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 χ = φ(N) F(N+1).

φ(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. Numerov 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 (φ = 0) 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,ℓ) → (j',ℓ')

For non-zero direct T-matrix element:

Parity condition: ℓ' + k + ℓ 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, J−1

J' = J = 0 not allowed

Integrated cross sections (ICS) have high energy behavior ℓn(E)/E

Other direct excitations ICS ~1/E

Exchange excitations ICS ~ 1/E3

Example: Ar(3p6) → Ar(3p54s)

J = 0; j = 1/2 or 3/2; ℓ = 1

J' = 0, 1(twice), 2; j' = 1/2; ℓ' = 0

J' = 1 levels give allowed excitations (k = 1)

J' = 0, 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

  level 3P23P1

3P01P1

30 eV ICS 2.210 5.812 0.445 16.2730° % 20 53 19 60

50 eV ICS 0.396 4.852 0.079 18.5224° % 20 76 20 78

100 eV ICS 0.037 3.834 0.007 15.2720° % 28 90 27 90

Percentage contribution to ICS (10-18cm2) from linear region of allowed DCS for excitation of Ar(3p54s) at various incident energies.

Example: Ar(3p6) → Ar(3p54p)

j = 1/2, 3/2; ℓ = 1; j' = 1/2, 3/2; ℓ' = 1

J' = 0, 1, 2, 3

From parity condition k must be even for non-zero direct term.

Since J = 0, J' = k and only final states with J' = 0 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.

State\Energy

4p1J=0

4p2J=1

4p3J=2

4p4J=1

4p5J=0

4p6J=2

4p7J=1

4p8J=2

4p9J=3

4p10J=1

30 eV 27.65 0.53 1.94 0.55 1.48 2.03 0.59 1.90 1.56 2.0650 eV 14.29 0.06 1.23 0.08 0.72 1.36 0.08 1.03 0.25 0.31

100 eV 6.90 0.01 0.74 0.01 0.36 0.83 0.01 0.58 0.04 0.07

ICS for excitation of the 4p levels of Ar(10-18cm2)

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.

NIST 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 (2004) J. Phys. B 37 247

Parcell L A, McEachran R P and Stauffer A D (1987) J. Phys. B 20 2307

Zuo T, McEachran R P and Stauffer A D (1991)

J. Phys. B 24 2853