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Application of correlated basis to a description of continuum states. 19 th International IUPAP Conference on Few-Body Problems in Physics University of Bonn, Germany 31.08 – 05.09.2009. Wataru Horiuchi (Niigata, Japan) Yasuyuki Suzuki (Niigata, Japan). Introduction.

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application of correlated basis to a description of continuum states

Application of correlated basis to a description of continuum states

19th International IUPAP Conference on Few-Body Problems in Physics

University of Bonn, Germany

31.08 – 05.09.2009

Wataru Horiuchi (Niigata, Japan)

Yasuyuki Suzuki (Niigata, Japan)

introduction
Introduction
  • Accurate solution with realistic interactions
    • Nuclear interaction
    • Nuclear structure
    • Some difficulties
      • Realistic interaction (short-range repulsion, tensor)
      • Continuum description

→ much more difficult (boundary conditions etc.)

  • Contents
    • Our correlated basis
    • Method for describing continuum states from L2 basis
    • Examples (n-p, alpha-n scattering)
    • Summary and future works
variational calculation for many body systems
Variational calculation for many-body systems

Hamiltonian

Realistic nucleon-nucleon interactions:central, tensor, spin-orbit

Generalized eigenvalue problem

Basis function

correlated gaussian and global vector
Correlated Gaussian and global vector

Correlated Gaussian

Global vector

x2

x1

x3

Global Vector Representation (GVR)

Parity (-1)L1+L2

advantages of gvr
Advantages of GVR

Variational parameters A, u

→ Stochastically selected

  • No need to specify intermediate angular momenta.
    • Just specify total angular momentum L
  • Nice property of coordinate transformation
    • Antisymmetrization, rearrangement channels

x2

y3

y1

y2

x1

x3

4 he spectrum
4He spectrum

Ground stateenergy

Accuracy ~ 60 keV.

H. Kamada et al., PRC64, 044001 (2001)

3H+p, 3He+n cluster structure appear

W. H. and Y. Suzuki, PRC78, 034305(2008)

P-wave

S-wave

3He+n

3H+p

Good agreement with experiment

without any model assumption

for describing continuum states
For describing continuum states
  • Bound state approximation
    • Easy to handle (use of a square integrable (L2) basis)
    • Good for a state with narrow width
    • Ill behavior of the asymptotics
  • Continuum states
    • Can we construct them in the L2 basis?
      • Scattering phase shift
formalism 1
Formalism(1)

The wave function of the system with E

Key quantity: Spectroscopic amplitude (SA)

A test wave function

Inhomogeneous equation for y(r)

U(r): arbitrary local potential (cf. Coulomb)

formalism 2
Formalism(2)

The analytical solution

G(r, r’): Green’s function

v(r): regular solution

h(r): irregular solution

SA solved with the Green’s function (SAGF)

Phase shift:

test calculations
Test calculations

Relative wave function

  • Neutron-alpha phase shift
  • Minnesota potential + spin-orbit
  • Alpha particle → four-body cal.
  • R-matrix
  • SAGF
  • Neutron-proton phase shift
  • Minnesota potential (Central)
  • Numerov
  • SAGF

The SAGF method reproduces phase shifts calculated with the other methods.

improvement of the asymptotics
Improvement of the asymptotics

Ill behaviors of the asymptotics are improved

n scattering with realistic interactions
α+n scattering with realistic interactions

Interactions:AV8’

(Central, Tensor, Spin-orbit)

Alpha particle → four-body cal.

Single channel calculation with α+n

S. Quaglioni, P. Navratil,

PRL101, 092501 (2008)NCSM/RGM

K. M. Nollett et al.

PRL99, 022502 (2007)

Green’s function Monte Carlo

  • 1/2+  → fair agreement
  • 1/2-, 3/2-→ fail to reproduce
    • distorted configurations of alpha
    • three-body force
summary and future works
Summary and future works
  • Global vector representation for few-body systems
    • A flexible basis (realistic interaction, cluster state)
    • Easy to transform a coordinate set
  • SA solved with the Green’s function (SAGF) method
    • Easy (Just need SA)
    • Good accuracy
  • Possible applications (in progress)
    • Coupled channel
      • Alpha+n scattering with distorted configurations (4He*+n, t+d, etc)
    • Extension of SAGF to three-body continuum states
      • E1 response function (cf. 6He in an alpha+n+n)
        • Complex scaling method (CSM)
        • Lorentz integral transform method (LIT)
    • Four-body continuum
      • Four-body calculation with the GVR
        • Electroweak response functions in 4He (LIT, CSM)
decomposition of the phase shift
Decomposition of the phase shift

Neutron-alpha scattering with 1/2+

Vc: central, tensor, spin-orbit