Nuclear structure versusα-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
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-decayGeiger-Nuttall law for half-lives • 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).
… and in 1930, two years after his explanation George Gamow in 1909, two years before the discovery of the G-N law
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 ↓ Internal region External region
External wave function describesa decaying state Decay law implies a wave function with complex energy Decay constant is proportional to decay width:
CONDITIONS OF A NARROW RESONANT STATE:1) External part is an outgoing wave It is called Gamow state ≡ spherical outgoing Coulomb-Hankel wave
2) Internal part is an “almost bound state” It can be normalized to unity in the internal region
Matching conditionbetween external and internal functions determines the scattering amplitude N It has very small values
Decay width is the fluxof outgoing particles The width does not depend on the matching radius R because both functions satisfy the same Schrödinger equation Half life is given by:
Decay widthcan be rewritten as a product between reduced width squared and penetrability on the matching radius R depending exponentially opon the Coulomb parameter
Geiger-Nuttall law in terms of the Coulomb parameter Geiger-Nuttal law supposes a constant reduced width
Geiger-Nuttall law for α-decay The generalization of the Geiger-Nuttall law is the Viola-Seaborg rule
Geiger-Nuttall law for cluster-decays Magic radioactivity “Pb decay” ZD ̴ 82 Viola-Seaborg rule generalized to cluster decays
Geiger-Nuttall law for proton emissionD.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 Which is the analog of the V-S rule ?
II. Beyond the Gamow approach D.S. Delion Universal decay rule for reduced widths Physical Review C80, 024310 (2009).
Evidence of the α-clusteringon nuclear surface • α-clustering phase diagram • Potential within the Two Center Shell Model • Pairing estimate of the formation amplitude
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 probabilitywithin the pairing approach
Two-center shell modeldouble humped barrier M. Mirea, Private communication
Cluster-daughter interactionshould be pocket-like on the nuclear surface
Conditions for an α-particle movingin a shifted harmonic oscillator potential The first eigenstate energy is the Q-value Its wave function is given by where the oscillator parameter is
Universal law for the reduced widths does not depend on the pocket radius and remains valid for any potential Viola-Seaborg rule is given by the fragmentation potential
Reduced width for cluster decays Blendowske rule for the spectroscopic factor can be explained In terms of the fragmentation potential
Reduced width for proton emissionis 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:
Bosonrepresentation 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 byprojecting out the intrinsic coherent state where Normalisation is given by integration
The expectation valueof 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
IV. Coupled channels descriptionof 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
Coupled channels method where the matrix of the system is given by: in terms of the channel reduced radii and momenta