andrey shirokov (moscow state univ.) in collaboration with alexander mazur (pacific national univ.)...
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Andrey Shirokov (Moscow State Univ.)
In collaboration with
Alexander Mazur (Pacific National Univ.)
Pieter Maris and James Vary (Iowa State Univ.)
INT, Seattle, June 8, 2011
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*Conventional: bound state energies are associated with variational minimum in shell model, NCSM, etc., calculations
*Is it also true for resonant states? Can we get resonance width from such calculations?
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*Resonant states: should we, probably, study excitation energies instead? Or the energies E = EA − EA − 1 for n−(A−1) scattering (or, generally, with A1 + A2 = A)?
*Is it important for them to be stable with respect to ħΩ or Nmax variation?
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*Would be nice to have a simple answer from conventional calculations without doing, say, complicated NCSM−RGM calculations.
*So, what are the general properties of eigenstates in continuum consistent with resonance at the energy Er and width Γ?
*Some observations, examples follow; this is still work in progress.
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*So, what are the general properties of eigenstates in continuum consistent with resonance at the energy Er and width Γ?
*Some observations, examples follow; this is still work in progress.
*I discuss some general properties for oscillator basis calculations; this is the only relavance to NCSM
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*O.Rubtsova, V.Kukulin, V.Pomerantsev, JETP Lett. 90, 402 (2009); Phys. Rev. C 81, 064003 (2010):
*I.M.Lifshitz (1947):
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*O.Rubtsova, V.Kukulin, V.Pomerantsev, JETP Lett. 90, 402 (2009); Phys. Rev. C 81, 064003 (2010):
*I.M.Lifshitz (1947):
So, the phase shift at the eigenenergies Ej can be easily calculated!
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*O.Rubtsova, V.Kukulin, V.Pomerantsev, JETP Lett. 90, 402 (2009); Phys. Rev. C 81, 064003 (2010):
*I.M.Lifshitz (1947):
Unfortunately, this does not work:The dimensionality of the matrix is small, the average spacing between the levels is not well-defined.One needs sometimes Dj value below the lowest Ej
0
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Direct and inverse problem
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*J-matrix inverse N-nucleus scattering analysis suggests values for resonant and non-resonant states that should be compared with that obtained in NCSM
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*J-matrix:
Let us try to extract resonance information from Eλ behavior only
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Eλ should increase with ħΩ
Within narrow resonance Eλ is nearly ħΩ-independent
The slope of Eλ(ħΩ) depends however on Nmax, l, Eλ value
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*Breit-Wigner:
*Simple approximation: φ=0
Derivatives calculated through
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*Breit-Wigner:
*Simple approximation: φ=0
Derivatives calculated through
Do not expect to get a reasonable result for Er or Γ if Γ/2Δ is small!
If |Γ/2Δ| is large, we get good results for Er, Γ and φ.
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*What can we do if we obtain Eλ in a non-resonant region above the resonance?
*We can extrapolate energies to larger (finite) Nmax value when Eλ is in the resonant region.
*Expected dependence is
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This works. However this extrapolation seems to be unstable and inconvenient
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More convenient is an exponential extrapolation.
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*We get stable Er and Γ; Γ is too small as compared with experiment.
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*I discussed general features of continuum states obtained in many-body calculations with oscillator basis.
*The best way to compare the calculated results with experiment is to use “experimental” phase shifts and get Eλ consistent with scattering data using simple inverse scattering technique.
*Studying ħΩ dependence of Eλ obtained in NCSM, one can get resonance energy and width. However, usually an extrapolation to a reasonable Nmax value is required.
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