Nucleonic molecular orbitals and Landau-Zener transitions in
heavy-ion reactions with the diabatic basis
() B. IMANISHI( 1 )and S. MISONO( 2 ) ( 1
) Ohto 1-3-1, Yono, Saitama 338, Japan (
2
) Tokyo Research Laboratory, IBM Japan - 1623-14 Shimotsuruma, Yamato-shi 242, Japan (ricevuto il 4 Agosto 1997; approvato il 15 Ottobre 1997)
Summary. — The validity of nucleon-molecular-orbital picture is discussed for the 12
C + 13
C system by using the orthogonalized coupled-reaction-channel rotating molecular-orbital (RMO) model. Specific RMO states, namely positive-parity ground RMO states, are well described by the picture of a covalent molecule, while the adia-baticity for RMO states is not satisfied in general because of the appearance of Landau-Zener radial couplings in the RMO basis. However, we show, by transforming the RMO basis to a diabatic basis, that the adiabatic molecular-orbital picture is again estab-lished. Momentum-dependent coupling interactions dominate transition phenomena in this diabatic basis, contrary to the assumption made in the classical Landau-Zener theory.
PACS 24.10 – Nuclear reaction models and methods.
PACS 24.10.Eq – Coupled-channel and distorted-wave models. PACS 01.30.Cc – Conference proceedings.
1. – Introduction
Direct reaction mechanisms in heavy-ion collisions describe a clustering aspect of re-action processes of systems where we assume the existence of two core nuclei and several valence nucleons or clusters consisting of nucleons. Among these reactions the process of the nucleon molecular-orbital formation which is studied by many authors so far [1-9] is particularly interesting. Based on the two-center shell model, Park, Scheid and Greiner have developed the molecular particle-core model [5-7].
In order to introduce the molecular orbital picture into the coupled-reaction chan-nel (CRC) method, Imanishi and von Oertzen have developed the orthogonalized CRC (OCRC) rotating molecular-orbital (RMO) approach [8, 9]. One advantage of this method
(
)Paper presented at the 174. WE-Heraeus-Seminar “New Ideas on Clustering in Nuclear and Atomic Physics”, Rauischholzhausen (Germany), 9-13 June 1997.
is that solutions of the RMO scattering equation, even those which contain singular in-teractions such as discussed in what follows, are easily obtained by using the equivalent OCRC equation in another kind of basis, and thus we can discuss the molecular-orbital picture by comparing the OCRC solution with experimental data.
However, when singular couplings exist in the RMO basis, we have the difficulty that a direct relation is missing between the coupling interactions and cross-sections obtained by the OCRC calculation, since we cannot actually solve the coupled equation in the RMO basis. In order to remove singularities of radial couplings in the RMO basis, we introduce a new basis, the diabatic basis into the OCRC approach. We show that these couplings get strongly reduced in the diabatic basis. However, the radial coupling interactions used in our diabatic basis are momentum-dependent, contrary to those assumed in the conven-tional Landau-Zener theory [10, 11].
2. – Definition of OCRC/RMO basis and the coupling interactions
We assume that a system consists of a valence nucleon n and two core nuclei C1and
C2and there are two mass partitions (C 1 +n)+C 2and C1 +(C 2 +n). We first construct
the non-orthogonal channel-basis wave functions ^
JM i
(r i
; )for each mass partition of
fragments from the eigenfunctions of these two nuclei and the wave functions for the an-gular part of the internuclear motion (the symbol represents a set of all coordinates to
describe the scattering system). By employing these non-orthogonal channel wave func-tions, we define the orthogonal basis wave functions
JM
(r
; )as the OCRC basis [9]:
JM (r ; )= X Z dr ^ JM (r ; )N ,1=2J (r ;r ); (1)
whereN is the overlap integral defined by the non-orthogonal basis. Then, we obtain
the OCRC radial coupled equation [9], which contains the interactionU totJ (r) =("+ R J (r)+U D;J (r)+ ^ M J (r))
, where the term R
J
(r)is the angular part of the kinetic
energy operator for the relative motion. The symbol"
denotes the energy of the intrinsic
state of the channel . The interactionU D;J
(r)is responsible for direct processes, while
the interaction ^ M
J
(r)is a local approximation of the non-local transfer interaction.
We define molecular orbital states JM p (r; )(RMO basis) [9], JM p (r; )= X JM (r; )A J p (r); (2)
and adiabatic potentialsV p
(r)so that in this basis the total interaction contained in the
radial OCRC equation is diagonalized:
V J p (r)= , (A J (r)) ,1 U tot;J (r)A J (r) pp : (3)
Thus, we have the following coupling interactions in the RMO basis:
V pq (r)=V (1) pq (r) d +V (2) pq (r); (4)
with V (1) pq (r)=, X h 2 A p (r)[ d dr A q (r)]; (5) V (2) pq (r)= X h 2 2 A p (r)[ d 2 dr 2 A q (r)]: (6)
It should be noticed that only pseudo-crossings exist in the RMO basis.
3. – Diabatic basis and the coupling interactions
At pseudo-crossings in the RMO basis radial coupling interactions have sharp peaks. In order to reduce the magnitudes of the singular coupling interactions, we make a con-version from the RMO basis to the diabatic basis around these crossings:
~ JM p (r; )= X JM (r; ) ~ A J p (r): (7) Here, ~ A J
(r)is the unitary transformation matrix [12], which is defined in the following
way. First, we smoothly connect the RMO statepat the right of the crossing distance (r =r
R
)with the RMO stateqat the left(r =r L
)to define the diabatic basis. We then
perform an interpolation in this range(r L
rr R
)to determine the components ~ A
p (r)
of diabatic states.
Thus, in the diabatic basis, we obtain the diabatic potentials ~ V p
(r)and the radial
cou-pling interactions ~ V pq (r) (p6=q)defined as, ~ V p (r)f ~ A ,1 (r)U tot (r) ~ A (r)g pp ; (8) ~ V pq (r)= ~ V (1) pq (r) d dr + ~ V (2) pq (r)+ ~ V (3) pq (r): (9)
The expression of coupling interactions in the diabatic basis is the same as the one in the RMO basis, except for the last term
~ V (3) (r)which is given by ~ V (3) pq (r)=f ~ A ,1 (r)U tot (r) ~ A (r)g pq : (10)
This term is considered as a leading term in the conventional Landau-Zener theory.
4. – Numerical calculation for the12 C+
13
Csystem
We made numerical calculations for the12 C+
13
Csystem at energies below and near
the Coulomb barrier, where we observe Landau-Zener transition. To construct the OCRC basis wave functions, we choose the channels: 12
C + 13 C(g.s.,1=2 , ), 12 C + 13 C (3.09 MeV,1=2 + ), 12 C+ 13 C (3.85 MeV,5=2 + ) and [12 C+ 13 C (8.2 MeV,3=2 + ); d 3=2
res-onance state] with the valence-neutron states, p 1=2, s 1=2, d 5=2 and d 3=2 in 13 C, respec-tively [9, 15]. Hereafter, we will use the following notations: 1)S(OCRC) denotes the S-matrix obtained by the exact OCRC calculation. 2)S(RMO) denotes theS-matrix
Fig. 1. – Top: Adiabatic potential diagram in theJ
=9=2 +
state for the12 C+
13
C system. The respective curves show values of the adiabatic potentialsV
J p
(r), which are measured on the basis of the ground OCRC diagonal potentialU
11
(r). Bottom: The fusion cross-sections of the 12 C+ 12 C and12 C+ 13
C systems [21]. The solid curve shows the four-channel OCRC calculation and the dashed curves the no-coupling calculation in the OCRC basis.
potentialV(r)and the perturbative interactionV(r). 3)S(DIAB) denotes theS-matrix
defined in the same way asS(RMO), except that we use the interactions ~
V(r)and ~ V(r)
in the diabatic basis.
4.1. Positive-parity states. – In the positive-parity states(=+), we observe strong
CRC effects due to the coupling interactions in the OCRC basis. Such strong CRC effects can be interpreted as due to the hybridization, i.e. different-parity mixing of single-particle orbits of the p andd (and alsod in particular) states [9], which gives
Fig. 2. –S-matrix elementsS J 21 in the J =7=2 +
state for the process of the inelastic scattering 13 C( 12 C; 12 C) 13 C (3.09 MeV,1=2 +
). The solid, dashed and dotted curves show the OCRC exact calculation, the one-step calculation with the diabatic basis and the one-step RMO calculation, re-spectively.
rise to the ground RMO states having a specific spatial configuration, i.e. the covalent molecule configuration,12
C+n+ 12
C. The adiabatic potentials for these ground RMO states are particularly lowered and isolated from those for the excited RMO states as shown in fig. 1. Due to the decrease of the potential barrier top, we observe there an enhancement of the fusion cross-section and the inelastic cross-section 12
C+ 13
C
(3.09 MeV,1=2 +
) at the sub- and near barrier energies [15]. The elastic scatteringS-matrix elementsS
11(RMO) calculated with the ground state
adiabatic potentials reproduce the exactS-matrix elementsS
11(OCRC) very well, while
the one-step perturbation calculation in the OCRC basis fails in the reproduction of the exact OCRC result [9]. This shows that the molecular orbital picture in the RMO basis is suited for the description of the elastic scattering processes.
The radial couplingsV (1) 12
(r)between the ground RMO state and the first excited
RMO state,p=1and2(forJ
=7=2 +
and9=2 +
) have broad shapes with non-negligible strength. However, we do not observe a pseudo-crossing between the RMO states,p=1
and2. Instead, a sharp crossing exists at a distance inside the potential barrier between
the RMO statesp=2and3. Thus, we perform the diabatic conversion only between the
RMO statesp=2and3. The energy dependence of a matrix elementS
12(DIAB) obtained
in this diabatic system is compared with that ofS
12(OCRC) and S
12(RMO) in fig. 2. Even
though the reproduction ofS
12(OCRC) by S
12(DIAB) is not satisfactory, it is still better
than that byS
12(RMO) in the RMO basis. The diabatic conversion at the crossing point
Fig. 3. – Adiabatic potential diagram and radial couplings in the RMO basis in theJ =7=2 , state for the12 C+ 13 C system.
4.2. Landau-Zener type transitions in negative-parity states. – Molecular orbital properties in negative-parity states are completely different from those in positive-parity states because of the core-symmetric requirement for the12
C+ 13
C system. The ground RMO adiabatic potentials in negative-parity states are neither lowered in the peripheral region, nor isolated from other potentials. Thus, we observe a crossing between the adia-batic potentials of the ground and first-excited RMO statesp=1and2[13, 14] in contrast
to the situation of the positive parity states. The RMO states p = 2and3also have a
crossing at a distance outside the potential barrier as shown in fig. 3. As shown in the lower part of this figure, the coupling interactionjV
(1) pq
(r)k(r)jtakes a very large value
at the pseudo-crossing betweenp=1and2, but just inside the Coulomb barrier. Thus,
this radial coupling cannot be treated as a perturbation.
The transition generated by this coupling is suppressed until the incident energy sur-passes the Coulomb barrier. However, at energies above the Coulomb barrier, the negati-ve-parity transition amplitudes grow up rapidly and interfere in the differential cross-sections with the positive-parity amplitudes, which are already enhanced at sub-barrier energies. This gives rise to a rapid change of the angular distributions in the inelastic scat-tering13 C( 12 C; 12 C) 13 C (3.09 MeV,1=2 +
) when changing the incident energy, as shown in fig. 4.
Fig. 4. – Differential cross-sections of the inelastic scattering,13 C( 12 C; 12 C) 13 C (3.09 MeV,1=2 + ). The solid and dashed curves show the OCRC calculation and the one-step OCRC calculation, re-spectively. The circles show the experimental data [14].
seen in fig. 5a), at a pseudo-crossing betweenp=1and2, the radial coupling ~ V
(1) (r)
is reduced to very small values. As well, we have the same result for the conversion betweenp=2and3. However, the momentum-dependent terms
~ V
(1) pq
(r)(p6=q)in the
diabatic basis still dominate the momentum-independent terms ~ V (2) pq (r)and ~ V (3) (r),
which Landau and Zener assumed to be dominant over ~ V (1) (r)and ~ V (2) (r)[10, 11].
The elastic scatteringS-matrix elementS
11(DIAB) obtained in the diabatic basis gives
a quantitatively good reproduction ofS
11(OCRC) in the energy range we use, while S
11
(RMO) fails in doing so. Figure 5b) shows the energy dependence of theS-matrix element S
21corresponding to the inelastic scattering 13 C( 12 C; 12 C) 13 C (3.09 MeV,1=2 + ). TheS -matrix elementS
21(DIAB) in the diabatic basis gives a good reproduction of the exact
oneS
21(OCRC), much closer than S
21(RMO). It also reproduces the pattern of the exact S-matrix element very well.
Fig. 5. – (a) Radial couplingsV (1)J pq
(r)((p;q)=(1;2)and(2;3)) in the RMO basis(solid curves) and those
~ V
(1)J pq
(r)in the diabatic basis (dashed curves) in theJ
=7=2 ,
state for the12 C+
13 C system. (b) The same as in fig. 2 but for theJ
=7=2
, state.
4.3. Properties of coupling interaction in the diabatic basis. – Landau and Zener intro-duced a diabatic basis by assuming the complete exchange of the adiabatic states [10, 11]. The only coupling interaction left in their diabatic basis is the momentum-independent one. Therefore, theS-matrix,S(DIAB) obtained by the perturbation calculation in the
di-abatic basis of Landau and Zener, should coincide with the full OCRC solutionS(OCRC)
at higher incident energies whereas theS-matrixS(DIAB) within our diabatic basis
dis-agrees. This feature comes from the incomplete exchange of the adiabatic RMO states, and thus momentum-dependent coupling interactions remain in the diabatic basis.
By adopting a typical example of the pseudo-crossing between the RMO states,p=2
and3(r X =7:8fm) atJ =7=2 ,
, we investigated, as a function of the distancer, the
behavior of the overlapjh p (r X ,r); q (r X
+r)ijbetween the RMO statep(p=2or 3) at the left of the pseudo-crossingr=r
Xand the RMO state
q(q=2or3) at the right of
the pseudo-crossing. The results show that two RMO states do not have a complete over-lap, no matter how we choose the distancer. Thus, in our diabatic basis, the radial
cou-plings, ~ V (1) (r)and ~ V (2)
(r), do not vanish. Furthermore, numerical results show that
the momentum-dependent interaction ~ V
(1)
(r) dominates the momentum-independent
ones, ~ V (2) (r)and ~ V (3) (r).
Thiel, Greiner and Scheid [16] and Thiel [17] have discussed the Landau-Zener transi-tion by making an exact quantum-mechanical calculatransi-tion in the molecular-orbital basis generated by the TCSM for the system12
C+ 17 O, 13 C+ 16
O. There exists a sharp radial coupling between the adiabatic ground and first excited states of17
O. They have repro-duced the experimental data reasonably. However, their result [16, 17] is completely dif-ferent from the results of the calculations carried out by Abe and Park [18], Park et al. [19] and Cha, Park and Scheid [20], who have artificially introduced momentum-independent radial couplings in their diabatic basis and discussed the transition. The latter calcula-tions [18-20] should be questioned, since the momentum-dependent terms are neglected there.
5. – Conclusion
We have discussed the nucleon molecular-orbital picture in heavy-ion reactions, with the use of the OCRC/RMO model, by assuming a direct reaction mechanism with cluster constituents for the12
C+ 13
C system. We summarize the results:
– In positive total parity states, we have a good molecular-orbital picture for the ground RMO states. Namely, the strong mixing of the statesp
1=2, s 1=2, d 5=2and d 3=2results in the
formation of a very stable molecular-orbital state, i.e. the covalent molecule12
C+n+ 12
C. This mixing causes the decrease of the potential barrier height for the ground state chan-nel, which explains the enhancement of sub- and near barrier fusion cross-sections ob-served in the experimental data. For the negative total parity states, we do not observe such phenomena.
– We have successfully made the diabatic conversion around the pseudo-crossings in the RMO basis, where we observe sharp radial couplings. Thus, we again establish a good molecular-orbital picture in the diabatic basis, especially for transition phenomena between molecular states.
We have pointed out that our diabatic basis significantly differs from the conventional diabatic basis introduced by Landau and Zener. The momentum-dependent coupling interactions are dominant over the momentum-independent interactions in our diabatic basis. This could be a general feature of Landau-Zener transitions in nuclear heavy-ion reactions.
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