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
𝑇𝑏𝑎 = 𝜒𝑏 |𝑉|𝐴𝜒𝑎
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
Percentage contribution to ICS (10-18cm2) from
linear region of allowed DCS for excitation of
Ar(3p54s) at various incident energies.
3P
3P
1P
level 3P2
1
0
1
30 eV ICS 2.210 5.812 0.445 16.27
30°
%
20
53
19
60
50 eV ICS 0.396 4.852 0.079 18.52
24°
%
20
76
20
78
100 eV ICS 0.037 3.834 0.007 15.27
20°
%
28
90
27
90
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.
ICS for excitation of the 4p levels of Ar(10-18cm2)
State\
Energy
30 eV
50 eV
100 eV
4p1
J=0
27.65
14.29
6.90
4p2
J=1
0.53
0.06
0.01
4p3
J=2
1.94
1.23
0.74
4p4
J=1
0.55
0.08
0.01
4p5
J=0
1.48
0.72
0.36
4p6
J=2
2.03
1.36
0.83
4p7
J=1
0.59
0.08
0.01
4p8
J=2
1.90
1.03
0.58
4p9
J=3
1.56
0.25
0.04
4p10
J=1
2.06
0.31
0.07
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