NNucleonic localisation and alpha radioactivity
J.-P. Ebran,
1, 2
E. Khan, and R.-D Lasseri
3, 4 CEA,DAM,DIF, F-91297 Arpajon, France Universit´e Paris-Saclay, CEA, Laboratoire Mati`ere en Conditions Extrˆemes, 91680, Bruy`eres-le-Chˆatel, France IJCLab, Universit´e Paris-Saclay, CNRS/IN2P3, 91405 Orsay Cedex, France ESNT, CEA, IRFU, D´epartement de Physique Nucl´eaire, Universit´e Paris-Saclay, F-91191 Gif-sur-Yvette (Dated: November 30, 2020)Relativistic energy density functional approaches are known to well describe nuclear states whichinvolve alpha clusters. Here, alpha emitting nuclei are analysed through the behavior of the spatiallocalisation of nucleonic states, calculated with an axially deformed RHB approach over the nuclearchart. The systematic occurrence of more localised valence states, having a n = 1 radial quantumnumber, allows to pinpoint nuclei in agreement with experimentally known alpha-emitters. Thecases of Po and
Te are investigated, showing the concomitant contributions of the pseudospinsymmetry and the presence of n = 1 states, on the alpha preformation probability. The impact ofthe localisation of valence states, on alpha preformation probability, is then analysed. It allows tostudy shell effects on this probability, over isotopic and isotonic chains. Finally, a phenomenologicallaw is also provided, relating this probability to the radial quantum number of the valence states. I. INTRODUCTION
The description of alpha radioactivity is a long stand-ing problem. For about one century, a large variety ofmodels have been devoted to this task, such as semi-classical approaches, or microscopic ones [1]. However,the identification of all nuclei, which are alpha-emitter,may not be achieved yet, as proven by the discovery ofalpha-emission of
Bi in 2003 [2], or the remaining ques-tion of the possible alpha radioactivity of
Pb [3]. Oneof the first successful model is the well-known Geiger-Nuttall law [4], and its description involving tunnelingeffect by Gamow [5] and the class of WKB models [1]. Inthese approaches, the alpha emission probability is de-composed into the product of an alpha preformation onewith the tunneling probability through the Coulomb bar-rier, taking into account the frequency of impact of thealpha particle on the barrier.The probability of tunneling effect depends on the Q-value. The study of the alpha preformation probabil-ity itself, has received significant attention since severaldecades [1, 6–8], although it has also often been approx-imated to 1. In addition, some points remain to be clar-ified, as first discussed by Buck and collaborators [9],such as the difference of behavior of the alpha emissionprobability for N smaller or larger than 126, respectively.This was first analysed by introducing a global quantumnumber in an alpha plus core description of the system,namely the Wildermuth condition [10, 11]. It points to-wards the possible role of quantum numbers in alpharadioactivity. It shall be relevant to further provide adescription of this shell effect at the nucleonic scale, in-volving nucleon quantum numbers. Relating this effectto nuclear structure, and studying the possible impactof the alpha preformation probability, would also be ofinterest, as already pointed out in Refs. [12, 13].The study of the discrepancy between an accurate phe-nomenological law (i.e. an enhanced Geiger-Nuttall law)[14] assuming an alpha preformation probability of 1, and the experimental alpha emission probability, also broughtrelevant investigations: the observed patterns, comparedto experiment, are imaging the preformation probability.For instance, the N=126 shell effect has an impact onthis probability [14], as well as the Z=82 one, as shownmore recently [15]. Considering a large variety of alpha-emitters, a variation up to a factor 30 of the preformationprobability can be inferred, in the N=126 case [14].More recently, the alpha emission probability was de-duced from the measurement of the alpha emission life-time in the
Te nucleus [16, 17], showing a larger emis-sion probability than in
Po, which remains to be fullyexplained.Nuclear structure properties could therefore help tobetter understand the alpha preformation probability.Indeed, light nuclei are known to exhibit alpha clusterstates. They are of course not alpha-emitters (exceptfor Be), because of their negative Q-value. However,the occurrence of alpha cluster states, and the alpha pre-formation probability in heavier nuclei shall be closelyrelated. A recent work has for example showed how todescribe alpha emission in
Te and alpha cluster statesin Ne on the same ground, using and alpha+core ap-proach [18].
Po alpha decay was also described witha significant alpha+core contribution [19]. Hence, a de-scription of alpha emission at the fully nucleonic levelcould be also interesting.In a previous work, we showed that the so-called lo-calisation parameter, at the nucleonic scale, was drivingthe occurrence of cluster states [20]: the radial quan-tum number n is the key quantity impacting spatial lo-calisation of nucleons, and hence, cluster occurrence innuclei. Therefore, nuclei with spatially localised valencestates should appear throughout the nuclear chart as fa-vored alpha-emitters, providing a universal and simpleapproach: a first link with alpha radioactivity was ex-plored in this work [20]. In the present work, we proposeto further investigate links between alpha cluster forma-tion (driven by localisation) in nuclei, and the alpha pre- a r X i v : . [ nu c l - t h ] N ov formation probability, using the criterion of the localisa-tion of the nucleonic wave function. The aim is twofold:i) study if localised valence nucleonic states allow to pin-point alpha decay emitters, and ii) analyse if the alphapreformation factor is favored by the occurrence of suchstates. Ultimately, a simple phenomenological law shallbe provided, relating the alpha preformation factor tothe radial quantum number of the valence state of theconsidered nucleus.In section II, we first define a criterion for localised nu-cleonic states. It is then used to predict alpha-emittersover the nuclear chart, by calculating the spatial dis-persion of valence states with the microscopic relativis-tic Hartree-Bogoliubov (RHB) approach: such a class ofmodels has proven to be a relevant tool, to analyse clusteroccurrence in nuclei on a general ground [21–26], as wellas to provide a sound comparison with measured excitedspectra of cluster states, such as in Ne isotopes [27] andfor the C Hoyle state [28]. It should be noted that itwas also recently shown to describe alpha radioactivity of
Te and
Xe [29]. In section III, the role of localisa-tion is analysed on the
Te and
Po benchmark cases,through their single-particle spectra. Section IV providesa study of the alpha preformation factor, related to nu-cleonic localisation. The shell effects are also analysedin this framework. A phenomenological law, relating theradial quantum number to the alpha preformation factor,is finally given.
II. ALPHA-EMITTERS OVER THE NUCLEARCHART
In order to predict the general behavior of alpha emit-ting nuclei over the nuclear chart, microscopic energydensity functional calculations are performed. Namely,the fully self-consistent relativistic Hartree-Bogoliubovapproach is used with the DD-ME2 functional for axi-ally symmetric nuclei [30]. Such an approach has provento be successful to describe on the same ground a largevariety of phenomena in nuclei [30], and the occurrenceof cluster states [21–28]. The pairing interaction used inthe RHB calculation is separable in momentum space,and driven by the bell-shape pairing gap in symmetricnuclear matter [31].Limitations of such an approach shall also be discussed.Regions of the nuclear chart involve nuclei with triaxial-ity or octupole degrees of deformation, which are nottaken into account here. Moreover, drip-line nuclei mayalso be delicate to describe with the present approach.Finally, the Wigner term, as well as proton-neutron pair-ing, could have an effect, especially for N=Z nuclei, butare not considered here. It should be noted that, moregenerally, quartetting is also known to have an impact onclustering [32]. All these effects can be relevant, but thepresent approach does not aim to provide a fully detailedprediction of all the alpha-emitters over the nuclear chart.It rather aims to investigate if the spatial localisation of valence states increases alpha preformation probability.It order to study the possible link between alpha-emitters and localised nucleonic valence states, it is firstnecessary to define a criterion for such a localisation. Ina second step, the corresponding nuclei fulfilling this cri-terion (calculated with the microscopic RHB approach),shall be compared to the experimentally known alpha-emitters.
A. Criterion for localised states
A first analysis of the occurrence of cluster states canbe investigated from the spatial localisation of the nucle-onic degree of freedom, through the localisation parame-ter, defined as [20–22]: α loc = 2∆ rr (1)where ∆ r = √ < r > − < r > is the typical spatialdispersion of the nucleonic wave function, and r (cid:39) . ρ (cid:39) .
16 fm − ).In order to better understand the role of the spatialdispersion, it has been shown, in the spherical harmonicoscillator approximation, that the dispersion of a givensingle-particle state is, to a good approximation, onlydriven by its radial quantum number n , and not the or-bital one [20]: α loc (cid:39) (cid:112) (cid:126) (2 n − mV r ) / A / (2)where V is the depth of the confining potential of theconsidered nucleus, composed of A nucleons of mass m.Microscopic calculations of the dispersion (using Eq. (1))show that the smallest dispersion pattern appears for sin-gle particle states with n = 1 [20], independently fromthe orbital quantum number. This key point opens thepossibility to pinpoint nuclei having spatially localisedvalence nucleons (namely n = 1 valence states), through-out the nuclear chart. Correlations between localisationand preformation of an alpha particle, in view of its emis-sion, can be investigated.Eqs (1) and (2) allow to determine a criterion for thespatial dispersion of valence state to be localised, in agiven nucleus. Using these two equations, in the har-monic oscillator approximation, leads to:∆ r.A − / (cid:39) √ (cid:126) r (32 mV ) / . (cid:112) (2 n −
1) (3)The prefactor value on the r.h.s of Eq. (3) is 0.4 fm,using a typical depth of the potential V =75 MeV [21].Taking into account the larger diffusivity of the nuclearpotential compared the the HO one, a factor 1.2 has to beconsidered [20], so the prefactor is about 0.5 fm. Hence,a relevant criterion over the nuclear chart to disentanglea localised n = 1 state (∆ r.A − / ∼ n = 2 state (∆ r.A − / ∼ √ (cid:39) (cid:104) ∆ r (cid:105) .A − / < . f m (4)The 0.7 fm value is therefore the condition to have asmall dispersion of the considered state (i.e. a n = 1state) in a given nucleus, taking into account the depen-dency on A. < ∆ r > is the average of the nucleonic spa-tial dispersion: in practice, in a microscopic calculationwhere pairing effects are at work, there is not a singlevalence state. Therefore, < ∆ r > is calculated as the av-erage dispersion of the valence states, weighted by theiroccupation probability. Since alpha preformation is un-der study, the number of valence states to be consideredis determined until a particle number (either neutrons orprotons) of 2 is reached from the sum of the consideredoccupation probabilities. The resulting mean dispersionallows then to pinpoint nuclei having a small spatial dis-persion, by using the criterion (4). We have checked,using the microscopic calculations, that this conditionallows to pinpoint n = 1 states over the nuclear chart.A complementary way to analyse the relevance of thiscriterion, is to study its impact on alpha radioactivity.This shall be done in the next subsection, through sys-tematic calculations over the nuclear chart, in order tocompare with known alpha emitting nuclei. This shallalso be undertaken in the last section of the present work,by the means of a phenomenological evaluation of the al-pha preformation factor. B. Calculation over the nuclear chart
The spatial dispersion of valence states is microscopi-cally calculated on the whole nuclear chart of even-evennuclei in the axially-symmetric RHB framework, takinginto account pairing and deformation effects.To investi-gate if spatially localised valence states increase the alphapreformation probability, the neutron or proton disper-sion is calculated with the RHB approach (depending onthe closest lower N or Z magic number, e.g. proton dis-persion for N or Z between 50 and 82, etc.) and thecondition (4) is applied, as well as Q α >
0, which is anecessary condition for alpha radioactivity. The Q α > n = 1 localisation condition is relevant for alpha-particleemission.Several nuclei on Fig. 1 are however predicted asalpha-emitters, but have not been experimentally tagged
50 60 70 80 90 100 110 120 130 140 150 160 N Z RHB and Q α >0 exp emittersRHB FIG. 1: Experimentally known alpha-emitters foreven-even nuclei (in blue), compared to the onepredicted to have a small dispersion (see text fordetails) in their valence state (dashed black) with theRHB calculation, and a positive Q α from availablemeasured masses. Removing the positive Q α condition(from available measured masses), allows to predictadditional nuclei (full purple dots) with the RHB model . so. This could of course be due to a limitation of thegeneral description of alpha radioactivity, into steps ini-tially assuming its localisation and preformation. An-other reason could be, that a majority of these nuclei arebeta-unstable, and it may be experimentally difficult tolook for alpha-emission if the partial beta-decay half-lifeis several orders of magnitude smaller than the possiblealpha one. Hence, these nuclei could be also considered aspossible predictions for alpha-emitters, which may havenot been detected yet. For instance, it could be interest-ing to experimentally look for nuclei which are predictedboth beta and alpha-emitters around the Gd region.Finally, a last reason could be the role of more elaboratedeformations than the axially symmetric one, such as tri-axiality and/or octupole deformations, as mentioned inthe introduction of section II.Figure 1 also displays nuclei having a small spatial lo-calisation from the RHB calculations, as discussed above,removing the Q α > α > n = 1states in these nuclei, compared to heavier one. Hence,the hindrance of alpha emission for these nuclei is largelydue to the Q α > / state, having a large spatial extension, due toits n = 3 value. On the contrary, Sn isotopes can have alarge alpha preformation probability, due to the presenceof n = 1 states in these lighter nuclei. However, most ofthem have a negative Q α value, and hence, cannot be de-tected as alpha-emitters. These effects shall be discussedin more details in the next section, with the comparisonof the Te and
Po cases.
III. ROLE OF LOCALISATION
In order to investigate more precisely the role of spatiallocalisation on alpha preformation and emission probabil-ity, two benchmark cases are compared:
Po and
Te.The former is a well-known alpha-emitter, whereas thelatter has recently been evidenced as an alpha emittingnucleus [16, 17]. Indeed, the deduced alpha preformationprobability has been found larger than in
Po. More-over, the alpha lifetime of
Te has also been recentlydescribed by an alpha+core approach [18], showing therelevance of connecting alpha cluster approaches to alphaemission.Figure 2 displays the single-particle spectrum of
Poobtained with the RHB calculations. Because of the pair-ing effect, the occupation probability of each state is in-dicated. For the valence neutrons states, not only n = 1states are involved, but also the 2g / state, which in-creases the spatial dispersion, because it has n = 2 [20].The contribution to the spatial localisation is detailedon the bottom of the figure, displaying the partial den-sities obtained from RHB calculations. The 1i / state,although of large (cid:96) value, is much more localised thanboth the 2g / and 3p / states, in agreement with ourmain point on spatial localisation (namely n dependencebut (cid:96) independence). It should be noted that the vicin-ity of the 2g / state with the 1i / one, is due to thepseudo-spin symmetry (PSS) [33], which plays an impor-tant role to describe the behavior of nuclei in this regionof the nuclear chart [34].In the case of the protons, the valence state of Pois almost only made of the 1h / state. It is also morelocalised than states with larger n , as seen on the partialdensities. Table I summarises the respective dispersionscalculated for the valence states of Po, showing thedecisive role of the n quantum number: a n >
Po with the RHB calculations. The states areKramers (Nilsson) states, with a degeneracy of 2, butare displayed with the quantum numbers of thespherical state from which they are stemming.drastically increases the dispersion, by about a factor 2or more: the localisation effect is decreased due to thepresence of n = 2 , n > n = 2 state close tothe 1i / neutron valence state. It is therefore expectedthat in lighter nuclei such as Te, the blurring of thespatial localisation disappears because of the absence of n >
Po 2g / / / / / / ∆r (fm) 1.95 1.17 2.40 1.08 2.45 1.91 TABLE I: Spatial dispersion of neutron (left part) andproton (right part) single-particle states of
Po,calculated with the RHB approach.
Te is therefore a specifically interesting nucleus, tostudy alpha preformation probability. It belongs to thelightest region where Q α remains positive. Moreover, thisN=Z nucleus would also correspond, in a simple picture,to an alpha particle on top of a doubly magic core. How-ever, this nucleus is close to the proton drip-line, mak-ing its description delicate. In the present approach, theaxially deformed RHB calculation finds its ground statewith a small deformation ( β =0.14) with a proton va-lence state at a slightly positive energy, by 140 keV. Thiscould be due to a limitation of the model to describenuclei close to the drip-line, as mentioned above. How-ever, due to the Coulomb barrier, the static descriptionof this proton quasi-bound state, could still be consideredFIG. 3: Same as Fig. 2 for Teas valid: our main goal is to focus on the spatial localisa-tion of the wave functions, and not to study the particleemission process itself. We also wish to consider a globalapproach, such as the relativistic Hartree-Bogoliubov,rather than using more dedicated models to accuratelydescribe a given set of nuclei. Finally, it should be notedthat the output of the RHB calculations, also shows acollapse of the pairing effect. This result favors a de-scription of
Te as a
Sn core + 4 valence nucleons,compared to a case with pairing, where the
Sn statesand the valence ones would be mixed, with occupationfactors between 0 and 1.Fig 3 shows that only n = 1 states are involved asvalence states, namely the 1g / state both for neutronsand protons: compared to Po,
Te is closer from thelightest nuclei, where clusters states can be found. Thecorresponding partial densities, as well as the one of the1g / state, are spatially localised, although the 1g / state shows more extension than the 1g / . In addition Te being a lighter nucleus than
Po, the dispersionof the n = 1 state is also smaller: for instance, the 1g / one is 0.98 fm, to be compared with the values for n = 1in Table I. It should be noted that the dispersion of the1g / state is about 1.5 fm, which is larger, probably dueto the difficulty to describe such a nucleus close to thedrip-line, involving quasi-bound states: the dispersion ofthe 1g / state is more representative of the typical spa-tial dispersion at work in this nuclei. Under this assump-tion, the neutron valence states of Po have a spatialdispersion in average, about 40 % larger than the 1g / of Te.It should be noted that the lowest neutron Kramers(Nilsson) states, originating from the spherical 2d / state in Te, are located at -11.2 MeV, showing thatthe degeneracy raising between PSS partner states [35]is much larger than in
Po. This is due to the effect ofdeformation in
Te, overcoming the one of the PSS.
IV. STUDY OF THE PREFORMATIONFACTOR
It could be useful to study the impact of spatial local-isation of the valence state, on the alpha preformationfactor. First, the localisation criterion (4), calculatedmicroscopically, shall be tested using the data on the al-pha emission half-life (through a phenomenological fit-like formula). This shall also allow to study the impactof shell effects on the alpha preformation factor. Finally,relating the alpha preformation probability to the radialquantum number of the valence state, can be of inter-est. On this purpose, a phenomenological law shall bederived.
A. Evaluation of the alpha preformation factor
1. Evaluation of the localisation criterion
In order to substantiate the above findings, it could beuseful to consider a complementary approach, and eval-uate the alpha preformation factor. A relevant way isto consider the following formula for the alpha emissionhalf-life:
Log T P heno / ( s ) = 9 . Z − . √ Q α − .
37 (5)where Q α is in MeV. In [14], it has been shown that Eq.(5) both accurately describes the experimental data andcompares well to theoretical WKB approximation infer-ring a preformation probability P=1. Therefore, discrep-ancies of the data with respect to this formula shall bedriven, as a first approximation, by the behavior of thealpha preformation factor, although other effects couldplay a role, such as triaxal or octupole deformation, res-onance densities or description of the alpha formationwith respect to the barrier penetration [36]. More pre-cisely, the ratio T P heno /T Exp =W Exp /W P heno , where Wis the total α emission probability, shall give an evalu-ation of the alpha preformation factor, as discussed in[14].In order to provide a quantitative relation betweenthe impact of the localisation criterion of Eq. (4) andthe alpha formation probability, Fig. 4 displays theT P heno /T Exp ratio as a function of the average dis-persion of the neutron valence state, for the Z= 84,86 and 88 isotopes. Large alpha preformation fac-tors (T
P heno /T Exp > ∼
1) involve small average nucle-onic dispersion of the neutron valence states ( < ∆ r >.A − / < ∼ P heno /T Exp < ∼
1) can involve large average nucle-onic dispersion of the neutron valence states ( < ∆ r >.A − / up to 1.4 fm). This shows the relevance of the lo-calisation criterion (4) on the alpha decay phenomenon.We have also performed a systematic calculation,showing that almost all the nuclei with the smallest pre- < ∆ r> n . A -1/6 (fm) T ph e no / T e xp FIG. 4: Ratio of the phenomenological to experimentalalpha emission half-life for Z=84,86,88 even-even nucleias a function of the average dispersion of the neutronvalence state (see Eq. 4 for details)formation factor are odd ones, in agreement with the hin-drance effect of the alpha preformation, known to occurin such nuclei [37, 38]. In the case of even-even nuclei,out of 140 nuclei where the T
P heno /T Exp ratio has beencalculated, 56 are predicted as localised by the RHB cal-culation. The mean value of the T
P heno /T Exp ratio is2.4 ± ±
2. Analysis of the shell effects
The above mentioned ratio can also be used to studythe impact of shell effects on alpha decay. Fig. 5 displaysthe T
P heno /T Exp ratio for Z > ∼
82 nuclei, where the exper-imental data is known on relatively large isotopic chains.It should be reminded that this ratio shall scales the al-pha preformation factor (see e.g. Fig. 4 of [14]). Onlyeven-even nuclei are displayed. Fig. 5 first shows thatthere is a sharp increase of this ratio for N ≥ T ph e no / T e xp
100 102 104 106 108 110 112 114 116 118 120 122 124 126 128 130 132 134 136 138 140 N Z=84Z=86Z=88
FIG. 5: Ratio of the phenomenological to experimentalalpha emission half-life for Z=84,86,88 even-even nuclei.Those predicted with a small dispersion (see text) aredisplayed in red.The present description also allows for a more detaileddiscussion. The N=126 shell effect is phenomenologicallyknown to impact the alpha preformation probability [14].In the present analysis, the strong increase of the prefor-mation probability, starting from N=128, comes from thefilling of the 1i / state (although partially, i.e. togetherwith the 2g / state, as seen on Fig. 2) whereas theregular decrease of the preformation probability beforeN=128 is related to the progressive filling from n = 1state (1i / ) to n=2,3 states such as 2 / , 3p / and3p / .The case of the smaller Z=82 shell effect, on the pre-formation probability [15], is more complicated to study:looking to the experimental alpha-emitters (Fig. 1), anisotopic chain of alpha-emitters nuclei through N=126involves many nuclei which are moreover close to thestability line (i.e. just above Pb). Hence, the fillingof the n = 1 state beyond N=126 happens in a similarway for the various isotopic chains, from Z=84 (Po) toZ=92 (U). On the contrary, following an isotonic chain ofalpha-emitters nuclei through Z=82, involves much fewernuclei, due in part to the proton drip-line: more intensenuclear structure effects are expected, and the occurrenceof the n = 1 state just above Z=82 is expected to be lesssystematic. B. Phenomenological alpha preformation model
Due to the complexity of the properties of the alphaemission [1], phenomenological models are often used, asillustrated by the successful Geiger-Nuttall law [4]. Fol-lowing this spirit, it could be finally useful to provide, asan alternative way, a phenomenological relation betweenthe alpha preformation probability and the spatial local-isation, over the nuclear chart. This law is expected togive only order of magnitudes, due to the simplicity ofthe approach, but relating the alpha preformation prob-ability to the radial quantum number of the valence statecan be of interest.The microscopic RHB calculations of the spatial dis-persion of the valence states of
Po (see Table I) and
Te, allows to calculate their average value, leading tothe following relation, as discussed in the previous sec-tion: < ∆ r ( P o ) > (cid:39) . < ∆ r ( T e ) > (6)Inspired by the form of the Geiger-Nuttall law, ouransatz, for the alpha preformation probability P, is apower law as a function of the localisation parameter: P = 10 − Bα loc + C (7)where B and C are constants to be determined. In a re-cent experiment [16], the alpha preformation probabilityin Te was deduced to be at least 3 times larger thanin
Po. We therefore take, as typical orders of magni-tudes, P(
Te)=1 and P(
Po)=0.1 to mimic this effect.This allows to determine B and C, namely with Eq. (2):
Log √ P = 1 − (cid:18) A (cid:19) / √ n − Log √ P (cid:39) − √ n − n is the radial quantum number of the valencestate of the considered nucleus. In the case of pairingeffect, n can be taken as the average of the n values ofthe valence states, weighted by their occupation proba-bilities. The alpha preformation probability, calculatedwith Eq. (9), is reduced by a factor 20-30 when the va-lence state switches from a n = 1 to a n = 2 state. Thisresult is in agreement with the typical observed varia-tion of the estimation of the preformation of the alphaprobability extracted from the data (see e.g. Fig. 4 ofRef. [14]). It should be noted that taking the upper limitP( Po)=0.3, instead of 0.1, would lead to a reductionof a factor 10 of the alpha preformation probability, froma n = 1 to a n = 2 state. Considering that approxima-tions are done on an exponential quantity, the presentapproach should remain qualitative. V. CONCLUSION
The alpha preformation probability has been analysedthrough the behavior of the spatial localisation of nucle-onic states. A criterion for localised states has been firstestablished. The systematically more localised n = 1states, independently of the orbital angular momentumvalue, allow to pinpoint nuclei which are more likely tohave a large preformation probability over the nuclearchart. In order to compare with experimentally knownalpha-emitters, axially deformed RHB calculations havebeen performed over the nuclear chart to provide mi-croscopic spatial dispersions. The systematic occurrenceof more localised valence states (which do have n = 1)shows a pattern which is in agreement with experimen-tally known alpha-emitters. The investigation of thesingle-particle spectra of Po and
Te, allows to un-derstand in more details why the alpha preformationprobability is larger in the latter than in the former. Thisis partly due to the PSS symmetry at work in
Po, in-volving a n=2 state, and to the fact that, being a lighternucleus,
Te involves almost only n = 1 states, each ofthem also having a bit smaller dispersion due to the masseffect on the localisation parameter.In order to study in a complementary way these results,a phenomenological evaluation of the alpha preformationfactor has been undertaken. It confirms the relevance ofthe criterion for localised states, and also shows that itis correlated to the enhancement of the alpha preforma-tion factor, especially after shell closure. Finally, a phe-nomenological law relating the preformation probabilityto the radial quantum number of the valence states, hasbeen extracted.All these results show the relevance of relativistic ap-proaches, not only to describe cluster states in nuclei, butalso to grasp the main properties of alpha radioactivity,as also shown in Ref. [29]. The present approach doesnot aim to be very accurate, especially in the difficultdomain of alpha radioactivity, where various orders ofmagnitudes are at stake. Effects of more advanced defor-mations could be studied, such as the role of triaxialityand/or octupole deformations. The description of Tecould also be improved with a more dedicated model,suited for nuclei close to the drip line. Finally, the build-ing of a four-body alpha wave function, together withquartetting correlations, could be considered in a nearfuture.
Acknowledgments