The fusion reaction rate density expression of Eq.(2.23) is very restrictive since all particles were taken to possess a constant speed and their motion was assumed to be monodirectional. However, the general case of an ensemble of particles possessing a range of speeds and moving in various directions can be introduced by extending Eq.(2.23) to include a summation over all particle energies and all directions of motion. The integral calculus is ideally suited for this purpose requiring, however, that we redefine some terms. Letting therefore the particle densities be a function of velocity v endows them with a range of energies and range of directions; that is, we progress from simple particle densities which give the number of particles per unit volume, to distribution functions describing how many particles in a considered position interval move with a certain velocity, according to

Na^> Na(Va)= Na Fa(Va)	(2.25a)
and
Nb^Nb(b)-NbFb(b)-	(2.25b)
The terms which have replaced the previous particle densities are distributions in
the so-called position-velocity phase space. Thus, functions Fa(va) and Fb(Vb) satisfy the normalization	the velocity distribution
II a > ‘■’O ^3 a > a	(2.26a)
’o as well as
J Fb( Vb)d3 Vb = 1 ■	(2.26b)

vb Here d3V() is to indicate integration over the three velocity components. Hence, though we show here only one integral, the implication is that for calculational purposes there will be as many integrals as there are scalar components for each of the vectors va and Vb.

Relative Speed, vr (106 m/s)

Fig. 2.4: Fusion cross section for a deuterium beam incident on a tritium target.

The relative speed of two interacting particles at a point of interest is, by its usual definition, given as

Vr = Va — Vi •

(2.28)

By first impressions, this double integral appears formidable, particularly when it is realized that the vectors va and vb possess, in general, three components so that a total of six integrations are required. However, aside from any algebraic or numerical problems of evaluation, this integral contains two important physical considerations. First, the cross section Ofa(lva — vbl) must be known as a function of the relative speed of the two types of particles, and second, the distribution functions Fj(vj) must be known for both populations of particles.

We note however that Eq.(2.28) possesses all the properties of an averaging process in several dimensions; that is, it represents averaging the product oab(lva — vb)lva — vbl with two normalized weighting functions Fa(va) and Fb(vb) over all velocity components of va and vb. Such averaging yields the definition

< C7V >ab ~ J ^Oab^ya-yi>f[ya-ybFa(ya)Fb(vb)d3 Vad.3 Vb — (2.29)

This parameter, here named sigma-v (pronounced "sigma-vee"), is often also called the reaction rate parameter. Note the implicit dependence on temperature via the distribution functions Fa(va) and Fb(vb) so that <Ov>ab is a function of temperature.

The reaction rate density involving two distinct types of particles a and b, Eq.(2.28), is therefore written in compact form as

(2.30)

This sigma-v parameter for the case of d-t fusion under conditions in which both the deuterium and tritium ions possess a Maxwellian distribution, that is F()() —» M ()( ) in Eq.(2.29), and where both species possess the same temperature, is

depicted in Fig.2.5. Note that generally <Ov>ab will also be a function of space and time because the velocity distributions may also depend upon these variables.

We make two additional comments about the reaction rate density expression, Eq.(2.30). First, the assumption of Maxwellian distributions, i. e.

Appendix C provides a tabulation of this and other <ov>ab parameters.

F()() —» M()() as discussed in Sec.2.3, is very frequently made in tabulations of sigma-v; the reason for this is because many approaches to the attainment of fusion energy rely upon the achievement of plasma conditions that are close to thermodynamic equilibrium. For cases where equilibrium conditions do not exist, the appropriate distribution functions Fa(va) and Fb(vb) must be determined and used in Eq.(2.29). For example, in some experiments, deuterium beams are injected into a tritiated target or into a magnetically confined tritium plasma to cause fusion by "beam-target" interactions. In such cases, one substitutes Fd(vd) by a delta distribution function at the velocity vb(t) characteristic of the instantaneous velocity of the beam ions slowing down in the plasma. However, Ft(vt) for the plasma target could well be assumed to be Maxwellian at the temperature of the tritium plasma. Then the averaged product of (7 and vr is often called the beam-target reactivity < ov >d! for d-t fusion and is displayed in Fig.2.6; it appears to be a function of both the target temperature and the instantaneous energy of the slowing-down beam deuterons.

Fig. 2.5: Sigma-v parameter for d-t fusion in a Maxwellian-distributed deuterium and tritium plasma.

Fig. 2.6: Beam target sigma-v parameter for the case of deuterium injection into a tritium target of various temperatures.

Secondly, Eq.(2.30) assumes that the а-type and b-type particles are different nuclear species; the case when they might be indistinguishable is treated in a subsequent chapter where, in addition to d-t fusion, the case of d-d fusion is also considered.

A demonstration of the occurrence of fusion reactions is readily accomplished in a simple experiment employing a small accelerator which bombards a tritiated target with deuterium ions of such a high energy that during slowing down in the target they pass through the most favourable energy range for fusion; consider, in particular, the tritium plasma target of Fig.2.6 for this purpose. The appearance of neutrons and alphas as reaction products at the proper energies is then the proof of fusion events. The objective of fusion energy research and development is, however, the attainment of a sufficiently high fusion reaction rate density under controlled conditions subject to the overriding requirement that the power produced be delivered under generally acceptable terms. This is a considerable challenge and to this end a variety of approaches have been and continue to be pursued.

Problems

2.1 Calculate the ratio of gravitational to electrical forces between a deuteron and a triton.

2.2 Determine the Coulomb barrier for the nuclear reactions d-t, d-h, and p — nB.

2.3 Confirm the correctness of Eqs.(2.17) and (2.19).

2.4 For Maxwellian distributed tritons at 9 keV, calculate

(a) the average kinetic energy,

(b) the average speed, and

(c) the kinetic energy derived from the average speed of (b). Compare the energies of (a) and (c), and explain any difference.

2.5 Transform M(v), Eq.(2.14), into M(E), Eq.(2.15), with the aid of the appropriate Jacobian.

2.6 Find M(E) from M(v) for the case of isotropy using spherical co­ordinates.

2.7 Consult appropriate sources to determine particle densities N (m’3), the corresponding energies kT (eV) and temperatures T (K) for the following plasmas: outer space, a flame, the ionosphere, commonly attainable laboratory discharges and values expected in a magnetically as well as inertially confined fusion reactor. Plot these domains on an N vs. kT plane.