The recognition that the Coulomb barrier for light ion fusion is in the 0.4 MeV range for the lightest known nuclides-p, d, and t-suggests that one approach to the attainment of frequent fusion reactions would be to heat a hydrogen gas up to a temperature for which a sufficient number of nuclei possess energies of relative motion in excess of U(Ro), Eq.(2.5). However, because of the tunneling effect, such excessive heating is not required and substantial rates of fusion reactions are achieved even when the average kinetic energy of relative motion of the reactant nuclei is in the tens-of-keV range. Note, however, that even providing such reduced conditions in a gas will be associated with heating up to temperatures near ~ 10s K, for which, in recognition of the ionization potential of hydrogen being only 13.6 eV, the gas will completely ionize. The result therefore is an electrostatically neutral medium of freely moving electrons and positive ions called a plasma. With the average ion energy in a fusion reactor plasma thus allowed to be substantially less than the Coulomb barrier, the energy released from fusion reactions must exceed-for a viable energy system-the total energy initially supplied to heat and ionize the gas, and to confine the plasma thus produced. In practice, this means that when a sufficiently high plasma temperature has been attained, we have to sustain this temperature and confine the ions long enough until the total fusion energy released exceeds the total energy supplied. In subsequent chapters, we will consider some details of the relevant phenomena and processes and identify parametric descriptions of energy balances.

On the basis of the above considerations then, it is apparent that selected aspects of the classical Kinetic Theory of Gases, augmented by electromagnetic force effects, can be used as a basis for the study of a plasma in which fusion reactions occur. Thus, for the case of N atoms of proton number Z„ complete ionization yields Nj ions and, in the case of charge neutrality, Z, Ni = Ne electrons; that is

N^Ni+ Ne= Ni + Zi Ni ■ (2.7)

For hydrogenic atoms, Z; = 1 and hence Ne = N;.

Often, the expression "Fourth State of Matter" is also assigned to such an assembly of globally neutral matter containing a sufficient number of charged particles so that the physical properties of the medium are substantially affected by electromagnetic interactions. Indeed, such a plasma may also exhibit collective behaviour somewhat like a viscous fluid and also possesses electrostatic characteristics of specified spatial dimension.

For a state of thermodynamic equilibrium, the Kinetic Theory of Gases asserts that the local pressure associated with the thermal motion of ions and electrons is given by

Pi = jN jiw,- v? (2.8a)

and

Pe = j Nemev2e (2.8b)

where the subscripts і and e refer to the ions and electrons respectively, and N( > refers to the subscript-indicated particle population densities; note that it is the average of the squared velocity which appears as the important factor. The average kinetic energy of the electrons and ions can be introduced by simple algebraic manipulation of the above equations:

= ^NiEi (2.9a)

and

= (2.9a)

Similar expressions may be written for the neutral particles in a plasma.

Accepting the kinetic theory of gases as a sufficiently accurate description allows for the use of well-known distribution functions. Implicit in this assumption is that the plasma under consideration is sufficiently close to thermodynamic equilibrium and that processes such as inelastic collisions, boundary effects and energy dependent removal of particles are of secondary importance.

The distribution functions for the particles of interest include dependencies on space, time and either velocity, speed or kinetic energy. Here we take a stationary ensemble of N* particles uniformly distributed in space-either neutrals, ions or electrons-allowing us to write

N(S)=N*M($) (2.10)

where £ is one of the independent characteristic variables of motion-v, v, or E — and M(Ej describes how the particles are distributed over the domain of this variable. Hence, N(EJ is the distribution function of the ensemble of relevant particles in ^-space and the symbol M() represents a normalized distribution function for the variable £ such that

J M( t, )d^ = 1 (2.11a)

with the integration performed over the entire definition range of the variable considered, i. e.

Jm ()d = jM(v)dv =Jm (E)dE =1.

—«> о 0

Note that the particle number of the ensemble is given by

]n(^=N )m(^=n •

For a gas or, for the case of interest here, a plasma in thermodynamic equilibrium and in the absence of any field force effect, its particles of mass m moving in a sufficiently large volume follow the Maxwell-Boltzmann velocity distribution function given by

Here к is the Boltzmann constant and T is the absolute temperature for the

Finally, the corresponding distribution of particles in kinetic energy space E is described by the Maxwell-Boltzmann distribution
0 < E < «>.	(2.15)
Velocity, vx
Fig. 2.2: Schematic depiction of the Maxwell-Boltzmann distribution functions for vx, v and E as independent variables; in each case, Ti < T2.