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Direct and dissipative modes in heavy-ion reactions: Incomplete deep inelastic collisions
Volume 119B, number 4,5,6
PHYSICS LETTERS
23/30 December 1982
DIRECT AND DISSIPATIVE MODES IN HEAVY-ION REACTIONS: INCOMPLETE DEEP INELASTIC COLLISIONS L. OEHME and R. REIF Technische Universitdt Dresden, Sektion Physik, 8027 Dresden, Mom msenstrasse 13, DDR
Received 6 July 1982 Revised manuscript received 7 September 1982
For the reaction 4°Ca(2°Ne, 160), E L = 260 MeV the angular dependence of the low-energycomponent in the inclusive kinetic energy spectrum of 160 is interpreted within a two-step reaction model, assuming a fast direct projectile break-up followed by a frictional interaction of the heavy projectile fragment with the target.
The experimental data on light particle emission in various heavy-ion induced reactions in the incident energy range of 5 - 2 0 MeV/nucleon give strong evidence, that a non-equilibrium reaction mode is involved, at least during the initial stage of the interaction [ 1]. As the dominating component, a direct transition from a bound to a continuum state of the relative motion of the light and heavy fragment of the projectile is expected to occur in the Coulomb and nuclear field of the target. For example, near the grazing angle the inclusive kinetic energy spectra of 160 emitted in the collision 20Ne(260 MeV) + 40Ca shows a direct peak at an energy corresponding to the beam velocity (see ref. [2]). A further striking feature of the spectra of the heavy projectile fragment is a well pronounced low-energy tail, which extends several tens of MeV below the direct peak. Such an extra component is a clear indication of a more complex reaction mechanism governed by a final state interaction in the three-body channel, into which the system enters after the direct reaction. As pointed out by Udagawa and Tamura [2] this effect can be understood successfully as a consequence of a break-up fusion process, in which a break-up of the projectile takes place first, but the light particle is fused into the target subsequently. In a more pragmatic way, Hussein et al. [3] decomposed the spectrum in three parts: a primary direct fragmentation component computed in a plane wave theory with local momenta (LPWBA), 0 031-9163/82/0000-0000/$02.75 © 1982 North-Holland
a semi-direct component resulting from the inelastic scattering of the observed fragment and an empirical non-direct background. The second component has been determined by folding a direct-fragmentation spectrum with an experimental inelastic scattering spectrum. The relative contribution of the first and the second part has been used as a free parameter, adjusted to fit the data. Apart from the very low-energy end, the tail of the experimental spectra for 208pb(160, 17N) and 208pb(160, 12C) at 315 MeV have been well reproduced for one reaction angle. In order to explain the low-energy component in the spectrum of the heavy projectile fragment and its angular behaviour the present paper proposes a twostep reaction model, in which a direct projectile breakup is succeeded by a deep inelastic interaction of the heavy projectile fragment with the target, so that the fast heavy projectile fragment is strongly damped to the final low kinetic energy. Following McVoy and Nemes [4] the one-step direct reaction component is calculated within a plane wave spectator model with local momenta. As in the approach of Hussein et al. [3] the cross section for the two-step interaction is estimated by folding a first order break-up cross section with a cross section for the inelastic target-heavy projectile fragment collision in the final channel. Different from ref. [3] the inelastic cross section is not taken from experiment, but calculated within a classical dynamical model for deep inelastic heavy ion col293
Volume 119B, number 4,5,6
PHYSICS LETTERS
lision (DIC), including statistical fluctuations. As collective coordinates the polar coordinates R, 0 have been chosen to describe the relative motion of target and projectile. The interaction of the two nuclei has been considered in the frame of a Fokker-Planck equation for the time development of the distribution function in the corresponding phase space. In this approach the mean values of the collective degrees of freedom and the conjugate momenta fulfil classical newtonian-type equations including frictional terms, governing the dissipation of energy and angular momentum of the relative motion. For their second moments a set of coupled first-order differential equations can be obtained. The fluctuations and the mutual correlations are determined by the Einstein dissipation-fluctuation theorem. In the model utilized in the present paper, the nuclear part of the conservative ion-ion force is the given by the proximity potential U, and for the formfactor .f(R) of the elements of the frictional tensor ( TRR = aR f(R ), Too = aeR 2f(R )) the expression f(R) = (OU/~R)2 has been used [5]. The present calculation starts from the following qualitative picture of the course of the heavy ion collision. During the initial stage of the reaction, in the field of the target nucleus an elastic break-up of the projectile into an a-particle and a heavy projectile fragment occurs, leading to a three-body channel with all participants in their ground state. In order to simplify the situation it is assumed, that this process is governed by the a-particle-target interaction, leaving the heavy projectile fragment as a spectator. After the first stage of the interaction we are encountered with a composite system of three particles in contact with a certain distribution of energy and momentum for their relative motion, as it originates from the fragmentation process. This system further developes in time. Now, let us assume, that for the spectrum of the heavy projectile fragment the dominating force is the target-heavy projectile fragment interaction, so that we have a two-body problem again. Then, in the second stage of the reaction the target and the heavy projectile fragment are forming a double nuclear system, for the evolution of which the state after the first direct break-up acts an initial state. Depending on the impact parameter in the ingoing channel and on the intermediate state after the break-up, this double nuclear system can fuse (incomplete fusion [6], massive transfer [7]) or undergo an "incomplete 294
23/30 December 1982
deep inelastic collision", in which the relative motion of both nuclei is statistically coupled to their intrinsic excitations. As a result of such an incomplete deep inelastic collision a strongly damped component emerges in the spectrum of the heavy projectile fragment with a characteristic dependence on the reaction angle. Because in DIC the kinetic energy loss is distributed among the fragments according to the mass number, in the final state mainly the heavy targets will be in an excited state with high probability. In the following we use the notation: 1(2) : heavy (light) fragment of the projectile (0), 3 : target. An index ik(i - kj) refers to the relative motion of particle i and k (particle i and the CM of particles k, j). Because we want to apply the model for the reaction Ne + Ca, the assumption of an infinitely heavy target has been dropped in the kinematical relations for energies and momenta. In order to compare with experiment the final results are transformed to the lab system. In order to reproduce the entire shape of the measured spectrum, the relative double differential cross section d2o/dEl_23 d~21_23 is constructed by superposition of a cross section o(1) for the (first order) elastic break-up and a cross section 0(2) for a succeeding (higher order) deeply inelastic interaction between the heavy projectile fragment 1 and the target 3 d2a/dEt_23 d~1_23 ~ d2o(1)/dEl_23 d~1_23 + a d2a(2)/dEl_23 d~21_23.
(1)
The coefficient a, which regulates the relative contributions of both cross sections, is used to fit the shape of the experimental spectrum. No attempt has been made to reproduce the absolute value of the summed cross section. The break-up cross section a (1) is estimated within a plane wave spectator model with local momenta, particle 1 appearing as a spectator. This approximation gives the T matrix element (see eq. (4) of ref. [4]) TLPWBA ~ ~b12(k~_23 -- kl_23)V23(k3_12 - k3_12 ).
(2) Here, ~b12 is the Fourier transform of the ground state wave function of the projectile, expressing the relative motion of particles 1 and 2 before break-up. The momenta before and after break-up are k and k', respectively, are measured from the top of the Coulomb
Volume 119B, number 4,5,6
PHYSICS LETTERS
barrier in the initial channel. V23 denotes the Fourier transform of the interaction between particle 2 and 3, which is responsible for the projectile fragmentation. Observing particle 1 only, we can integrate ITLPWBA12p(~E1-23) over particle 2. Neglecting the factor f d~23V23 , which should have little influence (see appendix B of ref. [4]), one gets d2o(1)/dEl_23 d~21_23 t
1(~12(k1_23 - kl_23)[2p(El_23).
(3)
Apart from the phase space factor O for the final states, the kinetic energy spectrum of particle 1 is determined by the momentum distribution in the projectile, which is assumed to be a gaussian
t~(k) 12 ~ exp(-~2k2/2~2).
(4)
The probability W(p(O) ~ Pl-23, P23) for a transition of the system from the initial state with projectile momentum p(0) to a final three-body state Pl-23, P23 via a two-step mechanism with intermediate states p(m)= (p~m_)23' p~r~)} (or alternatively ( p ~ ) , p~m)31}) is estimated from a folding procedure
W(P (0) -+ Pl-23, P23) = f dp(m) [4,,(1)(p(0) -~ p(m)) W(2)(p(m) ~ p 1- 23, P23)-
(5) Here, W(1) is the probability for the break-up transition from the initial state to the three-body channel with relative momenta p(m). W(2) gives the probability, that in the intermediate state m the relative motion of 1 and 3 is changed by a deep inelastic interaction between 1 and 3, leaving the particle 2 untouched. If in eq. (5) W(1) and W(2) are expressed by the fragmentation cross section (3) and the double differential cross section d2o(DIC)/dE13 d~13 , respectively, a multi-differential cross section d4o(2)/dEl_23d~l_23 X dE23 d~223 can be calculated by integrating p(m). Because the Q-value is fixed for the elastic break-up, one gets f ,UPl_23 (m) ,~i-23 .4~(m)
z.6 ['a w(m) .~n(m) .4~(m)~[ ~,(m) .~
= n ju~1_23ua~,l_23u~,23 t.,t~l_23.P . In Order to simplify the integration over the inter-
(6)
23/30 December 1982
mediate states m, it is assumed, that the break-up changes the relative energy of particles 1 and 3 without altering the direction of the relative momentum, ~(~) = ~(1~- Thus, the cross section d2o(DIC)/dE13 X d~13 is calculated for the systemA 1 +A 3 at the average relative energy ( E ~ ) ) after break-up. (E~r~)) __
+ le
)3_ 0_)231 Both stages of the reaction
are coupled via the kinetic energy of the relative motion of particles 1 and 3. In the folding integral according to eq. (5), V23 from eq. (2) can be integrated over ~2~m), which gives a factor approximately constant, so'that I~[ 2 may be integrated analytically over ~2~m_)~3.The remaining integral over the relative energy Elm_)23 can be performed, if one calculates d2o(DIC)/ dE13 d~13 for various incident energies. The model has been applied to analyse the reaction 40Ca(2°Ne, 160), E L = 260 MeV. For the width parameter in eq. (4) we used the same values as McVoy and Nemes [4] for the reaction 208pb(160, 15N), g = 140-315 MeV
"~2 ='~2A1A2/(A1 +A 2 _ 1),
"°0 = 90 MeV/c .
This choice is in agreement with value derived from 20Ne(860 MeV) 160 + ~ experiments on medium weight targets [8] or from fragmentation experiments at relativistic energies. The strength of the radial and tangential components of the friction tensor in the trajectory calculations is given by the parameters aR = 12 fm/c MeV, ao = 0.22 fm/c MeV, which are standard values extracted from an extensive analysis of fusion cross sections and various experimental data on DIC. The results of the calculations are presented in figs. 1 and 2. One can state, that the entire shape of the 160 spectrum at 6 ° is reproduced by proper choice of ce in eq. (1). In particular, the end point at low energies is given correctly. This means, that the standard friction force produces the correct amount of the energy loss in the final state interaction between the heavy projectile fragment and the target. For increasing reaction angle, it follows from the experimental data, that the elastic break-up peak goes down rapidly, the spectrum becomes more broad, with a dominating contribution located at the lowenergy end. This tendency is reproduced by the model, using only one value for the coefficient to combine both components of the spectrum for all reaction angles. For reaction angles larger than 8° , the over295
Volume 119B, number 4,5,6
PHYSICS LETTERS
23/30 December 1982
40Ccz ( ;ZONe' 16O}~Ti, 260 MeV
4°C0( 2°Ne,
160) 4Z'Ti, 260 MeV
3.0 5.0
2.0
>~
3.0
1.0
>. 1.0
cT •'o
2.0
1/.0
160
180
200 220 E, (MeV)
Fig. 1. Reaction 4°Ca(2°Ne, 160)44Ti, E L = 260 MeV. Experimental and calculated inclusive energy spectra of 160 for reaction angles 6 ° , 8° (lab system). Dashed curve: elastic break-up. Dotted curve: incomplete deep inelastic collision. Solid curve: combined spectrum. Dashed-dotted curve: breakup-fusion, from ref. [2]. The hatched area marks roughly the region covered by the fluctuating experimental data.
whelming part of the cross section 0(2) is connected with negative angle deflection. Furthermore, it must be emphasized, that one has the same normalization factor for all reaction angles to adjust the calculated summed spectrum to the absolute experimental cross section. So, the model reproduces simultaneously the shape of the spectrum and the angular distribution of the heavy projectile fragment. The detailed agreement between theory and experiment could be improved by adjusting the free parameters'o0, aR, ao to the particular reaction 20Ne + 40Ca, so that one can reach very similar results as with a breakup-fusion mechanism (compare fig. 1). The two-step reaction model suggested for incomplete deep inelastic heavy ion collisions gives some global insight in the process of heavy ion collisions, in which a direct mode is coupled to a mechanism statistical in nature. Different from the break-up fusion mechanism this approach emphasizes the interaction between the heavy projectile fragment and the target. It has the advantage, that actual calculations are simpler than in an elaborate DWBA treatment of a fluctuating component in the T-matrix element, 296
tat
2.0
-,o
1.0
1.0 1.0 ~
16" i i r i 140 160 180 200 220 240 I=I (MeV)
Fig. 2. Same as fig. 1, for reaction angles 10°, 12°, 14°, and
16° . caused by an interaction of a pair of fragments in the three-body state. Some further extensions of the model are of interest. First, the influence of the second stage of the reaction of the angular correlations between the emitted fragments could be predicted and compared with experiment. Second, because of a transfer of nucleons between the projectile fragments and the target during the second stage of the reaction, a transition to final channels other than a pure projectile fragmentation may proceed. The mass transfer may be taken into account employing an improved dynamical model [9] to compute the cross section for DIC. Such effects are indicated already in the experimental data. For example, in the collision 197Au + 160 (310 MeV) both final channels 12C + O and 13C + o~are populated [10]. Also for the reaction 40Ca + 160 (260 MeV) it was found in coincidence experiments, that a major part of a-singles strength does not accompany the 160 fragment (see ref. [2]). Valuable discussions with Dr. R. Schmidt and J. Teichert are gratefully acknowledged. The authors
Volume 119B, number 4,5,6
PHYSICS LETTERS
w o u l d like to thank J. T e i c h e r t for his support in performing the numerical calculations.
References [ 1 ] E. Betak and V.D. Toneev, Fiz. Elem. Chastits At. Yadra 12 (1981) 1432. [2] T. Udagawa and T. Tamura, Lecture Notes 1980 RCNP Kikuchi Summer School (Kobe, 1980) p. 171. [3] M.S. Hussein, K.W. McVoy and D. Saloner, Phys. Lett. 98B (1981) 162.
23/30 December 1982
[4] K.W. McVoy and M.C. Nemes, Z. Phys. A295 (1980) 177. [5] R. Schmidt and J. Teichert, Dubna-reports E4-80-527, E4-80-735. [6] K. Siwek-Wilczynska et al., Nucl. Phys. A330 (1979) t50. [71 D.R. Zolnowski et al., Phys. Rev. Lett. 41 (1978) 92. [8] J.B. Natowitz et al., Phys. Rev. Lett. 47 (1981) 1114. [9] R. Schmidt, V.D. Toneev and G. Wolschin, Nucl. Phys. A311 (1978) 247. [10] M. Bini et al., Phys. Rev. C22 (1980) 1945.
297
PHYSICS LETTERS
23/30 December 1982
DIRECT AND DISSIPATIVE MODES IN HEAVY-ION REACTIONS: INCOMPLETE DEEP INELASTIC COLLISIONS L. OEHME and R. REIF Technische Universitdt Dresden, Sektion Physik, 8027 Dresden, Mom msenstrasse 13, DDR
Received 6 July 1982 Revised manuscript received 7 September 1982
For the reaction 4°Ca(2°Ne, 160), E L = 260 MeV the angular dependence of the low-energycomponent in the inclusive kinetic energy spectrum of 160 is interpreted within a two-step reaction model, assuming a fast direct projectile break-up followed by a frictional interaction of the heavy projectile fragment with the target.
The experimental data on light particle emission in various heavy-ion induced reactions in the incident energy range of 5 - 2 0 MeV/nucleon give strong evidence, that a non-equilibrium reaction mode is involved, at least during the initial stage of the interaction [ 1]. As the dominating component, a direct transition from a bound to a continuum state of the relative motion of the light and heavy fragment of the projectile is expected to occur in the Coulomb and nuclear field of the target. For example, near the grazing angle the inclusive kinetic energy spectra of 160 emitted in the collision 20Ne(260 MeV) + 40Ca shows a direct peak at an energy corresponding to the beam velocity (see ref. [2]). A further striking feature of the spectra of the heavy projectile fragment is a well pronounced low-energy tail, which extends several tens of MeV below the direct peak. Such an extra component is a clear indication of a more complex reaction mechanism governed by a final state interaction in the three-body channel, into which the system enters after the direct reaction. As pointed out by Udagawa and Tamura [2] this effect can be understood successfully as a consequence of a break-up fusion process, in which a break-up of the projectile takes place first, but the light particle is fused into the target subsequently. In a more pragmatic way, Hussein et al. [3] decomposed the spectrum in three parts: a primary direct fragmentation component computed in a plane wave theory with local momenta (LPWBA), 0 031-9163/82/0000-0000/$02.75 © 1982 North-Holland
a semi-direct component resulting from the inelastic scattering of the observed fragment and an empirical non-direct background. The second component has been determined by folding a direct-fragmentation spectrum with an experimental inelastic scattering spectrum. The relative contribution of the first and the second part has been used as a free parameter, adjusted to fit the data. Apart from the very low-energy end, the tail of the experimental spectra for 208pb(160, 17N) and 208pb(160, 12C) at 315 MeV have been well reproduced for one reaction angle. In order to explain the low-energy component in the spectrum of the heavy projectile fragment and its angular behaviour the present paper proposes a twostep reaction model, in which a direct projectile breakup is succeeded by a deep inelastic interaction of the heavy projectile fragment with the target, so that the fast heavy projectile fragment is strongly damped to the final low kinetic energy. Following McVoy and Nemes [4] the one-step direct reaction component is calculated within a plane wave spectator model with local momenta. As in the approach of Hussein et al. [3] the cross section for the two-step interaction is estimated by folding a first order break-up cross section with a cross section for the inelastic target-heavy projectile fragment collision in the final channel. Different from ref. [3] the inelastic cross section is not taken from experiment, but calculated within a classical dynamical model for deep inelastic heavy ion col293
Volume 119B, number 4,5,6
PHYSICS LETTERS
lision (DIC), including statistical fluctuations. As collective coordinates the polar coordinates R, 0 have been chosen to describe the relative motion of target and projectile. The interaction of the two nuclei has been considered in the frame of a Fokker-Planck equation for the time development of the distribution function in the corresponding phase space. In this approach the mean values of the collective degrees of freedom and the conjugate momenta fulfil classical newtonian-type equations including frictional terms, governing the dissipation of energy and angular momentum of the relative motion. For their second moments a set of coupled first-order differential equations can be obtained. The fluctuations and the mutual correlations are determined by the Einstein dissipation-fluctuation theorem. In the model utilized in the present paper, the nuclear part of the conservative ion-ion force is the given by the proximity potential U, and for the formfactor .f(R) of the elements of the frictional tensor ( TRR = aR f(R ), Too = aeR 2f(R )) the expression f(R) = (OU/~R)2 has been used [5]. The present calculation starts from the following qualitative picture of the course of the heavy ion collision. During the initial stage of the reaction, in the field of the target nucleus an elastic break-up of the projectile into an a-particle and a heavy projectile fragment occurs, leading to a three-body channel with all participants in their ground state. In order to simplify the situation it is assumed, that this process is governed by the a-particle-target interaction, leaving the heavy projectile fragment as a spectator. After the first stage of the interaction we are encountered with a composite system of three particles in contact with a certain distribution of energy and momentum for their relative motion, as it originates from the fragmentation process. This system further developes in time. Now, let us assume, that for the spectrum of the heavy projectile fragment the dominating force is the target-heavy projectile fragment interaction, so that we have a two-body problem again. Then, in the second stage of the reaction the target and the heavy projectile fragment are forming a double nuclear system, for the evolution of which the state after the first direct break-up acts an initial state. Depending on the impact parameter in the ingoing channel and on the intermediate state after the break-up, this double nuclear system can fuse (incomplete fusion [6], massive transfer [7]) or undergo an "incomplete 294
23/30 December 1982
deep inelastic collision", in which the relative motion of both nuclei is statistically coupled to their intrinsic excitations. As a result of such an incomplete deep inelastic collision a strongly damped component emerges in the spectrum of the heavy projectile fragment with a characteristic dependence on the reaction angle. Because in DIC the kinetic energy loss is distributed among the fragments according to the mass number, in the final state mainly the heavy targets will be in an excited state with high probability. In the following we use the notation: 1(2) : heavy (light) fragment of the projectile (0), 3 : target. An index ik(i - kj) refers to the relative motion of particle i and k (particle i and the CM of particles k, j). Because we want to apply the model for the reaction Ne + Ca, the assumption of an infinitely heavy target has been dropped in the kinematical relations for energies and momenta. In order to compare with experiment the final results are transformed to the lab system. In order to reproduce the entire shape of the measured spectrum, the relative double differential cross section d2o/dEl_23 d~21_23 is constructed by superposition of a cross section o(1) for the (first order) elastic break-up and a cross section 0(2) for a succeeding (higher order) deeply inelastic interaction between the heavy projectile fragment 1 and the target 3 d2a/dEt_23 d~1_23 ~ d2o(1)/dEl_23 d~1_23 + a d2a(2)/dEl_23 d~21_23.
(1)
The coefficient a, which regulates the relative contributions of both cross sections, is used to fit the shape of the experimental spectrum. No attempt has been made to reproduce the absolute value of the summed cross section. The break-up cross section a (1) is estimated within a plane wave spectator model with local momenta, particle 1 appearing as a spectator. This approximation gives the T matrix element (see eq. (4) of ref. [4]) TLPWBA ~ ~b12(k~_23 -- kl_23)V23(k3_12 - k3_12 ).
(2) Here, ~b12 is the Fourier transform of the ground state wave function of the projectile, expressing the relative motion of particles 1 and 2 before break-up. The momenta before and after break-up are k and k', respectively, are measured from the top of the Coulomb
Volume 119B, number 4,5,6
PHYSICS LETTERS
barrier in the initial channel. V23 denotes the Fourier transform of the interaction between particle 2 and 3, which is responsible for the projectile fragmentation. Observing particle 1 only, we can integrate ITLPWBA12p(~E1-23) over particle 2. Neglecting the factor f d~23V23 , which should have little influence (see appendix B of ref. [4]), one gets d2o(1)/dEl_23 d~21_23 t
1(~12(k1_23 - kl_23)[2p(El_23).
(3)
Apart from the phase space factor O for the final states, the kinetic energy spectrum of particle 1 is determined by the momentum distribution in the projectile, which is assumed to be a gaussian
t~(k) 12 ~ exp(-~2k2/2~2).
(4)
The probability W(p(O) ~ Pl-23, P23) for a transition of the system from the initial state with projectile momentum p(0) to a final three-body state Pl-23, P23 via a two-step mechanism with intermediate states p(m)= (p~m_)23' p~r~)} (or alternatively ( p ~ ) , p~m)31}) is estimated from a folding procedure
W(P (0) -+ Pl-23, P23) = f dp(m) [4,,(1)(p(0) -~ p(m)) W(2)(p(m) ~ p 1- 23, P23)-
(5) Here, W(1) is the probability for the break-up transition from the initial state to the three-body channel with relative momenta p(m). W(2) gives the probability, that in the intermediate state m the relative motion of 1 and 3 is changed by a deep inelastic interaction between 1 and 3, leaving the particle 2 untouched. If in eq. (5) W(1) and W(2) are expressed by the fragmentation cross section (3) and the double differential cross section d2o(DIC)/dE13 d~13 , respectively, a multi-differential cross section d4o(2)/dEl_23d~l_23 X dE23 d~223 can be calculated by integrating p(m). Because the Q-value is fixed for the elastic break-up, one gets f ,UPl_23 (m) ,~i-23 .4~(m)
z.6 ['a w(m) .~n(m) .4~(m)~[ ~,(m) .~
= n ju~1_23ua~,l_23u~,23 t.,t~l_23.P . In Order to simplify the integration over the inter-
(6)
23/30 December 1982
mediate states m, it is assumed, that the break-up changes the relative energy of particles 1 and 3 without altering the direction of the relative momentum, ~(~) = ~(1~- Thus, the cross section d2o(DIC)/dE13 X d~13 is calculated for the systemA 1 +A 3 at the average relative energy ( E ~ ) ) after break-up. (E~r~)) __
+ le
)3_ 0_)231 Both stages of the reaction
are coupled via the kinetic energy of the relative motion of particles 1 and 3. In the folding integral according to eq. (5), V23 from eq. (2) can be integrated over ~2~m), which gives a factor approximately constant, so'that I~[ 2 may be integrated analytically over ~2~m_)~3.The remaining integral over the relative energy Elm_)23 can be performed, if one calculates d2o(DIC)/ dE13 d~13 for various incident energies. The model has been applied to analyse the reaction 40Ca(2°Ne, 160), E L = 260 MeV. For the width parameter in eq. (4) we used the same values as McVoy and Nemes [4] for the reaction 208pb(160, 15N), g = 140-315 MeV
"~2 ='~2A1A2/(A1 +A 2 _ 1),
"°0 = 90 MeV/c .
This choice is in agreement with value derived from 20Ne(860 MeV) 160 + ~ experiments on medium weight targets [8] or from fragmentation experiments at relativistic energies. The strength of the radial and tangential components of the friction tensor in the trajectory calculations is given by the parameters aR = 12 fm/c MeV, ao = 0.22 fm/c MeV, which are standard values extracted from an extensive analysis of fusion cross sections and various experimental data on DIC. The results of the calculations are presented in figs. 1 and 2. One can state, that the entire shape of the 160 spectrum at 6 ° is reproduced by proper choice of ce in eq. (1). In particular, the end point at low energies is given correctly. This means, that the standard friction force produces the correct amount of the energy loss in the final state interaction between the heavy projectile fragment and the target. For increasing reaction angle, it follows from the experimental data, that the elastic break-up peak goes down rapidly, the spectrum becomes more broad, with a dominating contribution located at the lowenergy end. This tendency is reproduced by the model, using only one value for the coefficient to combine both components of the spectrum for all reaction angles. For reaction angles larger than 8° , the over295
Volume 119B, number 4,5,6
PHYSICS LETTERS
23/30 December 1982
40Ccz ( ;ZONe' 16O}~Ti, 260 MeV
4°C0( 2°Ne,
160) 4Z'Ti, 260 MeV
3.0 5.0
2.0
>~
3.0
1.0
>. 1.0
cT •'o
2.0
1/.0
160
180
200 220 E, (MeV)
Fig. 1. Reaction 4°Ca(2°Ne, 160)44Ti, E L = 260 MeV. Experimental and calculated inclusive energy spectra of 160 for reaction angles 6 ° , 8° (lab system). Dashed curve: elastic break-up. Dotted curve: incomplete deep inelastic collision. Solid curve: combined spectrum. Dashed-dotted curve: breakup-fusion, from ref. [2]. The hatched area marks roughly the region covered by the fluctuating experimental data.
whelming part of the cross section 0(2) is connected with negative angle deflection. Furthermore, it must be emphasized, that one has the same normalization factor for all reaction angles to adjust the calculated summed spectrum to the absolute experimental cross section. So, the model reproduces simultaneously the shape of the spectrum and the angular distribution of the heavy projectile fragment. The detailed agreement between theory and experiment could be improved by adjusting the free parameters'o0, aR, ao to the particular reaction 20Ne + 40Ca, so that one can reach very similar results as with a breakup-fusion mechanism (compare fig. 1). The two-step reaction model suggested for incomplete deep inelastic heavy ion collisions gives some global insight in the process of heavy ion collisions, in which a direct mode is coupled to a mechanism statistical in nature. Different from the break-up fusion mechanism this approach emphasizes the interaction between the heavy projectile fragment and the target. It has the advantage, that actual calculations are simpler than in an elaborate DWBA treatment of a fluctuating component in the T-matrix element, 296
tat
2.0
-,o
1.0
1.0 1.0 ~
16" i i r i 140 160 180 200 220 240 I=I (MeV)
Fig. 2. Same as fig. 1, for reaction angles 10°, 12°, 14°, and
16° . caused by an interaction of a pair of fragments in the three-body state. Some further extensions of the model are of interest. First, the influence of the second stage of the reaction of the angular correlations between the emitted fragments could be predicted and compared with experiment. Second, because of a transfer of nucleons between the projectile fragments and the target during the second stage of the reaction, a transition to final channels other than a pure projectile fragmentation may proceed. The mass transfer may be taken into account employing an improved dynamical model [9] to compute the cross section for DIC. Such effects are indicated already in the experimental data. For example, in the collision 197Au + 160 (310 MeV) both final channels 12C + O and 13C + o~are populated [10]. Also for the reaction 40Ca + 160 (260 MeV) it was found in coincidence experiments, that a major part of a-singles strength does not accompany the 160 fragment (see ref. [2]). Valuable discussions with Dr. R. Schmidt and J. Teichert are gratefully acknowledged. The authors
Volume 119B, number 4,5,6
PHYSICS LETTERS
w o u l d like to thank J. T e i c h e r t for his support in performing the numerical calculations.
References [ 1 ] E. Betak and V.D. Toneev, Fiz. Elem. Chastits At. Yadra 12 (1981) 1432. [2] T. Udagawa and T. Tamura, Lecture Notes 1980 RCNP Kikuchi Summer School (Kobe, 1980) p. 171. [3] M.S. Hussein, K.W. McVoy and D. Saloner, Phys. Lett. 98B (1981) 162.
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[4] K.W. McVoy and M.C. Nemes, Z. Phys. A295 (1980) 177. [5] R. Schmidt and J. Teichert, Dubna-reports E4-80-527, E4-80-735. [6] K. Siwek-Wilczynska et al., Nucl. Phys. A330 (1979) t50. [71 D.R. Zolnowski et al., Phys. Rev. Lett. 41 (1978) 92. [8] J.B. Natowitz et al., Phys. Rev. Lett. 47 (1981) 1114. [9] R. Schmidt, V.D. Toneev and G. Wolschin, Nucl. Phys. A311 (1978) 247. [10] M. Bini et al., Phys. Rev. C22 (1980) 1945.
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