Alpha-cluster preformation factors in alpha decay for even–even heavy nuclei using the

cluster-formation model

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J. Phys. G: Nucl. Part. Phys. 40 (2013) 065105 (22pp) doi:10.1088/0954-3899/40/6/065105

Alpha-cluster preformation factors in alpha decay foreven–even heavy nuclei using the cluster-formationmodel

Saad M Saleh Ahmed, Redzuwan Yahaya, Shahidan Radimanand Muhamad Samudi Yasir

School of Applied Physics, Faculty of Science and Technology, Universiti Kebangsaan Malaysia,43600 Bangi, Selangor, Malaysia

E-mail: saadtm2000@gmail.com

Received 23 October 2012Published 3 May 2013Online at stacks.iop.org/JPhysG/40/065105

AbstractThe preformation factors of alpha clusters for fully-favoured alpha decay ofeven–even heavy nuclei, 82 < Z <114, 126 < N <170, were determinedusing the alpha preformation formula taken from the hypothesized cluster-formation model of our previous work. In this model, the formation of analpha particle inside the parent nuclei was considered as a quantum-mechanicalcluster-formation state. The eigenvalues of these states are determined from thedifferences between the binding energies. The total energy of the consideredtotal system state was investigated by adopting the alpha-decay energy and theenergy of the open-shell nucleons. The clustering amount was also calculatedon the basis of this model. The results showed a very good consistency withthe alpha-decay constants. This model presents a significant step forward in thedetermination of realistic preformation factors.

1. Introduction

For many years, the alpha-decay process has been widely used to study the nuclear structureof heavy nuclei. The alpha-decay widths of the nuclei have been measured with very goodaccuracy. The alpha-decay theory was mainly presented for the alpha-decay widths to becalculated either by the Wentzel–Kramers–Brillouin (WKB) or R-matrix [1–6] methods orby the general formula [7–8]. The main models for the nuclear structure of these nuclei haveselectively been adopted and adapted to reproduce the experimental widths/half-lives with thehope of presenting a unified approach for this process. Technically, the preformation factorwas theoretically considered and included within the alpha-decay theory, especially whenthe inclusion of alpha clusterization enhanced the calculations. The preformation factor isimportant due to its relationship to the dynamic states of the valence nucleons in the groundstate of nuclei that emit alpha particles. Therefore, the preformation factor may provide moreinformation about the nuclear structure if this factor is described and determined within amicroscopic model.

0954-3899/13/065105+22$33.00 © 2013 IOP Publishing Ltd Printed in the UK & the USA 1

J. Phys. G: Nucl. Part. Phys. 40 (2013) 065105 S M Saleh Ahmed et al

The preformation factor of an alpha cluster in a nucleus is defined as the probability offinding the alpha cluster inside the parent nucleus. This factor refers to the clusterization ofan alpha particle from four nucleons before the emission. Therefore, the value of this factorshould be less than or equal to one [9]. This factor was determined within many methods. Inthe R-matrix method it is called formation amplitude and can be calculated from the initialtailored wavefunction of the parent nucleus [1–5]. This formation amplitude is to express theamount of alpha formation inside the parent nuclei. In the WKB method the amount of alphaformation was included as a preformation factor to be multiplied by the assault frequencyand the penetrability. Some researchers ignored this factor (its value assumed one) whenthe clusterization was included within the shell model [10–13]. This inclusion and the useof overlapping alpha-daughter potentials of the nuclear force and the Coulomb force couldsignificantly enhance the reproduction of the alpha-decay widths. However, the deviations ofthe calculated logarithmic alpha-decay widths were still large by a factor of approximately 2–3.Many attempts proposed different nuclear alpha-daughter potentials and considered the valueof the preformation factor to be less than one; greater values were chosen for even–even nucleithan that for odd–even or odd–odd nuclei [10, 11, 14–20]. Other works fitted experimentalQ-values or experimental alpha-decay widths to determine the preformation factors for a groupof nuclei [7, 21–25]; however, the deviation was still considerable.

For a typical nucleus, 212Po, serious efforts were made by Varga et al (1992) [1, 5]to calculate the values of the alpha-decay preformation factors for some nuclei using theR-matrix method, the cluster model, and the shell model for the many-body system with aneffective NN interaction. The wavefunction of the ground state was well built to producethe experimental energy of the alpha decay of 212Po. This wavefunction was composed of alarge number of high configuration mixing bases of single particle states and merged with thecluster wavefunction [1, 5, 26]. These calculations led to a very good reproduction of the alpha-decay width, but for other nuclei with a larger number of nucleons, this complexity was avoided[7, 27]. Unfortunately, the accuracy of the calculations of the theoretical alpha-decay constantswithin the alpha-cluster postulate was not able to be well evaluated if the preformation factorwas not estimated microscopically [14, 28].

Recently, there has been an increasing desire to determine the preformation factor for eachnucleus with a model that can describe the clusterization of an alpha particle and distinguish theneighbouring nuclei. Potential work using the preformed cluster model (PCM) or a formula forthe preformation factor has been done to provide a better approach that individually determinesthe preformation factor and the penetrability to reproduce the alpha decay constant. In the PCMof Gupta and collaborators the preformation factor of alpha or any other clusters was calculatedby solving the Schrodinger equation for the dynamical flow of mass and charge [29–32]. Niand Ren (2009 and 2010) [33, 34] and Qian and Ren [27] used a formula based on a two-level model and obtained the value of the formation factor. This formula depends on theatomic number Z of the parent nucleus and on an adjustable parameter that was different fordifferent groups of nuclei. This formula is actually similar to the phenomenological formulaobtained from the nuclear calescence reactions [9, 35, 36], in which the spectroscopic factor ofthe alpha is only a chosen parameter. In the two-level model [37, 38], pairing interactions areconsidered for proton–proton (P–P), neutron–neutron (N–N), and proton–neutron pairs. Thesepairing energies are taken from the separation energies of the nucleons. Others have focused ondetermining a realistic preformation factor that can be used for calculating and reproducing thedecay constant using the WKB method. The preformation factor was determined by dividingthe experimental alpha-decay constant by the assault frequency and the penetrability, whosecalculation depended on the model of the alpha-core potential [25, 39–42]. Unfortunatelysuch a preformation factor was different when a different potential model for the calculated

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J. Phys. G: Nucl. Part. Phys. 40 (2013) 065105 S M Saleh Ahmed et al

penetrability was used. Such calculations reflect information about the clustered alpha but notabout the mechanism of alpha clustering.

In a previous work [43], we have proposed a new simple model, called the cluster-formation model (CFM), to determine the preformation factor and the clustering amount (CA)of an alpha cluster for 212Po. The determined alpha clustering value was in agreement with thatof Varga et al (1992) [1, 5], and the preformation factor was in agreement with the value thatNi and Ren [44] extracted by fitting the experimental alpha-decay width to their microscopiccalculations for a wide range of nuclei. This agreement motivated us to use the CFM todetermine the preformation factors of alpha decay for the even–even open-shell nuclei withalpha-decay mode intensity of 99%–100%. The even–even nuclei were chosen because themodel presented a good result for 212Po in a previous work; in addition, these nuclei are locatedbetween two magic numbers to avoid any rapid change in the values of the preformation factor.Because of these factors, the nuclei were expected to show a smooth and clear trend in thevalues of their preformation factors.

In our present work, the preformation factors within the CFM for even–even nuclei weredetermined. The formation energies of alpha clusters and the total energy of the parent nucleiwere used as in the previous work [43] and analysed based on the binding energy differences.For each energy calculation method, the preformation factors are calculated and comparedwith the experimental alpha-decay widths.

2. Theory

The process of clusterization in cluster radioactivity is considered to occur in two mechanisms;one mechanism is the formation of the cluster, and the second mechanism is the separationof the cluster from the daughter. This two-mechanism consideration is used to determine thepreformation factor. In the first of the following subsections, we show how the total Hamiltonianof the nucleus is split in accordance with the two mechanisms and how the clusterization stateis defined. In the second subsection, we show how the total wavefunction is defined in termsof clusterization states and the preformation factor. In the third subsection, we present thecluster-formation model to determine the preformation factor for any clusterization state.

2.1. Different Hamiltonians for different considerations of clusters

From the microscopic description of a quantum many-body system, the Hamiltonian operatorof a system of A nonrelativistic nucleons in the laboratory system can be written as [43]

H =A∑

i=1

p2i

2mi+

A∑i< j=1

Vi j (1)

where pi and mi are the momentum operator and the mass of the ith nucleon that interactswith each jth nucleon in the system by a two-body potential Vij. The total wavefunction ofthe system � can be obtained from the solution of the time-independent Schrodinger equation(TISE) using the symmetrized Hamiltonian of the system of total energy E as

H� = E�. (2)

In some nuclear reactions and decays, the system behaves like two groups (two clusters); thisbehaviour is called the clusterization effect. If the nucleons of the system are considered astwo groups with Ad nucleons and Ac nucleons, such that A = Ad + Ac, the Hamiltonian canthen be written as

H =⎛⎝ ∑

1�i�Ad

p2i

2mi+

∑i�Ad ,i< j�A

Vi j

⎞⎠ +

⎛⎝ ∑

Ad<i�A

p2i

2mi+

∑Ad<i�A,i< j�A

Vi j

⎞⎠ (3)

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J. Phys. G: Nucl. Part. Phys. 40 (2013) 065105 S M Saleh Ahmed et al

where Ad are the nucleons of i = 1 to Ad and Ac are the nucleons of i = (A− Ad) to A. Eachpair of terms inside each parenthesis in equation (3) corresponds to a group of nucleons,but the potential energy sum is expanded to all nucleons. To separate this interaction, eachgroup is considered to have two motions, the internal motion and the centre-of-mass motion.This interpretation requires that the kinetic energy of any nucleon is represented as a sumof two parts, Ki = Koi + K1i; the first term is due to the interaction of the ith nucleon withthe other nucleons in its group (internal motion), and the second term is the interactionwith the nucleons in the other group (for the centre-of-mass motion). For each group, thesummation over K1i represents the kinetic energy of that group’s centre of mass [45]. In terms ofoperators,

p2i = p2

oi + p21i. (4)

Substituting equation (4) in equation (3), merging the p21i terms in the first and the second

parentheses into one summation, splitting the range of the potential-energy sum in the firstparenthesis into i � Ad, i < j � Ad and i � Ad, Ad < j � A, and rearranging these terms, weobtain,

H =⎛⎝ ∑

1�i�Ad

p2oi

2mi+

∑i�Ad ,i< j�Ad

Vi j

⎞⎠ +

⎛⎝ ∑

Ad<i�A

p2oi

2mi+

∑Ad<i�A,i< j�A

Vi j

⎞⎠

+⎛⎝ ∑

1�i�A

p21i

2mi+

∑i�Ad ,Ad< j�A

Vi j

⎞⎠ . (5)

This equation contains three sets of parentheses. In each set, there are momentum operatorsand potential energies that could be separately used in the Schrodinger equation to obtain thequantum-mechanical states and the wavefunction that describe the nucleons; therefore, thesethree terms can be written in terms of different Hamiltonians as

H = Hf d + Hf c + Hdc. (6)

When each Hamiltonian is set in the TISE,Hf d�d(ξ ) = E f d�d(ξ )

Hf c �c(η) = E f c �c(η)

Hdc�dc(ρ) = Edc�dc(ρ).

(7)

�d(ξ ) is the wavefunction describing the internal dynamic quantum-mechanical state ofthe total energy Efd and the total spin due to the interactions among the nucleons (1,2, . . . ,Ad), represented and normalized in the spin and space coordinates ξ : ξ1, ξ2, . . . , ξAd . �c(η)

is the wavefunction describing the internal dynamic quantum-mechanical state of the totalenergy Efc and the total spin due to the interactions among the nucleons (Ad + 1, . . . , A),represented and normalized in the spin and space coordinates η: ηAd+1, ηAd+2, . . . , ηA. Thelast two wavefunctions are similar because each wavefunction describes a group of nucleonscorrelating to a cluster of total angular momentum resulting from the coupling of each nucleon’sspin and relative orbital angular momentum to that of another nucleon within the cluster itself.The wavefunction responsible for the coupling between the two clusters is not any of these lasttwo wavefunctions but is �dc(ρ). The wavefunction �dc(ρ) describes the dynamic quantum-mechanical state of the total energy Ecd due to the interactions between the nucleons (1,2, . . . , Ad) and the nucleons (Ad +1, . . . , A), represented and normalized in the spin andspace coordinates ρ: ρ1, ρ2, . . . , ρA. If the two clusters are considered as single particles, thewavefunction �dc should be a Slater determinant wavefunction. All these wavefunctions inequation (7) are orthogonal to each other because the wavefunctions are solutions of differentTISEs.

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J. Phys. G: Nucl. Part. Phys. 40 (2013) 065105 S M Saleh Ahmed et al

Note that only the first two Hamiltonians in equation (7) are invariant to the permutation.By considering the similarity of these Hamiltonians, they can be written together to represent allof the nucleons of the system as non-interacting clusters. The Hamiltonians responsible for theformation of these clusters are then written together as Hf = Hf d +Hf c, and the Hamiltonian ofequation (6) can be written as a sum of two separate Hamiltonians H = Hf +Hdc; one operatordescribes the clusters’ formation, and the other operator describes the interaction betweenthese clusters, which can be called the fragmentation term. Rewriting these Hamiltonians inthe TISE, we obtain

H� = (Hf + Hdc)� = E�. (8)

This equation can be separated in accordance with the Hamiltonians of equation (6), and thetotal wavefunction of the system, with a single total probability, can be written as a product ofthe wavefunctions of equation (7) as

� = �d(ξ )�c(η)�dc(ρ). (9)

These wavefunctions are eigenfunctions of the operators Hf d, Hf c and Hr with eigenvaluesEd, Ec, and Edc, respectively. The first two equations in equation (7) are related to the cluster-formation Hamiltonian Hf ; therefore, equation (8) can be written as

H� = (Ed + Ec + Edc)� = (E f + Edc)� = E�. (10)

The total energy E is the eigenvalue of the total Hamiltonian H of equation (6), and the totalenergy is

E = Ec + Ed + Edc. (11)

The reformulation of the Hamiltonian in equation (1) into equation (5) with different clusters ofdifferent numbers Ad of nucleons enables us to write the Hamiltonian in many different formsdepending on the consideration of the different clusters. Different wavefunctions can then beobtained for different values of Ad,; each wavefunction represents a different clusterizationstate.

2.2. Quantum-mechanical clusterization states

The observations of alpha and cluster decays in heavy and super-heavy nuclei and the cluster-preformation model consideration inside the parent nucleus before the emission signify theexistence of more than one preformation inside the parent nucleus. The evidence of differentpreformations in the parent nucleus is due to the different emitted clusters during cluster decay.The differences in preformations are due to the differences in the clusterizations inside theparent nucleus. For each preformation, there is a different wavefunction as in equation (9) anda different Hamiltonian as in equation (5). Therefore, we assume that, for each preformation orclusterization, there is a clusterization state represented by a wavefunction as in equation (9).This clusterization state wavefunction is from the solutions of the TISE as in equation (7) witha Hamiltonian as in equation (5).

If a nucleus of total energy E exhibits behaviours of N different clusterization states,there is a Hamiltonian Hi for the ith clusterization state defined by the wavefunction �i. TheHamiltonians are similar to that of equation (6) if the nucleus behaves like two clusters. Thewavefunctions are eigenfunctions of these Hamiltonians:

Hi�i = E �i i = 1, 2, . . . , N. (12)

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J. Phys. G: Nucl. Part. Phys. 40 (2013) 065105 S M Saleh Ahmed et al

Therefore, this nucleus can be described by a total time-independent wavefunction that is alinear combination of these clusterization orthonormalized wavefunctions:

� =N∑

i=1

ai�i. (13)

The constants ai are the amplitudes for the clusterization states of the complete set, and withinthe orthogonality condition,

N∑i

|ai|2 = 1. (14)

The total wavefunction should be also an eigenfunction of the total Hamiltonian H, which canbe set as

H =N∑

i=1

Hi. (15)

When this total Hamiltonian is set in the TISE, the eigenvalue is the total energy of the nucleusE as follows:

H� = E�. (16)

The linearity of this equation is set as long as the Hamiltonians of equation (12) are differentfor different wavefunction states; in Dirac notation,

Hi|� j〉 = δi jE|� j〉 (17)

where δi j is the Kronecker symbol. Therefore, the total wavefunction in equation (13) isnormalized. If equation (16) is multiplied by 〈�| from the left and integrated over the volumeelement, remembering the orthogonality and equation (17), we obtain

E =N∑

i=1

|ai|2E (18)

where |ai|2 is unitless and represents the probability of the ith clusterization state; this term isequivalent to the preformation factor, the determination of which was our goal.

2.3. Clusters in the hypothesized cluster-formation model

Mathematically, neither equation (18) nor equation (14) is sufficient to determine thevalues of the preformation factor for the clusterization states, even if the wavefunctionsof the clusterization states are obtained. It is still important to specify the interactionenergy responsible for the cluster formation to be able to determine the probabilities (thepreformation factors). This specification requires the definition of the cluster to obtain itssuitable wavefunction. Such a wavefunction is already mentioned in equation (8), foundin subsection 2.1, where the Hamiltonian for a clusterization state of two clusters is splitinto two types of Hamiltonians; one Hf is for the interaction responsible for the cluster-formation mechanism, and the other operator is for the relative motion between the clustersHdc. Therefore, equation (9) can be written as � = � f �dc where

� f = �d(ξ )�c(η). (19)

The cluster described by a wavefunction such as � f in equation (19) can be defined as agroup of particles that are tightly correlated with each other and act as one particle at rest.The wavefunction represents the total intrinsic energy, intrinsic total angular momentum, and

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J. Phys. G: Nucl. Part. Phys. 40 (2013) 065105 S M Saleh Ahmed et al

isospin of the cluster. Two clusters have the same cluster-formation state when the clusterconsists of the same particles and same intrinsic energy, angular momentum and isospin.Therefore, this state does not represent any interaction of a cluster with any other particle. Inaddition, the total wavefunction of a cluster is equivalent to its cluster-formation state whenthe cluster does not interact with another particle and has zero kinetic energy as a one-particlecluster. For this special case, the total energy and the total angular momentum are equivalentto the intrinsic energy (cluster-formation energy) and the intrinsic angular momentum of thecluster, respectively.

The clusterization inside a nucleus is not only the cluster formation but is also the behaviourof the formed cluster as one particle. This result is because the clusterization does not onlydescribe an agglomerated structure; this structure (the cluster) must also exhibit a dynamicstate with respect to other clusters (or particles). The clusterization depends on two values ofenergy: the intrinsic energy (cluster-formation energy) Ef and the energy of fragmentation Er

(see equations (8)–(11)). Therefore, the probability of the cluster formation increases whenthe cluster-formation energy decreases and the fragmentation energy increases, i.e., when thebinding energy among the cluster nucleons is greater than the binding energy of this cluster toanother. Hence, the assumption of the cluster-formation energy for the cluster-formation statedoes not ignore the fragmentation energy because Ef = E − Er (see equations (7)–(11)).

To connect between the cluster-formation model and the clusterization states,equation (18) can be rewritten in terms of the cluster-formation energy as

E =N∑

i=1

E fi. (20)

Comparing this equation with equation (18), the cluster-formation energy of the ith-clusterization state can be written in terms of the total energy as

E fi = |ai|2E. (21)

The probability of the ith clusterization state Pi is equivalent to |ai|2, which can be calculatedin accordance with equations (8)–(11) as

Pi = 〈�i|Hfi|�i〉〈�i|Hi|�i〉 . (22)

This equation can be written in terms of the cluster-formation and fragmentation Hamiltonians(or by remembering H = Hf + Hdc) as

Pi = 〈�i|Hfi|�i〉〈�i|Hfi|�i〉 + 〈�i|Hdci|�i〉 . (23)

3. Calculations, analysis, results, and discussion

The calculation of the cluster-formation probability (preformation factor) from equation (22)or equation (23) requires the substitution of a nuclear potential into the TISE and solvingthis equation to obtain the wavefunctions. The determination of these wavefunctions is notour goal in this work, although the evaluation of the cluster-formation approach requires thedetermination of two energy values; the total energy of the parent nucleus (or the consideredsystem) and the formation energy of the alpha cluster. As long as it is possible to use theexperimental value of these two energies, it is better to investigate the validity of this approachand the contribution of each to the alpha-cluster formation probability. The cluster-formationprobability of an alpha cluster inside the parent 212Po was found using the experimentalbinding energy [43]. Therefore, the experimental value of this probability can be found from

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J. Phys. G: Nucl. Part. Phys. 40 (2013) 065105 S M Saleh Ahmed et al

equation (22) when the equation is written in terms of the experimental total energy of theconsidered system E(exp) and the experimental alpha-cluster formation energy Eα (exp) as

P(exp)α = Eα(exp)

E(exp)

(24)

These two values of energy were determined by different proposed methods and considerationsto be investigated as shown in the next subsections. The CA is defined as the amount of alphaclustering in the nucleus with respect to the clustering of the free alpha particle:

CA = Eα(exp)

B(4, 2)(25)

where B(4,2) is the binding energy of free alpha particle.The preformation factor is not observed. To assess the results of the present work it is

important to choose the most appropriate quantity. In accordance with the recent calculationsof the alpha decay constant, the alpha decay constant is a multiplication of the preformationfactor by the assault frequency and the penetrability. For the chosen even–even heavy nuclei thedecay constant, the preformation factor, the assault frequency and the penetrability calculatedfrom the generalized liquid-drop model potential barriers are around 10−15–107 s−1, 10−1–10−2, 1019–1022 s−1 and 10−16–10−38 s−1 respectively [39, 40, 46]. It is clear that the largecontrast in the value of the decay constant is mainly due to the large contrast in the value of thepenetrability. Then the behaviour of the logarithm of the decay constant and the logarithm ofthe penetrability with the nucleon number are consistent. The most important quantity for thepenetrability and the assault frequency is the Q-value, of comparatively very small contrast,that also shows consistent behaviour as the logarithm decay constant with the nucleon numberof the nuclei [47, 48]. Therefore the assault frequency of comparatively small contrast alsoshows consistent behaviour as the logarithm of the decay constant [39, 40]. In spite of thelarge difference between the contrast of both the Q-value and assault frequency and the decayconstant [40], the use of the logarithm value for the decay constant to be compared withthem for consistency is a good procedure to show more details of quantitative gradients. Theconsistency of these three quantities with the decay constant is not only governed by thealpha decay formula but also the nucleon numbers (subshell and shell effect) when they aredepicted versus the nucleon number. The contrast of the extracted preformation factor is notsmaller than that of the Q-value and its behaviour versus the nucleon number shows goodconsistency with the logarithm of the decay constants [40]. In this work the penetration orthe assault frequency was not used because they are determined within methods based on thealready formed alpha particle. The Q-value was not chosen for the comparison because thisquantity is included in different quantities of the whole alpha decay process within differentmathematical models, which makes its behaviour complex to be understood. The amount thatexpresses the formation of alpha is certainly included within the value of the experimentaldecay constant. Although there is a very small contrast of the preformation factor with respectto the decay constant, it is conceptually expected that the decay lifetime decreases whenthe probability of alpha formation increases and vice versa. This behaviour was obtained byZhang and Royer (2008) [40] for Po isotopes when the preformation was extracted from thecalculated penetrability and the experimental alpha decay. Ismail et al (2010) [47] found thatthe decay constants of the Po isotopes slightly decreases with the decreasing values of thepreformation factor. Recent calculations showed that the preformation factor value remarkablychanges for closed shell and subshell nuclei [39, 40, 46, 47]. Therefore, the logarithmic decayconstants are a good choice to compare with the preformation factor because both of themhave consistent behaviour (increasing and decreasing) and show the same behaviour with thenucleon number. In accordance with the formula for the preformation factor, equation (24),

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J. Phys. G: Nucl. Part. Phys. 40 (2013) 065105 S M Saleh Ahmed et al

the total energy of the system decreases and the alpha-cluster formation energy increases withthe increasing preformation and subsequently the increasing alpha-decay constants. The totalenergy, the formation energy, the preformation factors, and the CA in equations (24)–(25) arecalculated and compared with the logarithm of the experimental alpha-decay constants foreven–even nuclei of mass number A in the next subsections.

3.1. Formation energy from previous work

In terms of the cluster-formation model, the preformation factor and the CA are energydependent. The formation energy Eα(exp) of an alpha cluster was determined from [43]

Eα(exp) = Eαd − Qα (26)

where Qα is the separation energy of the alpha particle and is defined as

Qα = B(A − 4, Z − 2) + B(α) − B(A, Z) (27)

with the alpha-decay energy Eαd defined as

Eαd = B(A, Z) − B(A − 4, Z − 2) (28)

where B(A, Z) is the binding energy of a nucleus of mass number A and atomic number Z.The experimental binding energies and the alpha-decay constants are taken from [49–51].

For the total system energy E(exp) in equation (24), we previously [43] investigatedtwo proposed values for 212Po and found that this energy was more realistic when the totalwavefunction was considered for only the last four nucleons; in other words, the daughtercluster is ignored in equation (13). Therefore, the total energy of the system made up ofthe last four nucleons from equation (28) was also adopted in the present work with twoconsiderations: (1) the total system in 212Po is only the last four nucleons, and (2) the totalsystem is all of the nucleons in the open model space above the closed shell because the212Po nucleus is a double-closed-shell nucleus. This consideration supported the effect ofalpha clustering at the surface and the investigation of the depth of the nuclear surface thatcontributes to the alpha clustering and decay.

(1) For the last-four-nucleons system (as in figure 1). The total energy is from equation (28),and the cluster-formation energy is the decay energy from equation (26). The excellentconsistency of the decay energy with the experimental decay constants shows a realisticcontribution to the preformation factor of equation (24). The contribution of the alpha-cluster formation energy from equation (26) to the experimental alpha-decay widths showsan excellent consistency; however, this formation energy shows an inverse behaviour withthe alpha-decay constants, whereas it is expected to be of the same harmony. When thepreformation factors from equation (24) are calculated, using the alpha-cluster formationenergy in equation (26) and the total energy in equation (28), and compared with thealpha-decay constants (see figure 1), the preformation factor of an alpha cluster in 212Pois reproduced as in our previous work [43] (Pα = 0.54), but with the opposite consistency.

(2) For the open-shell-nucleons system (as in figure 2). The energy of the open-shell nucleonsEs for any nucleus of atomic number Z and mass number A is the difference of the bindingenergies of this nucleus and of the nucleus of the double closed shell B(Acs, Zcs):

Es = B(A, Z) − B(Acs, Zcs). (29)

The cluster-formation energy is from equation (26). The comparison of this energy Es forthe even–even nuclei with the experimental alpha-decay constants (see figure 2) showsa very poor harmony between them. However, the preformation factors with the use ofthis energy from equation (29) and the alpha-formation energy from equation (26) are

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J. Phys. G: Nucl. Part. Phys. 40 (2013) 065105 S M Saleh Ahmed et al

212 216 220 224 228 232 236 240 244 248 252 256 260 264 268 272 276 280-20

-10

0

1016000

18000

20000

22000

24000

5000

10000

15000

200000.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8212 216 220 224 228 232 236 240 244 248 252 256 260 264 268 272 276 280

Log(

λ Exp

. sec

.-1)

A

Exp. decay constants

E (

keV

)

alpha-decay energy

alpha-cluster energy

E (

keV

)P

α

preformation factor

CA

clustering amount

Figure 1. Comparison of the experimental alpha-decay constants λExp. (squares) with theexperimental energies (circles); decay energy from equation (28) and formation energy fromequation (26), preformation factors from equation (24), and CA from equation (25).

less than one. The preformation factor of 212Po is reproduced as in the previous work.The preformation factors are generally in agreement with the experimental alpha-decayconstants for nuclei with 212 � A � 244. For heavier nuclei with A > 244, the reductionin the preformation factors clearly results from the greater contribution of this total energyin equation (29). This result indicates an exponential decrease (see figure 2), where a fewnucleons near the last four nucleons contribute to the clusterization of an interior alpha. Itwas reported and deduced that the alpha is emitted from the surface of the parent nucleus[3, 5, 26, 52–56], as this surface of nucleons is not as deep as the last closed shell in thenuclei of A > 244.

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J. Phys. G: Nucl. Part. Phys. 40 (2013) 065105 S M Saleh Ahmed et al

212 216 220 224 228 232 236 240 244 248 252 256 260 264 268 272 276 280-20

-10

0

10

0

200000

400000

0.0

0.2

0.4

0.6

212 216 220 224 228 232 236 240 244 248 252 256 260 264 268 272 276 280Lo

g(λ E

xp. s

ec.-1

)

A

Exp. decay constants

E (

keV

)

total system energy

preformation factorP

α

Figure 2. Comparison between the experimental alpha-decay constants λExp. (squares) with thetotal system energy from equation (29) (circles); the alpha particle is formed from clusterizationstates of nucleons above the double-shell closure, and the preformation factor is from equation (24),using this total system energy and equation (26) for the alpha-cluster formation.

3.2. Formation energy from the analysis of binding–energy differences

In terms of the cluster model of Wildermuth [54], the nucleus system of many particlescan be considered as clusters (subgroups) and/or particles of numbers depending on thenumber of protons and neutrons, the shell model configurations (the model space), and thetypes of interactions for the process in question. In the alpha-decay process, the alpha-core wavefunctions have been used in the alpha-decay theory to calculate the alpha-decaywidths. The total energy of this wavefunction for the two-body system is the Q-value (seeequation (27)), which is equivalent to the separation energy of an alpha particle from the surfaceof the nucleus. The calculations that included such wavefunctions have greatly improvedthe reproduction of the experimental alpha-decay widths [55–61]. The Q-value was used todetermine the alpha-cluster formation energy, but the values of preformation factors for theeven–even nuclei are not sufficiently satisfied (see figures 1 and 2).

In our present work here, the nucleon–nucleon energy was studied and used to determinethe alpha-cluster formation energy. In subsection 3.2.1, the nucleon–nucleon energy isdetermined and the pairing effect is investigated. In subsection 3.2.2, this nucleon–nucleonenergy is used to determine the alpha-cluster formation energy.

3.2.1. Nucleon–nucleon energy. The extraction of the last nucleon–nucleon energy wasconducted on the basis of the differences of the scalar total energies (binding energies of the

11

J. Phys. G: Nucl. Part. Phys. 40 (2013) 065105 S M Saleh Ahmed et al

(a) (b) (c)

Figure 3. The interactions (a) among the nucleus of A–2 and the two nucleons u and v, (b) betweenthe nucleus A–2 and the nucleon u, and (c) between the nucleus A–2 and the nucleon v.

same and different isovectors). If B(A) is the binding energy of a nucleus of A nucleons whoselast two nucleons are u and v, and if B(A–2) is the energy of another nucleus (w) without thetwo nucleons (u and v), the total exceeded energy due to the existence of the two nucleons isB(A)– B(A–2), which is equivalent to three interaction energies (see figure 3) represented bythe total energy between every two particles:

Eu−v + Eu−w + Ev−w = B(A) − B(A − 2) (30)

where Eu-w is the separation energy of the last nucleon (u) of the (A–1)-nucleons nucleus,and Ev–w is the separation energy of the last nucleon (u) of the (A–1)-nucleons nucleus.Equation (30) is the same as that in [62]. The separation energy of the last proton in a nucleusof A nucleons and with a proton number of Z is B(A, Z)–B(A–1, Z–1); for the last neutron, thisenergy is B(A, Z)–B(A–1, Z). If these two nucleons are non-identical, as u is a proton (p) andv is a neutron (n),

Ep−n = B(A, Z) + B(A − 2, Z − 1) − B(A − 1, Z − 1) − B(A − 1, Z) (31)

is the total energy of the interaction between the last proton and the last neutron in the nucleusof A nucleons and Z protons. In the same way, the correlation energy of the proton–proton(P–P) Ep–p and the neutron–neutron (N–N) En–n is derived as the following equations:

Ep−p = B(A, Z) + B(A − 2, Z − 2) − 2B(A − 1, Z − 1) (32)

and

En−n = B(A, Z) + B(A − 2, Z) − 2B(A − 1, Z). (33)

Equations (31)–(33) have been used to determine the experimental pairing energy of thenucleons in the nuclei [62, 63]. A comparison between the separation energy of the lastnucleon (proton or neutron), as shown in figure 4 and the pairing energies in equations (32)–(33) of the last two nucleons; the P–P and N–N results indicate a very good consistency of thepairing effect in even–even and odd–odd neighbouring (N−Z = 44) nuclei (within the range ofthe nuclei studied in the present work). The constancy of the pairing energies for even–even orodd–odd nuclei is generally a good indicator of the independence of the pair correlations fromthe other nucleons (see figure 4). One piece of evidence of the pairing effect is the energy gap,up to 1.5 MeV, in the energy-level spectra between the ground state and the first excited statein even–even nuclei [64]. For the even–even nuclei in our present work, the pairing energiesfrom equations (32)–(33) are depicted in figure 5, which presents the value of the energy gap.This evidence enables the use of equations (31)–(33) with a derivation method to extract theformation energy of an alpha cluster.

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82 8484 86 8888 90 9292 94 9696 98 100100 102

6000

9000

12000

E(k

eV)

Z

SP SN

126 128 130 132 134 136 138 140 142 144 146

6000

9000

12000

E (

keV

)

N

SP SN

8 0 8 2 8 4 8 6 8 8 9 0 9 2 9 4 9 6 9 8 1 0 0 1 0 2

-4 0 0 0

-2 0 0 0

0

2 0 0 0

-4 0 0 0

-2 0 0 0

0

2 0 0 0

8 0 8 2 8 4 8 6 8 8 9 0 9 2 9 4 9 6 9 8 1 0 0 1 0 2

E (

keV

)

Z

P -P

E (

keV

)

N -N

1 2 6 12 8 13 0 13 2 13 4 13 6 1 38 1 40 1 42 1 44 1 4 6

-40 00

-20 00

0

20 00

-40 00

-20 00

0

20 00

1 2 6 12 8 13 0 13 2 13 4 13 6 1 38 1 40 1 42 1 44 1 4 6

E (

keV

)

N

P -P

E (

keV

)

N -N

Figure 4. The experimental separation energy of last proton SP and last neutron SN and the pairingenergies from equations (32) and (33) versus the atomic number Z (on left) and the neutron numberN (on right) of the nuclei.

3.2.2. Formation energy of interior alpha cluster. The alpha-cluster formation is a correlationof four nucleons (see figure 6). In even–even nuclei, the alpha cluster is formed when the lastpair of protons and the last pair of neutrons are sufficiently correlated. This correlated energyE2p-2n can be included when a nucleus of mass number A and proton number Z is considered asa core of mass number A–4 and proton number Z–2 with four nucleons of total energy writtenas

B(A, Z) = B(A − 4, Z − 2) + Sα + Ep−p + En−n + E2p−2n. (34)

Substituting equations (27), (28) into equation (34) and with a given Sα (the alpha-particleseparation energy equivalent to the Q-value defined in equation (27)), the formation energy isobtained as

Eα = E2p−2n + Ep−p + En−n = Eαd − Sα. (35)

The right-hand side of this equation is the same as equation (26), which was already investigatedand discussed in section 3.1, but a more realistic formation energy of the alpha cluster is verymuch desired.

In accordance with the cluster-formation model, the alpha-decay energy is totallyresponsible for the transition from the state of the initial interior clustering alpha bound

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2 1 0 2 2 0 2 3 0 2 4 0 2 5 0 2 6 0 2 7 0 2 8 0 2 9 00

2 0 0

4 0 0

6 0 0

8 0 0

1 0 0 0

1 2 0 0

1 4 0 0

1 6 0 0

1 8 0 0

2 0 0 0

2 2 0 0

2 4 0 0

E (

keV

)

A

P -P N -N P -N

Figure 5. The experimental pairing energies of the even–even nuclei.

Figure 6. The correlations among the last four nucleons.

with the daughter nucleus to the free clustered-alpha particle and the daughter with Q energy.When this final state is considered for the calculation of the binding energy difference, theconservation law of energy redistributes the decay energy into two parts; one part is used forthe separation, and the other part is used for constructing the free clustered-alpha particle.Thus, the Q-value of equation (27) is made of two parts: the difference in energy for theseparation and the difference in energy for making the tight cluster of the alpha. To avoid theoverlap between these parts in the extraction of the formation energy of the interior alpha,the binding energy of the free alpha was not considered. Equation (34) can be rewritten (seefigure 6) as

B(A, Z) = B(A − 4, Z − 2) + Ep−p + En−n + E2p−2n

+ S2p(A − 2, Z) + S2n(A − 2, Z − 2). (36)

The separation energy of two protons, S2p, and of the neutrons, S2n, from the core of A–4nucleons can be defined as

S2p(A − 2, Z) = Sp(A − 3, Z − 1) + Sp′ (A − 3, Z − 1)

= B(A − 2, Z) − B(A − 4, Z − 2) − Ep−p (37)

and

S2n(A − 2, Z − 2) = B(A − 2, Z − 2) − B(A − 4, Z − 2) − En−n. (38)

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The last two equations are simplified after the use of

Sp(A − 3, Z − 1) = Sp(A − 2, Z) − Ep−p′

and

Sn(A − 3, Z − 2) = Sn(A − 2, Z − 2) − En−n′ .

Substituting equations (37)–(38) in equation (36), the total energy between the two pairsbecomes

E2p−2n = B(A, Z)+ B(A − 4, Z − 2)− B(A − 2, Z − 2) − B(A − 2, Z). (39)

Adding the energy of equations (32)–(33) to equation (39), the formation energy is

Eα = 3B(A, Z)+ B(A − 4, Z − 2) − 2B(A − 1, Z − 1) − 2B(A − 1, Z). (40)

Alternatively, the formation energy of equation (40) (see figure 6) is equivalent to

Eα = Ep−p + En−n + Ep−n + Ep′−n′ + Ep−n′ + Ep′−n. (41)

The formation energy of the alpha cluster, equation (40) or equation (41), depends on fourterms of the p–n interaction. Kaneko and Hasegawa [65] studied the significance of the p–n interaction on the alpha decay width and alpha condensation and found that the alphacondensation (the formation of the alpha) increases due to the strong p–n correlation fornuclei with Z > = 84.The alpha-cluster formation energy from equation (40) is comparedto the experimental alpha-decay constants, as shown in figure 7. The general trend of thisenergy shows a very good consistency with the decay constants despite the fact that someenergies were not found for some nuclei due to the unavailability of the binding energy forthe A–4 nucleus (see table 1). When this energy Eα and the system total energy Es fromequation (29) are used in equation (24), the calculated preformation factors are less thanone (see figure 7 or table 1). The calculated preformation factor of 212Po is 0.22, which issmaller than that (0.45) found using the alpha-cluster formation energy of equation (26) [43].Varga et al [1, 5] determined the preformation factor of 212Po using a combination of theshell and cluster models with a high configuration mixing and many shell-model bases. Theresearchers used two effective interactions that enabled the reproduction of the alpha-decayenergy and width. The value of the 212Po alpha-cluster preformation factor was between 0.23and 0.3. The calculated preformation factors of the other nuclei decrease slightly with themass number (see figure 7). This behaviour is similar to that found in figure 2, which is dueto the inclusion of a large contribution from the use of the open-shell nucleons energy Es

for the total system energy from equation (29). The alpha-decay energy, from equation (28),as a total system energy has shown a perfect consistency with the experimental alpha-decayconstants (see figure 1). Therefore, the preformation factors calculated using the energy valuesfrom equation (40) and equation (28) show an excellent consistency with the experimentalalpha-decay constants (see figure 7) and reproduce the preformation factor (0.23) determinedby Varga et al [1, 5]. The calculated preformation factors of the other even–even nuclei inour present work are so realistic because these factors are extracted from the experimentalbinding energies and their range of 0.1–0.27 is smooth and close to the value of 212Po. Asimilar behaviour is obtained for the CA from equation (25) of alpha particles when thisformation energy is used. These values (0.07–0.19) are slightly less than the values of thepreformation factors (see figure 7). This reduction between these two quantities reflects twopoints in accordance with the cluster-formation model: the preformation factor is correlatedwith the internal interactions of nucleons in the parent nucleus that lead to the formation ofa primary alpha in a process of clusterization that already occurs before the alpha-particleemission. However, the CA is correlated with some of the internal interactions that lead to atransition from the primary alpha to the free alpha during the alpha-decay process.

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Table 1. The calculated preformation factors (Pα) from equation (24) and the CA from equation (25) with different alpha- and system total energy.

Total system energy (keV) Eα (keV) Pα CA

A EL LOG(λ s−1) Eαd Es Equation (26) Equation (40) Equation (26)/Eαd Equation (26)/Es Equation (40)/Es Equation (40)/Eαd Using equation (26) Using equation (40)

214 Po 3.600 20 462.25 29 584.63 12 628.79 4369.75 0.62 0.43 0.15 0.21 0.45 0.15214 Rn 6.400 19 087.38 27 869.88 9878.97 4357.13 0.52 0.35 0.16 0.23 0.35 0.15216 Po 0.680 21 389.5 39 473.63 14 483.18 4412.75 0.68 0.37 0.11 0.21 0.51 0.16216 Rn 4.200 20 095.63 39 437.13 11 895.54 4755.13 0.59 0.30 0.12 0.24 0.42 0.17216 Ra 6.600 18 770 34 837 9244.31 4490 0.49 0.27 0.13 0.24 0.33 0.16218 Po −2.400 22 181 49 041.63 16 066.32 4735.75 0.72 0.33 0.10 0.21 0.57 0.17218 Rn 1.300 21 033.13 50 617.75 13 770.6 4925.63 0.65 0.27 0.10 0.23 0.49 0.17218 Ra 4.400 19 749.75 47 619.63 11 203.83 4771 0.57 0.24 0.10 0.24 0.40 0.17218 Th 6.800 18 446.75 40 331.63 8597.74 4621.75 0.47 0.21 0.11 0.25 0.30 0.16220 Rn −1.900 21 891 61 364.63 15 486.33 4832.25 0.71 0.25 0.08 0.22 0.55 0.17220 Ra 1.600 20 703.38 60 140.5 13 111 4950.88 0.63 0.22 0.08 0.24 0.46 0.17220 Th 4.900 19 342.5 54 179.5 10 389.47 4786 0.54 0.19 0.09 0.25 0.37 0.17220 U 7.100 17 891.13 44 150 7591.13 3759.88 0.42 0.17 0.09 0.21 0.27 0.13222 Rn −4.700 22 705.38 71 747 17 115.06 5241.13 0.75 0.24 0.07 0.23 0.60 0.19222 Ra −1.700 21 616.75 72 234.5 14 937.92 4302.75 0.69 0.21 0.06 0.20 0.53 0.15222 Th 2.500 20 168.75 67 788.38 12 041.8 4662.75 0.60 0.18 0.07 0.23 0.43 0.16222 U 5.700 18 874.38 59 206 9374.38 5064.88 0.50 0.16 0.09 0.27 0.33 0.18224 Ra −4.700 22 506.75 83 871.38 16 717.9 4142 0.74 0.20 0.05 0.18 0.59 0.15224 Th −0.180 20 997 81 137.5 13 698.5 4162.5 0.65 0.17 0.05 0.20 0.48 0.15224 U 2.900 19 676 73 855.5 11 056.18 4508 0.56 0.15 0.06 0.23 0.39 0.16226 Ra −11.000 23 425.13 95 172.13 18 554.5 4235.38 0.79 0.19 0.04 0.18 0.66 0.15226 Th −3.400 21 844.88 94 079.38 15 394.02 3983.88 0.70 0.16 0.04 0.18 0.54 0.14226 U 0.410 20 594.63 88 383 12 893.65 4246.63 0.63 0.15 0.05 0.21 0.46 0.15228 Th −7.900 22 775.5 106 646.88 17 255.42 4170.5 0.76 0.16 0.04 0.18 0.61 0.15228 U −2.900 21 492.25 102 629.75 14 688.75 4036.75 0.68 0.14 0.04 0.19 0.52 0.14228 Pu 1.800 20 346 94 201.5 12 396.35 ∗∗∗ 0.61 0.13 ∗∗∗ ∗∗∗ 0.44 ∗∗∗230 Th −13.000 23 525.63 118 697.75 18 755.66 4419.38 0.80 0.16 0.04 0.19 0.66 0.16

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Table 1. (Continued.)

Total system energy (keV) Eα (keV) Pα CA

A EL LOG(λ s−1) Eαd Es Equation (26) Equation (40) Equation (26)/Eαd Equation (26)/Es Equation (40)/Es Equation (40)/Eαd Using equation (26) Using equation (40)

230 U −5.400 22 302.88 116 382.25 16 310.14 4175.88 0.73 0.14 0.04 0.19 0.58 0.15230 Pu −2.200 21 115.63 109 498.63 13 935.73 4228.38 0.66 0.13 0.04 0.20 0.49 0.15232 U −9.500 22 882.13 129 529 17 468.5 3861.88 0.76 0.13 0.03 0.17 0.62 0.14234 Cm −1.900 20 930.75 130 429.38 13 565.71 3847.75 0.65 0.10 0.03 0.18 0.48 0.14236 U −15.000 23 722.63 153 978.63 19 149.53 3714.63 0.81 0.12 0.02 0.16 0.68 0.13236 Pu −8.100 22 428.5 151 957.5 16 561.43 3138.25 0.74 0.11 0.02 0.14 0.59 0.11238 Pu −9.600 22 702.63 164 838.25 17109.43 3292.38 0.75 0.10 0.02 0.15 0.60 0.12240 Pu −11.000 23 040 177 018.63 17 784.25 2977.5 0.77 0.10 0.02 0.13 0.63 0.11240 Cm −5.500 21 897.88 173 855.38 15 500.08 3030.13 0.71 0.09 0.02 0.14 0.55 0.11240 Cf −2.000 20 600 165 970 12 881.2 2748 0.63 0.08 0.02 0.13 0.46 0.10242 Pu −13.000 23 311 188 569.63 18 326.47 2972.5 0.79 0.10 0.02 0.13 0.65 0.11242 Cm −6.300 22 080 186 918.25 15 864.44 2698.75 0.72 0.08 0.01 0.12 0.56 0.10242 Cf −2.500 20 778.63 180 821 13 261.85 3283.38 0.64 0.07 0.02 0.16 0.47 0.12244 Pu −16.000 23 630.38 199 625.38 18 964.84 3373.13 0.80 0.10 0.02 0.14 0.67 0.12244 Cm −8.900 22 393.63 199 412.25 16 491.88 3231.13 0.74 0.08 0.02 0.14 0.58 0.11244 Cf −3.200 20 966.75 194 822.13 13 637.85 3109 0.65 0.07 0.02 0.15 0.48 0.11246 Cm −11.000 22 820.75 211 390.38 17 345.62 3235.25 0.76 0.08 0.02 0.14 0.61 0.11246 Cf −4.300 21 434.13 208 352.38 14 572.53 3323.88 0.68 0.07 0.02 0.16 0.52 0.12246 Fm −0.200 19 917.88 200 738.88 11 539.82 3747.63 0.58 0.06 0.02 0.19 0.41 0.13248 Cm −13.000 23 133.88 222 759.25 17 972.14 3164.63 0.78 0.08 0.01 0.14 0.64 0.11248 Cf −6.600 21 934.75 221 347 15 573.55 3081.25 0.71 0.07 0.01 0.14 0.55 0.11248 Fm −1.700 20 293.63 215 115.75 12 291.38 3059.38 0.61 0.06 0.01 0.15 0.43 0.11212 Po 6.400 19 341.5 19 341.5 10 387.38 4275 0.54 0.54 0.22 0.22 0.37 0.15250 Cf −8.800 22 167.13 233 557.5 16 038.69 3016.88 0.72 0.07 0.01 0.14 0.57 0.11250 Fm −3.400 20 738.63 229 091 13 181.7 3715.38 0.64 0.06 0.02 0.18 0.47 0.13252 Cf −8.100 22 078.75 244 838 15 861.88 3230.75 0.72 0.06 0.01 0.15 0.56 0.11252 Fm −4.100 21 142.75 242 489.75 13 990.05 3305.25 0.66 0.06 0.01 0.16 0.49 0.12252 No −0.550 19 745.75 234 861.5 11 195.98 3190.25 0.57 0.05 0.01 0.16 0.40 0.11254 Fm −3.200 20 988.13 254 545.63 13 680.64 2841.88 0.65 0.05 0.01 0.14 0.48 0.10254 No −1.900 20 069.88 249 160.88 11 844.09 3133.63 0.59 0.05 0.01 0.16 0.42 0.11256 No −0.620 19 714.25 262 204 11132.81 3104.25 0.56 0.04 0.01 0.16 0.39 0.11264 No −1.900 21 692 310 042 15 112 ∗∗∗ 0.70 0.05 ∗∗∗ ∗∗∗ 0.53 ∗∗∗

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Table 1. (Continued.)

Total system energy (keV) Eα (keV) Pα CA

A EL LOG(λ s−1) Eαd Es Equation (26) Equation (40) Equation (26)/Eαd Equation (26)/Es Equation (40)/Es Equation (40)/Eαd Using equation (26) Using equation (40)

264 Rf −2.700 20 084 306 874 11 944 3048 0.59 0.04 0.01 0.15 0.42 0.11264 Sg 0.240 19 212 300 802 10 002 3302 0.52 0.03 0.01 0.17 0.35 0.12264 Hs 3.100 17 704.75 290 305 7114.07 3323.25 0.40 0.02 0.01 0.19 0.25 0.12266 Rf −3.700 20 762 319 202 13 262 3346 0.64 0.04 0.01 0.16 0.47 0.12266 Sg −1.500 19 372 313 882 10 487.77 3250.5 0.54 0.03 0.01 0.17 0.37 0.11266 Hs 2.400 17 926 304 838 7590.38 3316 0.42 0.02 0.01 0.18 0.27 0.12268 Rf −2.700 20 112 330 154 12 012 ∗∗∗ 0.60 0.04 ∗∗∗ ∗∗∗ 0.42 ∗∗∗268 Sg −1.600 19 796 326 670 11 396 2028 0.58 0.03 0.01 0.10 0.40 0.07268 Hs −0.460 18 364 319 166 8464 3882 0.46 0.03 0.01 0.21 0.30 0.14268 Ea 3.800 16 265 306 570 4345 3123 0.27 0.01 0.01 0.19 0.15 0.11270 Sg −2.900 19 418 338 620 10 318 3386 0.53 0.03 0.01 0.17 0.36 0.12270 Hs −1.600 19 068 332 950 9770.1 3114 0.51 0.03 0.01 0.16 0.35 0.11270 Ea 3.600 17 042 321 880 5845.91 3362 0.34 0.02 0.01 0.20 0.21 0.12272 Sg −2.700 20 104 350 258 11 804 ∗∗∗ 0.59 0.03 ∗∗∗ ∗∗∗ 0.42 ∗∗∗272 Hs −1.800 18 148 344 818 8048 2316 0.44 0.02 0.01 0.13 0.28 0.08274 Hs −1.900 18 574 357 194 9074 3432 0.49 0.03 0.01 0.18 0.32 0.12274 Ea −0.460 16 846 349 796 5446 4506 0.32 0.02 0.01 0.27 0.19 0.16276 Hs −2.700 19 280 369 538 10 480 3192 0.54 0.03 0.01 0.17 0.37 0.11276 Ea −0.860 17 820 362 638 7220 2902 0.41 0.02 0.01 0.16 0.26 0.10278 Ea −1.200 18 262 375 456 8262 2930 0.45 0.02 0.01 0.16 0.29 0.10278 Ec 1.800 17 042 366 838 5662 4580 0.33 0.02 0.01 0.27 0.20 0.16280 Ec −0.160 17 772 380 410 7152 3786 0.40 0.02 0.01 0.21 0.25 0.13

(∗∗∗) means that the value is not calculated because the experimental value is not available.

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J. Phys. G: Nucl. Part. Phys. 40 (2013) 065105 S M Saleh Ahmed et al

212 216 220 224 228 232 236 240 244 248 252 256 260 264 268 272 276 280 284-20

-10

0

101000

2000

3000

4000

5000

6000

0.0

0.1

0.2

0.05

0.10

0.15

0.20

0.25

0.300.05

0.10

0.15

0.20

0.25212 216 220 224 228 232 236 240 244 248 252 256 260 264 268 272 276 280 284

Log(

λ Exp

. sec

.-1)

A

Exp. decay constants

E (

keV

)

alpha-formation energy

Pα

preformation factor with the use of Eq.(29)

Pα

preformation factor with the use of Eq.(28)

CA

clustering amount

Figure 7. Comparisons of the experimental alpha-decay constants (λExp.) with the alpha-clusterformation energy (circles) from equation (40) and the preformation factors (triangles) fromequation (24), using equations (29), (40); the alpha particle is formed from clusterization statesof the nucleons above the 208Pb, and the preformation factors (upside-down triangles) are fromequation (24), using equations (28), (40); the alpha particle is formed from the clusterization statesof the last four nucleons, and the CA (stars) are from equation (25), using equation (40).

4. Conclusion

The cluster-formation model is still valid for realistic preformation factors of a wide range ofheavy even–even nuclei without using any free parameter. This model confirms the existenceof an interior alpha cluster that is different from the free alpha of a higher nucleon-correlationenergy. The values of the formation energy of the interior alpha for the even–even nucleiconfirm the proposal of a primary alpha with a volume greater than that of the free alpha. Thisvolume expansion is reduced when the alpha is emitted in the decay process. The reductionoccurs rapidly due to the release of the interior alpha particle from the short-range force,not gradually and transiently during the alpha emission. The total energy among any numberof correlated nucleons, including the pairing energy and formation energy of alpha, inside

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a nucleus can remarkably be obtained from the binding-energy differences, as in equations(31)–(33), (35). The model confirmed the alpha-clustering surface effect and shows that thenucleons on the surface responsible for the clusterization of or condensation to the interioralpha are not deep. This result indicates that there is little deformation in even–even nuclei.The CA is a good factor that can be used with the Q-value when a transition ends withalpha-particle emission.

Like the Q-value, the formation energy is also important for the processes of thecluster emission because this energy indicates more detailed information about the nuclearstructure, especially the knockoff nuclear reactions that lead to cluster emission and the clusterradioactivity. The wide validity of this model for even–even open-shell heavy nuclei in thepresent work can motivate more future work for other heavy nuclei, which could be usefulfor the predictions of cluster radioactivity. The clustering amounts can be used as a correctionfactor for the clustering in any process that ends with the emission of an alpha particle.

The successful determination of preformation factors for the even–even nuclei opens anew line of research to theoretically investigate the nuclear structure models proposed forthe structure by the calculation of the values of the preformation factors. In addition, thedetermination of the cluster-formation wavefunction using a suitable interaction for the clusterformed or the determination of the wavefunction of the related motion of two clusters toreproduce the preformation factors from equations (22), (23) is a future study that may gainmore insight into the nuclear structure of heavy nuclei.

Acknowledgment

The authors would like to acknowledge the financial support from the Ministry of HigherEducation Malaysia (through the Fundamental Research Grant Scheme) UKM-ST-06-FRGS0248–2010.

References

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