Equilibrium deformations of neutron-rich nuclei in the A ≈ 100 region

Nuclear Phystcs A139 (1969) 269--276; (~) North-Holland Pubhshm# Co., Amsterdam N o t to be reproduced by photoprint or microfilm without written permission from the pubhsher

EQUII,IRRIUM

DEFORMATIONS

OF

NEUTRON-RICH

NUCLEI

I N THE A ~ 100 REGION D. A ARSENIEV Radwtechmcal Institute of the USSR Academy of Sctences, Moscow, USSR A SOBICZEWSKI Instltute for Nuclear Research, Warsaw, Hoza 69, Poland V G. SOLOVIEV Joint Institute for Nuclear Research, Laboratory of Theoretwal Physics, Dubna, USSR Received 8 September 1969

Abstract: Ground state eqmhbrmm deformatmns, deforrnatmn energies and quadrupole moments are calculated for even nuclei m the 28 < Z < 50, 50 < N < 82 region. It is found that the eqmhbrmm shapes of these nuclei are axially symmetric For most of them the oblate shapes are preferred. All nuclei in the investigated regton seem to be soft against both beta and gamma deformations 1. Introduction T h e r e g i o n 28 < Z < 50, 50 < N < 82 consists m a i n l y o f unstable, n e u t r o n - r i c h nuclei which c a n be o b t a i n e d as fission p r o d u c t s o f h e a v y elements. I n f o r m a t i o n a b o u t the structure o f these p r o d u c t s h a s been o b t a i n e d by the investigation o f the d e l a y e d g a m m a r a d i a t i o n f r o m t h e m 1 - a ) . T h e r e are lndxcatlons t h a t some o f t h e nuclei m the c o n s i d e r e d r e g i o n m a y have a stable d e f o r m a t i o n 1). A n estimate p e r f o r m e d in the s e m i - e m p i r i c a l w a y o f ref. 4) has s h o w n t h a t we m a y expect a stable d e f o r m a t i o n for the nuclei close to the r u t h e n m m isotopes. T h e expected values o f the d e f o r m a t i o n o f t h e r u t h e m u m i s o t o p e s have been g w e n there T h e a i m o f the p r e s e n t r e s e a r c h ~s to p e r f o r m m i c r o s c o p i c c a l c u l a h o n s o f the g r o u n d state e q m h b r m m d e f o r m a t i o n s o f even nucle~ m the whole 28 < Z < 50, 50 < N < 82 region. S o m e o f the results o f these calculations have been r e p o r t e d earher 5). I n sect. 2 we describe the calculations, in sect 3 we p r e s e n t a n d discuss t h e results a n d m sect. 4 we d r a w conclusions f r o m the results

2. Description of the calculation T h e c a l c u l a t i o n is p e r f o r m e d m close a n a l o g y with o u r previous m v e s t l g a h o n 6) p e r f o r m e d m the 50 < Z , N < 82 region. I n the p r e s e n t r e s e a r c h into the n e u t r o n r i c h r e g i o n we restrict ourselves o n l y t o t h e g r o u n d states o f even nuclei. W e use the M o t t e l s o n a n d N l l s s o n 7) p r o c e d u r e o f the c a l e u l a h o n o f the e q m h b r m m d e f o r m a t i o n s g e n e r a h z e d b y B~s a n d Szymafiskl 8) t o include the p a m n g rater269

D A ARSENIEVet al

270

action We adopt the Bardeen-Cooper-Schrlffer (BCS) wave function for the nuclear ground state, which is easy to operate with. The effect of using more exact PBCS wave functions corresponding to a definite number of particles was Investigated in ref 9) It was found there that for well deformed nuclei the use of PBCS wave functions mstead of the BCS ones leads to small changes of the equilibrium deformations The ground state energy of an even nucleus, we calculate, is of the form

= ~ p + ~ n + OPC,

(1)

where Ep and ~'n correspond to protons and neutrons, respectively, and they are given by d'p(n) = Z E ( v ) 2 ~ z - C2/G, (2) "o

with 2 ~ -= 1 - [ E ( v ) - A ] / ~ ( v ) a n d e(v) = ~/C2-t- [ E ( v ) - 2 ] 2. Here E(v) denotes the smgle-pamcle energy and G the pairing forces strength The chemical potential 2 and the energy gap 2C are calculated from the two equanons

21G = Z l/~(v), V

n = Y~ 2"~2.

(3)

V

The energy gp is obtained with all quantities in eq (2) calculated with the proton smgle-pamcle energies E(v) and with the pairing constant G z proper for protons The energy d~, is obtained with the corresponding quantities for neutrons. The Coulomb energy d c in eq (1) is calculated as the electrostatic energy of a uniformly charged ellipsoid deformed m a volume conserving manner a 0) In the numerical calculations we have used the recent NHsson potential with the ( l z) term 11). Two variants of the parameters of this potential have been used" 0) Parameters corresponding to the rare-earth regmn, i e. ~p = 0 0637 and Pp = 0.60 for protons and to, = 0 0637 and/~n = 0 42 for neutrons. The scheme IS not far from the one used in ref lZ) for the investigation of spherical nuclei in the region under discussion (u) Parameters extrapolated hnearly from the rare-earth and acnmde regmns to the region under investigation They are top = 0 0688 ~c, = 0 0638

and and

/tp = 0.558 for protons and /~,, = 0.491 for neutrons 6).

The pairing forces strength G taken in the calculanon is Gz = (28 5/A) MeV for protons and GN = (25/A) MeV for neutrons. These values correspond to the 24 levels, nearest to the Fermt level, allowing for nucleon scattering by the pmrxng forces and they are the same as in ref 6) 3. Results and discussion

We present here the results obtained with the variant (i) of the Nllsson potential parameters (i.e. adjusted to the rare-earth region) described in sect 2 The effects of using variant (n) instead of (1) are discussed.

271

EQUILIBRIUM DEFORMATIONS

W e a s s u m e a nucleus ( m o r e e x a c t l y the n u c l e a r p o t e n t m l ) to be an elhpsold, which can be d e f o r m e d in a v o l u m e c o n s e r v i n g m a n n e r . T h e shape o f such a n e l h p s o l d can be d e s c r i b e d b y two p a r a m e t e r s , e g. p a r a m e t e r s fl a n d y i n t r o d u c e d b y B o h r 13) A n e x a m p l e o f the d e p e n d e n c e o f the energy E(fl, ~), as c a l c u l a t e d f r o m eq. (1), on b o t h these p a r a m e t e r s is g w e n in fig. 1 for 108Sr W e see t h a t the energy m i n i m u m is o b t a i n e d for 7 = 60°, i.e for the axially s y m m e t r i c o b l a t e shape F o r some nuclei in the investigated r e g i o n the m i n i m u m is o b t a i n e d at 7 = 0°, 1 e. for an axially symm e t r i c p r o l a t e shape, b u t in all cases it occurs a t the axial s y m m e t r y o f a nucleus. Fig. 1 shows t h a t the surface o f the energy g(fl, ~) forms a valley w h i c h is r a t h e r s m o o t h with respect to the g a m m a d e f o r m a t i o n . T h e d e p e n d e n c e o f the m i n i m a l

© /-4

1o8,3~r7o .%.

0

Ol

02

03

04

05

[5

Fig. 1. A contour map of the ground state energy of 1°8Sr versus fl and? deformations. The numbers at the contour lines give values of the energy In MeV. The energy of the absolute mlmmum (occurring at fl = 0 26 and ? = 60°) is taken as zero p o i n t s o f this valley, each p o i n t r e p r e s e n t i n g the m i n i m a l energy m the y = c o n s t a n t plane, is given in fig. 2 W e see t h a t n o e n e r g y b a r r i e r occurs between o b l a t e a n d p r o late shapes when going via g a m m a d e f o r m a t i o n . It is r a t h e r characteristic for all the nuclei in the investigated region. A s stated the energy m i n i m u m o f a nucleus occurs at its axial s y m m e t r y , hence we s t u d y In the following only the energy o f the axially s y m m e t r i c nucleus 1 e. the energy d e p e n d i n g o n l y o n one d e f o r m a t i o n p a r a m e t e r . W e choose for th~s p a r a m e t e r the p a r a m e t e r e defined by N l l s s o n 14) a n d related to fl a p p r o x i m a t e l y by e ~ 0 95 ft. W e investigate the energy In a wide interval o f deformations: e = - 0 5(0 05)0.5, 1.e. b o t h for o b l a t e (e < 0) a n d for p r o l a t e (e > 0) shapes o f a nucleus. A n e x a m p l e o f the d e p e n d e n c e o f the energy d° on the d e f o r m a t i o n e is given in fig 3 for the nucleus l ° 2 K r W e see t h a t the energy has t w o r n m l m a : one for % < 0 a n d the second for e + > 0 W e have seen m the t w o - d i m e n s i o n a l case m fig 1 t h a t the

272

D A ARSENIEVet al

lower minimum at eo corresponds to the absolute minimum and thus to the eqmhbrium deformatmn of the nucleus From the upper mlmmum at e~- one can get the absolute mm~mum, going via gamma deformaUon, w~thout any (m general without any sigmficant) energy barrier Thus the upper minimum does not represent any stable nuclear state For l°2Kr the deformation energy for %, l.e the energy W ~ = ~ ( 0 ) - W ( % ) , is larger than the deformation energy ga+f for %+ But, as for some nuclei m the mvesUgated region the situation is opposite, Le ~-¢f < ~-~f and as the two energies W(%) and eC(e+) are close to each other, we are interested m the &fference between them -'t ~'~def ~---

=

+

6 [Mev]

I0

"6' 600

400

20"

!t

[Mev]

/

JO2k,r

36 '" 66

1.5 3

2

05

£i,f

I 0

0°

0

Fig 2. M i n i m u m energy projection along the g a m m a d e f o r m a t m n for ~°aSr Each point along the curve represents the energy m l m m u m with respect to beta d e f o r m a t i o n

-0

Fig

t;

3

-01

OI

%*

05

The g r o u n d state energy o f l ° 2 K r versus the deformaUon e

The contour maps presenting the values of the four quantities %, e0+, W~f and A 8de f in the whole region 28 < Z < 50, 50 < N < 82 are given in figs 4, 5, 6 and 7, respectively. We see from fig. 7 that an most of the region the oblate shapes are preferred. The largest preference is obtained for nuclei in the neighbourhood of 1°8Sr The energy corresponding to oblate shape of these nuclei is lower than the energy corresponding to prolate shape by about 1 5 MeV The prolate shape IS preferred only for some nuclei near the boundary of the region The deformation energy ~-e~ of these nuclei ~s as a rule less than 1 MeV and thus their deformaUons have probably a dynamic rather than a staUc character. A low value of A Wdefin the greater part of the investigated region gives a possablhty for an appearance in this region of shape isomers, discussed in ref. 6) for the neutrondeficient nuclei. The largest deformation energies are obtained for nuclei in the netghbourhood of 102Kr and they are about 3.5 MeV. Thus they are much lower than the corresponding energies obtained for neutron-defioent nuclei m the 50 < Z, N < 82 region 6) or

273

EQUILIBRIUM DEFORMATIONS

rare-earth region 15,16), which are a b o u t 8 MeV it m e a n s that the investigated nuclei are expected to be softer on the e (or fl) d e f o r m a t i o n t h a n the nuclei in the 50 < Z, N < 82 a n d rare-earth regions This is p r o b a b l y a consequence of the smaller (and, having a lower density) p r o t o n shell 28 < Z < 50 t h a n the shell 50 < Z < 82. The e q u l h b r i u m d e f o r m a t i o n s In the investigated region, as seen from figs. 4 a n d 5, are only slightly lower t h a n the d e f o r m a t i o n s in the 50 < Z, N < 82 [ref 6)] and rare-earth regions 15, 16) 48

-010

0

~.

44

-020

48

-010

44

40

40

36

36

32

321

28 50

2850

0

N 54

58

62

60

70

74

78

82

Fig 4 A contour map of the negative deformation versus proton Z and neutron N numbers The numbers at the contour lines give values of % . 4g

28

62

66

70

74

78

82

44 .0

4O

36 32

58

48 Z

44 40

54

Fig 5 T h e s a m e is in fig 2 for the p o s l t w e d e f o r m a t i o n e~

36 32

\ N

50 54 58 62 66 70 74 78 82 Fig. 6. The same as in fig 2 for the deformation energy d°Af= g(0) -- d'(e o ) The values of the energy are m MeV

N

2850

54

58

62

66

70

74

78

Fig 7 T h e s a m e as in fig. 2 for the difference A 8ae~ = , f ( e g ) -- @(e~) given m M e V

It can be seen from fig 6 a n d also from figs 4 a n d 5 that the subshell for Z = 38-40, o b t a i n e d at ~ = 0 in the single-particle p r o t o n scheme used by us, has almost n o effect o n the d e f o r m a t i o n s of nuclei wlth the n e u t r o n n u m b e r s somewhere in the middle of the shell 50 < N < 82. Fig. 6 shows also that even the nuclei with the magic n u m b e r of p r o t o n s Z = 28, b u t with n e u t r o n n u m b e r s N ~ 66, m a y be deformed. However, due to the low value of their d e f o r m a t i o n energy which is a b o u t 0.6 MeV, the deform a t i o n m a y be m o r e of a d y n a m i c t h a n a static character

82

D A ARSENIEVet al

274

Table 1 gives the detailed values of the equlhbrmm rupole moments

Qo(eg), d e f o r m a t m n

energies 8~f

d e f o r m a t i o n s eft, e l e c t r i c q u a d a n d t h e d i f f e r e n c e s AEde f f o r

nuclei with the deformation energies not lower than 2 MeV The electric quadrupole

moments

have been calculated from the formula

Q0 = Z q ~ 2 ~ ,

(4)

v

w h e r e qv~ a r e t h e d m g o n a l m a t r i x e l e m e n t s o f t h e q u a d r u p o l e m o m e n t the summanon

operator and

extends only over the proton states.

L e t u s d i s c u s s s h o r t l y w h a t a r e t h e effects o f u s i n g t h e s l n g l e - p a m c l e s c h e m e ( n ) i n s t e a d o f s c h e m e O) TABLE 1

Negative deformations eo, electric quadrupole moments Qo (e0) corresponding to these deformations, deformatmn energies d ~ f = o#(0)-- d~(eo) and dtfferences d gaff = g ( e o ) - - d~(e+) calculated for the ground state of nucleldes specified m the first column Nuclelde

eo

Qo (eo) (b)

X°SRu X~°Ru H2Ru

--0 25 --0 24 --0 24

1°4Mo 1°6Mo l°SMo ll°Mo 112Mo

gd~f (MeV)

A efaef (MeV)

Nuclelde

eo

Qo (So) (b)

~'def (MeV)

A gaer (MeV)

--2 3 --2 2 --2 3

20 2 1 20

--0 6 --0 7 --0 8

96Kr 9aKr l°°Kr 1°2Kr l°4Kr l°6Kr

--0 32 --0 32 --0 31 --0 30 --0 29 --028

--2 3 --2 3 --2 2 --2 2 --2 2 --22

27 32 34 34 32 26

--1 0 --1 2 --1 2 --1 3 --1 4 --14

--0 27 --026 --0 25 --0 24 --0 23

--2.3 --23 --2 2 --2 2 --2.1

23 26 27 26 22

--0 8 --09 --1 1 --1 3 --1 3

94Se 96Se l°°Se 1°2Se 1°4Se

--0 29 --0 29 --0 29 --0 28 --0 27 --026

--2 --2 --2 --2 --1 --1

0 0 0 0 9 9

23 28 30 31 29 24

--0 --0 --0 --1 --1 --1

--0 --0 --0 --0 --0 --0

28 28 28 27 26 24

--2 3 --2 3 --2.3 --2 3 --2 2 --2 1

22 26 29 30 28 2 3

--0 --0 --1 --1 --1 --1

9 9 0 2 4 5

94Ge 96Ge

--0 27 --027

--1 7 --1 8

22 24

--0 4 --05

9SGe

- - 0 26

--1 7

2 5

--0 6

--0 30 --030 --0 30 --0 29 --0 28 --0 26

--2 3 --23 --2 4 --2 3 --2 2 --2 1

24 29 31 32 30 24

--0 --1 --1 --1 --1 --1

9 0 0 2 4 6

l°°Ge l°2Ge

--024 --0 23

--1 6 --1 6

24 20

--06 --0 6

98Se

l°°Zr l°2Zr

x°~Zr l°6Zr l°SZr 11°Zr 9aSr

a°°Sr l°2Sr

1°4Sr l°6Sr 1°SSr

6 8 8 0 0 0

The equlhbrmm deformations are almost the same. Their absolute values are slightly l a r g e r i n s c h e m e ( u ) t h a n i n s c h e m e (i) b u t t h e d i f f e r e n c e b e t w e e n t h e m d o e s n o t e x c e e d 0 01 ( f o r n u c l e i w i t h t h e d e f o r m a t i o n e n e r g y Wd~f > 2 M e V ) . A l s o t h e p o s i n o n , on the (Z, N) chart, of the nuclei with the largest deformanons both schemes.

remains the same in

EQUILIBRIUM DEFORMATIONS

275

The deformation energies obtained with scheme (n) are larger than those obtained with scheme (1) The largest difference between them is about 1 MeV. The largest value of the deformation energies d ~ f ~ 4.5 MeV is obtained for nuclei in the neighb o u r h o o d of l°2Kr, 1 e. for the same nuclei as with scheme (1). Concerning the &fference A o~a~r we find that the number of nuclei with A Nd~f > 0 and with A d°def --- 0 lS larger when scheme (11) is used than the corresponding numb~ r when scheme (1) is taken. The extreme value of Ad°dCf is - 1 . 3 MeV (in comparison with - 1.6 MeV in scheme (a)) is obtained for nuclei m the nelghbourhood of i OSSr ' 1.e for the same nuclei as obtained with scheme (1) Finally, in table 2 we see the effect of the ___10 ~ change m the pairing forces strength G (both for protons and for neutrons) on the quantmes eft, e0+, ~ f and A Nd,f for 1 °2Kr and ~°SSr. A sensmvlty of these quantities, especially of the energies d ~ r and A d°d,f on the parameter 6° is visible. TABLE 2 Effect o f the ~ 1 0 ~ change m b o t h p r o t o n and n e u t r o n p a m n g force strengths G o n the q u a n t m e s eo, eo+, 8a~ r a n d A~'a¢ t The last two quantxtxes are m MeV 1°~Kr66 G

lAG

l°~Sr7o 0 9G

eo

--0 305

--0 282

--0 318

e+

0.257

0 248

gger

3 42

2 48

--1 28

--1 02

ASde t

G

1.1G

0.9G

--0 263

--0.245

--0.272

0,275

0 214

0,177

0 251

4,42

2.44

1.75

3.16

--1 49

--1 56

--1.16

--1.80

4. Conclusions The following conclusions m a y be drawn from the results of our calculations (1) Nuclei in the investigated region seem to be softer on both g a m m a and beta deformations than the nuclei m the rare-earth region. The largest deformatmn energies are about 4 MeV in comparison with 8 MeV obtained for the rare-earth regmn. They also seem to be softer on the beta deformatmn than the neutron-deficient nucle~ in the 50 < Z, N < 82 region. (n) Mxmmal energies of the nuclei are found at T = 60° or y = 0 °. Thus the equihbrlum shapes are axially symmetric. (in) For most of the nuclei the oblate shapes are preferred. The largest preference is obtained for nucleldes in the nelghbourhood of j °SSr. (iv) A sensitivity of the results on the single-particle scheme is observed The authors would like to thank Prof. Z. Szymafiskl for helpful discussions.

276

D

A

ARSENIEV et

al

References 1) S A E Johansson, Nucl Phys 64 (1965) 147 2) S E A Johansson and P Klemhemz, m Alpha-, beta- and gamma-ray spectroscopy, ed by K Slegbahn (North-Holland Publ C o , Amsterdam, 1965) p 805 3) L A Popeko, G A Petrov and D M Kammker, Report of the Institute of Physics and Technology, No 128, Leningrad, 1968 4) S A E Johansson, Ark Fys 36 (1967)599 5) D A Arsenxev, A Soblczewskl and V G Solovlev, Int Conf on propemes of nuclear states (Montreal, 25-30 August 1969) 6) D A Arsemev, A Soblczewskl and V G Solov~ev, Nucl Phys A126 (1969)15 7) B R Mottelson and S G Ndsson, Mat Fys Skr Dao Vld Selsk 1, No 8 (1959) 8) D R B~s and Z SzymafiskJ, Nucl Phys 28 (1961)42, Z SzymafiskJ, Nucl Phys 28 (1961)63 9) A Soblczewskl, Nucl Phys A93 (1967) 501, A96 (1967) 258 10) B C Carlsson, J Math Phys 2 (1961) 441 11) C Gustafson, I L Lamm, B Ntlsson and S G Nflsson, Ark, Fys 36 (1967) 613 12) L S Klsshnger and R A Sorensen, Revs Mod Phys 35 (1963)853 13) A Bohr, Mat Fys Medd Dan Vld Selsk 26, No 14 (1952) 14) S G Nflsson, Mat Fys Medd Dart Vld Selsk 29, No 16 (1955) 15) D A Arsemev, L A Malov, V. V Pashkevlch and V G Solovlev, Izv Akad Nauk SSSR (ser fiz ) 32 (1968) 866 16) I L Lamm, Nucl Phys A125 (1969)504