The general fusion reaction, Eq.(1.7) and Fig. 1.2, may be more completely characterized by noting that an unstable intermediate state may be identified in nuclear reactions. That is, we should write

a + b —> (яЬ) —^ d + є + Qafj

where (ab) identifies a complex short-lived dynamic state which disintegrates into products d and e. The energetics are determined according to nucleon kinetics analysis, with nuclear excitation and subsequent gamma ray emission known to play a comparatively small role in fusion processes at the energies of interest envisaged for fusion reactors.

Two-body interactions can be examined from various perspectives. For example, Newton’s familiar law of gravitational attraction applies to any pair of masses ma and ть to yield a force

(2.2)

effective on particle a. Here, G is the universal gravitational constant and r = ra — rb is the displacement vector between the two interacting particles, while r denotes its absolute value. While this force expression is universal, a simple calculation will show that for nuclear masses of common interest, this force is significantly weaker than the electrostatic and nuclear forces associated with nuclides and hence can be neglected.

The important electrostatic force between two isolated particles of charge qa and qb separated by a distance r in free space is determined by Coulomb’s law, given by

(2.3)

for the electrostatic force felt by particle a; here £<, is the permittivity of free space and the factor An is extracted from the proportionality constant by reason of convention. This force-repulsive for like charges and attractive for unlike charges-is of considerable importance in fusion.

From the definition of work and the phenomenon of energy stored in a conservative field, the work done in moving a particle of charge qa from a sufficiently distant point to within a distance r of a stationary charge of magnitude qb, is the potential energy associated with the resultant charge configuration. Specifically, this is given by

Г

U(r)=jFc, a(r’)dr’

і dnEo (r’f

_ 1 ЯаЯь

dnEo r

subject to the restriction that the particle distance of separation r satisfies r > Ra + Rb where Ra and Rb are the equivalent radii of the two charged particles. For nuclides of like charge, the potential energy at approximately the distance of "contact" R0 = Ra + Rb, is called the Coulomb barrier and, in view of Eq.(2.4), is given by

1 <la<lb

4nEo(Ra + Rb)

On the basis of electrostatic force considerations only, this then is the minimum kinetic energy an incident particle would have to possess in order to overcome electrostatic repulsion and come close enough to another particle for the short — range nuclear forces of attraction to dominate. For deuterium ions, this energy can be calculated to be about 0.4 MeV, depending upon the precise value for R0. A useful approximation is R0 « Rp(Aa/3 + A{,/3) where Rp = (1.3-1.7)Xl015 m denotes the radius of a proton which cannot be assigned a definite edge for quantum mechanical reasons.

Consideration of quantum mechanical tunneling provides for a non-vanishing probability of penetrating the Coulomb barrier with energies less than U(R0). The probability for this penetration varies as

Pr( tunneling ) °С — exp — Y

Vr V Vr )

where vr is the relative speed of the moving particles and у is a constant. Thus,
even at very low energy, a nucleus possesses a small, though finite, probability of compound formation with another nucleus. This compound can decay into fusion products and hence, some fusion reactions will also occur at room temperature, though at an insignificant rate.

At sufficiently small distances, r < R0, the attractive strong nuclear force dominates and a compound nuclear state is formed. The kinetic energy of the initiating particles together with the resultant nuclear potential energy is then shared by all the nucleons. Nuclear stability considerations thereupon determine if and how the nucleus disintegrates. Figure 2.1 provides a graphical representation of these effects.