A.2.1 Overview
The study of the interaction of two ions is a subject of growing interest in nuclear physics. The complex nature of the projectile makes it possible that a number of new reactions occur, and also when the projectile fuses with the target nucleus creating a compound nucleus one has consider the special features of the heavy-ion reaction due to the large angular momentum carried in by projectile.
At low energies two ions interact only through their Coulomb fields, and can scatter elastically or inelastically. Nuclear interactions can only take place if the
A.2.1 Overview
155 two-ion energy Ecm in their centre of mass is high enough to overcome the Cou-lomb barrier, and then the associated wave length (λ=h/ 2MEcm ) is much less than the nuclear dimensions. In such circumstances the interaction shows semi-classical features, and in particular it is appropriate to consider the ions moving along their classical orbits. This semiclassical nature of the heavy-ion interac-tions makes it possible to give an overall description in terms of the minimal dis-tance between the two interacting ions rmin, which is simply related to the impact parameter b by [132]:
where V(rmin) is the nuclear potential acting between the two ions. Even though such a description is only qualitative and a full treatment must take ac-count of the quantum mechanical nature of the process, it is possible to distin-guish four regions where the different reaction mechanisms predominate as the minimal distance between the two ions increase:
- the fusion region, with 0 ≤ rmin ≤ RF;
- the deep inelastic and the incomplete fusion region with RF ≤ rmin ≤ RDIC
- the peripheral region, with RDIC ≤ rmin ≤ RN;
- the Coulomb region, with rmin > RN, where RN is the distance above which nuclear interactions are negligible.
The ion orbits corresponding to these regions are schematically shown in Figure A.2.1.
In the fusion region the two-ion interaction leads predominantly to the for-mation of a composite nucleus which initially is quite far from statistical equilib-rium since a large part of its excitation energy is in the form of an orderly collec-tive translational motion of the nucleons of the projectile and the target. This or-derly motion is transformed into chaotic thermal motion by means of a cascade of nucleon-nucleon interactions.
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Figure A.2.1: Classical picture of heavy ion interactions showing the trajectories corre-sponding to close, grazing, peripheral and distant collisions.
This thermalization eventually leads to a compound nucleus in statistical equilibrium, but, before reaching this, a considerable fraction of the excitation energy may be carried off by fast particles or clusters of particles with energy considerably greater than the average thermal energy. The importance of these pre-equilibrium emissions increases with the two-ion relative energy, but cannot be completely neglected even at energies only slightly greater than that of the Coulomb barrier acting between the two interacting ions. The amount of pre-equilibrium emissions and the energy distribution of the emitted particles depend on the mean field acting between the two ions, the type of interacting ions and on the competition between the emission of particles into the continuum and nu-cleon-nucleon interactions which redistribute the energy of the two interacting nucleons. Once the compound nucleus is formed, its angular momentum may be so high that it is unstable toward fission into two fragments, and, even when this does not occur, the high angular momentum it possesses makes fission much more likely than in the case of light projectile reactions.
In the deep inelastic and incomplete fusion regions the overlap of the ions is much less than in the case of fusion, but it is sufficient to allow a strong interac-tion between the two ions which transforms a sizeable fracinterac-tion of the kinetic
en-A.2.1 Overview
157 ergy into internal excitation energy. This may occur by a process called deep inelastic collision in which the two ions form a dinuclear system which lasts for some time. The two ions are connected by a neck through which a substantial energy transfer occurs, but only a few nucleons are transferred from one to the other. When the dinuclear system breaks, two fragments fly apart which are similar to the projectile and the target and are called the projectile-like and the target-like fragments. In this impact parameter window another process may oc-cur in the case of light projectiles: the break-up fusion or incomplete fusion reac-tion in which the projectile breaks-up into two fragments one of which fuses with the target, while the other flies away almost undisturbed.
In the peripheral region the ions brush past each other. The processes which may occur are, with increasing impact parameter, the transfer of one or a few nucleons from one ion to the other, and elastic and inelastic scattering.
Finally in the Coulomb region the nuclear interaction between the two ions is negligible, but they may still interact via the Coulomb excitation.
The impact parameter b can be related to the corresponding orbital angular momentum L by the classical relation:
kp pb vb
Lh=µ = =h (A.4)
where µ is the two-ion reduced mass, and v and p the two-ion relative velocity and momentum. Previous considerations make it possible to assume, to a first approximation, that the different processes dominate in different L-windows and give approximate expressions for the reaction cross-sections in the different re-gions as:
where, to a good approximation, the transmission coefficient TL ≈ 1 for L ≤ Lmax, and i = 1, 2, 3, 4 corresponds, respectively, to fusion, to deep inelastic or break-up fusion, to peripheral and to Coulomb reactions and L = L1, L2, L3, L4, L5 equals, respectively 0, kRF, kRDIC, kRN, Lmax, with
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1
2
max = −
π σ k
L R (A.6)
where σR is the total cross-section including Coulomb excitation.
The total reaction cross-section to a good approximation is given by the semiclassical expression:
( )
⎥⎦
⎢ ⎤
⎣
⎡ −
=
CM
R E
R Rint2 1 V int π
σ (A.7)
where the interaction radius Rint and the interaction barrier V(Rint) may be ob-tained by plotting the product σRECM against the two-ion centre-of-mass energy.
The decomposition of the total cross-section into the contributions discussed above is schematically shown in Fig....
Figure A.2.2: Schematic picture of the contribution of different partial waves to the reac-tion cross-secreac-tion for a collision between two heavy ions. The relative proporreac-tions of the various non-elastic events vary with energy and masses of the ions.
A.2.2 Fusion reactions
Heavy ions reactions have been extensively studied because they offer the possibility of producing nuclei with high excitation energy and high spin thus
A.2.2 Fusion reactions
159 allowing the study of nuclear matter in conditions which are not easily formed in other reactions. Fusion reactions have also been used to try to produce super-heavy nuclei and to produce proton rich nuclei very far from the stability line.
We now consider a few important features of fusion reactions:
- for light projectiles and low incident energies the fusion cross-section may be a considerable fraction of the reaction cross-section;
- with increasing charge of the interacting ions the fusion probability falls abruptly, as show in Fig.
- the fusion cross-section at first increases linearly with 1/ECM, reaches a maximum thereafter decreases linearly with 1/ECM, as show in Fig... for the interaction of 16O with 27Al;
- the detailed energy dependence of the fusion cross-section may differ quite considerably for different interacting ions forming the same composite nu-cleus, as shown in Fig. for the interaction of 16O with 16B and 14N with 12C;
- in general the fusion cross-section varies quite smoothly with the energy, but in the case of light systems it may show quite large oscillations.
Figure A.2.3: Some features of the fusion cross section: ratio of the fusion and reaction cross-sections as a function of the projectile and target charges (left); fusion cross sec-tion of 16O wit 27Al.
These features may be explained, at least qualitatively, by considering (a) the
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dependence of the two-ion potential on the distance and the relative angular momentum, and (b) the limitation to the formation of the composite nucleus due to the absence of states below the yrast8 line which gives the minimum excita-tion energy a nucleus may have as a funcexcita-tion of the total angular momentum [133].
Figure A.2.4: Sum of the nuclear, Coulomb, and centrifugal potentials for 18O + 120Sn as a function of radial distance for various values of orbital angular momentum L. The nu-clear potential has the Saxon-Woods form with V = 40 MeV, r0 = 1.31, a = 0.45. The horizontal line at ECM = 87 MeV corresponds to an incident energy of 100 MeV. The turning points for various values of L are marked by dots
The effective potential acting between the two ions is assumed to be a com-plex one-body potential, the real part describing the refraction of the incident particle by the nucleus and the imaginary part the absorption by all non-elastic interactions. For the purpose of the following discussion we have to consider only the real part of this potential. The Coulomb potential may be approximated
8 In a plot of excited-state energy as a function of the orbital angular momentum, the yrast line is the line under which no more states are present.
A.2.2 Fusion reactions
161 with sufficient accuracy by that between two uniformly charged spheres, perhaps with some allowance for surface diffuseness. The strongly attractive nuclear po-tential represents all the complicated interactions between the two ions and for simplicity we use a Saxon-woods form:
( )
( ) Fi-nally the centrifugal term VN(r) is (ħ2/2µ) L(L+1)/r2. The relative contributions of these three potentials depend on the energy and on the masses and charges of the interacting ions. As an example, the predicted total potential for the interac-tion of 18O + 120Sn is shown for representative values of L in Figure A.2.4. This Figure shows that fusion may occur only up to a critical angular momentum Lcrit. The maximum value of the fusion reaction then occurs at the energy correspond-ing to the relative momentum:( ) ( )
If L is less than Lcrit fusion is possible. This justifies why for light ions the fusion cross section is an important fraction of the total one and why it decreases as the interacting ions charge increases. Indeed, the decrease is due to the larger influence of the repulsive Coulomb potential on high Z ions also at higher ener-gies, i.e. at lower distances; obviously if ions do not approach enough they can-not fuse in a compound nucleus. In terms of orbital angular momentum, the Cou-lomb potential reduces L, causing a decrement of fusion probability.
Moreover, also for high relative angular momentum, two ions could be un-able to make fusion; this happens when they are not un-able, together, to give to the compound nucleus the necessary energy for yrast line overcoming for the gained total angular moment J. When the two ions fuse, the composite nucleus is formed with the energy:
Q E
ECN = CM + (A.10)
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where ECM is the relative energy of the two ions in the centre of mass system and Q is the fusion Q-value, the energy which is gained or lost when the two ions fuse and is given by
(
M1 M2 M)
c2Q= + − CN (A.11) where M1, M2, and MCS are the ground state masses of the fusing ions and the composite nucleus.
Thermalization of the composite nucleus
When two heavy ions fuse they form a composite nucleus which is far from statistical equilibrium since a large fraction of its energy is in form of an orderly collective translational motion of the nucleons of the projectile and the target.
This orderly motion transforms into chaotic thermal motion through a cascade of nucleon-nucleon interactions during the thermalization of the composite nucleus.
This takes some time and before reaching thermal equilibrium nucleons or clus-ters which still have an energy considerably higher than their equilibrium ther-mal energy may be emitted into the continuum. These pre-equilibrium emissions must be taken into account to reproduce the multiplicity and the spectra of the ejectiles measured in heavy ion fusion reaction.
The ejectiles emitted in a fusion reaction at rather low incident energies are detected in coincidence with the residue, of mass near to the composite nucleus mass, emitted at a very forward angle with a velocity about equal to that ex-pected for a complete momentum transfer. This velocity, indicating by Mp and MT the projectile and the target mass and by vp the projectile velocity in the labo-ratory system is:
This is the velocity of the composite nucleus in a fusion reaction and the previous relation simply expresses the conservation of linear momentum and is valid on the assumption that the velocity of a heavy residue (often called the evaporation residue, ER) remains equal to that of the composite nucleus even
A.2.2 Fusion reactions
163 after its deexcitation by particle emission. This is a valid assumption when the ejectiles are emitted isotropically or symmetrically about 90° in the centre-of-mass system as is expected to occur for particles evaporated from the compound nucleus. It is no longer valid if pre-equilibrium particles are also emitted because they are preferentially emitted in the forward direction, thus reducing the veloc-ity of the residue. However, it is approximately valid if the ejectiles have mass and energy considerably smaller than those of the projectile, as occurs at not too high incident energies (Einc ≤ 25 AMeVucleon). The velocity of the evaporation residue is estimated by its time of flight from the target to the detector and its mass by measuring its energy.
At higher incident energies this procedure is no longer valid since the excita-tion energy of the composite nucleus is very high ant it emits, before reaching statistical equilibrium, so many particles that the velocity of the residue is con-siderably less than that given by the equation ...
In conclusion, the thermalization of the composite nucleus is due to a cas-cade of nucleon-nucleon interactions which starts as soon as the two ions over-lap.
Some calculations suggest that with increasing bombarding energy an in-creasing fraction of the excitation energy is dissipated by emitting pequilibrium particles (about 50% at an energy of about 30 AMeV), greatly re-ducing the probability of forming very excited equilibrated nuclei [134].
Evaporation from equilibrated nuclei
If the compound nucleus has both high excitation energy and high angular momentum, fission in two pieces may happen immediately after its formation.
Indeed, a large angular momentum causes, during the rotation, a deformation along the axis perpendicular to the nucleus rotation one; in this conditions, the equilibrium shapes are ellipsoids since these minimize the rotational energy. As the angular momentum J increases, these ellipsoids become more and more elongated until the nucleus becomes unstable against division into two parts. But in the decay process it is not possible to populate states of the residual nucleus below the yrast line. Hence, the compound nucleus mainly emits nucleons; then
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fission happens if energy and angular momentum are still high enough. These emissions are called evaporation because the nucleus comes off similarly to molecules of a liquid evaporation. Also alpha particles are emitted at the end of the evaporation chain, together with photon emission. Evaporation particles are less energetic than pre-equilibrium ones and are emitted in an isotropic way.
Moreover, these two kinds of emission happen in different moments: the first one persists up to 10-16 sec after the impingement between projectile and target nuclei, while second one stops after 10-21 sec. Finally, evaporation emission oc-curs as soon as Coulomb barrier is overcome, whereas pre-equilibrium emission appears if the relative energy between two ions increases.
A.2.3 Projectile fragmentation
In the projectile fragmentation process, nuclei accelerated to energies several times above the coulomb barrier break-up on a fixed target. The produced frag-ments have a velocity slightly lower than that of the incident beam and they are emitted in a rather small angular cone around zero degree.
Experimental and theoretical studies have been performed in order to de-scribe the reaction mechanism and to evaluate the production cross section for any projectile-target combination so that predictions could be made for future experiments. One of the proposed models for the projectile fragmentation is due to Serber, in 1947 [135]. According to this model, the peripheral highly ener-getic heavy ion reaction can be described as a two-step process in which each step occurs in clearly separated time intervals. The initial, and faster (time order of 10-23 seconds), process of interaction consists on the overlapping between pro-jectile and target nuclei. The spectator nucleons in the propro-jectile are assumed to undergo little change in momentum, and likewise for the spectators in the target nucleus. This step can lead to highly excited objects (pre-fragments) which are usually very different from the final observed fragments. Before being detected, the pre-fragment, which continues in the original beam direction with practically unaltered beam velocity, looses its excitation energy through the emission of particles (neutrons, protons and small clusters) and γ-rays and re-arranges itself
A.2.3 Projectile fragmentation
165 corresponding to the remaining numbers of protons and neutrons. This second step (de-excitation) occurs slowly relative to the first step and typically occurs at time scales on the order of 10-16 to 10-18 seconds (depending on the excitation energy of the pre-fragment). A simple portrayal of this process is shown in fig-ure 3.5.
Figure A.2.5.: A simplistic picture of the projectile fragmentation. The overlapping re-gion of the target and the projectile is shared off, leaving an excited pre-fragment. The pre-fragment de-excites through statistical emission and becomes the final observed fragment.
The presented scenario allowed to Goldhaber the developing of a statistical model of the fragmentation process as a rapid process in which the momentum distribution of the products is related to the momentum of the nucleons inside the projectile at the time of the reaction [136]. According to this model:
- fragments are produced with a velocity slightly lower than that of the beam;
- fragments are emitted in a rather small cone around the direction of the inci-dent beam;
- the momentum distribution has a Gaussian shape and it is isotropic in the centre-of-mass system. The width of the distribution is given by:
( )
is the Fermi momentum of the nucleons within the projectile.166
On the other hand the probability of formation of a given fragment is ex-pected to be exponential with a maximum in the projectile value [137]. This re-lation is very well reproduced by experimental data as reported in figure 3.6 for the Xe projectile fragmentation where the fragment rate decreases by approxi-mately one order of magnitude for each missing neutron.
Figure A.2.6. Isotope yields for projectile fragmentation of 124,129Xe (red and blue point) and fission of 238U (green points) [CDR01].
The incident energies used for projectile fragmentation experiments have changed with technological advances. The processes which occur at various en-ergies differ sensibly. The mechanism of production is intermediate between the transfer reactions observed at low energy and a pure fragmentation processes ob-served at higher energies up to several GeV per nucleon.
Low energy fragmentation was available for many years, and a large amount
Low energy fragmentation was available for many years, and a large amount