Two-body force in three-body
system: a case of (d,p) reactions
N.K. Timofeyuk
University of Surrey
11Be, 19C, 31Ne
etc.
Target
n
core
Description in terms of
three-body model seems to
be natural
Three-body Schrodinger equation:
T  V
core  n
 Vcore target  Vn target  E   0
Available methods to solve three-body Schrodinger equation:
• Faddeev
• CDCC
• Hyperspherical harmonics expansion
• Integral relations (Kievsky)
• Scattering theory on the momentum lattice (Kukulin)
• Weinberg basis expansion (Johnson-Tandy)
• Lagrange basis expansion (D. Baye)
• R-matrix
• Eikonal (high energies)
…
Do we have a right to solve this equation?
Three-body description would be correct if projection of the manybody function on a three-body channel dominated.
Microscopic theories say that a nuclear wave function is a mixture
of all possible channels with various core excitations.
Spectroscopic factors in different channels give a good idea about
their importance.
Examples:
11Be
(neutron binding energy is 0.5 MeV)
vibrational coupling model
Spectroscopic factors
10Be(0+)
10Be(2+)
0.84
0.16
(S. Fortier et al, PLB 461, 22 (1999))
no-core shell model NCSM
(Forssen et al, PRC71, 044312 (2009)
0.82
0.26
11Be(p,d)10Be,
E/A = 35 MeV
J.S. Winfield et al, Nucl. Phys. A683, 48 (2001)
n+9Be
6.812
20+
12+
6.263
6.179
5.959
5.958
2+
3.368
0+
0
Large cross sections
for population of
weakly-bound states
near 6 MeV: are
these configurations
important?
S~0.8
S~0.2
S=1.40
19C
(neutron binding energy is 580 keV)
Neutron knockout from 19C:
N.Kobayashi et al, PRC86, 054604 (2012)
Inclusive measurements: th = 164.8 mb exp = 163(12) mb
Core excitations play important role in 19C structure
Justification to neglect core excitations: Feshbach formalism
   g .s.  0 (r )   i  i (r )
Nucleon scattering:
A
r
i 0
N
0 is found from the
two-body equation:
Vopt   g .s.
Q
P
(T  Vopt  E )  0  0
1
vNA  vNAQ
Q  g .s.
e  QvNAQ
A
vNA   vNi
i 1
Q   i i
i 0
All core excitations are hidden into energy-dependent non-local nonhermitian optical potential.
Excluding core-excitations in the A + n + p three-body model
   g .s.  0 (r , R)   i  i (r , R)
n
Target
R
i 0
r
P
p
Q
Ground-state channel function can be found from three-body model
(T3  Vnp   g .s. Vopt  g .s.  E3 )  0  0
Vopt
Q
Q
 U nA  U pA  U nA U pA  U pA U nA  ...
e
e
U NA
Q
 vNA  vNA U NA
e
Optical potential for 3-body system has two-body and three-body terms
R.C. Johnson and N.K. Timofeyuk, PRC 89, 024605 (2014)
Two-body force in a three-body system
R.C. Johnson and N.K. Timofeyuk, PRC 89, 024605 (2014)
n
Target
R
r
(T3  Vnp   g .s. U nA  U pA  g .s.  E3 )  0  0
p
U NA  vNA  vNA
Q
U NA
E3  i 0  T3  Vnp  ( H A  E A )
Comparison to N-A optical potential:
U NA  vNA  vNA
Q
U NA
E N  i 0  TNA  ( H A  E A )
Two-body force in three-body system differs from two-body
optical potential!
N-A two-body force in A(d,p)B reactions
T( d , p )    p B Vnp (rnp )   d (rnp ) A
()
pB
d
p
rnp
n
R
B
A
T
()
dA
• The (d,p) amplitude contains projection
of the many-body function into the 3body A+n+p channel.
• The most important contribution to the
(d,p) reaction comes from small n-p
separations.
• In the adiabatic approximation the
distorted wave function satisfies the
equation:

A
A
()



V
(
U

U
)



E

R
d g . s . np
nA
pA
d g .s.
d
dA ( R )  0
What is needed for transfer reactions in the N-A force averaged over Vnp
Averaging operator U NA  vNA  vNA
Q
U NA
E3  i 0  T3  Vnp  ( H A  E A )
over Vnp and comparing it to two-body optical potential leads to
the following conclusion.
Two-body N-A potential for (d,p) reaction is equal to the non-local
energy-dependent complex optical N-A potential taken at the
effective energy
Eeff
1
1  d VnpTnp  d
 Ed 
2
2  d Vnp  d
n-p kinetic energy in deuteron equal approximately 57 MeV
Three-body problem for (d,p) reactions should be solved with energyindependent non-local nucleon potentials taken at effective energy
equal to half the deuteron energy plus another 57 MeV.
Available optical potentials:
Practically all available phenomenological systematics are local
and energy-dependent. Except:
• n-A non-local energy-independent potentials, F. Perey and
B.Buck, Nucl. Phys. 32, 353 (1962)
• Non-local energy independent nucleon potentials for Z=N
nuclei, M.M. Gianinni and G. Ricco, Ann. Phys. 102, 458 (1976)
• Non-local nucleon potentials with energy-independent real part
and energy-dependent imaginary part, M.M. Gianinni, G. Ricco
and A. Zucchiatti, Ann. Phys. 124, 208 (1980)
• Non-local energy-dependent dispersive optical potential for
N+40Ca, M.H. Mahzoon et al, Phys. Rev. Lett. 112, 162503
(2014). This potential simultaneously reproduces the data
above and below the Fermi energy.
• Energy- and mass independent non-local optical potential,
Y.Tian, D.-Y. Pang and Z.-Y. Ma, IJMP 24, 1550006 (2015)
Summary
• Application of few-body models for complex systems has to be
justified.
• The three-body model for the A+n+p system contains two-body
and three-body contribution. The latter describes interaction of
n and p through intermediate recoil excitation of the target.
• Two-body n-A and p-A potentials in the three-body A+n+p
system differ from those in free n-A and p-A systems.
• For A(d,p)B reaction the averaging procedure provides a recepi
for n-A and p-B forces. This involves knowing a non-local energy
dependent N-A potential at the energy equal to half the
deuteron incident energy shifted by the n-p kinetic energy
averaged over the short-ranged n-p interaction.