- Email: [email protected]
Shell model in the continuum analysis of analogue resonances in 89Y(p, p′)89Y
1.E.7:2.L[
Nuclear Physics A210 (1973) 1--18, (~) North-Holland Pubhshmg Co, Amsterdam Not to be reproduced by photoprmt or mlcroffim without written permlssmn from the pubhsher
SHELL MODEL
OF
IN THE CONTINUUM
ANALOGUE
RESONANCES
IN
ANALYSIS
89y(p, p,)89y,
B J. DALTON Rt 1, Box 266, Ltzella, Georgia 31052 t and D ROBSON ~t Department of Phystcs, Flortda State Unwerslty, Tallahassee, Florida, 32306 t Received 22 January 1973 Abstract. Excitation functtons for the reactmns 8 9 y ( p , p , ) a 9 y * leading to the second and third excited states of agy are calculated The levels of 90y are calculated using a sample configuration-mixing shell model and converted into analogue resonances using a many-level multi-channel R-matrix approach The non-analogue component of the inelastic scattering is descrlbed via Hauser-Feshbach theory and the effects of rmxmg of lsospm on ttus compound contribution are mvestlgated Comparison with the previously published experimental results ymlds tentative spin and parity assignments for the analogue resonances in 9°Zr
1. Introduction The i s o b a r i c span a n a l o g u e states c o r r e s p o n d i n g to the b o u n d excited states o f the nucleus 9 0 y have been observed b y L o n g a n d F o x 1) t h r o u g h the inelastic scattering r e a c t i o n 8 9 y ( p , p , ) 8 9 y , T h e y observed the excitation functions (at 70 °) for the cases m which the residual nucleus 8 9 y , was left in the ~+, 3 - , ~ - , a n d ~+ excited states respectively This r e a c t i o n is Illustrated m fig 1 The f o u r excitation functions o b served consisted o f a n u m b e r o f resonances s u p e r i m p o s e d u p o n a r a t h e r s m o o t h b a c k g r o u n d w h i c h increased slowly with energy The m a j o r i t y o f the resonance structure a p p e a r e d in the ~ - a n d ~ - residual nucleus cases The existence o f the a n a l o g u e r e s o n a n c e features in the inelastic channels suggests a p p r e c i a b l e configuration r m x m g an the excited states o f 9 o y I n their analysis, L o n g a n d F o x 1) have suggested t h a t these resonance features could be explained b y the w e a k - c o u p l i n g m o d e l 2) in w h i c h the single-particle orbltals are c o u p l e d to the excited states o f 8 9 y as a core This coupling via the residual i n t e r a c t m n ~s a s s u m e d to be w e a k so t h a t the core states o f 8 9 y are n o t s~gnlficantly altered I n this p a p e r , we describe calculations o f the excitation functions for the r e a c t i o n 8 9 y ( p , p , ) 8 9 y , for the ~ - a n d 3 - cases These calculations involved the c o m b i n e d t Research supported m part by the National Scmnce Foundatmn grants NSF-GP-15855 and NSF-GJ-367 tt Alfred P Sloan Research Fellow 1 August 1973
2
B J DALTON A N D D ROBSON
use of a configuratmn-mlxmg shell model calculation together with the reduced Rmatrix 3) and generalized Hauser-Feshbach 4) scattering formulations, and were originally intended for comparison with the experimental results of Long and Fox Our objectwe m doing these calculatmns was to see ff some of the resonance features such as positions and widths, could indeed be described by using the weak-couphng model as suggested by these authors 89
89
y (p, p') y*
89
89
59Y50
39Y50 9o 40Z r50
F~g 1 Schematic diagram of the inelastic scattering channels for 89y(p, p , ) 8 9 y , indicating those which are relevant to this work
In our model description, we first describe the low-energy negative-panty states of 90y by using a configuration-mixing shell model calculation (described in sect 2) For these calculations, a hmlted partMe-hole basis is used The single-pamcle neutron orbltals used are hmlted to the set {2dg_, 3s~, 2d~_, lg~} We assumed that the full g~ neutron shell is part of the inert core for the energy range considered (up to about 3 MeV in 9oy) This assumption is in agreement with the 91Zr(p, d)9°Zr reaction studied by Ball and Fulmer 5) m whach the l = 4 pick-up contribution did not appear until an excitation energy of about 4 2 MeV m the residual nucleus The hole states for the 89y core are hmlted to the proton orbltals p~ 1 p~ 1, and f~ 1 corresponding to the ground and first two excited states of 89y respectively The ground state configuration of 8 9 y is assumed to be pure p~ although Courtney and Fortune 6) suggest that a possible 10 ~ mixture of the [(2p~)-Z(2p~)-l(lg~) 2] configuration is needed to explain the lifetime of the first excited 0 + state of 9 ° Z r As we discuss an sect 2, one possible effect of ignoring this 10 ~ ground state mixing in 8 9 y is the discrepancy between the calculated and experimental values for 7 + - 2 (gxound state) energy splitting in 90y Using a pure p~ configuration for s 9 y and the measured value of 0 897 MeV (see fig 1) for the g~ smgle-partMe energy of 8 9 y , w e
sgy(p, p,)agy* ANALOGUE RESONANCES
3
obtained a value for this sphttmg which as from 200 to 300 keV too large for several sets of residual interaction parameters This same splitting discrepancy was obtained in zero-range calculations by Adams s) In his calculations, the g~ single-particle energy had to be changed from the "experimental" value of 0 897 MeV to a value of 0 52 MeV in order to fit the experimental separation For the single-particle basis states, we used the Coulomb plus Woods-Saxon potentials with a derivative-type spin-orbit term The well depth for each orbit was adjusted to give ihe specified binding energies, which we used as parameters in fitting the known energy levels of 9Oy [refs 7, 9)] For the residual Interaction between neutron and protons, we used a finite-range Gausslan form with Wlgner and Bartlett terms The rank-one tensor strength (1 e , the coefficient of the ap trn term) was determined by fitting the known doublets of 9Oy [refs 7, 9)] The set of parameters used for the negative-parity states was in turn used to calculate the positive-panty eigenvalues of 9Oy These states were described in these calculations as mixed configurations of the g~ first excited proton state of s 9 y coupled with the neutron orbltals specified above The energy ordering of the first six o f these positive-parity states agreed with the ordering obtained in the calculations of Pandya 10), and with the recent s 9y(d, p~)90y experimental results of Eros et al ~~) In taking the zero-range limit, the calculated order of the first 3 +, 5 + levels inverted, agreeing with the ordering obtained in the zero-range calculations of Adams 8) In the scattering part of this calculation, we separated the reaction amphtude into the energy averaged and fluctuation parts The cross section corresponding to the fluctuation amphtude was described by using the generalized Hauser-Feshbach 4) method The analogue resonances are included in the energy averaged amplitude In describing the resonance part of the energy averaged amplitude, we used the reduced R-matrix formulation of Teichmann and Wlgner 3) In this reduced R-matrix formulation, the scattering contributions for channels other than the elastic and melastic p-C and n-A channels [in the notation of Robson 12)] are reduced out This reduction is convenient in that it allows the eliminated channels to be described separately from the retained p-C and n-A channels The reduced widths for the analogue (T>) contribution are obtained from the configuration mixing amphtudes and surface value of the wave functions used in the shell model calculations of 90y In this way, the calculated resonances for the reaction 89y(p, p,)89y ~ are constructed to correspond to the isobaric spln analogues of the excited shell model states of 9oy In sect 5, the calculated excitation functions (at 70 °) for the ~ - and 3 - cases are shown for proton energies between 6 3 and 8 2 MeV Through comparisons with the Long and Fox work and with the recent work of Spencer et al ~3) we are able to suggest total angular momentum assignments for several resonances in the region between 6 7 and 7 5 MeV Above this region, the comparison with experiment is poor. The number of resonance peaks at these higher energtes m the recent experimental data far exceeds the
4
B J DALTON AND D ROBSON
number that can be possably obtained from the hmated configuration basis used here This is strong evidence that other configuration maxtures are apprecmble above 7 5 MeV The overall calculated resonance peak heaghts were lower than those seen in the experiments, lndacatmg a need of stronger configuration mixing of all levels with the p~ hole confgurations
2. Configuration-mixing shell model calculations for 9Oy In the first part of this sechon, we briefly outline the essentml aspects of the shell model formulation used m these calculations In the second part, we discuss the parameters used in descrablng the excited states of 9Oy The physical wave functaon used to describe the negatwe-panty states of 9oy as expanded as a hnear combination of pamcle-hole configurataons, 1 e, ~b(E~, J - ) = Z Ac(E~J-)ICJ-> ,
(2 1)
C
where each pamcle-hole configuratmn IC J - > has the form (m j-j coupling)
ICJ-> = I{A.u¢zJ"+~q52Jb} ® J2, J - M }
(2 2)
The subscripts a and b refer to the proton hole shells connected by the protonneutron residual anteractlon For thas calculatmn, these shells are hmlted to the set {2p~ 1, 2p~ 1 lf~ 1} The antasymmetrlzataon ln&cated by AN m eq (2 2) is thus taken over all of the N = 2A + 1 + 2Jb pamcles in the two andacated shells The function [Jzm2 ) amphcat m eq (2 2) refers to the neutron orbltals whach we llmat to the set {2d_~, 2d~, 3s~, lg_~} The configuraUons appearing m eq (2 2) are determined by those pamcle-hole states whach can be coupled to a given value of the total angular momentum J The physical sagmficance of using the above hmlted configurataon basas as discussed an sect 5 in conjunctaon wath the experamental results of the reactaon 89y(p, p,)S 9 y , In calculatmg the single-particle orbatals, we used the Coulomb and Woods-Saxon potential (including a derivative spin-orbit term) to represent the anteractlon of the resadual nucleons wath the Ferma sea For the resadual interaction, we used the fimterange form
V~2 = 4~ Utz(rtz)[A + BPr], V~2 = exp
(-~lr, zlZ),
Pr
= ¼(l--a," a2)
(2.3)
Introducing the resadual interactaon into the Hamdtoman and usang the expansion (2 1), we obtain an the usual manner the secular equation N
Z Ac[(ec - E,)3c,c +] = 0 C
t=l
(2 4)
89y(p, p,)89y* ANALOGUE RESONANCES
5
In tl~s equation, the energy ec is given by the sum ec = ec(1) + ec(2)
(2 5)
Here, ec (2) represents the neutron single-particle energy for a given orbital and ec (1) represents the single-proton-hole energy relative to the reference state, 1 e , ec(1 ) is just the energy needed to separate a proton from the reference state In order to obtain the amplitudes Ac, the matrix elements in eq (2 4) were evaluated using the standard multipole expansion technique ~4, 15) The Slater integrals involved in these formulas were calculated numerically using Woods-Saxon wave functions with the Gausslan residual interaction form of eq (2 3) It is important to point out here that these wave functions were normalized to unity only over the range r = 0 to the matching radius R an accordance with the R-matrix resonance analysis discussed an sect 3 Using the calculated matrix elements and energies ~c, the resulting elgenvalues E~ and amphtudes A c (E~, J - ) were found by numerically solving the secular equation (2 4) For each total angular momentum value J and elgenvalue E~, the sum of the squares of the amphtudes was normalized to unity As we discuss m sect. 3, the reduced widths for the resonance reactions are obtained from these amplitudes and values of the single-particle wave functions at the R-matrix matching radius In our calculations for the posture-parity states of 9Oy, the physical wave function was expanded in the same manner as in eq (2 1), but with the proton hole states {p~ 1, p~ ~, f~ ~} replaced by the g~ proton particle state The corresponding positiveparity amplitudes Ac(E~, J+) were obtained by following a procedure Identical to that outlined above fol the negative-parity case The negative- and posmve-parity calculations here are thus made with respect to two different reference states, namely the full and empty 2p~ shell The energy difference AEo between these two reference states is essentially the energy of the two 2p~ protons relative to the 8SSr core A determination of this energy difference is beneficial in estabhshing the positive-parity energy levels relative to the negative-parity ones Instead of directly calculating AEo, however, we attempted to establish the relative positions of the J - and J + energy spectra by directly calculating the well-known 2 - ( g r o u n d state)-7 + energy sphttlng m 9 o y In making this splitting calculation we assumed that the 2 - ground state of 9Oy was a pure p~ d~ configuration and that the 7 + state was a pure g~d~ configuration By treating the 2p~ state as a particle state (rather than a hole state as outlined above) and using for the g~ single-particle energy the experimental value of 0 897 MeV, we obtained an energy splitting which was from 0 2 to 0 3 MeV too large for several sets of residual interaction parameters This disagreement would seem to indicate that either the lg~ excited state of s g y is more involved than a smgle-pamcle state outside an inert aSSr core, or that contiguratlon mixing is playing a stronger role in the 7 + and 2 - states, or both As the calculations below show, these two states are apparently not composed of pure configurations
6
B J DALTON AND D ROBSON
In calculating the single-particle wave functions for 9oy, we used a fixed WoodsSaxon radius of R = 5 5 fm and a diffuseness of a = 0 5 fm The well depth U was adjusted m the computer program to gwe the specified binding energy for each partial wave These single-particle binding energies (see table 1) were used as parameters m fitting the known energy levels of 9 0 y [refs 7, 9)] TABLE 1
Binding energies used for single proton and neutron orb]tals m negative-panty calculations 2p{ 87
2p~. 10 35
l f{ 10 536
2d{ 7 20
3s{ 5 85
2d{ 4 75
1g{ 4 45
F o r the proton hole energies, the calculations were first made using the "experimental" values of - 1 499 MeV and - 1.736 MeV for the p~ and f÷ hole states, respectively However, a better fit to the 89y(p, p,)S9y reaction data was obtained by using values of - 1 65 MeV and - 1 836 MeV respectively m place of the two experimental values i
9Oy (j_)
i
l
40
CO =-12
o
35
2 I
--3
3.0
2.5
~
3
--4
--2
4
r j
t
l
4
{2o Ld
~
.
~ o
I --I -0
1.0 --7+
05 ~
0 0
--z EX'P
-Oil
3
C~-*"
a -013
F]g 2 Calculated positive-panty energy levels for 9oy versus C1 for Co = --1 2 and 7 = 0 3 shown wlth the first slx experimental levels of LlnS et al tl) and the calculated results of Pandya
89y(p,p,)sgy* ANALOGUE RESONANCES
7
A first estimate o f the neutron single-particle energies was obtained by fitting the centrold o f the well-known doublets formed here from the p~ 1 hole with the four n e u t r o n orbltals specified above The centrold sphttmg was made without including the pamcle-hole configurations corresponding to the f~ 1 and p~ ~ hole states The sphttmgs o f the doublet levels were used as a means to determine a first estimate o f the rank-one tensor strength m the residual interaction In calculating the J + and J energy spectra for 9 o y , we adjusted the rank-zero ( C o ) and rank one (C1) tensor strengths separately These strengths are gwen m terms o f A and B in eq (2 3) by Co = A +¼B,
C 1 = --¼B
F o r our choice o f wave funcUons, we used the values o f Co = - 1 2, a range parameter o f ~ -- 0 3 and plotted m figs 2 and 3 the J ÷ and J - energy levels versus C~ f r o m C~ = - 0 1 to C 1 = - 0 3 This range o f C1 m these figures corresponds to a range for the spin-dependent to spin-independent strength ratio e = CI/Co o f e = 0 083 to c¢ = 0 25 I n a h g m n g the calculated resonance p o s m o n s for the reactmn S9y(p, p,) 8 9 y , with the experimentally determined resonance p o s m o n s we used a value o f C~ = - 0 25 This value corresponds to a value o f ~ = 0 21 which is shghtly larger
4.0 COy(j+) 55
CO =-I 2
30
T LI.I
2
f
25
:3
2O
1.5
1.0
0.5
- - 4 --;5
.
--2
--2
0.0
3
~ 7
PANDYA
EXP
2
I
-0 1 C I'->
17
-O.3
Fig 3 Calculated negative-panty energy levels for 9Oy versus Ct for Co = --1 2 and 7 = 0 3 Also shown are the known experimental doublets 7.9)
23O1401213234212431230-
0 000 0 201 1 144 1 318 1 562 1 580 1 808 1821 1 876 2 039 2086 2 192 2 315 2 551 2 724 2 945 3 005 3 154 3 324 3 476 3633 3 855
0 007 -0019
- 0 006
- - 0 052
0 202 0 117
0 976 0 950 - 0 079
- 0 017 -0011 - 0 029
0 020
0100
--0153
- - 0 80
- - 0 007
p~-~d~
0 025 0119 - - 0 164
0 986 0 999
p½-~d_~
Cl = --1 2, Co = - - 0 2 5 , 7 = 0 30
J-
E (MeV)
TABLE 2
0 098
- 0 126
0 056
- - 0 168
- - 0 484 0 399
- - 0 870 0 889
p½-~s~
0031
0 988 - 0 993
0 002 - 0 064
0 000
0 006
- - 0 005
p ~ - ~g~_
- - 0 478 0965 0 785 0 992 --0130 - - 0 118 0 045 - - 0 116 0 175 - - 0 132 - 0 005 - 0 001 0 012 0 010 -0015
0 352 0 999
0 143 0 012
p~-~d÷
032 035 118 024
- 0 022 - 0 116 - 0 038 -0019 - 0 993
0 052 0 156 0 037
0 012 - - 0 012 0011 - - 0 044 - - 0 033 0021 0 009
0 --0 --0 0
p_4- ~d~
- 0 964 - 0 005
0 057 0 173 0 984
0039
- - 0 076 0136 - - 0 015
- - 0 124
0 071
p~-~s~
- 0 065 0006
- - 0 005 0 087 - 0 104
- - 0 001 0029 0 004 - - 0 013 0 024
0010
0 003
0 004 0 001
p..k- ~g~
0 027 0 029 - - 0 479 0 246 0 043 0 875 - - 0 776 0105 - - 0 571 0 115 0979 0 986 - - 0 995 - - 0 125 - - 0 068 - - 0 047 - 0 070 - 0 001 0 030 0 104 -0119 0 068
f ~ - ~d~
Calculated energy levels a n d m i x i n g a m p h t u d e s for n e g a t i v e - p a n t y states o f 9 o y
- - 0 060 0018 - 0 029 0 005 0001 0 012 - - 0 053 0 044 - - 0 004 0 014 0 085 - 0 009 0 008 - 0 169 0046
0 047 0 006
0 005 0 004
f ~ - ~d~
- 0 970 0991
0 031
- 0 025
0 091
0 028 0112 0 115
0030
0 008 0 022
f~-~s~
- - 0 011 0012 - - 0 028 - - 0 002 0012 - - 0 003 - - 0 010 0 042 0 058 0 006 0 081 - 0 038 - 0 017 - 0 022 0003
0 010 0 008
0 007 0 001
f~-~g~
~Z
*'] O ~Z
OO
89y(p, p,)e9y* A N A L O G U E RESONANCES
t h a n t h e v a l u e o f ~ = 0 16 a s s u m e d b y P a n d y a to) T h e p o s l t w e - p a r l t y e x p e r m a e n t a l levels s h o w n in fig 2 w e r e o b t a i n e d f r o m t h e 8 9 y ( d , p 7 ) 9 0 y r e a c t m n o f Lxns
et al 11)
T h e o r d e r i n g o f t h e first six J + levels is in a g r e e m e n t w i t h t h e e x p e r i m e n t , a l t h o u g h the c a l c u l a t e d level p o s m o n s are m g e n e r a l t o o l o w TABLE 3 Calculated energy levels and mixing aroplltudes for positive-parity states of 9o y
E (MeV)
J
g_~dff
0 000 0 089 0 234 0 293 0 398 0 459 1 598 1 688 2 098 2 451 2 606 2 674 2 792 2 891 2 894 3 015 3 083 3 081 3 157
7 2 3 5 4 6 5 4 1 3 2 6 4 5 3 4 5 6 7
0 999 0 999 0 995 0 984 0 997 --0 999 --0 178 --0 077
C=
--0 101 --0 043 0 027 0 006 0 020 0 021 --0 008 --0 002 --0 000 --0 017
g~_s~
0 175 0 077 0 979 0 988
g~d~
0 103 0 037 0 002 --0 026 0 099 0 130 0 964
--0 134 0 106 0 013 0 006
--0 0 --0 --0 --0 --0 --0
972 928 963 246 348 249 233
g~_g~ 0 017 0 043 0 005 0 011 0 008 --0 007 0 019 0 034 1 000 0 247 0 999 --0 233 0 347 --0 248 0 969 0 937 0 969 0 972 0 999
--1 2, D = --022, ;~ = 0 3 0
T h e c a l c u l a t e d m i x i n g a m p h t u d e s c o r r e s p o n d i n g to t h e J -
and J + energy spectra
o f figs 2 a n d 3 are s h o w n m tables 2 a n d 3 r e s p e c t i v e l y T h e s e a n a p h t u d e s are disc u s s e d in sect 5 in c o n j u n c t i o n w i t h t h e c a l c u l a t e d r e s o n a n c e f e a t u r e s for t h e inelastic s c a t t e r i n g m e n t i o n e d a b o v e F i n a l l y it s h o u l d be p o i n t e d o u t t h a t t h e r e s o n a n c e energies o b s e r v e d in the i n e l a s t i c e x c i t a t i o n f u n c h o n o f t h e 8 9 y ( p , p,)S 9 y ~ d a t a n e e d n o t a h g n in a o n e - t o - o n e w a y w i t h t h e e~genvalues o f t h e s t r u c t u r e c a l c u l a t i o n s d e s c r i b e d a b o v e T h e level shlfts a r i s i n g in t h e R - m a m x f o r m u l a t i o n c a n be as large as 100 k e V
3. Energy averaged amphtude I n this s e c t i o n , we d e s c r i b e the s c a t t e r i n g a m p l i t u d e u s i n g t h e r e d u c e J R - m a t r i x f o r m u l a t i o n o f T e l c h m a n n a n d W l g n e r 3) F o l l o w i n g p r e v i o u s w o r k 12), we first describe t h e i s o b a r i c spin f o r m o f t h e R - m a t r i x a n d t h e n r e l a t e It to the n e u t r o n - p r o t o n form of the R-matrix W e use h e r e t h e n o t a t i o n o f R o b s o n 12) m w h i c h the t a r g e t C has i s o b a r i c spin
10
B J DALTON AND D ROBSON
designated by To The isobaric analogue of C is designated by A, and the isobaric analogue of the ( n + C ) system is designated by IT>) where T> = To+½ The state I T < ) with T< = T o - ½ has the same isobaric spin as the ground state of the nucleus (p+C) F o r this calculation, we assumed no internal mixing between the 7"> and T< states. The isobaric spin basis functions (which &agonahze the nuclear potential) are defined by IT>) = (1/a)[pC) + (b/a)[nA), IT<) = (b/a)[pC)- (1/a)[nA),
(3 1)
where a = ( 2 T o + 1)~ and b = (2T0) ~ The relations In (3 1) are used tz) to express the R - m a m x in the neutron-proton basis m terms of the R - m a m x in the isobaric spin basis With the assumption of no Internal mixing between the 7"> and T< components, the R-matrix in the isobaric span basis may be written in the form
0)
Re
(3 2)
Here, R "> (R e) represents the scattering among the 7"> (T<) channels F o r a target nucleus C, the reactions C(p, p')C* and C(p, n')A* can proceed via both the R "> and 114 parts, as can be seen by inverting the relations in eq (3 1) However, reactions involving the channels n + B, where T B = T o - 1, e g the reaction 89y(p, n)89Zr, proceed only through R e The analysis here is specific to those reactions for which the entrance channel is of the p + C type It is possible through standard methods a) to ehrmnate all nonentrance channels from the explicit R-matrix However, we retain here all elastic and inelastic channels of type p + C and n + A but ehmmate all other T< channels in R e, 1 e
oi
=
:1
,
Rear RJe]
A=
(,o Rr
C ~ (0, l~er), e
O = Reee
(3 3)
Here e stands for the ehmmated channels (1 e , the n + B type) and r stands for the retained channels in the T< set With the usual time reversal and causality assumption, one has following Telchmann and Wlgner C = B x where T -= matrix transpose, and the reduced R-matrix takes the form
89y(p, p,)S9y* ANALOGUE RESONANCES
11
where (3 5)
~ < = Rrr+R~eLe( 1-R
The second term of eq (3 5) has the same dimension as the first term and represents the effect of the ehmmated exit channels on the scattering If we have N angular m o mentum channels for each J~ value in the proton scattering, there will be N channels o f the n-A type retained an the R~ part By definition, the matrix ~ r satisfies the equation (U:)
= (R;>
O<)(D~),
(36)
where U > (U <) are column vectors containing only the retained T,. (T<) channel wave functmns g~> (X~< ) respectively The elements In the column vectors D > and D < are gwen by
D~>
= ac
> .... 8Z;~r
--Bczc, >
D~<
< = a c ~~Z< - r ~ ,, - B c Zc, =
(3 7)
e
where we have set B2 = B~ = B c for convenience Following previous work x2), we set the channel constants B c equal to the log-derivative of the single-neutron wave function m the n + C system In our calculations, these constants are just the logderivatives of the Woods-Saxon neutlon wave functions used m sect 2 to describe the nucleus 9Oy Following the earher work of Robson i2) for the single-channel case, we use the relations m (3 1) to express the corresponding many-channel reduced R-matrix in the neutron-proton basis in terms of R "> and ~ < The resulting reduced R-matrix for the particle basis is gwen by
= e, + eo
(3 8)
where
~/'
1 ( R'> = ~
bR ~>
bR~>'~ b2R ">]'
Of°= 1 ( ba~< a~
-b~
<
-b~<) ~<
(39) ,
and as before, a = ( 2 T o + 1)~ and b = (2To) ~ The matrices ~ ' and ~ o each have dimension 2N In eq (3 8) the 7"> and T< R - m a m x contributions have been completely separated This separation is convenient since the two are calculated m a different way In describing an experiment with finite resolution, one can separate the scattering matrix into two parts, one describing the energy averaged part ( U ) and the other describing the fluctuations about the average (UF), 1 e U=
(U)+U ~
(3 10)
In sect 4 we describe the fluctuation part U v by the generalized Hauser-Feshbach 4) statistical method
t2
B J DALTON AND D ROBSON
The enelgy averaged part ( U ) separates into two parts, correspondmg to the separation m (3 8), i e (u)
= u°+u
(311)
'
Here, U o describes the average scattering contrlbutlon from G ° [the so called "farlevel" contribution 16)] and U ' descr%es the exphclt average contribution of ~ ' modified by terms involving the contribution G ° as explained by Lane and Thomas [ref 16)] In terms of our calculation, for the reaction S9y(p,p,)S9y,~ the matrix U ° describes the average ~sobanc analogue T> scattermg The effects of the T< states on the T> scattering are mcluded in U ' through the reverse of terms hke ( 1 - R ° L ) which arise m the separation of U into U o and U ' In the explicit description of the T> scattering, the matrix ~ ' is expanded m the usual way in terms of the level m a m x and reduced widths The reduced width amphtudes for the ~ ' > expansion of (3 9) are given by
7~'>c= (hZ/21~ac)-~AxcUac,
(3 12)
where the coefficients A~c are just the shell model configuration mixing amplitudes Ac(E~, J+) of sect 2 at the elgenenergy E~ = E z for channel e and uz~ are the appropriate neutron stogie-particle wave functions for the corresponding 9Oy states evaluated at r = a~ The T< contribution to the scattermg (m the energy average) is described by the optical model The optical model used corresponds to the T< component of a Lane potential U o + Ult To The energy averaged off-diagonal contribution is not explicitly described That is, we replace ~ in eq (3 9) by R °pt, i e Go The diagonal elements of
- b R °pt]
1 ( bZR°pt = p \--bg°P t
R °pt a r e
(3 12)
Ropt ]
by definmon given by 1
R°~' --
,
(3 13)
f~-B~ wherefc is the log derivative of the optical model wave functmn evaluated at the channel radH ac 4. Cross section
We describe the analogue resonance features in the S9y(p, p,)Sgy~, reaction via the standard reduced R-matrix formulation discussed in sect 3 The smooth background in the melastlc scattering is described via the generahzed Hauser-Feshbach 4) methcd For this inelastic scattermg case where both mcomang and outgoing proJectile spins ale ½, we use the cross-section formula da~ df2~,
_
_
1 2k~(2I + 1)
V
HL(C(, c0PL(cos 0~,)
(4.1)
sgy(p, p,)sgy, ANALOGUE RESONANCES
t3
Here, 0~, as the angle between the incident and scattering dlrecnon, I is the target spin, and PL as the regular Legendre polynomial The quantity H z (a, a') is given m terms of the scattering matrix [lnj-j couphng 1)] by
HL(cd, ~) = ¼
E
(-1)I-r+s't-sl+J'z-s2Y(Ji J1JzJ2, IL)
C1CtlC2C'2JiJ2
x Y(j'~ Ji J'2.12, I'L)T~'~ TJ?~ *,
(4 2)
where c~ = a' l~J[ I ' , c, = ald,I Here the element TS~, is given m terms of the scattering m a m x Uc~. by T~, = 6c~'e2 . . . . U~c,, (4 3) where coc as the relative Coulomb phase The Y-coefficients are the same as in ref 1) As discussed an sect 3 the scattering matrix is separated up Into the energy averaged part and a part representing the fluctuations about the average Using eq (3 10) an eq (4 1) the energy averaged cross section separates into two parts ( a ) a = f f + a F,
(4 4)
where ~ is obtained by using the energy averaged amphtude directly in (4 2) The fluct u a n o n cross section qV IS obtained by replacing (U~,~,~(U~c,~)) v F , In a v by theleadmg term in the generalized Hauser-Feshbach 4) expansion, 1 e , for each J-value
v
v
•
s TJJS s,
(4.5)
where the transmlssmn coefficients are given by Ts = 1 - Z I
(4 6)
Ct
SJ = Z Ts
(4 7)
C
The sum in eq (4 7) as over all outgoing channels including those ot the type n + S9Zr m the reaction 8 9y(p, n)8 9Zr Since these latter channels are not exphcitly descrabed, we use the standard stanstical factor to represent their contribution to S s, that is, we rewrite S s in the form K
S s = ~ TS+p(2J+l)exp ( - J ( J + 1)/2a2),
(4.8)
C=1
where p is the level density and a is the span cut-off parameter The sum in the first term of eq (4 8) is over all K channels which are explicitly retained
5. Results and conclusions The particular calculations discussed in this section were made for direct comparison with the pubhshed experimental excitation function of Long and Fox i), for the reaction s9y(p, p , ) 8 9 y . where the residual nucleus a 9 y , is m either the J~ = } - or - excited state These experimental excitation functmns were taken at 70 ° and con-
14
B J DALTON AND D ROBSON
slsted of a large number of resonances superimposed upon a rather smooth background with a slow energy dependence For this experiment, the accumulative resolution was about 20 keV and the observed resonance widths varied from 30 to 100 keV In order to descrlbe the smooth background, we used the generalized Hauser-Feshbach *) method The amphtude for the elastic part of the T< scattering was obtained from optical model calculations as discussed m sect 3 For the optical model wave functions, we used the Woods-Saxon potential form with a diffuseness and radius o f 0 6 and 5 6 fm respectively The absorptive strength of the potential was taken to be 7 5 M e V - 0 33 Ep where Ep is the proton energy (in MeV) relative to a given state o f 8 9 y , The log derivative of the wave function described in eq (3 13) was calculated at a matching radius of 7 fm The contribution of the eliminated channels (such as the n + s 9Zr type) m the generahzed Hauser-Feshbach denominator term S s of eq (4 8) was parametnzed by the spin cut-off parameter a and the level density p. We used the value of 2 for ~r and for p we used the constant-temperature Fermi gas formula p ( E ~ ) = Po exp(aE,) ~ ruth Po = 0 2 and a = 2 0 MeV -1 Here, Ex is the excitation energy m the c o m p o u n d nucleus 9°Zr In the Long and Fox work, the major resonance structure began at about 6 3 MeV m both the f~ 1 and p~ 1 cases The corresponding calculated excitation functions are shown in figs 4 and 5 with the angular m o m e n t u m labels indicated for the various level contributions The experimental and calculated excitation functions show reasonable agreement (within the expectations of the hmited configuration basis used) between the energies of 6 3 and 7 4 MeV Several resonances which appeared between 6 7 and 7 2 MeV m the calculated excitation function were not seen in the original work of Long and Fox These resonances however, have been recently seen by Spencer et al 13) who repeated the experimental work of Long and Fox with an improved resolution of 5 keV Their experimental resonance positions in this regmn hne up reasonably well with the calculated ones In figs 4 and 5 the Hauser-Feshbach component a v is plotted by itself in order to show that in general it also resonates at an analogue resonance energy For most resonances the effect is small because of the strong neutron competition factor described parametrically here by the quantity S s As can be seen, the largest resonance effects in av occur for the larger values of J~ This is to be expected because S s becomes weaker rapidly as J exceeds the spin cut-off parameter G (equal to 2 in these calculations) via the exponential term in eq (4 8) It is important to point out here that because of level shift contributions the calculated resonance posmons do not line up in a one-to-one way wath the corresponding calculated energy levels of 9 0 y These level shifts, like the level widths, depend upon the amplitudes of the configuration involved m the physical wave fnnctlon at a given energy, as well as upon the orbital angular m o m e n t u m values in each configuration These calculated level shifts varied from 20 to 100 keV for the various resonances, being m general smallest for the larger total angular m o m e n t u m values 0 e , the 4 - levels).
89y(p, p , ) e 9 y ,
ANALOGUE
RESONANCES
15
89y (p,p,.)B9y (3/2 o) HAUSER-FESHBACH
"E ¢; 13
"G~2
{
08
O0
I
I
I
RESONANCE
"o
I
PLUS
i
I
1
I
HAUSER-FESHBACH
11,62
I
64
I
66
I
68
I
"/4
70 712 E(MeV)
7~6
I
78
I
80
F i g 4 Calculated excitation functions for t h e reaction s 9 y ( p , p , ) s g y , ( ~ - state) T h e u p p e r curve is the result for a F alone a n d t h e lower curve ~s t h e result for ~ + a r
'
'
'
H;US
R';PES;0,c'.
'
'
RESONANCE PLUS HAUSER-FESHBACH
A
' i
3-
1°~2-
oo
i-62
F 64
o
I-
I 66
2-
I 68
2-
I i 70 72 E(MeV)
2-
i 74
I 76
i 78
I 80
F~g 5 Calculated excitation functions for t h e reaction s g y ( p , p , ) 8 9 y , ( ~ - state) T h e u p p e r curve is the result for a F alone a n d t h e lower curve is the result for ~-t-~ v
For the region 6 3 to 7 5 MeV we now d]scuss the individual resonances in conj u n c n o n with the configuration mixing amphtudes of table 2 in a one-to-one comparison with the corresponding expenmental data In thls way, we are able to suggest total angular momentum assignments for several of the experimentally observed resonances Near the energies of 6 4 and 6 48 MeV, two resonances are observed by Long and Fox 1) m both the 3 - and -~-- cases The calculanons here show a possible 0 - and 4 ~:ontrlbut]on m th]s region From table 2, we see that the 4 - contribution is weak because of the weak entrance configuranon (1 e , 0 006 p~- t g~) The 0 - contribution shows up rather strongly m the }- case From table 2 again, we see that thas contribution ~s presumably due to the - 0 48 p~ts~(entrance) 0 875 f~ad_:(exlt) configuranons at
16
B J DALTON
AND
D. ROBSON
1 58 MeV excltatlon energy Because of their positions, we believe that the two observed levels near 6 48 MeV correspond respectively to the 0 - and 4 - levels just discussed The weakness of both the 0 - and 4 - levels m the ~-- calculated case indicates a need for stronger mixing between configurations involving the p~ t and p~ 1 holes The next major resonance structure observed is two resonances symmetrically centered around 6 61 MeV Both resonances have appreciable observed strength m both the ~ - and ~ - cases These two resonances show up m the ~ - and ~ - calculated energies at almost the same energies as in the observed cases The calculations indicate possible contribution from two 1 - levels and a 2 - level m this region From table 2, one can see that the 2 - level has appreciable entrance strength, 1 e , - 0 153 p~ 1 d~ and + 0 119 p~ 1 d~ However, because of the destructive interference of the amphtudes for these two configurations, the net 2 - contribution is small The entrance configuration amplitudes m the two 1- levels are constructively coherent with both levels showing appreciable exit strength m both the p~-i d~_ and f~ a d~ configurations The calculated resonance height m the ~ - case of the first level (the one just below 6 6 MeV) is larger than the second resonance just above 6 6 MeV. This situation appears just reversed m the corresponding observed resonances for the 5-- case This indicates that the calculated mixing amplitudes should be reversed (at least m the f~ ~ case) Because of the closeness of these two 1 - levels, a small adJustment m the shell model parameters could bring about this reverse m amphtude mixtures With the above comparison, we suggest that these two observed resonances near 6 6 MeV are due to two 1- levels with each having appreciable strength m the p~ 1 d~ and f~ 1 d~ configurations In the region between 6 65 and 7 2 MeV, the results of Long and Fox show httle resonance structure m either the ~ - or the ~ - case F r o m figs 4 and 5, one can see that the calculations indicate the presence of several resonances m this region At about 6 8 MeV, we find an appreciable 3 - resonance contribution m the k - case but none in the ~ - case at this energy F r o m table 2, we see that this contribution results from the - 0 017 p21 d~(entrance)0 992p~ ~d~(exit)configurations A level near this energy has likewise been recently seen by Spencer et al 13) in the -~- case but not in the { - case Near 6 85 MeV, the calculations (figs 4 and 5) show a 2 - resonance contrlbutlon in the ~ - case, with little strength in the ~ - case This resonance does not show up strongly m the Long and Fox work A level has been observed by Spencer et al ~3) at this energy However, the observed strength appears in the ~ - case and not m the 5 - c a s e I f our calculated 2 - resonance is to correspond to the observed resonance (at 6 85 MeV) of Spencer et al t3) then a stronger mixture of the p~. ~ hole is again needed These calculations show a 3 - level near 6 94 MeV, with a small contribution in the v5-- case but with little strength m the ~ - case A resonance does not occur at this energy m the Long-Fox work nor in the work of Spencer et al 13) A weak 4 - contribution appears in the calculated ~ - case at about 7 08 MeV.
sgy(p, p,)s9y, ANALOGUE RESONANCES
17
Spencer et al 13) observed a weak resonance in b o t h the 3 - and -}- cases at this energy. Although the calculated 3 - strength is small, it is possable (with Increased p~ 1 hole mixture as suggested above) for this 4 - level to correspond to the observed level at this energy Above 7 2 MeV the observed resonances of Long and Fox have each been resolved into two or more resonances m the amproved-resolutlon experament of Spencer et al [ref 13)] The number of separate spakes seen above 7 2 MeV as far greater t h a n t h e number of mdwadual resonances one can obtain from our truncated configurataon basas We discuss below however m what regaon the J - levels obtained m our calculations should contribute The inclusion of other configurations of comparable energy would m general produce a sphttang of the given levels presently calculated into two or more levels In thas case, the present level strength would be dastnbuted among these new levels of the same Y~ as an mtermedlate structure resonance Near the energy 7 3 MeV, the calculataons gave a wade 2 - level m both the ~ - and - cases, being weakest m the latter A corresponding strong wide peak has been seen at thas energy by Long and Fox 1) in the -~- case Thas peak is resolved into two peaks m the 3 - case In thear amproved-resolutaon experament, Spencer et al 13) resolved thear peak (m both the 3 and ~ cases) into three partially overlapping resonances We beheve that one (or more) of these three resonances corresponds to the calculated 2 resonance formed mainly from the p~ 1 d~ entrance configuration This large entrance configuration at thas energy could account for the observed large resonance strength At the energy of 7 4 to 7 48 MeV Long and Fox 1) observe a broad resonance wath consaderable strength an the 3 - case and a shght contribution an the -~- case Spencer et al 13) as well as Courtney and Moore 17) resolve thas broad resonance anto two resonances an both the ~ - and 3 - cases The calculaUons here show a large 1 - contrlbutaon at thas energy an the ~ - case F r o m table 2, we hkewlse see that thas 1- level as composed mainly of the p~ 1 d~ entrance configurataon The strength of thas 1 - level as lower than the 2 - level strength just dascussed (for the ~ - case), even though the latter is hkewlse mainly an entrance configurataon In the 2 - case, the exat as mainly through the f~ 1 s~ configurataon Thas f_~ - 1 s~ configurataon is not allowed in the 1 level because of angular m o m e n t u m couplang The largest resonance observed by Long and Fox as well as by Spencer et al occurs in the ~ - case at about 7 82 MeV [ref 1)] Spencer et al 13) have resolved this peak into at least three partmlly overlapping resonances In the calculated excitation funcUon, the largest resonance also occurs at thas energy in the ~ - case Thas resonance as a 4 - level formed mainly of the p~ 1 g~ entrance configuration The fact that the calculated and observed resonance strengths are largest at this energy suggest that the present model as a reasonably good one for describing the source of lnelastac strength The next Iargest resonance an the calculataons appears near 7 96 MeV This 3 - level shows up qmte strongly an both the 3 - and -}- cases Next to it as a 1- level wath apprecmble strength an the ~ - case, but w~th almost no strength an the ~ - case We beheve that these 3 - and 1 - levels correspond to the two reasonably larger peaks recent-
18
B J DALTON AND D ROBSON
ly resolved at this energy by Spencer et al 13) The height of the observed resonance at a b o u t 7 96 MeV increases in the 3 - case from 0 9 mb/sr at 90 ° [Spencer et al t3)] to 2 15 m b / s r at 160 ° [Courtney a n d M o o r e 17)] The present type of calculation appears to describe the gross features o f the inelastic scattering strengths b u t the sensitivity to shell model parameters a n d the rapid mcrease m complexity when more complicated configurations are included seems to preclude a more definitive analysis of such experiments at the present t~me The authors would hke to t h a n k Drs D D Long, J Fox, R J Phllpott, J A d a m s , J E Spencer, a n d M H a s m o f f for their c o o p e r a t i o n a n d m a n y helpful discussions Special thanks go to D r G Love for helping to r u n some test calculations for the c o m p u t e r routines a n d to D r W J C o u r t n e y for p o i n t i n g out several recent experim e n t a l developments a r o u n d the mass-90 region
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) I1) 12) 13) 14) 15) 16) 17)
D D Long and J D Fox, Phys Rev 167 (1968)1131 R D Lawson and J L Uretskl, Phys Rev 108 (1957) 1300 T Telchmann and E P Wlgner, Phys Rev 87 (1952) 123 G R Satchler, Phys Lett 7 (1963) 55 J B Ball and C B Fulmer, Phys Rev 172 (1968)1199 W J Courtney and tt T Fortune, Phys Lett 41B (1972) 4 C Watson, C F Moore and R K Shehne, Nucl Phys 54 (1964)519 J Adams, Ph D dissertation, Florida State Umverslty, Tallahassee, Florida J R Comfort and J P Schlffer, Phys Rev C4 (1971)803 S P Pandya, Phys Lett 10 (1964) 178 W Lms, J Ernst, N Takahashl, E Grosse and D Proetel, Nucl Phys A179 (1972)161 D Robson, Phys Rev 137 (1965)B535 J E Spencer, M Hasmoff, B Small and S S Hanna, Stanford annual report, 1972, and private commumcatxons Y E Ktm, Phys Rev 131 (1963)1712 V Gdlet and N V Man, Nucl Phys 54 (1964) 321, and references quoted thereto A M Lane and R G Thomas, Rev Mod Phys 30 (1958)257 W J Courtney and C F Moore, private commumcahon
Nuclear Physics A210 (1973) 1--18, (~) North-Holland Pubhshmg Co, Amsterdam Not to be reproduced by photoprmt or mlcroffim without written permlssmn from the pubhsher
SHELL MODEL
OF
IN THE CONTINUUM
ANALOGUE
RESONANCES
IN
ANALYSIS
89y(p, p,)89y,
B J. DALTON Rt 1, Box 266, Ltzella, Georgia 31052 t and D ROBSON ~t Department of Phystcs, Flortda State Unwerslty, Tallahassee, Florida, 32306 t Received 22 January 1973 Abstract. Excitation functtons for the reactmns 8 9 y ( p , p , ) a 9 y * leading to the second and third excited states of agy are calculated The levels of 90y are calculated using a sample configuration-mixing shell model and converted into analogue resonances using a many-level multi-channel R-matrix approach The non-analogue component of the inelastic scattering is descrlbed via Hauser-Feshbach theory and the effects of rmxmg of lsospm on ttus compound contribution are mvestlgated Comparison with the previously published experimental results ymlds tentative spin and parity assignments for the analogue resonances in 9°Zr
1. Introduction The i s o b a r i c span a n a l o g u e states c o r r e s p o n d i n g to the b o u n d excited states o f the nucleus 9 0 y have been observed b y L o n g a n d F o x 1) t h r o u g h the inelastic scattering r e a c t i o n 8 9 y ( p , p , ) 8 9 y , T h e y observed the excitation functions (at 70 °) for the cases m which the residual nucleus 8 9 y , was left in the ~+, 3 - , ~ - , a n d ~+ excited states respectively This r e a c t i o n is Illustrated m fig 1 The f o u r excitation functions o b served consisted o f a n u m b e r o f resonances s u p e r i m p o s e d u p o n a r a t h e r s m o o t h b a c k g r o u n d w h i c h increased slowly with energy The m a j o r i t y o f the resonance structure a p p e a r e d in the ~ - a n d ~ - residual nucleus cases The existence o f the a n a l o g u e r e s o n a n c e features in the inelastic channels suggests a p p r e c i a b l e configuration r m x m g an the excited states o f 9 o y I n their analysis, L o n g a n d F o x 1) have suggested t h a t these resonance features could be explained b y the w e a k - c o u p l i n g m o d e l 2) in w h i c h the single-particle orbltals are c o u p l e d to the excited states o f 8 9 y as a core This coupling via the residual i n t e r a c t m n ~s a s s u m e d to be w e a k so t h a t the core states o f 8 9 y are n o t s~gnlficantly altered I n this p a p e r , we describe calculations o f the excitation functions for the r e a c t i o n 8 9 y ( p , p , ) 8 9 y , for the ~ - a n d 3 - cases These calculations involved the c o m b i n e d t Research supported m part by the National Scmnce Foundatmn grants NSF-GP-15855 and NSF-GJ-367 tt Alfred P Sloan Research Fellow 1 August 1973
2
B J DALTON A N D D ROBSON
use of a configuratmn-mlxmg shell model calculation together with the reduced Rmatrix 3) and generalized Hauser-Feshbach 4) scattering formulations, and were originally intended for comparison with the experimental results of Long and Fox Our objectwe m doing these calculatmns was to see ff some of the resonance features such as positions and widths, could indeed be described by using the weak-couphng model as suggested by these authors 89
89
y (p, p') y*
89
89
59Y50
39Y50 9o 40Z r50
F~g 1 Schematic diagram of the inelastic scattering channels for 89y(p, p , ) 8 9 y , indicating those which are relevant to this work
In our model description, we first describe the low-energy negative-panty states of 90y by using a configuration-mixing shell model calculation (described in sect 2) For these calculations, a hmlted partMe-hole basis is used The single-pamcle neutron orbltals used are hmlted to the set {2dg_, 3s~, 2d~_, lg~} We assumed that the full g~ neutron shell is part of the inert core for the energy range considered (up to about 3 MeV in 9oy) This assumption is in agreement with the 91Zr(p, d)9°Zr reaction studied by Ball and Fulmer 5) m whach the l = 4 pick-up contribution did not appear until an excitation energy of about 4 2 MeV m the residual nucleus The hole states for the 89y core are hmlted to the proton orbltals p~ 1 p~ 1, and f~ 1 corresponding to the ground and first two excited states of 89y respectively The ground state configuration of 8 9 y is assumed to be pure p~ although Courtney and Fortune 6) suggest that a possible 10 ~ mixture of the [(2p~)-Z(2p~)-l(lg~) 2] configuration is needed to explain the lifetime of the first excited 0 + state of 9 ° Z r As we discuss an sect 2, one possible effect of ignoring this 10 ~ ground state mixing in 8 9 y is the discrepancy between the calculated and experimental values for 7 + - 2 (gxound state) energy splitting in 90y Using a pure p~ configuration for s 9 y and the measured value of 0 897 MeV (see fig 1) for the g~ smgle-partMe energy of 8 9 y , w e
sgy(p, p,)agy* ANALOGUE RESONANCES
3
obtained a value for this sphttmg which as from 200 to 300 keV too large for several sets of residual interaction parameters This same splitting discrepancy was obtained in zero-range calculations by Adams s) In his calculations, the g~ single-particle energy had to be changed from the "experimental" value of 0 897 MeV to a value of 0 52 MeV in order to fit the experimental separation For the single-particle basis states, we used the Coulomb plus Woods-Saxon potentials with a derivative-type spin-orbit term The well depth for each orbit was adjusted to give ihe specified binding energies, which we used as parameters in fitting the known energy levels of 9Oy [refs 7, 9)] For the residual Interaction between neutron and protons, we used a finite-range Gausslan form with Wlgner and Bartlett terms The rank-one tensor strength (1 e , the coefficient of the ap trn term) was determined by fitting the known doublets of 9Oy [refs 7, 9)] The set of parameters used for the negative-parity states was in turn used to calculate the positive-panty eigenvalues of 9Oy These states were described in these calculations as mixed configurations of the g~ first excited proton state of s 9 y coupled with the neutron orbltals specified above The energy ordering of the first six o f these positive-parity states agreed with the ordering obtained in the calculations of Pandya 10), and with the recent s 9y(d, p~)90y experimental results of Eros et al ~~) In taking the zero-range limit, the calculated order of the first 3 +, 5 + levels inverted, agreeing with the ordering obtained in the zero-range calculations of Adams 8) In the scattering part of this calculation, we separated the reaction amphtude into the energy averaged and fluctuation parts The cross section corresponding to the fluctuation amphtude was described by using the generalized Hauser-Feshbach 4) method The analogue resonances are included in the energy averaged amplitude In describing the resonance part of the energy averaged amplitude, we used the reduced R-matrix formulation of Teichmann and Wlgner 3) In this reduced R-matrix formulation, the scattering contributions for channels other than the elastic and melastic p-C and n-A channels [in the notation of Robson 12)] are reduced out This reduction is convenient in that it allows the eliminated channels to be described separately from the retained p-C and n-A channels The reduced widths for the analogue (T>) contribution are obtained from the configuration mixing amphtudes and surface value of the wave functions used in the shell model calculations of 90y In this way, the calculated resonances for the reaction 89y(p, p,)89y ~ are constructed to correspond to the isobaric spln analogues of the excited shell model states of 9oy In sect 5, the calculated excitation functions (at 70 °) for the ~ - and 3 - cases are shown for proton energies between 6 3 and 8 2 MeV Through comparisons with the Long and Fox work and with the recent work of Spencer et al ~3) we are able to suggest total angular momentum assignments for several resonances in the region between 6 7 and 7 5 MeV Above this region, the comparison with experiment is poor. The number of resonance peaks at these higher energtes m the recent experimental data far exceeds the
4
B J DALTON AND D ROBSON
number that can be possably obtained from the hmated configuration basis used here This is strong evidence that other configuration maxtures are apprecmble above 7 5 MeV The overall calculated resonance peak heaghts were lower than those seen in the experiments, lndacatmg a need of stronger configuration mixing of all levels with the p~ hole confgurations
2. Configuration-mixing shell model calculations for 9Oy In the first part of this sechon, we briefly outline the essentml aspects of the shell model formulation used m these calculations In the second part, we discuss the parameters used in descrablng the excited states of 9Oy The physical wave functaon used to describe the negatwe-panty states of 9oy as expanded as a hnear combination of pamcle-hole configurataons, 1 e, ~b(E~, J - ) = Z Ac(E~J-)ICJ-> ,
(2 1)
C
where each pamcle-hole configuratmn IC J - > has the form (m j-j coupling)
ICJ-> = I{A.u¢zJ"+~q52Jb} ® J2, J - M }
(2 2)
The subscripts a and b refer to the proton hole shells connected by the protonneutron residual anteractlon For thas calculatmn, these shells are hmlted to the set {2p~ 1, 2p~ 1 lf~ 1} The antasymmetrlzataon ln&cated by AN m eq (2 2) is thus taken over all of the N = 2A + 1 + 2Jb pamcles in the two andacated shells The function [Jzm2 ) amphcat m eq (2 2) refers to the neutron orbltals whach we llmat to the set {2d_~, 2d~, 3s~, lg_~} The configuraUons appearing m eq (2 2) are determined by those pamcle-hole states whach can be coupled to a given value of the total angular momentum J The physical sagmficance of using the above hmlted configurataon basas as discussed an sect 5 in conjunctaon wath the experamental results of the reactaon 89y(p, p,)S 9 y , In calculatmg the single-particle orbatals, we used the Coulomb and Woods-Saxon potential (including a derivative spin-orbit term) to represent the anteractlon of the resadual nucleons wath the Ferma sea For the resadual interaction, we used the fimterange form
V~2 = 4~ Utz(rtz)[A + BPr], V~2 = exp
(-~lr, zlZ),
Pr
= ¼(l--a," a2)
(2.3)
Introducing the resadual interactaon into the Hamdtoman and usang the expansion (2 1), we obtain an the usual manner the secular equation N
Z Ac[(ec - E,)3c,c +
t=l
(2 4)
89y(p, p,)89y* ANALOGUE RESONANCES
5
In tl~s equation, the energy ec is given by the sum ec = ec(1) + ec(2)
(2 5)
Here, ec (2) represents the neutron single-particle energy for a given orbital and ec (1) represents the single-proton-hole energy relative to the reference state, 1 e , ec(1 ) is just the energy needed to separate a proton from the reference state In order to obtain the amplitudes Ac, the matrix elements in eq (2 4) were evaluated using the standard multipole expansion technique ~4, 15) The Slater integrals involved in these formulas were calculated numerically using Woods-Saxon wave functions with the Gausslan residual interaction form of eq (2 3) It is important to point out here that these wave functions were normalized to unity only over the range r = 0 to the matching radius R an accordance with the R-matrix resonance analysis discussed an sect 3 Using the calculated matrix elements and energies ~c, the resulting elgenvalues E~ and amphtudes A c (E~, J - ) were found by numerically solving the secular equation (2 4) For each total angular momentum value J and elgenvalue E~, the sum of the squares of the amphtudes was normalized to unity As we discuss m sect. 3, the reduced widths for the resonance reactions are obtained from these amplitudes and values of the single-particle wave functions at the R-matrix matching radius In our calculations for the posture-parity states of 9Oy, the physical wave function was expanded in the same manner as in eq (2 1), but with the proton hole states {p~ 1, p~ ~, f~ ~} replaced by the g~ proton particle state The corresponding positiveparity amplitudes Ac(E~, J+) were obtained by following a procedure Identical to that outlined above fol the negative-parity case The negative- and posmve-parity calculations here are thus made with respect to two different reference states, namely the full and empty 2p~ shell The energy difference AEo between these two reference states is essentially the energy of the two 2p~ protons relative to the 8SSr core A determination of this energy difference is beneficial in estabhshing the positive-parity energy levels relative to the negative-parity ones Instead of directly calculating AEo, however, we attempted to establish the relative positions of the J - and J + energy spectra by directly calculating the well-known 2 - ( g r o u n d state)-7 + energy sphttlng m 9 o y In making this splitting calculation we assumed that the 2 - ground state of 9Oy was a pure p~ d~ configuration and that the 7 + state was a pure g~d~ configuration By treating the 2p~ state as a particle state (rather than a hole state as outlined above) and using for the g~ single-particle energy the experimental value of 0 897 MeV, we obtained an energy splitting which was from 0 2 to 0 3 MeV too large for several sets of residual interaction parameters This disagreement would seem to indicate that either the lg~ excited state of s g y is more involved than a smgle-pamcle state outside an inert aSSr core, or that contiguratlon mixing is playing a stronger role in the 7 + and 2 - states, or both As the calculations below show, these two states are apparently not composed of pure configurations
6
B J DALTON AND D ROBSON
In calculating the single-particle wave functions for 9oy, we used a fixed WoodsSaxon radius of R = 5 5 fm and a diffuseness of a = 0 5 fm The well depth U was adjusted m the computer program to gwe the specified binding energy for each partial wave These single-particle binding energies (see table 1) were used as parameters m fitting the known energy levels of 9 0 y [refs 7, 9)] TABLE 1
Binding energies used for single proton and neutron orb]tals m negative-panty calculations 2p{ 87
2p~. 10 35
l f{ 10 536
2d{ 7 20
3s{ 5 85
2d{ 4 75
1g{ 4 45
F o r the proton hole energies, the calculations were first made using the "experimental" values of - 1 499 MeV and - 1.736 MeV for the p~ and f÷ hole states, respectively However, a better fit to the 89y(p, p,)S9y reaction data was obtained by using values of - 1 65 MeV and - 1 836 MeV respectively m place of the two experimental values i
9Oy (j_)
i
l
40
CO =-12
o
35
2 I
--3
3.0
2.5
~
3
--4
--2
4
r j
t
l
4
{2o Ld
~
.
~ o
I --I -0
1.0 --7+
05 ~
0 0
--z EX'P
-Oil
3
C~-*"
a -013
F]g 2 Calculated positive-panty energy levels for 9oy versus C1 for Co = --1 2 and 7 = 0 3 shown wlth the first slx experimental levels of LlnS et al tl) and the calculated results of Pandya
89y(p,p,)sgy* ANALOGUE RESONANCES
7
A first estimate o f the neutron single-particle energies was obtained by fitting the centrold o f the well-known doublets formed here from the p~ 1 hole with the four n e u t r o n orbltals specified above The centrold sphttmg was made without including the pamcle-hole configurations corresponding to the f~ 1 and p~ ~ hole states The sphttmgs o f the doublet levels were used as a means to determine a first estimate o f the rank-one tensor strength m the residual interaction In calculating the J + and J energy spectra for 9 o y , we adjusted the rank-zero ( C o ) and rank one (C1) tensor strengths separately These strengths are gwen m terms o f A and B in eq (2 3) by Co = A +¼B,
C 1 = --¼B
F o r our choice o f wave funcUons, we used the values o f Co = - 1 2, a range parameter o f ~ -- 0 3 and plotted m figs 2 and 3 the J ÷ and J - energy levels versus C~ f r o m C~ = - 0 1 to C 1 = - 0 3 This range o f C1 m these figures corresponds to a range for the spin-dependent to spin-independent strength ratio e = CI/Co o f e = 0 083 to c¢ = 0 25 I n a h g m n g the calculated resonance p o s m o n s for the reactmn S9y(p, p,) 8 9 y , with the experimentally determined resonance p o s m o n s we used a value o f C~ = - 0 25 This value corresponds to a value o f ~ = 0 21 which is shghtly larger
4.0 COy(j+) 55
CO =-I 2
30
T LI.I
2
f
25
:3
2O
1.5
1.0
0.5
- - 4 --;5
.
--2
--2
0.0
3
~ 7
PANDYA
EXP
2
I
-0 1 C I'->
17
-O.3
Fig 3 Calculated negative-panty energy levels for 9Oy versus Ct for Co = --1 2 and 7 = 0 3 Also shown are the known experimental doublets 7.9)
23O1401213234212431230-
0 000 0 201 1 144 1 318 1 562 1 580 1 808 1821 1 876 2 039 2086 2 192 2 315 2 551 2 724 2 945 3 005 3 154 3 324 3 476 3633 3 855
0 007 -0019
- 0 006
- - 0 052
0 202 0 117
0 976 0 950 - 0 079
- 0 017 -0011 - 0 029
0 020
0100
--0153
- - 0 80
- - 0 007
p~-~d~
0 025 0119 - - 0 164
0 986 0 999
p½-~d_~
Cl = --1 2, Co = - - 0 2 5 , 7 = 0 30
J-
E (MeV)
TABLE 2
0 098
- 0 126
0 056
- - 0 168
- - 0 484 0 399
- - 0 870 0 889
p½-~s~
0031
0 988 - 0 993
0 002 - 0 064
0 000
0 006
- - 0 005
p ~ - ~g~_
- - 0 478 0965 0 785 0 992 --0130 - - 0 118 0 045 - - 0 116 0 175 - - 0 132 - 0 005 - 0 001 0 012 0 010 -0015
0 352 0 999
0 143 0 012
p~-~d÷
032 035 118 024
- 0 022 - 0 116 - 0 038 -0019 - 0 993
0 052 0 156 0 037
0 012 - - 0 012 0011 - - 0 044 - - 0 033 0021 0 009
0 --0 --0 0
p_4- ~d~
- 0 964 - 0 005
0 057 0 173 0 984
0039
- - 0 076 0136 - - 0 015
- - 0 124
0 071
p~-~s~
- 0 065 0006
- - 0 005 0 087 - 0 104
- - 0 001 0029 0 004 - - 0 013 0 024
0010
0 003
0 004 0 001
p..k- ~g~
0 027 0 029 - - 0 479 0 246 0 043 0 875 - - 0 776 0105 - - 0 571 0 115 0979 0 986 - - 0 995 - - 0 125 - - 0 068 - - 0 047 - 0 070 - 0 001 0 030 0 104 -0119 0 068
f ~ - ~d~
Calculated energy levels a n d m i x i n g a m p h t u d e s for n e g a t i v e - p a n t y states o f 9 o y
- - 0 060 0018 - 0 029 0 005 0001 0 012 - - 0 053 0 044 - - 0 004 0 014 0 085 - 0 009 0 008 - 0 169 0046
0 047 0 006
0 005 0 004
f ~ - ~d~
- 0 970 0991
0 031
- 0 025
0 091
0 028 0112 0 115
0030
0 008 0 022
f~-~s~
- - 0 011 0012 - - 0 028 - - 0 002 0012 - - 0 003 - - 0 010 0 042 0 058 0 006 0 081 - 0 038 - 0 017 - 0 022 0003
0 010 0 008
0 007 0 001
f~-~g~
~Z
*'] O ~Z
OO
89y(p, p,)e9y* A N A L O G U E RESONANCES
t h a n t h e v a l u e o f ~ = 0 16 a s s u m e d b y P a n d y a to) T h e p o s l t w e - p a r l t y e x p e r m a e n t a l levels s h o w n in fig 2 w e r e o b t a i n e d f r o m t h e 8 9 y ( d , p 7 ) 9 0 y r e a c t m n o f Lxns
et al 11)
T h e o r d e r i n g o f t h e first six J + levels is in a g r e e m e n t w i t h t h e e x p e r i m e n t , a l t h o u g h the c a l c u l a t e d level p o s m o n s are m g e n e r a l t o o l o w TABLE 3 Calculated energy levels and mixing aroplltudes for positive-parity states of 9o y
E (MeV)
J
g_~dff
0 000 0 089 0 234 0 293 0 398 0 459 1 598 1 688 2 098 2 451 2 606 2 674 2 792 2 891 2 894 3 015 3 083 3 081 3 157
7 2 3 5 4 6 5 4 1 3 2 6 4 5 3 4 5 6 7
0 999 0 999 0 995 0 984 0 997 --0 999 --0 178 --0 077
C=
--0 101 --0 043 0 027 0 006 0 020 0 021 --0 008 --0 002 --0 000 --0 017
g~_s~
0 175 0 077 0 979 0 988
g~d~
0 103 0 037 0 002 --0 026 0 099 0 130 0 964
--0 134 0 106 0 013 0 006
--0 0 --0 --0 --0 --0 --0
972 928 963 246 348 249 233
g~_g~ 0 017 0 043 0 005 0 011 0 008 --0 007 0 019 0 034 1 000 0 247 0 999 --0 233 0 347 --0 248 0 969 0 937 0 969 0 972 0 999
--1 2, D = --022, ;~ = 0 3 0
T h e c a l c u l a t e d m i x i n g a m p h t u d e s c o r r e s p o n d i n g to t h e J -
and J + energy spectra
o f figs 2 a n d 3 are s h o w n m tables 2 a n d 3 r e s p e c t i v e l y T h e s e a n a p h t u d e s are disc u s s e d in sect 5 in c o n j u n c t i o n w i t h t h e c a l c u l a t e d r e s o n a n c e f e a t u r e s for t h e inelastic s c a t t e r i n g m e n t i o n e d a b o v e F i n a l l y it s h o u l d be p o i n t e d o u t t h a t t h e r e s o n a n c e energies o b s e r v e d in the i n e l a s t i c e x c i t a t i o n f u n c h o n o f t h e 8 9 y ( p , p,)S 9 y ~ d a t a n e e d n o t a h g n in a o n e - t o - o n e w a y w i t h t h e e~genvalues o f t h e s t r u c t u r e c a l c u l a t i o n s d e s c r i b e d a b o v e T h e level shlfts a r i s i n g in t h e R - m a m x f o r m u l a t i o n c a n be as large as 100 k e V
3. Energy averaged amphtude I n this s e c t i o n , we d e s c r i b e the s c a t t e r i n g a m p l i t u d e u s i n g t h e r e d u c e J R - m a t r i x f o r m u l a t i o n o f T e l c h m a n n a n d W l g n e r 3) F o l l o w i n g p r e v i o u s w o r k 12), we first describe t h e i s o b a r i c spin f o r m o f t h e R - m a t r i x a n d t h e n r e l a t e It to the n e u t r o n - p r o t o n form of the R-matrix W e use h e r e t h e n o t a t i o n o f R o b s o n 12) m w h i c h the t a r g e t C has i s o b a r i c spin
10
B J DALTON AND D ROBSON
designated by To The isobaric analogue of C is designated by A, and the isobaric analogue of the ( n + C ) system is designated by IT>) where T> = To+½ The state I T < ) with T< = T o - ½ has the same isobaric spin as the ground state of the nucleus (p+C) F o r this calculation, we assumed no internal mixing between the 7"> and T< states. The isobaric spin basis functions (which &agonahze the nuclear potential) are defined by IT>) = (1/a)[pC) + (b/a)[nA), IT<) = (b/a)[pC)- (1/a)[nA),
(3 1)
where a = ( 2 T o + 1)~ and b = (2T0) ~ The relations In (3 1) are used tz) to express the R - m a m x in the neutron-proton basis m terms of the R - m a m x in the isobaric spin basis With the assumption of no Internal mixing between the 7"> and T< components, the R-matrix in the isobaric span basis may be written in the form
0)
Re
(3 2)
Here, R "> (R e) represents the scattering among the 7"> (T<) channels F o r a target nucleus C, the reactions C(p, p')C* and C(p, n')A* can proceed via both the R "> and 114 parts, as can be seen by inverting the relations in eq (3 1) However, reactions involving the channels n + B, where T B = T o - 1, e g the reaction 89y(p, n)89Zr, proceed only through R e The analysis here is specific to those reactions for which the entrance channel is of the p + C type It is possible through standard methods a) to ehrmnate all nonentrance channels from the explicit R-matrix However, we retain here all elastic and inelastic channels of type p + C and n + A but ehmmate all other T< channels in R e, 1 e
oi
=
:1
,
Rear RJe]
A=
(,o Rr
C ~ (0, l~er), e
O = Reee
(3 3)
Here e stands for the ehmmated channels (1 e , the n + B type) and r stands for the retained channels in the T< set With the usual time reversal and causality assumption, one has following Telchmann and Wlgner C = B x where T -= matrix transpose, and the reduced R-matrix takes the form
89y(p, p,)S9y* ANALOGUE RESONANCES
11
where (3 5)
~ < = Rrr+R~eLe( 1-R
The second term of eq (3 5) has the same dimension as the first term and represents the effect of the ehmmated exit channels on the scattering If we have N angular m o mentum channels for each J~ value in the proton scattering, there will be N channels o f the n-A type retained an the R~ part By definition, the matrix ~ r satisfies the equation (U:)
= (R;>
O<)(D~),
(36)
where U > (U <) are column vectors containing only the retained T,. (T<) channel wave functmns g~> (X~< ) respectively The elements In the column vectors D > and D < are gwen by
D~>
= ac
> .... 8Z;~r
--Bczc, >
D~<
< = a c ~~Z< - r ~ ,, - B c Zc, =
(3 7)
e
where we have set B2 = B~ = B c for convenience Following previous work x2), we set the channel constants B c equal to the log-derivative of the single-neutron wave function m the n + C system In our calculations, these constants are just the logderivatives of the Woods-Saxon neutlon wave functions used m sect 2 to describe the nucleus 9Oy Following the earher work of Robson i2) for the single-channel case, we use the relations m (3 1) to express the corresponding many-channel reduced R-matrix in the neutron-proton basis in terms of R "> and ~ < The resulting reduced R-matrix for the particle basis is gwen by
= e, + eo
(3 8)
where
~/'
1 ( R'> = ~
bR ~>
bR~>'~ b2R ">]'
Of°= 1 ( ba~< a~
-b~
<
-b~<) ~<
(39) ,
and as before, a = ( 2 T o + 1)~ and b = (2To) ~ The matrices ~ ' and ~ o each have dimension 2N In eq (3 8) the 7"> and T< R - m a m x contributions have been completely separated This separation is convenient since the two are calculated m a different way In describing an experiment with finite resolution, one can separate the scattering matrix into two parts, one describing the energy averaged part ( U ) and the other describing the fluctuations about the average (UF), 1 e U=
(U)+U ~
(3 10)
In sect 4 we describe the fluctuation part U v by the generalized Hauser-Feshbach 4) statistical method
t2
B J DALTON AND D ROBSON
The enelgy averaged part ( U ) separates into two parts, correspondmg to the separation m (3 8), i e (u)
= u°+u
(311)
'
Here, U o describes the average scattering contrlbutlon from G ° [the so called "farlevel" contribution 16)] and U ' descr%es the exphclt average contribution of ~ ' modified by terms involving the contribution G ° as explained by Lane and Thomas [ref 16)] In terms of our calculation, for the reaction S9y(p,p,)S9y,~ the matrix U ° describes the average ~sobanc analogue T> scattermg The effects of the T< states on the T> scattering are mcluded in U ' through the reverse of terms hke ( 1 - R ° L ) which arise m the separation of U into U o and U ' In the explicit description of the T> scattering, the matrix ~ ' is expanded m the usual way in terms of the level m a m x and reduced widths The reduced width amphtudes for the ~ ' > expansion of (3 9) are given by
7~'>c= (hZ/21~ac)-~AxcUac,
(3 12)
where the coefficients A~c are just the shell model configuration mixing amplitudes Ac(E~, J+) of sect 2 at the elgenenergy E~ = E z for channel e and uz~ are the appropriate neutron stogie-particle wave functions for the corresponding 9Oy states evaluated at r = a~ The T< contribution to the scattermg (m the energy average) is described by the optical model The optical model used corresponds to the T< component of a Lane potential U o + Ult To The energy averaged off-diagonal contribution is not explicitly described That is, we replace ~ in eq (3 9) by R °pt, i e Go The diagonal elements of
- b R °pt]
1 ( bZR°pt = p \--bg°P t
R °pt a r e
(3 12)
Ropt ]
by definmon given by 1
R°~' --
,
(3 13)
f~-B~ wherefc is the log derivative of the optical model wave functmn evaluated at the channel radH ac 4. Cross section
We describe the analogue resonance features in the S9y(p, p,)Sgy~, reaction via the standard reduced R-matrix formulation discussed in sect 3 The smooth background in the melastlc scattering is described via the generahzed Hauser-Feshbach 4) methcd For this inelastic scattermg case where both mcomang and outgoing proJectile spins ale ½, we use the cross-section formula da~ df2~,
_
_
1 2k~(2I + 1)
V
HL(C(, c0PL(cos 0~,)
(4.1)
sgy(p, p,)sgy, ANALOGUE RESONANCES
t3
Here, 0~, as the angle between the incident and scattering dlrecnon, I is the target spin, and PL as the regular Legendre polynomial The quantity H z (a, a') is given m terms of the scattering matrix [lnj-j couphng 1)] by
HL(cd, ~) = ¼
E
(-1)I-r+s't-sl+J'z-s2Y(Ji J1JzJ2, IL)
C1CtlC2C'2JiJ2
x Y(j'~ Ji J'2.12, I'L)T~'~ TJ?~ *,
(4 2)
where c~ = a' l~J[ I ' , c, = ald,I Here the element TS~, is given m terms of the scattering m a m x Uc~. by T~, = 6c~'e2 . . . . U~c,, (4 3) where coc as the relative Coulomb phase The Y-coefficients are the same as in ref 1) As discussed an sect 3 the scattering matrix is separated up Into the energy averaged part and a part representing the fluctuations about the average Using eq (3 10) an eq (4 1) the energy averaged cross section separates into two parts ( a ) a = f f + a F,
(4 4)
where ~ is obtained by using the energy averaged amphtude directly in (4 2) The fluct u a n o n cross section qV IS obtained by replacing (U~,~,~(U~c,~)) v F , In a v by theleadmg term in the generalized Hauser-Feshbach 4) expansion, 1 e , for each J-value
v
v
•
s TJJS s,
(4.5)
where the transmlssmn coefficients are given by Ts = 1 - Z I
(4 6)
Ct
SJ = Z Ts
(4 7)
C
The sum in eq (4 7) as over all outgoing channels including those ot the type n + S9Zr m the reaction 8 9y(p, n)8 9Zr Since these latter channels are not exphcitly descrabed, we use the standard stanstical factor to represent their contribution to S s, that is, we rewrite S s in the form K
S s = ~ TS+p(2J+l)exp ( - J ( J + 1)/2a2),
(4.8)
C=1
where p is the level density and a is the span cut-off parameter The sum in the first term of eq (4 8) is over all K channels which are explicitly retained
5. Results and conclusions The particular calculations discussed in this section were made for direct comparison with the pubhshed experimental excitation function of Long and Fox i), for the reaction s9y(p, p , ) 8 9 y . where the residual nucleus a 9 y , is m either the J~ = } - or - excited state These experimental excitation functmns were taken at 70 ° and con-
14
B J DALTON AND D ROBSON
slsted of a large number of resonances superimposed upon a rather smooth background with a slow energy dependence For this experiment, the accumulative resolution was about 20 keV and the observed resonance widths varied from 30 to 100 keV In order to descrlbe the smooth background, we used the generalized Hauser-Feshbach *) method The amphtude for the elastic part of the T< scattering was obtained from optical model calculations as discussed m sect 3 For the optical model wave functions, we used the Woods-Saxon potential form with a diffuseness and radius o f 0 6 and 5 6 fm respectively The absorptive strength of the potential was taken to be 7 5 M e V - 0 33 Ep where Ep is the proton energy (in MeV) relative to a given state o f 8 9 y , The log derivative of the wave function described in eq (3 13) was calculated at a matching radius of 7 fm The contribution of the eliminated channels (such as the n + s 9Zr type) m the generahzed Hauser-Feshbach denominator term S s of eq (4 8) was parametnzed by the spin cut-off parameter a and the level density p. We used the value of 2 for ~r and for p we used the constant-temperature Fermi gas formula p ( E ~ ) = Po exp(aE,) ~ ruth Po = 0 2 and a = 2 0 MeV -1 Here, Ex is the excitation energy m the c o m p o u n d nucleus 9°Zr In the Long and Fox work, the major resonance structure began at about 6 3 MeV m both the f~ 1 and p~ 1 cases The corresponding calculated excitation functions are shown in figs 4 and 5 with the angular m o m e n t u m labels indicated for the various level contributions The experimental and calculated excitation functions show reasonable agreement (within the expectations of the hmited configuration basis used) between the energies of 6 3 and 7 4 MeV Several resonances which appeared between 6 7 and 7 2 MeV m the calculated excitation function were not seen in the original work of Long and Fox These resonances however, have been recently seen by Spencer et al 13) who repeated the experimental work of Long and Fox with an improved resolution of 5 keV Their experimental resonance positions in this regmn hne up reasonably well with the calculated ones In figs 4 and 5 the Hauser-Feshbach component a v is plotted by itself in order to show that in general it also resonates at an analogue resonance energy For most resonances the effect is small because of the strong neutron competition factor described parametrically here by the quantity S s As can be seen, the largest resonance effects in av occur for the larger values of J~ This is to be expected because S s becomes weaker rapidly as J exceeds the spin cut-off parameter G (equal to 2 in these calculations) via the exponential term in eq (4 8) It is important to point out here that because of level shift contributions the calculated resonance posmons do not line up in a one-to-one way wath the corresponding calculated energy levels of 9 0 y These level shifts, like the level widths, depend upon the amplitudes of the configuration involved m the physical wave fnnctlon at a given energy, as well as upon the orbital angular m o m e n t u m values in each configuration These calculated level shifts varied from 20 to 100 keV for the various resonances, being m general smallest for the larger total angular m o m e n t u m values 0 e , the 4 - levels).
89y(p, p , ) e 9 y ,
ANALOGUE
RESONANCES
15
89y (p,p,.)B9y (3/2 o) HAUSER-FESHBACH
"E ¢; 13
"G~2
{
08
O0
I
I
I
RESONANCE
"o
I
PLUS
i
I
1
I
HAUSER-FESHBACH
11,62
I
64
I
66
I
68
I
"/4
70 712 E(MeV)
7~6
I
78
I
80
F i g 4 Calculated excitation functions for t h e reaction s 9 y ( p , p , ) s g y , ( ~ - state) T h e u p p e r curve is the result for a F alone a n d t h e lower curve ~s t h e result for ~ + a r
'
'
'
H;US
R';PES;0,c'.
'
'
RESONANCE PLUS HAUSER-FESHBACH
A
' i
3-
1°~2-
oo
i-62
F 64
o
I-
I 66
2-
I 68
2-
I i 70 72 E(MeV)
2-
i 74
I 76
i 78
I 80
F~g 5 Calculated excitation functions for t h e reaction s g y ( p , p , ) 8 9 y , ( ~ - state) T h e u p p e r curve is the result for a F alone a n d t h e lower curve is the result for ~-t-~ v
For the region 6 3 to 7 5 MeV we now d]scuss the individual resonances in conj u n c n o n with the configuration mixing amphtudes of table 2 in a one-to-one comparison with the corresponding expenmental data In thls way, we are able to suggest total angular momentum assignments for several of the experimentally observed resonances Near the energies of 6 4 and 6 48 MeV, two resonances are observed by Long and Fox 1) m both the 3 - and -~-- cases The calculanons here show a possible 0 - and 4 ~:ontrlbut]on m th]s region From table 2, we see that the 4 - contribution is weak because of the weak entrance configuranon (1 e , 0 006 p~- t g~) The 0 - contribution shows up rather strongly m the }- case From table 2 again, we see that thas contribution ~s presumably due to the - 0 48 p~ts~(entrance) 0 875 f~ad_:(exlt) configuranons at
16
B J DALTON
AND
D. ROBSON
1 58 MeV excltatlon energy Because of their positions, we believe that the two observed levels near 6 48 MeV correspond respectively to the 0 - and 4 - levels just discussed The weakness of both the 0 - and 4 - levels m the ~-- calculated case indicates a need for stronger mixing between configurations involving the p~ t and p~ 1 holes The next major resonance structure observed is two resonances symmetrically centered around 6 61 MeV Both resonances have appreciable observed strength m both the ~ - and ~ - cases These two resonances show up m the ~ - and ~ - calculated energies at almost the same energies as in the observed cases The calculations indicate possible contribution from two 1 - levels and a 2 - level m this region From table 2, one can see that the 2 - level has appreciable entrance strength, 1 e , - 0 153 p~ 1 d~ and + 0 119 p~ 1 d~ However, because of the destructive interference of the amphtudes for these two configurations, the net 2 - contribution is small The entrance configuration amplitudes m the two 1- levels are constructively coherent with both levels showing appreciable exit strength m both the p~-i d~_ and f~ a d~ configurations The calculated resonance height m the ~ - case of the first level (the one just below 6 6 MeV) is larger than the second resonance just above 6 6 MeV. This situation appears just reversed m the corresponding observed resonances for the 5-- case This indicates that the calculated mixing amplitudes should be reversed (at least m the f~ ~ case) Because of the closeness of these two 1 - levels, a small adJustment m the shell model parameters could bring about this reverse m amphtude mixtures With the above comparison, we suggest that these two observed resonances near 6 6 MeV are due to two 1- levels with each having appreciable strength m the p~ 1 d~ and f~ 1 d~ configurations In the region between 6 65 and 7 2 MeV, the results of Long and Fox show httle resonance structure m either the ~ - or the ~ - case F r o m figs 4 and 5, one can see that the calculations indicate the presence of several resonances m this region At about 6 8 MeV, we find an appreciable 3 - resonance contribution m the k - case but none in the ~ - case at this energy F r o m table 2, we see that this contribution results from the - 0 017 p21 d~(entrance)0 992p~ ~d~(exit)configurations A level near this energy has likewise been recently seen by Spencer et al 13) in the -~- case but not in the { - case Near 6 85 MeV, the calculations (figs 4 and 5) show a 2 - resonance contrlbutlon in the ~ - case, with little strength in the ~ - case This resonance does not show up strongly m the Long and Fox work A level has been observed by Spencer et al ~3) at this energy However, the observed strength appears in the ~ - case and not m the 5 - c a s e I f our calculated 2 - resonance is to correspond to the observed resonance (at 6 85 MeV) of Spencer et al t3) then a stronger mixture of the p~. ~ hole is again needed These calculations show a 3 - level near 6 94 MeV, with a small contribution in the v5-- case but with little strength m the ~ - case A resonance does not occur at this energy m the Long-Fox work nor in the work of Spencer et al 13) A weak 4 - contribution appears in the calculated ~ - case at about 7 08 MeV.
sgy(p, p,)s9y, ANALOGUE RESONANCES
17
Spencer et al 13) observed a weak resonance in b o t h the 3 - and -}- cases at this energy. Although the calculated 3 - strength is small, it is possable (with Increased p~ 1 hole mixture as suggested above) for this 4 - level to correspond to the observed level at this energy Above 7 2 MeV the observed resonances of Long and Fox have each been resolved into two or more resonances m the amproved-resolutlon experament of Spencer et al [ref 13)] The number of separate spakes seen above 7 2 MeV as far greater t h a n t h e number of mdwadual resonances one can obtain from our truncated configurataon basas We discuss below however m what regaon the J - levels obtained m our calculations should contribute The inclusion of other configurations of comparable energy would m general produce a sphttang of the given levels presently calculated into two or more levels In thas case, the present level strength would be dastnbuted among these new levels of the same Y~ as an mtermedlate structure resonance Near the energy 7 3 MeV, the calculataons gave a wade 2 - level m both the ~ - and - cases, being weakest m the latter A corresponding strong wide peak has been seen at thas energy by Long and Fox 1) in the -~- case Thas peak is resolved into two peaks m the 3 - case In thear amproved-resolutaon experament, Spencer et al 13) resolved thear peak (m both the 3 and ~ cases) into three partially overlapping resonances We beheve that one (or more) of these three resonances corresponds to the calculated 2 resonance formed mainly from the p~ 1 d~ entrance configuration This large entrance configuration at thas energy could account for the observed large resonance strength At the energy of 7 4 to 7 48 MeV Long and Fox 1) observe a broad resonance wath consaderable strength an the 3 - case and a shght contribution an the -~- case Spencer et al 13) as well as Courtney and Moore 17) resolve thas broad resonance anto two resonances an both the ~ - and 3 - cases The calculaUons here show a large 1 - contrlbutaon at thas energy an the ~ - case F r o m table 2, we hkewlse see that thas 1- level as composed mainly of the p~ 1 d~ entrance configurataon The strength of thas 1 - level as lower than the 2 - level strength just dascussed (for the ~ - case), even though the latter is hkewlse mainly an entrance configurataon In the 2 - case, the exat as mainly through the f~ 1 s~ configurataon Thas f_~ - 1 s~ configurataon is not allowed in the 1 level because of angular m o m e n t u m couplang The largest resonance observed by Long and Fox as well as by Spencer et al occurs in the ~ - case at about 7 82 MeV [ref 1)] Spencer et al 13) have resolved this peak into at least three partmlly overlapping resonances In the calculated excitation funcUon, the largest resonance also occurs at thas energy in the ~ - case Thas resonance as a 4 - level formed mainly of the p~ 1 g~ entrance configuration The fact that the calculated and observed resonance strengths are largest at this energy suggest that the present model as a reasonably good one for describing the source of lnelastac strength The next Iargest resonance an the calculataons appears near 7 96 MeV This 3 - level shows up qmte strongly an both the 3 - and -}- cases Next to it as a 1- level wath apprecmble strength an the ~ - case, but w~th almost no strength an the ~ - case We beheve that these 3 - and 1 - levels correspond to the two reasonably larger peaks recent-
18
B J DALTON AND D ROBSON
ly resolved at this energy by Spencer et al 13) The height of the observed resonance at a b o u t 7 96 MeV increases in the 3 - case from 0 9 mb/sr at 90 ° [Spencer et al t3)] to 2 15 m b / s r at 160 ° [Courtney a n d M o o r e 17)] The present type of calculation appears to describe the gross features o f the inelastic scattering strengths b u t the sensitivity to shell model parameters a n d the rapid mcrease m complexity when more complicated configurations are included seems to preclude a more definitive analysis of such experiments at the present t~me The authors would hke to t h a n k Drs D D Long, J Fox, R J Phllpott, J A d a m s , J E Spencer, a n d M H a s m o f f for their c o o p e r a t i o n a n d m a n y helpful discussions Special thanks go to D r G Love for helping to r u n some test calculations for the c o m p u t e r routines a n d to D r W J C o u r t n e y for p o i n t i n g out several recent experim e n t a l developments a r o u n d the mass-90 region
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) I1) 12) 13) 14) 15) 16) 17)
D D Long and J D Fox, Phys Rev 167 (1968)1131 R D Lawson and J L Uretskl, Phys Rev 108 (1957) 1300 T Telchmann and E P Wlgner, Phys Rev 87 (1952) 123 G R Satchler, Phys Lett 7 (1963) 55 J B Ball and C B Fulmer, Phys Rev 172 (1968)1199 W J Courtney and tt T Fortune, Phys Lett 41B (1972) 4 C Watson, C F Moore and R K Shehne, Nucl Phys 54 (1964)519 J Adams, Ph D dissertation, Florida State Umverslty, Tallahassee, Florida J R Comfort and J P Schlffer, Phys Rev C4 (1971)803 S P Pandya, Phys Lett 10 (1964) 178 W Lms, J Ernst, N Takahashl, E Grosse and D Proetel, Nucl Phys A179 (1972)161 D Robson, Phys Rev 137 (1965)B535 J E Spencer, M Hasmoff, B Small and S S Hanna, Stanford annual report, 1972, and private commumcatxons Y E Ktm, Phys Rev 131 (1963)1712 V Gdlet and N V Man, Nucl Phys 54 (1964) 321, and references quoted thereto A M Lane and R G Thomas, Rev Mod Phys 30 (1958)257 W J Courtney and C F Moore, private commumcahon