Semiquantal approximations to heavy ion reactions

PHYSICS REPORTS (Section C of Physics Letters) 11, no. 1(1974) 1—28. NORTH-HOLLAND PUBLISHING COMPANY

SEMIQUANTAL APPROXIMATIONS TO HEAVY ION REACTIONS R. A. BROGLIA, S. LANDOWNE, R. A. MALFLIET*, V. ROSTOKIN~and Aa. WINTHER The Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen 0~ Denmark Received July 1973

Contents: 1. 2. 3. 4.

Introduction The semiclassical scattering amplitude Semiclassical treatment of absorption Illustrations

3 4 9 14

5. Conclusion Appendix A Appendix B References

21 21 27 28

Abstract: Generalizations of the semiclassical theory of heavy ion reactions are studied in which the reaction amplitude associated with a given trajectory is interpreted as a partial wave reaction matrix element corresponding to the angular momentum of that trajectory. By summing these amplitudes with the correct quantum mechanical geometrical factors and phases a semiquantal scattering amplitude is obtained which takes proper account of some of the interference and diffraction phenomena in heavy ion reactions. The different role played by the absorption in the quantal and semiclassical descriptions is investigated.

Single orders for this issue PHYSICS REPORTS (Section C of PHYSICS LETTERS) 11, No. 1(1974)1—28. Copies of this issue may be obtained at the price given below. All orders should be sent directly to the Publisher. Orders must be accompanied by check. Single issue price Dfl. 12.50, postage included.

On leave from K.V.1. Nuclear Physics Accelerator Institute, University of Groningen, Holland. j On leave from Moscow Physical Engineering Institute, Moscow, USSR.

*

SEMIQUANTAL APPROXIMATIONS TO HEAVY ION REACTIONS

R.A.BROGLIA, S.LANDOWNE, R.A.MALFLIET, V.ROSTOKIN and Aa.WINTHER The Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen 0, Denmark

I

NORTH-HOLLAND PUBLISHING COMPANY

—

AMSTERDAM

R. A. Broglia etaL, Heavy ion reactions

3

1. introduction A characteristic feature of heavy ion reactions is that the wavelength in the relative motion is very short compared to the nuclear radii or, below the Coulomb barrier, to the distance of closest approach. This allows for a semiclassical description on the reaction [1] where the amplitudes on different reaction channels are evaluated as functions of time. One assumes that the relative position of the centers of mass of the two nuclei is a classically well-defined quantity whose time-dependence is found by solving the classical equations of motion in the combined field of Coulomb repulsion and nuclear attraction. While this is known to be a very good approximation for Coulomb excitation [71 and for reactions somewhat below the Coulomb barrier, the simple semiclassical theory meets difficulties in describing angular distributions for reactions above the Coulomb barrier if the nuclear attraction is strong enough to produce a maximum in the classical deflection function. In this case more than one classical trajectory leads to a given scattering angle and interference phenomena occur. To deal with such situations in elastic scattering, methods have been devised [3, 5], which are classical in the sense of geometrical optics. In the present work we explore such methods by deriving the semiclassical equations of motion from the quantal coupled equations for inelastic processes through the use of WKB-type approximations. This derivation, which has earlier been carried through for the case of Coulomb excitation, is given in Appendix A. It contains a variety of different results which can be used as basis for approximations intermediate between the exact quantum mechanical and the simple semiclassical expressions. In section 2 we discuss an especially useful semiquantal approximation of this type where the quantal partial wave reaction matrix element is substituted by the semiclassical reaction amplitude for a trajectory with an impact parameter corresponding to the partial wave angular momentum. The scattering amplitudes are obtained by adding these amplitudes after they have been multiplied with the proper geometrical factors and phases. In this way the interference and diffraction effects which are especially important wheii the deflection function is non-monotonic are properly taken into account. At the same time, the simplicity of the semiclassical coupled equations is retained, as well as the simple mechanical picture they convey of the reaction. Another characteristic feature of heavy ion reactions is the importance of multistep processes, where several quanta carrying mass, charge, energy and angular momentum are transferred between the ions. For the peripheral collisions where elastic scattering still dominates, the feedback which these processes have on the classical trajectory can in principle not be incorporated accurately. This is because the time at which a givers transition takes place is principally undetermined. The best one can do is to use for each term in the coupled equations a trajectory which is the average of the two trajectories appearing as initial and final states in the corresponding matrix element. This prescription fixes, through the knowledge of energy, angular momentum, masses and charges, the shapes of the trajectories, but leaves the orientation in space undetermined. The derivation given in Appendix A leads to an unambiguous prescription for this orientation. Thus for peripheral collisions where elastic scattering dominates, all trajectories should be chosen with a common symmetry axis determined from the trajectory in the entrance channel. For close collisions the number of quanta exchanged may become so large that the average transfer of mass, energy etc. becomes relatively well defined. In that case an average trajectory can be defined which takes the average dissipation of energy and angular momentum and the average exchange of mass and charge into account in a continuous way. The trajectory corresponding to a definite channel should then be adjusted to approximately osculate the average trajectory at the point where the channel is most strongly populated. It is in this limit of multistep processes that macroscopic dissipation functions like friction have a well-defined meaning, and the actual

4

R. A. Broglia etaL, Heavy ion reactions

values of these quantities can in principle be determined directly from the semiclassical equations of motion. A third characteristic feature of the description of direct reactions between heavy ions is the need of including an absorption to account for those channels, like compound nuclear formation, that cannot, for principal or practical reasons, be included in the calculation. The absorption is most commonly described by introducing a phenomenological imaginary potential, iW, in the equations. In the semiclassical limit the only effect of this potential is an exponential attenuation of all amplitudes proportional to the time integral of W along the trajectory. Quantum mechanically the imaginary potential gives rise to a real as well as an imaginary phase shift. While the diffraction phenomena associated with the onset of the imaginary phase shift are included in the semiquantal description, outlined above, the real phase shift corresponds in the limit of geometrical optics to a reflection. As is discussed in section 3 and illustrated in section 4, we have not been able to include this reflection in a simple way in the determination of the classical trajectory. Actually it may in practice be difficult to distinguish this effect from the effects of a repulsive component of the real ion—ion potential which is expected to appear when the two ions approach distances closer than the sum of the mean square radii of the two nuclei. Also it is felt that the treatment of the absorption in heavy ion collisions in terms of a local, channel independent imaginary potential is unsatisfactory. The long range part of the absorption is due to direct reactions not included in the actual calculations. This part of the absorption may be strongly dependent on spectroscopic details like Q-values of neglected channels. In so far as the neglected channels do not feed back to the original channels, the absorption can be calculated directly from the semiclassical coupled equations by estimating the depopulation of the channels included in the equations. Also the short range part of the absorption leading to the formation of compound nuclei, will depend on the importance of direct, multistep, processes populating states from which fusion is possible.

2. The semiclassical scattering amplitude 1n’a is defined by the asymptotic For a scattering process the partial wave reaction matrix R~1~ form of the channel wavefunction (see Appendix A)

~

(r)

=

~,

a)

~

~

l~)exp~—i~j~(r)} —

Rf~~

exp{i~j~ (r)}

(2.1)

where 5(13, a) isthe Kronecker symbol and S is the channel spin. The entrance channel*, o~,is specified by the orbital angular momentum quantum number i~,and the total nuclear spin I~,while the quantities l~and I~are the corresponding quantum numbers for the exit channel (cf. eq. (All)). The asymptotic phase ~pis given by k~r—r~ ln 2k~r—~-irl~ +~j~

(2.2)

where k~is the wavenumber of relative motion in channel f3. In the definition of the phase (2.2) we have included the phase shift due to the diagonal part of the real potential U~for the interaction between the heavy ions in channel j3. This interaction is the sum of the nuclear ion-ion potential VN and the Coulomb potential VC. The r dependent part of the Coulomb phase shift has been displayed explicitly, i~ being the Coulomb parameter. *

In this paper we use the same notation as in ref. [1] except that we indicate the channels by greek letters.

R. A. Broglia et al., Heavy ion reactions

5

The scattering amplitude for the reaction ct—s’ j3 is related to the reaction matrix by (cf. eq. (A21)) ~ ~/~+

1 ~

~3 1~l~jS X Yi~m~O3,p)

[6(a, i~)t5(l~,l~) exp{i(j3i~j~ + i3l~I~~)} R1~1~ lnI~]~ (2.3) —

The entrance and exit channels are here specified by the total nuclear spins and the corresponding magnetic quantum numbersM0 and M~in a coordinate system in which the incoming wave vector kn is along the z-axis. The polar coordinates s3 and ~pindicate the direction of the wave vector k0 In this laboratory system, Yi~m~being a spherical harmonics. An approximate expression for the reaction matrix R can be obtained by identifying it with a reaction amplitude, c, calculated by means of the semiclassical coupled equations [1]. One finds [4, 6, 7] (see Appendix A) Rl~1~1~10C~M~I0M~ (p, oo) (—1 )1~ -

(2.4)

where c is the solution of the equations ih~J~M~I0M~ (p, t)

=

~ (I0M~IV—iW—U~IL~M> exp~(E~—E7)t}CI~M4I0M~(P~ t)

(2.5)

with the initial condition CJ~M’J0M~ (p, —oc)

=

o(L~,4) ö(M~,M~).

(2.6)

In eqs. (2.4)—(2.6) we have specified the impact parameter p, which should be evaluated in terms of l~by the classical relation l~+~-pk0.

(2.7)

Since the matrix element in (2.5) contains the magnetic quantum numbers M’ the c’s depend on the choice of coordinate system while the R-matrix is rotationally invariant. Eq. (2.4) only holds if (2.5) is solved in the focal coordinate system, A, where the z-axis is chosen perpendicular to the plane containing k0 and k~and the x-axis is the symmetry axis of the trajectory in the entrance channel. The axes should be oriented such that the z-component of ‘n is positive and that x is positive at the distance of closest approach (cf. fig. 2.1). In this coordinate system the magnetic quantum numbers M~and M~of the total spins are defined by S=l~+M~=l~+M~.

(2.8)

One can understand eq. (2.8) by noting that when, in the classical limit, S and 1 are large the difference between them is approximately given by the projection of! on!, I being perpendicular to the plane of the orbit. 1a and relatively large Equation (2.4) is often quite accurate evenItfor low angular momenta wavelengths (see e.g. ref. [6] section II B.6). is therefore indicated that one may obtain an accurate expression for the scattering amplitude (2.3) by evaluating the summations in (2.3) with the prescription (2.4). Further simplifications can be obtained when small angular momenta do not contribute significantly to (2.3). One can then utilize the asymptotic formula (l 1m1XjiIl2m2)ii Dj~12,_~(O,ct, 0) (2.9)

6

R. A. Broglia et al., Heavy ion reactions

with cosa=m1/(11+~-),

0~a~ir

(2.10)

for the Clebsch-Gordan coefficients in (2.3). Furthermore, using eq. (2.8) one can express the summation over l~and S as a summation over ML and M~,i.e. ~ 0Cit l~M~M~

(~

x D~M~

Cj~

2

IaM~ ((la +

~)/k0,

oo)

exp{i(~j~ +

2, ~i~in)}

Y1~,MnM~ ~

~

(2.11) +

XC

A

=

Ziab

Fig. 2.1. A classical trajectory in the combined field of Coulomb repulsion and nuclear attraction, and the êoordinate systems A and C used for the evaluation of the amplitudes. Also shown is the laboratory system where the polar coordinates of the final velocity are ~3and ~,.

We have here left out the first term in (2.3) since we do not consider forward elastic scattering (s~= 0). The strong dependence of the phase shift ~3on 1 has been taken into account to first order in the transfer angular momentum by using the expansion +

(J~ 1~)

=

—

+

~

(M~ M~)O~(la) —

(2.12)

I= ~

where O~(l)= 2 ~3~!’~J3~’

(2.13)

In the classical limit this is the deflection function as can be seen from the WKB approximation for the phase shift

f r0 ~

(kr(r)—ko)dr.

(2.14)

R. A. Brogiva et al., Heavy ion reactions

7

The radial wavenumber kr is given by kr(r)k(r)Pka[l —(l+4)2/k~r2—U

0(r)/E(a)]4,

(2.15)

and one finds 2~’° 2(1+4) ~ dr (216) 2/k~r2 dl k0 ~ r2 ~/i (1 + ~) The quantity r 00 is the classical turning point in channel a, while E(a) is the energy of relative motion in this channel. Assuming that cA has a smooth dependence on ~ the summation over M, and ML in (2.11) can be performed. This summation implies a rotation of the classical amplitude c from the coordinate system A to a system C where the z-axis is along the incoming beam and the x-axis is in the plane of the scattering such that the impact parameter corresponds to positive x (cf. fig. 2.1). We thus obtain the result —

—

(~~) =

• I~M~

~

~M0 -M~~

IaMo

((1

+

4)/k0,

x exp{i(flij~t6110)}

Do)

(~,p).

~M0—M~

(2.17)

This expression is the one used for the numerical calculations presented in section 4. In the remainder of this section we study the expression (2. 17) in the, case where the summation can be carried out by the method of stationary phases. First we utilize the asymptotic relation 11~ I e’ ~ (2.18) which is valid for large values of l and for values of ~ such that

ImI/l<’t9<7r ImI/l (2.19) where mI/i~1. Substituting the summation over lby the Poisson formula [8] as a sum of integrals we obtain

~, ~

I~M~

=

i~2~k~k~ sin

x exp~i(~31j~ + i3ii~)+ i(M0

—

f

~

M~)(cP + +

dl(l +

4)4 CI~M~1

1+

4)/k

0M0((

~)}{exp(_i [(1+ 4) ~

exp(i[(l+4)~+ 2irpl

—

—~+

2~rpl

—

0,

no)

+

(M0

—

M~)

(M0 —M~~)~j)}.(2.20)

Assuming again a smooth behaviour of c as a function of/ one may evaluate (2.20) by the method of steepest descent. We use the expression fdlA(1) exp{iB(l)}

~

A(l)~,/~ex~{i[B(l)+

~i;~~])

(2.21)

where lis determined from the condition dB(1)/dlj1 = 0

(2.22)

8

R. A. Broglia et ci., Heavy ion reactions

while y is given by y(l) = 4 d2B(l)/d12.

(2.23)

Considering the first term in (2.20) the condition (2.22) leads to the relation 0(1) =

—

2irp,

0 < ~1
(2.24)

where 0(l) =4(0~(l)+ 0~(l)),

(2.25)

0~(l)being the deflection function for the channel 3. Since several values of! may satisfy this relation the integral (2.20) may receive contributions from several points of steepest descent 1. The integral may also receive contributions from the second term where (2.22) leads to the relation

(2.26) The result may be written —

1

4 cJ~Mfl,JOMO((1+~)/kO,no)

~

~

~

(2.27)

with s(l)

+1

if0<0(l)+2p(l)ir~9
—1

if—7r <0(l)

-

-

+

2p(l) iT

=

—~

<0.

(2.28)

In this expression the amplitude c is measured in the same coordinate system asf, i.e. in a system where the z-axis is along the incoming beam, while the polar coordinates of the final momentum are ~ and p. The summation should be extended over all those values of 1 which satisfy either (2.24) or (2.26). Note that the result (2.27) can be obtained [12] directly from the semiclassical channel wavefunction of ref. [1] by calculating the phase multiplying the wavepackets ensuring that they have the same phase at t = —°o (see Appendix B). It may happen that the solution of the coupled equations (2.5) leads to amplitudes whose phases depend rather sensitively on 1. Extracting this dependence as an exponential function, one can interpret the phase as an additional phaseshift to be added to ~ In the semiclassical limit (2.27) this would lead to a different deflection function. The additional phaseshift may arise partly from the nondiagonal matrix elements in (2.5) of the total interaction V— iJ4~,or from the diagonal matrix elements of W. The latter effect will be discussed in section 3. It is noted that the inclusion of the phase due to c leads to a stability of the semiclassical scattering amplitude against the particular choice of the potential U~.If instead of U~we would have chosen a central potential (J 13 + LXU to define the phaseshift 3, the coupled equations (2.5) would contain the extra term ~Uc4 0(p, t) on the right-hand side. The solution c’ of these equations can therefore be expressed in terms of the solutions c of (2.5) through the relation c~,0(p,

on)

=

c~,0(p, no) exp~

$

~U(r(t)) dt}.

(2.29)

R. A. Broglia etaL, Heavy ion reactions

9

The phaseshift ~3/J arising from the potential U~+ ~U is, according to (2.14) given by +

=

L?~Udr= I3ij

~~‘r(’)

—

5

~

(2.30)

~U(r(t)) dt

r

00

—

to lowest order in ~U The product c~,0(p, no) exp(!3/10 + pJ~~) appearing in (2.27) is thus independent of z~Uto first order in this quantity. If only one 1 value contributes to (2.27) one obtains the crude semiclassical result for the reaction cross section

(~)

=

2!c~

=

If0

(~,~)l

0(p,00)12

(2.31)

where ~ d&2

(2.32)

pdp sin O~d0~

The result (2.27) is not applicable close to the angle where the deflection function 0(l) has a maximum or a minimum, i.e. 0’ = 0. Such scattering angles are known as rainbow angles. For situations where rainbow scattering occurs several methods [3, 51 have been devised to perform the integrals (2.20). In many cases one may approximate the function 0(l) by a parabola 0(l) = O~+~q(l_/~)2 where 0~= 0’(lR)

=

(2.33)

0. If the amplitudes c vary slowly withl one may evaluate the integrals (2.21)

in terms of the Airy-function [31. The result can be written 9~ ~ fI0M0-. I~Md(5

~ ic 2i9) 121r(1R+—)\z

exP(i~I31RIO+~31RIP~_(lR

~ exP{i~lRI0+

q ~c

~lRI~

—~

—

(iR

1

—I

0~sin

00 ((lR + ~)/k0,

no)

+4)~}Ai(q~(0R—~Y)), +

4) (2OR—

for0R >0for °R <0. (2.34)

0R)),

Ai(q~)

(~—

To this amplitude should be added the contributions of the type (2.27) from other possible branches. This method has been used for analysing heavy ion elastic [15 1 and inelastic [12] scattering data. Other analytic expressions which improve upon the stationary phase method have also been applied [16]. 3. Semiclassical treatment of absorption An imaginary potential may quantum mechanically give rise to three different types of phenomena; namely, absorption, diffraction, and reflection. In the crudest semiclassical theory [1] such a potential only gives rise to a mean free path type of absorption and has no influence on the classical trajectory. When the semiclassical amplitudes are used in the more refined reaction theory given in section 2 above (see e.g. eq.

10

R. A. Broglia et a!., Heavy ion reactions

(2.17)) the absorption may also give rise to diffractive effects. However reflection phenomena are still alien to the theory since wave mechanical reflection corresponds, in a classical picture, to a modification of the orbit. In the present section we especially take up the discussion of this feature of the imaginary potential. The reaction amplitudes entering e.g. in eq. (2.17) are solutions of the coupled equations (2.5). In these equations the monopole part W0(r) of the imaginary potential may be eliminated by the substitution [1] c~,0(p,

t)

=

c~,0(p, t) exP(_~

f

W0 (r(t’)) dt’)

(3.1)

provided that W0 is channel independent. The coupled equations for c’, i.e. ih~,0(p,t)-~ ~ ~

c0(p,t)

(3.2)

are then independent of the monopole part of W. Writing the exponential in (3. 1) as a spatial path integral it is seen that the absorption acts like a position dependent mean free path X(r) proportional to the velocity v(r) i.e. X(r)

=

h v(r)/W(r).

(3.3)

In the semiclassical limit the exponential factor in (3.1), being real, will have no influence on the conditions (2.24)—(2.26) of stationary phases and thus no influence on the deflection function 0(l). In the case where only one impact parameter contributes to a given scattering angle, and where c has a smooth dependence on 1, one finds the cross section

(~)

=

Ic~,0(p,00)12

(3.4)

where (da/d&2)~is the elastic cross section in channel 3, i.e.

(M~

~

f

Wo(r(t))dt}

ex~(_-~

(3.5)

where da,3/d~2is given by (2.32). If the exponential factor in (3.1) is rapidly changing withl the cross section obtained from (2. 1 7) will show diffractive phenomena. The monopole absorption plays the same role in the coupled equations (A25) as in the equations (2.5) provided that the second derivative in (A25) is neglected. One may try to include the effect of the monopole absorption in a similar way as the effect of the average value of V was included in the radial equations (A 12). In this case the equations would read the same but U~ would be substituted by U~ iW~on both sides of the equation. The phaseshift ~ in (A 19) would now be the complex phaseshift due to the potential U~ iW~while hf~18in (A23)--(A25) would be the solution of the homogeneous equation (A22) corresponding to this complex potential. If in equation (A25) we neglect the second derivative we obtain, in the limit of large 1, the coupled equations —

—

~ (I~M~l V— U~—i(W— W~)II~M) exp{*(E~_E7)t) c~,0

(3.6)

R. A. Broglia eta!., Heavy ion reactions

ii

which are formally similar to (2.5). However the full dependence of the scattering amplitude (2.3) on the monopole absorption is now included in the complex phaseshift f~j.Since the real part of this phaseshift, which determines the deflection function, depends on U~and on Wd this description differs fundamentally from the one given in section 2. With the coupled equations (3.6) and the corresponding complex phaseshifts we have a complete prescription for calculating scattering amplitudes including some basic quantal effects of the absorptive potential. Note however that the reflective properties of the absorptive potential is not included in the classical trajectory determining the amplitudes in (3.6). In practice this may give rise to serious discrepancies with a quantal calculation, as shown in section 4 below. To illustrate these features we calculate the complex phaseshift by the WKB method. For the radial SchrOdinger equation 2 1 Ed I ~ + k2(1 V(r) + iW(r))I u(r) = 0 (3.7) Ldr —

where V(r)1(U~(r)+~~~1~) E 2m 0r

(3.8)

and (3.9)

W(r) = Wa(r)/E the WKB solution is

4 exp{/c~ f u(r)

(1

—

(V

—

iW

—

1)4 dr},

r

V+ iW~~ x r>r0~.

sin(k~5(1— V+iW)4dr+~), The lower limit

T~in

(3.10)

the integrals is the complex zero of the integrand i.e.

l—V(P,,~)+iW(F~)=0

(3.11) to

3i~I~ corresponding

while r0~is the classical turning point (V(r0~)= 1). The complex phaseshift the complex potential U~ iW~is then given by

I

—

~

f

=(l~+4)~—k~~~+ k~ (~/l V+ iW— i) dr+k~ —

f

~/i

—

V+iW dr.

(3.12)

rod

For small values of the imaginary potential we can estimate the last term in (3.12). In that case, the complex zero (3.11) lies close to the real turning point, i.e. r~ + i W(ro~)/{V’(rod) iW’(r —

0~)}

(3.13)

where V’ and W’ denote the derivatives of Vand W. The last term in (3.12) can thus be approximated by ~ ~2 kd(iW(rod))2/tlW (rod)— V(rod)J. (.p

(3.14)

12

R. A. Brog!ia etaL, Heavy ion reactions

If we retain only terms of order W, we can write the real and imaginary part of 13 in the form: Re13IdJd ~~(ld Im~dJd

+4)—kdrod

+

kd

5 (~—~(r)— i) dr r~ d

~ k~~

dr.

(3.15)

It is then seen that Rej3 is identical to (2. 14) while the imaginary part can be written

$

ImI3ldId_~.~Th

Wd(r(t))dt.

(3.16)

Inserting (3.16) in (2.3) with the amplitude c’ determined from (3.6), we obtain the same results as by the prescription (3.1) (with real phaseshifts) with the bonus that we can deal with different imaginary potentials in entrance and exit channel. In order to illustrate the role of the absorptive potential we consider the special case of a particle moving in a purely imaginary potential W(r) = S~/r2.This case can be solved analytically. The radial equation

~

+k2_a 2th)U(T)=O

(3.17)

with al(l+1)

(3.18)

and

b

2

=

(3.19)

2moSwIh

can be solved in terms of Bessel functions of complex order. The regular solution is u(r) ~/‘~J~(kr)

(3.20)

with —ir
(3.21)

For kr ~ 1 this solution has the asymptotic form u(r)~cos(kr—~iTviT).

(3.22)

It is noted that the WKB solution (3.10) of equation (3.17) is u(r)

J~/i (a’

=

sin (k

=

sin (s~/(kr)2 (a’

—

—

—

—

ib)/(kr)2 dr +

ib)

~

—\/~‘~12 arc cos (.~~/kr)

+~

(3.23)

where a’(l+4)2.

(3.24)

R. A. Broglia et a!., Heavy ion reactions

13

For kr ~‘ 1 the expression (3.23) coincides with (3.22). The complex phaseshift (3.12) can thus be evaluated explicitly to give Re13~~47r\/4(~J~’2+b2_+a’)+4ir(l+4) Im13=47T\4(~/~ + b2 _a’)*iTbk/~7al2

+

b2

Expanding (3.25) in powers of b it is seen that Re 13

+

a’).

(3.25)

b2 while

Im~3~-~.iTb/(l+4)—O(b3).

(3.26)

The first term in (3.26) is identical to the integral along a straight line trajectory of 4S~/r2in agreement with (3.16). The correction term in (3.26) diminishes the absorption. This is connected with the fact that the imaginary potential also gives rise to some reflection which manifests itself in a real phaseshift. In fact we may compute the deflection function corresponding to this phaseshift. One finds 0(l)=2 dRe~ dl

rL1

i~

41

(327)

~ 2(1 +b2/(l+4)4) ~l +~i~b2/(l+4)4\

J-

The deflection function (3.27) is illustrated on fig. 3.1. It shows that a considerable repulsive

vr~~

2.0

1.5

1.0

05

I

It

~I2

0(1) Fig. 3.1. Deflection function 0(1) given by eq. (3.27) arising from a purely imaginary potential W(r)

=

h2b/(2m 0r2)

14

R. A. Broglia eta!., Heavy ion reactions

effect of the imaginary potential is present when (1 +

4)2

~

h.

4. illustrations In this section we present numerical results for elastic and inelastic scattering calculated by means of the methods discussed in the previous sections. Since the inelastic cross sections are calculated by first order perturbation theory the aim is not to make detailed comparison to experiments. Such analyses are performed more conveniently, and accurately, by available DWBA computer programs. The purpose is partly to gain insight into the physical effects that are implied by different optical potentials commonly used and partly to investigate how reliably one may calculate scattering amplitudes for heavy ion reactions above the Coulomb barrier by means of the semiclassical theory in view of future applications of the theory to more complicated situations where many channels are involved. In the semiclassical description the scattering is governed by the deflection function 0(l). In the combined field of Coulomb repulsion and nuclear attraction three situations can arise [21 which are displayed in fig. 4.1. In case (I) the nuclear attraction is so weak that the deflection function is changed only slightly from the Coulomb deflection function. In case (II) the nuclear attraction is strong enough to provide a maximum °R in the deflection function and thus to a singularity in the classical cross section known as rainbow scattering. In case (III) the nuclear attraction is so strong as to compensate, for a given i-value, l~,the Coulomb plus centrifugal potentials such that orbiting occurs (0 (la) —* —on) The above classification depends on the bombarding energy. Let us consider a situation where the nuclear potential is strong enough to produce a Coulomb barrier, i.e. a maximum in the effective potential for 1 = 0 (cf. fig. 4.2). For low bombarding energy (well below the barrier) the deflection function is of type I. For energies close to the barrier it is of type II; a rainbow angle developing close to l = 0. For a bombarding energy E, slightly above the barrier one will find an i-value where the effective potential has a maximum just at the energy E, and the radial motion stops for infinitely long time giving rise to orbiting, i.e. a deflection function of type III. For increasing bombarding energy the critical angular momentum 10 will shift towards larger values until finally for high bombarding energies orbiting becomes impossible and one is back in case II (or eventually case I). The absorptive potential plays a very different role in the three cases. In the crude semiclassical approximation, the effect of the absorption is given by (3.5). The factor A

=

exp~-

f

W(r(t))

dt~

(4.1)

is illustrated in fig. 4. 1 on horizontal scales. The rather different behaviour of the absorption for the three cases is due to the fact that for stronger nuclear attraction the projectile is pulled deeper into the absorptive potential. This can be seen from the vertical scale marked r0 indicating the distance of closest approach in fermi. The angular distribution for elastic scattering is markedly different for the three cases. In a potential corresponding to case I the crudest semiclassical approximation is in rather good agreement with the corresponding quantum calculation. The cross section (cf. fig. 4.3) is a smooth function of angle or energy and the exponential drop of the cross sections for large angles is governed by the absorption [2]. In case II or III the angular distribution is dominated by the existence of the first rainbow

p(fm)

L~0 \\

7O-~

\

-60

V0=2MeV

r0 =1.65 a5=0.6

W02OMeV r~1.25 a~0.5l.

10 — 0

r0(fm)

60—

~

I

I

I

I

I

I

I

125

7.3

/ I

-10

G(p)

-70

\

-60 10

15

—

V0 7MeV r0

=

1.51 a0

=

0.6

70-

-

rb(fm)

W0=2OMeV r~=1.25a~=0.5~ I

I

I

I

I

~ O(p)

-~c

\

60 10 •

\\

~5t

V02OMeV r~=1.39 a~=0.6 W020MeVr~=1.25 a~=0.54

~40

r0(fm)

60-

~

I

125

~ ~ ~

70-

~

1~5Op55 25

35

65

75

85 °CM

I

I

I

e_kStdt

50— 40

~ ‘~

0.2

~

0.4

06

IL)]

0.8

A

Fig. 4.1. The classical deflection function 0(l) (eq. (2.16)), the classical absorption coefficient (eq. (4.1)) and the quantal reflection coefficient i~(l)are displayed as functions of the angular momentum I for the scattering of 160 on °~Ni at 60 MeV bombarding energy in the laboratory system. Scales corresponding to the impact parameter p and to the distance of closest approach r 0 are also given. The deflection functions were calculated utilizing Saxon-Woods potentials having the same diffuseness, av = 0.6 fm, but different values of the depth lz~,and radius parameter r~.The values are indicated on the corresponding figures. All three potentials have the same tail for large r, which is given by the tail of the standard potential [1, 2]. The same imaginary Saxon-Woods potential was utilized to calculate the absorption and reflection coefficients. The corresponding parameters are W0 = 20 MeV, rs,x~’= 1.25 fm, and aw = 0.54 fm. Again for large values of r, this potential has a tail which is similar to the tail of the standard potential of ref. [1]. The quantal calculations of the reflection coefficients were carried out utilizing the code DWUCK [13]. For case III the deflection function tends to large negative angles, and—0(!) is indicated by a dotted curve.

16

R. A. Brog!ia etaL, Heavy ion reactions

angle in the deflection function. The crudest semiclassical theory breaks down and one should use e.g. the expression (2.34). For angles less than °R two trajectories contribute to each scattering angle and the interference between them produces oscillations in the angular distribution which increase in amplitude and wavelength as one approaches the rainbow angle. For angles

-36

I .30

\

It

I .22.

I1/

Fig. 4.2. Effective potential Veff(r) due to the combined action of the nuclear potential, the centrifugal barrier, and the Coulomb field charge, for different values of!. The nuclear potential is typical for scattering of 160 ions 58Ni for the generated case III ofby fig.a point 4.1. The horizontal lines indicate the centre of mass bombarding energies where a transition on occurs between deflection functions corresponding to the cases I, II, and III displayed in fig. 4.1.

larger than 0R the cross section decreases rapidly in a way that is governed by the curvature of the deflection function. In contrast to case I the steepness of the fall off of the cross section in this region is rather insensitive to the absorptive potential [2]. An overall agreement with experimental data (cf. e.g. ref. [9]) on elastic scattering can be obtained with potentials oftype I. However the oscillations of the cross section present in most data, cannot be accounted for. If one tries to produce oscillations by a rapid onset of absorption one may fit experiments for small angles. However, the large angle drop off of the cross section is then grossly underpredicted. This effect is due to the quantal reflective property of the absorptive potential, which can only be compensated for by an increase in the real attractive potential thus changing the situation into one of type II or III. All experimental data [9] have been accounted for by potentials of this type. For inelastic scattering one expects, for potentials of type I, a smooth angular distribution displaying a single minimum due to the cancellation between the nuclear and Coulomb contribution

Z~~

4+

05.~u

I—I 0

.L

-.~

/ /

=5

I

/

Z 000 ~ .00 0

—

..— ——

2

,7~ H

/—

~I

(0

..—

...

)

‘~ ~

=

+

o~ r

I

i2))

.0

-

-~

.

0

II

.0

(0UJ

.0

‘.0

—

Q

—

0’. —

0) 0—0.0

+

uJ

‘.5

~0 ___

0i~_~

.~

°‘.E

H7 II

II

~~:>

1/

(

0’.

_II -~

~uJ6 w

~w•~ z

___

=

=—

H

/

I-H

~g 0.5~c-.~ 0

;

0

)

II

if

0 0.— 0)

. .-

I.

)

~ 0

,~

-.~-=--

/V~~

If

°

~00

/

I1

=

~Z

0

o

50)

-~ II

~

—

~

~

o

=~

.-

.0 2

0

— (1)6

U

00)

-~

Ui

0

on’. I

0 -‘.

0 0

‘.J(

i.,

0)5)0)

~

o

‘

0) -0

bib b0’~

.0 0

~ ~

~

‘.SUi(,_I 0 0

18

R. A. Broglia eta!., Heavy ion reactions

to the inelastic form factor [21. As is shown in fig. 4,3 this expectation is born out by a quantal calculation at low bombarding energies (40 MeV). At higher bombarding energies (50 MeV and 60 MeV) where the deflection function is still of type I the quantal calculations display oscillations around the classical angular distribution. The origin of the oscillations can be traced to the rapid variation of the nuclear excitation amplitude at the higher bombarding energies which is evident from the corresponding classical angular distribution. This explanation is confirmed by the fact that the elastic angular distribution, also shown in the figure, has a smooth behaviour. For inelastic scattering, in cases where the deflection function is of type II or III, oscillations in the cross section will occur for angles less than °R. where two trajectories contribute. In the “dark” region, i.e. for angles larger than the angular distribution drops off exponentially. The cross section for excitation of the state 13 in the “lit” region (0 < OR) can approximately be written (according to (2.27)) as (4.2) du~ A> c~a(p>, no) exp(iö>) + A< c~0(p<,no) exp(iö<)]2. The labels > and ,no) = c~,0(p<, on) = 1, and persist even if there would be no cancellation in the inelastic formfactor. The competition between the nuclear and the Coulomb contributions is instead reflected in a phase rule [11]. Thus, one may assume that the amplitude c~0(p->, no) is mainly due to Coulomb excitation, while c~0(p<,on) is mainly due to excitation via the nuclear potential. Since these two amplitudes have opposite sign the oscillations present in the inelastic scattering will be 180 degrees out of phase with oscillations in the elastic scattering. In the remainder of this section we shall illustrate the improvements to the crude semiclassical approximation (2.27) that can be obtained by the partial wave expansion (2.1 7). An illustration of the results that can be obtained by the simple stationary phase approximation (2.34) is given in fig. 4.5. The angular distributions obtained by eq. (2.17) for the three cases discussed above are shown in fig. 4.4 together with quantum mechanical calculations. In all calculations only 1 50 partial waves were included, which is enough for the present comparison purposes. The figure shows that the applicability of the semiclassical treatment by eq. (2.1 7) has been extended to all three cases. The discrepancies which still exist between the quantal and semiclassical descriptions are due to the different way in which the absorption is dealt with in the two descriptions. This has been checked by calculating the cross sections without absorption (cf. fig. 4.5). As in seen in fig. 4.4 the semiclassical calculation reproduces especially well the oscillations in the cross section for small scattering angles. In particular the oscillations present in case I, which were discussed in connection with fig. 4.3, are reproduced in detail. The discrepancies for large angles in case I can be diminished by utilizing the improved WKB approximation (3. 12) for calculating the complex phaseshift. It is a striking feature of the quantal calculations that cases II and III give very similar results. However, the semiclassical calculations are markedly different from each other and, in III, do not reproduce the quantum calculations at large angles. The major difference between the semi°R,

*

The energy and angular momentum dependence of the phases 5 are not expected to change the qualitative arguments given below.

R. A. Broglia eta!., Heavy ion reactions

I I

I

I

~

I

19

~T

~

I

I I

.EIic~o+58Ni’\\\~ QUANT. MEd-I.

\

SEMI-CLASSICAL

I

S

II

I

I I

d~ ldQ

~

~ 2*i

fm~

P

I

III

-..+~.-. -±~

~‘.~f\ Ii

I

]I

1

±--~~~

1

III

‘.1

~

58Ni

2’.

INELASTIC °0+ EL/B=6OMeV

-

‘

.

II

\l_,~ I] i] I ‘~IjI [,

I

QUANT. MECH. SEMI- CLASSICAL _______~L..~ L 26 35 2.5 55

~

°CM

55

I 75

85

26

35

I

I

45

55 °CM

I

55

75

55

25

35

45

I 55

65

75

85

°CM

Fig. 4.4. Semiclassical and quantal calculations of the elastic and inelastic scattering of 60 MeV 160 ions on °°Nifor the three different types of potentials discussed in fig. 4.1. The quantal calculations were carried out with the code DWUCK [13], utilizing 150 partial waves. The integration step was chosen equal to 0.1 fm, and the maximum radius was 40 fm. The semiclassical calculations were done with 150 partial waves in eq. (2.17). The amplitudes c were calculated in first order perturbation theory, and the phases d in the WKB approximation (eq. (2.14)). Exact classical trajectories were utilized in working out both the c’s and the d’s.

1.0

~-

ELASTIC

—

-‘

160+°8Ni

+

~QUANT.MECH. ELAB=6OMeV SEMI—CLASSICAL PHASE STATIONARY

da~ I

fm2

‘I

II

-,

/

1 0

—

‘

—

0.1

~ 2~

INELASTIC +

/ .1, 1;”.?

\

\

E~=6OMeV

~

fl’~( (Ic] IJ

.‘

I ]~‘...

\

I

7

I

58Ni

]I[

t

I 1!1.

.

I

—

(I II

I

1

I.!

-

U

-

QUANT. MECH. SEMI-CLASSICAL STATIONARY

I

/

PHASE

I

I 25

I 35

45

I 55 0CM

I~.

65

~I 75

________ _____________ 85

25

35

45

55

65

I 75

I 85

°CM

Fig. 4.5. Elastic and inelastic scattering without absorption. The real potential was that used in cases II and III of fig. 4.4 and the Q-value of the inelastic scattering was set equal to zero. Both semiclassical and quantal calculations were made with 150 partial waves. For case II the results obtained utilizing eq. (2.34) are also indicated by dash—dot curves. The inelastic cross section for case III is multiplied by a factor 1/10.

R. A. Broglia eta!., Heavy ion reactions

21

classical calculations is due to the occurrence of orbiting in case III for partial wave l = 31. For the elastic scattering, the sudden onset of absorption produces sharp oscillations in the angular distribution. These also appear in the inelastic scattering but the effect is magnified by the excitation amplitude for 1 = 3 1, which, because of the long collision time, is an order of magnitude larger than the amplitudes for other partial waves. In the quantal calculation it is apparent that repulsion arising from the imaginary potential eliminates the orbiting phenomena. 5. Conclusion In this paper we have tested different approximate methods to calculate the scattering amplitude for reactions among heavy ions. The reaction mechanism is in all cases described by the classical reaction amplitudes, while the geometry is dealt with in different degrees of approximation. It has been shown that one may in this way carry out quantitative calculations for a wide variety of nuclear potentials, and in particular to include diffractive and interference phenomena. Although we have mainly been concerned with elastic and inelastic processes the method can readily be generalized to describe reactions involving transfer of mass and- charge. The main discrepancies between the quantal calculation and the classical approximations are associated with the imaginary potential commonly used in order to account for the absorption in heavy ion scattering. In particular it remains an open problem how to include the reflective effects associated with imaginary potentials in the calculation of the classical trajectory. The authors would like to acknowledge discussions with F. Videbaek, P. R. Christensen, U. Götz, M. Ichimura and B. S. Nilsson. Appendix A The present derivation of the connection between the quantal reaction matrix for inelastic scattering and the classical excitation amplitude is a generalization of a method given in ref. [7]. A version of this method has been published [4]. We start from the quantal description of an inelastic scattering of heavy ions and shall investigate the approximations which apply when, in the relative motion, the wavelength is small and the angular momentum large. The Hamiltonian for inelastic scattering in the center of mass system is given by (Al)

HHa+HA+TaA+V

where Ha and HA are the intrinsic Hamiltonians of projectile and target, respectively. The total interaction between a and A is denoted by V, while TaA is the kinetic energy of relative motion. We look for stationary solutions of the Schrödinger equation Hlk1a.)

=

Etotlka)

(A2)

with the asymptotic behaviour 1(a)

110M0) expti(k0. r+ r~ln (k0r—k0. r))}

=

2kdr))}. +

~

fp,a (t~, ~)IIdMp)~-exp{i(kdr—i~d ln (

(A3)

22

R. A. Broglia et al., Heavy ion reactions

In (A2) the total energy E101 in the centre of mass system is given by 2k~/2m 2k~/2m = E0 + h 0 Ed + h 0

(A4)

where Ea and Ed are the total nuclear energies of the states llaMa) and II!IMd) in entrance and exit channels respectively while k0 and kd are the relative wavenumbers in these channels, m0 being the reduced mass. The relative position of the centres of mass of a and A is denoted by r. The quantity ~ is the Coulomb parameter ~

fla

(AS)

where Va is the asymptotic relative velocity in channel a, i.e. k0 = mova/h. The scattering amplitude is related to the cross section 2 by (A6) (dOId~Z)a_*d= (ka/kd)lfd, a(t~, p)I where ~ and ~pare the polar coordinates of the vector r. The nuclear state vector llaMa) is related to the state vectors llaMa) and llAMA) of projectile and target by the equation llaMa) = l(1JA)1aMa) =

~

(taMa1i~.M~& 1~M

0)llaMa) llAMA).

(A7)

MaMA

The T-matrix for the reaction between the uncoupled states is (IbMbIBMB

IT]IaMaIAMA) = ~ (JaMaIAMAIIOMOXJbMbIBMB lIdMp)(JdMdlTlIaMO)

(A8)

‘a ‘JO

where

2 2orh (A9) m ~ (~,~0). 0 In order to calculate the scattering amplitude we introduce the radial wavefunctions in the channel spin representation where we write an arbitrary solution of the Schrödinger equation in the form (IJOMJOIT]IaMa)

!i1’)- ~ —

q!JOIJOSN

=

A~ (q) __~~__g(1JOIJO)s(r)l(lpId)SN)

where the channel-spin wavefunction is given by 1JOmJOIJOMJOISN) IIJOMJO) YlJOmJO(1~,Ip). l(lJOIJO)SN) = MJO~rn) (

(AlO)

(All)

3

The radial wavefunction g is independent of the magnetic quantum number N of the channel spin S while the initial condition is labeled by the index q. Inserting (Al 0) in (A2) one finds that the functions g must satisfy the coupled equations

(~______ 2 _ld(lp_+ r2

+

k~ 2m0 Ud(r)) g~~J) s(r) — -~~—

dr

=

((l~l~)SI V—iW— U 8l(L~I7)S)g~7)J)~(r).

(A12)

R. A. Broglia et aL, Heavy ion reactions

23

Since in practice many channels will be left out we have included on the right-hand side a local imaginary potential W. The matrix element of the interaction can be written in terms of the reduced matrix elements of the multipole components V~of the potential V~,= fVY~(?)d~2

(Al3) 2l~+l)(217+ l)(2A+ l)(ld l~ X){Sl~ 17) (_l)s+Ip-X(,llV lI) ~ J( (A 14)

The quantity U)3(r).is the monopole component of the diagonal matrix element of V in channel /3, ~2I~ + 1 ~ (IJOIIVOIII)3) = (I)3M)3I ~ Vd~lI)3M)3). (Al5)

$

The imaginary part W of the potential on the right-hand side of(Al2), for which a similar multipole expansion exists, also contains the monopole component. It is noted that the matrix element in (A 14) can be evaluated in terms of matrix elements of the two nuclei a and A by expanding ~ as V~(~a, ~ r) = ~ (VJ’(~a)VJ(~’A))(J’J)x~if(J’J)x(r). (A16) JJ,

One finds then ((IbIB)I)3Il V~ll(’c’C )I’y)

_____________________ ~/(2X+

(Ic ‘C

l)(21)3+ 1)(2L~+1) ~ JJ’Ir ‘‘b

L,

J

A

r ‘B

T

(IbllVJ’(~a)llIc)UBIlVJ(~A)lIIC)f(J’f)x(r).

i~

(A17) The equations (A 12) have as many regular solutions as the number of states included. The special solution which fulfills the boundary condition (A3) has the asymptotic form g~j~(r)= 6(a, 13)6(1)3, l~)exp{—iIp!)3J)3(r)} —R~ 1~~1 exp{+iIpj.~J)3(r)}

(A18)

where R’~is the reaction matrix in the channel spin representation, and the phase Ip(r) is defined by =

k)3r—fl)3 ln (2k)3r)—~7rl0+13l)3I)3~

(A19)

/3 being the real phaseshift due to the potential U~. The superposition (A 10) which produces the state (A3) from the states (A 18) is given by 2~~1) exp(if3laja) (laOIaMalSMa). (A20) 1

=

o(N, Ma)’~

24

R. A. Brog!ia et aL, Heavy ion reactions

With these coefficients one finds the following expression for the scattering amplitude in terms of the reaction matrix laldS

x (l)3m)3I)3M)3ISMO)

+

l(laOIaMalSMa)

~f3rn)3(~~ 1,0) {6~a, j3) 6(la, 1)3)

~

—

exp{i(j3iaja + 131)3I)3)}

R~J)3lI}.

(A2l)

In the general case the reaction matrix has to be determined by finding a complete set (q) of regular solutions to (Al 2) making a linear combination of these to satisfy (Al 8) in the asymptotic region. We shall here indicate a different method which is useful for small wavelengths. We thus introduce first the incoming and outgoing solutions of the uncoupled equations

~

—

1)3(1)3+ 1)

+

k~

~

LJ)3(r)) h(~~J~)(r) = 0

(A22)

ln (2k)3r) + i3i~i~)}.

(A23)

with the asymptotic behaviour

—4 iil~

h~= exp ~±i(k)3r

—

T1d

We define furthermore two functions c~j)3J)3)s(r)such that g(

1a

~

1)3J)3)5(r)

c~j~s(r)

(A24)

1)3

h~)3J)3)(r)(_1)

are solutions of the coupled equations (A 12) i.e. -~—~

d ln hp~j (r) d

c(~J~) 5(r)+

2

(~JO)—

1717

~

((1JO4)Sl V

—~—

±

C(l)3J)3)S

(r) 1

=

~

—

iW

—

U)3 1(1711 )S)(—l )~d ~ h~7J7)(r)[h~J)3)(r)V -

c( 1717)5(r).

(A25) The solutions are specified by the asymptotic conditions c~J13I)3)S(r)r

~

6(a, 13)6(l~,l~)

(A26)

and c(l)3I)3)s(r)r.

0

(A27)

c(~I)3)s(r).

One thereby obtains a unique solution since one may integrate (A25) for c from r = no with the asymptotic condition (A26) (and dc/dr = 0) towards r = 0 taking at small distances r the value of c and dc/dr as initial conditions for the integration o1 the c~equation. The function g~~s(’~g~J)5(r)

(A28)

—g~J)5(r)

satisfies the asymptotic condition (A 18) provided that the function

C~)3J)3)S(r)

approaches a

R. A. Broglia eta!., Heavy ion reactions

25

constant. Furthermore, since J )(r)

JOJO

r-+ij

h~J)(r)

.

(A29)

.

0)3

the function (A28) is regular at the origin provided that the function c~010)5(r)is well-behaved close to the origin. Under these circumstances Rf,11

=

(A30)

c~0J0)5(r = no)(_l)1d_~.

The necessary conditions for the fulfilment of the above conditions of smoothness need further investigation. They are fulfilled if the wavelength 1/k is so small that one can neglect the second derivative in (A25). In this limit the functions h ±are given by the WKB solutions i.e.

[i

ex~[±i

2

h~0J0)(r)

m 0~i~)dr’

1r(_~_’.j)

+~).

(A3l)

By v0(r) we denote the radial velocity at the distance r v0(r)

=

‘.‘2T

‘1

‘~~

______

—

m0

+

JO

(A32)

2 ~

m0r

where T0(r) is the kinetic energy 7~(r)= E~01 E0 —

—

(A33)

U0(r).

The quantity r00 is the classical turning point in channel /3. Provided that the differences between the classical turning points in the different channels are small one can write the ratio of the functions h~appearing in (A25) in the following way (i~I~)’!) c \]1

(1717)’!) 1 \

(V7 v0(r)\2 ~,‘.V0V7 (r))

2~

x exp The quantity

12 _______

\

where ~

is

—

m~

T vZr)

—

(17

1~(17+1)3

+

1 )h ~

(r))).

(A34)

r

21

(A35)

2 2

m0r

/

the average angular momentum quantum number

~-(l~ + 17).

1)37

—

denotes the average velocity

/ =

T7

V 07 (r)

l)137(r)

~ (rot

(A36)

The quantity r007 is the classical turning point corresponding to the trajectory defined by the radial velocity (A35). We now introduce + —

—

r r r 0137

dr

V07 (r)

(A3 7)

26

R. A. Broglia et aL, Heavy ion reactions

which is the time measured along a trajectory where the radial velocity ~isV07(r). We utilize the classical equation T2~ ~

(1)37

=

1 + ~-)

(A38)

to define the azimuthal angle ~07(t07) in the plane of the orbit for the average trajectory and neglect the difference U0(r) U7(r). Assuming that the matching (A27) is fulfilled already at the classical turning point r0137-the quantity C~(no)can be written —

c(!I)S(r

1717

= no) =

(f

1)3~7((1 dt07(—1)

x exp[i((E0 —E7)t07/h —(17

0l0)Sl V— jW — U01(1717)S) lo) ~07(t07))]

—

c~1j5 (r(t7))~+ C~J0)S(r07).

(A39)

Defining —

C(1 I )S(t)

(C(10J0)S (r(t)),

0

t>

-

—

° 1Cuo1~s(r(t)), t <0 and changing all integration variables to t we find the result

(A40) 10~7

,f

dt 1717 ~ ((14)SIV—iW— UI(17I7)S)(H)

c(~I0)S(no)o’.~- —00

x exp[i((E 0

E7)t/fl —(17 — l~)Ø07(t))] C(lJ)S(t)

—

+

6(la, lo) 6(a, 13).

(A41)

Having a common variable tin (A4 1) the position vector r is different for each term given by the average position in the channels 13 and y at the common time t. A further simplification is achieved when the angular momenta 1~and 17 are so large that one can use the asymptotic formula (2.9) and the relation (1 A I

(_l)2172b)3 ___

(17 S 10

x/217

+

I

/ I

A

I

\

~ I ‘1,1)3 — S 17 — 1)3 S 17/

i

(A42)

—

in evaluating the matrix element (A 14). Using the notation S—ia =M~,

5—17 =M,~,,

171)3

p

(A43)

and formulating (A4 1) as differential equations one finally obtains the result 11 V~,— ~ V ih C(~J0)5(t)= ~ (10M13 0 6(7~.,0)JI7M,~) —

17M,,A.p

x

(_1)M

~

i~,l~)37(t)) exp{i(E0

—E7)t/h} c(IJ)s(t).

(A44)

Utilizing the multipole expansion for V and W, i.e. V—iW= ~

(VxM~iWxM)Yx~(?)

(A45)

2.46

we see that (A44) is identical to eq. (2.5) in the coordinate system A, which proves eq. (2.4).

R. A. Broglia et al., Heavy ion reactions

27

Appendix B In the semiclassical description of nuclear reactions [11 the channel wavefunctions are written in the form 1,11

lI1/,~(~a)L’,~(~A)exp~i(5o

exp(—~E0t) =

0(t)

~

(BI)

where i~i~and ~L’~are the intrinsic wavefunctions of projectile and target nuclei respectively, E0 being the total energy of the two nuclei. The phase 6~is given by 6~ -~(m0v0(t) ra —E(a)t)

+

a0(t)

(B2)

where

4 (f m0(u0(t’))2 dt’ —m0v0(t)

a0(t)

.

(B3)

Ra(t)}

and where E(a) is the total energy in the relative motion E(a)

U0(R0(t))

+

4 m~(v~(t))~.

(B4)

The quantities Ra and ma are the classical relative position vector and the reduced mass in channel a. The classical velocity isv0 (t) R0(t), while Ua(Ro) is the potential energy. As a function of the quantum mechanical centre of mass coordinate ra the channel wavefunction (B 1) is a plane wave where the momentum changes in time. Actually the semiclassical wavefunction is obtained from (B 1) by multiplication with a function which is nonvanishing only along the classical trajectory and which describes the shape of the wavepacket for the relative motion. At times ~ ±no the phase a0 becomes constant and the difference a0 (no) — a0 (— no) signifies the phaseshift between the incoming plane wave and the plane wave scattered at the angle 0(p) where p is the impact parameter corresponding to the classical trajectory. The total phase change can be written in the form (t)

2a0

~2a0(oo)a0(oo)_a0(—oo) =

~[La(~ 0(p)) — 2m0V0R0(0)

=

—

—

f

~

m0R0

(t)

.

7L~

(t)

dt

~

+

5 f

(m0k0(t)

F0

.

—

m0V0)V0(t)

dt~

R0 dt

(B5)

where L0 is the angular momentum L0

=

pm0 V0

(B6)

and V0 the velocity at large distances (V0 V0(no)), while R0 = 1R0 I and F0 = —PU0. The connection between the total phase change (B5) and the phaseshift (2.14) in the WKB approximation is seen to be ~

‘I3loIa

—-~.(l~ +~)O(l)

(B7)

28

R. A. Broglia et al., Heavy ion reactions

if we substitute L0 by the quantity (l~+ 4) h. While /3~is the radial phaseshift, the second term in (B7) may be interpreted as the angular phaseshift.* In the crude semiclassical description of the reaction the phase change a0 is irrelevant. However, if several different trajectories lead to the same scattering angle the reaction amplitudes c0(p, no) should be multiplied by the phase factor exp{2ict0}. Furthermore they should be weighted with the square root of the classical elastic cross section for the different impact parameters, i.e. with -

____

sin 0(p) (dO(p)/dp)

B8

before they are summed and squared to give the reaction cross section. With an appropriate choice of the complex sign of the square-root in (B8) one arrives directly at the result (2.27), since the total phase change in a reaction is composed of the entrance channel phaseshift up to time 0 and the exit channel phaseshift from ~ 0, onwards. This derivation indicates that the result (2.17) can be used not only for elastic and inelastic scattering but also for nuclear reactions where mass and charge are transferred. References Broglia and A. Winther, Nucl. Phys. Al 82(1972)112; Broglia and A. Winther, Phys. Reports 4C (1972) 153. [2] R.A. Broglia, S. Landowne and A. Winther, Phys. Lett. 40B (1972) 293. [3] K.W. Ford and J.A. Wheeler, Annals of Phys. 7 (1959) 259. [1]

R.A. R.A.

[41 K. Alder and

H.K.A. Pauli, NucI. Phys. A128 (1969) 193. MV. Berry and K.E. Mount, Rep. Prog. Phys. 35 (1972) 315. K. Alder, A. Bohr, 1. Huus, B. Mottelson and A. Winther, Rev. Mod. Phys. 28 (1956) 432. K. Alder and A. Winther, Electromagnetic Excitations (North-Holland, Amsterdam) in print. P. Morse and H. Feshbach, Methods of Mathematical Physics (McGraw Hill, New York, 1953). R. Siemssen, in: Nuclear Spectroscopy II, ed. J. Cerny (Academic Press, New York) in preparation. A. Winther and J. de Boer, in: Coulomb Excitation, eds. K. Alder and A. Winther (Academic Press, New York, 1966). R. Malfliet, S. Landowne andY. Rostokin, Phys. Lett. 44B (1973) 238. R. Maifliet, Contribution to: Symposium on Heavy Ion Transfer Reactions, Argonne Natl. Lab., March 1973. PD. Kunz and B. Nilsson, private communication. M. Samuel and U. Smilansky, Computer Phys. Comm. 2(1971)455. ML. Halbert and A. Zucker, Nucl. Phys. 16 (1960) 158; A. Zucker, Ann. Rev, of Nod. Science 10(1960) 27; B.N. Kalinkin, T.P. Kochkina and B.I. Pustylnik, Reactions Between Complex Nuclei, Asilomar 1963, eds. A. Ghiorso, R.M. Diamond and HE. Conzett (University of California Press, 1963) p. 69. [16] R. da Silveira, Phys. Lett. 45B (1973) 211; R. da Silveira and Ch. Leclercq — Willain, preprint IPNO/TH 73—52. [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

*

Actually the angular phaseshift should be l(1 + 1)(ir —0), but the term 1(1 + 4)ir is customarily included in the radial phaseshift.