Spectroscopic factors and Asymptotic Normalization Coefficients from the Source Term Approach and from (d,p) reactions N.K. Timofeyuk University of Surrey

Can shell model be used to calculated spectroscopic factors (SFs) and Asymptotic Normalization Coefficients (ANCs)? Source Term Approach to SFs and ANCs Measuring SFs and ANCs: what (d,p) theories are available in the market ? Non-locality in (d,p) reactions and its influence on SFs and ANCs

Standard shell model works with the wave function  P defined in a subspace P (H PP – E)  P = 0 where H PP is the projection of the many-body Hamiltonian into subspace P. This means that some effective NN interactions should be used. Effective NN interactions are found phenomenologically to reproduce nuclear spectra. If such shell model is to be used to calculate other quantities, the operators that define these quantities should be renormalized. Shell model SFs are calculated using overlaps constructed from wave functions  P (A-1) and  P (A) so all the information about missing subspaces is lost.

Is it possible to recover information about missing subspaces for SF calculations staying within the subspace P? One possible way to do it is to use inhomogeneous equation for overlap function with the source term defined in the subspace P.

SF is the norm of the radial part of the overlap function I lj (r) Overlap function: I lj (r)  C lj W - ,l+½ (2  r)/r C lj is the asymptotic normalization coefficient (ANC), W is the Whittaker function,  = (2με) 1/2, ε is the nucleon separation energy The asymptotic part of the overlap functions I lj (r) is given by

The overlap integral can be found from the inhomogeneous equation source term For radial part of the overlap function the inhomogeneous equation gives:

This matrix elements contain contributions from subspace P and from missing subspace Q. Important!!! Solution of inhomogeneous equation has always correct asymptotic behaviour Overlap integrals and SFs depend on matrix elements Solution of the inhomogeneous equation is (N.K. Timofeyuk, NPA 632 (1998) 19) where G l (r,r ’ ) is the Green’s function. N.K. Timofeyuk, Phys. Rev. Lett. 103, (2009) N.K. Timofeyuk, Phys. Rev. C 81, (2010)

Effective interactions: Two-body M3YE potential from Bertsch et al, Nucl. Phys. A 284 (1977) 399 Where the coefficients V i,ST and a i,ST have been found by fitting the matrix elements derived from the NN elastic scattering data (Elliot et al, NPA121 (1968) 241)

experiment: Shell model Reduction factor (e,e’p) 1.27 ± ħ  (non TI) ± 0.07 p knockout 1.12 ± ħ  (TI) 2.13 (p,d) 1.48 ± ħ  (non TI) O S STA = 1.45 Wave functions: Independent particle model Harmonic oscillator s.p. wave functions Oscillator radius is chosen to reproduce the 16 O radius

Model wave functions are taken from the 0ħ  oscillator shell model Oscillator radius is fixed from electron scattering Centre-of-mass is explicitly removed A  16

SFs: Comparison between the STA and the experimental values

S STA Comparison to variational Monte Carlo calculations

A  16 Spectroscopic factors: STA v SM

Removing one nucleon Adding one nucleon Double closed shell nuclei A  16 Oscillator IPM wave functions are used with ħ  = 41A -1/3 - 25A -2/3 and the M3YE (central + spin-orbit) NN potential N.K. Timofeyuk, Phys. Rev. C 84, (2011)

A A-1 lj S exp /(2j+1 ) STA CBFM SCGFM CCM 16 O 15 N p 1/  p 3/  O 23 N p 1/ O 23 O s 1/  Ca 39 K d 3/  s 1/  Ca 47 K s 1/  d 3/  d 5/  Ni 56 Ni p 3/  Pb 207 Tl s 1/  d 3/  d 5/  g 7/  h 11/  S STA /S IPM : Comparison to other theoretical calculations and to knockout experiments

132 Sn(d,p) 133 Sn, E d = 9.46 MeV ( N. B. Nguyen et al, Phys. Rev. C84, (2011)) Adiabatic model + dispersion optical potential Neutron overlap function Standard WS DOM 132 Sn+n S DOM = 0.72 S STA = 0.68 C 2 DOM = 0.49 C 2 STA = 0.42

Conclusion over Part 1 The reason why shell model SFs differ from experimental ones is that they are calculated through overlapping bare wave functions in the subspace P. Model spaces that are missing in shell model wave function can be recovered in overlap function calculations by using an inhomogeneous equation with shell model source term. The source term approach gives reasonable agreement between predicted and experimental SFs.

A n p d B Transfer reaction A(d,p)B (d,p) reactions help to deduce spin-parities in residual nucleus B to determine spectroscopic factors (SFs) to determine asymptotic normalization coefficients (ANCs) by comparing measured and theoretical angular distributions:

Theory available for  theor (  ). Exact amplitude Where  (+) is exact many-body wave function  B is internal wave function of B  p is the proton spin function  pB is the distorted wave in the p – B channel obtained from optical model with arbitrary potential U pB

Alternative presentation of exact amplitude Where  (+) is exact many-body wave function  pB is exact solution of the Schrodinger equation, in which n-p interaction is absent, and has scattering boundary condition in the p – B channel. However,  pB is obtained not with the p-B potential but with the p-A potential.

The simplest approximation for Distorted wave Born approximation (DWBA):  (+) =  dA  d  A where  dA is the distorted wave in the d – A channel obtained from optical model. In this approximation deuteron is not affected by scattering from target A.

Failure of the DWBA: 116 Sn(d,p) 117 Sn E d =8.22 MeV R.R. Cadmus Jr.,and W. Haeberli, Nucl. Phys. A327, 419 (1979) Deuteron potential:

p A n R d r np Including deuteron breakup: is obtained from three-body Schrodinger equation: U nA and U pA are n-A and p-A optical potentials taken at half the deuteron incident energy E d /2

Solving 3-body Schrödinger equation in the adiabatic approximation. Johnson-Soper model. R.C. Johnson and P.J.R. Soper, Phys. Rev. C1, 976 (1970) Adiabatic assumption: Then the three-body equation becomes Only those part of the wave function, where r np  0, are needed: The adiabatic model takes the deuteron breakup into account as  Anp (R,0) includes all deuteron continuum states. The adiabatic d-A potential

Other methods to solve three-body Schrodinger equation: Johnson-Tandy (expansion over Weinberg state basis) R.C. Johnson and P.C. Tandy, Nucl. Phys. A235, 56 (1974 ) Continuum-discretized coupled channels (CDCC) Faddeev equations

Optical potentials are energy-dependent Optical potentials are non-local because interactions with complex quantum objects are non-local. Projectile can disappear from the model space we want to work with and then reappear in some other place Optical potentials depend on r and r

Non-locality and energy dependence of optical potentials Non-local optical potential: Link between WFs in local and non-local models:... and equivalent local potential: F. Perey and B.Buck, Nucl. Phys. 32, 353 (1962)   0.85 fm is non-locality range

Three-body Faddeev calculations of (d,p) reactions with non-local potentials A. Deltuva, Phys. Rev. C 79, (R) (2009)  c.m. (deg)

Non-locality in DWBA Perey factor distorted waves from local optical model Perey factor reduces the wave function inside nuclear interior.

Adiabatic (d,p) model with non-local n-A and p-A potentials (N.K. Timofeyuk and R.C. Johnson, Phys. Rev. Lett. 110, (2013) and Phys. Rev. C 87, (2013)) U dA = U nA + U pA U C is the d-A Coulomb potential M 0 (0)  0.8  d  0.4 fm is the new deuteron non-locality range  d is the d-A reduced mass

For N = Z nuclei a beautiful solution for the effective local d-A potential exists: where E 0  40 MeV is some additional energy ! ~ 0.86 ~ 1

Where does the large additional energy E 0  40 MeV come from? E 0 to first order is related to the n-p kinetic energy in the deuteron averaged over the range of V np Since V np has a short range and the distance between n and p is small then according to the Heisenberg uncertainty principle the n-p kinetic energy is large.

Effective local adiabatic d-A potentials obtained with local N-A potentials taken at E d /2 (Johnson-Soper potentials) and with non-local N-A potentials U 0 loc

Adiabatic (d,p) calculations with local N-A potentials taken at E d /2 and with non-local N-A potentials

Ratio of (d,p) cross sections at peak calculated in non-local model and in traditional adiabatic model

All above conclusions were obtained assuming that non-local potentials are energy-independent non-local potentials have Perey-Buck form zero-range approximation to evaluate d-A potential applications to N=Z nuclei Work in progress: To extent the adiabatic model for energy-dependent non- local potentials Corrections beyond zero-range To be able to make calculations for N  Z nuclei

Conclusions for Part 2 Non-local n-A and p-A optical potentials should be used to calculate A(d,p)B cross sections when including deuteron breakup Effective local d-A potential is a sum of nucleon optical potentials taken at energies shifted from E d /2 by  40 MeV. Additional energy comes from Heisenberg principle according to which the n-p kinetic energy in short-range components, most important for (d,p) reaction, is large. Non-locality can influence both absolute and relative values of SFs

Conclusions: Overlap functions calculations within a chose subspace must include renormalized operators to account for contributions from missing model spaces. Reaction theory used to extract SFs and ANCs must be improved

Double magic 132 Sn Fully occupied shells: Neutrons: 0s 1/2, 0p 3/2, 0p 1/2, 0d 5/2, 1s 1/2, 0d 3/2, 0f 7/2, 1p 3/2, 0f 5/2, 1p 1/2, 0g 9/2, 0g 7/2, 1d 5/2, 1d 3/2, 2s 1/2, 0h 11/2 Protons: 0s 1/2, 0p 3/2, 0p 1/2, 0d 5/2, 1s 1/2, 0d 3/2, 0f 7/2, 1p 3/2, 0f 5/2, 1p 1/2, 0g 9/2 Final nucleus J  E x (MeV) S STA /S IPM 131 Sn 3/2 + g.s / / / In 9/2 + g.s / / Sn 7/2  g.s /2 

SF of double-closed shell nuclei obtained from STA calculations: Oscillator IPM wave functions are used with ħ  = 41A -1/3 - 25A -2/3 and the M3YE (central + spin-orbit) NN potential A A-1 lj S IPM S exp ( e,e’p ) S STA S STA /S IPM 16 O 15 N p 1/ (13) p 3/ (22) Ca 39 K d 3/ (19) s 1/ (7) Ca 47 K s 1/ (7) d 3/ (16) d 5/ (49) Pb 207 Tl s 1/ (9) d 3/ (22) d 5/ (28) g 7/ (20) A  16 nuclei

DWBA calculations of the 10 Be(d, p) 11 Be Reaction at 25 MeV Optical potentials used don’t reproduce elastic scattering in entrance and exit channels! B. Zwieglinski et al, NPA 315, 124 (1979) When optical potentials reproduce elastic scattering in entrance and exit channels we obtain: N.K. Timofeyuk and R.C. Johnson, PRC 59, 1545 (1999)

Spectroscopic factors obtained using Global systematic of nucleon optical potentials CH89 DOM Woods-Saxon potential used for neutron bound state DOM used for neutron bound state STA

N.K. Timofeyuk, Phys. Rev. C 84, (2011) S STA /S IPM