A surface-delta description of analyzing power measurements for collective states of Ni and Zn isotopes

A SURFACE-DELTA DESCRIPTION OF ANALYZING POWER MEASUREMENTS FOR COLLEC'T'IVE STATES OF N1 AND Zn ISOTOPESt

M. J. THROOP, Y. T. CHENG, A. GOSWAML O. NALCIOGLUtt and D. K McDANI 4 University of Oregon, Etpme, Oregon 97403 I,. W. SWENSON Oregon State Unfttrsity, Corvai7is, Oregon 97331

and NELSON JÀRMIE, J. H. MIT, P. A. LOVOI, D. STUPIN, GERALD G. OHLSEN and G. C. SAIZMAN Los Alamos Scient(lie Laboratory, Los Alamos, New Mexico 87544 Received 4 May 1976 (Revised 21 February 1977) Abstract : Analyzing powers and differential cross sections for the inelastic scattering of 15 MeV polarized protons to the stronger collective states of ss " 6 °Ni and s`Za have been measured. The data were compared to detailed microscopic reaction calculations using a central plus spin-dependent effective interaction. The nuclear structure wave functions used were obtained from a quasiparticle random-phase appoximation calculation using a spin-dependent surface delta interaction and a basis set containing both neutron and proton configurations. The overall quality of the fits strongly suggests that the surface-delta wave functions provide a good representation for spherical nuclei in this intermediate mans range. The effects of exchange and further modifications to the microscopic effective interaction are discussed. NUCLEAR REACITON

a`Zn(p,p'), E- 15 McV; measured o(B), A(9) . Enriched target, 99 .0 %,.

I

1. Introdnction The description of the almost spherical nuclei in the intermediate mass region of A = 50-100 is complicated by the lack of any successful phenomenological model. A macroscopic collective treatment 1) has had limited success in describing the properties of the low energy 2+ and 3- oollective states and the 0+, 2*, 4* "two-phonon" triplet. Even here it is common ') to utilize one parameter per state. To remedy this Kisslinger and Sorenson s) performed extensive microscopio calculations for this region using the pairing interaction to describe the quasiparticles to simulate the longer range part of the effective interaction. Reasonable success was obtained in describing the properties of excited states in nuclei with A > 100, but not for lighter t Work supported in part by a grant from the National Science Foundation. tt Now at the University of Wisconsin, Madison, Wisconsin. 475

476

M. J. THROOP

nuclei . With an increasing amount of data accumulating for these latter nuclei, there exists a strong need for a more suitable wave function description. A common approach for evaluating nuclear structure wave functions is to calculate electromagnetic transition probabilities and compare them withexperiment However, a more sensitive test of nuclear structure wave functions is provided by inelastic proton scattering at finite momentum transfer. This can be particularly valuable if morethan one type of datais available at a givenincident proton energy . A simultaneous fit to data such as the differential cross section, Q(9), asymmetry, A(9) and spinflip probability, S(B) puts an extremelystringent requirement uponthewave functions. The availability of high-intensity polarized proton ion sources a) makes it possible to routinely obtain this type of information. Comparisons of differential cross-section data with the predictions of microscopic calculations have met with considerable success. This has not been the vase for asymmetry data where microscopic fits have been rather unsatisfactory'), even for a closed shell nucleus like a°Ca where a good wave function description is expected 6). The situation does not appear to have been substantially altered in the more recent literature. The sensitivity of the A(9) comparisons lies in their dependence on reaction amplitude differences whereas c(B) depends on amplitude sums. To obtain improved asymmetry fits one can include exchange effects, introduce more complex effective interactions and utilize better wave functions . One possibility for improving the microscopic description which has seen considerable usage in recent years is to calculate the nuclear structure wave functions using a surfa,oe-delta interaction (SDI) '). These calculations have shown that this effective interaction is as successful in describing nuclear spectra as other commonly suggested effective nuclear forces such as pairing plus quadrupole or Gaussian with exchange . An extremely attractive feature of the SDI is thatthe resulting ealoulational simplicity permits large basis sets to be used in the shell-model diagonalization. Goswami and Lin s) utilized this simplification in a doubly open shell calculation for the A = 50-90 mass range. They found that good B(E2) results could be obtained for isotopes of Ni and Zn. They also used these waves functionsto predict the existence of bound spinquadrupole states which might exhibit enhanced inelastic proton scattering transitions 9). The search for these states provided the initial impetus for the present investigation. The specific motivation for the present study was to test the validity of the SDI wave functions by checking how well they describe inelastic proton cross sections and asymmetries for collective states in Ni and Zn isotopes. SDI wave functions were used earlier 1 °) to describe inelastic proton scattering by the Ni isotopes, but an assessment of the validity of this wave function description was difficult because. of the lack of asymmetry data. The collective states were chosen because inelastic proton scattering to them is well described by a macroscopic reaction calculation. A further advantage of scattering to collective states is that in a microscopic description exchange contributions can be approximately accounted for as a cross-section scale

ss .60Ni, 64Zn COLLECTIVE STATES

477

factor 11). A comparison of the microscopic and macroscopic fits to the asymmetry data serves as a measure of the overall quality of the microscopic description. This paper presents asymmetry and differential cross-section data for 15 MeV polarized protons scattered from ss, "Ni and s4Zn. Data were obtained at lab angles between 20° and 160° for each nucleus. Excitation of states through angular momentum transfer of L = 2-5 was observed The experimental procedure and data analysis are described. The asymmetry and cross-section data have been fitted using a microscopic finite range central interaction in the distorted-wave Born approxmation (DWBA) with SDI wave functions. The data have also been fitted using a macroscopic deformed optical potential interaction for purposes of comparison. It is shown that the microscopic description for inelastic scattering to collective states is at least as good as the macroscopic. A preliminary report of this work has been published elsewhere 14). 2 . Experimental procedure Protons from the Los Alamos Scientific Laboratory Lamb-shift polarized ion source 4) were accelerated in the tandem Van de Graaff to an energy of 15.00 MeV. Polarized proton beams of up to 150 nA intensity with a typical polarization of > 90 % were scattered from isotopically separated (> 98 %) so- 6 Ni and 64Zn foilst of sr 1 mg/cml thickness. Scattered protons were detected with a cooled Si surfacebarrier detector in a 76 cm diameter scattering chamber 13). The rectangular detector aperture subtended an angle of 0.65 in the scattering plane and a solid angle of 0.224 msr. Resolution for the scattered proton groups was about 40 keV FWHM . Pulse height spectra were obtained at 10° lab intervals between 20° and 160°. A typical spectrum of 64Zn is shown in fig. 1 . Additional data were collected in short runs at'5° intervals to provide a large number of points for the elastic asymmetry and differential cross-section angular distributions, since these were used to determine the best optical model parameters. At each angle the yield was measured for a particular spin orientation (up or down) with respect to the scattering plane, immediately followed by a yield measurement for the opposite orientation. The beam polarization was measured to an accuracy of 1 % or better at the beginning of each yield measurement. Proton groups corresponding to known levels in the target nucleus were identified on the basis of their kinematic behavior. Contaminant groups due to inelastic proton scattering, primarily from C and O, were similarly identified. The only other charged particles which could contaminate proton groups in the energy range of interest were alphas from the (p, a) reaction. These were weak and easily identified by their kinematic behavior and relatively large energy spread. No analysis was carried out at angles where inelastic proton and contaminant groups were unresolvable . Analysis of the spectra of the collective states was simplified since peaks corresponding to levels below 5 MeV had sufficient statistics to permit fits with a Gaussian plus linear background computer routine. Background subtractions were small except t Supplied by Micromatter, Inc ., Seattle, Washington.

478

s

25 4

M. J. THROOP 64 7-n (Pj) Ep-15 MsV ; Bp-9Cr

W

zs

tfl

.V .I ;. L-kW am 4100 CHANNEL .NUMBER

- -

"

N

64

Y

z a

CHANNEL NUMBER

Fig. .1 . A typical spectrum for

° 4Zn(p, p') with

E, - 15.0 MeV and 9, - 90°.

for a few weak groups, which were analyzed by hand. The yield from partially resolved peaks was estimated by summing only over that portion of the peak which was uncontaminated by neighboring peaks. The channels for the peak sum and the background subtraction were kept identical for the spin-up and spin-down yield measurements at a particular angle. In this way, the resulting asymmetry, which depends on a ratio ofyields, does not depend on the choice of channels over which the sum was taken. Since the cross-section results required knowledge of the total yield, these data were obtained assuming a triangular peak shape of known half-width. The minimum absolute cross-section error is f 15 % due to uncertainty in the value of the target thickness. 3. Ileoretical analysis The inelastic scattering amplitude in the distorted-wave Born approximation (DWBA) for a one-step process is given as tnwa+ °c

f

X;K (kr, r)«Ek" I «lt(r)I V(r r)~

J,(~))I

W" )xl u~'(ki~

r)dr .

(1)

e* .6ONi,

ssya COLLECTIVE STATES

479

The distorted waves X;~,~,(kr, r) and Xi+,,(ki, r) were generated by solving the elastic scattering problem in the incoming and outgoing channels using an optical potential which included a spin-orbit term. Information about the nuclear excitation is isolated in the nuclear matrix element. The latter includes a sum over both neutron and proton excitations. Following customary practice the interaction was approximated as a onebody effective operator resulting in the nuclear form factor being expressible as a linear combination of elementary single-particle form factors 14). The DWBA reaction amplitude of eq. (1) is a direct one in which exchange effects have not been included. The success of macroscopic calculations (which inherently exclude exchange) for nuclear scattering from collective states implies that the direct and exchange amplitudes must have very nearly the same angular dependence. This has been verified by microscopic calculations including exchange i l. ' 5) for the excitation of the 2+ state in "'Sri by 16 MeV protons and the excitation of the 3 - and 5states in "'Ca by 20 MeV protons. In the present microscopic treatment the effect of exchange was initially accounted for through a renormalization of the effective interaction strength. The direct/exchange amplitude ratio is known 11) to decrease with increasing angular momentum transfer L so that this renormalization is L-transfer dependent. Renormalization of the direct amplitude to account for exchange will likely be least successful for very large L-transfers. During the course of this investigation a microscopic reaction program including exchange 15) (DWBA-70) became available, permitting some preliminary estimates of explicit exchange effects. The effective interaction between the projectile and target nucleus was taken as a sum of one-body operators : V(rp, r) -

d

(Vo+Viap - tri) exp [ - PRJrp-r1J 2],

tal

(2)

where the scalar strength VO and the spin-spin strength Vl have an isospin dependence . Initial values for them as well as for the Gaussian range parameter ßß were taken from a phenomenological fitto nucleon-nucleon scattering 1'). The nucleon-nucleon values are Vó = -23 McV, Vi° = -2.6 MeV, lam= -7.8 MeV, Vïp = 7.8 MeV and ßR = 0.2922 fin'. In fitting the cross-section data the only change made in the effective strengths was to multiply each one by the same scale factor. While evidence has accumulated showing that inclusion of spin-orbit, tensor and other terms in the effective interaction can lead to improved fits to the data in some oases 1a), the simpler effective interaction given by eq. (2) was acceptable in the present analysis since the prime objective of this study was to test the validity of the SDI wave funotions. If good fits to the asymmetry data can be obtained with a simplified effective interaction, then real credence can be given to the SDI structure calculation. The microscopic reaction formalism used for the major portion of the analysis in the present work closely follows that of Glendenning and Veneroni 1 `), except for the modification required by the use of the quasiparticle randomphase approximation (QRPA) in the nuclear structure calculation . Specifically, eq. (36)

480

M. J. THROOP

of ref. 14) beoomes #~uAr)

_ - 1: [X,b +( -i+zZ i][U.Vb +( -YV.Ub7Fâ4r) . .2b

(3)

In this equation FL' j (r) is the form factor for a single-particle transition and the X,b, are the structure coefficients arising from the QRPA calculation using the surfaceX.b delta interaction. The Condon-Shortly phase convention has been adhered to, and the coupling rule !+a = j followed. The factor it was included in the definition of the spherical harmonics in order to maintain the property of time-reversal invariance. The single-particle form factors were evaluated with the usual choice of harmonic . For a Gaussian interaction form in eq. (2) and oscillator parameter v = 41M/A=A} harmonic oscillator wave functions, the required radial integrals can be evaluated analytically . The DWBA calculations were carried out with the reaction code of Sherif where the macroscopic calculation included the full-Thomas form of the spin-orbit interaction. The microscopic calculation was carried out with the same reaction code, with the microscopic form factor of eq. (3) substituted for the macroscopic form factor. Coulomb excitation was included in both macroscopic and microscopic calculations. The nuclear structure calculationleading to the X,b and%b coefficients is described in the appendix where the equations of ref. s) are generalized to states of spin and parity other than 2+. The influence of single-particle energies on the results is discussed . The B(E2) values are given for the Ni and Zn isotopes . 4. Results The nine parameter potential used to generate the distorted waves was of the standard Wood-Saxon form, with surface absorption, plus a Coulomb term. The optical model parameters used in the present analysis are given in table 1. These parameters were determined by starting with the Beoohetti-Greenlees set s°) . and adjusting the potentials until a good fit was obtained for the elastic asymmetry data. The resulting fits for 64Zn asymmetry and cross-section data are shown in fig. 2; fits of similar quality were obtained for e°Ni. The fits obtained for -"Ni were not quite TAM 1 Optical model parameters Tarpat

va

We

rs

aí

V..,.

4.0.

$Ni 6.Ni 64Zo

54.0 54.0 54.0

7.8 9.1 9.5

1.36 1.30 1.31

0.57 0.58 6.58

5.5 6.0 5.7

0.55 0.60 0.65

ro = r. s 1.17 fm; r.,,, e 0.95 fm; so = 0.75 fm.

8P

Fig 2. Optical model fits the elastic scattering differential cross section and asymmetry data for . The ninepsrameter optical potential used was of the standard Woods-Saxon foam, with sur"7m face absorption, plus a Coulonmb term.

Fig. 3. Differential crow sections and asymmetries for inelastic scattering to the collective 2+ states of °9"40 Ní and s"Zn. The dashedlines show macroscopic fits with a deformed spin-orbit potential: solid lines give the predictions of the microscopic reaction calculation performed as described in the text .

482

M . J. THROOP

12 s

-Z 1=g 1

-8 -

iO

1*

(o

0

9

M

em

l

is/qw `UP/op as/qw `U p/DP

is/qw `U p/Dp

! _aI

~

1 Lt

%r-

a +

O

Od

as/qw

`u. PIDp

+

O

Od

q

+ .

O

Rd

as/qw ` U p/Dp i s/Qw ` u p/_°p

i M

Y _. p Ó Ttt

le

9e

t% i

. .

es .solli, 6 Zn COLLECTIVE STATES

ep

483

ep

Fig. 6. Differential cross sections and asymmetries for inelastic scattering to the collective 5- states of ss" 6 °Ni and 64Zn. The dashed lines show macroscopic fits with a deformed spin-orbit potential; solid lines give the predictions of the microscopic reaction calculations performed as described in the text .

as good ; this may be due to the presence of compound-nuclear effects which are still not negligible in the P+ "Ni system at 15 MeV. No systematic search was made for optical parameters which would give a better simultaneous fit to the elastic asymmetry and cross-section data since the parameters of table 1, chosen for a good asymmetry fit, were entirely satisfactory for the present purpose. Figs . 3-6 show asymmetry and cross-section data for the 2+ , 3 - , 4+ and 5- collective states in the nuclei sa- "Ni and 64Zn. The 2+ and 3- collective states are well known and were cleanly resolved with good statistics (except for the "Ni 3state at 4.48 MeV). The known 4+ states at 4.75 MeV in 'eNi and 3 .09 MeV in 64Zn were strongly excited with an L = 4 angular distribution. A state at 5.12 MeV in '*Ni has been shown to have spin-parity 4+ in (p, p'7) and (p, t) measurements 21). The known 5- state in 64Zn at 4.15 MeV was strongly excited. The 5.59 MeV state in "'Ni was also strongly excited and probably has spin-parity 5 - [ref. 22 )]. The 5.02 MeV state in 6*Ni was assumed to have spin-parity 5- as its angular distri-

484

M. J. THROOP

bution was similar-to those for the -"Ni and 64Zn 5 - states . The macroscopic deformed optical model predictions are indicated by, dashed lines and include a fullThomas spin-orbit term with a strength = 0.75. This value was chosen to give good agreement with the 2 + asymmetry data, although it is somewhat smaller than the values 1 .0-2.0 usually used t 9) . The microscopic predictions using the parameters and transition amplitudes described in sect . 3 are indicated by solid lines. In both calculations the peak differential cross sections were normalized to the data and the required values of ßa or Von are shown in table 2. 5. Disconion Figs. 3-6 show the experimental asymmetries and cross sections for the inelastic proton excitation of collective states in sa, 6 oIVi and 64Zn compared with the microscopic predictions using SDI wave functions. While the 2* microscopic asymmetry fits are perhaps not quite as good as the macroscopic, excellent agreement is obtained between the 3 - microscopic fits and the data ("Ni excepted). The 4' and 5 - m_croscopic asymmetry fits for 64Zn are as good as or better than the macroscopic fits. TA= 2 Deformation parameters and effective interaction strengths for collective excitations - VOP

(MêV)

1'

Mass number

E, (MeV)

PL

2'

58 60 64

1.454 1.332 0.992

0.234 0.247 0.268

44 46 48

3-

58 60 64

4475 4,040 3.002

0.141 0.186 0.218

70 67 74

4'

58 60 64

4.754 5.132 3.085

0.147 0.143 0.132

107 125 116

5-

58 60 64

5.589 5.020 4152

0.160 0.105 0.142

190 63 86

The -18,6 IN 4 + and 5 - states (as well as the "Ni 3 - ) are at higher excitation energies so that substantial background subtractions and possible contamination by nearby peaks are likely causes of the poor agreement between the data and either the microscopic or the macroscopic predictions . Generally, the microscopic predictions are poorer than the macroscopic predictions at very forward angles.

s8.69I,j1 , 64Za

COLLECTIVE STATES

485

An important test of the validity of the SDI wave functions can be made by determining the effective interaction strengths required to fit the absolute cross sections. The relative values for Vo, Vl for the p-p and p-u interactions were obtained from an analysis l') of the low-energy nucleon-nucleon data for an assumed singlet-even to triplet-even ratio of 0.6. We have chosen to renormalize each microscopic interaction. value by the same constant in order to match the observed peak cross section. The renormalized values of VOP' are tabulated in table 2 for the collective states studied here ; the free nucleonnucleon interaction for Vó is -23 MeV. Two important features of table 2 should be noted, both of which arise from the large basis space used. First, the values of the effective interactions for the 2+ excitations are only a factor of two larger than the free nucleon values. This can be contrasted with early calculations 14) for ss . 6oNi which were done in the Tamm-Dancoff approximation and with only neutron configuraticr s where values of 65-75 MeV were required for ìó°. Second, the effective interaction strength required for the three nuclei tested is constant to f 10 % for states involving a particular L-transfer (the 5 - states excepted). This implies that the nucleon structure calculation correctly predicts variations in collectivity with changing nucleon number . Also, the required strength increases with larger L-transfer, as would be expected for the present calculation where exchange effects have notbeen included . The required Tó° for the 5- excitations fluctuate widely, but the 5- amplitudes are very sensitive to small adjustments in the single-particle energies. The ss. "Ni 5- data are probably not of sufficient quality to justify any changes in the assumed single-particle energies . It should be noted that for the well established 64Zn 5- state a quite reasonable value for Vó of -86 MoV was obtained. Exchange effects were not included in the main body of microscopic reaction calculations in this paper. As pointed outearlier, this neglect was not expected to be serious for the description of inelastic scattering from the collective states (except for a renormalization of the cross section). To study this in more detail, calculations with and without knock-on exchange were done with the microscopic reaction code DWBA-70 for scattering from 2+ states . The microscopic interaction included only the central and spin-spin terms and used the same potential strengths to permit a ready comparison with our previous results. The DWBA-70 utilizes a Yukawa interaction between nucleons and a range parameter of 1.416 fm was chosen in keeping with the value suggested from fits to inelastic proton scattering data at 30 MeV [ref sa )]. The results of this calculation verified our expectation; the calculated differential cross section with direct plus exchange was roughly twice that calculated with direct only and this ratio was independent of angle to IL good approximation. This means that the values of V$° tabulated in table 2 for 2+ states should be reduced by a factor of about 1 .4 to about -32 MeV. The exchange corrected effective interaction strengths are now only about 40 % greater than the value of -23 MeV obtained from the nucleon-nucleon analysis. As shown in fig. 7 the analyzing power results are unaltered by the inclusion of exchange, except at very backward angles,

486

M. J. THROOP

where exchange improves the situation by smoothing out the A(9) fluctuations. Thus, use of SDI wave functions, calculated in a large shell model basis, accounts for the majority ofthe 2' scattering strength . The core polarization contribution (in part due to a large number of small core excitation amplitudes) can be accounted for with an encouragingly small renormalization of the effective interaction strength.

ft 7.

Comparison of the microscopic predictions of the asymmetry for the scattering of 15 MeV polarized protons to the 2+ state of s'Za . The dashed line show the prediction without exchange and the solid line shows the effect of including exchange. All calculations were done with DWRA70.

Ace) -Y

Evi---

.6

~1 f '11

t

A(e) ---10 40

M, go

v BD

N

o 0

40

e0

M

eo

Fig. 8. Asymmetries for the 2+ and 3 - states of °8NJ and "La at 30 MeV. The dashed lines show macroscopic fits with a deformed spin-orbit potential, solid lines give the predictions of the microscopic reaction calculations as described in the text.

6"-60m , 64za

couw

NE STATBs

487

In order to demonstrate that the success obtained at 15 MeV was not fortuitous, similar microscopic calculations have been made for 30 MeV cross-section and asymmetry data 1 . 24). The A(B) results for the 2+ and 3 - states of s8 Ni and 64Zn are summarized in fig. 8. The general oscillatory character of the asymmetry is predicted quite well, even though the magnitude of the 2+ A(0) is not. The spin-orbit contribution is known to increase at higher energies 19), so that poorer fits at forward angles might be expected ; this appears to be the case. As at 15 MeV, the microscopic fits to the 3- asymmetry data are better than the macroscopic particularly for the 64Zn. In the case of "Ni the 3 - A(9) data may still be plagued by resolution problems ; however the microscopic predictions do agree well with the data between 1106 and 1656. Exchange contributions are expected to rapidly decrease with incident proton energy 11, "). This is borne out by the fact that at 30 MeV a rough estimate of the effective interaction strength Vó required for normalization of Q(9) is -31 MeV in contrast with a value of -46 MeV required to fit the 15 MeV data . Since the 30 MeV effective strength is only 1 .4 times the assumed nucleon-nucleon value of -23 MeV, we interpret this as a further indication that core-polarization contributions must be small. Except for the discussion above about the effect of exchange, all of the microscopic calculations so far utilized a Gaussian spatial dependence for the effective interaction. The DWBA-70 uses a Yukawa spatial dependence, but no significant changes in either the cross sections or asymmetries were found when the microscopic calculations were performed with this program and with the relative effective forces the same as for the Gaussian calculations. The 2+ A(B) fits were still rather poor at forward angles, regardless of whether or not the microscopic spin-orbit term was included. However, a strong dependence upon the range of the interaction was found. By increasing the Yukawa potential range from the nucleon-nucleon value of 1 .06 to 1.416 fm [ref. ")], the improved fit at forward angles for the 2+ asymmetry shown in fig. 7 was obtained ; the rest of the microscopic calculations were done using a range of 1.85 fm for the Gaussian shape as suggested from the nucleon-nucleon analysis of ref. 1 °). Ofcourse, changing the range parameter also changes the absolute normalization of the differential cross sections. The good agreement of the SDI predictions for the 3- state asymmetry and cross section with the observed data is particularly pleasing . In contrast, the macroscopic predictions are not as good . The present SDI nuclear structure calculations utilized neutrons and protons in the 2p-If subshell and the g} subshell. Because of the negative parity for the 3- state, in the microscopic reaction description only single quasiparticle transitions between the odd parity 2p-lfstates and the even parity gt. subshell can take place. In the wave function language, this means that the number of amplitudes X,6 andX,r describing the negative parity states is small. In the present case, the lowest 3 - state arising from the nuclear structure calculation has only four important components, including both neutron and proton excitations. Thus, the agreement obtained here with the microscopic calculations confirm that these components are

488

M. J. THROOP

the dominant ones in the description of this moderately collective octupole state. Throughout this paper we have stressed that the most important feature of the SDI wave functions is that they permit a large basis set to be used in the microscopic calculations . As summarized in the appendix, this results in both the proper collectivity for the 2' -" 0* transition strength along with a reasonable behavior with shell filling. In view of the success obtained for the 3- collective states where only a few amplitudes contribute, it is logical to ask how well one can do for the other states if only a few dominant amplitudes are included . This hypothesis was tested for the 2* states and the answer is that the asymmetry results are not changed appreciably if only two to four dominant amplitudes are included rather than the full 36 component SDI wave function. Naturally, one loses the collective enhancement in the cross sections obtained with the complete wave function as well as the reasonable behavior with occupation probability. Thus, for the collective states where all the amplitudes are inphase theasymmetries depend mainly upon radial wave function and the optical parameters . This last statement will not be true when calculations are made for noncollective states where the various amplitudes are no longer in phase. Compound nuclearcontributions are estimated to be small for most of the collective state data . This is suggested primarily by the good quality of the macroscopic fits which assume a direct process. Further confirmation is found in the agreement between the asymmetries measured in the present 15 MeV experiment for the 2* and 3 states of -"'Ni and the earlier measurements at 18.6 MeV [ref. ")]. The consistency of the SDI microscopic fits for the 15 and 30 MeV data strengthens this belief. The fits at 15 MeV are perhaps even better than those at the higher energy, even though any compound nuclear contributions are expected to be larger at the lowerproton energy. Recent asymmetry measurements 27) on s8Ni at the University of Washington have evidenced strong fluctuations when the incident proton energy was varied by only 60 keV near Ep - 15 MeV t. However, we still assert that the 15 MoV data is mostly direct based upon the above evidence and upon the similarity for the cross section and asymmetry behavior for all three nuclei, ss, 'ONi and s'Zn at 15 MeV. A strong compound nuclear component would most likely cause quite different results for the three nuclei . The authors would like to express their thanks to the operations personnel at the tandem accelerator for their considerable assistance, particularly Dr. R. Hardekopf for his assistance with. the operations of the polarized-ion source. Prof. V. Madsen of Oregon State University made several useful suggestions during the course of this investigation. We would also like to thank Prof. S. M. Austin of Michigan State University for providing us with a copy of the DWBA-70 computer code. Finally, t B. Guengo has made careful analyzing power measurements on °"Ni with the University of Washington tandem accelerator from 14.7 to 15.3 . MeV. These data show considerable fluctuations in the analyzing power data, presumably due to compound nuclear effects.

° 8 " 6 °NI, 64Za COLLECTIVE STATES

489

we greatfully acknowledge the interest and useful suggestions of Prof. J. B. Blair of the University of Washington . Appeaf The nuclear structure coefficients X,b and 8,b are the coefficients of the RPA quadrupole phonon operator aim = Y, fX,*4M+(-)L+'-Mgj&Aj-M} .

(A.1)

The pair creation operator is defined as the angular momentum two-quasiparticle combination, The structure coefficients are then obtained from the solution of the RPA secular equation, X,b X, b A

N11+6.& 8*

B

(-B -A) where

a 03

vi +S,b

1+S,b

(A.2)

1+8, b

(X2 ,b -g3b) 9 41ib

1

It is in the calculation of A and B that the detailed nuclear structure calculation entets. Inasmuch as the equations of ref. a) are strictly valid only for 2+ states the relevant equations for arbitrary spin-parity are summarized here. The surface-delta interaction generalized to include both neutron and proton excitations, can be written as °), V = V,+ V,

a

-4n

.1-61 (x

*

Q=

+Xt

3+471 -&2 )

x

S(rl -R)S(rzrlr2

R)S(eis)'

(A.3)

In addition to representing well the pairing properties of the effectivenucleon-nucleon interaction, the surface-delta interaction simplifies the radial integrals to a constant J at the nuclear surface. In terms of this, X. is then related to the pairing strength Go and atomic number A by, (A.4) X.f - Go/A. For Ni and Za isotopes Go = 24 MeV. Tlie triplet X, can be treated as an adjustable parameter; in the present investigation it was chosen so as to closely fit the first excited 2+ state energy. The neutrons and protons were assumed to fill five subshells beyond a closed *°Ca core, namely the ft, ft, pf, p} and gt shells .

490

M. J. THROOP

In order to express the matrix elements A and B, it is first necessary to evaluate the antisymmetrized particle-particle and particle-hole matrix elements . For like particles, <(nalasaja)(nblbsbJb) IV.I(n .lrS.JJ(nildSrjd)J>.

<(n.l s. .1.) -1(nblbSbJb)JIV.I(n.lcSaiJ -1(nelrsdJOJ> = 2X.[(-)~°

where

M = ("I) 4.i2

N

f(

+" -JM-

(A.5)

j'V] ,

(A.6)

-)r .+'`C~-.j- C}i

=(-L 41 2

~C

jj ; j Ja*J

J=

2ji +1 .

(M)

For unlike particles the only matrix element needed is <(n.l.s .1.). 1 (nblbSbJb).JI VI (n.IS,ja); 1 (njjrsdja),J>

= [(Xs+2(-)~ +l` -JX,)M+X.M](-)b+4-j+(Xe-X.)N.

(A.8)

The matrix elements for A and Bare then obtained in terms of the above by expressing the quasiparticles in terms of the usual Bogolyubov transformation giving, - ~E..+EJ8., bd+X,(1+S,b)-}(1+Sa)-} +ì,-JM x (1 +(- y° +4-~M] + RV-Iva + + v;;v ;x(- ) °

-N)]+ [(v1vd-v;;v ;a)( -)'° +i.(M+N)]},

x(-Y,+10(1+(-Ya+1° J)(1 +( -)~+ ,d - ] +[(vâvâ +v;;v;,x-)~+4(M+N)] +[(vlvd-v;;vd) x ((-)~° +" -JM-N)]},

x [(Xe+2(-)i°+trJX,+X.Ì(-)~°+4-JM+(X,-XJN] + [~(ví+bvo+e- fl.bfld)][(Xt+2(-~°+~-JXe+X.)(-~°+(`M

(A.10)

58-6ONi, "Zo COLLECTIVE STATES

61

- (1+3.b)-*(I+Sd)-*{[i(viv,+d-v;v ;)] (A.12)

x

+ [«PIP.+d + v;;v;ffl&,+2(-t+ 4-Jxt + LX-)~+§'M . «z,-LX-)~*+"N]}.

where u', u - , v" and v - are defined the same the way as in ref. 11). Table 3 shows the values of X.b and X.,, for "*Zn 2' state at 0.992 MeV. The X are, of course, absent in the Tamm-Dancoff approximation. The importance of the X is greatly dilminisbed for higher spin states.

T~ 3 Nuclear wave f~on amplitudes 1)for tbc 15'Z& 2` atate calcuIatedwithazpin-depe~ surfitcí>~Jnter~ Il, I: 2> il, #; 2> il. 1; 2> li, 1; 2> li, 1; 2> i#, ì; 2> li, 1; 2> li, ì; 2> .11, 1; 2> Z,

x@

x. Zp

0.0866 0.0992 0.0324 0.0290

0.0963 0.1073 0.0527 0.0838

-0.3006 -0.4202 -0.1576 -0.2236

0.4220 0.1879 0.1677 0.0729

0.3009 0.2265 0.1191 0.1021

-0.3162 -0.2025 -0.1484 -0.0775

0.3268 0.4520 0.1294 0.1564

0.2497 0.2796 0.1229 0.1315

0.0530 0.0138 0.0854 0.0492

11) The basis state is denoted as 1J.J& ; J> whom a, b are quasiparticles. TAinz 4 The B(E2t) values (in el - W) for doublya

Isotopes for Ni and Za

Isotope

B(E2t)zm')

B(E2t)n ')

B(Mt).,

SeNi "Ni 62 Ni 6'Ni 64z' 6'Zn 6'Zn

0.101 0.100 0.106 0.084 0.137 0.125 0.124

0.017 0.051 0.100 0.092 0.264 0.245 0.164

0.072 0.091 0.083 0.087 0.170 0.145 0.125

'I) Effective charges of e. ~ 1.4 and e. = 0.4 were used for the proton and neutron respectively. 'I) Ref. 1).

The surface-delta QRPA calculations of ref. ') used single-particle energies taken from Kisslinger and Sorenson 3). These were satisfactory for the 2' collective state calculations because the 2' predictions are not critically dependent on the singleparticle energies. On the other hand, the surface-delta QRPA results for 3- states are sensitive to neutron and proton single-particle energies for the gi orbital. The g4 proton single-particle energy was adjusted to get reasonable . 3- strengths in inelastic

proton scattering . In terms of the notation of ref. '), A4 = 0.55 (Z-28) MeV was added to the proton single-particle energy. The B(E2) calculations reported in ref. ') were wrong due to numerical errors . These have been recalculated for the present choice of single-particle energies and are summarized in table 4, where the results of ref. 1), are included for comparison. It can be seen that the B(E2) results are in reasonable accord with experiment, ifneutron andproton effective charges of0.4 and 1.4 respectively are used. These numbers are in excellent agreement with the renormalization factor 1.4 used in the microscopic exchange calculations for the 2' collective states and represent a substantial improvement over the values from ref 1), where effective charges of 1.0 and 2.0 respectively were use& Rderemmm

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