Nuclear structure versus
α-clustering and α-decay
Doru S. Delion
Alexandru Dumitrescu
(Bucharest)
Collaboration: R.J. Liotta (Stockholm), P. Schuck (Orsay)
IFIN-HH
Depart. Theor. Phys.
Outline
I. Gamow picture of the α-decay
II. Beyond the Gamow approach: α-clustering &
universal law for reduced widths
III. Coherent state model (CSM) for
description of even-even nuclei
IV. Coupled channel description of α-transitions
to excited states within CSM
V. Surface α-clustering in 212Po
VI. Conclusions
Creation of heavy nuclei by
supernovae explosion
Most of nuclei are unstable:
nuclear decay modes
induced by strong interaction:
proton emission
two-proton emission
neutron emission
α-decay
cluster decay (C,O, Ne, Mg, Si)
binary & ternary fission
I. Gamow picture of the α-decay
Geiger-Nuttall law for half-lives
ZD
log10 T  a
b
E
 H. Geiger and J.M. Nuttall "The ranges of the α particles from
various radioactive substances and a relation between range and period
of transformation," Philosophical Magazine, Series 6, vol. 22, no. 130,
613-621 (1911).
 H. Geiger and J.M. Nuttall "The ranges of α particles from
uranium," Philosophical Magazine, Series 6, vol. 23, no. 135, 439-445
(1912).
George Gamow in 1909,
two years before
the discovery of the G-N law
… and in 1930,
two years after
his explanation
G. Gamow "Zur Quantentheorie des Atomkernes" (On the quantum theory of
the atomic nucleus), Zeitschrift für Physik, vol. 51, 204-212 (1928).
The first probabilistic interpretation
of the wave function
Rext
↓
int 
Internal region
 Next
External region
External wave function describes
a decaying state
Decay law
ext ( R, t )  n(t )  n0 e  t
2
implies a
wave function
ext ( R, t )  ext ( R)e  i ( E /  )t
with complex
energy

E  E0  i
2
Decay constant is
proportional to
decay width:



Narrow decaying resonance (Γ<<E0)
is a quasi-stationary process
CONDITIONS OF A NARROW
RESONANT STATE:
1) External part is an outgoing wave
It is called Gamow state
≡ spherical outgoing
Coulomb-Hankel wave
ext ( R) 
H
()
l
i ( kR l )
(kR)
e
R


R
R
2) Internal part is an “almost bound state”
It can be normalized to unity
in the internal region
Rext
 R
int
0
( R) dR  1
2
Matching condition
between external and internal functions
Ψint (R)  NΨ ext (R )
determines the scattering amplitude N
It has very small values
int ( R )
10
N 
 10
ext ( R )
Decay width is the flux
of outgoing particles
int ( R )
  v N   v
ext ( R )
2
2
The width does not depend on the matching radius R
because both functions satisfy
the same Schrödinger equation
Half life is given by:
 ln 2
T

Decay width
can be rewritten
as a product between
reduced width squared
and penetrability on
the matching radius R
Γ  2γ 2 P
2

2
2
γ 
int ( R)
2m R
κR
 dχ
P

ce
2
()
H 0 ( , kR)
depending exponentially
opon the Coulomb parameter
2Z D Z C
2Z D Z C


v
 2E / m
Geiger-Nuttall law
in terms of the Coulomb parameter
log10 Γ  log10 P  log10 2γ 2
Geiger-Nuttal law supposes
a constant reduced width
log10 P  aχ  b
ZD
 c
E
Geiger-Nuttall law for α-decay
The generalization of
the Geiger-Nuttall
law is the
Viola-Seaborg rule
a1Z D  a2
 b1Z D  b2
E
 log10  ln 2
log10 T 
 log10 P  log10 2 2
Geiger-Nuttall law for cluster-decays
Magic radioactivity
“Pb decay”
ZD ̴ 82
Viola-Seaborg rule
generalized to
cluster decays
log10 T 
a1Z D Z C  a2
 b1Z D Z C  b2
E
Geiger-Nuttall law for proton emission
D.S. Delion, R.J. Liotta, R. Wyss, Systematics of proton emission,
Physical Review Letters 96, 072501 (2006)
Reduced half-life
corrected by the
centrifugal barrier
Tred
T
 2
Cl
Which is the analog
of the V-S rule ?
log10 Tred  a
ZD
 b( Z )
E
II. Beyond the Gamow approach
D.S. Delion
Universal decay rule for reduced widths
Physical Review C80, 024310 (2009).
Evidence of the α-clustering
on nuclear surface
 α-clustering phase diagram
 Potential within the Two Center Shell Model
 Pairing estimate of the formation amplitude
Phases of the nuclear matter
Phase diagram for deuteron
and α-particle
G. Ropke, A. Schnell,
P. Schuck, P. Nozieres,
Four-particle condensate
in strongly coupled
fermion systems,
Phys. Rev. Lett. 80,
3177 (1998).
Microscopic α-particle formation probability
within the pairing approach
P   D
F ( R)    D P
D , P  BCS
  e
D,P
  ( r2  r2  r2 ) / 2
1/ 2 1/ 2
Two-center shell model
double humped barrier
M. Mirea,
Private communication
Cluster-daughter interaction
should be pocket-like on the nuclear surface
Conditions for an α-particle moving
in a shifted harmonic oscillator potential
The first eigenstate
energy is the Q-value
Its wave function is given by
where the oscillator parameter is
1
Q  E  
2
 ( R )  A0 e
m


  ( R  R0 ) 2 / 2
Universal law for the reduced widths
2
2 2
log
e

A0
2
10
log10   
V frag  log10

2emRB
does not depend on the pocket radius
and remains valid for any potential
Viola-Seaborg rule is given by the
fragmentation potential
V frag
Z D ZC

Q
RB
Reduced width for α-decay
from even-even nuclei
n
Z
N
1
Z<82
50<N<82
2
3
82<N<126
Z>82
82<N<126
4
126<N<152
5
N>152
Reduced width for cluster decays
Blendowske rule
for the spectroscopic
factor
( AC 1) / 3
S  S
can be explained
In terms of the
fragmentation
potential
S  c
2
Reduced width for proton emission
is divided in two regions:
dependence on fragmentation potential (b) explains
the two regions in the Geiger-Nuttall law
Reduced width
has two regions
with respect to the
quadrupole deformation
and has two regions
with respect to the
fragmentation potential
III. Coherent states
were introduced to describe collective excitations of deformed nuclei by
P.O. Lipas and J. Savolainen, Nucl. Phys. A130, 77 (1969).
Coherent State Model (CSM) was introduced by
A.A. Raduta and R.M. Dreizler, Nucl. Phys. A258, 109 (1976).
CSM was successfully used as a formalism
to describe low-lying, as well as high-spin states
in spherical, transitional and rotational even-even nuclei.
Generalised CSM was used to describe proton-neutron
excitations like scissors M1 and M3 modes by
A.A. Raduta and D.S. Delion, Nucl. Phys. A491, 24 (1989).
Coherent State Model (CSM)
Nuclear surface is described in terms of collective coordinates
A deformed wave function can be expanded
in terms of quadrupole collective coordinates:
Boson representation
leads to a state of a coherent type
in the intrinsic system of coordinates
where the deformation parameter is proportional
to the standard quadrupole deformation
Ground state band is given by
projecting out the intrinsic coherent state
where
Normalisation is given by integration
The expectation value
of the number of phonons operator
is written in terms of the following ratio:
Energy versus deformation parameter d
has a vibrational shape for small d
and rotational behavior for large d
Electromagnetic transitions
Transition operator
B(E2) values
Effective charge
Daugher nuclei
for even-even
α-emitters
with known
branching ratios
to excited states
Deformation parameter fitting energies versus
(a) standard quadrupole deformation
(b) Casten parameter
Hamiltonian parameter versus
deformation parameter
Rigidity parameter versus
deformation parameter
Energy ratios versus deformation parameter
in are universal functions
Effective charge versus
deformation parameter
IV. Coupled channels description
of α-transitions to excited states within CSM
Schrodinger equation
has a Hamiltonian containing the sum of
kinetic, daughter and α-daughter terms:
α-daughter potential
Spherical term is given by the double folding
and repulsion (simulating Pauli principle) terms
Quadrupole term is given by QQ interaction
between daughter nucleus and α-particle
Double folding plus repulsion
(simulating Pauli principle) potentials
Wave function
has the total angular momentum = 0
Coupled channels method
where the matrix of the system is given by:
in terms of the channel reduced radii and momenta
Reduced matrix element
of the α-daughter potential
is given in terms of the CSM normalisation factors
The effective coupling strength
dependends linearly on deformation parameter:
Fundamental outgoing resonant states
In the internal region
asymptotics at small distances is regular:
In the external region
asymptotics at large distances is given
by Gamow (Coulomb-Hankel) outgoing waves:
General solution is given
by the superposing fundamental resonances
Matching between internal and external solutions
at some radius R1
and their derivatives
leads to:
The secular equation for
outgoing resonant states
because the regular waves are much smaller
than the regular ones inside the barrier.
Resonant states are normalized in the internal region:
Channel decay widths
By using continuity equation one obtains
total width as a sum of partial widths:
where channel velocity is given by:
Channel intensities
define the strength of α-transitions
to some excited state with spin J
The only free parameter is the α-daughter coupling strength
which can be determined by the I2 value for each transition
α-daughter coupling strength reproducing I2 versus
deformation parameter (a)
and mass number (b)
α-daughter coupling strength versus
reduced width squared
α-daughter coupling strength versus
the difference N-Nmagic
Nmagic = 126, N>126
= 82, N<126
α-transition intensities versus
decay label in the table of α-emitters
Hindrance factors
α-transition intensities for chains
with worse predictions for I4
versus deformation parameter (a), (c)
and rigidity parameter (b), (d)
V. Surface α-clustering in 212Po
D.S. Delion, R.J. Liotta, P. Schuck, A. Astier, and M.-G. Porquet
Phys. Rev. C85, 064306 (2012)
Positive parity states
Negative parity states
Single particle basis
contains two components
where the cluster component is given
by a Gaussian centered on the nuclear surface
containing components with larger
principal quantum number N~8,9,10
Transition operator is proportional
to the principal quantum number
Surface clustering states
have large values
of the ang. momentum
and N > 8
Bound states
have low values
of the ang. momentum
and N < 8
Surface α-clustering explains
large electromagnetic transitions in 212Po
B(E2:J+2J)-values
B(E1:I-J+)-values
Surface α-clustering in 212Po
explains decay width between ground states
Formation probability
versus cm radius
(a): total
(b): cluster component
Log (width / exp. )
versus cm radius
VI. Conclusions
1) A pocket-like α-daughter interaction leads to
the universal law for reduced widths:
Log reduced width depends linearly on the fragmentation potential
which is fulfilled by all cluster and particle emission processes
2) CSM is able to decribe energy levels and B(E2)-values in
vibrational, transitional and rotational even-even nuclei
in terms of the deformation parameter d.
3) CSM describes α-transitions to excited states these nuclei
by using an universal α-daughter QQ interaction,
predicting a linear dependence of the α-daughter strengths
on the deformation parameter d.
4) The α-daughter strength is proportional to reduced width squared
and has the largest value in the region above 208Pb, where
the α-clustering explains large B(Eλ) values in 212Po.
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