Heterogeneous Reactors and Wigner-Seitz Cells

Abstract

When deterministic neutron transport methods are applied to lattice or whole-core problems, the multi-group approximation is usually applied to the cross-sectional treatment for the energy domain. Due to the complicated energy behavior of resonance cross sections, the weighting spectrum for collapsing multi-group cross sections is very dependent on energy and space, which becomes a crucial challenge when analyzing a lattice or full-core configuration.

Problems

Problem 8.1

Cadmium has a resonance for neutrons of energy 0.178 eV, and the peak value of the total cross section is about 7000 b. Estimate the contribution of scattering to this resonance.

Problem 8.2

A nucleus has a neutron resonance at 65 eV and no other resonances nearby. For this resonance, Γ_n = 4.2 eV, Γ_γ = 1.3 eV, and Γ_α = 2.7 eV, and all other partial widths are negligible. Find the cross section for (n, γ) and (n, α) reactions at 70 eV.

Problem 8.3

Neutrons incident on a heavy nucleus with spin JN = 0 show a resonance at an incident energy E_R = 250 eV in the total cross section with a peak magnitude of 1300 barns, the observed width of the peak being Γ = 20 eV. Find the elastic partial width of the resonance.

Problem 8.4

The diffusion length of beryllium metal is determined by measuring the flux distribution in an assembly 100 cm × 100 cm × 65 cm high, at the base of which is a plane source of thermal neutrons. The corrected saturation activities in counts per minute of foils located at various vertical distances above the base are given in the table below. Similar measurements in a horizontal direction gave a value of 2.5 cm as the extrapolation distance, i.e., the distance from the assembly at which the neutron flux extrapolated to zero. Using the figure below, what would be the diffusion length of thermal neutron in beryllium? (Fig. 8.5)

Fig. 8.5

Data for evaluation of diffusion length of beryllium

Vertical distance (z cm)		Saturation activity (c/m)
10	.....................................	310
20	.....................................	170
30	.....................................	90
40	.....................................	48
50	.....................................	24
60	.....................................	12

Problem 8.5

A reactor core contains enriched uranium as fuel and beryllium oxide as moderator (Σ_s = 0.64 cm⁻¹ and ξ = o.17 at a neutron energy of 7 eV). The thermal neutron flux is 2 × 10¹² neutrons/(cm²) (s), and Σ_a for these neutrons in the fuel is 0.005 cm⁻¹; for each thermal neutron absorbed, 1.7 fission neutrons are produced. Neglecting all absorption during slowing down, estimate the epithermal neutron flux at 7 eV per unit lethargy interval. Hint: Use Eq. 3.75 that is given in Chap. 3 for lethargy or the so-called logarithmic energy decrement as well.

Problem 8.6

An experiment was performed to measure the Fermi age of fission neutrons slowing down to indium resonance in an assembly of beryllium oxide blocks, using a fission plate as the neutron source. The corrected indium foil saturation activities, normalized to 1000 at the source plate, are given in the table below. Using Eq. (a) below for the age of neutron of specified energy in a given medium, determine the neutron age at the indium resonance energy.

Distance from fission plate (r cm)	Relative activity (A0)	A 0 r 2	A 0 r 4
0	1000	–	–
4.0	954	1.53 × 104	2.44 × 105
7.9	828	5.16 × 104	5.16 × 105
11.7	660	9.04  × 104	1.24 × 107
19.35	303	1.13 × 105	5.59 × 107
27.15	103	7.59 × 104	4.24 × 107
34.5	29.3	3.55 × 104	4.3 × 107
42.7	7.2	1.31 × 104	2.4 × 107
50.5	1.6	4.08 × 103	1.04 × 107
62.2	0.45	1.74 × 103	6.7 × 106
77.8	0.15	9.08 × 102	5.5 × 106

Problem 8.7

A reactor consists of natural uranium rods of 1 inch diameter, arranged in a square lattice with a pitch of 6 inches in heavy water. If the thermal utilization f is given by the following equation:

$$ \frac{1}{f}=1+\left(\frac{V_1{\Sigma}_{a1}}{V_0{\Sigma}_{a0}}\right)F+\left(E-1\right) $$

where F and E for a cylindrical fuel rod are given as follows:

$$ F=\frac{\kappa_0{r}_0}{2}\cdot \frac{I_0\left({\kappa}_0{r}_0\right)}{I_1\left({\kappa}_0{r}_0\right)} $$

and

$$ E=\frac{\kappa_1\left({r}_1^2-{r}_0^2\right)}{2{r}_0}\left[\frac{I_0\left({\kappa}_1{r}_0\right){K}_1\left({\kappa}_1{r}_1\right)+{K}_0\left({\kappa}_1{r}_0\right){I}_1\left({\kappa}_1{r}_1\right)}{I_1\left({\kappa}_1{r}_1\right){K}_1\left({\kappa}_1{r}_0\right)-{K}_1\left({\kappa}_1{r}_1\right){I}_1\left({\kappa}_1{r}_0\right)}\right] $$

I₀ and K₀ are being zero-order modified Bessel functions of first and second kinds, respectively, and I₁ and K₁ are the corresponding first-order function.

Note: The modified Bessel functions may be expanded in series form for cases where the ratio of moderator to fuel is fairly high and absorption in the fuel is weak. When κ₀r₀ is less than 1 and κ₁r₁ is less than 0.75, the results may be approximated to give the following results [5]:

$$ F\approx 1+\frac{{\left({\kappa}_0{r}_0\right)}^2}{8}-\frac{{\left({\kappa}_0{r}_0\right)}^4}{192} $$

and

$$ E\approx 1+\frac{{\left({\kappa}_1{r}_1\right)}^2}{2}\left[\frac{r_1^2}{r_1^2-{r}_0^2}\ln \frac{r_1}{r_0}+\frac{1}{4}{\left(\frac{r_0}{r_1}\right)}^2-\frac{3}{4}\right] $$

Then calculate the thermal utilization for this lattice. The diffusion length of thermal neutron in natural uranium metal is 1.55 cm; the effective macroscopic absorption cross section is 0.314 cm⁻¹.

Problem 8.8

Calculate the resonance escape probability for the natural uranium-heavy water lattice in Problem 8.7. Use the following table of data as well (Table 8.1):

Table 8.1 Resonance neutron data for moderators