Consider an infinite mass of natural uranium, re­presenting the simplest conceptual design for a nu­clear reactor. Assume nj fast neutrons (i. e., with energy of fission neutrons — 2 MeV) are introduced into the reactor. To determine the value of k® for the reactor it is necessary to calculate the number of neutrons in the generation following the absorption in the uranium of all the original generation of ni fast neutrons.

In considering the possible fates of the nj neutrons it is important to recall that natural uranium consists overwhelmingly of U-238 (U-238 : U-235 = 138 : 1). There are three possibilities.

4.3.1 Fission

Figure 1.6 (c) shows that fast neutrons can induce fission in U-238 provided they have energy in excess of the threshold 1.1 MeV, the fission cross-section being much the same value as for U-235. Thus the fission events taking place in the reactor can be re­garded as being predominantly of U-238 nuclei and giving rise to the next generation of fast neutrons, ni say.

4.3.2 Capture

Some of the n і neutrons will undergo direct capture in the natural uranium, i. e., non-fission absorptions in the U-238 and U-235.

4.3.3 Scatter

Some of the n) neutrons will be scattered but only those undergoing inelastic scattering are significant.

In this case the neutrons will emerge from the inelastic scattering event with energy less than the threshold value of 1.1 MeV necessary for U-238 fission and will subsequently be captured in the resonance capture peaks of the U-238. In this context therefore inelastic scattering may be regarded as leading in­directly to capture.

Elastic scatter events may be ignored because the neutrons are effectively unchanged by the collision and are still identified as being of the original nj neutron generation.

4.3.4 koo for natural uranium and fast neutrons

The initial n і neutrons will therefore either cause fission or be captured (directly or indirectly via in­elastic collision). The fraction of ni causing fission may be calculated using the cross-section values for the nuclear reactions and hence the next generation of neutrons, ndetermined by

Of n і

П 2 — ———————- 1%

<7f + + a,

where v is the average number of neutrons released per fission = 2.55 for 2 MeV impinging neutrons. Using the values for cross-sections given in Table 1.2:

n2 = [0.29п[/(0.29 + 0.04 + 2.47)] 2.55 — 0.26n,

Therefore k® = 0.26 < 1. Hence natural uranium, no matter what the geometry, cannot of itself sustain a chain reaction.

4.4 To achieve к OO > 1

The value of k® is determined by the balance be­tween neutron production in the reactor fuel and neutron loss by absorption in the reactor materials and by leakage out of the reactor for a finite system.

Production : Absorption + Leakage.

In Section 4.3 of this chapter, natural uranium was exposed to fast neutrons. It was found that к® < 1 and thus this is not a viable nuclear reactor design. There are two ways by which k« may be increased:

• Change the properties of the fuel.

• Change the properties of the impinging neutrons.