DECAY THEORY BEYOND THE GAMOW PICTURE

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1 Dedicated to Academician Aureiu Sanduescu s 8 th Anniversary DECAY THEORY BEYOND THE GAMOW PICTURE D. S. DELION Horia Huubei Nationa Institute for Physics and Nucear Engineering, P.O. Box MG-6, Bucharest, Romania E-mai: deion@theory.nipne.ro Received September 3, 211 In the custer emission theory the decay width is a product between the penetrabiity and reduced width. The first component gives the so-caed Gamow description of the decay process, eading to the Geiger-Nutta aw. The second component is connected to custering features and satisfies an anaytic universa aw, which is the next step beyond the standard Gamow rue. As a consequence, the reduced width can be described microscopicay by using a mixed singe partice basis with two components. The first part describes the usua spectroscopic properties, whie the second one is connected to custering features. The universa aw impies that the harmonic osciator parameter of the custer part is proportiona to the Couomb parameter. Key words: α-decay, custer decay, proton emission, penetrabiity, reduced width. PACS: 21.6.Gx, e, j. 1. INTRODUCTION In the odest Gamow approach of the α-decay process the α-partice was considered a preformed custer, moving around the core and penetrating quantum mechanicay the Couomb barrier [1]. This may justify the so-caed extreme custer modes, where the decay process is described by the penetration of the α-core Couomb potentia [2 4]. We wi show that this picture is abe to provide more informations, namey to provide a simpe anaytica reation, connecting the reduced width to the fragmentation potentia. It is vaid for any kind of decay process induced by the strong interaction, except fission, and it is the next natura step beyond the Gamow picture. This phenomenoogica aw has a direct consequence on the microscopic decay theory. The she mode provides the theoretica too to compute microscopicay the reduced width, proportiona to the α-custer ampitude inside the parent wave function [5, 6]. First cacuations evidenced that the absoute decay widths were much smaer than the experimenta vaues. In the ate 197 s it was shown that the incusion of many configurations is abe to increase the vaue of the absoute decay width by more than four orders of magnitude [7]. This was not enough and ony a theory where a combined she- and custer-mode configurations were considered coud reproduce the absoute decay width [8,9]. We wi show that this mixed representation is abe to simutaneousy describe α decay widths and eectromagnetic transitions. (c) Rom. RJP Journ. 57(Nos. Phys., Vo. 1) 57, Nos. 1, 212 P , Bucharest, 212

2 15 D. S. Deion 2 2. ANALYTIC RELATION FOR THE REDUCED WIDTH For spherica emitters the decay width of the process P (arent) D(aughter)+ C(uster) is defined in terms of the scattering ampitude N as foows [1] Γ = v N 2 Rf (int) (R) 2 = v, (1) G (R) where v is the veocity of the custer-daughter system at infinity and G (R) is the monopoe irreguar Couomb function. The above ratio does not depend on the custer-daughter radius R, because both interna f (int) (R) and externa G (R)/R wave functions shoud satisfy the same Schrödinger equation. Traditionay the decay width is rewritten in terms of the penetrabiity and reduced width squared P = Γ = 2P γ 2, (2) κr G (R) 2, γ2 = 2 (int) f (R) 2, (3) 2µR where κ is the momentum and µ the reduced custer-daughter mass. The decay processes can be schematicay described by the foowing custerdaughter pocket-ike spherica potentia [1] V (r) = ω β(r R ) 2 + v, R R B 2 = Z DZ C e 2 (4) V C (R), R > R B R where R is the surface pocket radius. Notice that the radia equation of the shifted harmonic osciator (ho) potentia is simiar with the equation of the one-dimensiona osciator. By considering Q-vaue as the first eigenstate in the shifted ho we Q v = 1 2 ω, together with the continuity condition at the top of the barrier R B, one obtains the foowing reation ω β(r B R ) 2 = V frag (R B ) + 1 ω, (5) 2 2 where it was introduced the so caed fragmentation (or driving) potentia, as the difference between the top of the Couomb barrier and Q-vaue [ ] χ V frag (R B ) = V C (R B ) Q = Q ρ 1, (6) in terms of the Couomb parameter and reduced radius, respectivey χ = 2Z DZ C v (c) RJP 57(Nos. 1) , ρ = κr. (7)

3 3 Decay theory beyond the Gamow picture 151 For a shifted ho we one has for the ground state By using (5), one obtains the foowing reation f (int) (R) 2 = A 2 e β(r R ) 2. (8) og 1 γ 2 (R B ) = og 1e 2 ω V frag(r B ) + og 1 2 A 2 2eµR B. (9) Notice that the sope in (9) has a negative vaue and it is connected with the shape of Fig. 1 The ogarithm of the α-decay reduced width squared versus the fragmentation potentia (6) for regions of the nucear chart described by (1). the interaction potentia (ho energy ω), whie the free term gives information about the ampitude of the custer wave function. Our cacuation has shown that the inear reation (9) but with different coefficients, remains vaid in the most genera case of the doube foding pus repusive interaction between fragments, used in [11, 12]. Most of experimenta data refer to the α-decay. Therefore we anayzed (9) in α-decays connecting ground states of even-even nucei. In figure 1 the data are (c) RJP 57(Nos. 1)

4 152 D. S. Deion 4 divided into five regions of even-even α emitters as foows 1) Z < 82, 5 < N < 82 Fig. 1 (c), stars ; 2) Z < 82, 82 < N < 126 Fig. 1 (a), crosses ; 3) Z > 82, 82 < N < 126 Fig. 1 (b), circes ; 4) Z > 82, 126 < N < 152 Fig. 1 (c), squares ; 5) Z > 82, N > 152 Fig. 1 (d), trianges. In cacuations it was used the vaue of the touching radius, i.e. R B = 1.2(A 1/3 D + A1/3 C (1) ). (11) Notice that the regions 1 contain rather ong isotopic chains, whie in the ast region 5 one has not more than two isotopes/chain. This is the reason why, except for the ast region 5, the reduced width decreases with respect to the fragmentation potentia, according to the theoretica prediction given by (9). The inear dependence of og 1 γ 2 versus the fragmentation potentia (9) remains vaid for any kind of custer emission. This fact is nicey confirmed by heavy custer emission processes in figure 2 (a). Here it is aso potted a simiar dependence for α-decays corresponding to the same heavy custer emitters. The straight ine is the inear fit for custer emission processes, except α-decays og 1 γ 2 =.586(V C Q) (12) The above vaue of the sope og 1 e 2 / ω in (9) eads to ω 1.5 MeV, with the same order of magnitude as in the α-decay case. Let us mention that a reation expressing the spectroscopic factor (proportiona with the reduced width) for custer emission processes was derived in [13] S = S (A C 1)/3 α, (13) where A C is the mass of the emitted custer and S α 1 2. As can be seen from figure 2 (b) between A C and V frag there exists a inear dependence and therefore the above scaing aw can be easiy understood in terms of the fragmentation potentia. Concerning the reduced widths of proton emitters in [14, 15] it was pointed out the correation between the reduced width and the quadrupoe deformation. This fact can be seen in figure 3 (a), where the region with Z < 68 corresponds to β >.1 (open circes), whie the other one with Z > 68 to β <.1 (dark circes). The two inear fits have obviousy different sopes. Notice that the two dark circes with the smaest reduced widths correspond to the heaviest emitters with Z > 8. At the same time one sees from figure 3 (b) that the same data are custered into two regions, which can be directy reated with the fragmentation potentia (6). Here, the two inear fits in terms of the fragmentation potentia, corresponding to the two regions of charge numbers, have roughy the same sopes, but different vaues (c) RJP 57(Nos. 1)

5 5 Decay theory beyond the Gamow picture Fig. 2 (a) The ogarithm of the reduced width squared versus the fragmentation potentia (6). Different symbos correspond to custer decays in figure 2. The straight ine is the inear fit (12) for custer emission processes, except α-decay. (b) Custer mass number versus the fragmentation potentia. in origin. Thus, the two different ines seen in proton emission systematics [14] can be directy connected with simiar ines in figure 3 (b). They correspond to different orders of magnitude of the fragmentation potentia, giving different orders to wave functions and therefore to reduced widths. 3. SHELL-MODEL DESCRIPTION OF DECAY PROCESSES The ampitude to find an α-partice in a she-mode wave function of the parent nuceus is given by the foowing overap F (R α ) = Ψ D ψ α Ψ P, (14) depending on the α-daughter distance R α. Here, we denoted by Ψ D/P the manybody wave function of the daughter/parent nuceus buit in terms of singe partice orbitas generated by the nucear mean fied. On the other hand, ψ α is the intrinsic wave function of the α-partice. Its radia part is a product of two-proton and twoneutron Gaussians, whie the spin component is a singet state [5]. By using as a residua interaction the pairing force, the formation ampitude becomes peaked on the nucear surface. (c) RJP 57(Nos. 1)

6 154 D. S. Deion Fig. 3 (a) The ogarithm of the reduced width squared versus the quadrupoe deformation. By open circes are given emitters with Z < 68, whie by dark circes those with Z > 68 for proton emission. The two regression ines fit the corresponding data. (b) The ogarithm of the reduced width squared versus the fragmentation potentia (6). The symbos are the same as in (a). The binding energy per nuceon for an α-partice is much arger with respect to their neighbors. Due to a smaer nucear density at surface, an α-custer structure is energeticay more favorabe in this region. Therefore, custering is a surface effect and in the previous Section we have shown that it can be simuated by assuming that the α-partice moves in a pocket-ike potentia centered at some radius R on the nucear surface. Thus, in order to propery describe custering properties at the singe partice eve one has to consider two components of the singe partice orbitas, i.e. ψ (r) = N (SM) ψ (SM) (r) + N (cus) ψ (cus) (r). (15) The she mode part has a standard spherica harmonic osciator representation ψ (SM) (r) = n b n R (β ) n (r), (16) where β = M N ω/ is the standard harmonic osciator parameter and the principa quantum number has the vaues N = 2n + 6. (c) RJP 57(Nos. 1)

7 7 Decay theory beyond the Gamow picture Fig. 4 (a) Gaussian distribution centered on the nucear surface (soid ine) and various expansion terms (dot-dashed ines). (b) Expansion coefficients of a Gaussian centered on the nucear surface given by (17). For the custer part one uses a wave function of the shifted osciator ψ (cus) (r) = e βc(r r ) 2 /2 = n c n R (β) n (r), (17) where N is spread around a arger vaue N 8 1, as can be seen from figure 4. Notice that in [9] the harmonic osciator parameter of the custer part has a smaer vaue, i.e. β < β, in order to reduce the dimension of the singe partice basis (15). Here a coefficients are found by a diagonaizing procedure of the Woods-Saxon mean fied. Actuay, according to (5), the custer parameter β shoud be proportiona to the fragmentation potentia, or to the Couomb parameter χ in (6), in order to achieve a sef consistent description of the emission process [16]. The typica exampe of a heavy nuceus where the α-custering effects are very important is 212 Po. Here, the structure of ow-ying states can be expained in terms of two proton and two neutron orbitas above the doube magic inert core 28 Pb [17]. Anyway, by using the standard components of the two protons and two neutrons the decay width is underestimated by two orders of magnitude. It is necessary to consider the second custer component in (15), with N (cus) =.3, in order to describe the absoute vaue of the α-decay width, as can be seen in figure 5 (b) (soid ine). In (c) RJP 57(Nos. 1)

8 156 D. S. Deion 8 x Fig. 5 (a) α-partice formation probabiity for 212 P o 28 P b + α. Dashed ine: she mode, Soid ine: she mode + α-custer. (b) Logarithm of the ratio between the decay width and the experimenta vaue as a function of the daughter-α-partice radius. spite of the fact that in figure 5 (a) the difference between the formation probabiity within the she mode approach (dashed ine) and mixed approach (soid ine) is rather sma, the enhancement of the decay width is about two orders of magnitude. Notice that the same vaue was obtained in [8], by using a diagonaizing procedure of the residua interaction. This picture is aso supported by anayzing eectromagnetic transitions. The eectric transition probabiity of the mutipoarity λ is proportiona to the reduced matrix eement of the transition operator squared, i.e. B(Eλ : J J ) = 1 J 2J + 1 ˆT λ J 2, (18) where J and J denote initia and fina states and the transition operator is proportiona to spherica harmonics ˆT λµ = r λ Y λµ. (19) The radia part of the quadrupoe transition matrix eement is proportiona to the (c) RJP 57(Nos. 1)

9 9 Decay theory beyond the Gamow picture Fig. 6 B(E2 : J J 2 vaues (W.u.). She-mode vaues in 21 Po and she-mode + custer vaues in 212 Po (soid ines). By dashes are given experimenta vaues. principa quantum number n 1 1 r 2 n 2 2 N = 2n +. (2) As it is shown in figure 4, the custering components are centered around the region N 8 1 and therefore they shoud enhance transition probabiity. It is known that B(E2) vaues are satisfactory described within the she mode for 21 Pb, but they are by one order of magnitude ess than the experimenta vaues in 212 Po. They are of the same order of magnitude as those in 21 Pb, given by the ower soid ine in figure 6 [18]. In order to expain this discrepancy we used the mixed representation (15) with the same α-custer ampitude. In figure 6 one indeed obtains a very good agreement of the computed vaues (upper soid ine) with respect to experimenta data (upper dashed ine). This is a very convincing evidence in favor of the α-custering structure of this nuceus. 4. CONCLUSIONS Based on a simpe mode of the custer-core dynamics, namey a shifted harmonic osciator potentia surrounded by the Couomb interaction, it was derived an (c) RJP 57(Nos. 1)

10 158 D. S. Deion 1 universa anaytica reation expressing the ogarithm of the reduced width squared as a inear function in terms of the fragmentation potentia, defined as the difference between the Couomb barrier and the Q-vaue. It is fufied with a reasonabe accuracy by a experimenta decay data, describing transitions between ground states. In heavy nucei this custer component cannot be buid from usua singe partice orbitas because protons and neutron ie in different major shes. Thus, an additiona Gaussian centered on the nucear surface, predicted by the above mentioned universa aw, is necessary to be incuded in the singe partice basis. The ho parameter of the custer part is proportiona to the Couomb parameter. We described both α-decay width and B(E2) vaue in 212 Po by using a mixed singe partice basis, containing she-mode and α-custer components. Acknowedgments. This work was performed in the frame of the project PN-II-ID-PCE of the Romanian Ministry of Education and Research. REFERENCES 1. G. Gamow, Z. Phys. 51, 24 (1928). 2. A. Sănduescu, R.Y. Cusson, W. Greiner, Lett. Nuovo Cim. 36, 321 (1983). 3. B. Buck, A.C. Merchant, S.M. Perez, Phys. Rev. Lett. 72, 1326 (1994). 4. S. Ohkubo, Phys. Rev. Lett. 74, 2176 (1995). 5. H.J. Mang, Phys. Rev. 119, 169 (196). 6. A. Sănduescu, Nuc. Phys. 37, 332 (1962). 7. I. Tonozuka, A. Arima, Nuc. Phys. A 323, 45 (1979). 8. K. Varga, R.G. Lovas, R.J. Liotta, Phys. Rev. Lett. 69, 37 (1992). 9. D.S. Deion, A. Insoia, R.J. Liotta, Phys. Rev. C 54, 292 (1996). 1. D.S. Deion, Theory of partice and custer emission (Springer-Verag, Berin, 21). 11. S. Petonen, D.S. Deion, J. Suhonen, Phys. Rev. C 75, 5431 (27). 12. S. Petonen, D.S. Deion, J. Suhonen, Phys. Rev. C 78, 3468 (28). 13. R. Bendowske, T. Fiessbach, H. Waiser, Z. Phys. A 339, 121 (1991). 14. D.S. Deion, R.J. Liotta, R. Wyss, Phys. Rev. Lett. 96, 7251 (26). 15. E.L. Medeiros, M.M.N. Rodrigues, S.B. Duarte, O.A.P. Tavares, Eur. J. Phys. A 34, 417 (27). 16. D.S. Deion, A. Sănduescu, W. Greiner, Phys. Rev. C 69, (24). 17. D.S. Deion, J. Suhonen, Phys. Rev. C 61, 2434 (2). 18. A. Astier, P. Petkov, M.-G. Porquet, D.S. Deion, P. Schuck, Eur. Phys. J. A 47, 165 (21). (c) RJP 57(Nos. 1)