Thermal-group Cross Sections

Detailed cross-section curves in the low-energy range are presently available for a number of the more important reactor materials (see, for example. Fig. 4.29), and the general scheme described in Sec. 4.7b may be used to compute the thermal-group values. As a rule the integration indicated in (4.209) must be carried out numerically however, in a few instances, notably in the case of the fissionable materials U, U , and Pu, some of the preliminary computation has already been performed. Thus for these fuels data are available which permit one to compute the thermal-group cross sections by applying a suitable correction factor to the values obtained directly from the cross-section curves. One definition of such a factor has been... 

The critical-mass calculation for any one of these reactors entails a trial-and-error procedure. One can either assume a size and compute the critical concentration, or vice versa. We will do the latter. The next step then is to evaluate the macroscopic cross sections of the reactor medium for the entire lethargy range, i.e., Sa(i ), /( )i ( )> well as the thermal-group cross sections at the operating temperature of the system. These data may now be applied directly to the computation of the various parameters which appear in the critical equation, i.e., nh, Pth, 17) /) and L. In the first trial calculation it is convenient to assume a value for the fast effect (perhaps c = 1) and check it later. All the necessary data having been collected, B may be computed from the appropri-... 

When working with the thermal-neutron population, the proper definition of ri is given in terms of the thermal-group cross sections, and not in terms directly of the measured values as presented in cross-section curves. It is clear then that the present calculation will involve the various f factors derived previously in Sec. 4.7g. 

Calculations used a two-dimensional model based on the KARE diffusion theory code and WOXX three-group cross sections. Homogenization (self-shielding) factors were determined from fine structure calculations using the Tranvar thermal transport code. Several other cross section schemes and transport codes in common usage were also Investigated, but no significant differences were noted. 

To provide validation of the calculational model, two critical assemblies were analyzed with the 27-enetgy-group cross-section library and the ANISN-W code. Both a thermal and a fast system were studied to ettsure the accuracy of the calculations over a wide range of conditions. The calculated eigenvalue (keff) for the benchmark thermal critical system was 1.016, and was 0.987 for the fast system. 

Criticality analyses were carried out using KENO-IV (Ref. 7) Monte Carlo calculations, and few-group PDQ (Ref. 8) diffusion calculations were benchmarked against these. The KENO-IV calculations employed a 123-group cross-section set that has sufficient fine groups to model thermal and fast neutron disadvantage effects, but in the... 

We find then that the computation of the thermal-neutron group cross sections may be carried out by integrating the measured cross-section data weighted by the neutron distribution in speed space (4.200) over the range 0 < v < . It would appear that the details of the computation introduce some ambiguity from a physical viewpoint since the integration... 

The derivative of the microscopic cross sections r is obtained from the definition of the thermal group (4.209) and depends, of course, on a detailed knowledge of the energy dependence of a in the vicinity of kT. Unless otherwise noted, we will assume in this treatment that (1) all absorption cross sections aa vary as 1/v in the thermal range and (2) scattering and transport cross sections a, and cross section is also /v [see (4.234)] therefore, in the case of the... 

The quantity which appears in (o) is to be chosen so that is a good estimate of the slowing-down density of fast neutrons out of the fast group at r. The source of fast neutrons is taken to be the neutrons produced by fissions caused by thermal neutrons vXf2 r). The parameters Di and 2) are the thermal-diffusion coefficient and thermal-absorption cross section in the core, respectively, computed in the usual way. The source term for the thermal neutron equation (h) is taken to be the number of neutrons slowing down out of the fast group, reduced by the number lost in actual absorption during slowing down. We define... 

Application to Thermal Reactor. A simple, direct illustration of the use of the results obtained above is demonstrated in the analysis of the reactor in which all fissions are due to thermal neutrons alone. For this example we make the following choice of group cross sections ... 

The symbol 2 ° denotes the thermal absorption cross sections of the nonfuel components of the core and the thermal absorption cross section of the fuel. Thus the present method permits a direct solution for Nf and avoids entirely the trial-and-error computations required for the standard two-group method of Sec. 8.4. It should be evident from the procedure outlined above that the computational effort involved in the Feynman-Welton two-group model is only a small fraction of that required in the standard method. This fact may be more clearly demonstrated by considering a numerical example. 

Reference has already been made to the METHUSELAH code which has formed the basis for SGHWR nuclear design. The code uses a five-group model of the neutron physics and relies on an extensive library of effective group cross sections deduced from nuclear data. The first version came into operation towards the end of 1961 (refs 2, 5). Some revisions of the nuclear data libraries and Improvements in the representation of the thermal spectrum were... 

The HELIOS model in Fig. 4.16 was used to estimate the reactivity change due to this level of burnup. A reactivity loss of 0.00065 (about 10 cents) was calculated. The model was also used to generate few group cross sections for use in the coupled neutrons/thermal analysis program described elsewhere in this report. 

The subscripts /, s, c, and 6 refer, respectively, to the fast flux, slow flux, core region, and blanket region D is the diffusion coefficient is the neutron flux 2/ refers to the effective cross section for removing neutrons from the fast group and 2, refers to the thermal absorption cross section. Other symbols have the same meaning given previously, with k calculated on the basis that (ep)fuei= 1. 

Boron [7440-42-8] B, is unique in that it is the only nonmetal in Group 13 (IIIA) of the Periodic Table. Boron, at wt 10.81, at no. 5, has more similarity to carbon and siUcon than to the other elements in Group 13. There are two stable boron isotopes, B and B, which are naturally present at 19.10—20.31% and 79.69—80.90%, respectively. The range of the isotopic abundancies reflects a variabiUty in naturally occurring deposits such as high B ore from Turkey and low °B ore from California. Other boron isotopes, B, B, and B, have half-Hves of less than a second. The B isotope has a very high cross-section for absorption of thermal neutrons, 3.835 x 10 (3835 bams). This neutron absorption produces alpha particles. 

Figure 3.5. TEM cross-sectional image of a SnSli4Se0.5 film prepared by spin coating from a hydrazine-based solution. The final thermal treatment for this film was at 270 °C for 20 minutes. [Reproduced with permission from [Ref. 31]. Copyright 2004 Nature Publishing Group.]...

---