A shell-model representation to describe radioactive decay

In this talk I will describe the formation and radioactive decay of nuclear clusters by using a microscopic formalism. This is based in the shell model. I will show the advantages and the success of the standard shell model in this subject, but also its limitations and ways of improving it. The starting point of all microscopic descriptions of cluster decay is by using the expression of the decay width formulated by Thomas in Thomas obtained his famous expression by evaluating the residues of the R-matrix in a profound and very difficult paper (Prog. Theor. Phys. 12 (1954) 253). Since in the microscopic treatment I will present the Thomas expression is fundamental, it is important to understand all the elements that enter in it. I will therefore start by presenting a clear and easy derivation of the Thomas formulae by using simple quantum mechanics arguments. The first feature to be noticed is that a decaying cluster feels only the centrifugal and Coulomb interactions outside the surface of the daughter nucleus. Therefore the corresponding (outgoing) wave function in that region has the form Radioactive Decay width

. The detector of the decaying particle can be considered to be at that distance The probability rate per second that the particle goes through a detector surface element dS=r 2 sinθ dθ dφ is At very large distances, where both the centrifugal interaction (depending upon 1/r 2 ) and the Coulomb one (1/r), are negligible, the wave function is a plane wave, i. e.

integrating over the angles, the decay probability per second becomes 1/T=|N lj | 2 v. Matching the out and the inner solution at R one gets RΨ lj (R)=N lj [G lj (R)+iF lj (R)] and Since

Maglione,Ferreira, RL, PRL81, 538 (1998) This is the famous Thomas expression for the decay width, which he obtained as the residues of the R-matrix. Ψ lj (R) is the cluster formation amplitude and kR/(F 2 lj (R)+G 2 lj (R)) is the penetrability through the centrifugal and Coulomb barriers. It is important to notice that the width should NOT depend upon R if the calculation of the formation amplitude is properly performed. For the decay process B  A+C the formation amplitude F is

ρ=kr Proton decay Therefore, T red (χ)=T 1/2 /|H l (+) | 2 is independent upon l.

Delion, RL, Wyss, PRL 96, (2006) Z>67 Z>50

Universal decay law in cluster radioactivity Geiger-Nuttall law As before, we have For l=0 transitions  cos 2 β << 1

T 1/2 depends upon the formation amplitude, but the quantity should not depend upon R, i. e. Therefore

C.Qi, F.R.Xu,RL,R.Wyss, PRL103, (2009)

The analysis of the formation amplitude can provide important Information on the mother nucleus B. For instance in the decay 212 Po(gs)  208 Pb(gs)+α, one can write within the pairing vibration approximation

______ ______ ______ ______ ______ ______ _______ _______ _______ _______ _______ G. Dodig-Crnkovic, F. A. Janouch and RJL, PLB139, 143 (1984) 212 Po(gs) 208 Pb(gs)

protons neutrons F. Janouch and RJL, PRC 25, 2123 (1982)

F. Janouch and RJL, PRC 27, 896 (1983) 210 Pb(gs) 210 Po(gs)

In 212 Po (N=128, Z=84) there are two neutrons and two protons outside the 206Pb core. In 210 Po (N=126, Z=82). There are only two protons outside the core To extract the formation amplitude from experiment one notices that the half life is Therefore, and

C. Qi, Andreyev, Huyse, RL, VanDuppen Wyss, PRC C81, (2010) N=128, Z=84 N=126, Z=84

It is highly unsatisfactory that the standard shell model cannot reproduce absolute decay width. It is even more intriguing to realize that the clustering on the nuclear surface is well described by the shell model representation but not the subsequent cluster decay. The shell model is, more than a model, a device that provides a very good representation to describe nuclear spectroscopy. It can describe, e. g., the magic numbers, which would be impossible to do by using more common representations, like plane waves. In the same fashion, with the standard shell model representation one can describe the clustering of the nucleons forming the decaying cluster, but not the motion of the cluster leaving the mother nucleus. This indicates that it is the high lying configurations in the representations, which are important at large distances, which are not properly included.

There have been successful microscopic attemps to describe alpha decay. But in these treatments the mother nucleus was described in terms of shell model plus cluster components, i. e. the wave function of the decaying nucleus was written as (K. Varga, R. Lovas, RJL, PRL 69, 37 (1992)) The important feature here is that the cluster component is assumed to take care of the high lying configurations and therefore the shell model component is evaluated within a major shell only. The cluster component is in this approach written in terms of shifted Gaussian functions. This method was recently applied to describe simultaneously anomalous large B(E2) values and alpha-decay half lives in transitions from 212 Po. By using a shifted Gaussian component in the single-particle wave functions it was possible to describe the alpha decay half life, while the shell model component describes the B(E2) (D. S. Delion, RJL, P. Schuck, A. Astier, M. G. Porquet, PRC 85, (2012)).

A shifted Gaussian single-particle wave function can be very well represented by a harmonic oscillator eigenstate with radial quantum number n=0 and orbital angular momentum l large. With the principal quantum number N=2n+l one gets (this and what follows is from D. S. Delion, RJL, PRC (R), in press).

Since the spectroscopic properties of nuclei are well described by the standard shell model, while the alpha cluster moves outside the nuclear surface, we proposed that the mean field generating the representation should include the standard one, e. g. a Woods-Saxon potential, and on top of that a pocket-like potential just outside the surface, i. e.

216 Rn

Thus the potential has the form where For heavy nuclei (lead isotopes or heavier) we choose b=1 fm, r c =1.3(4 1/3 +A 1/3 ) and the parameters V clus are required to satisfy These parameters will be chosen such that the calculated decay widths do not depend upon the matching point R, as will be seen.

In the lead isotopes the single-particle states (representation) have, as a function of V clus, the form

Since the decay width should not depend upon the matching radius, the following quantity should vanish This happens if the condition is fulfilled with the values of the cluster potentials following the trends shown in the next figure

220Ra 222Ra 224Ra V clus (protons)

220 Ra decay

Proton-proton (solid line) and neutron-neutron formation probability

The reason why the standard shell model representation fails to reproduce radioactive decay is that the behavior of the single particle wave functions do not fulfill proper asymptotic conditions. To correct this deficiency we have proposed as a mean field the standard shell model plus an attractive pocket potential localized just outside the nuclear surface. We have shown that the eigen- vectors of this new mean field preserve the low lying states of the standard shell model (thus keeping all its benefits) while providing high-lying states that induce a large overlap between neutron and proton wave functions. Within this new representations we could explain alpha-decay processes in heavy nuclei. Summary