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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 Next 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 ( r2 r2 r2 ) / 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+2J)-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. THANK YOU !