## Four Factor Formula:

Four factor formula is defined as-

k = ∈pfη Where, k = Effective multiplication factor. ∈ = Fast fission factor. p = resonance escape probability. f = Thermal utilisation factor. η = Number of fast neutrons released/fission in next generation. |

(i) Consider a system with enriched uranium and moderator.

(ii) Let one thermal neutron induced fission and result in the release of ‘n’ fast neutrons.

(iii) Let out of these ‘n’ fast neutrons, some will cause fission in the uranium nucleus and additional fast neutrons are produced; say by an amount given by factor ‘∈’ called fast fission factor.

(iv) Therefore, the number of fast neutrons is n∈. But for those fast neutrons, the fission cross-section is low for U^{238}. Thus it is necessary to convert fast neutrons into slow neutrons (using a suitable moderator).

(v) During this process of slowing down the energies, some neutrons may be captured by U-nucleus (called resonance absorption).

(Vi) Let ‘p’ be the probability of neutrons survived i.e. resonance escape probability.

(vii) Therefore, the number of neutrons surviving and reaching to thermal energies i.e. ‘n∈p’. But out of these, only fraction ‘f’ is used for fission called thermal utilisation factor. (Other thermal neutrons are absorbed by the moderator, coolant and other materials in the reactor).

(viii) Therefore number of thermal neutrons (and hence a number of fission reactions) are n∈pf.

(ix) Now let η be the number of fast neutrons released/fission,

∴ total number of fast neutrons = n∈pfη |

(x)

∴ k = n∈pfη/n = ∈pfη |