## Model calculations

Using this formalism, we have determined the fuel volume necessary to obtain ks = 0.98 as a function of R1, for a reactor using a U-Pu fuel with the following characteristics.

вҖў The relative volumic fractions were 0.5 for lead, 0.08 for iron, 0.39 for the fuel and 0.03 for vacuum.

вҖў The fuel was 88% uranium 238 and 12% plutonium in molar fractions. Both uranium and plutonium were in the dioxide form.

вҖў The relative amounts of plutonium isotopes were 62.7% 239Pu, 24.3% 240Pu, and 13% 241Pu, corresponding to the concentration of spent PWR reactor fuel.

вҖў The cross-sections were one-group cross-sections extracted from the MCNP Monte Carlo calculation which is described in the next section.

The results are shown in figure 10.1. The values shown in the figure correspond to a 1 MW proton beam, each proton being assumed to produce 30 neutrons. For an internal radius R1 = 0.15 m, the maximum neutron flux reached is 3.6 x 1015n/cm2/s, corresponding to a maximum specific power of 280W/cm3. The total thermal power was 120 MW. The Internal radius fi, metres Figure 10.1. Variations of the volume, maximum flux, and ratio of maximum to minimum flux, for a three-zone spherical reactor, as a function of the internal radius of the fuel zone. The multiplication factor was ks = 0.98. The neutron source was a 1 mA 1 GeV proton beam producing 30 neutrons per proton. The y axis labels are shown in the inset key.

volume of the fuel zone is around 0.7 m3 for a fuel weight of 3.5 tons. This configuration can be considered as the smallest possible demonstration design able to:

вҖў reach a multiplication factor of 0.98

вҖў approach the maximum acceptable specific power

вҖў reach representative neutron fluxes, so that fuel evolution can be studied in realistic conditions.

The 1 m internal radius could be representative of an energy producing reactor. The maximum specific power is 50W/cm3 for the 1mA beam. A 10 mA beam would lead to an acceptable 500 W/cm3 specific power and a 1200 MWth reactor, for a fuel zone volume of 2.7 m3, and a fuel weight of 14 tons. In this configuration, the ratio of maximum to minimum flux is only 1.25, a very reasonable value. The thickness of the fuel zone is less than 20 cm.

A realistic reactor could be neither homogeneous nor spherical. The fact that the proton beam has to penetrate inside the reactor leads to truncated cylindrical shapes, rather than spherical, which are not apt for a simple analytical treatment, even in the one-group diffusion approach.

More realistic calculations are necessary, and we give an example of one of these.