Continuum Shell Model Alexander Volya Florida State University This work was done with Vladimir Zelevinsky Supported by DOE and NSF.
Outline Continuum Shell Model Basic Theory Reaction Formalism One-body and Two-Body decays Applications of CSM Two-body decay of 11 Li Study of isotope chains Other applications
Basic Theory [1] C. Mahaux and H. Weidenmüller, Shell-model approach to nuclear reactions, North-Holland Publishing, Amsterdam 1969
Non-Hermitian eigenvalue problem Breaking of T-invariance Right eigenvalue Left Eigenvalue Same Energy Momentum k ~ Momentum k=-k T-conjugated states are not degenerate (left and right) Q-space outgoing waves Normalization and expectation values
Complex energy: meaningful or meaningless? Eigenvalue problem has only complex (E>threshold) roots but E is real? Definitions of resonance Gamow: poles of scattering matrix Eigenvalue proglem with regular inside outgoing outside boundary condition discrete resonant states + complex energies Breit-Wigner: Find roots on real axis Cross section peaks and lifetimes
Interpretation of complex energies For isolated narrow resonances all definitions agree Real Situation Many-body complexity High density of states Large decay widths Result: Overlapping, interference, width redistribution Resonance and width are definition dependent Non-exponential decay Solution: Cross section is a true observable (S-matrix )
Continuum Shell Model Hamiltonian PHP Internal, one-body + two-body Original shell model, adjusted, tested (USD and others) Exact shell model for bound states E<threshold QHQ - External, one-body (presently) kinetic energy + long range interaction, (plane waves or Bessel functions, Coulomb functions) QHP Cross space, one-body + two-body Responsible for coupling of spaces
One-body decay Continuum channel State α in N-1 nucleon daughter Particle in continuum state j Energy E=E α +ε j Transition Amplitude
Single-particle scattering problem The same non-hermitian eigenvalue problem Internal states: External states: Single-particle decay amplitude Single particle decay width: (requires definition and solution for resonance energy)
Study of square well Poles of scattering matrix Evolution of poles Low-energy limit: (l>0)
Scattering calculation using Woods-Saxon with size parameters adjusted for 16 O One-body decay realistic one-body potential
Properties on W General properties Energy dependent Not reducible to one or two body operator Due to symmetry one-body is diagonal (general matrix W is factorizable) If decay dominated by one final state If decay is weak do perturbation theory for W (diagonalize real part only)
Two-body decay: sequential Mediated by s.p. part of QHP Virtual N-1 state from previous solution Width at low energy
Two-body decay: Direct Mediated by 2-body part of QHP V(r,r ) effective pair potential Width at low energy
11 Li model Dynamics of two states coupled to a common decay channel Model H Mechanism of binding by Hermitian interaction
11 Li model; energy-dependent widths Coupling to continuum is parameterized Squeezing of phase-space volume in s and p waves, Threshold E c =0 Model parameters: ε 1 =100, ε 2 =200, A 1 =7.1 A 2 =3.1 (red); α=1, β=0.05 (blue) (in units based on kev) Upper panel: Energies with A 1 =A 2 =0 (black)
Scattering and cross section near threshold Scattering Matrix Solution in two-level model Cross section
Realistic Shell Model Example Interaction PHP Shell model Hamiltonian, USD interaction Assume that USD includes Hermitian QHQ and QHP one-body Woods-Saxon potential QHP two-body phenomenological parameterization Solution From 4 He up to 10 He From 16 O up to 28 O Given guess energy E for state α, W(E) is constructed by considering all open channels. Non-Hermitian Hamiltonian is solved for iteratively for new E (Breit-Wigner resonance condition)
CSM calculations for He Interaction in PP space CKIHE PRC37, 2220 (1988).
Oxygen Isotopes Continuum Shell Model Calculation sd space, USD interaction single-nucleon reactions
Comparison with experiment A J p Experiment CSM calculation E(MeV) G (kev) E(MeV) G (kev) 17 O 3/2 + 1.646 96 1.646 102 18 O 1 + 8.817 70 10.820 86 19 O 3/2 + 6.119 110 5.530 90 Features No free parameters Bound states identical to SM Decaying states shift and acquire width
Features of Continuum Shell Model Extension of traditional shell model Non-Hermitian and energy-dependent effective Hamiltonian: Self-consistency Decay chains Open channels Many-body structure Reaction theory New phenomena Superradiance and narrow resonances
References A. Volya and V. Zelevinsky, Phys. Rev. C 67 (2003) 054322 N. Auerbach, V. Zelevinsky, and A. Volya, Phys. Lett. B 590 (2004) 45 V. Zelevinsky and A. Volya, nucl-th/0406019 A. Volya and V. Zelevinsky, J. Opt. B. 5 (2003) S450