Conservation conditions are fundamental aids in the quantitative assessment of nuclear processes. Of paramount relevance here is the joint conservation of nucleon number and energy associated with an initial ensemble of interacting nuclear species of type a and type b which, upon a binary collisional interaction, yield two particles of type d and e:

a + b —» d + e. (1.7)

Note that the details of the highly transient intermediate processes are not listed— only the initial reactants and the final reaction products are shown.

An accounting of all participating nucleons is aided by the notation ^nuclear species named X containing^

A nucleons of which Z are protons J

and therefore yields the more complete statement for the reaction of Eq.(1.7) in a form which lists the number of nucleons involved in this nuclear rearrangement: £Xa + bXb^Xd+£X’. (1.9)

Recall that Aj is the sum of Zj protons and Nj neutrons in the nucleus, Aj = Zj + Nj. Nucleon number conservation therefore requires

Aa + Аь — Ad + Ae (1-Ю)

and, similarly for charge conservation we write

Za + Zb = Zd + Ze • (1-11)

Characterization of energy conservation for reaction (1.7) follows from the knowledge that the total energy E of an ensemble of particles is given by the sum of their kinetic energies Ek and their rest mass energies Er= me2; here m is the rest mass of the particle and c is the speed of light in free space. The total energy of an assembly of particles, for which we add the asterisk notation, is

therefore

E = ^(Ek, j + Er, j)=^(Ek, j + mjc2) • (1-12)

j j

For the nuclear reaction of Eq.(1.7) we write the total energy-which must be conserved-as

* *

Ebefore ~ Eafter

and hence

(Ek, a + maC ) + (Ekj? + тьс )-(Ek, d + mdc )+(Ek, e + mec ) ■

Rearrangement of these terms yields

(Ek, d + Ek, e)'(Ek, a + Ekj>)= [(ma + m)-(md + me)]c2 ■

This important equation relates the difference in kinetic energies-before and after the collision-to their corresponding differences in rest masses; thus, as shown, a change in kinetic energy is related to a change in rest masses. Particle rest masses have been measured to a very high degree of accuracy allowing therefore the ready evaluation of the right-hand part of this equation. This defines the Q-value of the reaction

Qab= [(ma + rm)-(md + me)]c2 = -[(md + me)-(ma + ть)]c2 ^

and represents the quantity of energy associated with the mass difference before and after the reaction. Hence, we may write more compactly

Qab = (-^)abC2 (115b)

where Am is the mass decrement (i. e. Am = mafter — mbefore) for the reaction. Evidently, Qab is positive if (Ат)аь < 0 and negative otherwise; the former case­involving a decrease of mass in the process-constitutes an exoergic reaction and the latter may be termed endoergic. Further, for the case of Qab < 0, the kinetic energy of the reaction-initiating particle must exceed this value before a reaction can be induced; that is, a threshold energy has to be overcome before the reaction will proceed.

Equation (1.15b) is a form of the famous relation

E = me2 (1.16)

and asserts-as first proposed by Einstein-that matter and energy are equivalent. As a consequence, we may assert that if processes occur which release energy of amount E, then a corresponding decrease in rest mass of amount (-Am) must have taken place.