We show in chapter 8 that, if control rods are to be avoided, the multiplica­tion factor keff should be limited to about 0.98 for the standard fast-neutron hybrid reactors and 0.95 for the slow-neutron ones. This limitation on keff also limits the energy gain G0/(1 — keff) accordingly. An interesting sugges­tion was made as early as 1958 by Avery [131], in order to increase the energy gain of a hybrid system; it was reactivated by Abalin et al. [124] and Daniel and Petrov [132]. It consists of coupling two multiplying systems in such a way that neutrons produced in the first one can penetrate the second while those produced in the second cannot penetrate the first. We quantify the possible gain which can be obtained in this way.

Let one neutron be created in a multiplying medium. If absorbed it produces kx new neutrons. However, in a finite system it only produces keff neutrons.[43] Since keff = Pcapk1 the escape probability is

Pesc = 1 — k2 • (6.12)

k1

If we consider a system with N0 injected neutrons and multiplication keff, the number of escaping neutrons will be

N0 ki keff 1 — keff ki

Now, we consider two multiplying media which communicate. Let

k !21

K =

!12 k2 _

be the matrix of efficiencies: a neutron born in medium 1 gives k1 progeny in medium 1, and i12 in medium 2. Let и(1) and n(2) be the number of generation i neutrons in media 1 and 2. The number of neutrons of the next generation in each medium is

that is,

ni +1 = Knt

and the final number of neutrons as a function of the initial one:

which yields, for

while, if one neutron is created in medium 2, the final numbers are

If one could define a system where ^12 ^ 0 and ^21 = 0, one would get

Abalin et al. [124] propose that the first medium could be a fast-neutron multiplier but a strong thermal-neutron absorber while the second medium would be both a slowing down and thermal-neutron multiplier. The fast neutrons created in medium 1 could, eventually, reach medium 2 and be slowed down and multiplied. In contrast, slow neutrons from medium 2 could not reach medium 1 without being immediately absorbed in the strong thermal-neutron absorber (for example a gadolinium nucleus). As suggested by equation (6.12), an estimate of!12 is

which shows that it is interesting to maximize k1M, and thus to use as pure fissile material as possible. !12 will, in general, be of the order of unity. It is mandatory that (1 — k1)(1 — k2) > !21!12 ~ !21. In order for the system to be of interest as compared with standard ones, k1 and k2 should be close to 0.95. This means that the coupling term, !21, should be less than 2 x 10—3. A serious safety problem might arise from an unwanted decrease of the amount of absorber in medium 1.

Rather than using a difference between the neutronic properties of medium 1 and 2, it is possible to play on the relative geometrical arrangement of the two media. For example, consider that the first medium is a sphere with radius R1 surrounded by a spherical shell between R1 and R2 comprising medium 2. A neutron exiting medium 1 has, evidently, a unit probability of entering medium 2 so that!12 is given by equation (6.18). A neutron emitted from the inner surface of the shell has probability 1 — 1 — (R2/R2) of

entering medium 1. In the absence of medium 1, neutrons lost by medium 2 exit by the external surface of the shell. If the shell is not too thick, the number of neutrons crossing the inner surface of the shell should be approxi­mately equal to this last quantity. It follows that

The minimization of!21 implies a minimization of (k2l — k2)/k2l in agree­ment with other constraints. Typically, for breeder reactors, (k2l — k2)/k2l is close to 0.1. It follows that the condition on the product!12!21 implies that Rj/R2 < 0.2. With R1/R2 = 0.1, !12!21 — 5 x 10—4 and

n2F) = 500N0 (6.20)

for k1 = k2 = 0.95 and!12 = 1. In these conditions, if a neutron is created in medium 2, the final number of neutrons will be n1 = 0.25 and n2 = 25, so n1 + n2 = 25.25. The large amplification difference when neutrons are created in medium 1 and when they are created in medium 2 shows that very high neutron multiplication can be obtained in the case being discussed, the system remaining, however, safely far from criticality. This could give the possibility of reducing by almost one order of magnitude the power require­ment for the accelerator.

Figure 6.16 shows the result of a very simple Monte Carlo calculation which illustrates the preceding discussion, and shows how a very high multiplication can be obtained, while staying very far from criticality. The model reactor is made of a central plutonium sphere with a radius of 4.62 cm surrounded by a plutonium shell with an inner radius of 10 m and a thickness of 1.54 cm. Each single component is characterized by keff = 0.95. The very high values of k; for small i reflects the high value of p for pure plutonium. The sharp decrease is due to the large escape prob­ability of neutrons created in the inner sphere. After 20 generations, the multiplication process takes place essentially in the outer shell. The simulated value of ks = 0.997 is to be compared with the analytically calculated value of ks = 0.9964. Figure 6.16 also illustrates the consideration of section 4.1.3.

Finally Daniel and Petrov [132] have proposed the use of the difference in fissile concentration in zone 1 (booster) and zone 2 to obtain a high value of ks while keeping a reasonably small value of keff. They did a one-group

diffusion calculation for a two-zone fast reactor with a subcriticality p2 = 1 — keff2 = 0.03, for the external zone and kXl = 1.2, corresponding to keff1 = 0.98. For the spherical geometry, with a volume ratio between the two zones of 103, they obtained a booster gain of 3.6, allowing a corresponding decrease in the beam power.