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al reactions release energies on the order of a few evs e g the binding energy of the electron to hydrogen is 13 6 ev nuclear fission reactions typically release energies on the order of hundreds of millions of evs two typical fission reactions are shown below with average values of energy released and number of neutrons ejected u 235 neutron fission fragments 2 4 neutrons 192 9 mev pu 239 neutron fission fragments 2 9 neutrons 198 5 mev displaystyle begin aligned ce 235 u neutron text fission fragments 2 4 text neutrons 192 9 text mev ce 239 pu neutron text fission fragments 2 9 text neutrons 198 5 text mev end aligned 18 note that these equations are for fissions caused by slow moving thermal neutrons the average energy released and number of neutrons ejected is a function of the incident neutron speed 18 also note that these equations exclude energy from neutrinos since these subatomic particles are extremely non reactive and therefore rarely deposit their energy in the system reactor physics edit prompt neutron lifetime edit the prompt neutron lifetime l displaystyle l is the average time between the emission of a neutron and either its absorption or escape from the system 19 the neutrons that occur directly from fission are called prompt neutrons and the ones that are a result of radioactive decay of fission fragments are called delayed neutrons the term lifetime is used because the emission of a neutron is often considered its birth and its subsequent absorption or escape from the core is considered its death for thermal slow neutron fission reactors the typical prompt neutron lifetime is on the order of 10 4 seconds and for fast fission reactors the prompt neutron lifetime is on the order of 10 7 seconds 18 these extremely short lifetimes mean that in 1 second 10 000 to 10 000 000 neutron lifetimes can pass the average also referred to as the adjoint unweighted prompt neutron lifetime takes into account all prompt neutrons regardless of their importance in the reactor core the effective prompt neutron lifetime referred to as the adjoint weighted over space energy and angle refers to a neutron with average importance 20 mean generation time edit the mean generation time λ displaystyle lambda is the average time from a neutron emission to a capture that results in fission 18 the mean generation time is different from the prompt neutron lifetime because the mean generation time only includes neutron absorptions that lead to fission reactions not other absorption reactions the two times are related by the following formula λ l k e f f displaystyle lambda frac l k_ mathrm eff in this formula k e f f displaystyle k_ mathrm eff is the effective neutron multiplication factor described below effective neutron multiplication factor edit the effective neutron multiplication factor k e f f displaystyle k_ mathrm eff is most often quantified as the ratio of the rate of neutron production to the rate of neutron loss in a nuclear system and it is often described using the six factor formula k e f f rate of neutron production rate of neutron loss displaystyle k_ mathrm eff mbox rate of neutron production over mbox rate of neutron loss using k e f f displaystyle k_ mathrm eff and the prompt neutron lifetime l displaystyle l the following differential equation can be used to describe the time rate of change of the neutron population d d t n t k e f f 1 l n t displaystyle d over dt n t left k_ mathrm eff 1 over l right n t when solved for n t displaystyle n t this equation represents the neutron population n displaystyle n at any given time t displaystyle t given an initial neutron population n 0 displaystyle n 0 at t 0 displaystyle t 0 n t n 0 e k e f f 1 l t displaystyle n t n 0 e left k_ mathrm eff 1 over l right t when describing a nuclear reactor where neutron population is directly proportional to thermal power the following equation is used p p 0 e t τ displaystyle p p_ 0 e t tau where p displaystyle p is the reactor power at time t displaystyle t given an initial power p 0 displaystyle p_ 0 and τ displaystyle tau the reactor period the value of τ displaystyle tau can be calculated as τ l k e f f 1 displaystyle tau l over k_ mathrm eff 1 six factor formula edit the effective neutron multiplication factor k e f f displaystyle k_ mathrm eff can be described using the product of six probability factors that describe a nuclear system these factors traditionally arranged chronologically with regards to the life of a neutron in a thermal reactor include the probability of fast non leakage p f n l displaystyle p_ mathrm fnl the fast fission factor ε displaystyle varepsilon the resonance escape probability p displaystyle p the probability of thermal non leakage p t n l displaystyle p_ mathrm tnl the thermal utilization factor f displaystyle f and the neutron reproduction factor η displaystyle eta also called the neutron efficiency factor the six factor formula is traditionally written as follows k e f f p f n l ε p p t n l f η displaystyle k_ mathrm eff p_ mathrm fnl varepsilon pp_ mathrm tnl f eta where p f n l displaystyle p_ mathrm fnl describes the probability that a fast neutron will not escape the system without interacting the bounds of this factor are 0 and 1 with a value of 1 describing a system for which fast neutrons will never escape without interacting i e an infinite system also written as l f displaystyle l_ f ε displaystyle varepsilon is the ratio of total fissions to fissions caused only by thermal neutrons fast neutrons have a small probability to cause fissions in uranium specifically uranium 238 the fast fission factor describes the contribution of fast fissions to the effective neutron multiplication factor the bounds of this factor are 1 and infinity with a value of 1 describing a system for which only thermal neutrons are causing fissions a value of 2 would denote a system in which thermal and fast neutrons are causing equal amounts of fissions p displaystyle p is the ratio of the number of neutrons that begin thermalization to the number of neutrons that reach thermal energies many isotopes have resonances in their capture cross section curves that occur in energies between fast and thermal if a neutron begins thermalization i e begins to slow down there is a possibility it will be absorbed by a non multiplying material before it reaches thermal energy the bounds of this factor are 0 and 1 with a value of 1 describing a system for which all fast neutrons that do not leak out and do not cause fast fissions eventually reach thermal energies p t n l displaystyle p_ mathrm tnl describes the probability that a thermal neutron will not escape the system without interacting the bounds of this factor are 0 and 1 with a value of 1 describing a system for which thermal neutrons will never escape without interacting i e an infinite system also written as l t h displaystyle l_ th f displaystyle f is the ratio of number of thermal neutrons absorbed in by fissile nuclei versus the number of neutrons absorbed in all materials in the system this factor describes the efficiency of thermal neutron utilization in the system hence the name thermal utilization factor the bounds of this factor are 0 and 1 with a value of 1 describing a system for which the entire system is made of fissile nuclei i e thermal neutrons can only react with fissile materials similarly a value of 0 5 describes a system for which reactions with fissile and non fissile nuclei are equal for a conventional nuclear power reactor this factor is the only one that can be directly controlled by the operator with manipulations to the control rods you can increase the amount of neutrons being absorbed in non fissile nuclei while simultaneously decreasing the amount of neutrons absorbed in fissile nuclei η displaystyle eta describes the probability that a neutron absorbed will cause a fission reaction this factor describes the behavior of the fissile material specifically if a neutron is absorbed how likely is it to cause a fission and how many neutrons does the fission produce the multiplication factor is sometimes calculated with a simplified four factor formula which is the same as described above with p f n l displaystyle p_ mathrm fnl and p t n l displaystyle p_ mathrm tnl both equal to 1 and is used when an assumption is made that the reactor is infinite in that neutrons are very unlikely to leak out of the system this value k displaystyle k_ infty is often used in safety evaluations of reactor designs criticality edit because the value of k e f f displaystyle k_ mathrm eff is directly related to the time rate of change of the neutron population in a system it is convenient to classify the state of the nuclear system with regards to the critical value of the neutron population equation the point at which the behavior of a nuclear system shifts is when k e f f displaystyle k_ mathrm eff is exactly equal to 1 this point is called criticality and describes a system in which the production rate and loss rate of neutrons is exactly equal when k e f f displaystyle k_ mathrm eff is less than or greater than one the terms subcriticality and supercriticality are used respectively to describe the system k e f f 1 displaystyle k_ mathrm eff 1 subcriticality the neutron population in the system is decreasing exponentially k e f f 1 displaystyle k_ mathrm eff 1 criticality the neutron population is maintaining a constant value k e f f 1 displaystyle k_ mathrm eff 1 supercriticality the neutron population in the system is increasing exponentially in a practical nuclear system like a fission reactor if criticality is intended it is likely that k e f f displaystyle k_ mathrm eff will actually oscillate from slightly less than 1 to slightly more than 1 primarily due to thermal feedback effects the neutron population when averaged over time appears constant leaving the average value of k e f f displaystyle k_ mathrm eff at around 1 during a constant power run both delayed neutrons and the transient fission product burnable poisons play an important role in the timing of these oscillations reactivity edit the value of k e f f displaystyle k_ mathrm eff is generally not easy to calculate or use practically instead a system s 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 ρ displaystyle rho as a function of the neutron multiplication factor k e f f displaystyle k_ mathrm eff ρ k e f f 1 k e f f displaystyle rho k_ mathrm eff 1 over k_ mathrm eff or when comparing the reactivity differences between two nuclear systems with multiplication factors k 1 displaystyle k_ 1 and k 2 displaystyle k_ 2 δ k k k 2 k 1 k 1 k 2 displaystyle delta k over k k_ 2 k_ 1 over k_ 1 k_ 2 for most systems the reactivity ρ displaystyle rho has a very small range making any value difficult to qualitatively describe or interpret like k e f f displaystyle k_ mathrm eff often it is expressed in units of δ k k displaystyle delta k k per cent mille or almost solely in the united states with the derived units of dollars and cents note that ρ displaystyle rho is often also expressed as δ k k displaystyle delta k k δ k k ρ 100 displaystyle delta k over k rho times 100 p c m ρ 10 5 displaystyle mathrm pcm rho times 10 5 ρ β e f f displaystyle rho over beta _ mathrm eff the value β e f f displaystyle beta _ mathrm eff 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 β displaystyle beta which is the fraction of neutrons in the system that are 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 β e f f displaystyle beta _ mathrm eff subcritical multiplication edit when a nuclear system is subcritical an introduction of neutrons to the system will result in that population decaying away however if neutrons are introduced at a constant rate i e from a neutron source a nuclear system can appear critical while not actually maintaining true criticality this is called source criticality and due to a phenomenon called subcritical multiplication the neutron population equation can be modified to be written as follows d d t n t k e f f 1 l n t s t displaystyle d over dt left n t right left k_ mathrm eff 1 over l right n t s t this is a much more difficult differential equation to solve in this case we assume that all neutrons are from the source and that each generation of neutrons is of equal magnitude in this case we can approximate using a geometric series n t n 0 n 0 k e f f n 0 k e f f 2 n 0 k e f f 3 n 0 i 0 k e f f i 1 1 k e f f n 0 displaystyle n t n_ 0 n_ 0 k_ mathrm eff n_ 0 left k_ mathrm eff right 2 n_ 0 left k_ mathrm eff right 3 cdots n_ 0 sum _ i 0 infty k_ mathrm eff i left 1 over 1 k_ mathrm eff right n_ 0 graphical representation of subcritical multiplication we take the above equation and define a new factor m displaystyle m called the subcritical multiplication factor m 1 1 k e f f displaystyle m 1 over 1 k_ mathrm eff multiplying this factor by the source strength in neutrons sec will give the stable neutron population as long as k e f f displaystyle k_ mathrm eff is known n s 0 m displaystyle n_ infty s_ 0 times m much more commonly this equation is used to estimate k e f f displaystyle k_ mathrm eff as the stable neutron population is easy to measure but it is difficult to know the strength of a neutron source to get around this as a system approaches criticality m displaystyle m approaches infinity therefore it is much more practical to measure 1 m displaystyle 1 m which approaches zero as a system approaches criticality 1 m displaystyle 1 m can be approximated by the ratio of count rates before and after a reactivity addition c r 0 c r 1 m lim c r 0 c r 0 k e f f 1 displaystyle cr_ 0 over cr approx 1 over m quad therefore quad lim _ cr_ 0 cr rightarrow 0 left k_ mathrm eff right 1 most neutron sources are a combination of an alpha particle emitter and beryllium beryllium 9 the only naturally occurring stable isotope of beryllium is capable of emitting a neutron when an alpha particle is absorbed this α n displaystyle alpha n binary reaction is what generates neutrons the most common of these are americium beryllium ambe plutonium beryllium pube and polonium beryllium pobe sources be 4 9 α 2 4 n 0 1 c 6 12 displaystyle ce 9 _ 4 be 4 _ 2 alpha rightarrow 1 _ 0 n 12 _ 6 c antimony 124 is also used in conjunction with beryllium to generate neutrons as the gamma ray emitted by antimony 124 is at a uni...