Study of the inertial functions for rare-earth nuclei

Nueaear Physics A283 (1977) 394-412 ; © Narth-Hofland Publfdünp Co., Amsterdam Not to be reproduced by photopriat m mirro®lm WMout writtm permission óvm the pablirhee

STUDY OF THE MERTIAL FUNCTIONS FOR RARE-EARTH NUCLEI InrNtM

of Physies, 71w

KRZYSZTOF POMORSKI Maria Sklodowako-Ctvls University, Lublin, Poland

TIMESA KANIOWSKA and ADAM SOBICZEWSKI Institute for Nuclear Research, Hoza 69, Oä681 Warssawa, Poland STANISLAW 0. ROHOZßiSKI Institute of Theoretical Physics, Warsaw University, Hoza 69, 00-681 Warszawa, Poland Received 19 August 1976 (Revised 17 January 1977) Abstruct : Vibrational (Bpp, Bh, B,yy) and rotational (Bs , B B,) inertial functions of Bohr's collective Hamiltonian are studied microscopically as functions ofdeformation. The study is eutmded to the inertial functions of the Hamiltonian appropriate for the phonon mtpansion treatment . The cranldng approximation is used. The results are compared with those of the pairing-phtsquadrupole model . Noun-adiabatic effects are discussed.

A study of dynamic properties of nuclei based on Bohr's collective Hamiltonian 1, `) requires knowledge of the parameters of this Hamiltonian. These are the potential energy and the inertial functions. The potential energy has already been well explored Relatively little, however, is known about the inertial functions. In dynamic studies, a general case of only the quadrupole motion has been considered up to now. The corresponding Bohr Hamiltonian contains six inertial functions j). Two methods of the dynamic calculations are used. One is the direct numerical solution of the Schrödinger equation 3). The other is the diagonalisation of the Hamiltonian in the basis of the five-dimensional quadrupole oscillator 4-6). Two . forms of the Hamiltonian, each convenient for the respective method, are used. In calculations using the diagonalization (phonon expansion) method, only a simplified form of the inertial functions has been employed up to now. All the functions have been assumed constant as functions of deformation and some of them have been simply put zero . In the numerical solution of the Schrddinger equation, the inertial functions as well as the potential energy have been calculated in the framework of the pairing-plus-quadrupole (ppQ) model 3. 7-9). In that model, a nucleus is divided into two parts. One is a cloud of "active", valence nucleons, the contribution of which to the inertial functions is calculated microscopically. The other is a core of "inactive" nucleons, the contribution of which is treated phenomenologically . It renormalizes the contribution of the valence nucleons. The study of ref. r e) has 394

INERTIAL FUNCTIONS

395

shown that the renormalization is large. Namely, to reproduce the collective properties of 15OSm and "'Sin one has to assume 1°) the renormalization factor of the order of 3-5 . As an alternative to the PPQ model, the cranking model can be used for the oaloulation of the inertial functions. In this model, there is no need for the separation of a nucleus into the valence and the core parts. The model treats all nucleons in the same way and is completely microscopic. Due to this, possible discrepancies with experiment are easier to interpret. They give more direct information about the model. In the previous paper 11), we have studied the properties of the inertial functions obtained in the framework of the cranking model . In particular, we have shown that the oranking functions based on the modified harmonic oscillator potential satisfy the proper symmetry conditions. Much attention was given to the comparison between the cranking and the liquid-drop inertial functions. Numerical calculations have been performed for nuclei in the barium region . In the present paper, we aim at extending the study to the race-earth nuclei . In particular, we aim at comparing the cranking functions with those of the PPQ model and experimental indications. We also intend to calculate the inertial functions of the Hamiltonian appropriate for the phonon expansion method . It is interesting to see how close is the form of these functions used in the dynamic calculations, performed up to now 4-6), to that obtained microscopically. The last point we would like to study is an estimate of non-adiabatic effects on the inertial functions. The study is performed for a number of spherical, transitional and deformed doubly even nuclei of the rare-earth region. The modified harmonic oscillator potential is taken for description of the single-particle structure of a nucleus. In sect. 2, we find the relations between the inertial functions appearing in the two forms of the collective Hamiltonian mentioned above. In seot. 3, we describe the oaloulations of the functions and in seot. 4 we present and discuss the results. Seat. 5 gives the conclusions drawn from the research . 2. Two forms of the collective kinetic energy operators Throughout the paper, we assume an ellipsoidal shape of a nucleus. The semi-axes of the ellipsoid are (2.1) (ß, y) = &(ß, y)(1 +N/5/4nß cosy.), where y, = y-i;fx with i, = 1, 2, 3 corresponding to x = x, y, z, respectively. The axes Ox, Oy, Oz are the principal axes of the ellipsoid and ß, y are the Bohr deformation parameters 1). The dependence of Ra on ß and y is obtained from the volume conservation condition for the ellipsoid (of. e.g. ref, 11)) . Two forms of the collective kinetic energy operators are used in the calculations of the collective properties of nuclei. One is the form which is obtained by the quantization of the classical kinetic energy 1-3). The other is the form which is obtained

396

IG POMOR4KI at aL

directly from the invariance conditions imposed on the already quantal Hamil4-6) . It is the purpose of this section to find the relations between the inertial functions (or equivalently : inertial or mass parameters) appearing in the two forms. The classical, explicitly rotationally invariant, form of the kinetic energy is

tonian

T - j Y, Bjw.ct,.dj,.,

(2.2)

where a, (it - 0, f 1, f2) are the collective variables and B. are components of the inertial tensor, both in the laboratory system . The components BW are isotropic functions of a. Expressing a by the deformation parameters ß, y and the Euler angles 0,, we get T = «B0(ß. y)ft2+2pB,,0, y)R~+ß2B77(ß, y»2+ 1 .1Z(ß9 yknC }. : x

(2.3)

where m are components of the angular velocity of the intrinsic system with respect to the laboratory system. The moments of inertia may be written in the form (2.4) J.(ß, y) = 4ß2B.(ß, y) sins ?R " Thus, we have six inertial functions, three vibrational: BAR, BR,, B and three rotational : B, B. , B, m the classical kinetic energy. They remain, naturally, the parameters of the kinetic energy after the quantization of it. All of them depend on both ß and y. The other form of the collective kinetic energy is obtained in a direct way by a construction ofan invariant of the collective variables a,. and the momentanr = -RD/ âac conjugate to them . We get then the general expression

T = J(P(Oo)(xx]o+P(j1) [a[7m]s]o +P22) [[C ]4[ ]4]0 0]2[XX]21o+Pi4 )[[MIVX

(2"5) ]4]0+P44)IIC ]2L ]Z]4L ]4]o+h"C'}, where pú0), p12 ), . . . pa4) are scalar functions of a and h.c. stands for the Hermitean conjugate part. The symbol [ ]s demotes the tensor coupling of the quantities inside to the multipolarity A. The relations between the inertial functions p of eq. (2.5) and the functions B of eqs. (2.3) and (2.4) are +P34~ [[° [ ]2]4[

P(Oo) = J MPIR+P +3Po), , -y 44PR,+Pi sin 2y+P2 cos 2y), Pi2) = 4ß sin 3y P?) - ~- [2PRR-P) sin 3y+4Pp, cos 3y-Pl sin y+P2 cos y], 4ß2 sin 37

INERTIAL FUNMONS

397

( Pâo) -

1 pp sny 6+P cos7-o y 3P) [3(Pi =3+Pi ,9y sny 2ßs sias 37 -Pl cos 47-Ps sin 4y], ß3~~3y. r[3Ph sin 3Y +3(Pn -P) 1 P34) = o cos 37-P, cos 7-P2 sin Y].

P44) = where

4

[3(P-Po)-PI cos 2y+P z sin 27], s z 3Y

Po - «Ps+P,+P.),

Pl = 2P.-P. -P,, hs lis py --, P,-- . B, Bs

lis p.-,, Bs

ha Bn pop a W

9

p$7

.

-hs BR, ~ W

(2.6)

P2

=

'j3(Ps-Ps),

Rp hs B Prr ' W

and W = Bpp B-BÌ,. The relations (2.6) are obtained by equating the corresponding terms in the kinedo energy operator obtained by the Pauli `quantization 14) of eq. (2.3) and in the kinetic energy operator of eq. (2.5). It may be mentioned, by the way, that the transformation ofone ofthe two operatorsto the formdirectly comparable withthe other results inan appearance of additional terms not containing the differential operators. This means a redefinition of the potential energy in the corresponding Hamiltonian 13). This is important to keep in mindwhen making acomparison between thewhole Hamiltonians corresponding to the two forms of the kinetic energy operators considered. Up to now, only very simple forms of the inertial functions p have been used in the dynamic calculations. Cineuss and Chviner s) set two of the functions, and pI(L2) , constant (i.e. independent of deformation) and the other zero . Thus, they have 2 P(O0) = C 1, (2.7) Pil`) = Cz, Pi ) = Pi4) = P34) = P44) = U" Dussel and Bès 4) and v. Bernus et al. e) assume

pr)

co) Po . C1 +Ca~'.

Al) - C3. Pl

Pa<_) = C 4~

(4) C Ps = s.

P3(4)

-0 _ P4(4) 10

(2.8)

where C1, i = 1, . . . 5, are constants. As we will see in soot. 4, the microscopic calculations lead to the functions p which are very different from those of eqs. (2.7) and (2.8). 3. Mtcnoscopic calculations of the ine:tW f The calculations are performed in the same way as in ref. for details. Here, we specify only the main points .

11)

to which we refer

398

K . POMORSKI et aL

The inertial parameters are calculated in the cranking approximation. Modified harmonic oscillator potential 14-16) is used for description of the single-particle motion . Pairing interaction is taken into account by the BCS formalism. The formula for the moment of inertia is

z

fR = 2Az F, (u,v,.-u v )2, .." E,+E,.

where J. is the single-particle angular momentum operator and x = x, y, z. The formula for the vibrational mass parameter connected with collective variables a, and aj is D«,j

=

2h 2 Y,


âH

ôa,

IvXvI

ôH

Iv>

ôaJ

(UV., +u,.v,)z+P'J,

(3.2)

where His the single-particle Hamiltonian, a, and aj stand for ß or y and DRp - BR'O, Dpy --_ fiBar , j)  = p2B.. In both eds. (3.1) and (3.2), K, and v, are the variational parameters of the BCS wave function and E, is the quasipartiole energy corresponding to the single-particle state Iv>. The term PJ in eq. (3.2) is due to the coupling between the collective motion and the pairing interaction. Numerical calculations are performed for the following ten nuclei from the rare earth region : 148 -1 "Nd, 148-i s6Sm' ' s o- is'Gd and 166Er. They represent all three kinds of nuclei : spherical, transitional and deformed . The results are obtained for the following 36 grid points : ß = 0 (0.1) 0.5, ,y = 0° (10°) 60°. The "A = 165" parameters 17) of the modified-oscillator potential are employed, i.e. Kp - 0.0637, yp = 0.60 for protons and K. = 0.0637, Et = 0.42 for neutrons. Seven oscillator shells, N = 0-7, both for protons and for neutrons are taken into account. The isospiu dependent oscillator frequency l') (hbo)p(a) = 41 (1( )

3

N

A

A-} MeV

(3.3)

is taken and also the isospin dependent pairing force strength 17) Gp(,,) =

9.2(±)7.4

N

A

A-1 MeV,

(3.4)

with %IISZ(N) levels above and the same number of levels below the Fermi level taken for solving the pairing equations, are used. 4. Resalo and dlacuesion 4 .1 . RESULTS

We illustrate here the results by giving examples of threenuolei: 15 OSm, 152Sm and 166Er. The first one is transitional, almost spherical, the second is rather deformed and the third is well deformed .

INERTIAL FUNCTIONS

399

Fig. 1 shows the potential energy V for them. The energy is calculated by the maorosoopio-microsoopio method (liquid-drop plus shell-oorreotion energy), exactly as in ref. 1 s) . The results are taken from ref. 19) . We can see in fig. l. that the minimum of V is very shallow, both in ß and y, for 1 s oSm, less shallow for "2Sm and very deep, again both in ß and y, for "'Er (of also ref. le)). The inertial functions are presented in figs . 2-7. Rather large fluctuations of their values are seen. They are effects of the single-particle structure of a nuoleus. The fluctuations are larger for vibrational than for rotational functions. Their amplitude oomes to about 40 % of the average value for BRR. Larger fluctuations of the vibrational functions are probably connected with the less collective character of them as compared to the rotational functions. In fact, an investigation") of the microstructure of the moment of inertia J of 1"Ba has shown that the largest contribution of one pair of the single-particle states to f is about 15 %. The most important six pairs give around 50 % and 20 pairs 77 % of J. Similar study for the vibrational parameter BIO, performed by us for 1 soSm and 1 s2Sm, gives about 30 % for the largest contribution of one pair. Only 3 most important pairs give already about 60 % and 10 pairs about 85 % of BRO. Thus, we can say that the vibrational functions are about twice less collective than the rotational ones. This fact is probably mainly due to the higher power of the energy denominator in the formula for them, eq. (3.2) (higher than for the rotational functions, eq. (3.1)) . Due to this, only few pairs of states, eaoh pair being close in energy to the Fermi level and having, simultaneously, large matrix element
IL POMORSSI et aL

INERTIAL FUNCTIONS

w

401

402

K. POMORSKI et oL

INERTIAL FUNMON3

403

404

IL POMORSKI et aL

0°

20°

40°

1f

60°

Fl& 8. Dependenoe of the inertial functionsp, eq. (2.5), on the deformation y for P - 0.30f keY1

30 10

~

Ô ~°

~

I " Y ry

~~r~

..

W1 r

152

-30

62

05

~

Sm 90

03

Ó~ tO°

~e

I?

0i

01

03 ß a5

Fia. 9. Dependence of the inertial functions p on the deformation ß fory - 10° and 50°.

tion of

pr', on the deformation ß. The strongest dependenoe is obtained for piz) and

p4e1. The reason for whioh we plotted the figure for y = 10° and 50° and not for y = 0° and 60° is that eqs. (2.6) lead to indefinite expressions forp of the "0f0" type for the last two values of y. Due to this, all the funotions p, exoept pú0) , have to be oaloulated as limits of the ratios of small values, what leads to large inaoouraoies. Figs. 8 and 9 tell us that the approximations for p, eqs. (2.7) and (2.8), used up to now ~' 6 ), are far from what one gets in the miorosoopio oaloulations. Only the approximation for the funotion J0) seems to be pretty good. 4.2. COMPARISON wl'l'H THE REsuLTs OF THE PAIRING-PLUS-QUADRUPOLE OPPQ) MODEL

Results of the PPQ model oaloulations of the inertial functions are reported by Kumar 10, 22) for 150Sm and 152Sm. Although the PPQ formulae for the inertial

MERTIAL FUNCI'ION3

405

functions s, 1) are formally identical with those of the cranking model, used by us, the calculations differ in few points. The main two are the following. One is that in the PPQ model all the single-particle states of a nucleus are divided into two classes. One corresponding to "active", valence nucleons, the contribution ofwhich to an inertial parameter is calculated microscopically. The other representing a core of "inactive" nucleons which contribute to the parameter only via a simple renormalization factor. In the cranking model, there is no such division. All singleparticle states are treated in the same, microscopic, way. The second differenoe consists in different single-particle schemes used in the two calculations . The schemes are already different for the spherical shape, but then also the deformation dependence of the levels is different. The last comes from two sources. One of them is that in the PPQ model there is nó volume conservation condition for the single-particle potential, while this condition is imposed on the Nilsson potential 15 " 16, 13 ). The other is that in the diagonalization of the singleparticle Hamiltonian no mixing of the oscillator shells is taken into account in the PPQ model, while it is accounted in the Nilsson model via the transformation to the stretched coordinates 23). When comparing our results with those ofrefs. ", 11), we should also keep in mind the difference in the parametrization of the deformation. The relation between the Bohr deformation parameters ß and y, eq. (2.1), used by us, and those of Kumar ß, and ye is a an

ßi =

ß

16 sin y+2k(6-k2) sin 2y-k3 sin 4y 16 cos y-2k(6-0) cos 2y-k3 cos 4y

4 cos y+k cos 21 2(2+k) z cos Y,+4k cos (y+y+)+kz cos (2y - Yj'

(4.1)

where k = ,/5/4aß. For example; for (P, y) = (0 .50, 0°), (0.50, 30°), (0 .50, 60°), we get from eqs. (4.1): (Ps , 7j) - (0.39, 0°), (0.51, 43°), (0.61, 60°), correspondingly. Direct comparison between the results of the two calculations shows the following. The values corresponding to the valence states (as these, the states of N = 4 and 5 for protons and N = 5 and 6 for neutrons are taken) are not far from each other. The cranking values are larger by about 30 %. The contribution of the core calculated explicitly in the cranking model is very small. It only amounts to less than 1 % of the total value of Bpp for the deformation around the equilibrium point. This is because a significant contribution is obtained only from the states close to the Fermi level (i.e. from the valence states), as discussed in the previous subsection. Thus, the cranking calculations suggest the following. The large phenomenologic renormalization of the valence inertial parameters, found in ref. 111), cannot be interpreted as due to the core contribution. It should be rather ascribed to interactions not included in the model.

406

K. POMORSKI et al.

50

0

0. 20'

9

Ar

60'

Fig. 10. Dependence of the inertial functions B on the deformation y for P = 0.30.

Fib. 11 . Dependence of the inertial functions B on the deformation P for y - 0° and 60°.

In addition, we can say that for large deformations the division of the oscillator shells into valence and core shells looses sense as the shells strongly mix. The inolusion and explicit consideration of all shells, up to N = 7 or more, is important then. Concerning the deformation dependence of the inertial functions, the comparison shows that it is different in the two models . This probably comes from the differences in the two calculations described above. Only few general properties, like these that the fluctuations of the functions with deformation are larger in the vibrational than in rotational case and that the amplitude of these fluctuations comes to about 4050 % of the average value, are similar. All this can be seen from the comparison between the oontour maps given here and in refs. 1°. 22). The comparison between the cross sections of the maps along ß = oonst. (dependence on y) and y = 0° and 60° (dependence on ß) is also instructive. The oross sections are shown in figs . 10 (ß =

INERTIAL FUNCTIONS

40 7

0.30) and 11. Our rotational parameters seem to be more smooth functions of deformation than those of the PPQ model. 4.3 . EXPERIMENTAL INDICATIONS FOR THE VALUES OF THE INERTIAL FUNCTIONS

(i) A simple estimate of the values of the inertial functions may be obtained from ananalysis ofthe energy E2 + of thefirst 2+ state and the reduoodtransitionprobability B(E2; 0 -+ 2) for the excitation of it. Assuming the harmonic approximation for the collective Hamiltonian and the zero-deformation approximation for the inertial functions Bpy =0, Bpp=B."=B =B,=B,, -B, (4.2) one gets 2) 2

B = fie2 3 ZeRo) 1 4n E2+B(E2 ; 0 -+ 2)

(4.3)

for the case of a spherical nucleus (fivefold degenerate 2+ level). The calculation of the ratio of such B to the hydrodynamioal value B" has been performed in ref. 2) for all nuclei for which the experimental data for E2 + and B(E2) exist. For Sm isotopes, it gives BIB'"" k-i 10-13. In our microscopic calculations, we get B"°'/ B" ;ts 5. Thus, the microscopic values are about 2.0-2.6 times too small. (ii) Another estimate may be obtained from experiment via. a relation of the type of the energy weighted sum rule. For example, using eq. (2.5), we can write for the function póo) (4.4) ~3 E [4, [T, af = 2í00)S or, as the potential energy V depends only on the coordinates a,, and not on the momenta n", (4.4a) -~j3 Y, [a.* , [H., arll = 2Pó°~, where Ho = T+ V is the collective Hamiltonian. For the average value ofpóo) in the ground state 10>, we get directly from eq. (4.4a) 1 2 V5 (3 ZeRo) 4n

F, (Epe - Eo)B(E 2;

0 -" IV) = (2I+1)-} <011P (oo)110i,

(4.5)

where the electric quadrupole transition operator, appearing in the transition probability B(E2), is taken in the form

In eq. (4.5), the summation runs over all excited states of the. complete set of the eigenstetes ofthe Hamiltonian Ho. Here, I' is the spin and T' am the remaining quantum numbers of the state 1I'T'> .

408

pr

K. POMORSKI et aL

To get an estimate of ) in the ground state of 1 s 2 Sm, we Can use the experimental data for the energy E2 + and the reduced transition probability B(E2) for the first 2+ state of the ground, beta- and gamma-vibration bands. They are: Es + = 0.122, 0.811 and 1.086 MeV and B(E2, 0+ -. 2+) = 3.37, 0.023 and 0.090 e2 - b2, respectively (of. ref. 1°)). Eq. (4.5) gives then

r=

;e; 6.4 keV. (p O(c) Thus, the microscopic value, which is around 15 keV (of. figs. 8 and 9), is about 2 .3 times too large. Transformed to the functions B, this means that the microscopic B are about 2.3 times too small. So, the result almost coincides with the previous one. One should stress here that the sum rule (4.5) is not the classical sum rule discussed e.g. in ref. 2). It is a "collective" sum rule as only the collective part Ha of the Hamiltonian is used to construct it. Thus, only transitions between collective states appear in the sum, in contrast to the classical sum rule. Also in contrast to the classical sum rule, the transitions to the lowest states exhaust the rule (4.5) approximately, Thus, the above value 6.4 keV, although obtained from rule (4.5) with omission of the higher transitions, i.e. obtained as a lower limit for pó° ), is expected to be close to the real value. An estimate following below supports this expectation and gives evidence that the higher, tmharmonio transitions, omitted in the present estimate, are really small. For a deformed nucleus, considered as a good rotor and a harmonic beta- and gamma-vibrator, the sum is only over transitions to the first three 2+ states. ` For a spherical nucleus, considered as a harmonic oscillator, it reduces to only one transition. In this case, with the zero-deformation approximation for the inertial functions, eq. (4.2), the sum rule of eq. (4.5) becomes eq. (4.3). (iii) The most accurate test for the inertial functions is supplied by the dynamical calculations of nuclear properties . Assuming that the potential is established, the functions which reproduce experimental results may be considered as a good test for the microscopic functions. As such test we can use the inertial functions of Kumar obtained in his dynamicalcalculations 1 °)for 1 s *Sm and 1 s2 Sm.As mentioned earlier, the functions arecalculated in the framework of the PPQ model and then fitted to experiment by one free parameter (renormalization factor). Comparison between our cranking functions and those of Kumar shows that the cranking functions are smaller by a factor 2-3. The direct comparison is legitimate as the potential energies (ours and Kunmar's) are close to each other. We have also performed an independent dynamical calculations for 1 s2Sm using directly our cranking functions and have also found that the renormalization factor 2-3 is needed . Thus, all the three estimates imply that the cranking inertial functions are about 2-3 times too small to reproduce experiment. This agrees with the results of ref. 11) obtained for nuclei around barium. The factor 2-3 should be considered as rough and incomplete. One of the reasons for this is that we have only used a rather limited experimental data for the estimates.

INERTIAL FUNCTIONS

409

Another is that we have not included the hexadeoapole deformation in the research, which appears important 1e), Some extension of the above estimates may be provided by the results of re: 18). The cranking rotational functions calculated in that reference for well deformed (or, better, with well-defined deformation) nuclei are about 1.5 times smaller than the experimental ones. The hexadecapole deformation has been included there. So, aocounting these results, we can say that the cranking functions are about 1 .5-3.0 times too small. In connection with the rotational functions, we should probably add that aooording to a number of papers, among them ref. 1s), for many nuclei the cranking functions differ from experiment less than by the factor 1.5. However, in all these cases the interpretation of the first 2' state as a pure rotational state may be not correct. In these oases, even if the axial deformation ß of a nucleus is well defined, the nucleus is relatively soft to the y-deformation. Thus, we are somewhere between the case of a pure axial rotor and the case of a complete -f-instability (Wilets-Jean model 24)). In these two oases, the same experimental energy E2 . implies the rotational parameters differing by the factor 2. For example, if the microscopic parameter agrees with the experimental on : when the pure rotational interpretation of E2+ is applied, it is twice too small when the complete y-instability limit is assumed. Concluding, we may say that for not very well deformed nuclei (not large deformation energy or not large prolate oblate energy difference), only full dynamic calculations can tell us what inertial functions are required to reproduce experiment. 4.4. EFFECT OF NONADIARATICITY

The cranking formulae of Inglis, which are used by us for the oaloulation of the inertial functions, are obtained in the adiabatic approximation. Thus, they are valid only for the vase when the collective energy is small compared to the single-paitiole energies. Let us examine the effects of non-adiabatioity on the vibrational inertial functions We will admit the axial and non-axial vibrations of arbitrary energy, assuming only the hsrmonicity of the vibrations in time. We choose the quadrupole axial Q20 and non-axial Q22 moments as the collective variables. They correspond to the singleparticle operators 420 = 2,Z2-X2-y2,

422 =

X2 _y2'

respectively . The operator for the whole system of A nucleons is

Q

A

= E 0). !_1

(4 .6)

(4.7)

If the nucleons interact via the quadrupole-quadrupole interaction, the collective energy fin is obtained from the dispersion equation 23,26) 1

= 2xQY, II2(dk-do) koo (dk-do)2 -(i~co)2

(4.8)

41 0

K. POMORSKI et al.

where xQ is the interactionstrength and dro and fk are the energies of the ground 10> and the excited Ik> states of the system, respectively. The inertial function is given by s s. 26) B

2ftsxz

Kk1QI0>h(dk-Oro)

4.9

Using the BCS wave function for the ground state 10> and the two-quasipartiole wave functions for the excited states Ik>, eqs. (4.8) and (4.9) become 1 = 2KQ

and

Boa = 2 x

Eve (E,+E,.)Z-(ftco)z

s II (E,+E,.)

(Urar,+U,

, II?(E,+E?)~

0

V[(E,+Ee) -(h ) ] ,

.v,)2

(U,v"+U,,v,)2 .

(4.8a) (4.9a)

In both equations, the summation runs over both neutron and proton states . The interaction strengths between protons, neutrons and protons and neutrons are put equal. Coupling between the oolleetive motion and the pairing interaction, term PJ in eq. (3.2), is disregarded here . To get the dependence ofthe function Bag on the collective energy hw, for each Aw the strength KO in eq. (4.9a) is taken such as to fulfil eq. (4.8a). The resulting BQQ, calculated for i66Er, are presented in fig. 12 for both Q = Q20 and Q = Qsz. Adiabatic results (as the results of our paper are) correspond to the limit Am = 0. We can see that important effects of non-adiabaticity appear only for relatively high

15

10

5

0 00

0.4

08

i2

2s

Fig. 12. Dependence of the inertial function Boo on the energy Am of the collective vibration for Q - 026 and Q - Qsa" Positionsof the experimental values of the energy are shown by arrows. The value for the beta vibration (dashed arrow) is taken as an average of those for 166Er and 1"Er.

INERTIAL FUNCriONS

41 1

Atw. For the empirical values #m, shown in fig. 12 by arrows, they are of the order of 2 % for BQ .Qss and 15 % for BQ,oQw, i.e. not large. The non-adiabatic corrections make the inertial functions smaller and thus even more different from the values implied by experiment. S. Conci~ The following conclusions may be drawn from our research : (i) Single-particle structure effects are important for the inertial functions. The functions display strong fluctuations as functions of deformation. The amplitude of the fluctuations is larger for vibrational than for rotational functions and comes to about 40 % of the average value. The larger fluctuations of the vibrational functions are connected with the less collective character of them as compared with the rotational functions. (ii) hCorosoopic inertial functions B are about 1.5-3.0 times too small to reproduce experiment . One may expect that an inclusion of additional residual interactions into the model will increase their values s7). (iii) Almost all contribution to B comes from the valence shells (in the terminology of the PPQ model) . For example, for Bpp, the contribution of the core is smaller than 1 ,/o".

(iv) Non-adiabatic effects on the inertial functions are rather small. They are of the order of 15 % and reduce the inertial functions. (v) The miarsooopie inertial functions p, appearing in the collective Hamiltonian appropriate for the boson expansion treatment, eq. (2.5), differ significantly from those assumed iD the dynamic calculations performed up to now. The authors would like to thank Professor K. Kumar and Drs. 1. Ragnarsson and J. Srebrny for very useful discussions. Very helpful discussions and comments by Professors A. Bohr and B. R. Mottelson are gratefully acknowledged . Refer 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)

A . Bohr, Mat . Fys. Medd. Dan . Vid . Selsk. 26 (1952) no. 14 A Bohr and B. R Mottelson, Nuclear structure (Benjamin, Reading, Mass., 1975) vol . 2, chap . 6 K. Kumar and M. Baranger, Nucl. Pups. A92 (1967) 608 0. 0. Dusel and D . R Bès, Nurl. Phys. A143 (1970) 623 0. 0neusa and W. 0reiner, NucL Phys. A171 (1971) 449 L. v. Bernus, U. Schneider and W. 0reiner, Nuovo Cim. Lett . 6 (1973) 527 M. Baranger and K . Kumar, NucL Phys. 62 (1965) 113 ; A110 (1968) 490 K. Kumar and M. Baranger, Nucl . Phys. A110 (1968) 529 M. Baranger and K . Kumar, Nucl . Phys. A122 (1968) 241 K. Kumar, Nucl. Phys. A231 (1974) 189 T. Kaniowska, A . SobiczewaU, K. Pomorski and S. 0. Rohozindv, Nucl. Phys. A274 (1976) 151 W. Pauli, Handbuch der Physik (Springer, Berlin, 1933) vol. 24/1, p. 120 ; B. Podolsky, Phys. Rev . 32 (1928) 812

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