Spectroscopic factors and Asymptotic Normalization Coefficients

from the Source Term Approach and from (d,p) reactions

N.K. TimofeyukUniversity 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

(HPP – E) P = 0

where HPP 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 Ilj(r)

Overlap function:1

†1 2 1 2 1 1( ) ... ( ,... ) ( ,... )A

AA A AI d d r x x x x x x

Ilj(r) Clj W-,l+½ (2r)/r

Clj 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 Ilj(r) is given by

The overlap integral can be found from the inhomogeneous equation

1

21

1/ 2

( ) ( ) ( )

ˆˆ( )A A

A

Al lj lj

lj l J Jj J

Z eT I r v r

r

v r Y r

V

1 1

1

1 1

ˆA A

i A A ANN i A

i Ai i

e e Z e eV r r

r r r

V= =

source term

1 1 1 1 1A A A A A A A A A AT T E E V V

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 Gl (r,r’) is the Green’s function.

N.K. Timofeyuk, Phys. Rev. Lett. 103, 242501 (2009)N.K. Timofeyuk, Phys. Rev. C 81, 064306 (2010)

Effective interactions:

Two-body M3YE potential from Bertsch et al, Nucl. Phys. A 284 (1977) 399

Where the coefficients Vi,ST and ai,ST have been found by fitting the matrix elements derived from the NN elastic scattering data (Elliot et al, NPA121 (1968) 241)

3 3( ) ( ) ( ) ( ), , , ,

1 1

32 ( ) ( )

, , 121

ˆ ˆ( ) ( ) ( )( )

ˆ ˆ( )

ST c c ls lsi ST i ST ST i T i T T

i i

t ti T i T T

i

V r V Y x P V Y x l s P

r V Y x S P

( ) exp( ) / , j ji iY x x x x a r

experiment: Shell model Reduction factor (e,e’p) 1.27 ± 0.13 0ħ (non TI) 2.0 0.64 ± 0.07p knockout 1.12 ± 0.07 0ħ (TI) 2.13(p,d) 1.48 ± 0.16 4ħ (non TI) 1.65

16O

SSTA = 1.45

Wave functions:• Independent particle model• Harmonic oscillator s.p. wave functions• Oscillator radius is chosen to reproduce the 16O 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

A 16

SFs: Comparison between the STA and the experimental values

SSTA

0.6000.319

Comparison to variational Monte Carlo calculations

A 16Spectroscopic 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, 054313 (2011)

A A-1 lj Sexp /(2j+1) STA CBFM SCGFM CCM

16O 15N p1/2 0.64 0.07 0.73 0.89 0.8 0.9 p3/2 0.56 0.06 0.65 0.89 0.8 0.9 24O 23N p1/2 0.59 0.61-0.65 24O 23O s1/2 0.87 0.10 0.83 0.92 40Ca 39K d3/2 0.650.05 0.76 0.85 0.8 s1/2 0.520.04 0.58 0.87 0.8 48Ca 47K s1/2 0.540.04 0.69 0.84 0.36 d3/2 0.570.04 0.68 0.86 0.59 d5/2 0.110.02 0.71 0.85 57Ni 56Ni p3/2 0.580.11 0.59 0.65 208Pb 207Tl s1/2 0.490.74 0.74 0.85 d3/2 0.580.06 0.72 0.83 d5/2 0.490.05 0.73 0.83 g7/2 0.260.03 0.61 0.82 h11/2 0.570.06 0.48 0.82

SSTA/SIPM: Comparison to other theoretical calculations and to knockout experiments

132Sn(d,p)133Sn, Ed = 9 .46 MeV (N. B. Nguyen et al, Phys. Rev. C84, 044611 (2011))

Adiabatic model + dispersion optical potential

Neutron overlap function

Standard WSDOM

132Sn+n

SDOM = 0.72SSTA = 0.68

C2DOM = 0.49 C2

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

np

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:

exp

theor

( )

( )

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 UpB

( )( , )d p pB p B np ip pB

i B

T V V U

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.

( )( , )d p pB npT V

,pB pB p B B

The simplest approximation for

Distorted wave Born approximation (DWBA):

(+) = dAdA

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.

( )( , )d p pB p B npT V

( , )DWBAd p pB p B np dA d AT V

Failure of the DWBA: 116Sn(d,p)117Sn Ed=8.22 MeV

R.R. Cadmus Jr.,and W. Haeberli, Nucl. Phys. A327, 419 (1979)

Deuteron potential:

p

A

n

R

drnp

( , ) 0R np np nA pA Anp npT T V U U E R r

Including deuteron breakup:

is obtained from three-body Schrodinger equation:

UnA and UpA are n-A and p-A optical

potentials taken at half the deuteron

incident energy Ed /2

( ) ( , )Anp np A R r

( , )Anp np R r

( )( , ) ( ) ( , )d p pB p B np np Anp np AT V r R r

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 rnp 0, are needed:

0),( npAnpdnpnp VT rR

( , ) 0,R nA pA d Anp np d dT U U E E E R r

( ( ) ( ) ) ( ,0) 0R nA pA d AnpT U U E

R R R

The adiabatic model takes the deuteron breakup into account as Anp(R,0)

includes all deuteron continuum states.

The adiabatic d-A potential

( ) ( ) 0R dA d dAT U E R R

2

2 2( ) ( ) ( ) ( ) ( )np np

dA np d np np np nA pAU d V U U

r r

R r r r R R

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

( )NL NLT E d V r r' r,r' r'

Non-locality and energy dependence of optical potentials

Non-local optical potential: ( )NL NLT E d V r r' r,r' r'

Link between WFs in local and non-local models:

32/3

22 /)(exp

2)(

r'rr'rr'r, NUV

( )L L LT E U r r r

)(2

exp2

2

rUrUErU NLL

... and equivalent local potential:

F. Perey and B.Buck, Nucl. Phys. 32, 353 (1962)

0.85 fm is non-locality range

2

2exp ( )

4NL L Lr U r r

Three-body Faddeev calculations of (d,p) reactions with non-local potentialsA. Deltuva, Phys. Rev. C 79, 021602(R) (2009)

0 60 120 180 c.m. (deg)

2 2

2 24 4( , ) e e

p p d dpB dAU r U r

DWBAd p pB p B np dA d AT V

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, 112501 (2013) and Phys. Rev. C 87, 064610 (2013))

0

d dA loc dAT E U r r r

UdA = UnA + UpA

UC is the d-A Coulomb potential

M0(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

E0 40 MeV is some additional energy !

~ 0.86~ 1

Where does the large additional energy E0 40 MeV come from?

E0 to first order is related to the

n-p kinetic energy in the deuteron averaged over the range of Vnp

Since Vnp 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 Ed /2 (Johnson-Soper potentials) and with non-local N-A potentials U0

loc

Adiabatic (d,p) calculations with local N-A potentials taken at Ed /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 Ed /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 132Sn

Fully occupied shells: Neutrons: 0s1/2, 0p3/2, 0p1/2, 0d5/2, 1s1/2, 0d3/2, 0f7/2, 1p3/2, 0f5/2, 1p1/2, 0g9/2, 0g7/2, 1d5/2,

1d3/2, 2s1/2 , 0h11/2

Protons: 0s1/2, 0p3/2, 0p1/2, 0d5/2, 1s1/2, 0d3/2, 0f7/2, 1p3/2, 0f5/2, 1p1/2, 0g 9/2

Final nucleus J Ex (MeV) SSTA/SIPM

131Sn 3/2+ g.s. 0.80 1/2+ 0.332 0.83 5/2+ 1.655 0.81 7/2+ 2.434 0.75

131In 9/2+ g.s. 0.64 1/2+ 0.30 0.74 3/2+ 1.29 0.74

133Sn 7/2 g.s. 0.68 3/2 0.854 0.72

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 SIPM Sexp (e,e’p) SSTA SSTA/SIPM

16O 15N p1/2 2.0 1.27(13) 1.45 0.73 p3/2 4.0 2.25(22) 2.61 0.65 40Ca 39K d3/2 4.0 2.58(19) 2.90 0.73 s1/2 2.0 1.03(7) 1.15 0.58 48Ca 47K s1/2 2.0 1.07(7) 1.38 0.69 d3/2 4.0 2.26(16) 2.70 0.68 d5/2 6.0 0.683(49) 4.21 0.71 208Pb 207Tl s1/2 2.0 0.98(9) 1.48 0.74 d3/2 4.0 2.31(22) 2.88 0.72 d5/2 6.0 2.93(28) 4.38 0.73 g7/2 8.0 2.06(20) 4.88 0.61

A 16 nuclei

DWBA calculations of the 10Be(d, p)11Be 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

0.52

0.67

0.68

0.64

N.K. Timofeyuk, Phys. Rev. C 84, 054313 (2011)

SSTA/SIPM

0.620.610.75 1.10 0.63 0.77