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Semiclassical approximation with complex trajectories
Volume 52B, number 2
PHYSICS LETTERS
30 September 1974
SEMICLASSICAL APPROXIMATION WITH COMPLEX TRAJECTORIES J. KNOLL * and R. SCHAEFFER
Service de Physique Th~orique, Centre d'Etudes Nucl~aires de Saclay, BP n° 2 - 91190 Gif-sur-Yvette, France Receive.d 1 August 1974 The semiclassicaldescription of elastic scattering is extended to include complex trajectories. Initial results are presented. A unified description of rainbow, diffraction, and interference phenomena is achieved, leading to a quantitative agreement with quantum mechanical calculations. Many kinds of experiments in nuclear physics are performed in a regime where semiclassical concepts are applicable [1-5] such as, e.g. the scattering of two heavy ions below the Coulomb barrier. But at higher energies the rapidly varying nuclear interaction is more important than the Coulomb one. Among various effects, this leads to singularities of the classical problem called caustics. In addition dispersive processes exist. So in the framework of an optical model the interaction potential becomes complex and gives rise to diffraction effects due to absorption. In order to get a semiclassical description in classical forbidden regions or for complex potentials an extension of the theory to include complex paths is necessary [ 6 - 8 ] . We have developed this extension for the three dimensional W.K.B. theory [7]. First we shall sketch the theory. We start with the Schr6dinger equation for the optical wave X(r).
variables (the azimuthal angle plays no role for central potentials *~ and use the integral approximation [1,9] for the partial wave series X(r) = ~ il(2l+l)×t(r)Pl(cosO) 0 **
(2)
,f db A(r,O,b)exp[i/hS(r,O,b)]
~--,o
,
_oo
•
2 ~
A(r,O,b) =-~rPb V i n t ~ n O
¸
(n2(r)-b2/r2)-l/4 r
S(r,O,b) = P[b(O-O') + f x/n2(p)-b21p2 dp] , (3) r'
nE(r) = 1 - V(r)/E; p2/2M = E .
In the limit/[ ~ 0 of this equation one gets the three dimensional Hamilton-Jacobi equations which define all possible trajectories arriving at point r. But these equations do not define the way these trajectories have to be combined in order to get x(r), in particular which ones should have been retained and which ones should not. We therefore prefer to take the limit ~ ~ 0 directly on the wave function as given by its quantal partial wave expansion. Thus, we solve the HamiltonJacobi equation for the separated angular and radial
The constants r ' and 0' (usually r ' = 0% 0'= n) of the radial integration (3) specify the boundary conditions chosen. The problem of the proper integration path from r ' to r is nicely solved by Pokrovskii and Kha, latnikow [11] who have extended the turning point problem in order to describe the reflection of waves at complex turning points. Since for a given impact parameter b in general more than one turning point exists, the integral expression (2) contains an additional summation over all those turning points which, according to ref. [ 11 ], contribute. Neglecting multiple reflections (a wave refected at one turning point will not get reflected at an other one) one can avoid this multivalued problem in taking instead of b the turning
* On leaveof absence from University of Freiburg, Germany; research follow of the "Stiftung Volkswagenwerk", Germany.
,1 Except for glory scattering where even non-classicalpaths contribute (see ref. [ I0] ).
(2~---MA + E -
V(r))x(r)=O .
(1)
131
Volume 52B, number 2
PHYSICS LETTERS
point r 0 given by r 0 n(r O) = b as a variable. The restriction to single reflexions prevents us from being able to treat resonances in potential wells. In the limit ~ ~ 0 the whole Integral (2) can be expressed by a few contributions ~
]/
47ri
×(r) tt--,O saddles V ~b2S/Ob 2
A(r, bs)exp[i/l~S(r, bs)]. (4)
coming from the stationary points b s of the exponent S, the saddles, which are determined as an implicit function o f t ' and r by /.
This is known to be the equation of a classical trajectory describing the motion of a particle from r', 0' to r, 0 with impact parameter b s and classical action S along this path. These are the ordinary W.K.B. equations, except that we allow for every contribution constructing the wave (4), even those which arise from complex values of b s and therefore from "complex classical trajectories", complex solutions of the Hamilton Jacobi eq. (5), with complex values for the classical action S. According to the laws of the saddle point method (c.f. Morse Feshbach [6] ) the relative hights and positions of the saddles tell which ones have to be retained in the sum (4) and which not ,2. This extension enables us to consider contributions of the order of exp ( - Im Sfli), describing waves damped by absorption or waves propagating into the classical shadow region. It therefore unifies the description of the classical allowed and forbidden region, and allows the use of complex potentials. It extends the validity into the direction of wave optics, allowing a proper description of all kinds of diffraction phenomena, as they become important if the variation of the potential becomes sizeable within a small number of wave lengths. The examples below show that the contribution of these new trajectories is important, more important than the corrections in higher order of/i. Singularities arise at the confluence of two saddles, that is at the caustics of the classical problem where ~2S/Ob 2 = 0. In this case a regular solution can be obtained by refining the integration method. For real •2 In our case it can be shown that at least the condition Im S(bs) I> 0 has to be fulfilled. 132
30 September 1974
potentials this problem has been solved by Berry [12, 5] starting from the Airy approximation of Ford and Wheeler [1]). The method was first applied to heavy ions by Silveira [13]). Its extension to complex potentials is straighforward. The branches to be chosen for the argument of the Airy function c.f. ref. [12] are in principle given by the saddle point limit:/i ~ 0, they are obvious in the lit region and can be traced to other regions by continuity. In this picture Fresnel diffraction is nothing but a particular case of rainbow scattering where the caustic is pushed into the complex plane by the absorbtive part of the potential. Fraunhofer diffraction is caused by two trajectories passing the scattering center at opposite sides. It occurs if the classical deflection function as given by (5) varies rapidly, regardless whether this is caused by a strong variation of the real or the imaginary part of the potential. We shall now compare elastic scattering cross sections obtained by our approximation with the exact partial wave results [ 15]. The appearance of new contributions from complex trajectories can be nicely demonstrated by considering a purely repulsive (real) potential which has a monotonic deflection function. Here only one "true" classical path exists for every angle. Thus the semiclassical description in the usual sense (i.e. real paths) will show a structureless cross section (dotted curve in fig. 1a) in contrast to the quantal result (full curve). However there is another path which is complex and describes a wave bending around the core into the shadow region. It interferes with the first one yielding almost the rig,ht pattern except in the very forward direction where glory-type effects [1,10] dominate. Another case where complex trajectories must be used for real potentials is the rainbow phenomenon where a complex trajectory describes the damping of the quantal wave in the shadow region. As practical examples we have chosen the scattering of 160 by 160 at 50 MeV (fig. lb), at 300 MeV (fig. lc), and the scattering of 139 MeV o~particle.s by 58Ni (fig. ld). The dashed curves represent our calculations with complex trajectories. They follow very closely the exact partial wave results (full curves). It is interesting to compare our method with the more usual one [2] (dotted curves of figs. lb, c, d) which uses only the real part of the potential to define the trajectories, and takes the imaginary part as an attenuation factor along this path. The usual method is actually a first-order
Volume 52B, number 2
PHYSICS LETTERS
30 September 1974
100,1
i,.~,
Soft repulsive sphere
Position of comptex turning po/n~ re teaclingto rear ongtese
Jm re ifm]
10 50HeV
-~. ~,®_=. ~
~
i
.
-80 =
~,/~ "~t'~/ ....... " " " J
,"" ":.
~ .,~:1
O
."~%,/
°"IV /V I '11
"N ,,o.-o /
V V;' -'\
20"
20"
0 •
10"
Vept tH.eVl . . . . LW(r)
.. . . . . . . .
%...
30" 0 O"
.
.
~
.
J
1~2¢'Re re [fm]
~ , lb r ~ [fm]
/ @IINI+(I 139 HeY
4He + SeNi
10( /.0"
60" O
Fig. 1. Elastic cross sections calculated by our approximation (dashed line) compared to the exact results (full line) and the usual semiclassical method with paths given by the real part of the potential (dotted line), a) Soft repulsive sphere of Wood Saxon type with VIE = 4, k R = 18.7 and k :z = 1.24. b) 16O + usO at 50 MeV with V = 50 MeV, W= 20 MeV, R = 1.2 641/3 + A 1/3)fro, a = 0.6 fro. c) the same as in b) at 300 MeV. d) a + SSNi at 139 MeV. The parameters of the potential and the W.K.B. phase shift calculation (dashed-dotted line) are taken from ref. [14]. perturbation theory in Im VIE [7] and is good if Fresnel scattering dominates and at the more forward angles of Fraunhofer scattering. But as soon as one reaches the strong absorption region (large angles) large deviations occur whereas the complex trajectory method remains accurate. Moreover, to obtain such agreement with the "real potential approximation" we were already obliged to include the complex trajectory o f the rainbow which occurs aroung 25 °, 5 ° and 3 ° respectively for the three cases. A strict restriction to real paths would lead to an agreement only for angles smaller than the above. The dashed-dotted curve in fig. 1 d finally illustrates another kind of semiclassical result obtained by using W.K.B. phase shifts in a partial wave series (taken from ref. [14] and discussed by Malfliet [4]). Since the turning points are-kept real in this calculation, the result becomes rapidly worse when the imaginary part o f the potential starts to be important. The practical calculation o f the scattering amplitude procedes in the following way. One has to find trajec-
Fig. 2. Real and imaginary part of the opticalpotential (below), and the positions of the complex turning points leading to real scattering angles, for the case in fig. ld. The Coulomb rainbow occurs at 2.5 ° . At angles of - 4 0 ° three trajectories give sizeable contributions.
tories which start at infinity and lead to given angle. These trajectories can either be classified by their impact parameter b s or, as actually used here, by their turning points ros, together with b s = rosn(ros ). One therefore has to look for the values ofros which satisfy the deflection equation (5). In practice, important contributions to the amplitude come only from turning points close to the real axis. This is illustrated in fig. 2, where for the c~ + 58Ni case o f fig. 1 the positions of the turning points leading to different angles 0 are shown. Once the turning points are known for one angle, they can be traced to other angles by A0 = (aO/Oro) Ar o = (a2s/ab2)(db/dro)Aro. Finally for every trajectory found in this way the classical action S (or equivalent the W.K.B. phase shifts) and a2s/ab 2 are needed to construct the scattering amplitude. It should be mentioned that at most two or three trajectories lead to a given angle (counting positive and negative angles separately). Investigations underway on inelastic processes [16] indicate that there is almost 133
Volume 52B, number 2
PHYSICS LETTERS
only one trajectory at each side of the nucleus will contribute, This will even more simplify the method. Summarising, for all cases considered the semiclassical calculations including complex paths reproduce the exact results with considerable accuracy. This is even true in cases where the W.K.B. approximation is generally believed to fail, for instance for 50 MeV protons and c~particles. The oscillations in the cross sections are quantal effects in the sense that they vanish in the strict limit h ~ 0, b u t they are obtained in terms o f purely classical quantities. A semiclassical analysis can lead to a better explanation and a more physical insight into the process as compared to quantal calculation. We thank R. Balian for many discussions and strong encouragement to use complex trajectories.
References [11 K.W. Ford.and J.A. Wheeler, Ann. Phys. NY 7 (1959) 239,287.
134
30 September 1974
[2] R.A. Broglia and A, Winther, Phys. Rev. 4C (1972) 112. [3] D.M. Brink, lecture notes, Orsay, march 1972. [4] R.A. Malfliet, Extended Seminar on Nuclear Physics, I.C.T.P. Trieste, Italy, Sep. 1973. [5] M.V. Berry and K.E. Mount, Rep. Progr. Phys. 35 (1972) 315. [6] J. Stine and R.A. Marcus, Chem. Phys. Lett. (1972)536. [7] J. Knoll and R. Schaeffer, Extended Seminar on Nuclear Physics, I.C.T.P. Trieste, Italy, Sep. 1973, to be published in the proceedings. [8] R. Balian and C. Bloch, Ann. Phys. 63 (1971) 592, and to be published. [9] W.H. Miller, J. Chem. Phys. 53 (1970) 1949. [10] P. Pechukas, Phys. Rev. 181 (1969) 166. [11] V.L. Pokrovskii and I.M. Khalatnikov, J.E.T.P. 40 (1961) 315. [12] M.V. Berry, Proc. Phys. Soc. 89 (1966) 479. [13] R. da Silveira, Phys. Lett. 45B (1973) 211. [14] D.A. Goldberg and S.M. Smith, Phys. Rev. Lett. 29 (1972) 500. [15] J. Raynal, Programme MAGALI, ref.: DPh.T/69-42, Saclay. [16] P.M. Morse and H. Feshbach, Methods of theoretical physics I, Me Graw-Hill, p. 437. [17] J. Knoll and R. Sehaeffer, to be published.
PHYSICS LETTERS
30 September 1974
SEMICLASSICAL APPROXIMATION WITH COMPLEX TRAJECTORIES J. KNOLL * and R. SCHAEFFER
Service de Physique Th~orique, Centre d'Etudes Nucl~aires de Saclay, BP n° 2 - 91190 Gif-sur-Yvette, France Receive.d 1 August 1974 The semiclassicaldescription of elastic scattering is extended to include complex trajectories. Initial results are presented. A unified description of rainbow, diffraction, and interference phenomena is achieved, leading to a quantitative agreement with quantum mechanical calculations. Many kinds of experiments in nuclear physics are performed in a regime where semiclassical concepts are applicable [1-5] such as, e.g. the scattering of two heavy ions below the Coulomb barrier. But at higher energies the rapidly varying nuclear interaction is more important than the Coulomb one. Among various effects, this leads to singularities of the classical problem called caustics. In addition dispersive processes exist. So in the framework of an optical model the interaction potential becomes complex and gives rise to diffraction effects due to absorption. In order to get a semiclassical description in classical forbidden regions or for complex potentials an extension of the theory to include complex paths is necessary [ 6 - 8 ] . We have developed this extension for the three dimensional W.K.B. theory [7]. First we shall sketch the theory. We start with the Schr6dinger equation for the optical wave X(r).
variables (the azimuthal angle plays no role for central potentials *~ and use the integral approximation [1,9] for the partial wave series X(r) = ~ il(2l+l)×t(r)Pl(cosO) 0 **
(2)
,f db A(r,O,b)exp[i/hS(r,O,b)]
~--,o
,
_oo
•
2 ~
A(r,O,b) =-~rPb V i n t ~ n O
¸
(n2(r)-b2/r2)-l/4 r
S(r,O,b) = P[b(O-O') + f x/n2(p)-b21p2 dp] , (3) r'
nE(r) = 1 - V(r)/E; p2/2M = E .
In the limit/[ ~ 0 of this equation one gets the three dimensional Hamilton-Jacobi equations which define all possible trajectories arriving at point r. But these equations do not define the way these trajectories have to be combined in order to get x(r), in particular which ones should have been retained and which ones should not. We therefore prefer to take the limit ~ ~ 0 directly on the wave function as given by its quantal partial wave expansion. Thus, we solve the HamiltonJacobi equation for the separated angular and radial
The constants r ' and 0' (usually r ' = 0% 0'= n) of the radial integration (3) specify the boundary conditions chosen. The problem of the proper integration path from r ' to r is nicely solved by Pokrovskii and Kha, latnikow [11] who have extended the turning point problem in order to describe the reflection of waves at complex turning points. Since for a given impact parameter b in general more than one turning point exists, the integral expression (2) contains an additional summation over all those turning points which, according to ref. [ 11 ], contribute. Neglecting multiple reflections (a wave refected at one turning point will not get reflected at an other one) one can avoid this multivalued problem in taking instead of b the turning
* On leaveof absence from University of Freiburg, Germany; research follow of the "Stiftung Volkswagenwerk", Germany.
,1 Except for glory scattering where even non-classicalpaths contribute (see ref. [ I0] ).
(2~---MA + E -
V(r))x(r)=O .
(1)
131
Volume 52B, number 2
PHYSICS LETTERS
point r 0 given by r 0 n(r O) = b as a variable. The restriction to single reflexions prevents us from being able to treat resonances in potential wells. In the limit ~ ~ 0 the whole Integral (2) can be expressed by a few contributions ~
]/
47ri
×(r) tt--,O saddles V ~b2S/Ob 2
A(r, bs)exp[i/l~S(r, bs)]. (4)
coming from the stationary points b s of the exponent S, the saddles, which are determined as an implicit function o f t ' and r by /.
This is known to be the equation of a classical trajectory describing the motion of a particle from r', 0' to r, 0 with impact parameter b s and classical action S along this path. These are the ordinary W.K.B. equations, except that we allow for every contribution constructing the wave (4), even those which arise from complex values of b s and therefore from "complex classical trajectories", complex solutions of the Hamilton Jacobi eq. (5), with complex values for the classical action S. According to the laws of the saddle point method (c.f. Morse Feshbach [6] ) the relative hights and positions of the saddles tell which ones have to be retained in the sum (4) and which not ,2. This extension enables us to consider contributions of the order of exp ( - Im Sfli), describing waves damped by absorption or waves propagating into the classical shadow region. It therefore unifies the description of the classical allowed and forbidden region, and allows the use of complex potentials. It extends the validity into the direction of wave optics, allowing a proper description of all kinds of diffraction phenomena, as they become important if the variation of the potential becomes sizeable within a small number of wave lengths. The examples below show that the contribution of these new trajectories is important, more important than the corrections in higher order of/i. Singularities arise at the confluence of two saddles, that is at the caustics of the classical problem where ~2S/Ob 2 = 0. In this case a regular solution can be obtained by refining the integration method. For real •2 In our case it can be shown that at least the condition Im S(bs) I> 0 has to be fulfilled. 132
30 September 1974
potentials this problem has been solved by Berry [12, 5] starting from the Airy approximation of Ford and Wheeler [1]). The method was first applied to heavy ions by Silveira [13]). Its extension to complex potentials is straighforward. The branches to be chosen for the argument of the Airy function c.f. ref. [12] are in principle given by the saddle point limit:/i ~ 0, they are obvious in the lit region and can be traced to other regions by continuity. In this picture Fresnel diffraction is nothing but a particular case of rainbow scattering where the caustic is pushed into the complex plane by the absorbtive part of the potential. Fraunhofer diffraction is caused by two trajectories passing the scattering center at opposite sides. It occurs if the classical deflection function as given by (5) varies rapidly, regardless whether this is caused by a strong variation of the real or the imaginary part of the potential. We shall now compare elastic scattering cross sections obtained by our approximation with the exact partial wave results [ 15]. The appearance of new contributions from complex trajectories can be nicely demonstrated by considering a purely repulsive (real) potential which has a monotonic deflection function. Here only one "true" classical path exists for every angle. Thus the semiclassical description in the usual sense (i.e. real paths) will show a structureless cross section (dotted curve in fig. 1a) in contrast to the quantal result (full curve). However there is another path which is complex and describes a wave bending around the core into the shadow region. It interferes with the first one yielding almost the rig,ht pattern except in the very forward direction where glory-type effects [1,10] dominate. Another case where complex trajectories must be used for real potentials is the rainbow phenomenon where a complex trajectory describes the damping of the quantal wave in the shadow region. As practical examples we have chosen the scattering of 160 by 160 at 50 MeV (fig. lb), at 300 MeV (fig. lc), and the scattering of 139 MeV o~particle.s by 58Ni (fig. ld). The dashed curves represent our calculations with complex trajectories. They follow very closely the exact partial wave results (full curves). It is interesting to compare our method with the more usual one [2] (dotted curves of figs. lb, c, d) which uses only the real part of the potential to define the trajectories, and takes the imaginary part as an attenuation factor along this path. The usual method is actually a first-order
Volume 52B, number 2
PHYSICS LETTERS
30 September 1974
100,1
i,.~,
Soft repulsive sphere
Position of comptex turning po/n~ re teaclingto rear ongtese
Jm re ifm]
10 50HeV
-~. ~,®_=. ~
~
i
.
-80 =
~,/~ "~t'~/ ....... " " " J
,"" ":.
~ .,~:1
O
."~%,/
°"IV /V I '11
"N ,,o.-o /
V V;' -'\
20"
20"
0 •
10"
Vept tH.eVl . . . . LW(r)
.. . . . . . . .
%...
30" 0 O"
.
.
~
.
J
1~2¢'Re re [fm]
~ , lb r ~ [fm]
/ @IINI+(I 139 HeY
4He + SeNi
10( /.0"
60" O
Fig. 1. Elastic cross sections calculated by our approximation (dashed line) compared to the exact results (full line) and the usual semiclassical method with paths given by the real part of the potential (dotted line), a) Soft repulsive sphere of Wood Saxon type with VIE = 4, k R = 18.7 and k :z = 1.24. b) 16O + usO at 50 MeV with V = 50 MeV, W= 20 MeV, R = 1.2 641/3 + A 1/3)fro, a = 0.6 fro. c) the same as in b) at 300 MeV. d) a + SSNi at 139 MeV. The parameters of the potential and the W.K.B. phase shift calculation (dashed-dotted line) are taken from ref. [14]. perturbation theory in Im VIE [7] and is good if Fresnel scattering dominates and at the more forward angles of Fraunhofer scattering. But as soon as one reaches the strong absorption region (large angles) large deviations occur whereas the complex trajectory method remains accurate. Moreover, to obtain such agreement with the "real potential approximation" we were already obliged to include the complex trajectory o f the rainbow which occurs aroung 25 °, 5 ° and 3 ° respectively for the three cases. A strict restriction to real paths would lead to an agreement only for angles smaller than the above. The dashed-dotted curve in fig. 1 d finally illustrates another kind of semiclassical result obtained by using W.K.B. phase shifts in a partial wave series (taken from ref. [14] and discussed by Malfliet [4]). Since the turning points are-kept real in this calculation, the result becomes rapidly worse when the imaginary part o f the potential starts to be important. The practical calculation o f the scattering amplitude procedes in the following way. One has to find trajec-
Fig. 2. Real and imaginary part of the opticalpotential (below), and the positions of the complex turning points leading to real scattering angles, for the case in fig. ld. The Coulomb rainbow occurs at 2.5 ° . At angles of - 4 0 ° three trajectories give sizeable contributions.
tories which start at infinity and lead to given angle. These trajectories can either be classified by their impact parameter b s or, as actually used here, by their turning points ros, together with b s = rosn(ros ). One therefore has to look for the values ofros which satisfy the deflection equation (5). In practice, important contributions to the amplitude come only from turning points close to the real axis. This is illustrated in fig. 2, where for the c~ + 58Ni case o f fig. 1 the positions of the turning points leading to different angles 0 are shown. Once the turning points are known for one angle, they can be traced to other angles by A0 = (aO/Oro) Ar o = (a2s/ab2)(db/dro)Aro. Finally for every trajectory found in this way the classical action S (or equivalent the W.K.B. phase shifts) and a2s/ab 2 are needed to construct the scattering amplitude. It should be mentioned that at most two or three trajectories lead to a given angle (counting positive and negative angles separately). Investigations underway on inelastic processes [16] indicate that there is almost 133
Volume 52B, number 2
PHYSICS LETTERS
only one trajectory at each side of the nucleus will contribute, This will even more simplify the method. Summarising, for all cases considered the semiclassical calculations including complex paths reproduce the exact results with considerable accuracy. This is even true in cases where the W.K.B. approximation is generally believed to fail, for instance for 50 MeV protons and c~particles. The oscillations in the cross sections are quantal effects in the sense that they vanish in the strict limit h ~ 0, b u t they are obtained in terms o f purely classical quantities. A semiclassical analysis can lead to a better explanation and a more physical insight into the process as compared to quantal calculation. We thank R. Balian for many discussions and strong encouragement to use complex trajectories.
References [11 K.W. Ford.and J.A. Wheeler, Ann. Phys. NY 7 (1959) 239,287.
134
30 September 1974
[2] R.A. Broglia and A, Winther, Phys. Rev. 4C (1972) 112. [3] D.M. Brink, lecture notes, Orsay, march 1972. [4] R.A. Malfliet, Extended Seminar on Nuclear Physics, I.C.T.P. Trieste, Italy, Sep. 1973. [5] M.V. Berry and K.E. Mount, Rep. Progr. Phys. 35 (1972) 315. [6] J. Stine and R.A. Marcus, Chem. Phys. Lett. (1972)536. [7] J. Knoll and R. Schaeffer, Extended Seminar on Nuclear Physics, I.C.T.P. Trieste, Italy, Sep. 1973, to be published in the proceedings. [8] R. Balian and C. Bloch, Ann. Phys. 63 (1971) 592, and to be published. [9] W.H. Miller, J. Chem. Phys. 53 (1970) 1949. [10] P. Pechukas, Phys. Rev. 181 (1969) 166. [11] V.L. Pokrovskii and I.M. Khalatnikov, J.E.T.P. 40 (1961) 315. [12] M.V. Berry, Proc. Phys. Soc. 89 (1966) 479. [13] R. da Silveira, Phys. Lett. 45B (1973) 211. [14] D.A. Goldberg and S.M. Smith, Phys. Rev. Lett. 29 (1972) 500. [15] J. Raynal, Programme MAGALI, ref.: DPh.T/69-42, Saclay. [16] P.M. Morse and H. Feshbach, Methods of theoretical physics I, Me Graw-Hill, p. 437. [17] J. Knoll and R. Sehaeffer, to be published.