Theories for large amplitude collective motion and constrained Hartree-Fock

Nuclear Physics A265 (1976) 301 --314; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

THEORIES FOR LARGE AMPLITUDE COLLECTIVE MOTION AND CONSTRAINED HARTREE-FOCK K. G O E K E

Institut fiir Kernphysik, Kernforschungsanlage Jiilich, D-5170 Jiilieh, West Germany Received 8 March 1976 Abstract: For a large amplitude collective motion the adiabatic time dependent Hartree-Fock theory (ATDHF), the infinitesimal generator approach (IG), the constrained random phase approximation (CRPA) and the cranking model (CR) are compared, the only assumption made being that the system moves along a collective path generated by constrained HartreeFock (CHF). Without restricting to a certain model it is shown, that C R P A and I G yield identical expressions for the collective mass, which depend explicitly on the constraining operator and are therefore not unique. A T D H F is shown to depend only on the collective path rather than on the constraining operator as well. The nuclear monopole vibrations in ~eO are taken as an example to demonstrate the magnitude o f the effect. The masses are calculated in the different theories taking the Skyrme III interaction and r 2 as constraining operator. The theories are unified by using a constraining operator, which is projected on its particle-hole subspace at each value o f the time dependent collective coordinate. This choice is justified and compared with various local prescriptions for the constraining operator.

I. Introduction

In recent years there has been growing interest in the description of nuclear large amplitude collective motion. Typical processes of this sort are nuclear fission, heavy ion reactions and vibrations of soft nuclei. Usually one views the nuclear motion as taking place along a one dimensional path through a multidimensional potential energy surface. If the system moves with sufficiently small collective velocity, the motion may be considered as adiabatic and one can assume the collective energy to separate into potential and kinetic parts, the potential term q/'o(q) depending on a collective coordinate q and the kinetic part being quadratic in the corresponding velocity, ~ ( q , tj) = ]rM(q)~ 2, giving rise to the definition of the collective mass M(q). The kinetic energy may be quantized leading to a Schr6dinger equation in q. It is generally believed, that the collective path of an adiabatic motion may be constructed by constrained Hartree-Fock methods (CHF), i.e. one assumes, that the wave function, which describes the moving nucleus at a certain value q of the collective coordinate, is close to the static Slater determinant Iq). The Iq) may result from a Hartree-Fock variation subjected to the subsidiary condition (ql~[q) = q with an appropriately chosen constraining operator 0~. Under these conditions for all approaches discussed below the collective potential is given by ~¢/'o(q) = (ql/C/Iq) with / / b e i n g the total Hamiltonian without any constraint. The collective mass, however, 301

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K. GOEKE

is described differently in the various approaches. It is the aim of this article to investigate the different formulae for M(q) for the adiabatic time dependent Hartree-Fock theory (ATDHF) of Baranger and V6n6roni 1), for the method of infinitesimal generators (IG) of Villars z), the constrained RPA of Pal 3) (CRPA) and the cranking model (CR). A comparison between various approaches for the collective mass has been performed already by Holzwarth 4) and Krieger and Goeke s). In contrast to these investigations the present study will be general and not restricted to a certain model. Particular emphasis will be put on the question, how far the collective masses are dependent on the choice of the constraining operator rather than on the collective path alone. Eventually such a dependence will affect also the kinetic zero point energy corrections and therefore the final collective potential, ~g'(R). In sect. 2 we will briefly recall some basic properties of CHF in order to fix the nomenclature. In sects. 3-6 we will derive the essential equations of the various theories, which allow a direct comparison. In sect. 7 we will solve the considered theories for a practical example, namely the monopole vibrations in 160, using Skyrme's interaction. The choice of the constraining operator and the unification of the above approaches will be discussed in sect. 8. Concluding remarks will be given in sect. 9. It should be mentioned that all following calculations and conclusions can easily be generalized to constrained Hartree-Fock-Bogoliubov states by going to the 2n dimensional representation 2, s). Also the generalization to several constraining operators is straightforward.

2. Constrained Hartree-Foek theory We assume a two body Hamiltonian

tl = E <~ltl[3)c+~c#+¼ E (~PIV[~"5)c+~c~ c, cr • a#

a#~8

(2.1)

In CHF one usually minimizes A in the trial space of glater determinants IO) subjected to the subsidiary condition <¢1(~1¢> = q, where (~ = Z Q ( 0 is an appropriately chosen one body constraining operator: _ O.

(2.2)

The/~ is a Lagrange multiplier. Essentially eq. (2.2) determines states [2), however one often prefers to label the states by a collective coordinate q with a clear geometric interpretation. There is no need to use q = <,~1~12), as in principle one can define the states Iq) by any correspondence one wishes to establish between 12) and the appropriately chosen collective coordinate q. Since we are going to use Iq) in the theories of collective motion, the q = q(t) is to be considered as time dependent. Eq. (2.1) is equivalent to requiring the commutator between the CHF Hamiltonian Wcnv = Wo-3.Q,

(2.3)

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303

and the single particle density operator Po = p2 = p~- to be zero: [WcaF, Po] = 0.

(2.4)

Wo = t +TrVPo.

(2.5)

The term Wo is given by The self-consistent basis is usually obtained by diagonalizing Wcar. This leads to occupied single particle states I ~ ) , IN>,... and unoccupied ones I~), I~) . . . . defining the single particle energies, for example

=

CHFe 8vt ous,

(MlWcavlm> = 0.

(2.6)

However, it will be convenient for our purpose, to use another representation obtained by performing unitary transformations among the occupied and unoccupied states, respectively, in order to diagonalize in the subspaces W0 rather than WCI~F. This defines other occupied states I M ) = aS 10), IN) and unoccupied states Ira), In) with the single particle energies, for example (MIWolN) = e° 6Ms,

(MlWcaFIm) = 0.

(2.7)

This representation, which diagonalizes, in the respective subspaces, Wo rather than WcaF will be used throughout this article. Obviously the ,o depend on ~. or q via Po, but not on the constraining operator explicitly as the ecaF do. For a given value 2 or q the positive semidefinite CHF stability matrix reads in this representation "~'CHF = ( A .

B A*)crlr'

(2.8)

with ACaV nNkK = (8o_ ~o)~5~~SNx+ (nKI V I N k ) - (nl2QIk)6sx + (KI2QIN)fit,, BCna~ = (nklVINK>.

(2.9)

We will distinguish the two representations by calling them the WcaF representation and Wo representation, depending on which operator is diagonalized.

3. Adiabatic time dependent Hartree-Fock theory (ATDHF) In time dependent Hartree-Fock (TDHF) one assumes the nuclear wave function q~(t) at all times to be a Slater determinant. The corresponding equation of motion for the single particle density operator, p = p2 = p+, is it~ = [W,p],

(3.1)

W = t + r r Vp.

(3.2)

with Baranger and V6n6roni l) extract from p = p(t) the part corresponding to classical coordinates (time even) and the part corresponding to their momenta (time odd) by

304

K. GOEKE

an ansatz, which is chosen in such a way that the coordinates still correspond to a determinant

p(t) = eiZtOpo(t)e-'Xt°,

(3.3)

with the properties p2 = Po, P : = Po and Z + = Z. The Po and Z are assumed to be invariant under time reversal. The adiabatic limiting case is derived by assuming the dimensionless operator Z to be small so that it may serve as an expansion parameter 6, 7):

p(t) = Po(t) +pl(t) +p2(t) + . . . .

(3.4)

p,(t) = i[z, Po],

(3.5)

with

P2(t) = --½ [Z, Ix, Po]]To second order pl is the time odd part of p. The total time independent energy E = ( ~ ( t ) l / / l ~ ( t ) ) = Tr(Wp)

(3.6)

may be calculated through second order in Z, giving E = :¢fTv + 3¢'o,

(3.7)

aT'TV(Po, Z) = ½ Tr (Z/~o)

(3.8)

~e'o(Po) = Tr tpo +½Tr Trpo VPo.

(3.9)

with the kinetic energy

and the potential energy

The equations to be solved are given by Brink et al. 7) as

with ao

=

1

ipo IJo ao = Po {[Wo, P l ] + [ W I , Po]} O'o,

(3.10)

Po f~ tro = po{Wo+W2 +i [W1, •]-½[[Wo, Z], Z]} O'o,

(3.11)

- P o and the definitions W o = t + T r VPo,

(3.12)

W, = Tr gp~,

(3.13)

W2 = Tr Fp2.

(3.14)

In the following we will assume Po to be dependent on a collective parameter q = q(t) and constructed by C H F as described in sect. 2. If we further assume, that at each time the particle-particle and bole-hole matrix elements of Z are zero with respect to any basis in which Po = po(q(t)) is diagonal, then z = i[po, p d ,

(3.15)

and X = x(q, ~) is uniquely determined by eq. (3.10) since/~o = q ~po/Oq is known.

LARGE AMPLITUDE COLLECTIVE MOTION

305

Therefore we will neglect eq. (3.11) for a moment and develop eq. (3.10) further. Using in the nomenclature of sect. 2

Po(q) = ~ IK)(KI,

(3.16)

K

one can write eq. (3.10) explicitly

i(nlkolN> = (s°-e°)(nlpalN> + (nl WAIN>.

(3.17)

With the explicit expression (3.13) and the complex conjugate of (3.17) we can write eq. (3.10) in the form

i ~

[A.N~, B.N~,~

[(klPalK> '~ -i(nlkolN)*] = ~ \B.*Nk* A*NkK]ro\(kIp,IK)*] '

(3.18)

with TD

a.skxr° = (8o_ eo)6.k 6NK+ (nKI VINk>,

BnNkK ~ -

(nkIVINK>.

(3.19)

Eq. (3.18) can be written in the simpler form (3.20) with

[ATD BTD~

(3.21)

Eq. (3.21) is a linear equation determining p, in terms of Po and/~0. The kinetic energy can be transformed as well to o~g'ra = ½i Tr {[Po, Pa]/~o} = ½i ~, (Nlplln>(nl~olN)-(nlPalN)(Nh6oln)}

(3.22)

Nn

or

'-:'~TO =

½(P*Pl),~'rv p

,

clearly showing t h a t ~ ' r v is a second order term, since p, is of first order. To ,A(rD there corresponds a RPA matrix efrrv = J ~ ' r v ,

(3.24)

J=

(3.25)

with (10 0 1 ) .

The JV'TD has the eigenvalue problem X¢

X¢

If the inverse of.A/'TD exists, one can bring (3.22) into a more familiar form, using the

K. GOEKE

306

representation ,~/.~ = ~ {1 (X,) ~>o ~ Y¢

1 (~:~ (Y¢-X¢)}

(X~- Yg)+ r..o-~¢\XJ

rD'

(3.27)

where the sum goes over one set of eigenvectors only. With eq. (3.26) we can rewrite eq. (3.23) and obtain ::TD =

+

( _~b° :, )

,

(3.28)

and furthermore with eq. (3.27) explicitly

/Ro

fO~ D

It~*X~D--Po YcrVl2} •

(3.29)

~>0

If we denote the eigenstates of the eigenvalue problem (3.26) by I~ o ) then TD

+

TD

aNl~o ) = (X~)N,, TD + TD (~¢ laNa,l~t'o ) = (Y~D)Nn.

<~¢ la,

(3.30)

Inserting (3.30) into eq. (3.29) we obtain with-~:rv = ½Mrv(q) ~z the final expression for the collective mass: Re co~D MTD(q)= 2 ~ I(~u~DI0~I~PoXD)I 2 (3.31) ,:>o l P:'l 2

with c3+having only p-h elements and being defined as

0 IK)*a~a,. O+= ~ (nl ~q IK)a,+ax--(nl-~q

(3.32)

Eq. (3.31) tells us that with a given path constructed by CHF the ATDHF is equivalent to performing a RPA calculation at each point of the path and then using the cranking like formula (3.31). However, the RPA calculation is not performed with an RPA matrix connected with the corresponding CHF stability matrix (2.9) by J'c~CHF" This may cause complex excitation energies, however, it allows an important consequence: As one can explicitly see in eq. (3.19), the matrices.A'ra and Jffrv depend on the constraining operator 0. only in so far as 0. determines the collective path;they do not show an explicit dependence as d¢'cnv does. This means, that in ATDHF one obtains the same mass for the same collective path, even if this one may be created by several constraining operators. 4. Theory of infinitesimal generators Villars 2) starts from a more general CHF scheme = 0,

(4.1)

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307

where ~ is the one particle operator describing the collective aspect of the system and P is a "weakly" conjugate operator i (2/~I[P, (~]12#) = 1.

(4.2)

The infinitesimal generators/~ and S are defined by 12+62,/2> = (1 +i62/~) I 2,/z~>,

(4.3)

12,/z+6#) = (1 +i6pS)12,/z).

(4.4)

The adiabatic approximation is done by calculating the generators/~ and S only at /t = 0, i.e. along the collective path described in sect. 2, with q = (2, 01512, 0). The energy

Po(q)

E = <2#IH-A0.-#P12#>

(4.5)

is expanded for small values of/t = ~ up to second order giving rise to the definition of the collective kinetic energy ~ I G ---- q2½i( 2,/~ = 011"P, g]l 2,/t -~- 0>

(4.6)

or equivalently fflG = _~2 ~ (2, ~ = 0 I [[//-2(.~, g], g]12, t~ = 0>.

(4.7)

For # = 0 the relationships between R, ~ and S, P are

(,,.)__~c~(s),

~4.~,

where . . ~ ' ~ is the CHF stability matrix and given by eqs. (2.8) and (2.9). By inserting eq. (4.9) into (4.6) one obtains ~IG

=

-½i(12(S*S)(_ep.)

(4.10)

or

Here .A'~cHF is the corresponding RPA matrix, given by .A/'cHr = J.~cHF. If one chooses in agreement with (4.2) P = --/~ (0q/02)-',

(4.12)

with q = (2, 01~]2, 0), then one can bring eq. (4.11) into the form =

•

308

K. G O E K E

Eq. (4.13) is identical to the corresponding eq. (3.28) of ATDHF, apart from the different definition of .A/'CHF. We can follow the lines of sect. 2 ending with the expression for the collective mass:

Mm(q) = 2~>o E ~---~--F 1 I<~"rlO~l ~ocnF>12,

(4.14)

with ~A/'cnFl~ar> = to: - c ~ ,tx"¢ , , c . r ,/ and with aq given by (3.21). Since eq. (4.13) is only dependent on time derivation, the a~ may be written in terms of any collective coordinate, which has a clear correspondence to 12>. There is no need to use in the final expression (4.14) for q the expectation value of 6. Obviously Mia(q) is not identical to MTD(q), since ~¢t'TD#~gCnr, of. (2.9) and (3.19). The difference is made exclusively by the h-h and p-p matrix elements of the constraining operator. This means, that Mm depends explicitly on the constraining operator and not only on the collective path. This feature can be traced back to eq. (4.5), where in IG the expectation value of the constrained Hamiltonian is expanded, whereas in A T D H F the free Hamiltonian is considered, cf. (3.6). This leads then to eq. (4.7) where Q appears explicitly. We will see in sect. 7 how large the difference between MTD and Mm can be in a practical case and we will see in sect. 8 how the constraint dependence of IG can be avoided.

5. Constrained random phase approximation (CRPA) Pal 3) starts from the time dependent CHF problem ~
i~ti~(t)> -- 0.

(5.1)

The solutions Iq'(t)) of (5.1) are expanded using the Thouless theorem IO(t)> = exp ( - iEo t) exp [ ~ Ckr(t)a~ ar]lq),

(5,2)

kK

where Iq> is the solution of the time independent CHF with the same constraining operator, see sect. 2. After replacing O/Ot = :t O/Oq, adiabacity is introduced by assuming the C~x(t) to be small quantities of first order, like t~. Under these assumptions eq. (5.1) is equivalent to

with ,.KcHr being the CHF stability matrix, see eq. (2.8). The kinetic energy is obtained by expanding the expectation value of the constrained Hamiltonian E = <~(t)l//-2~14~(t)>

(5.4)

up to second order in the coefficients C~x (t), giving for the second order term < H - 2 Q > ~2,

½ (C'C) .~'cHF (Cc.) .

(5.5)

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AMPLITUDE

COLLECTIVE

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309

Due to the adiabacity assumed, one can identify ~kx = 4(kl ~ IK) - (kl/~olK).

(5.6)

The collective kinetic energy may be obtained from eq. (5.5) by defining ,/ffCl~r = J Jr'c. F, using eq. (5.3) and proceeding along similar lines as given in eqs. (3.21) to (3.32). The results concerning the collective energy and the collective masses are identical to those in the I G approach, see eq. (4.13) and (4.14). Therefore CRPA also yields expressions for the collective mass which are explicitly dependent on the p-p and h-h matrix elements of the constraining operator and one can repeat the arguments of sect. 4. In CRPA like in IG the expectation value of the constrained Hamiltonian it/' = [ I - ; t ~ rather than of the free Hamiltonian / / is expanded with respect to the collective velocity. 6. The cranking model (CR) The cranking expression for the collective mass can be obtained in ATDHF by omitting in ,,KrD the matrix elements of the residual interaction. Then eq. (3.31) transforms to the simple expression

M,~(q) =

2~ kK

I(kldqlg)12 0 0

(6.1)

8k m 8K

Obviously eq. (6.1) also depends only on the collective path. In the Wo representation the corresponding expression of IG and CRPA is not of the simple structure of (6.1), however in the WCnFrepresentation one obtains M~

¢

~'~ M C R P A

-- 2 E II2 CHF

kK 8 k

CHF "

(6.2)

~ 8K

Here the 2Q dependence is now hidden in the ecaF. By now the cranking model has not yet been explicitly used with CHF wave functions and ecHF or 8°, but only with wave functions and single particle energies resulting from a single particle Hamiltonian with an appropriately parametrized single particle potential.

7. An example: Monopole vibrations It is interesting to see for a practical example the difference between CRPA, IG and ATDHF caused by the different treatment of the p-p and h-h elements of the constraining operator. For this purpose we will consider the nuclear monopole vibration in 160. The constraining operator ~ = r 2 is generally believed to be reasonable for this mode. The Skyrme-III force is taken as interaction and the calculations 7,1 i) are performed in a basis spanned by spherical harmonic oscillator states limited by N = 2 n + l < 14. The nucleus is assumed to be spherical for all values of

310

K. GOEKE

the Lagrangian multiplier 2. The oscillator parameter of the basis is determined by minimizing the energy of the unconstrained Hartree-Fock solution with respect to it, and this value is also taken for $=~0. The collective coordinate is taken to be the rms radius R = (/A)~ and the derivative with respect to R is calculated numerically with spline techniques using a sutYieiently dense mesh of R-points. Detailed studies and calculations will be published elsewhere 11). The results for the collective masses calculated by ATDHF, IG and CRPA are presented in fig. 1. For $ = 0 the three

Logmnge Rammeter ~. -1~ -1.2 -o.6 -0.3 oo ols I I I I I I 160 MonopoleV i b r .

-~_

= o.s

I /

~GRPA

/

-

-

~ .~

:~Eg -110

ATDHF

/

Collective

% %.%..~..............

.-*'°°°'°°° s / sCRI~

-13o-

-140

~" :E

~

I

30

i'" / /s p s

Collective~

ATDI-IF

" , , ~.

I

2,4

~ ~ - ~ -'~

1

2.5

--

13

'° l

,',,/Io CRPA IG ~

~

O

~*

I

2.6

I

2.7

rms-mdius R (fm)

I

2.8

=,

Fig. I. Collective masses for ATDHF, IG and CRPA. The collective masses for the nuclear monopole vibration in 160 are calculated by the adiabatic time dependent Hartree.Fock theory (ATDHF), the theory o f infinitesimal generators (IG) and the constrained random phase approximation (CRPA). The collective path is constructed by constrained Hartree-Fock with Skyrme III interaction and using H" = H - - 2t 2. The masses are given in terms of the nucleon mass. Here, "f'o (R) is the collective potential not corrected for zero point motion, and ~ is the corrected one, in units o f MeV.

theories give identical results, since ..¢t'rD = "#CUP However, with increasing 2 there is an increasing disagreement between A T D H F on one side and I G and CRPA on the other. The disagreement between the corresponding cranking expressions is similar. At R = 2.9 fm there is even a factor of three difference. This example, which

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AMPLITUDE

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311

is not particular, shows that without modification ATDHF and CRPA, IG yield essentially different results for the collective masses. It will be discussed in the next section, how one nevertheless can obtain a unified picture. Not only the collective masses, but also the final collective potentials turn out to be different in the various approaches: As described in sect. 2 the states Iq) are not necessarily eigenfunctions of some collective operator but wave packets in the collective space. Therefore the collective potential ~e'o(q) inevitably contains a contribution from the zero point energy ~'o(q)of the wave packet. If d'o(q) varies noticeably with q, it may influence the shape of the potential and therefore the total dynamics. The energy ~'o(q) separates into the potential zero point energy ~o, and the kinetic zero point energy J ' o , originating essentially from the folding of the potential and kinetic energy with the wave packet, respectively. If one assumes Gaussian wave packets with the width t, (~/Iq) = (~) ÷exp [ _ ½fl(~/_q)2],

(7.1)

and neglects the coupling between collective and intrinsic motion (adiabacity) one can write, following Reinhard's paper i o), 1

1

p2

p + ~1p 2)} e_½at~_,O~dq'

(7.2)

with P = -iO/d~land ~r(~) being the unfolded collectivemass. By carrying out the integraland performing the defolding one obtains for small flthe collectivepotential corrected for zero point motion

(7.3)

3e'(q) ~ ~9"o(q)--~0~o--3"o,

with 1

oCq) =

02

oCq),

(7.4)

"~-o(q) = fl/4M(q).

(7.5)

The form of J ' o is intimately connected with the quantization prescription, such that J ' o would look different from eq. (7.5), if, for example, the Pauli prescription had been chosen instead. However, in all cases ~'o turns out to depend on the collective mass and therefore to be dependent on the constraining operator if calculated with CRPA and IG. The present quantization prescription (7.2) follows naturally from the generator coordinate method 4) and in addition eq. (7.5) is rather simple compared with those from other quantization prescriptions. In IG and CRPA the J ' o can be written explicitly as .37-0 = _ 4 ~ ~>0

,~>o

CHF CHF 2 CHF - I I(~', laql~'o )I (o~)

'

(7.6)

312

K. GOEKE

and analogously in ATDHF. In fig. 1 the uncorrected collective potential ~l/'o(R ) and the corrected one ~¢'(R) are given, see ¢q. (7.3). One realizes a difference between CRPA and IG on one side, and A T D H F on the other. Also the minima are not at the same positions. By unifying ATDHF, CRPA and IG in the next section, the difference will be made to vanish.

8. The choice of the constraining operator The above theories are designed for large amplitude collective phenomena such as that occurring in the description of fission or heavy ion reactions. Often one has some preconceived notion about the behaviour of the nucleus. In those cases one describes the shape of the nucleus in terms of one or a few choosen variables q~, related to constraining operators 0.~, each with a clear geometric interpretation. It should be pointed out, that the shapes or the nuclear wave functions are the only physical quantities during the process, and the constraining operators and CHF are only a tool, which is believed to create the different shapes in an adiabatic way. However, since the variation (2.2) leads to

,~K~(q,l(Fl- AQ)a~ arl~ > = o, (8.1) with arbitrary infinitesimal 6rk, one realizes that only the p-h elements of ~ determine the path. Therefore the only physical content of 0. relevant for the nuclear wave function during the collective motion are represented by the p-h elements of 0.. Here particles and holes are defined with respect to any 2 or time dependent basis, which diagonalizes Po, in particular the Wo representation or the Wcnv representation. The p-p and h-h elements affect only the single particle energies ~cnv. This means that the constraining operators Q,~ depending on some parameters ~ and fl and constructed from a given 0.,

Q,~ = ctPo QPo+fl(1--po) 0 . ( I - p 0 ) + P o 0.(1--P0)+(1-Po) 0Po

(8.2)

all create the same collective path and therefore the same wave functions, but different single particle energies ecnF. Since the e° are independent on ~ and fl they are the quantities which are physical, rather than ecnF. It is just the different treatment of the single particle energies, which distinguishes A T D H F from CRPA and IG. The matrix ~ c n F looks in the WCHFrepresentation like Jr'To in the Wo representation if one replaces ecnv by e °. Therefore one should use a constraining operator, which is at each time projected on its particle-hole subspace

(~ph = P0 ~ ( 1 - - p o ) + ( I - - p 0 )

(~ PO-

(8.3)

Obviously (~ph depends on either the time t, q or 2. For such a choice of the constraining operator the matrices .A'TD and "~¢CnF are identical, see (2.9) and (3.19). Only under this condition do ATDHF, CRPA and I G yield the same collective masses. Furthermore, because the zero point energies are identical, these theories describe the dynamics of the collective motion in the same way.

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Recently there have been several methods worked out, which locally allow the constraining operator to be calculated rather than choosing it. It is therefore to be investigated if the suggested choice of 0~ = (~phis contradictory to these approaches. The A T D H F itself provides eq. (3.11) which may be used for such a purpose. If one replaces Wo by WcnF+20~ and using Po WcnF t r o = 0, eq. (3.11) turns out to be an equation for P0 2Q a o, since ~ is known from eq. (3.10). One realizes that (3.11) provides no information about the p-p or h-h elements of (~. Holzwarth and Yukawa s) obtained a prescription within the framework of the generator coordinate theory by variation of the generator coordinate trial function with respect to the (collective) path of integration. The constraining operator generally varies as a function of the constraining parameter and leads to a path which lies at the bottom of a potential energy surface valley. This means that the determinants which constitute a collective path can be obtained by CHF with a constraining operator which represents one of the normal modes of the system in the same constrained basis. Again this is only a prescription for the p-h elements of the constraining operator. RecentlyVillars s) gave a prescription for ~ in the framework of a generalized generator coordinate method, closely related to ATDHF. The constraining operator is essentially given as the difference between the p-h and the h-p part of the operator P, defined by Plq) = i(O/Oq)lq), cf. eq. (3.32). Again this does not contradict the above choice of ~ = 0.ph. It should be mentioned that also (q[ liP, 0.][q) = 1 does not restrict the choice of the p-p and h-h elements of (~, since Po P Po = (1 - P o ) p (1 - P 0 ) = 0. Actually there is no technical difficulty in handling the operator (8.3) rather than Q. After reaching the selfconsistent solution of WcnF = Wo - 2Q, one has in principle to start a new iteration scheme with (8.3). However, since the wave function is bound to be the same, the iteration scheme simply reduces to a diagonalization of Wo in the subspaces of occupied and unoccupied single particle states, respectively, i.e. in a change to the Wo representation, and forgetting the p-p and h-h elements of ~. There is also no difficulty in defining the collective coordinate. Eventually all above theories can be brought into a form which contains time derivation rather than coordinates, cf. (3.28), (4.13). The time, however, can be connected with any coordinate, in particular with the h-h elements of (~, if one wishes. 9. Summary and conclusions In this paper we have studied four different theories for collective nuclear motion, namely the adiabatic time dependent Hartree-Fock theory (ATDHF) in the formulation given to it by Baranger and Veneroni 1), the theory of infinitesimal generators (IG) of Villars z), Pal's formulation 3) of the random phase approximation (CRPA) and the different versions of the cranking model (CR). The investigation has been concerned with the theoretical and numerical comparison of the resulting collective masses. This has been done in general without using a certain model, the only assumption being that the nuclear wave functions are Slater determinants during the collective process depending on a time dependent parameter, and are obtained by con-

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strained Hartree-Fock (CHF). The A T D H F , C R P A and I G can be brought into a form, which allows a direct comparison. It turns out, that C R P A and I G lead to the same expression for the collective mass. This expression depends explicitly on the hole-hole and particle-particle elements of the constraining operator 5. Since within C H F the collective path is determined exclusively by the particle-hole elements of 0., the particle-particle and hole-hole elements of ~ can be varied without changing the path, but with large spurious effects on the collective masses and therefore on the dynamics of the system. In contrast to C R P A and I G the A T D H F is independent of the particle-particle and hole-hole elements of 0. yielding therefore nonspurious results. Furthermore, C R P A and I G yield the same expression for the collective mass as A T D H F , if one chooses a (time dependent) constraining operator, which is projected on its particle-hole subspace at each point of the collective path. Since in all considered theories the collective kinetic energy can be formulated in terms of time derivatives, the definition of a collective coordinate is not affected by this choice of the constraining operator 5. In such a ease also the cranking formulae derived f r o m A T D H F , C R P A and I G agree. The use of such a constraining operator is justified, since the particle-particle and hole-hole matrix elements of ~ bear no physical significance with respect to the nuclear wave function in the collective process, in contrast to the particle-hole elements, which determine the collective path. This choice is compatible with the theories discussed and with various explicit prescriptions in order to obtain locally the constraining operator. In order to demonstrate the effect of the particle-particle and hole-hole elements, the collective masses resulting from IG, C R P A and A T D H F have been calculated for nuclear monopole vibrations. The interaction used was the Skyrme I I I force, the collective path was created by WcHv = W o - A r 2 and the corresponding expressions in the above theories were solved numerically in a large finite basis without further approximations. It turned out that the masses of C R P A and I G differ sometimes by a factor of three from the nonspurious A T D H F value, demonstrating that the effect discussed is not negligible.

References 1) M. Baranger and M. Vrnrroni, to be published 2) F. M. H. Villars, Prec. Int. Conf. on dynamic structures of nuclear states, Mont Tremblant, 1971, ed. D. J. Rowe, p. 3 3) M. K. Pal, Trieste lectures 1973, in Theory of nuclear structure, eel. 2 (IAEA, 1975) p. 59; M. K. Pal, D. Zawiseha and J. Speth, Z. Phys. A272 (1975) 387 4) G. Holzwarth, Nucl. Phys. A207 (1973) 545 5) S. J. Krieger and K. Goeke, Nucl. Phys. A234 (1974) 269 6) D. M. Brink, M. J. Giannoni and M. Veneroni, Nucl. Phys. A2.58 (1976) 237 7) Y. M. Engel, D. M. Brink, K. Goeke, S. J. Krieger and D. Vautherin, Nucl. Phys. A249 (1975) 215 8) G. Holzwarth and T. Yukawa, Nucl. Phys. A219 (1974) 125 9) F. M. H. Villaxs, Lecture presented at Int. Conf. on Hartree-Fock and self-consistent field theories, Miramare-Trieste, March 1975 10) P. G. Reinhard, Nucl. Phys. A252 (1975) 120 11) K. Goeke, Prec. Int. Workshop IV on gross properties of nuclei and nuclear excitations, Hirschegg, Austria, Jan. 1976, ed. W. D. Myers, p. 26