2. Results and discussion

We find that for low to moderate energies (below 100 keV) a description of photoionization accurate within $1\%$ must include as many as the first three multipole screening contributions to observables U (i.e. $\Delta_jU$, as defined below, where j corresponds to the 2^j-pole screening contribution, for j<4). As energy increases all $\Delta_j$ will approach zero. For total cross sections $\sigma$ (i.e. integrated over angles) only dipole and quadrupole screening corrections are needed. We describe below how $\Delta_j$ depends on the n and l quantum numbers of the bound state which is photoionized, on Z (neutral atom), and on ejected photoelectron energy E. Beyond dipole $\Delta_j$ are greatest in photoionization of heavy atom inner shells, but they are significant even for outer shells and for quite light elements. While the results we present here are for independent particle approximation (IPA) matrix elements and the consequent predictions for observables, these conclusions may also be used to assess the significance of screening effects when such matrix elements are linked through electron-electron correlations [3]. Our calculations are based on our relativistic IPA code, calculating states in a potential of Dirac-Slater type, including all significant multipole and partial wave contributions.

Screening contributions enter the multipole decomposition of the total IPA photoionization matrix element M in three ways [[4]: (1) in the bound state normalization constant, i.e. the strength of the bound state wave function at the nucleus, (2) in continuum partial wave phase shifts, and (3) in reduced radial matrix elements (bound state normalization constant factored out), which determine the magnitude of the $\Delta_j$ (magnetic, unlike electric, matrix elements do not contribute significantly to $\Delta_j$ because they are very small in the energy range for which screening effects beyond bound state normalization enter the description of photoionization). The $\Delta_jU$ characterize the size of the multipole screening corrections to the physical observables U specified by the square of the total matrix element M. In this letter we will give examples of the size of non-dipole $\Delta_j$ in physical observables.

Consider physical observables $U = \sum_{l,m} U_{l,m}Y_{l.m}$defined on a scale of unit magnitude, and the corresponding observables U_l,m of their expansion in spherical harmonics. Such observables include angular distributions $A(E,\theta)$ $\equiv$ $[d\sigma /d \Omega ](E,\theta)$ averaged over photon polarization and electron spin, shape functions $S(E,\theta )$$\equiv$ $A (E,\theta )/\sigma (E)$ with $\sigma (E)$ the total cross section integrated over angles, and photon electron polarization correlations C_ij. Corresponding to U, define U^(j) as U obtained with the reduced multipole matrix elements through order j (and all associated phase shift differences in the observable) calculated in the screened potential, the higher multipole reduced matrix elements (and associated phase shift differences) replaced by nuclear point Coulomb values. Then we define the screening contributions

$$\Delta_jU\equiv U^{(j)}-U^{(j-1)}=\sum_{lm}(\Delta_jU_{lm})Y_... ...,\phi) \ \ \ \ \ \ \Delta_jU_{lm}=U_{lm}^{(j)}-U_{lm}^{(j-1)}.$$ (1)

We now illustrate and describe more systematically the importance of the multipole screening corrections to the shape of the photoelectron angular distribution $\Delta_jS(E,\theta )$. In Fig. 5.1 we show the domains in (Z,n,l) (also illustrating E_b) in which dipole and quadrupole screening contributions suffice (for all energies and angles) to obtain a specified level of accuracy in the shape function. The domain in which Coulomb properties plus dipole screening contributions suffice for a $2\%$ level of accuracy in the shape function is represented by the dark grey areas in the two panels. In the light grey areas dipole plus quadrupole $\Delta_jS$ suffice for this level of accuracy. In the white areas octupole screening contributions can exceed $2\%$. The next higher multipole contribution, $\Delta_4S(E,\theta )$, never exceeds $1\%$; we show the boundary of its $0.3\%$ domain. (The d-states and f-states generally show quadrupole screening contributions which exceed $2\%$, octupole screening corrections which can exceed $2\%$ when $Z>\approx 50$.)

Next we discuss the magnitudes of the $\Delta_jS(E_{gm},\theta_{gm})$(global maximum value of $\Delta_jS$, occuring at energy E_gm and angle $\theta_{gm}$) as a function of (Z,n,l). In Fig. 5.2 we show the quadrupole and octupole magnitudes for s-state photoionization (the trends are quite similar for higher l). As Z increases, these screening contributions increase somewhat linearly. This trend is consistent with the simple analytic quadrupole retardation effects calculated by Bechler and Pratt [5] for the K and L shells (Z=6-40). (For the K-shell the $\Delta_2$calculated here are within about $20\%$ of predictions from the analytic nonrelativistic formulas for all Z. The L-shell results do not agree as well.) Maximum quadrupole screening contributions are a few percent for light atoms and increase to over $40\%$ for inner shells of heavy atoms. Octupole screening effects are smaller, less than $1\%$ until Z exceeds 25, but can reach nearly $10\%$ for inner shells of heavy atoms (Fig. 5.2). In general maximum screening effects decrease with increasing n. However they do not approach zero with increasing n, but instead saturate, in Uranium at about $10\%$ for quadrupole contributions and $4\%$ for octupole contributions. The trends with regard to (Z,n,l) dependence of dipole screening contributions are the opposite to those for higher multipoles. Of course the dipole effects are dominant in the case of outer shells.

Next we discuss how higher multipole screening effects (maximized for $\theta$) depend on E. Fig. 5.3 shows examples (for s-states in Uranium) of how $\Delta_2S(E,\theta_{gm})$ and $\Delta_3S(E,\theta_{gm})$(global maxima for $\Delta_jS$ as a function of $\theta$ for a given E) vary with E. At low energy (less than 1 keV ) $\Delta_2S(E,\theta_{gm})$ is large for inner shells but decreases with increasing n. At energies greater than 1 keV as n increases (for constant Z) the curves for different n merge to a common curve (reflecting the fact that all ns wave functions have the same shape at the small distances which are probed at high energy), whose magnitude at fixed energy is approximately proportional to Z^4/3. (The increased oscillations for outer shells at low photoelectron energies appear to reflect the additional outer lobes of higher n states which are being probed at low energies.) For all subshells $\Delta_2S(E,\theta_{gm})$ remains significant until near 100 keV, while at very high energy it slowly approaches zero. These features are also seen in $\Delta_3S(E,\theta_{gm})$, of smaller magnitude, which approaches zero more rapidly with n at low energy and stays somewhat significant at higher energies. Similar behaviors are seen in all atoms, though the magnitude of higher multipole screening effects decrease at all E as Z decreases. Note that E_gm (the energy at which the maximum multipole screening contribution occurs) is near threshold for $n\leq 3$ but may be comparable or larger above 1 keV when n>3. (It should be noted that in the full cross section, summed over multipoles, there are small non-normalization screening corrections of relativistic origin which do not go to zero in the high energy limit [6].)

Finally, we discuss the angular dependence of higher multipole screening contributions, as in shape functions and polarization correlations. Fig. 5.4 shows an example of the $\theta$ dependence of $\Delta_2S(E,\theta )$ and $\Delta_3S(E,\theta )$ for U 4s. The curves cover a wide range of photoelectron energies, showing changes in amplitude but little change of shape. This common shape, including the angles at which multipole screening effects are maximum, or near zero, is characteristic of all s-state cases, while for p states and higher the shape has some energy dependence.

We may understand these s-state features, as in the shape function, beginning from Eq. (1), which states that $\Delta_jS$ is a sum over $Y_{lm}(\theta,\phi)$ with coefficients which are proportional to $(\Delta_jS_{l m})$. [In this case only m = 0 terms are present and one usually writes $S = \sum_l B_l P_{l}(\theta)$]. Since higher multipole screening contributions are small in comparison to dipole terms, they enter observables such as the shape function mainly through cross terms (when present) of dipole and higher multipole matrix elements. The cross term of the dipole and jth multipole matrix elements will yield non-vanishing terms in the coefficients B_l of the shape function expansion for l through l_max=j+1 (see for example [7]); there will be non-vanishing terms for even or odd l as j+1 is even or odd. Knowing that the only higher multipole screening corrections which are significant are for j = 2,3 the only terms which need be considered are l = 1,3 (for j=2) and l = 2,4 (for j=3).

For s-states we can approximate the quadrupole contribution to B₁ as equal and opposite to that in B₃, likewise for the octupole contributions to B₂ and B₄. (See for example [2], which notes that for s-states the dipole amplitude is proportional to $\epsilon \cdot p$, the quadrupole amplitude to the same multiplied by $p \cdot k$, yielding a cross term, averaged over polarization, of the stated form). The s-state shapes of the quadrupole and octupole screening angular distributions are therefore (normalized to their maximum values) given by N_quad(P₁ - P₃) = [(27)^1/2/2]x(1-x²); [ N_quad = (27)^1/2/ 5] for j=2, with zeroes at x = 0,$\pm$1 ($\theta$= 0^o, 90^o and 180^o) and extrema at x = $\pm (1/3)^{1/2}$ ($\theta$=54.7^o and 125.3^o), corresponding to the magic angle (where dipole terms vanish, but note not octupole terms), and N_oct(P₂ - P₄) = (1 - 5x²)(1 - x²); [ N_oct=-8/7] for j=3, with zeroes at x = $\pm$1, $\pm (1/5)^{1/2}$ ($\theta =$ 0^o, 63.4^o 116.6^o and 180^o) and extrema at x = 0, $\pm (3/5)^{1/2}$ ($\theta =$ 39.2^o, 90.0^o and 140.8^o) opposite in sign, the magnitude at x = 0 is 5/4^th of the other two maxima. Note both quadrupole and octupole effects vanish in this approximation at forward and backward angles; only dipole effects survive. For bound states with angular momentum l>0 we can no longer approximate the quadrupole contributions to B₁ and B₃ as equal and opposite, or likewise for the octupole contributions to B₂ and B₄. As a result the positions in $\theta$ of maxima, minima, and zeros of these contributions to the shape function will be energy dependent.

We can summarize, and better understand, the importance of higher multipole screening contributions to physical observables in terms of several simple functions, for which we can offer qualitative empirical analytic expressions. We define R(E,E_b) as an approximation to the ratio R_j+1,jof reduced electric field matrix elements $M_{j\l _b \l _c }$ that differ by one multipole (where $\l _b$ and $\l _c$ correspond to bound and continuum state quantum numbers),

$$R_{j+1,j}=\frac{M_{j+1,l_b,l_c+1}}{M_{j,l_b,l_c}}\approx R(E,E_b) =\frac{E+(E_b/E_b+1)^{2/3}}{E+b},$$ (2)

where b is a parameter that depends on the angular momentum l of the bound state (b=5 for l_b=0 and b=20 for l_b=1, taking energies in keV). The behavior of R(E,E_b) as a function of E for Uranium s state photoionization is illustrated in Fig. 5a. Near threshold $R(E,E_b)\approx (1/b)[E_b/(E_b+1)]^{2/3}$, which for large E_b approaches 1/b and for small E_b is proportional to E_b^2/3. When E exceeds 1 keV, R(E,E_b) becomes nearly independent of E_b (or shell). The high energy maximum in higher multipole $\Delta_j$ occurs at about this energy. As E continues to increase, R(E,E_b) goes to unity (at high energy all multipole matrix elements become similar in magnitude).

We define Q(E,Z) as an approximation to the ratios of screened $M^{sc}_{j\l _b \l _c}$ and Coulombic $M^{coul}_{j\l _b \l _c}$ for j>1

$$Q_j(E,Z)=\vert 1-\vert\frac{M^{sc}_{jl_bl_c}}{M^{coul}_{jl_bl_c}}\vert \vert\approx Q(E,Z)=(\frac{Z^2}{aE+Z^2})^{2/3}$$ (3)

using a function of E,Z having only one fit parameter a=15000 (again measuring energies in keV). Q(E,Z) ranges from one (for maximum screening, at threshold) to zero for no screening at very high energy. The approach to zero occurs at a lower E for light atoms than for heavy atoms (Fig. 5.5b). (This expression is not appropriate for dipole $\Delta_jU(E,\theta )$, because Q_j is generally much smaller than unity near threshold, especially for inner shells.)

The general approximation for the relative magnitude of a higher multipole screening contribution may be written, in the case of an observable U such as the shape function, for which higher multipole matrix elements enter linearly in interference terms,

$$\Delta_jU(E,E_b,Z,\theta )=[R(E,E_b)]^{j-1}Q(E,Z)T(\theta ).$$ (4)

The function T reflects the angular dependence of the screening contributions; the s-state form is being used to approximate for all subshells. The first two terms in (4) are due to the reduced electric field matrix element and determine the magnitude of the screening contribution. Screening effects also depend on $\theta$ [ $T(\theta )$) in equation (4)], examples of this will be given. The product of R(E,E_b) and Q(E,Z) (for j=2) as a function of E is given in Fig. 5.5c. The general behavior of $\Delta_2S(E,\theta_{gm})$ is produced. The threshold amplitude becomes very small for low binding energy. Similar results can be obtained for octupole screening effects. In the total reduced cross section, higher multipole contributions would enter quadratically, corresponding to squaring (4) with T=1. If for E near E_gm Q is near 1, we can expect the magnitude of quadrupole screening effects in total cross sections to correspond to the magnitude of octupole screening effects in angular distributions.

We should caution that we have only estimated over-all magnitudes. For example, we have not considered the phase shift differences which multiply reduced matrix elements. These can modify the size of $\Delta_jU$at specific energies, causing oscillations with energy, but will not greatly affect the magnitudes more globally. Specific calculations are required for a comparison with any given specific experiment; what we have given here are cruder estimates which can focus attention on possible parameter domains which merit examination.