Horia Hulubei Institute of Physics and Nuclear Engineering

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a- half-lives of ground and excited states of
the heaviest elements
Horia Hulubei Institute of Physics and Nuclear
Engineering RO-077125, Bucharest-Magurele,Romania
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Contents

• Introduction
• Reaction theory for cluster decay
• 2.1 Single channel
• 2.2 Coupled channels
• 3. Details of the calculations
• 3.1 Cluster formation amplitude (CFA)
• 3.2 Reaction energies (Qa)
• 3.3 Eigenvalue finding
• a-decay rates of SHEs 294118 , 291116,290116
• 5. a-decay above the closed shells
• 6. Summary and outlook

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1. Introduction
Challenges in the study of SHEs The increase
of the proton number Z leads to the following
major difficulties - rise of Q-reaction
energies - appearance of many competing decay
channels (a-SF, a-p,? a-cluster ?) - the a-decay
is the dominant decay mode - Fine structure
measurements (GSI) (additional information
about the nuclear structure)
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1. Introduction

• Microscopic study of a-decay rates
• Improving the structure models in order to
  describe essential features and obtain
  spectroscopic information that can then tested
  against data.
• Extending the range of applicability of reaction
  models by using accurate reaction channels
  methods (including antisymmetrisation,
  deformation, resonance scattering, etc)
• Including microscopic fine structure information
  in coupled channel reaction models.

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2. Reaction theory for a-decay(Decay of a
resonance level )
2.1 Single channel The decay width (a-decay of
a single resonance state k into a single decay
channel n)
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2. Reaction theory for cluster decay
- the radial motion of the fragments at
large and small separations - the kinetic
energy of emitted particle FA is the
antisymmetrized projection of the parent wave
function F1(?1) and F2(?2) - the internal
(space-spin) wave functions of the fragments,
is the wave function of the angular motion, is
the inter-fragment antisymmetrizer, r connects
the centers of mass of the fragments, and the
symbol ltgt means integration over the internal
coordinates and angular coordinates of relative
motion. The diagonal elements Vnn of the
potential are given by a sum of nuclear and
Coulomb terms
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2. Reaction theory for cluster decay
For the nuclear potential we use the Woods-Saxon
parameterization The Coulomb potential is
taken of the usual form
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2. Reaction theory for cluster decay
Boundary conditions The lower limit in
integrals is an arbitrary small radius rmin gt 0,
while the upper limit rmax is close to the first
exterior node of . To avoid the usual
ambiguities encountered in formulating the
potential for the resonance tunneling of the
spherical barrier we iterate directly the nuclear
potential in equations of motion. The one-body
(o.b.) resonance width in the single channel
problem can be expressed only with the
eigenvalues and eigenfunctions of the system
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2. Reaction theory for cluster decay
2.2 Coupled channels The relative motion of
the fragments can be strongly influenced by
couplings of the relative motion of the fragments
to several nuclear intrinsic motions. The usual
way to address the effects of coupling between
the intrinsic degrees of freedom and relative
motion is to numerically solve the coupled
channel equations, including all the relevant
channels. The total decay width for the
multichannel decay of the state k into a set
of n different channels is
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2. Reaction theory for cluster decay
The matrix elements Vnm of the coupling
Hamiltonian consist also of nuclear and Coulomb
components. The nuclear component can be
generated by changing the target radius in the
nuclear potential to a dynamical
operator The nuclear coupling term is thus
given by
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2. Reaction theory for cluster decay
The resulting nuclear coupling matrix elements
between states n gt I0 gt and m gt I'0 gt
are Similarly, the Coulomb matrix elements
are then given by In the case in which all
exit channels are open the boundary conditions
should be
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2. Reaction theory for cluster decay
A special type of eigenvalue solution will be
considered here for which the behavior of the
solution in each separate channel is similar to
that of Gn in the one channel problem. The
present analysis of nuclear decay rates is based
on the following key points the shell model and
resonance treatment of cluster formation
amplitudes, account the couplings between the
relative motion of the fragments and several
nuclear collective motions, account the effects
of non-linear couplings to all orders,
self-consistence of the scattering potential, the
rotational excitation of nuclei by the cluster
transfer and the existence and convergence of the
resonance scattering solution. Such an analysis
includes the most important aspects of nuclear
structure and reaction dynamics and permits to
derive conclusions concerning basic nuclear
properties mass, spins, moments, lifetimes.
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3. Details of the calculations
3.1 Cluster formation amplitudes Our goal is
to make microscopic (shell model) estimations for
a-decay rates at the SHEs those nuclear
structure have been investigated extensively in
microscopic approaches such as Skyrme
Hartree-Fock (SHF) theory and Relativistic
Mean-Field (RMF) theory or Macroscopic-Microscopic
(MM) method. The most notable success of the MM
approach has been the Finite-Range Droplet Model
(FRDM) which allows to identify the major magic
numbers in the region of SHEs. The FRDM predicts
a magic proton number at Z114, while
experimental data give little support to this
magic number. All these approaches use the
Strutinsky shell correction method for the
microscopic single-particle shell model
components. To explore the occurrence of magic
number in the region of SHEs, extensive
shell-corrections calculations have been
performed by Kruppa et al. for a set of RMF and
Skyrme forces. We have calculated
microscopically the a-CFA using the single
particle wave functions extracted from
self-consistent calculations of Kruppa. In this
purpose we use the method which has been first
developed by Mang and Rassmusen, for the harmonic
oscillator s.p. wave functions and later extended
to Woods-Saxon wave functions.
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3. Details of the calculations

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3. Details of the calculations

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3. Details of the calculations

According to the shell model of the nucleus the
w. f. of an individual nucleon in the
self-consistent field can be represented in the
form of a product of a space function, a spin
function and an isospin function.
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3. Details of the calculations

The wave function of a system of A nucleons is
represented in the form of a product of a
one-particle w.f. (with space, spin and isospin
components), which, however, should be
anti-symmetrized with respect to interchanges of
any pair of nucleons. This antisymmetrization
will be achieved if we write the w.f. of the
system of A nucleons in the form of a Slater
determinant
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3. Details of the calculations

Protons 2f 7/2, 1j 13/2, 2f 3/2 and 3p 3/2 for
106ltZlt126 Neutrons 2g 7/2 , 2d 5/2, 3p ½ and 1j
13/2 for 164ltNlt184 A.T.Kruppa et al.
Phys.Rev.C61 034313 (2000). Skyrme-Hatree-Fock (
SHF) Relativistic Mean Field (RMF) Macroscopic-mic
roscopic (MM) models (P. Müller et al, At. Data
and Nucl. Data Tab. 66, 131 (1997).)
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3. Details of the calculations
3.2 Reaction Energies Qa Experimental values
Yu. Ts. Oganessian et al, Phys. Rev. C 74,
044602(2006).
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3. Details of the calculations
Calculated Qa values
P. Müller et al, At. Data and Nucl. Data Tab. 66,
131 (1997).
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3. Details of the calculations
3.3 Eigenvalue finding The boundary value
problem has solution only for some particular
values of the parameter chosen as eigenvalue
(denoted by ?). In our approach this adjustable
parameter is the depth of the Woods-Saxon
potential (the energies are fixed)
M. Rizea, Comp. Phys. Commun. 143 (2002) 83-99
Routines DOPRIN,COULCC-Thompson and Barnett
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3. Details of the calculations
The eigensolutions of the coupled channel
Schrödinger equation.
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4. a-decay rates of SHEs 294118 , 291116,290116
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4. a-decay rates of SHEs 294118 , 291116,290116
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4. a-decay rates of SHEs 294118 , 291116,290116
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4. a-decay rates of SHEs 294118 , 291116,290116
Shell moddel calc Empirical calc.
The Gamow plot of calculated for a-halflives of
294118 observed in the 249Cm 48 Ca reactions.
The left plot shows our results, while the right
one presents the Viola-Seaborg empirical
estimates.
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4. a-decay rates of SHEs 294118 , 291116,290116
Shell moddel calc Empirical calc.
The Gamow plot of calculated for a-halflives of
291116 observed in the 249Cf 48 Ca reactions.
The left plot shows our results, while the right
one presents the Viola-Seaborg empirical
estimates.
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4. a-decay rates of SHEs 294118 , 291116,290116
Calculated shell model (full symbols) and one
body (open symbols) Gamow plot for the a-decay
chain of 291116.
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5. a-decay above the closed shells
a
a
a
Experimental a-halflives of the trans-tin(1) and
the trans-lead(2)nuclei and the calculated
a-halflives for the trans- 298114 nuclei,versus
Casten-Zamfir factor P Np Nm/(Np Nn)
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5. a-decay above the closed shells
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6. Summary and outlook

• SHELL MODEL ALPHA HALF-LIVES CALCULATIONS
• comparison to experim.- reasonable
  agreement
• comparison to empirical estimates-
  (Viola-Seeborg) large discrepancies
• acquisition of spectroscopic information,
  fine structure, clustering
• - APPLICATIONS OF THE METHODS IN
• Resonant particle spectroscopy
• Recoil decay tagging (in
  beam-experiments)
• - ALPHA DECAY PROPERTIES are strongly connected
  with VALENCE NUCLONS
• HOMOLOGS from Mendeleev Table HAVE THE SAME
  Chemical and
• Alpha decay properties
• Thanks A. Sandulescu
• W. Scheid
• M. Rizea
• A. Silisteanu

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Thank you ! CSSP 2007 - Sinaia