CLUSTERIZATION PROBABILITY IN ALPHA-DECAY A NEW APPROACH

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1 Romanian Reports in Physics, Vol. 65, No. 4, P , 2013 CLUSTERIZATION PROBABILITY IN ALPHA-DECAY 212 Po NUCLEUS WITHIN CLUSTER-FORMATION MODEL; A NEW APPROACH SAAD M. SALEH AHMED 1, REDZUWAN YAHAYA and SHAHIDAN RADIMAN University Kebangsaan, Faculty o Science and Technology, School o Applied Physics, Bangi, Selangor, Malaysia 1 saadtm2000@gmail.com Received June 12, 2012 Abstract. A clusterization theory is presented to describe the clusterization probability (preormation actor). The clusterization states are described as a quantum-mechanical cluster-ormation state in a proposed cluster-ormation model to determine the preormation actor o alpha-decay process in radioactive nuclei. The ormation o alpha particle inside the parent nuclei is considered within two postulates; the compound nucleus o Bohr s assumption and the surace eect. The total and ormation energy are obtained rom the rom the binding energies dierences. This model is tested or 212 Po or the alpha-cluster decay. The calculated preormation actor 0.54 has shown good agreements with that o some others as reported. As such, this model could give more insight to the understanding o the nuclear structure in the radioactive nuclei. Key words: alpha-cluster decay, clusterization probability. 1. INTRODUCTION One o the most important processes that have been used to study the nuclear structure o heavy nuclei is the alpha-decay process because o its domination, high-accuracy experimental measurements, and the availability o its microscopic theory. This process spontaneously occurs in most heavy and super-heavy nuclei and is one o the most dominant modes. The simplicity o this process is due to the small number o observables, the decay width, the energy o emitted alpha particle, and some properties o its transition rom ground state to another ground state with the same parity. These reasons have pushed researchers to microscopically modulated the shell closure, and stability o nuclei that ormed islands o stability, around Z ~ 114 and N ~ 184, by dierent theories and models [1 10]. Theoretically, Gamow and Condon and Gurney (1928) were the irst to present an explanation o α-decay as a quantum tunneling eect. Usually, the

2 1282 Saad M. Saleh Ahmed, Redzuwan Yahaya, Shahidan Radiman 2 α-decay process is described as a preormed α-particle tunneling through a potential barrier between the cluster and the daughter nucleus. The penetrability or the penetration probability through the barrier can be determined by the well known Wentzel-Kramers-Brillouin (WKB) semiclassical approximation. The absolute α- decay width is mainly determined by R-matrix [11 15], the general ormula [16-17], or by direct product o the preormation actor and the penetration probability calculated by using WKB approximation [6,18 25]. In the R-matrix ormula, the decay width is a product o the preormation actor (ormation amplitude at coulomb radius r c ) o the alpha cluster inside the parent nucleus and the α-cluster penetration probability. However, in the general ormula the compound nuclei are considered in resonance within quasi-bound states and the alpha-decay process is a transition rom a state represented by a parent waveunction governed by its Hamiltonian to a state o a daughter with the alpha particle represented by a waveunction governed by another Hamiltonian. The preormation actor is sometimes included to be multiplied by the decay width [16, 17]. The preormation actor (cluster ormation probability and also presented in term o the spectroscopic actor) is deined as the quantum-mechanical probability o inding the cluster inside the parent nucleus [4-5, 21, 26]. The preormation actor is very important because it relects inormation about the nuclear structure, since it is a good indicator o deormation in the nuclei. This is explained as the high probability o alpha-cluster ormation rom the last-shells nucleons that leads to deorm the sphericity o the nuclear surace [6]. With the use o R-matrix to calculate the alpha-decay width o 212 Po, Tonozuka and Arima (1979) [11] calculated the ormation amplitude alpha cluster in 212 Po using high coniguration mixing up to 13 ω bases o harmonic-oscillator shell model. The improved calculation was still short o the experimental width by a actor o 23. Dodig-Crnkovic et al. (1985,1989) [12 13] recalculated and improved the ormation amplitude using the multistep shell model method in which the interaction among the valence nucleons o 212 Po were included in dierent considerations. When high-lying states were considered with some pairing interactions, the alpha-decay width was less than the experimental by a actor o 10 [12]. In an attempt to reduce the dimension o matrix elements (using surace delta interaction within truncated model space), the calculation led to enhance the alphadecay width to be less rom the experimental data by a actor o several times[13]. Varga et al. (1992) [14 15] combined the shell model with cluster model within bases o 538 dimensions and included the interaction among the our nucleons within the model space up to the next magic number or 208 Pb. They conirmed the existence o alpha-core structure. Varga and Lovas (1994) repeated the work o Varga et al. (1992) [14 15] but with Gaussain-bases shell model to reduce the dimension o the bases. They included 400 bases and 120 conigurations. The

3 3 Clusterization probability in alpha-decay 212 Po nucleus 1283 results o alpha-decay width were about same as that o Varga et al. (1992) in which 631 bases and 193 conigurations were included to get the best agreement between the calculated and experimental binding energy. For groups o heavy nuclei, the general orm and WKB approximation were adopted to reproduce the alpha-decay widths or hal-lives by using dierent alphacore potential to provide more understanding o the nuclear structure. In some calculations the WKB approximation was adopted, missing the eect o the cluster ormation by assuming a value o 1 or the preormation actor when the alphadecay width were calculated or even-even nuclei and or 212 Po [27] but a value less than one or odd-even, even-odd, or odd-odd [18 20, 27]. In other calculations the preormation actor was assumed as a quantum-mechanical probability with a value less than one and dierent or each nucleus. This value could easily be extracted by dividing the experimental value o the alpha-decay rate by the penetration actor o the nucleus. This led to a phenomenological ormula that could be used to determine the preormation actor or each nucleus. This assumption improved the reproduction o alpha-decay widths [28], but, did not add any more understanding to the microscopic theory o the models used in this process since the potentials o the nuclear structure had contributions only in the penetration actors. The parameters o the suggested potential were mostly determined by itting to the experimental values o alpha-particle energy, thereore each dierent potential can produce dierent preormation actors. In actual act, the preormation actor has neither been calculated or each nucleus by using the current models o nuclear structure, nor its realistic value have been determined [1, 4 6, 21, 23, 25, 29 31]. Recently, Qian and Ren (2011) [24] used a Z-dependent ormula based on two-level model and obtained the value o 210 Po preormation actor. This ormula is actually similar to the phenomenological ormula in which the spectroscopic actor is only the chosen parameter. In the two-level model [32 33] the pairing interactions are considered or proton-proton, neutron-neutron, and proton-neutron with the pairing energies taken rom the separation energies o the nucleons. In our work we present a new quantum-mechanical theory or the clusterization eect. The clusterization eect is interpreted as a ormation o clusters inside a nucleus. Any type o cluster is considered as a quantummechanical cluster-ormation state. The eigenvalues, ormation energies, o these states can be calculated rom the binding energies. The ormation or preormation actor is derived by using Schrödinger equation. This model in its simplest orm is tested or 212 Po nucleus in which only alpha-cluster is considered. In section (2) the theory o clusterization and the cluster-ormation model is presented to derive the preormation actor. An application or the theory is presented or alpha decay o 212 Po in section (3), the method o ormation energy calculation or alpha cluster and results are presented in section (4), and the discussion or 212 Po are given in section (5).

4 1284 Saad M. Saleh Ahmed, Redzuwan Yahaya, Shahidan Radiman 4 2. THEORY OF CLUSTERIZATION AND CLUSTER-FORMATION MODEL When a system exhibits a behavior o more than one cluster, there is more than one state o clusterization. The total wave unction o the system is a linear combination o these clusterization states. These states have same energy and angular momenta but dierent probability densities. Each clusterization state is due to the dierent Hamiltonian or the system. I the system o A nucleons is considered as one cluster, its Hamiltonian operator in the lab system is A 2 ˆ A pi H0 ( r) = ( + Vij ), (1) 2m i= 1 i j= 1 ( j > i) where p ˆi and m i are the momentum operator and the mass o the i th nucleon which interacts with each j th nucleon in the system by a two-body potential V ij. In the second summation notation the parentheses (j > i), are used to show that this summation is only or j index. This Hamiltonian is invariant to the permutations o position vectors r i. When this Hamiltonian is set to the Time-Independent Schrödinger Equation (TISE) as H Ψ ( r) = EΨ ( r), (2) where Ψ 0 is the normalized waveunction o the system. I the system is considered in two groups with A d nucleons and A c nucleons, then A = A + A, (3) d where A d are the nucleons o i = 1 to A d and A c are the nucleons o i = ( A d +1) to A. Then the Hamiltonian can be written as pˆ pˆ H = ( + V ) + ( + V ) 1 c Ad 2 A A 2 A i i ij ij. (4) i= 1 2mi j= 1 i= A 1 2 d + mi j= 1 ( j> i) ( j> i) The two outer summations are or two groups o nucleons but the potential energy terms are expanded to all nucleons. To separate this interaction it is possible to consider the kinetic energy o any nucleon as a sum o two parts, Ki = Koi + K1 i; the irst is due to the interaction o i th nucleon with the other nucleons in its group and the second is or the nucleons in the other group. In terms o operators, pˆ = pˆ + pˆ1. (5) i oi i

5 5 Clusterization probability in alpha-decay 212 Po nucleus 1285 Substituting Eq.(5) in Eq.(4) and splitting the potential-energy terms o the other group rom each sum and rearrange them, we obtain Hamiltonian or a certain value o A d, A d A d ˆ A ˆ A Ad ˆ A p ˆ oi p oi p p 1i 1 j H1( u) = V ij V ij Vij. i 1 2m = i j= 1 i= A 1 2m (6) d + i j= Ad + i= m i j= A m d + j ( j> i) ( j> i) The use o the dierent space coordinates u is due to the redistribution o nucleons in two groups. When H 1 is set in TISE as H Ψ ( u) = EΨ ( u). (7) Ψ 1is obtained to be the normalized waveunction o the system when it behaves as a two-cluster system. So it is to describe one o the clusterization states. In Eq.(6), it is possible to consider dierent values o A d to obtain dierent Hamiltonians H 2, H 3,.., or this equation can be rearranged to be written or three groups (three-cluster system) to obtain more dierent Hamiltonians. Then, the total waveunction is a linear combination o these dierent clusterization states as as Ψ= a 0Ψ 0 + a1ψ 1 + a 2Ψ a nψ n. (8) These waveunction are a complete set o n coordinate space. Each waveunction o clusterization state is deined with dierent space coordinate, so, any is orthogonal to the other, and ound rom the solution o TISE; H Ψ = EΨ, i= 0,1,..., n. (9) i i i The constants (a 0, a 1,.., a n ) are the amplitudes o each the waveunctions states and subjected to The total Hamiltonian o the system is o n a + a a = 1. (10) H = H 0 + H H n (11) which should be linear or the eigenunction o Eq.(8) as HΨ = EΨ. (12) The linearity can be set as long as the states are orthogonal and the Hamiltonian o each state is with dierent space coordinate. The expectation value o energy E rom the total waveunction, in Dirac notation, is E = Ψ H Ψ. (13)

6 1286 Saad M. Saleh Ahmed, Redzuwan Yahaya, Shahidan Radiman 6 when Eq.(8) and Eq.(11) are substituted in Eq.(13) we obtain o 1... n E = a E + a E + + a E. (14) Ψ ο is a o and in. There is a probability or each clusterization The probability o inding the system in the clusterization state the clusterization state Ψ 1is a 1 2 state which is energetically avored, i.e. there should be implicitly speciic energy E responsible or each clusterization state, so Eq.(14) can be written as E = E + E E. (15) 0 1 n Then the probability o i th clusterization state or the preormation actor (clusterization probability) P i can be written as E i a i = Pi =. (16) E From Eq.(16), the clusterization-state probability depends on the total energy o the system and another energy which is desired to be determined within a model that well describes the clusterization eect in the structure o the system. The inclusion o clusterization eect in the structure o the nuclear system within a model requires the considerations o many clusterization states: the system is one cluster, the system is made o two clusters, made o another two dierent clusters, three clusters, etc. Magic-number light and medium nuclei in ground states are examples o the one-cluster system. Radioactive nuclei that emit clusters are examples o one and two-cluster nuclei (more than cluster may be called sub clusters or sibling clusters). For any one-cluster (or mono-cluster) system, the cluster is ormed due to high binding energy among the nucleons; hence there is only one clusterization state. The energy responsible or this one cluster ormation is the total energy o the system, so, in accordance to Eq.(16), the probability o this state is one. For the a two-cluster (di-cluster) system, Eq.(6) can be considered, or a certain value o A d, to determine the total energy o the system and the energy responsible or its clusterization. This equation contains three main sums. In each one there are momentum operators and potential energy written in dierent space coordinates ( u : ξ i, ηi and ρ i), which could be substituted in TISE to obtain the quantummechanical states and the waveunction that describe the nucleons, so these three terms can be written in term o dierent Hamiltonians as H = H A ( ξ ) + H A ( η ) + H A A ( ρ ). (17) d c d c The irst Hamiltonian H A d ( ξ) is or the irst cluster and will be replaced by H, 1 and the second is or the second cluster and will be replaced by H. However the 2 2

7 7 Clusterization probability in alpha-decay 212 Po nucleus 1287 third is or relative motion between the two clusters and will be replaced by H r. Then Eq.(17) and its corresponding energy equation can be rewritten as H = H + H + H, (18) 1 2 r E = E + E + E. (19) 1 2 r The Hamiltonian o Eq.(18) can be written as a sum o two H = H + H r ; one or the clusters ormation ( H = H + H 1 ) and the other or the interaction between 2 the two clusters. Writing them in TISE, we obtain HΦ = ( H + H ) Φ. (20) This equation can be separated in accordance to the Hamiltonians o Eq.(17) and the total waveunction Φ o the system can be expanded as Φ=Φ ( ξ) Φ ( η) Φ ( ρ ) (21) 1 2 r The three waveunctions are supposed to be deined on their space coordinates ξη, and ρ, and ound rom the solution o TISEs; H Φ ( ξ ) = E Φ ( ξ ), (22) H r Φ ( η ) = E Φ ( η ), (23) H rφ r( ρ ) = E rφ r( ρ ). (24) These waveunctions are eigenunctions to the operators; H, H and 1 H 2 r with eigenvalues E, E, and 1 E 2 r respectively. The irst two Eqs.(22, 23) are related to the cluster-ormation Hamiltonian H = H + H, so, Eq.(20) can be 1 2 HΦ= ( E + E + E ) Φ= ( E + E ) Φ= EΦ. (25) 1 2 r r The total energy E is the eigenvalue o the Hamiltonian o Eq.(17). The system described in the waveunction Φ in Eq.(25) is considered to be a twocluster system, i.e. Eq.(23) is only to describe one clusterization state. There are three energies in Eq.(25): E, E r, and E. To determine the amount o energy that is responsible or the cluster ormation to be set in Eq.(16), it is important to mention that the agglomeration o a part o nucleons o the system to a cluster occurs when the agglomerated nucleons are bound enough to each other and not bound enough to the other nucleons. This mean, in terms o total binding energy, the cluster ormation occurs when the total binding energy o the cluster E is large and its binding energy E r to the others is small. In addition, the clusterization increases when the total energy o the system E is large because clusterization eect is so much observable in heavy nuclei. Since, the energy responsible or the

8 1288 Saad M. Saleh Ahmed, Redzuwan Yahaya, Shahidan Radiman 8 cluster ormation is directly proportional to the absolute values o E and E, and inversely to E r, we assume that this energy is E because E = E E r (see Eq.(25)). As a result, Eq.(16) can be rewritten or the clusterization state o Eq.(25) as P = Φ E. (26) The preormation actor o the cluster ormation probability P Φ can theoretically be calculated as P Φ E Φ H Φ =. (27) Φ H Φ When the cluster ormation waveunction is diicult to obtain, it is possible to use E = E - E r which can be written in terms o the expectation values or the calculation o P Φ as P Φ = Φ H Φ Φ H r Φ Φ H Φ. (28) The derivation o the cluster-ormation probability or the preormation actor o a cluster rom Eq.(27) or Eq.(28) is based on the clusterization theory that enables to write the total waveunction in more generality and based on presenting the cluster-ormation model or the structure o nuclear system. For the clusterization state described in Eq.(21) the system is considered in the clusterization state o two clusters. Each cluster is described by the clusterormation eigenunctions: Φ, Φ 1, as in Eqs.(22, 23), and normalized on its sub 2 space. These two waveunctions are the cluster ormation states and should be considered when the system exhibits the clusterization eect especially when the preormation actor or the cluster-ormation probability is required to be calculated rom Eq.(27). The cluster-ormation waveunction is always reerred in works as an intrinsic waveunction. 3. PREFORMATION FACTOR IN ALPHA DECAY One o the approaches o the alpha-decay process is the preormation o alpha cluster inside the parent nucleus by a certain interaction which is in charge o only this preormation. In compound nuclei, there may be many possibilities o ormations o dierent clusters [34], but or simplicity, we assume that there are two possible clusterization states; one Ψ α is or two cluster: the alpha-cluster ormation state and the daughter-cluster ormation state, and the second Ψ 0 is or

9 9 Clusterization probability in alpha-decay 212 Po nucleus 1289 all other possible cluster-ormation states. The total clusterization wave unction o the parent nucleus Ψ can be written as Ψ = a 0Ψ 0 + a α Ψ α, (29) where a 0 is the amplitude o the parent clusterization state without the alphaormation (or alpha clusterization), and a α is the amplitude o the clusterization state o the alpha-cluster ormation state. The cluster ormation inside the parent nucleus was potentially studied with the alpha decay and emphasized in literature review [35], and the clustering eect played a signiicant role in the improvement o the calculations o alpha-decay widths [4 5, 21, 28, 35]. The ormation Hamiltonian operator H or these states consists o the potential responsible or the ormation o the clusters. In many-body system, it is so diicult to get a orm or this potential and any other nuclear potential rom the available nuclear model [4 5, 21]. In the preormed cluster model the cluster states are derived rom the ragmentation theory in which the cluster states are unctions o dynamical collective coordinates o mass and charge asymmetries [2, 26]. The cluster-ormation probability o alpha cluster (as in Eq.(27)), P α is deined as the probability o inding the ormed-alpha particle inside a nucleus with the daughter, as P α = Ψ α Ψ H H Ψ Ψ α. (30) This equation can also be derived using a projection operator ˆPα or alphadaughter cluster-ormation as ˆP α = Ψ α Ψ α, (31) with ψ α is the normalized waveunction o alpha-cluster ormation. Lovas et al. (1998) [28] deined the preormation actor as a projection o the parent nuclei waveunction on the alpha-cluster waveunction. Applying this to the total clusterization waveunction, we again obtain the preormation actor, 2 P = Ψ Pˆ Ψ = Ψ Ψ Ψ Ψ = a, (32) which is equivalent to Eq.(16). α α α α α 4. CALCULATIONS AND RESULTS To calculate the alpha-ormation probability theoretically we need to ind the total waveunction and the cluster-ormation (intrinsic) waveunction o the

10 1290 Saad M. Saleh Ahmed, Redzuwan Yahaya, Shahidan Radiman 10 clustering alpha or Eq.(30). These waveunctions are supposed to be determined rom the solution o TISE in Eq.(20) and Eqs.(22, 23). The determination o the waveunctions are not our target in this work although these are required in Eq.(30). The target o the present work is the evaluation o the cluster-ormation model by the determination o two values o energy; the total energy o the parent nucleus and the ormation energy o the alpha cluster. As long as it is possible to use the experimental value o these two energies it is better to investigate the validity o this approach and the contribution o each to the alpha-cluster ormation probability FORMATION-CLUSTER STATE ENERGY OF 212 Po The ormation energy o a cluster-ormation state is the total intrinsic energy o the cluster and it is equivalent to the total energy o a cluster when the cluster is at rest. This energy is responsible or the cluster ormation in any system rom its constituents. When the cluster is considered ormed rom some o the system nucleons, its cluster ormation energy is equivalent to the sum o the binding energies among the cluster nucleons, or it is the subtraction o the interaction energy between the cluster nucleons and the other nucleons rom the sum o the cluster-nucleons energies. The experimental ormation energy o alpha cluster ( E ) can easily be obtained rom the calculations o the binding energy dierences, the mass deect between the alpha particle and the our ree nucleons. It can ound as ollows; E = E Q, (33) α d α where E α d is the alpha decay energy given as E = d B( A, Z) B( A α 4, Z 2), (34) and Q α is equivalent to the separation energy o alpha particle, and is deined as Q = B( A 4, Z 2) + B(4,2) α B( A, Z). (35) The Q-values and the alpha-decay energy were used to extract the alpha-decay preormation actor [36]. The second energy that contributes to the clusterormation probability o Eq.(30) is the total energy E or the parent nucleus. It can also be ound rom the binding energy dierences, so Eq.(30) can written as E P = α E. (36)

11 11 Clusterization probability in alpha-decay 212 Po nucleus 1291 The theory mentioned above is presented on the basic o the possibility o existence an interaction between any two nucleons inside the nucleus. Thereore, two assumptions are adopted to determine this energy. One is Bohr s assumption o compound nucleus [37] which leads a proposal o this energy as the total energy o the whole parent nucleus, thereore the total binding energy o the parent nucleus is E = E = B( A, Z), (37) Bohr where B(A,Z) is the binding energy o a nucleus o atomic mass number A and atomic number Z. Secondly, or nuclei that undergo the alpha-decay process, the emitted alpha is rom the nuclear surace [11, 34, 38 40]. To consider the energy on the surace, the considered system is not the whole parent nucleus but it is or only the surace nucleons. So the total waveunction Ψ in Eq.(29) is or the surace nucleons. For 212 Po, there are our nucleons above the shell closure. Then, in accordance to the surace assumption, the surace-nucleons energy E sur. is the energy responsible to orm the alpha cluster and the alpha emission. This energy is equal to the alpha decay energy E α given in Eq.(34), or d = =. (38) E E sur. E α d 4.2. FORMATION PROBABILITY OF ALPHA IN 212 Po NUCLEUS The calculations o the ormation energies o 212 Po nucleus were done using experimental data rom Re. [41]. The total energy (E) o the considered system have been ound using Eq.(37) or Bohr assumption o and Eq.(38) or the surace eect in which the total waveunction is considered without the part o the core (the daughter intrinsic waveunction). For each calculation, the ormation-cluster probability has been ound rom Eq.(22). The results are shown in Table 1. The major dierence between two calculated values o cluster ormation P α is the signiicant dierences in the considerations system. The irst value o P α is or two ormation o clusters (two cluster-ormation states) whereas in the second the value is or only the alpha-cluster state. The large value o P α = based on Bohr assumption relects the role o the double-shell closure in the daughter o 212 Po, which gives more chance to the alpha clustering or the alpha decay. The value o P α = 0.54 based on the surace eect is smaller because it is or the alpha-cluster only and it is relatively large because it is the probability rom all possibilities o clustering in only the last our nucleons within the interaction rom the daughter.

12 1292 Saad M. Saleh Ahmed, Redzuwan Yahaya, Shahidan Radiman 12 Table 1 The experimental total energy o the parent 212 Po nucleus E, alpha-cluster ormation energy E, cluster ormation probabilities P α Basic o the method Bohr assumption E (kev) E (kev) P = E E E Bohr = Surace eect E sur. = α 5. DISCUSSION The ormation o any cluster inside the nucleus is proposed as a quantummechanical ormation-cluster state with its eigenvalues o ormation energy is calculated rom the measured binding energies o nuclei and energy separation. 212 Po nucleus was chosen to test this postulate because this typical nucleus consists o two protons and two neutrons in its model space ater the shell closure. Calculations or this nucleus were done within R-matrix theory o decay by others to test the shell model, the hybrid model (shell and cluster model), and the Bardeen, Cooper, and Schrieer (BCS) model. Within some o these models the alpha-decay energy and the ground state binding energy were reproduced to test the type o two-nucleon orce, and the alpha-cluster ormation was presented in dierent orms depending on the alpha-decay theory used. In addition, the preormation actor o 212 Po was also extracted by others who reproduced the experimental alpha-decay widths o certain groups o nuclei using WKB approximation and the general orm o decay width FORMATION ENERGY OF ALPHA CLUSTERIZATION IN 212 Po The calculated ormation energy o alpha E or 212 Po (as given in Table 1) is kev. The value indicates a bound system, the alpha cluster. Comparison with the ormation energy o the ree alpha ( B α = MeV the total binding energy o alpha particle), indicates that the alpha cluster is already preormed with all our nucleons but its intrinsic energy is less than that o the ree alpha points to two eects; the volume expansion and the transient eect. Considering the r-dependence o the nuclear orce it will be easy to deduce that there is a volume expansion in the interior alpha with the nucleon density less than that o the exterior alpha. Brink and Castro (1973) [38] studied the stability o nuclei through the density o the nuclear matter due to dierent nucleon-nucleon orces and ound that the alpha cluster is ormed when the density o the nucleon becomes one third the density o nuclear center density. Considering the alpha-

13 13 Clusterization probability in alpha-decay 212 Po nucleus 1293 decay process in 212 Po and the dierences between the interior and exterior alpha cluster indicates that there is a transient eect terminates with exterior alpha particle with less volume. This transient property o alpha-cluster ormation inside the parent nucleus was reported by Hodgsona and Betak (2003) [34]. This ormation energy could be a very good indicator to clusterization that could happen inside nuclei ALPHA-CLUSTER FORMATION PROBABILITY OF 212 Po There is no experimental preormation actor or comparison and any estimated values o preormation were either extracted rom itting experimental data o alpha-decay width within a model with some adjustable parameters or determined by adopting the shell and the cluster models within complicated calculations. Comparison o the preormation actor o our approach with those o others that use microscopic theories and methods to determine a speciic preormation actors or each nucleus was also conducted. These methods included the use o microscopic models or the structure o the parent nuclei. There were many attempts, as in Table 2, to calculate the preormation actor o 212 Po in the eighties and nineties. All these attempts used the shell-model conigurations or 212 Po as a core o 208 Pb+4 nucleons, and used the R-matrix ormula in which the ormation amplitude is a unction o radial distance, and the value o preormation actor was ound at the surace where the radius is called the Coulomb radius r c, about 8.2 m or 208 Pb. The calculation o the proposed model in our present work is based on experimental data and ree rom parameter which may lead to realistic values o preormation actors. As in Table 2, Tonozuka and Arima (1979) [11] used the R-matrix to calculate the ormation amplitude o alpha cluster in 212 Po using high coniguration mixing up to 13 ω bases o harmonic-oscillator shell model neglecting the clusterization process through the exclusion o the nucleon-nucleon interactions. The improved calculation was still less than the experimental width by a actor o 23. The value o the ormation amplitude was ound at 9.6 m o Coulomb radius. In our approach the clusterization is the main eect and the energy required or this can microscopically be described as the sum o all nucleon-nucleon interactions among the nucleons that orm the alpha cluster. When these interactions were partially considered using the multistep shell model method in the interaction among the valence nucleons o 212 Po the ormation amplitude was improved [12-13]. High-lying states with some pairing interactions were considered but the alpha-decay width was ound to be less than the experimental by a actor o 11 while the ormation amplitude was 0.74 at the Coulomb radius r c = 8.2 m [12]. In the truncated model space using surace delta interaction the calculation led to enhance the alpha-decay width to be less rom the experimental data by a actor o several times [13]. Varga et al. (1992) [14 15] used the R-matrix in combination o

14 1294 Saad M. Saleh Ahmed, Redzuwan Yahaya, Shahidan Radiman 14 the shell model with cluster model within bases o 538 dimensions and included the interaction among the our nucleons up to the next magic number 208 Pb. The Volkov (V1) orce and the Brink-Boeker B1 orce were tested and ound the clustering amount at the same coulomb radius to be conirming the existence o alpha-core structure. The V1 and B1 eective interactions were chosen to be suitable to determine the binding energies o the last our nucleons using the shell model bases or the single particle and to reproduce the alpha-decay energy. Similar work was conducted by Varga and Liotta (1994) [42] but with Gaussian-bases shell model to reduce the dimension o the bases. Bases o 400 and 120 conigurations were included. The ormation amplitude at about r c = 8 m was about 0.1 m -1/2 but their results o alpha-decay width were about same o Varga et al. (1992). Bases o 631 and 193 conigurations were needed to get the best agreement between the calculated and experimental binding energy. Varga et al. (1992 [15], 1994 [42]) achieved good results when they reproduced the alphadecay width with a dierence rom the experimental by a actor less than one. This combination o the two models relects very good description on the eect o clusterization expressed in clustering amount o about The Bardeen, Cooper, and Schrieer (BCS) method was also used to reproduce the alpha-decay width which was two-third the experimental value [28]. The model used in the present work adopted the binding energy dierences as eigen energies or the cluster states to give direct contribution to the more realistic value o preormation actor. Table 2 The amount o alpha-clustering in 212 Po represented in dierent orms and expressions and calculated by the use o R-matrix and microscopic theories Re. Tonozuka and Arima (1979) [11] Dodig-Crnkovic et al. (1985) [12] Varga et al. (1992) [14 15] Varga and Liotta (1994) [42] Our work Expression o clustering Dimensionless Reduced width ormation amplitude (or reduced width) Amount o clustering Notice at Coulomb radius r c o 9.6 m r c = 8.2 m r c = 9 m Γ exp. / Γ cal. clustering amount r c = 8.2 m 2/3-1 ormation amplitude 0.1 About r c =8 m 1/2 Cluster ormation probability Total energy rom Eq.(38) Total energy rom Eq.(37) According to the proposed model two values o alpha-cluster ormation probabilities were calculated (Table 2) to indicate to the contribution o the parent-nucleus total energy (Table 1). Comparing the large value o P α =0.994 by

15 15 Clusterization probability in alpha-decay 212 Po nucleus 1295 Bohr assumption with all previous calculations (Table 2 and 3) has shown that the probability o alpha clustering is great but still smaller than one, and smaller than that o Dodig-Crnkovic et al. (1985) [12], 0.74 at small Coulomb radius 8.2 m. The small preormation actor o Tonozuka and Arima (1979) was due to ignoring the interaction among the nucleons that orm the alpha cluster. In the present work, the large value o the preormation actor is due to the inclusion o the interaction among these nucleons and the total energy o the parent nucleus based on Bohr assumption which shows the contribution to both alpha and the daughter clusterization. When the surace nucleons interactions were considered in the determination o the total energy o the parent nucleus the value o preormation actor was more realistic where P α = 0.57 is within the values ( ) as given in Table 2, except that or Tonozuka and Arima (1979) [11], which include the contribution rom the surace eect. The small dierences may be due to the contrasts o the suracenucleons-interactions contributions. Table 3 The preormation actor P α o alpha-decay in 212 Po calculated by dierent methods Reerence P α. Theory o decay Method o extraction Buck et al. (1993) [18] 1.00 WKB Assumed as a large value or even-even nuclei. Buck et al. (1994) [27] 1.00 WKB Assumed as a large value or even-even nuclei. Hoyler et al. (1994) Adjusted and chosen to reproduce the WKB [23] experimental alpha-decay width. Ni and Ren (2009) The general From itting to experimental width to even-even 0.56 [17] orm nuclei with 126 < N 176. Routray et al. (2009) 1.00 WKB Assumed as a large value or even-even nuclei. Our work 0.54 Clusterormation Cluster ormation or alpha model Cluster ormation or alpha and the daughter. Many researchers potentially reproduced the alpha-decay widths or hal-lives or dierent groups o radioactive nuclei to provide uniied microscopic insight to the nuclear structure o these nuclei. The consideration o alpha-core system yielded a remarkable reproduction o the decay widths, within a deviation actor o a value less than 3. In such calculations (Table 3) the preormation actor was included because o uncertain ormation o alpha-core system especially in the initial state o the parent nucleus. The calculated preormation actor completely depended on the calculation o the penetrability, the WKB approximation, except or the work o Ni and Ren (2009) [17] because the general orm o decay width was used. All the calculations were perormed by adopting the Coulomb potential and a nuclear potential or a system o two-body, alpha-core system, but the

16 1296 Saad M. Saleh Ahmed, Redzuwan Yahaya, Shahidan Radiman 16 various nuclear potentials used in the calculations o alpha-decay width do not have large dierences in the value o the preormation actor [4]. The importance o such calculations is due to the consideration o alpha-core system because within such assumption the interaction o nucleons that orm the alpha cluster is already mostly included. Thus it is worth to compare the results o these works with the preormation actor or 212 Po (Table 3) in the present work. Buck et al. (1993, 1994) [18, 27] and Routray et al. (2009) [30] calculated the alpha-decay width or alpha-decay nuclei. The preormation actor was assumed to be one, the largest or the even-even nuclei and showed that cluster model was good in reproducing the experimental alpha-decay width with a small dierence. However, or other nuclei, the preormation actors extracted were less than one, the existence o alpha-core system beore the emission o alpha particle was strongly conirmed. Hoyler et al. (1994) [23] used the double density olding potential that was relevant or the elastic and inelastic alpha scattering or the alpha-core system with large volume integral and relatively high global quantum number G o alpha oscillatory. The itting to the experimental hal-lie o alpha decay led to a small preormation actor o The remarkable attempts aim to get good values o preormation actor (Table 2) with Ni and Ren (2009) [17] showed a notable value o 0.54 or the preormation actor through a itting or only 212 Po (Table 3). In addition, Ni and Ren (2009) used the general ormal instead o WKB method and the potential was the Wood-Saxon interaction. The preormation actor o 0.54 calculated (as shown in Table 3) is in a very good agreement with that o Ni and Ren (2009) [17]. A similar value or 218 U preormation actor (about 0.3) was obtained by Qian and Ren (2011) [24] when a ormula based on two-level model was used. This ormula is a parameter and Z dependent, which is actually similar to the phenomenological ormula in which the spectroscopic actor is the only chosen parameter. In the two-level model [32 33] the interactions o pairing are considered or proton-proton, neutron-neutron, and proton-neutron. These pairing energies are taken rom the separation energies o the nucleons. As a general noticeable remark in Table 2 and 3, one can consider the value o the preormation actor o alpha decay in 212 Po is between 0 < P α 0.7. Our calculated preormation o 0.54 is in agreement with that o 0.56 recently determined by Ni and Ren (2009) [17]. 6. CONCLUSION The cluster-ormation model proposed in the present work is a good step in inding a realistic value o the preormation actor or nucleus, but urther works need to be conducted to conirm its validity or all nuclei. The value obtained also has shown the particle existence o alpha cluster, which can be used or prediction o cluster radioactivity.

17 17 Clusterization probability in alpha-decay 212 Po nucleus 1297 The magnitude o the ormation energy is a relection to the cluster-volume expansion beore the penetration process. This may be taken into account when the penetrability is calculated. In addition, this magnitude indicates to the deormation on the surace o the parent nucleus. Acknowlodgements. The authors would like to acknowledge the inancial support rom the Ministry o Higher Education Malaysia (through the Fundamental Research Grant Scheme) UKM- ST-06-FRGS APPENDIX A MANY MICROSCOPIC HAMILTONIANS OF MANY CONSIDERTIONS OF CLUSTERIZATIONS The Hamiltonian operator o a system o A-nucleons nucleus in the laboratory system can be written as A 2 pˆ i H = + 2m A V ij, (A-1) i= 1 i j= 1 j > i Where pˆ i and m i are the momentum operator and the mass o the i th nucleon which interacts with each j th nucleon in the system by a two-body potential V ij. This Hamiltonian is invariant to the permutations o position vectors r i. In some nuclear reactions and decay, the system exhibits a behavior o two groups (two clusters) in what is called the clusterization eect. I the nucleons o the system are considered in two groups with A d nucleons and A c nucleons, where d c A = A + A, (A-2) where A d are the nucleons o i = 1 to A d and A c are the nucleons o i = ( A A d ) to A. Then the Hamiltonian can be written as H = + V + + V Ad A A A 2 2 pˆ ˆ i p i ij ij i 1 2m i j 1 i A A 2m = = = c i j= 1 j> i j> i. (A-3) The two outer summations are or two groups o nucleons but the potential energy terms are expanded to all nucleons. To separate this interaction it is possible to consider the kinetic energy o any nucleon as a sum o two parts, K i = K oi + K 1i; the irst is due to the interaction o i th nucleon with the other nucleons in its group and the second is or the nucleons in the other group. In terms o operators,

18 1298 Saad M. Saleh Ahmed, Redzuwan Yahaya, Shahidan Radiman 18 pˆ = pˆ + pˆ. (A-4) i oi 1i Substituting Eq.(A-4) in and splitting the potential-energy terms o the other group rom each sum in Eq.(A-3) and rearrange them, we obtain a non-invariant Hamiltonian to the permutations, Ad Ad A A Ad A pˆ ˆ ˆ ˆ oi p oi p p 1i 1 j H = + V ij + + V ij V ij. i 1 2m = i j= 1 i= A A 2m c i j= A Ac i= m i j= A A m c j j> i j> i (A-5) This equation contains three main sums. In each one there are momentum operators and potential energies which could be substituted in Schrodinger Equation (S.E.) to obtain the quantum-mechanical states and the waveunction that describe the nucleons, so these three terms can be written in term o dierent Hamiltonians as H = H ( A ) + H ( A ) + H ( A A ). (A-6) d c r d c It is important to notice that only the irst two Hamiltonians are invariant to the permutation. It is possible to consider the similarity o the irst two Hamiltonians as being o clusters, so they can be written together to represent the whole nucleons o the system. Then, the Hamiltonian responsible or the ormation can be set as H = H ( A ) + H ( A ). (A-7) d c The Hamiltonian o Eq.(A-6) can be written as a sum o two H = H + H r ; one or the clusters ormation and the other or the interaction between the two clusters. Writing them in the Time-Independent Schrodinger Equation (TISE), we obtain HΨ = ( H + H ) Ψ. (A-8) This equation can be separated in accordance to the Hamiltonians o Eq.(A-6) and the total waveunction o the system can be written as Ψ=Φd( ξ) Φc( η) Φdc( ρ ). (A-9) The three waveunctions are supposed to be deined on their space coordinates ξη, and ρ, and ound rom the solution o TISEs; H Φ ( ξ ) = E Φ ( ξ), H Φ ( η ) = E Φ ( η), H Φ ( ρ ) = E Φ ( ρ ).(A-10) d d d d c c c c r cd r dc These waveunctions are eigenunctions to the operators; H d, H c and H r with eigenvalues E d, E c, and E dc respectively. The irst two equations in r

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