TOWARDS A SELFCONSISTENT CLUSTER EMISSION THEORY … · NUCLEAR PHYSICS TOWARDS A SELFCONSISTENT CLUSTER EMISSION THEORY D. S. DELION National Institute of Physics and Nuclear Engineering,

NUCLEAR PHYSICS

TOWARDS A SELFCONSISTENT CLUSTER EMISSION THEORY

D. S. DELION

National Institute of Physics and Nuclear Engineering,POB MG-6, Bucharest-Mãgurele, Romania,

A. SÃNDULESCU

Center for Advanced Studies in Physics,Calea Victoriei 125, Bucharest, Romania,

W. GREINER

Institut für Theoretische Physik, J.W.v.-Goethe Universität,Robert-Mayer-Str. 8-10, 60325 Frankfurt am Main, Germany

Received December 10, 2004

We propose a selfconsistent theory of the α-particle decay, which can beextended to the heavy cluster emission. The strong dependence of the Q-value versusthe Coulomb term and the more bound α-like configurations suggest that preformedclusters should exist on the nuclear surface. This is confirmed by the fact that thederivative of the shell-model preformation amplitude is practically a constant alongany α-chain, while that of the outgoing wave function changes exponentially uponthe Coulomb parameter. Thus, an α-cluster additional term in the preformation factoris necessary for a selfconsistent description of the decay width.

1. INTRODUCTION

The even-odd pair staggering of binding energies found along the α-lineslines, with N – Z = const, can be nicely explained in terms of a “pairing” in theisospin space between proton and neutron pairs, considered as bosons [1, 2].This suggest that α-particles are already preformed at least in the low densityregion of the nuclear surface. On the other hand the α-particle energy (Q-value),computed as the difference between the binding energies of initial and finalsystems, is directly connected with the decay width. The linear dependencebetween the logarithm of the decay width and the square root of the Q-value wasexplained by G. Gamow by supposing that the preformed α-particle moves insome attractive potential and penetrates the surrounding Coulomb barrier [3].The half-lives of α-particle emitters are well described by using an equivalentlocal potential [4]. The attractive depth and the radius of the repulsive core

Rom. Journ. Phys., Vol. 50, Nos. 1–2 , P. 165–176, Bucharest, 2005

166 D. S. Delion, A. Sãndulescu, W. Greiner 2

determines the energy and wave function of the decaying state, understood as anarrow resonance [5, 6].

The R-matrix theory [7, 8] makes a step forward and expresses the decaywidth as a product between the particle preformation probability and thepenetration through the barrier [9, 10, 11, 12]. Due to the antisymmetrisationeffects between the α-particle and daughter wave functions the interactionbecomes non-local in the internal region [13]. It was shown that the usual shell-model space using N = 6–8 major shells underestimates the experimental decaywidth by several orders of magnitude [14, 15], due to the exponential decrease ofbound single particle wave functions [16]. The inclusion of narrow singleparticle resonances is not able to cure this deficiency [17]. Only the backgroundcomponents in continuum can describe the right order of magnitude ofexperimental decay widths [18, 19, 20, 21].

Anyway, the shell model estimate of the α-particle preformation factor isnot consistent with the decreasing behaviour of Q-values along any neutronchain [22, 23]. In our previous papers [24, 25] we analyzed this feature bytreating the α-decaying state as a resonance, namely by using the matchingbetween logarithmic derivatives of the preformation amplitude and Coulombfunction. It turns out that this condition is not satisfied along any neutron or αchain if one uses the standard shell model estimate for the preformation factor.We corrected the slope of the preformation amplitude by changing the harmonicoscillator (ho) parameter of single particle components. These components areconnected with an α-cluster term, not predicted by the standard shell model [26].Recently a similar idea was used in Ref. [27].

The aim of this paper is to stress on the fact that this behaviour is stronglyconnected with the structure of the Q-value. Namely the Coulomb repulsive termgives the main linear behaviour between closed shells and therefore it should bealso recovered in the preformation factor. We will show that in order to fulfil theso-called plateau condition it is necessary to use an additional α-clustercomponent, depending upon the Coulomb parameter.

2. THEORETICAL BACKGROUND

As we pointed out in Introduction the decay width is directly connectedwith the Q-value, computed as follows

( 2 2 ) (2 2 0) ( )E B Z N B B Z Nα = − , − , β + , , − , , β , (2.1)

where ( )B Z N, , β is the binding energy, depending upon the charge, neutronnumbers and quadrupole deformation parameter. This quantity is given by theWeizsäker type relation, like for instance in Ref. [28]

3 Towards a selfconsistent cluster emission theory 167

2 3 2 1 3 1 2( ) ( )

( ) ( )vol surf Coul sym pair

def shell

B Z N a A a A a Z A E A I a A

E Z N E

/ − / − /, , β = − − − , − +

+ , , β + β .(2.2)

Along any α-line with I = N – Z = const the Coulomb term has a much strongervariation versus Z (quadratic) than the other ones. Therefore the Q-value,depends linearly upon the charge number and the shell model dependencepractically disapears.

We will show that this feature is also reflected by the shell-model estimateof the α-particle preformation factor. The standard procedure to estimate thedecay width within the microscopic approach was described in several papers,like for instance [18, 19, 20, 21]. In a phenomenological approaches one definesan equivalent local α-core interaction for any distance. By expanding thesolution of the corresponding Schrödinger equation in spherical waves, i.e.,

( ) ˆ( ) ( )lm lm

l

g rY r

rΨ = ,∑r (2.3)

one finds the energy of a decaying resonant state by matching the internal( ) ( )intlg r and external outgoing components ( ) ( )ext

lg r at some radius r = R. Thedecay width can be derived from the continuity equation as follows

2( )r ll

v lim g r→∞Γ = | | ,∑ (2.4)

where v is the cm velocity at infinity.The external components in a deformed Coulomb field were derived by

Fröman within the WKB approach [6]. It turns out that the major effect is givenby the quadrupole deformation of the barrier [19, 29]. The decay width can beestimated by using the following ansatz

2( )0 2

0 00

( )( ) ( ) ( )

( )

int

ll

g Rv D R R D R

G kR

⎧ ⎫⎧ ⎫⎡ ⎤⎪ ⎪⎪ ⎪Γ = ≡ Γ ,⎢ ⎥⎨ ⎬⎨ ⎬χ,⎢ ⎥ ⎪ ⎪⎪ ⎪⎣ ⎦ ⎩ ⎭⎩ ⎭∑ (2.5)

where the deformation matrix prllD with 0l′ = is given in terms of the so-called

Fröman matrix [6]. By 0 ( )G kRχ, we denoted the monopole irregular Coulombfunction, depending upon the product between the momentum k and matchingradius R. Here χ is the Coulomb parameter

21 22

Z Z ev

χ = . (2.6)

Thus, the decay width contains a ratio between the internal and externalsolutions. It does not depend upon the matching radius R within the local


potential approach, because the internal and external wave functions satisfy thesame equation and therefore are proportional. This is the so-called “plateaucondition”.

The situation becomes different when the value of the internal wave

function ( )0 ( )intg R is given by an independent microscopic approach. It is

replaced by the so-called preformation amplitude, defined as follows

( )0

0( )

( ) d d ( ) ( ) ( )int

A A A B Bg R

RR

∗ ∗α α α≡ = ξ ξ Ψ ξ Ψ ξ Ψ ξ ,∫F (2.7)

where the integration is performed over internal coordinates. The structure of afree α-particle is given by one pair of protons in a singlet state and a similar pairof neutrons [12]. Each particle lies in the ground state 0s of an ho well with theparameter 20 5 fm−

αβ ≈ . .The most important ground state correlations are given by the pairing

interaction. We use the Bardeen-Cooper-Schrieffer (BCS) approach for motherand daughter wave functions. In order to estimate the overlap integral (2.7) weexpand the mother wave function in terms of sp states, multiplied by thedaughter wave function, as follows

0 01 1 1[ ] [ ]2 2 2B j j j j j j A

j j

j P j Pπ π π ν ν ν

π ν

π νΨ = + ψ ⊗ψ + ψ ⊗ψ Ψ .∑ ∑ (2.8)

We use the short-hand index notation ( ),j ljτ ≡ τε where τ = π, ν denotes isospin,

ε sp energy, l angular momentum and j total spin. Otherwise jτ has the usualmeaning of the single particle spin. The expansion coefficients are given in termsof BCS occupation amplitudes as follows

( ) ( )A Bj j jP u vτ τ τ= . (2.9)

In order to perform the integral (2.7) analytically we expand sp wave functionsin the ho basis, i.e.,

12

20

0

ˆ( ) ( ) ( ) ( )maxn

j m nj nl l j mn

s c r Y r sτ τ

τ=

⎡ ⎤ψ , = β ⊗χ , τ = π, ν.⎣ ⎦∑r R (2.10)

The radial ho wave function is defined in terms of the Laguerre polynomial. Thesp parameter β0 is connected with the standard ho parameter by using a scalingfactor f0 as follows

00 0 0 1 3

NN

M ff f

A /ω

β = β = ≈ , (2.11)


where A is the mass number. By performing the recoupling of proton andneutron pairs in (2.8) to relative and cm coordinates the preformation amplitudebecomes

20

0 0

4 2 1 2 20 0 0 0

( )

( ) (4 ) (4 )max min

RN max min N N

N

n P R

W n P L R− β / /

β , , ; =

= β , , β β .∑F

ε N (2.12)

We stress on the fact that the exponential term is similar to the cm α-particlewave function, but it depends upon the single particle ho parameter β0. Theexpansion coefficients are given in terms of recoupling Talmi-Moshinskybrackets as in Ref. [19]. We consider in our sp basis only those states with Pτlarger than the minimal value Pmin, taken as a parameter.

3. NUMERICAL ANALYSIS

The most important ingredient, governing the penetrability of the α-particlethrough the barrier, is the Coulomb parameter χ. The irregular Coulomb function

0 ( )G kRχ, depends exponentially on it

( )1 20

21 22

00

( ) ( ) e

cos

sin cosG kR ctg

Z Z ekR R RR E

χ α− α α/

α

χ, = α ,

α = = , = .χ

(3.1)

The decay width has also an exponential dependence upon the quadrupoledeformation. As it was shown in Ref. [25] the function D(R) in Eq. (2.5)practically does not depend upon the radius. The largest correction gives a factorof three for heavy nuclei and a factor of five in superheavy ones.

The preformation amplitude, given by Eq. (2.12), is very collective andtherefore the transitions between ground states are not sensitive to the mean fieldparameters. Thus, in our analysis we used the universal parametrisation of theWoods-Saxon potential [30] and we considered the gap parameter estimated by

12 BAτ∆ = / [31], where AB is the mass number of the mother nucleus. Thequadrupole deformation parameters in the Fröman matrix are taken fromRef. [32].

The preformation factor is very sensitive with respect to the maximal spradial quantum number nmax, the sp ho parameter β0 and the amount of sphericalconfigurations taken in the BCS calculation, given by { }.minP min Pτ= It turnsout that beyond nmax = 9 the results saturate if one considers in the BCS basis spstates with 0 02.minP P≥ = . We improved the description of the continuum by


choosing a sp scale parameter 0 1f < in Eq. (2.11). This parameter is notindependent from Pmin. It turns out that the common choice of f0 and Pmin

ensures not only the right order of magnitude for the decay width, but also theabove mentioned continuity of the derivative.

The logarithm of the decay width can be approximated by the followinglinear ansatz

10 0 1( )exp

Rlog R

⎡ ⎤Γ = γ + γ .⎢ ⎥Γ⎣ ⎦(3.2)

In the ideal case the coefficients should vanish, i.e., 0 1 0,γ = γ = in order to havea proper description of the decay width. In other words we can in principle findthe Coulomb parameter χ by solving the equation

1( ) 0,γ χ = (3.3)

for given parameters nmax, β0, Pmin and in this way to predict Q-valueindependently, based only on the microscopic factor.

We analysed α-decay chains from even-even nuclei with N > 126, given inthe Table 1.

Table 1

Even-even α-decay chains in the region Z > 82, N > 126.In the first column of each table is given the isospinprojection I = N – Z. In the next columns are given theinitial neutron and proton numbers, the number of

states/chain and the reference

I N1 Z1 No Ref.

38 130 92 1 [4]

40 130 90 2 [4]

42 130 88 3 [4]

44 130 86 6 [4]

46 132 86 8 [4]

48 134 86 12 [4]

50 136 86 9 [4]

52 142 90 7 [4]

54 146 92 5 [4]

56 150 94 4 [4]

58 154 94 2 [4]

60 172 112 3 [33]

It turns out that the values nmax = 9, f0 = 0.8 and Pmin = 0.025 give the bestfit concerning the parameters γ0 and γ1. From Fig. 1.a we see that the quantity


0 10 ( )explogγ ≈ Γ/Γ has a variation of one order of magnitude around 0 0,γ = but

the description of the slope γ1, given in Fig. 1.b, is by far not satisfactory. Thereason for the variation of the slope parameter γ1 is the relative strong dependenceof the Coulomb parameter χ upon the neutron number along α-chains. In Fig. 1.cwe give the values of this parameter for the even-even chains, which is in anobvious correlation with the slope parameter γ1. Therefore the derivative of themicroscopic preformation amplitude changes along α-chains much slower incomparison with that of the Coulomb function. As we pointed out the term givenby the shell correction disapears in the Q-value (except the magic numbers) and itremains a linear in Z dependence. Thus, indeed the most important effect is givenby the Coulomb repulsion. In order to stress on this dependence we performed thesame analysis in the region Z > 82, 82 < N <126, given in the Table 2.

Fig. 1. – (a) The ratio parameter γ0, defined by Eq. (3.2), versus the neutron numberfor f0 = 0.8, Pmin = 0.025 and different even-even α-chains in Table 1. (b) The slopeparameter γ1, defined by Eq. (3.2), versus the neutron number. (c) The Coulomb

parameter χ, defined by Eq. (2.6), versus the neutron number.


Table 2

Even-even α-decay chains in the region Z > 82, 82 < N < 126.The quantities are the same as in Table 1

I N1 Z1 No. Ref.

28 114 86 1 [4]

30 116 86 2 [4]

32 118 86 3 [4]

34 120 86 3 [4]

36 122 86 2 [4]

38 124 86 1 [4]

In Figs. 2.a,b we plotted the parameters γ0, γ1 depending upon the neutronnumber. We used the same parameters, i.e., nmax = 9, f0 = 0.8, Pmin = 0.025. Onecan see that indeed their values are very close to zero. The decay widths arereproduced within a factor of two. We point out the small decrease of parametersalong considered α-chains is correlated with a similar behaviour of the Coulombparameter χ in Fig. 2.c.

Our estimate shows that the linear correlation coefficient between γ1 and χis larger than 0.7. This allows us to introduce a supplementary, but universal,correcting procedure for the preformation factor. Thus, let us define a variablesize parameter f by a similar to (2.11) relation, namely

.Nfβ = β (3.4)

The parameter χ enters in the exponent of the Coulomb function (3.1). This factsuggests a similar correction of the preformation factor, i.e.

2

0(1 2)4 2 2

0 0

( )

e ( ) (4 ) (4 )

m max min

RN m max min N m mN

N

n P R

W n P L R/− β /

β, β , , ; =

= β , , β β .∑F

N (3.5)

We suppose a linear dependence of the size parameter f upon the Coulombparameter

1( ) ( ) .m m N m Nf f fβ −β = − β = χ − χ β (3.6)

The above relation (3.5) can be written as follows24( ) 2

0 0

0 0

( ) e ( )

( 0 0 ) ( )

m Rm max min m max min

m m max min

n P R n P R

R n P R

− β−β /β, β , , ; = β , , ; == β −β , , ; β , , ; ,

F FF F

(3.7)

i.e., the usual preformation amplitude is multiplied by a cluster preformationamplitude with 0.maxn = Thus, one has to multiply the right hand side of theexpansion (2.12) by this factor.


Fig. 2. – (a) The ratio parameter γ0, defined by Eq. (3.2), versus the neutron number forf0 = 0.8, Pmin = 0.025 and different even-even α-chains in Table 2. (b) The slopeparameter γ1, defined by Eq. (3.2), versus the neutron number. (c) The Coulomb

parameter χ, defined by Eq. (2.6), versus the neutron number.

We choosed a strategy to determine the parameters connected with themaximal value of the ratio parameter γ0. As we will show later this choice has aphysical meaning connected with the α-clustering picture. We remark fromFig. 1.a that the maximal value of the ratio parameter γ0 corresponds to amaximal value of the Coulomb parameter. By using a constant ho parameterwith the size parameter fm = 0.83 for all analyzed even-even emitters the Fig. 1.ais pushed down and one obtains for the maximal value of the ratio parameter

0 ( ) 0maxγ = . In this way we suppose that in this point the α-clustering isdescribed entirely by the pairing correlations. In this way for other decays theα-clustering process increases by decreasing the Coulomb parameter, becausethe ho parameter β in (3.6) is smaller and therefore the tail of the preformationfactor increases.


From Figs. 1.a.b we can see that the pure α -clustering should be enhancedin the region above N = 126 and in superheavy nuclei. This is agreement withseveral calculations pointing out on a very strong clustering process in Po, Rnand Ra isotopes. Our calculations predict a similar feature for superheavy nuclei.

Therefore in our calculations we used the parameters fm = 0.83, χm = 55.For the proportionality coefficient in Eq. (3.6) the regression analysis gives thevalue f1 = 8.0⋅10–4. The situation in the superheavy chain can be described byassuming a quadratic dependence of the coefficient f1 upon the number ofclusters 0( ) 2N N Nα = − / with N0 = 126, namely

21 1 2f f f Nα→ + . (3.8)

A quadratic in Nα dependence of the Q-value was also empirically found inRef. [1]. The final results are given in Fig. 3.a,b. We considered a correcting term

Fig. 3. – (a) The parameter γ0 versus the neutron number for different even-even α-chains in Table 1. The preformation parameters are fm = 0.83, f1 = 8.0⋅10–4,

f2 = 1.28⋅10–6, Pmin = 0.025. (b) The same as in (a), but for the slope parameter γ1.


with f2 = 1.28⋅10–6. The improvement of the slope parameter is obvious. Themean value of this parameter and its standard deviation for even-even chains is

1 0 001 0 034.γ = − . ± .The quadratic dependence in Eq. (3.8) can be also interpreted in terms of

the total number of interacting clustering pairs, namely 2 2 ( 1) 2N N Nα α α≈ − / .Thus, our analysis based on the logarithmic derivative continuity, shows veryclearly that the effect of the α-clusterisation becomes much stronger forsuperheavy nuclei.

4. CONCLUSIONS

We proposed in this paper a selfconsistent theory of the α-decay. Weanalysed the decay widths for deformed even-even emitters with Z > 82. Theα-particle preformation amplitude was estimated within the pairing approach.We used the universal parametrisation of the mean field and the empirical rulefor the gap parameter 12 A∆ = / . The penetration part was computed within thedeformed WKB approach. It is possible to satisfactorily describe all α-decaywidths from even-even nuclei by using a constant, but smaller ho parameter

0 80 Nβ = . β and Pmin = 0.025.It turns out that the slope of the decay width versus the matching radius has

a strong variation for N > 126, in an obvious correlation with the Coulombparameter. Thus, the relative amount of the α -clustering here cannot bedescribed only within the pairing approach and an additional mechanism isnecessary. We supposed a cluster factor, multiplying the preformation amplitude.It contains exponentially an ho parameter, proportional to the Coulombparameter.

The method improves simultaneously the ratio to the experimental widthand the slope with respect to the matching radius. The relative increase of theα-clustering is related to the decrease of the Coulomb parameter. It is strongerfor two regions, namely above N = 126 and in superheavy nuclei. It has aminimum around N = 152.

An additional dependence upon the number of interacting α-particlesimproves the plateau condition for superheavy nuclei. This additional clustering,which seems to be very strong, may affect the stability of nuclei in this region.

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Documents

TOWARDS A SELFCONSISTENT CLUSTER EMISSION THEORY … · NUCLEAR PHYSICS TOWARDS A SELFCONSISTENT CLUSTER EMISSION THEORY D. S. DELION National Institute of Physics and Nuclear Engineering,