Direct Transfer Reactions

Figure 1.
Compound-nucleus mechanism.

Experiments at the NPL studied many reactions of the type A(a,b)B (or A+a → B+b) in which a fragment x is transferred between the target nucleus A and the projectile a. Niels Bohr discussed only those nuclear reactions that occur via formation of a compound nucleus (CN) which subsequently breaks up into a final nucleus B and an ejectile b, as shown in Fig. 1. Because the compound nucleus lives long enough to "forget" how it was formed, the CN mechanism results in the c.m. angular distribution of b having a symmetrical U-shaped form, as shown in Fig. 2.

Figure 2.
Typical compound-nucleus angular distribution.

However, the success of models introduced by Serber [SER47] in 1947 (see Fig. 3) and Butler [BUT51] in 1951 showed that at higher energies some transfer reactions can occur "directly" in one step, without formation of a compound nucleus. The simplest direct transfer reactions are (d,p) "stripping" and (p,d) "pickup", in which it is a neutron that is transferred.

Many more direct transfer reactions were identified at the NPL as well as other laboratories. Diagrams showing the simplest direct mechanisms for generalized stripping and pickup reactions are shown in Figs. 4 and 5. In these diagrams the horizontal dashed lines symbolize the effect of elastic scattering of a and A (or b and B) through the appropriate optical potential U_a (or U_b). These diagrams correspond to what is called the Distorted-Waves Born Approximation (DWBA) [AUS70] for the transition amplitude. DWBA treats the iterated interaction that binds x and a (or x and b) as a perturbation; this is found to be a successful first approximation, even though strong interactions are involved.

Figure 3.
Suppose the deuteron d, consisting of a proton (red) and a neutron (green), is peripherally incident on the target nucleus. It may happen that the neutron hits the nucleus and is absorbed by it, while the proton flies on by and eventually enters the detector. This model clearly suggests that the proton angular distribution will tend to be peaked in the forward direction.

In Figs. 3 and 4 the vertices involving a, b, and x can be described quite well for our purposes from knowledge of the structure of the fairly simple particles a, b, and x. In contrast, the nuclear vertices involving A, B, and x depend on the structure of the nuclei A and B under investigation. The nuclear vertex (and hence the transition amplitude) is proportional to a certain spectroscopic matrix element:

For pickup, transition amplitude

For stripping, transition amplitude

Here Φ_Aiis the initial state of the target and Φ_Bf is the state in which the final nucleus is detected. The operator a^↑_{N ljm} is the creation operator for fragment x in a single-particle state with radial quantum number n, orbital angular momentum l, and total angular momentum jm, while its Hermitian conjugate a_Nljm is the corresponding destruction operator. The cross sections dσ/dΩ are proportional to spectroscopic factors S^fi_nljm, which are the absolute squares of the spectroscopic matrix elements:

For pickup,

For stripping,

Figure 4.
b = a - x

Figure 5.
b = a + x

It follows that, provided the direct reaction mechanism is valid, it is possible to use transfer reactions to measure spectroscopic factors. Because the cross section is proportional to the spectroscopic factor, an equation of the type

is used for practical analysis of experiments, with calculated by DWBA. A particularly important quantity is l, the orbital angular momentum of the transferred particle. The angular distribution of the cross section depends on the value of l. This can be seen from a simple version of the plane-wave Butler theory, in which the reaction is assumed to occur only at the surface of the target nucleus, which is taken as a sphere of radius R. This gives for the angular distribution

Figure 6.
Unlike CN reactions, direct reactions have cross sections that depend smoothly on the bombarding energy. The direct reactions represented by Figs. 4 and 5 tend to be peripheral, and therefore typically have diffraction-like forward-peaked angular distributions very different from the CN angular distribution of Fig. 2. Fig. 6 schematically shows a typical direct transfer angular distribution.

Here j_l(qR) is a spherical Bessel function, p_a and p _b are the momenta of the projectile and the ejectile, respectively, and θ is the scattering angle. A more physical explanation of the l dependence is given on page 353 of the book [COH71] by B. L. Cohen. The l dependence of the angular distribution can be used to determine l, as illustrated in Fig. 7.

Spectroscopic factors give information that is very useful in determining the structure of individual nuclear states. As a simple example, consider the stripping reaction ¹⁶O(d,p)¹⁷O beginning with the target nucleus ¹⁶ O in its ground state. Suppose that (as specified by the simplest version of the shell model) the ground state of ¹⁷ O consists of an ¹⁶ O ground state core with a neutron in an N = 2, l = 2, j = 5/2 state orbiting around it. Then the corresponding spectroscopic factor should be exactly 1, and measurements of the stripping cross section (combined with calculations of the DWBA cross section) should confirm this. If, however, the ground state of ¹⁷ O contains appreciable components in which the ¹⁶ O core is not in its ground state, the spectroscopic factor will be appreciably less than 1, and the measurements should be able to reveal this. As always happens in nuclear physics with strongly interacting probes, there are uncertainties in the reaction mechanism and in the optical potentials that should be used in the DWBA calculations, so care must be exercised in analyzing the results of experiments.

Figure 7.
The reaction is ⁵⁸Ni(d,p)⁵⁹ Ni with 15 MeV deuterons. For different final states in ⁵⁹Ni different transferred orbital angular momentum values l dominate. The curves are from DWBA theory. The points are experimental cross sections on a logarithmic scale. In these cases the value of l is easily inferred from the angular distribution. (Taken from B. L. Cohen et al. (1962) [COH62])