University of Groningen
Seperable approximations and three- and five-body systems.
Fournier, Anton
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1976
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Fournier, A. (1976). Seperable approximations and three- and five-body systems. s.n.
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INTRODUCTION
Since the discovery of the neutron, about four decades ago,
the determination
of the nucleon-nucleon
interaction
has been one
of the outstanding problems in physics. Despite much effort,
the
question has been today only partially
resolved. Attempts to
(NN) potential
derive the nucleon-nucleon
from relativistic
field
theory have so far converged to a unique answer only in the region
beyond x 2 fm, where one pion exchange dominates the interaction.
guided by boson exchangeAt the same time, and increasingly
theories,
there have been attempts to generate semi-phenomenologica1 potentials
which fit
the knonn N-N bound state and scatËering
properties.
But also this procedure does not yield a unique solupotentials
tion. Very different
can be constructed which fit
the
data equally we11. In order to discriminaEe bet\reen various models
for the interaction,
one has to perform calculations,
involving
more than tvuo particles,
such as the binding energy and density
of nuclear matter, Brem-strahlung,
or nuclear reactions.
In this respect the three-body
system has been very popular,
since we have here a mathematically
rigorous theory, and if one
quantum mechanics, no
accepts the lirnitations
of non-relativistic
further
approximations
have to be made. Three-body forces,
which
are generally not taken into account, can also be treated
involving
with realistic
exactly.
Èhree particles,
Calculations
local potentials,
are very complicated, and require large ánd fast
computers, which have become available
only in the last few years.
For separable potentials,
or separable approximations to
local potentials,
the three-body equations reduce to a set of
coupled two-body equations,
thereby reducing the computational
complexity
by a large amount.
In ehapter 3 of this thesis we discuss some separable approxipotentials.
maÈions tolocal
In order to investigate
the applicability
of these approximations in three-body calculations,
we used
them to calculat.e the binding
energy and low energy phase-shifts
for a system of three identical
bosons. The potentials
r ^ 7 ec o n s i d e r ed are by no means realistic,
since they only contain an S-wave
part, and no hard core. One of the approximations,
the unitary pole
expansion, appeared to be favourable,
since it has a siÍnple energy
independent form, and gives a good convergence to the exact solution of the local potential.
If one compares three-body properties
calculated with different phenomenological potentials,
it is hard to interpret
the
results,
since the difference
The
may be due to various effects.
N-N phase-shifts
are in general not exacÈly equal, especially
in
the high energy region, where a potential
model is known to be incorrect;
furthermore
the potentials
may have a very different
forrn
at sx0a1l distances. Therefore it is useful to perform rnodel calculation with families
of potentials
in only one respecE.
which differ
Here one can take advantage of the possibility
of solving the
problernl in certain cases one can construct poinverse scattering
which exactly yield a prescribed phase-shift.
tentials
In chapter
4 we give a description
of Fiedeldeyts solution
of the inverse
problem for rank two separable potentials.
scattering
We used the
method to investigate
the sensitivity
of the triton
binding energy
and the n-d scattering
lengths, with respect to changes in the
potenËia1s. These calculations
were performed for a simpl-e model
of a separable potential
limited
to the singler
and uncoupled
triplet
S-wave channels. hre studied here the off-she11 effects,
and the effects
of variations
in the high energy two-body phaseshi.fts. It turned out that it is possible to get variations
of
about 2.5 MeV in the triton
binding energy due to different
offshe11 behaviour of the potential.
However, the interactions
which
yield the largest effects
appear to be pathological
in a certain
sense, and must be rejected on physical grounds. The variation
in
ET due to accepËable potentials
is about 1.5 MeV and can primarily
be ascribed to different
singlet
interactions,
The doublet scattering lengths calculated with these potentials
show a linear relationship to the triton
binding energy; all points are nearly on the socaIIed ynr-II].Ds Irne.
We also iourrd a considerable effect of different
high energy
two-body phase-shifts
on E1 and 2a. The quartet scattering
lengiit
is essentially
not influenced by different
off-she11 and high energy behaviour of the poÈentials.
The variation
in E1 and 2a due to
different
triplet
interactions
appeared to be strongly correlated
to the deuteron wave functíon.
By means of a variational
procedure
potentials
we constructed different
which gave identical
deuteron
wave functions,
and it turned out that this is a strong constraint
on the possibility
of obtaining off-she11 effecrs.
In order to solve three-body
equations one needs the two-particle
T*matrix for energies varying from the three-body centre-ofmass energy to minus infinity.
Calculations
at higher energies involve consequently a larger part of the two-body interactions.
The
neutron deuteron breakup cross-sections
are related to the threebody off-she11 elastic
scattering
amplitudes. Therefore one may
expect to find a higher sensitivity
to details
of the nuclear force
in the cross-section
above breakup threshold
than that found in the
triton
binding energy. However, three-body calculations
above the
threshold are very complicated, even for separable interactions,
since the effective
potentials
have logarithmic
singularities
in
this region, which is a reflection
of the fact that final
states
with three free nucleons are possible.
In section 5 we present a method to perform breakup calculations, which we used to extend the investigations
of chapter 4. It
appeared that, at l4.l MeV, the cross-sections
are more sensitive
than the triton
binding
energy to off-shel1
and high energy variations only in a few selected regions of phase space. In general
the three-body
system is largely determined by the low energy N-N
phase-shifts.
The calculations
of cross-section
at sti11 higher
energies are hard to perform, since many partial
waves are needed,
furthermore relativistic
effects becorne important,
and up to now
)
there
is no accurate r'ray to take these into account.
The results
of the three-nucleon
investigation
of the last
decade have been a great success and, in a certain sense, a failure
at the same time. It appeared to be possible to fit
the three-body
data with different
and rather simple potentials,
which means that
the three-body system cannot provide very detailed
inforrnation
about the nucleon-nucleon interaction.
However, the three-nucleon results have greatly
stimulated
interest
in systems with four or more particles,
since on the one
hand one may hope that in light nuclei more sensitivity
can be
found, and on the other hand, one can expect that simple models
will
also work in this case. Various integral equations for the
N-body problem have been proposed, all sharing the property that
they are much too complicated to solve exactly.
However, in general
one is not interested
in the complete solution,
but only in the
transition
operators between t\.íoor three-particle
channels, whích
are N-body operators integrated
over the internal
coordinates of
the particles.
In chapter 6 we give a discussion of the approximations which
are necessary to reduce the N-body equation of Redish and Bencze
to an effective
two-body equation. Making a few more assumptions,
we arrive at a model for scattering
of complex nuclei which is
very analogous to the Amado model for the n-d system. It is hard
to give general estimates of the quality
of the approximations,
especially in the energy region where four or more particles
can
be free. The assumption that we made in the derivation
of the
model, namely that the main part of the interactions
between
two nuclei is due to one-particle
exchange, appeared to be
questionable.
In chapter 7 we give the results of the neutron-alpha scattering calculations
that we performed with this mode1. We found a
qualitative
agreement with the experimental phase-shifts
belor+
20 MeV, but especially
the observed splitting
of the P I l2 and
P 3/2 phases could not be reproduced. In the discussion of these
results we give arguments which indicate
that double exchange
processes cannot be neglected. Furthermore we calculated breakup
cross-section for the reactio.t p 4H" - 3He n prat about 47 MeV.
I n t h i s e n e r g y r e g i o n v , / ec a n n o t a c c u r a t e l y
calculate
the required
/ ^ c F - ^J rrr ^s r 1! / 1 \ E^r1d D- L- r,L: ^ d^u-l P- rrr.L:utusE .È ,, r ^ ^ *L lLr E^r - e f o r e W e h a d t O i n t f O d u c e t \ . d o
\urr
adjustable coefficients
in order to get some agreement wiÈh the
experimental data. However, it is clear that by using the reaction
mechanism, provided by the particle
exchange rnodel, we can explain
qualitatively
the observed cross-secCions.
Long before the N-body integral
equations were available,
methods for treating
nuclear reactions r^:eredeveloped. One can for
instance solve the Schródinger equation by means of variaÈiona1
techniques. Neutron-alpha scattering
calculations
based on the
princple,
Kohn-Hulthén variational
have been performed, yielding
much better results
than we obtained. However, ic is ríorthwhile
to improve the approximations
equations
to the N-body integral
since here one can start
from an exact theory,
and furthermore
three-, or more-body thresholds can, in principle,
be taken into
account correctly,
a problem which has up to now found no satisfactory solutions in other theories.
Finally
chapter 2 of this thesis contains the theoretical
framework we need in the subsequent chapters.