DECOUPLING OF THE COLLECTIVE SUBMANIFOLD IN THE ATDHF THEORY

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DECOUPLING OF THE COLLECTIVE SUBMANIFOLD IN THE ATDHF THEORY

G. Do Dang

To cite this version:

G. Do Dang. DECOUPLING OF THE COLLECTIVE SUBMANIFOLD IN THE ATDHF THEORY.

Journal de Physique Colloques, 1987, 48 (C2), pp.C2-87-C2-90. �10.1051/jphyscol:1987214�. �jpa-

00226479�

JOURNAL DE PHYSIQUE

Colloque C2, supplement au n o 6, Tome 48, juin 1987

DECOUPLING OF THE COLLECTIVE SUBMANIFOLD IN THE ATDHF THEORY

G. DO DANG

Laboratoire de Physique Theorique et Hautes Energies, Universitk d e Paris-Sud, Bat. 211, F-91405 Orsay Cedex, France

Rgsum6.- On montre qu'une exploitation rationnelle des criteres de d6couplage permet de d6terminer compl6tement les sous-espaces collectifs.

Abstract.- It is shown that when properly exploited, the de- coupling conditions allow a unique determination of the collec- tive submanifold of any dimension.

This contribution is addressed to a specific question, namely do the equationsderived from the adiabatic time-dependent Hartree-Fock theory (ATDHF) provide a sufficient basis for a unique determination of the collective submanifold of more than one dimension

The issue has been raised recently1 and, to the above question, it is claimed that the answer is negative unless some constraint based on intuitive arguments is artificially imposed. In view of the fact that, for a full description of nuclear phenomena, more than one collective coor- dinate are often necessary, it would seem worthwhile to clarify the situation. We shall show below that, when supplemented by constraints derived from the decoupling conditions, the equations obtained from the ATDHF theory give a definite recipe for the determination of the collective subspace

.

As it is well-known that the TDHF theory can be transformed into a problem of classical mechanics governed by Hamilton's equations of motion3, we shall in the following base our discussionswithin the framework of the latter. In terms of the set of coordinates

6

( c 1 t 2 . . . c N

and momenta 2

(n1n2.. .rN) , we therefore suppose

-

that the system is described by the hamiltonian

The analog of the assumption in TDHF that at every instant, the system is describable by a determinantal state corresponds in this case to the assumption of point transformations

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987214

JOURNAL DE PHYSIQUE

1 2

N from the initial coordinates L to the new coordinates q = (q q . . . q

We shall also require the conjugated momenta E of

The exact decoupling of a submanifold X of dimension K < N means that, if at the time t

0, the system point is on C (q a

for a > K) and its

velocity in the tangent plane TC (pa = 0 for a > K), then it remains far ever on X. Obviously, the conditions for this to be realized are

= =

0 for a > K. The motion of the system in this case is a

governed by the hamiltonian

which depends only on the collective coordinates. Using Hamilton's equations, the decoupling conditions fia

qa

0 give

where i,j ... refer to the collective coordinates (=1,2, ... ^{K) and}

a,b.. . to the non-collective ones

K+1, . . ^{.N) and} c,a z ac/aqa . . .

The exact decoupling therefore requires that

The physical interpretation of these equations is obvious

ifrat the time t=O, condition (11) is satisfied and the system point is on X

(p, = qa

0 ) then the exact decoupling requires the absence of

dynamical and centrifugal (kinematical) forces (eqs. (I) and (111) respectively). It is important to notice that within the assumption of point transformations, eqs. (I, 11, 111), under the above or other equivalent forms, are all that are derivable from the expansion to zeroth-, first- and second- order of adiabaticity. The question then is to see whether they are sufficient for the determination of the collective submanifold.

Let us consider first the case of one dimension for which the

collective space is just a path (valley path). In this case, it has been shown4 that the simultaneous satisfaction of the zeroth-, first- and second- order conditions gives a unique solution. The non-

imposition of eq. (111) i.e. of the vanishing of the centrifugal force, as proposed initially by villars5, leads to instabilities, as is clear from the study of the landscape model 6 .

The form of eq. (111) of the second-order condition, in contrast to the previously derived form 4 , has the attractive feature that it

gives a physical interpretation to the missing ingredient in Villar's equations for a complete specification of the collective path. Further- more, as we shall now show, it is also precisely this decoupling

condition which, when properly exploited, allows a unique determination of the collective submanifold of more than one dimension.

Let ("X be the sequence of point functions on C defined as

one has the theorem

"if eqs. (I, I1 and 111) are satisfied on C then, for any a, $ ( ' ) X lies in TC, the tangent plane to C at any point". The proof that the components of the gradient vectors ortho- gonal to C vanish, ( ' ) X

0, requires that (a-1)%, = 0. But, as

(0) - 'a a

X,a = VIa

0 is just eq. (I), the others follow by induction.

The above theorem provides an explicit determination of the col- lective submanifold. As the collective space is assumed to have dimension K, the K+1 vectors $(')X, a

0, 1, . . . K which are all tangent to C at any given point must be linearly dependent, i.e.

For K

1, eq. (6) defines the valley path

it is a path that is orthogonal both to the equipotential (V = constant) and the equi- gradient

( ' ) X

U

- 1 ₂ V r a BaB V,@ = constant) lines. The genera-

lization to K > 1 defines what may be termed generalized valleys.

Their determination may be done as follows. With eqs. (6) being a system of N equations, the elimination of the K "Lagrange multipliers"

R a leaves N-K equations of the form F (5) a -

0. Just for the purpose

of defining C , any set of coordinates, ~ l , c 2 . . . cK may be chosen

C2-90 JOURNAL DE PHYSIQUE

1 2 K

to be q q ...q , the above N-K equations give the remaining ones in the form ca ⁼ ga(q1q2.. .qK) which completely specify C.

From the above, it may seem at this point that it is always pos- sible to decouple a submanifold of any dimension K < N. Actually, the possibility of finding the transformation g" (Q) does not imply exact decoupling and in the above formalism, the deviation from this limit will manifest itself in the following way. Eqs. (6) provide the functions g " ( ~ ) from which the tangent vectors ag"/aqi may be obtained by direct differentiation. On the other hand, using the chain rule, one may derive the following system of equations

where use is made of the contravariant forms of the gradient functions.

Eqs (6) are the conditions for the system (7) , taken as a linear system for g : i to have solutions. Thus, using the results obtained from eqs. (6) for ( a ) ~ f a and a new set of tangent vectors gIi

may be calculated. The deviation from exact decoupling will manifest itself in the non-vanishing of the quantities

from which an invariant measure may be defined

The goodness of decoupling is obtained whenever D1 < < 1.

1. A.K. Mukherjee, Phys. Lett. g=, 129 (1986).

2. G. Do Dang and A. Klein, Phys. Rev. Lett. 55, 2265 (19853.

3. A. Kerman and S.E. Koonin, Ann. Phys. 110, 332 (1976).

4. A.K. Mukherjee and M.K. Pal, Nucl. Phys. *, 289 (1982).

5. F. Villars, Nucl. Phys. E , 269 (1977).

6. K. Goeke, P.G. Reinhard and D.J. Rowe, Nucl. Phys. m, 408 (1981).