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Antisymmetry in distorted-wave calculations of inelastic nucleon scattering
I
2.F
I !
Nuclear Physics A103 (1967) 657--676; (~) North-Holland Publishing Co., Amsterdam N o t to be reproduced b y p h o t o p r i n t or microfilm without written permission f r o m the publisher
ANTISYMMETRY IN DISTORTED-WAVE CALCULATIONS OF INELASTIC NUCLEON SCATTERING K. A. AMOS t Department of Physics, University of Pittsburgh, Pittsburgh, Pennsylvania *t Received 19 June 1967 Abstract: Distorted-wave calculations of the inelastic scattering of 18.9 MeV protons to the first
excited state of 8sy have been performed using bound-state wave functions consistent with spectroscopic information and using projectile wave functions generated from a seven-parameter optical-model potential that is determined by elastic scattering. Exchange terms appear in the calculations because the initial-state wave function is antisymmetrized and because the twonucleon effective interaction has an exchange character. The effective interaction is taken to have a central static form, that is a linear combination of three Yukawa functions. Two dissimilar parametrizations of this interaction are considered, each of which is determined from aspects of free two-nucleon scattering. Comparison with limited experimental data encourages belief that these "no-parameter" microscopic distorted wave calculations are credible and that this reaction should help to calibrate the matrix elements needed to analyse more complicated reactions. 1. Introduction Direct-reaction inelastic scattering that excites collective modes o f vibration or rotation in nuclei is often well described by a distorted-wave ( D W ) theory in which the optical-model potential that represents the projectile-nucleus interaction is assumed to reflect the mass deformation of the target 1,2). In particular, this theory has yielded g o o d agreement with the shapes and magnitudes of experimental cross sections for low-energy quadrupole and octupole excitations in m a n y nuclei 2). These analyses, however, are " m a c r o s c o p i c " in that all details of nuclear structure are hidden in form factors. In view o f the success o f these " m a c r o s c o p i c " D W calculations, attempts 1,3) have been made to unfold their f o r m factors in terms of individual nucleon motion. Such microscopic formulations generally assume that the independent-particle model adequately describes the m o t i o n of individual target nucleons, and that the interaction promoting an inelastic transition is a sum of individual interactions between the projectile and target nucleons. With these assumptions, the transition amplitude for inelastic scattering f r o m the target nucleus becomes a coherent sum of single-particle (SP) matrix elements 3). Each SP element describes the scattering of the projectile f r o m one b o u n d nucleon in the presence of the remaining target core. In the usual D W approach 3), this remaining target core is assumed to be solely the source of potentials that affect the m o t i o n of the two interacting particles. t Present Address: University of Georgia, Athens, Ga. tt Work supported by the National Science Foundation. 657
658
K . A . AMOS
However, a microscopic calculation of a reaction that has been successfully analysed by the "macroscopic" DW theory, generally requires the evaluation of a coherent sum of numerous SP matrix elements. Therefore, it is essential that these SP elements be carefully calibrated. Hence, attention first should be focussed upon reactions whose transition strengths are divided among as few configurations as possible. Ideally, we would like to treat reactions for which just one SP element is significant. Spectroscopic considerations 4, 5) indicate that inelastic scattering from 89y, especially to its first excited state, is adequately described this way. Further, this nucleus is particularly appealing in that its core 88Sr is sufficiently heavy to make credible the representation of the projectile with optical-model wave functions. Also, for energies of current interest ( ~ 20 MeV), we do not need an excessively large number of partial waves to generate these wave functions. However, an adequate DW calculation must not only use good representations of the projectile and bound nucleon and their effective interaction but also should be antisymmetrized. This then requires evaluation of exchange terms that arise from antisymmetrization of the wave functions and from an exchange character associated with the effective interaction. The difficulties involved, especially in handling exchange effects, are minimized if the projectiles are nucleons. In particular, their existence as such in the region of strong nucleon-nucleus interaction is credible, and the simplicity of the coordinate system facilitates the calculation and interpretation of the exchange terms. Further, we can assume that a two-nucleon interaction is the mechanism that promotes this reaction. But practical limitations of D W calculations require the use of oversimplified forms for this interaction. However, a recent survey of inelastic scattering a) indicates that such forms are sufficient to understand the reactions. Even so, the few calculations 3,6) that include space-exchange terms have not used the best prescriptions of the nucleon wave functions..In particular, the preliminary calculations 3) of inelastic proton scattering to the first excited state of 89y used a simple four-parameter optical-model potential to generate the projectile wave functions and used harmonic-oscillator wave functions for the extra-core bound nucleon. In this paper, improved nucleon wave functions and an alternative parametrization of the effective two-nucleon interaction are used. These are detailed in sect. 3. The mathematical expressions for the transition amplitude and cross section are derived in sect. 2, and the results of our calculations are presented and discussed in sect. 4. 2. Differential cross section
The differential cross section for inelastic nucleon scattering in which only one SP matrix element is important is given by da -- K ~ Idt'] 2, dQ .v~
(1)
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659
where K = lO(k'/k)(pp/2n2h2) 2, 2 = [(2j + 1)(2s + 1)]-~
Z
,
(2)
mjm'jmsm's
ave
and the units are mb/sr. The primes denote final-state quantities and k, pp, s and ms are respectively the wave number, reduced mass, spin quantum number and spin projection of the projectile. The total angular momentum j and its projection mj are associated with the bound nucleon. The antisymmetrized SP matrix element is defined by ~g = (~'i ~b~ltlzl~ka~ 2 - ~ / 2 ~ 1 ) ,
(3)
where t12 is the effective two-nucleon interaction. We choose this interaction to have the form 7) t, 2 = ¢/'12(r)[W + BP ° + M P s + Hp°pS]. (4) Here the operators po and pS exchange the spin and spatial coordinates of the interacting nucleons, respectively. The isospin exchange operator P ' has been replaced by the product _popS. This is permissible because the initial state is antisymmetrized. Then pSpop~ --- _ 1. ~ (5) In eq. (3), the bound states are denoted by 4. For the reaction of interest, namely inelastic scattering of protons from 89y, q~ and 4' represent single protons bound to a SSSr core and in 2p½ and lg¢ states, resPectively. Hence # has the form • (kmJ) = F"tJ(rk) ~ (1½; rnvll, ½; j, m j ) Ytm(f2k)Z~v(Sk)Z(tk) v
= (Rk% + Skflk)%,
(6)
where the subscripts denote the coordinates and the superscripts the single-particle quantum numbers. The isospin wave function is denoted by • a n d the spin wave functions X~*(S) are represented by a for spin up (v ½) and by fl for spin down (v = -½). The spatial wave functions R and S involve the same radial function F. The forms of this function F to be used in our calculations are discussed later. The functions ~ in eq. (3) describe the relative motions of the projectile and target core in the initial and final states. As we consider the inelastic scattering of protons from S9y, direct reaction theory s) implies that ~ is a proton optical-model wave function whose asymptotic behaviour is consistent with the elastic scattering of protons from SSSr. In general, elastic scattering analyses require the use of a spinorbit coupling term in the optical-model potential. However, practical considerations preclude the retention of this spin-orbit term in analyses of inelastic scattering. Hence the continuum wave functions in eq. (3) can be written as
= X,(~ i or fli)z(fi).
(7)
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K.A. AMOS
The spatial wave function X i is independent of the spin of the projectile and is the distorted wave function associated with a spin-independent optical potential, parametrized to fit elastic scattering. Using eqs. (6) and (7), the antisymmetrized initial state in the matrix element [(eq. (3))] can be expressed as li> = 1 ~ 1 ¢ 2 - ¢ 2 ¢ , >
= I~5 or IBS,
(8)
where 1,4) and IB) are associated with spin-up and spin-down incident projectiles, respectively. These two wave functions therefore make incoherent contributions to the cross sections. Defining [6, 7]+ = 6~7z---627~, (9) with subscripts again denoting coordinates, we find
I.~> = [x, R ] _ ~ In> :
+½[x, s]_ [~,/U+ +½[x, s]+[~,/~]_,
IX, S]_fll]~2-aL½[X, R]_[(z, f l ] + - ½ [ X , R]+[~,/~]_.
(10)
The isospin wave functions have been omitted and can be ignored since the bound and continuum particles in both the initial and final channel are protons. The three terms in both 1.4> and IB> are eigenvectors of the coordinate-exchange operators in eq. (4) with eigenvalues -4- 1. Hence t121,4> = ¢/'121'i>, qzlB> =
¢r,zl/~>,
(11)
where I~> = a l l [X, R]_~I~ 2 +½all [X, S]_ [~, 1~]+ +½alo [X, S]+ [~, p]_,
(12)
]B> = al 1[X, S]_/~1p2 + ½al i [X, R]_ [~,/~] + - ½al o [X, R] + [~,/if]_.
(13)
The isospin-spin coefficients a r s are defined by all = W - M + B - H , alo = W+M-B-H, aol = W + M + B + H , aoo = W - M - B + H .
(14)
Eqs. (12) and (13) can now be used to re-express cross section (1) by using the orthogonality of the spin wave functions and by taking the incoherent sum of the spin projections ms and m~. This gives 4 da _ g [ 2 ( 2 j + l ) ] _ 1 ~ ~ i/il 2, (15) dO mj~'j i= I
DW CALCULATIONS
661
where
11 = all(X'~ R'2[C/'121[X, R]_> +½, I2 = ½(X'~ R'21"Y121a11[X, S]_ - a l o r X , S]+>,
(16)
and where the matrix elements 13 and 14 are related to 12 and 11, respectively, by interchanging R and S. The latter relationship holds for all cases to be considered here and is to be assumed whenever 11 and 12 are specified. Hence, use of a static, central effective interaction and neglect of spin-orbit coupling in the optical-model potential result in differential cross sections amenable to calculation. The matrix elements I i mix only two types of basic integrals since the functions R and S involve the same radial function F. These we define as the direct (dr'D) and space exchange (~'E) matrix elements. The former contains the same set of coordinates in the continuum wave functions of both the initial and final states. The latter contains an initial-state continuum wave function that has the same coordinates as the final-state bound wave function. For the particular reaction on S9y, these matrix elements are •-'~'D = ~ <4,½; m', v'l~, m',)(1, ½; m, vi½, m,)
×, •--¢t'r = ~ <4, ½; m', v'l~, m'~) 'v¥'
×. The space-exchange matrix elements in eq. (16) arise not only from the use of exchange operators in the two-nucleon effective interaction but also from the antisymmetrization of the wave functions. To isolate the role of antisyrnmetrization, calculations are also made in which antisymmetry is omitted, but the exchange character of the interaction is not. The preceding theory can be followed to derive expressions for the cross section, except that the identity of eq. (5) can no longer be used. This requires that the isospinprojection operator P~ be used explicitly. The differential cross sections, in which antisymmetry has been omitted, are again given by eq. (15), where now
11 = ( X'l g'21"C/'~21½a11[X, R]_ + ~QI[X, R ] + ) + ( X'l S'21"C/~121~Q2[X,S]_ +,~Q3I-X, S]+ ), I2 = XaQ4(X'1S'21~/'12[[X, R]_ + [ X , R]+).
(17)
The coefficients Qi are defined by
Q1 = 4 ( W - H + B+ M) = 2all + 2alo + 2aox-2aoo , Q2 = 4 ( W - H - M ) = 3a11+ a l o - a o l + aoo, Q3 = 4 ( W - H + M ) = a l l + 3 a l o + a o l - aoo, Q1 = 4(B)
=
a11- alo+ a o l -
aoo.
(18)
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K.A. AMOS
Comparison of eqs. (16) and (17) shows that the coefficients that mix the direct and space-exchange matrix elements in the antisymmetrized development are not related in a simple manner to those that appear in the non-antisymmetrized development. Hence antisymmetry introduces interference effects between the direct and space-exchanged matrix elements that cannot be obtained from the non-antisymmetrized development. This is illustrated most simply in the case of a Serber interaction (W = M = 0.5, B = H = 0.0; alo = ao~ = 1.0, a l l = aoo = 0.0). In this case the matrix elements in the antisymmetrized development [eq. (16)] become I,
=
~_
!
¢
,
2
S]+>,
I2 = -½< X'~ R'zIW, zI[X, S] + >.
(19)
The reduction of the matrix element 12 of eq. (17) to the form given in eq. (19) is impossible. T w o other calculations are reported. They relate to previous work and illustrate the significance of the direct and space-exchange matrix elements. The first, which we call the standard finite-range case, uses the values W = 1, B = M = H = 0 and omits antisymmetry. The matrix elements for this case can be obtained from eq, (17) and are
i, = + , I z = 0.
(20)
The second, which we call the knock-on case, uses the values W = B = H = 0, and M = 1 and omits antisymmetry. The matrix elements for this case can be obtained from eq. (17) and are
It = + , I 2 = 0.
(21)
3. Model prescriptions To calculate the cross sections for inelastic proton scattering from s 9 y , the wave functions, effective interaction and isospin-spin coefficients ars involved in eq. (16) are specified by elastic scattering analyses, bound state spectroscopy and two-nucleon data. Hence, to the extent that our specifications are complete, we have a "noparameter" theory. 3.1. P R E S C R I P T I O N OF T H E PROJECTILE WAVE F U N C T I O N S
Implicit in the D W theory for inelastic scattering is the assumption that elastic scattering is dominant. Therefore, the projectile wave functions (X, Y) should be determined from an optical-model potential that is parametrized to best fit the elastic scattering data 8). From recent investigations 9), it is known that this potential should involve complex central and spin-orbit terms and should be made non-local. However,
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DW CALCULATIONS
at present, it is not feasible to retain such a complete parametrization in D W calculations and still to include space-exchange effects. Therefore, in our calculations, the spin-orbit term and explicit non-locality are neglected, so that our optical-model potential has the form - ( V o + iWo)f°)(r)+aiWDav d f ( 2 ) ( r ) + Vc(r),
q/ore(r)
(22)
dr
where
f(i)(r) = f(r, =
ai, roi)
[1 + exp ( ( r - %, A+)/a,)]- 1.
(23)
The Coulomb potential is that for a charged sphere of radius rcA ~.
A
Q
B
1.0
0 l.i_ w -t" I--
•
-o-o
7"
r
'
"-
r
'
0.1
O.Oi 0
i
I 40
i
i 80
i
I 120
i
,
0
I
40 Oc.m. ( D E G R E E S )
i
I 80
i
I 120
L
Fig. 1. C o m p a r i s o n o f o p t i c a l - m o d e l analyses w i t h the data f o r the elastic scattering o f 18.9 M e V
protons from 89y. The continuous lines are the analyses using the best-fit parameters of set 1 in table 1. The analyses using the parameters of set 2 and set 3 in table 1 are shown by the dashed lines in A and B, respectively.
Neglecting spin-orbit coupling is not expected to affect significantly the evaluation of the cross sections for inelastic scattering, although the parameters of eq. (22) appropriate for elastic scattering may be modified if a spin-orbit term is included. To see if such a modification is significant for the particular case of 18.9 MeV protons o n S 9 y considered in this paper, analyses of the elastic scattering are shown in fig. 1. The parameters of the optical-model potential, eq. (22), are given by sets 1-4 in
664
K.A. AMOS
table 1 t. In both the A and B curves of fig. 1, the data of the Colorado group a 0) and the results of analysis with their best-fit parameters (set 1) are shown by the triangles and the continuous line, respectively. The dashed curve in fig. IA gives the results for elastic scattering using the optical-model parameters of set 2. These differ from the best-fit values of set 1 simply by omitting the spin-orbit term. As can be seen, the comparison of these results with the data is quite good. Hence the parameters of set 2 are appropriate for use in calculating differential cross sections for inelastic scattering. TABLE 1 Optical-model potential parameters
set set set set set
1 2 3 4 5
Vo
Wo
ro
a
re
WD
aD
rD
(MeV)
(MeV)
(fro)
(fro)
(fro)
(MeV)
(fm)
(fro)
0.7 0.7 0.6 0.6 0.63
1.25 1.25 1.25 1.25 1.25
9.8 9.8
0.65 0.65
1.25 1.25
10.0
1.2 1.2 1.25 1.25 1.25
9.8
0.6
1.25
52.6 52.6 50.0 50.0 57.4
Vso (MeV)
a~o (fm)
rso (fro)
5.7
0.7
1.2
0.63
1.25
10.0 a)
a) T h e p a r a m e t e r s o f set 5 were used to c o m p u t e radial b o u n d state functions. T h e p r o g r a m s p e r f o r m i n g this were supplied by Dr. R. Drisko o f O a k Ridge in whose n o t a t i o n Vso = 31.85.
The dashed curve in fig. 1B gives the results for elastic scattering using the opticalmodel potential parameters of set 3 of table 1. These parameters were used in previous D W calculations 3) of inelastic scattering and differ from the best-fit values not only by omission of the spin-orbit terms but also by use of volume absorption. In addition small changes have been made from the best-fit values of the other parameters. The fit to the elastic scattering data is not as good as with parameter sets 1 and 2. However, no attempt was made to search for a best set of parameters for this volume absorption form of the optical-model potential. Instead, set 3 was regarded as a convenient set of compromise parameters with which to see the effect that the shape of the optical-model absorption has upon the calculated cross sections for inelastic scattering. This is achieved by comparing the inelastic scattering results using the parameters of set 3 with those using the parameters of set 4. The latter set of parameters are identical with those of set 3 except that the volume absorption has been replaced by a surface absorption. The analysis of elastic scattering with the parameters of set 4 is not shown, but it results in a compromise similar to that associated with the use of the parameters of set 3. Parametrizing the optical-model potential by eq. (22) also neglects the effects of non-locality. However, in view of the "Percy effect" 11,12), the role of such nont T h e spin-orbit term Vs°(r) :
Vs°(h/mnC)2r
is included in analyses using set 1 p a r a m e t e r s .
1 d dr f~3~(r)~r" 1
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CALCULATIONS
locality in D W calculations of inelastic scattering can be estimated by altering the relative contributions of the nuclear interior and exterior to the direct- and spaceexchange matrix elements. The appropriate prescription to estimate these effects is not considered here. Instead, we investigate how much the nuclear interior contributes to our calculations by weighting each wave function by ~(r)
=
f, 1,
r < gy, r > R s-
(24)
The cut-off radius used is 6.0 fm, which is a value near the Saxon radius, and the two extremes of volume (Rz = 6.0, f = 1) and surface (R s = 6 . 0 , f = 0) calculations are reported.
°7// /',
2
1
4
r(fm)
8
I0
12
Fig. 2. Moduli of the radial functions for a 2p~_proton bound in an harmonic-oscillator potential with hto0 ~ 9.2 MeV (continuous line) and in the Woods-Saxon potential with the parameters of set 5 of table 1 (dashed line). 3.2. PRESCRIPTION OF THE BOUND STATES The bound extra-core nucleon most often is represented by harmonic-oscillator wave functions. This description is known to be inaccurate, especially at large distances and for small binding energies, as ex-emplified by calculations of certain quadrupole matrix elements 13) and by D W calculations of nucleon-transfer reactions i4). In particular, the latter found that use of appropriate Woods-Saxon bound eigenfunctions significantly improved agreement with both the magnitude and shape of the experimental cross sections.
666
K.A. AMOS
However, for not too small binding energies, Woods-Saxon eigenfunctions can have quite good overlap with harmonic oscillator functions in the nuclear interior. Therefore, inelastic scattering of nucleons for small Q-values should not reflect the gross changes found in the two-nucleon transfer analyses 14). Nevertheless, there are differences between the wave functions of the two bound-state bases and therefore in our calculations, the ground and first excited states of 8 9 y a r e represented by Woods-Saxon eigenfunctions. In particular, these states are represented by a 2p~ and lg~ wave function, respectively. The radial forms of these wave functions are I~\
I
I
I
I
I
I
I
I
0.1
II' O.O5
j),[
, , , 2
4
r(fm)
~
,~ , , 8
I0
12
Fig. 3. Moduli o f the radial functions for a lg?r proton bound in a harmonic-oscillator potential with he% ~ 9.2 MeV (continuous line) and in the Woods-Saxon potential with the parameters of set of table 1 (dashed line).
shown by the dashed curves in figs. 2 and 3. Their values were obtained using the parameters of set 5 of table 1. These parameters give binding energies of - 7.06 MeV and -6.19 MeV, respectively, in agreement with the experimental proton separation energy and Q-value. In addition, calculations were made using "equivalent" harmonic oscillator wave functions. Their radial forms are depicted by the continuous lines in figs. 2 and 3. The wave functions are equivalent in that they have the same root-mean-square radii (4.3 and 4.8 fm) as the 2p, and lg~ Woods-Saxon functions, respectively. The oscillator constant for both of these functions is 9.2 MeV, in agreement with that judged appropriate for electron scattering is) (hco ~ 9.3 MeV) for this nucleus.
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3.3. PRESCRIPTION OF THE EFFECTIVE INTERACTION
The "microscopic" formulations of inelastic scattering 1, 3) reduce the manybody problem to a three-body problem composed of projectile, struck target nucleon and target core. This target core is assumed to supply potentials that bind the struck nucleon (U2) and that describe elastic scattering of the projectile (U1). If the core is otherwise inert, the effective interaction for an inelastic scattering process 1,3) is t12 = V 1 2 + V12[E+ie-H]-lV12 .
(25)
H = TI+ U I + T 2 + U2+ V12,
(26)
The Hamiltonian is and in the present application, V12 is the proton-proton potential. The presence of the binding and distorting potentials in the energy denominator of eq. (25) necessitates the replacement of t12 by a pseudo-potential. This we choose to have the form of eq. (4). The specification of the coefficients of the exchange operators, or equivalently of the isospin-spin coefficients ars is given in the next subsection. Here we specify the parametrization of the function 3¢/'12(r). The parametrization of this function is based upon the success of the DW impulse approximation 15) calculations for high incident energies. In such calculations t12 is replaced by the free two-nucleon t-matrix. Hence an appropriate starting point is to parametrize our pseudo-potentials to aspects of the free two-nucleon scattering data. Further, to simplify the proposed DW calculations, it is convenient to choose the pseudo-potential as a central, static, linear combination of Yukawa potentials 3 "~12(r) = $ " 0 X A j exp F - # j r ] / # j r . j=l
(27)
Two sets of parameter values are considered. The first set was chosen phenomenologically to fit the free two-nucleon scattering cross sections at 90° in the centre-ofmass system over a large energy range. This defines the ALM potential. The second set was found by fitting the two-nucleon singlet-even phase shifts 17) and define the Reid potential. The parameter values of these pseudo-potentials, together with those of the single-Yukawa pseudo-potential used by Satchler 1) are given in table 2. TABLE 2 Parameter values of the pseudo-potentials Type ALM Reid Satchler
~o (MeV)
Ax
#1
A2
/zz
--83.0 --10.46 --205.0
1.0 1.0 1.0
0.73 0.7 1.0
--5.0 631.0
1.5 2.8
Aa 20.0 --4338.0
/~z 3.0 4.9
The ALM and Satchler pseudo-potentials are attractive everywhere, whereas the Reid pseudo-potential is repulsive at short distances. This difference is reflected in
668
K.A. AMOS
the structure of the Fourier transforms of these pseudo-potentials (fig. 4). As the D W matrix elements of eq. (16) can be considered as distorted Fourier transforms of the pseudo-potential, the inelastic scattering cross sections should reflect this difference in the character of the pseudo-potentials. 3.4. S P E C I F I C A T I O N O F T H E M I X T U R E C O E F F I C I E N T S
When two protons scatter in the presence of a core, the antisymmetrized DW expressions for the inelastic scattering cross-sections (15) and (16) involve directI
\ ~\\
,o
I
I
o• • ° I°
join i oJ I • I • o l • •
'E F t.c
,?
0
I I i Q2(fm-2) 2
I 3
Fig. 4. The Fourier transforms of the pseudo-potentials, whose parameters are given in table 2, as: a function of the square of the momentum transfer. The Satchler, ALM and Reid forms are givert by the continuous, dashed and dotted lines, respectively. and space-exchanged matrix elements admixed by the singlet-even (alo) and tripletodd (all) coefficients. From spectroscopy as), we choose alo = 0.6 and vary all between 0 and - 0 . 6 , normalized by keeping aol = 1.0.
4. Results The results of D W calculations of the inelastic scattering of 18.9 MeV protons to the first excited state of 89y are discussed in four subsections. In order, they present the results for variation of the optical-model parameters, of the bound-state radial wave functions and of the effective interaction, and the results with and without antisymmetry.
DW CALCULATIONS
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In the first three subsections, three types of calculation are discussed. The results of these calculations are labelled by D, E and S in the diagrams. The curves D are the results of calculations in which antisymmetry is omitted and a Wigner force (W = 1, B = M = H = 0) is used. These purely direct or standard finite-range calculations use the matrix elements of eq. (20). The curves E are the results of calculations in which antisymmetry is omitted, and a Majorana force (M = 1, W = H = B = 0) is used. These purely space-exchange or knock-on calculations use the matrix elements of eq. (21). The third type of calculation, whose results are labelled by S, include antisymmetry and use the matrix elements of eq. (16) with the values alo = 0.6, all = 0.0. In the fourth subsection the influence of antisymmetry is discussed. The results of calculations using the matrix elements of eqs. (16) and (17) are presented for various values of the coefficients aTS. 4.1. V A R I A T I O N O F T H E O P T I C A L - P O T E N T I A L P A R A M E T E R S
Previous papers ~' 3,6) have considered the role of many of the parameters of the optical-model potential in inelastic scattering. Here we consider the effect of the type of absorption. To investigate this, D W calculations were made using the WoodsSaxon bound-state eigenfunctions and the ALM potential with the optical-potential parameters given by set 3 and set 4 in table 1. The results are given in fig. 5 for volume calculations (R I = 6.0,f = 1) and in fig. 6 for surface calculations (R I = 6.0, f = 0), where the labels A and B refer to the use of parameters set 3 and set 4, respectively. The structures of the resulting cross sections are relatively independent of the type of absorption used except at large scattering angles. However, their magnitudes are not. The cross sections using a surface-absorption potential are larger than those using a volume-absorption potential unless the nuclear interior (r < 6.0 fm) does not contribute to the calculations. Such behaviour is consistent with the fact that surface absorption yields optical-model wave functions whose magnitudes inside the 6.0 fm radius are larger than those associated with volume absorption. Although the parametrizations considered here yield compromise fits to the elastic scattering, the differences observed in figs. 5 and 6 are significant, because the relative magnitudes of the surface and volume-absorption results strongly depend upon the extent to which the nuclear interior should contribute. Hence if magnitudes of inelastic scattering are to be used a criteria for determining the appropriate effective interaction, the absorption character of the appropriate optical-model potentials must be chosen carefully as must other factors which affect the contribution from the nuclear interior. If such cut-out effects are physical, comparison of figs. 5 and 6, especially of the antisymmetrized calculations S, indicate that they should be observable in the structure of the cross sections. 4.2. V A R I A T I O N O F T H E B O U N D STATE
As stated earlier, it is not expected that the gross changes observed in the two-
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180
(Degrees)
Fig. 6. Surface calculations (R1 = 6.0, f = 0) of the inelastic scattering cross sections with the opticalmodel potential parameters of set 3 and set 4 of table 1. The results using volume absorption (set 3) in the optical-model potential are shown in A and those using surface absorption (set 4) in B.
DW CALCULATIONS
671
nucleon stripping calculations 14) will be as evident in inelastic nucleon-scattering analyses when Woods-Saxon eigenfunctions are used instead of the simpler harmonicoscillator eigenfunctions. To see the quantitative effects, calculations were made using the optical-model parameters given by set 3 and the ALM potential. The results using harmonic-oscillator eigenfunctions have essentially the same structure as those found using the Woods-Saxon eigenfunctions and therefore are not illustrated in a diagram. However, the cross sections evaluated using these two bound-state prescriptions differ in magnitude. The harmonic-oscillator results are 10--20 ~ larger for volume calculations (R: = 6 . 0 , f = 1) and 30-40 ~o smaller for surface calculations (R: = 6.0, f = 0) than the corresponding Woods-Saxon results. These differences reflect the different properties of the radial wave functions at large radii. This is most clearly seen in the surface-calculation results where only the tails of the bound states are permitted to contribute. The difference between the volume calculations are consistent since the dominant contribution to these results come from the nuclear interior. In this region, the harmonic-oscillator wave functions are larger than the corresponding Woods-Saxon eigenfunctions to compensate for their more rapidly decaying tails. 4.3. EFFECT OF THE INTERACTION The dependence of the cross Sections upon the parametrizations of the interaction is shown in figs. 7 and 8 for volume (R: = 6 . 0 , f = 1) and surface (R: = 6 . 0 , f = 0) calculations, respectively. The curves labelled A were found using the ALM potential, and those labelled B were found using the Reid potential. In all cases, the opticalmodel parameters set 2 and the Woods-Saxon bound-state eigenfunctions were used. The experimental data 1o) are also shown in fig. 7. From figs. 7 and 8, it is dearly seen that the cross sections for the two pseudopotentials differ in shape and magnitude. In particular, the cross sections found using the Reid potential are enhanced at backward scattering angles when compared with those found using the ALM potential. Also the Reid potential gives cross sections with an average magnitude larger than those found using the A L M potential. The differences reflect the dissimilarities between the Fourier transforms of the two pseudo-potentials. Hence inelastic scattering cross sections are sensitive to the prescription of the effective interaction. The sensitivity of the cross section to the range of the effective interaction is most clearly seen by comparing the structures of the cross sections in fig. 7. The strong short-range term in the Reid potential forces the shapes and magnitudes of the direct (D) and knock-on (E) cross sections in fig. 7 B to be quite similar (in the limit of a zero-range interaction these cross sections are indistinguishable, as are their matrix elements). Such similarity between direct and knock-on cross sections is far less evident in the results found using the ALM potential (fig. 7A). Hence the range of the effective interaction strongly affects the evaluation of the direct and space-exchanged matrix elements. In particular their interference properties are range depend-
I
I
°
A
/
/
o
iiII \
...\ill
~ ~ b
ee
• ;~...'s
•
•
"-Y0 ICra
"(3
1 90
0
I
90
0
180
e C M. ( D e g r e e s )
Fig. 7. Volume calculations (Rs = 6.0,f = 1) of the inelastic scattering cross sections with the ALM and Reid pseudo-potentials. The results found by (A) using the ALM pseudo-potential and (B) using the Reid pseudo-potential. Experimental data are also shown. I
I
A
B
I0-~-
D~
03 [33
"O b
//.."/
~\
/
-...
...
..~.
"""
I0 - ~ -
r
°jooe
7
7 0
• I 90
1 0
90
180
8 C.M. ( D e g r e e s ) Fig. 8. Surface calculations (Rf = 6 . 0 , f = 0) o f the inelastic scattering cross sections with the A L M
and Reid pseudo-potentials. The results are those found by (A) using the ALM pseudo-potential and (B) using the Reid pseudo-potential.
DWCALCULATION$
673
ent. Hence antisymmetrized cross-section calculations should reflect in their structure these range-dependent interference effects. Comparison of the antisymmetrized results S in fig. 7 shows that they do. At present there is no satisfactory agreement with the structure we might infer from experiment. However, more data are required before such inference can be made safely. The magnitude of the cross sections are also sensitive to the prescription of the interaction. Their dependence upon the strength of the interaction is trivial. However range effects are also important in determining the magnitudes of cross sections since the interference properties of the direct and space-exchanged matrix elements are range dependent. This can be seen by comparing the relative magnitudes of the direct (D), knock-on (E) and antisymmetrized (S) calculations for the two pseudopotentials. These effects are essentially unchanged, if, for reasons of non-locality in the optical potential or off-shell behaviour of the effective two-nucleon interaction, the contributions from the nuclear interior should be weakened or even removed. The results in an extreme case (R/ = 6 . 0 , f = 0) are shown in fig. 8. It is interesting to note that these "no-parameter" calculations can predict the experimental magnitudes, and to this extent, the microscopic prescription of this reaction is credible. 4.4. EFFECTS OF EXCHANGE AND ANTISYMMETRY In an antisymmetric calculation, the interference of the direct and space-exchanged matrix elements is influenced by the coefficients ars. If both the projectile and struck nucleon are protons, we have seen that antisymmetry requires only the coefficients ato and a l l to be specified. The values ato = 0.6 and all = 0.0 used to obtain the curves S in figs. 5-8 are one possible combination consistent with spectroscopy. The results associated with the use of a Serber force (W = M = 0.5, B = H = 0.0, alo = 1.0, a l l = 0.0) are therefore identical to the curves S but with magnitudes (1.0/0.6) 2 greater. Hence even a simple change in the specification of the coefficients ars can produce a change in magnitude by a factor of 3. However, spectroscopy also permits a range in the values of a11. The effect of varying this coefficient is illustrated in figs. 9A and 10A. Fig. 9 exhibits the results for volume (Rj, = 6 . 0 , f = 1) calculations and fig. 10 for surface (R s = 6.0, f = 0) calculations. The continuous lines correspond to coefficients alo = 0.6, a H = 0.0 whereas the dashed lines correspond to alo = 0.6, at~ = - 0 . 5 . Calculations using a H = - 0 . 2 5 were intermediary. In all cases, the ALM potential, Woods-Saxon bound state eigenfunctions, and the optical-model parameters set 2 were used. As can be seen there is a small increase in the structure and magnitude of the cross sections as the triplet-odd state coefficient a 11 becomes more negative. These increases are more pronounced if the nuclear interior is removed. If antisymmetry is omitted in the calculation whose results are shown in figs. 9A
F A
IO-I r'r" o9
I
/ \
.rn
-0
b iO-Z
-0
I 0
I
90
0
90
180
8 C.M. (Degrees) Fig. 9. Volume calculations (R 1 = 6 . 0 , f = 1) of the inelastic scattering cross sections with and without antisymmetry. The results are (A) those calculated from matrix elements derived with antisymmetry and (B) those calculated from matrix elements derived without antisymmetry. The continuous lines are the results using a n = 0.0 and the dashed lines are the results using al~ ~ --0.6.
f~\ ~" 10-3 60 ff]
-E) b -~ tO-4
I 0
I
90
0
8
C.M. (Degrees)
90
l
180
Fig. 10. Surface calculations (Rf = 6.0,J = 1) o f the inelastic scattering cross sections with and without antisymmetry. The results are (A) those calculated fi'om matrix elements derived with antisymmetry and (B) those calculated from matrix elements derived without antisymmetry. The continuous lines are the results using a l l = 0.0 and the dashed lines are the results using alt -- --0.6.
DWCALCULATION$
675
and 10A, the cross sections become those shown in figs. 9B and 10B, respectively, for a particular choice of the coefficients aoo and aol. Normalization requires aol = 1 and, we choose aoo = 0.0 for convenience. Comparison of the un-antisymmetrized results with the antisymmetrized results shows that if the nuclear interior contributes appreciably (fig. 9), antisymmetry produces small but noticeable changes in the structure of the cross sections and their variations with the coefficient a11. However, if the nuclear interior does not contribute to the matrix elements, antisymmetrization produces significant changes in the resulting cross sections. 5. Conclusions
The variations in cross section found in these DW analyses of inelastic nucleon scattering indicate the need to use good prescriptions of the projectile and target nucleons in antisymmetrized single particle matrix elements. The results of our calculations show strong dependence upon the parametrization of the effective interaction. In particular, the range of the effective interaction is important in determining the magnitudes and interference properties of the direct and space-exchange matrix elements and therefore in determining the magnitude and structure of the cross section. Hence the omission of the space-exchanged terms in a recent survey of inelastic scattering 1) should in part account for the variation in the strength of the effective interaction required in that survey. In addition the differential cross sections are influenced by the type of effective interaction assumed and by antisymmetry in the wave functions. These factors determine the selection and value of the coefficients ars that also affect the relative contributions of the direct and space-exchanged matrix elements. For the reaction investigated these effects primarily influenced the magnitudes of the cross sections. However, if contribution from the nuclear interior should be neglected, antisymmetry also influenced the cross-section structure. Hence, for ranges of the effective interaction currently in use, space-exchange terms arising from antisymmetrization and the exchange character of the effective interaction have important consequences in DW analyses of inelastic scattering. Therefore, their inclusion is necessary before and adequate calibration of the singleparticle matrix elements is possible. The experimental data available at this time are insufficient to either resolve the variations found in our study or test the applicability of our model prescription. Therefore, the 89y experiment for more angles and different incident energies is of great interest. In addition, other experiments such as the (p, n) reaction on 89y and inelastic scattering to the 5- level of 90Zr are important since the correlation of their analyses with that of the 89y experiment discussed here is necessary both to test the reaction model and to probe other aspects of the effective interaction. The author wishes to thank Professor N. Austern for most helpful discussions and Drs. E. Baranger and F. Tabakin for useful comments. I also thank Dr. R. M. Drisko
676
K . A . AMOS
for the code to c o m p u t e W o o d s - S a x o n b o u n d - s t a t e eigenfunctions an d Mr. F. R o s e n z w e i g f o r his assistance in solving c o m p u t a t i o n a l problems. C o m p u t a t i o n s f or this w o r k were p e r f o r m e d with the U n i v e r s i t y o f P i t t s b u r g h c o m p u t e r which is s u p p o r t e d by the N a t i o n a l Science F o u n d a t i o n under G r a n t n u m b e r G-11309.
References 1) G. R. Satchler, Nuclear Physics 77 (1966) 471 2) J. S. Blair in Nuclear spectroscopy with direct reactions, ANL-6878, Vol. II, p. 143; T. Stovall and N. M. Hintz, Phys. Rev. 135 (1964) B330 3) K. A. Amos, V. A. Madsen and I. E. McCarthy, Nuclear Physics A94 (1967) 103 4) N. Auerbach and I. Talmi, Nuclear Physics 64 (1965) 458 5) M. M. Stautberg, thesis, University of Colorado (1966) unpublished 6) A. Agodi and G. Schiffrer, Nuclear Physics 50 (1964) 337; T. Une, S. Yamazi and H. Yoshida, Prog. Theor. Phys. 35 (1966) 1010 7) L. Rosenfeld, Nuclear forces (North-Holland Publ. Co., Amsterdam, 1948) 8) N. Austern, in Selected topics in nuclear theory, ed. by F. Janouch (IAEA, Vienna, 1963) 9) F. G. Perey, in Direct interactions and nuclear reaction mechanisms, ed. by E. Clementel and C. Villi, Padua (1962); G. R. Satchler, Nuclear Physics A100 (1967) 481 10) M. M. Stautberg, J. J. Kraushaar and D. W. Ridley, Phys. Rev., to be published 11) N. Austern, Phys. Rev. 137 (1965) B752 12) G. R. Satchler, in Lectures in theoretical physics, Vol. VIII C, University of Colorado, Boulder, (1965) 13) A. Faessler and R. K. Sheline, Phys. Rev. 148 (1966) 1003 14) R. M. Drisko and F. Rybicki, Phys. Rev. Lett. 16 (1966) 275 15) S. A. Moszkowski, Handbuch der Physik 39 (1957) 411 16) R. M. Haybron and H. McManus, Phys. Rev. 140 (1965) B638 17) H. Bethe, lectures at Carnegie Institute of Technology (1966) 18) J. P. Elliot and A. M. Lane, Handbuch der Physik 39 (1957) 337; J. Soper, Phil. Mag. 2 (1957) 1219
2.F
I !
Nuclear Physics A103 (1967) 657--676; (~) North-Holland Publishing Co., Amsterdam N o t to be reproduced b y p h o t o p r i n t or microfilm without written permission f r o m the publisher
ANTISYMMETRY IN DISTORTED-WAVE CALCULATIONS OF INELASTIC NUCLEON SCATTERING K. A. AMOS t Department of Physics, University of Pittsburgh, Pittsburgh, Pennsylvania *t Received 19 June 1967 Abstract: Distorted-wave calculations of the inelastic scattering of 18.9 MeV protons to the first
excited state of 8sy have been performed using bound-state wave functions consistent with spectroscopic information and using projectile wave functions generated from a seven-parameter optical-model potential that is determined by elastic scattering. Exchange terms appear in the calculations because the initial-state wave function is antisymmetrized and because the twonucleon effective interaction has an exchange character. The effective interaction is taken to have a central static form, that is a linear combination of three Yukawa functions. Two dissimilar parametrizations of this interaction are considered, each of which is determined from aspects of free two-nucleon scattering. Comparison with limited experimental data encourages belief that these "no-parameter" microscopic distorted wave calculations are credible and that this reaction should help to calibrate the matrix elements needed to analyse more complicated reactions. 1. Introduction Direct-reaction inelastic scattering that excites collective modes o f vibration or rotation in nuclei is often well described by a distorted-wave ( D W ) theory in which the optical-model potential that represents the projectile-nucleus interaction is assumed to reflect the mass deformation of the target 1,2). In particular, this theory has yielded g o o d agreement with the shapes and magnitudes of experimental cross sections for low-energy quadrupole and octupole excitations in m a n y nuclei 2). These analyses, however, are " m a c r o s c o p i c " in that all details of nuclear structure are hidden in form factors. In view o f the success o f these " m a c r o s c o p i c " D W calculations, attempts 1,3) have been made to unfold their f o r m factors in terms of individual nucleon motion. Such microscopic formulations generally assume that the independent-particle model adequately describes the m o t i o n of individual target nucleons, and that the interaction promoting an inelastic transition is a sum of individual interactions between the projectile and target nucleons. With these assumptions, the transition amplitude for inelastic scattering f r o m the target nucleus becomes a coherent sum of single-particle (SP) matrix elements 3). Each SP element describes the scattering of the projectile f r o m one b o u n d nucleon in the presence of the remaining target core. In the usual D W approach 3), this remaining target core is assumed to be solely the source of potentials that affect the m o t i o n of the two interacting particles. t Present Address: University of Georgia, Athens, Ga. tt Work supported by the National Science Foundation. 657
658
K . A . AMOS
However, a microscopic calculation of a reaction that has been successfully analysed by the "macroscopic" DW theory, generally requires the evaluation of a coherent sum of numerous SP matrix elements. Therefore, it is essential that these SP elements be carefully calibrated. Hence, attention first should be focussed upon reactions whose transition strengths are divided among as few configurations as possible. Ideally, we would like to treat reactions for which just one SP element is significant. Spectroscopic considerations 4, 5) indicate that inelastic scattering from 89y, especially to its first excited state, is adequately described this way. Further, this nucleus is particularly appealing in that its core 88Sr is sufficiently heavy to make credible the representation of the projectile with optical-model wave functions. Also, for energies of current interest ( ~ 20 MeV), we do not need an excessively large number of partial waves to generate these wave functions. However, an adequate DW calculation must not only use good representations of the projectile and bound nucleon and their effective interaction but also should be antisymmetrized. This then requires evaluation of exchange terms that arise from antisymmetrization of the wave functions and from an exchange character associated with the effective interaction. The difficulties involved, especially in handling exchange effects, are minimized if the projectiles are nucleons. In particular, their existence as such in the region of strong nucleon-nucleus interaction is credible, and the simplicity of the coordinate system facilitates the calculation and interpretation of the exchange terms. Further, we can assume that a two-nucleon interaction is the mechanism that promotes this reaction. But practical limitations of D W calculations require the use of oversimplified forms for this interaction. However, a recent survey of inelastic scattering a) indicates that such forms are sufficient to understand the reactions. Even so, the few calculations 3,6) that include space-exchange terms have not used the best prescriptions of the nucleon wave functions..In particular, the preliminary calculations 3) of inelastic proton scattering to the first excited state of 89y used a simple four-parameter optical-model potential to generate the projectile wave functions and used harmonic-oscillator wave functions for the extra-core bound nucleon. In this paper, improved nucleon wave functions and an alternative parametrization of the effective two-nucleon interaction are used. These are detailed in sect. 3. The mathematical expressions for the transition amplitude and cross section are derived in sect. 2, and the results of our calculations are presented and discussed in sect. 4. 2. Differential cross section
The differential cross section for inelastic nucleon scattering in which only one SP matrix element is important is given by da -- K ~ Idt'] 2, dQ .v~
(1)
DW CALCULATIONS
659
where K = lO(k'/k)(pp/2n2h2) 2, 2 = [(2j + 1)(2s + 1)]-~
Z
,
(2)
mjm'jmsm's
ave
and the units are mb/sr. The primes denote final-state quantities and k, pp, s and ms are respectively the wave number, reduced mass, spin quantum number and spin projection of the projectile. The total angular momentum j and its projection mj are associated with the bound nucleon. The antisymmetrized SP matrix element is defined by ~g = (~'i ~b~ltlzl~ka~ 2 - ~ / 2 ~ 1 ) ,
(3)
where t12 is the effective two-nucleon interaction. We choose this interaction to have the form 7) t, 2 = ¢/'12(r)[W + BP ° + M P s + Hp°pS]. (4) Here the operators po and pS exchange the spin and spatial coordinates of the interacting nucleons, respectively. The isospin exchange operator P ' has been replaced by the product _popS. This is permissible because the initial state is antisymmetrized. Then pSpop~ --- _ 1. ~ (5) In eq. (3), the bound states are denoted by 4. For the reaction of interest, namely inelastic scattering of protons from 89y, q~ and 4' represent single protons bound to a SSSr core and in 2p½ and lg¢ states, resPectively. Hence # has the form • (kmJ) = F"tJ(rk) ~ (1½; rnvll, ½; j, m j ) Ytm(f2k)Z~v(Sk)Z(tk) v
= (Rk% + Skflk)%,
(6)
where the subscripts denote the coordinates and the superscripts the single-particle quantum numbers. The isospin wave function is denoted by • a n d the spin wave functions X~*(S) are represented by a for spin up (v ½) and by fl for spin down (v = -½). The spatial wave functions R and S involve the same radial function F. The forms of this function F to be used in our calculations are discussed later. The functions ~ in eq. (3) describe the relative motions of the projectile and target core in the initial and final states. As we consider the inelastic scattering of protons from S9y, direct reaction theory s) implies that ~ is a proton optical-model wave function whose asymptotic behaviour is consistent with the elastic scattering of protons from SSSr. In general, elastic scattering analyses require the use of a spinorbit coupling term in the optical-model potential. However, practical considerations preclude the retention of this spin-orbit term in analyses of inelastic scattering. Hence the continuum wave functions in eq. (3) can be written as
= X,(~ i or fli)z(fi).
(7)
660
K.A. AMOS
The spatial wave function X i is independent of the spin of the projectile and is the distorted wave function associated with a spin-independent optical potential, parametrized to fit elastic scattering. Using eqs. (6) and (7), the antisymmetrized initial state in the matrix element [(eq. (3))] can be expressed as li> = 1 ~ 1 ¢ 2 - ¢ 2 ¢ , >
= I~5 or IBS,
(8)
where 1,4) and IB) are associated with spin-up and spin-down incident projectiles, respectively. These two wave functions therefore make incoherent contributions to the cross sections. Defining [6, 7]+ = 6~7z---627~, (9) with subscripts again denoting coordinates, we find
I.~> = [x, R ] _ ~ In> :
+½[x, s]_ [~,/U+ +½[x, s]+[~,/~]_,
IX, S]_fll]~2-aL½[X, R]_[(z, f l ] + - ½ [ X , R]+[~,/~]_.
(10)
The isospin wave functions have been omitted and can be ignored since the bound and continuum particles in both the initial and final channel are protons. The three terms in both 1.4> and IB> are eigenvectors of the coordinate-exchange operators in eq. (4) with eigenvalues -4- 1. Hence t121,4> = ¢/'121'i>, qzlB> =
¢r,zl/~>,
(11)
where I~> = a l l [X, R]_~I~ 2 +½all [X, S]_ [~, 1~]+ +½alo [X, S]+ [~, p]_,
(12)
]B> = al 1[X, S]_/~1p2 + ½al i [X, R]_ [~,/~] + - ½al o [X, R] + [~,/if]_.
(13)
The isospin-spin coefficients a r s are defined by all = W - M + B - H , alo = W+M-B-H, aol = W + M + B + H , aoo = W - M - B + H .
(14)
Eqs. (12) and (13) can now be used to re-express cross section (1) by using the orthogonality of the spin wave functions and by taking the incoherent sum of the spin projections ms and m~. This gives 4 da _ g [ 2 ( 2 j + l ) ] _ 1 ~ ~ i/il 2, (15) dO mj~'j i= I
DW CALCULATIONS
661
where
11 = all(X'~ R'2[C/'121[X, R]_> +½
(16)
and where the matrix elements 13 and 14 are related to 12 and 11, respectively, by interchanging R and S. The latter relationship holds for all cases to be considered here and is to be assumed whenever 11 and 12 are specified. Hence, use of a static, central effective interaction and neglect of spin-orbit coupling in the optical-model potential result in differential cross sections amenable to calculation. The matrix elements I i mix only two types of basic integrals since the functions R and S involve the same radial function F. These we define as the direct (dr'D) and space exchange (~'E) matrix elements. The former contains the same set of coordinates in the continuum wave functions of both the initial and final states. The latter contains an initial-state continuum wave function that has the same coordinates as the final-state bound wave function. For the particular reaction on S9y, these matrix elements are •-'~'D = ~ <4,½; m', v'l~, m',)(1, ½; m, vi½, m,)
×
×
11 = ( X'l g'21"C/'~21½a11[X, R]_ + ~QI[X, R ] + ) + ( X'l S'21"C/~121~Q2[X,S]_ +,~Q3I-X, S]+ ), I2 = XaQ4(X'1S'21~/'12[[X, R]_ + [ X , R]+).
(17)
The coefficients Qi are defined by
Q1 = 4 ( W - H + B+ M) = 2all + 2alo + 2aox-2aoo , Q2 = 4 ( W - H - M ) = 3a11+ a l o - a o l + aoo, Q3 = 4 ( W - H + M ) = a l l + 3 a l o + a o l - aoo, Q1 = 4(B)
=
a11- alo+ a o l -
aoo.
(18)
662
K.A. AMOS
Comparison of eqs. (16) and (17) shows that the coefficients that mix the direct and space-exchange matrix elements in the antisymmetrized development are not related in a simple manner to those that appear in the non-antisymmetrized development. Hence antisymmetry introduces interference effects between the direct and space-exchanged matrix elements that cannot be obtained from the non-antisymmetrized development. This is illustrated most simply in the case of a Serber interaction (W = M = 0.5, B = H = 0.0; alo = ao~ = 1.0, a l l = aoo = 0.0). In this case the matrix elements in the antisymmetrized development [eq. (16)] become I,
=
~_
!
¢
,
2
S]+>,
I2 = -½< X'~ R'zIW, zI[X, S] + >.
(19)
The reduction of the matrix element 12 of eq. (17) to the form given in eq. (19) is impossible. T w o other calculations are reported. They relate to previous work and illustrate the significance of the direct and space-exchange matrix elements. The first, which we call the standard finite-range case, uses the values W = 1, B = M = H = 0 and omits antisymmetry. The matrix elements for this case can be obtained from eq, (17) and are
i, =
(20)
The second, which we call the knock-on case, uses the values W = B = H = 0, and M = 1 and omits antisymmetry. The matrix elements for this case can be obtained from eq. (17) and are
It =
(21)
3. Model prescriptions To calculate the cross sections for inelastic proton scattering from s 9 y , the wave functions, effective interaction and isospin-spin coefficients ars involved in eq. (16) are specified by elastic scattering analyses, bound state spectroscopy and two-nucleon data. Hence, to the extent that our specifications are complete, we have a "noparameter" theory. 3.1. P R E S C R I P T I O N OF T H E PROJECTILE WAVE F U N C T I O N S
Implicit in the D W theory for inelastic scattering is the assumption that elastic scattering is dominant. Therefore, the projectile wave functions (X, Y) should be determined from an optical-model potential that is parametrized to best fit the elastic scattering data 8). From recent investigations 9), it is known that this potential should involve complex central and spin-orbit terms and should be made non-local. However,
663
DW CALCULATIONS
at present, it is not feasible to retain such a complete parametrization in D W calculations and still to include space-exchange effects. Therefore, in our calculations, the spin-orbit term and explicit non-locality are neglected, so that our optical-model potential has the form - ( V o + iWo)f°)(r)+aiWDav d f ( 2 ) ( r ) + Vc(r),
q/ore(r)
(22)
dr
where
f(i)(r) = f(r, =
ai, roi)
[1 + exp ( ( r - %, A+)/a,)]- 1.
(23)
The Coulomb potential is that for a charged sphere of radius rcA ~.
A
Q
B
1.0
0 l.i_ w -t" I--
•
-o-o
7"
r
'
"-
r
'
0.1
O.Oi 0
i
I 40
i
i 80
i
I 120
i
,
0
I
40 Oc.m. ( D E G R E E S )
i
I 80
i
I 120
L
Fig. 1. C o m p a r i s o n o f o p t i c a l - m o d e l analyses w i t h the data f o r the elastic scattering o f 18.9 M e V
protons from 89y. The continuous lines are the analyses using the best-fit parameters of set 1 in table 1. The analyses using the parameters of set 2 and set 3 in table 1 are shown by the dashed lines in A and B, respectively.
Neglecting spin-orbit coupling is not expected to affect significantly the evaluation of the cross sections for inelastic scattering, although the parameters of eq. (22) appropriate for elastic scattering may be modified if a spin-orbit term is included. To see if such a modification is significant for the particular case of 18.9 MeV protons o n S 9 y considered in this paper, analyses of the elastic scattering are shown in fig. 1. The parameters of the optical-model potential, eq. (22), are given by sets 1-4 in
664
K.A. AMOS
table 1 t. In both the A and B curves of fig. 1, the data of the Colorado group a 0) and the results of analysis with their best-fit parameters (set 1) are shown by the triangles and the continuous line, respectively. The dashed curve in fig. IA gives the results for elastic scattering using the optical-model parameters of set 2. These differ from the best-fit values of set 1 simply by omitting the spin-orbit term. As can be seen, the comparison of these results with the data is quite good. Hence the parameters of set 2 are appropriate for use in calculating differential cross sections for inelastic scattering. TABLE 1 Optical-model potential parameters
set set set set set
1 2 3 4 5
Vo
Wo
ro
a
re
WD
aD
rD
(MeV)
(MeV)
(fro)
(fro)
(fro)
(MeV)
(fm)
(fro)
0.7 0.7 0.6 0.6 0.63
1.25 1.25 1.25 1.25 1.25
9.8 9.8
0.65 0.65
1.25 1.25
10.0
1.2 1.2 1.25 1.25 1.25
9.8
0.6
1.25
52.6 52.6 50.0 50.0 57.4
Vso (MeV)
a~o (fm)
rso (fro)
5.7
0.7
1.2
0.63
1.25
10.0 a)
a) T h e p a r a m e t e r s o f set 5 were used to c o m p u t e radial b o u n d state functions. T h e p r o g r a m s p e r f o r m i n g this were supplied by Dr. R. Drisko o f O a k Ridge in whose n o t a t i o n Vso = 31.85.
The dashed curve in fig. 1B gives the results for elastic scattering using the opticalmodel potential parameters of set 3 of table 1. These parameters were used in previous D W calculations 3) of inelastic scattering and differ from the best-fit values not only by omission of the spin-orbit terms but also by use of volume absorption. In addition small changes have been made from the best-fit values of the other parameters. The fit to the elastic scattering data is not as good as with parameter sets 1 and 2. However, no attempt was made to search for a best set of parameters for this volume absorption form of the optical-model potential. Instead, set 3 was regarded as a convenient set of compromise parameters with which to see the effect that the shape of the optical-model absorption has upon the calculated cross sections for inelastic scattering. This is achieved by comparing the inelastic scattering results using the parameters of set 3 with those using the parameters of set 4. The latter set of parameters are identical with those of set 3 except that the volume absorption has been replaced by a surface absorption. The analysis of elastic scattering with the parameters of set 4 is not shown, but it results in a compromise similar to that associated with the use of the parameters of set 3. Parametrizing the optical-model potential by eq. (22) also neglects the effects of non-locality. However, in view of the "Percy effect" 11,12), the role of such nont T h e spin-orbit term Vs°(r) :
Vs°(h/mnC)2r
is included in analyses using set 1 p a r a m e t e r s .
1 d dr f~3~(r)~r" 1
DW
665
CALCULATIONS
locality in D W calculations of inelastic scattering can be estimated by altering the relative contributions of the nuclear interior and exterior to the direct- and spaceexchange matrix elements. The appropriate prescription to estimate these effects is not considered here. Instead, we investigate how much the nuclear interior contributes to our calculations by weighting each wave function by ~(r)
=
f, 1,
r < gy, r > R s-
(24)
The cut-off radius used is 6.0 fm, which is a value near the Saxon radius, and the two extremes of volume (Rz = 6.0, f = 1) and surface (R s = 6 . 0 , f = 0) calculations are reported.
°7// /',
2
1
4
r(fm)
8
I0
12
Fig. 2. Moduli of the radial functions for a 2p~_proton bound in an harmonic-oscillator potential with hto0 ~ 9.2 MeV (continuous line) and in the Woods-Saxon potential with the parameters of set 5 of table 1 (dashed line). 3.2. PRESCRIPTION OF THE BOUND STATES The bound extra-core nucleon most often is represented by harmonic-oscillator wave functions. This description is known to be inaccurate, especially at large distances and for small binding energies, as ex-emplified by calculations of certain quadrupole matrix elements 13) and by D W calculations of nucleon-transfer reactions i4). In particular, the latter found that use of appropriate Woods-Saxon bound eigenfunctions significantly improved agreement with both the magnitude and shape of the experimental cross sections.
666
K.A. AMOS
However, for not too small binding energies, Woods-Saxon eigenfunctions can have quite good overlap with harmonic oscillator functions in the nuclear interior. Therefore, inelastic scattering of nucleons for small Q-values should not reflect the gross changes found in the two-nucleon transfer analyses 14). Nevertheless, there are differences between the wave functions of the two bound-state bases and therefore in our calculations, the ground and first excited states of 8 9 y a r e represented by Woods-Saxon eigenfunctions. In particular, these states are represented by a 2p~ and lg~ wave function, respectively. The radial forms of these wave functions are I~\
I
I
I
I
I
I
I
I
0.1
II' O.O5
j),[
, , , 2
4
r(fm)
~
,~ , , 8
I0
12
Fig. 3. Moduli o f the radial functions for a lg?r proton bound in a harmonic-oscillator potential with he% ~ 9.2 MeV (continuous line) and in the Woods-Saxon potential with the parameters of set of table 1 (dashed line).
shown by the dashed curves in figs. 2 and 3. Their values were obtained using the parameters of set 5 of table 1. These parameters give binding energies of - 7.06 MeV and -6.19 MeV, respectively, in agreement with the experimental proton separation energy and Q-value. In addition, calculations were made using "equivalent" harmonic oscillator wave functions. Their radial forms are depicted by the continuous lines in figs. 2 and 3. The wave functions are equivalent in that they have the same root-mean-square radii (4.3 and 4.8 fm) as the 2p, and lg~ Woods-Saxon functions, respectively. The oscillator constant for both of these functions is 9.2 MeV, in agreement with that judged appropriate for electron scattering is) (hco ~ 9.3 MeV) for this nucleus.
DW CALCULATIONS
667
3.3. PRESCRIPTION OF THE EFFECTIVE INTERACTION
The "microscopic" formulations of inelastic scattering 1, 3) reduce the manybody problem to a three-body problem composed of projectile, struck target nucleon and target core. This target core is assumed to supply potentials that bind the struck nucleon (U2) and that describe elastic scattering of the projectile (U1). If the core is otherwise inert, the effective interaction for an inelastic scattering process 1,3) is t12 = V 1 2 + V12[E+ie-H]-lV12 .
(25)
H = TI+ U I + T 2 + U2+ V12,
(26)
The Hamiltonian is and in the present application, V12 is the proton-proton potential. The presence of the binding and distorting potentials in the energy denominator of eq. (25) necessitates the replacement of t12 by a pseudo-potential. This we choose to have the form of eq. (4). The specification of the coefficients of the exchange operators, or equivalently of the isospin-spin coefficients ars is given in the next subsection. Here we specify the parametrization of the function 3¢/'12(r). The parametrization of this function is based upon the success of the DW impulse approximation 15) calculations for high incident energies. In such calculations t12 is replaced by the free two-nucleon t-matrix. Hence an appropriate starting point is to parametrize our pseudo-potentials to aspects of the free two-nucleon scattering data. Further, to simplify the proposed DW calculations, it is convenient to choose the pseudo-potential as a central, static, linear combination of Yukawa potentials 3 "~12(r) = $ " 0 X A j exp F - # j r ] / # j r . j=l
(27)
Two sets of parameter values are considered. The first set was chosen phenomenologically to fit the free two-nucleon scattering cross sections at 90° in the centre-ofmass system over a large energy range. This defines the ALM potential. The second set was found by fitting the two-nucleon singlet-even phase shifts 17) and define the Reid potential. The parameter values of these pseudo-potentials, together with those of the single-Yukawa pseudo-potential used by Satchler 1) are given in table 2. TABLE 2 Parameter values of the pseudo-potentials Type ALM Reid Satchler
~o (MeV)
Ax
#1
A2
/zz
--83.0 --10.46 --205.0
1.0 1.0 1.0
0.73 0.7 1.0
--5.0 631.0
1.5 2.8
Aa 20.0 --4338.0
/~z 3.0 4.9
The ALM and Satchler pseudo-potentials are attractive everywhere, whereas the Reid pseudo-potential is repulsive at short distances. This difference is reflected in
668
K.A. AMOS
the structure of the Fourier transforms of these pseudo-potentials (fig. 4). As the D W matrix elements of eq. (16) can be considered as distorted Fourier transforms of the pseudo-potential, the inelastic scattering cross sections should reflect this difference in the character of the pseudo-potentials. 3.4. S P E C I F I C A T I O N O F T H E M I X T U R E C O E F F I C I E N T S
When two protons scatter in the presence of a core, the antisymmetrized DW expressions for the inelastic scattering cross-sections (15) and (16) involve directI
\ ~\\
,o
I
I
o• • ° I°
join i oJ I • I • o l • •
'E F t.c
,?
0
I I i Q2(fm-2) 2
I 3
Fig. 4. The Fourier transforms of the pseudo-potentials, whose parameters are given in table 2, as: a function of the square of the momentum transfer. The Satchler, ALM and Reid forms are givert by the continuous, dashed and dotted lines, respectively. and space-exchanged matrix elements admixed by the singlet-even (alo) and tripletodd (all) coefficients. From spectroscopy as), we choose alo = 0.6 and vary all between 0 and - 0 . 6 , normalized by keeping aol = 1.0.
4. Results The results of D W calculations of the inelastic scattering of 18.9 MeV protons to the first excited state of 89y are discussed in four subsections. In order, they present the results for variation of the optical-model parameters, of the bound-state radial wave functions and of the effective interaction, and the results with and without antisymmetry.
DW CALCULATIONS
669
In the first three subsections, three types of calculation are discussed. The results of these calculations are labelled by D, E and S in the diagrams. The curves D are the results of calculations in which antisymmetry is omitted and a Wigner force (W = 1, B = M = H = 0) is used. These purely direct or standard finite-range calculations use the matrix elements of eq. (20). The curves E are the results of calculations in which antisymmetry is omitted, and a Majorana force (M = 1, W = H = B = 0) is used. These purely space-exchange or knock-on calculations use the matrix elements of eq. (21). The third type of calculation, whose results are labelled by S, include antisymmetry and use the matrix elements of eq. (16) with the values alo = 0.6, all = 0.0. In the fourth subsection the influence of antisymmetry is discussed. The results of calculations using the matrix elements of eqs. (16) and (17) are presented for various values of the coefficients aTS. 4.1. V A R I A T I O N O F T H E O P T I C A L - P O T E N T I A L P A R A M E T E R S
Previous papers ~' 3,6) have considered the role of many of the parameters of the optical-model potential in inelastic scattering. Here we consider the effect of the type of absorption. To investigate this, D W calculations were made using the WoodsSaxon bound-state eigenfunctions and the ALM potential with the optical-potential parameters given by set 3 and set 4 in table 1. The results are given in fig. 5 for volume calculations (R I = 6.0,f = 1) and in fig. 6 for surface calculations (R I = 6.0, f = 0), where the labels A and B refer to the use of parameters set 3 and set 4, respectively. The structures of the resulting cross sections are relatively independent of the type of absorption used except at large scattering angles. However, their magnitudes are not. The cross sections using a surface-absorption potential are larger than those using a volume-absorption potential unless the nuclear interior (r < 6.0 fm) does not contribute to the calculations. Such behaviour is consistent with the fact that surface absorption yields optical-model wave functions whose magnitudes inside the 6.0 fm radius are larger than those associated with volume absorption. Although the parametrizations considered here yield compromise fits to the elastic scattering, the differences observed in figs. 5 and 6 are significant, because the relative magnitudes of the surface and volume-absorption results strongly depend upon the extent to which the nuclear interior should contribute. Hence if magnitudes of inelastic scattering are to be used a criteria for determining the appropriate effective interaction, the absorption character of the appropriate optical-model potentials must be chosen carefully as must other factors which affect the contribution from the nuclear interior. If such cut-out effects are physical, comparison of figs. 5 and 6, especially of the antisymmetrized calculations S, indicate that they should be observable in the structure of the cross sections. 4.2. V A R I A T I O N O F T H E B O U N D STATE
As stated earlier, it is not expected that the gross changes observed in the two-
i B
4
lO-Ii &
I//
\
/
\
'
**°
/i
\
i
"\.L-
/
J
°'i
/ t
0
90
0
i 9O
I F80
8 C M. (Degrees) Fig. 5. Volume calculations ( R + = 6.0,3"= 1) of the inelastic scattering cross sections with the optical-model potential parameters of set 3 and set 4 of table 1. The results using volume absorption (set 3) in the optical-model potential are shown in A and those using surface absorption (set 4) in B.
cr" o9
I
I
A
B
[0-3
?
£13
S -(D b ¸ 10-4
I 0
I
90
0
8 C,M.
90
180
(Degrees)
Fig. 6. Surface calculations (R1 = 6.0, f = 0) of the inelastic scattering cross sections with the opticalmodel potential parameters of set 3 and set 4 of table 1. The results using volume absorption (set 3) in the optical-model potential are shown in A and those using surface absorption (set 4) in B.
DW CALCULATIONS
671
nucleon stripping calculations 14) will be as evident in inelastic nucleon-scattering analyses when Woods-Saxon eigenfunctions are used instead of the simpler harmonicoscillator eigenfunctions. To see the quantitative effects, calculations were made using the optical-model parameters given by set 3 and the ALM potential. The results using harmonic-oscillator eigenfunctions have essentially the same structure as those found using the Woods-Saxon eigenfunctions and therefore are not illustrated in a diagram. However, the cross sections evaluated using these two bound-state prescriptions differ in magnitude. The harmonic-oscillator results are 10--20 ~ larger for volume calculations (R: = 6 . 0 , f = 1) and 30-40 ~o smaller for surface calculations (R: = 6.0, f = 0) than the corresponding Woods-Saxon results. These differences reflect the different properties of the radial wave functions at large radii. This is most clearly seen in the surface-calculation results where only the tails of the bound states are permitted to contribute. The difference between the volume calculations are consistent since the dominant contribution to these results come from the nuclear interior. In this region, the harmonic-oscillator wave functions are larger than the corresponding Woods-Saxon eigenfunctions to compensate for their more rapidly decaying tails. 4.3. EFFECT OF THE INTERACTION The dependence of the cross Sections upon the parametrizations of the interaction is shown in figs. 7 and 8 for volume (R: = 6 . 0 , f = 1) and surface (R: = 6 . 0 , f = 0) calculations, respectively. The curves labelled A were found using the ALM potential, and those labelled B were found using the Reid potential. In all cases, the opticalmodel parameters set 2 and the Woods-Saxon bound-state eigenfunctions were used. The experimental data 1o) are also shown in fig. 7. From figs. 7 and 8, it is dearly seen that the cross sections for the two pseudopotentials differ in shape and magnitude. In particular, the cross sections found using the Reid potential are enhanced at backward scattering angles when compared with those found using the ALM potential. Also the Reid potential gives cross sections with an average magnitude larger than those found using the A L M potential. The differences reflect the dissimilarities between the Fourier transforms of the two pseudo-potentials. Hence inelastic scattering cross sections are sensitive to the prescription of the effective interaction. The sensitivity of the cross section to the range of the effective interaction is most clearly seen by comparing the structures of the cross sections in fig. 7. The strong short-range term in the Reid potential forces the shapes and magnitudes of the direct (D) and knock-on (E) cross sections in fig. 7 B to be quite similar (in the limit of a zero-range interaction these cross sections are indistinguishable, as are their matrix elements). Such similarity between direct and knock-on cross sections is far less evident in the results found using the ALM potential (fig. 7A). Hence the range of the effective interaction strongly affects the evaluation of the direct and space-exchanged matrix elements. In particular their interference properties are range depend-
I
I
°
A
/
/
o
iiII \
...\ill
~ ~ b
ee
• ;~...'s
•
•
"-Y0 ICra
"(3
1 90
0
I
90
0
180
e C M. ( D e g r e e s )
Fig. 7. Volume calculations (Rs = 6.0,f = 1) of the inelastic scattering cross sections with the ALM and Reid pseudo-potentials. The results found by (A) using the ALM pseudo-potential and (B) using the Reid pseudo-potential. Experimental data are also shown. I
I
A
B
I0-~-
D~
03 [33
"O b
//.."/
~\
/
-...
...
..~.
"""
I0 - ~ -
r
°jooe
7
7 0
• I 90
1 0
90
180
8 C.M. ( D e g r e e s ) Fig. 8. Surface calculations (Rf = 6 . 0 , f = 0) o f the inelastic scattering cross sections with the A L M
and Reid pseudo-potentials. The results are those found by (A) using the ALM pseudo-potential and (B) using the Reid pseudo-potential.
DWCALCULATION$
673
ent. Hence antisymmetrized cross-section calculations should reflect in their structure these range-dependent interference effects. Comparison of the antisymmetrized results S in fig. 7 shows that they do. At present there is no satisfactory agreement with the structure we might infer from experiment. However, more data are required before such inference can be made safely. The magnitude of the cross sections are also sensitive to the prescription of the interaction. Their dependence upon the strength of the interaction is trivial. However range effects are also important in determining the magnitudes of cross sections since the interference properties of the direct and space-exchanged matrix elements are range dependent. This can be seen by comparing the relative magnitudes of the direct (D), knock-on (E) and antisymmetrized (S) calculations for the two pseudopotentials. These effects are essentially unchanged, if, for reasons of non-locality in the optical potential or off-shell behaviour of the effective two-nucleon interaction, the contributions from the nuclear interior should be weakened or even removed. The results in an extreme case (R/ = 6 . 0 , f = 0) are shown in fig. 8. It is interesting to note that these "no-parameter" calculations can predict the experimental magnitudes, and to this extent, the microscopic prescription of this reaction is credible. 4.4. EFFECTS OF EXCHANGE AND ANTISYMMETRY In an antisymmetric calculation, the interference of the direct and space-exchanged matrix elements is influenced by the coefficients ars. If both the projectile and struck nucleon are protons, we have seen that antisymmetry requires only the coefficients ato and a l l to be specified. The values ato = 0.6 and all = 0.0 used to obtain the curves S in figs. 5-8 are one possible combination consistent with spectroscopy. The results associated with the use of a Serber force (W = M = 0.5, B = H = 0.0, alo = 1.0, a l l = 0.0) are therefore identical to the curves S but with magnitudes (1.0/0.6) 2 greater. Hence even a simple change in the specification of the coefficients ars can produce a change in magnitude by a factor of 3. However, spectroscopy also permits a range in the values of a11. The effect of varying this coefficient is illustrated in figs. 9A and 10A. Fig. 9 exhibits the results for volume (Rj, = 6 . 0 , f = 1) calculations and fig. 10 for surface (R s = 6.0, f = 0) calculations. The continuous lines correspond to coefficients alo = 0.6, a H = 0.0 whereas the dashed lines correspond to alo = 0.6, at~ = - 0 . 5 . Calculations using a H = - 0 . 2 5 were intermediary. In all cases, the ALM potential, Woods-Saxon bound state eigenfunctions, and the optical-model parameters set 2 were used. As can be seen there is a small increase in the structure and magnitude of the cross sections as the triplet-odd state coefficient a 11 becomes more negative. These increases are more pronounced if the nuclear interior is removed. If antisymmetry is omitted in the calculation whose results are shown in figs. 9A
F A
IO-I r'r" o9
I
/ \
.rn
-0
b iO-Z
-0
I 0
I
90
0
90
180
8 C.M. (Degrees) Fig. 9. Volume calculations (R 1 = 6 . 0 , f = 1) of the inelastic scattering cross sections with and without antisymmetry. The results are (A) those calculated from matrix elements derived with antisymmetry and (B) those calculated from matrix elements derived without antisymmetry. The continuous lines are the results using a n = 0.0 and the dashed lines are the results using al~ ~ --0.6.
f~\ ~" 10-3 60 ff]
-E) b -~ tO-4
I 0
I
90
0
8
C.M. (Degrees)
90
l
180
Fig. 10. Surface calculations (Rf = 6.0,J = 1) o f the inelastic scattering cross sections with and without antisymmetry. The results are (A) those calculated fi'om matrix elements derived with antisymmetry and (B) those calculated from matrix elements derived without antisymmetry. The continuous lines are the results using a l l = 0.0 and the dashed lines are the results using alt -- --0.6.
DWCALCULATION$
675
and 10A, the cross sections become those shown in figs. 9B and 10B, respectively, for a particular choice of the coefficients aoo and aol. Normalization requires aol = 1 and, we choose aoo = 0.0 for convenience. Comparison of the un-antisymmetrized results with the antisymmetrized results shows that if the nuclear interior contributes appreciably (fig. 9), antisymmetry produces small but noticeable changes in the structure of the cross sections and their variations with the coefficient a11. However, if the nuclear interior does not contribute to the matrix elements, antisymmetrization produces significant changes in the resulting cross sections. 5. Conclusions
The variations in cross section found in these DW analyses of inelastic nucleon scattering indicate the need to use good prescriptions of the projectile and target nucleons in antisymmetrized single particle matrix elements. The results of our calculations show strong dependence upon the parametrization of the effective interaction. In particular, the range of the effective interaction is important in determining the magnitudes and interference properties of the direct and space-exchange matrix elements and therefore in determining the magnitude and structure of the cross section. Hence the omission of the space-exchanged terms in a recent survey of inelastic scattering 1) should in part account for the variation in the strength of the effective interaction required in that survey. In addition the differential cross sections are influenced by the type of effective interaction assumed and by antisymmetry in the wave functions. These factors determine the selection and value of the coefficients ars that also affect the relative contributions of the direct and space-exchanged matrix elements. For the reaction investigated these effects primarily influenced the magnitudes of the cross sections. However, if contribution from the nuclear interior should be neglected, antisymmetry also influenced the cross-section structure. Hence, for ranges of the effective interaction currently in use, space-exchange terms arising from antisymmetrization and the exchange character of the effective interaction have important consequences in DW analyses of inelastic scattering. Therefore, their inclusion is necessary before and adequate calibration of the singleparticle matrix elements is possible. The experimental data available at this time are insufficient to either resolve the variations found in our study or test the applicability of our model prescription. Therefore, the 89y experiment for more angles and different incident energies is of great interest. In addition, other experiments such as the (p, n) reaction on 89y and inelastic scattering to the 5- level of 90Zr are important since the correlation of their analyses with that of the 89y experiment discussed here is necessary both to test the reaction model and to probe other aspects of the effective interaction. The author wishes to thank Professor N. Austern for most helpful discussions and Drs. E. Baranger and F. Tabakin for useful comments. I also thank Dr. R. M. Drisko
676
K . A . AMOS
for the code to c o m p u t e W o o d s - S a x o n b o u n d - s t a t e eigenfunctions an d Mr. F. R o s e n z w e i g f o r his assistance in solving c o m p u t a t i o n a l problems. C o m p u t a t i o n s f or this w o r k were p e r f o r m e d with the U n i v e r s i t y o f P i t t s b u r g h c o m p u t e r which is s u p p o r t e d by the N a t i o n a l Science F o u n d a t i o n under G r a n t n u m b e r G-11309.
References 1) G. R. Satchler, Nuclear Physics 77 (1966) 471 2) J. S. Blair in Nuclear spectroscopy with direct reactions, ANL-6878, Vol. II, p. 143; T. Stovall and N. M. Hintz, Phys. Rev. 135 (1964) B330 3) K. A. Amos, V. A. Madsen and I. E. McCarthy, Nuclear Physics A94 (1967) 103 4) N. Auerbach and I. Talmi, Nuclear Physics 64 (1965) 458 5) M. M. Stautberg, thesis, University of Colorado (1966) unpublished 6) A. Agodi and G. Schiffrer, Nuclear Physics 50 (1964) 337; T. Une, S. Yamazi and H. Yoshida, Prog. Theor. Phys. 35 (1966) 1010 7) L. Rosenfeld, Nuclear forces (North-Holland Publ. Co., Amsterdam, 1948) 8) N. Austern, in Selected topics in nuclear theory, ed. by F. Janouch (IAEA, Vienna, 1963) 9) F. G. Perey, in Direct interactions and nuclear reaction mechanisms, ed. by E. Clementel and C. Villi, Padua (1962); G. R. Satchler, Nuclear Physics A100 (1967) 481 10) M. M. Stautberg, J. J. Kraushaar and D. W. Ridley, Phys. Rev., to be published 11) N. Austern, Phys. Rev. 137 (1965) B752 12) G. R. Satchler, in Lectures in theoretical physics, Vol. VIII C, University of Colorado, Boulder, (1965) 13) A. Faessler and R. K. Sheline, Phys. Rev. 148 (1966) 1003 14) R. M. Drisko and F. Rybicki, Phys. Rev. Lett. 16 (1966) 275 15) S. A. Moszkowski, Handbuch der Physik 39 (1957) 411 16) R. M. Haybron and H. McManus, Phys. Rev. 140 (1965) B638 17) H. Bethe, lectures at Carnegie Institute of Technology (1966) 18) J. P. Elliot and A. M. Lane, Handbuch der Physik 39 (1957) 337; J. Soper, Phil. Mag. 2 (1957) 1219