HAL Id: jpa-00214829
https://hal.archives-ouvertes.fr/jpa-00214829
Submitted on 1 Jan 1971
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
SEMICLASSICAL TREATMENT OF HEAVY ION REACTIONS
A. Winther
To cite this version:
A. Winther. SEMICLASSICAL TREATMENT OF HEAVY ION REACTIONS. Journal de Physique Colloques, 1971, 32 (C6), pp.C6-83-C6-86. �10.1051/jphyscol:1971611�. �jpa-00214829�
JOURNAL DE PHYSIQUE Colloque C6, supplkment au no 11-12, Tome 32, Novembre-Dkcembre 1971, page C6-83
SEMICLASSICAL TREATMENT OF HEAVY ION REACTIONS
A. WINTHER
The Niels Bohr Institute, University of Copenhagen, Denmark
Rhumb. - Une thbrie serni-classique des reactions du transfert est present&, qui inclut les principaux effets de recul dus a l'khange d'energie, de masse et de charge. Les equations du mouvement contiennent les effets de non-orthogonalite des fonctions d'onde et dkrivent les processus d'excitation et de transfert a n'irnporte quel ordre.
Abstract. - A semiclassical theory of transfer reactions is presented which includes the main recoil effects due to the exchange of energy, mass, and charge. The equations of motion contain the effects of non-orthogonality of the wave functions and describe the excitation and transfer processes to any order.
In the present talk I would like to present some of the main features of the semiclassical theory of nuclear reactions based on a recent paper together with R. Broglia [I]. While the use of a semiclassical treat- ment of transfer reactions goes back to Breit and Ebel [2], the present theory is formulated in analogy to the theory of Coulomb excitation and is directly inspired by the results of Trautman and Alder 131 on the semiclassical limit of DWBA in the formulation given by Buttle and Goldfarb [4].
The basis for the use of the semiclassical picture is the fact that in heavy ion reactions the wave lenght A
in the relative motion of the two colliding nuclei is very small compared to the nuclear radii so that we can talk about a well-defined trajectory of the centers of mass. In a direct reaction with relatively small exchange of energy, mass, and charge between the two nuclei, the centers of mass preserve their identity and we may follow the trajectory during the whole collision. In fact it is possible to include first order terms in the energy (AE), mass (Am) and charge (AZ) transfer i. e. in the quantities
where E, is the bombarding energy, ma the mass of the projectile, or rather the lightest of the two colliding nuclei, and 2, the charge of this nucleus.
In the collision between the two nuclei we consider a number of different channels which we denote by the indices r and s. Thus in the reaction
the index r indicates the configuration a, A as well as the states m and 11 in which the two nuclei are found.
The index s may similarly indicate b, B and the state
indexes p and q. It is understood that the (target) nucleus A is transformed into B (and a into b). In the semiclassical picture there is no interference between this reaction and the exchange reaction where A is transformed into b and a into B.
For each channel s one defines the classical trajectory
for the motion of the first nucleus with respect to the second. It is the solution of the classical equations of motion
where ms is the reduced mass while Us is the expecta- tion value of the total interaction between the two nuclei in channel s. Below the Coulomb barrier we would thus have
where Zb and 2, are the charge numbers of nuclei b and B.
The amplitude cs(t) for being in channel s at time t is determined by a set of coupled differential equations analogous to those used in Coulomb excitation, i. e.
The wave functions $, and o, are the channel wave function and its adjoint, respectively, while E, and E, are the total energies of the nuclei in the channels s and r, e.g.
E, = E:, f E,, A
.
(6)Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971611
C6-84 A. WIN
The difference Es - Er is the Q-value for the reac- tion s + r. The interaction energy appearing in the matrix element is the difference between the total channel interaction
Vr =
2
V(ri - r j )i e a
and the interaction Ur used to calculate the classical orbit.
We define the channel wavefunction $, by the expression
where $:, and I//: are the intrinsic eigenstates of the nuclei a and A while 6 , is given by
In equation (9) ur(t) is the classical relative velocity
vr(t> = k ( t ) while
is the dynamical variable indicating the relative center of mass position. The factor exp(iSr) takes into account the momentum and energy of the nucleons ( i ) due to the relative motion of the nuclei.
Since the wave functions (8) belonging to different channels are non-orthogonal, care must be taken in forming the adjoint wave functions or so that they satisfy the relation
If this equation holds for all states included in the coupled equations the summation may be extended over an arbitrary number of non-orthogonal channels.
The wave functions or may be constructed from the overlap matrix
mrs = ($r, $.s>
.
(1 3) It is seen thatsatisfies (12) if is the reciprocal of the a matrix i. e.
In evaluating the overlap (13) as well as the matrix elements in (5) the relative position of the centers of mass should be given by the classical trajectory through the relation
If the classical trajectories are all chosen to corres- pond to a given scattering angle the differential cross section for the reaction r + s is given by
where dors is the average elastic cross section for the channels rand s, while c,(co) is the solution (for t = co of ( 5 ) , which should be solved with the initial condition
The semiclassical approximation obviously only makes sense if the two relative center of mass posi- tions r, and rs in (16) are not too different. Let us for definiteness consider a stripping reaction where
We may then write
where r,, is the relative position of the cores, while M is the total mass ma
+
rn, = mb+
m,. The mass of the transferred cluster is denoted by m, while r,, is the position of the center of mass of the cluster with respect to the pointIt is seen that r, lies on the line connecting the centers of the two cores closest to the lightest one. Therefore the quantity
is small not only because m, 6 ma, but also because the main part of the integrals in ( 5 ) and (13) comes from values of r, w r,.
The coupled equations (5) reduce to the coupled equations of Coulomb excitation if the overlap a,, can be neglected. In this case
and the phase difference 6 , - b , can be neglected.
For higher bombarding energies one may try to solve (5) by taking the overlap (13) and the transfer matrix elements into account to first order only.
To first order one finds
where a:, is the nondiagonal part of r , and we may write ( 5 ) in the form
SEMICLASSICAL TREATMENT O F HEAVY ION REACTIONS
and
i a%(t) =
z
(i::,(v,,.
- ub.) $:t,j exp[x
i (E:+
E: - E:. - E:) t ] c i t . ( f )+
p'q'
where P"" is the projection operator on all the states m and 12 included in channel aA :
Equation (25) describes direct excitations in the entrance channel and the first term in (26) describes direct excitation in the exit channel while the last term describes the transfer. Tt is interesting to note that as more and more states are included in the entrance channel which may fccd the exit channel the effective interaction which causes the transfer [V, P] is reduced.
Thus if we consider only one state no H Z , we find
where UaA is the expectation value of V,, in tlic state
t7z0 no. Tf a large number of states aA were included, P
would approach the unit operator and the matrix element would vanish. This aspect of non-orthogonality has not been included in recent coupled channel calculation of transfer reactions.
Until now we have studied mostly the simple first order theory in which the reaction amplitude is given explicitly by the formula
where the AE is Q-value for the reaction.
The expression (29) differs from the corresponding expression in DWBA not only by the substitution of an integration over the distorted waves by a time integra- tion but also through the fact that recoil effects are concentrated in the phases 6 in the channel wave functions. For a stripping reaction (19) we may thus write the amplitude (29) in the form
x exp
[;;
(dEt - y,,)] dt ( 3 0 )where
and
r l
Y$I'(') = o(ua,4(R(t')) - ubE(R(tt)) -
- f (in,, - t?zbR) ( ~ ( t ' ) ) ~ ) d t ' . ( 3 2 ) The separation of the phase difference 6,
-
6, into a and y is somewhat arbitrary but it is seen that the approximation in DWBA of neglecting recoil effects, i. e. setting r,, equal to rbB is equivalent to setting a,, = 0 (cf. E q . ( 2 3 ) ) .The main dependence of (30) on the Q value of the reaction enters through the time dependent phase AEt - y(t). Thus although the tails of the nuclear wave functions change as, one changes the Q value, the matrix element in (30) does not change drastically since the reaction anyway takes place at the nuclear surface. A change in Q-value may however drastically change the value of the timeintegral since this integral vanishes exponentially for large values of AE.
The detailed comparison of the semiclassical amplitude (30) with the corresponding results of DWBA has especially been studied for subcoulomb stripping [ 3 ] . Utilizing the expansion of the tails of the nuclear wave functions as given in ref. [4] one may give closed expressions for the stripping cross sections.
For large angle scattering they depend on the Q-value essentially through a factor
where a, and a, are half the distances of closest approach in the entrance and exit channels, while ii is the average wave-length in the relative motion at infinite distance. The maximum cross section is obtained if a, = a,, i. e. if [l]
The steepness with which the cross section vanishes as a function of the Q-value is determined by the quantity p which is defined by the wavenumber, K in the tail of the nuclear wave function
p = ua (35)
a being the average of a, and a,. As a consequence of (33) we note that for proton transfer where the eflective K is large the cross section is much less sensitive to the Q-value than for neutron transfer.
It should be mentioned that the result (30) also
C6-86 A. WINTHER
shows the so-called post-prior symmetry, in that the It is seen that the post-prior symmetry only holds interaction V,, - U,, in the entrance channel can be for c(co).
substituted by the interaction Vb, - U,, in the exit Finally I would like to mention that it seems rather channel. The general symmetry relation which follows straightfoward to include absorption in the coupled from unitarity is equations (5) in a similar way as it is done in DWBA
simply by allowing Ur to be complex.
L($,
> (Vr - 'Jr) $ r ) - ($s 9 (Vs -us)
$ r ) ] XUr -+ U,
+
iW,.x cxp
[l
(E, - E,.) r] =h This ansatz which destroys the unitarity of the coupled
equations seems to work very well for the few cases we
(E, - E.) t ]
)
. (36)dt have studied until now.
References
[I BROGLIA (R.) and WINTHER (A.), Nuel. Phys. (in press).
[2] BREIT (G.) and EBEL (M.), Phys. Rev., 1956, 103, 679.
131 TRAUTMANN (D.) and ALDER (K.), Helv. Phys. Acta, 1970, 43, 363.
[4] BUTTLE (P. J. A.) and GOLDFARB (L. J. B.), NucI. Phys.
1966, 78, 409.
