If the energy dependence of a cross section is known, the total rate at which neutrons react with a nuclide is obtained by integrating the flux, cross-section product over all possible energies:

Total reactions with neutrons per unit volume per unit time = f ф{Е)о(Е2) dE

‘o

(2.54)

It is convenient to determine an effective cross section a for the nuclide, so that when a is multiplied by the total thermal flux фм the proper reaction rate is obtained:

If the cross section is one that varies inversely with the neutron speed, as in the case with many of the absorption cross sections, then

o(E) = o(E0)(^^J for l/v absorbers (2.56)

U2 e-£/*• d (^-

where o(E0) is arbitrarily chosen to be the cross section at the energy E0 = kT corresponding to the most probable neutron speed. We shall first assume that all neutrons are in a Maxwell-Boltzmann thermal equilibrium, so that ф(Е) = Фм(Е). The integral in Eq. (2.55) is then transformed to the variable E/E0, and Eqs. (2.52) and (2.56) are substituted to yield

The integral can be evaluated in terms of the gamma function, which in this case has a value of Vrr/2:

— _ g(£~o)s/ff for yjv abSorbers in a Maxwell-Boltzmann distribution (2.58)

Tables (see App. C) usually list values of the thermal absorption cross sections for mono­energetic neutrons of speed 2200 m/s. Because this happens to be the most probable speed for neutrons in thermal equilibrium at 293.2 K, the effective cross section at temperature T (K) can be obtained from

Figure 2.10 Neutron density and flux distributions with respect to speed ratio.

The effective cross section obtained from Eq. (2.59), when multiplied by the total flux of thermal neutrons, will give the proper value of reaction rate with thermal neutrons for a 1/u absorber. However, many of the most important nuclides entering into reactor calculations (e. g., the fissile nuclides) are not 1/u absorbers, and the integration of Eq. (2.54) must consider dependence of the neutron spectrum and the cross section on neutron energy (or speed). In refined calculations this integration is done stepwise by dividing the energy scale encountered in reactors (0 to ~ 12 MeV) into energy groups. An effective cross section is determined for each group and is multiplied by the flux of neutrons in that group to determine the group reaction rate. Digital computers are normally employed.

For simplified reactor calculations a “one-group” approximation can be employed. Westcott [W4] has developed a convention such that the total reaction rate with Nt atoms of nuclide і is given

Total reactions per unit volume per unit time = Nj$o (2.60)

where $ is defined in terms of some arbitrary reference speed v as

where n is the total density of neutrons in the reactor.

The reference speed v is arbitrarily chosen as 2200 m/s, which is the most probable speed for a Maxwell-Boltzmann distribution at temperature T = 293.2 K. The cross section о is now the specially defined effective cross section that, when multiplied by the “2200 m/s flux” $, gives the proper reaction rate constant.

From Eqs. (2.54), (2.60), and (2.61),

/ 0(£M£) dE

In the Westcott formulation the energy distribution ф(Е) is treated as a Maxwell-Boltzmann energy distribution Фм(Е) of thermalized neutrons on which is superimposed an epithermal distribution Фе(Е) of nonthermalized neutrons, so that

The epithermal flux distribution фЕ{Е) can be approximated by a 1 /Е energy dependence above some lower cutoff energy of fjkT, and it can be normalized to the integrated thermal flux фм by a factor 0. Then

фЕ{Е)ЛЕ = фмЩ-<Ш (2.66)

where Д is the unit step function at цкТ energy. A typical value of ц for a well-moderated reactor is 5.

By substituting Eqs. (2.64) and (2.66) into (2.63),

m dE = фм (ф^г е-Е/*т + dE (2.67)

which will be used in solving the integral of Eq. (2.62).

To solve Eq. (2.62) we also need to formulate the total neutron density n as the sum of the densities of Maxwell-Boltzmann neutrons nM and epithermal neutrons nE

n = nM + nE

To obtain n from a flux distribution,

-/

Jo

where Eq. (2.46) was used to change from v to E.

Using Eq. (2.64) in (2.69) to obtain nM:

(2.70)

and using Eq. (2.66) in (2.69), we obtain nE:

JfikT

From Eqs. (2.70) and (2.71),

»e _ 40 пм ч/щї

We now substitute Eqs. (2.67), (2.68), and (2.72) into (2.62) and perform the integration. The results can be written in the form
and

The fraction nE/n in Eq. (2.75) is a parameter specified by the reactor designer. For a purely thermal spectrum nE — 0, so that r = 0 and a = a^oog — For the pressurized-water reactor consiaered in Sec. 6.4 of Chap. 3, the epithermal ratio r is estimated to equal 0.222. When a varies inversely with v, a(E) = o22ao/kf/E, so that g = 1 and s = 0.

The factor g is called the “non-1 /и correction factor.” It becomes greater than unity for a cross section that decreases with increasing neutron speed less rapidly then 1 jv, and it becomes less than unity when the cross section decreases more rapidly than 1 /v.

Values of g and s for 333 U, 335 U, and 239 Pu, as a function of the thermalization temperature T, are listed in Table 2.7. More detailed compilations are available in published reports [Cl, Wl, W4, W5].

The Westcott g and s factors can also be used to determine the effective thermal cross section a, such that when multiplied by the integrated Maxwell-Boltzmann thermal flux <pM the proper reaction rate with a nuclide is obtained, as already defined by Eq. (2.55). From Eqs. (2.55) and (2.62), a is related to a by

Фм

By substituting Eqs. (2.45), (2.68), and (2.70) into (2.77),

(2.78)

or, using Eq. (2.75) to introduce the spectrum parameter r:

(2.79)

3 is the effective cross section defined by Eq. (2.79), which is used later in this text (cf. Sec.

4 and Chap. 3).

The Westcott formulation for the effective cross sections a and a is useful only for well-moderated thermal reactors, where the approximations of the neutron spectra are more reasonable. Even in such reactors, more detailed calculations of actual neutron spectra and effective cross sections are necessary for precise reactor design. The Westcott cross sections are not applicable to fast-spectrum reactors, where neutron moderation and thermalization are suppressed.

T, °С	g (abs)	s (abs)	g (fiss)	s (fiss)
20	0.9983	233 u* 1.286	1.0003	1.216
40	0.9979	1.330	1.0005	1.256
60	0.9976	1.372	1.0007	1.295
80	0.9973	1.412	1.0009	1.333
100	0.9972	1.452	1.0011	1.370
120	0.9971	1.490	1.0014	1.406
140	0.9971	1.527	1.0016	1.440
160	0.9971	1.562	1.0019	1.474
180	0.9972	1.597	1.0022	1.507
200	0.9973	1.631	1.0025	1.539
220	0.9975	1.664	1.0029	1.570
240	0.9978	1.697	1.0032	1.600
260	0.9980	1.728	1.0036	1.630
280	0.9984	1.759	1.0040	1.659
300	0.9987	1.789	1.0044	1.688
330	0.9993	1.833	1.0051	1.730
360	1.0000	1.876	1.0058	1.770
390	1.0007	1.918	1.0065	1.809
420	1.0015	1.958	1.0073	1.847
450	1.0024	1.998	1.0081	1.885
480	1.0033	2.036	1.0090	1.921
510	1.0042	2.074	1.0099	1.956
540	1.0052	2.111	1.0108	1.991
570	1.0062	2.147	1.0118	2.025
600	1.0072	2.182	1.0128	2.058
20	0.9771	235 |j§ 0.1457	0.9781	-0.0263
40	0.9723	0.1595	0.9735	-0.0178
60	0.9678	0.1729	0.9692	-0.0096
80	0.9636	0.1856	0.9650	-0.0017
100	0.9597	0.1977	0.9611	0.0058
120	0.9560	0.2092	0.9573	0.0131
140	0.9526	0.2201	0.9538	0.0197
160	0.9494	0.2302	0.9505	0.0260
180	0.9465	0.2396	0.9474	0.0317
200	0.9438	0.2484	0.9445	0.0368
220	0.9413	0.2565	0.9418	0.0416
240	0.9391	0.2640	0.9392	0.0459
260	0.9370	0.2711	0.9369	0.0496
280	0.9351	0.2774	0.9347	0.0530
300	0.9334	0.2833	0.9327	0.0559

(See footnotes on page 52.)

Table 2.7 Westcott parameters for 233 U, 235 U, and 239 Pu (Con­tinued) Г, °С g (abs) s (abs) g (fiss) s (fiss) 330 0.9312 0.2913 0.9299 0.0597 360 0.9292 0.2987 0.9274 0.0629 390 0.9275 0.3054 0.9252 0.0655 420 0.9261 0.3117 0.9232 0.0680 450 0.9248 0.3180 0.9214 0.0703 480 0.9237 0.3242 0.9197 0.0727 510 0.9228 0.3304 0.9182 0.0750 540 0.9219 0.3370 0.9169 0.0776 570 0.9211 0.3439 0.9156 0.0805 600 0.9204 0.3510 0.9143 0.0837 239 Pu§ 20 1.0723 2.338 1.0487 1.794 40 1.0909 2.369 1.0623 1.820 60 1.1117 2.389 1.0777 1.835 80 1.1350 2.396 1.0952 1.840 100 1.1611 2.390 1.1150 1.836 120 1.1903 2.373 1.1373 1.822 140 1.2227 2.343 1.1623 1.797 160 1.2582 2.298 1.1898 1.761 180 1.2970 2.239 1.2200 1.713 200 1.3388 2.166 1.2528 1.653 220 1.3836 2.077 1.2880 1.581 240 1.4313 1.974 1.3255 1.497 260 1.4817 1.857 1.3653 1.402 280 1.5345 1.727 1.4071 1.297 300 1.5895 1.586 1.4507 1.182 330 1.6758 1.356 1.5193 0.996 360 1.7658 1.110 1.5910 0.796 390 1.8588 0.854 1.6651 0.588 420 1.9539 0.594 1.7410 0.376 450 2.0505 0.334 1.8182 0.166 480 2.1477 0.081 1.8959 -0.040 510 2.2451 -0.163 1.9738 -0.238 540 2.3419 -0.395 2.0514 -0.426 570 2.4377 -0.614 2.1281 -0.604 600 2.5321 -0.817 2.2037 -0.770 is chosen here as the r2 parameter in the Westcott formulation, consistent with the use of a cutoff energy for epithermal neutrons, as in Eq. (2.66). * From Westcott [W4]. ® From Critoph [Cl ].