2
Nuclear Reactions

2.1 Cross Sections

The cross section, s, is a quantitative measure ofthe probability that an interaction will occur. In the following, we define several quantities that are displayed in Figure 2.1. Suppose that a beam of particles per unit time t, covering an area A, is incident on a target. The number of nonoverlapping target nuclei within the beam is . We assume that the total number of interactions that occur per unit time,, is equal to the total number of emitted (nonidentical) interaction products per unit time, . If the interaction products are scattered incident particles, then we are referring to elastic scattering. If the interaction products have an identity different from the incident particles, then we are referring to a reaction. The number of interaction products emitted at an angle ? with respect to the beam direction into the solid angle dO is . The area perpendicular to the direction ? covered by a radiation detector is given by dF = r2 dO. The cross section is defined by

We will use this general definition to describe reaction probabilities in astrophysical plasmas and in laboratory measurements of nuclear reactions. In the latter case, two situations are frequently encountered: (i) if the beam area, A, is larger than the target area, At, then

and the number of reactions per unit time is expressed in terms of the incident particle flux,, the number of target nuclei,, and the cross section; (ii) if the target area, At, is larger than the beam area, A, then

Figure 2.1 Typical nuclear physics counting experiment, showing a beam of particles per unit time, nonoverlapping target nuclei within the beam area A, interaction products and a detector of area dF. The detector is located at an angle of ? with respect to the incident beam direction. The two situations are as follows: (a) the target area is larger than the beam area; and (b) the beam area is larger than the target area.

and the number of reactions per unit time is expressed in terms of the incident particle current,, the total number of target nuclei within the beam per area covered by the beam,, and the cross section. For a homogeneous target, is equal to the total number of target nuclei divided by the total target area At. The latter quantity is easier to determine in practice. We can also express the total cross section, s, and the differential cross section, ds/dO, in terms of the number of emitted interaction products

If we define , that is, the number of emitted interaction products per target nucleus, then we obtain

With the definition of a flux or current density j as the number of particles per time per area, we can write for the beam and emitted interaction products

For the total and differential cross section, one finds

These quantities are related by

Common units of nuclear reaction and scattering cross sections are

1b = 10-24 cm2 = 10-28 m2

1 fm2 = (10-15 m)2 = 10-30 m2 = 10-2 b

In this chapter, all kinematic quantities are given in the center-of-mass system (Appendix C), unless noted otherwise.

2.2 Reciprocity Theorem

Consider the reaction A + a B + b, where A and a denote the target and projectile, respectively, and B and b are the reaction products. The cross section of this reaction is fundamentally related to that of the reverse reaction, B + b A + a, since these processes are invariant under time-reversal, that is, the direction of time does not enter explicitly in the equations describing these processes. At a given total energy, the corresponding cross sections are not equal but are simply related by the phase space available in the exit channel or, equivalently, by the number of final states per unit energy interval in each case. The number of states available for momenta between p and p + dp is proportional to p2 (Messiah, 1999). Hence

The linear momentum and the de Broglie wavelength are related by ? = h/p. The wave number k of the free particle is defined in terms of the de Broglie wavelength by Hence, we have p = mv = hk. It follows (Blatt and Weisskopf, 1952) that

This expression, called the reciprocity theorem, holds for differential as well as total cross sections. The factors (1 + dij) appears because the cross sections between identical particles in the entrance channel are twice those between different particles, other factors being equal.

When particles with spin are involved in the reactions, then the above equation must be modified by multiplying the density of final states by their statistical weights. Since there are (2ji + 1) states of orientation available for a particle with spin ji, we can write for unpolarized particles

It follows that the cross section sBbAa can be easily calculated, independently from any assumptions regarding the reaction mechanism, if the quantity sAaBb is known experimentally or theoretically. Equation (2.15) is applicable to particles with rest mass as well as to photons. In the former case, the wave number is given by and the linear momentum can be expressed as p2 = ?2k2 = 2mE, where E denotes the (nonrelativistic) center of mass energy and m is the reduced mass (see Appendix C). In the latter case, the wave number is defined as k = E/(?c), and the linear momentum can be expressed as p2 = ?2k2 = E2/c2, where E denotes the photon energy; furthermore, (2j?+ 1) = 2 for photons. See also Eqs. (3.27) and (3.28). It must be emphasized that the symbols A, a, b, and B do not only refer to specific nuclei but, more precisely, to specific states. In other words, the reciprocity theorem connects the same nuclear levels in the forward and in the reverse reaction.

The reciprocity theorem has been tested in a number of experiments. An example is shown in Figure 2.2. Compared are differential cross sections for the reaction pair 24Mg(a,p)27Al (open circles) and 27Al(p, a)24Mg (crosses), connecting the ground states of 24Mg and 27Al. Both reactions were measured at the same center-of-mass total energy and angle. The differential cross sections exhibit a complicated structure, presumably caused by overlapping broad resonances. Despite the complicated structure, it can be seen that the agreement between forward and reverse differential cross section is excellent. Such results support the conclusion that nuclear reactions are invariant under time-reversal. See also Blanke et al (1983).

Figure 2.2 Experimental test of the reciprocity theorem for the reaction pair have 24Mg(a,p)27Al (open circles) and 27Al(p,a)24Mg (crosses), connecting the ground states of 24Mg and 27Al. The differential cross sections of both reactions are shown for the same total energy and detection angle in the center-of-mass system. The cross sections also been adjusted to compensate for differences in spins. (Reprinted with permission from W. von Witsch, A. Richter and P. von Brentano, Phys. Rev. Vol. 169, p. 923 (1968). Copyright (1968) by the American Physical Society.)

2.3 Elastic Scattering and Method of Partial Waves

2.3.1 General Aspects

The interactions between nucleons within a nucleus and between nucleons participating in nuclear reactions have to be described using quantum mechanics. The fundamental strong interaction is very complicated and not precisely known. We know from experiments that it is of short range. Furthermore, it exhibits a part that is attractive at distances comparable to the size of a nucleus and another part that is repulsive at very short distances. Because of the complexity of this nucleon-nucleon interaction it is necessary to employ approximations. Instead of calculating all the interactions between all nucleons exactly, one frequently resorts to using effective potentials. These describe the behavior of a nucleon, or a group of nucleons (such as an a -particle), in the effective (average) field of all the other nucleons. Because of the approximate nature of this approach, the resulting effective potentials are usually tailored to specific reactions and energies and thus lack generality. The most widely used approximate potentials are called central potentials. They depend on the magnitude of the radius vector, but not on its direction,

Figure 2.3 Schematic representation of the scattering...