TOWARDS A SELFCONSISTENT CLUSTER EMISSION THEORY

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1 NUCLEAR PHYSICS TOWARDS A SELFCONSISTENT CLUSTER EMISSION THEORY D. S. DELION National Institute of Physics and Nuclear Engineering, POB MG-6, Bucharest-Mãgurele, Romania, A. SÃNDULESCU Center for Advanced Studies in Physics, Calea Victoriei 15, Bucharest, Romania, W. GREINER Institut für Theoretische Physik, J.W.v.-Goethe Universität, Robert-Mayer-Str. 8-10, 6035 Frankfurt am Main, Germany Received December 10, 004 We propose a selfconsistent theory of the α-particle decay, which can be extended to the heavy cluster emission. The strong dependence of the Q-value versus the Coulomb term and the more bound α-like configurations suggest that preformed clusters should exist on the nuclear surface. This is confirmed by the fact that the derivative of the shell-model preformation amplitude is practically a constant along any α-chain, while that of the outgoing wave function changes exponentially upon the Coulomb parameter. Thus, an α-cluster additional term in the preformation factor is necessary for a selfconsistent description of the decay width. 1. INTRODUCTION The even-odd pair staggering of binding energies found along the α-lines lines, with N Z = const, can be nicely explained in terms of a pairing in the isospin space between proton and neutron pairs, considered as bosons [1, ]. This suggest that α-particles are already preformed at least in the low density region of the nuclear surface. On the other hand the α-particle energy (Q-value), computed as the difference between the binding energies of initial and final systems, is directly connected with the decay width. The linear dependence between the logarithm of the decay width and the square root of the Q-value was explained by G. Gamow by supposing that the preformed α-particle moves in some attractive potential and penetrates the surrounding Coulomb barrier [3]. The half-lives of α-particle emitters are well described by using an equivalent local potential [4]. The attractive depth and the radius of the repulsive core Rom. Journ. Phys., Vol. 50, Nos. 1, P , Bucharest, 005

2 166 D. S. Delion, A. Sãndulescu, W. Greiner determines the energy and wave function of the decaying state, understood as a narrow resonance [5, 6]. The R-matrix theory [7, 8] makes a step forward and expresses the decay width as a product between the particle preformation probability and the penetration through the barrier [9, 10, 11, 1]. Due to the antisymmetrisation effects between the α-particle and daughter wave functions the interaction becomes non-local in the internal region [13]. It was shown that the usual shellmodel space using N = 6 8 major shells underestimates the experimental decay width by several orders of magnitude [14, 15], due to the exponential decrease of bound single particle wave functions [16]. The inclusion of narrow single particle resonances is not able to cure this deficiency [17]. Only the background components in continuum can describe the right order of magnitude of experimental decay widths [18, 19, 0, 1]. Anyway, the shell model estimate of the α-particle preformation factor is not consistent with the decreasing behaviour of Q-values along any neutron chain [, 3]. In our previous papers [4, 5] we analyzed this feature by treating the α-decaying state as a resonance, namely by using the matching between logarithmic derivatives of the preformation amplitude and Coulomb function. It turns out that this condition is not satisfied along any neutron or α chain if one uses the standard shell model estimate for the preformation factor. We corrected the slope of the preformation amplitude by changing the harmonic oscillator (ho) parameter of single particle components. These components are connected with an α-cluster term, not predicted by the standard shell model [6]. Recently a similar idea was used in Ref. [7]. The aim of this paper is to stress on the fact that this behaviour is strongly connected with the structure of the Q-value. Namely the Coulomb repulsive term gives the main linear behaviour between closed shells and therefore it should be also recovered in the preformation factor. We will show that in order to fulfil the so-called plateau condition it is necessary to use an additional α-cluster component, depending upon the Coulomb parameter.. THEORETICAL BACKGROUND As we pointed out in Introduction the decay width is directly connected with the Q-value, computed as follows Eα = B( Z, N,β) + B(,, 0) B( Z, N,β ), (.1) where BZ (, N,β ) is the binding energy, depending upon the charge, neutron numbers and quadrupole deformation parameter. This quantity is given by the Weizsäker type relation, like for instance in Ref. [8]

3 3 Towards a selfconsistent cluster emission theory / 13 / 1 / vol surf Coul sym pair BZ (, N,β ) = a A a A a Z A E ( AI, ) a A + + E ( Z, N,β ) + E ( β ). def shell (.) Along any α-line with I = N Z = const the Coulomb term has a much stronger variation versus Z (quadratic) than the other ones. Therefore the Q-value, depends linearly upon the charge number and the shell model dependence practically disapears. We will show that this feature is also reflected by the shell-model estimate of the α-particle preformation factor. The standard procedure to estimate the decay width within the microscopic approach was described in several papers, like for instance [18, 19, 0, 1]. In a phenomenological approaches one defines an equivalent local α-core interaction for any distance. By expanding the solution of the corresponding Schrödinger equation in spherical waves, i.e., gl () r Ψ m() r = Y lm() r ˆ, (.3) r l one finds the energy of a decaying resonant state by matching the internal ( int ) ( ext gl () r and external outgoing components gl ) () r at some radius r = R. The decay width can be derived from the continuity equation as follows Γ= v lim g () r, (.4) l r where v is the cm velocity at infinity. The external components in a deformed Coulomb field were derived by Fröman within the WKB approach [6]. It turns out that the major effect is given by the quadrupole deformation of the barrier [19, 9]. The decay width can be estimated by using the following ansatz ( int) g0 ( R) l G 0 0 0( χ, kr) l l Γ= v D ( R) Γ ( R) D( R), (.5) where the deformation matrix D pr ll with l = 0 is given in terms of the so-called Fröman matrix [6]. By G0( χ, kr) we denoted the monopole irregular Coulomb function, depending upon the product between the momentum k and matching radius R. Here χ is the Coulomb parameter 1 ZZe χ=. v (.6) Thus, the decay width contains a ratio between the internal and external solutions. It does not depend upon the matching radius R within the local

4 168 D. S. Delion, A. Sãndulescu, W. Greiner 4 potential approach, because the internal and external wave functions satisfy the same equation and therefore are proportional. This is the so-called plateau condition. The situation becomes different when the value of the internal wave ( int) function g0 ( R ) is given by an independent microscopic approach. It is replaced by the so-called preformation amplitude, defined as follows g ( int) 0 ( R) F 0( R) = dξ d A ( ) α ξ Ψα ξα ΨA( ξa) ΨB( ξ B), R (.7) where the integration is performed over internal coordinates. The structure of a free α-particle is given by one pair of protons in a singlet state and a similar pair of neutrons [1]. Each particle lies in the ground state 0s of an ho well with the parameter β 05fm α.. The most important ground state correlations are given by the pairing interaction. We use the Bardeen-Cooper-Schrieffer (BCS) approach for mother and daughter wave functions. In order to estimate the overlap integral (.7) we expand the mother wave function in terms of sp states, multiplied by the daughter wave function, as follows (.8) Ψ = 1 j + 1 P [ ψ ψ ] j + 1 P [ ψ ψ ] Ψ. B π j j j 0 j j j 0 A π π π ν ν ν ν j j π ν We use the short-hand index notation jτ ( τε lj), where τ = π, ν denotes isospin, ε sp energy, l angular momentum and j total spin. Otherwise j τ has the usual meaning of the single particle spin. The expansion coefficients are given in terms of BCS occupation amplitudes as follows ( A) ( B) j j j P = u v. (.9) τ τ τ In order to perform the integral (.7) analytically we expand sp wave functions in the ho basis, i.e., n max ψ ( r, s) = c R ( β r ) Y ( rˆ ) χ 1 ( s), τ=π,ν. (.10) jm τ njτ nl 0 l n= 0 The radial ho wave function is defined in terms of the Laguerre polynomial. The sp parameter β 0 is connected with the standard ho parameter by using a scaling factor f 0 as follows M ω f β = β =, N 0 0 f0 N f0 A 13 / jm τ (.11)

5 5 Towards a selfconsistent cluster emission theory 169 where A is the mass number. By performing the recoupling of proton and neutron pairs in (.8) to relative and cm coordinates the preformation amplitude becomes F0( β, 0 nmax, Pmin ; R) = = β,, β β. ε 4β 0R / W ( n P ) N (4 ) L1 / (4 R) (.1) N N 0 max min N0 0 N 0 We stress on the fact that the exponential term is similar to the cm α-particle wave function, but it depends upon the single particle ho parameter β 0. The expansion coefficients are given in terms of recoupling Talmi-Moshinsky brackets as in Ref. [19]. We consider in our sp basis only those states with P τ larger than the minimal value P min, taken as a parameter. 3. NUMERICAL ANALYSIS The most important ingredient, governing the penetrability of the α-particle through the barrier, is the Coulomb parameter χ. The irregular Coulomb function G0( χ, kr) depends exponentially on it 1 / 0 χα ( sin αcos α) G0 ( χ, kr) = ( ctgα ) e, ZZe 1 cos α= kr = R, R0 =. χ R E α (3.1) The decay width has also an exponential dependence upon the quadrupole deformation. As it was shown in Ref. [5] the function D(R) in Eq. (.5) practically does not depend upon the radius. The largest correction gives a factor of three for heavy nuclei and a factor of five in superheavy ones. The preformation amplitude, given by Eq. (.1), is very collective and therefore the transitions between ground states are not sensitive to the mean field parameters. Thus, in our analysis we used the universal parametrisation of the Woods-Saxon potential [30] and we considered the gap parameter estimated by τ = 1/ AB [31], where A B is the mass number of the mother nucleus. The quadrupole deformation parameters in the Fröman matrix are taken from Ref. [3]. The preformation factor is very sensitive with respect to the maximal sp radial quantum number n max, the sp ho parameter β 0 and the amount of spherical configurations taken in the BCS calculation, given by Pmin = min{ P τ }. It turns out that beyond n max = 9 the results saturate if one considers in the BCS basis sp states with P P min = We improved the description of the continuum by

6 170 D. S. Delion, A. Sãndulescu, W. Greiner 6 choosing a sp scale parameter f 0 < 1 in Eq. (.11). This parameter is not independent from P min. It turns out that the common choice of f 0 and P min ensures not only the right order of magnitude for the decay width, but also the above mentioned continuity of the derivative. The logarithm of the decay width can be approximated by the following linear ansatz Γ( R) log10 =γ 0 +γ 1R. (3.) Γexp In the ideal case the coefficients should vanish, i.e., γ 0 =γ 1 = 0, in order to have a proper description of the decay width. In other words we can in principle find the Coulomb parameter χ by solving the equation γ1( χ ) = 0, (3.3) for given parameters n max, β 0, P min and in this way to predict Q-value independently, based only on the microscopic factor. We analysed α-decay chains from even-even nuclei with N > 16, given in the Table 1. Table 1 Even-even α-decay chains in the region Z > 8, N > 16. In the first column of each table is given the isospin projection I = N Z. In the next columns are given the initial neutron and proton numbers, the number of states/chain and the reference I N 1 Z 1 No Ref [4] [4] [4] [4] [4] [4] [4] [4] [4] [4] [4] [33] It turns out that the values n max = 9, f 0 = 0.8 and P min = 0.05 give the best fit concerning the parameters γ 0 and γ 1. From Fig. 1.a we see that the quantity

7 7 Towards a selfconsistent cluster emission theory 171 γ0 log10 ( Γ/Γ exp ) has a variation of one order of magnitude around γ 0 = 0, but the description of the slope γ 1, given in Fig. 1.b, is by far not satisfactory. The reason for the variation of the slope parameter γ 1 is the relative strong dependence of the Coulomb parameter χ upon the neutron number along α-chains. In Fig. 1.c we give the values of this parameter for the even-even chains, which is in an obvious correlation with the slope parameter γ 1. Therefore the derivative of the microscopic preformation amplitude changes along α-chains much slower in comparison with that of the Coulomb function. As we pointed out the term given by the shell correction disapears in the Q-value (except the magic numbers) and it remains a linear in Z dependence. Thus, indeed the most important effect is given by the Coulomb repulsion. In order to stress on this dependence we performed the same analysis in the region Z > 8, 8 < N <16, given in the Table. Fig. 1. (a) The ratio parameter γ 0, defined by Eq. (3.), versus the neutron number for f 0 = 0.8, P min = 0.05 and different even-even α-chains in Table 1. (b) The slope parameter γ 1, defined by Eq. (3.), versus the neutron number. (c) The Coulomb parameter χ, defined by Eq. (.6), versus the neutron number.

8 17 D. S. Delion, A. Sãndulescu, W. Greiner 8 Table Even-even α-decay chains in the region Z > 8, 8 < N < 16. The quantities are the same as in Table 1 I N 1 Z 1 No. Ref [4] [4] [4] [4] [4] [4] In Figs..a,b we plotted the parameters γ 0, γ 1 depending upon the neutron number. We used the same parameters, i.e., n max = 9, f 0 = 0.8, P min = One can see that indeed their values are very close to zero. The decay widths are reproduced within a factor of two. We point out the small decrease of parameters along considered α-chains is correlated with a similar behaviour of the Coulomb parameter χ in Fig..c. Our estimate shows that the linear correlation coefficient between γ 1 and χ is larger than 0.7. This allows us to introduce a supplementary, but universal, correcting procedure for the preformation factor. Thus, let us define a variable size parameter f by a similar to (.11) relation, namely β= f β N. (3.4) The parameter χ enters in the exponent of the Coulomb function (3.1). This fact suggests a similar correction of the preformation factor, i.e. F0 ( β, β, n, P ; R) = = e 4 W ( β, n, P ) (4 β ) L (4 β R). m max min β R / (1/ ) N m max min N N0 m N 0 m N (3.5) We suppose a linear dependence of the size parameter f upon the Coulomb parameter β β = ( f f ) β = f ( χ χ ) β. (3.6) m m N 1 m N The above relation (3.5) can be written as follows 4( β β m ) R / 0 m max min F0 m max min F ( β, β, n, P ; R) = e ( β, n, P ; R) = (3.7) = F0( β β m, 0, 0 ; R) F0( β m, nmax, Pmin; R), i.e., the usual preformation amplitude is multiplied by a cluster preformation amplitude with n max = 0. Thus, one has to multiply the right hand side of the expansion (.1) by this factor.

9 9 Towards a selfconsistent cluster emission theory 173 Fig.. (a) The ratio parameter γ 0, defined by Eq. (3.), versus the neutron number for f 0 = 0.8, P min = 0.05 and different even-even α-chains in Table. (b) The slope parameter γ 1, defined by Eq. (3.), versus the neutron number. (c) The Coulomb parameter χ, defined by Eq. (.6), versus the neutron number. We choosed a strategy to determine the parameters connected with the maximal value of the ratio parameter γ 0. As we will show later this choice has a physical meaning connected with the α-clustering picture. We remark from Fig. 1.a that the maximal value of the ratio parameter γ 0 corresponds to a maximal value of the Coulomb parameter. By using a constant ho parameter with the size parameter f m = 0.83 for all analyzed even-even emitters the Fig. 1.a is pushed down and one obtains for the maximal value of the ratio parameter γ 0( max) = 0. In this way we suppose that in this point the α-clustering is described entirely by the pairing correlations. In this way for other decays the α-clustering process increases by decreasing the Coulomb parameter, because the ho parameter β in (3.6) is smaller and therefore the tail of the preformation factor increases.

10 174 D. S. Delion, A. Sãndulescu, W. Greiner 10 From Figs. 1.a.b we can see that the pure α -clustering should be enhanced in the region above N = 16 and in superheavy nuclei. This is agreement with several calculations pointing out on a very strong clustering process in Po, Rn and Ra isotopes. Our calculations predict a similar feature for superheavy nuclei. Therefore in our calculations we used the parameters f m = 0.83, χ m = 55. For the proportionality coefficient in Eq. (3.6) the regression analysis gives the value f 1 = The situation in the superheavy chain can be described by assuming a quadratic dependence of the coefficient f 1 upon the number of clusters Nα = ( N N0 )/ with N 0 = 16, namely f1 f1+ fn α. (3.8) A quadratic in N α dependence of the Q-value was also empirically found in Ref. [1]. The final results are given in Fig. 3.a,b. We considered a correcting term Fig. 3. (a) The parameter γ 0 versus the neutron number for different even-even α- chains in Table 1. The preformation parameters are f m = 0.83, f 1 = , f = , P min = (b) The same as in (a), but for the slope parameter γ 1.

11 11 Towards a selfconsistent cluster emission theory 175 with f = The improvement of the slope parameter is obvious. The mean value of this parameter and its standard deviation for even-even chains is γ 1 = ± The quadratic dependence in Eq. (3.8) can be also interpreted in terms of the total number of interacting clustering pairs, namely N α Nα( Nα 1) /. Thus, our analysis based on the logarithmic derivative continuity, shows very clearly that the effect of the α-clusterisation becomes much stronger for superheavy nuclei. 4. CONCLUSIONS We proposed in this paper a selfconsistent theory of the α-decay. We analysed the decay widths for deformed even-even emitters with Z > 8. The α-particle preformation amplitude was estimated within the pairing approach. We used the universal parametrisation of the mean field and the empirical rule for the gap parameter = 1/ A. The penetration part was computed within the deformed WKB approach. It is possible to satisfactorily describe all α-decay widths from even-even nuclei by using a constant, but smaller ho parameter β= 080. β N and P min = It turns out that the slope of the decay width versus the matching radius has a strong variation for N > 16, in an obvious correlation with the Coulomb parameter. Thus, the relative amount of the α -clustering here cannot be described only within the pairing approach and an additional mechanism is necessary. We supposed a cluster factor, multiplying the preformation amplitude. It contains exponentially an ho parameter, proportional to the Coulomb parameter. The method improves simultaneously the ratio to the experimental width and the slope with respect to the matching radius. The relative increase of the α-clustering is related to the decrease of the Coulomb parameter. It is stronger for two regions, namely above N = 16 and in superheavy nuclei. It has a minimum around N = 15. An additional dependence upon the number of interacting α-particles improves the plateau condition for superheavy nuclei. This additional clustering, which seems to be very strong, may affect the stability of nuclei in this region. REFERENCES 1. G. Dussel, E. Caurier, and A. P. Zuker, At. Data Nucl. Data Tables, 39, 05 (1988).. Y. K. Gambhir, P. Ring, and P. Schuck. Phys. Rev. Lett. 51, 135 (1983).

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