Transcript Slides - Agenda INFN

Non-locality in adiabatic model
of (d,p) reactions
N.K. Timofeyuk and R.C. Johnson
(University of Surrey)
Transfer reaction A(d,p)B
d
p
n
A
B
Why do we measure transfer (d,p) reactions?
• To deduce spin-parities in residual nucleus B
• To determine spectroscopic factors (SFs)
• To determine asymptotic normalization coefficients (ANCs)
(probabilities to find bound neutron far away from A, which is
important for nuclear reactions of astrophysical interest)
How do we
• deduce spin-parities in residual nucleus B?
• determine SFs and ANCs?
By comparing measured and theoretical angular distributions:
 exp ( )
 theor ( )
Breaking news:
We have updated theory to predict  theor( ) for (d,p) reactions
(N.K. Timofeyuk and R.C. Johnson, Phys. Rev. Lett. 110, 112501 (2013))
What theory for  theor( ) is available on the market?
T( d , p )  pB p  B Vnp  (  )
No deuteron breakup:
Distorted wave Born approximation (DWBA): (+) = dAdA
(developed in 1960s)
Deuteron breakup:
( )  
Anp ( R, rnp )  A
Adiabatic model (1970) : approximate solution for 
Anp ( R, rnp )
Continuum-discretized coupled channels (CDCC) (1980s):
equation for 
Anp ( R, rnp ) are solved exactly
Three-body Schrodinger equation:
T


T

U

U

V

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 Ed /2
R
A
Why at Ed /2 ?
Its legitimacy has not been investigated
Energy dependence of optical potentials can be a consequence of
non-locality
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    U L (r ) L  r 
  2

U L r  exp  2 E  U L r   U N (r )
 2

Then outside the nuclear interior NL  L
F. Perey and B.Buck, Nucl. Phys. 32, 353 (1962)
Three-body equation with non-local potential in adiabatic
approximation reduced to two-body model for
0
T

E

r


U

d  dA  
loc  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 comes
from?
E0 in a first order is related to
n-p kinetic energy in deuteron averaged over the range of Vnp
Since Vnp has short range, 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
Conclusions
• All optical potentials are non-local because the target and/or
projectile have internal structure.
• Non-local n-A and p-A optical potentials should be used to
calculate A(d,p)B cross sections when including deuteron breakup
• In non-local model effective local d-A potential can be derived
• Effective local d-A potential is a sum of nucleon optical potentials
taken at energies shifted from Ed/2 by some additional energy.
• 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.
N.K. Timofeyuk and R.C. Johnson, Phys. Rev. Lett. 110, 112501 (2013)
N.K. Timofeyuk and R.C. Johnson, submitted to Phys. Rev. C (2013)