Cross section, macroscopic

Thus the parameter S s 1/X, the reciprocal of the average distance to collision (mean free path) or, saying it in another way 

S s average number of collisions per unit path of travel (2.22)  

A physical argument which displays the basis for this definition of S may be obtained from the simple experiment discussed in Sec. 2.2b. According to Eq. (2.1) 

Z s fraction of first collisions in dx about x where n(x) is the density of uncollided neutrons at x. Or, rewriting 

Since the derivative at the left of this relation gives the change in the neutron density n(x) over the distance dx, it must correspond also to the number of collisions which occur in dx. Thus a verbal statement of (2.23) might be as follows  

---

The analysis of steady-state and transient reactor behavior requires the calculation of reaction rates of neutrons with various materials. If the number density of neutrons at a point is n and their characteristic speed is v, a flux effective area of a nucleus as a cross section O, and a target atom number density N, a macroscopic cross section E = Na can be defined, and the reaction rate per unit volume is R = 0S. This relation may be appHed to the processes of neutron scattering, absorption, and fission in balance equations lea ding to predictions of or to the determination of flux distribution. The consumption of nuclear fuels is governed by time-dependent differential equations analogous to those of Bateman for radioactive decay chains. The rate of change in number of atoms N owing to absorption is as follows ... 

Xab appears here as a universal parameter. However, it was found experimentally to depend on a number of factors [30-33] such as temperature, molecular weight, composition, inter-monomer distance (and therefore on the scattering vector Q), isotopic constitution, tacticity, microstructure, etc. These dependencies are shortcomings of the crude RPA description. The scattered intensity (macroscopic cross section d (Q)/d 2) is given by ... 

If scattering processes are left out of consideration, is equal to the absorption coefficient ft for certain particles or photons. 1 is then equal to the mean free path of the particles or photons in the absorber. If absorption and scattering are taken into account, the absorption coefficient fi is given by the total macroscopic cross section... 

This is illustrated in Fig. 11.1. Fission of may be taken as an example v is the average number of neutrons liberated in nuclear fission of by thermal neutrons and F[/Z a is the ratio of the macroscopic cross sections for fission and for absorption of neutrons. For pure nuclides (e.g. pure iTf/iTa = The fission factor... 

Whilst the term macroscopic cross-section is used for the phenomenological cross-section defined here, it should be noted that it is also used to mean the quantity a[M] which is the effective cross-section of all the target particles in 1 cm and has dimension of cm". ... 

The cross section cr (b) is called the microscopic cross section. Another form of the cross section, also frequently used, is the macroscopic cross section S (m ), defined by the equation... 

The macroscopic cross section is analogous to the linear attenuation coefficient of -y rays (Sec. 4.8.4). If a parallel beam of monoenergetic neutrons with intensity 1(0) impinges upon a material of thickness t, the number of neutrons that emerges without having interacted in the material is (see Fig. 4.24)... 

Example 4,20 What are the macroscopic cross sections Xj, X , and X, for thermal neutrons in graphite The scattering cross section is = 4.8 b and the absorption cross section is o- = 0.0034 b. What is the mean free path ... 

For a mixture of several isotopes, the macroscopic cross section X,- is calculated by... 

What are the capture, fission, and total macroscopic cross section of uranium enriched to 90 percent in for thermal neutrons (p = 19.1 X 10 kg/m. )... 

Answer The factor exp ( — 2,/) should be calculated. For the worst case, consider / = length of the counter. The total macroscopic cross section is... 

With regard to E- V energy transfer reactions, we report primarily the vibrational (microscopic) energy content of the quencher molecules, and no effort was made to measure total (macroscopic) cross-sections. The data obtained for the relaxation of 0(2 i)2) and Na(3 / y) by CO have demonstrated unambiguously the mechanisms of these two important reactions. The former occurs by a complex-forming mechanism and the latter is impulsive in nature. 

Calculate the macroscopic cross-section for reaction of natural uranium with thermal neutrons. See Figures 16.1 and 19.5. 

System perturbations can usually be expressed as perturbations in the macroscopic cross sections. The operators of the perturbation integrals of the different formulations depend on these cross-section perturbations the functional dependence varies with the formulation. The simplest dependence is found in the integrodifferential formulation in which the perturbation operators are the cross-section perturbations. Conversely, the integral transport theory formulations include kernel perturbations that do not depend linearly on cross-section perturbations. Consequently, it is necessary to evaluate the perturbation in the kernels before applying integral pertur-... 

Atomic Nnmber Z Name Otemical Atonic Weight A Density (/>) gm/cc. 20°C 760 mm Hg Nuclei per ec (u 10-24) N (1-h) Avg. Lxtg Energy Loss f Tl Neut lermal (0.025 ev). ron Cross-Sections, Mean Free Path cm Slowing (town Moderating Ratio... 

Atomic- Number Z Name Chemical Atomic Weight A Denaity (P) gjyfcc, 20 C 760 an Hg Nuclei per cc (X 10-34) N (l-b) Arg. Lag Energy Ipsa ( n Neut ermal ron Cr cr) in (0.025 ev) MS Sections )sms Ibermal (0.025 ey) Macroscopic Cross Sections, oN Ibermal (0.025 er) Mean Free Path cm Slowing Down Power No-gf Moderating Ratio (crs/cr ) Xf... 

Figure 2. Accurate values and Inonii s diffusion theoretic values for the nonescape probability from an infinite slab of thickness 2a, in which the scattering is isotropic and has a macroscopic cross section S.

Here dS is a surface element of the lump, n its inner normal, is the unit vector in the direction of the solid angle element dQ, The integration is over the solid angle to the inside of the body and over its total siuface. So is the total macroscopic cross section in the lump formerly designated by 2, and f is the mean chord length in the absorber. Finally, To is defined as... 

---