Clear evidence of a clusters in the ground state of heavy nuclei

We obtained clear evidence of α clustering at the surface of heavy nuclei by measuring the cross sections for quasi-free α-knockout reactions along the tin isotopic chain. The α-cluster appearance at the nuclear surface could not only be a natural explanation for the α-preformation in the α-decay theory, but also provides the modification to the relation between neutron-skin thicknesses Δτηρ and a slope parameter L in the nuclear equation of state.

The formation of α clusters is an essential ingredient in heavy nuclei, pre-requisite to describe the α decay as proposed 90 years ago by G. Gamow -the quantum tunnelling of preformed α particles through a Coulomb barrier (1) (Figure 1).
However, a consistent description of α clusters and nucleons in one model is challenging from a theoretical perspective. Clusters should get dissolved in the saturated nuclear matter and contribute to the mean-field as independently moving nucleons. This contradictory situation was overcome by decay theoretical studies -α particles exist only in nuclear surface (2,3), the dilute nuclear matter. So far there is no clear evidence of α clustering especially in the ground state of heavy nuclei except for α decay.  Recently, the formation of α clusters is described by the generalised relativistic mean-field (gRMF) model with explicit cluster degrees of freedom (5). Figure 2 a is the density distribution of the α particles as a function of radius, the amplitudes are locally located around the surface, and a significant difference between 112 Sn and 124 Sn is predicted. Figure 2 b is the isotopic dependence of the number of α particles, which decreases by increasing the mass number.

Relation to nuclear equation of state
One important aspect of α clusters in nuclear surface is their relation to the nuclear equation of state(EOS) (7). The knowledge of the nuclear EOS for neutron rich matter is important for nuclear physics as well as for the understanding of properties of cosmic objects like neutron stars. In the nuclear EOS, a parameter L is introduced in the symmetry energy term, which is the derivative of the symmetry energy, and thus called the slope parameter. Mass-dependent neutron-skin thicknesses for thin isotopes (9). Both a and b are the theoretical calculations It is known that the neutron skin thickness has a close correlation to the slope parameter L (10). Figure 4 a shows the linear correlation of L and neutron skin thickness for 208 Pb (8). By measuring the neutron-skin thickness one can predict the L parameters by using this relation, which triggered many experimental projects to obtain the neutron-skin thickness. However the theoretical calculations in figure 4 a are based on the conventional mean-field models, which do not take surface α clustering into account. Figure 4 b is the theoretical calculations of neutron skin thickness of tin isotopes (9). The calculations without α clusters(open circles) and with α clusters (filled points) give 15% 44% differences to neutron-skin thicknesses. Thus, the existence of α clusters in nuclear surfaces requires the revision of the theoretical predictions to include α clustering.

Quasi-free α knockout -Sn(p, pα) experiment
The most direct proof to see the amplitude of α clusters in nuclei should be quasi-free alpha knockout reactions, figure 6 is the schematic view of a quasi-free alpha knockout reaction. A projectile proton comes from the left side and hits an alpha cluster at the surface of a tin isotope. A proton knocks out an alpha particle and a cadmium isotope remains as a spectator.

A-4 Cd
The scattered angles of protons and alpha particles satisfy the quasi-free condition.   The experiment was performed, at the Research Center for Nuclear Physics in Osaka University using double-arm magnetic spectrometer Grand Raiden spectrometer (GR) (11) and Large acceptance spectrometer (LAS). Scattered protons and α particles are detected with drift chambers and the plastic scintillators at the focal planes.  Considering the energy conservation law, the missing mass of the residue M X is given by where E is total energy, m is mass, T is kinetic energy, T p in is the incoming beam energy 392 MeV. Figure 7 shows the positions of coincident protons and alpha particles in GR (x-axis) and LAS (y-axis), respectively. A clear loci is seen, which corresponding to constant T α − T pout term in Eq. 2; therefore, the missing mass spectrum will make a peak structure.   The fact that α particles from the most outer shells are knocked out in this transition, implies that they are from the surface of the tin nuclei. Furthermore the (p, pα) probe is sensitive only to the surface region in this energy because of the absorption effects of α particles inside the nuclei (12). Figure 9 is the isotopic dependence of the mea- Thus, we confirmed the trend of isotopic dependence of α formation amplitude via α knockout reactions.