Program for Calculating Effective Microscopic Cross Sections
Abstract Generated abstract
This paper presents a computational procedure for determining effective microscopic neutron absorption cross sections for materials irradiated in a reactor spectrum. It formulates the calculation using a neutron energy distribution composed of Maxwellian thermal and epithermal components, derives expressions for the effective cross section through the parameters g, s, and r, and relates the spectrum parameters to neutron scattering, absorption, and slowing down quantities. The authors describe a digital computer program that interpolates tabulated energy dependent cross sections and numerically evaluates the required integrals over specified temperature, epithermal fraction, and joining energy ranges. The reported general calculation error does not exceed 0.01 percent.
Full Text
UDC 621.039.54
PROGRAM FOR CALCULATING EFFECTIVE MICROSCOPIC CROSS SECTIONS
E. M. BELOV, S. M. NEELOV, V. K. YASELSKY, M. S. SUKHANOV
(Presented by the scientific seminar of the Faculty of Physics and Technology)
When carrying out physical experiments on the irradiation of investigated samples with neutrons in a reactor spectrum, it is often necessary to know the effective microscopic absorption cross section of various materials, \(\hat{\sigma}\). The effective absorption cross section can be determined as follows.
In the slowing-down process, the distribution of neutron density over energies can be written in the form [1]
\[ n(E)\,dE = n(1-a)N(E)\,dE + naf(E)\,dE, \tag{1} \]
where \(N(E)\) and \(f(E)\) are the distribution functions of thermal and slowing-down neutrons with respect to energies, respectively; \(a\) is the fraction of the neutron density belonging to the epithermal part of the spectrum.
The distribution function of thermal neutrons (the Maxwell distribution) \(N(E)\) has the form
\[ N(E)\,dE = \frac{2}{\sqrt{\pi}} E_T^{-3/2}\sqrt{E}\,e^{-E/E_T}\,dE, \tag{2} \]
and for epithermal neutrons—
\[ f(E)\,dE = \sqrt{\frac{\mu E_T}{2E}}\,\Delta\,\frac{dE}{E}, \tag{3} \]
where the energy \(E_T\) is determined by the formula
\[ E_T=\kappa T. \]
Here \(T\) is the temperature of the neutron gas, and \(\kappa\) is Boltzmann’s constant.
The function \(\Delta\) is introduced into expression (3) in order to limit the spectrum of epithermal neutrons from below by the energy \(\mu \kappa T\). The simplest form of this function is the step function, convenient in carrying out numerical calculations on a digital computer,
\[ \Delta = \begin{cases} 0, & E < \mu\kappa T,\\ 1, & E \ge \mu\kappa T. \end{cases} \tag{5} \]
The rate of the neutron absorption reaction by a nucleus with absorption cross section \(\sigma(E)\), for a neutron flux determined by expression (1), is equal to
\[ R=\int_{0}^{\infty} n(E)\sqrt{E}\,\sigma(E)\,dE. \tag{6} \]
The effective absorption cross section \(\hat{\sigma}\) in this case is determined by the expression
\[ \hat{\sigma}= \frac{\displaystyle \int_{0}^{\infty} n(E)\sqrt{E}\,\sigma(E)\,dE} {\displaystyle \sqrt{E_0}\int_{0}^{\infty} n(E)\,dE}, \tag{7} \]
where \(E_0=0.253\ \text{eV}\) is the energy of thermal neutrons at \(T_0=239^\circ\mathrm{K}\).
After substituting into (7) the relation for the neutron flux (1), taking into account equations (2), (3), and carrying out a number of transformations, the effective absorption cross section can be written as [1]
\[ \hat{\sigma}=\sigma_0(g+rs), \tag{8} \]
where \(\sigma_0\) is the absorption cross section at energy \(E=E_0\), and the parameters \(g\) and \(s\) are determined according to the formulas
\[ g=\frac{1}{\sigma_0 E_T^2}\sqrt{\frac{4T}{\pi T_0}} \int_{0}^{\infty} E e^{-E/E_T}\sigma(E)\,dE, \tag{9} \]
\[ s=\frac{1}{\sigma_0}\sqrt{\frac{4T}{\pi T_0}} \int_{\mu kT}^{\infty} \left[\sigma(E)-\sigma_0 g\sqrt{\frac{E_0}{E}}\right]\frac{dE}{E}. \tag{10} \]
The parameter \(r\) characterizes the ratio of the neutron density in the epithermal region to the total neutron density
\[ r=\alpha\sqrt{\frac{\pi\mu}{16}}. \tag{11} \]
The quantity \(\alpha\) depends on the neutron-physical characteristics of the materials and is determined by the formula
\[ \alpha=\frac{2\gamma\sqrt{1/\mu}}{1+2\gamma\sqrt{1/\mu}}, \tag{12} \]
where
\[ \gamma=\frac{l_s}{\xi l_a} \tag{13} \]
is a quantity reciprocal to the slowing-down coefficient.
The parameter \(\mu\) characterizes the neutron “joining” energy and depends on the coefficient \(\gamma\). It can be determined, depending on the numerical value of \(\gamma\), from a transcendental equation of the form
\[ e^{-\mu}\mu^2=\frac{\sqrt{\pi}}{2}\gamma. \tag{14} \]
Thus, to determine the neutron spectrum it is necessary to know the following neutron-physical quantities:
\(l_s\)—the scattering mean free path;
\(l_a\)—the absorption mean free path;
\(\xi\)—the average logarithmic loss of neutron energy per collision.
According to the algorithm given, a program was compiled for calculating effective microscopic absorption cross sections. As input data, in addition to the quantities indicated above, the values \(\sigma(E)\) were also specified. The error of the calculations in the general case does not exceed 0.01%.
Program for calculating effective microscopic cross sections
begin integer i, j, n, m;
real array E1[1:38], σ1[1:38], σ2[1:38], E2[1:1998],
A[1:2], C[1:2], Д[1:2];
P0042(E1, σ1); j := 2; E2[1] := E1[1]; σ2[1] := σ1[1];
for i := 1 step 1 until 37 do begin
n := (E1[i + 1] — E1[i])/0.005;
if n = 1 then begin
E2[j] := E1[i + 1]; σ2[j] := σ1[i + 1]; j := j + 1;
end;
else begin
C[1] := E1[i]; C[2] := E1[i + 1]; Д[1] := σ1[i]; Д[2] :=
= σ1[i + 1];
for m := 1 step 1 until 2 do
A[m] := Д[m] × sqrt(C[m] × ((C[m] — 0.08418)↑2 +
(0.5 × (0.9572 + 0.06788 × sqrt(C[m])))↑2); A[1] := A[2] — A[1]
for m := 1 step 1 until n do begin
real a; a := E1[i] + 0.005 × m;
σ2[j] := σ1[i] + m × A[1]/sqrt(a)/
((a — 0.08418)↑2 + (0.5 × (0.09572 + 0.06788 ×
sqrt(a)))↑2; E2[j] := E1[i] + m × 0.005; j := j + 1;
end;
end;
end;
begin real array E3[1:198], σ3[1:198],
for j := 1 step 1 until 198 do begin
σ3[j] := σ2[j]; E3[j] := E2[j]; end; P1041(E3, σ3);
end;
begin real g, s, v, w, r, μ, T, σ4, σ5, Em, a, β;
for T := 250 step 50 until 1200 do begin
Em := 8.6167₁₀ — 5 × T; a := 4 × T/(3.1416 × 293.16);
v := sqrt(a)/2.652₁₀6;
w := v/Em ↑ 2;
for r := 0 step 0.1 until 0.8 do begin
for μ := 3 step 1 until 6 do begin
real B1, B2; B1 := B2 := 0;
for i := 1 step 1 until 1998 do begin
\[ \begin{aligned} &\alpha := -E2[i]/Em;\\ &B1 := B1 + E2[i]\times \exp(\alpha)\times \sigma2[i]\times 0.005;\\ &\text{end;}\\ &g := w\times B1;\\ &\text{for } n := \mu\times Em/0.005 \ \text{step } 1 \ \text{until } 1998 \ \text{do begin}\\ &\alpha := 0.0253/E2[n];\\ &B2 := B2 + (\sigma2[n]-2.652_{10}\,6\times g\times \operatorname{sqrt}(\alpha))/E2[n]\times 0.005;\\ &\text{end;}\\ &s := v\times B2;\quad \alpha := 3.1416\times \mu;\quad \beta := 10/(\mu\times Em);\\ &\sigma4 := 2.652_{10}\,6\times (g+r\times s);\\ &\sigma5 := \sigma4/(v\times 2.652_{10}\,6)/(1+4\times r/\operatorname{sqrt}(\alpha)\times \operatorname{sqrt}(\mu)/4\times \ln(\beta)-1);\\ &P1041(g,\ s,\ \sigma4,\ \sigma5);\\ &\text{end;}\\ &\text{end;}\\ &\text{end;}\\ &\text{end; end.} \end{aligned} \]
REFERENCES
- V. Artamkin, O. Tuzhavina, Effective cross sections for the neutron spectra of thermal reactors. Atomic Energy, 1959, vol. 6, No. 2.