application of correlated basis to a description of continuum states 19 th international iupap...
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Application of correlated basis to a description of continuum
states19th International IUPAP Conference on Few-
Body Problems in Physics
University of Bonn, Germany31.08 – 05.09.2009
Wataru Horiuchi (Niigata, Japan)Yasuyuki Suzuki (Niigata, Japan)
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
Hamiltonian
Basis function
Realistic nucleon-nucleon interactions: central, tensor, spin-orbit
Generalized eigenvalue problem
Correlated Gaussian and global vector
Global Vector Representation (GVR)
x1 x3
x2
Correlated Gaussian
Global vector
Parity (-1)L1+L2
Advantages of GVR
• No need to specify intermediate angular momenta.– Just specify total angular momentum L
• Nice property of coordinate transformation– Antisymmetrization, rearrangement channels
Variational parameters A, u → Stochastically selected
x1 x3
x2 y1 y2y3
4He spectrum
Good agreement with experiment without any model assumption
3H+p, 3He+n cluster structure appearW. H. and Y. Suzuki, PRC78, 034305(2008)
P-wave
S-wave3H+p
3He+n
Ground state energy Accuracy ~ 60 keV.H. Kamada et al., PRC64, 044001 (2001)
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)
Key quantity: Spectroscopic amplitude (SA)
The wave function of the system with E
A test wave function
U(r): arbitrary local potential (cf. Coulomb)
Inhomogeneous equation for y(r)
Formalism(2)The analytical solution
G(r, r’): Green’s function
SA solved with the Green’s function (SAGF)
Phase shift:
v(r): regular solutionh(r): irregular solution
Test calculations
Neutron-proton phase shiftMinnesota potential (Central)• Numerov• SAGF
Neutron-alpha phase shiftMinnesota potential + spin-orbitAlpha particle → four-body cal.• R-matrix• SAGF
The SAGF method reproduces phase shifts calculated with the other methods.
Relative wave function
α+n scattering with realistic interactions
Interactions: AV8’ (Central, Tensor, Spin-orbit)Alpha particle → four-body cal. Single channel calculation with α+n
1/2+ → fair agreement1/2-, 3/2- → fail to reproduce
• distorted configurations of alpha• three-body force
K. M. Nollett et al.PRL99, 022502 (2007)Green’s function Monte Carlo
S. Quaglioni, P. Navratil,PRL101, 092501 (2008)NCSM/RGM
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)
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