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SZ
ORNU-persy DTIE
*
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;
AUG 1 1 1964
2.
MASTER
Some Applications of the Distorted Wave Approximation
for Direct Nuclear Reactions
71
by
2
R. H. Bassel
--LEGAL NOTICE

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rusy,
c omo, or weten odorathon utmed no report, or the
wa taformaktan, warna, method, or proceso deslowed to this roport may not undertege
Mirely od ra;
1. Anou may Identities with ropect to the www oh, or har du multing in the
olm, formation, nou, wood, or procene checloud i s mort.
As
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much porno mamtructor at the Commuter, omployee connector romero,
datumawe, or krom m, meg bormation per me employ er cauract
m t Coll
.or Moonboyu vid much concor.
Lectures to be presented at Hercegnovi, Yugoslavia
July 13-29, 1964

Facsimile Price $_8./0
Microfilm Price $_ 2 2
.
Available from the
Office of Technical Services
Department of Commerce
Washington 25, D. C.
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1.
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ini
The subject which I will discuss is the distorted-waves theory'
of direct reactions and its application to the study of nuclei. I shall
mainly be concerned with a study of the reaction theory, its range of
validity, and its ability to give quantitative information about nuclear
structure.
con
It is necessary to state what is meant by a direct reaction. The.
definition commonly in use, due to Austern', is that a direct reaction
involves only a few degrees of freedom of the system, as opposed say to
reactions which proceed via a compound nucleus. These few degrees
of freedom can be explicitly treated in a reaction calculation while the
remaining processes can be completely ignored, in which case we have
site video. *****
a plane wave theory, or lumped into a reaction channel which leads,
ultimately, to the distorted wave theory.
To make these ideas explicit, it is useful to formally derive the
distorted wave matrix element and to show where the various approximations
enter in order to do a practical calculation.
The theory will be formulated in a general way, so that
rearrangement collisions such as stripping and inelastic scattering
appear on the same footing.
Consider the reaction
a + A ab + B
where, say for inelastic scattering,
b
= a
B = A* an excited state of the aggregate A,
*
.-
E-

1
while for stripping
a :d
b=p
B = A + n.
The Hamiltonian for this reaction can be written,
H = H + HA+ V&A = H, + V; ,
= H. + H3 + VbB = H + Ve ,
where H, and H, are the Hamiltonians for the internal motion of aggregates
a and A, and V. is the interaction between them. Similar definitions
hold for Hp, Hg, and VbB
The solution to the non-relativistic Schrodinger equation
(H - E) y = 0
(2)
is formally straightforward and is easily found using the operator
techniques.
If the theory is formulated in the initial system (a, A), the
solution of interest has incoming and outgoing waves in the incident .
channel and only outgoing waves in all other channels. This solution is
usually denoted :', the subscript i referring to incoming and outgoing
waves in the incident channel, while the superscript + means only
outgoing waves in all other channels.
We proceed by adding and subtracting a potential U, to the
Hamiltaonian, that is rewrite H as
H = H. + (V; - U;) + Vi
(1')

.
and define the "distorted" wave x:') as the solution of
(H; + V; - E) X(+)=0.
(3)
+
If elastic scattering is important U, is taken as the potential which describes
the elastic scattering of a by A and, correspondingly, x') contains
the elastic scattered part of the total wave function. x. (+) is written as a
product solution
X" (FaA
a lea! A (EA) = x++)
(4;
where x\*) describes the relative motion of a and A, and g, 1)
& lea) are wave functions of internal motion of a and A. This product
will henceforth be contracted to $,$. = $ . Then with
.
?+) = x (+) + 48!
(5)
loc, contains all scattered waves other than the elastic wave), application
of the Schrodinger equation gives
(H - E) Og; = - (V; - ;} *;(+).
Defining the Green's function
(H - E) G = -1, = - 010 - ',
where G is the total Green's function, the solution of the total Hamiltonian
with an inhomogeneous point source at p = 0', with p representing all the
coordinates in the problem, Ő = (i, e ).
IR
11
1
.
-
.
-
-
celine
region wi
interna
BE
TAS
Then
p! = G(V; - 0°) xx(+)
and integration over the primed coordinates is implied..
Symbolically,
E-H + ie
1
and the positive continuation e insures outgoing scattered waves.
In the limit UA - 0, x:(+) - Y, wherein the relative motion is
described by a plane wave, and Og, -ºg which now contains the elastic
scattered parts.
We are interested in that part of y1) in the channel (b, B).
To find this note that G can be expanded in eigensolutions of the final
system f :(b, B), which we now show. Using the operator relation
1 2
3 4 5
110)
we find
G = GE+GE VG
(11)
where
(12)
LL
For our particular final system
Vt
Greeting car732! t'oblets (ºpplads ('58)
(13)
LU
Oy

1ST
je
.
...*.; ondorio now.
with G', the free Green's function defined by
(T-K,90F = -967B FB,
(14)
with outgoing solution
di Me
0's - Zona Foto
K; F6B - 7'6B
(15)
21h
:
Here Me is the reduced mass of the colliding fragments in the final channel.
Since we are interested in the wave function at infinity for a particular
final channel f, we project out the bound aggregate and take the
limit as IbB , that is
ike FbB
fbb
- iK, IBB
ik, ' "bB ro* (1 + V, G) (V; - vy) *}{*). (16)
- le
196B0%)=-
PBB
POB
But
• .K
bB
*
o bB (1 + VEG) =
ut (-)
(17)
by a similar analysis.
. Here y") is the solution to the total Hamiltonian with plane wave
parts in the final channel but with incoming scattered parts in all other channels.
This is necessary to satisfy time reversibility.
A counter at infinity which has been biased to count particles of
type b and the proper energy, sees only the spherical wave given by
*
*
.
i
-.
-
.
..
-
-
equation (16), with amplitude
.
:
(18)
2h
{{K, - K) = (-/V; - V: 1 x F#).
It is more convenient to work with the T matrix, the element of which
is of interest is defined by the relation,
Me
f(K; -K) = TIK: - Kę).
(19)
2ch
Obviously, the total wave function y's can be expressed as the sum
of a distorted wave and a scattering solution with incoming waves in all
channels except f; i.e.,
= [1 + 6* [V - Upt ]x!=)
(20)
so that
T(K: - K) = <!-!IV; - ;) x:{t}} + (84-')(V- Uç) G (V; - 0,7) x<. (21)
Again from time reversibility
T(K; - Ke) = T(-K --K;)
= (x"}|{ve - 03/x+),
(22)
which is apparent from equation (8) for yk"). Strictly speaking, this result
for the scattering amplitude holds only if the distorting potential is a
fur.ction only of the relevant relative coordinate."
Although these results are exact, and completely general, the
theory has been deliberately formulated to single out the elastic scattering
channel which is presumed to be strong. The remaining interaction
F
23

remo comecom esse............dont nombre
(V; - Ui), or (V. - U) is the residual interaction, that part of the
interaction over and above the interaction which gives elastic scattering,
and which is responsible for transitions between states. If our
assumptions are correct and the reaction amplitude is weak relative
to the elastic amplitude, then it is reasonable to treat the reaction in
perturbation theory. This amounts to taking
Texact TOWBA
and setting AT = (X - Up) G (V; - U///x)) equal to zero.
The remaining matrix element TowRA can be written in either
the prior form
.
.
.
.
.
.
foWBA = (!!/«v; - 0,1}x(+),
(23-1)
.
*DWBA
.
.
.
-.
or the post form
Ipwba = {x{"|ve - Up) /xt).
(23-2)
The two are formally equivalent, but computationally quite different. The
particular choice one uses depends on the physics of the problem.
So we can see that DWBA considers matrix elements of de residual
interaction between elastic scattering states. In practice, the relative
motion parts of X
are generated from optical potentials which
describe the elastic scattering in initial and final channels, (if such data
is available). As you know the optical potential is complex, the imaginary
part of the potential accounts for the flux taken out of the incident beam
and fed into other reaction channels. This flux goes mostly into the
compound nucleus channels, but some goes into the direct channels,
TS
.
Y
:
7
.
4
.
RE
including the reaction channel which is being considered explicitly.
12
Similarly, the real part of the optical potential has contributions from the
reaction channels which lead to the non-locality of the potential. Thus it
SP
is "slightly" inconsistent to use elastic scattering parameters to generate
distorted wave functions for reaction calculations. However, it is
hoped, and in some cases has been demonstrated, that this causes
negligible error.
Remembering that x:(+) and X-) are chosen as product solutions,
equation (23) is written
-
.
..
-
-
-
TowBA-
= s faran ldibb xl-s* , FbB) (b.B/Vred) A, a) x+) (Fan (24)
-
-
-
-
-
-
.
...
with J the Jacobian of the transformation to the relative variables ron, and
.
.
FbB
m
The matrix element written in this way displays the factorization
a
.
into a relative motion part consisting of the distorted waves, and a nuclear
ms
.
matrix element
,.
n
e
.
*
*
.
.
(b, B Vres, A, a
** $B Vresa A
(25)
where the € are the remaining variables independent of the relative
coordinates.
This factorization has been much discussed but is worthy of
emphasis. The nuclear matrix element contains all the detailed reaction
information. It includes the angular momentum selection rules, and its
.
24
information. It includes the angular momentum selection rules, and its

.

amplitude yields "directly" the nuclear iníormation sought. Its functional
aA S
dependence on the channel coordinates ra and he depends on the reaction
being studied, differing, for example, for stripping and inelastic scattering.
On the other hand, the treatment of the distorted waves, the dynamical
part, is much the same for all reactions and leads to computational
simplicity.
It is possible to make very general statements about the nuclear
matrix element (23). Following Satchler', we expand the matrix element
into terms where a definite angular momentum J is transferred to
nucleus A to form the residual nucleus B.
part s, where
and
1=5.5.
Then
J (b, BI VresA, a) = J (5 mp, JB MBI Vresh JA MA Samad
= { 1334,96 * * 0,5u, mbad: Fog bb, ah) x
ess
(JAJ MA: MB - MalJB MB) (Sa So ma-mols mamy)
-
(esm, mi - my {J Mg - MA
(26)
(26)
.
-
.
: ;
-
*
-
.
-
1
.
.
10
.
where the Clebsch-Gordan coefficients take care of the angular momentum
coupling, and m = Mg + mp + Mama
By inversion
22+1
Gess,m
H
) (S. m. JB MB IV res! JA MA Sa May x
2J3 + 1
Sb
mo Saº
(JAJ MA' MB - Mal JB MB) x
(Sa So mamos mamy) (eSm, mm J MAMB. (27)
The possible dependence of G,SL m on the nuclear and projectile quanrum
numbers is denoted by the appearance of B, b, and A, a as arguments
of this function. From the form of G it is seen to have angular momentum
properties like Y, m. It also carries the parity of the transition,
(a)(A) (b)(B) through the matrix element (23).
In general the function G can be factored into an amplitude and a
form factor,
Gess, m = Arst, m fisi, mlad' *bb?.
(28)
*
**a... ..-
where A is often chosen to include spectroscopic information (e. g. fractional
-
.-
-..
parentage coefficients) and the interaction strengths.
This factorization, while not necessary, is convenient, so that one
can study standard form factors and incorporate the normalization at the
end. For example, in the study of neutron hole states, by say the (p, d)
reaction or by the (He, a) reaction, the form factor will be the same but the
amplitude, A,51. mi different.
Going back to equation (24), the total amplitude may be rewritten
.
in"
.
:
:.:.
--
-
*
..
.
TowBA = {(25 + 2;1/2 (SAJ MA, Mg-MA / JB Mpy Begin my ma (K. Ky 1291
with
emmma
Bej
lije
(i)
(23 +11°C m' m'm
(eSm', m'a-m'b/Jm-m
+ m)
(5.8m -m's /m. -m'y) 1995 m. sparen Sarios
*l*)(Fosfasu, m ( a +bB] xlt! (K;, Fan!.
(30)
In terms of which the differential cross section for unpolarized
projectiles and targets becomes
do . M.M. Ky _2J5+1__ [Asr bem m mb12
dn. (27 h
(2JA + 1) (25a * J m mma
13
viesu
In (31) the sum over l, the orbital angular momentum transfer, and S, the
spin transfer is coherent, but the sum over J is incoherent. That is
different values of l and S leading to the same J can interfere, but
different J's cannot. If the amplitude depends only on m and my through
the angular momentum coupling coefficient, that is if the overlap integral
involving the distorted waves is not dependent on these quantum numbers,
these sums can be performed and the cross section becomes incoherent in
ISJ and m.
To complete our treatment of the general theory, we must discuss
the properties of the distorted waves x(+) and x{").

12
.
29)
As we have stated, the distorted wave theory generally assumes
the elastic scattering to be strong and attempts to take this into account
l'exactly".
5
In practice this means generating these functions from the Schrödinger
equation with the optical potential which explains the elastic scattering.
The local optical potentials currently in vogue have forms for the
real potential which are constant for small separations of the colliding
fragments and fall smoothly to zero for large separations. The most
popular shape is the Woods-Saxon form
V(r) = - V./11+ety
(31)
x = (1-P. 4!/3)/
(32)
with r A
s the "nuclear" radius and 4. 4a the skin thickness.
The absorbtive potential can have this same form, or be surface
peaked or a mixture of the two, e.g.
(33)
W(r) = - W.111+ e*, - 4w, et/11+ e*;?
wich
x' = (r - IwA")/aw.
If the particles are charged, the potential includes a Coulomb
interaction, which, for a point charge incident on a uniform charge
distribution with radius R., is
U-221 22
TOR
#
!
F
-
Sin
13
22'e? (3 -?/R.3) ESR
(34)
2Rc
For scattering in which projectile and target have finite charge distributions,
the Coulomb interaction can be simulated with this form with a judicious
choice of the charge radius parameter R.
form
Voor het te jetes trata tot 4 (O. L)
(35)
is added.
Neglecting, for the time being spin orbit coupling in the optical
potentials, the distorted waves can be expanded as
***) (X, For-AX2, (K, Fan! Y CRI YA EPA) (36)
with the X,
the solutions of the Schrodinger equation
The on **,? - LAYLA + 18-3A CV (a) ti w(7) + 0.67]}^2 = 0. (37)
aA
Asymptotically, where the nuclear potentials have vanished, the
partial waves solutions have the form
YLA
(HL, (K;, tad) - MLA #L, (ky, tal]o"LA
(38)
where H, = (G+ i Fl), ihe Coulomb functions irregular and regular at
the origin and o, is the Coulomb phase shift, and nis the reflection
coefficient for the L'th partial wave.
14
The time reversed solution xl-'(Xq. PbB) catiofies ingoing boundary
conditions and is the solution of the Schrodinger equation with the complex
conjugate of the potential. However, X
lowever. x l) is the solution of interest and
is related to the outgoing solution by
xls) (K, FOB) = x64) (-KE, POR
(39)
and is expanded accordingly.
Except for the trivial case of zero nuclear potentials, the
Schrodinger equation must be solved numerically for each of the partial
wave solutions
• The distorted wave matrix element is then seen
A
to be a six-dimensional integral, which is in general a rather complicated
and difficult calculation. To bypass this difficulty, the zero-range
approximation is often introduced, and the six dimensional integral is
reduced to a much more manageable three-dimensional integral. In this
approximation particle b is assumed to be produced at the point that
particle a is absorbed, that is fest, man fbb? - 8/7%
B A an!
Mg
1,5 m aal. The merits of this approximation depend on t?:e reaction.
For inelastic scattering without exchange, and if the interaction is local,
the zero range approximation is exact. For other reactions Austern et al.
have suggested the validity of the zero-range approximation can be
MA
assessed by transforming the matrix element (22) into momentum space,
separately Fourier analyzing the product of the distorted waves and the
form factor fist m. If only thr small momentum components of the
distorted waves are important, such as found in surface reactions, then
only the small momentum components of the form factor will contribute
and zero-range approximation will be quite good.
-
--
-
**
*
*.*
Taking this approximation, we are able to handle the angular
momentuin variables of the matrix elements exactly. With the expansion
(24) for the relative motion parts (x*17, xit), and remembering istim
behaves as Y,-", the reduced amplitude Bs,
case of no spin orbit),
*m
im mamb becomes (in the
" bie..
m
inutit
emm mo
,21 + !
BSJ
Beste 2 mb =(-1** **B 22 , (lom, m.-my/Jm+ m -my) *
2J 1
.
(s., mam sma
sem
LBLA
m loj
LB 'LBLA
(40)
LLB
“А
в
with
er en ska 2194 je np + 1* [cz - m:/462 + my}
(1 + 0,01, 0) (1 + m-m 1999
(41)
and
MA al fesj
(20)1/2
M.
LBA K; KE MA
5) fesj(r) XL. (K;. r) dr
(42)
MB
are the radial integrals which must, in most cases, be evaluated numerically.
An important property carried by the lº coefficient is that the parity
change in the reaction (-) A B is equal to (-) { through the Clebsch-Gordan
coefficient (Lot 0,0 L 0) which vanishes unless the sum of the angular
momenta La + LB + is even.
Let us pass on to the study of single nucleon stripping or pickup
which has been the subject of intensive investigation. In particular we shall
specialize to deuteron stripping, which, if we ignore the D-state of the
in? -
**
16
deuteron and tensor forces, is the simplest and best understood
7
.
example of a nuclear re-arrangement collision.
The coordinate system for the reaction is shown in Fig. 1. In
terms of our previous notation
#*
-
Faard
P
.
*bB **PB
LA-LB
LE -L-p .
int
netts
The proton is considered elementary and for these purposes has
-
no internal structure, therefore
Oples) = Y1/2, my fool
(43)
the normalized spin function. The deuteron is an s state of relative
motion and may be described say, by the product of the Hulthen wave
function
polimp) = nie
np - Binp
(44)
Inp
.
.
.
..
-
-
and the spin function
(44')
Here a
with
the binding energy of the deuteron, and M is the
4.722 -
nucleon mass. The current value of B is 7a.
12
The nuclear wave function g may be expanded into a fractional
parentage expansion, over the eigenstates of the target nucleus A, as
discussed by McFarlane and French*, i.e.,
*
**
JBMB ir na
J'AM'Aie
m
*m
,
v
YB
* TE
2
Jom
Ys, m. (n) {J'A J M'A, Mg-MA lug Mg) (18 m, m, \ J M'A-MB). (45)
.**
.***
in
-
Where the Clebsch-Gordan expresses the coupling of J to J', to form
the nucleus B with angular momentum Jg. Usi(rn) is the radial wave
function of the captured neutron and ya mlon) is the spin wave function of
the neutron.
Ies; is the coefficient of fractional parentage, and n is the
number of equivalent nucleons in the orbit into which the nucleon is captured.
The spectroscopic amplitude S, a; defined by French' and McFarlane
and French" is related to these numbers by
-
-
-
n
(
Is= زیر
از
(46)
and is a measure of the probability that the final nucleus is made up of
the configuration of the initial nucleus and the captured nucleon, and is
normalized to n if there are n particles outside an inert core.
The precise determination of S is the object of stripping
experiments and theories, for it is this number which yields information
about the structure of the nucleus. Once extracted from the data it can
be compared with the predictions of a nuclear model.
ts

.
.
.
.
For the (d, p) problem, the initial interaction is
H
*
*
-.-
Vi = VA = VnA+VPA
(47)
and the final interaction is ·
(48)
Ve = VpB = Vnp + VPA
of the matrix elements (21) the post form is chosen, and the residual
interaction
Vres =Ve-V = Vmp + V
(49)
'np' pa · URB
is approximated as
-
-
-
-
Vres = Vnp
(50)
on the basis that there is cancellation between the proton-target interaction
and the optical potential in the final channel. This cancellation is certainly
not complete, in particular the non-diagonal terms of V.A correspond to
target excitation. Additionally, VA is considered to be the sum of two
body interactions; it is real while VoA is complex. Thus our choice of
the neutron-proton interaction is based on the idea that it is the dominant
part of the interaction for stripping, and can be viewed as a model to be
affirmed or refuted by experiment. Lack of success may then be attributed
PA
to our neglect of the other terms, as well as the higher order processes
neglected.
This choice of Von as the interaction and the method of expanding
the nuclear wave functions to separate off one particle indicate that the
stripping reaction only gives information about single particle states of
-
t - -
:
*
'
+
2
•
-
+
.
r
.
-
-
7
2
-
•
nuclei. The role of the other nucleons is contained in the neglected parts
of the amplitude and in the treatment of the distorted waves.
Thus the nuclear matrix element after the required integrations
is
(b, BIV[eo! A, a) = { 1-3 *5*15 (SĄJ MA, Mg-MA 13-Mg)
18J
(lsm ma-mol J MB-MA) (Sa So ma-mols ma-my)
25 + 1
*m
125.
In kerja Uenyeind , tm vnp @glampe
(51)
2S + 1
which by comparison with equation (26) yields
4:56. m
'van yet Vinpealeine
מת
2S + 1
We take the zero range approximation by writing
Vnpoglimp) = D. Oltp
where
D. - f np @da (t) di
- Svoza300 ()
Wtih the Hulthen wave function this integral is evaluated to be
3/2
Dr. Voyal conte 12
- 17.5 x 102 Mev 23/2
(54)
From equation (28) we choose
1 25 + 1
112 Do
A SJ, m
(n
T 25 + 1
-
DF
.
20
-
PE
and f, s; (r) =U, (r), the neutron bound state radial wave function.
In the calculations on which I shall report the single particle wave
-SE
functions U, 4F) are normalized eigensolutions of the Schrodinger equarior.
E
with a well of the Woods -Saxon shape. The well used has two parameters,
the radius r, and diffusivity a. These parameters are not well known and
in practice are usually taken the same as the optical parameters for the
proton scattering. The depth is adjusted until an eigenstate of the proper
energy is found. The functions so generated are similar to oscillator
functions in the interior and asymptotically are Hankel functions.
Armed with this apparatus, how good is the theory? A good theory
is required to reproduce at least two things, the shape of the angular
distribution and the absolute magnitude. For surface reactions the shape
of the differential cross section is not too dependent on the shape of the
form factor f (equivalently, the nuclear matrix element), but the absolute
magnitude is. The shape of the angular distribution is dependent on the
angular momentum transfer, and the distorted waves. Often it is possible
to find good agreement with the differential cross-section shape, but poor
agreement in magnitude. This is taken as a failure of the theory.
There have been many distorted wave analyses of stripping
reactions, mostly for nuclei for which the spectroscopic amplitude is
not well known, and for which elastic scattering data in either entrance
or exit channels were not available. Most of these analyses give quite
reasonable fits, but at the expense of arbitrarily varying the parameters
of the optical model. It is important to gauge the theory by studying a
case in which the elastic scattering data is available, as well as the
reaction data and for which a reasonable estimate of the spectroscopic
-
mm
-
=
-2°C
21
-
-
amplitude is available a priori. Recent measurements by Lee, Schiffer,
and Zeidmann at Argonne provide such data'. In their experiments they
have measured the elastic scattering of deuterons from 40ca, and the
stripping of neutrons to form states in 4. Ca. Since 40 Ca is doubly magic,
these states in Ca are assumed to be states of a single neutron coupled
to an inert core. Thus the spectroscopic amplitude should be unity. There
is some evidence in this data that this is not completely correct. There are
two Ps/2 states at 2 MeV and 2.5 MeV excitation. Also the MIT group
has found several ? = 1 groups at higher excitations.
Other appealing reasons for the study of these reactions are:
(1) calcium is sufficiently heavy so that the difficulties associated with
optical model analyses of scattering from light nuclei might not appear,
aná (2) these experiments have been done at deuteron bombarding energies
of 7, 8, 9, 10, 11, and 12 MeV, allowing a study of the energy dependence.
There is some irformation lacking, namely the elastic data of
proton scattering from Ca, which cannot be measured. To bypass this
difficulty, use can be made of the systematics of the proton parameters
found by Perey. Alternatively, there is data available for the scattering
of protons from argon-40 from 8 to 14 MeV which has been fitted' and
later applied to the scattering of protons from Ca with equally good
results. Samples of these fits are shown in Fig. 2. The parameters are
similar to the Perey parameters except for a smaller real radius
parameter. The analysis I shall discuss makes use of the fitted parameters.
..
.
The deuteron elastic scattering data was analyzed at the
appropriate energies'. The optical potentials are a function of the
relative coordinate measured from the center-of-mass of the deuteron.
-
V
-
C.
-
.-
.
2 -
INN
W11
TI
mm.com intern de calidad located
i
n
descono
22
Asymptotically this is correct, aside from small Coulomb effects, but close
to the nucleus it is probably in serious error. Here the deuteron can break
up, which for scattering is compensated for by the imaginary potential.
Thus the wave functions in the interior are in serious doubt, but the error
caused in distorted wave calculations must be assessed by comparison
with experiment.
As is known from the work of Halbert and Perey and Perey", the
deuteron optical potentials are characterized by a small real parameter
and a surface imaginary potertial peaked at a radius somewhat larger than
the real radius. The potentials found in this work are of the same form,
differing however in having a smaller strength and a larger radius for
the imaginary well.
Unfortunately, the elastic scattering can be fitted with a multitude
of potentials, all 0.1 which give essentially the same phase shifts, and
therefore the same scattering. The reasons for this are understood
Briefly, it follows from the fact, shown by Austern", that for strongly
absorbed particles, the low partial waves can be treated in WKB
approximation. Austern demonstri üs that the reflection coefficient
can be written as the sum of two parts. One arising from the reflection
from the nuclear surface, affecting the higher partial waves which are
weakly attenuated, the other arising from reflection from the angular
momentum barrier in the interior of the potential. Thus, the nuclear
volume influences the angular distribution through the low partial waves
that are not too attenuated. Neglecting reflection from the surface the
23
me = exp (21 Se (72)]
(55)
(55)
with
5,478) = 0518,69 413
and r, the classical turning point. Here K, (r) is the local momentum
in the well and is given by
KG ? (3) -
Fem • V6+) - V_(*) - i W() hele
"] (56)
2 Mr
and C, is a constant independent of the potentials.
If now V and W are replaced by V' and W' so that
Se 18') = 5,(r) + nt, n = 1,2,3.....
it follows that this contribution to n, is unchanged. Physically this means
we can add an integer number of half wave lengths on the interior without
changing the asymptotic behavior of the wave function, for those partial
waves for which reflection from the surface is negligible.
(Optical potentials for protons and neutrons do not show these
ambiguities, except for isolateu cases, because nucleons are not strongly
absorbed. In any case, since the optical potential is the positive energy
continuation of the shell model potential, a choice can be made. )
It turns out that there are potentials with real well depths from
30 to 450 MeV, see Table I, and deeper ones could be found. The scattering
given by several of these wells for 11 MeV deuterons is shown in Fig. 3.
? -
V
24
Except for X the scattering is the same for all potentials and comparison
of S, (r) for low partial waves given by these potentials show that they
indeed differ by almost a.
It is interesting to look at the wave functions generated by
potentials as they enter diroctly into the stripping problem. Figure 4
shows the total wave functions plotted along the beam axis for potentials
X, Y, and 2. The additional multiples of a wavelength added in the
interior change the wave function inside while asymptotically they are
much the same. Clearly they will give much different results for the
stripping unless the interior contribution is eliminated. This is
illustrated in Fig. 5, which shows the distorted wave cross sections for
the ground state transfer, with the various deutercn wave functions. The
radial integrations are done from the origin to infinity. The data of
Lee et al. is also shown. Potential X is clearly unsatisfactory, and
Y is little better. The remaining potentials adequately reproduce the
peak pusition but require different spectroscopic factors. G and J
especially underestimate the cross section and require a spectroscopic
amplitude greater than 1. The same kind of calculations were carried
out for the first excited state which is P, and the results are shown in
Fig. 6. Here the situation is much better, all potentials give the correct
peak position and with strengths which do not exceed unity. The deeper
potentials, especially 2, are to be preferred because they give beiter
agreement with the large angle data.
These curves exhibit the dependence of the calculations on the
interior contributions. For potential X, the wave function is fairly broad
-
-
- *
-
reise
sv.-
-
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-
....
ereem isus
.
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t-
.
.
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-
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inside, while for G and J the wave functions are rapidly oscillating and
have small contributions from the interior. For the capture of the
neutron into the 18 or bit, these contributions are fairly important. For
capture into the 2p orbit, they are less important because the bound state
function has a node on the interior. Since the wave function changes sign
-
.--*
1
2
:
.
T
.
2.
there is cancellation in the interior and consequently the surface dominates,
-
-
where the deuteron wave functions are similar. Of course, if the radial
integrals are carried out from some lower cutoff radius where the
relative wave functions are the same, the results for all potentials are
identical, as shown in Fig. 7,
This multiplicity of potentials forces a choice as to which potential
is most physically me an ingful, if any. The arguments against the validity
of the deuteron wave functions might imply that any of these with a suitable
cutoff radius on the integrals might suffice. However, this gives results
for the ground state for which the magnitude is too small, comparable to
the predictions of G and J in Fig. 5. On the basis of Figs. 5 and 6, the
shallow potentials are di scarded be cause of poor fits, the deeper potentials
because of the large magnitude necessary for agreement with experiment.
This leaves potential Z which has a depth of about 100 MeV, and which
incidentally gives the best agreement in share. We might have argued that
the deuteron optical well is in some sense the superposition of the neutron
and proton optical wells averaged over the internal motion of the deuteron.
This then implies that the very de ep wells are unphysical and suggests
a well depth of about 10 0 MeV (or shallower!).
---

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26
Figures 8 through 11 show the results of the distorted wave
calculations for stripping to the ground state, and the first three excited
states. The solid curves are calculated without a cutoff and the dashed
curves for a radial cutoff of 4 fermis. These calculations were made with
deuteron potentials which give the best fit to elastic scattering at the
energies measured. The linear plots serve to emphasize the small angle
data, although it is apparent that the detailed fits to large angles is not
very good. Some of this large angle discrepancy may be attributed to
the theory, but the data show marked changes as a function of energy,
especially the second peak in the ground state angular distributions. For
.
the most part there is little necessity for a radial cutoff. The spectroscopic
amplitudes found as a function of energy from normalizing the theory to
the main peak are listed in Table II. With the best fit potentials they
show marked variations due to fluctuations in the elastic data. By using
an average deuteron potential which gives reasonable fits to the elastic
data, the energy variation of the spectroscopic factors is smoother. From
this table note that the f, /2 and P, 2 strengths are somewhat less than
one, while the full P3/2 strength is contained in the two levels measured.
As was mentioned there are p states at higher excitation whose J is
unknown, tentatively they may be assigned as Pv2 and could supply some
of the remaining P, 12 strength. Note that the radial cutoff makes the
spectroscopic factor greater than one. The errors associated with these
numbers arise from uncertainties and fluctuations in the stripping data,
as well as the analyses. The fact that the f2/2 strength is less than unity
is probably due to uncertainties in these analyses and in the data, although
it may reflect a property of Ca.

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27
The energy variation is shown on the next figure (12), where the
peak cross section is plotted as a function of energy. The black dots
are experiment, the open triangles are the calculations using the best
fits to the elastic data, while the line represents the average deuteron
potential. Except for the 11 MeV points where the measured cross
section for the ground state suddenly rises, the average potential gives
quite good agreement with the data.
To point out the necessity of having some idea of the elastic
scattering, calculations were performed using deuteron potentials
recommended by Perey and Perey' and by Hodsons. The predictions
these potentials give for the elastic scattering are shown in Fig. 13,
labelled PB and H, and are poor compared to the fit found with the
2-potential. On the other hand, these potentials are quite adequate in
predicting the stripping angular distribution shape, Fig. 14, but again
poor in absolute magnitude. From this study we conclude that a
knowledge of the elastic data is essential if quantitative spectroscopic
10
13
information is the goal of the investigation. On the other hand our
previous remarks suggest that too close attention to the data can also
be misleading. Fluctuations in the elastic data gives parameter fluctuations
which are reflected in the spectroscopic factors.
It would seem that ideally measurements should be made in a
smooth portion of the excitation curve, or experiments performed over
a small energy range so that an average can be obtained. Less
pessimistically, if there is scattering data available from neighboring
nuclei taken at energies close to that of the stripping experiment, this
information will be sufficient to extract meaningful spectroscopic
information.
28
This then is an exhaustive study of the zero-range approximation
for stripping. The analyses give fair agreement with the data, yielding
spectroscopic factors within 80% of what would be expected on a simple
model of the Ca nucleus. The situation is not wholly satisfactory;
there are still discrepancies in fit at large angles which are more clearly
seen in a logarithmic plot such as Fig. 14, which shows the 12 MeV
data with the calculations using the 2-potential. .
These discrepancies might arise from a rumber of sources. For
example, the neglected interactions, higher order terms, spin effects,
and spurious contributions from the interior.
Let us first consider the interior contributions and how to
modify the theory to take better account of them. The simplest
procedure is to introduce a radial cutoff on the radial integrals.
Examination of Figs. 8-11 and Fig. 15 shows that this sometimes
qualitatively improves the fit, but at the same time overestimates the
spectroscopic factors.
Nevertheless, we do not believe in the contributions from the
interior and seek better methods than a cutoff to eliminate them.
There are several corrections which do this in a natural way.
The iirst of these is to relax the zero-range approximation and allow
the neutron-proton force to have a finite range. This means that we
consider both variables r, and r. To accomplish this the form
factor f,simligi ro) can be expanded in multipoles of both arguments
(Satchler)
T
Trapillio Ximet MS
fisu, mlrd. 5)
, 21, L2 (Pas pod
ܕܢܐܙܝܐ
Eto
29
M
-m
tu
. (@
P;) (L, L, M, m-mlim)
(57)
where the Clebsch-Gordan coefficient ensures that the original f behaves
like the spherical harmonic Y,m.
The angular integrations can then be done, yielding Ly - Lg, and
L2 - Lp The remaining problem is then to compute the Kernal functions
F . (ry, which is by no means a trivial task. The derivation is
Fej
quite lengthy and full details are given in the paper of Austern et al.
For our purposes it is sufficient to say that it is do-able. It turns out
that for stripping, since the deuteron is an internal S-state of motion, that
the angular momentum algebra described in zero-range can be carried
over to the finite range case. A flexible code has been constructed by
Drisko and Satchler and the effects of finite range can be calculated
Some qualitative features are easily discussed. For stripping
consider the product: V p go which previously was set equal to a delta
function to go to the zero-range approximation. Now we take
V moplamp og omot - Hip? aj od
.
(58)
and for the Hulthen function takes the Yukawa form
Vnplanp) og (mp) = N(a? - ps o PP
The Fourier transform of this function is
with D as before. The zero-range approximation is taken as K = 0.
30
This function is peaked at K = 0 and falls off with increasing K.
On the surface where the potentials are weak
Ka = E
VA - ,31/2
and
:
KP = [ 2MP (V, - E);}/2
approach their asymptotic values. There the momentum transfer
(1/2 Kd - Kp) is small for medium energy projectiles, and therefore
only the small momentum components of G(K) are sampled. This indicates
that for reactions confined to the surface the zero-range approximation
will be good. In the interior where the local momenta are large, there
will be a net suppression over the zero-range approximation. However,
small momentum transfer components are still present in the interior
which allows small momentum transfers to be important even at large
scattering angles.
(As an aside, if only low momentum components turn out to be
important, the range function of the Yukawa form can be replaced by a
Gaussian with a range and normalization chosen to match the Yukawa
form for small values of the momentum variable. This is attractive for
computational purposes, because of the simple expansion properties of
the Gaussian.)
The effects of the finite range correction are illustrated in
Fig. 16, which compares the zero-range approximation, with and
without a cutoff, to the finite range calculations without a cutoff, for
the ground state transition at a deuteron energy of 11 MeV. Two sets of
curves are shown, computed with the 2 potential and the Y potential.
To be noted is that the finite range effecis are bigger for the shallower
potential because the wave function for the shallower potential is less
oscillatory in the interior than for the deep potential. The overall effect
is to uniformly decrease the overall magnitude of the curves without
appreciably changing the shape, arising from the dominance of small
momentum transfers.
On the plane wave Born approximation, the finite range effects
are more drastic. There the zero-range angular distribution is
modified by multiplying the usual Blitler expression by
Gekº put the
(63)
with Kd and Kp taking on their asymptotic values. For the present case
this factor varies from 94% at 0° to 56% at 180°.
Finite range effects on the p-states, illustrated in Fig. 17, are
not as large, in fact they lead to a cross section slightly larger than the
zero-range cross section. · This result again follows from the change in
sign of the 2p functions in the interior -- the negative contribution to the
amplitude is diminished by the finite-range averaging.
Non-local Effects
Another effect which dampens the contribution from the interior
is due to the non-locality of the potential. As you know from the work of
Brueckner and co-workers , the self-consistent Hartree-Fock potential
is momentum. dependent. This effect manifests itself in the local optical
model treatment of scattering by making the potential energy dependent.
32
However, there is an additional effect, which is unimportant for scattering,
which causes the relative wave functions to be reduced in the interior.
Loosely speaking, this comes about because the mor.entum dependence
manifests itself in coordinate space by introducing surface corrections
to the potential. These give rise to increased reflections and diminish
the wave function in the interior. The asymptotic behavior is unaffected.
14
Perey, by comparing the total wave functions from the local and non-local
potentials for nucleons, has suggested that the relationship between them
s
YNL(r) = -
[1 +iB-14 U, (r)]''C
(64)
where B 18.85 fm is the range of the non-locality and
Uç (t) = (VIP) + i W(r)] .
2M
(More recently, Perey and Saxon have derived this result.) Since for
the scattering problem Yı, and you are equal asymptotically, C = 1. For
the bound state wave function we must use the normalization condition :
sluefp)/dr = 1
(65)
which gives
C>1.
Thus all wave functions are smaller on the interior than the local
corresponding wave functions.

33
This form of correction can be trivially applied to the zero-range
approximation by defining a new form factor as a product of the old form
factor times the three non-local corrections.
Perey has applied these ideas to the 40cald, p) and 9ºZr(d,p)
stripping reactions. He finds that the contributions from the interior
are indeed diminished but the magnitude is relatively unchanged from
the calculations without a cutoff, due to the non-locai enhancement of the
bound state tail. These calculations will be published shortly. Here
we demonstrate the effect by dampening the interior by 50% with a
smooth transition to no dampening with a function of the Saxon shape,
as illustrated in Fig. 18. The cross section is indeed diminished but
the shape is unchanged relative to the no cutoff calculations. The
calculation with a cutoff of 4 fms is shou'n for comparison.
Spin Effects
Thus far we have neglected the spin effects. These can be
included by the addition of spin-dependent terms in the optical model
wave functions and the bound state wave function. For the neutron and
proton the interaction is of the vector spin-orbit type
Vsulr) ~(s. L) OV(r)
(66)
For spin-one particles Satchler has suggested the vector spin-orbit
force or tensor forces. Calculations to date have only made use of the
vector spin-orbit form.
34
The optical model wave functions are now matrices in spin-space,
e. g.,
x(+) -- xf+) s
167)
m', in
where the subscripts m' allow for a spin-flip by the spin dependent force.
.
The partial waves are now J dependent as well as L dependent
XI (7) - XJ, L
(68)
with
J = L + S.
The angular momentum albegra is somewhat more complicated and
will not be discussed here, a full discussion can be found in reference 3.
Spin effects can be divided into two types, effects on the magnitude,
and changes in the shape of the angular distributions. In this section we
consider the effect on magnitude. The change in magnitude arises mainly
from the change in shape of the bound state wave function. This occurs
because the Saxon well which binds the neutron is of the form
V(F) = V. (1) + a(L) 4
vo(r)
(69)
r dr
where
a(L) = -
L if J = L + 1/2
= (1 + 1), J = L - 1/2
.
Thus for neutrons captured into the J = L + 1/2 orbit, the spin-orbit
interaction is attractive and surface peaked. The bound state wave function
is expanded. For the opposite spin state, J = L - 1/2, the wave function
is contracted. If the spin-orbit well is 8 MeV (pion units) deep the effect
35
is to increase the calculated cross section by some 25% for the ground
state transition (J = 7/2, L = 3). If an 8 MeV spin orbit interaction is
included in the proton channel, the cross section is further increased -
to some 30% greater than the calculation without spin-orbit. Adding a
5 MeV spin orbit to the deuteron well compensates for the proton spin-
orbit interaction, bringing the cross section to only 21% greater than
the spinless case. These values of the spin-orbit strengths assigned
to the proton and neutron well are close to those found to fit experimental
data. The deuteron spin-orbit strength is not as well established. In
general it is not needed to fit deuteron elastic data. Thüs inclusion of
spin Orbit coupling changes our estimate of the strength S by some 20%
(smaller).
A similar study on the excited p states showed the magnitude
difference to be almost negligible, some 5% greater for the P312 state
and 5% smaller for the P12 state.
If now finite range and spin orbit coupling are included, they tend
to compensate, as can be seen from the table of spectroscopic amplitudes
in the rows labelled Zs.
To summarize the results that have been discussed thus far, we
cay say that the zero-range approximation without corrections has given
a fairly good description of the *° Cald, p)*'Ca stripping reaction. The
spectroscopic amplitudes found from this work are within 20% of what is
expected on the shell model. This agreement might be considered fortuitous
inasmuch as we have included contributions from the interior. For the
ground state transition we have seen that relaxing the zero-range
36
approximation by allowing the n-p force to have a finite range does not change
the shape of the theoretical cross section but does decrease its magnitude,
thereby leading to an increased spectroscopic amplitude S. Spin-orbit
coupling, especially in the bound state function, has the opposite effect
on the ground state transition, yielding a 20% increase in the absolute
magnitude. A combination of these two corrections gives magnitudes
comparable to those found in the zero-range approximation without a
.
cutoff. On the cther hand, both the finite-range correction and the neutron
spin-orbit correction are negligible for the excited states. A rough
summary of these effects is given in Table II.
Another source of uncertainty are the optical parameters. As
we have seen, equivalent potentials which give the same elastic
scattering give different results for stripping, especially in magnitude.
Additionally, potentials which do not fit the elastic data, correctly
predict the shape of the angular distribution around the main peak, but
again with incorrect magnitude. Further, the shape of the angular
distributions at large angles is only roughly reproduced and is critically
dependent on the optical parameters chosen. This point will be discussed
again.
For heavier nuclei the uncertainty in which set of optical
parameters to use has been somewhat alleviated by the systematic
parameter surveys of Pereyº for protons and by Perey and Perey" for
deuterons. A word f caution is in order here, these parameters should
not be extrapolated, either in energy or in A, beyond the bounds
considered by these papers. Interpolation, on the other hand, is reasonably
safe.
37
Finally it should be pointed out that relative magnitudes of peak
cross sections are less dependent on these uncertainties, except
perhaps for the spin-orbit dependence, and should be more reliable
than absolute values.
J Dependence of the Angular Distributions
These measurements show quite different angular distributions
at large angles for stripping with the same é transfer but with different
J-values. In Fig. 15 this is illustrated for the 12 MeV measurements.
The data (black circles) for the P angular distribution shows a deep
minimum at 100', while this minimum does not appear in the P212
differential cross sections. (The filling in the minimum after the main
stripping peak is a Q effect. ) Schiffer and Lee have found that this
systematic difference persists for other (d, p) reactions in this mass
region and have used it to identify the J-value of the neutron orbital,
Qualitatively, the origin of this difference can be explained, if
the spin-orbit distortions are treated as perturbations. Johnson's
has shown that then the cross section can be written as the sum of two
terms,
do = A(e) + (-.7-4-1/2 B(e)
(70)
d2
with Ald) arising from the central well distortions and B(o) proportional
to the spin-orbit strengths of the optical wells. Thus we may have
constructive or destructive interference at some large angle depending
on the J of the captured neutron.
. Distorted wave calculations have been made to see if the theory can
explaii. this effect. It was found that spin-orbit interaction in the bound
state well produced no shape changes, but that spin-orbit distortions in
either channel do indeed produce such changes (shown in Fig. 15). Here
the theory gives a shallow minimum for the Pvz case but also predicts
a minimum at large angles for the P3/2 groups which is not observed
experimentally. Additionally, examination of other data indicates that
this effect is not reliably reproduced, either as a function of energy or
target mass, if best fit elastic parameters are used. However, Rost
has been able to find this eftect in many cases by allowing only proton
spin-orbit distortion, but the other proton parameters are allowed to be
relaxed from their best fit value 8.
In short, the distorted wave approximation is capable of
explaining such phenomena, but is as yet incapable of giving a detailed
account. Indeed, as can be seen from Fig. 8, the theory does not
well reproduce the 2nd peak in the ground state angular distribution
which is much larger than the J -dependent effecis. Until this can be
explained, it is premature to ask more detailed questions about small
effects.
Effective Binding Energy
The calculations discussed thus far make use of a bound state
function, w. (r), defined by the overlap of the target and residual nuclei.
Here n is the principal quantum number and again l the orbital angular
momentum of the captured neutron. As has been mentioned, this function
is taken as an eigensolution of the Schrödinger equation with a well of
39
the Woods -Saxon shape. We have no a priori knowledge of the shape or
depth of this well. We can postulate that the wave function from the true
"single particle" well might have a different shape than we are using.
For the states of a single nucleon added to an inert core, our
simple prescription might be reasonable, this well could then be
interpreted as the shell model well. If, however, the core is polarized
by the additional nucleon, or if there are several particles outside
a closed shell, deviations from che shell inodel might be expected --
in the first case through the re-arrangement of the core nucleons, in
the second case through the residual interactions of the extra-core
nucleons. A direct manifestation of this effect is a shift in binding
energy away from that given by the shell-model for a particular orbital.
The question arises as to how to best represent this function.
We know that asymptotically the wave function behaves like
e-Kr
(71)
with the wave number K proportional to the square of the separation
energy.
@z = + Q.
(72)
One school of thought then argues that the wave function should be an
eigensolution with this eigenvalue.
Alternatively, the true wave function can be expanded in a complete
set of shell model wave functions, including the continuum of this set.
It is then argued that one should use the shell model orbital with the
.
same quantum numbers as the true single particle orbital, on grounds,
that it is the largest component. Thus we would solve for a function
with some effective energy, Eorf e. In the spirit of the shell model
this would imply describing all the single particle functions of a given
nucleus as eigensolutions of a well with constant depth.
The point is important because the magnitude of the wave
function in the region of strong overlap with the distorted waves directly
affects the measure of the spectroscopic amplitude S. For offer
the wave function expands. For E Eurf the wave function contracts.
Most distorted wave stripping calculations have used the
separation energy. The evidence for the other prescription is mixed.
The clearest evidence for this concept was found in the (p, d)
reactions on the isotopes of Fe.' This reaction excites the T 1/2
states of the same shell model configuration which differ by several
MeV. To obtain agreement with theoretical spectroscopic factors,
it was necessary to use the same binding energy for both transitions.
A shape difference was recently pointed out by Rost''. Two
states in "Fe are excited by the (p, d) reaction on "Fe. Both are
{ = 3, but with J = 712 and 5/2. They differ in excitation by only
.4 MeV. To obtain agreement with the shapes of the angular distributions,
Rost adjusted the depth of the central well binding the f5/2 particle until
it agreed in depth with the central well binding the fq/2 particle. The
agreement is quite good as is seen in Fig. 19.
.,..-
. .
.
.
.
On the other hand Satchler has found contrary evidence in an
analysis of 'Cald, t) reactions. He obtains good agreement with
theoretical predictions by using the separation energy. On the other
hand, if a constant well is used for all transitions, he finds the ground
state spectroscopic amplitude a factor of two too large.
For the present case *° Cald, p). Ca, the two Pq2 states are
candidates for such a study. Here for the state at 2.5 MeV, the separation
energy for the 2 MeV state was used, (6.42 MeV), resulting in a 15%
decrease in cross sections which, incidentally, makes the sum of the
Sz/2 strengths exceed unity.
This conflicting evidence indicates that no simple prescription
can be given, each nucleus must be examined separately. More work,
both theoretical and experimental, is needed.
Other Single Nucleon Transfer Reactions
We have discussed most of the current ideas on the theory of
stripping reactions and have seen that the theory has given reasonable
results, although some uncertainties and discrepancies in fit still remain.
The disagreement of the theory with the experimental second maxi mum of
the ground state group is especially significant and indicates that we are
neglecting an important part of the amplitude.
The situation for heavy nuclei is somewhat better as Fig. 20
shows, for the (d, p) reactions on cadmium-114. These data were
taken by Silva et al.'' at Oak Ridge and were analyzed using the
parameters of the Perey's" which give a good account of the deuteron
elastic scattering. No parameters were juggled and the curves
shown did not use any cutoffs of the radial integrals. The spectroscopic
amplitudes are in fair agreement with pairing theory calculations.
This is not always the case, for example Miller et al. were
unable to find a set of deuteron parameters which could simultaneously
explain the elastic scattering of deuterons and the stripping angular
distributions for 206Pb.
The situation for light nuclei is as yet unclear, no systematic
analyses of elastic scattering and stripping have been completed although
one is in progress at Oak Ridge. Some preliminary results for the
oln, d)''N ground state transition, le = 1), are available. The
measurements were taken at Zagreb with 14 MeV neutrons and the analyses
done at Oak Ridge. Again this is a favorable case for testing the ability
of the DWBA to give absolute spectroscopic factors. On the shell
N
model "ºo is doubly magic and the expected spectroscopic factor is two.
Data is available for 14 MeV neutron scattering from "ºo and
parameters similar to these found by Pereyº give a good account of the
angular distribution as seen in Fig. 21.
For the exit channel parameters which fit the elastic scattering
of 8 MeV deuterons from 150 were used, Fig. 22, since there isn't
any data for the scattering of 4 MeV deuterons from ''N. Several
variations of the deuteron optical parameters were also tried.
The agreement with the pickup data is very good provided one
allows spin-orbit coupling in the well which binds the proton and uses the
finite range form of the theory. On the other hand, to find agreement
with the data using the zero-range approximation we must invoke a radial
cutoff, even then Sis reduced to 1.63.
. . .
.
.
.
.
.
.
.
hehe
.
.
.
posted
in
43
Figures 23 and 24 show the results of these calculations with
the Zagreb data. The first of these uses a deuteron potential with the
imaginary potential surface peaked, the second uses an imaginary potential
of the Woods -Saxon shape. Although there are differences either one
gives a reasonable prediction to the shape and the correct magnitude
with S = 2. Still another calculation was tried varying the deuteron
parameters slightly from those shown in Fig. 22. In particular, a
was changed from 0.9 fms to 0.8 fms, r' from 1.85 fms to 1.8, and
V increased by 10 MeV. The predicted shape is quite good but the magnitude
Csanged by 50%, yielding S = 3, illustrating again that incorrect conclusions
can be drawn if the wrong optical parameters are used.
Single nucleon stripping and pickup reactions involving other
ions, e. 8., ('He, d) or (d, 'He), are treated in much the same way.
Aside from the uncertainties of treating the relative wave functions in
the optical model approximation and what to take as the interaction,
there is an additional complication, namely the unknown overlap of the
internal wave functions of the particles. To date, this knowledge is
unavailable and the distorted wave theory is left with an unknown
normalization. Until more details of the configurations of these ions
are known, this normalization must be extracted from experiment,
using target nuclei where the spectroscopic factors are reasonably certain.
Some work has been done on the 40 Cald, 'He) 39K and 40 Calºhe, d)
reactions, and the theory works quite well. The normalizations extracted
from these calculations agree to within 30% and it appears that we will
be able to use these reactions to extract absolute spectroscopic factors.
Much work remains to be done.
44
Inelastic Scattering
This subject has been covered in some detail ia the literature,21
and in the proceedings of various summer schools. In these notes only
the main ideas will be reviewed and some recent applications to light
nuclei discussed.
From the previous considerations on stripping we have seen
that stripping and pickup reactions give information about single particle
(hole) states of nuclei. The approximation that is taken and which
has proven fairly good is that only the interaction between the stripped
particle and the outgoing particle need be considered.
Inelastic scattering is somewhat different. In this case many
between target and projectile can be expanded in multipoles of ángular
momentum transfer L
vii. e) - Elity VZ (, e) yokim (7)
Vir,
(73)
L, M
and the nuclear matrix element becomes
10A I VL (r, €) 1 A} .
(74)
where target A has been excited to A*. From this point two approaches
have been considered. In the first it is assumed that there is detailed
knowledge of the nuclear wave functions a, and Axe, and the interaction
of the incoming projectile with the constituent nucleons of the target, i.e.;
VL (, 6) =2 V (- 1}}
(75)
where i labels the coordinate of the ith nucleon, and the sum extends over
all the active nucieons.
This method has been discussed in detail by Levinson and
Bannerjee for the "2cp, p'!2* C reaction, and more recently by
Funsten and Rost' for inelastic scattering of protons from nuclei in the
1912 shell. In the first case, only nucleons in the p shell were considered,
in the second, only f particles were included. Agreement with data
is found only if the interaction strength is taken some two to three
times that needed to explain nucleon-nucleon scattering at low energies.
Additionally, Levinson and Bannerjee were forced to depart from
parameters which fitted elastic proton scattering from carbon. On
the other hand, Fun tea and Rost used parameters consistent with
those found by Perey.
The discrepancies in strength are attributable to neglecting
the many other possible configurations which can participate in the
reaction, Calculations which use better nuclear wave functions are
underway at Oak Ridge.
Another approach which avoids the question of which nucleons
are participating is to be described by the collective modef,* wherein the
surface is deformed (or deformable) and is characterized by the
24.
radius parameter
R = R0 (1 + E aka Yu 9165.6")
= Ro + R :
(76)
Here the age are the collective coordinates and o' and ' are polar angles
relative to body fixed axes.
46
tai deres familie.
It is reasonable to assume that the interaction between the
nam..
projectile and the target is also non-spherical and depends on the distance
from the surface R as defined above.
tak z
We then take the potential Vir - R) and expand it in a Taylor's
series
V12 - R) - V - Ro) - Speed Wes - RJ|R=R * ar”)R=R#...
Vir - R) = Vlo - Ro) - sR
t
...
17
(77)
dr
0
2
-
.
In the perturbation limit the first term. gives elastic scattering,
-
the second term taken once leads to single excitation of the nucleus. The
second term taken twice gives double excitation but is not computable
in distorted wave approximation, since it refers to a two-step process.
The third term also yields double excitation, and is treatable in first
approximation, but it is inconsistent to treat the direct term without
taking into account two-step processes.
The reactions which can be treated in the usual DW8A manner
are, therefore, single excitations and the interaction is proportional to
(78)
For collective rotations, the age can be written as
azo = ß cos y
42+1 = 0
Q2+2 = ß sin y
with B the usual deformation parameter, and y the assymetry. For axially
47
symmetric nuclei y = 0. Then the interaction is,for a quadrupole deformation,
v11) = - ROB a
(r - Ro) X2
(79)
with respect to body fixed axes. Thus one number is extracted from
comparison of theory with experiment, the deformation parameter B.
For collective vibrations, the area are interpreted as a linear
superposition of boson destruction and creation operators
banco [mxq + (-190-a]
(80)
Here hW, is the phonon energy and Cy is the restoring parameter. The
interaction tren is
dVIr-Rol
v!!! = il Roa
*
em
(81)
and the nuclear matrix element yields
11W
{PA*lem) A
(82)
20 e
which can be interpreted as an effective (rms) deformation
ß = vzetih
202
(83)
It is impossible to distinguish between rotational and vibrational excitation
in first order.
A literal interpretation of these formulae suggests that the
parameters of the form factor are the same as the optical potential which
explains elastic scattering. All calculations to data, in which DWBA is
expected to apply, have agreed with this supposition. Thus once the
elastic scattering has been fitted, the inelastic scattering angular
48
distribution is predicted and agrees quite well with measurements. This
theory has been used with fair success to explain collective excitations
in medium light nuclei.
Recently, the theory has b-en applied to the excitation of "c by
126 MeV "Cions and by 168 MeV oxygen ions. Figures 25 and 26
show the optical model fits to the relevant elastic data, and Figs. 27
and 28 the DWBA predictions for the excitation of the 2+ (4.43 MeV) state
in C. The deformation B listed in Fig. 27 should be replaced by 0.8.
Analyses of the excitation of this state by a-particles yields B = 0.5.
So there is major disagreement between the deformations extracted from
different experiments.
Our previous discussion has shown that the nuclear matrix element
has amplitude
B R, V
Alss, m=
-
Blair has suggested that since the interaction radius Ro is not uniquely
determined, these measurements do not give B but the combination B Ro.
Applying this we find for the excitation of the 2+ state by a-particles
B RO = 2 fms, while the excitation by C and by oxygen give B Ro = 1.8 fms,
and 1. 73 fms, respectively.
In the oxygen + carbon experiment cited above, the 3- state in
oxygen is also excited. Again the DWBA theory gives a reasonable
prediction of the cross section, as shown in Fig. 29. It appears that
inelastic scattering from light nuclei at sufficiently high projectile energies
can be easily interpreted and gives consistent spectroscopic information.
Acknowledgments
The author gratefully acknowledges his colleagues, G. R. Satchler
and R. M. Drisko, for helpful conversations and their advice.
REFERENCES
1. N. Austern in Fast Neutron Physics, I, ed. by J. B. Marion and
J. L. Fowler (Interscience Publishers, Inc., New York, 1963);
selected topics in Nuclear Theory, ed. by F. Janouch (I. A. E. A.,
Vienna, 1963).
2. E. Gerjuoy, Annals of Physics 5, (1958).
3. G. R. Satchler, to be published in Nuclear Physics.
. R. H. Bassel, R. M. Drisko, and G. R. Satchler, "The Distorted
Wave Theory and the Code Sally," ORNL Report 3240.
4. N. Austern, R. M. Drisko, E. C. Halbert, and G. R. Satchler,
Phys. Rev. 133, B3 (1964).
M. H. McFarlane and J. B. French, Rev. Mod. Phys. 32, (1960).
J. B. French in Nuclear Spectroscopy, B, ed. by F. Ajzenberg-Selove
(Academic Press, New York, 1960).
7. L. L. Lee, J. P. Schiffer, B. Zeidman, and G. R. Satchler,
R. M. Drisko, and R. H. Bassel, to be published in Phys. Rev. .
8. F. G. Perey, Phys. Rev. 131, 745 (1963).
9. G. R. Satchler, to be published.
10. E. C. Halbert, Nucl. Phys. 50, 353 (1964); C. M. Perey and F. G. Perey,
Phys. Rev. 132, 755 (1963).
11. R. M. Drisko, G. R. Satchler, and R. H. Bassel, Physics Letters 5,
347 (1963).
12. N. Austern, Ann. Phys. 15, 299 (1961).
13. P. E. Hodson, Direct Interactions and Nuclear Reaction Mechanisms,
ed. by E. Clementel and C. Villi (Gordon and Breach, New York, 1963).
14.
F. G. Perey, Direct Interactions and Nuclear Reaction Mechanisms,
15. R. C. Johnson, private communication.
16. E. Rost, B. F. Bayman, and R. Sherr, Bull. Am. Phys. Soc. 9, No. 4,
1964.
17.
R. Sherr, E. Rost, and M. E. Rickey, to be published in Phys. Rev.
Letters.
18. G. R. Satchler, et al., to be published.
19. R. J. Silva and G. E. Gordon, to be published in Phys. Rev.
20. D. W. Miller, H. E. Wegner, and W. S. Hall, Phys. Rev. 125,
2054 (1962).
21. R. H. Bassel, G. R. Satchler, R. M. Drisko, and E. Rost,
Phys. Rev. 128, 2693 (1962), and references therein.
22. C. Levinson and M. K. Bannerjee, Ann. Phys. 2, 471 (1957);
and Ann. Phys. 3, 67 (1958).
23. H. O. Funsten, N. Roberson, and E. Rost, Phys. Rev. 134, 1B,
117 (1964).
24.
A. Bohr and B. Mottelson, Kgl. Danske. Videnskab, Selskab, Matt-
fys. Medd. 27, No. 16 (1953).
TABLE I
Optical Parameters for li MeV Deuterons
on Calcium-40
Potential
W(MeV)
w(F)
aw(F)
V(MeV)
32.5
rolf)
. 943
a(F)
.905
Wp(MeV) Or(mb)
5.6 1353
1.7
.691
72.4
.936
. 943
1.5
. 542
11.8
1163
120.7
1133
. 966
1.002
1.04
846
.769
1.48
1.47
.492
.453
.453
176.9
1.47
1117
240.0
. 707
. 415
1104
1.46
1.47
16.4
21.0
26.4
34.5
37.8
51.5
1.09
.651
1092
307.0
406,0
460.0
1.07
1.154
1095
.633
.573
.368
.359
.304
1.47
1.49
V
1085
TABLE IL
Spectroscopic Factors
? MeV
8 MeV
2 MeV
10 MeV
1.2 MeV
12 MeV
Average
(MeV)
.6.14
5.37
.894
4.2
.928
.742
.813
1.54
.876
•943
5.15
.925
.891 :
.901
1.70
31.5
.866
.934
.888
1.52
25.0
.676
..745
.732
.856
1.68
.91 .03
.87 .07
.86 3.07
1.67 1.26
1.99
4.19
22.5
.695
27.5
.654
34.8
• 788
.795
.843
.830
840
Peak (mb/st)
S (Av. z)
S (Best z)
S (Best zs)
S (Av. zase
Peak (mb/st)
S (Av. z) a
S (Best z)
S (Best zs)"
Peak (mb/st)
S (Av. 2)
S (Best z)
S (Best zş).
S (Av. z),
S (Av. z).
S (Best z)
S (Best zs)"
.676
.71 $.06
.73 $.09
2 I .
.532
H
3.67
.306
13.5
.299
.307
.289
.32 $.03
.276
.301
.299
.292
.952
1.046
1.031
5.65
.832
.756
1.63
30.0
.662
.566
.536
18.0
.369
.320
.299
.411
1.031
.886
.835
(24.0)
1.78)
1.7)
(.67)
12.7
.296
.316
.316
.315
1.084
1.146
1.156
16.5
.351
.319
.294
.381
1.146
1.032
.958
.324
1.001
1.218
1.285
3.67 + 14.19
953
.32 $ .02
.32 $.04
.34 1.04
1.03 1.07
1.05 1.10
1.03 + .15
921
2.19
Peak (mb/st) 12,5
S (Av. za .572
S (Best z)
.691
S (Best zs) .. .732
18.0
.721
20.5
760
820
.838
(21.9)
(.77)
1.8)
1.78).
.784
(23.5)
(.81)
1.7)
1.72)
.68 1.08
.77 5.05
.779
.78 1.04
a. Zero-range approximation without spin-orbit coupling
b. Finite-range, with spin-orbit coupling
c. Radial cut-off at 4.1F
TABLE III
Variations of Peak Cross Sections for a Deuteron Energy
of about U Mev
Elect
282/2
24.1/2
Q - 6.14 Mev
203/2
Q ~ 4 Mev
Q = 2.19 MeV
--
.
.
• 15%
+
- 45%
+ 3%
- 6%
+ 15%
H
..15%
Finite Range
Cut-off, 4F
Bert S 80.5 MeV
Neutron spin-orbit, 8 MeV
Proton spin-orbit, 8 Met
Deuteron spin-orbit, 5 MeV
Neutron radius 1.25
. + 25%
- 5%
+
+
-7%
3% ;
- 2% .
+ 20%.
...+ 15%

1
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Figure 2
UNCLASSIFIED
ORNL-OWG 64-3860

doldw (mb/steradian)
*..20
.
40
.:
60
:...
80 100
Oc.m. (deg)
120
.
140
160
Figure 3
UNCLASSIFIED
ORNL-DWG 63-530R
I ca 40 (0,0)
Eg = 11 Mev
• EXPERIMENTAL
-OPTICAL MODEL Z -
OPTICAL MODEL Y-
do/dw (mb/sizrodian)
an ido
no ā
no dono a ou no as a ..

30
60
120
150
180
90
Oc.m. (deg)
Figure 4
UNCLASSIFIED
ORNL-OWG 64-1452

Ca+d
11 Mev
8 = 0°
..... POT X
--POTY
POT Z
-10
-8
-6
-4
-2
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2
4
6
8
10
-
Figure 5
UNCLASSIFIED
ORNL-OWG 64-1460

Ca 40 (d,p)
E = 11 Mev
Q = 6.14 Mev
d=3
doldw (mb/steradion)
NO CUT-OFF
POT S
0.90
0.94
1.00
1.64
1.90
1
20
40
60
80 100
Oc.m. (deg)
120
140
160
180
.
..
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.
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C
.
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W
..
"
MW".
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UNCLASSIFIED
ORNL-DWG 64-1459

Ca 40 id,p)
Figure 6
E = 11 Mev
Q = 4.19 Mev.
b=1
shpes
doldw (inb/steradion)
NO CUT-OFF
POT
0.95
0.78
0.74
0.75
Y
-F---
04
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0
20
40
60
80
100
120
140
160
180
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Figure 7
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Ca(d,p) -
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20
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160
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121 122 12
V = 44.2 Mev –
W = 28.3 Mev
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Rc = 1.6 fermis
o= 0.683 fermi

• EXPERIMENT
- CALCULATION
5x10° C
·
10
15
20
35
40
45
25 30
Oc.m. (deg)
.
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UNCLASSIFIED
ORNL-DWG. 63-4206
500

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• EXPERIMENTAL
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(2012, c1212
EXPERIMENT
- COUPLED EQUATIONS
Ro = 2.03 , B = 0.64
---DWBA, B = 0.52
DWBA, WITH COULOMB
EXCITATION, B = 0.54 ----

Ozt (mb/sr)
NC
10
15
20
25 30
c.m.Ideg)
35
40
45
.
-
.
-
3
Figure 28
UNCLASSIFIED
ORNL-DWG 63-1208

ola) (mb/sr )
016 (C12, C12* ) 016
E oxy = 168 Mev .
b = 2
Q= -4.43 Mev
.
.
.
. .
.
0.1
B = 0.62
.
SALLY, NUCLEAR EXCITATION
BRO = 1.67F
- -
-
.
SALLY, NUCLEAR + COULOMB EXCITATION
B = 0.64 BRO = 1.73F
.
.
. .
• EXPERIMENTAL
0.014
10
20
30
Bc.m. (deg?
.
7
2:
1
2
UNCLASSIFIED
ORNL-DWG 63-1207
.

-
-
I
F0161¢'2, C12; 016*
Eoxy = 168 Mev
l= 3 .
Q = -6.14 Mev
-
--
old) (mb/sri
-
- SALLY; B = 0.46,
BR.= 1.35 F
• EXPERIMENTALS
Finn
S
. 1.-
.
é
Figure 29
o
10
15
20
35
40
45
25 30
Oc.m. (deg)

-
-
END
UN