EFFECTS OF ANTISYMMETRIZATION ON POTENTIALS IN HEAVY ION SCATTERING

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EFFECTS OF ANTISYMMETRIZATION ON POTENTIALS IN HEAVY ION SCATTERING

J. Park, W. Scheid, W. Greiner

To cite this version:

J. Park, W. Scheid, W. Greiner. EFFECTS OF ANTISYMMETRIZATION ON POTENTIALS IN HEAVY ION SCATTERING. Journal de Physique Colloques, 1971, 32 (C6), pp.C6-245-C6-247.

�10.1051/jphyscol:1971655�. �jpa-00214873�

JOURNAL DE PHYSIQUE

Colloque C6, supplbment au no 11-12, Tome 32, Novembre-Dkcembre 1971, page C6-245

EFFECTS OF ANTI SYMMETRIZATION ON POTENTIALS IN HEAVY ION SCATTERING (*)

J. Y. PARK

Physics Department, North Carolina State University (**), Raleigh, U. S. A., and Institut fur Theoretisch Physik der Universitat FrankfurtIM., Germany

and W. SCHEID and W. GREINER

Institut fur Theoretische Physik der Universitat FrankfurtIM., Germany

Rksumk. - Les effets de I'antisymetrisation dans la diffusion des ions lourds ont Btk ktudies en dkveloppant la fonction d'onde totale en fonction du nombre de nucleons kchanges. Une formule donnant la contribution additionnelle au potentiel effectif due aux effets de l'antisymktrisation a kt6 obtenue.

Abstract.

Antisymmetrization effects in heavy-ion scattering are studied by expanding the total wavefunction according to the number of exchanged nucleons. A formula for the additional contribution to the effective potential due to antisymmetrization effects is obtained.

In the scattering of identical nuclei, it is customary to symmetrize the wavefunction with respect to the exchange of the two nuclei and to neglect the full antisymmetrization of the wavefunction with respect to all possible exchanges of individual nucleons.

For convenience we consider each nucleus in terms of a core and extra-core particles and describe the relative motion of the two colliding nuclei by the relative coordinate between the cores. In this particle- core model we only consider the antisymmetrization of the wavefunctions of the extra-core particles. An important advantage of the particle-core model is that the interchange of the extra-core particles leaves the relative coordinate of the scattering problem unchanged.

The total wavefunction of the system can be written

of the total angular momentum on a space-fixed axis is a good quantum number during the scattering, we construct wavefunctions from the two-center wave- functions and rotate them to a space-fixed axis (say z-axis).

We specify the intrinsic system by three Euler angles, two of which are the angles of the relative coordinate 4

and 8, and the third, $, describing a rotation around the intrinsic 2'-axis. Then,

Inserting this wavefunction into the ansatz (I), we obtain a wavefunction which has asymptotically the orbital angular momentum L and the intrinsic angular momentum J of the nuclei.

as

R x j ~ ( r ) AM~LJI DE'M,(~, 0,

@I.JM,(~, b)

(3) 'Y

RILJl(r) [iLyL(8, v ) 63 &(a, b ) I l ~ , (1)

ALJI

where

where R,,,,(r) is the wavefunction describing the

AM, L J I =

iL 2-L+1 ^{( L I O Mf I I M r ) . (4)}

relative motion of the colliding nuclei, and $,,(a, b) 4

is the in.trinsic wavefunction for the total system.

This wavefunction can be used to solve the scattering Here, /Z denotes a set of intrinsic quantum numbers.

If the scattering is described in terms of the two- problem since, as in ansatz (I), asymptotically the angular momentum splits into the orbital and the center shell model, the quantization axis is always

intrinsic angular momenta.

taken along the direction connecting the two centers,

Since it is convenient for some problems to describe and the projection M does not have the same value

the scattering process in a basis in which nucleons in the initial and final directions. Since the projection

are bound to individual centers of the colliding - nuclei, we investigate the antisymmetrization effects by

(*)

This work

been supported

the Bundesministerium

fiir bildung

Wessenschaft and

the Deutsche Forschungs- expanding the wavefunction according to the

gemeinschaft. number of exchanged nucleons. The antisymmetrized

(* *)

Permanent address. and normalized intrinsic wavefunction can be expanded,

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971655

C6-246 J. Y. PARK A N D W. SCHEID A N D W. GREINER

as follows, according to the number of nucleons exchanged.

Here, f is the normalization constant and is given by

In most cases of application the two cores of the colliding nuclei are composed of the same ground- state, even-even nucleus with zero spin. In this case the normalized direct (non-exchange) wavefunction,

@,,

may be conveniently written as

- -

1

--

xo(U xo(W

J<

I

>

X

9d0, 2, ..., Na ; Nu + 1, ..., N) , (7) with

where

In equation (7) ~ ~ ( 1 ) and ~ ~ ( 1 1 ) denote the internal wavefunctions of the cores I and I1 in the ground state with zero spin respectively. The second term in (8) corresponds to the exchange of all core-particles between the two colliding nuclei. In equation (9)

represent the single particle wavefunctions with respect to the nucleus a. They are not necessarily mutually orthogonal.

The exchange wavefunction @,,,, corresponding to the exchange of nucleons can be similarly expressed.

For example, for one-exchange of the nucleon 2 in the nucleus a with the nucleon N, + 1 in the nucleus b, we have

a:!,l

Dd(1,Na + 1 , 3 , ..., N,;2,N, + 2, ..., N ) . (10) The effective Hamiltonian H(r) for the relative motion is obtained by calculating the expectation value of the Hamiltonian with intrinsic wavefunctions.

We obtain, using (5),

--

=

J-'- Na ! N b !f 1 ^a$ Ha:", dr

Since the wavefunctions are antisymmetrized within one nucleus, the matrix elements < @, I HI

>

are independent of i for all values of I. Hence,

This result is in agreement with the similar expansion given by Goldberger and Watson [I] for the scattering amplitude.

Additional contribution to the effective potential due to antisymmetrization effects is calculated as the diffe- rence

In the special case when the extra-core particles of two identical colliding nuclei are in the same state, i. e. when I,

I,

I, and Na

Nb

N/2, the wave function for the extra-core particles given by (8) simplifies to

1 + (- l)L+J+NIZ

9 d =

2 [a; (I, ..., ;) ^@

Hence, the sum of the angular momentum quantum numbers L and J has to be even or odd for the scatter- ing of identical nuclei depending whether the mass number A is even or odd.

The formalism simplifies considerably when the single-particle wavefunctions

which are already orthogonal around each center, are also mutually orthogonal. In this case all overlap integrals between the direct and exchange parts vanish and hence, f

1.

The Hamiltonian can be expressed as a sum of n-body operators in the coordinates of the extra-core particles,

H = Ho + H1 + H z .

All matrix elements of the n-body operator H,, between the direct and I-nucleon exchange states vanish for n d 2, i. e.

< Gd 1 H, 1 >

0 for I >, n . (14) This is because, for example, one-exchange of two nucleons necessarily involves a two-body force, and one-body force cannot exchange particles. For a one- body operator H I , equation (14) implies

<

I H~ I

>

^{0 for all I .}

EFFECTS OF ANTISY MMETRIZATION O N POTENTIALS C6-247

Hence, which is symmetric in all particle coordinates, the

--

Pauli principle is already fulfilled when simple product

H l ( r )

<

I H I 1 4, >

< 4 I H I 1 4 > ^{, (15)} wavefunctions 4 in which each particle occupies different eigenstate are used. For a two-body operator

where H , only one transition matrix element,

-

N / 2

4

~ ~ ( 1 ) ~ ~ ( 1 1 ) [n ^{.p;(k) 8}

_(: ^{+ k ) ] <}

^I

^I

>

k = 1 J

.

does not vanish.

Thus, we obtain the well-known result. Namely, Applications of this formalism to various scattering when the Hamiltonian consists of one-body operators, and reactions and its extensions are in progress.

References

[I] GOLDBERGER (M. L.)

WATSON (K. M.)

(Wiley, New York, 1964), 164.