Determining (n,f) cross sections for actinide nuclei indirectly: An examination of the Surrogate Ratio Method Page: 4 of 20
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2
in the present report. In particular, we will explain the
Surrogate method, review the Weisskopf-Ewing approxi-
mation to the method, and introduce the Ratio approach.
In Section III we describe the simulations we are using
and outline the logic of the tests we are carrying out in
order to assess the validity of the Ratio method. In Sec-
tion IV we present calculations that test the primary as-
sumption underlying the Ratio approach, the validity of
the Weisskopf-Ewing approximation. We then compare
cross section predictions of both the Surrogate Method
in the Weisskopf-Ewing limit and the Surrogate Ratio
method to predetermined reference cross sections. Our
findings and conclusions are summarized in Section V.
II. SURROGATE APPROACHES
This section introduces the main concepts employed
in this study. The Surrogate idea is explained and the
challenges associated with carrying out, analyzing, and
interpreting a Surrogate experiment are outlined. TheWeisskopf-Ewing limit of the full Hauser-Feshbach theory
is briefly reviewed in the context of Surrogate reactions
and the Ratio Method, which makes use of the Surrogate
idea and assumes the validity of the Weisskopf-Ewing
approximation, is introduced.
A. The Surrogate Idea
The Surrogate nuclear reaction technique is an indirect
method for determining the cross section for a particular
type of "desired" reaction, namely a two-step reaction,
a + A -> B* -> c + C, that proceeds through a com-
pound nuclear state B*, a highly excited state in statis-
tical equilibrium (see Figure 1).
The formalism appropriate for describing compound-
nucleus reactions is the statistical Hauser-Feshbach the-
ory (see, e.g., chapter 10 of Ref. [16]). The average cross
section per unit energy in the outgoing channel X' for
reactions proceeding to an energy region in the final nu-
cleus described by a level density is given by:(1)
J11 '5J Z TTi{Es )T ,8, f,1Ex >)p1(U')
Here ac denotes the entrance channel a + A with energy
Ea and reduced wavelength Aa. The spin of the inci-
dent particle is i, the target spin is I, the channel spin
is s = + I, and the compound-nucleus angular mo-
mentum and parity are J, r. The excitation energy of
the compound nucleus, Ee , is related to Ea via the
separation energy Sta(B) of the particle a in the nu-
cleus B: Ee = Sa(B) + Em. The transmission coef-
ficient associated with the entrance channel is denoted
Tj i and the statistical-weight factor 4el is given by
(2J+ 1)/[(2i + 1)(2I+ 1)]. Quantitites associated with
the exit channel of interest are denoted by primed sym-
bols: X' c' + C', i' is the spin of the outgoing particle
c', I' is the spin of the residual nucleus C', s' = + I' is
the channel spin and EX, the energy for X'. The energy
of the decaying nucleus, Ee , is related to EX, via the
relation Ee Sc' (B) + EX, where Sc, (B) is the separa-
tion energy of c' in B. The transmission coefficients for
this channel are written as TX 1,,,(EX,) and p1,(U') de-
notes the density of levels of spin I' at excitation energy
U' in the residual nucleus C'. All energetically possi-
ble final channels X" have to be taken into account, thus
the denominator includes contributions from decays to
discrete levels in the residual nuclei (given by the sum
Z') as well as contributions from decays to regions of
high level density in the residual nuclei (given by the
second sum in the denominator which involves an energyintegral of transmission coefficients and level densities in
the residual nuclei). The relevant quantitites in these fi-
nal channels X" are denoted by double-primed symbols,
in analogy to the particular channel of interest, X'. In
writing Eq. 1, we have suppressed the parity quantum
number except for that of the compound nucleus. The
level density depends in principle on parity (even though
this dependence is weak in practice), and all sums over
quantum numbers must respect angular-momentum and
parity conservation.
The above Hauser-Feshbach formula neglects correla-
tions between the incident and outgoing reaction chan-
nels that can be taken into account formally by includ-
ing width fluctuation corrections. These correlations en-
hance the elastic scattering cross section and reduce the
inelastic and reaction cross sections, although this de-
pletion rarely exceeds 10-20% (even at energies below
approximately 2 MeV) and becomes negligible as the ex-
citation energy of the compound nucleus increases. In the
remainder of this study we will neglect width fluctuation
corrections and rewrite the Hauser-Feshbach formula as:dojX (Ba)
dECN (Ee, J, <)GCN(Ee , J, 4), (2)
1wwhere uCN (Ee, J, r) denotes the cross section for form-
ing the compound nucleus at excitation energy Ee with
angular-momentum and parity quantum numbers Jr indojF (Ea)
dEX
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Escher, J E & Dietrich, F S. Determining (n,f) cross sections for actinide nuclei indirectly: An examination of the Surrogate Ratio Method, article, May 22, 2006; Livermore, California. (https://digital.library.unt.edu/ark:/67531/metadc890483/m1/4/: accessed April 24, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.