Consider the nuclear reaction
Where = kinetic energy of the projectile x
= rest energy of the projectile x
And similarly and are kinetic energy and restmass energy of product y and and are kinetic energy and restmass energy of product y
The target nucleus is assumed to be at rest
The quantity Q represents the difference between the KE of product of reactions and that of reactant
= Rest mass energy of reactant – Rest mass energy of product …(2)
The quantity Q is called the energy balance of the reaction or Q-value of the reaction.
If Q is positive then nuclear reaction is called exoergic reaction
If Q is negative then nuclear reaction is called endoergic reaction
The term in equation -(1) represents the recoil energy of the product nucleus. It is usually small and hard to measure but can be eliminated by taking into account the conservation of momentum.
The Q Equation : The analytical relationship between the kinetic energy of the projectile and outgoing particle and the nuclear distintegration energy Q is called the Q-equation
Applying law of conservation of momentum
This is known as Q-value equation
Solution of the Q-equation
Q-value equation can be seen as quadratic equation in . So, solution can be put as
when is real and positive, the reaction is energetically possible
Endoergic Reaction : Here Q < 0
For example :
has Q = -10 MeV
has Q = -MeV
(i) For very low energy projectiles
So, is imaginary and hence, no reaction occurs. is insufficient to start the reaction.
(ii) Threshold Energy :
The smallest value of projectile energy (bombarding energy) at which an endoergic reaction can take place is called the threshold energy of that reaction.
Reaction becomes possible when is large enough to make
has its minimum possible value at which is
The value of is much less than the masses of particles in a nuclear reaction and so
Exoergic Reactions : Here Q > 0
(i) For very Low energy projectiles e.g. thermal neutrons
V = 0,
is same for all value of