SLAG.m-409
April 1968
*t
CALCUL4TIONS OF THE THREE-NUCLEONLCW ENERGYPARAMETERS
H. Pierre
Stanford
Linear
Accelerator
Center,
Noyes
Stanford
University,
Stanford,
California
and
H. Fieldeldey
Atomic Energy Board,
Pretoria,
South Africa
ABSTRACT
Since exact calculations
using
separable
models assume nuclear
mental two-nucleon
forces
parameter
culations
problem to clarify
and if
After
what physical
existing
this
input
calculations
and quartet
viewed from this
of view.
*
t
a review
with
can be used for
on which features
is needed, and a review
is available
of the binding
scattering
So far
of the mathematical
quantum mechanical
information
and of the n-d doublet
point
"realistic"
type of calculation
shed some light
of the non-relativistic
so to what accuracy
experiments,
of either
with
experi-
must be known in order to make reliable
principles.
structure
which do not reproduce
However, the exact calculations
interaction
from first
low energy parameters
calculations
comparison
and these studies
studies,
forces
and variational
is premature.
the two-nucleon
physical
data,
have not converged,
experiment
of the three-nucleon
of
caland
three-body
of whether
from two-nucleon
energy of the triton
lengths,
as two-nucleon
et,
a2 and a4, are reinput
goes, new
Work supported by the U. S. Atomic Energy Commission and the South
African Atomic Energy Board, Private Bag 256, Pretoria,
South Africa.
Invited paper presented at the Conference on Three-Particle
Scattering
in Quantum Mechanics, Texas A and M, April 4-6, 196%
-2-
experimental
confirming
information
has removed a serious
the theoretical
prediction
source of difficulty
from charge-independence
by
that
the
n-p 'So effective
range rs = 2.73 If: 0.03 F, but a comparable sensitivity
to the percentage
D state
least
4-$-7&as for-the
as good for
by Phillips
will
culations
require
conventional
extensive
further
in the models arising
nucleon-nucleon
behavior
of the phases at infinite
or in conjunction
also larger
potentials)
that
larger
F.
effects
properties
on extracting
(wave function)
off-shell
few MeV in et between equally
to exist
even after
converged to the same unique
far from the energy shell.
either
error
that,
variational
separately
value of et
are one to two orders
in either
forces
quantity,
information,
models for
system with
an eye
a discrepancy
of a
the nuclear
speaking)
from
work is
force
and Faddeev calculations
(mathematically
and
arising
unless further
of the two-nucleon
realistic
to
(which are quite
in the calculated
likely
fit
due
due to different
due to three-body
done on electromagnetic
models.
in their
Both uncertainties
They also make it
meson exchange.
continue
between effects
such differences
than the experimental
than estimated
cal-
near but off the energy
behavior
can lead to differences
MeV, and of a2 of l-2
Existing
from differences
in asymptotic
However, they do show clearly
out
investigation.
clearly
is now at
of 7%) pointed
momentum, differences
and separable
and differences
of magnitude
value
S phases up to 300 MeV, differences
large between local
of l-4
(where the evidence
of E and a2 do not discriminate
t
to differences
shell,
in the deuteron
answers for
will
have
these
-3-
1.
Theoretical
nuclear
INTRODUCTION
physics
has developed
an impressive
phenomenology over the past three decades without
the nuclear
mately
fic
force
other
than that
charge-independent,
systems.
the nuclear
force,
menological
models from first
to believe
physics
Planck's
subtle
about speci-
all
of these pheno-
as one believes
calculable
constant,
effects
the three
that
their
all
knowing the
the velocity
in the exclusion
principle
a basis
is still
in nuclear
missing
Two places
the nuclear
forces
next decade.
the binding
exist
spins).
of light,
Yet this
been exactly
that
represented
with
all
principles
of our understanding
nuclear
of
off
Calculation
effects
mass formula
there
the
of
But this
matter.
and surface
by the semi-empirical
is harder,
Such
rests.
enough, is the calculation
coulomb effects
force.
of chemistry
some hope of success during
of infinite
once these are turned
pendence of the nuclear
tests
surprisingly
energy and density
Helium gave con-
physics.
where quantitative
One of these,
and that
as the two particle
on which all
may be attempted
sumes at the outset
as well
the success in understanding
fidence
from first
input
in the case of atomic systems came about because of quanti-
in particular,
nuclei,
just
and approxi-
given a knowledge of-
are in principle
masses (and for
success in calculating
systems;
short-range
that,
principles,
charge and mass of the.electron,
tative
is strong,
one should be able to calculate
of atomic and molecular
confidence
knowing much about
aided by generous empirical
Yet one would like
and the nuclear
it
body of
as-
have
for finite
is exact charge-inde-
of the three-nucleon
but has the advantage that
there
system
are
I
-4-
more parameters
effects
to be compared with
can be tested
think
experimentally
192 .
them interesting
if
thing
with
aspects.
clearly
important
some feature
It
can about how this
of the interaction
is obviously
for nuclear
measurable meson effects;
there
useful
physics
until
this
be treated
particle
Unfortunately,
now attempts
Fortunately
by now, however,
merical
answers for
can be used with
sensitivity
rather
disturbing,
the three-
problem and begins to bring
confidence,
problem
(or even if
a part
that
until
have led only to ambiguous answers.
different
calculational
three-body
and mathe-
side,
to estimate
uncertainties.
one is concerned with
with
solving
nu-
Hence these models
models.
on the numerical
if
in
of where nuclear
and Faddeev) have led to identical
the simplest
try
we
is also
problem is so complicated
utterly
problem and not just
therefore
to
cannot be set.
(variational
at least
tied
It
systems.
is known, the limits
of the problem to various
three-nucleon
We will
physics,
is
which clashes
to know at what point
to answer these questions
techniques
But it
and where it becomes necessarily
the three-body
matical
and even imi-
to glean any information
as a non-relativistic
is ever such a situation),
of elementary
conservation
particles.
works out in more complicated
can safely
reason to
(in which case it would be exact)
nucleon problem ceases to be a three-body
physics
the partial
subtle
to have a symmetry which is not directly
and hence must represent
other
quite
is theoretical
for elementary
a symmetry of the system of description
with
and that
has tended to breed acceptance,
in the proliferating-models
a very peculiar
there
Familiarity
law of charge-independence
tation
experiment,
These results
the physics
a mathematical
to keep our focus on the physics
the
are
of the
puzzle.
of the situation,
-5-
and leave to other
aspects
speakers the discussion
of the problem,
more practical.
plicated
and whether
For our own part,
three-nucleon
culations
reproduced
we will
at least
mathematical
approach will
prove
to believe
any com-
hesitate
until
mathematical
the same numerical
only a prelude
one or another
calculation
using different
of more technical
two independent
and numerical
techniques
calhave
answer for the same model; but that
is
to the physics.
MATHEMATICALAND PHYSICAL STRUCTURE
II.
OF THE THREE-PARTICLE SCHROEDINGER
EQUATION
We consider
three
distinguishable,
ml7 m2’ m3 interacting
we consider
Pi =R-j
forces
with
forces
particles
depend only on the magnitude
from the three
to non-local
and/or
If the total
stage.
Schroedinger
equation
energy operator,
j at R. and particle
-J
local potentials
Vi(oi),
velocity-dependent
for
interactions
energy eigenvalue
this
the equation
of masses
between each pair.
- gk between particle
be derived
later
only through
spinless
Initially
of the distances
k at zk, and which can
but the generalization
becomes trivial
of the time-independent
system is Z, and K is the usual kinetic
we wish to solve is
(K -I- Vl + V2 + V3)y = Zy
For three
scattering
y(~l&1~3)
free
particles
wave function
= XI(~&‘R3)
at a
in the final
state,
given by Goldberger
(11.1)
the general
stationary
state
and Watson 3 becomes
/ d3Pl I d3P, / d3P3"3@-$-$-~3)
+ +
Pd
i(.Pl._Rl+ I&.%+
e
P3*R3)
- < ~l~~3~T(Z)JI
Z + ie - (P:/2ml+
P",/2m2+ P;/2m3)
(11.2)
>
I
-6-
where I is any initial
state
normal laboratory
experiment,
of the particles,
so in that
we have to start
and a plane wave for
be a stationary
state
the third
solution
if we choose the bound state
tion
to elastic
T contains
needed to insure
from this
kinematic
will
only these two-particle
survive,
and if
three-particle
thing
for T, we find
sufficient
plus
by
6-functions
these states.
single
We
so if
Z is nega-
free particle
terms
none of these are allowed,
we
corresponding
set of boundary
in the world
to apply.
conditions
that
the mathematical
on (11.1)
to
If we insert
(11.2)
is not the
into
free,
into
while
channels
the other
number of iterations
(11.1)
the Lippmsnn-Schwinger
led to integral
was asympto-
to interact,
equations
This
the separaticn
in which one of the particles
two could continue
and
equaticn
problem so posed is ambiguous.
problem was solved by Faddeev 475 , who found that
of the T matrix
tically
for
are
in the system,
the appropriate
of the homogeneous equation
go over to momentum space, thus obtaining
difficult
there
bound states.
This complicated
easiest
and if
breakup threshold,
that
wave cor-
can be accomplished
relations
bound state
Z is so negative
look for bounded solutions
this
= ZxI
Then, in addi-
an outgoing
bound state,
terms with
assume Z to be zero at three-particle
tive,
of two
it will
(K + Vl)xI
2 and 3.
or any other pair
Formally,
additional
the right
equation
we must include
between this
these must also be included.
assuming that
a bound state
In other words,
to be of particles
scattering
any other bound states
particle.
of the simpler
to the term given in (11.2),
responding
with
For a
case x will correspond to that bound state
I
a plane wave for the motion of the center of mass of that
wave function,
state,
E --+O+ is implied.
and the limit
after
a
of the Fredholm
-7-
type.
In configuration
three
space, this
amounts to replacing
(11.1)
by the
equations
(K + Vl- z)Y, = - vl(y2
+ y3)
(11.3)
(K f V2- z)Y2 = - v2(y3 + yl)
(K + V3- Z)Y3 = - V3(Yl + Y2)
with
Y = Yl I- Y2 + Y3 and the boundary,condition
stationary
the right
only if
for
solutions
of the three
hand sides of (11.3)
the right
short-range
in 9 variables
Clearly
asymptotically,
this
by setting
will
work
which we now prove
of mass coordinate
six coordinates
can be reduced to two continuous
First
quantum numbers as follows.
(ml$
momentum p, and the corresponding
remaining
obtained
forces.
and a set of discrete
out the center
the Yi approach
equations
equal to zero.
hand sides vanish
These equations
variables
separate
that
take
+ m2F& + m3g3)/M with
energy by defining
can be represented
conjugate
2
z = Z - P /2M. The
by the yector
between any
two particles
p, and the vector r from the center of mass of that pair
in three different
ways; these vectors, and their
the third particle,
conjugate
momenta are
2-i = R+ - (mjgj
p'
-1
=R-j
+ %gk)/(mj+
- ok; 9i = (Tgj
2 = p:/2Mi
If ri
to
%);
Pi = C(mj+ mk)~i - mi(gj+
(11.4)
- mjEk)/ cmj+ mk)
cmj' mk)mi
-t q:/2pi
Mi =
M
cli =
mjmk
mj+ ?k
and Q make angles OiOi and eicpi respectively
can immediately
s)I/M
reduce the left
hand side of (11.3)
with
space-fixed
to two radial
we
variables
I
by introducing
by Blat-t
the orthonormal
and Weisskopf6
=
angular
momentum functions
and the expansion
c
ei,‘Pi)
(@pi;
JQ
Since J and M are constants
possible
the right
in the following
of the motion,
equations.
hand side of (11.3)
In order to calculate
space of two continuous
variables
Jacobian
these to be the three
Euler
for definiteness
exercise
body-fixed
we take this
in rotation
in these coordinates
matrices
of course,
is that
hand side onto our
s $&dRj,
we make a
new angles.
Following
angles CX, S, y of a body-fixed
defined
the angle c between 5 and p, and a constant
angle between this
omit them whenever
of the right
to a set of four
of the triangle
>
in terms of the wrong variables.
by the operator
Gmnes' we pick
(II-5
The difficulty,
the projection
of unit
in the plane
we will
is expressed
transformation
axis lying
Ja,, as defined
YM
by the three particles,
parameter
6 which gives the
axis and one of the vectors
It
to be the angle to Lo.
to show that
the angular
become, in the notation
in the plane;
is then a simple
momentum functions
of Edmonds',
(11.6)
Since the angles a, S, y are common to all
of the four
left
with
a series
integrations
a single
integral
of plane rotations
in one coordinate
just
three
coordinate
give an orthogonality
over 5.
are all
system to that
Since all
that
integral,
vectors
now lie
is needed to relate
in another,
and all
systems,
angular
three
and we are
in a plane,
the angle 5
and radial
-9-
coordinates
to express '2,
rlj
pl,
f,,
we consider
To be explicit,
in one system to the other.
how
and cl2 (the angle between 21 and ~2) in terms of
This is
P19 and (1.
22
2
+ b20; - 2abrlplcos
'2 =ar 1
2
p2 = rl
2
22
+ c Pl + 2crlolcos
f;l
51
(I-1.7)
r2p2cos
cl2 = a.21 - brlr2cos
rlr2cos
The corresponding
2 -+3,
The final
--d2
drf
+ Vl(Pl)
expressions
&,
3 +l,
result
c = m&2+
"3)
p3, c3 and cl2 are obtained
r3,
by the cyclic
1 -+2 in a, b, c and by cl--+- (1.
is the differentio-integral
equation
Q&l)
2
'1
(11.9)
L
3 ]
d(cos il)$;$,h'(rl~l~os
-1
= ,
and the corresponding
expressed
for
51
5,
b = m3M/(m2+ ml) b3+ m2)
a = ml/ b2+ m3)
permutation
c2 = arz - bcp: + (ac-b)rlplcos
equations
in terms of rl,
for
2
U
and u.3
51) riF1u',$rS'ps)
Here rs and p, are to be
pl and cos 5, by means of (II.7),
and the kernel
is given by
(k+512,0,5+~12+~2,0)
or the obvious generalizations.
Since our purpose at this
point
is to
(11.9)
I
-lO-
investigate
the structure
of the source term, we d-o not attempt
further
simplification.
We see immediately from (11.8) that if Vl vanishes asymptotically
-pi/R
like,
that the source has the necessary compactness in this
say, e
coordinate.
resulted
that
This is to be contrasted
with
to exist
the p, r plane which runs to infinity;
space.
totically
for
rl
to formulate
three-particle
It might be hoped that
free,
and is outside
> R, that
not the case.
In fact,
1
.
for
This long-range
since particle
from angular
earlier
channel is asymp-
we see from (11.7)
factors
interpretation
which is illustrated
the range of forces,
energy and momentum individually
and since us is also
off
like
and rl
large)
in Figure
state,
l/rl.
we are conhas a ready
1.
If
all
three
they would have to have their
connected as they are for
the final
no further
free particles,
scattering
would be
and the source would have to be zero (which is guaranteed
the vanishing
of VI).
However, if
range of force,
their
as they are for
free particles,
particles
2 and 3 are still
energy and momentum are not necessarily
and if
that
of order unity,
of the source in the limit
physical
in
of the interacting
pair
-y/R
like e
, but this is
the source only falls
(pl less than the range of forces
possible
off
strip
in configuration
1 in this
the range of force
sidering
and if we are discussing
frustrated
plVl(ol)us(arl,rl)/arl,
character
were outside
fact
pl < R and r1 large,
asymptotically,
since in
along a diagonal
boundary conditions
Hence, apart
the source term approaches
of order unity
that
the source would also fall
r 2-tar 1 and p2-+r
particles
which would have
if we had not made the Faddeev channel decomposition,
case the source term continues
attempts
the situation
they can pick
by
within
the
connected
up momentum from
-ll-
somewhere, can still
supplied
spherical
hand side of (11.8),
integral
large
provide
physical
origin
for
it
us of very similar
such an approach will
to that
solve for the scattering
(11.2)
and introduce
of (11.2)
formal
in configura-
forms and phases in
simple
singularities
source terms in coordinate
approach.
by assuming that
XI is a plane wave (i.e.
free particles
any other
case),
(11.1)
to three
particles
to z and 2 respectively,
and transformation
the Lippmann-Schwinger
free
take out the center-of-mass
momenta 2 and CJ conjugate
into
the
in p and r may
structure
Faddeev that
to handle than l/r
of three
from which we can construct
these equations
asymptotic
But we agree with
We
extend over such a
in order to clarify
space.
immediately
by Sasakawa9 .
the result
If we specialize
the insertion
to those for
We have presented
in clarifying
not
the left
problems even with
in momentum space are easier
motion,
for
three-body
of solving
like
to the equations
formally
that
off
Were this
structure
integrals
looks unlikely
of the difficulty
space, and now turn
this
channels,
solving
have some advantages
configuration
gives
and applying
space, and also because the explicit
still
wave only falls
down the Green's function
z)-',
method for
forces.
of this
but the resulting
that
of rl,
This momentum can be
wave in one of the two other
as has been discussed
do this
a practical
short-range
tion
(K -i- Vl-
scattering,
region
write
equations
can, of course,
state.
the source term has been explained,
of
the case, we could readily
two-particle
the final
and since the amplitude
the behavior
obtain
into
from the outgoing
as illustrated,
l/Q'
scatter
equation
to momentum space
-
I
or,
in operator
form
T(Z) =
As noted above, this
matical
problem,
as it
V + V GO(Z)T(Z)
stands does not pose a well-defined
but if we write
it
(11.11)
T = CiTi,
is equivalent
V = CiVi,
we see immediately
that
(1 - VIGo(d)Tl(i‘)
= Vl + VlGO(z)[T2(z) + T3(d1
(1 - V2G,,W))T2(z) = v2 + V2G0(z)[T3(z)
(1 - V3Go(z))T3(z)
stationary
construct
state
solutions
the integral
amounts to solving
three-particle
the two-particle
Hilbert
to the system of equations
space treatment
for
space, i.e.
< q' ti(2
-I
(11.12)
that
if we knew the
z)Yi = 0, we could immediately
the three-body
wave function.
Lippmann-Schwinger
equation
This
in the
solving
(1 - ViGO(Z))ti(Z)
or
terms
+ T2(z)1
of (K + Vi-
equations
and rearrange
+ Tl(z)l
= v3 + V3GO(z)[Tl(z)
We saw above in our configuration
mathe-
(11.13)
= Vi
2
- k$) CJ,> = < 9' 'i 9 '
I I
i I
(11.14)
+ /d3q"
< d'\Vili'
>
'
2 -
<q"V 1 ii g>
I
-13-
where we have factored
out E3(g1- p) from both sides.
This differs
the usual equation
two-particle
in that
energy is'z
for
- p2/2Mi rather
q' can differ
(11.13)
defined
into
discuss
as well
further
gives
below,
obtain
by applying
(a) the
and (b) that
This equation
so the operator
the expression
(1 - ViGO(z))
for V,I given in
immediately
(1 - ViGo(Z) )Ti = (1 - ViGo(z))ti[l
so that
value q2/2pi
as in direction.
Substituting
inverse.
(11.12)
scattering
than the on-shell
from q in magnitude
can be solved as we will
has a well
elastic
from
the inverse
+ Go(z)(Tj+
(1 - ViGo)
7
Tk) ‘J
(11.15)
-1 to both sides we immediately
the Faddeev equations
= ti
Ti
+ tiGo(Tj
(11.16)
+ Tk)
or more explicitly
2
- %)I3
i
> s'(~'-E)
(II. 1-7)
I
< ~"cJ" Tj(z
Clearly
) + Tk(z+
we could now reduce these equations
by using the orthonormal
route
followed
follow
Wes7
corresponding
to two continuous
variables
momentum functions,
which was in fact the
10
route is to first
by Ahmadzadeh and Tjon . An alternative
angular
by introducing
to the total
the three
discrete
quantum numbers J, M, A
momentum and its projection
on space11
2
i; Osborn and Noyes
axes, and the three energies w.=
1 Pi/2m
and body-fixed
showed that
>
this
angular
system can then be reduced to two variables.
The final
-14-
as given by Osborn is
result
FrL)
12
(E’, ei) =
tf)(E-r.e.,
1 1
E’-r.e! 11 , z-riei)
-s& k Fez
,
p dE”
rsei
(II.l!)
tr)(E’
-r i e’i, Et’- riei ; z -riei)
E”-z
c
pm”
Kf&$lm,,
(E’, ei ; E1’, e;)
where we have made the expansion
J
<iZt~AIITi(‘,\ZJh,=g
s
(2Q+l)w1
8=0 m=-J
F’)
with
dJh,,(+~f)P~(cos
7’;)
m.19)
(E’, e\)
e!1 = wi ,
and the kernel_ is
K~~~,lm,,
,
(E’, e;; El’, el) = (2Q1’+1) $$$$
rnl’
p,,' (cos r’;)
cos’p(E”,e;,
. Pp”
cosrl))
dkrnl, (Of’s) @ E” _ r s elfs r [
(11.20)
-15-
where cos Ts is given by
cos rs (ei, E”, e: )= f
where the + applies
(11.21)
for
s = j and the - for
0:: must be determined
The argument of the d function
E I I , e".S
This can be done by taking
sin2 8
iS = 4zTZJ)
sin2ys
the ratio
~s(“i+~s+usl~
mi + m
(1+ m>mJ
(E - rs usI
#i
S’
mS’
mi+m ,
S
A number of points
The first
is that
use only for
states
the kernel
t-matrix
such that
require
over a range of variables
experiments.
(11.23)
w i(E” 9e;, r,)
system of equations.
so the reduction
a small number of angular
interactions;
at low energy.
of the equation
by
(E” - rse’$ )
sin’ c (ei, Elf, e; )
the sum over & runs to infinity,
interactions
force
is determined
need to be made about this
dominate the two-body
the nuclear
us)
(11.22)
Thus the argument to be used in dL,,(Bfi)
sin’ 8 ys =
of sin2eis
in terms of
2
and sin y,.
4mimsuius
mS’
=
s = k, and r i= M(/mj+ mk).
fortunately
The second is that
this
momentum
is true
the driving
a knowledge of the off-shell
inaccessible
If we have some model for
to two-body
the potential,
is of
of
term and
two-body
scattering
or more generally
-16-
for
the off-shell
on empirical
T-matrix,
input,
phase shifts"*.
this
It
this
is easy to show13 that
to the phase shifts
forces
equivalently
full
off-shell
function
Low equation).
system
that,
tt(p,q;z)
interactions
from the start,
that
to follow
the assumption
assumption
forces.
insensitive
it
obviously,
of a separable
to this
required
assumption,
either
or that
check on the compli-
system.
we see explicitly
inside
the range of nuclear
to show that
it
as was
commits us to a specific
about the form of the wave function
We are therefore
interactions
to separable
is possible,
is a useful
to note is
momentum states,
separable,
ourselves
interaction
point
angular
More importantly,
both routes.
mation about the two-nucleon
true.
two-body
derivation
(the
for the two-nucleon
and in particular
a simpler
shown long ago by Mi.tra14;
by a quadrature
The third
we were to restrict
the range of
from which the
information
the explicit
if
Of course,
cated algebra
of this
to non-local,
is trivial.
inside
= tt(q,p;p*/*p))
in the next section.
now we have introduced
what we need to know
can be constructed
The availability
the generalization
in fact
is the wave function
tt(p,q;p*/2p)
be discussed
w:ill
but if we have to rely
means we must know much more than the two-body
in addition
(or,
is no problem,
the results
is compatible
Unfortunately,
with
neither
other
are
infor-
assumption
is
-17-
THE NUCLF,ON-NUCLEON
INTERACTION
III.
We now know the nucleon-nucleon
the entire
elastic
tion
is vain,
that
any reasonable
until
and that
this
shifts.
hope is destroyed,
any reasonable
high-momentum behavior
given this
data,
will
do.
a static
each isospin
for p-p,
time that
a simple
found that
static
5 for
description
ferential
data.
Christian
titatively
outside
acceptable.
the static
spin,
approach,
from the phase
which requires
10 functions
Thus Christian
plus tensor
forces
and Noyes 17
did not lead
of the n-p and p-p high energy difshowed that
this
a hard core,
but Gatmnel,
was not quan-
20 found they had to go
order momentum21
as had first
been suggested by Case and Pais .
dependent terms (LS),
22
, and was
But as soon as the first
phase shift analysis became available
made unique by taking
framework
charge-independence
type of model still
Hence Gatnmel and Thaler
potential
scattering
has been known for a long
by introducing
and Thaler 19 found that
nuclear
and 12 for n-p scattering;
work.
Jastrow13
restored
potential
we need to specify
potentials
to a charge-independent
could be approximately
n-n,
model wouldn't
simple monatonic
cross section
local
But even so, it
state).
the
go on the assump-
about the on-shell
by the nucleon
even if we assume charge independence,
(5 for
we will
The obvious non-relativistic
is to construct
5 functions
problem
threshold,
model for
over
phenomenology for the short-range
This problem is complicated
us to specify
the three-nucleon
non-relativistic
we pow have complete information
amplitude,
If
to high accuracy
on what happens above pion production
hope of constructing
physics
range 15~6 .
scattering
depends crucially
phase shifts
the highest
partial
and include
first
waves from one-pion-exchange
23 ,
-1%
it
became clear
that
(for
that
solution
even this
2) it
phases at 310 MeV with
would take a region
9.7 F in radius
was insufficient.
was impossible
any simple
in order
to achieve
agreement.
Noyes 25 showed that
the singlet
at all
tigation
by Reid
26
energies
confirmed
this
Once we have been forced
the elastic
scattering
what model to use.
and Yale
28
spin-orbit
inves-
conclusion.
in order to take care of this
angular
I momentum dependent by fitting a
their
condition
parameter
model has a different
angular
so has the necessary
differences
models agree with
concentrate
not mean that
attention
ween them; often,
all
separately
the momentum dependence arising
interaction
flexibility
And
result.
in each orbital
built
in.
Al-
in the adequacy with which these dif-
the elastic
on any particular
the elastic
which is fitted
produces the desired
of course any separable
ferent
Feshbach, Lomon
model to the S-waves.
and Wong'l found that
are large
Bryan and Scott 29 avoid
wave.
meson exchange automatically
though there
Hamada-Johnston 27
interactions
model to each partial
momentum state,
us to determine
momentum dependent terms in the
"static"
from vector
allow
assumption,
second-order
separate
Scotti
potential
the phenomenological
makes his model orbital
to each state.
data
above 50 MeV, and an exhaustive
26
have a boundary
more extensive
with
Reid
the problem by not applying
it
of the order of
was a general
to abandon the static
For example,
24
S2 and +)2
difficulty
effect.
and Tubis3'
this
1
noted
model, and that
After
data do not in themselves
models introduce
form of quadratic
potential
the
in the interaction
became available,
state
to reconcile
static
of non-locality
It was originally
scattering
these differences
scattering
angular
data,
especially
momentum state,
if we
this
data can be used to distinguish
reflect
does
bet-
is the amount of care the
-19-
authors
fit
have devoted to that
state
in their
fit,
since usually
could be improved by adding more phenomenological
the three-nucleon
problem depended only on the fit
we could therefore
use, provided
the data.
pick
worse, we will
see in the next section
that
characteristics
the three-nucleon
ciple
The special
require
that
if
theory
there
to nucleons,
they will
of the pion,
and determination
therefore
this
determine
ways32' 33 .
partures
problem.
theorists;
point
spin,
parity
range part
length
of intermediate
range,
confirmed
confirmed
Whether this
(a meson), a virtual
or simply uncorrelated
the effect
is settled,
The 'P splitting,
state
on the nuclear
the parameter
which switches
fitting
coupled
The discovery
and coupling
of the nuclear
and effective
prin-
strongly
force.
constant
force,
and
in a number of
range then show an
by the
from OPE, and shown to be spin-independent
of two pions,
more extensively
and the uncertainty
to a short-range
of its
The 'So scattering
force
is to rely
has been quantitatively
component of the 3P waves'.
resonance
Consequently,
if we are to
are other massive particles
the longest
OPE prediction
attractive
nucleon parameters
of the model.
of relativity
give rise
to
in most cases, but even
the three
One way to narrow these possibjlities
on theory.
data,
of the model to
some way to narrow down these possibilities
get anywhere with
If
to the on-shell
the adequacy of the fit
has not been investigated
-do depend on the off-shell
we must find
parameters.
any of these models which was most convenient
only we insured
Even that
the
1
D2 and 'G4 de-
by the central
force
O+ boson exchange is an actual
(ABC effect),
a correlated
271 exchange,
force
will
from tensor
will
J=O state
is still
debated by
be similar,
and until
the
have to remain phenomenological.
to L-S signature
at 210 MeV
-2o-
is evidence for
the exchange of a heavy vector
range repulsion
evidenced
attraction
to repulsion
pattern31i,
and if
there
by the change in sign of the ISo phase from
near 250 MeV.
one takes into
The u-meson fits
account n-p data,
be important 35 , but careful
relations
analysis
reveals
that
Hence the very complicated
nucleon-nucleon
spin,
phase shifts
Unfortunately,
be fitted
description
combination
how to go from this
force.
One procedure
marily
a longer
central
models.
as these expec-
review
It
using several
from the indirect
range central
force
of these one
Conference 35 .
paper for the Gatlinburg
the nucleon-nucleon
interaction
of boson exchanges in a covariant
to a non-relativistic
can
S-matrix
model for the nuclear
velocity-dependent
interaction
is perhaps significant
that
to give a better
to tensor
insofar
suggested by Wong33 has been followed
and leads to a non-local
potential,
and energy dependence of the
can be done in a number of ways 35), d oes not
specify
claimed39
by 7 exchange.
For an excellent
see Bryan's
(and even this
only the p2 terms.
nucleon-nucleon
up (incorrectly)
known bosons,
simply knowing that
by this
to
is in complete accord with what we expect
can be made quantitative.
boson exchange 37 models,
this
3n exchange does make a significant
isospin,
from the exchange of the lightest
tations
into
the p meson is also
of the forward
36 ; some models mock this
contribution
neatly
The 7 meson is not expected
in about the expected strength35.
dispersion
meson, as is the short-
fit
to nuclear
fewer parameters;
cause that
force
the fit
matter
this
by Ingber 39
of which he keeps
the resulting
than the Reid L-dependent
improvement results
to the two-nucleon
(lower mass 0 meson), and a higher
in the triplet
model is
even states
pri-
data allows
ratio
than do the "static"
of
-21-
But this
approach requires
a phenomenological
tances or high momenta, and more or less ignores
A more phencmenological
40
and Bouricius
approach,
, expressed
in current
wave length
entrance
for
channel will
be lost
two nucleons will
entrance
derivative
constant;
if
this
this
local
is a static,
two-pion-exchange
and this
In its
and the wave function
of the
by fixing
controversies
condition
falls
continues
the
data as does the covariant
in this
44
respect
.
calculation
to be true
outside
42
and
Bryan and Marshak
tail"),
version 43 , it
uses OPE and
two phenomenological
parameters
about the "derivation"
at 0.7 F, the vector
but are included,
This model leads to as good a fit
potentials.
potentials
latest
with
changes do not make a major contribution,
logical
be
knowledge of the elastic
("potential
by Saylor,
potential
the theoretical
since the boundary
are superior
will
a pion Compton
the independence
interaction
Lomon and Tubis 30 .
static
the large
as a consequence of the Wigner
on the phase shift,
by Feshbach,
term;
all
phenomenologically
radius
This model was revived
to represent
about half
radius,
to zero,
the radius.
a static
is to note that
to some energy-independent
41
is assumed, Hoenig and Lomon have shown that the wave
condition
there
of Breit
of the wave function
must be zero inside
causality
even if
rapidly
this
channel being approximately
logarithmic
function
fall
processes.
coupled to nucleons,
approach within
inside
dis-
to the many bosons and boson
Hence essentially
of each other.
inelastic
language,
resonances which we now know to be strongly
once the two nucleons
at short
which stems from a suggestion
number of degrees of freedom corresponding
excited
cutoff
of Scotti
of this
meson exagain as
to the two-nucleon
and Wong, and both models
to the Hamada-Johnston and Yale pehnomeno-
Faced with
two models 'in
agreement wit
-22-
we must look still
farther
We know in advance that
to find
ways to distinguishing
they will
give different
nucleon problem since the boundary
wave function
of about 0.7 F in radius,
(or phenomenological
region
condition
hard -cores)
of about 0.5 F in radius,
while
between them.
results
in the three-
model has a 'hole'
vector
meson repulsion
only keep the two nucleons
which is nearly
in the
three
out of a
times smaller
in volume.
One way in which the models differ
Non-relativistically,
admixture
4s.
one expects
in the deuteron
However, if
integrates
the intermediate
state.
to fit
part
length,
at,
from meson currents,
of a believable
effect
that
discrepaZXy
it
the model45,
the scattering
at higher
and quadrupole
is larger
but no
fit
than has usually
has usually
that
shows that
discrepancy
by electrons,
usually
7% D state
but examination
most of the contribution
meson cur-
at threshold,
described
of the deuteron,
etc.,
to the
coherent
and
and not on
meson currents
Evidence for
ed,
been "calculated"
Since the corresponding
been thought
only 4 i$$ D state.
energy,
about 7% D
depends only on the low energy parameters
claimed from photodisintegration
from deuterium
and adjusts
the binding
one has pushed this
contradiction.
energy)
moment, Q, one finds
However, the Feshbach-Lomon model just
same data with
range force,
in assuming a hard core at about 0.5 F
in n -I- p -+y + d is 1% in cross section
rent
large.)
OPE as the longest
of the wave functionto
(This discrepancy
point
to be able to compute the D-state
one takes a model with
(which is required
properties.
from the n, p, and d magnetic moments, and gets
the wave function
scattering
is in electromagnetic
are that
can fit
the
has often
n
0
been
production
of these calculations
comes from the overlap
between
-23-
the deuteron
state,
this
wave function
and the scattering
wave function
the comparison being made with models that
long-range
ventional
region.
have shown that
did not have OPE in
Since the Feshbach-Lomon model agrees with
models outside
to be irrelevant
a fermi
or so, most of this
in the current
context.
low energy e-d scattering
shifts
see below that
both the calculated
is sensitive
n-d scattering
length,
nucleon-nucleon
between theory
with
and experiment
scattering
off-shell
question
results
available
differences
obviously
discrepancies
are in reasonable
43
is in
were due to a neglected
agreement
can be computed from the
, so much more precise
would be required
between models.
The quantity
behavior
the large
reported
matrix
low energy n-p bremsstrahlung
p-p
to spot
There is a possibility
might be more promising,
according
to
which is measured is tl(p,q;q2/2p),
notation 50 , the quasi-phase
-iSg
tJ/(p,q;d/2li)
Y/e
+ (q2- P') /
0
models-.
and the doublet
in off-shell
most of the effect
significant
or in Cromer's
et,
amounts, this
originally
than those currently
McGuire and Cromer 49 .
energy,
differences
and recent
nucleon-nucleon
to the difference
(7% D state)
Unfortunately,
experiments
that
binding
bremsstrahlung.
experiment 47 ; further,
on-shell
46
study.
to look for
term in the calculation,
is likely
between 4% and 7% D state
a2, by singificant
deserves much more careful
A second place
the difference
triton
'evidence"
con-
However, Casper and Gross
between the Feshbach-Lomon model and conventional
Since we will
in the final
dr Fj,(Pr)[u,j$qr)
= +t$P,S)
= (p/q)
- cos Et(q)F&(qr)
&
sin 6-e
- sin &,p,(q)Gt(qr)l
(111.1)
-24-
where to first
order in the off-shell
Ag(P,q)
Their
calculations
= (g4in
Eg(s) [l
show significant
at low energy when A,, varies
pectively
range,
from 0 to 2 F.
different
even this
at least
first
difficulty
for
the three-nucleon
sitive
to this
case,
However, until
n-p effective
a zero-range
with
recently,
there
correcting
a re-evaluation
1.11 and 4.22 F resorder off-shell
variation
Clearly
models.
at low
looks promising.
calculations
arises
of three-nucleon
because of our un-
range.
force
Since Thomas
leads to infinite
in advance that
see shortly,
et will
this
has been a discrepancy
binding
be sen-
is indeed the
between the
measured in low energy n-p scattering
measurements,
dependence (2.35 F, or,
Fortunately,
and as we will
range directly
the most accurate
nuclear
system, we expect
parameter,
length
up to 300 MeV, and not just
from two-nucleon input
1
about the n-p and n-n So effective
showed long ago that
Since for exponential,
and non-local
low energy parameters
certainty
in n-p bremsstrahlung
of 1.20,
should be compared, but the result
A somewhat different
(111.2)
to the same scattering
between local
models which agree on-shell
energy,
fitted
they quote values
for Ae we see that
is significantly
+ ,s]
differences
square or Yamaguchi interactions
and effective
parameter .(p2- q2)
and the value predicted
for
the fl'-fi'
51 of the experiment
ported
last
year 52 taken together
report
last
year5? removes this
with
difficulty,
using
from charge in-
mass difference,
2.73 F)
of Houk and Wilson
a new experiment
by Koester
as we now show.
realso
45
.
-25-
The route
leading
from p-p scattering
r np
= 2.73 t 0.03 F is a little
S
first
for
necessary
to show that
the 'So phase shift
zation
corrections
coulomb-nuclear
a direct
in other
interference
angular
to assume that
(to an accuracy
effect,
and since
models based on vector
all
assumption,
the remaining
and hence a fotiriori
effective
accuracy
cross section
the values
the tensor
force
available
combination
by one-pion-exchange
is about a quarter
assume it
of the tensor
justifies
to better
of 5@,
then confirms
is
of the
(especially
to be short
range,
it
at 10 MeV and below.
it
is possible
than the requisite
energies,
accuracy,
Having estab-
a shape-dependent
(to about the 3% experimental
in the shape parameter)
to
at 9.69 MeV, which analysis
them at 3 MeV and below.
of the S phase at single
OPE. Since the OPE shape effect
force
force
.55 showed that
of p-p scattering
assumptions
range analysis
gives
6 1o+36
11+Sj6
172
,
?
analysis of the p-p data
models of the nuclear
Noyes and Lipinski
a unique analysis
lished
The
energy at which the L.S effect
meson exchange),
should only be a few percent
confirms
momentum states.
of 5%)
measured is about 25 MeV, where it
Under this
to under-
of the P phases is less than 5% of the
The lowest
combination.
is
combination
of Ref. 22), but the original
the L'S conbination
force
It
is also necessary
the !D2 phase was also given by OPE to an accuracy
(OPE), that
actually
here.
of vacuum polari-
term in the differential
of the P phases was given
obtain
but it
measurement of the central-force
and that
it
Taking account
unambiguously.
below 3 MeV54 had in addition
tensor
so we review
that
the p-p data below 3 MeV can be analysed
scattering
(in the notation
tensor
involved,
is straightforward,
stand the nuclear
data to the prediction
the prediction
56,57
given by
55 and 25 25 MeV, the
is also seen at 10
I
-26-
theoretical
value can be adopted,
Since electromagnetic
coulomb repulsion
allows
effects
and a:'
can be shown to be negligible,
45 that
nn
r S = 2.846 + 0.02 F, where the errors
of the reaction
ai"
nn
r.
include
there
a generous estimate
the n-n scattering
= 16.4 + 1.9 F is in excellent
unfortunately
charge symmetry then
= - 16.96 k 1 F and
of at most 1 F, the experimental
uncertainty
that
for
a:
is no direct
in disagreement
as has been known for
with
length
evidence
Independent
any slight
to account for
we showed
45
that
range
the predicted
charge-dependent
correction
could account for this
the model phenomenologically
60 show that
value for the effective
of other possible
that
the value of rip
of Ed' "nH' ~npw
by Engelke,
is compatible
with
this
effects
a correction
2,25,45,61,62
at
at most half
, there
= 2.44 k 0.11 F obtained
and the two n-p total
Benenson, Melkonian
range is rsnp =
charge-dependent
they could contribute
As we have emphasized repeatedly
5% probability
values
asnP
for
this
2.73 + 0.03 F; estimates
as big.
of
discrepancy with the predicted
scattering
lengths, and
+ 0
in the OPE part of the model,
the known n -JI mass difference
including
shorter
of how we adjust
on the value
value of - 23.7146 F, but
(which we expect but do not know how to calculate)
discrepancy.
the prediction;
the same values
the experimental
a generation,
the anaiysis
had a theoretical
agreement with
experimental
of the
of Haddock, et al. 59
result
Exact charge independence would predict
(lnd .:I,
and the
Since Bander 53 has shown that
of the result.
d(rc-,y)2n
from the data.
other than vacuum polarication
the unambiguous prediction
model-dependence
and rfP extracted
cross sections
is less than
from accepted
measured
and Lebowitz 63 at 0.4926 and 3.205 MeV
prediction.
stead from one or more of the other
That the discrepancy
ingredients
might rise
of the analysis
was
in-
-27-
emphasized by Breit,
64
Friedmann,
and Seamon .
anH by Koester 53 and the preliminary
ported
at'Gainsville5'
value
Taken together
with
this
gives
that,
it
that
the triplet
scattering
scattering
can be seen immediately
rs,
the uncertainty
This same analysis
from higher
triplet
cross section
showed that,
partial
At this
error
level
analysis 65
of that
coming from the error
measurements in the 0.5-5.0
cancel.
-I- 0.0127 F
from the quoted errors
in the same energy range,
nearly
is at =
= - 23.7146
I"Btl I:asi.
arising
waves is almost negligible,
shape corrections
a
from our previous
in ed, at7 and as is at most a few percent
n-p total
length
length
< SatEas > = - 0.9566
in calculating
in published
situation.
by CJ (0) = 20.442 + 0.023 b.
w
result 53 that anH = - 3.719 t 0.002 F,
Koester's
correlation
of accuracy
this
should be replaced
immediately
error
alter
as re-
of the data of Houk and Wilson has shown51 that
5.4255 f. 0.0043 F and the singlet
with
of Houk and Wilson
did not significantly
Recent analysis
the published
result
The new measurement of
and that
The effective
MeV range.
the contribution
the single
and
range is then
given by
r
= 2(@-7
S
+- $)/k2
(111.3)
S
S
where
u
S
3n
-=
%
The error
analysis
-
=U
u
9
i
--+ l(
"t
is clearly
t
-
Y#o
yat-
trivial.
l)k
(111.4)
+ k2
Results
for the individual
cross
-29-
section
basis,
measurements,
previously
are given in Figure
different
2, and the weighted
data selections,
both with
2.73 * 0.03 F, in Table I.
theoretical
prediction
assumption
that
and triplet
insures
a cancellation;
weighted
singlet
is in fact
S-waves,
shape correction,
for
n-p analysis
sign of as and at then
shape correction,
consistent
would no longer
it
with
If we use
is more serious.
rises
by 0.11 F, while
by 0.13 F.
the prediction,
Both sets of
but barely
If the old ratio
Hence, continued
attention
con-
1 and Table I.
out to Houk and Wilson 52 , Koester's
accepted value for
value
aC/aH
were used, rs would fall
by 0.15 F, and the Columbia measurements would still
needed.
be true,
occurs at about 5 MeV.
the change of the previously
charge-independence.
the
Note that
by the same amount.
as can be seen from Figure
deviations.
the
by only 0.029 F, and if we omitted
measurements,
as has been pointed
is the
the same in the
Columbia measurements 63 , rs falls
each other,
standard
has been documented
the triplet
to the data selection
separately
anH requires
by five
since the opposite
shape correction
if we use only the older
with
in this
data above 5 MeV, this
only the two published
data are still
rs =
to the assumptions.
is approximately
it would fall
accurate
The sensitivity
Further,
involved
if we omitted
as the maximum'triplet
sistent
of the result
the OPE shape correction
average of rs would rise
if we included
the restriction
complete agreement between the
the shape correction
singlet
r s using three
and the data has now been achieved.
Our reason for believing
that
average for
and without
Clearly,
We now examine the sensitivity
above, but all
by Hafner 65 on an experimental
selected
imply a failure
to this
of
problem is still
-29-
According
sections,
to Houk
66
, if we use only the two published
plus two unpublished
MeV, and treat
the singlet
again in agreement with
shape parameter
the prediction.
ticularly
in order to include
Lebowitz
at 7.942 MeV, careful
the contribution
exploit
At this
attention
from higher
partial
Additional
evidence
than 2.3 F is provided
Friedman,
66
elsewhere
Halt
for
smaller
waves will
overdetermine
periments
the expectation
tions,
will
and that
stand for
this
and par-
measurement by
and to
be needed in order to
Careful
analysis
is now in progress
of the
and
rs is closer
as the test
this
out t0
where coulcmb corrections
p-p data already
and the necessary
numbers shifting
reconfirmation
of charge inde-
could be carried
We hope, however,
to perform.
to 2.7
of data up to 350 MeV by
So far
high precision
with basic
that
analysis
and Seamon67 .
the I=1 phase shifts,
again be confronted
energy,
.
in the 20-30 MeV region,
are easier
higher
available.
by a recent
than at low energy,
rs = 2.704 + 0.095 F,
to the shape correction
pendence goes, Breit 33 has also shown that
high precision
at 3.204 and 5.374
unpublished
of Houk and Wilson and of Lebowitz
be reported
Breit,
as free,
the accurate
the accuracy now potentially
experiments
will
measurements by Lebowitz
Columbia cross
exist
are
which
spin-dependent
that
by several
we will
not soon
standard
of charge independence presented
some time at the current
level
of accuracy.
ex-
deviahere
-3o-
Iv.
Although
alternative
n-d cross section
sets of doublet
the transmission
Alfimenkov,
that
CALCULATIONSCX?'et, a2, AND a4
and quartet
of neutrons
Lusichikov,
the set with
measurements at low energy allow
through
Nikolenko,
a polarized
Taran,
a4 = 6.14 + 0.06.
In the quartet
from approaching
sonably easy to obtain
the right
the correct
not be discussed
dependence we expect for
simple calculations
will
discuss
this
The behavior
and Shapiro
68
demonstrated
experiments
state,
the exclusion
at Figure
keeps
and it
is rea-
value for a4 from any model that
on-shell
nucleon-nucleon
reproduce
a4 quite
elastic
scattering
reproduce
scattering
dominates
with
well
and need
the low energy region,
quite
and
fairly
well,
as Amado
afternoon.
of k ctn s2 at low energy is quite
3.
The fitted
effective
another
sharpened.
Aaron,
matter,
as
69 which we
curve of the form k ctn s2 =
at first
range expansion,
gives a plausibility
considerably
scattering;
no bound state,
the low energy behavior
pole in k ctn s2 very close to threshold
Delves"
has
Also k ctn E4 has the usual type of energy
- A + Bk2 - C/(1 + Dk2) looks very strange
to the two-nucleon
and analy-
principle
can be seen from the curve given by van Cers and Seagrave
reproduce
target,
the values a2 = 0.11 f 0.07 F and
model calculations
further.
since the quartet
for
by measuring
deuterated
too close to the deuteron,
low energy behavior
hence the separable
lengths,
The latest
a4 > a2 is correct.
sis by van Oers and Seagrave 69 yields
the neutron
scattering
two
sight
to those used
but the occurence
in fact
argument for this
of a
has a simple explanation.
pole,
which can now be
71 and Bander72 both
Amado, and Yam
found
I
-3l-
that
three
identical
bosons interacting
Yamaguchi or square-root
range and.strength
to fit
the average of the singlet
(as we will
excited
of the pole in T corresponding
of the residue
imply a "ghost"
the residue
state.
to this
of the triton
state
is not so obviously
The situation
and exponential
interactions
state,
it
and that
diffuse
"orbit"
Although
acting
state
via central
interaction
ficient
brings
(i.e.
forces,
reasonably
S-matrix
language,
surface.)
the T matrix
"wrong" sign for
for this
system.
that
local
weakly bound first
state
Yukawa
excited
In fact,
wave function.
corresponds
bound core consisting
triton
wave function
the actual
in admixtures
force
this
system would
in a three-body
which would be obtained
to move the first
scattering
which in a two-body
sign to
as
to a very
of the other two
to the much more compact and nodeless ground state
9% of the actual
symmetric
has the opposite
also lead to this
about a tightly
in contrast
particles,
the residue
73
by Osborn , who finds
one might guess, the wave function
somewhat more
Bander noted that
probability;
has a perfectly
nucleon-
but also a
a difficulty
has been clarified
the right
below),
state
pole,
of imaginary
discuss
of the
and triplet
not only give a bound state
bound than the triton
very weakly bound first
that
potentials
Yamaguchi type and of approximately
nucleon S-wave interactions
tightly
via separable
excited
it
corresponds
for three
identical
to the fully
bosons inter-
spin dependence of the two-nucleon
of other
state
states,
and apparently
above the threshold
up to the point
where it
for
is Itvirtual",
is sufn-d
or in
move the pole onto the second sheet of the Riemann
However, between the triton
must go through zero,
-1
is real)
(where k ctn 6 - ik = T
pole in T and this
and if
this
this
will
virtual
pole,
happens below threshold
guarantee
a pole in k ctn 6.
I
-32-
That this
actually
now that
this
happens is shown by the fit
situation
be aimed at getting
the four
that
constants
the precise
to be quite
is understood,
this
sensitive
of a2 obtained
should
as a2, or better
still,
by a calculation
to the details
zero-range
sensitive
nuclear
low energy nucleon-nucleon
forces
to the shape-independent
authors 75 more than a decade ago showed that
central
forces
to the radial
overbind
the bound state
of three
S
fitted
these give a binding
potentials
commenting on this
result,
calculations
of
and that
to the parameters
first
the result
attempt
is
for
mass interacting
a S = - 23.6%
F,
For two Yukawa potentwo exponential
two Yamaguchi potentials,
we will
attractive
has been reconfirmed
energy of 12.76 MeV, for
10.50 MeV, and for
for
of the Faddeev equations
F -1 .
radial
approximation
purely
bosons of nucleonic
F, y = 0.231608
= 2.7251 F, at = 5.4039
tials,
This result
solution
identical
via the sum of two potentials
r
the triton,
form assumed.
by Noyes ans Osborn76 by direct
and to its
The variational
various
sensitive
would give
we must also expect calculations
partieters.
considerably
We also see
force which is assumed.
to the range of the force
in marked contrast
fit.
can be expected
of the nuclear
energy to the triton,
of et to be quite
variation,
as well
Clearly,
3.
calculations
A, B, C, D of the van OersSeagrave
value
binding
three-nucleon
zero in T right,
Since Thomas74 showed that
infinite
in-Figure
11.24 MeV. Before
to establish
the accuracy
of the numbers.
The first
used gaussian
the results
(to better
question
quadrature
to ask is about numerical
for
good to at least
than this
accuracy)
the double integrals,
$6.
More convincing
with
a calculation
accuracy.
The code
and many checks showed
is the exact agreement
by Ball
and Wong77 which
I
-33-
used an expansion
next talk.
The code also
variable
calculations
much higher
lished
in Stbmian
(as a two-variable
the latter
interactions,
the neglect
interaction
a contribution
but is is not clear
on this
point.
8 = 2 states
agreed exactly
Ball
that
calculation
by Humberston,
find
with
angular
pub-
were used.
momentum
For gaussiam
and Wills 75
Gammel, Hill
the calculation
and Wong77 in fact
in their
new calculation
of higher
of Baker,
the single-
parameters
of about 3% of the binding
states,
in the
which can be done to
is justifiable.
the exact calculation
discuss
code) agreed with
calculations
is whether
in the two-nucleon
indicates
will
of Siten'ko and Kharchenco when their
The second question
states
whi&Wong
of the Yamaguchi interaction,
accuracy;
results
functions
energy from & > 0
is completely
reliable
only -$$ correction
using Yukawa potentials.
from
Further,
and Osborn79 using variational
Hall
a
tech-
niques which give both upper and lower bounds which are very close together,
and which include
all
angular
curves 73 very closely
in the region
mentioned
shows that
calculation
found by Osborn 73 at higher
linear
variation
We conclude that
via purely
calculated.
potentials
instead
that
lie
1
of Ed with
local
Unfortunately,
with
short-range
of separable
of interest.
the apparent
coupling
constants
potential
strength
the binding
attractive
momentum states,
energy of three
potentials
this
potentials.
nearly
as close together
reason to expect the opposite.
collapse
the last
of the gound state
was spurious,
continues
identical
into
and that
this
for
local
about that
is no evidence
by the variational
in the case of hard-core
region.
and unambiguously
or I would be talking
there
the
bosons interacting
not been demonstrated
In particular,
the upper and lower bounds obtained
Osborn's
Incidentally
can be accurately
has still
repulsion,
bracket
methods will
potentials,
and good
-34-
The implication
are equivalent
nearly
of these numerical
in the case of the local
two decades old)
are quite
Loiseau
end Nogami
due to meson exchanges will
Of course,
this
new physics
local
tive
potential
result
result
(or radial
see below,
the experimental
phase at all
energies
given by Newton
local
81
for
potential
a local
models is likely
as well
simply fitting
forces
of this
The
model is closer
local
to
potential
would
the Yamaguchi singlet
Bargmann potential),
at short
distance,
than the exponential
Yamaguchi potential,
of off-shell
the high energy
done using the procedure
a generalized
thanthe
energy to
is even more disturbing.
to fit
repulsive
But the
is significant.
attractive
potential
is slightly
to et.
the effec-
the binding
central
(which can be readily
Yet the non-local
the difference
off-shell
that
behavior
constructing
force
and experiment.
of the interaction)
the on-shell
the three-body
than an MeV in order to get
Yamaguchi interaction
gives 0.74 MeV more binding
that
better
between theory
expected to give even less binding
below).
that
we
number, but does show that we will
S phases than a purely
If we construct
give.
this
variation
results
about 1.5 MeV (attraction)
contribute
- even for local
for the non-local
As we will
to variational
is not enough to determine
accuracy
noted
To put the 2.26 MeV dis-
have estimated
shows immediately
range parameters
behavior
90
to considerably
it
as already
and the Yukawa models in context,
out of a discrepancy
the requisite
potentials
is not a very reliable
have to calculate
(which,
distrubing.
crepancy between the exponential
note that
results
exponential
so would be
potential
as we have just
potential,
(see
seen,
indicating
behavior
between local
and non-local
to be more than an MeV.
Hence reliable
models for
as for the on-shell
two-nucleon
interaction
will
the
have
-35
to be developed before
in the last
we will
section,
we can trust
theory
have to extract
calculations
of et.
is not a very reliable
As discussed
guide for this,
as much as we can from electromagnetic
and
experi-
ments.
This suspicion
binding
energy calculations
results
we have just
still
nucleon potential,
received
there
if
the two-nucleon
at all
energies
Bargmann potential
plete
agreement with
local
potential
tering
length
potential
mentioned
and binding
that
for
it with
Hulthen
that
bound by an average
gives rise
potential
.to a single
similar
equivalent
to the
agreement with
the
which gives the genera-
In all
three
should be good to at least
1%.
cases the scatare the same.
which binds
the triton
binds it with
14.352
generalized
The two local
shows that
Bargmann
potential
results
the variational
on the basis
of the on-shell
in the calculated
value
calculations
of the two-nucleon
scattering,
of et.
are
of Osborn and of Ball
Hence whether we ask for the local
of the Yamaguchi potential
or on the basis
potential
com-
in which case the
the exact results
potentials
bound
The second is to require
the equivalent
10.689 MeV.
and Wong for
4 MeV difference
particles
a Yamaguchi interaction
but comparison with
function,
Concentrating
wave function,
potential.
variational,
valent
group.
energy of the two-body bound state
G. Erens 93 finds
binds
by preliminary
ccmplete
above.
the bound state
MeV, the equivalent
12.263
a local
in triton
confirmed
from zero to infinity,
is the Hulthen
L . P. ?Sok'2 finds
force
One is to require
lized
MeV, while
identical
are two ways to define
phase shift
behavior
from van Wageningen's
of three
Yamaguchi interaction.
with
has been dramatically
on the bound state
state,
of off-shell
about the importance
equiwave
leads to nearly
This destroys
at one
-36-
blow many commonly expressed
is to fit
the scattering
two-nucleon
just
assumptions,
length
bound state
and binding
wave function.
how we take the two-nucleon
body problem has more effect
body forces!
right,
if
such as-that
energy,
It
all
is needed
or to get the right
also shows immediately
scattering
matrix
off
shell
on et than the anticipated
Hence we must get the physics
we exe to draw any fundamental
that
of this
physical
that
in the three-
effect
off-shell
of threeextension
conclusions
from these
calculations.
Since we have just
results
quite
overbind
this
fact
were mainly
to include
in the model used for the nuclear
local
potential
variational
calculations
for
local
important
problem
Yamaguchi models give
potentials
g4,~5,~6,~7
calculations
(i.e.
which
in demonstrating
the ease
could be solved in the separable
and we turn to the more recent
which have attempted
force
attractive
the pioneering
with which the three-body
mation,
central
comparable to purely
the triton),
established
seen that
tensor
force.
and more ambitious
forces
and/or
comparison.
After
calculations
short
Unfortunately,
approxi-
range repulsion
here we have no
herculean
efforts,
the
calculations
of Et using the Hamada-Johnston potential
had
33
to about 5.7
improved the 2-3 MeV binding obtained by Blatt and Delves
MeV, and as of last
summer Delves 39 estimated
that
by 1970 they might
converge to about 7 MeV leaving
body forces
by
a progress
report
trast,
that
IDiSeaU
on this
simple separable
the 1.5 MeV estimate of attractive
three30
to take up the slack.
We will have
and Nogemi
program from Delves this
models overbind
by making the models more 'realistic",
reduced.
the triton,
afternoon.
In con-
and the hope has been
the binding
energy would be
-37-
The simplest
way to present
a very instructive
the value for
plot
and is reproduced
et,
in the last
or the n-n value
scattering,
nucleon problem is obvious
report
above that
is it
tivity
of the results
etc.
to see if
= - %
probably
any more light
whether
effective
from p-p
by charge-symmetry
uncertainty
in the three-
However, the sensi-
a worry.
probability
remains,
e-d scattering,
can be shed on this
for
time,
and we are happy that we could
no longer
of re-analyzing
1
of this
at a glance,
The model used by Phillips
vs(P,q)
an open question
of 2.35 MeV predicted
to the D-state
again the importance
At that
4.
45 by charge independence
The seriousness
from p-p scattering.
in Figure
value of 2.44 MeV for the singlet
of 2.73 MeV predicted
the value
is to compare them on
it was still
section,
to use the n-p experimental
range,
results
90 which compares simultaneously
given by Phillips
a2 and for
as we discussed
the latest
this
and we emphasize
photodisintegration,
problem.
calculation
was
1
(p2+ Bs2) (q2+ ps2)
VJp,q)
with
= - %
the singlet
meters adjusted
tP2S12(8) /
1
1
(p2+ FE)' 1 \ q2+ St2
1P2+ Bt2)
parameters
adjusted
tds,,m
cs2+ ZJ2 1
as indicated,
to change the percentage
D state while
to at = 5.397 F, ed = 2.2245 MeV, and & = 0.282 F2.
sive parameter
study on the effect
(but not the percentage
Kharchenco,
Petrov
D state)
of varying
was carried
and Petrov 91 , and further
and Storozhenco 92 more recently.
and the triplet
details
maintaining
A still
the singlet
out earlier
published
paraa fit
more exteneffective
range
by Site&o,
by Kharchenco,
In order not to confuse the plot,
'
I
-3%
we give only two representative
It
The difference
and 3,.
value by Phillips
points
from these .calculations,
between the It point
is due solely
and the corresponding
work of Mitra
in potential
is all
Corresponding
- 23 F.
shifted
strength
35 .
and collaborators
ot as is due to the fact
tivity
over from the values
that
that
points
in a S’ as can be seen
to the difference
from the study of the dependence on as contained
the earlier
labeled
in those papers,
Of course this
insensi-
a change of only a percent
values
given by Phillips
or so
as from - 17 to
is needed to shift
for differing
or from
of rs would be simply
by approximately
the same
amount.
A different
point
labeled
effect
3, which replaced
in both the singlet
maintaining
studied
is indicated
the Yamaguchi cen'iral
and the triplet
the same fit
has as much effect
by the same authors
interaction
to the two-nucleon
as changing the percentage
force
form factor
while
by (p2+ S2)-3,
Clearly,
parameters.
D state
Surprizingly,
shift,
there
little
Mitra93,
force
more ambitious
and the results
change in the on-shell
calculation
less realistic
in that
phase
has been made by Schrenk and
in Figure
model is the same as above for the points
for
an in-
21 and S from 1.13 to 5.22.
are also presented
uses the Naqvi 94 parameters
phase
5.
as can be seen from Figure
A still
let
is remarkably
this
In order to try
by 1%.
to see why, we have computed the change in the on-shell singlet
2 -3
which requires
shift for the still
more extreme case (p2+ S )
crease in A from 0.15 X 10' to 2.96 X 10
by the
4.
Here the trip-
labeled
(C + T)N; the second triplet
the original
which has simply been dropped here.
fit
was made including
The main change is that
(C + T)y, but
force
is
an L*S term,
the singlet
-39-
potential
is of the form
22
t'P q
(q2+ B;) - (P2+- Q2(q2+
1
1
Vs(P,,9) = - %
i (p2+ B;)
where the second (replusive)
term causes the singlet
The point
sign at high energy.
which is somewhat unrealistic
short
which are still
best with
that
publi.shed
to
is quite
due to Gupta 96
Clearly,
in changing the form,
Whether introducing
produce
to see how they agree
little
effective
however,
the models can produce differences
would finally
fits
so we did not have time
lengthy,
for the other Gupta parameters
know how to interpret.
this
G are several
There seems to be surprizingly
D state.
range is much too
the Naqvi model are very close to the Gupta curve.
from 2.33 to 2.7 F in the singlet
et al.
due to Naqvl* 95,
N is the fit
Over the same range (50-330 MeV),
good.
the calculation
compute values
on-shell.
phase to change
we compare the one which is claimed to be
I
values of the So phase in Figure 5 and agree
is reasonably
phases for
Unfortunately
labeled
I
unpublished;
experimental
the fit
labeled
because the effective
The points
(2.33 F).
$j2
change in et and a2 in going
range,
the on-shell
as important
or by Phillips
repulsion
"agreement with
a result
we do not
changes between
as those found by Sitenko,
in changing the percentage
into
the triplet
experimentll
model as well
is anybody's
guess at
point.
Two calculations
and by Borysowicz
forces,
which bear on this
end Dabrowski 93.
point
Unfortunately,
have been made by Tabakin
neither
but Tabakin has taken the care to make sure that
compares agree on-shell
to 0.01 radians
includes
tensor
the models he
in so over the entire
elastic
97,
I
-4o-
scattering
range.
He uses two second rank separable
a smooth type similar
"hard shell"
to that
potential
used by Naqvi ad
of the type invented
most (but because of the non-locality
of the wave function
ferent
like
short-range
-tic,
behavior,
infinite
Since Tabakin fitted
smaller
radius
it
parison
with
'(or different
find
accuracy,
than the estimate
if
the
and Dabrowski
raised
in a similar
version
.
S
for
scattering,
models.
r C = 0.4 F, as comcomparison.
Unfor-
as can be seen from the com-
given in Figure
by Tabakin
local
of the Puff model, has, as
repulsion,
behavior
to the same on-shell
$3
energy of 2.7 MeV
model to a Puff model with
1 So phase shifts
from
and the triplet
a decrease of binding
too much short-range
asymptotic
off-shell
came out to be only 0.2 F, which is con-
the Borysowicz-Dabrowski
to answer the question
fitted
larger
the 0.93 MeV Tabakin finds
stands,
tl~at.
showing
(separable)
than one would expect from hard-core
when going from a no-core
tunately,
to the system,
an average between the singlet
and Dabrowski
pared with
momentum. He finds
as has been shown by Borysajicz
phases, his hard-shell
Borysowicz
which goes to infinii;;;
of the two models differs.
could be considerably
calculation,
siderably
model, not all)
not lead to the ssme value of et to that
The effect
Tabakin's
, and a phase shift
agreement to 330 MeV and similar
momentum behavior
keeps
and hence produces dif-
model givs 0.4 MeV less binding
will
The latter
of the separable
than going to zero at infinite
even on-shell
one of
Gupta, and the second a
by Puff 99.
the core radius,
correlations
rather
the hard-shell
that
outside
potentials,
5.
It
also does not attempt;
about short-range
correlations
S), since the two models were ncjt
as Tabakin was careful
to do.
-4l-
And the absolute
value
of the binding
been questioned
by Jaffe
and Reiner 100 , who obtain
supposed to be an identical
Another
is that
it
interesting
includes
a charge-dependent
aspect
the effect
of the Borysowicz-Dabrowski
of splitting
the presence
tion
= - 16.6, rr
introduces,
S' state
= 2.8);
no-core,
This would seem to indicate
of a core than without
of Schrenk and Mitra
indicate
the singlet
how much isospin
discussed
it,
parameters
cause.
a rather
this
aw =
S
drops the binding
sensitivity
as was indicated
this
in
and from 8.9 to 8.3 MeV
a smaller
above.
3/Z admixture
but they do find
due to this
calculation
way (from as = - 23.69, rs = 2.5 to the pair
energy from 11.6 MeV to 10.7 MeV for
a core.
8.7 MeV for what is
calculation.
- 23.69, rtf" = 2.5 plus a:
with
(3.3 MeV) has
energy obtained
Unfortunately,
to rs in
by the calculathey do not
charge-dependent
effect
small change in the percentage
The percentage
of S' state
of
changes from 1.1
to 2.0 when the core is added, or from 1.4 to 2.4 when the core is added
in the charge-dependent
model.
-42-
WAVEFUNCTIONS OF H3 AND He3
v.
An extensive
review
the three-nucleon
of the work on the ground state
on "The Nuclear
on Light
Nuclei
at Brela.
clusions
and report
V.l
If
Three-Body
We shall
somewhat more fully
the nucleon-nucleon
must be wholly
the work which has been done since.
interaction
is charge symmetric the Coulomb
ascribed
C
found to be
= 0.764 MeV
to electromagnetic
much work has been done, the question
been settled.
Direct
variational
if
this
although
is really true has not yet
101
of nEc, assuming charge
lead to an estimate
which is likely
to be low,
0.6 MeV
since the trial
A number of authors"'
wave functions
have followed
electron
by Delves has been that
scattering
it
Coulomb energy of 0.76 MeV.
The result
The conclusion
Calculations
have been more or less consistent
not impossible
of very careful
Oksmato and Lucas 103 has been a discrepancy
in the Coulomb energy.
if
of other
with
approach.
to
as reported
to obtain
calculations
the
They
which have been fitted
form factors.
is very difficult
used underbind
a different
the Coulomb energy with wave functions
the experimental
tials
Unfortunately,
effects.
calculations
AEc-
calculated
summary of his con-
The Coulomb Energy of He3
f&
triton.
Problem" given at the Symposium
only give a short
energy of 'He, which is experimentally
symmetry,
of
1967 has been given by L. M. Delves 39
system up until
in his lectures
properties
a
by
of 0.13 MeV, on the low side,
authors
this.
104 using local
If this
poten-
is true we can
-43-
conclude that
that
the n-n interaction
charge asymmetric
is stronger
three-body
forces
than the p-p interaction,
exist.
Interpreting
a charge asymmetry of the two nucleon potential,
corresponds
according
to about l$ in the strength
the result
the discrepancy
as
of 0.13 MeV
of the n-n and p-p interaction,
of Okamoto103 .
to a rough calculation
There are counter
or
arguments against
this
One is that
interpretation.
no charge asymmetry has been found in the Coulomb energy of the heavier
mirror
nuclei.
interpreting
This argument is somewhat weakened by the difficulty
small quantitative
effects
ment came from the work of Mitra
separable
which approximately
potentials,
with
had not been included
experiment
after
in their
inclusion
Since then more attempts
separable
potentials.
of treating
mations
best approximation
testing
perburbation
is just
calculation
3He combined with
separable
given by Alessandrini,
of Alt,
Grassberger
this
the validity
et al.
bit
a triton
energy,
a
Since repulsive
core
better
have been made with
concerned with
the problem
of separable
of two different
potentials.
approxi-
model triton.
The
than the straightforward
wave function.
This best value
for
to be much too high because of the
two-nucleon
10s .
question
in a simplified
An exact treatment
involved.
binding
they expected agreement
in the framework
function
AEc, which is 1.3 MeV, is expected
approximations
potentials
were mostly
a little
with
the triton
than too low.
to settle
the Coulomb interaction
to the Coulomb Green's
They found for those
of such effects.
Some authors
Adya 107 is concerned with
106.
fit
Coulomb energy which was too high rather
effects
The second argu-
in complex nuclei.
and his group
of
of the Coulomb interaction
potentials
To do this
and Sandhas log (AGS).
in
of rank one has been
they employed the formalism
They found that
in the case that
-44-
there
is a non-separable
the AGS equations
into
core effects
was estimated
by using the Sitenko
110
and Kharchenko
the triton.
106 Naqvi potential,
agreement with
of particles,
them to take the
any approximations.
Of the tensor
parameters
which underbinds
the experimental
However, no
forces
in the
Using only the central
0.63 and 0.84 MeV for @.Ec. Although
obtained
in reasonable
This allowed
The contribution
This overbinds
of the (CN+ SN)
tively
account without
were included.
Yamaguchi potential.
part
between only one pair
can be solved exactly.
Coulomb interaction
repulsive
interaction
value,
it,
they respec-
the average value
is
hard core effects
are
expected to lower it by 15%.
106
and was recently
was also used by Gupta and Mitra
111
. They reject the subtraction
of the
by Okamoto and Lucas
This procedure
criticized
tensor
since it produces
part,
the Yamaguchi potential
singlet
effective
'experimentaltr
nantly
value
with
.
state
local
this
because they were fitted
point
potentials
fitted
the same Coulomb energy AEc = 0.85 MeV.
hard core radii.
potential
They then find
of 0.436 fm and conclude that
value of r s is probably
Okamoto and Lucas make variational
nearly
an exponential
to be predomi-
C
data as the Yamaguchi potential
with
to a
of AE .
tering
culation
are believed
the use of the correct
exponential
of
of rs = 2.320 rt. 0.044 F, the
Because the two protons
for the calculation
To investigate
tions
112
The parameters
potential.
were also criticized
range of 2.15 fm instead
in the singlet
important
an incomplete
to the same low energy scat-
used by Gupta and Mitra,
fitted
and found
Next they repeated
to r S = 2.83
AEc = 0.613 MeV for
Gupta and Mitra's
calcula-
F
the cal-
and with
various
the hard core radius
calculation
probably
contains
-45-
a large
due to the wrong choice of r s and the neglect
error
A separable
potential
of LQc with
calculation
reported
by Jaffe
and Reiner
100
They fitted
.
justice
with
the repulsive
finite
core of the Coulomb energy,
conclusion
concerning
view of the approximations
also reported
in six dimensions,
three-nucleon
bound state
the question
local
form factors
as
the importance
does not allow
of
a de-
forces
in
without
of the Coulomb
wave expansion
an exact solution
potentials
scattering
potential
of the
a repulsive
with
used was extremely
core.
the gap between the calculations
forces
has
with
separ-
has been somewhat narrowed.
and Radii
of electrons
of3H and$e
from
3H and 'He can be expressed
of the momentum transfer
of the target
of the
agreement was obtained
of the charge dependence of nuclear
although
in terms of two functions
Although
the central
The Charge Form Factors
The elastic
tic
problem.
being a square well
able and with
V.2
which allows
data to 5-l@,
not yet been settled,
value
and Reiner found
a calculation
method of partial
wave function
Clearly
but still
not
involved.
energy of 'He using a promising
unrealistic,
Jaffe
the charge dependence of nuclear
Simonov and Badalyan113
the experimental
was
a procedure
expectation
This confirms
values of 0.71 and 0.73 MeV respectively.
account,
On variational
to the Coulomb potential.
and He wave functions.
trition
central
Puff potentials
grounds the Coulomb energy should be bounded by its
calculated
into
but also to the p-p interaction,
only to the n-n interactions,
which does not do full
separable
which takes hard core effects
of the Puff type,
recently
a rank-two
of the hard core.
particle.
q, the charge and magne-
Rather accurate
experiments
have
-46-
out on e - 3H and e - 3He scatteringf
been carried
The resulting
of the spatial
form factors
114 are identified
distributions
of the electric
of the bound states
practically
absent and the magnetic
of the three-nucleon
provide
the most reliable
bound state
ch
rapidly
a contradiction
rather
-&ch
ween the like
weaker that
weak, the 'H and 'He
2
'
of the charge distribution
rch(He3)
= 1.97 * 0.05 F
rch(H3)
= 1.70 * 0.05 F
the 3He charge form factor
of the
continues
However, this
to fall
off more
is not necessarily
since the charge form factor
measures the charge distribution
116
114
and Dalitz and Thacker
Schiff-
than the mass distribution.
have pointed
much, since the Coulomb potential
2
=l
than the 'H form factor.
with
the wave functions
114
From experiment
energies
to be large,
For small q2 we can write
where r ch is the root mean square radius
At higher
Although
are relatively
are different.
likely
are
data on the properties
wave function.
charge dependent effects
F
nucleus.
transform
charge and magnetic moment
exchange effects
of 3H and 3He are not expected to differ
charge form factors
with the Fourier
of 3H and 3He. 114 Since charge exchange effects
the charge form factors
and other
and analysed 114,115 .
out that
particles,
that
this
is likely
to be true,
which are predominantly
between unlike
to spread out over a bigger
particles.
region
since the interaction
bet-
in the singlet
is
This causes the like
in space than the unlike
state,
particles
particles.
-47-
This effect
factors
gibes in the right
even if
direction
the wave functions
to explain
the difference
in form
of 3H and 'He are substantially
the
same.
Delves"
has given a careful
the 'H and 'He form factors
wave function.
present
states
than the principal
the most important
If
tential.
to estimates
admixtures
to an unlikely
The 3He wave function
by Gibson 1x7 failed
vestigate
with
116
uncomfortably
the result
neutron
with
large
of the various
and found that
form factor
contain
The first
contributed,
of 4%. The inclusion
leading
of the D-states
However, the necessary
by the fact
that
S'
the variational
e - 3H scattering
1-15.
is not taken to be identically
the Dalitz-Thacker
of about 1.59. TO in113
included
in 'He, Gibson
it was possible
Psl = 2% and P3/2 = 0.25%.
from inelastic
will
states.
only the S' state
to an S' probability
compared with
are nearly
in terms of these admixtures,
of the induced T = 3/2 state
in his calculations
are
on the assumed shape of the S' wave
demonstrated
corresponds
the role
experiment
P(S')
strongly
state
induced by the Coulomb po-
to remove the discrepancy.
which is clearly
wave function
it
115 assumed that
depends quite
function,
can be analyzed
high probability
symmetric
is expected
are expected to be the S' state
for the probability
by Schiff
calculation
probability
S totally
T = 3/2 mixed symmetry S state,
The form factors
leading
of the triton
the 'He and 3H wave functions
of mixed symmetry, and the D-states.
an additional
between
between the 3H and 'He form factors
in the wave functions.
identical,
of the difference
in terms of the classification
A difference
as soon as other
discussion
to reach agreement
However, Ps, = $ is still
101
= l.% and
result
ps '
This can be reduced if
zero
106,119 .
With a
the
-4%
positive
form factor
possible
to obtain
somewhat diminished
on the asymptotic
for
as given by experiment 120 it
the neutron
a Ps, = 1%.
The reliability
of these conclusions
116.
of the wave function
is
dependence of P especially
S'
by the strong wave function
part
is
They hold only for
a
given assumed shape of the wave function.
Recently
tials
of Ps, with
a new calculation
of the Puff type,
including
repulsive
rank-two
separable
core effects
poten-
has been reported
and Dabrowski 98 who find
for
P = 2.0 - 2.4% respectively
S'
charge independent and charge dependent forces, which is more than twice
121
. This seems
the value of Ps, = 0.31 - 0.96% found by Bhakar and Mitra
by Borysowicz
to show that
Although
with
inclusion
of the hard core increases the discrepancy again.
122
favors a Ps, = 1.2% with rs = 2.7 fm and PD = 7$,
Phillips
a difference
include
in the charge radii
repulsive
In a recent
electromagnetic
central
principal
clearly
This means that
Dalitz
the 'H wave function
is then found that
in the inside
112.
assuming local
is exactly
mainly by the asymptotic
is quite
important,
A very small probability
of 0.001 - 0.006% is found in the He3 nucleus.
alone is
in the electro-
also comes to the conclusion
region
spin
given by the
the Coulomb repulsion
the observed difference
The author
in the
having no spin and isobaric
a hard core,
to explain
and Thacker
the variation
the differences
due to the Coulomb potential,
are not determined
the wave function
state
It
form factors.
charge radii
with
without
insufficient
magnetic
Ohmura123 discusses
form factors
S-state.
out that
large.
article
potentials
dependence.
He does also point
core effects.
of PS’ with rs is fairly
of 0.12 to 0.16 fm, he does not
for
region,
that
the
but that
in disagreement
the T = 3/2
The Schroedinger
I
-4g-
equation
for the three-body
expansion
higher
bound state
of the wave function
order terms.
predicted
A negligible
in the previously
Alessandrini,
with
et al.
109
From the previous
is solved approximately
Legendre polynomials
admixture
discussed
work with
coming into
discussion
we see that
recent
nucleon
solute
with
magnitude
article,
potentials
by
= 1 and the initial
giving
evidence
easier
without
concerning
the three-
out that
to fit
the ab-
than the
since we leave the initial
slope as determined
the correct
It
between the
explained,
is rather
of PS,
from 0% - 0.25%.
Delves $9 points
This is not surprising
Hence any wave function
is also
estimates
the difference
experimental
of the form factors
?H - 'He difference.
F(0)
other
In his review
systems.
condition
that
of 'H and 'He can be satisfactorily
conflict
separable
the
.
not yet been demonstrated
charge radii
omitting
of T = 3/2 states
range from 2.4% - 0.3% and the T = 3/2 probability
has clearly
by an
by the charge radius.
charge radius
will
fit
the small
Only at momentum transfers
momentum transfer
part of the form factor.
-2
They then drop
some structure
appears in the form factors.
q* 2 7-3 fm
below the straight
can be explained
It
potential.
line
by the presence
is found that
good can give quite
hand, it
reasonable
yielding
for a Gaussian charge distribution,
of a repulsive
even wave functions
is not enough to fit
wave functions
V.3
predicted
nearly
fits
core in the nucleon-nucleon
which are not particularly
to the form factors
the triton
binding
the same binding
which
energy,
115.
On the other
since different
energy give different
form
The Beta Decay of the Triton
In his review
article
89 Delves discusses
the information
on the 3,
-5o-
wave function
which may be contained
In particular,
decay with
interactions
It
124,125
Blin-Stoyle
neutron
the ft
=
value for
m5c4
2fl3fl7hl2
Where Gv and GA are the polar
and M the matrix
A
of O+ to O+ decays
of weak
probabilities.
is given by
+ G;1MA121
vector
coupling
constants
weak Hamiltonians.
of Gv and GA using
The measured ft
value for
i%j2
= 1, 1MAi2 = 3
= 1137 +_ 20 sees.
H
are nearly
equal,
it
that
For the axial
3H wave function.
of the
vector
matrix
2
IMAI
where he neglected
Delve:'
An
'H is125
the 'H and 'He wave functions
which is independent
and
126,127 combined with the measured value of (ft)
WI3
If we assume that
of the PCAC theory
of the corresponding
for neutrons 12' leads to values
for the neutron.
[G;jyI12
3He.
a comparison of this
an allowed beta-decay
and the axial
elements
3H into
of
on the S'- and D-states
and give information
can be shown that
analysis
has suggested that
decay should lead to a test
(ft)-1
%
in the beta-decay
we find
= 3[P,
according
to Blatt
130
- $ Ps' + ; PDl
any T = 3/Z states.
Inserting
finds
2
IMAI
- 2.70
variational
values
follows
-5l-
In any case
II
2
MA
calculations.
< 3, independent
Even that
of the neutron
124,131 .
crepancy by taking
theory.
there
2
M , increasing
A
I
I
2.
assumption
3.
the experimental
the measured ft
mentioned
Until
account,
value
to remove the dis-
using CVC and PCAC
instead
of an enhancement
Delvesgg
the discrepancy.
reasons for this
the ft
points
out
discrepancy:
charge dependent effects,
of CVC and PCAC theories,
results
and Tint125,
pendent reasons for
with
tried
a reduction
of T = 3/2 and other
by Blin-Stoyle
mation
possible
neglect
Stoyle125.
into
they obtained
are several
The last
and Tint14
exchange effects
1.
4.
leads to contradiction
Blin-Stoyle
Unfortunately,
in the value of
that
value
of the accurady of the variational
values
of the neutron
the ft
new experiments
the beta decay of 'H.
are in error,
p + p -+d + IX, used
and
and/or
of 'H are in error.
reason seems to be the most likely.
doubting
on the three-nucleon
for the reaction
value
of the neutron
have been performed
bound state
wave function
Some inde-
are given by Blin-
no additional
infor-
can be deduced from
-52-
CONCLUSIONS
VI.
TWOpoints
will
emerge clearly
be necessary
difference
to understand
the wave function
discriminate
inside
distances
elementary
particle
the correct
to understand
and rr
in that
has an important
bearing
on this
but differs
condition
models.
Work on momentum dependent models is still
the three-nucleon
matter
meaningful
l-2
F level
in
been removed.
potentials,
in one way for
boundary
model, and in another
problem,
problem
Again,
indicates
current
comparison with
for
the non-local
but the work of Ingber
this
because of the difference
model.
due to uncertainty
This is pretty
question.
to local
force
this
it will
be
of the two-nucleon
we stick
directly,
number of
T
than we do now, and the size of the hole in the wave function
better
nuclear
as sug-
inside
has probably
behavior
-
or 0.7 F in radius
we have also seen that
the off-shell
un-
be able to
model and the large
the comparable uncertainty
The second is related
the
0.5 F in radius
meson repulsion,
condition
value to use for r:
it
have to understand
degrees of freedom which can be excited
Fortunately,
important
we will
between a hole in the wave function
as suggested by the boundary
enough to tell
of 1 F, and in particular
gested by hard core models and vector
radius.
well
is that
of E and of around 1 fermi
t
In order to do this
of a2'
The first
if we are to avoid a basic
of around 1 MeV in calculations
in calculations
matrix
review.
the deuteron
and 7% D state,
between 44
certainty
from this
it
allows
in a2 calculations.
separable
territory
at the l-4 MeV level
in
related
at least
in the two-nucleon
on these points
defined
the non-local
on the closely
may also be crucial,
uncertainties
experiment
virgin
well
in-
central
frustrate
in et and the
if
-53-
Other conclusions
separable
models can be brought
phase shifts,
angular
that
are more controversial.
that
a rather
momentum states
to the triton
potentials.
hard-core
low cutoff
are included,
therefore,
much error,
calculations
be necessary
It
there will
be a 2 MeV or so discrepancy
is our guess that
models which give the same fit
Hence, it
to 300 MeV.
information
before
with
give more
local
froma variety
or not there
close bounds
calculations
has been carried
between separable
we will
elastic
and local
scattering
have to extract
electromagnetic
are three-body
out,
data up
off-shell
experiments
forces
in the three-
system.
The point
of view taken here is to treat
a problem in physics
rather
We mean in no way to belittle
than in mathematics
the real
the three-nucleon
or in nuclear
accomplishment
very simple
reasonable
of both the two- and the three-nucleon
description
the lo-1576 level.
us to the fact
that
But we hope that
a much harder
between nuclear
of these studies.
interactions
these real
and equations
successes will
task remains before
and elementary
particle
physics
system as
phenomenology.
made by the separable
models in showing that
nection
such
or methods giving
to two-nucleon
of two-nucleon
hard-core
to also claculate
when this
is expected that
we can say whether
nucleon
and less certainly,
in order to be sure the variational
have converged.
still
nucleon-nucleon
such models will
models by other methods (Faddeev,
on et from both sides)
that
in the number of two-nucleon
than variational
It will,
seems likely
good agreement with
can be used without
when the small effects
binding
into
It
can give a
systems at
not blind
a convincing
con-
can be forged
out
-54-
We are indebted
is
crucial
SO
preliminary
for
to Houk for
Delves for
prior
to his publication.
sending us his Brela
have taken the liberty
Chapter V.
of relying
who have kept us informed
Phillips
Nuttall
this
fruitful
of their
in particular
and Wong.
for
lectures
heavily
We are also most grateful
communications,
discussion
this
prior
new result,
to quote it
and his
to publication,
on his review,
to the many workers
work through
which
We are most grateful
preprints
to Amado, Erens, Kok, Mitra,
Most of all,
organizing
us ofhis
and for permission
our subject,
analysis
informing
and we
particularly
in this
conference
and interchange
of ideas.
in
field
and private
Osborn,
we wish to express our gratitute
interesting
to
to
and making possible
-55-
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authors
Analysisll,
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Northeastern
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preprint;
I am indebted
prior
to publication.
A. H. Cromer, "Nucleon-Nucleon
Bremsstrahlung
Theorem:,
Northeastern
sending this
a(p,q)
preprint;
paper prior
useful
to the
and the Low Energy
I am indebted
to the author
In the notation
to publication.
Brems-
for
of Rev. 13
= sin %l)f(P,&
T. Houk, private
I am indebted
prior
to
subject.
J. H. McGuire and A. H. Cromer, 'Low Energy Nucleon-Nucleon
strahlung
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9,
611 (1967)
communication
and Erratum,
to Dr. Houk for permission
Rev. Mod. Phys. (in press);
to present
this
new result
to publication.
52.
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53.
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(1967),
New York (1966),
p. 3.
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G. Breit,
K. A. Friedmann
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etc.,
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Dr. Lebowitz
analysis
G. Breit,
2025 (1963);
the data selection
f or the error
see Table 1.
< 6as6at > has the wrong sign in Eq. 3; this
correlation
Dr. Houk points
does not change
in the paper.
T. L. Houk, private
joint
Physics,
de Physique Nucleaire,
of the Symposium on Light
Gordon-Breach,
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K. A. Friedman,
J. M. Holt
and R. E. Seamon, preprint
entitled
-6o-
"Short
and Long Range Charge Independence",
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Jr.,
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I am much indebted
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B. J. Hill
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(1966).
Moscow
R. W. McAllister,
Rev. Letters
Schiff,
B. F. Gibson and B. P. Carter,
Of the International
Dubna, 1964.
L. I.
R. J. Blin-Stoyle,
Theoretical
Nuclear
Appendix A.5.
Phys. Rev. Letters
-13, 55 (1964).
Physics,
Wiley
Table I.
Values of the 'So n-p Effective
Range for Various
Data Selections
Present Analysis
unp(o) = 20.442
at
-+ 0.023 b
= 5.4235
(errors
Data Selection
r
am = - 3.719 * 0.002 F
a
F
'd
negligible
S
Previous
S
=-
Analysis"
unp(0) = 20.33 f .Oy B
&ri.H=
-3.741
+ ,011 F
23.7146 F
= 2.22452 MeV
in determination
of rs)
D.F.
x2
4 Prob
and as codetermined
("t
r
S
D.F.
with
x2
rs)a
$ Prob
ALLb
2.758
* 0.053
7
5.942
54.9
2.52
-I- 0.10
6
4.45
62.1
EBMLC
2.646
f 0.072
1
0.060
80.8
2.44
t- 0.11
1
0.15
68.5
ALL - EBML
2.891
+ 0.078
5
2.449
79.4
2.64 t- 0.13
4
1.27
86.5
Adding constraint
r = 2.73 * 0.03 F
S
ALL
2.736 + 0.026
E3
5-993
64.8
2.71 + 0.03
7
8.40
33.8
EBML
2.713 + 0.028
2
0.232
89.1
2.71 k 0.03
2
6.31
4.25
ALL - EBML
2.750
+ 0.028
6
2.750
81.7
2.73 + 0.03
5
1.77
87.!3
a H. P. Noyes, Nucl.
b
'
Phys. 2,
508 (1965),
Table 2.
H. P. Noyes, Phys. Rev. 130, 2025 (1963), Table 1. The attentive
reader will note that the degrees of freedom
do not add up for the prxous
analysis.
On checking the computer output I find that for some reason which
now escapes me, the cross section at 2.54 MeV (Table 1, No. 6) was omitted; if this is done in the current
analysis,
the value for rs for the remaining seven points becomes 2,734 in embarrassing good agreement with
the prediction.
C. E. Engelke,
R. E. Benenson, E. Melkonian
and J. M. Lebowitz,
Phys. Rev. 129, 324 (1963).
-66-
FIGURE CAPTIONS
Figure
1.
Possibility
that
the outgoing
3 in the illustration)
the direct
inside
channel
can always scatter
2.
the range of forces,
like
(channel
the two particles
if
in
they are
which gives a source term in channel
pl/rl
for
rl
computed from n-p total
Values of rip
channel
(2 and 3 in the illustration)
1 which only vanishes
Figure
wave from another
large.
cross section
below 5 MeV compared with the prediction
I!z 0
corrected for the 'II - fi mass difference
measurements
from charge-independence,
in the OPE term,
that
r w = 2.73 + 0.03 F.
S
Figure
3.
Fit
to the low energy n-d doublet
S phase shift
of the form
k ctn s2 = - A + Bk2 - C/(1 + Dk2) as given by van Oers and
Seagrave,
Figure
4.
Phys. Letters
24~, 562 (1967).
Comparison of calculations
Phillips,
Nuclear
of et and a2 as given by A. C.
Physics A107, 209 (196Q
N. M. Petrov and S. A. Storozhenko,
096%
and G. L. Shrenk and A. N. Mitra,
Symposium.
The much more extensive
second group in the reference
lication
Physics
by A. G. Sitenko,
Letters
shifted
cited
Physics A106, 464
preprint
results
and Brela
obtained
by the
and in the earlier
pub-
V. F. Kharchenko and N. M. Petrov,
21, 54 (1966) h ave mostly been omitted
not to confuse the plot.
slightly
Nuclear
V. F. Kharchenko,
They use only 4$ D state,
from Phillips'
results
(open circle,
in order
and are
solid
-67-
circle,
for
open triangle)
because of the.slightly
as, as is illustrated
It.
Phillips
if
labeled
shifted
by these authors
guchi form factor
in the central
the interaction.
Results
directly
comparable,
potential
fitted
meters,
since they include
let
as Phillips
The designation
refers
from Naqvi and Gupta by Shrenk and Mitra,
same order
labeled
Figure
5.
along both dotted
are not
use
% D
use the Naqvi triplet
model used (N,Gl,G;,G~,G;,G2,G3)
for
curves;
of
a second rank singlet
interaction
the L*S term.
parts
The (C+T)y points
by Naqvi and Gupta.
omitting
by cubing the Yama-
but not the tensor
The (C+T)N points
points.
The value
from Shrenk and Mitra
the same (Yamaguchi) triplet
state
of rs also agree with
by about the same amount.
3t is obtained
value
r S = 2.7 F by the open square
for
Values for other values
labeled
different
para-
of the sing-
to parameters
taken
and occurs in the
clarity
the points
are
only along the (C+T)N curve.
Comparison of various
separable
between 50 and 330 MeV.
the n-p Yale IV fit
R. D. Haracz,
singlet
The dotted
models with
experimental
curve is from
(R. E. Seamon, K. A. Frideman,
J. M. Halt
and A. Prakash,
(1968))
and the experimental
points
results
(M. H. MacGregor, R. A. Arndt
G. Breit,
Phys. Rev. s,
the latest
p-p
and R. M. Wright,
of the Nucleon-Nucleon
.(p,p) Analysis
from 0.400 MeV" UCRL 70075 (Part VII)
corrected
1579
Livermore
"Determination
Rev. (in press))
experiment
Scattering
Matrix
by the n-p to p-p difference
VII.
and Phys.
used
-6%
by the Yale group;
sentially
identical
procedures
in spite
and different
the separable
form factor
those form factors
criteria
for
the parameters
surprisingly,
of the Ysmaguchi
to the 9th power.
The Gupta result
used by
curve is the same as
is computed from the parameters
and Dabrowski.
and n=3
The Puff potengiven by Borysowicz
has been computed by us from
for the Gl model as given by Shrenk and Mitra;
the Naqvi parameters
give a curve between 30
and 330 MeV which lies
almost on top of Gl, according
phase shifts
by Naqvi,
ficantly
is es-
fitting
data selection
n=l is the Yamaguchi form,
raised
fit
of the very different
model using the square root
(Bander).
result
energy-dependent
The shape-independent
the two groups.
tial
the Livermore
published
although
at low energy due to the different
it will
differ
effective
to the
signirange.
R
Fig. 1
-
-
T
,LL-EBML
‘T
I
I
$ 2.7
m
Q 2.6
I
r
- 2.5 _
0
A
I
I
1
2
3LABORATORY ENERGY IN MeV
Fig. 2
I
4
:BML
5
99481
Fig. 3
8.0
EXPERIMENT
8.5
-
-
-
(C+T);;
0
as=
- 20.34
rs = 2.50
A
as = - 20.34
rs = 2.70
0
as = - 23.69
l
as=
N
a
G;
GlG;G;1G2G3
-20.34
= - 23.7
a,“=as=-18
20
rs = 2.70
rs=2.85
rs =2.355
r
r
s
s
= 2. 7
= 2.7
m
tz
0
-
0‘5
0
n-d
DOUBLET
0.5
SCATTERING
Fig. 4
LENGTH
1.0
IN FERMIS
I.5
50
40
L
n = l/2
30
(Shape Independent)
-
20
IO
rY E
(ExDeriment 1
0
=\
7
-10
-20
-30
50
100
I50
LABORATORY
200
250
300
ENERGY IN MeV
101981
Fig. 5
i