Scattering of identical nuclei, exchange symmetry and molecular resonances

To best see the quantum interference of identical particles it is useful to stay below the Coulomb barrier (sub-Coulomb scattering), i.e. where nuclear effects can be kept small. Then one has to deal with calculable (Rutherford) amplitudes only. In addition to the forward scattering Rutherford cross section there is a corresponding recoil Rutherford term plus an interference term between both. In the scattering of identical particles a detector at the center-of-mass (c.m.) angle θ is unable to distinguish whether it registers forward-scattered ejectiles under θ or, under the angle $\pi -\theta$, backward-emitted recoils. This is shown in figure 13.1. The formal scattering theory (see below) shows that the angular distributions must be symmetric around $\pi /2$ and therefore must be described by even-order Legendre polynomials. Quantum-mechanically, in addition, it is to be expected that the forward- and backward-scattered particle waves interfere. In this case no classical description of the scattering process is possible. In addition, the details of the interference depend on the spin structure of the interacting particles: identical bosons behave differently from identical fermions and when the particles have spin $\ne 0$ (i.e. always for fermions) the spin states must be coupled and superimposed in the cross section with their spin multiplicities as weighting factors. The following examples, which can be tested experimentally, will explain this.

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13.1.1. Identical bosons with spin I = 0

Here

$$\begin{eqnarray}&&{\left[{\rm{d}}\sigma /{\rm{d}}{\rm{\Omega }}(\theta )\right]}_{{\rm{B}}}={\left|{f}_{1}(\theta )+{f}_{2}(\pi -\theta )\right|}^{2}.\end{eqnarray} \tag{ 13.1 }$$

13.1.2. Identical fermions with spin ${\boldsymbol{I}}{\boldsymbol{=}}{\bf{1}}{\boldsymbol{/}}{\bf{2}}$

For the fermions the spin singlet cross section

$$\begin{eqnarray}&&{\left[{\rm{d}}\sigma /{\rm{d}}{\rm{\Omega }}(\theta )\right]}_{{\rm{s}}}={\left|{f}_{1}(\theta )-{f}_{2}(\pi -\theta )\right|}^{2}\end{eqnarray} \tag{ 13.2 }$$

and the triplet cross section

$$\begin{eqnarray}&&{\left[{\rm{d}}\sigma /{\rm{d}}{\rm{\Omega }}(\theta )\right]}_{{\rm{t}}}={\left|{f}_{1}(\theta )+{f}_{2}(\pi -\theta )\right|}^{2}\end{eqnarray} \tag{ 13.3 }$$

in the total (integrated) cross section must be added incoherently, each weighted with their spin multiplicities:

$$\begin{eqnarray}&&{\left[{\rm{d}}\sigma /{\rm{d}}{\rm{\Omega }}(\theta )\right]}_{{\rm{F}}}=\frac{1}{4}{\left|{f}_{1}(\theta )+{f}_{2}(\pi -\theta )\right|}^{2}+\frac{3}{4}{\left|{f}_{1}(\theta )-{f}_{2}(\pi -\theta )\right|}^{2}.\end{eqnarray} \tag{ 13.4 }$$

In these two cases the interference has opposite signs which, e.g. at $\theta =\pi /2$, has the consequence that in the case of two bosons there is an interference maximum, for fermions a minimum. Under the special assumption that there is no spin–spin force acting (${f}_{{\rm{s}}}={f}_{{\rm{t}}}=f$), and with $f(\theta )=f(\pi -\theta )$ one obtains for identical fermions a decrease, for identical bosons an increase each by a factor of 2 compared to the classical cross section.

For pure (sub-)Coulomb scattering (meaning: Coulomb scattering at energies sufficiently below the Coulomb barrier) of identical particles the scattering amplitudes can be calculated explicitly (i.e. also summed over partial waves) since we deal with the Rutherford amplitude known from scattering theory, see section 2.1.3:

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{\rm{d}}\sigma }{{\rm{d}}{\rm{\Omega }}}\right)}_{\mathrm{Coul}}={\left(\displaystyle \frac{{Z}^{2}{e}^{2}}{4{E}_{\infty }}\right)}^{2}\left[\displaystyle \frac{1}{{\mathrm{sin}}^{4}\displaystyle \frac{\theta }{2}}+\displaystyle \frac{1}{{\mathrm{cos}}^{4}\displaystyle \frac{\theta }{2}}+\displaystyle \frac{2{(-1)}^{2s}\mathrm{cos}\left({\eta }_{{\rm{S}}}{\mathrm{ln}\mathrm{tan}}^{2}\displaystyle \frac{\theta }{2}\right)}{(2s+1){\mathrm{sin}}^{2}\displaystyle \frac{\theta }{2}{\mathrm{cos}}^{2}\displaystyle \frac{\theta }{2}}\right].\end{eqnarray} \tag{ 13.5 }$$

In addition to the forward scattering Rutherford cross section there is a corresponding recoil Rutherford term plus an interference term between both. Figure 13.2 shows this behavior, which is analogous to that of light in Young's double-slit experiment, but additionally shows the influence of spin and statistics. Two fermions show destructive interference at π/2 whereas two bosons interfere constructively. In fact the experiments could be used to determine the spins of the particles involved from the 'amplitude' of the interference pattern. Above the Coulomb barrier, additional terms including interference terms arise from the hadronic interaction. A special example is low-energy proton–proton scattering in which, for S-waves, one nuclear phase shift ${\delta }_{0}$ must be considered for which, e.g. a 'nuclear' scattering length ${a}_{\mathrm{pp}}$ may be obtained (see section 12.3.2).

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The first and key experiments designed specifically to study the scattering of identical bosons and at the same time details of the heavy-ion interactions and structure were performed by Bromley et al [4, 5]. Scattering of identical spin-zero nuclei was investigated. Larger tandem Van de Graaff accelerators had become available as well as compact solid-state detectors. Both are especially suited to studying heavy-ion reactions, the accelerators because of the possible high energies for heavy ions due to the multiple charge states of the ion beams, excellent energy definition and stability, and ease of changing energy and targets, the latter because of the possibility of using many small detectors providing good angular resolution and relatively thin depletion layers. Thus, measurements of the highly structured cross sections, both in angle and energy, were facilitated. Figure 13.3 shows the scattering chamber set-up used in [5]. Figure 13.4 shows a selection of the angular distributions for ${}^{12}{\rm{C}}$–${}^{12}{\rm{C}}$ and ${}^{16}{\rm{O}}$–${}^{16}{\rm{O}}$ elastic scattering at energies below the Coulomb barriers such that the Mott cross section applies. The interference pattern for identical boson scattering is evident. One of the earliest measurements of elastic scattering between identical fermions (${}^{13}{\rm{C}}$–${}^{13}{\rm{C}}$) was performed by Voit et al [9], see also figure 13.2 for details of the spin and statistics. Figure 13.5 compares excitation functions for ${}^{12}{\rm{C}}$–${}^{12}{\rm{C}}$ and ${}^{16}{\rm{O}}$–${}^{16}{\rm{O}}$ elastic scattering. It is remarkable that the excitation functions for the two systems are quite different, i.e. relatively smooth for the ${}^{16}{\rm{O}}$ case, but with (quasi-)periodic structures for the ${}^{12}{\rm{C}}$ scattering (intermediate or gross structures with superimposed fine structure). These would be typical signatures for a doorway phenomenon. In fact, these oscillations have been interpreted as rotational states of nuclear molecules which partly decay into CN fine-structure states. The excitation functions follow the Mott cross section but with increasing energy there is a relatively sharp onset of strong absorption (leading to compound-nucleus formation). The r dependence of the combined potential of the Coulomb potential, an absorptive (attractive) nuclear potential and the orbital angular-momentum barrier at higher could form a shallow minimum with rotational molecular states as shown in figure 13.6. In contrast to these early measurements more detailed studies, also at higher energies, exhibit (quasi-)periodic structures, partly with marked fine-structure oscillations, also in other systems such as ${}^{16}{\rm{O}}$–${}^{16}{\rm{O}}$, ${}^{12}{\rm{C}}$–${}^{16}{\rm{O}}$, ${}^{14}{\rm{C}}$–${}^{14}{\rm{C}}$, ${}^{18}{\rm{O}}$–${}^{16}{\rm{O}}$, and ${}^{18}{\rm{O}}$–${}^{18}{\rm{O}}$, see [16]. Three cases are shown in figure 13.7. Later measurements showed intermediate structures in excitation functions also in several combinations of nuclei, even in ${}^{16}{\rm{O}}$–${}^{16}{\rm{O}}$, but also between non-identical nuclei, ${}^{12}{\rm{C}}$–${}^{16}{\rm{O}}$, ${}^{18}{\rm{O}}$–${}^{16}{\rm{O}}$ and ${}^{18}{\rm{O}}$–${}^{18}{\rm{O}}$, partly with superimposed (CN) fine structure. An example for a precise measurement of the fine structures is shown in figure 13.8. For a detailed discussion of nuclear molecular states and attempts to describe them, e.g. in the framework of the optical model with shallow or deep optical potentials, see [2]. For general reading on heavy-ion nuclear physics see the selected references [1, 3, 6, 8, 10–13, 15].

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