# Application of simple ramsauer model to neutron total cross sections Page: 4 of 6

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2 Analysis

A more recent analysis by Anderson and Grimes examined the effects of including an

isospin dependent interaction. Precision total cross sections were fit to a few percent. A

paper is currently being prepared for submission to Nuclear Science and Engineering5 which

examines the effectiveness of a Glauber model including refraction. It is concluded that over

a significant energy range from 5 to 50 MeV, the chord length seen by a neutron traversing a

nucleus is about twice the radius due to refraction. Thus we conclude that the total cross

section should be well represented by

a- = 2n (R+X)2 (1-a cosp). (1)

R is the radius of the nuclear potential, X is the reduced wave length, a is a parameter which

is 1 if there is no absorption and is less than 1 if absorption occurs, and R denotes the phase

change in passing through the nucleus. Clearly, the "black nucleus" limit corresponds to

a=0, in which case the total cross section is 21c (R+))2. For smaller absorption, the a

parameter will multiply the cosy term and allow oscillations to occur.

Finlay and collaborators6 have reported total neutron cross sections of a number of elements

from 5 to 500 MeV. The following fits are based on these data. The data were first

normalized by dividing the total cross section by 2n (R+-)2 where R=r A'f and ro was 1.37

as compared to 1.35 as found by Peterson2. If P reflects the phase difference between

transmitted and outside waves, it would have the form of a constant multiplying the

difference E+Va - fI where the constant included the radius of the nucleus. V0 is the

depth of the nuclear potential, and E is the kinetic energy of the neutron outside the nucleus.

We, therefore, initially tried fits with p expressed in the form:

=Q=K{ 4a+bE-[E } (2)

Using this form and parameters from standard optical models7 we obtain agreement with the

experimental data at about the 2.5% level. Somewhat better fits are obtained with the form:

(3= k { (9 a+b-E -4E)+ k, ( a+bE- )2 } (3)

Values for k, a, b, and k, were obtained by fitting this form to the lead total cross section

values. a is then determined by finding how far above and below 2n (R+1)2 the data go

(typically a=0.11 to 0.13). Using k,=0.07, a=38 and b=0.85 (obtined from the lead data),

the other individual elements were fit by varying k only. The variation of the best fit values

of k under these constraints as a function of A' is shown in Fig. 1. Note the approximately

linear variation between these quantities as expected. In Fig. 2, we show comparisons of the

reduced cross section aT/2n (R+-)2 with the fit provided by Eq. (1) with R expressed as in

the form of Eq. (3). Fig 2a shows the fit for the element tin, and Fig. 2b the fit for lead. In

each case only k was varied, but in these cases and in other elements measured by Finlay et

a16 , the k values fall on a straight line relative to A'n within 1.5%.

3 Summary

The recent analysis supporting the use of Ramsauer models in parameterizing neutron total

cross sections has been buttressed by fitting experimental data over a wide range of elements

directly. The expression used, however, does not have any parameters reflecting

deformation, closed shells or isospin. Although each of these characteristics undoubtedly

influences neutron total cross sections, fits to the experimental data are achieved at the 1.5%

level. This form should, therefore, be useful in estimating cross sections for targets where

experimental data are unavailable. The principal limitations of these fits is that they apply

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Bauer, R.W.; Anderson, J.D.; Grimes, S.M. & Madsen, V.A. Application of simple ramsauer model to neutron total cross sections, article, April 29, 1997; California. (https://digital.library.unt.edu/ark:/67531/metadc691950/m1/4/: accessed April 23, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.