Nuclear Reactions

Nuclei can be transformed into others of a different kind by interactions with each other. Since nuclei are all positively charged, a high-energy collisions in necessary between two nuclei if they are to get close enough together to react. Because it has no charge, a neutron can initiate a nuclear reaction even if it is moving slowly. In any nuclear reaction, the total number of neutrons and the total number of protons in the products must be equal to the corresponding total numbers in the reactants.

Nuclei are bombarded with known projectiles and the final products are observed. Isotopes of nuclei with atomic numbers as high as Z ≈ 80 are used as projectiles. Incident particles are many, some of them are:

Helium-4 (alpha particle) Notation Particle n neutron proton deuteron triton Helium-3 Helium-4 (alpha particle)

A nuclear reaction is written as: Target Reaction Products Projectile Or alternatively

Q-value Q= (mass of reactants -mass of products) c Reaction Q-value represents the energy balance, or mass change, that occurs between a projectile and a target in a nuclear interaction and the products of the reaction Q= (mass of reactants -mass of products) c When Q> 0, energy is released as kinetic energy –exoergic or exothermic When Q<0,energy must be supplied for the reaction to occur -endoergic or endothermic 2

Low-energy Reaction kinematics It can be assumed that velocities of the nuclear particles are sufficiently small that non-relativistic kinematics can be used. We consider a projectile x moving with momentum Px and kinetic energy Kx. The target is at rest, and the reaction products have momenta Py and PY, and kinetic energies Ky and KY. The particles y and Y are emitted at angles θy and θY with respect to the direction of the incident beam. We can also assume that the resultant nucleus Y is not observed in the laboratory

( usually loses all its kinetic energy by collisions with other atoms) if it is a heavy nucleus, moving relatively slowly, it generally stops within the target. As we did in the case of radioactive decay, we use energy conservation to compute the Q value for this reaction (assuming X is initially at rest . Using energy conservation to compute the Q value for this reaction (assuming X is initially at rest)

initial energy = final energy The m’s in the above equation represent the nuclear masses of the reacting particles. Adding equal numbers of electron masses to each side of the equation, and neglecting the electrons binding energies, the nuclear masses become atomic masses with no additional corrections needed. Rewriting the equation, yields:

The rest energy difference between the initial particles and final particles is defined to be the Q value of the reaction as explained before:

Laboratory and Center-of-Mass systems Experimental nuclear reactions are often analyzed in what is called the center-of-mass system. This system moves at constant velocity with respect to the laboratory system in such a manner that in it the colliding particles (and final particles) have zero total momentum. If the target nucleus is at rest in the laboratory system, the velocity vcm of the centre-of-mass system will be along the direction of the incident bombarding particle.

Therefore, with respect to the centre-of-mass, the magnitudes of the velocity of the target nucleus vcm and the incident particle v =v-vcm are respectively (in a non-relativistic treatment). Here v is the velocity of the incident particle (threshold speed) as measured in the laboratory, requiring the sum of the momenta of the target nucleus (mass m(X)) and the incident particle (mass m(x)) to be zero in the centre-of-mass system, we obtain the centre-of-mass velocity system from

m(x) m(X) m(x) m(X) Before Collision v v =v-vcm vcm Laboratory System Centre-of-mass System

Where (m(X)+m(x)) Vcm is the momentum of the centre of mass in the laboratory System . From the above equations, the target nucleus and incident particle have, before the reaction takes place, respective velocities in the center-of-mass system of:

We must conserve total relativistic energy K+mc in the reaction, here we restrict our discussion to small velocities v<<c so that the classical expression for the kinetic energy can be used. Energy conservation in the centre-of-mass frame gives: 2

From the above and previous equations, we can find the threshold kinetic Energy (in the laboratory reference frame) of the incoming particle and its relation to the Q value of the endothermic reaction.

Threshold energy is the kinetic energy that the incident particle must have for the reaction to be possible.

Radioisotope Production in Nuclear Reactions A stable (nonradioactive) isotope X is irradiated with the particle x to form the radioactive isotope Y; the outgoing particle y is of no interest and is not observed. The activity of the isotope Y that is produced from a given exposure to a certain quantity of the particle x for a certain time t. Let R represent the constant rate at which Y is produced; this quantity is related to the cross section and intensity of the beam of x as explained before.

The rate R at which Y is formed is identical to the rate R at which y is emitted. In a time interval dt, the number of Y nuclei produced R dt. Since the isotope Y is radioactive, the number of nuclei of Y that decay in the interval dt is λNdt, where λ is the decay constant and N is the number of Y nuclei present. The net change dN in the number of Y nuclei is:

The solution is: The activity is:

For large irradiation times t>>t1/2,this expression approaches the constant value R. When t is small compared with the half-life t1/2, the activity increases linearly with time: