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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
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 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 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
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.
.
lim | r (r) | | Nlj |
r
out
lj
2
2
The detector of the decaying particle can be considered to be at that
distance The probability rate per second that the particle goes
2 sinθ dθ dφ is
through
a
detector
surface
element
dS=r

r 2
Plj |  ( r ) | vdS
out
lj
v  hk / 
Since
lim | rljout (r) |2 | Nlj |2
r
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
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/(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 BA+C
the formation amplitude F is

F(R) 
*
d

d


 A C B (B )A (A )C (C )
Proton decay
ρ=kr
2(Z 1)e 2

hv
(+)|2 is independent upon l.
Therefore,
T
(χ)=T
/|H
red
1/2
l

Z>67
Z>50
Delion, RL, Wyss, PRL 96, 072502 (2006)
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 decay
212Po(gs)  208Pb(gs)+α, one can write within the pairing vibration
approximation

|212 Po(gs) |210 Po(gs) 210Pb(gs) 
_______
_______
_______
_______
_______
______
______
______
______
______
______
______
______
______
212Po(gs)
208Pb(gs)
G. Dodig-Crnkovic, F. A. Janouch and RJL, PLB139, 143 (1984)
209
Pb
209
Bi
neutrons
protons
F. Janouch and RJL, PRC 25, 2123 (1982)
210Pb(gs)
212
212
210Po(gs)
Po   
208
Pb
Po Pb Po
210
210
F. Janouch and RJL, PRC 27, 896 (1983)
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
, and
Therefore
N=128, Z=84
N=126, Z=84
C. Qi, Andreyev, Huyse, RL, VanDuppen Wyss, PRC C81, 064319 (2010)
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 212Po. 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, 064306 (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.
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 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 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 the
trends shown in the next figure
220Ra
222Ra
224Ra
Vclus(protons)
220Ra
decay
Proton-proton (solid line) and neutron-neutron formation probability
Summary
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 eigenvectors 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.