cut-off radius. Near the equal-energy condition, the predicted matrix-elements multiplied by (E1 - E2)2, are a smooth function which has a discontinuous derivative. Because they are obtained by analytic continuation from the bound-bound (and bound-free) matrix-elements, they have the best possible consistency with those. The free-free matrix-elements satisfy the matrix f-sum of Eq. (1). The coefficients of the delta-function singularities in R(E1,l1 --> E2,l2) at equal energies are adjusted to satisfy the sum-rule. For the test with 20 energies (shown in Figures 1-3), forty numbers are adjusted to satisfy the sum-rule for 190 cases. The results shown in Figures 1-3 are very satisfactory. With the corrector coefficients U(E), V(E) shown in Fig. 4, it is possible to evaluate integrals over the free-free matrix-elements. This paper does not yet calculate the effective attenuation (opacity) for comparison to the foil attenuation experiments of reference 4, however it solves a key mathematical problem for the evaluation of the two-photon absorption cross-section given by Eq. (2). A recent paper by Livermore scientists(28) has tried to estimate the overall opacity by calculating absorption by 2s electrons on neon-like Fe16+ and finds a small result. Even if that calculation is correct there are many other ions and transitions to consider. Two-photon absorption is also studied in section 10 above by a time-dependent Schroedinger equation calculation using the AC Stark code. Introducing Gaussian wave-packets for the X-ray pulses has greatly improved cross-sections from the AC Stark code. A second advance described here was to compare the code to hand calculations using the same atomic data; this was possible because the data-set is not too large. The agreement gives persuasive evidence that the two methods agree. The AC Stark code takes a million time-steps per cross-section. It is too slow to calculate opacities but can be used to check various detailed aspects of the perturbation calculation. The main physics limitation on the AC Stark code is its lack of coupling to the continuum: free electron states are not included as final states or even as intermediate states. The good agreement for the specific cross-sections tested is a strong, independent support for the perturbation theory approach to the calculation of two-photon absorption phenomena.

APPENDIX A.) Detailed Balance for one-color 2-photon processes A number of researchers have tried to calculate the MGM cross-section of Eq. (2) for the case of equal photon energies h1 = h2. This is the "one color" calculation. The results are the smallest possible cross-sections as easily seen from Figures 5, 6. This is not only the smallest cross-section but also the smallest opacity contribution because the integral over directions and energies of

second photon gives a great enhancement of the absorption rate. In addition, the one-color two-photon process has three interesting peculiarities, maybe not universally recognized. One is that for given initial and final states (i, j), energy conservation for two-photon absorption reads 2 h = Ej - Ei This equation fixes the photon energy as does one-photon line absorption. For the 2-color two-photon processes only the sum (or difference) of the two photon energies is constrained so they give a continuum absorption (or emission), typically over a broad range of energies. A second important difference for the cross-section is that if the two photons are really identical (same energy, same direction, same polarization) there is no way to distinguish the orders of absorption. This means the amplitude should only be half the amplitude of the nearby transitions with photons that are not identical. A third difference involves the photon creation and annihilation operators. When two different photons are absorbed the amplitude is proportional to the matrix-element of photon annihilation operators a1 a2 whose square is n1 n2, the photon population in the initial state of the radiation field. However when the photons are identical, the matrix-element (squared) is n1(n1 - 1). For emission of two identical photons, the photon population factors (matrix elements of photon creation operators a+) give (n1+1)(n1+2). At first sight it seems difficult to satisfy detailed balance with these population factors. The factors above refer to the case where the n's are integer photon populations. For detailed balance we must consider thermal averages over a black-body field. It is known that for Bose-Einstein statistics(30) < nk,s2 > =

(2 2 + )

This equation is derived by the Einstein method for fluctuations. When applied to the one-color detailed balance factors, the ratio of emission to absorption is found to be

(n +1)(n + 2) emiss = absorb n(n -1)

2 æ 1 ö ÷÷ = e2hn /kT = çç1+ n ø è

This is exactly the ratio required to compensate for the LTE population ratio of lower and upper states for the emission/absorption processes; their energy difference is 2h. Of course it is mainly an academic point to show this consistency of the two-photon formulas. The overwhelming majority of two-photon processes cannot involve identical photons. Even for an x-ray laser beam, the beam divergence and frequency spread guarantee that capture of two identical photons is an extraordinary event. However the power of detailed balance is that it applies to all processes, even the most unlikely.

APPENDIX B.) X-Ray Wavelength correction to matrix-elements In section 2 it is mentioned that the fundamental formula for the radiative matrix-elements should include an exponential factor exp(iq • r) involving the X-ray wavelength (see Equation 5 in section 2). For visible light this factor can be ignored because | q • r | << 1 but for X-rays and for excited states it might be important. This factor behaves like a density effect because it becomes important for excited states that have larger radial extension. If this exponential factor were expanded it would lead to a series of quadrupole, octopole, ... transitions and would change the selection-rules which decide the intermediate states. This would require a much more elaborate calculation. However a simpler approximate calculation may succeed to capture the main effect of the exponential factor. This approximate calculation makes a spherical average (over directions of the X-ray wave-vector q) of the exponential factor. The modified matrix-elements can be evaluated using the explicit radial wave-functions for the hydrogen excited states. The calculation shows little or no change for lower quantum numbers but a reduction by about 50% for n = 6 matrix-elements of the Fe 16+ ion with X-ray wavelength of ~ 7 Å. A simple approximate curve-fit agrees with the numerical calculations to about 10% accuracy for X-ray wavelengths ≥ 7 Å. This curve-fit is

-b[qR(0)]2 R(q) = R(0) e where R(0) = Rn,ln',l' is the long-wavelength limit (q --> 0) of the same matrix-element in atomic units, and the parameter  = 0.275 (a0/Z*)2.

APPENDIX C.) Hand calculation This appendix describes a hand calculation using the two-photon perturbation theory formula to compare to the AC Stark code. The calculation uses the same states, energies and matrix-elements as the solution of the time-dependent Schroedinger equation. The time and frequency ranges affect the inferred cross-sections as described in sections 9, 10. With the hand calculation there is no issue of integration time or spectral widths. (The "hand calculation" uses a pocket calculator!) The hand calculation has been installed in a one-page computer code, the "Bridge" code. It was then easily extended by adding matrix-elements and energy levels and gave results shown in Figures 5, 6. Atomic data used in the AC Stark code includes one-electron energy levels for the Fe16+ ion: E1s = - 7,561 eV

E2s = - 1,348.1 E3s = - 464.56 E4s = - 203.39 E5s = - 93.654

E2p = - 1,256.4 E3p = - 435.25 E4p = - 191.43 E5p = - 87.755

E3d = - 394.84 E4d = - 175.82 E5d = - 80.119

E4f = - 166.24 E5f = - 75.210

E5g = - 72.549

The hydrogen matrix elements (atomic units) are:

R2s2p = - 5.196152 R2s3p = 3.064815 R2s4p = 1.282277 R2s5p = 0.773952 R3s2p = 0.938404

R2p4d = 1.709702 R3p4d = 7.565411 R4p4d = -20.784610 R5p4d = 3.045320 R2p3d = 4.747992

R3s3p = -12.727922 R3p3d = -10.062306 R3s4p = 5.469336 R4p3d = 1.302254 R3s5p = 2.259575 R5p3d = 0.482798

These BB matrix-elements are multiplied by ao/Z*, where ao = .529177 10-8 cm and Z* = 16). The matrix elements can be checked against formulas in section 4 above and agree nicely, including the sign of the n = 0 transitions. The angular factor for absorption s-p-d is ang = .088889 when the two photons have the same polarization. If the polarizations are perpendicular, ang = .06666. The formula from second-order perturbation theory (equivalent to Eq. (2) in section 2 above) is 2 d Ein Rind Enf Rnf 8p 3a 2 MGM s ( w1, w 2 ) = G ang I( w1 + w 2 ) å ( w1 )( w 2 )(2l +1) n Ei - En The line-profile function I() is an energy-conserving delta function so the sum of photon energies should equal the energy difference of upper and lower level. We consider the case 2s --> np --> 4d and include states n = 2, 3, 4, 5 as does the AC Stark code. For detuning by E = 50 eV from "resonance", i.e., at 1 = 962.85 eV, 2 = 209.43 eV, the formula gives a cross-section of 4.3726 10-52 cm4sec-eV (this is an MGM cross-section per 2s electron, integrated across the absorption profile). The cross-section is accurately proportional to 1/E2, for moderate values of E < 50 eV. At different photon energies, different intermediate states are most important. As written above the MGM cross-section has both photon energies specified. It should be clear these are not final predictions of the opacity. Many other initial, final and intermediate states must be included to predict the opacity. The opacity cross-section is the integral of MGM over the photon spectrum for 2 and that will give a much larger answer. The important benefit of this calculation is that the comparison to the time-dependent Schroedinger equation (AC Stark code) verifies the handling of intermediate states (the two-photon orders), signs and other details of the perturbation theory calculation. That is a powerful mutual verification of the two methods of calculation.

Conflict of Interest

All authors have participated in (a) conception and design, or analysis and interpretation of the data; (b) drafting the article or revising it critically for important intellectual content; and (c) approval of the final version. This manuscript has not been submitted to, nor is under review at, another journal or other publishing venue. The authors have no affiliation with any organization with a direct or indirect financial interest in the subject matter discussed in the manuscript

ACKNOWLEDGEMENTS: The authors appreciate helpful discussions with Drs. A. Starace (U. Nebraska), J. Colgan (Los Alamos National Laboratory), Y. R. Shen (U. California at Berkeley), R. Marrus (U. California at Berkeley) and I. Murakami (National Institute for Fusion Science).

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FIGURES

Figure 1.) The matrix f-sum for energies E1, E2 (atomic units) for the case (E1, s) --> (E2, s) via an intermediate state (E", p). The diagonal terms (E1 = E2) are divergent integrals and are not plotted. The RMS f-sum is also indicated. Inspection will show (1) very small values of the f-sum, which should be zero, and (2) slight influence of the singularity coming from (E1 - E2).

Figure 2.) The matrix f-sum for energies E1, E2 (atomic units) for the case (E1, p) --> (E2, p) via an intermediate state (E", d). The diagonal terms (E1 = E2) are divergent integrals and are not plotted. The RMS f-sum is also indicated. Inspection will show (1) very small values of the f-sum, which should be zero, and (2) slight influence of the singularity coming from (E1 - E2).

Figure 3.) The matrix f-sum for energies E1, E2 (atomic units) for the case (E1, d) --> (E2, d) via an intermediate state (E", f). The diagonal terms (E 1 = E2) are divergent integrals and are not plotted. The RMS f-sum is also indicated. Inspection will show (1) very small values of the f-sum, which should be zero, and (2) slight influence of the singularity coming from (E1 - E2).

Figure 4.) Plot of the corrector functions U(E), V(E) discussed in section 6 for the cases shown in Figures 1-3.

Figure 5.) MGM cross-sections from the Bridge code described in Appendix C. This perspective drawings shows that many of the cross-sections have peaks when one (or the other) photon energy matches a line transition of the ion in question. These peaks artificially line up, because of the crude one-electron energy-level spectrum of the average-atom model. With more accurate excited-state energies, the curves would not all coincide and the resulting oscillations in the total opacity would be considerably reduced. The diagonal terms (h1 = h2) are one color cross-sections and have the smallest cross-sections.

Figure 6.) This figure compares calculations from the AC Stark code and from the Bridge code based on formulas in Appendix C. The solid and dashed curve are Bridge code calculations for 2s-->4d and 2p-->4f respectively. The plots look similar. Absorption starting from the 2p state must be weighted by the 6 bound electrons in the Neon-like ion, so the 2p contribution is larger. The point symbols (dots, triangles, +'s) come from the AC Stark code. Where the calculations overlap the agreement is generally excellent (a few percent). The apparent left-right symmetry is not accidental because these cross-sections are plotted as functions of 1 and 2 is the complement E - 1.