# a shell-model representation to describe radioactive decay

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A shell-model representation to describe radioactive decay. Radioactive Decay width. 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 - PowerPoint PPT PresentationTRANSCRIPT

Influence of the work of Berggren on the Stockholms group

A shell-model representation to describe radioactive decayIn 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 formRadioactive 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 sin d d is

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.

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

Rlj(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.lj(R) is the cluster formation amplitude and kR/(F2lj(R)+G2lj(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

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

Delion, RL, Wyss, PRL 96, 072502 (2006)Z>67Z>50

Universal decay law in cluster radioactivityGeiger-Nuttall law

As before, we have For l=0 transitions

cos2