Accuracy of the Relativistic Distorted-Wave Approximation (RDW) A. D. Stauffer York University Toronto, Canada Formula for the RDW T-matrix element for electron excitation of an atom from state a to state b where = (N) ) F(N) +1). (N) ) is a Dirac Fock configuration-interaction wave

function for the N) -electron target atom calculated from the GRASP program with a definite value of the total angular momentum J. F(N) +1) represents the scattered electron which is calculated as a solution of the Dirac equations with a distortion potential U which is normally chosen as the static potential of the upper state and includes non-local exchange. A is the antisymmetrizing operator to allow for

exchange of the incident electron with one of the bound electrons and V is the interaction potential between the incident electron and the target. The Dirac equations for F are solved using an integral form of these equations (see Zuo et al, 1991, for details). This allows a larger step size for a given accuracy as well as being more stable than solving the differential equations directly (e.g. N) umerov method for Schroedinger equations).

Exchange is included via antisymmetrization of the total wave function leading to non-local potential terms. The T-matrix includes an integration to infinity along the real axis of an oscillatory function. This is avoided by replacing the integral in the asymptotic region ( = 0) where all the functions are known ) where all the functions are known analytically by an integration in the complex plane. This integration can be carried out accurately using a few points in a Gaussian integration (see Parcell et al

(1987) for details). Why relativistic? Target wave functions have distinct values of J including fine-structure energy differences. Cross sections have different magnitudes and energy variations which depend on J. Oscillator strengths available for transitions between fine-structure levels.

RDW method is a first-order theory: One electron excitations (j,) (j',') For non-zero direct T-matrix element: Parity condition: ' + k + even (J', k, J) and (j', k, j) satisfy triangular inequalities k is the order of the multipole term in the expansion of V in spherical harmonics. Exchange T-matrix element always non-zero. Dipole allowed excitations (k = 1):

Initial and final states have opposite parity. J' = J+1, J, J1 J' = J = 0) where all the functions are known not allowed Integrated cross sections (ICS) have high energy behavior n(E)/E Other direct excitations ICS ~1/E Exchange excitations ICS ~ 1/E3 Example: Ar(3p6) Ar(3p54s) J = 0) where all the functions are known ; j = 1/2 or 3/2; = 1

J' = 0) where all the functions are known , 1(twice), 2; j' = 1/2; ' = 0) where all the functions are known J' = 1 levels give allowed excitations (k = 1) J' = 0) where all the functions are known , 2 levels give exchange excitations (metastable states) LS - coupling not valid here since one of the J' = 1 levels would be a triplet state giving a forbidden transition Percentage contribution to ICS (10) where all the functions are known -18cm2) from linear region of allowed DCS for excitation of

Ar(3p54s) at various incident energies. 3 3 1 level 3P2 P1 P0) where all the functions are known P1 30) where all the functions are known eV ICS 2.210) where all the functions are known 5.812 0) where all the functions are known .445 16.27 30) where all the functions are known

% 20) where all the functions are known 53 19 60) where all the functions are known 50) where all the functions are known eV ICS 0) where all the functions are known .396 4.852 0) where all the functions are known .0) where all the functions are known 79 18.52 24 % 20) where all the functions are known 76

20) where all the functions are known 78 10) where all the functions are known 0) where all the functions are known eV ICS 0) where all the functions are known .0) where all the functions are known 37 3.834 0) where all the functions are known .0) where all the functions are known 0) where all the functions are known 7 15.27 20) where all the functions are known % 28 90) where all the functions are known 27 90) where all the functions are known

Example: Ar(3p6) Ar(3p54p) j = 1/2, 3/2; = 1; j' = 1/2, 3/2; ' = 1 J' = 0) where all the functions are known , 1, 2, 3 From parity condition k must be even for non-zero direct term. Since J = 0) where all the functions are known , J' = k and only final states with J' = 0) where all the functions are known or 2 have non-zero direct terms and, therefore, larger cross sections. Final states with J' = 1 or 3 are exchange excitations with smaller cross sections.

ICS for excitation of the 4p levels of Ar(10) where all the functions are known -18cm2) State\ Energy 30) where all the functions are known eV 50) where all the functions are known eV 10) where all the functions are known 0) where all the functions are known eV 4p1 J=0) where all the functions are known

27.65 14.29 6.90) where all the functions are known 4p2 J=1 0) where all the functions are known .53 0) where all the functions are known .0) where all the functions are known 6 0) where all the functions are known .0) where all the functions are known 1

4p3 J=2 1.94 1.23 0) where all the functions are known .74 4p4 J=1 0) where all the functions are known .55 0) where all the functions are known .0) where all the functions are known 8

0) where all the functions are known .0) where all the functions are known 1 4p5 J=0) where all the functions are known 1.48 0) where all the functions are known .72 0) where all the functions are known .36 4p6 J=2

2.0) where all the functions are known 3 1.36 0) where all the functions are known .83 4p7 J=1 0) where all the functions are known .59 0) where all the functions are known .0) where all the functions are known 8 0) where all the functions are known .0) where all the functions are known 1

4p8 J=2 1.90) where all the functions are known 1.0) where all the functions are known 3 0) where all the functions are known .58 4p9 J=3 1.56 0) where all the functions are known .25

0) where all the functions are known .0) where all the functions are known 4 4p10) where all the functions are known J=1 2.0) where all the functions are known 6 0) where all the functions are known .31 0) where all the functions are known .0) where all the functions are known 7 Accuracy The shape of the differential cross sections (DCS) is

different depending on whether the excitation has a non-zero direct term or is an exchange excitation. DCS for direct excitations have a large peak in the forward direction similar to the Born approximation which contributes almost all of the ICS at higher energies. The value at zero degrees is proportional to the oscillator strength for the excitation. DCS for exchange excitations are flatter in the

forward direction and may actually decrease towards zero degrees. Significant contributions to the ICS come from most of the angular range. Thus wave functions that produce accurate oscillator strengths will produce accurate DCS for direct excitations, at least in the forward direction. There is no direct connection between oscillator strengths and exchange excitations but we assume

accurate wave functions will produce accurate DCS values in these cases. Since the largest contribution to the ICS for direct excitations comes from the forward direction, accurate oscillator strengths will produce accurate ICS. ICS for exchange excitations are generally less accurate than for direct excitations. Problem: How to judge accuracy of oscillator strengths

produced by target wave functions. N) IST ASD has extensive tables of measured oscillator strengths for allowed transitions including estimated errors. Can use these to judge accuracy of wave functions. In cases where the excitation does not correspond to an allowed transition use intermediate states to obtain these. Example: Ar(3p6) Ar(3p54p) Calculate wave functions including 4s orbitals. GRASP

will produce dipole allowed oscillator strengths for 3p6 3p54s and 3p54s 3p54p transitions which can be compared to measured values. Almost all of the 32 values fall within the error bounds specified. These detailed transitions are another justification for a relativistic treatment. Conclusions The RDW method is capable of producing accurate results at medium and high energies for direct

excitations provided the wave functions used produce accurate oscillator strengths. Exchange excitation DCS somewhat less accurate but ICS more reliable than DCS. Relativistic treatment important to resolve fine structure of initial and final states which provides detailed information on behavior of ICS as well as oscillator strengths. Only modest computational effort is required to obtain

RDW results so there is considerable scope for undertaking more elaborate calculations. References Khakoo et al (20) where all the functions are known 0) where all the functions are known 4) J. Phys. B 37 247 Parcell L A, McEachran R P and Stauffer A D (1987) J. Phys. B 20 230) where all the functions are known 7 Zuo T, McEachran R P and Stauffer A D (1991) J. Phys. B 24 2853