Novel Direct Transfer Reactions at NPL
1966-1984

Many pioneering studies of nuclear reactions and their mechanisms have been made at NPL. Notable were Daehnick's discoveries of new and unusual direct transfer reactions, to form states that would otherwise be inaccessible, and to study the shell-model structure of these states by applying direct reaction ideas or by other experimental methods such as gamma ray spectroscopy.

Multi-Nucleon Transfer

A 1966 paper published with L.J. Denes and R. M. Drisko [DEN66] showed that at bombarding energies near 15 MeV, the "alpha pickup" reaction (d,⁶Li) on targets ranging in mass from 12 to 19 has the diffraction-like, forward peaked angular distribution and the comparative insensitivity to energy changes that are characteristic of direct reactions. A DWBA calculation, treating the transferred α particle as a cluster, agreed satisfactorily with the data. This confirmed the direct-reaction mechanism, and showed, surprisingly, that the weakly bound (2.2 MeV) deuteron can snatch a more massive α particle away from the target nucleus that contains it, without the deuteron disintegrating.

In 1967 NPL experiments on the reactions (d,⁷Li), (d,⁷Be), (d,⁹Be) at 15 MeV were described [DEN67]. These were analyzed in terms of the direct transfer of ⁵He, ⁵Li, and ⁷Li, respectively. States of residual nuclei with masses ranging from 8 to 19 were formed. Though the DWBA calculations used zero-range approximation and were comparatively primitive by later standards, reasonable values of spectroscopic factors were extracted. (Zero-range approximation simplifies the DWBA calculation by treating the interaction between the transferred particle x and the projectile or ejectile a or b to which it is bound as being of negligibly short range. This means that, in the DWBA diagram shown in Fig. 4 of the page on Direct Transfer Reactions, the vertex involving x, a, and b is treated as a constant, usually denoted by D ₀.) That such fragile clusters, containing one or two particles outside an α particle core, could be transferred directly seemed even more surprising than the direct transfer of an α particle.

Another multi-nucleon direct transfer reaction, (p,α), was reported in 1970 [DIT70]. It was analyzed as the direct transfer of a triton, by a non-local, finite-range DWBA calculation (made using P.D. Kunz's code DWUCK before finite range calculations became routine). This calculation reproduced the experimentally observed dependence of the l = 1 angular distribution on the j value of the transferred triton.

Two-Nucleon transfer

Perhaps slightly less exotic than these multi-nucleon transfers were the two-nucleon transfer reactions (p,t), (³He,p), (p,³He), (α,d), and (d,α). The first [???] (d,α) reaction at NPL was reported in 1966 [PAR66]. This reaction [and its inverse (α,d)] developed into a major theme in reactions at the NPL. By 1977 [DAE77] (d,α) had figured in 13 refereed papers and 4 conference presentations by Daehnick and collaborators, while from 1976 [DAE76] to 1984 [SPI84] there were 4 refereed papers and 4 conference presentations on (α,d).

Spin and Parity Assignments From (d,α)

In 1968 Daehnick and Park (1968) [DAE68] gave a forward-looking account of the usefulness of the (d,α) reaction in spectroscopic studies of odd-odd nuclei. Since both d and α have isospin T = 0, it follows by isospin conservation that T_f = T_i. Compared with single-nucleon transfer, (d,α) reduces the ambiguity of the assignment of the total angular momentum J of the final state, because the target nucleus is even-even and has J = 0⁺. Therefore J_f = j_x, the total angular momentum of the transferred particle (denoted by x in the DWBA diagram shown in Fig. 4 of the page on Direct Transfer Reactions). The value of j_x is not always easily identified, but since the deuteron has total intrinsic spin 1, the only possibilities are j_x = l_x,l_x + 1, and l_x - 1 (unless this is negative) where l_x is the orbital angular momentum of the transferred particle. The angular distribution is strongly dependent on l_x, though sometimes two l_x values contribute to the cross section. However, even if one can identify l_x = 0 as only one of the values that contributes, it follows that j_x = 1, and J_f^π = 1⁺. The parity can be assigned as π_f = π_i(–1)^lx if even one value of l_x is known. (These parity assignments assume that the transferred particle is bound in an s state in the α particle.) States for which π_f = (-1)^Jf must have l_x = J_f uniquely, so these states have angular distributions clearly characteristic of their l_x values. However, a state for which π_f = (-1)^Jf+1 can have contributions from two values l_x = J_f ± 1, and this mixing leads to a less typical angular distribution lacking pronounced diffractive minima. Nevertheless, it is possible for a state with π_f = (-1)^J to be mistaken for one with π_f = (-1)^f, if one of the values l_x = J_f ± 1 makes an unusually small contribution which is not noticed.

A (d,α) Case Study

Click on the thumbnail to see the full image.

Figure 1.
Differential cross sections for ⁶⁸Zn(d,α)⁶⁶Cu and resulting J_f^π assignments. The dotted curves and the curves for l = 2 are DWBA predictions. The remaining solid curves are empirical angular distributions for unique l_x values, or combinations of angular distributions for different l_x values.

Fig. 1 shows angular distributions for ⁶⁸Zn(d,α)⁶⁶Cu with 12 keV total experimental resolution. The underlined assignments of J_f^π = 1⁺, 2⁺, 3⁺, and 4⁺ were considered reliable, based on a high resolution experiment on ⁶⁵Cu(d,p)⁶⁶Cu. Since the angular distributions shown all had clear structure, it was possible to infer many of the l values. Unique l_x assignments were for J_f^π = 1⁺, l_x = 0; for J_f^π = 2⁺, l_x=2; and for J_f^π = 4⁺, l_x = 4; while for J_f^π = 3⁺, l_x = 0 and l_x=2 were both possible.Thus the angular distributions characteristic of l_x= 0, 2, and 4 could each be identified empirically, and the good agreement of DWBA calculations with these confirmed the identifications. These characteristic angular distributions were then used to identify the pure or mixed l_x values for other states in ⁶⁶Cu. Thus the 0.724 MeV state was l_x = 4 and l_x = 2, and so was assigned J_f^π = 3⁺, while the 0.819 MeV state was l=2, and so was assigned J_f^π = 2⁺ (or, if a small l_x = 0 or l_x = 4 component was missed, 1⁺ or 3⁺, respectively).

Two-Step Contributions in (α,d)

In 1978 Daehnick et al.[DAE78] reported a ground breaking experimental and theoretical study of two-step contributions to the mechanism of (α,d). Experiments on the reaction Pb(α,d) were carried out at 48 MeV using the Princeton cyclotron and at 33 MeV using the MP Tandem of the Max Planck Institute. Their analysis showed that the mechanism of (α,d) can be more complex than simple direct one-step transfer, even in cases such as this, where the target ground state is a good closed shell core and not prone to vibrational excitation, and where the one-step amplitudes are allowed and large. They studied the negative parity multiplet of two-particle states in ²¹⁰Bi corresponding to an h_9/2 neutron and a g_9/2 proton coupled to total angular momentum values J_f = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

Figure 2.
Direct diagram for (α,d)

Figure 3.
Two-step diagrams for (α,d)

Fig. 2 shows the diagram for the usual direct 1-step transition amplitude corresponding to DWBA. To reproduce the experimental data, it was necessary to augment the 1-step amplitude with two 2-step amplitudes of the sequential transfer type, one corresponding to (α,t) followed (t,p), the other to (α,³He) followed by (³He,d), as indicated in Fig. 3. Another representation of these paths is shown in Fig. 4.

Figure 4.
Energy level representation of 1-step and 2-step mechanisms

The 2-step effect was most clearly seen in the dependence of the angle-integrated (α,d) cross section on the angular momentum J_f of the final state. The 1-step amplitude gives a smooth dependence on J_f, but including the 2-step amplitudes reproduces the "saw tooth" dependence of the angle-integrated cross section, as shown in Fig 5. The states of even J_f [i.e. the so-called "unnatural parity" states, those with π_f = (1)^{J f +1}] have larger integrated cross sections than the neighboring "natural parity" states, by factors ranging from 2 to 8.

Figure 5.
Angle-integrated (α,d) cross sections.

Experimental points compared with 1-step+2–step theory, using intermediate - channel optical potentials fixed by other reactions (solid lines) or modified (dashed lines).

The effects are due the interference of the 2-step amplitudes with the 1-step amplitude. The total 2-step amplitude is roughly 0.6 of the one-step amplitude, so the relative magnitude of the cross section can range between about |1–0.6|² = 0.16 and about |1+0.6|² = 2.56. The saw tooth dependence results from a relative phase (–1)^Jf between the 1-step amplitude and one of the 2-step amplitudes. This phase comes from a symmetry property of the Clebsch-Gordan coefficient for coupling 9/2 and 9/2 to make J_f.

These results provided a salutary check on the limitations of the usual 1-step picture as implemented through DWBA calculations. The warning is sharpened by noting that the angular distributions are little affected by 2-step effects, so that angular distributions cannot be relied on to reveal the presence of important 2-step contributions.