- Email: [email protected]
The role of the giant resonances in deep inelastic collisions between heavy ions
Volume 61B, number 2
PHYSICS LETTERS
THE ROLE OF THE GIANT RESONANCES
15 March 1976
IN
DEEP INELASTIC COLLISIONS BETWEEN HEAVY IONS
R.A. BROGLIA, C.H. DASSO and Aa. WINTHER The Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen O, Denmark Received 30 January 1976 A unified description of heavy ion reactions which includes aU the most important nuclear degrees of freedom in an average way is attempted, in terms of the semiclassical coupled equations. In heavy ion collisions one may distinguish two characteristic times. The first is the time interval 7"coll during which the two nuclei are in contact i.e. •
~
o)11,
1--E~cM
MeV -1 ,
(1)
where Elab/A is the bombarding energy per nucleon in MeV, while EB/EcM is the ratio of the Coulomb barrier energy to the center of mass energy. For collisions not too far above the Coulomb barrier this time is of the order o f 0 . 5 - 1 MeV -1 . This time is important for the strength of the interaction, which is measured by X ~ (rcoll/h) f ( b ) ,
(2)
where f ( b ) is the formfactor in MeV at the distance of closest approach b (cf. ref. [1] eq. (2.54) ft.). The second one, which measures the degree o f adiabaticity of the process, is associated with the time interval rchar over which the formfactor changes by a factor of two. For grazing collisions rchar and '/'coil coincide. For reactions taking place at shorter distances where the formfactor is not monotonically increasing rchar may however become an order of magnitude smaller than rcoll. The quantum transition that can take place during the collision are thus limited adiabatically to have an energy difference A E ~ h/rchar , which for energies not too high above the Coulomb barrier corresponds to 1 0 - 2 0 MeV. Since this number is of the same order of magnitude as the Fermi energy, a description o f the collision process in terms of an adiabatic two-center shell model is not very appropriate. It is thus natural to treat the reaction in terms o f the degrees of freedom of the two colliding nuclei. In view of the large strength parameters (2) this should be
done in a coupled channel treatment as indicated in ref. [1 ]. A simplified formulation of this approach in which the main average features of the reaction can be calculated was described in ref. [2]. The realization that the nuclear transitions are limited by rchar instead of 7coli , implies that states up to 20 MeV in the two nuclei may be important. In view of the fact that a major fraction of the sum rule for inelastic scattering lies between 1 0 - 2 0 MeV associated with giant resonances, one expects these states to play a decisive role in deep inelastic processes. To investigate this, we have endevoured in a calculation along the lines of ref. [2] including all the different inelastic degrees of freedom and realistic formfactors, leaving out again transfer processes. The different states of the nuclear spectrum were described as harmonic vibrations including multiple excitations. In this case, the average result of the quantal calculation coincides with the completely classical calculation. In such calculations one can include the fact that the oscillator strength of a given multipolarity ~, is spread over an energy interval F x by introducing a damping term in the equation of motion for this mode Bx&x u + Cxax u + 7x&xu =fxu(r(t)),
(3)
where fxu(r(t)) is the inelastic formfactor. The coefficients B x and C x are the mass parameter and restoring force respectively of the mode of either target or projectile nucleus. We have used the linear approximation in the amplitudes axu , thus limiting the interaction to be of monopole-multipole type with the following expression for the formfactor (e.g. for target excitation)
fxu(r(t)) -- R h(0) ~U(r(t)) ..... + fCE(r(t)) ~r(t) .xXu~r(t))
(4) 113
Volume 61B, number 2
PHYSICS LETTERS
15 March 1976
Table 1 Level scheme and oscillator strengths used to describe the six nuclei involved in the reactions displayed in fig. 1. The energy ~co h and the percentage of the isoscalar energy weighted sum rule i.e. R h = hcohB(Eh)/S(Eh), where S(EX) = (k(2~.+ 1)2)/4n) (~2/2M) X (Z2e 2/A) (rZh-2), are given for each state of spin h and parity ( - 1 ) h. The numbers given in (a) are taken either from experiment or systematics. These modes and their multiple excitations describe the structure of the corresponding nuclei at low excitation energies. The numbers in (b) were obtained as discussed in the text and describe, together with their multiple excitations, the average response of the nucleus, even at high excitation energies. O
Ni
~co h
(MeV)
h
R~
?'ltox
(%)
(MeV)
6.9 6.1 8.1
2 15 3 13 5 3
23.0
2 80
0.7 2.3 2.4 4.7 4.8 6.0 10.6
Ar Rk
t~ h
(%)
(MeV)
7~coh
(%)
(MeV)
1.5 2.8 3.7
2 4 3
9 2 9
4.3 15.0
5 3 2 80
4.5 17.0
5
1
2.2 4.8 2.8 3.1 10.4 13.1
2 13 5 3 3 18 4 3 6 2 2 62
2.6 3.2 4.1 4.3 5.7 9.8 14.0
3 5 2 4 6 2 8
14 2 18 18 1 60 20
at0 tion. The q u a n t i t y R(A() is the radius o f the target while U ( r ) is the ion-ion p o t e n t i a l for which we have everywhere used the p r o x i m i t y p o t e n t i a l o f ref. [3]. The damping coefficient is c o n n e c t e d w i t h the spreading width F x b y the relation 7x = 2BxFx/PL T h r o u g h eq. (3) we can describe in an accurate way the energy and angular m o m e n t u m loss to m a n y nuclear states if the total strength B(EX; 0 --> ~) h
(5)
2x/~xB x ,
has a Gaussian distribution of width F x, centered at the energy hcox= h(Cx/Bx) 1/2. This was checked numerically by splitting one harmonic oscillator state with damping, into ten harmonic oscillator states without damping according to the prescription above. Agreement between the resulting energy and angular momentum of the two calculations was found to be better than 114
3 ( 5 2 ( 4
Rh
~co h
(%)
(MeV)
20 20 80 20
X
8.6
17.2
3 {5 2 4 6
20 20 8O 20 5
Rh
hco~.
(%)
(MeV)
h
R~
(%
10.6
( 53
20 20
12.0
3 ( 5
20 20
21.3
2 4 6
80 20 5
24.0
2 4 6
80 20 5
6.9
(5
9.3
18.6
182
3
20
{5
20
I 2 4
80 20
8
2
[65
Kr
3 2
13.8
20 2o
4
80 20
8
2
[6,
Bi
Co)
where f ~ ,E(r(t)) is the formfactor for Coulomb excita-
)2(R(A0))2X
32.5
Ag
(a)
(3
16.3
k
Ar
2 90
Ar
= (2X+l) ~-~nZAe
Ni
Rh
10 3 10 4 3 2
Kr
Ag
h
4.5 2.5
1.5
2 13 5 2 3 7 6 2 8 4 4 8 2 64
h
O
5% even for Px ~ 0.7 h cox. In figure 1 we present the results of the calculations done with the present model. The characteristic feature of these results is the clear distinction that can be made between the grazing collisions and the deep inelastic collisions. The results for the quasielastic grazing collisions are sensitive to the details of the low-lying spectrum of the two nuclei. The calculation were here performed utilizing the levels indicated in table l(a), and with an energy dependent damping strongly suppressing the multiple excitation of the lowest states. The calculations for the deep inelastic collisions are insensitive to these low lying levels but are controlled by the main features of the distribution of strength in the two nuclei. In these collisions, where each nucleus absorbes an energy of 3 0 - 1 0 0 MeV, the correlations responsible for the existence of low-lying collective states vanish, so that only the strength distribution associated with the unperturbed particle-hole excitation are expected to remain. In the calculations we have here used the levels given in table l(b) with a width F x
Volume 61B, number 2
80
PHYSICS LETTERS
ecM
ec~
0 °Ni
15 March 1976
Ar • Ag
ecM
60
40
Kr
÷ Bi
60 120
I
,
20
I
/ il
40
60
80
20
~_
[ 200
100
40
[
I
-20
12°f
EcM (MeV)
384
120 IvleV.
8° I
f
L
60
/
525 MeV
300
100 k
725 Mev
,
!
40O
/
60 MeV
40
]-
625 MeV
288 MeV
./~'\
ZOO
20
(MeV)
600
500
MeV
336 MeV
F
90 MeV
~
I
400
/
f ....
o ..................
f
EcM
EcM (MeV)
}00 i
__1
200
,
[
I
i
100
200
[f
//
200 -
I
=
I
600
/
[
I
400
200 [(
600
L
:
400
/
f
100
200"
~
I
// / I
t
100
200
_ _
[
200
400
I
600
t
Fig. 1. Average deflection functions, fmal energy and final angular momenta for the reactions O + Ni, Ar + Ag and Kr + Bi. The f'trst row of figures show the average deflection functions. The dotted, broken and full drawn curves correspond to three different bombarding energies as indicated in the second row of figures, where the average final energy in the center of mass system is plotted as a function of the angular momentum l in the entrance channel. In the last row we indicate with the same type of curves the associated angular momentum of relative motion If in the exit channel, also as a function of l. The thin 45 ° line indicates no loss of angular momentum. The curve for the reaction O + Ni was not included because it shows no significant deviation from this line.
equal to half of the excitation energy for the 2 + states and equal to the excitation energy for the other states. The average excitation energy (= h6ah) of a given multipolarity was taken to be AN(41[A1/3 MeV) with AN= 1 or 2, while the widths were estimated from the strength distribution (response function) of unperturbed particle-hole excitations. The isovector modes were neglected since the isovector part of U is much smaller than the isoscalar part. We view the results given in fig. 1 as displaying the main features of heavy ion reactions. The present calculation was carried out using basically the full multiple
particle-hole spectrum and realistic formfactors. In this situation the inclusion of the neglected transfer processes is not expected to modify the picture in any essential way, as the simple addition of the transfer degrees of freedom would imply double counting. Transfer channels can be introduced mainly at the cost of inelastic processes and vice-versa, as discussed in ref. [1 ], in connection with the problems of non-orthogonality. Although the results of fig. 1 display many features observed experimentally, a detailed comparison with the data would require the inclusion of the transfer channels, as well as of the subsequent decay of the reaction products. 115
Volume 61B, number 2
PHYSICS LETTERS
The main feature of fig. 1 is the classification of heavy ion reactions into four regimes. The first corresponds to compound nucleus formation at low relative angular momenta, where the two nuclei remain within the interaction region for a time that is expected to be long enough to establish statistical equilibrium. The second is that of strongly damped (deep inelastic) collisions where the two nuclei come apart within a time 7coll having lost a major part of the relative kinetic energy. The deflection function in this region shows a concentration of cross section at an angle similar to the grazing angle. The calculations show, especially at high bombarding energies, that a sizeable part of the cross section correspond to reactions where the outgoing particle leaves the interaction region with an energy well above the Coulomb barrier. In all cases the loss of angular momentum is less than 3 0 - 4 0 units of h. The third regime corresponds to impact parameters between grazing collisions and deep inelastic collisions where the two nuclei stay within the range of the interaction for a period long compared to 7co11. In general, the angular momentum of this composite system is well above the critical angular momentum for fission, which is thus expected to be the main mode of decay. Experimentally this regime may be distinguished from the fission of the compound system in the first regime by the high angular momentum of the fragments and distinguished from the deep inelastic by the same feature as well as by the expected 1/sin 0 angular distribution. The fourth regime corresponds to the quasi-elastic processes. In the first three regimes the quadrupole giant resonance and the AN= 1 and 2 particle-hole excitations of higher multipolarities play a decisive role in the mechanism of energy absorption. Since on the average at most one or two quanta of each of these modes is excited, the energy spread around the average energy loss
116
61B, number 2
is large. Preliminary estimates indicates energy spreads of the order of magnitude of 20-30%. Insofar as the boundary separating the different regimes is determined mainly by the energy loss, this spread in excitation energy would reflect itself in a spread of the boundary angular momentum. This would for instance lead to a rather smooth angular momentum distribution of the compound nucleus formation, as opposed to the sharp triangular distributions obtained with small fluctuations. The inclusion of transfer reactions will imply a flow of mass and charge in e.g. the deep inelastic reaction, the direction of which can be estimated from the condition of optimum Q-value [1]. Thus the transfer processes populate, on the average, states of neighbouring nuclei where the quantity E~ + U;~- ~ m~v 2 is conserved, E#, U#, m# and o being the intrinsic energy, the ion-ion potential, the reduced mass and the relative velocity of the ions respectively. Since the level density increases with E#, the mass and charge flow between the two ions will tend to diminish U¢ and increase m# [4]. For reactions of lighter projectiles on heavier targets the outgoing scattered particle will thus tend to have a smaller charge with a tendency towards heavy isotopes, which becomes more pronounced for higher energies. We would like to acknowledge the help of P.F. Bortignon.
References [1 ] R.A. Broglia and A. Winther, Phys. Rep. 4C (1972) 153. [2] R.A. Broglia, C.H. Dasso and A. Winther, Phys. Lett. 53B (1974) 301. [3] J. Randrup, W.J. Swiatecki, C.F. Tsang, LBL reprint 3603, Dec. 1974. [4] J.P. Bondorf et al., J. de Phys. 32C6 (1971) 145.
PHYSICS LETTERS
THE ROLE OF THE GIANT RESONANCES
15 March 1976
IN
DEEP INELASTIC COLLISIONS BETWEEN HEAVY IONS
R.A. BROGLIA, C.H. DASSO and Aa. WINTHER The Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen O, Denmark Received 30 January 1976 A unified description of heavy ion reactions which includes aU the most important nuclear degrees of freedom in an average way is attempted, in terms of the semiclassical coupled equations. In heavy ion collisions one may distinguish two characteristic times. The first is the time interval 7"coll during which the two nuclei are in contact i.e. •
~
o)11,
1--E~cM
MeV -1 ,
(1)
where Elab/A is the bombarding energy per nucleon in MeV, while EB/EcM is the ratio of the Coulomb barrier energy to the center of mass energy. For collisions not too far above the Coulomb barrier this time is of the order o f 0 . 5 - 1 MeV -1 . This time is important for the strength of the interaction, which is measured by X ~ (rcoll/h) f ( b ) ,
(2)
where f ( b ) is the formfactor in MeV at the distance of closest approach b (cf. ref. [1] eq. (2.54) ft.). The second one, which measures the degree o f adiabaticity of the process, is associated with the time interval rchar over which the formfactor changes by a factor of two. For grazing collisions rchar and '/'coil coincide. For reactions taking place at shorter distances where the formfactor is not monotonically increasing rchar may however become an order of magnitude smaller than rcoll. The quantum transition that can take place during the collision are thus limited adiabatically to have an energy difference A E ~ h/rchar , which for energies not too high above the Coulomb barrier corresponds to 1 0 - 2 0 MeV. Since this number is of the same order of magnitude as the Fermi energy, a description o f the collision process in terms of an adiabatic two-center shell model is not very appropriate. It is thus natural to treat the reaction in terms o f the degrees of freedom of the two colliding nuclei. In view of the large strength parameters (2) this should be
done in a coupled channel treatment as indicated in ref. [1 ]. A simplified formulation of this approach in which the main average features of the reaction can be calculated was described in ref. [2]. The realization that the nuclear transitions are limited by rchar instead of 7coli , implies that states up to 20 MeV in the two nuclei may be important. In view of the fact that a major fraction of the sum rule for inelastic scattering lies between 1 0 - 2 0 MeV associated with giant resonances, one expects these states to play a decisive role in deep inelastic processes. To investigate this, we have endevoured in a calculation along the lines of ref. [2] including all the different inelastic degrees of freedom and realistic formfactors, leaving out again transfer processes. The different states of the nuclear spectrum were described as harmonic vibrations including multiple excitations. In this case, the average result of the quantal calculation coincides with the completely classical calculation. In such calculations one can include the fact that the oscillator strength of a given multipolarity ~, is spread over an energy interval F x by introducing a damping term in the equation of motion for this mode Bx&x u + Cxax u + 7x&xu =fxu(r(t)),
(3)
where fxu(r(t)) is the inelastic formfactor. The coefficients B x and C x are the mass parameter and restoring force respectively of the mode of either target or projectile nucleus. We have used the linear approximation in the amplitudes axu , thus limiting the interaction to be of monopole-multipole type with the following expression for the formfactor (e.g. for target excitation)
fxu(r(t)) -- R h(0) ~U(r(t)) ..... + fCE(r(t)) ~r(t) .xXu~r(t))
(4) 113
Volume 61B, number 2
PHYSICS LETTERS
15 March 1976
Table 1 Level scheme and oscillator strengths used to describe the six nuclei involved in the reactions displayed in fig. 1. The energy ~co h and the percentage of the isoscalar energy weighted sum rule i.e. R h = hcohB(Eh)/S(Eh), where S(EX) = (k(2~.+ 1)2)/4n) (~2/2M) X (Z2e 2/A) (rZh-2), are given for each state of spin h and parity ( - 1 ) h. The numbers given in (a) are taken either from experiment or systematics. These modes and their multiple excitations describe the structure of the corresponding nuclei at low excitation energies. The numbers in (b) were obtained as discussed in the text and describe, together with their multiple excitations, the average response of the nucleus, even at high excitation energies. O
Ni
~co h
(MeV)
h
R~
?'ltox
(%)
(MeV)
6.9 6.1 8.1
2 15 3 13 5 3
23.0
2 80
0.7 2.3 2.4 4.7 4.8 6.0 10.6
Ar Rk
t~ h
(%)
(MeV)
7~coh
(%)
(MeV)
1.5 2.8 3.7
2 4 3
9 2 9
4.3 15.0
5 3 2 80
4.5 17.0
5
1
2.2 4.8 2.8 3.1 10.4 13.1
2 13 5 3 3 18 4 3 6 2 2 62
2.6 3.2 4.1 4.3 5.7 9.8 14.0
3 5 2 4 6 2 8
14 2 18 18 1 60 20
at0 tion. The q u a n t i t y R(A() is the radius o f the target while U ( r ) is the ion-ion p o t e n t i a l for which we have everywhere used the p r o x i m i t y p o t e n t i a l o f ref. [3]. The damping coefficient is c o n n e c t e d w i t h the spreading width F x b y the relation 7x = 2BxFx/PL T h r o u g h eq. (3) we can describe in an accurate way the energy and angular m o m e n t u m loss to m a n y nuclear states if the total strength B(EX; 0 --> ~) h
(5)
2x/~xB x ,
has a Gaussian distribution of width F x, centered at the energy hcox= h(Cx/Bx) 1/2. This was checked numerically by splitting one harmonic oscillator state with damping, into ten harmonic oscillator states without damping according to the prescription above. Agreement between the resulting energy and angular momentum of the two calculations was found to be better than 114
3 ( 5 2 ( 4
Rh
~co h
(%)
(MeV)
20 20 80 20
X
8.6
17.2
3 {5 2 4 6
20 20 8O 20 5
Rh
hco~.
(%)
(MeV)
h
R~
(%
10.6
( 53
20 20
12.0
3 ( 5
20 20
21.3
2 4 6
80 20 5
24.0
2 4 6
80 20 5
6.9
(5
9.3
18.6
182
3
20
{5
20
I 2 4
80 20
8
2
[65
Kr
3 2
13.8
20 2o
4
80 20
8
2
[6,
Bi
Co)
where f ~ ,E(r(t)) is the formfactor for Coulomb excita-
)2(R(A0))2X
32.5
Ag
(a)
(3
16.3
k
Ar
2 90
Ar
= (2X+l) ~-~nZAe
Ni
Rh
10 3 10 4 3 2
Kr
Ag
h
4.5 2.5
1.5
2 13 5 2 3 7 6 2 8 4 4 8 2 64
h
O
5% even for Px ~ 0.7 h cox. In figure 1 we present the results of the calculations done with the present model. The characteristic feature of these results is the clear distinction that can be made between the grazing collisions and the deep inelastic collisions. The results for the quasielastic grazing collisions are sensitive to the details of the low-lying spectrum of the two nuclei. The calculation were here performed utilizing the levels indicated in table l(a), and with an energy dependent damping strongly suppressing the multiple excitation of the lowest states. The calculations for the deep inelastic collisions are insensitive to these low lying levels but are controlled by the main features of the distribution of strength in the two nuclei. In these collisions, where each nucleus absorbes an energy of 3 0 - 1 0 0 MeV, the correlations responsible for the existence of low-lying collective states vanish, so that only the strength distribution associated with the unperturbed particle-hole excitation are expected to remain. In the calculations we have here used the levels given in table l(b) with a width F x
Volume 61B, number 2
80
PHYSICS LETTERS
ecM
ec~
0 °Ni
15 March 1976
Ar • Ag
ecM
60
40
Kr
÷ Bi
60 120
I
,
20
I
/ il
40
60
80
20
~_
[ 200
100
40
[
I
-20
12°f
EcM (MeV)
384
120 IvleV.
8° I
f
L
60
/
525 MeV
300
100 k
725 Mev
,
!
40O
/
60 MeV
40
]-
625 MeV
288 MeV
./~'\
ZOO
20
(MeV)
600
500
MeV
336 MeV
F
90 MeV
~
I
400
/
f ....
o ..................
f
EcM
EcM (MeV)
}00 i
__1
200
,
[
I
i
100
200
[f
//
200 -
I
=
I
600
/
[
I
400
200 [(
600
L
:
400
/
f
100
200"
~
I
// / I
t
100
200
_ _
[
200
400
I
600
t
Fig. 1. Average deflection functions, fmal energy and final angular momenta for the reactions O + Ni, Ar + Ag and Kr + Bi. The f'trst row of figures show the average deflection functions. The dotted, broken and full drawn curves correspond to three different bombarding energies as indicated in the second row of figures, where the average final energy in the center of mass system is plotted as a function of the angular momentum l in the entrance channel. In the last row we indicate with the same type of curves the associated angular momentum of relative motion If in the exit channel, also as a function of l. The thin 45 ° line indicates no loss of angular momentum. The curve for the reaction O + Ni was not included because it shows no significant deviation from this line.
equal to half of the excitation energy for the 2 + states and equal to the excitation energy for the other states. The average excitation energy (= h6ah) of a given multipolarity was taken to be AN(41[A1/3 MeV) with AN= 1 or 2, while the widths were estimated from the strength distribution (response function) of unperturbed particle-hole excitations. The isovector modes were neglected since the isovector part of U is much smaller than the isoscalar part. We view the results given in fig. 1 as displaying the main features of heavy ion reactions. The present calculation was carried out using basically the full multiple
particle-hole spectrum and realistic formfactors. In this situation the inclusion of the neglected transfer processes is not expected to modify the picture in any essential way, as the simple addition of the transfer degrees of freedom would imply double counting. Transfer channels can be introduced mainly at the cost of inelastic processes and vice-versa, as discussed in ref. [1 ], in connection with the problems of non-orthogonality. Although the results of fig. 1 display many features observed experimentally, a detailed comparison with the data would require the inclusion of the transfer channels, as well as of the subsequent decay of the reaction products. 115
Volume 61B, number 2
PHYSICS LETTERS
The main feature of fig. 1 is the classification of heavy ion reactions into four regimes. The first corresponds to compound nucleus formation at low relative angular momenta, where the two nuclei remain within the interaction region for a time that is expected to be long enough to establish statistical equilibrium. The second is that of strongly damped (deep inelastic) collisions where the two nuclei come apart within a time 7coll having lost a major part of the relative kinetic energy. The deflection function in this region shows a concentration of cross section at an angle similar to the grazing angle. The calculations show, especially at high bombarding energies, that a sizeable part of the cross section correspond to reactions where the outgoing particle leaves the interaction region with an energy well above the Coulomb barrier. In all cases the loss of angular momentum is less than 3 0 - 4 0 units of h. The third regime corresponds to impact parameters between grazing collisions and deep inelastic collisions where the two nuclei stay within the range of the interaction for a period long compared to 7co11. In general, the angular momentum of this composite system is well above the critical angular momentum for fission, which is thus expected to be the main mode of decay. Experimentally this regime may be distinguished from the fission of the compound system in the first regime by the high angular momentum of the fragments and distinguished from the deep inelastic by the same feature as well as by the expected 1/sin 0 angular distribution. The fourth regime corresponds to the quasi-elastic processes. In the first three regimes the quadrupole giant resonance and the AN= 1 and 2 particle-hole excitations of higher multipolarities play a decisive role in the mechanism of energy absorption. Since on the average at most one or two quanta of each of these modes is excited, the energy spread around the average energy loss
116
61B, number 2
is large. Preliminary estimates indicates energy spreads of the order of magnitude of 20-30%. Insofar as the boundary separating the different regimes is determined mainly by the energy loss, this spread in excitation energy would reflect itself in a spread of the boundary angular momentum. This would for instance lead to a rather smooth angular momentum distribution of the compound nucleus formation, as opposed to the sharp triangular distributions obtained with small fluctuations. The inclusion of transfer reactions will imply a flow of mass and charge in e.g. the deep inelastic reaction, the direction of which can be estimated from the condition of optimum Q-value [1]. Thus the transfer processes populate, on the average, states of neighbouring nuclei where the quantity E~ + U;~- ~ m~v 2 is conserved, E#, U#, m# and o being the intrinsic energy, the ion-ion potential, the reduced mass and the relative velocity of the ions respectively. Since the level density increases with E#, the mass and charge flow between the two ions will tend to diminish U¢ and increase m# [4]. For reactions of lighter projectiles on heavier targets the outgoing scattered particle will thus tend to have a smaller charge with a tendency towards heavy isotopes, which becomes more pronounced for higher energies. We would like to acknowledge the help of P.F. Bortignon.
References [1 ] R.A. Broglia and A. Winther, Phys. Rep. 4C (1972) 153. [2] R.A. Broglia, C.H. Dasso and A. Winther, Phys. Lett. 53B (1974) 301. [3] J. Randrup, W.J. Swiatecki, C.F. Tsang, LBL reprint 3603, Dec. 1974. [4] J.P. Bondorf et al., J. de Phys. 32C6 (1971) 145.