Shapes and sizes of nuclei with 50 < Z, N < 80 / Bożena Nerlo-Pomorska, Beata Skorupska - Biblioteka UMCS

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ANNALES

U NIV E RS IT A TIS MARIAE C U RI E-S К Ł 0 D O W S К A LUBLIN — POLONIA

VOL. XLV, 10 SECTIO AAA 1990 Instytut FizykiUMCS

Bożena NERLO-POMORSKA, Beata SKORUPSKA

Shapes and Sizes of Nuclei with 50 < Z, N < 801

Kształt i rozmiary jąder o 50 < Z, N < 80

INTRODUCTION

The neutron deficient nuclei with 50 < Z, N < 80 nucleon numbers have been already investigated theoretically [1,2] but the last development of the laser measurement techniques [3,4] as well as the new possibilities of the microscopic analysis enable now the better explanation of the diversing sizes and shapes mechanisms in this region. The Te-Gd isotopes far from the ß stability line show some interesting features. They are well deformed, with the deformation energy up to ~ 10 MeV. Their potential energy reaches minimal values not only for the prolate shapes but also for the oblate ones, especially for isotopes with N > 74. They have also some hexadecapole deformation at the equilibrium point [5].

For a long time there was a shortage of the single-particle levels scheme parameters here. The extrapolated from the rare earth and actinide regions, Nilsson potential parameters, were not sufficient to describe the subtle effects in the neutron deficient nuclei. Now we have used the universal set of parameters [6] depending only on the average mass number of the whole region: A ~ 126 in our case.

We have performed the dynamical calculation on the basis of the collective hamiltonian, obtained in the generator coordinate method (GCM).

This hamiltonian consists of the GCM mass parameters and the potential

1 Paper supported by the Polish Ministry of Education, project CPBP 01.09.

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B. Nerlo-Pomorska, В. Skorupska so

energy improved by the zero point correction term. The many-body hamiltonian used here contains in spite of the Nilsson type single particle hamiltonian the pairing interaction and the long range two-body forces in the local approximation [7]. The quadrupole (e) and hexadecapole (€4) deformation parameters are taken as generator coordinates, and BCS ground state as the generator function. The BCS wave function was approximately projected on a good particle number [8] and the average strength of pairing forces was taken from ref. [9].

We have calculated the potential energy surfaces, equilibrium deforma

tions, mass parameters and dynamical multipole moments Qx(X = 0,2,4) for all the even-even nuclei with nucleon numbers 50 < Z < 80, Z < N < 80.

The chief points of the theory are presented in chapter 1. The numerical re

sults of the calculation are illustrated in section 2. As the isotopic shifts of the mean square radius do not agree with the experimental data the more accurate investigation of their pairing and deformation parameters depen

dence is presented in chapter 3. The conclusions and proposals of further investigations are drawn in the last part of the paper.

THEORETICAL MODEL

The calculation was done on the two dimensional grid of deformation parameters —0.4 < €2 = € < 0.5, —0.12 < c4 < 0.Г2.

The single particle (hamiltonian Hsp) eigen problem was solved with the Nilsson potential with the new correction term parameters proposed by Seo in [6] kq = 0.021, Ki — 0.90,70 = 0.062

= en|rç). (1)

The many body hamiltonian H consists of the mean field hamiltonian Hq of Nilsson type taken in the given grid points {c^} = {c, c4], the pairing forces and the long range two body correlations in local approximation [10]

Я + Я

о

({«

а

})-|

E fc(A>A-G((S+)S + (Ś)Ś+), (2)

where

Я» = £е,С?С, (3)

ч

with the Cr),C+ fermion annihilation and creation operators.

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Shapes and Sizes of Nuclei with 50 < Z, JV < 80 SI

The strength of the long range forces is obtained from the selfconsi

stency condition

(4) The F„ operators are taken in the form

= (5)

The operator S is

S=Ecjc+

The pairing strengths for protons (Gp) and neutrons (Gn) are equal

GPZ2'3 = GnN2'3 = 0.29 h Zo . (6) The eigen-function of the hamiltonian (2) is approximated by the BCS wave function depending on the single particle coordinates {z} and parametrically on the collective ones a:

I«) = !«,{*)> = 1Ж + v,c+ct,|0), (7)

u

where V2 is the pair occupation probability, U2 = 1 — |0) is the particle vacuum state. The function |a) will be used as a generator function and the deformation parameters are taken as the collective generator coordinates

«1 = c,a2 = f4-

The collective hamiltonian 7/соц is given in the generator ^coordinate method as

«011 = 7+9. (8)

The kinetic term T is

Т=-2^^2'/5Й7(В'1)^’ (9)

where the mass parameters are

(«•)

jk The tensor overlap width is

♦— —*

™=(o1 hh|o> (11)

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92 В. Nerlo-Pomorska, В. Skorupska

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Shapes and Sizes of Nnclei with 50 < ^, < 80 93

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04

Tab.1.Theequilibriumfeaturesoftheeven-evennucleiwith50<Z<80,Z<N<80

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Shapes and Sizes of Nuclei with 50 < Z, JV < 80

55

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В. N erlo-Pomorska, В.Skompska 96

and jhk is a linked matrix element of the many-body hamiltonian

4— —♦

jhk = {a^êiïTk

|a) ■ <a lÊ la>™ -

The potential term of the collective hamiltonian V consists of the two parts

V = (а|Я|а) - Eo (13)

where Eo, so called zero point correction is equal here

& = (u)

The expectation value of the many body effective hamiltonian (2) is evaluated within the Strutinsky [11] prescription (Estrut)

(а|Я|а) « Estrut (15)

The Strutinsky energy reproduces well the nuclear masses and consists of the macroscopic liquid droplet Eld [12] part and the shell correction AEshell

describing the shell and pairing effects on the potential energy

Estrut = Eld = AEshell (16)

The eigen problem of the collective hamiltonian 7YCO// (7) is solved by diagonalisation in the two dimensional harmonic oscillator wave functions base

Kollier = (17)

The full many-body wave function фа describing a nucleus in a state a is given within the GCM approximation by the integral:

j fQ(a)\a,{x})da (18)

This function will serve to calculate the dynamical values of the multipole moments. The weight function /Q(a) in (18) is directly connected [13] with the collective wave function Фа from eq. (17).

The multipole moments operators are defined as follows

Qo = г2,Qi = r2P2(cosi?),Q4 = r4P4(costf), (19) where r, i? are the single particle coordinates, Рд — the Legendre polyno

mials.

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Shapes and Sizes of Nuclei with 50 < ^, JV < 80

It was proved in [13] that the expectation value of Q\ operator between the ground state functions V’o is equal to

Qx = j $5(a|QA|a)$oda, (20)

where the integral is evaluated in the collective space only.

In order to compare the theoretical estimates of the quadrupole moments to the experimental data, obtained from the reduced quadrupole transition probabilities B(E2,2+ 0+) [14] should use the sum rule

^B(E2,2? $oQhoda (21)

. 1О7Г J

t

where the sum goes over all the possible 2* states and is model independent.

There is usually one transition B2 favoured in this sum [15] so it is reasonable to compare the available experimental data obtained from B(£2) transitions [14] with the quadrupole moments calculated as follows

г f -11/2

(<2г)1/2 = [/ Whode] (22)

The mean square radii of nuclei (r2) are usually not given in experiment straighty, we deal rather with the isotopic shifts of (r2) between various mass numbers [3,4]

a(r’) = (>V - (rV (23)

They are related to the electric monopole moments Qq by

(r2)z+N = Qo/Z (24)

If the mean square radius for the magic spherical nucleus was unmeasured we took its liquid drop estimate, designing the experimental Q^p up to this constant.

2. THE EQUILIBRIUM DEFORMATIONS, POTENTIAL ENERGIES AND MOMENTS

In the table 1 are listed: the equilibrium values of the deformation parameters c0,^, potential energies V and deformation energies

E(le/ = V(€O,^)-V(0,0), (25)

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B. Nerlo-Pomorska, В. Skorupska

98

Then the electric monopole moments Qo and the isotopic shifts of mean square radius

d(r2) = (r2)N+z - (T2)N+Z~2 (26) are compared with their available [3,4] experimental values d(r2)exp. The electric quadrupole moments are calculated as {Q2} and (Q^)1^2 The second ones are closer to experimental data [14] Q^- The corresponding reduced quadrupole transitions

B2=^Qtf (27)

107Г 5 -

B2 = _IbT w

and the experimental values B2eip are also presented. The hexadecapole moments can be seen in the last column of the Table 1.

The results are printed for all the even-even combinations of 50 < Z < 80 and Z < N < 80.

For the most isotopes the prolate equilibrium shapes are favoured. Only the Те nuclei and the heaviest isotopes of other elements have a chance to be oblate. The deformation energies, thanks the influence of zero point vibration to became larger by up to 1.5 MeV in our calculation, and the far from magic numbers nuclei are usually very well deformed. It means that the nonaxial 7 instability in this region would be not so important as it was suggested in ref. [1].

Presented in Fig. 1 the mean square radius (points) calculated in fact up to the value of the spherical lightest isotopes radius do not reproduce well the data (crosses) in our calculation, specially their isotopic shifts show sometimes quite opposite behaviour to the experimental data. For example the theoretical shifts of the heaviest Те, Xe, Ba isotopes grow with A while their experimental values decrease. The more detailed analysis of our model is needed and will be presented in the next chapter.

In Fig. 2 [16] the quadrupole moments are drawn. The agreement with experimental data (crosses) of the (Q2)172 values (circules) is in general better than {Q2) (points) specially for Те and heavy Xe, Ba, Ce isotopes. The negative moments (Q2) яге obtained for the heaviest Ce-Gd isotopes.

The hexadecapole moments are shown in Fig. 3. The systematic smaller values are obtained for N = 70, because of the shell effects.

In general the shapes of the nuclei are well described but their sizes demand the more careful description.

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Shapes and Sizes of Nuclei with 50 < Z, JV < 80 29

Fig. 1. The microscopic electric mean square radii (r2) in b (points) compared with the experimental data [4] (crosses) for even-even Te-Gd isotopes

-4 -4

111111111111U111111111IJ1111111IJ1111 l-LLlJ_i_L

I

i lI i.1A

106 108 130 112 120 132 118 134 126 136 130 138 136 138

Fig. 2. The electric quadrupole moments Q2 in b calculated as {Q2) (points) and (circles) compared with experimental data [14] (crosses) for even-even Te-Gd

isotopes

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B. Nerlo-Pomorska, В. Skorupska

Те Хе Ba Ce Nd Sm Gd

0.8 Q8

106 118 130 112 120 132 118 134 126 136 130138 136 138142

Fig. 3. The microscopic electric hexadecapole moments Q« in b2 for even-even Te-Gd isotopes

3. THE DEPENDENCE OF MEAN SQUARE CHARGE RADII ON THE MODEL PARAMETERS

In order to improve the reproduction of the mean square radius we analyse their dependence on the diversing parameters [17].

The presented in Fig. 1 experimental data [4] are obtained assuming the radius of the magic isotope 132Ba and 140Xe from spherical liquid drop model. It is seen that the microscopic (r2) grow quicker with A then the experimental ones, even when their average values are close to each other.

The mean square radii depend mostly on the quadrupole deformation c.

In Figs. 4 a,b we can see this dependence around equilibrium deformation e° marked by the arrows for Xe and Ba isotopes. Also the hexadecapole deformation influences the (r2) much as one can see on Figs. 5 a,b for Xe, Ba isotopes. Note the almost one order of magnitude difference in the (r2) scale of Figs. 4 and 5, 6. Because these collective variables are not sufficient to give a good behaviour of i{r2) we should take next the pairing

— gap parameter A as a dynamical (collective) parameter. As it is shown in Figs. 6 a,b the (r2) value of stable 130Xe or 126Ba nuclei grows strongly with △.

It is almost impossible to get the values of (r2) and Q2 — close to the experimental data including the €,€4 parameters only. We have evaluated

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Shapes and Sizes of Nuclei with 50 < Z, .V < 80 Ш

Fig. 4. The dependence of the microscopic (r2) in fm2 on the quadrupole deformation t for the I30Xe (a) and 126Ba (b) nuclei. The equilibrium deformations t° are signed by

arrows

Fig. 5. The dependence of the microscopic (r2) in fm2 on the hexadecapole deformation

<4 for the 130Xe (a) and 126 Ba (b) nuclei. The equilibrium deformations e° are signed by arrows

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102 B. Nerlo-Pomorska, В. Skorupska

Fig. 6. The dependence of the microscopic (r2) in fm2 on the pairing energy gap △ (in MeV) for the 130Xe (a) and 126 Ba (b)

within the liquid drop model the values of (cLD€4D) (squares in Fig. 7 a,h) which would reproduce the experimental data

(<r2)exp,QD-(«LD,àD)#(e0,^)

We have assumed the favored by the microscopic calculation prolate deformation of the nucleus when solving the above equation. The solutions are far from the microscopic (c0,^) (stars in Figs. 7 a,b) equilibrium deformations both for Xe and Ba isotopes, as it is seen in Figs. 7 a,b respectively.

Let us try to reproduce the experimental data (r2)eTp, Q?xp with the quadrupole deformation c and the energy gap A varying them free within the microscopic BCS model

(<r2)~₽, Q~p) -S(<>) * (A A0) / (ЛАт)

As you can see in Figs. 8 a,b the extracted from experimental ((r2)exp, (fLD, △) values (squares) do not agree either with the microscopic (e°,A°) (stars) equilibrium points, nor even with the values (c°, Am) obtained from the experimental masses [18] of nuclei (circles). The values

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Shapes and Sizes of Nuclei with 50 < Z, JV < 80 103

Fig. 7. The microscopic equilibrium points (e°,cj) (stars) and their estimates deduced from the experimental values of ({r2)e,₽,QjXp) [3,4,14] within the liquid drop model

(cLD,c}*D) (squares) for the Xe (a) and Ba (b) isotopes

of A which reproduce the experimental (r2)exp and Q|rp are almost half of their estimates from masses Am It is necessary to use the dynamical pairing model [19] which would give the most probable A values smaller than the equilibrium value corresponding to the minimal potential energy.

Such calculation is prepared now.

CONCLUSION

The following conclusions can be drawn from our calculations:

1. The dynamical treatment should be used in order to describe the nuclear shapes.

2. The quadrupole moments should be calculated as rather than as (Q2) in order to compare them with the data experimental transition probabilities.

3. The nuclear radius description for the far from the ß stability line nuclei demands the dynamical treatment of pairing forces, especially the inclusion of the energy gap as a dynamical variable.

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104 В. Nerlo-Pomorska, В. Skorupska

Fig. 8. The microscopic values of (e°, Д°) (stars) compared with the evaluated from exper

imental masses (c°,ùm) (circles) and the (eLD, △) (squares) reproducing in microscopic model the experimental ((r2)'xp, Qjlp) for Xe (a) and Ba (b) isotopes

REFERENCES

1. Arseniev D. A., Sobiczewski A., Soloviev V. G., NucL Phys. A126 (1969), 446.

2. Nerlo-Pomorska B., NucL Phys. A259 (1976), 481.

3. Aufmuth P., Heilig K., Steudel S., Atomic Data and Nuclear Data Tables 37 (1987), 455.

4. Otten E. W., Treatise on Heavy Ion Science, Vol. 8 (1989).

5. Nerlo-Pomorska B., Pomorski K., Annales UMCS, Sec. AAA, 43/44, 21 (1988/89), 211.

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Kształt i rozmiary jąder o 50 < Z, N < 80 1Q5

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STRESZCZENIE

Przeprowadzono mikroskopowe dynamiczne rachunki elektrycznych momentów o po- lowości A = 0,2,4 parzysto-parzystych jąder neutrono-deficytowych. Zaproponowano nową interpretację teoretyczną doświadczalnych momentów kwadrupolowych otrzymywa

nych ze zredukowanych prawdopodobieństw przejść B(E2,2+ —» 0+). Przeprowadzono dokładniejsze badania zależności kwadratów średnich promieni jąder od parametrów de

formacji wyższych multipolowości i oddziaływania „pairing”.

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