Fission barriers of odd-mass nuclei and odd nuclei with 100 ⩽ Z ⩽ 111

Nuclear Physics @ North-Holland

A444 (1985) Publishing

FISSION AND

1-12 Company

BARRIERS ODD

S. CWIOK,

OF ODD-MASS

NUCLEI Z. LOJEWSKI

WITH

NUCLEI

lOO< Z<

111

and V.V. PASHKEVICH

Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, USSR Received 2 January (Revised 26 March

1985 1985)

Abstract: The height and structure of fission barriers for odd-mass nuclei and odd nuclei with 100s 2 G 111 are calculated by the Strutinsky method and using a realistic Woods-Saxon potential. Non-axial variations of the nuclear shape are taken into account. The influence of the blocking effect and of taking into account the conservation of spin and parity on the height of the barriers is considered. As in the adjacent even-even nuclei, in heavy isotopes of the heavy elements (Z = 107-111, N = 160-165) the occurrence of a shape isomer is possible. In the nuclei considered the fission barrier heights do not show a tendency of decreasing with increasing mass ndmber.

1. Introduction

In connection with experiments to synthesize the nuclei of new heavy and superheavy elements and with the search for them in nature le3), the fission barriers of even-even nuclei with 100 G 2 c 110 were calculated in ref. “) using the Strutinsky method 5,6). The fission barriers of heavy elements were recently also investigated in refs. 7-9). In comparison with even-even nuclei, odd-mass nuclei are somewhat more stable with respect to spontaneous fission than their even-even neighbours and therefore experimental data about their properties are more copious. In particular, experimental studies of nuclei with 2 = 107 and 109 are currently in progress lo-13). The theoretical assessment of the properties of these elements is of some interest. The main factors which determine the nuclear fission probability are the height and structure of the fission barrier. In the present paper, which may be considered to be an extension of ref. 4), the fission barrier heights and the landscape of deformation energies of the odd-mass nuclei and odd nuclei with 2 = lOO- 111 are calculated. The barriers of odd-mass nuclei were obtained earlier in refs. 9,14,15), where the modified harmonic oscillator potential 16) was used. In the calculation of the barriers 14), or the difference between the barriers of odd-mass and even-even nuclei 15), it was supposed that in the axially- and mirror-symmetric nucleus the odd particle occupies, during the fission process, a level with definite spin projection on the symmetry axis, 0, and definite parity, r. The barriers thus calculated exceed by several MeV in height the barriers obtained by interpolating the barriers of the adjacent even-even nuclei. This difference, termed a “specialization energy” 17),

2

S. i?wiok et al. / Fission barriers

results in a several MeV increase in the spontaneous fission half-lives. It is known, however, that the heavy actinides lose their axial symmetry in the vicinity of the first fission barrier so that the spin projection ceases to be a good quantum number and the forbiddenness associated with the constraint 0 = const should not necessarily be strictly fulfilled 15). In ref. 9, a large number of even-even and odd-mass nuclei with 76 6 2 5 100 were investigated taking into account the possible non-axiality of nuclear shape, the odd particle being assumed to occupy the lowest free level on the Fermi surface. In ref. ‘) use was made of the thoroughly adjusted parameters entering into the Nilsson potential and the liquid-drop model, as well as the pairing strength constants. However the very heavy elements with 2 2 101 were not considered there. In the present paper, as in ref. 4), calculations of the fission barriers are carried out using the realistic Woods-Saxon-type potential with the parameters of refs. i**“) and taking into account the non-axial variation of nuclear shape. The influence of the blocking effect and the account of the conservation of R and rr on the fission barrier height is considered. Calculational details and the choice of parameters are discussed in sect. 2. The main results are given in sect. 3,

2. method

of calculating

deformation

Dividing the deformation energy into components, Strutinsky method, we obtain

E($) = ELD(b) + 6E,~~U(6) +

energy

as is usually done in the

6Epair(P^)

*

(1)

We will calculate, in terms of the droplet model *‘,*l), the part E,n(& which smoothly depends on particle number and deformation. The details of the calculation and model parameters, such as those of the single-particle potential that were used to calculate the shell correction SE shelland the pairing correction SEpair, are given in ref. “). For the reader’s convenience the main parameters of the Woods-Saxon potential are listed in table 1. All the calculations presented in this paper have been carried out using the level spectrum of the nucleus 256102.In formula (l), @ denotes the set of parameters that specify the nuclear surface shape (in our case these are ,&, Y and P4 [ref.

“)I. TABLE

Parameters Central

[ protons neutrons

fz,

1.275 1.347

1

of the single-particle

potential

4, Spin-orbit

potential

interaction

(d*.O. rf:I 0.70 0.70

IM:v,

lfml

rr:1

-58.3 -40.9

0.919 1.261

0.70 0.70

A 18.19 32.21

5

S. Cwiok et al. / Fission barriers

Some comments should be given on the method of calculating the shell correction =&l&l in an odd-mass nucleus. As in the even-even nuclei, GEshell is calculated as a difference between the single-particle energy and the averaged energy 5*6),i.e. (N--1)/2 ~&h‘sh,ll

=

2

c i=l

Ei+&(N+1)/2-E(~)

-

(2)

Here ci is a single-particle level spectrum and E is a “smooth” energy calculated, as usual 5*6),with the Fermi energy i, defined on condition that IV=1 F?i(li),

(3)

where N is the (odd) number of nucleons of one kind and & is the “smooth” occupation number ‘“). It can be seen that in such an approach J!? is a uniform (smooth) function of the particle number iV both for even and odd N-values 23). Therefore in the case of odd N, E is close to the value obtained by inte~olating through the points corresponding to even values of the argument N [refs. 14*15)]. In an alternative method the energy and number of particles corresponding to the ‘“smooth” level spectrum are calculated taking into account an effect that is analogous to the blocking effect in the nuclear system with pairing 24). Namely, in obtaining the averaged quantities in the level spectrum the level occupied by the odd particle is deleted and the energy of this particle is added to the total energy and unity is added to the number of particles. The “smooth” energy thus obtained differs from the interpolated values by approximately 1 MeV and the masses of the odd-mass nuclei systematically lie below the masses of the neighboring even-even nuclei. In the present paper this latter method has not been used but one can say that in both methods the barrier values are close to each other because the difference between the two deformation energies weakly depends on deformation. The pairing correction was calculated in the usual way, taking into account the blocking effect 23,24)if not stated otherwise. 3. Results The general nature of the potential energy landscape changes only slightly with the addition of one particle and therefore the calculation of the deformation energy of heavy odd-mass nuclei was carried out on the same coordinate mesh, as in the case of neighboring even-even nuclei “). The nuclei were assumed to be axiallysymmetric in the ground state and in the vicinity of the second minimum (&== 0.7-0.X), and the non-axial variation was taken into account only in the region of the barrier between them. It was assumed that the nuclei were mirror-symmetric. The violation of mirror symmetry is important near the second minimum and it may not be taken into account in calculating the fission barriers for the heavy nuclei considered.

4

S. C?wiok et al. / Fission barriers

Fig. 1. Deformation energy of the 258105 nucleus along the fission valley as a function of pz. The solid line shows the calculation with the blocking effect *3.24) taken into account and assuming the odd particle to occupy the lowest free level. The dashed line represents the case without taking account the blocking effect. The dash-dotted line corresponds to the calculation done as in ref. ‘s), with the odd particle occupying the lowest free level of definite spin projection and parity.

As was already noted in the introduction, it was supposed in refs. “,15) that for the calculation of the fission barrier height the odd particle occupied the lowest possible state with definite 0 and rr. In our calculation, as in ref. 9), in the vicinity of the barrier LI has no well-defined meaning and the odd particle is supposed to occupy the lowest possible free level. The influence of this supposition on the fission barrier and also the role of the blocking of the odd-particle state 23724)are demonstrated in figs. 1 and 2. The fission barrier of the *“105 nucleus calculated under different assumptions is presented in fig. 1. The blocking effect is seen to lead to a 0.3-0.6 MeV increase

in the barrier

height,

and the hypothesis

that 0. and rr should

2.

o-> 3 2 -2 " -EL -4.

Fig. 2. Deformation energy of the 259Fm nucleus along the fission valley as a function of &. The solid line shows the calculation with the odd particle occupying the lowest free level. The dashed line represents the case with the odd particle on the lowest free level with definite parity.

5

S. Cwiok ef al. / Fission barriers

be constant

during

value resulting are compared. difference

the fission

process

gives a 1.0-1.5 MeV increase

in the barrier

in a value which seems to be too large. In fig. 2 the two calculations In one of them r is conserved an-d in the other it is not. The energy

in this case is seen not to exceed 0.4 MeV. In other nuclei this value varies

within the range 0.0 to 0.3 MeV being negligibly small. Therefore in our subsequent calculations, as in ref. 9), it was assumed that the odd particle should occupy the lowest free state irrespective of its parity. The potential energy values for all the nuclei are presented in figs. 3-14 in which for the odd number are also given. In the consideration between the odd proton and the odd neutron nuclei considered are seen to be deformed in

-4

-

0.2

I

03

I

I

I

04 ,,, 05

considered, along the fission valley, values of Z nuclei with odd neutron of odd-odd nuclei the interaction was not taken into account. All the the ground state though in the very

L

1

I

06

I

IA

07

Fig. 3. Deformation energy along the fission valley as a function of quadrupole deformation for fermium isotopes (2’ 100). Dashed lines indicate the case with y=O. The left-hand scale corresponds to the lightest isotope (bottom curve). Other curves are shifted upwards by 6 MeV. The curves are labelled with respective

neutron

numbers.

6

- 32 9 r 23 3 2 24 L xfM-

Fig. 4. As in fig. 3, but for 2 = 101. The curves

are shifted

upwards

by 4 MeV.

heavy isotopes of the elements with 2 = 110 and 111 the tendency towards decreasing deformation becomes apparent. The deformation of odd-mass nuclei in the ground state is close to that of their even-even neighbors and, as can be seen from fig. 15, the ground-state energy of the former is higher than the energy of the latter by approximately the magnitude of the pairing gap. In fig. 15 the subshell N = 152 manifests itself very clearly. In the heavier nuclei the IV = 162-164 subsheil begins to show up. In the heavier isotopes of the elements considered the barrier is split into two subsidiary peaks, the depth of the well between them reaching a value of about

S. &ok

et al. I Fission

barriers

7

1’ ’ ’ ’ ’z=103 ’ ’ ’ ’ ’ ’ ‘I 42

b0

4

c II

I

02

I

03

I,

‘,

04

05

R,

-4

11

06

11

I

07

Fig. 5. As in fig. 4, but for z = 102.

Fig. 6. As in fig. 4, but for Z = 103.

36

z=104

32

_’

,'-L

‘\

i

16

-4 t

r,-

02

03

04

T)I

05

06

07

Fig. 7. As in fig. 4, but for Z = 104.

S. biok

et aL / Fission barriers

S. Cwiok et al. / Fission barriers

44

Z=108

40

zat

-4 02

04

03

05

11

07

06

02

0, Fig.

Fig.

11. As in fig. 3, but for z = 108.

1

02

03

I

04

I

IL

05

/

,

06

,

t /

1

?kbim

82

05

I

I

06

12. As in fig. 4, but for Z = 109.

,

07

Fig. 13. As in fig. 3, but for Z = 110. The curves are shifted

upwards

by 5 MeV.

1

07

Fig. 14. As in fig. 3, but for Z = 111.

1

10

S. cwiok et al. / Fission barriers

18. 16. 14. 12.

-.:.

108,

% 36. ‘-^.

:“/S;,:, 142 I46 150N154

158

162 16

Fig. 15. Ground-state potential energy V, as a function of neutron number. The energy of the spherical liquid drop was taken as a reference point. Curves are labelled by corresponding values of Z. The zero-point vibration energy was supposed to be equal to 0.5 MeV. The leftlhand scale corresponds to the bottom curve (Z = 100). Other curves are shifted upwards by 2 MeV. The values for even-even nuclei are taken

from ref. 4).

4 MeV for the heaviest elements. The position of the minimum (&= 0.4, p4== 0.08-0.12) remains almost unchanged in different odd-mass nuclei and coincides with that for the neighboring even-even nuclei “). The possible existence of shape isomers and the influence of this intermediate minimum on the heavy-ion-induced fusion reactions at low excitation energies have already been discussed earlier4). The fission barriers of the odd-mass nuclei and odd nuclei are presented in figs. 16 and 17, together with the results of ref. “). Such a comparison is quite legitimate as both calculations were carried out with the same set of parameters and using the same method of calculation. The heights of the fission barriers of the odd-mass nuclei are seen to exceed by, on average, 0.5-1.0 MeV the corresponding values for the neighboring even-even nuclei. Thus in the calculation is reflected the fact that odd-mass nuclei are more stable against spontaneous fission than their even-even neighbors [see e.g. refs. 1o-13)]. The local maxima of the fission barriers are attained for neutron numbers equal to 151 and 161 and this is due to the above-mentioned ground-state subshells and the odd-even effect. The most remarkable fact is that the fission barrier of nuclei with 2 = 108-l 11 and N = 155-165 is high enough to enable one to speak about the possible production and investigation of these nuclei in heavy-ion reactions 25).

4. Conclusions The height and structure of fission barriers for odd-mass nuclei and odd nuclei with 100 G 2 =G111 and 140 s N s 165 have been calculated by the Strutinsky method

11

S. Cwiok et a/. / Fission barriers

28;: 26.3 242 22. 20.

106

18.

-^r

105

16. x34

14.

103 102 101

I

.

L

142

146

150

154

N

158

162

16

Fig. 16. The fission barriers as functions of neutron number. The left-hand scale corresponds to the bottom curve (Z = 100). Other curves are shifted upwards by 2 MeV. Curves are labelled by corresponding values of Z. The values for even-even nuclei are taken from ref. “).

~lOOjlO1~102~103~104]105~106~107~1~~l09~110~1II l4215.8l

14.71

j

1

j

1

1

1

147 8.1 7.5 6.0

6.4

146 7.4 6.9 6.6

68

149807675

66

lsn7L71fi~

” _. _ _ _ 75179/73/6.61711

1

1

72

164

8.3 7.7

I

7.2 5

165

7.4 a

166

6.7 7.0

Fig. 17. The fission barriers

in MeV, the values

for even-even

nuclei are taken

from ref. 4).

12

S. i?wiok et al / Fission barriers

and by taking into account the non-axial variation of nuclear shape. The set of results obtained, together with analogous data for even-even nuclei from ref. 4), permits studies of variations of the main features of the deformation energy landscape in a wide range of neutron and proton numbers. In the heavier isotopes of the heavier elements (2 = 107-l 11, N = 160-165) the appearance of shape isomers is possible at deformation p2 F=0.4. The fission barriers of odd-mass nuclei exceed by OS-l.0 MeV those of the adjacent even-even nuclei. The influence of the blocking effect and of the conservation of R and rr on the fission barrier height has been considered. In the nuclei investigated fission barriers do not exhibit a tendency of decreasing with increasing mass number, thus providing evidence for the comparative stability of these nuclei against spontaneous fission. The authors are grateful to V.G. Soloviev for continuous interest in the work and to YuTs. Oganessian for stimulating discussions.

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25)

G.N. Flerov and G.M. Ter-Akopian, Pure and Appi. Chem. 53 (1981) 909 Yu.Ts. Oganessian et a& Nucl. Phys. A294 (1978) 213 G. Miinzenberg et at, Z. Phys. 300 (1981) 107 S. Cwiok et al., Nucl. Phys. A410 (1983) 254 V.M. Strutinsky, Nucl. Phys. A95 (1967) 420 V.M. Strutinsky, Nucl. Phys. Al22 (1968) 1 J. Randrup et al., Phys. Rev. Cl3 (1976) 229 K. Junker and J.Z. Hadermann, Z. Phys. A282 (1977) 391 W.M. Howard and P. Moller, At. Nucl. Data Tables 25 (1980) Yu.Ts. Oganessian et al. Nucl. Phys. A273 (1976) 505 G. Miinzenberg et al., Z. Phys. A309 (1978) 89 G. Miinzenberg et al, Z. Phys. A315 (1984) 145 Yu.Ts. Oganessian, Proc. Int. School Seminar on heavy ion physics, Alushta, 1983: Dubna D7-X3-644, p. 55 S.G. Nilsson et al., Phys. Lett. 30B (1969) 437 J. Randrup et aZ., Nuci. Phys. A217 (1973) 221 S.G. Nilsson et al., Nucl. Phys. A131 (1969) 1 J.A. Wheeler, Physica 22 (1956) 103 J. Dudek and T. Werner, J. of Phys. G4 (1978) 1543 J. Dudek et al., J. of Phys. G5 (1979) 1359 W.D. Myers and W.J. Swiatecki, Ark. Fiz. 36 (1967) 343 W.D. Myers and W.J. Swiatecki, Ann. of Phys. 55 (1969) 395 M. Brack et al., Rev. Mod. Phys. 44 (1972) 320 D.A. Arseniev et al.. Izv. Akad. Nauk SSSR (ser. fiz.) 32 (1968) 866 V.G. Soloviev, Teoria slozhnykh yader (Nauka, Moscow, 1971) [English transl.: Theory of complex nuclei (Pergamon, Oxford, 1976)] Yu.Ts. Oganessian ef a!., Z. Phys. 319 (1984) 215