Resonances in ^{12}C and ^{24}Mg: what do we learn from a microscopic cluster theory?
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Resonances in C and Mg : what do we learn from a microscopiccluster theory? P. Descouvemont a Physique Nucl´eaire Th´eorique et Physique Math´ematique, C.P. 229, Universit´e Libre de Bruxelles (ULB), B 1050 Brussels,Belgium Received: date / Revised version: date
Abstract.
We discuss resonance properties in three-body systems, with examples on C and Mg. We usea microscopic cluster model, where the generator coordinate is defined in the hyperspherical formalism. The C nucleus is described by an α + α + α structure, whereas Mg is considered as an O + α + α system.We essentially pay attention to resonances. We review various techniques which may extend variationalmethods to resonances. We consider 0 + and 2 + states in C and Mg. We show that the r.m.s. radiusof a resonance is strongly sensitive to the variational basis. This has consequences for the Hoyle state (0 +2 state in C) whose radius has been calculated or measured in several works. In Mg, we identify two 0 + resonances slightly below the three-body threshold. PACS.
XX.XX.XX No PACS code given
Clustering is a well established phenomenon in light nuclei(see Refs. [1,2,3] for recent reviews). In nuclear physics,most cluster states involve the α particle. Due to its largebinding energy, the α particle tends to keep its own iden-tity, which leads to the α cluster structure [4,5]. Thecluster structure in α nuclei (i.e. with nucleon numbers A = 4 k , where k is an integer number) was clarified byIkeda et al. [6] who proposed a diagram which identi-fies situations where a cluster structure can be observed.The α cluster structure is essentially found for nuclei near N = Z . The α model and its extensions were utilized bymany authors to investigate the properties of α -particlenuclei such as Be, C, O, etc. In particular, the in-terest for α -cluster models was recently revived by thehypothesis of a new form of nuclear matter, in analogywith the Bose-Einstein condensates [7,8].Nuclear models are essentially divided in two cate-gories: (1) in non-microscopic models, the internal struc-ture of the clusters is neglected, and they interact by anucleus-nucleus potential; (2) in microscopic models, thewave functions depend on the A nucleons of the system,and the Hamiltonian involves a nucleon-nucleon interac-tion. Recent developments in nuclear models [9,10,11] aimto find exact solutions of the A -body problem, but theypresent strong difficulties when the nucleon number in-creases. To simplify the problem, cluster models assumethat the nucleons are grouped in clusters. The simplestvariant is a two-cluster model and is being developed since a Directeur de Recherches FNRS more than 40 years [12,13]. Multicluster microscopic mod-els are more recent (see, for example, Ref. [3]). They allowto extend the range of applications.In the present paper, we focus on two α nuclei: Cand Mg, which are described by three-body structures(3 α and O + α + α , respectively). Over the last 20 years,there was a strong interest on C, and in particular on the0 +2 resonance, known as the Hoyle state [14]. The unboundnature, however, make theoretical studies delicate.With C and Mg as typical examples, we discussmore specifically the determination of resonance proper-ties in cluster models. As resonances are unbound, a rig-orous treatment would require a scattering model, withscattering boundary conditions. There are, however, var-ious techniques aimed at complementing the much sim-pler variational method which is, strictly speaking, validfor bound states only. In a variational method, negativeenergies are associated with physical bound states. Thepositive eigenvalues correspond to approximations of thecontinuum. For narrow resonances, a single eigenvalue isin general a fair approximation. We show, however, thatthe calculation of physical quantities, such as the r.m.s.radius, should be treated carefully.The paper is organized as follows. In Sec. 2, we presentthe microscopic three-body model, using hypersphericalcoordinates. Section 3 is devoted to a brief discussion ofdifferent techniques dealing with resonances. The C and Mg nuclei are presented in Sects. 4 and 5, respectively.Concluding remarks are given in Sect. 6.
P. Descouvemont: Resonances in C and Mg: what do we learn from a microscopic cluster theory?
In a microscopic model, the Hamiltonian of a A -nucleonsystem is given by H = A X i =1 T i + A X i Coordinates in the microscopic three-cluster system. where the hyperradius ρ and the hyperangle α are definedas ρ = ~x + ~y α = arctan( y/x ) . (4)The hyperspherical formalism is well known in non-mi-croscopic three-body systems [22,21], where the structureof the nuclei is neglected. This formalism makes use offive angles Ω = ( Ω x , Ω y , α ) and of an hypermomentum K which generalizes the concept of angular momentum intwo-body systems. The hyperspherical functions are [23] Y ℓ x ℓ y KLM ( Ω ) = φ ℓ x ℓ y K ( α ) (cid:2) Y ℓ x ( Ω x ) ⊗ Y ℓ y ( Ω y ) (cid:3) LM , (5)where ℓ x and ℓ y are angular momenta associated with theJacobi coordinates xxx and yyy . Functions φ ℓ x ℓ y K are given by φ ℓ x ℓ y K ( α ) = N ℓ x ℓ y K (cos α ) ℓ x (sin α ) ℓ y P ℓ y + ,ℓ x + n (cos 2 α ) . (6)In these definition, P α,βn ( x ) is a Jacobi polynomial, N l x l y K is a normalization factor, and n = ( K − ℓ x − ℓ y ) / Ψ JMπ = X ℓ x ℓ y ∞ X K =0 A φ φ φ Y ℓ x ℓ y KJM ( Ω ) χ Jπℓ x ℓ y K ( ρ ) , (7)where A is the A -nucleon antisymmetrizor, and φ i are thecluster wave functions, defined in the shell-model. For the α particle, the internal wave function φ is a Slater deter-minant involving four 0 s orbitals. Definition (7), however,is valid in a broader context where φ is a linear combi-nation of several Salter determinants. A recent example[24] is the Li nucleus described by a Li+n+n structure,where the shell-model description of Li involves all Slaterdeterminants (90) which can be built in the p shell. Theoscillator parameter b is taken identical for the three clus-ters. Going beyond this approximation raises enormousdifficulties due to center-of-mass problems, even in two-cluster calculations [25]. In Eq. (7), the hypermoment K . Descouvemont: Resonances in C and Mg: what do we learn from a microscopic cluster theory? 3 runs from zero to infinity. In practice a truncation value K max is adopted. The hyperradial functions χ Jπℓ x ℓ y K ( ρ ) areto be determined from the Schr¨odinger equation (2).As for two-cluster systems, the RGM definition clearlydisplays the physical interpretation of the cluster approxi-mation. In practice, however, using the Generator Coordi-nate Method (GCM) wave functions is equivalent, and ismore appropriate to systematic numerical calculations [3].In the GCM, the wave function (7) is equivalently writtenas Ψ JMπ = X γ Z dR f Jπγ ( R ) Φ JMπγ ( R ) , (8)where we use label γ = ( ℓ x , ℓ y , K ). In this equation, R isthe generator coordinate, Φ JMπγ ( R ) are projected Slaterdeterminants, and f Jπγ ( R ) are the generator functions (seeRef. [3] for more detail). In practice, the integral is re-placed by a sum over a finite set of R values (typically10-15 values are chosen, up to R ≈ − 12 fm).After discretization of (8), the generator functions areobtained from the eigenvalue problem, known as the Hill-Wheeler equation, X γn (cid:20) H Jπγ,γ ′ ( R n , R n ′ ) − E Jπk N Jπγ,γ ′ ( R n , R n ′ ) (cid:21) f Jπ ( k ) γ ( R n ) = 0 , (9)where k denotes the excitation level. The Hamiltonian andoverlap kernels are obtained from 7-dimension integrals in-volving matrix elements between Slater determinants (seeRefs. [20,3] for detail). They are given by N Jπγ,γ ′ ( R, R ′ ) = h Φ JMπγ ( R ) | Φ JMπγ ′ ( R ′ ) i H Jπγ,γ ′ ( R, R ′ ) = h Φ JMπγ ( R ) | H | Φ JMπγ ′ ( R ′ ) i . (10)The non-projected matrix elements are computed withBrink’s formula [4], and the main part of the numericalcalculations is devoted to the two-body interaction whichinvolves quadruple sums over the orbitals. The projectionover angular momentum, which requires multidimensionintegrals, makes the calculation very demanding. A moredetailed description is provided in Ref. [24]. From the wave function (7,8), various properties can becomputed. We discuss more specifically the r.m.s. radius,defined as h r i = 1 A h Ψ JMπ | A X i =1 ( r i − R c . m . ) | Ψ JMπ i , (11)which, in the GCM is determined from h r i = X γn X γ ′ n ′ f Jπγ ( R n ) f Jπγ ′ ( R ′ n ) × h Φ JMπγ ( R n ) | A A X i =1 ( r i − R c . m . ) | Φ JMπγ ′ ( R n ′ ) i . (12) The matrix elements between Slater determinants are ob-tained as in Eqs. (10). Notice that these calculations arerigorous for bound states, i.e. states with energy E Jπk lowerthan the three-cluster breakup threshold E T = E + E + E , E i being the internal energy of cluster i , computedconsistently with the same Hamiltonian. In this case, therelative functions χ Jπγ ( ρ ) [see Eq. (7)] tend rapidly to zero,and the sum over ( n, n ′ ) in (12) converges. For resonances( E Jπk > E T ), the convergence of (12) is not guaranteed.We discuss this issue in more detail in Sect. 3.The energies and r.m.s. radii discussed in the previoussubsection involve all generator coordinates. It is, however,useful to analyze the energies of the system for a single R value. This quantity is referred to as the energy curves.Two variants can be considered. In the former, only thediagonal matrix elements of the Hamiltonian are used, andthe energy curves are defined as E Jπγ ( R ) = H Jπγ,γ ( R, R ) N Jπγ,γ ( R, R ) − E T . (13)This definition ignores the couplings between the channels.At large distances, they tend to E Jπγ ( R ) → Z γγ e R + ¯ h m N ( K + 3 / K + 5 / R + 14 ¯ hω, (14)where Z γγ e is a diagonal element of the Coulomb three-body interaction (see, for example, Ref. [26]), m N is thenucleon mass, and ¯ hω is the residual energy associatedwith the harmonic oscillator functions (¯ hω = ¯ h /m N b ).In the alternative approach, the energy curves stemfrom a diagonalization of the Hamiltonian for a fixed R value. They are given by the eigenvalue problem X γ (cid:20) H Jπγ,γ ′ ( R, R ) − E Jπk ( R ) N Jπγ,γ ′ ( R, R ) (cid:21) c Jπ ( k ) γ ( R ) = 0 . (15)At large distances, the coupling elements ( γ = γ ′ ) tend tozero and both definitions (13) and (15) are equivalent. Inthree-body systems, however, the couplings are known toextend to large distances, even for short-range interactions(see, for example, Refs. [27,26,28]).The energy curves cannot be considered as genuine po-tentials. However, they provide various informations, suchas the existence of bound states or of narrow resonances,the level ordering, the cluster structure, etc. The eigenvalue problem (9) is, strictly speaking, valid forbound states only. The variational principle guaranteesthat an upper limit of the exact solution is found, andthe wave function tends exponentially to zero. The situ-ation, however, is different for positive-energy states. In P. Descouvemont: Resonances in C and Mg: what do we learn from a microscopic cluster theory? that case, the lowest energy is zero, i.e. the optimal solu-tion, according to the variational principle, corresponds toa system where the clusters are at infinite distance fromeach other.For narrow resonances, the bound-state approximation(BSA), which is a direct extension of (9), usually providesa fair approximation of the energy, even with finite bases.If the width is small, the energy is fairly stable when thebasis changes. The situation is different for the wave func-tion. The long-range part may be sensitive to the choice ofthe basis, and matrix elements using these wave functionsmay be unstable. A typical example will be shown withthe 0 +2 resonance of C.For broad resonances, there are various techniques whichcomplement the variational method. The idea is to avoidscattering calculations, such as in the R -matrix theory[29], where resonance properties are derived from an anal-ysis of the phase shifts (or scattering matrices). The com-plementary methods have solid mathematical foundations,and are, in principle, relatively simple to implement invariational calculations. We briefly summarize below someof them. – The complex scaling method (CSM) is based on therotation of the space and momentum coordinates [30,31,32]. In other words, the space coordinate rrr and mo-mentum ppp of each particle are transformed as U ( θ ) rrrU − ( θ ) = rrr exp( iθ ) ,U ( θ ) pppU − ( θ ) = ppp exp( − iθ ) , (16)where θ is the rotation angle. Under this transforma-tion, the Schr¨odinger equation reads H ( θ ) Ψ ( θ ) = U ( θ ) HU − ( θ ) Ψ ( θ ) = E ( θ ) Ψ ( θ ) , (17)and the solutions Ψ ( θ ) are square-integrable providedthat θ is properly chosen [32]. They can be expandedover a finite basis, after rotation of the Hamiltonian. Ofcourse the potential should be available in an analyticform to apply transformation (16).The ABC theorem [31] shows that the eigenvalues E k ( θ )are located on a straight line in the complex plane, ro-tated by an angle 2 θ . Resonant states are not affectedby this angle and correspond to stable eigenvalues E k ( θ ) = E R − iΓ/ , (18)where E R is the energy and Γ the width of the reso-nance.Recently, the CSM has been extended to the calcula-tion of level densities [33,34] and to dipole strengthdistributions [32]. Of course the resonance properties(18) derived from the CSM should also be consistentwith those derived from a phase-shift analysis. – In the complex absorbing potential (CAP) method, animaginary potential is added to the Hamiltonian ker-nel. The first applications were developed in atomicphysics [35] and in non-microscopic nuclear models[36]. Ito and Yabana [37] have extended the method to microscopic cluster calculations within the GCM.The Hamiltonian kernel (10) is replaced by H ( R, R ′ ) → H ( R, R ′ ) − iηW ( R ) δ ( R − R ′ ) , (19)where η is a positive real number. The absorbing po-tential is usually taken as W ( R ) = θ ( R − R )( R − R ) β , (20)where R is an arbitrary radius, larger than the rangeof the nuclear force and θ is the step function. In mostcalculations, β is taken as β = 4.This method provides the energy and widths of reso-nances as in (18), even for broad states. In Ref. [37],it was shown, however, that the method needs manygenerator coordinates. In their microscopic study of α + He scattering, Ito and Yabana use 100 generatorcoordinates, up to R = 50 fm. For computational rea-sons, this method is difficult to apply to three-bodysystems, owing to the strong couplings between thechannels, and to the long range of the potentials. – The analytic continuation in a coupling constant (ACCC) method has been proposed by Kukulin et al. [38] toevaluate the energy and width of a resonance. Themain advantage is that the ACCC method only re-quires bound-state calculations, much simpler than scat-tering calculations involving boundary conditions. Someapplications to a microscopic description of two- andthree-cluster models can be found in Ref. [39]. Themethod has been applied to a non-microscopic descrip-tion of C in Ref. [40].To apply the ACCC method, one assumes that theHamiltonian can be written as H ( u ) = H + u H , (21)where u is a linear parameter. The linear part H issupposed to be attractive so that, for increasing u val-ues, the system becomes bound. For u = u , the energyis zero, and we have E ( u ) = 0.The problem is to determine the resonance proper-ties for the physical value u < u . In the bound-stateregime ( u > u ), the wave number k is imaginary, andis parametrized by a Pad´e approximant as k ( x ) = i c + c x + · · · + c M x M d x + · · · + d N x N , (22)where x = √ u − u , and ( M, N ) define the degree ofthe Pad´e approximant. The ( M + N + 1) coefficients c i and d j are calculated in the bound-state region byusing u i values such that E ( u i ) < 0. Going to thephysical u value ( u < u , x imaginary), one determines k from (22). The energy E R and width Γ are obtainedfrom E = ¯ h k m = E R − iΓ/ . (23) . Descouvemont: Resonances in C and Mg: what do we learn from a microscopic cluster theory? 5 The ACCC is, in principle, a simple extension of thevariational method. However, it was pointed out [38,39] that this method requires a high accuracy in thenumerical calculation. In particular, the u value mustbe determined with several digits. It is therefore notrealistic for microscopic three-body systems. – The box method [41] can be used for narrow reso-nances. This method has been applied essentially inatomic physics. The idea is to search for positive eigen-values of the Schr¨odinger equation (2) inside a box.Then, looking at the eigenvalues as a function of thebox size, a narrow resonance appears at a stable energy(see, for example, Fig. 1 of Ref. [41]). This method issimpler than other approximate techniques and, there-fore, permits the use of large bases. The method hasbeen recently extended to the determination of reso-nance widths [42]. However, it requires the numericalcalculation of the first and second derivatives, whichmeans that, in practice, many generator coordinatesmust be used for a good accuracy of the resonanceparameters. C The C nucleus has attracted much attention in recentyears, in particular for the 0 +2 resonance, known as theHoyle state (see Ref. [14] for a recent review). The Hoylestate, located just above the 3 α threshold ( E R = 0 . α reaction rate. Its exis-tence was predicted by Hoyle [43] on the basis of the ob-served abundance of C.There is an impressive literature about the Hoyle state,and we refer to Refs. [14,8] for an overview. One of its char-acteristics is to present a marked α + Be cluster structure[44]. This α clustering is well established in many lightnuclei, such as Li, , , , Be, , , O, , , Ne [5]. Thespecificity of the C nucleus is that the second cluster Be also presents an α cluster structure. As in all excitedstates located near a breakup threshold, the Hoyle statepresents an extended density, which means that the den-sity at short distances is decreased if compared to wellbound states. This natural property, common to all nu-clei, lead some authors to refer to the concept of ”dilutegas” [45,46] and of Bose-Einstein Condensates [7,8].Our aim here is not to perform new calculations onthe C nucleus. The first microscopic 3 α calculation wasperformed by Uegaki et al. in 1977 [47], and improved indifferent ways [48,44,7,49,45,50]. The ab initio calcula-tion of Ref. [49] works rather well for the ground state,but needs the introduction of specific 3 α configurationsto reproduce the Hoyle state. One of the frequent issuesabout the Hoyle state is the determination of its r.m.s. ra-dius, which is expected to be large (see references in Ref.[14]).We adopt the same microscopic 3 α model as in Ref.[50], where the Minnesota nucleon-nucleon interaction [18]with u = 0 . -15-10-50510152025 0 5 10 15 20 25 (cid:1) = 0 (cid:4) (cid:5) (fm) (cid:11) (cid:12) (cid:13) (cid:4) (cid:5) ( M e V ) 08 64 Fig. 2. Energy curves (13) for J = 0 + in C. The curvesare labeled by the K values. The dashed line represents theasymptotic behaviour (14) for K = 0. In Fig. 2, we display the energy curves (13) for J = 0 + and for different K values. Note that in C, a full sym-metrization of the wave functions for α exchange leads tothe cancellation of the K = 2 component (see, for exam-ple, Ref. [26]). The dashed line illustrates the asymptoticbehaviour (14) with Z = 28 . 81 [26]. The energy curvesfor K = 0 , , R = 4 fm. Atshort distances, the antisymmetrization between the nu-cleons makes these curves equivalent. This is not true atlarge distances, where the effective three-body Coulombinteraction is different, and where the centrifugal termplays a role.Figure 3 presents the binding energy (with respect tothe 3 α threshold) of the 0 +1 and 0 +2 states for increasingsize of the basis. We define R max as the maximum R valueincluded in the basis. The upper and lower panels presentthe energy and r.m.s. radius, respectively. As expected fora bound state, the 0 +1 energy and r.m.s. radius convergerapidly. With R max ≈ √ < r > = 2 . 21 fm is too smallcompared to experiment (2.48 fm [51]) but this differenceis due to the overbinding of the theoretical state.More interesting results are obtained for the 0 +2 state.If the energy remains almost constant above R max ≈ R max = 24 fm, which representsa huge value, much larger than in standard calculations(see, for example, Ref. [50]). This is, however, necessaryto illustrate convergence issues. As we may expect fromthe unbound nature of the 0 +2 state, the r.m.s. radius di-verges. Any (large) value can be obtained, provided thatthe basis is large enough. This explains why calculationsin the literature are quite different [14].The same quantities are plotted in Fig. 4 for J = 2 + .The existence of a 2 +2 resonance, second state of a bandbased on the Hoyle state, is well established [48,44], but P. Descouvemont: Resonances in C and Mg: what do we learn from a microscopic cluster theory? -15-10-5051015 0 5 10 15 20 25 (cid:1) = 0 (cid:4) (cid:5) ( M e V ) (cid:12)(cid:4) (cid:13)(cid:4) 〈 (cid:15) (cid:13) 〉 ( f m (cid:13) ) (cid:12)(cid:4) (cid:13)(cid:4) (cid:19) (cid:20)(cid:21)(cid:22) (fm) Fig. 3. Total energy (top) and r.m.s. radius (bottom) of the0 +1 and 0 +2 states in C. Energies are given with respect to the3 α threshold. this state should be broad. Even the energy is not sta-ble, and does not present a plateau with R max . The cor-responding r.m.s. radius strongly diverges. Notice that amicroscopic 3 α model does not predict any 0 +3 or 2 +3 res-onance.Let us briefly comment on non-microscopic descrip-tions of the 3 α system. Two α + α potentials, the shallowAli-Bodmer potential [52] and the deep Buck potential [53]reproduce very well the experimental α + α phase shifts upto 20 MeV. When applied to N α systems ( N > C ground state isstrongly underbound [54], and the 0 +2 resonance is farabove the 3 α threshold, in contradiction with experiment.This problem may be partly addressed by adding a phe-nomenological 3-body force, but this technique introducesspurious 3 α resonances [26].The deep Buck potential raises the question of the two-body forbidden states which should be removed in the 3 α model. This is usually done by using a projection tech-nique but, here also, several problems remain (see the dis-cussions in Refs. [55,26]). An efficient alternative would beto use non-local α + α potentials [50], but the simplicityof non-microscopic models is lost. A fully satisfactory de-scription of the 3 α system within non-microscopic modelsremains an open issue. Mg The O + α + α system is more complex than C sincethe level density is much higher. The Mg nucleus wasstudied within an α + Ne multicluster model in Ref. [56],where Ne is described by an α + O structure. In Ref. -10-50510 0 5 10 15 20 25 (cid:1) = (cid:4) (cid:5) ( M e V ) (cid:12)(cid:4) (cid:13)(cid:4) 〈 (cid:15) (cid:13) 〉 ( f m (cid:13) ) (cid:12)(cid:4) (cid:13)(cid:4) (cid:19) (cid:20)(cid:21)(cid:22) (fm) Fig. 4. See caption to Fig. 3 for J = 2 + . [57] the authors use a microscopic O + α + α model tosearch for 0 + resonances in a stochastic approach. Thebasis, however, is more limited since a fixed geometry isadopted. The present work contains a more extended ba-sis, owing to the use of the hyperspherical coordinates.The oscillator parameter is chosen as b = 1 . 65 fm,which represents a compromise between the optimal valuesof α and of O. We use the Volkov force V2 [19] witha Majorana exchange parameter M = 0 . O + α + α threshold ( − . 05 MeV).The calculations of the matrix elements are much longerthan for C. Since most of the computer time is devotedto the quadruple sums involved in the trwo-body inter-action, the ratio between the computer times is approxi-mately given by 6 / = 16, since Mg involves 6 orbitals( s and p ), whereas C involves 3 orbitals ( s only).As for C, we use a large basis with 10 R -valuesfrom 1.2 fm to 12 fm, complemented by larger values R = 13 . , , , 19 fm. In Fig. 5, we present the bindingenergies and r.m.s. radii of J = 0 + states as a function of R max , the maximum R value included in the basis.The upper panel shows that four states are bound withrespect to the O + α + α decay, two of which are abovethe α + Ne threshold ( − . < r > ≈ , whereas the 0 +3 and 0 +4 radii present aslower convergence and a larger radius < r > ≈ 10 fm .The radii corresponding to the four negative-energy curvesare linked by solid lines.The right side of the upper panel shows the experi-mental 0 + states. The ground-state energy is adjusted bythe nucleon-nucleon interaction. The 0 +2 energy is in fair . Descouvemont: Resonances in C and Mg: what do we learn from a microscopic cluster theory? 7 -15-10-5051015 0 5 10 15 20 25 (cid:1) = (cid:4) (cid:5) ( M e V ) 〈 (cid:13) (cid:14) 〉 ( f m (cid:14) ) (cid:18) (cid:19)(cid:20)(cid:21) ( fm) Fig. 5. Energies with respect to the O + α + α threshold(top) and r.m.s. radii (bottom) of J = 0 + states in Mg fordifferent sizes of the basis. The color schemes are identical inboth panels. In the upper panel, the right side shows exper-imental energies. In the lower panel, the r.m.s. radii for thethird and fourth eigenvalues are almost superimposed. agreement with experiment. In the high-energy part of thespectrum, several 0 + experimental states are present.Figure 5 suggests two important properties. Around E ≈ E x ≈ Mg spectrum. As R max increases,the label of the eigenvalue varies. The radii correspond-ing to the plateau therefore show up as individual pointsaround < r > ≈ . . The dip near R max = 5 fm isdue to a crossing between the energy curves. The secondinformation concerns the radii. The lower panel of Fig.5 shows a clear distinction between physical states andapproximations of the continuum. The former present astable radius, whereas the latter are characterized by di-verging radii. A careful study of the radii, and in particu-lar of their stability against the extension of the basis, istherefore an efficient way to make a distinction betweenphysical states and pseudostates.Figure 6 displays the energy curves for J = 2 + . Thelevel density is still higher than for J = 0 + and it isdifficult to make a clear link between theory and exper-iment. The 2 +1 and 2 +3 energies are in fair agreement,but the GCM 2 +2 energy is too low by about 2 MeV.This is probably due to the lack of a spin-orbit force in a O + α + α model. The model predicts seven states be-low the O + α + α threshold. The r.m.s. radii are notpresented as they qualitatively follow those of J = 0 + .The converged radii for the 2 +1 and 2 +2 states are around < r > ≈ . , which is close to the radius of the groundstate. -15-10-50510 0 5 10 15 20 25 (cid:1) = 2 (cid:4) (cid:5) ( M e V ) (cid:12) (cid:13)(cid:14)(cid:15) (fm) Fig. 6. See caption to Fig. 5 for J = 2 + . Only energies aredisplayed. The goal of the present work is to illustrate the calculationof resonance properties in cluster models, and more espe-cially in multicluster models. A rigorous treatment of theresonances would require a scattering theory, with exactboundary conditions. If this approach is fairly simple intwo-cluster models, it raises strong difficulties when morethan two clusters are involved. The bound-state approxi-mation is commonly used owing to its simplicity. We haveshown, however, that positive-energy eigenvalues shouldbe treated carefully, and that, even for narrow resonances,the wave function may be sensitive to the basis. A directconsequence is that some properties, such as the r.m.s. ra-dius, are unstable. The stability against the basis shouldbe assessed.Several method exist to complement the bound-stateapproximation, such as the CSM or the ACCC. They per-mit to determine the energy and width of a resonance.In practice, however, they are difficult to apply to mi-croscopic calculations since they usually need very largebases. We have used an alternative of the box method,where the number of basis functions is progressively in-creased. We have shown that a stability of the energy canbe obtained, but the corresponding r.m.s. radii are un-stable. The application to the Hoyle state in C is anexcellent example which explains the variety of the valuesin the literature. 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