Comparison of collective mass parameters from the generator coordinate method and the cranking model

Volume 131B, number 4,5,6

PHYSICS LETTERS

17 November 1983

C O M P A R I S O N OF C O L L E C T I V E MASS P A R A M E T E R S FROM THE G E N E R A T O R C O O R D I N A T E M E T H O D AND THE C R A N K I N G M O D E L

S.J. W A N G 1, W. C A S S I N G GSI Darmstadt, POB 110 541, D-6100 Darmstadt, Germany

and W. N C ) R E N B E R G Institut fiir Kernphysik, Technische Hochschule Darmstadt, D-6100 Darmstadt, Germany and GSI Darmstadt, POB 110 _541, D-6100 Darmstadt, Germany Received 8 June 1983

Mass parameters for nuclear vibration and rotation as well as superfluid nuclear vibration and rotation are investigated in the deformed harmonic oscillator basis in the framework of the semiclassical cranking approach and the quantal generator coordinate method (GCM). We find that both methods yield approximately the same results in the case of nuclear vibration while for rotation GCM appears to underestimate experimental data if pairing correlations are included in contrast to the cranking approach.

The evaluation of collective mass parameters in large-amplitude collective nuclear motion has led to a variety of approaches in the past, which range from macroscopic models like irrotational flow [1] to microscopic models like the cranked shell model [2] with various degrees of sophistication (cranked H a r t r e e - F o c k - B o g o l i u b o v [3]) or the generator-coordinate method (GCM) [4,5]. It is well known that the mass p a r a m e t e r in the semiclassical cranking model (CM) [2]

B ~ = 2h 2 Z

~_-~

(1)

nO0

does not work due to quasi-crossings of singleparticle levels if BCM is evaluated within adiabatic shell-model states [6,7]. O n the other hand G C M is a fully quantum-mechanical approach and appears to be free from the levelcrossing problem,

In the formulae (1) and (2) for BCM and BOCM the operator P denotes the infinitesimal generator for the collective variable under consideration while &,, &0 and E,, E0 correspond to many-body states and excitation energies, respectively. The relation between BCM and BGCM has been discussed by several authors [5,8], but apart from qualitative aspects a more quantitative investigation is still needed. It is thus the aim of this letter to study in addition to general inequalities between BCM and BOOM the expressions (1) and (2) explicitly in the single-particle and quasi-particle picture for two cases of nuclear collective motion, i.e. for quadrupole vibration and collective rotation. The results are compared to the corresponding mass-parameters for irrotational flow. We start proving the inequality BCM > BGCM

Boc~

2h ~

[Y'" J(~" IP1~°)1212 Y~n ( E n - Eo)[(&,,lPl,~o)l :"

t On leave from University of Lanzhou, China.

(2)

(3)

in the independent-particle model for the ground-state band. In this case for x. := I12~0.

0 031-9163/83/0000-0000/$03.00 © 1983 North-Holland

a. := E.

E,,~0,

(4)

265

Volume 131B, number 4,5,6

PHYSICS LETTERS with

eq. (3) reads

~'(~x.)2/(~a.x.).

(5)

Eq. (5) can easily be proved by comparing

= ~'~ x~ + ~ n

(adam + a , d a , ) x , , x , ,

(6)

m>n

with

(? / x,

(7)

: ~'~ x 2 + ~'~ 2x,,x,. n

17 November 1983

m>n

O~x = O ~ o ~ q ,

o.,, = oJo/q,

ensuring volume conservation. This model has been used earlier [6,9,10] for the solution of eqs. (1) and (9) and yields transparent analytical formulae. For the actual comparison between BCM and BGCMwe additionally evaluate eqs. (2) and (10) within the same basis. For quadrupole motion eqs. (1) and (2) then yield the simple expressions 1 2 BCM = gh(2kxNdo)x + k~Nz/w~),

(12)

and

and considering (adam + am/a,)/> 2 for adam 0. This yields

(~.)(~a.x.).(~x.),

1 2 + o)~k~Nz) -l BGCM 4h (2k 2xNx + k .2N , ) 2 (2wxkxNx =

(8)

(13) with Nx = Ny and A

which is equivalent to (5). It should be noted that the equality in eq. (5) holds only if the excited states are degenerate, i.e. E , - E0 = const. Thus BGCM is always smaller than BCM except for the case of highly degenerate excited states. In order to describe the mass parameter for nuclear superfluid motion we evaluate eqs. (1) and (2) in the quasi-particle representation, i.e. for rotation B ~:ar~= 2h 2 ~

I(klPlk')[2

(UkVk'-

Uk'l)k) 2

(9)

k,k' Ek + Ek, " B ~ , ~ = 2h 2

[Xk'k'](klPlk')12Kukvk'--

u~'vk)2]2

(10)

× Zk,k' (Ek + Ek,)t(klPIk')t2(UkVk ' - Uk'Vk)2'

where [k) and Ek denote quasi-particle states and energies respectively {Ek =[(ek - A)2+ A2]m}. For vibration, additional terms [see (20)] appear in (9) and (10). In order to get a more specific idea about the quantitative aspects of BOCM we apply formulae (1), (2), respectively (9), (10) of quadrupole motion and rotation in the deformed harmonic oscillator model, with the sp hamiltonian given by 2+ y2) + wZz2], H ( q ) : ( - h 2 / 2 m ) A + ~m[wZ(x ~

(11) 266

(14)

Nj = ~'~ (nj + ½)~, k i = (do)/dq)/wj

with

j = x, y, z .

(15)

While wi and kj are continuous functions of the collective parameter q this does not hold for Nj(q) if I&0) in eqs. (1) and (2) describes the nuclear ground state for fixed q. The quantity N j ( q ) which corresponds to the total number of oscillator quanta in x-, y- and z-direction, changes discontinuously at level crossings [6[ for the Hamiltonian (11). If additional coupling terms like spin-orbit interaction are included in (11), level crossings change to quasi-crossings and Ni(q) becomes a rapidly changing function. Thus BCM a s well a s BGCM are discontinuous or rapidly changing functions of q close to crossings or quasi-crossings of sp levels. There are only two limits which yield continuous mass parameters BcM(q) and BocM(q). One case is the continuous approximation [11] for an infinite nuclear system requiring that for slow collective motion the microscopic nuclear density should follow the shape of the deformed harmonic potential, i.e. for our model Wx(q)Nx(q) = w y ( q ) N y ( q ) = o ) z ( q ) N . ( q ) .

The other limit corresponds to diabatic sp motion which should apply for the range of

(16)

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In the case of superfluid nuclear vibration we encounter a serious problem when directly evaluating the expressions (1) and (2) because BcM as well as BccM show large irregular oscillations as a function of q. These oscillations can be traced back to matrix elements of the type

i

5O

z~O

30

(kklP[0) = (d/dq)(ek - a ) - A/2E 2 , 2O

L

10 11 I I I I I I ~ ~h~ ~I~ ~i~ ~I~ ~I ~ 06

08

1.0

1.2

14

16

18

q= COo/c0Z Fig. 1. Mass parameters for vibration in the cranking approach (full line) and by GCM (dashed line) for diabatic single-particle motion. collective kinetic energy per nucleon of [12] 0.04 MeV < Eco,/A < 40 M e V .

(17)

In this limit quantum numbers of the sp states are frozen which implies (for a spherical nucleus initially) Nx = Ny = Nz - No = const.

(18)

The mass-parameters evaluated with (18) correspond to a state [&) of the nuclear system with nonzero intrinsic excitation energy for q # 0. The actual values for B o c M (dashed line) and BcM (full line) in case of diabatic sp motion (18) are shown in fig. 1. The G C M mass p a r a m e t e r deviates only slightly from the cranking result in the physically interesting region 0.6 < q < 1.8. A.s has been pointed out in refs. [10,13] BcM is identical to the mass p a r a m e t e r for irrotational flow Birr = trt f IVdpl2pmicr(r) dr 3 ,

(19a)

with the velocity potential

cb = (1/8q)(x2 + y2_ 2z2).

(19b)

In eq. (19a) the microscopic density distribution

Pm,cr(r)= Z Ig,o(r)r a

17 November 1983

(19c)

occ.

has to be used where qs~ denote the occupied single-particle states. The relation BCM = Birr also holds if condition (16) is used for Nj(q) but for different q,~,(r) in (19c).

(20)

for the creation of a quasi-particle pair [9]. According to Schfitte [9] pair excitations should not be included in the mass-formula because this type of transition does not allow for a perturbative treatment with respect to first order in 0. Due to the drastic variation of (20) with respect to q, i.e. the implication of a strong time dependence of the matrix element, the transition is adequately described by the L a n d a u Z e n e r formula J = exp{- rr A2/[c)(d/dq)(ek - a )1},

(21)

which is not analytic in q, and in particular not proportional to c~2 as required for formulae (1) and (2). Apart from the extensive discussion about the neglect of matrix elements (20) in the mass formulae there is a recent argument by Mosel [14] which excludes these matrix elements from a more general point of view. Instead of calculating the collective mass p a r a m e t e r directly via (1) the current j = (gl/2q)(x, y, - 2 z ) is investigated in the deformed oscillator model (11) which also includes matrix elements of the type (20). The current has to change sign with respect to time reversal which does not hold for contributions from pair creations. Thus pair creations correspond to real excitations of the nuclear system [9] while only virtual excitations should contribute to the collective mass. Eliminating matrix elements of the type (20) in eqs. (1) and (2) we obtain rather smooth mass parameters B ~ ( q ) (dashed-dotted line in fig. 2) and B~EM(q) (dashed line in fig. 2) which are approximately 40% less than the corresponding mass parameters BCM determined from (12) (full line in fig. 2) and BC,CM determined from (13) (dotted line in fig. 2) for condition (16). We again observe that the generator coordinate method as well as the cranking model yield ap267

Volume 131B, number 4,5,6

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t,030

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proximately the same mass parameters for nuclear quadrupole vibration. However, the situation changes drastically in the case of nuclear rotation around the x-axis. Again the expressions (1) and (2) can be evaluated analytically, i.e. (h/2o),coz){[(o),

- coD2/(co,

+ ¢0~)](N, + N~)

+ [(coy+ co~)2/(w, - co~)](N_,- N,)},

(22)

and BOCM = (h/2coycoz) x {(coy - coz)2(N, +/N/z) + (COy + COz)2lNy - N~I}2

x {(CO, + CO~)(CO, - CO~)~(% + N~)

CO l(CO,+ CO~)2lNr- NIF'.

~

06

(23)

We now address ourselves to the question how the moment of inertia changes if we deform the mean field of a given nucleus. This is actually a more academic question because experimentally the ground-state deformation is fixed for a given number of protons Z or neutrons N [15]. Nevertheless the variation in q for fixed A = N + Z allows for a qualitative and quantitative comparison of the various limits. In the continuum limit (16) the cranking model is known to yield the rigid body moment of inertia [11,16] (dashed line in fig. 3),which constitutes an upper bound for the inertial parameter. The lower bound, however, which is given by the irro-

08

-

/

)Du

so \ ~,, ° ° ~ 2 ° II t/ ~'i ~ ~

particle representation; dashed line: quasi-particle representation).

268

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200

"~ ~ ~'< ~':"~-"~ ~ ---..Z-Z

+ ICO, -

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Fig. 2. Mass parameters for vibration in the cranking model (CM, full line: particle representation; dashed~:toned line: quasi-particle representation) and by O C M (dotted line:

=

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q=%/Caz

BCM

i

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2s0

~'xN,

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17 November 1983

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1.2

-

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1.4

1

I

I

1.6

I

1.8

I

I

20

q:~o/WZ Fig. 3. Mass parameters for rotation: rigid rotor (dashed line), irrotational flow inertia (full line), GCM result in the continuum limit (16) (open square), microscopic inertias in the CM (open triangle), microscopic inertias given by GCM (open dot).

tational flow inertia Birr (full line in fig. 3), is achieved for condition (18), where N0 is determined for the spherical nucleus. If we evaluate Nj(q) microscopically from (14), i.e. by occupying the lowest levels for a given deformation (adiabatic limit), the cranking model (22) yields B~rr for approximately spherical shapes due to shell effects and values close to the rigid-rotor limit Brig for larger deformations [open triangles in fig. (3)]. The fact that BCM in this case even may exceed Br~g is due to an actual larger deformation of the microscopic density than the deformation of the mean-field which is used as variable here. The generator coordinate method on the other hand yields BGCM= BCM = Bir~ for condition (18), and in the continuum limit (16) inertias which are significantly lower than the rigid-rotor limit (open squares in fig. 3). In the microscopic adiabatic limit for Nj(q) described above, eq. (23) for BGCM gives inertia parameters again identical to Bier for spherical shapes and values well in between the two limiting cases of Birr and Bng (open dots in fig. 3) which follow approximately experimental systematics. In case of nuclear superfluid rotation, i.e. the evaluation of eqs. (9) and (10) with a gap parameter • - 0.7 MeV, we again find inertias close to B~r~for spherical shape and values well

Volume 131B, number 4.5,6 i

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/

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~= COo/Wz Fig. 4. Mass parameters for rotation: rigid rotor (dashed

line), irrotational flow inertia (full line), CM result for superfluid motion (open triangle), GCM result for superfluid ,notion (open dot). in between Brig and Bir~ for larger deformations I B ~ : open triangles in fig. 4, B~M: open dots in fig. 4). While BF:~[ is known to yield inertias close to experimental values, B~rM seems to underestimate these values according to fig. 4. Summarizing our studies we like to point out that the quantal mass formula (2) given by G C M always yields smaller mass parameters than the corresponding semiclassical cranking expression (1). The equality holds only for a highly degenerate spectrum. This is a general property and also holds if the nuclear wavefunctions contain m a n y - b o d y correlations. The reason for this inequality is as follows. According to the definition of the mass p a r a m e t e r in the cranking model (1) excited states contribute with a weight inversely proportional to the excitation energy, i.e. in a non-uniform way. On the other hand different excitations contribute tO BGCM more or less uniformely. Thus BCM will be significantly different from BGCM if the excitations are highly non-degenerate as in the independent particle model. The inclusion of pairing correlations reduces the contributions of low lying states more drastically in the cranking model than for BocM and consequently the difference between BcM and BGCM decreases. In the case of the deformed harmonic oscillator model the collective generator for the

17 November 1983

vibration P = -iO/Oq, which is a linear function of the operators xO/Ox, yO/Oy and zO/Oz, can only excite the same type of phonon pairs, i.e. + + Since the excitation energy a x+a x+, a y+a y+ or azaz. (2ho2j) is approximately the same for reasonable deformations for x- or y- and z - p h o n o n pairs, the spectrum is nearly degenerate and BCM as well as BocM yield almost the same results. For rotation, however, the collective generator induces two different types of excitation, i.e. yand z-phonon pairs and a y - z - p h o n o n exchange a~.az, which have significantly different excitation energies, i.e. 2hwj and hl~o~- o)zl respectively. Thus for rotation the excitations are highly non-degenerate and the difference between Bcu and BGcu is significant. The inclusion of pairing effects in addition reduces the nondegeneracy of the excitation spectrum and also decreases the difference between Bcu and Bacu. A n o t h e r interesting aspect of our investigation is that the cranking model yields the irrotational flow value for the collective mass p a r a m e t e r in the limit of diabatic single-particle motion (18) which is more closely studied in the framework of dissipative diabatic dynamics [17]. This is because the nodal structure of all occupied single-particle states is kept fixed throughout the collective motion and only a net collective flow induced by the changing meanfield remains. In contrast to this limit the evaluation of Bcu in the adiabatic limit involves drastic changes of single-particle contributions at quasi-crossings for a tiny change of the meanfield which induces additional rotational flow patterns on top of the collective flow generated by the changing mean-field. In this respect our studies support early suggestions of Griffin [18] with respect to the "principle of kinetic dominance" for rapid collective motion, which states, that fast collective motion should proceed along a path where the collective mass p a r a m e t e r is minimized. From a principal point of view the evaluation of mass parameters within G C M appears to be natural because they result from a fully quantum mechanical approach. On the other hand it is well known that BCM fits experimental data quite well if pairing correlations are included 269

Volume 131B, number 4,5,6

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w h i l e BGCM is a p p r o x i m a t e l y 2 0 % - 3 0 % s m a l l e r . The conflict how to compromise between the t h e o r e t i c a l c o n c e p t , i.e. a fully q u a n t u m mechanical theory (GCM), and the apparent p r a c t i c a l s u c c e s s of t h e s e m i c l a s s i c a l a p p r o a c h ( c r a n k i n g m o d e l ) still r e m a i n s o p e n . T h e a u t h o r s a c k n o w l e d g e v a l u a b l e discussions with Professor U. Mosel.

References [1] J.R. Nix, Ph.D. Thesis (University of California, LBL, 1964); Nucl. Phys. A15 (1965) 1. [2] D.R. Inglis, Phys. Rev. 96 (1954) 1059. [3] A. Faessler, K.R. Sandhya Devi, F. Gr/immer, K.W. Schmid and R.R. Hilton, Nucl. Phys. A256 (1976) 106; J.L. Egido and P. Ring, Nucl. Phys. A383 (1982) 189. [4] D. Brink and A. Weiguny, Nucl. Phys. A120 (1968) 59. [5] G.O. Xu and S.J. Wang, Intern. Conf. on Selected aspects of heavy-ion reactions; Communications (Saclay, May 82).

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[6] P. M611er and J.R. Nix, Nucl. Phys. A296 (1978) 289. [7] V.M. Strutinsky, Z. Phys. A280 (1977) 113. [8] F. Villars, in: Nuclear self-consistent field, eds. G. Ripka and M. Porneuf (North-Holland, Amsterdam, 1975). [9] G. Sch/itte, Phys. Rep. 80 (1981) 113. [10] J. Kunz and U. Mosel, Relation between current and inertial parameter in the cranking model, Preprint Universit~it Giessen. [11] P. Ring and P. Schuck, Nuclear many-body problems (Springer, Berlin, 1980) p. 134. [12] W. N6renberg, Phys. Lett. 104B (1981) 107; Preprint GSI-82-36. [13] H.C. Pauli and L. Wilets, Z. Phys. A277 (1976) 83. [14] U. Mosel, private communication. [15] S.G. Nilsson and O. Prior, Mat. Fys. Medd. Dan. Vid. Selsk 32 (1961) No. 16. [16] A. Bohr and B.R. Mottelson, Mat. Fys. Medd. Dan. Vid. Selsk. 30 (1955) No. 1. [17] W. Cassing, A. Lukasiak and W. N6renberg, Proc. Intern. Workshop on Gross properties of nuclei and nuclear excitations XI (Hirschegg, Austria, January 1983) p. 107-111. [18] J.J. Griffin, Physics and chemistry of fission (International Atomic Energy Agency, Vienna, 1969) p. 3.