As seen from equation (6.11), the number of secondary neutrons is kN0/(1 — k). Each of these neutrons is produced by a fission (we disregard (n, xn) reactions), which itself produces v neutrons. Thus, the number of secondary fissions in the system is kN0/v(1 — k). Since each fission releases about 0.18 GeV energy,[28] the thermal energy produced in the medium will be 0.18kN0/v(1 — k). This energy has to be compared with the energy of the incident protons Ep to define an energy gain of the system:

G — v(1 — k)Ep — T~—k ■ (4:3)

The CERN FEAT experiment [122] gave a constant value of G0k — 3, for incident proton energies larger than 1 GeV and for a uranium target.

The proton beam is produced with a finite efficiency which is the product of the thermodynamic efficiency for producing electricity from heat (typically 40% in foreseen reactors^) by the acceleration efficiency. For high intensity accelerators, most of the power is used for the high-frequency cavities, at variance with low-intensity accelerators where most of the power is spent in the magnetic devices. High intensities, therefore, are expected to allow 40% efficiencies (see Appendix III) [2, 76]. Finally, the total efficiency for proton acceleration is expected to be in the vicinity of 0.16. This leads to a minimum value of the multiplication factor for obtaining a positive energy production (ignition), km — 0.68. For a value k — 0.98 a net energy gain of 16 is achieved.