Can cross sections be accurately known for priori? Page: 4 of 7
This article is part of the collection entitled: Office of Scientific & Technical Information Technical Reports and was provided to UNT Digital Library by the UNT Libraries Government Documents Department.
Extracted Text
The following text was automatically extracted from the image on this page using optical character recognition software:
III CROSS SECTION SENSITIVITY B S-matrix formalism
A. Spherical optical potential
The optical model for nucleon-nucleus interaction is the
starting ingredient in calculating cross sections. This model
allows us to determine neutron elastic scattering as well as ab-
sorption cross sections. The spherical optical model potential
is usually defined as
U(r, E) -V,(r, E) - iW (r, E) - iWV (r, E) +
Vso(r, E){ - s + iWo(r, E){ - s.
(2)
Here, all components are separated into energy-dependent
well depths and energy-independent radial parts according to
V, V,(E)f(r,R,av) ,
W = W, (E)f (r, RV, aV),
d
Ws =-4as W.~(E)-f (r R8, as) , (3)
f '1 dd
V8 = V80(E) m f (r, Rso, aso),
fi '1 d
Wso Wo(E) (h) f (r, Ro, aso),
where the indices v, s, and so refer, respectively, to volume-
central, surface-central, and spin-orbit potential. The form-
factor is given by the frequently used Woods-Saxon shape
f(r, R,, a,) = {1 + exp[(r - R2)/a2]}-1, (4)
where the geometric parameters are the radius R, = rA1/3
and the diffuseness, a1, with A being the atomic mass number.
B. S-matrix formalism
The optical potential allows us to compute the energy-
averaged S-matrix, or equivalently the complex phase shifts
r/ . These are related by
S -(E) =e2ne (E) - af (E) ed 3 E (5)
where a, are real. The superscript refers to those ele-
ments where the possible values of f for a given j are j 1/2.
In term of these and with E a 1/A2, the total and absorption
cross sections are given by
Cxm
atot(E) = 27rA2 ( + 1)(1 -Re[S ]) + f( -Re[S,]) ,
f-o
Qabs(E) - TrA2 (e + 1)(1 -S I2t) + (1 -S I2) ,
f-o
respectively. The elastic cross section can be obtained by sub-
tracting nabs from the total cross section.C. Cross section uncertainties
The uncertainty of any cross section a due to uncertainties
in a set of parameters, p = {pi, . .. Pp.. , p,,. ... }, is given
in linear approximation by the square root of((3 )2) yy (6p p) ,
"where (8p &p~) is an element of the covariance matrix of the
parameters, and where the sensitivity coefficients, &a/0pp,
are to be calculated with the best estimates of p . In our
work the sensitivity coefficients were numerically computed
as first-order partial derivatives assuming a linear dependence
of the parameters on the cross sections for small perturbations
Ap .
III. CROSS SECTION SENSITIVITY
We carried out sensitivity calculations by varying several
of the optical model parameters. In this paper, we concentrate
on the results obtained for the uncertainties in the depth and
radius of the real volume potential. Both V and r~ contribute
significantly to the results for total and absorption (or absorp-
tion) cross sections, and we will show their correlation, which
is significant.
We quantify the effect of the perturbation of the model pa-
rameter p on the cross section via the dimensionless ratio(6)
S(E,p) (Ep) -(Bp )
a(E,p)(7)
where Q(E, p) is the cross section calculated for the central
value of p, whileo(E, p) = o(E; p Ap)
(8)
are the cross sections calculated with the value of the parame-
ter p perturbed by the quantity Ap. A very similar analysis
for dimensionless sensitivity parameters was recently carried
out by Fessler et al. [5].
As an example, in Fig. 1, we discuss neutron reactions on
56Fe and the response of the (n,tot) and (n,abs) cross sections
to the variation of the volume nuclear radius (rV) and the real
volume depth (V) of the Koning-Delaroche optical poten-
tial [4]. There are remarkably different levels of sensitivity
between two reactions, and strong energy dependences. All
sensitivities change sign several times between 1 keV and 200
MeV. The immediate consequence of this behavior is that at
these zero-crossing points the parameter uncertainties (even if
arbitrarily large) will not contribute to uncertainty in the cross
section.
The sensitivities plotted in Fig. 1 for (n,tot) depict the
strong correlation between rV and V (continuous black and
red lines) as well as the principal dependence of the structure
of the total cross sections, i.e., Vrz above ~ 10 MeV. Fig. 1
also shows the cross section sensitivities of (n,abs). We note
that the parameter correlation is still seen although it is limitedIII CROSS SECTION SENSITIVITY
B S-matrix formalism
Upcoming Pages
Here’s what’s next.
Search Inside
This article can be searched. Note: Results may vary based on the legibility of text within the document.
Tools / Downloads
Get a copy of this page or view the extracted text.
Citing and Sharing
Basic information for referencing this web page. We also provide extended guidance on usage rights, references, copying or embedding.
Reference the current page of this Article.
Pigni, M. T.; Dietrich, F. S.; Herman, M. & Oblozinsky, P. Can cross sections be accurately known for priori?, article, June 24, 2008; United States. (https://digital.library.unt.edu/ark:/67531/metadc898696/m1/4/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.