- Email: [email protected]
Shell model study of N = 50 nuclei
1.D.1
l
Nuclear Phystcs A220 (1974) 477-490, (~) North-Holland Pubhshing Co, Amsterdam
$
Not to be reproduced by photoprmt or m~crofilmwithout writtenpermissionfrom the pubhsher
S H E L L M O D E L S T U D Y O F N ffi 50 N U C L E I t D H GLOECKNER and F J D SERDUKE Argonne Nattonal Laboratory, Argonne, Ilhnots 60439 Received 11 October 1973 Measured E2 and M4 transltzons and energy levels m the N = 50 tsotones wath Z :> 38 are satisfactorily reproduced by a shell model with proton configurations (2P½, lg])" Four sets of two-body matrix elements are found by least-squares fitting to energy levels and transition rates m the N = 50 ~sotones These interactions include one that exactly conserves seniority and another that fits the strongly mh~bRed 8+--> 6 + transition m 94Ru Slight seniority nonconservatton is reqmred to predict inhibited N = 50 transiUon rates The sign of the offdmgonal matrix element is determined by comparison with experimental M4 rates The semonty conserwng mteractlon is found to be equivalent to the best semonty nonconserwng interaction for prediction of energy levels
Abstract
1. I n t r o d u c t i o n
Nuclear properties of the N = 50 lSOtones have been described i - 5 ) by a shell model with SSSr as a n inert core a n d valence p r o t o n s restricted to the (2p½, lg~) single-particle levels M a n y new experimental energy levels have been identified recently a n d a few E2 hfetlmes measured Most of the new data agree well with prediction based on the older models However, the 8 + ~ 6 + t r a n s m o n In 94Ru c a n n o t be explained on the basis of these older models I n a n effort to u n d e r s t a n d the n a t u r e of the p r o t o n - p r o t o n effective interaction a n d the Inhibited E2 rate in 94Ru, we have constructed, by least-squares fitting, four effective i n t e r a c h o n s with the (2p~, lg~) p r o t o n model space I n a p r e h m m a r y report 6) of this work we have discussed a t w o - b o d y interaction (the "rates fit" in this work) that does adequately describe the k n o w n t r a n s i t i o n rates m the N = 50 lsotones, including the 8 + ~ 6 + t r a n s i t i o n m 94Ru. I n that report we concentrated o n E2 t r a n s I h o n rates a n d c o m p a r e d the results with those o b t a i n e d on the bas~s of a n older interaction 2) Here we l o o k at other nuclear properties of the N --- 50 lSOtones a n d c o m p a r e that "rates fit" with three other interactions I n sect 2 we discuss the c o n s t r u c t i o n of four different effectwe interactions between p r o t o n s All four interactions are the result of a least-squares fitting procedure One interaction requires that the two-body interaction conserve seniority A n o t h e r reqmres a fit to the k n o w n E2 transition rates i n the N = 50 lSOtones The other two fit only experimental energy levels, these differ m the c o n s t r u c t i o n of the f u n c t i o n to be minimized We also find that the lifetimes of the M 4 Isomers in this t Work performed under the auspices of the US Atomic Energy Commission 477
ot -
O÷
SENIORITY
o+
O+
TOTAL ENERGY
*
,
,
EXPT
OL-----*
0÷
2+
5-
5~ 4-/
4*
8* 6÷ 2*
LEVEL
0"
0"--
5-
2L
4-...
8+, 6t--
RATES
O*
0÷
5-"
2t.
4-
6t 4~ -
8÷
Fig l Expcrlmelltal and theoretical energy levels for 9°Zr All energle~ are m MeV and refer to the experimental g r o u n d state F o r the "level" and "rates" fits only exc~tahon energ,es are s h o w n , ~ e , the theoretical spectrum is shifted so that experimental and theoretical ground states hne up States included m the fit are marked with an asterisk All theorettcal levels are shown Experimental levels were taken f r o m S S Ghcksteln et al, Phys Rev C4 0 9 7 1 ) 1818
0
2c
2~
2+ 5-
4-¸¸¸
4-
2÷ 5-
4*
4*
3c
8+ 6+
35 - 8* 6*
9°Zr
-
0+ 0+
2*
4÷
6* 5~
4~ 4-"
7-
2÷
9-
Iv
2~
SENIORITY
Or__--TOTAL ENERGY
Z+
8" '" 6+ 50÷ 4+
2* ,4-" 4-
3-
2"~
""
2+ - •
9---'-----
II-
EXPT
4+
2+
9-
Ir
*
.
*
.
LEVEL
0+
2*
4÷
, 8 * 6÷--~'--
4-
4~___
x 2t~
K-
*
*
*
RATES
0÷
2*
4+~--
8+ 6*...
44~
2÷
3-
2+
9-
II-
Fig 2 Experimental ,and theoretical energy levels for 92Mo All levels are s h o w n to 3 MeV Selected levels are s h o w n above 3 MeV See also caption for fig 1 The experimental data were taken f r o m D C Kocher et al, Nucl Data B7 (1972)299, Ledereretal, Nucl Phys A169 (1971) 449 and from ref 13)
I
20-
8÷ 6+ 25-5--0+ 4+
2÷ 43C-4-
3 5 --4* 3-
4 C -- 2*
9-
4 5_11-
92Mo
m
m 7~
Z
7~
O t© m (3
-T-
~7
--
I
SENIORITY
TOTAL ENERGY
_
0+.__
"-
0+
2+
4÷
EXPT
8+ 5- --.
0t
2+
4 +_
O~
0+
....
.
'
= ~
4------42~
0 +-
....... - , ~
9-
*
~
~
LEVEL
0+
2+
O r . .
6+
442,~
7----
o.
'99-
-
RATES
0+
2+
4 +-
4-.-4---Zt.. 6+~ • 0+ 8
99-
Fig 3 Experlmeiltal and theoreUcal energy levels for 94Ru Selected levels only are shown above 2 9 MeV Sec caphon for fig l The experimental data were taken from J M Jaklewc e t a l , Phys Lett 29B (1969) 179 and from J B Ball et a l , Phys Rev Lett 29 (1972) 1014
i~
2*
4+
6_
2~
44-
7J
99-
7-~
'....
o\
99-
or.
Y_
-
2
3C-
45
94Ru -
TOTAL ENERGY
I/-~-9/÷ - - -
5/--
9/~ 9/£ 7/÷--3/---
I i/31'1" - It/ ~ - 13/~---. 7C-51"1. - 17/15/~--
9/~-
15/~ - -
_ _ - -
-
_ _ _
I/9/+-_
5,'-~
5/9//~/ 31-, 37/+ /~
EXPT
(I I / ~ . ~ (I I/-) - 5 / L 17/- ~ - - ~ (5/-} I ~ / + 7/- - 13/~+
7/++ 9Ix
SENIORITY
I/---9 / L - -
51-----
9/~ 9/L - 7 / L - 3/-
I I/~ 3/i - l I/~ 13/~ - 7/~ 5 / L 17/~1 3 / ~ -
--
5 / - ~ 17/+ --
3C
2t/+
9~---~\--
I ~
15/+ 17/~---
2l/+
-
-
LEVEL
it, I / - - ~. 9 / L -
N 5/"
: 9~ it 7'~----w- 5 / ~
3/+ ill+ 13// - 7/- - 51+--15/~. 17/----9/------
,/--_
195~ /
15./'t _ _ ..t7/+~-
N21/+
-
-
3,';-----
9/'1"RATES
5/:----
----- - 7/+ 3/=_-
9/~
7/~1:51+ - 1715/'+- " - 1:5/--- _ _ 9/-
I II zI I/"---
15/:- - 91+_
17/~ - 15V~.
21/+
Fig 4 Experimental and theoretical energy levels for 91Nb The complete theoretical spectrum is shown Sec caption for fig 1 The experimental data were taken from F Rauch, Z Phys 243 (1971) 105, J L Horton et a l , Nucl Phys A190 0972) 362 and from ref 21)
0
O5
1
2
2~
3C
15/~ 17/+
3 5 - 21/+
91Nb
4~
t'*
O
r~ r', r"
II
480
D
H
GLOECKNER
AND
F
J
D. SERDUKE
region lead to an unambiguous choice for the sign of the off-diagonal matrix element of the effective interaction In sect 3 we discuss the even N = 50 nuclei, and m sect 4 ~ve discuss odd nuclei W e present our conclusions in sect 5
2. The effective proton-proton interaction There are nine proton-proton t w o - b o d y matrix elements and two proton singleparticle levels within the (2p~, lg~) m o d e l space Thesv eleven parameters are varied m a least-squares search that includes 45 energy levels in the N = 50 lsotones Levels included in the fits are indicated by asterisks in figs 1-5 We consider four separate fits which differ in the additional lestrlctions placed on the least-squares fit The "total-energy fit" includes only energy levels, but m i n i m i z e s the differences in the experimental and theoreUcal total energies relative to the SaSr lnelt core The 93Tc 51 +
21'1•
5/+ ~ ~ _ 15/ °/
_ _ --'15/+
i~1/+
17115/~--
17/+ -
( 15/+}
7/+
- -
91"
-
9/-
- -
919/* 3/÷
1 51 III÷ 5151 ÷
II/~
- - ----
- _ _
5/-
- - *
- -
- -
(3/-)
1:3./*
- -
9/t
9/+
-
91-
9/"
- -
-
91-
57"
31 + 3/-
- - -
5/-
_ _
___
tlf,~
- -
lip
_ _
i,~d4-
_ _
--wITW+
515151 +
5/1.
2:11+ 17/* 151" 17/7/+
7Z'~
91*
- ~ *
- -
I~V-
9/( .31- )~ it/+
- -
131+
- -
17/+
-
3/-
S/-
5/+ 21/+ I 7/-
17/-* - -
9: --9/91+ - - 3/+
20
5/+
15/"i"
17/" I- 17I+ T 13If- 7/+
--
51 +
,o~ 7/+
I 7/+ 0 5~ I/-
O~
E
II-
9/÷
9/I. TOTAL
ENERGY
- -
7/+ - - *
7:
II-
- - *
1t-
9/+
- - *
91+
SENIORITy
EXPT
- -
7/+
- -
II-
91 ~" - LEVEL
RATES
Experimental and theoretical levels for 93Tc All levels through the first a~ - level ( ~ 2 3 MeV) are shown Selected levels at higher energy are shown See caption for fig 1 The experimental data Fig 5
~ere taken from P
J
Rdey
e t a l , P h y s R e v C 3 ( 1 9 7 1 ) 186, f r o m A 1 9 7 ( 1 9 7 2 ) 211 a n d f r o m ref 2 t )
Y
Shamai
et al,
Nucl
Ph'~s
"seniority fit" has the additional constraint that the interaction c o n s e i v e seniority The "'level fit" includes only energy levels, but m i n i m i z e s the s u m of the difference between experimental and theoretical excitation energies The "rates fit" IS similar to the "level fit" but in addition requires that the k n o w n N = 50 E2 transition rates be fitted
N = 50 SHELL MODEL
481
A p r o b l e m c o m m o n to all f o u r least-squares fits ts the d e t e r m i n a t i o n o f the sign o f the off-dmgonal m a t r i x element, ((2p½)21V, rfl(lg~)2)s= o The energy o f all states m the N = 50 nucle~ ~s i n d e p e n d e n t o f this sign M o r e o v e r , E2 t r a n s i t i o n rates are I n d e p e n d e n t o f th~s sign because the effective reduced E2 trans,tlon rate m a t r i x element Q2 involves only the l g l single-particle level where
Q2 = eeff(g~llrZY2[lg~) F o r t u n a t e l y , t w o - p a r t , t i e transfer rates a n d M 4 t r a n s t l o n rates are sens~twe to the s~gn o f the off-dmgonal m a t r i x element Since there are m a n y ambiguities assocmted w~th the analys~s o f two-particle transfer reactions, we have chosen to d e t e r m i n e the TABLE 1 M4 trans~tmns m the N = 50 lSOtones
Nucleus
Trans,tlon
E~, Internal Branching (MeV) conother/ version") M4 b)
~+1 --->½g's
0911
91Nb 93Tc
½-a --> 9+ 2gs .1'-1 --->~;s
0391
034
18/82
95Rh
a- 1--~'~; s
0543
011
10/90 ~) 17
166
1/99 3/97
Theoretical ~) r,. +(sec)
s9y
0 105
00084
Exp b) rm (sec)
Smgleparticle rm- (sec) estimate a) "rm (sec)
23 1 7 7× 106 37
60 11 6× 106 1 0~ 109 1 9 25"<105
69
106
103
74×103
103
102~)
4 4 102 2 3 . 103 1 1 l02
~) L A Sllv and I M Band, Alpha, beta, and gamma ray spectroscopy, ed K Slegbahn, vol 2 (North-Holland, Amsterdam, 1965) p 1639 b) C M Lederer, J M Hollander and I Perlman, Table of isotopes (Wdey, New York. 1967) p 224 ¢) Total-energy fit used for both positive (r,. +) and negatwe (rm-) sign of the 0 + off-diagonal mamx element (p2IVlg2) normahzed to fit the trans,tlon m soy and adjusted for branching and internal conversion d) S A Moszkowskl, Alpha, beta. and gamma ray spectroscopy, ed K S,egbahn, vol 2 (NorthHolland, Amsterdam, 1965) p 863 e) Ref 22) sign f r o m e x p e r i m e n t a l M 4 hfetlmes in N = 50 nuclei There Is an isomeric M 4 t r a n s i t i o n between the first excited state a n d the g r o u n d state in each o f the o d d - A N = 50 nuclei This i s o m e r is a .~+ to ½- t r a n s i t i o n in 8 9 y a n d a ½- to ~+ t r a n s i t i o n in 9 I N b , 93Tc a n d 95Rh U n d e r the usual a s s u m p t i o n t h a t the M 4 t r a n s i t i o n proceeds via a o n e - b o d y o p e r a t o r , the lifetime o f the isomer d e p e n d s on the effective r e d u c e d M 4 m a t r i x element W e d e t e r m i n e this m a t r i x element e m p m c a l l y f r o m o u r smgleparticle nucleus 8 9 y T a b l e l gwes the M 4 lifetimes calculated for the two choices o f the sign o f the ((2p~)21Verf[(lg~t) 2) m a t r i x element. T h e results for the " t o t a l - e n e r g y fit" are presented, all f o u r interactions give s~mllar results T h e ewdence ~s clear A g r e e m e n t with e x p e r i m e n t ~s far superior ff the off-diagonal m a t r i x element is chosen to be positive W e note in passing t h a t th~s choice agrees w~th the s~gn p r e d i c t e d by a surface delta interaction
482
D. H G L O E C K N E R A N D F J D S E R D U K E
We have found the same result for a shell-model effective lnteractton of the stronttum tsotopes where neutron holes fill the 2P½, lg~ levels Thts result m strontmm, however, d~sagrees with the result extracted from two-pamcle transfer given m ref 7) 2 I THE "TOTAL-ENERGY FIT"
In the fit, exctted states and ground states are gtven equal wetght m the constructmn of the functton to be mmtmlzed The functmn that is mmtmtzed ts then 45 __ Eexpt] / 2 = E ]-~theory L--~ --* _t2
where here the E, refer to the total energy of the state TABLE 2 T ~ o - b o d y matrix elements and the smgle-parUcle levels found by least-squares fitting ~) Quant~t)
Total energy
Semont~
Level
((p+)2 Vj(p½)2 J=o
- 0542
-0544
11p½)2[ l/i(g~) 2 s=o
0 853
0 853
0 877
0 893
((p~g~)'~V~(p½g~) s=-~
0 714
0 716
0 686
0 699
((p½g-)~ V[(pxg~) ~S-5
0 195
0 194
0 215
0 220
((g3)21Vl(g~)2)s=o
- 1 707
--1 705
--1 728
--1 702
-0480
Rates --0458
(g~)2~V](gg-) 2 .t=2
--0 613
--0 615
--0 623
--0 644
< (gg_)2,Vl(g~)Z)a,=4.
0 144
0 155
0 149
0 243
((g~)2, p l(ger)z/s=6
0 450
0 437
0 475
0 351
0 565
0 570
0 549
0 588
--7 125
--7 124
--7 129
--7 136
g3 s m g l e - p a r h d e energ3,
6 247
-6 248
--6 244
- 6 250
rms de~latton per level
0 0519
(g~)2 l,
(g~)2 ~J=8
p+ single-particle energ}
0 0522
") All u m t s are MeV b) Thls n u m b e r does not include the c o n t n b u t m n to the
/z
0 0594
0 0799 ~)
from the t r a n s m o n rates
TgBLE 3 Binding energies relative to 8aSr core Nucleus
Exp a)
Total energy
Semont~
Level
Rates
89y 9°Zr °*Nb °2Mo °3Tc 9~Ru °SRh 96pd
7 068 15 434 20 592 28 058 32 162 38 397 41 568
7 125 15 398 20 654 27 993 32 182 38 394 41 513 46 638
7 124 15 399 20 654 27 988 32 178 38 388 41 512 46 638
7 129 15 392 20 638 28 002 32,175 38 405 41 503 46 640
7 136 15 397 20 642 27 989 32 174 38 409 41 514 46 656
All energies are m MeV
a) A H W a p s t r a and N B Gove, Nucl Data Tables 9 (1971) 265
N = 50 SHELL MODEL
483
The matrix elements and smgle-parncle levels that emerged from the total-energy least-squares fit are given in column 2 of table 2 The resulting spectra pre&cted are shown m figs 1-5 Binding energies relative to the SSSr core are gwen m table 3 The rms dewatlon for energy levels obtained m this fit is superior to that obtained in the other three fits considered (see table 2) We consider this mteracnon to be the best, we have thus used this interaction m our forthcoming study of the N --- 49 lsotones s) 2 2 THE "SENIORITY FIT"
The seniority of a state is the number of unpaired pamcles ~t contains A pa~r here ~s defined as two protons in the same orbtt coupled to angular m o m e n t u m zero The total seniority of a state ~s the sum of the semorlt~es in the 2p~ and lg~ orb~tals For j" configurations w l t h j < ~ the seniority, v, is automatically a good quantum number 9) For j = ~ it can be shown that wolanon of semorlty conservanon ,s propornonal to 1o) m = ((g~)3, S = ~, v = llVeffl(g~) 3, S = ~, v = 3) 1 - 2 0 x / 4 ~ [ - 6 5 E 2 + 3 1 5 E 4 - 4 0 3 E 6 + 153Ea]' where E a = <(g~)2lVerfl(g~)2>a The X2 function was ldenncal to that of the "total energy fit" but the fit was performed w~th the ad&tlonal constraint that M vanish The most s~gmficant result that emerged here ~s that a semonty conserwng effectwe lnteractxon is entirely consistent with the known experimental levels m the N = 50 lsotones (compare the rms dewatlons m columns 2 and 3 of table 2) 2 3 THE "LEVEL FIT"
Experimentally the techniques of measurement of binding energies and of excitation energles are &fferent In this fit (and m the "rates fit") we have considered contnbunons from excltanon energies, and from binding energies separately The function to be minimized ~s then 39 6 Z 2 ~--- E (Etlhe°ry-EteXpt) 2"{- E ( Egs the°ry--Egs expt)2 t=l ;=1
The "total-energy" and "level" fits differed only m the construction of their respective Z2 functions The "level fit" followed the generally accepted procedure that appears to have been used m previous investigations 1-5) of this region By contrast, the X2 function of the "total-energy fit" did not & s c n m m a t e between binding energies and excited states A prtort, there appears to be no reason to prefer one approach over the other In this case, the "total-energy" procedure did lead to a shghtly better rms deviation per level, 52 keV compared to 59 keV for the "level fit"
484
D H GLOECKNER AND F J D S E R D U K E
The average o f the differences of c o r r e s p o n d i n g fitted p a r a m e t e r s b e t w e e n the " t o t a l - e n e r g y " a n d t h e " l e v e l " fit is 20 k e V T h e l a r g e s t s u c h d i f f e r e n c e is 62 k e V T h e d i f f e r e n c e s b e t w e e n t h e s e t w o sets o f m a t r i x e l e m e n t s m a y b e i n t e r p r e t e d as a m e a s u r e o f h o w well t h e f i t t i n g p r o c e d u r e h a s d e t e r m i n e d t h e m o d e l s p a c e p a r a m e t e r s for the body of experimental data that was considered TABLE 4 E2 trans~tton rates
Nucleus
Transition
E, (keV)
Internal conversion ')
Mean lifetime (ns) exp b)
°°Zr 92Mo
8+ ~ 6+ 2+ --0 + 11- ~ - 9 -
O'*Rn
141 5 2t82
0 32 0
235
005
semonty
180 ± 9 (014±001)
10 -3
224 032":.10 -3
10 - 3
249 2 29 062 10 -3
279 2 15 061 10 -3
103
43
97
127 5:07
8 + ~6 + 6÷ ~ 4* 2~ ~0 +
148 330 1509
0 29 0 017 0
8 ~ -~ 6*
145
0 33
(102
6-- ~ 4 + 4÷-~2 + "+ -~-0 +
31 ! 755 1428
0 024 0 0
107
222 03
10 - 3
11 1
275 ±10 2 2 2 ~ 0 07 (045-_008)
±7
rates
)
10
11 1
10 ~ 50 042
064
[0 - 3
103
106 042 063
10 - 3
a) L A Shv and 1 M Band, Alpha, beta, and gamma ray spectroscopy, ed K Slegbahn, vol 2 (North-Holland, Amsterdam, 1965) p 1639 b) Data for the 2 + ~- 0 + transitions m 9 ° Z r and in 92Mo were taken from F R Metzger, Nucl Ph)s A182 (1972) 213 and P H Stelson et al, Nucl Data 1 (19651 21, respectively Data for the 8 ÷ ~ 6 + transition m 9°Zr were taken from K G Lobner, Nucl Phys 58 (1964) 49 Data for the 8 + ~ 6 +, 6 + ~ 4 + In 92Mo were from S Cochavl et al, Phys Rev C3 (1971) 1352 Data for the 11- >- 9 - ~ere from C M Lederer et al, Nucl Phys A169 (1971) 449 Lifetimes m 94Rtl were from J M Jaklevic et al Phys Lett 29B (1969) 179
2 4 THE ' RATES FIT' T h e " r a t e s f i t " i n c l u d e s e i g h t E 2 t r a n s i t i o n r a t e s in a d d i t i o n t o t h e level e n e r g i e s ( m a r k e d b y a s t e r i s k s m c o l u m n 3 o f figs 1 - 5 ) m t h e c o n s t r u c t i o n o f t h e f u n c t t o n t o b e m i n i m i z e d T h e /(2 f u n c t i o n is t h e n 39
6
8
Z 2 = E w,[E,-E~,~P'] 2+ E w,[ Eg~ - E g ~ cxP'] z + E w~[B(E2)tfl¢°W-B(E2)~'P'] z, *=1
*=1
k=l
w h e r e w, a n d w k a r e w e i g h t s a s s i g n e d t o e x p e r i m e n t a l d a t a p o i n t s [see r e f
6) f o r a
d i s c u s s i o n o f t h e r e l a t i v e w e i g h t s a s s i g n e d t o e n e r g y levels a n d E 2 t r a n s m o n r a t e s ] , E , a r e t h e e n e r g i e s r e l a t i v e t o g r o u n d s t a t e s , a n d E~ s a r e t h e g r o u n d s t a t e e n e r g i e s T h e e x p e r i m e n t a l E 2 t r a n s i t i o n r a t e s i n c l u d e d m t h i s fit a r e c o m p a r e d p r e d l c t t o n s f r o m t h i s fit a n d wxth t h o s e f r o m t h e " s e m o n t y f i t " in t a b l e 4
wtth the
N = 50 SHELL MODEL
485
An additional parameter was Introduced for this least-squares fit This parameter as the reduced E2 transition rate matrix element, Q2 (given above), from which an effective proton charge eelf = 1 72e is obtained 6) The "rates fit" can predict the inhibited 8 + ~ 6 + transition rate in 94Ru The other three interactions considered here predict this rate too fast by a factor of about 20 (see, for example, results for the "seniority fit" in table 4) Other transition rates are predicted shghtly better with the "rates fit" than with the other interactions determined in this study However, the agreement between theory and experiment for energy levels is slightly poorer, the rms deviation per level is 80 keV compared to 59 keV for the "levels fit" The "rates fit" has shown that contrary to previous expectation [ref ~x)] the inhibited 8 + --* 6 + E2 rate in 94Ru may be explained within the (2p½, lg~) model space This result does not deny the probable role of configuration mixing from outside the (2p~, lg~) model space in the inhibition of this transition The "rates fit" has been constructed in such a way that the contribution to the rate f r o m the large components of the 8 + and 6 + wave functions cancel so that the transmon proceeds through small components in these wave functions It appears unlikely that small components of configurations from outside our model space do not s~gmficantly alter the details of the inhibition of th~s rate
3. Even N = 50 nuclei
3 ! THE NUCLEUS 9°Zr The nucleus 9°Zr is described in our model as two protons outside of the aaSr inert core The entire theoretical spectrum is shown in fig 1 The binding energy of the ground state is given in table 3 The most distinguishing feature of the 90Zr spectrum is that its first excited state is a 0 + state This xs predicted from even the simplest of two-body lnteracUons (e g , the surface-t5 interaction) given the 2 p u l g ~ single-particle splitting revealed in the 89y spectrum The 3 - octopole excitation at 2 748 MeV and the 2 + state at 3 33 MeV are the only states below about 4 MeV without model space analogues Collective 3- states at approximately this excitation energy are seen throughout the A = 90 region, and in particular in SSSr at 2 734 MeV and In °2Mo at 2 849 MeV. All four theoretical interactions invert the order of the 2 + and 5- states One explanation of this is that the experimental 2 + state contains significant admixtures of the configurations {(p~)-2(p,)2(g~)2)s=2, Ip~lp,(g~)2)s=2 and If~Ip~(g~)2)s= 2 The position of the model space 2 + state in 9°Zr may be estimated (independently of our calculations) from the 92M0 and 94Ru spectra If seniority were conserved, the spacing between the 8 +, 6 +, 4 + and 2 + states in 94Ru, 92M0 and 9°Zr should be the same 5, 6) The fact that seniority is an excellent quantum number for the N = 50 nuclei has been noted by many investigators 2, ,, 5)
D H GLOECKNER
486
AND F J D SERDUKE
8÷
- -
_
142
148
145
372
330
312
775
755
92M0
94Ru
6*
4
*
--
899
/ /
2+
/
90Zr
Fig 6 The e x p e r . n e n t a l 8 +, 6 +, 4 + and 2 + states o f 9°Zr, 92Mo and 94Ru showing the sphttmgs (m keV) o f adjacent levels These sphttmgs are the same m all the nucle~ for a semorlty conserwng interaction in the 2p~-lg 3 model space This figure partlcularlv illustrates the depression o f the 9°Zr 2 + state
TABLE 5 Spectroscopic factors for smgle-pamcle transfer
Target nucleus
sg~
9°Zr 92Mo
Target spin
1_,-
0+ 0+
Final state j~r (MeV)
CoInpllrlSOll ~tth Mrlpplng reacttolt; 0 + (g s ) 1 32
0 ÷ (l 76)
0 68
4 - (2 73) 5- (2 32)
10 10
92Mo
0+ O*
* "Total-energy fit" used ~) R e f 12) ~) R e f 15)
exp
1 31 a) 0 52 a) 1 0 (norm) ") I 24 a)
~+ ( g s )
034
0 39 b), 0 42 D, 0 3 a)
½- (0 105)
0 93
0 88 b), 0 87 ~), 0 9
~+ ( g s )
074
0 64*), 0 52t), 0 67 d), 0 5 ' )
0 30
0 205 *), 0 29 r), o 3 d) 0 2 8 ' )
- (0 38)
9°Zr
C2S
theory*
Compartson i~tth ptcAup reacttons ½- (g s ) 1 32
1 32 (norm) c)
~+ (0 914)
0 68
0 76 c)
~+ (g s )
2 51
2 6 b), 2 7 d)
} - (0 105)
1 40
1 7 h), 1 4 d)
~) Ref x6)
a) R e f t-)
e) R e f is)
r) R e f lOj
a)
g) Ref zo)
N ~ 50 SHELL MODEL
487
and is apparent from the "seniority fit" shown in figs 1-5 Taking the model space 2 + state to be the same energy below the 9°Zr 8 + state as the 8 + ~ 2 + splitting in 9 2 M o o r 9 4 R u would put the 9 ° Z r 2 + approximately 200 keV higher than ]t is seen experimentally This is illustrated in fig 6 which shows the experimental 8 +, 6 +, 4 + and 2 + states of Zr, M o and Ru Further ewdence of the Influence of other configurations m the 9 ° Z r 2 + state comes from its anomalously short hfetlme as indicated m table 4 There is good agreement w]th available spectroscopic data 12) for states m 9 ° Z r Table 5 gives experimental and theoretical single-particle transfer results 4-
30
o*.. 21/2"
25
5"---
17/2+-..
6" 4*
15/2-=--
4* ....
20 9/2*
5/2-
ii/z+.. 5/2:---
15
1312./
2*
s/2*..
lO
7/2* 05
_
112-
112-
V2÷
9/2*
EXPT
TOTAL ENERGY
95Rh
O* ....... TOTAL ENERGY
96pd
Fig 7 Theoretical predictions for 95Rh and 96pd using the total energy fit In 9SRh, all levels to 1 5 MeV are shown as are selected levels that might be populated m some (heavy ion, vn) reactions The experimental results are from ref zz) In 96pd, all levels through the 8 + are shown 3 2 THE NUCLEUS 92Mo
The nucleus 9 2 M o lS described here as four protons filling the 2p~, lg~ shell-model orbltals Excellent agreement with experimental energy levels is obtained for all four interactions (fig 2) Recently two 4 - states predicted by all four models to be near 3 0 MeV have been seen m a 9 2 M o ( n , n ' 7 ) experiment 13) The lowest of these states is predicted in all four 4models to have the structure [(lg~)~+ × (p½)] Jv==4 Such a seniority-four state would be difficult to populate in a direct process The lowest 5- state, however, is a senioritytwo state in all our models. A discussion of transition rates m thls nucleus, and of the tentaUve 12 +, has already been made m ref 6)
488
D H GLOECKNER AND F J D SERDUKE
3 3 THE NUCLEUS 94Ru The nucleus ' 4 R u is described here as (2p~, lg~) 6 A comparison between experimental and theorencal energy levels for this nucleus is gwen m fig 3 All models predict a second unobserved 4 + state near 2 4 MeV This state is predominantly semority-two (g~_)6 The "rates fit" is the only interacnon that correctly predicts the 8 + ~ 6 + E2 transition m this nucleus 6) The requirement that th~s transmon rate be severely inhibited (1 e , the theoretical B(E2) be small compared to the single-particle estimate) most d~stmgmshes the "rates fit" from the three other interactions The major differences m the 94Ru spectrum is a shift of the lowest 6 + and second 4 + states upward for the rates fit The agreement w~th experiment, however, remains excellent 3 4 THE NUCLEUS 96pd The model space configuration for 96pd 1s (2p~, lg~) s This proton-rich nucleus has not yet been seen experimentally It might be posstble to populate states in this nucleus by (~, 4n) or (3He, 3n) on 96Ru In antlclpatmn of such an experiment, the low lying model space states are shown in fig 7 Our "total-energy fit" predicts the binding energy to be 46 638 MeV (table 3) The suggested reactmns typically populate yrast levels, so that we expect to see a cascade of stretched E2 terminating at the ground state, the lowest 8 +, 6 +, 4 + and 2 + states are shown in fig 7 Goodness of seniority m the effective interactmn and the systematlcs of 8 + ~ 6 + ~ 4 + ~ 2 + states m the other even-Z nuclm (fig 6) suggest strongly that this cascade should yield y-rays of approximately 140, 320 and 760 keV Barring an unusual mhlbmon, the 8 + -~ 6 + transmon should have a mean life of approxtmately 200 ns
4. Odd N -- 50 nuclei
4 1 THE NUCLEUS 91Nb The nucleus 91Nb is described here as three protons filhng the 2p~, lg~ orbttals The spectrum is shown m fig 4 An energy gap occurs in this nucleus due to the fact that the 2p, smgle-parttcle level is lower than the lg~ smgle-pamcle level Only one positive p a n t y state can have a filled 2p½ orbital - namely the ~+ ground state All other positive p a n t y states wdl he approximately twice the slngle-pamcle sphttmg above the ground state The lowest negatwe p a n t y state is the ½- state at 0 105 MeV which has the configuration (2p~)x (lg~)~=o) The next lowest negative parity states will then be higher by roughly the difference between ((lg~)2]Veffl(lg~)2)l and ((lg,)ZlVl(lg~)2)s=o The ~z- state at 1 842 MeV is seen in proton pickup 12) on 92M0 to have most of the lf~_ smgle-pamcle strength The 2p~ single-hole strength is more fragmented, though most of it is seen m the -~- state at 1 606 MeV Both of these states are outside our model space and are not considered here
N = 50 SHELL MODEL
489
Excellent agreement with experiment i4-17) Is found for spectroscopic factors for single-particle transfer to states m this nucleus (table 5) 4 2 THE NUCLEUS 9aTc
The nucleus 93Tc 1s described in th~s model as five protons outside an inert SSSr core The spectrum is shown m fig 5 Low-lying states in thas nucleus are described well by all four fits Experimental 17-2o) and theoretical spectroscopic factors for singleparticle stripping to the lowest two states are given in table 5 A few high-spin states in this nucleus have recently been identified 21) from 92M0(0~, p2n)93Tc These states are in excellent agreement with all four models except for a tentative (~s_+) state at 4 256 MeV This state appears nearly 1 MeV higher in all four models 6) 4 3 THE NUCLEUS 95Rh Our model space description of 95Rh has seven protons filhng the 2P½-1g~ orbltals Recently, the spins and parities of ~ts lowest two states were ldenufied by Welffenbach et al zz). These are the $+ ground state and an isomeric ½- state at 543 keV This ½- state is an M4 isomer with a mean hfe of 173 sec Again, this M4 transition further supports our assignment of the posltwe sign for the off-diagonal matrix element (table 1) The low-lying states of °SRh predicted by our total energy fit are shown m fig 7 5. Conclusion
We have deduced four sets of matrix elements for the proton-proton effective mteracUon m the 2p½-1g~ model space. All of the mteracUons were fit to 45 energy levels m the N = 50 nuclei. One of them, the "rates fit", was also fit to the E2 transition rates in these nuclei. The resulting matrix elements are not qualltatwely different from sets of matrix elements deduced in previous studies 2- s) Our interest in tlus study has been to explore the limits of the 2p,-lg~ model to accommodate existing energy-level and transition-rate data with and without the constramt of seniority conservatlon. There is one off-diagonal matrix element m this model space, all energy levels and E2 hfet~mes are independent of its sign. However, a calculation of the hfetlmes of the M4 transitions m the odd-Z N = 50 nuchdes unambiguously in&cares that ((2p½)21V, rfl(lgi)2)s= o lS positive The nature of the proton-proton effective interaction in this region is most clearly revealed by contrasting the results of the various fits. In particular, the near equality of the "semonty fit" whach was forced to conserve semonty and the "total-energy fit" which was not so constramed indicates that semorlty ~s a better quantum number within the identical nucleon 2p~r-lg I configurations than earlier studies 2) indicated. In fact, the rms dewatlon per level for the "semority" and "total-energy" fits differ
490
D H GLOECKNER A N D F J D S E R D U K E
only by a fraction of a kdovolt and this alone enables us to conclude that the current body of experimental energy-level data does not call for any semonty breaking m the model space effective interaction Contrasting the "rates" and "level" fits m&cates that only moderate adjustments m the two-body matrix elements are required to reproduce the xnhlb~ted E2 transitions in 9 4 R u The resulting small vtolatlon of sentonty conservation Is, however, essenttal to the explanation of these inhibitions wlthtn the model space Since the reqms~te mhlbmon mechamsm could arise through admixtures of configurations from outside the model space, th~s reqmrement for semorlty breaking from the transition-rate data should be wewed as weaker than the previous re&cation of seniority conservation from the energy-level data The "total-energy" and "level" fits were the results of identical parametenzatlons and were fit to ldent~cal sets of data The only difference was the construction of the function that was minimized in the fitting process A comparison of the resulting fits could reasonably be viewed as a scale by which the slgmficance of &fferences m twobody matrix elements may be measured We would hke to thank Profs R Lawson and M Macfarlane for their participation m the early phase of this work and their continued adwce throughout the course of this investigation References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22)
I Talml and 1 Unna, Nucl Phys 19 (.1960) 225 S Cohen, R D Lawson, M H Macfarlane and M Soga, Phys Lett 10 (1964)195 J B Ball, J B McGrory and J S Larsen, Phys Lett 41B (1972)581 J Vervler, Nucl Phys 75 (1966) 17 N Auerbach and I Talmb Nucl Phys 64 (1965) 458 D H Gloeckner, M H Macfarlane, R D Lawson and F J D Serduke, Phys Lett 40B (1972) 597 J E Kttchlng, W G Davies, W J Darcey, W McLatchJe and J Morton, Nucl Phys A177 (1971) 433 D H Gloeckner, R D La,sson and F J D Serduke, to be pubhshed G Schwartz and A de-Shaht, Phys Rev 74 (1954) 1257 G Racah and I Talml, Physlca 18 (1952) 1097, J B French, Nucl Phys 15 (1960)393 J B Ball, J B McGrory, R L A u b l e a n d K H Bhatt, Phys Lett 29B (1969)182 G Vour~opoulos and J O Fox, Phys Rev 177 (1969) 1558 G P Glasgow, J D Brandenberger, K Slnram and M T McElhstrem, Report of the Experimental Nuclear Physics Group, Umv of Kentucky, December 1972, p 17 R Chapman, .I E Kltchmg, W McLatchle, Nucl Phys A196 (1972) 347 K T Knopfle, M Rogge, C Mayer-Borlcke, J Pedersen and D Burch, Nucl Ph~s A159 (1970) 642 M R Cates, J B Ball and E Newman, Phys Rev 187 (1969)1682 J Plcard and G Bassam, Nucl Phys A131 (1969) 636 P J Rdey, J L Horton, C L Hollas, S A A Za~dJ C N Jonesand.] L Ford, Ph3s Rev C4 (1971) 1864 Y Shamal, D Ashery, A I Yavm, G Bruge and A Chaumeaux, Nucl Phys A197 (1972) 211 R L Kozub and D H Youngblood, Phys Rev C4 (1971) 535 M Grecescu, A Ndsson and L Harms-Rmgdahl, Nucl Phys A212 (1973) 429 C Welffenbach, S Gularth and J K P Lee, private cornrnunlcatlon
l
Nuclear Phystcs A220 (1974) 477-490, (~) North-Holland Pubhshing Co, Amsterdam
$
Not to be reproduced by photoprmt or m~crofilmwithout writtenpermissionfrom the pubhsher
S H E L L M O D E L S T U D Y O F N ffi 50 N U C L E I t D H GLOECKNER and F J D SERDUKE Argonne Nattonal Laboratory, Argonne, Ilhnots 60439 Received 11 October 1973 Measured E2 and M4 transltzons and energy levels m the N = 50 tsotones wath Z :> 38 are satisfactorily reproduced by a shell model with proton configurations (2P½, lg])" Four sets of two-body matrix elements are found by least-squares fitting to energy levels and transition rates m the N = 50 ~sotones These interactions include one that exactly conserves seniority and another that fits the strongly mh~bRed 8+--> 6 + transition m 94Ru Slight seniority nonconservatton is reqmred to predict inhibited N = 50 transiUon rates The sign of the offdmgonal matrix element is determined by comparison with experimental M4 rates The semonty conserwng mteractlon is found to be equivalent to the best semonty nonconserwng interaction for prediction of energy levels
Abstract
1. I n t r o d u c t i o n
Nuclear properties of the N = 50 lSOtones have been described i - 5 ) by a shell model with SSSr as a n inert core a n d valence p r o t o n s restricted to the (2p½, lg~) single-particle levels M a n y new experimental energy levels have been identified recently a n d a few E2 hfetlmes measured Most of the new data agree well with prediction based on the older models However, the 8 + ~ 6 + t r a n s m o n In 94Ru c a n n o t be explained on the basis of these older models I n a n effort to u n d e r s t a n d the n a t u r e of the p r o t o n - p r o t o n effective interaction a n d the Inhibited E2 rate in 94Ru, we have constructed, by least-squares fitting, four effective i n t e r a c h o n s with the (2p~, lg~) p r o t o n model space I n a p r e h m m a r y report 6) of this work we have discussed a t w o - b o d y interaction (the "rates fit" in this work) that does adequately describe the k n o w n t r a n s i t i o n rates m the N = 50 lsotones, including the 8 + ~ 6 + t r a n s i t i o n m 94Ru. I n that report we concentrated o n E2 t r a n s I h o n rates a n d c o m p a r e d the results with those o b t a i n e d on the bas~s of a n older interaction 2) Here we l o o k at other nuclear properties of the N --- 50 lSOtones a n d c o m p a r e that "rates fit" with three other interactions I n sect 2 we discuss the c o n s t r u c t i o n of four different effectwe interactions between p r o t o n s All four interactions are the result of a least-squares fitting procedure One interaction requires that the two-body interaction conserve seniority A n o t h e r reqmres a fit to the k n o w n E2 transition rates i n the N = 50 lSOtones The other two fit only experimental energy levels, these differ m the c o n s t r u c t i o n of the f u n c t i o n to be minimized We also find that the lifetimes of the M 4 Isomers in this t Work performed under the auspices of the US Atomic Energy Commission 477
ot -
O÷
SENIORITY
o+
O+
TOTAL ENERGY
*
,
,
EXPT
OL-----*
0÷
2+
5-
5~ 4-/
4*
8* 6÷ 2*
LEVEL
0"
0"--
5-
2L
4-...
8+, 6t--
RATES
O*
0÷
5-"
2t.
4-
6t 4~ -
8÷
Fig l Expcrlmelltal and theoretical energy levels for 9°Zr All energle~ are m MeV and refer to the experimental g r o u n d state F o r the "level" and "rates" fits only exc~tahon energ,es are s h o w n , ~ e , the theoretical spectrum is shifted so that experimental and theoretical ground states hne up States included m the fit are marked with an asterisk All theorettcal levels are shown Experimental levels were taken f r o m S S Ghcksteln et al, Phys Rev C4 0 9 7 1 ) 1818
0
2c
2~
2+ 5-
4-¸¸¸
4-
2÷ 5-
4*
4*
3c
8+ 6+
35 - 8* 6*
9°Zr
-
0+ 0+
2*
4÷
6* 5~
4~ 4-"
7-
2÷
9-
Iv
2~
SENIORITY
Or__--TOTAL ENERGY
Z+
8" '" 6+ 50÷ 4+
2* ,4-" 4-
3-
2"~
""
2+ - •
9---'-----
II-
EXPT
4+
2+
9-
Ir
*
.
*
.
LEVEL
0+
2*
4÷
, 8 * 6÷--~'--
4-
4~___
x 2t~
K-
*
*
*
RATES
0÷
2*
4+~--
8+ 6*...
44~
2÷
3-
2+
9-
II-
Fig 2 Experimental ,and theoretical energy levels for 92Mo All levels are s h o w n to 3 MeV Selected levels are s h o w n above 3 MeV See also caption for fig 1 The experimental data were taken f r o m D C Kocher et al, Nucl Data B7 (1972)299, Ledereretal, Nucl Phys A169 (1971) 449 and from ref 13)
I
20-
8÷ 6+ 25-5--0+ 4+
2÷ 43C-4-
3 5 --4* 3-
4 C -- 2*
9-
4 5_11-
92Mo
m
m 7~
Z
7~
O t© m (3
-T-
~7
--
I
SENIORITY
TOTAL ENERGY
_
0+.__
"-
0+
2+
4÷
EXPT
8+ 5- --.
0t
2+
4 +_
O~
0+
....
.
'
= ~
4------42~
0 +-
....... - , ~
9-
*
~
~
LEVEL
0+
2+
O r . .
6+
442,~
7----
o.
'99-
-
RATES
0+
2+
4 +-
4-.-4---Zt.. 6+~ • 0+ 8
99-
Fig 3 Experlmeiltal and theoreUcal energy levels for 94Ru Selected levels only are shown above 2 9 MeV Sec caphon for fig l The experimental data were taken from J M Jaklewc e t a l , Phys Lett 29B (1969) 179 and from J B Ball et a l , Phys Rev Lett 29 (1972) 1014
i~
2*
4+
6_
2~
44-
7J
99-
7-~
'....
o\
99-
or.
Y_
-
2
3C-
45
94Ru -
TOTAL ENERGY
I/-~-9/÷ - - -
5/--
9/~ 9/£ 7/÷--3/---
I i/31'1" - It/ ~ - 13/~---. 7C-51"1. - 17/15/~--
9/~-
15/~ - -
_ _ - -
-
_ _ _
I/9/+-_
5,'-~
5/9//~/ 31-, 37/+ /~
EXPT
(I I / ~ . ~ (I I/-) - 5 / L 17/- ~ - - ~ (5/-} I ~ / + 7/- - 13/~+
7/++ 9Ix
SENIORITY
I/---9 / L - -
51-----
9/~ 9/L - 7 / L - 3/-
I I/~ 3/i - l I/~ 13/~ - 7/~ 5 / L 17/~1 3 / ~ -
--
5 / - ~ 17/+ --
3C
2t/+
9~---~\--
I ~
15/+ 17/~---
2l/+
-
-
LEVEL
it, I / - - ~. 9 / L -
N 5/"
: 9~ it 7'~----w- 5 / ~
3/+ ill+ 13// - 7/- - 51+--15/~. 17/----9/------
,/--_
195~ /
15./'t _ _ ..t7/+~-
N21/+
-
-
3,';-----
9/'1"RATES
5/:----
----- - 7/+ 3/=_-
9/~
7/~1:51+ - 1715/'+- " - 1:5/--- _ _ 9/-
I II zI I/"---
15/:- - 91+_
17/~ - 15V~.
21/+
Fig 4 Experimental and theoretical energy levels for 91Nb The complete theoretical spectrum is shown Sec caption for fig 1 The experimental data were taken from F Rauch, Z Phys 243 (1971) 105, J L Horton et a l , Nucl Phys A190 0972) 362 and from ref 21)
0
O5
1
2
2~
3C
15/~ 17/+
3 5 - 21/+
91Nb
4~
t'*
O
r~ r', r"
II
480
D
H
GLOECKNER
AND
F
J
D. SERDUKE
region lead to an unambiguous choice for the sign of the off-diagonal matrix element of the effective interaction In sect 3 we discuss the even N = 50 nuclei, and m sect 4 ~ve discuss odd nuclei W e present our conclusions in sect 5
2. The effective proton-proton interaction There are nine proton-proton t w o - b o d y matrix elements and two proton singleparticle levels within the (2p~, lg~) m o d e l space Thesv eleven parameters are varied m a least-squares search that includes 45 energy levels in the N = 50 lsotones Levels included in the fits are indicated by asterisks in figs 1-5 We consider four separate fits which differ in the additional lestrlctions placed on the least-squares fit The "total-energy fit" includes only energy levels, but m i n i m i z e s the differences in the experimental and theoreUcal total energies relative to the SaSr lnelt core The 93Tc 51 +
21'1•
5/+ ~ ~ _ 15/ °/
_ _ --'15/+
i~1/+
17115/~--
17/+ -
( 15/+}
7/+
- -
91"
-
9/-
- -
919/* 3/÷
1 51 III÷ 5151 ÷
II/~
- - ----
- _ _
5/-
- - *
- -
- -
(3/-)
1:3./*
- -
9/t
9/+
-
91-
9/"
- -
-
91-
57"
31 + 3/-
- - -
5/-
_ _
___
tlf,~
- -
lip
_ _
i,~d4-
_ _
--wITW+
515151 +
5/1.
2:11+ 17/* 151" 17/7/+
7Z'~
91*
- ~ *
- -
I~V-
9/( .31- )~ it/+
- -
131+
- -
17/+
-
3/-
S/-
5/+ 21/+ I 7/-
17/-* - -
9: --9/91+ - - 3/+
20
5/+
15/"i"
17/" I- 17I+ T 13If- 7/+
--
51 +
,o~ 7/+
I 7/+ 0 5~ I/-
O~
E
II-
9/÷
9/I. TOTAL
ENERGY
- -
7/+ - - *
7:
II-
- - *
1t-
9/+
- - *
91+
SENIORITy
EXPT
- -
7/+
- -
II-
91 ~" - LEVEL
RATES
Experimental and theoretical levels for 93Tc All levels through the first a~ - level ( ~ 2 3 MeV) are shown Selected levels at higher energy are shown See caption for fig 1 The experimental data Fig 5
~ere taken from P
J
Rdey
e t a l , P h y s R e v C 3 ( 1 9 7 1 ) 186, f r o m A 1 9 7 ( 1 9 7 2 ) 211 a n d f r o m ref 2 t )
Y
Shamai
et al,
Nucl
Ph'~s
"seniority fit" has the additional constraint that the interaction c o n s e i v e seniority The "'level fit" includes only energy levels, but m i n i m i z e s the s u m of the difference between experimental and theoretical excitation energies The "rates fit" IS similar to the "level fit" but in addition requires that the k n o w n N = 50 E2 transition rates be fitted
N = 50 SHELL MODEL
481
A p r o b l e m c o m m o n to all f o u r least-squares fits ts the d e t e r m i n a t i o n o f the sign o f the off-dmgonal m a t r i x element, ((2p½)21V, rfl(lg~)2)s= o The energy o f all states m the N = 50 nucle~ ~s i n d e p e n d e n t o f this sign M o r e o v e r , E2 t r a n s i t i o n rates are I n d e p e n d e n t o f th~s sign because the effective reduced E2 trans,tlon rate m a t r i x element Q2 involves only the l g l single-particle level where
Q2 = eeff(g~llrZY2[lg~) F o r t u n a t e l y , t w o - p a r t , t i e transfer rates a n d M 4 t r a n s t l o n rates are sens~twe to the s~gn o f the off-dmgonal m a t r i x element Since there are m a n y ambiguities assocmted w~th the analys~s o f two-particle transfer reactions, we have chosen to d e t e r m i n e the TABLE 1 M4 trans~tmns m the N = 50 lSOtones
Nucleus
Trans,tlon
E~, Internal Branching (MeV) conother/ version") M4 b)
~+1 --->½g's
0911
91Nb 93Tc
½-a --> 9+ 2gs .1'-1 --->~;s
0391
034
18/82
95Rh
a- 1--~'~; s
0543
011
10/90 ~) 17
166
1/99 3/97
Theoretical ~) r,. +(sec)
s9y
0 105
00084
Exp b) rm (sec)
Smgleparticle rm- (sec) estimate a) "rm (sec)
23 1 7 7× 106 37
60 11 6× 106 1 0~ 109 1 9 25"<105
69
106
103
74×103
103
102~)
4 4 102 2 3 . 103 1 1 l02
~) L A Sllv and I M Band, Alpha, beta, and gamma ray spectroscopy, ed K Slegbahn, vol 2 (North-Holland, Amsterdam, 1965) p 1639 b) C M Lederer, J M Hollander and I Perlman, Table of isotopes (Wdey, New York. 1967) p 224 ¢) Total-energy fit used for both positive (r,. +) and negatwe (rm-) sign of the 0 + off-diagonal mamx element (p2IVlg2) normahzed to fit the trans,tlon m soy and adjusted for branching and internal conversion d) S A Moszkowskl, Alpha, beta. and gamma ray spectroscopy, ed K S,egbahn, vol 2 (NorthHolland, Amsterdam, 1965) p 863 e) Ref 22) sign f r o m e x p e r i m e n t a l M 4 hfetlmes in N = 50 nuclei There Is an isomeric M 4 t r a n s i t i o n between the first excited state a n d the g r o u n d state in each o f the o d d - A N = 50 nuclei This i s o m e r is a .~+ to ½- t r a n s i t i o n in 8 9 y a n d a ½- to ~+ t r a n s i t i o n in 9 I N b , 93Tc a n d 95Rh U n d e r the usual a s s u m p t i o n t h a t the M 4 t r a n s i t i o n proceeds via a o n e - b o d y o p e r a t o r , the lifetime o f the isomer d e p e n d s on the effective r e d u c e d M 4 m a t r i x element W e d e t e r m i n e this m a t r i x element e m p m c a l l y f r o m o u r smgleparticle nucleus 8 9 y T a b l e l gwes the M 4 lifetimes calculated for the two choices o f the sign o f the ((2p~)21Verf[(lg~t) 2) m a t r i x element. T h e results for the " t o t a l - e n e r g y fit" are presented, all f o u r interactions give s~mllar results T h e ewdence ~s clear A g r e e m e n t with e x p e r i m e n t ~s far superior ff the off-diagonal m a t r i x element is chosen to be positive W e note in passing t h a t th~s choice agrees w~th the s~gn p r e d i c t e d by a surface delta interaction
482
D. H G L O E C K N E R A N D F J D S E R D U K E
We have found the same result for a shell-model effective lnteractton of the stronttum tsotopes where neutron holes fill the 2P½, lg~ levels Thts result m strontmm, however, d~sagrees with the result extracted from two-pamcle transfer given m ref 7) 2 I THE "TOTAL-ENERGY FIT"
In the fit, exctted states and ground states are gtven equal wetght m the constructmn of the functton to be mmtmlzed The functmn that is mmtmtzed ts then 45 __ Eexpt] / 2 = E ]-~theory L--~ --* _t2
where here the E, refer to the total energy of the state TABLE 2 T ~ o - b o d y matrix elements and the smgle-parUcle levels found by least-squares fitting ~) Quant~t)
Total energy
Semont~
Level
((p+)2 Vj(p½)2 J=o
- 0542
-0544
11p½)2[ l/i(g~) 2 s=o
0 853
0 853
0 877
0 893
((p~g~)'~V~(p½g~) s=-~
0 714
0 716
0 686
0 699
((p½g-)~ V[(pxg~) ~S-5
0 195
0 194
0 215
0 220
((g3)21Vl(g~)2)s=o
- 1 707
--1 705
--1 728
--1 702
-0480
Rates --0458
(g~)2~V](gg-) 2 .t=2
--0 613
--0 615
--0 623
--0 644
< (gg_)2,Vl(g~)Z)a,=4.
0 144
0 155
0 149
0 243
((g~)2, p l(ger)z/s=6
0 450
0 437
0 475
0 351
0 565
0 570
0 549
0 588
--7 125
--7 124
--7 129
--7 136
g3 s m g l e - p a r h d e energ3,
6 247
-6 248
--6 244
- 6 250
rms de~latton per level
0 0519
(g~)2 l,
(g~)2 ~J=8
p+ single-particle energ}
0 0522
") All u m t s are MeV b) Thls n u m b e r does not include the c o n t n b u t m n to the
/z
0 0594
0 0799 ~)
from the t r a n s m o n rates
TgBLE 3 Binding energies relative to 8aSr core Nucleus
Exp a)
Total energy
Semont~
Level
Rates
89y 9°Zr °*Nb °2Mo °3Tc 9~Ru °SRh 96pd
7 068 15 434 20 592 28 058 32 162 38 397 41 568
7 125 15 398 20 654 27 993 32 182 38 394 41 513 46 638
7 124 15 399 20 654 27 988 32 178 38 388 41 512 46 638
7 129 15 392 20 638 28 002 32,175 38 405 41 503 46 640
7 136 15 397 20 642 27 989 32 174 38 409 41 514 46 656
All energies are m MeV
a) A H W a p s t r a and N B Gove, Nucl Data Tables 9 (1971) 265
N = 50 SHELL MODEL
483
The matrix elements and smgle-parncle levels that emerged from the total-energy least-squares fit are given in column 2 of table 2 The resulting spectra pre&cted are shown m figs 1-5 Binding energies relative to the SSSr core are gwen m table 3 The rms dewatlon for energy levels obtained m this fit is superior to that obtained in the other three fits considered (see table 2) We consider this mteracnon to be the best, we have thus used this interaction m our forthcoming study of the N --- 49 lsotones s) 2 2 THE "SENIORITY FIT"
The seniority of a state is the number of unpaired pamcles ~t contains A pa~r here ~s defined as two protons in the same orbtt coupled to angular m o m e n t u m zero The total seniority of a state ~s the sum of the semorlt~es in the 2p~ and lg~ orb~tals For j" configurations w l t h j < ~ the seniority, v, is automatically a good quantum number 9) For j = ~ it can be shown that wolanon of semorlty conservanon ,s propornonal to 1o) m = ((g~)3, S = ~, v = llVeffl(g~) 3, S = ~, v = 3) 1 - 2 0 x / 4 ~ [ - 6 5 E 2 + 3 1 5 E 4 - 4 0 3 E 6 + 153Ea]' where E a = <(g~)2lVerfl(g~)2>a The X2 function was ldenncal to that of the "total energy fit" but the fit was performed w~th the ad&tlonal constraint that M vanish The most s~gmficant result that emerged here ~s that a semonty conserwng effectwe lnteractxon is entirely consistent with the known experimental levels m the N = 50 lsotones (compare the rms dewatlons m columns 2 and 3 of table 2) 2 3 THE "LEVEL FIT"
Experimentally the techniques of measurement of binding energies and of excitation energles are &fferent In this fit (and m the "rates fit") we have considered contnbunons from excltanon energies, and from binding energies separately The function to be minimized ~s then 39 6 Z 2 ~--- E (Etlhe°ry-EteXpt) 2"{- E ( Egs the°ry--Egs expt)2 t=l ;=1
The "total-energy" and "level" fits differed only m the construction of their respective Z2 functions The "level fit" followed the generally accepted procedure that appears to have been used m previous investigations 1-5) of this region By contrast, the X2 function of the "total-energy fit" did not & s c n m m a t e between binding energies and excited states A prtort, there appears to be no reason to prefer one approach over the other In this case, the "total-energy" procedure did lead to a shghtly better rms deviation per level, 52 keV compared to 59 keV for the "level fit"
484
D H GLOECKNER AND F J D S E R D U K E
The average o f the differences of c o r r e s p o n d i n g fitted p a r a m e t e r s b e t w e e n the " t o t a l - e n e r g y " a n d t h e " l e v e l " fit is 20 k e V T h e l a r g e s t s u c h d i f f e r e n c e is 62 k e V T h e d i f f e r e n c e s b e t w e e n t h e s e t w o sets o f m a t r i x e l e m e n t s m a y b e i n t e r p r e t e d as a m e a s u r e o f h o w well t h e f i t t i n g p r o c e d u r e h a s d e t e r m i n e d t h e m o d e l s p a c e p a r a m e t e r s for the body of experimental data that was considered TABLE 4 E2 trans~tton rates
Nucleus
Transition
E, (keV)
Internal conversion ')
Mean lifetime (ns) exp b)
°°Zr 92Mo
8+ ~ 6+ 2+ --0 + 11- ~ - 9 -
O'*Rn
141 5 2t82
0 32 0
235
005
semonty
180 ± 9 (014±001)
10 -3
224 032":.10 -3
10 - 3
249 2 29 062 10 -3
279 2 15 061 10 -3
103
43
97
127 5:07
8 + ~6 + 6÷ ~ 4* 2~ ~0 +
148 330 1509
0 29 0 017 0
8 ~ -~ 6*
145
0 33
(102
6-- ~ 4 + 4÷-~2 + "+ -~-0 +
31 ! 755 1428
0 024 0 0
107
222 03
10 - 3
11 1
275 ±10 2 2 2 ~ 0 07 (045-_008)
±7
rates
)
10
11 1
10 ~ 50 042
064
[0 - 3
103
106 042 063
10 - 3
a) L A Shv and 1 M Band, Alpha, beta, and gamma ray spectroscopy, ed K Slegbahn, vol 2 (North-Holland, Amsterdam, 1965) p 1639 b) Data for the 2 + ~- 0 + transitions m 9 ° Z r and in 92Mo were taken from F R Metzger, Nucl Ph)s A182 (1972) 213 and P H Stelson et al, Nucl Data 1 (19651 21, respectively Data for the 8 ÷ ~ 6 + transition m 9°Zr were taken from K G Lobner, Nucl Phys 58 (1964) 49 Data for the 8 + ~ 6 +, 6 + ~ 4 + In 92Mo were from S Cochavl et al, Phys Rev C3 (1971) 1352 Data for the 11- >- 9 - ~ere from C M Lederer et al, Nucl Phys A169 (1971) 449 Lifetimes m 94Rtl were from J M Jaklevic et al Phys Lett 29B (1969) 179
2 4 THE ' RATES FIT' T h e " r a t e s f i t " i n c l u d e s e i g h t E 2 t r a n s i t i o n r a t e s in a d d i t i o n t o t h e level e n e r g i e s ( m a r k e d b y a s t e r i s k s m c o l u m n 3 o f figs 1 - 5 ) m t h e c o n s t r u c t i o n o f t h e f u n c t t o n t o b e m i n i m i z e d T h e /(2 f u n c t i o n is t h e n 39
6
8
Z 2 = E w,[E,-E~,~P'] 2+ E w,[ Eg~ - E g ~ cxP'] z + E w~[B(E2)tfl¢°W-B(E2)~'P'] z, *=1
*=1
k=l
w h e r e w, a n d w k a r e w e i g h t s a s s i g n e d t o e x p e r i m e n t a l d a t a p o i n t s [see r e f
6) f o r a
d i s c u s s i o n o f t h e r e l a t i v e w e i g h t s a s s i g n e d t o e n e r g y levels a n d E 2 t r a n s m o n r a t e s ] , E , a r e t h e e n e r g i e s r e l a t i v e t o g r o u n d s t a t e s , a n d E~ s a r e t h e g r o u n d s t a t e e n e r g i e s T h e e x p e r i m e n t a l E 2 t r a n s i t i o n r a t e s i n c l u d e d m t h i s fit a r e c o m p a r e d p r e d l c t t o n s f r o m t h i s fit a n d wxth t h o s e f r o m t h e " s e m o n t y f i t " in t a b l e 4
wtth the
N = 50 SHELL MODEL
485
An additional parameter was Introduced for this least-squares fit This parameter as the reduced E2 transition rate matrix element, Q2 (given above), from which an effective proton charge eelf = 1 72e is obtained 6) The "rates fit" can predict the inhibited 8 + ~ 6 + transition rate in 94Ru The other three interactions considered here predict this rate too fast by a factor of about 20 (see, for example, results for the "seniority fit" in table 4) Other transition rates are predicted shghtly better with the "rates fit" than with the other interactions determined in this study However, the agreement between theory and experiment for energy levels is slightly poorer, the rms deviation per level is 80 keV compared to 59 keV for the "levels fit" The "rates fit" has shown that contrary to previous expectation [ref ~x)] the inhibited 8 + --* 6 + E2 rate in 94Ru may be explained within the (2p½, lg~) model space This result does not deny the probable role of configuration mixing from outside the (2p~, lg~) model space in the inhibition of this transition The "rates fit" has been constructed in such a way that the contribution to the rate f r o m the large components of the 8 + and 6 + wave functions cancel so that the transmon proceeds through small components in these wave functions It appears unlikely that small components of configurations from outside our model space do not s~gmficantly alter the details of the inhibition of th~s rate
3. Even N = 50 nuclei
3 ! THE NUCLEUS 9°Zr The nucleus 9°Zr is described in our model as two protons outside of the aaSr inert core The entire theoretical spectrum is shown in fig 1 The binding energy of the ground state is given in table 3 The most distinguishing feature of the 90Zr spectrum is that its first excited state is a 0 + state This xs predicted from even the simplest of two-body lnteracUons (e g , the surface-t5 interaction) given the 2 p u l g ~ single-particle splitting revealed in the 89y spectrum The 3 - octopole excitation at 2 748 MeV and the 2 + state at 3 33 MeV are the only states below about 4 MeV without model space analogues Collective 3- states at approximately this excitation energy are seen throughout the A = 90 region, and in particular in SSSr at 2 734 MeV and In °2Mo at 2 849 MeV. All four theoretical interactions invert the order of the 2 + and 5- states One explanation of this is that the experimental 2 + state contains significant admixtures of the configurations {(p~)-2(p,)2(g~)2)s=2, Ip~lp,(g~)2)s=2 and If~Ip~(g~)2)s= 2 The position of the model space 2 + state in 9°Zr may be estimated (independently of our calculations) from the 92M0 and 94Ru spectra If seniority were conserved, the spacing between the 8 +, 6 +, 4 + and 2 + states in 94Ru, 92M0 and 9°Zr should be the same 5, 6) The fact that seniority is an excellent quantum number for the N = 50 nuclei has been noted by many investigators 2, ,, 5)
D H GLOECKNER
486
AND F J D SERDUKE
8÷
- -
_
142
148
145
372
330
312
775
755
92M0
94Ru
6*
4
*
--
899
/ /
2+
/
90Zr
Fig 6 The e x p e r . n e n t a l 8 +, 6 +, 4 + and 2 + states o f 9°Zr, 92Mo and 94Ru showing the sphttmgs (m keV) o f adjacent levels These sphttmgs are the same m all the nucle~ for a semorlty conserwng interaction in the 2p~-lg 3 model space This figure partlcularlv illustrates the depression o f the 9°Zr 2 + state
TABLE 5 Spectroscopic factors for smgle-pamcle transfer
Target nucleus
sg~
9°Zr 92Mo
Target spin
1_,-
0+ 0+
Final state j~r (MeV)
CoInpllrlSOll ~tth Mrlpplng reacttolt; 0 + (g s ) 1 32
0 ÷ (l 76)
0 68
4 - (2 73) 5- (2 32)
10 10
92Mo
0+ O*
* "Total-energy fit" used ~) R e f 12) ~) R e f 15)
exp
1 31 a) 0 52 a) 1 0 (norm) ") I 24 a)
~+ ( g s )
034
0 39 b), 0 42 D, 0 3 a)
½- (0 105)
0 93
0 88 b), 0 87 ~), 0 9
~+ ( g s )
074
0 64*), 0 52t), 0 67 d), 0 5 ' )
0 30
0 205 *), 0 29 r), o 3 d) 0 2 8 ' )
- (0 38)
9°Zr
C2S
theory*
Compartson i~tth ptcAup reacttons ½- (g s ) 1 32
1 32 (norm) c)
~+ (0 914)
0 68
0 76 c)
~+ (g s )
2 51
2 6 b), 2 7 d)
} - (0 105)
1 40
1 7 h), 1 4 d)
~) Ref x6)
a) R e f t-)
e) R e f is)
r) R e f lOj
a)
g) Ref zo)
N ~ 50 SHELL MODEL
487
and is apparent from the "seniority fit" shown in figs 1-5 Taking the model space 2 + state to be the same energy below the 9°Zr 8 + state as the 8 + ~ 2 + splitting in 9 2 M o o r 9 4 R u would put the 9 ° Z r 2 + approximately 200 keV higher than ]t is seen experimentally This is illustrated in fig 6 which shows the experimental 8 +, 6 +, 4 + and 2 + states of Zr, M o and Ru Further ewdence of the Influence of other configurations m the 9 ° Z r 2 + state comes from its anomalously short hfetlme as indicated m table 4 There is good agreement w]th available spectroscopic data 12) for states m 9 ° Z r Table 5 gives experimental and theoretical single-particle transfer results 4-
30
o*.. 21/2"
25
5"---
17/2+-..
6" 4*
15/2-=--
4* ....
20 9/2*
5/2-
ii/z+.. 5/2:---
15
1312./
2*
s/2*..
lO
7/2* 05
_
112-
112-
V2÷
9/2*
EXPT
TOTAL ENERGY
95Rh
O* ....... TOTAL ENERGY
96pd
Fig 7 Theoretical predictions for 95Rh and 96pd using the total energy fit In 9SRh, all levels to 1 5 MeV are shown as are selected levels that might be populated m some (heavy ion, vn) reactions The experimental results are from ref zz) In 96pd, all levels through the 8 + are shown 3 2 THE NUCLEUS 92Mo
The nucleus 9 2 M o lS described here as four protons filling the 2p~, lg~ shell-model orbltals Excellent agreement with experimental energy levels is obtained for all four interactions (fig 2) Recently two 4 - states predicted by all four models to be near 3 0 MeV have been seen m a 9 2 M o ( n , n ' 7 ) experiment 13) The lowest of these states is predicted in all four 4models to have the structure [(lg~)~+ × (p½)] Jv==4 Such a seniority-four state would be difficult to populate in a direct process The lowest 5- state, however, is a senioritytwo state in all our models. A discussion of transition rates m thls nucleus, and of the tentaUve 12 +, has already been made m ref 6)
488
D H GLOECKNER AND F J D SERDUKE
3 3 THE NUCLEUS 94Ru The nucleus ' 4 R u is described here as (2p~, lg~) 6 A comparison between experimental and theorencal energy levels for this nucleus is gwen m fig 3 All models predict a second unobserved 4 + state near 2 4 MeV This state is predominantly semority-two (g~_)6 The "rates fit" is the only interacnon that correctly predicts the 8 + ~ 6 + E2 transition m this nucleus 6) The requirement that th~s transmon rate be severely inhibited (1 e , the theoretical B(E2) be small compared to the single-particle estimate) most d~stmgmshes the "rates fit" from the three other interactions The major differences m the 94Ru spectrum is a shift of the lowest 6 + and second 4 + states upward for the rates fit The agreement w~th experiment, however, remains excellent 3 4 THE NUCLEUS 96pd The model space configuration for 96pd 1s (2p~, lg~) s This proton-rich nucleus has not yet been seen experimentally It might be posstble to populate states in this nucleus by (~, 4n) or (3He, 3n) on 96Ru In antlclpatmn of such an experiment, the low lying model space states are shown in fig 7 Our "total-energy fit" predicts the binding energy to be 46 638 MeV (table 3) The suggested reactmns typically populate yrast levels, so that we expect to see a cascade of stretched E2 terminating at the ground state, the lowest 8 +, 6 +, 4 + and 2 + states are shown in fig 7 Goodness of seniority m the effective interactmn and the systematlcs of 8 + ~ 6 + ~ 4 + ~ 2 + states m the other even-Z nuclm (fig 6) suggest strongly that this cascade should yield y-rays of approximately 140, 320 and 760 keV Barring an unusual mhlbmon, the 8 + -~ 6 + transmon should have a mean life of approxtmately 200 ns
4. Odd N -- 50 nuclei
4 1 THE NUCLEUS 91Nb The nucleus 91Nb is described here as three protons filhng the 2p~, lg~ orbttals The spectrum is shown m fig 4 An energy gap occurs in this nucleus due to the fact that the 2p, smgle-parttcle level is lower than the lg~ smgle-pamcle level Only one positive p a n t y state can have a filled 2p½ orbital - namely the ~+ ground state All other positive p a n t y states wdl he approximately twice the slngle-pamcle sphttmg above the ground state The lowest negatwe p a n t y state is the ½- state at 0 105 MeV which has the configuration (2p~)x (lg~)~=o) The next lowest negative parity states will then be higher by roughly the difference between ((lg~)2]Veffl(lg~)2)l and ((lg,)ZlVl(lg~)2)s=o The ~z- state at 1 842 MeV is seen in proton pickup 12) on 92M0 to have most of the lf~_ smgle-pamcle strength The 2p~ single-hole strength is more fragmented, though most of it is seen m the -~- state at 1 606 MeV Both of these states are outside our model space and are not considered here
N = 50 SHELL MODEL
489
Excellent agreement with experiment i4-17) Is found for spectroscopic factors for single-particle transfer to states m this nucleus (table 5) 4 2 THE NUCLEUS 9aTc
The nucleus 93Tc 1s described in th~s model as five protons outside an inert SSSr core The spectrum is shown m fig 5 Low-lying states in thas nucleus are described well by all four fits Experimental 17-2o) and theoretical spectroscopic factors for singleparticle stripping to the lowest two states are given in table 5 A few high-spin states in this nucleus have recently been identified 21) from 92M0(0~, p2n)93Tc These states are in excellent agreement with all four models except for a tentative (~s_+) state at 4 256 MeV This state appears nearly 1 MeV higher in all four models 6) 4 3 THE NUCLEUS 95Rh Our model space description of 95Rh has seven protons filhng the 2P½-1g~ orbltals Recently, the spins and parities of ~ts lowest two states were ldenufied by Welffenbach et al zz). These are the $+ ground state and an isomeric ½- state at 543 keV This ½- state is an M4 isomer with a mean hfe of 173 sec Again, this M4 transition further supports our assignment of the posltwe sign for the off-diagonal matrix element (table 1) The low-lying states of °SRh predicted by our total energy fit are shown m fig 7 5. Conclusion
We have deduced four sets of matrix elements for the proton-proton effective mteracUon m the 2p½-1g~ model space. All of the mteracUons were fit to 45 energy levels m the N = 50 nuclei. One of them, the "rates fit", was also fit to the E2 transition rates in these nuclei. The resulting matrix elements are not qualltatwely different from sets of matrix elements deduced in previous studies 2- s) Our interest in tlus study has been to explore the limits of the 2p,-lg~ model to accommodate existing energy-level and transition-rate data with and without the constramt of seniority conservatlon. There is one off-diagonal matrix element m this model space, all energy levels and E2 hfet~mes are independent of its sign. However, a calculation of the hfetlmes of the M4 transitions m the odd-Z N = 50 nuchdes unambiguously in&cares that ((2p½)21V, rfl(lgi)2)s= o lS positive The nature of the proton-proton effective interaction in this region is most clearly revealed by contrasting the results of the various fits. In particular, the near equality of the "semonty fit" whach was forced to conserve semonty and the "total-energy fit" which was not so constramed indicates that semorlty ~s a better quantum number within the identical nucleon 2p~r-lg I configurations than earlier studies 2) indicated. In fact, the rms dewatlon per level for the "semority" and "total-energy" fits differ
490
D H GLOECKNER A N D F J D S E R D U K E
only by a fraction of a kdovolt and this alone enables us to conclude that the current body of experimental energy-level data does not call for any semonty breaking m the model space effective interaction Contrasting the "rates" and "level" fits m&cates that only moderate adjustments m the two-body matrix elements are required to reproduce the xnhlb~ted E2 transitions in 9 4 R u The resulting small vtolatlon of sentonty conservation Is, however, essenttal to the explanation of these inhibitions wlthtn the model space Since the reqms~te mhlbmon mechamsm could arise through admixtures of configurations from outside the model space, th~s reqmrement for semorlty breaking from the transition-rate data should be wewed as weaker than the previous re&cation of seniority conservation from the energy-level data The "total-energy" and "level" fits were the results of identical parametenzatlons and were fit to ldent~cal sets of data The only difference was the construction of the function that was minimized in the fitting process A comparison of the resulting fits could reasonably be viewed as a scale by which the slgmficance of &fferences m twobody matrix elements may be measured We would hke to thank Profs R Lawson and M Macfarlane for their participation m the early phase of this work and their continued adwce throughout the course of this investigation References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22)
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