Transcript test

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
(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.

†
Overlap function: I (r )  dx1...dx A2  A1 ( x1 ,... x A2 ) A ( x1 ,... x A1 )
SF is the norm of the radial part of the overlap function Ilj(r)
The asymptotic part of the overlap functions Ilj(r) is given by
Ilj(r)  Clj W-,l+½ (2r)/r
Clj is the asymptotic normalization coefficient (ANC),
W is the Whittaker function,
 = (2με)1/2 , ε is the nucleon separation energy
The overlap integral can be found from the inhomogeneous equation
 A1 TA  TA1  E A  E A1  A   A1 VA1  VA  A
For radial part of the overlap function the inhomogeneous equation gives:
Z A1e2
(Tl 
  ) Ilj (r )  vlj (r )
r
source term
vlj (r )   Yl  rˆ   1/ 2    J A1 
j

 J
A
Vˆ  J A
A1
ˆ=
V=


i 1
VNN  ri  rA  
A1

i 1
ei e A
Z
e e
 A1 A
ri  rA
r
Solution of the inhomogeneous equation is (N.K. Timofeyuk, NPA 632 (1998) 19)
where Gl (r,r’) is the Green’s function.
Important!!!
• Solution of inhomogeneous equation has always correct asymptotic behaviour
• Overlap integrals and SFs depend on matrix elements
This matrix elements contain contributions from subspace P and from
missing subspace Q.
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
3
V (r )  V Y ( x
ST
i 1
(c)
i , ST
(c)
i , ST
3
) PˆST  Vi ,(Tls )Y ( xi(,lsT ) )(l  s ) PˆT
i 1
3
  r 2Vi ,(Tt )Y ( xi(,tT) )Sˆ12 PˆT
i 1
Y ( x)  exp(x) / x,
xij  aij r
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)
16O
SSTA = 1.45
experiment:
(e,e’p)
1.27 ± 0.13
p knockout 1.12 ± 0.07
(p,d)
1.48 ± 0.16
Wave functions:
• Independent particle model
• Harmonic oscillator s.p. wave
functions
• Oscillator radius is chosen to
reproduce the 16O radius
Shell model
0ħ (non TI) 2.0
0ħ (TI)
2.13
4ħ (non TI) 1.65
Reduction factor
0.64 ± 0.07
A  16
• Model wave functions are taken from the 0ħ oscillator shell model
• Oscillator radius is fixed from electron scattering
• Centre-of-mass is explicitly removed
SFs: Comparison between the STA and the experimental values
A  16
Comparison to variational Monte Carlo calculations
SSTA
0.600
0.319
A  16
Spectroscopic factors: STA v SM
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
Removing one nucleon
Adding one nucleon
N.K. Timofeyuk, Phys. Rev. C 84, 054313 (2011)
SSTA/SIPM:
Comparison to other theoretical calculations
and to knockout experiments
A
A-1
lj
16O
15N
24O
23N
24O
23O
40Ca
39K
48Ca
47K
p1/2 0.64 0.07 0.73
p3/2 0.56 0.06 0.65
p1/2
0.59
s1/2 0.87 0.10 0.83
d3/2 0.650.05 0.76
s1/2
0.520.04 0.58
s1/2
0.540.04 0.69
d3/2 0.570.04 0.68
d5/2 0.110.02 0.71
p3/2 0.580.11 0.59
s1/2
0.490.74 0.74
d3/2 0.580.06 0.72
d5/2 0.490.05 0.73
g7/2 0.260.03 0.61
h11/2 0.570.06 0.48
57Ni
56Ni
208Pb
207Tl
Sexp /(2j+1) STA
CBFM
SCGFM
0.89
0.89
0.8
0.8
0.85
0.87
0.84
0.86
0.85
0.8
0.8
0.36
0.59
0.65
0.85
0.83
0.83
0.82
0.82
CCM
0.9
0.9
0.61-0.65
0.92
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 WS
DOM
132Sn+n
SDOM = 0.72
SSTA = 0.68
C2DOM = 0.49 C2STA = 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.
Transfer reaction A(d,p)B
d
p
n
A
(d,p) reactions help
B
• 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
T( d , p )    p  B Vnp  Vip  U pB 
 
pB
()
iB
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
Alternative presentation of exact amplitude
T( d , p )  
 
pB
Vnp 
()
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.




pB
pB  p B ,
B 
However, pB is obtained not with the p-B potential but with the
p-A potential.
 
()
T




V

The simplest approximation for ( d , p )
pB
p B np
Distorted wave Born approximation (DWBA):
(+) = dAdA
where dA is the distorted wave in the d – A channel obtained
from optical model.
 

T( DWBA




V

d , p)
pB
p B np
dA  d  A
In this approximation deuteron is not affected by scattering from
target A.
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:
Including deuteron breakup:
( )  
Anp ( R, rnp )  A

Anp ( R, rnp ) is obtained from three-body Schrodinger equation:
T


T

V

U

U

E


R
np
np
nA
pA
Anp ( R, rnp )  0
p
d
rnp
n
UnA and UpA are n-A and p-A optical
potentials taken at half the deuteron
incident energy Ed /2
R
)
T( d , p )  pB p  B Vnp (rnp )  (Anp
( R, rnp ) A
A
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:
T




V



np
np
d
Anp( R, rnp )  0
Then the three-body equation becomes

T

U

U

E

 R nA pA d  Anp ( R, rnp )  0,
Ed  E   d
Only those part of the wave function, where rnp  0, are needed:

(TR  U nA ( R)  U pA ( R)  Ed ) Anp
( R, 0)  0
The adiabatic d-A potential
The adiabatic model takes the deuteron breakup into account as Anp(R,0)
includes all deuteron continuum states.
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)
TR UdA (R)  Ed  
dA ( R)  0
rnp
rnp 

U dA ( R)   drnp  d (rnp ) Vnp (rnp ) U nA ( R  )  U pA ( R  ) 
2
2 

2
• 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
T  E   NL  r     dr' V (r,r' ) NL  r' 
Non-locality and energy dependence of optical potentials
F. Perey and B.Buck, Nucl. Phys. 32, 353 (1962)
Non-local optical potential:

T  E   NL  r     dr' V (r,r' ) NL  r' 
 r  r'  exp  (r  r' ) 2 /  2

V (r, r' )  U N 
3/ 2 3
2





... and equivalent local potential:
  0.85 fm is non-locality range
T  E  L  r   UL (r )L  r 
  2

U L r  exp 2 E  U L r   U N (r )
 2

Link between WFs in local and non-local models:
  2

 NL  r   exp  2 U L  r   L (r )
4

Three-body Faddeev calculations of (d,p) reactions
with non-local potentials
A. Deltuva, Phys. Rev. C 79, 021602(R) (2009)
0
60
120
c.m. (deg)
180
Non-locality in DWBA
 p 2p
DWBA
(d , p)
T
 e
4
2
U pB  r 

pB
  p  B Vnp e
d d2
4
2
U dA  r 

dA  d  A
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))
T  Ed dA  r    Uloc0  r dA  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
~1
~ 0.86
E0  40 MeV is some additional energy !
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 U0loc
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 nonlocal 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+
1/2+
5/2+
7/2+
g.s.
0.332
1.655
2.434
0.80
0.83
0.81
0.75
131In
9/2+
1/2+
3/2+
g.s.
0.30
1.29
0.64
0.74
0.74
133Sn
7/2
3/2
g.s.
0.854
0.68
0.72
A  16 nuclei
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
SIPM
Sexp (e,e’p)
SSTA
p1/2 2.0
p3/2 4.0
40Ca 39K
d3/2 4.0
s1/2 2.0
48Ca 47K
s1/2 2.0
d3/2 4.0
d5/2 6.0
208Pb 207Tl
s1/2 2.0
d3/2 4.0
d5/2 6.0
g7/2 8.0
1.27(13)
2.25(22)
2.58(19)
1.03(7)
1.07(7)
2.26(16)
0.683(49)
0.98(9)
2.31(22)
2.93(28)
2.06(20)
1.45
2.61
2.90
1.15
1.38
2.70
4.21
1.48
2.88
4.38
4.88
16O
A-1
15N
lj
SSTA/SIPM
0.73
0.65
0.73
0.58
0.69
0.68
0.71
0.74
0.72
0.73
0.61
DWBA calculations of the 10Be(d, p)11Be Reaction at 25 MeV
B. Zwieglinski et al, NPA 315, 124 (1979)
N.K. Timofeyuk and R.C. Johnson, PRC
59, 1545 (1999)
When optical potentials reproduce
elastic scattering in entrance and exit
channels we obtain:
Optical potentials used don’t
reproduce elastic scattering in
entrance and exit channels!
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
SSTA/SIPM
0.62
0.61
0.75
1.10
0.63
0.77
N.K. Timofeyuk, Phys. Rev. C 84, 054313 (2011)