Editing Nuclear chain reaction (section)

==== Reactivity ====
The value of <math>k_{\mathrm{eff}}</math> is generally not easy to calculate or use practically. Instead, a system's [[Reactivity of a nuclear reactor|reactivity]] is quantified instead. The reactivity of a nuclear system is qualitatively described as the departure from criticality. The equation below describes the pure reactivity <math>\rho</math> as a function of the neutron multiplication factor <math>k_{\mathrm{eff}}</math>:

<math>\rho = {k_{\mathrm{eff}} - 1 \over k_{\mathrm{eff}}}</math>

or when comparing the reactivity differences between two nuclear systems with multiplication factors <math>k_1</math> and <math>k_2</math>,

<math>{ \Delta k \over k } = {k_2 - k_1 \over k_1 k_2}</math>

For most systems, the reactivity <math>\rho</math> has a very small range, making any value difficult to qualitatively describe or interpret, like <math>k_{\mathrm{eff}}</math>. Often, it is expressed in units of <math>%\Delta k/k</math>, [[per cent mille]], or (almost solely in the United States) with the derived units of [[Dollar (reactivity)|dollars and cents]]. Note that <math>\rho</math> is often also expressed as <math>\Delta k / k</math>

<math>% { \Delta k \over k} = \rho \times 100</math>

<math>\mathrm{pcm} = \rho \times 10^5</math>

<math>$ = {\rho \over \beta_{\mathrm{eff}}}</math>

The value <math>\beta_{\mathrm{eff}}</math> is known as the effective delayed neutron fraction, and it describes the fractional contribution of delayed neutrons to the fission rate of the system and is quantified as the ratio of the total number of fissions caused by delayed neutrons to the total number of fissions in a system. This number is slightly different than the delayed neutron fraction <math>\beta</math>, which is the fraction of neutrons in the system that are [[Delayed neutron|delayed]], because delayed neutrons are generally born at lower energies, and thus are easier to thermalize, meaning they are more likely to cause a fission than a prompt neutron. This weighting effect is given in the derivation of <math>\beta_{\mathrm{eff}}</math>.