Spontaneous fission of double-odd nuclei in [omega]-nonconserving model - Biblioteka UMCS

ANNALES

U N I V ER S I T A T I S MARIAE CURIE-SKLODOWSKA LUBLIN —POLONIA

VOL. XITXLI, 3 SECTIOAAA

1985/1986

• Instytut Fizyki UMCS

A. BARAN, Z. ŁOJEWSKI

Spontaneous Fission of Double-odd Nuclei in O-nonconserving Model

Spontaniczne rozszczepienie jąder

podwójnie nieparzystych w

modelu

niezachowującym

Спонтанное деление

дважды нечетных

ядер в Q

-несохраняющеймодели

Dedicated to Professor Stanisław Szpikowski on occasion of his 60th birthday 1. INTRODUCTION

Calculating fission half life times one accepts the con­

servation of both the energy and the angular momentum of the nucleus. The third component of the angular momentum is also as­

sumed to be constant during the barrier penetration process and it determines the third angular momentum component of the whole system. In our previous paper [1] we assumed the same. However, it seems that there is no reason to keep the angular momentum projection Л of the odd particle constant.

As it was shown earlier, the inclusion of axial and asym­

metric distrotions leads to a mixing of $L quantum numbers as

16 A. Baran, Z. Łojewski

well as parities 7Г in odd particle state [2]. This in conse­

quence decrease the specialization energy.

Such a treatment of fission process leads also to modifica­

tions in the mass tensor. Consequently the mass tensor and the fission barrier modifications lead to new estimations of fission half lifes and hindrance factors which were studied extensively in our previous paper [1] where the comparison to existing ex­

perimental data was given.

The aim of presented paper is to show the effect of non- conserving' on the hindrance factors and to compare them to pre­

viously calculated ones.

2. METHOD OF CALCULATION OF si

To generate the single particle orbits we use the Nilsson potential [3] with deformation parameters 8g, 8^. The pairing correlations were included in the BOS model. The deformation energy was calculated according to the Strutinsky prescription [4]. The smooth part of the energy was the same as in ref. [5].

The mass parameters entering the probability of barrier pe­

netration is given in the adiabatic approximation by the formula [6]

В Ы-У N>

[(uv7w + uwvv)2(1 -Цц) + S^(uvuw- Vv.vw)2j

(1)

Spontaneous Fission ot Double-odd Nuclei 17

where

av 31 л e^-i эд2

(2)

The superscript at single particle state

' B£i€j for which

is the designation of the odd the mass parameter is calculated.

The states , | CO ) are Nilsson single particle states. The parameter Л is the Fermi energy of the system, and Д is the BCS gap parameter.

The fission process is treated as a penetration through a one-dimensional barrier taken along an effective statical path

8 A £g) determined from.the condition V ( 8 2’ 64) = min.

2=const (5)

The penetration probability P in а «ГКВ approximation reads

P = (1 + exp{s}) , (4)

where S is an action integral S =

4

J д/ 2 V(s) - В B(s)’ ds

calculated along the trajectory specified above. The mass

(5) B(s) is called an effective static mass parameter and has the follow­

ing form

d6 d£

’<•) = £ —

i,j ds ds

Here s is the length of the trajectoiy and s1 and s^ are entrance and exit points respectively and are determined from the equation E = V(s), where E = 0.5 MeV [5, 7].

The half life time is given now by the relation In 2 1

= “n~ ? (7)

where n is a number of assaults barrier in a time unit. The value to 1020-58/s<

of the nucleus on the fission

0.5 MeV for E makes n equal

IB A. Baran, Z. Łojewakl

The change in the action integral Л s calculated as a difference between the action S for odd-even/odd system and the neighbouring even-even one is simply connected to the hin­

drance factor h defined by о-е/ о

Tsf ÄS

h = ■-£ё~ = e • (8 )

Tsf

q

e-e

Here Tg£ and Tgf are spontaneous fission half life times of odd-even/odd and even-even systems, respectively.

Since in our model we use only L2 ar(^ ^4 degrees of free­

dom the absolute values of T f are certainly not good enough to reproduce the experimental data.

In order to have better agreement one has to include other deformations such as L_c, £, and T. On the other hand if one calculates the value of Д S in a way described above, one can believe that the effect of other degrees of freedom does not enter the result very much.

The action Sa~^0 for the odd-even/odd system being the sum of properly calculated S®"6 of even-even system [7] and appro­

ximate value of △ S is used then to calculate spontaneous fis­

sion half life time T sf In the action S®”eall effects con- p nected to £35» Ł g and deformations are fully included. Spon­

taneous fission half life time of odd-even and/or odd-odd nucleus is given by the expression

o-e/o e-e e-e Tsf ~ h Tsf (Sp

Such a procedure makes it possible to compare with ex­

perimental data.

The problem of nonconserving the total projection 2 of angular momentum was discussed in ref. [2] and is connected to the T degree of freedom. The inclusion of T into the calcula­

tions makes the Nilsson state to be a mixture of components with different Q, values. It was shown that Nilsson single par­

ticle states of this kind lower the potential barrier of the

nucleus on an amount of 1-2 MeV [б].

Spontaneous Fission of Double-odd Nuclei ¹⁹

Sable 1. Hindrance factors for odd-even and odd-odd nuclei.

Upper number corresponds to 'nonconserving model.

The lower numbers are those presented early 1j.

К

Г X z i

"Г“

1

! 1 rt

105

1 i I 1 1 -41

1 97 ! i

i

99 101

1 103 I Î

1

107

1 1 1 1 1

109 0.1

¹ •

0.1

¹ 1

1.5 0.3 ; 0.1 0.2

^{1 1}

0.1

^{f 1}

146 1.5

1

I I

0.3

^Î I

t

4.8 0.6 i 2.5 0.2

^ł Î

«

2.8

¹ 1

—I

4.2

^{Î Î}

3.8

¹ 1

4.3 1.1 i 1.0 0.5

^{1 1} ₁

0.9

¹ ₁¹

147 8.0

^I _Î

1

7.0

¹ 1

1

11.0 з.з !

«. 2.4 0.7

¹ I

1

3.1

¹ 1

—J

0.1

^! i

0.4

^I

1.5 0.4 ; 0.2 0.2

^{1 1}

0.0

¹ 1

148 1.6

ⁱ ₁

0.4

ⁱ ₁

3.6 0.6 î 2.4 0.3

¹ t

2.7

¹₁

-4

149

2.6

¹ i

2.9

¹ 1

3.4 2.2 ! 1.8 0.9

^! 1

1.7

1¹

6.1

^»I

1

5.0

^{1 1}₁

7.8 2.5 J 2.9 1.0

₁¹

3.3

¹ ₁

мЦ

0.0

^Î t

0.6

¹ I

0.7 0.5 ! 0.3 0.2

¹I

0.1

ⁱ I

150 1.7

^i?_!

0.6

^{1 1}

1

3.3 0.6 } 1.9 0.4

¹ ₁

2.7

ⁱ _i

3.4

¹ _î^<

4.0

_ł¹

3.4 1.5 ! 1.2 0.2

¹ ₁

o.a

ⁱ i

151 5.8

¹ 1

1

4.5

^{1 1} ₁

7.2 1.9 J 2.1 0.3

^{1 1}₁

3.0

_1/

152 0.1

^{ł i}₁

0.3

^I1 ₁

0.8 0.4 ! 0.4 0.3

¹ I

0.1

ⁱ

1.7

^{1 1}

1

0.3

^{1 1}

1

3.3 0.5 } 1.8 0.2

^{1 1}

1

2.8

1 \

153 1.3

^{i 1} _i

1.0

^{1 1} ₁

2.2 1.4 } 1.2 0.2

¹ ₁

0.9

₁

5.7

¹

1.8

^{! 1}

1

5.4 1.8 !

1 2.5 1.3

^{1 1}

1

2.9

154 0.2

^!

0.1

^{1 1}

1.2 0.4 ; 0.4 0.1

¹₁¹

0.1 1.9

¹ 1

i

0.2

1

1

3.4 0.6 ! 1.6 0.2

¹1

1

2.8

^I1

1

155

1.8

^{1 1} _j

2.2

^{1 1} ₁

2.6 0.9 j 1.6 0.9

^{1 1} _J

1.4

^{1 1} ₁

5.7

^I ł

i

2.3

¹ I

1

5.0 1.8 ! 1.8 0.9

^r 1

3.0

¹ 1 1

156 0.3

¹¹

0.2

^{I 1}

1.0 0.4 ’ 0.5 0.1

¹ ₁

0.1

¹₁¹

2.1

¹ ₁

0.5

¹ 1

1

3.6 0.4 !

Î 1.5 0.2

¹₁

1

2.8

1¹

157 214 i 3.2

¹ 1

2.0 1.1 0.9 0.4

¹¹

0.9

¹

6.0

^{i i}_!

3.4

^I 1

6.4 1.2 ! 1.9 1.4

¹ 1

1

3.1

¹ 1

___J

158 0.4

¹ 1

0.2

¹ 1

1.0 0.1 I 0.5 0.0

¹ 1

0.0

¹¹

2.2

^{1 1}

1

0.5

¹ _ł

4.3 0.4 !

1 1.5 0.2

¹ 1

3.0

¹ ₁

1

159 3.5

^Î 1

2.8

¹₁

1.5- 1.0 ! 1.1 0.1

¹ 1

0.2

^{1 1}

5.1

¹1

__L_

3.0

¹ 1

___L

6.6 1-з 1.8

— 0.8

^{1 1}

—L

2.6

^{1 r}

—I

20 A. Baran, Z. Łojewskl

In order to reproduce the nonaxiality of single particle states we postulate the nonconservation of 52 during the barrier penetration process. The odd particle is placed on the lowest empty axially symmetric single particle Nilsson orbit and the specialization energy (a difference between the energy of an odd system and the corresponding energy of even-even one) is calcula­

ted for the whole fission bax-rier. During the fission process the particle changes the orbits and correspondingly its value but having the lowest possible energy.

In the minimum of the potential energy and behind the po­

tential barrier the odd nuclear system has a well defined 3-rd angular momentum component 52 which reproduces the ground state value of 52 . The last property depends on the choice of И , p, parameters of the Nilsson model and was discussed in our pre­

vious paper [1].

3. RESULTS

Fig. 1 presents the barriers of «°LId as calculated before and in the presented paper. For odd-even nuclei the high of the barrier is about 0.5rO.S MeV smaller as compared to the bar­

riers calculated with conserved Q. . For odd-odd nuclei the cor­

responding numbers lie in the interval 1.0?1.5 MeV. The 52 nonccn- serving model leads also to changes in mass parameters B. This may be seen in Fig. 2 where mass parameters of 2;>^Md are presented.

Both effects lead to decreasing spontaneous fission half lives and corresponding -hindrance factors. The latter are summarized in Table 1. One sees that the new hindrance factors in 52 "noncon- serving model" (the upper numbers) are smaller from the old ones

$1 conserving model (lower numbers) by about 3 units.

In the case of odd-odd nuclei we observe decreasing of hin­

drance factors with increasing atomic number Z for a given isotope

(N = const). It is caused by the fact that the fission barriers

for a larger Z are shortened. For even-odd nuclei the hindrance

factors are relatively small.

Spontaneous Fission of Double-odd Nuclei .. 21

Jig. 1. Fission barriers for 2^6Md. (Iqp) denotes the barrier ob­

tained for the case of two particle occupied the lowest Nilsson orbits.

Fig. 2. The mass parameters for 256Kd>

22 A. Baran, Z. Ło jew ski

Fig. 3. Spontaneous fission half life times for Md isotopes.

The even-even nuclei are represented by the lowest curve. The curve designed by 'old' gives our former results and the

curve 'lop' gives results obtained here.

We want to stress here that presented values of hindrance factors have the lowest possible values which can be obtained in the accepted model.

Results presented here are complementary to those presented in paper [1], where the experimental data are also shown.

REFERENCES

1. Ł o j e w s к i Z., B a r a n A.: Z. Phys. A, 1935, 322, 695.

2. R a n d r u p J. et al.: Nucl. PhysJl^973, 217, 221.

3. Nilsson S. G., Tsang C. F., Sobiczewski A., Szymański Z., Wycech S•, Gustafs­

son G.» Lamm I. L., Möller P., Nilsson

B.: Nucl. Phys. A, 1969, 131, 1,

Spontaneous Fission of Double-odd Nuclei ... 23

4. S t r u t Ł n 5. M y e r s ".

NY, 1974, 84, ô.îomorsk 7. Baran A., P

Sobie zews 8. Ć w i о к S., Ł V. V.: Nucl. Phys

sky V. M.j Nucl.

D., S w i a t e c 186.

i K.: Nukleonika, о га о r s к i к i A.: Nucl.

o j e и s к i A, 1985, 444,

Phys. A, 1968, 122, 1 к i 7.’. J . : Ann. Phy s.

1973, 25, 125.

К., Łukasiak Phys. A, 1981, 561, 85 Z., Pashkevic 1.

STRESZCZENIE

W pracy zasugerowano możliwość rozszczepienia nieparzys­

tych jąder atomowych z niezachowaniem momentu pędu oraz parzy­

stości nieparzystej cząstki 'wewnątrz * bariery potencjału ją­

drowego. Wyliczono czasy życia i współczynniki wzmocnienia ją­

der nieparzystych i nieparzysto-nieparzystych oraz porównano je z wynikami opublikowanymi poprzednio.

РЕЗЮМЕ

В работе предложена возможность деления нечетных атомных ядер с несохранением момента количества движения и четности НО' четной частицы "внутри" барьера ядерного потенциала. Вычислены времена жизни и коеффициенты усиления для нечетных и нечетно- -нечетных ядер и приведено их сравнение с результатами опубли­

кованными раньше.

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’'W . qs...