A shell-model representation to describe

radioactive decay

In this talk I will describe the formation and radioactive decay of nuclear clustersby 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, butalso 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 1954. 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 formulaeby 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=r2 sinθ dθ dφ is

€

v = hk /μ

€

Plj =|ψ ljout (

r r ) |2 vdS

At very large distances, where both the centrifugal interaction (depending upon 1/r2) and the Coulomb one (1/r), are negligible, the wave function is a plane wave, i. e.

€

limr→∞

| rψ ljout (r) |2=| N lj |2

€

limr→∞

| rψ ljout (r) |2=| N lj |2

integrating over the angles, the decay probability per second becomes 1/T=|Nlj|2v.

Matching the out and the inner solution at R one gets

RΨlj(R)=Nlj[Glj(R)+iFlj(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.Ψlj(R) is the cluster formation amplitude and kR/(F2

lj(R)+G2

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 BA+C the formation amplitude F is

€

F(R) = dξ A∫ dξCψ B* (ξ B )ψ A (ξ A )ψC (ξC )

€

→

ρ=kr

Proton decay

Therefore, Tred(χ)=T1/2/|Hl(+)|2 is independent upon l.

€

χ =2(Z −1)e2

hv

Delion, RL, Wyss, PRL 96, 072502 (2006)

Z>67

Z>50

Universal decay law in cluster radioactivity

Geiger-Nuttall law

As before, we have

For l=0 transitions

cos2β << 1

T1/2 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, 072501 (2009)

The analysis of the formation amplitude can provide important Information on the mother nucleus B. For instance in the decay212Po(gs) 208Pb(gs)+α, one can write within the pairing vibration approximation

€

|212Po(gs) >=|210Po(gs)⊗210Pb(gs) >

________________________

______________________________

___________________________________

€

209BiPb209

G. Dodig-Crnkovic, F. A. Janouch and RJL, PLB139, 143 (1984)

212Po(gs)

208Pb(gs)

protons

neutrons

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

PoPbPo 210210212

PbPo 208212

F. Janouch and RJL, PRC 27, 896 (1983)

210Pb(gs) 210Po(gs)

In 212Po (N=128, Z=84) there are two neutrons and two protons outside the 206Pb core. In 210Po (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, 064319 (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 torealize 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. Itcan describe, e. g., the magic numbers, which would be impossibleto do by using more common representations, like plane waves. In the same fashion, with the standard shell model representation onecan describe the clustering of the nucleons forming the decayingcluster, 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 interms of shell model plus cluster components, i. e. the wave functionof 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 assumedto take care of the high lying configurations and therefore the shellmodel 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 212Po. By using a shifted Gaussian component in thesingle-particle wave functions it was possible to describe the alphadecay half life, while the shell model component describes the B(E2)(D. S. Delion, RJL, P. Schuck, A. Astier, M. G. Porquet, PRC 85, 064306 (2012)).

A shifted Gaussian single-particle wave function can be very wellrepresented by a harmonic oscillator eigenstate with radialquantum number n=0 and orbital angular momentum l large. Withthe 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 bythe standard shell model, while the alpha cluster moves outsidethe 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-likepotential just outside the surface, i. e.

216Rn

Thus the potential has the form

where

For heavy nuclei (lead isotopes or heavier) we choose b=1 fm,rc=1.3(41/3+A1/3) and the parameters Vclus are required to satisfy

These parameters will be chosen such that the calculated decaywidths 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 Vclus, 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 thetrends shown in the next figure

220Ra

222Ra

224Ra

Vclus(protons)

220Ra decay

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

The reason why the standard shell model representation fails toreproduce radioactive decay is that the behavior of the singleparticle wave functions do not fulfill proper asymptotic conditions.To correct this deficiency we have proposed as a mean field thestandard shell model plus an attractive pocket potential localizedjust outside the nuclear surface. We have shown that the eigen-vectors of this new mean field preserve the low lying states of thestandard shell model (thus keeping all its benefits) while providinghigh-lying states that induce a large overlap between neutron andproton wave functions. Within this new representations we couldexplain alpha-decay processes in heavy nuclei.

Summary