38
I. J. Thompson
2 Phenomenological R-matrix
The “phenomenological R-matrix method” that is followed in the remainder of this
paper does not start from a Hamiltonian and does not have an infinite series of poles.
Rather it uses a finite number P of R-matrix pole energies e p , with reduced width
amplitudes γ pα as parameters in the familiar finite sum
R αα (E) =
P
p=1
γ pα γ pα
e p − E
,
(2)
to be adjusted to fit experimental scattering data. Positive-energy poles are again
aligned with scattering resonances. Other poles are “background poles” at higher
positive energies to attempt to represent the effects of all the remaining terms
missing in comparison with expression (1).
Both the exact and phenomenological R-matrix expressions yield (a) unitary
S-matrix at each energy, and (b) orthogonal scattering wave functions at different
energies. When we come to the approximations often used in R-matrix theory, they
should only be accepted if at least they still yield those features. Both conditions
derive from having a Hermitian and energy-independent Hamiltonian.
The Reich–Moore approximation [2], by contrast, has imaginary damping widths
for missing channels, so condition (a) is not satisfied. It is perhaps satisfactory if a
specific meaning is given to the missing flux, e.g., capture or fusion.
Another convenient approximation changes the boundary conditions in the Bloch
operator, so B is not constant but is set equal to the shift function at each energy:
B = S(E). But now condition (b) is not satisfied since H = T + V + ˆ
B is energydependent.
The “alternative parametrization” of Brune [3] is much better than using B =
S(E) for making R-matrix pole energies close the energies of cross-section peaks
and resonances, since the Brune basis is transformable to and from the Lane and
Thomas formalism.
3 Verification of R-matrix Codes
An inter-comparison of the capabilities of the R-matrix codes AMUR [4], AZURE2
[5], EDA (LANL), FRESCOX [6], GECCCOS (TU Vienna), SAMMY [7], and
CONRAD [8] was performed [9] following a series of IAEA consultants meetings
since 2015 [10–13]. As the codes were developed initially for the solution of
different problems, each one has its particular features, strengths, and weaknesses,
an inter-comparison is particularly valuable.
I. J. Thompson
2 Phenomenological R-matrix
The “phenomenological R-matrix method” that is followed in the remainder of this
paper does not start from a Hamiltonian and does not have an infinite series of poles.
Rather it uses a finite number P of R-matrix pole energies e p , with reduced width
amplitudes γ pα as parameters in the familiar finite sum
R αα (E) =
P
p=1
γ pα γ pα
e p − E
,
(2)
to be adjusted to fit experimental scattering data. Positive-energy poles are again
aligned with scattering resonances. Other poles are “background poles” at higher
positive energies to attempt to represent the effects of all the remaining terms
missing in comparison with expression (1).
Both the exact and phenomenological R-matrix expressions yield (a) unitary
S-matrix at each energy, and (b) orthogonal scattering wave functions at different
energies. When we come to the approximations often used in R-matrix theory, they
should only be accepted if at least they still yield those features. Both conditions
derive from having a Hermitian and energy-independent Hamiltonian.
The Reich–Moore approximation [2], by contrast, has imaginary damping widths
for missing channels, so condition (a) is not satisfied. It is perhaps satisfactory if a
specific meaning is given to the missing flux, e.g., capture or fusion.
Another convenient approximation changes the boundary conditions in the Bloch
operator, so B is not constant but is set equal to the shift function at each energy:
B = S(E). But now condition (b) is not satisfied since H = T + V + ˆ
B is energydependent.
The “alternative parametrization” of Brune [3] is much better than using B =
S(E) for making R-matrix pole energies close the energies of cross-section peaks
and resonances, since the Brune basis is transformable to and from the Lane and
Thomas formalism.
3 Verification of R-matrix Codes
An inter-comparison of the capabilities of the R-matrix codes AMUR [4], AZURE2
[5], EDA (LANL), FRESCOX [6], GECCCOS (TU Vienna), SAMMY [7], and
CONRAD [8] was performed [9] following a series of IAEA consultants meetings
since 2015 [10–13]. As the codes were developed initially for the solution of
different problems, each one has its particular features, strengths, and weaknesses,
an inter-comparison is particularly valuable.