Heterogeneous Reactors and Wigner–Seitz Cells

Abstract

When deterministic neutron transport methods are applied to lattice or whole-core problems, the multigroup 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 multigroup 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, ${\Gamma}_n=4.2\kern0.24em \mathrm{eV}$, ${\Gamma}_{\gamma }=1.3\kern0.24em \mathrm{eV}$ and ${\Gamma}_{\alpha }=2.7\kern0.24em \mathrm{eV}$, and all other partial widths are negligible. Find the cross section for (n, γ) and (n, α) reactions at 70 eV.

• Problem 8.3 Neutron incidents on a heavy nucleus with spin JN = 0 show a resonance at an incident energy ${E}_R=250\kern0.24em \mathrm{eV}$ in the total cross section with a peak magnitude of 1300 barns, the observed width of the peak being $\Gamma =20\kern0.24em \mathrm{eV}$. 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 x 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?

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

  Fig. P8.4

  

  Data for evaluation of diffusion length of beryllium

• Problem 8.5 A reactor core contains enriched uranium as fuel and beryllium oxide as moderator (${\varSigma}_s=0.64\;{\mathrm{cm}}^{-1}$ and $\xi =0.17$ at a neutron energy of 7 eV). The thermal neutron flux is 2 × 10¹² neutrons/(cm²) (s), and Σ _a for these neutrons, 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 neutron 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 the equation (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 (A 0)	(A 0 r)	(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-in. diameter, arranged in a square lattice with a pitch of 6 in. in heavy water. If the thermal utilization f is given, the following equation is

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

  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

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

  I ₀ and K ₀ are being zero-order modified Bessel functions of first and second kinds, respectively, and I ₁ and K ₁ are the corresponding first-order function.

  Note that the modified Bessel functions may be expanded in a series form for cases where the ratio of moderator to fuel is fairly high and absorption in the fuel is weak. When κ ₀ r ₀ is less than 1 and κ ₁ r ₁ 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 P8.8 Resonance neutron data for moderators