Collective vibrations of the α particle in the GIPQ method

Volume 107B, number 6

PHYSICS LETTERS

31 December 1981

COLLECTIVE VIBRATIONS OF THE a PARTICLE IN THE GIPQ METHOD

S. DRO2~D2~ and J. OKO~OWlCZ Institute of Nuclear Physics, 31-342 Cracow, Poland Received 12 July 1981 Revised manuscript received 16 October 1981

Collective monopole and quadrupole vibrations of the a particle are discussed as solutions of the gauge invariant periodic quantization (GIPQ) method. A non-determinant wave function is assumed throughout the calculation. One solution fulfilling the GIPQ condition has been found for each vibrational mode.

An application of the time-dependent HartreeFock (TDHF) theory to the description of large amplitude collective motion has been reported previously by Kan and co-workers [1]. In this method, called the gauge invariant periodic quantization (GIPQ) method, a quantization condition arises from the gauge invarlance, i.e. the invariance under the change of the zero point of the energy. An alternative method, based upon the functional integral technique and the stationary phase approximation, has been formulated by Reinhardt [2] and Levit et al. [3]. It leads to the same quantization condition as the GIPQ method. Recently Kan [4] extended the GIPQ method, originally based on Slater determinants, to more general manifolds of trial wave functions 4~. The GIPQ condition requires that ~ should be periodic, ~P(t + T) = q~P(t),

implies that a correct description of the dynamics of the a particle requires separation of the center-of-mass wave function. Such a separation is possible in the harmonic oscillator approximation. One can do it by introducing new coordinates as follows: 4

Rc=l~ri, -~i=1

(1)

and T f 0

(~bPli/~0tlqSP) dt = 2nzrh

o

(2)

In this letter we apply the GIPQ method to collective vibrations of the a particle using the B r i n k Boeker nucleon-nucleon interaction. An antisymmetrized product of harmonic oscillator eigenfunctions is chosen as the wave function ¢ ( r l , r2, r3, r 4). The large effect of spurious center-of-mass motion in the determinant approximation for light nuclei 0 03 I-9163/81/0000-0000/$ 02.75 © 1981 North-Holland

~i=Rc-ri

.

(3)

The internal coordinates r i are not independent, since they must satisfy the relation E4= 1 ri = 0. The wave function takes the form ~b(rl, r2, r3, r4) = ~ba(rl, r2, r3) Z ( R c ) ,

(4)

where ~bc~is the nondeterminant internal wave function and Z ( R c ) the c.m. wave function. ~bc~is assumed to depend on time only through a single time dependent parameter q(t). A description of the dynamics requires the introduction of another, momentum-hke parameter p(t). This can be achieved by a phase factor eip S [5], where S is the velocity field appropriate for the collective oscillations o f interest. Hence, the wave function takes the form

(9 = ~bc~(q) e ipS Z ( R c ) .

(5)

The validity of such a restricted parametrization in the TDHF context was discussed in the fiterature [6]. The equations of motion for q5 are obtained from the action-integral variational principle in the following way: 403

Volume 107B, number 6

PHYSICS LETTERS

OE/ap, BlJ = - O E / O q ,

Bil =

(6)

31 December 1981

The spurious energy contribution connected with

Z ( R c) appears to be a time independent additive term where

E(p, q) = <¢lnl~)

(7)

and B

2/i l m [ - ~

=

gp ].

(8)

For monopole vibrations q = a = m~/2/~

(9)

and 4

s =E

(lO)

i=l

Then the wave function becomes 4

~ = N - 1 exp [ - ( a - i p ) i ~ = l , 2 ] Z ( R c ) .

(11)

For the quadrupole vibration parameter we choose the ratio/3 between the frequencies perpendicular and parallel to the symmetry axis :

q = 3 = 091/0,)11 = a.l./ali •

in <¢1H1¢) that does not influence the dynamics. Therefore, this term can be subtracted from (q~IHI~), to obtain the correct binding energy of the ~ particle. The most convenient way of exciting the mode under consideration is to choose, as initial condition, the equilibrium value o f q and a non-zero value o f p to "kick" the nucleus away from its ground state. An equivalent excitation mechanism is to start from nonequilibrium q and p equal to zero. Figs. 1 and 2 show the dependence of the vibration period and action integral on excitation energy for monopole and quadrupole vibrations, respectively. Solid curves correspond to the parameter set B1 of the Brink-Boeker nucleon-nucleon interaction [7]. The monopole vibration period increases with excitation energy and tends to infinity when the excitation energy reaches the binding energy of the a particle. This is connected with four-nucleon break-up which is allowed within the parametrization. Different behavior manifests itself in the quadrupole vibration case because of the volume conservation condition which does not allow break-up. As one can see the quantization condition gives one level for monopole

(12)

The volume conservation condition

alia 2 = a 3

(13)

gives

r ~ 1.0 '0

all =/3-2/3a0,

a± = 31/3a0

(14)

/!

I---

and 4

S =E

(~2 + ~/2y _ 2~/2z) .

i

(15)

i= 1 tx

I

i

}

i

z z

Then

10

~_~o -rz 0.5

(b=N-lexpI-(/3-2/3ao

-- (13l/3a 0 + 2ip)

- iP) i~l ) ,(r2x= + r'2ty

} zl .=

16

I

20

,~i,

36

E" [Mov] Z(Rc) ,

(16)

where N is a normalization factor. In all these formulae r4 = --(rl + r2 + r3). 404

t

Fig. 1. Period and action integral (divided by 21rfi)vezsus excitation energy of the ~ particle for monopole vibrations. The two curves correspond to different types of nucleon-nucleon interaction (see text), Arrows indicate level energies obtained.

Volume 107B, number 6

PHYSICS LETTERS

31 December 1981

,--.~1.25

~I.00 075 ,

I

I

/j

2,0

1.0 G -I'~, O5

1 [10'~s] 1.0

G

// /

[fm-~] 0.4

///

1.5

0.5

I

0.3 0.2

/

0.1 __

0

10

20 E" NoV]

I

1.0

2'.0

310

4.JO

30

Fig. 2. Same as fig. 1 for quadrupole vibrations. vibrations, at 24.1 MeV, and another one for quadrupole vibrations, at 21.5 MeV. These levels are found in the single-particle continuum region. Periodicity of the solutions in our case is guaranteed by the restricted parametrization which makes the motion appear as bound. However also in unrestricted TDHF calculations for heavier nuclei one obtains periodic solutions in the analogous region [8]. The B1 n u c l e o n - n u c l e o n interaction gives too small an rms radius (1.49 fro) when it is calculated with the separation of spurious motion. Therefore, we repeated the calculation with the depth parameters in the B1 interaction modified in such a way that they give an rms radius in agreement with Hofstadter's value of 1.61 fm [9]. These modified values are S 1 = - 1 7 5 . 7 MeV and S 2 = 576.6 MeV. The results for the modified B1 interaction are shown by dashed curves in figs. 1 and 2. They result in slightly shifted excitation energies of 23.2 MeV for monopole and 19.6 MeV for quadrupole vibrations. A similar excitation energy for monopole vibrations was obtained in ref. [10] using the hyperspherical functions method with the Volkov n u c l e o n - n u c l e o n interaction [11 ]. In our GIPQ calculations the Volkov result is very close to the modified B1 one. Finally in fig. 3 the time dependence of the collective parameters is presented for excitations corresponding to the levels found. One can see that the vibrations have a large amplitude and so conventional small amplitude approximations should be inadequate.

Fig. 3. The time dependence of the collective parameters a and fl for monopole and quadrupole vibrations, respectively.

We are grateful to Professors Z. Bochnacki and A. Budzanowski for encouragement and helpful discussions.

References [ 1] K.-K. Kan, J.J. Griffin, P.C. Lichtner and M. Dworzecka, Nucl. Phys. A332 (1979) 109; K.-K. Kan, Phys. Rev. C22 (1980) 2223; J.J. Griffin, M. Dworzecka, P.C. Lichtner and K.-K. Kan, Phys. Lett. 93B (1980) 235; M. Dworzecka, K.-K. Kan and J.J. Griffin, Proc. Intern. Workshop on Gross properties of nuclei and nuclear excitations (Hirschegg, Austria, 1980) (Inst. far Kernphysik, Technische Hochschule Darmstadt, Germany, 1980) p. 66; K.-K. Kan, Phys. Rev. C24 (1981) 789. [2] H. Reinhardt, Nucl. Phys. A331 (1979) 353; A346 (1980) 1. [3] S. Levit, J.W. Negele and Z. Paltiel, Phys. Rev. C21 (1980) 1603. [4] K.-K. Kan, Phys. Rev. C24 (1981) 279. [5] A.K. Kerman and S.E. Koonin, Ann. Phys. (NY) 100 (1976) 332; K.-K. Kan and J.J. Griffin, Phys. Rev. C15 (1977) 1126. [6] A.S. Jensen and S.E. Koonin, Phys. Lett. 73B (1978) 243; J.S.K. Anderson, J. Blocki and A.S. Jensen, Proc. Intern. Workshop on TDHF (Saclay, France, 1979). [7] D.M. Brink, B. Boeker, Nucl. Phys. 91 (1967) 1. [8] J. Blocki, M. Dworzecka, S. Dro~d~ and J. Okofowicz, to be published. [9] R. Hofstadter, Ann. Rev. Nucl. Sci. 7 (1957) 231. [10] K.V. Shiticova, Nucl. Phys. A331 (1979) 365. [11] A.B. Volkov, Nucl. Phys. 74 (1965) 33. 405