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N U C L E A R

P H Y S I C S A

ELSEVIER Nuc lea r Phys ic s A 586 (1995 ) 445-4 56

Sh e l l m od e l ca lcu la t ions fo r neu t ron r ich nu c le i inthe Oflp she l l

W . A . R i c h t e r a , M . G . Va n d e r M e r w e a, B . A . B r o w n ba Physics Department University o f Stellenbosch Stellenbosch 7600 South Africa

b National Su perconducting Cyclotron Laboratory and Departm ent o f Physics and AstronomyMichigan State University Ea st Lansing M I 48824 USA

Received 4 August 1994; revised 22 November 1994

A b s t r a c t

A new two-b ody interac tion recent ly der ived for nucle i in the 0 f lp she l l by f it ting two-bod ymatrix elem ents to 494 e nerg y levels in A = 41 -6 6 n uclei , is used to invest igate the neutron-richnuclei in the vicini ty of the do ubly clo sed nuclide 48Ca. This study is o f fundam ental interest inprovidin g a test for the new effective interaction awa y from the stabili ty l ine. M asses and bind ingenergies are calculated fo r a variety of neutron-rich nuclei and comp ared with experimental data,wh ere available. In ad dition level sche m es fo r ~-52Ca, sl-52Sc and sl-52Ti have b een calcu latedand are com par ed with available experimental data. In general a go od correspond ence betw eentheory and experiment is found, but some systematic discrepancies are apparent .

1 I n t r o d u c t i o n

A n e w t w o - b o d y i n te r a c t io n w a s r e c e n t l y d e r i v ed f o r n u cl e i in a l a r ge p a r t o f t h e 0 f l ps h e ll A = 4 1 - 6 6 ) b y c o n s i d e r i n g a t r u n c a te d fp c o n f i g u r a t io n s p a c e i n c o r p o r a t i n g l p l h

e x c i t a t i o n s f r o m t h e f 7/ 2 sh e ll , i .e . 0 ~ / 2 l p 3 / z O f s / 2 1 p l / 2 )m + O ~ l l p s / 2 0 f s / 2 1 p l / 2 ) m +lc o n f i g u r a t io n s , w i t h m t h e m i n i m u m n u m b e r o f n u c l e o n s o u t s id e t h e f i ll ed o r p a rt i al l yf i ll e d f7 /2 sh e l l [ 1 , 2 ] . T h e f i n a l i n te r a c t i o n , d e n o t e d b y T B L C 8 , wa s o b t a i n e d b y f i t t i n g

t o a s et o f 4 9 4 l e ve l e n e rg i es . G o o d a g r e e m e n t w a s o b t a i n e d w i t h e x p e ri m e n t a l v a l u esf o r th e o b s e r v a b l e s c a l c u l a te d w i t h t h e T B L C 8 w a v e f u n c t io n s , i.e . e n e r g y l e v el s a n d

s t at ic e l e c t r o m a g n e t i c m o m e n t s [ 1 ] .I n v i e w o f t h e g o o d r e s u l ts o b t a i n e d w i t h o u r T B L C 8 i n t er a c ti o n , i t i s a p p r o p r ia t e t o

i n v e s t i g a t e t h e a p p l i c a b i l i t y o f t h e i n t e r a c t io n f o r n u c l e i f u r t h e r a wa y f r o m t h e l i n e o fs t a b il i ty. Se c t i o n 2 c o n t a i n s a b r i e f r e v i e w o f th e T B L C 8 i n t e r a c ti o n . I n Se c t i o n 3 we

c a l c u la t e t h e m a s s e s o f a n u m b e r o f n e u t r o n - r i c h nu c l e i w i t h t h e T B L C 8 i n t er a c t io n a n d

037 5-94 74/9 5/ 09 .5 0 (~) 1995 Elsevier Science B.V. Al l rights reservedSSDI0 3 7 5 - 9 4 7 4 ( 9 4 ) 0 0 8 0 2 -7

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W.A. Richter et al./Nuclear Physics A 586 1995) 445-456

3 . M a s s e s o f n e u t r o n r i c h n u d e i

447

T h e m e a s u r e m e n t o f n u c l e a r m a s s e s i s o f f u n d a m e n t a l i m p o r t a n c e f o r a s s e s s i n g t h ep r e d i c t i ons o f nuc l e a r mod e l s f o r g r o s s p r ope r ti e s , e. g . g r ound - s t a t e b ind ing e ne rg i es , o rde t a i le d p r o pe r t i e s , e .g . l e ve l st r uctu r e, pa i r i ng e ne rgy a nd Co u lom b e ne rgy sy s t e ma t i c s ,a nd de c a y p r ope r t i e s . I n th i s s e c t ion m a sse s on the ne u t r on - r i c h s ide o f t he va l l e y o ff l - s t a b i l i t y c a l c u l a t e d w i th ou r T BL C8 in t e r a c t i on a r e c ompa r e d w i th t he mos t r e c e n ta va i l a b l e me a su r e m e n t s c ons i s t i ng o f va lue s f r om S e i f e r t e t a l. [ 5 ] , b a se d on m e a su r e -me n t s w i th a t ime - o f - f l i gh t i soc h r ono us TO F I ) spe c t r ome te r, a nd va lue s f r om the 1 993m a s s c o m p i l a t i o n o f A u d i a n d W a p s t ra [ 6 ] . T h e a d v a n t a g e o f c o m p a r i n g w i t h o n e s e tof expe r imenta l da ta l ike tha t o f Se i fe r t e t a l . i s tha t gene ra l t r ends a re le ss l ike ly tobe obsc u r e d a s a r e su l t o f sy s t e ma t i c e r r o r s w h ic h ma y va r y f r om one e x pe r ime n t t oanothe r.

T he Cou lomb d i sp l a c e me n t e ne rg i e s r e qu i r e d i n t he she l l - mode l c a l c u l a t i ons ha v ebe e n ob ta ine d f r om the e ne rgy d i f f e r e nc e s o f a na logue s t a t e s , one ge ne r a l l y be ing ag r ound s t a t e . T he Cou lomb c on t r ibu t ions o f t he s t a t e s u se d in t he f i t s t o ob t a in t h eT B L C 8 in t e r a c ti on ha ve be e n ob ta ine d in t h is w a y, a nd a r e d i s c us se d in R e f . [ 1 ] . I nc a se s w he r e one pa r tne r i s no t know n e xpe r ime n ta l l y e s t ima te s ha ve be e n ma de u s inge x t r a po la t i ons o f l i ne a r l e a s t - squa r e s f i t s t o know n d i sp l a c e me n t e ne rg i e s . Be c a use w euse a 4 Ca c o r e , w e c o m pa r e c a l c u l a te d a nd e xpe r ime n ta l b ind ing e ne rg i e s Br e l r e l a ti veto 4 Ca, w he r e Br el = B - B 4 C a ) a nd B i s t he a bso lu t e b ind ing e ne rgy. T he she l l - mode lb ind in g e ne rgy r e l a t i ve t o 4 Ca c a n be e xp r e s se d a s BrTM = Bint - C , w he re B int is thein t e r a c t i on e ne rgy be tw e e n the e x t r a - c o r e nuc l e ons c a l c u l a t e d i n t he i so sp in f o r ma l i smw i th ou r T B L C 8 in t er a c t ion , a nd C i s t he Cou lo m b d i sp l a c e me n t e ne rgy, e s tima te d a sde sc r ibe d a bove .

A co m par ison w i th the r e su l t s Se i fe r t e t a l . [5 ] fo r the mass de fec ts d , i s g iven inTa b le 1 . T he ma sse s f o r t h r e e ve r y ne u t r on - r i c h nuc l e i no t me a su r e d by S e i f e r t e t a l .a r e t a ke n f r om the c omp i l a t i on o f A ud i a nd W a ps t r a [ 6 ] . S e ve r a l c a se s in the r e g iono f N = 31 to N = 37 a r e a b se n t f r om Ta b le 1 , be c a use t he she l l -mod e l d ime n s ione xc e e ds a bou t 3500 a nd c a l c u l a t i ons ha ve no t ye t be e n c a r de d ou t . We ma ke he r e af e w c omme n t s a bou t t he mos t r e c e n t da t a f r om S e i f e r t e t a l . i n c ompa r i son w i th t h ep r e v ious T O F I da t a o f Tu e t a l. [ 7 ] T he va lue s f r om Tu e t a l. a nd S e i f e r t e t a l. a r egene ra l ly qu i te c lose to each o th e r fo r N < 36 , the la rges t d i f f e renc e be in g for 54Scw he r e t he va lue o f Tu e t a l . is l e s s bound tha n the va lue o f S e i f e r t e t a l. by a b ou t0 .9 Me V. F o r t he mos t ne u t r on - r ic h i so tope s o f V, Cr, M n a nd F e N ~> 36 ) t he r e a r esys t e ma t i c de v ia t i ons be tw e e n the tw o se t s , t h e va lue s f r om Tu e t a l . a lw a ys be ing l e s sbound . T he l a rge s t de v i a t i ons i n t h i s g r oup oc c u r f o r59 60V w i th r e spe c t ive de v ia t i on so f a bou t 2 a nd 1 .6 Me V. r e spe c t ive ly ) , f o r61 62Cr dev ia t ion s of about 1 .2 and 1 .7 MeV.r e spe c t ive ly ) , f o r63 64Mn d e v i a t io n s o f a b o u t 0 .9 M e V ) a n d f o r66Fe de v ia t i on a bou t

1 . 5 M e V ) .T he d i f f e r e nc e s ATM --Aexp a re show n in F ig . 1 . A neg a t ive va lue for th i s quant i ty ind i -

c a t e s t ha t e xpe r ime n t i s mor e bound tha n the o r y. T he c i r c l e s c o r r e spond to e xpe r ime n ta lda t a f r om the c omp i l a t i on o f A ud i a nd W a ps t r a [ 6 ] a nd the c r o s se s to e xpe r ime n ta l da t a

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448 W.A. Richter et al./Nucle ar Physics A 586 1995) 445-4 56

Table 1Binding energies and mass defects f or neutron-rich isotopes. Brel is the binding energy relative to 4Ca andA is the mass defect, both in units o f Me V. SM stands for shell model, C for the estimated C oulom b dis-placem ent energy relative to Ca. The experimental values are from Seifert et al. [5] (a) or from the masscompilation [6] (b ). T he spin values giv en are predicted by ou r calculations

A Element N 2J BTM C A xp Ref. ATM A TM A xprel

51 Ca 31 3 89 .20 -36 .3(0 .4) a -35 .26 -1 . 0(0 .4)52 32 0 94 .03 -32 .5(0 .5) b -32 .0 1 -0 . 5(0 .5)52 Sc 31 6 101 .39 7 .12 - 40 .38 ( 0 .23 ) a - 40 .16 - 0 .22 ( 0 .23 )53 32 7 106 .75 7 .10 - 38 .96 ( 0 .26 ) a - 37 .44 - 1 .52 ( 0 .26 )54 33 6 109 .95 7 .09 - 34 .4 ( 0 .5 ) a - 32 .58 - 1 .82 ( 0 .5 )55 34 7 t14 .88 7 .08 -28 .5(1 .0) b -29 .43 0 .9(1 .0)54 Ti 32 0 122 .10 1 4 .3 8 - 45 .76 ( 0 .23 ) a - 45 .52 - 0 .24 ( 0 .23 )55 33 1 1 2 5 .8 3 14.34 - 41 .81 ( 0 .24 ) a - 41 .17 - 0 .64 ( 0 .24 )56 34 0 132.34 14.30 -39 .13 (0.2 8) a -39 .61 0.48(0.28)5 7 3 5 5 1 3 4 . 1 8 1 4 . 2 5 - 3 3 . 3 ( 1 . 0 ) b - 3 3 . 3 7 0 . 1( 1 .0 )

59 V 36 7 1 5 3 .4 9 21.79 - 37 .91 ( 0 .35 ) a - 37 .32 - 0 .59 ( 0 .35 )60 37 2 156.36 21 .73 -33 .1(0 .6) a -32 .13 -1 .0(0 .6)58 Cr 34 0 1 6 0 .2 8 29 .60 -51 .93 (0 .24) a -52 .97 1 .04(0 .24)60 36 0 170.30 29.39 -46 .83 (0.2 6) a -4 6.8 4 0.01(0.26)61 37 5 172.86 29 .29 -42 .7 7(0 .2 8) a -41 .33 -1 .4 4(0 .28)62 38 0 178.32 29 .17 -41 .2(0 .4) a -38 .72 -2 .4 8(0 .4)62 Mn 37 4 187.54 37.24 -48 .47 (0.2 6) a -48 .73 0.26(0.26)63 38 7 192.64 37 .08 -46 .75 (0 .28) a -45 .76 -0 .9 9(0 .28)64 39 2 195.76 36 .93 -43 .10 (0 .35) a -40 .80 -2 .3 0(0 .35)65 40 7 1 9 9 .7 1 36.80 - 40 .9 ( 0 .6 ) a - 36 .68 - - 4 .2 ( 0 .6 )63 Fe 37 5 201.84 45.33 -55 .49 (0.2 7) a -55 .73 0.24(0.27)64 38 0 209.16 45 .09 -55 .08 (0 .28) a -54 .98 -0 .1 0(0 .28)

65 39 5 212.43 44 .90 -51 .2 9(0 .2 8) a -50 .18 -1 .1 1(0 .28)66 40 0 217.72 44 .77 -50 .3 2(0 .3 5) a -47 .30 -2 .9 2(0 .35)

f r o m S e i f e r t e t al . [ 5 ] . T h e f i l l e d c i r c l e s f o r N ~< 3 0 i n d i c a t e t h e m o r e s t a b l e n u c l e i

o r i g i n a l l y i n c l u d e d i n t h e A = 4 1 - 6 6 l e a s t -s q u a r e s f it f o r o u r T B C L 8 i n t e r a c t io n [ 1 ] .

I t i s e v i d e n t t h at t h e T B L C 8 i n t er a c t i o n g i v e s a g o o d a c c o u n t o f t h e m e a s u r e d b i n d i n g

e n e r g i e s o r m a s s e s , w i t h t h e t h e o r e t i c a l v a l u e s g e n e r a l l y w i t h i n 1 M e V o f t h e m e a s u r e d

v a l u e s . T h e l a r g e s t d i s c r e p a n c y w i t h t h e T B L C 8 i n t e r a c ti o n o c c u r s f o r 6 5 M n , w h e r e t h e

d e v i a t i o n i s a b o u t 4 M e V. T h e s p i n s o f t h e n u c l e i p r e d i c t e d b y o u r c a l c u l a ti o n s a r e a l s o

i n d i c a t e d i n Ta b l e 1 .

I n R e f . [ 7 ] T u e t al . c o m p a r e t h e d i f f e r e n c e s b e t w e e n t h e th e o r e t i c a l l y p r e d i c t e d

m a s s e s o f f o u r m i c r o s c o p i c - m a c r o s c o p i c m o d e l s a n d t h e e x p e r i m e n t a l m a s s ( s e e F ig . 3

o f R e f . [ 7 ] ) . T h e m o d e l s a r e t h o s e o f M 0 1 1 e r - N i x [ 8 ] , Ta c h i b a n a [ 9 ] , t h e s e m i -

e m p i r i c a l s h e ll m o d e l o f L i r a n - Z e l d e s [ 1 0] a n d J ~ i n e c k e - M a s s o n s r e c a lc u l a t i o n o f t h e

G a r v e y - K e l s o n m a s s r e l a t i o n s h i p s [ 11 ] . T u e t al . f o u n d l a r g e s y s t e m a t i c d e v i a t i o n s f o r

5 1 C a a n d 5 2 -5 4S c w h e r e t h e b i n d i n g e n e r g i e s a r e s i g n i f i c a n t l y u n d e r p r e d i c t e d . I n F i g . 1

a n d Ta b l e 1 w e n o t e t h e s a m e t e n d e n c y, b u t o u r d e v i a t io n s o f a b o u t 1 M e V a r e s l i g h t l y

l e s s t h a n t h e a v e r a g e o f t h e f o u r m o d e l s i n e a c h c a se . A s i n R e f s . [ 7 , 5 ] w e a l s o f in d t ha t

g o o d g e n e r a l a g r e e m e n t b e t w e e n e x p e r i m e n t a n d t h e o r y i s r e s t o r e d f o r h e T i is o t o p e s .

T h i s a p p e a r s t o s u p p o r t t h e s u g g e s t i o n o f l o c a l iz e d r e g i o n s o f e n h a n c e d b i n d i n g f o r t h e

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B i n d i n g e n e r g y d i f f e r e n c e s

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F i g . I . A T - A c x p v e r s u s n e u t r o n n u m b e r N f o r n e u t r o n - r i c h n u c l e i . h e c r o s s e s c o r r e s p o n d t o e x p e r i m e n t a l

m a s s e s f r o m S e i f c r t e t al . [ 5 ] a n d t h e c i r c l e s t o c o m p i l a t i o n v a l u e s f r o m R e f . [ 6 ] w h e r e t h e f i l l ed c i r c le s

i n d i c a t e i s o t o p e s w h i c h w e r e i n t h c o r i g i n a l e a s t - s q u a re s i t o f t h e T B L C 8 i n te r ac t io n . r r o r b a r s a r e s h o w n

w h e n t h e y a r e l a r g e r t h a n 0 . 4 M e V a n d t h e v a l u e s f o r a g i v e n i s o t o p e a r e c o n n e c t e d b y l in es .

Ca a nd S c i so tope s [ 7 ,5 ] . I t i s a l so f ound tha t t he b ind ing o f t he mos t ne u t r on - r i c hi s o to p e s t e n d t o b e o v e r p r e d i c te d b y m o s t o f t h e f o u r m o d e l s , w h e r e a s o u r m i c r o s c o p i cc a l c u l a t i ons t e nd to und e r p r e d i c t t he b ind ing o f t he se i so tope s by sm a l l e r a moun t s .

I n F ig . 1 i t c a n a l so be s e e n tha t t he r e a ppe a r s t o be a sy s t e ma t i c pa t t e r n f o r the t h r e ei so tope s a bove V, w h ic h c a n be mor e r e a d i ly s e e n f o r F e a nd Cr w he r e t he da t a i s mor ec omple t e . A f a i r l y smoo th pos i t i ve de v ia t i on i s f o l l ow e d by a d r a ma t i c ne ga t ive d iv ef o r t he mos t ne u t r on - r i c h spe c i e s . I n t he c a se o f t he da t a f r om Tu e t a l . t h e de v ia t i o nf o r t he mos t ne u t r on - r i c h nuc l e i i s ge ne r a l l y sma l l e r t ha n f o r t he da t a f r om S e i f e r t e ta l. , im p ly in g le s s b ind ing . W he n the l p3 /2 subshe l l i s p r ima r i l y be ing f il le d N = 29 to32 ) , a ve r y good r e p r oduc t ion o f t h e e xpe r ime n ta l ma sse s i s obse r ve d , a nd w he n the0f5 /2 subsh e l l is p r im ar i ly be in g fi l led N = 33 to 38) , the she l l m ode l ove rp red ic t sthe b ind ing e ne rgy, w h ic h r e a c he s a ma x im um a t a bou t the midd le o f t he subshe l l. Bu tw h e n the l p l / 2 subshe l l is p r im a r i l y be ing f i ll e d N = 39 to 40 ) t he r e i s a n i nc r e a s in gunde r p r e d i c t i on w i th N . T he sy s t e ma t i c de v ia t i ons s e e m to i nd i c a t e t ha t t he p r o b le mm a y l i e w i th t he de sc r ip t i on o f the s ing l e - pa r t ic l e e ne rg i es , o r m a y ind i c a t e t he p r e se nc eo f the 0g9 /2 i n t r ude r o r b i t a l. T he r a the r l a rge pos i t i ve de v ia t i on 1 .4 M e V ) f ound b y Tue t a l . fo r 59V becomes a sma l l nega t ive dev ia t ion wi th the da ta o f Se i fe r t e t a l . I t hasbe e n sug ge s t e d i n Re f . [ 7 ] t ha t t he p r e se nc e o f a n unknow n i som e r m a y ha ve a f f e c te dthe me a su r e me n t f o r 59V.

I n some c a se s t he c a l c u l a t i ons c a n be c a r r i e d ou t f o r t he f u l l 0 f lp mode l sp a c e ,

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em ploy in g the F P D 6 in t e rac t ion o f R ef . [4 ] . In t he fu ll space t he d imens ions o f theshel l -model s ta tes increase rap id ly wi th A and calcu la t ions can usual ly be carr ied outup to about mass 52 for the Ca i so topes . As the 0f lp shel l i s p rogress ively f i l l ed i tbecomes again poss ib le to car ry out fu l l -space ca lcu la t ions for some nucle i wi th masseslarger than A = 60 , e .g . 62Cr. Fo r masses near A = 50 the fu l l -space ca lcu la t ion genera l lydoes somewhat bet ter than the ca lcu la t ion in the FPV model space , e .g . for 51Ca thec a lc u l at e d m a s s d e f ec t s a r e - 3 5 . 5 6 a n d - 3 5 . 2 6 M e V f o r F P D 6 a n d T B L C 8 , re s pe c ti ve ly,c o m p a r e d t o th e e x p e ri m e n ta l v a lu e o f - 3 6 . 3 ( 0 . 4 ) M e V [ 5 ] a n d f o r 5 2C a t h e s a m eq u a n ti ti e s a re - 3 2 . 8 2 a n d - 3 2 . 0 1 M e V f o r F P D 6 a n d T B L C 8 , re s pe c ti ve ly, c o m p a r e dt o - 3 2 . 5 ( 0 . 5 ) M e V [ 6 ] f o r e x p e ri m e n t. H o w e v e r, f o r t h e r e gi o n a r o u nd A = 6 2 - 6 6where t he fu l l - space ca lcu l a ti ons a re aga in pos s ib le , t he t heo ry becomes 10 -20 M eVunderbound compared t o exper imen t . Th i s i s pe rhaps no t su rp r i s i ng because t he F P D6in t e rac t i on was on ly de t e rmined f rom energy da t a i n t he l ower pa r t o f t he 0 f l p she l l( A = 4 1 - 4 9 ) .

W e have a l so ca l cu l a ted a num ber o f masses o f neu t ron - r ich nuc le i fo r wh ich noexper imenta l in format ion i s avai lab le as yet . The predic ted b inding energ ies and massdefect s are g iven in Table 2 .

4 . L e v e l s t r u c t u r e o f t h e N = 3 0 3 2 i s o t o n e s

P rev ious t heo re ti ca l work i n t h i s reg ion main ly con cen t ra ted on t he u se o f a c l o sed0f7 /2 neu t ron she ll [12 , 1 3 ] . H owever, Hu ck e t a l. [14 ] i nc luded l p lh j um ps f rom

the 0f7/2 subshel l to the lp3/2, 0f5/2 and lpl /2 subshel ls . They invest igated the iso-t opes 52K , 52ca and 52S c by us ing t he r ea l is t ic K uo -B row n in t e rac t ion wi th a f ewm ono po le changes [15 ] . The i r ca l cu la t ions showed t ha t t he s truc tu re o f t hese nucl e i

n02-n , ~ (n = 4 , 3 and 2 resp ect ivelyan be descr ibed by the dominant conf igura t ions VlT/2

a n d r = ( l p 3 / z 0 f s / 2 1 p u 2 ) ) a n d th o s e r e a c he d th r o u g h l p l h j u m p s , o e ll -n - ~ + l T h i s isv l 7 / 2 It he s ame mode l space u sed t o ob t a in t he TB LC 8 in t e rac t i on . The mod i f i ed Kuo-B rowninteract ion a l so reproduced the re la t ive energy level s reasonably wel l .

The exper imen ta l spec t ra fo r t he l ow- ly ing no rmal -pa r it y s ta te s a re com pared wi ththose o f t he T B L C 8 in t e rac t i on in F igs . 2 -8 . The exper imen ta l in fo rmat ion on spec t ro -scopic quan t i ti es in th is reg ion i s in genera l very l imi ted . The data used fo r a comp ar i sonwi th t he t heo re t i ca l r e su lt s ob t a ined wi th t he T B L C 8 in t erac t ion were ob t a ined f rom theNu c l ea r Da t a S hee t s compi l a t ions ( s ee F igs . 2 -8 ) , t he Tab le o f I so topes o f Led ere r andShi r ley [ 16] , and f rom the s tudy of the A = 52 iso topes by H uck e t al. [ 14] .

4 1 The N=3 0 isotones

5Ca: T he calcu la t ion g ives an exci ta t ion energy o f the f irs t 2 + s ta te of 1 .12 M eVcompared t o t he exper imen ta l va lue o f 1 . 026 MeV. The exper imen ta l l eve l spec t rum

sugges t s tha t there i s a l arge gap to the second exci ted s ta te as pred ic ted by the TBLC8interact ion .

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W.A. Richter et aL /Nuclear Physics A 586 1995) 445-456

Ta b le 2

P r e d i c te d s p i n s , b i n d i n g e n e r g i e s a n d m a s s d e f e c t s f o r n e u t r o n - r ic h n u c l e i . B r e l i st h e b i n d i n g e n e r g y r e la t iv e t o 4 C a a n d z l is t h e m a s s d e f e c t , b o t h i n u n i t s o f M e V.S M s t a n d s f o r s h e l l m o d e l a n d C f o r t h e e s t i m a t e d C o u l o m b d i s p l a c e m e n t e n e r g y

451

A Z N 2 J B ~ C d TM

5 3 C a 3 3 5 9 5 .4 6 - 2 5 . 3 75 4 3 4 0 1 0 0 . 1 0 - 2 1 . 9 55 5 3 5 5 1 0 0 .7 5 - 1 4 . 5 25 6 3 6 0 1 0 4 . 1 4 - 9 . 8 55 7 3 7 5 1 0 4. 01 - 1 . 6 458 38 0 106 .30 4 .1159 39 1 104 .75 13 .766 0 4 0 0 1 0 6 .9 4 1 9 .6 45 6 S c 3 5 6 11 7 .2 5 7 . 0 7 - 2 3 . 7 35 7 3 6 7 1 2 0 .5 2 7 . 0 6 - 1 8 . 9 35 8 3 7 1 0 1 2 1 .5 1 7 . 0 5 - 1 1 . 8 5

5 9 3 8 7 1 2 4 .1 0 7 . 0 4 - 6 . 3 760 39 5 124 .13 7 .03 1 .6736 1 4 0 7 1 2 5 .7 4 7 .0 2 8 .1 35 8 Ti 3 6 0 1 3 9 .2 7 1 4.21 - 3 0 . 3 95 9 3 7 5 1 4 0 .4 4 1 4 .1 6 - 2 3 . 5 06 0 3 8 0 1 4 4 .1 5 1 4 .1 2 - 1 9 . 1 36 1 3 9 1 1 4 3 .8 0 1 4 .0 8 - 1 0 . 7 16 2 4 0 0 1 4 6 .7 7 1 4 .0 4 - 5 . 6 16 1 V 3 8 7 1 6 0 .0 8 2 1 . 6 8 - 2 7 . 7 76 2 3 9 5 1 6 1 . 3 4 2 1 . 6 2 - 2 0 . 9 66 3 4 0 7 1 6 4 .0 8 2 1 . 5 3 - 1 5 . 6 36 3 C r 3 9 1 1 7 9 .3 3 2 9 . 0 4 - 3 1 . 6 6

6 4 4 0 0 1 8 3 .6 9 2 8 . 9 2 - 2 7 . 9 4

5 . 5

5 0

4 5

4 0

3 5

~ 3 . 0

~ 2 . 5

2 . 0

1 . 5

1 . 0

0 . 5

0 . 0

- 0 . 50 . 0 0

T L C R ~ p

I { I

dal i l l

i

0

5 0 C a e n e r g y l e v e l s IT = 5

F i g . 2 . C o m p a r i s o n o f t h e o r e t ic a l a n d e x p e r i m e n t a l e n e r g y s p e c t r a f o r l o w - l y i n g n a tu r a l -p a r i ty l e v e l s i n 5 C a .T h e e x p e r i m e n t a l s p e c t r u m f r o m R e f . [ 1 7 ] ) i s c o m p a r e d w i t h t h e s p e c t r u m b a s e d o n t h e T B L C 8 i n te r a c ti o n ,w i t h t h e e x p e r i m e n t a l g r o u n d - s t a t e e n e r g y t a k e n a s z e r o . J i s in d i c a t ed f o r A e v e n a n d 2 J f o r A o d d .

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452 W.A. Richter et al,/Nuclear Physics A 586 1995) 445-456

4 . 0

3 . 5

3 .0

2 . 5

~ 2.0

1 . 0

0 . 5

0 . 0

0 . 5

0 . 0 0

T B L C S ~ X P

~3 . 5 -1. - /

I 3 . 5 - 1

13-)1(1-1]

?

- - l l l - J

1

1717

51Sc energy le vel s T = 4 .5 )

Fig. 3. Energy levels of 51Sc. ConvenfionsarethesameasinFig. 2. The expefimentalspectrumis fromRef. [18].

51Sc: The calculation gives excitation energies of the first 2- and ~ - states of0.84 and 1.02 MeV, compared to the experimental values of 0.860 and 1.062 MeV,respectively. The theoretical value obtained for the first - level of 2.59 MeV alsocompares favourably with the experimental value of 2.347 MeV. The tentative spinassignments and positions of the experimental energy levels appear to indicate that theygenerally have counterparts in the theoretical spectrum.

52Ti: The calculation gives excitation energies of the first 2 and 4 states of 1.06 and

4 . 5

4 . 0

3 . 5

3 . 0

; 2 .5

x 2.0w

1.5

1.0

0.5

0.0

-0 .50.0

3

4

g

i i i

5 2 T i e n e r g y l e v e l s T = 4 )

Fig. 4. Energy leve~ of 52Ti. Convenfionsarethe same asinFig. 2 Theexperimentalspeetrum~ fromRef. [19].

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W.A. Richter et al./Nuclear Physics A 586 1995) 445-456 453

4 0

3.5

3.0

2.5

~ 2.0

1.0

0 . 5

0.0

0 . 5

0. O

TBLC~, ~ p

.__a_.._9

i i i

5 1 C a e n e r g y l e v e l s T = 5 . 5 )

Fig. 5. Energy levels of 51Ca. Conventions ar et he same as in Fig. 2. The experimental spectrum is ~omRef. 118].

2.43 MeV, compared to the exper imenta l va lues of 1 .050 and 2 .425 MeV, respect ively.Th e theore t ica l va lue obta ined for the fi rs t 6 l evel of 3 .147 MeV al so com paresfavourab ly wi th t he exper imen ta l va lue o f 3 . 028 MeV. The measu red ene rgy l evel s

ag ree r easonab ly we l l on t he who le wi th t he p red i c ted ones .

TBLCR, ~ p

3.0

2 . 5

2 . 0

1.0

0.5

0.0

-0 . 50.00

1L

5 2 S c e n e r g y l e v e l s T = 5 )

Fig. 6. Energy levels of 52Sc. Convenfionsare the same as in Fi g. 2. The expefi ment alsp ectr umis fro mRe~. [19 14l.

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454 W.A. Richter et al./Nuclear Physics A 586 1995) 445-456

TBLCB {~p

2 . 5

2.0 ~

1 o

0 . 5

0 0 ~ (~ l

- 0 . 5

0 00

53Ti energy le ve ls T = 4.5)

Fig. 7. Energy levels of 53Ti. Conventions are the same as in Fig. 2. The experimental spectrum is fromRef. [201.

4 2 The N=31 isotones

5 1C a : T h e e x p e r i m e n t a l i n f o r m a t i o n i s v e r y l im i t e d a n d o n l y t h e g r o u n d s t a te h a s

b e e n a s s i g n e d a t e n t a t iv e sp i n o f ~ . T h e c a l c u l a t i o n a l so p r e d ic t s a 3 - g r o u n d st a te .

T h e e x c i t a t i o n e n e rg y o f t h e fir st - s t a te i s c a lc u l a t e d t o b e a t 1 . 1 3 Me V .

>

5 5 ~5 . 0 -

4.5

4.0

3.5

3.02.5

2.0

1.5

1.0

0.5

0.0o 5 1

0.00

T B L q E ~ P

_ L . _

(~+~

i i i

5 2 C a e n e r g y l e v e l s ( T = 6 )

Fig. 8. Energy levels of 52Ca. Conventions are the same as in Fig. 2. The experiment lspectrum is fromRef. [19].

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W.A. Richter et al./Nu clear Physics A 586 1995) 445-456 455

5 Sc: H u c k e t a l. c a l c u l a t e d t h e 52 Sc g r o u n d s t a te i n t h e m o d e l sp a c e 0 ~ / 2 ( l p 3 /2 0 t'5 /2 1 p l / 2 ) 3 d u e t o c o m p u t e r l i m i t a t i o n s . H o we v e r, i n o u r c a l c u l a t i o n s we m a d e u se o f

the fu l l O ~ / 2 l P 3 / 2 0 f s / 2 1 p l / 2 ) 3 +0 ~ / 2 ( l p 3 / 2 0 f s / 2 1 p l / 2 ) 4 m o d e l s p a c e . T h e c a l c u l a ti o np r e d i c t s a 3 + g r o u n d s t at e . T h e e x c i t a t i o n e n e rg y o f t h e f i r st 2 + s t a t e i s c a l c u l a t e d t o b ea t 0 . 9 8 Me V . E x p e r i m e n t a l l y a 3 + g r o u n d s t a te i s a l so i n d i c a t e d , w h i l e t h e f i r st 2 + s t a tei s e x p e r i m e n t a l l y f o u n d t o b e a t 0 . 6 8 M e V [ 1 4 ] .

5 3 Ti : T h e e x p e r i m e n t a l i n f o r m a t i o n i s v e r y l i m i t e d i n t h a t o n l y t h e g r o u n d s t a t e h a sb e e n a s s i g n e d a t e n t a ti v e sp i n v a l u e o f 3 . T h e c a l c u l a t io n a l so p r e d i c t s a 3 - g r o u n d

s t at e . T h e e x c i t a t i o n e n e rg y o f th e f i rs t - s t at e i s c a l c u l a t e d t o b e a t 0 . 5 5 M e V.

4 .3 . T h e N = 3 2 i s o t o n e s

5 2 C a : T h e c a l c u l a t i o n g i v e s a n e x c i t a t i o n e n e rg y o f t h e f i rs t 2 + s t at e o f 1 . 85 M e V,wh i c h i s h i g h e r t h a n t h e t y p i c a l 2 + s t a t e e n e rg y b u t l o w e r t h a n t h e su g g e s t e d e x p e r i -

m e n t a l v a l u e o f 2 . 5 6 3 M e V. T h i s h i g h e n e rg y s i g n i f ie s a p a r ti a l P3 /2 su b sh e l l c l o su r ea t N = 3 2 . T h e d o m i n a n t c o n f i g u r a t i o n s i n t h e g r o u n d - s t a t e w a v e f u n c t i o n a r e 7 7 %

2 2 2~ / 2 P~/ 2 , 1 4 % ~ / 2 P3 / 2 ~ / 2 a n d 7 % ~ / 2 P3 /2 P l /2 H u c k e t a l. [ 1 4 ] o b t a i n e d a g r o u n d - s t a t e

w a v e f u n c t i o n w i t h a l a r g e r ~ / 2 p4 / 2 c o m p o n e n t o f 9 4 % ~ / 2 p4 / 2 .

4 . 4 . C o n c l u s i o n

T h e s h e l l - m o d e l c a l c u l at i o n s p r e s e n t e d s h o w s t h a t t h e s t ru c t u r e o f t h e n e u t r o n - r i c hn u c l e i i n t h e v i c i n i t y o f t h e d o u b l y c l o se d 4 8 C a n u c l e u s , sp e c i f i c a ll y t h e N = 3 0 , 3 1 a n d3 2 i so t o n e s c a n b e a d e q u a t e l y d e sc r i b e d b y u s i n g a c o n f i g u r a t i o n sp a c e t h a t i n c o r p o r a t e s

l p l h e x c i t a t i o n s f r o m t h e 0 f 7 /2 sh el l. T h i s su p p o r t s t h e f i n d i n g o f Hu c k e t al . [ 1 4 ] .

A ckno w ledgemen t s

W. R . w i s h e s t o a c k n o w l e d g e t h e h o s p i t a l it y a n d s u p p o r t o f t h e N a t i o n a l S u p e r c o n -d u c t i n g L a b o r a t o r y, U n i v e r s i t y o f M i c h i g a n S t at e. T h i s w o r k w a s s u p p o r t e d i n p a rt b y

N a t i o n a l S c i e n c e F o u n d a t i o n G r a n t 9 4 - 0 3 6 6 6 .

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[15] A. Poves and A. Zuker, Phys. Reports 70 19 81) 235.[ 16] C.M. L edere r and V.S . Shirley, eds., Table of isotopes W iley, New York, 1978).[17] T.W. Burrows, Nucl. Data Sheets 61 19 90) 1 .[18] Zhou Chunmei , Nucl . Da ta Shee ts 63 199 1) 229 .[ 19] H uo Junde, Hu Dail ing, Sun Huibin, You Janm ing and Hu Baoh ua, Nucl. Data Sheets 58 198 9) 677.[20] Huo Junde, Hu Dall ing, Nucl. Data Sheets 61 199 0) 47.