The geometrical mapping of a nuclear vibron model

Transcription

1 REVISTA MEXICANA DE FÍSICA 49 SUPLEMENTO 4, AGOSTO 003 The geometrical mapping of a nuclear vibron model P.O. Hess a, H. Yépez a, and Ş. Mişicu b a Instituto de Ciencias Nucleares, UNAM, Circuito Exterior, C.U., Apartado Postal , México, D.F., Mexico b Institut für Theoretische Physik, J.W.v.-Goethe Universität, Robert-Mayer-Str. 8-10, 6035 Frankfurt am Main, Germany Recibido el 13 de enero de 003; aceptado el 3 de marzo de 003 A nuclear vibron model for two clusters is presented. The dynamical symmetry of the relative motion is U R 3, generally known as the vibrational limit. The geometrical mapping is given. We show that a minimum for the relative nuclear potential at r 0 can be obtained. Starting from a known internuclear potential, the parameters of the model can be deduced, however, with a final ambiguity, generating two classes of spectra. We show that the geometrical information is essential in order to determine not only the allowed range of the parameters of the model but also the structure of the Hamiltonian. Thus, although a fit to a known spectrum can be done with some Hamiltonian, the resulting potential makes not always sense. The procedure is explained for the 1 C+ 1 C system. Keywords: Algebraic cluster model; nuclear molecules. Un modelo nuclear de vibrones para dos cúmulos es presentado. La simetría dinámica para del movimiento relativo es U R 3 generalmente conocido como límite vibracional. El mapeo geométrico es dado. Mostramos que se puede obtener para el potential nuclear relativo un mínimo a r 0. Partiendo de un potencial internuclear dado, se pueden deducir los parametros del modelo, sin embargo, con una ambiguedad final, generando dos clases de espectros. Mostramos que la información geométrica es esencial para poder determinar no solo el rango de los valores de los parameteros permitidos, sino también la estructura del Hamiltoniano. Aunque un ajuste a un espectro dado se puede obtener con un Hamiltoniano dado, el potencial resultante no siempre tiene sentido. El procedimiento esta explicado para el caso de 1 C+ 1 C. Descriptores: Modelo algebraico de cúmulos; moléculas nucleares. PACS: 1.60.Fw; 1.60.Gx 1. Introduction In the past decades several phenomenological models for nuclear molecules have been developed for a nearly complete overview see Ref. [1]. Geometrical [,3] and algebraic models [4, 5] have been developed for two cluster molecules. Both were applied to the system 1 C+ 1 C [6, 7]. Recently, this field of activity received a boost due to the supposed observation of a three cluster molecule [8]. The first attempts to describe these new systems were given in Ref. [9], all being geometrical models. The use of a geometrical description found its limits very soon due to the restriction to very particular configurations. For example, the principal configuration was a linear one all clusters lined up in a row with small oscillations around this equilibrium position. Triaxial configurations, where the triangular points give the position of a cluster, where also tried within a geometrical description, however, with no inner structure of the nuclei spherical clusters [10]. For the interaction the harmonic approximation was assumed. As a next step, algebraic models with inner structure nuclear vibron models where tried out. Algebraic models are more flexible and allow the description of very complicated structures and interactions in simple terms, especially when a dynamical symmetry is involved. First, models for three cluster molecules where investigated [11] within a vibron model [1] without an inner structure of the clusters. Afterwards, we went back to the problem of two cluster molecules in order to understand basic concepts related to the structure of the Hamiltonian and its geometrical content. The intention of this contribution is to present a nuclear vibron model within a particular dynamical symmetry, i.e. U R 3 generally known as the vibrational limit. We will not emphasize the fitting of the parameters to a given spectrum, which can normally not be done for heavy systems due to lack of experimental information. The interesting part is to deduce the parameters of the model starting from a known internuclear potential, which can be determined by various means. This also implies that one has to investigate carefully the geometrical relation of the Hamiltonian to a potential. We will find out that a geometrical input is essential in order to determine not only the correct values of the parameters but also the algebraic structure of the Hamiltonian. Without that, some Hamiltonians, though able to adjust the spectrum, do not make any sense on the geometrical side and vice versa. Geometric constraints will influence the structure of the Hamiltonian and give relations between the different parameters. The ideal system, where the method can be tested, is 1 C+ 1 C, because both the spectrum and the internuclear potential are known. This allows to check the results obtained, starting from a known potential and determining the parameters, which in turn give the structure of the spectrum. The results reported are preliminary and detailed changes can still occur, i.e. we will report on work in progress.

2 40 P.O. HESS, H. YÉPEZ, AND Ş. MIŞICU. The Model.1. The Algebraic Model In order to simplify the investigation, the problem will be restricted to a dynamical symmetry, given by U C1 6 U C 6 U R 4 SU C1 3 SU C 3 SU R 3 [N 1 ] [N ] [N R ] λ 1, µ 1 λ, µ n R, 0 SU C 3 SU R 3 SU3 SO3 ρ c λ C, µ C n R, 0 λ, µ κ L, 1 where N i gives the number of bosons of cluster i in the IBA model, n R the total number of bosons in the relative motion, λ i, µ i denotes the SU3 representation of the i-th cluster, λ C, µ C is the SU C 3-irrep to which the two clusters are coupled, λ, µ is the total SU3 irrep, L the angular momentum and ρ C and κ are multiplicity indices. No multiplicity appears in the reduction of the SU C 3 SU R 3 to the total SU3 group because of the symmetric irrep n R, 0. The κ value can be approximately related to the projection of the angular momentum L on the intrinsic z- axis [13]. In case of a symmetric system, the additional condition λ + µ + n R =even has to be observed [14]. Note that for symmetric systems where no inner structure is taken into account λ = µ = 0 only n R = even is allowed, which implies L = even and positive parity. However, when the inner structure is included the final λ and µ values are in general different from zero. In this case the n R can be odd and consequently L = odd too and the negative parity is also allowed. The use of the U R 3 dynamical chain is suggested by Ref. [15] and convincingly applied in Ref. [16]. There, the results were compared to the ones of the SO R 4 model without inner structure of the clusters [7], concluding that the data recommend a U R 3 dynamical symmetry. The structure of the individual clusters is described via an Interacting Boson Approximation IBA. For heavy light clusters the IBA-1 IBA-4 is used [17, 19]. A possible Hamiltonian, which has the dynamical symmetry 1, can be given by H=χ 1 C SU C1 3 +χ C SU C 3 +χ 1 C SU C 3 + χ ω n r + χ R C SU R 3 +χ T C SU3 + χ N C SU C 3 C SU C 3 θ0 n r + al + ck, where χ ω is the parameter of n R and the other χ-values give the strength of the different quadrupole-quadrupole interactions, L is the angular momentum operator and K is the operator introduced in Ref. [13] whose eigenvalue is approximately the square of the projection of the angular momentum onto the intrinsic symmetry axis. The second order Casimir operators are a function of the angular momentum operators and the quadrupole operators. The significance of the term proportional to χ N will be explained later on. Its origin lies in the possibility to adjust the position of the minimum for a relative orientation of the symmetry axis different from the ground state orientation. The expression < C SU C 3 > θ0 contains geometric information and we will show further below that, in order to obtain a suitable Hamiltonian, its use is necessary, otherwise inconsistencies with the calculated internuclear potential appears. The first two terms in the first line of describe the structure of the individual nuclei in the SU3 rotational limit of the Interacting Boson Approximation IBA [17]. The eigenvalues of the second order Casimir operators are C SU3 = C λ, µ = λ + λµ + µ + 3λ + µ, from which the eigenvalues of can be deduced. For later use, it is important to note that the second order Casimir operator C SU C 3, describing the relative coupling of the two clusters, appears not only multiplied by the parameter χ 1 but also within the operator C SU3 of the total coupling and in a term which is multiplied by χ N. Thus, χ 1 + χ T + n 0 χ N, where n 0 is the number of relative oscillation quanta in the ground state, determines the strength of the relative cluster coupling and, therefore, the scale of these excitations. The same holds for χ T + χ R which determines the strength of the coupling to the relative vibrational motion and the scale of excitation in n R. Note also, the number of n R = n 0 in the ground state is not necessarily zero. This reflects the nonzero position of the relative minimum, which in an oscillator basis requires an n 0 > 0... The geometrical mapping The geometrical mapping of the Hamiltonian is performed, using as a trial state the coherent state [18] N R, α 1 = s + α 1 p 0 N R NR!1 + α 1 N R 0, 3 where α 1 is the parameter of the normalized coherent state and N R is the total number of bosons, comprised by the number of p-bosons, describing the relative motion, and the auxiliary s-boson. Because in the intrinsic molecular system the z-axis is defined by the line connecting the center of masses of the clusters, only the z-component p 0 of the p-boson is

3 THE GEOMETRICAL MAPPING OF A NUCLEAR VIBRON MODEL 41 necessary. The geometrical potential is obtained, determining the expectation value of the Hamiltonian to this coherent state. We obtain a function in the variable α 1 which has to be related to the relative distance. We define the relation via [18] r m = ra m 4 s s with r a m = b p ms + s p m. 5 as the distance variable. The upper index a refers to algebraic and the subindex m to the spherical component of the distance operator. In the molecular system, the z-axis is defined along the axis connecting both center of masses. In that case the only distance variable of importance is r 0 which we denote from here on as r, the internuclear distance. Note, that in 5 the oscillator length b appears which defines a scale in the model. With this, the Hamiltonian is mapped to the geometrical potential H = V [ V = χ 1 + χ 1 + χ T 10N 1 + 4N 1N 1 1β1 1 + β1 ± β ] 4 β 1 ± β + 1 ] 4 β + χ R + χ T b r + 1 4b 4 r4 1 ± β 3 cos θ 1 θ χ T + χ ω 1 [ + χ + χ 1 + χ T 10N + 4N N 1β 1 + β 8N 1 N β 1 β b r + χ 1 + χ T 1 + β1 1 + β 1 ± β 1 N 1β 1 b r 1 + β1 1 ± β 3 cos 1 4N 1 N β 1 β θ 1 + χ N b r 1 + β1 1 + β 1 ± β 1 3 cos θ χ N β T b r 1 + β 1 ± β 1 1 ± β [ 3 cos θ 1 θ 1 3 cos θ 10 θ 0 1 ], 6 where the upper lower sign refers to prolate oblate nuclei [19]. The β i i = 1, are the deformation of the clusters within the IBA model [17] and b is the oscillator length. The last term in the square bracket of the last line comes from the expectation value < C SU C 3 > θ0 at a particular angle θ 0, consisting actually of the two orientation angles θ 10 and θ 0, giving the orientation of the two symmetry axis in the equilibrium position of the nuclear molecule, with respect to the molecular z-axis. It guarantees that the minimum of the potential at these orientations is at the desired position. Otherwise, the structure of the Hamiltonian is too rigid and would produce a dependence in the orientation which contradicts severely the calculated relative nuclear potential and leads to inconsistencies see conditions below. Suppose, we know only the relative nuclear potential and its dependence on the relative orientation angle. The question is: How far we can get in deducing the parameters of the model? In order to answer this question, we first have to define the conditions the potential and the energies have to fulfill, giving relations between the parameters of the model. The conditions are: i The position of the minimum should agree to the calculated one, i.e. V r = 0, 7 r0,θ 0 where r 0 is the position of the minimum at the equilibrium orientation θ 0. ii The stiffness C r at the minimum as obtained in the double folding calculation, i.e. for r 0 and a given orientation angle θ 0, should be the same V r r 0,θ 0 = C r. 8 iii The slope of the potential at the minimum position r 0 for a different orientation than the equilibrium one C r,θ1 should be reproduced or somehow given, i.e. V r r 0θ 1 = C r,θ1, 9 this condition can also be substituted by requiring that the position of the minimum at θ 1 is reproduced. iv The expectation value of < r > is equal to r 0, which gives and r 0 = b n 0 10 v taking this value of n 0 for the number of relative quanta n R, one has to obtain the lowest energy using the algebraic expression of the model En R = n 0 = min., 11 where we assumed, that the lowest irreducible representation irrep is at the maximal coupling, i.e. λ, µ = λ C + n 0, µ C

4 4 P.O. HESS, H. YÉPEZ, AND Ş. MIŞICU of the cluster irrep with the relative motion, where λ C, µ C is obtained by a coupling of the individual cluster states. Conditions 10 and 11 cannot be satisfied without the interaction term in the Hamiltonian which contains the geometrical information. If this term would be skipped, the n 0 value deduced in 10 will not give the lowest state in energy, i.e. in contradiction to Application to 1 C+ 1 C The system 1 C+ 1 C is particular useful because both the energy spectrum and the relative nuclear potential is known. A list of the molecular states can be obtained in Ref. [1] and the relative potential can be obtained via a double folding calculation [, 4], where we use the procedure outlined in Ref. [3]. In this manner, the resulting spectrum, through fitting the parameters to the potential, can be compared to the measured one and the method can be checked. This is particularly important for systems where no experimental information is available, as for example heavy systems where quasi-molecules with three clusters may occur. The deformation value of 1 C is deduced from the BE-value of Ref. [5], taking the relation of β phys [0] to the BE-value as given in Ref. [6]. Here the corrections due to the large deformation is important. The equilibrium orientation is assumed such that both symmetry axes are perpendicular to the molecular axis. This configuration minimizes the Coulomb energy and is generated by coupling the irrep of SU3 to the maximal one. The orientation angle for that system is now defined to be θ 1, the last being the orientation of the symmetry axis of the carbon nuclei. Using the analytical separation of the angular variables in the double folding calculation [4] we determined a cut through the potential at θ 1 = Applying the above constraints, we arrive at a relation of all parameters as a function of one parameter left, e.g. χ ω. The explicit description of the details will be published elsewhere [7]. In Fig. 1 we plot the dependence of χ T on χ ω, where the lines indicate the allowed solutions of the above posed conditions. Note that several branches are allowed. However, only one single branch makes physical sense: either the number of oscillation quanta n 0 in the ground state is negative, or χ T + χ R, the factor in front of r 4, is negative or zero i.e. the potential is unstable. Only the lower line corresponding to negative χ ω fulfills all conditions. Restricting the possible χ ω values to this physical branch, in Fig. we plot the dependence of χ T, χ R +χ T, χ 1 +χ T and χ 1 + χ T + n 0 χ N on the parameter χ ω. In Fig. 3 the n 0 and the b are plotted versus χ ω. We have not found a further condition to determine the value of χ ω. Of course, the measured spectrum could be used in order to fix the scale of χ ω. However, we supposed that we do not know it and we want to deduce the molecular spectrum. Unfortunately, the χ-parameters cannot be completely FIGURE 1. Dependence of the parameter χ T on χ ω. The lines correspond to the solutions satisfying the conditions posed in the last section. The physical lower left with negative χ ω and unphysical branches are indicated. FIGURE. χ-parameters and combinations as a function on χ ω, for the physical branch. fixed and we can only discuss the types of spectra to be expected. Note, that there are two distinct regions: one where χ T + χ R is small left hand side of Fig. and the other one where it is large right hand side of Fig.. The combination determines the position of the relative vibrational excitations different n R. The χ 1 + χ T + n 0 χ N varies slowly over a wide range, approaching 0 for large χ ω, and it determines the scale of excitation of λ C, µ C. If χ 1 + χ T + n 0 χ N is negative positive the lowest SU C 3 irrep in energy is the largest smallest irrep λ C, µ C. On the other hand, when χ T + χ R, which is the factor of n r, is small, the relative vibrational states dominate at low energy while for χ T + χ R they are at large energy. For small χ ω, the minimum of the potential corresponds to a large n 0 while for large χ ω the n 0 is of the order of 10. In order to compare to measured energy spectrum, in the Fig. 4a the spectrum of the two possible limits are shown. The left panel corresponds to χ T + χ R small χ ω =.5

5 THE GEOMETRICAL MAPPING OF A NUCLEAR VIBRON MODEL 43 FIGURE 3. The number of oscillation quanta n 0 of the ground state and the square of the oscillator length b as a function on χ ω. In order to reproduce the ground state at the same position, a smaller oscillation length stiffer basis states implies a larger number of relative oscillations in the basis. In this sense, the absolute number of oscillation quanta does not have an immediate physical interpretation, except that the equilibrium distance of the two clusters r 0 = b n 0 is different from zero. and the right panel to χ T +χ R large χ ω = 40. In the right panel of the Fig. 4b we show the experimental spectrum while in the right panel the SO R 4 dynamical symmetry without inner structure of the clusters is presented [7]. The potential, as deduced for the first case upper part of Fig. 4, left panel, is given in Fig. 5. When the term proportional to χ N is skipped, an inconsistency appears. For that case, the number of relative oscillation quanta n 0 in the ground state will be very different using either condition 10 or 11. Taking any one of these conditions a minimum will show up at a value θ 1 which is very different from the one obtained in a double folding calculation. This shows that one has to be careful in proposing Hamiltonians in an algebraic model and one has to take into account the potential it corresponds to. As can be seen, the relative nuclear potential obtained in a double folding calculation is qualitatively reproduced and in good agreement near the minimum. For the other case the potential is practically the same, showing that for a wide range of χ ω the form of the potential does not change the other χ-parameters depend on χ ω and change accordingly and that the knowledge of the potential alone is not sufficient for determining all parameters of the algebraic model. Also the spectra in both cases are comparable. Of course, due to having started from a potential, the spectrum is not completely reproduced but the characteristics are the same. This also indicates that just reproducing the spectrum is not sufficient an old knowledge because the interpretation of the states are different in both limits. We also tried to adjust the spectrum and to map the resulting Hamiltonian to a potential and as it seems work in progress the potential obtained is consistent with the one in Fig. 5. Also here we have to use the geometrical information that the n 0 is related to the position of the minimum somehow obtained from geometrical considerations, like the sum of the two cluster radii. We also deduced the potential within the SO R 4 model, obtaining a potential which is too soft and very shallow. 4. Conclusions We have presented a nuclear vibron model within the dynamical symmetry group U R 3 for the relative motion. A Hamiltonian was suggested, which included terms with geometrical information, otherwise no consistency in the mapping to a geometrical potential could be obtained. A geometrical mapping was given and we showed that a minimum of the geometric potential can be obtained. In determining the parameters of the model, several constraints had to be applied in order to obtain consistency between the algebraic FIGURE 4. In a part of the figure the left panel gives the spectrum for χ ω =.5 and the right one for χ ω = 40. In b part of the figure the left panel shows the experimental spectrum and the right panel the result of the SO4 model [7].

6 44 P.O. HESS, H. YÉPEZ, AND Ş. MIŞICU FIGURE 5. The potential as obtained in the geometrical mapping, for χ ω =.5. The solid and the dashed dotted line are from the double folding calculation at θ 1 = 0 0 and θ 1 = 15 0 respectively. The corresponding lines from the geometrical mapping carry additional symbols. Hamiltonian, its spectrum, and the resulting geometrical potential. We showed that geometric information is necessary. Only adjusting the parameters to the energy is not sufficient. The spectrum might be reproduced but the potential in general does not make sense or vice versa, because the consistency constraints are not always fulfilled. This is particularly important the more parameters in the model appear, as is in general the case for the vibron model with two clusters, including inner structure. It will be even more important for three cluster molecules. That is why the investigation of the nuclear vibron model with two clusters is so important before going back to the three cluster case. One important result is that the use of the dynamical chain 1 does not produce a strong Coriolis coupling, as obtained in [11]. This will be maintained also for the three cluster case. It indicates that one has to be very careful in choosing the interaction of an algebraic Hamiltonian. Geometrical information, like to be consistent with a double folding calculation, is important. Acknowledgment The authors acknowledge very fruitful discussions with J. Cseh and G. Lévai from the ATOMKI Debrecen, Hungary. This work was supported by the CONACyT-MTA and CSIC- MTA exchange programs. H.Y.M. acknowledges financial support from DGEP-UNAM and Ş.M. from the European Community through a Marie Curie fellowship. Financial help from DGAPA, project No. IN11900, is acknowledged.. On leave of absence from the National Institute for Nuclear Physics, Bucharest, P.O. Box MG6, Romania. 1. W. Greiner, J. Y. Park and W. Scheid, Nuclear Molecules Singapore: World Scientific, P.O. Hess, W. Greiner and W.T. Pinkston, Phys. Rev. Lett ; P.O. Hess and W. Greiner, Il Nuovo Cimento E. Uegaki, Prog. Theor. Phys. Suppl F. Iachello, Nucl. Phys. A c. 5. H.J. Daley and F. Iachello, Ann. Phys ; H.J. Daley and B.R. Barrett, Nucl. Phys. A P.O. Hess and P. Pereyra, Phys. Rev. C K.A. Erb and D.A. Bromley, Phys. Rev. C A.V. Ramayya, J.K. Hwang, J.H. Hamilton, A. Sandulescu, A. Florescu, G.M. Ter-Akopian, A.V. Daniel, Yu.Ts. Oganessian, G.S. Popeko, W. Greiner, J.D. Cole and GANDS95 Collaboration, Phys. Rev. Lett ; A.V. Rammaya, J.H. Hamilton, J.K. Hwang and GANDS95 Collaboration, Rev. Mex. Fís P.O. Hess, W. Scheid, W. Greiner, and J.H. Hamilton, J. Phys. G L139; Ş. Mişicu, P.O. Hess, A. Săndulescu, and W. Greiner, J. Phys. G L147; P.O. Hess, Ş. Mişicu, W. Greiner, and W. Scheid, J. Phys. G 6 000, 1; P. O. Hess, S. Misicu, and W. Greiner, Rev. Mex. Fís. 46 S Ş. Mişicu, P.O. Hess, and W. Greiner, Phys. Rev. C R. Bijker, P.O. Hess and Ş. Mişicu, Heavy Ion Phys ; P.O. Hess, R. Bijker, and Ş. Mişicu, Rev. Mex. Fís. 47 S R. Bijker and L. Leviatan, Few-Body Systems H. Naqvi and J.P. Draayer, Nucl. Phys. A J. Cseh, Phys. Lett. B ; J. Cseh and G. Lévai, Ann. Phys. N.Y J. Cseh, Phys. Rev. C ; J. Cseh and J. Suhonen, Phys. Rev. C J. Cseh, G. Lévai, and W. Scheid, Phys. Rev. C F. Iachello and A. Arima, The Interacting Boson Model, Cambridge Monographs on Mathematical Physics, Cambridge, H. Yépez, P.O. Hess, and S. Misicu, Heavy Ion Physics D. Bonatsos, The Interacting Boson Models of Nuclear Structure Oxford University Press, Oxford, J.N. Ginocchio and W. Kirson, Nucl. Phys. A U. Abbondanno Report INFBM/BE-91/11 Trieste, M. Seiwert, W. Greiner, and W.T. Pinkston, J. Phys. G L1. 3. Ş Mişicu, A. Sandulescu, F. Carstoiu, M. Rizea, and W. Greiner, Il Nuovo Cimento A F.Carstoiu and R.J.Lombard, Ann.Phys S. Raman, C.H. Malarkey, W.T. Milner, C.W. Nestor, and P.H. Stelson, Atom. Data and Nucl. Data Tab J.M. Eisenberg and W. Greiner, Nuclear Theory I: Nuclear Models North-Holland, Amsterdam, H. Yépez, P.O. Hess, and Ş Mişicu, in progress 00.