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SEMI-MICROSCOPIC THEORY OF HEAVY ION SCATTERING
W. Greiner, W. Scheid
To cite this version:
W. Greiner, W. Scheid. SEMI-MICROSCOPIC THEORY OF HEAVY ION SCATTERING. Journal de Physique Colloques, 1971, 32 (C6), pp.C6-91-C6-112. �10.1051/jphyscol:1971613�. �jpa-00214831�
JOURNAL DE PHYSIQUE Coffoque C6, suppf~ment au no 11-12, Tome 32, Novembre-Dhcembre 1971, page C6-91
SEMI-MICROSCOPIC THEORY OF HEAVY ION SCATTERING
W. GRElNER and W. SCHEID
Institut fiir Theoretische Physik der Universitiit FrankfurtIMain, Germany
Rhurnk. - C'interaction entre noyaux cornplexes peut Ctre calculee en unifiant le mod& de la goutte liquide gkneralise (ELDM) et le modkle des couches deux centres (TCSM). Les approxi- mations soudaine et adiabatique sont discutees LC potentiel imaginaire est calcule a partir de la probabilite de transition des Ctats moleculaires vets les Ctats du noyau compod. Dans la fission et dans la diffusion d'ions lourds, les masses semblent dependre d'une facon trks importante de la distance entre les fragments. Les fonctions d'excitation dcs diffusions elastiques et inklastiques
1 6 0 - 1 6 0 et 1 *C-l2C sont calculCes et comparees B I'experience. La large structure d'ensemble
est interpret& comme etant due k des etats molCculairea virtuels tandis qu'une grande part de la structure fine semble Ctre reliee a I'excitation de molecules nucleaires quasi liees.
Abstract. - The interaction between complex nuclei can be calculated by unifying the extended liquid drop model (ELDM) and the two center shell model (TCSM). The sudden and adiabatic approxiniations are discussed. The imaginary potential is calculated via the transition probability of the molecular states into the compound states. The dependence of the masses in fission and heavy ion scattering on the distance between the fragments seems to be very important. The
0 16-0lQnd C'2-C1* elastic and inelastic excitation functions are calculated and compared with experiments. The gross structure is interpreted as being due to virtual molecular states while a great part of the fine structure seems to be related to the excitation of quasibound nuclear molecules.
I. Introduction. - Heavy ion scattering and heavy ion reactions have become both experimentally and theoretically very interesting. There are several reasons for that : First, such experiments can be regarded as predecessor experiments for the fusion reactions leading to superheavy elements. The various reaction mechanisms which might lead to such elements and which are new and typical for reactions between complex particles (such as multi-nucleon or heavy cluster transfer) can already be studied. Into this topic, for example, the quartet structures belong, which are heavily discussed at this Conference. Second, and related to this general topic, are the possibilities of creating short-lived nuclear molecules. This type of reactions came especially into focus after the obser- vation of pronounced structures in the elastic 016-016 cross section by the Yale group [I]. These cross sec- tions serve to test theoretical predictions (models) on the forces, e. g. potentials between complex nuclear clusters, kinetic energy operators and coupling of inelastic molecular channels.
If nuclei are scattered on each other, there exists the possibility, that the two-nucleus-system forms a nuclear molecule. It is a system of two nuclei, which are attached to each other on their surfaces. The two nuclei are bound together due to the long range part of the nuclear forces. They do not amalgamate, because the molecule rotates and the repulsive centri- fugal forces are in equilibrium with the attractive nuclear forces. There might also exist - depending
on the energy - a hard core due to the Pauli principle (compression effect). We will show that some struc- tures in the elastic scattering cross sections can be
understood in terms of nuclear molecules.
Let us first review briefly the experimental situa- tion : The elastic 900 scattering cross sections for the C12-ClZ, N ~ ~ - N ~ ~
ot6-ot6
and 0'"-0'~ SYS-tems [I], [2] are shown in figure 1. All of these follow the following pattern : At low energies the Ruther- ford cross section is reproduced followed by a decrease of the cross sections a t the Coulomb barrier by a fac- tor 30-100 and typical gross structures
( r
z 2-3 MeV) in all of these elastic excitation functions above the Coulomb barrier. Also some intermediate structure( r
z 0.2-0.3 MeV) is noticed, especially in the016-016 system and much more pronounced in the soft C12-C12 system. Moreover, the decrease of the cross sections above the Coulomb barrier is espe- cially pronounced in the 018-0'' and - t o a lesser extend - in the N14-N14 system, both of which also seem to show no intermediate structure.
Maher [2] argues that it is unlikely that the inter- mediate structures in the Ot6-016 cross sections are of compound elastic nature. The situation in the C12-Ct2-system should be similar. However, fine structure with a width of 0.1 MeV may also arise from statistical fluctuations. This latter interpretation is especially put forward by Vandenbosch [3].
We take the following stand point : It might very well be, that some of the intermediate structures are
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971613
C6-92 W. GREINER A N D W. SCHEID
FIG. 1. - The experimental elastic scattering cross sections for the C12-Cl2, N14-N14, 0 1 6 - 0 1 6 and O I ~ - O ~ 8 - ~ y s t e m s .
due to compound states and also due to statistical fluctuations. The strong coupling to the molecular channels (channels which are characterized by a nucleus-nucleus system) suggests, however, that most of these structures are of quasimolecular origin. We therefore try to develop a phenomenological model for nucleus-nucleus scattering (extended liquid drop model (ELDM) [4]), find out what it can predict in comparison with experiments and also try to put the model on a more microscopic basis by using the two-center shell model [5]. Several qualitatively new features occur in these heavy ion processes. So, for example, the mass parameters become explicitly dependent on the distance between the nuclei [ 6 ] . They approach only asymptotically constant valucs (e. g. reduced mass). Furthermore, both, the masses and the potentials become explicitly energy dependent, which reflects partly the non-locality of the nucleus- nucleus interaction. This all makes a careful study of the kinetic energy operator necessary. Another promising approach is the particle-core model for the individual nuclei. It has been applied originally
by v. Oertzen [7] to exchange reactions and extended recently by Park, Scheid et al. [8] into a general theory of nucleus-nucleus interactions. The underlying ideas can be formulated in both models, in the individual center shell model as in the two center shell model.
Due to the very complicated problem of heavy ion collisions and due to the complicated nature of a full microscopic theory we necessarily have to make several simplifying (and sometimes crude) assump- tions which lead to a greater transparency of the problem, but which still have to be justified in forth- coming investigations.
11. The interaction between complex nuclei. -
The channels in nucleus-nucleus collisions are only well defined as long as the two clusters d o not overlap.
We call this the asymptotic region. In this region the dynamics can be characterized by the relative motion of the two nuclei and by their individual internal excitation.
In the overlap region, however, the system is charac- terized by a totally antisymmetric wave function depending on all nucleons. In the case of the 016-016
collision we are obviously dealing with a S3'-system in the overlap region. The 32 particles can be split up in many partitions, one of them is 0 1 6 - 0 1 6 .
Hence it is clear, that no unique definition of channels (in particular elastic channel) in the overlap region is possible.
The time dependence of a heavy ion collision may be characterized by two extreme cases :
a ) The densities of the individual nuclei overlap reversibly.
b ) The shell structure of the individual nuclei is destroyed and a common compound shell model is generated.
These two extreme cases may also be called sudden and adiabatic approximation respectively. One expects the sudden approximation to be valid for very high collision energies, since one needs about 130 MeV to compress S32 to the volume of 0 1 6 [4], and since compressional modes are expected to occur at excita- tion energies of about 30-40 MeV [4, 91. In the actual case of 0'6-0'6 scattering the collision time is z,,,, z 5 x s while the orbiting time for a 30 MeV nucleon is so,, z 3 x S. Hence both times are comparable and none of the above extreme approximations will be valid ( I ) . Two models, which - as we will see - supplement each other, are useful for the description of the nucleus-nucleus collisions.
11. I PHENOM~NOLOGICAL EXI'EN1)EI) LIQUID DROP MODEL (ELDM). - Its starting point is the following
(1) Since the collective surface modes (vibrations, rotations) are much lower in energy, the typical tirnzs involvcd a r e
l v i b N-- 10-21 S. Thus, with respect to these modes the collisions can be considered as sudden.
SEMI-MICROSCOPIC THEORY OF HEAVY ION SCATTERING C6-93
expression for the energy as a function of the density
P
P I
(see [lo]).+ >-
(2: - I)'1
p2 d z .2 Po
This energy expression consists of a term propor- tional to the nucleon number, a compression term, a Yukawa term (which is attractive), a Coulomb term (which is repulsive) and a symmetry energy. It can easily be verified that the compressibility is
The functional (1) is able to predict the binding energies of nuclei (Bethe-WeiGcker formula) and this allows a proper description of the break-up and fusion processes : The energy relations between ori- ginal and final fragmentations are properly incorpo- rated. This is in complete analogy with the liquid drop model in fission processes. In fact, the model (I) may be considered as extended liquid drop model allowing also for compression and symmetry effects. The sudden approximation within this model is given by a superposition of the unperturbed densities of the two nuclei (Fig. 3)
which leads to the sudden nucleus-nucleus potential as function of the relative coordinate r
which shows a Lenard-Jones type behaviour contai- ning a hard core (Fig. 2).
The density in such a nucleus-nucleus collision is also illustrated in figure 3. The compression effect which leads to the core, is noticed. One can, of course, also obtain from formula (I) the adiabatic potential.
This is achieved, similarly as in fission calculations, by prescribing the fission shape under the important requirement that matter is nowhere compresscd (volume conscrvation of equipotentials or equidensity- surfaces).
Both the shapes and the corresponding potentials, are compared for the sudden and adiabatic case in figure 4. The centrifugal potentials for the various
I
-2501
i s3'(-2718 MeV) -...* L - - I . .
0 2 5 6 8 1 0 I Z
Fro. 2. - The sudden nucleus-nucleus potentials obtained from the extended liquid drop, eq. (4).
partial waves have also been added. The important difference is the disappearence of the hard core in the case of the adiabatic (fission-) potential. At r = 0 one has the spherical ground state of S32.
The 1 = 0-potential is precisely the symmetric- fission-potential for S32.
FIG. 3. -The density of the nucleus-nucleus system in a sudden nucleus-nucleus collision for various distances bctween the two
nuclei [44].
W. GllEINER AND W. SCHElD
a) SUDDEN APPROXIM.
~ A ~APP I
(
C 0 2 1 6 8 1 0FIG. 4. - Comparison of sudden and adiabatic 0 1 6 - 0 1 6 poten- tials. The centrifugal potential has been added for the various
partial waves.
11.2 Two CENTER SHELL MODEL. - The two center shell model [5] furnishes a very convenient basis for a microscopic formulation of heavy ion collisions [l 1, 81, in which also our extreme cases (sudden, adiabatic) and also all intermediate processes can be formulated and calculated. The Hamiltonian of the two center shell model is [5]
with
~ ~ ( Z - Z , ) ~ + O ~ ~ ~ for z > 0 V(z, P ) = -
2 (6)
o ~ ( z + z 0 ) ' + o , 2 p 2 fbr z < 0 and
Here 2z, is the distance between the two centers ; o, and o, are the oscillator frequencies in z-and p-direction respectively. The term
i2
stands for N(N + 3, where N is identical with the principal2
oscillator quantum number for zo = 0, c~ and has a similar meaning for all other values of 2, [5]. If the I-dependent terms, i. e. U(p, s), are neglected, one can obtain analytical solutions for the double- oscillator wave functions and energies [5]. The ener- gies are given by ( o = o, = o,)
Typical single particle spectra for symmetric and asymmetric break up (as recently obtained by Maruhn et al. [5]) including the I-dependent terms are shown in figures 5 and 6 respectively. The occurrence of the new fragment shells (doubly degenerate in the symme- tric break up) and the occurrence of the two different
FIG. 5. - The single particle levels for the symmetric break-up in the two center shell model.
fragment shell models in the asymmetric break-up is evident.
We assume that during the collision the concept of a shell model potential remains valid, i. e. that the shell model potentials of the colliding nuclei combine in such a way that they act as a common instantaneous shell model potential for the nucleonic movement.
It is then possible that all nucleons are always in eigenstates of the instantaneous shell model Hamil- tonian, which is the two-center-Hamiltonian (5). On the other hand, also more complicated configurations can occur, where several nucleons undergo transitions.
The latter process will require more time than the former, because more elementary nucleon-nucleon interactions are involved. As they lead to an appre-
SEMI-MICROSCOPIC THEORY OF HEAVY ION SCATTERING C6-95
For each two center distance R = 2 2, the single particle energies ei(R) of the two-center-Hamil- tonian (5) are calculated. Filling the lowest states with the appropriate number of particles, we get an instantaneous internal energy corresponding t o the elastic configuration as (2)
A
U ( r ) = ei(R)
.
i = 1 (9)
The difference
V ( r = R) = U ( R )
-
U ( m ) (10) can then be identified as the real nuclear part of the effective ion-ion potential at r = R.In the double-center-oscillator (5) the frequency plays the role of a radius parameter. The potential V(z,p) may differ from the original (free nuclei) potential with respect to the radius parameter. We have thus also here the possibility to describe sudden and adiabatic processes. If we take o = o,, i. e. if the oscillator constants are not changed during the collision, we get obviously compression effects and are describing a sudden collision. If o = o,(R), wherz the index v stands for volume conservation
olume conservation of the equipotentials) we get the adiabatic processes. Figure 7 illustrates the rear- rangement of shells in a sudden collision.
0 4 8 12 '6 FIG. 7 . - Instantaneous shell model potentials for the nucleonic movement in the 016-016-collision (hw, = 13.22 MeV). The
Zo , CENTER SEPARATION single particle levels correspond to tc = 0.08 and are assigned FIG. 6. -The single particle lcvels for the asymmetric break-up with the quantum numbers N O ; mn. Regions of close lying
in the two center shell model. levels are hatched.
ciable change in the internal structure of the colliding nuclei, they will only have a small width for decay back to the entrance channel. In a never-come-back- upproximation they will merely absorb flux from the entrance channel and will thus give rise to an imagi- nary potential. Besides, they also produce fine struc- ture in the excitation functions (see later). It is pos- sible, however, that intermediate states of very special structure are formed, as e. g. quasimolecular states, which may have relatively large widths for decay back into the entrance channel. In our pictorial des-
We are now in a position to parametrize the adia- baticity or nonadiabaticity of the process by defin- ing the instantaneous oscillator frequency at dis- tance R
o(R) = o,(R) f
f
iE, 1) (w, - w,(R)). (1 1) Here o,(R) is the frequency meeting the requirement of volume conservation during the ion-ion collision.The function f (E, I) serves to parametrize the time behaviour of the collision. It depends on energy E, angular momentum I and possibly also on the dis- cription these intermediate states correspond to very
(2) It is somewhat questionable what procedure one should
silnple excited states in the instantaneous apply to calculate thc total cnergy of a system, which is not in
potential. As they are coupled strongly to the elastic its equilibrium shape. See the section on renormalization of the
channel, they must be treated explicitly [4], [8], [I 11. potential.
C6-96 W. GREINER AND W. SCHEID
tance R. This function is at present not known except for two limiting cases : For slow collisions volume conservation is expected to hold, thus f (E, I) + 0 for small E (it is quite possible that this limit is of no practical consequence, because the Coulomb barrier may inhibit nuclear interactions before it is approached). For fast collisions, f (E, 1) will approach unity. The question arises, how f (E, I) can be calcu- lated. Let
be the true Hamiltonian of the A-nucleon system.
Furthermore, let cp,(R, f, E) be the trial function obtain- ed for the relative motion of the two clusters as a solution of the model Hamiltonian
Here 1 denotes the angular momentum of the rela- tive motion and V(R, f ) is the nucleus-nucleus poten- tial calculated according to eqs. (9) and (10) (with renormalization - see next section) for a given adia- baticity parameter f (see eq. (1 1)). If R is treated as a collective coordinate namely as the relatlve dis- tance r, one obtains a two center many-body wave function by identifying R = r :
where %(I, ..., A ;f, R) is the Slater determinant built up from the two center-shell model single particle states, which, of course, also depend on J The rela- tive coordinate is the distance between the two centers of masses
A^
denotes the antisymmetrization operator, which is necessary because cp, is not completely antisymmetric.This is due to the fact, that a n exchange from a nucleon of cluster 1 with a nucleon of cluster 2 changes r (14).
The wave function (13) contains the relative motion of the two clusters properly and depends as well on the adiabaticity parameter f as on the scattering energy E.
Using the exact A-body Hamiltonian (12), the adia- baticity parameter may then be obtained by, e. g.
the variational problem
for any given I and E. Thus one obtains in general f,(E), i. e. an adiabaticity parameter which depends on the relative angular momcnturn 1 and on the rela- tive energy E. This in turn leads via relations (ll), (9) and (10) to I-and E-dependent real nucleus-nucleus potentials. Calculations of the adiabaticity along these
lines are in progress.
One can also treat the coordinate R as a redundant coordinate. Then, in this approximation, the anti-
symmetrization operator in eq. (13) can be deleted because
$1 = 9l(R,f, E) ~ ( 1 , ..-7 A ;.A R) (16) is already antisymmetric, as long as R is treated as an independent (redundant) coordinate, or as a para- meter. It is worthwhile to note that it has been shown recently by Park, Scheid et al. [8] that the wave func- tions of type (16) can be written in complete analogy to the strong coupling wave functions of the collec- tive (unified) model. One has in the most general casc
which becomes in the asymptotic region
Here 0,
4
are the polar angles which rotate the intrin- sic axis into the laboratory axis (see Fig. 8) and $FIG. 8. -The intrinsic two center axis connects the two centers of mass of the nuclci. The polar angles O,$ rotate the intrinsic
2'-axis into the z-axis of the laboratory.
describes a rotation around the intrinsic axis. The indices in brackets like (LJ) or (J) indicate that these quantities arc only asymptotically good quantum numbers. Thc interesting feature of the strong cou- pling wave function (17) is that it goes over asympto- tically into the wave function (18) which couples explicitly the channel spin of the particles to the orbital angular momentum L. This is what one expects when the two ions can be treated indivi- dually described with respect to the intrinsic axis by XiJM', and with respect to the laboratory axis by
C
9;,,,,(4O$) zj.,,. We also note that obviously the.w '
SEMI-MICROSCOPIC THEORY O F HEAVY I O N SCATTERING
radial wave functions with various M' quantum numbers become asymptotically equal up to a Clebsch-Gordan-coefficient
Let us come back to the two extreme cases of sudden ( j = 1) and adiabatic (f = 0) potentials. The single particle energies for 0 1 6 - 0 1 6 collisions in these two cases are shown in figurc 9. The corresponding nucleus-
FIG. 9. - Single particle levels for Ol~-O~6-collisions with h a , - 13.22 MeV and spin orbit strength K
-
0.08 as a func- tion of relative distance R. The sudden and adiabatic cases arc depicted on the left and right hand side, respectively. The levels are assigned in the order they split a t large distance. The shape of the nucleus-nucleus systcm is shown on the top of the figure.nucleus potentials for various parameter choices are shown in figure 10.
When the two centers approach each other from infinity some states in every shell are lowered while others rise energetically. The higher lying states extend over a larger volun~e and hence change thcir energies already at larger distances compared to the lower lying states. For R + 0 most of the states rise steeply in energy in the sudden approximation. This is the conzpression eflect (3). The sudden nuclear potentials calculated from these spectra according to eq. (9) have characteristic minima around R = 2 d, where d is the nuclear radius. At this relative distancc the highest occupied levels in the two potential wells begin to interact with each other. This minimum appearing in the sudden nuclear potentials can thus be considered as a binding effect due to the nucleons near the Fermi surface. If one starts with two magic nuclei, as e. g. 016-016, one has to fill particles into the lowering as well as into the rising states in order
( 3 ) We like to mention in this connection the recent work of Fliessbach [44], who also obtains a compression effect in a somewhat diferent semi-microscopic treatment.
FIG. 10. - The real nucleus-nucleus potentials for the 0 1 6 - 0 1 6
and Cl2-C12-systems. The sudden 016-016-potentials have at R
-
0 the values 264.40,239.02 and 21 1.73 MeV for K = 0,0.08 and 0.166 rcspectivcly. The sudden CI Z-C12-potentials have at R = 0 the value 185.04 MeV independent of K . T h e Coulombenergy has to be added of course.
to fulfill the Pauli principle. Thus one will in such a case only get a very shallow and narrow dip in the sudden potential. On the other hand, for two C t 2 - nuclei one puts more particles into lowering than into rising states, thus taking more advantage of the par- tial lowering of statcs. It is seen from these examples that our prescription of calculating sudden potentials will necessarily produce a dip becoming more shallow and narrow when one approaches a magic number (less polarizability of magic nuclei). As the inclusion of a spin orbit potential makes 0 1 6 less magic and C12 more magic, it will deepen the potential of the former and flatten that of the latter. This is especially evident for K = 0.166, which corresponds to C L Z being more magic than 016. The considerable influence of shell structure on nucleus-nucleus potentials is thus revealed.
When R approaches zero, the fast potentials rise steeply because of the compression efTect. In the
016-016 case, for example, the configuration goes over into that of a S3' nucleus with all particles being in their lowest states, but in a compressed shell model potential, namely that of 016. This configuration corresponds to a cornpresscd and hence highly cxcited
C6-98 W. GREINER A N D W. SCHEID
S3' nucleus, which could be made evident by expanding the eigenfunctions in terms of those of the adiabatic S32 shell model potential. In our shell model approach the compression energy, i. e. the energy required to compress a S32 nucleus to the size it would have in an OL6-potential, is over 200 MeV (depending on the spin orbit strength K), which is quite high compared to theoretical predictions and experimental estimates [4], [lo], [12]. This may be a shortcoming of this model and needs further investigation. The quite weak and shallow attractive part of the so obtained sudden potentials as compared to the statis- tical calculations and fits (see the foregoing paragraph) may also be a consequence of the overestimation of compression energy within the shell model.
The adiabatic potentials must, as R 4 0, approach - 35.4 MeV in the 0 1 6 - 0 1 6 case and - 25.6 MeV in the C1'-C12 case. This could be achieved by adjusting the spin orbit potential strength IC, which is, however, already fixed by the single particle levels in the free nuclei. Thus the behaviour R -+ 0 of the potentials calculated according to eq. (9) is incorrect as well in the sudden as in the adiabatic case. This can be corrected by a renormalization procedure.
11.3 RENORMALIZATION PROCEIXJRE. - The incor- rect asymptotic behaviour of the nucleus-nucleus potentials, calculated within the two center shell model, in as well the sudden as adiabatic case, is due t o the fact, that the shell model is generally not able at present to predict properly the binding ener- gies of nuclei. This, however, is precisely achieved within the generalized liquid drop model (I), which is an extension of the standard liquid drop mass for- mula to configurations which may contain compressed matter. As outlined in section 11.1 it is possible with the ansatz (1) to calculate a total energy E(R) of the instan- taneous configuration and a real part of the nucleus- nucleus potential as the difference E(R) - E(co).
The shell model approach to the nucleus-nucleus interaction is then used to correct the extended liquid drop model (ELDM) potential for shell effects.
Following the method described
-
by Str~~tinsky [I31 one calculates a total energy U(R) of the instantaneous configuration after smearing out the shell model levels ei(R). An extended liquid drop model potential including shell corrections is then given byThis procedure contains via the shell corrections, which depend on the adiabaticity parameter f, eq.
(15), the adiabaticity of the process. It has up to now been applied only for adiabatic (fission) potentials.
As a most recent example we show the potential energy surface of U236 in figure 11 [14]. The equal individual fragments can also have a deformation in their shape, which is denoted by the ratio of the
24 (
,
I I I I .----
. ,FIG. 11. -Potential energy surface for 2 3 6 U in the two spheroid model. 1 gives hcre directly the distance between the two cen- ters in fermis. The abscissa si /3 - aib (see Fig. 8).
longer axis, a, to the shorter axis, b, i. e. by alb, I is the two center separation in fermis.
11.4 THE IMAGINARY POTENTIAL-ELEMENTS OF FOR- MAL R E A C ~ I O N THEORY. - The imaginary part of the nucleus-nucleus potential can be calculated again in both, the ELDM and the two center shell model.
The results of the ELDM are well known. One considers the outflow time T(r) from the instable configuration containing compressed matter (see Fig. 3) which depends on the distance r and obtains
The result of this type of calculation is shown in the lower part of figure 26. For details we refer to the literature [4].
Much more interesting is the microscopic calcu- lation of the imaginary potential. Formulas for the calculation of the optical potentials are obtained within a modification of the Feshbach-formalism [IS], [16], [17]. Let us briefly mention a few essential steps : the Hanliltonian of the total A-nucleon sys- tem without the center-of-mass part TcM is
For a fragmentation (A/2, A/2) one can describe the elastic channel through the relative coordinate
q /A:2 A
SEMI-MICROSCOPIC THEORY O F HEAVY ION SCATTERING C6-99
and one may rewrite H a s follows [I61 into U(r,) since the U,,(ra), eq. (29), are restricted to the overlap region of the two nuclei.
H = T(ra) + Ho(a1) + Ho('~2)
+
rl] ''i,,, j z Z.
i z l 9 j a 2 The integral equation for U(ra) reads
(24) EU(r,) = Here cc and ( a , , a,) respectively characterize the par-
- H I > < V " , I H I ticular way in which the A nucleons are split up into =
(
+,(x1) 4b(a2) H -1 ->'L---
E n 1 - E I
the partition ( A / 2 , A/2). There are
($1
such pos- sibilities ! No denotes the intrinsic Hamiltonian for the separated fragmentswith the eigenstates
It is clear that the knowledge of the overlap function
~ I o ( ~ I ) $0 = Ea $0
-
(26) U(ra) furnishes all information on the elastic scatter--
ing process. It is also clear, that the exact equation If one introduces the projection operator P, ( 3 3 ) has to be considerably approximated and sim- which projects the part the antis~mme- plified to allow for further physical interpretations trized wave function Y of the total A-nucleon systemand We do not follow here all the approxi- PY' = { u(rx) @ , ( a l ) Ob(a2)
1
(27) rnationswhich are to be made in detail, but merelydiscuss the essential points :
one can show [ S ] that pr&n be written as a) The states
I
c p ,>
are assumed to be compound states which may decay with a certain T. Thus their= A
(
l +.(aI Ob(a2)> <
b a ( r , ) 4 s ( 0 2 )I
x eigenvalues Em, eq. (31), are replaced byEn,
- iT.1 b) The states 1 9 ,
>
may be counted by the density (1 -- 3 1
l a> <
( )I )
8 ) of states p, where the index s denotes subgroups of these states (certain angular momenta, parities, etc.) Here the functions Un(ra) are solutions of the inte- F~~~ both assumptions it follows that~ r a l equation
-
un(ra) = i n a ' f a
t
( - 1)'.1
4 : ( a , ) 4 ; ( @ 2 ) Un(ra,) x5
-H I V m > < p m ' H
-El" - E + i n ~ p s ( ~ ) ~ I l o s > s < c p s I H . (34) The Schrodinger equation for the total systemcan be rewritten by multiplying with the projection operators
?
and Q = 1 -P^
and one obtainsHere QY denotes the non-elastic part of the total wave function Y , which can be expanded in terms of the wave functions defined by the eigenvalue problem
c) The wave functions
4,
of the individual nuclei are antisymmetrized. The mutual antisymmetrization of the nucleons in both nuclei is neglected. We sym- metrize, however, if both nuclei are in the same states. Since we are dealing with elastic scattering of nuclei in their ground states, the projection operator simplifies tod ) As an approximate solution of the problem (31) we suggest a product ansatz
. .
where With these functions one can eliminate the non-
elastic channels from the equations (30). One intro- k = ( ~ ~ 1 ~ 1 ) .
duces the following abbreviation : These functions are solutions of the approximate U(ra) =
<
4 , ( a 1 )4 d . 2 ) p
YI>
= #@a) f HamiltonianThis overlap function depends solely on the relative
distance r, and goes asymptotically (large r,) over x
1
Y I M ~> <
YIhf,I ) I
Vs>
= 1 Vs>
( 3 7 )C6- 100 W. GREINER AND W. SCHEID
where
e) The interaction between the nuclei is expanded in multipoles
f ) The intrinsic wavc functions 4,(a) depend on 3(A/2) - 3 intrinsic coordinates. The difference bet- ween intrinsic coordinates and particle coordinates is neglected and the functions +,(cr) are replaced by functions $,(ri) which depend on 3 A/2 coordinates again. This leads to a spurious state problem as it is known from the concept of collective variables in nuclear structure physics [18].
g ) The kinetic energy of the radial motion is neglected in the compound states ((( Mitbewegungs- effekt B). This assumption means that one calculates as if the compound states fission adiabatically.
Under all these assumptions and simplifications the Hamiltonian (33) reduces to the optical model for elastic scattering between two nuclei [16]
= T
+
U - iC
W, I Y,,,,> <
YI,,, 1 (39)1.m
where
and
This reproduces for the real potential (40) essen- tially the formula (9) discussed earlier. The imaginary part is nothing else than the spreading width of the molecular states into the compound states, which is proportional to the transition probability from the molecular to compound configurations. Since this is proportional to the density p,,(E*) of the states cp,,, it is obvious that the imaginary potential becomes strongly I-dependent and E-dependent.
In fact, the formula (41) for the imaginary poten- tial may be derived directly from thc simple physical picture that the molecular state may decay into the compound states of density p(E*). The transition probability is then given by the golden rule
= -- 2 n p l ( ~ ' )
I <
comp.I
V 1 elast.>
h h
(42) and the absorptive potential is
This is completely analogue to the calculation of the spreading width of photonuclear giant reso- nances, where also the decay into compound states is studied 1191. Helling et al. [20] have calculated the imaginary potential on the basis of (42), (43). The following formula for the density of states has been used
-- 2 1 + 1 (1
+
1 i 2 ) ~ pl(E*) = C exp[2 J o ~ * l - - 2 a 2 - expL
- - 2 a 2-1 .
The excitation energy of the compound states E*
is given for the 0'6-016-sy~tem by
E* = E
+
2 B016 - B S 3 2 = E 4- 16,6 MeV, (45) where B denotes the binding energies of the corres- ponding systems. We encounter here an additional problem : for every excitation energy E* exists a maximal angular momentum l,,,(E*) (yrast-level).No compound states are possible at E* which have higher angular momenta. It is usually assumed that this maximal angular momentum I,,, is obtained if E* is completely rotational energy
We therefore choose the cut-off parameter a of eq. (44) such, that the exponential function has decreased by a factor lle for 1 = I,,,. This leads to the following linear dependence of oZ on E *
= OE* - A 2 '
and the imaginary potential (43) becomes by use of (42) and (43)
- ( 2 1 + W L = - icr exp[2 JOE*] -.
2 PE*
x cxp
[
-]
x p d r (48)overlap
where fi = 0/h2 and the parameter 1211 a = 0.035 ( A - 12) MeV-'. The square of the average matrix element into thc compound states has been set equal to the number of nucleons in the overlap region
1
p dr. The fact, that it is energy independentoverlap
seems to be a good approximation as known from the investigations of the spreading width 1191. The strength parameter cc and the angular momentum cut-off parameter /3 are adjusted to the experiment. If applied to the 016-016 gross structure one finds with the sud- den real potential
P
= 1.9 MeV-',
which corresponds to a moment of inertia of a ho~nogeneous S32-sphere with a radius of about 2.5 fm. The radial dependence of W , is shown in figure 12 while figure 13 gives the fit to the elastic scattering cross section. It is seenSEMI-MICROSCOPIC THEORY OF HEAVY ION SCATTERING C6-I01
FIG. 12. - Radial dependence of the imaginary potential, which is assumcd to be proportional to the number of nucleons
in thc overlap region.
CM ENERGY ( M e V )
FIG. 13. - Excitation function for thc elastic 0 1 6-016-scatte- ring. The theoretical curve is obtained with the rcal sudden potential of figure 2 and the angular momentum dependent imaginary potcntial of eq. (48). Thc parameters of the absorp-
tive potential are r
-
I .1 x 10-3 MeV and /? = 1.9 MeV-].that the maxima-minima ratio in the theoretical excitation function is very pronounced and that the decrease of the cross section a t large energies is pro-
perly described. It should finally be mentioned that the importance of the I-dependent imaginary poten- tials in heavy ion scattering has been first discussed by Chatwin et al. [ 2 2 ] . This treatment here can be considered a s a kind of justification of their findings.
TI. 5 VARIABLE MASSES IN FISSION AND HEAVY ION COLLISIONS. - The effect of a variable mass has been studied by Hofmann and Dietrich [ 2 3 ] in a one- dimensional model using several phenomenological forms for the mass variation. Similarly Updegraff and Onley [ 2 4 ] have included this effect in a three- dimensional collective model. The advantage of the double center shell model, however, is its ability t o describe the complete fission process to the stage of two separated fragments and, furthermore, its appli- cability to heavy ion scattering. Lichtner, Drechsel et al. [ 6 ] have recently performed calculations of the mass within the symmetric double center oscillator (without 1-s coupling) which is described by two variables [ 5 ] , the coordinate R, which is the separation of the centers of the two potentials, and P, which describes the deformation of the two fragments (see Fig. 8).
One allows for the rotation of the system by inclu- ding two angles O and 4, describing the orientation of the axis of symmetry. To the second order in the velocities the classical kinetic energy then will have the form
where the inertial parameters are functions of the coordinates R and /?. We calculate the inertial para- meters with the cranking model which gives, for two collective coordinates X and Y, the mass
<ildH/dXIj> <jJaHldYli>
B , , = 2 h 2
z
- .. .. . . .-i filled ( c j - c ~ ) ~
j unfilled
(5'3) and similarly, for the moment of inertia I, the usual formula of lnglis [ 2 5 ] . The two collective coordinates R and p, although they describe somewhat restricted shapes, should give the main features of fission : as the initial stage of fission is expected to be mainly a /?-deformation, while the coordinate R allows one to describe the fission process even up to final stage of separation of the two fragments.
At large deformations these two coordinates clearly describe different shapes. However, even at small deformations these coordinates are never equivalent in that the shape given by small values of R contains all even multipoles (i. e. to first order in R all even multipole moments are present) while in the limit of small /? only the quadrupole moment is nonzero.
We note that a removal of the restriction to cylindri- cal symmetric shapes will modify eq. ( 4 9 ) by introduc- ing three different moments of inertia.
In high energy heavy ion scattering the restriction
C6-102 W. GREINER AND W. SCHEID
t o quadratic terms in the velocity may not be very good and fourth and higher order terms need to be studied. They correspond to a velocity dependence of the variable mass.
Figure 14 shows the calculated mass parameters plotted against R for two values of
P
for the case of':;u.
AS expected, the mass B K R approachesFIG. 14. - The mass parameters as function of the separation of the fragments.
the reduced mass for large separation. One can in fact easily show analytically that the double oscilla- tor used here produces asymptotically the correct reduced mass.
Below the scission point, however, the effective mass becomes considerably larger than the reduced mass and shows strong fluctuations due to the varia- tion of the ground state pairing structure, i. e. the variation of the BCS occupation probabilities. This was first pointed out by Belyaev [26]. Beyond the scission point, however, the occupation probabilities become constant and the mass is solely determined by the variation of the single particle wave functions with functions with deformation. The latter contribution gives rise to a smooth background and approaches the reduced mass.
The mass parameter BRR is the mass in terms of the coordinates R and jl of the double center model (fig. 8).
It should be kept in mind, however, that the coordi-
nate R does not coincide with the distancc between the centers of mass of the two fragments. In fact, this distance increases much slower than R below the scission point and only asymptotically the two dis- tances coincide.
The mass parameter BBB which is given in figure 146 shows fluctuations about a value higher than the irrotational value by a factor of about 5-10. For large R the mass BBg reaches a constant value, twice the mass parameter for p-vibrations in the individual fragments, because the two center model describes simultaneous P-vibrations in both fragments. The interference term BgK shown in figure 14c approaches zero past the scission point, where the motion in R and /? becomes decoupled.
As is well known, the pairing force has a big effect on the moment of inertia, a t least for small values of R.
The moment of inertia may be written as the sum of two terms, one describing the rotation of the two potential centers and the other describing the rotation of the two fragments about their centers. For small values of R the moment of inertia approaches that of the latter term and for large separations the first term, which approaches the value mAR2/4, gives the main contribution (see Fig. 15a). Analogous to the one-center shell model the second term is reduced considerably by the pairing force, while the first term is essentially unchanged.
For applications to fission or heavy ion collision the fourdimensional problem of eq. (49) may be reduced to three dimensions by choosing one particular fission (reaction) path. In order to study the effects of quan- tization we choose a path of ground state deforma- tion (p = Po), which corresponds to the scattering of two identical ions with constant deformation. Using the prescription of Pauli and Podolsky [27] to quantize eq. (49), we obtain the Schrodinger-equation.
where B = BRK.
This expression may be simplified by techniques which are a combination of the methods of [23] and [24]. We replace the wave function Y by ( p / ~ ) ' / ~ d, and change to a new coordinate by the transforma- tion (p = mA/4)
Eq. (5 1) then becomes
SEMI-MICROSCOPIC THEORY OF HEAVY ION SCATTERING C6- 103
FIG. 150. - Thc moment of inertia as a function of the separa- tion R of the fragments and the rescaled coordinate s, respec- tively. Pairing forces are included in the calculation. b. - The potential energy [28] along /3 = /lo as function of K and x. The
potential V 1 ( x ) includes the additional potential.
with
Eq. (53) looks like a Schrodinger equation for a system with constant mass except for the variable moment of inertia and the additional potential pro- duced by the quantization procedure. Further, because of the simple form of the volume element, the effects of the variable inertial parameters is now completely contained in the potential energy.
In figure 15a, b we show the moment of inertia I(x) and the potential energy
as given in eq. (53). There are three main effects : 1 ) The change of scales produces a larger poten- tial barrier.
2) The centrifugal energy is changed because of the change in the moment of inertia due t o both scal- ing and internal rotation (difference of I ( R ) from
p R 2 ) . The latter effect being particularly important for small values of R.
h2 d 2
41
3) The additional potential -- ----F
41
is gene-2 p dx
-1
rally small except, again, for small values R (or x).
It is clear from the above discussion and from eq.
(53) that the effect of the variable mass BRR can be taken into account by a straightforward modification of the potential and a simultaneous use of the asymp- totic mass value BRR = p, the reduced mass of the fragments. The change in the potential increases the tunnel path (see fig. 15h) and also modifies the zero point energy. In general the net effect will be an increase of the fission life times by an order of magni- tude.
111. Quasimolecular structures. - We have seen in the last paragraph that the interaction between nuclei of both adiabatic and sudden type allows the possibility of nuclear molecules. We will now inves- tigate the properties of such quasimolecular states more closely and show in particular, which structures in the elastic scattering cross section can originate from nuclear molecules.
111. I QUASIBOUND AND VIRTUAL MOLECULAR STA- TES. -The nucleus-nucleus system may perform stable rotations and vibrations in the quasimolecular potential. This is illustrated in figure 16 where the centrifugal potential has been added to the quasimo- lecular potential. In order that there occur stable rotations for given angular momentum I a t a dis- tance r = b, the following requirement has t o be fulfilled.
Thus the maximal angular momentum for which this condition is fulfilled is related to the maximum of the positive r-derivative of V(r), i. e. to max (aV/dr). In our example it is classically possible that stable rotations occur up to 1 = 18h. Quantum mechanically this is somewhat different because of the zero point vibrations in the potential minima.
Therefore the potential contains only up to a certain angular momentum states, which are still within the minimum of the potential. We call these states qua- sibound (full horizontal lines in Fig. 16). In contrast to that we call those states (resonances) which are not anymore lying within the potential minimum (dashed in Fig. 16) virtual states.
In the quasibound states the two nuclei build a nuclear molecule. Both nuclei are attracted to each other on their surfaces. In the microscopic two center model these molecular states can also be understood : The outer, most loosely bound nucleons are taken out of the individual shell model and instead belong to
C6- 104 W. GREINER A N D W. SCHEID
t o look at figure 17, which shows the phase shifts of the potential of figure 16 as a function of energy.
The widths of the resonances are a measure for the
FIG. 17. - The nuclear phase shifts of the elastic 016-016- scattering for the real potential shown in figure 16. N o imaginary
potential has been applied.
mean life time and thus also for the mean (( rotational angle )) of the virtual molecule, which turn out to be z = 3 x s and 4, % 600, respectively. We note that these life times are of the same order of magnitude as the scattering time of the nuclei.
In the calculation of elastic cross sections the vir- tual resonances lead to the main gross structure.
This is especially true as long as no imaginary poten- tial is turned on (see Fig. 18). But even with absorptive
FIG. 16. - The real potential for 016-016-scattering in the sudden approximation. The centrifugal potential is included.
The virtual states (I > 12) are shown as dashed horizontal lines, while the quasibound states (I ,( 10) are full horizontal lines. The position of these statcs has been chosen such that the nuclear phase shift has the value 6r = n12 a t these energies.
The phase shifts are obtained from a phase shift analysis.
FIG. 18. - The 90n-excitation-function for elastic 0 1 6 - 0 1 6 -
scattering. The full curves are calculated for the real sudden potential shown in figure 2. The absorptive potential seen in the lower part of figure 2b has been used. It has been put to zero in the excitation function seen in the upper part of this figure. The angular momenta of the mainly contributing partial
wavcs are indicated.
a state in which they orbit around both nuclei (homeo- polar binding). The two nuclei do not fission because of the repulsive centrifugal and eventually hard core forces.
In the virtual states the two nuclei are also buil- ding a molecule, but only for a very short time. One can also understand these types of states as inter- ference phenomena : The incoming wave is reflected as well on the outer potential barrier as on the inner barrier (caused by the centrifugal well and eventually by the hard core). In the casc of the orbiting reso- nances (virtual states) one has a standing wave with nodes at the inner and outer barrier. It is interesting
potential this statement is correct, because the virtual states are only slightly above the Coulomb barrier.
At this energy there are yet not many compound states with the same angular momentum and thus those partial waves which resonate with the virtual states feel onlj? a small imaginary potential. The situation is quite different at higher energies when par- tial waves are resonating with higher virtual states
SEMI-MICROSCOPIC THEORY O F HEAVY ION SCATTERING C6-105
which lie quite above the Coulomb barrier. In those states the natural and the absorptive width become quite large which leads to a stronger damping.
We can thus say that the main structures in the experimental scattering cross sections are related to virtual nuclear molecules. Accordingly we can attri- bute to the main structures at 21, 24 and 29 MeV seen in the 016-016 elastic excitation function the vir- tual resonances with 1 = 12, 14 and 16 respectively (this depends on the underlying real potential). For higher energies the gross structures are washed out (see Fig. 13).
111.2 THE EXCITATION OF QUASIBOUND MOLECULES.
- It is obvious that quasibound states cannot be excited in the elastic scattering directly, because of the inpenetrability of the potential barrier. Only via an indirect, i. e. inelastic, mechanism it seems possible to circumvent the Coulomb barrier and to excite the quasibound molecules with sufficient strength.
111 . 2 . 1 The Double-Resonance-Effect. - In an ine- lastic excitation the relative motion of the two nuclei , loses energy and angular momentum. Therefore the two nuclei <( fall )) deeper into the potential minimum.
It is then possible that a quasibound nuclear mole- cule is formed if the residual energy and angular momentum of the relative motion meets the charac- teristica of quasibound states (see Fig. 19) [4], [16].
FIG. 19. -The excitation of quasibound states. Thc two nuclei scatter elasticly at the energy Ei. Due to the nuclear forces low energy collective statcs in the individual nuclei are excited.
After this excitation the energy available for rclative orbital
Then the elastic scattering cross section shows typical structures stemming from the quasibound molecular states. The following conditions have, however, to be fulfilled :
a) The inelastic and elastic channel must be stron- gly enough coupled to each other.
b ) The partial wave of the entrance channel by which the quasibound state is excited, has also to resonate with a virtual state.
c) The imaginary potential for the quasibound states has to be sufficiently weak.
The first point is supported by the experimental C1 2-C12-~ross section compared with the O t 6 - 0 ' case. The former shows namely much more structure than the latter which is probably due to the relative softness of the C12-nuclei which in turn leads to a strong coupling between elastic and inelastic channels (which are surface vibrations in CL2).
The experimental excitation function of the 0'6-016- scattering shows also that the intermediate structures are related to the gross structure (Fig. 20). In addi-
C M ENERGY [ M e V ) -
motion is EL-E*. If this energy agrees with that of the quasibound FIG. 20. - The experimental 016-0l"scattering cross section state, the latter will be excited. at various angles [ I ] .
