aa r X i v : . [ nu c l - t h ] J u l Universal decay rule for reduced widths
D.S. Delion , Horia Hulubei National Institute of Physics and Nuclear Engineering,407 Atomi¸stilor, Bucharest-M˘agurele, 077125, Romˆania Academy of Romanian Scientists, 54 Splaiul Independent¸ei, Bucharest, 050094, Romˆania
Emission processes including α -decay, heavy cluster decays, proton and di-proton emission areanalyzed in terms of the well known factorisation between the penetrability and reduced width. Byusing a shifted harmonic oscilator plus Coulomb cluster-daughter interaction it is possible to derive alinear relation between the logarithm of the reduced width squared and the fragmentation potential,defined as the difference between the Coulomb barrier and Q-value. This relation is fulfilled witha good accuracy for transitions between ground states, as well as for most α -decays to low lying2 + excited states. The well known Viola-Seaborg rule, connecting half lives with the Coulombparameter and the product between fragment charge numbers, as well as the Blendowke scallingrule connecting the spectroscopic factor with the mass number of the emitted cluster, can be easilyunderstood in terms of the fragmentation potential. It is shown that the recently evidenced tworegions in the dependence of reduced proton half-lives versus the Coulomb parameter are directlyconnected with the corresponding regions of the fragmentation potential.PACS numbers: 21.10.Tg, 23.50.+z, 23.60.+e, 23.70.+j, 25.70.Ef
1. INTRODUCTION
The family of emission processes triggered by thestrong interaction contains various decays, namely par-ticle (proton or neutron) emission, two-proton emission, α -decay, heavy cluster emission and binary or ternaryfission. There are also other nuclear decay processes in-duced by electromagnetic ( γ -decay) or weak forces ( β -decays). The purpose of this work is to investigateonly the first type of fragmentation, where the emittedfragments are left in ground or low-lying excited states.They are called cold emission processes and are presentlyamong important tools to study nuclei far from the sta-bility line. Nuclei close to the proton drip line are inves-tigated through proton emission, while the neutron dripline region is probed by cold fission processes. On theother hand superheavy nuclei are exclusivelly detectedby α -decay chains [1]. Actually the first paper in theo-retical nuclear physics applying quantum mechanics [2]was devoted to the description of the α -decay in termsof the penetration of a preformed particle through theCoulomb barrier.There are two goals of this paper. The first one is toexplain the well known Viola-Seaborg rule [3], valid forall kinds of cold emission processes. It turns out that itis possible to give a simple interpretation of this rule interms of two physical quantities, namely the Coulomb pa-rameter, connected with the penetrability, and the frag-mentation potential, connected with the reduced width.An universal linear dependence between the logarithmof the reduced width and fragmentation potential willbe derived. It will be shown that this interpretation isvalid not only for transitions between ground states, butalso for transitions to excited states. On the other hand,the scalling dependence of spectroscopic factors in heavycluster decays versus the mass numbers of the emittedcluster can be also understood in terms of the fragmen-tation potential.
2. EXPERIMENTAL DECAY RULES
Let us consider a binary emission process P → D + C from a parent ( P ) to the daughter nucleus ( D ) andthe lighter cluster ( C ), which can be in particular an α -particle or a proton. The total decay width is the sum ofpartial decay widths corresponding to different angularmomenta, given by [4]Γ l = 2 P l ( E l , r ) γ l ( β, r ) , (2.1)where it was introduced the standard penetrability andreduced width squared [5] P l ( E l , r ) = κ l r (cid:12)(cid:12)(cid:12) H (+) l ( χ l , κ l r ) (cid:12)(cid:12)(cid:12) ,γ l ( β, r ) = ~ µr | s ( int ) l ( β, r ) | . (2.2)The outgoing spherical Coulomb-Hankel function H (+) l ( χ l , κ l r ) depends upon two variables, namely theCoulomb (or twice the Sommerfeld) parameter χ l = Z D Z C e ~ v l , (2.3)and the reduced channel radius ρ l = κ l r . Here v l = ~ κ l /µ and κ l = √ µE l / ~ are the asymptotic relative velocityand momentum between the emitted fragments, respec-tivelly, in terms of the reduced mass of the daughter-cluster system µ . It is also defined the center of mass(cm) channel energy E l = Q − E ( ex ) l of emitted frag-ments, in terms of the difference between the total energy(Q-value) and the excitation energy of the daughter nu-cleus E ( ex ) l . The internal component s ( int ) l ( β, r ) at a cer-tain radius r inside the Coulomb barrier, is for deformedemitters a superposition of different Nilsson componentsmultiplied by the propagator matrix [4], depending ondeformation parameters β , i.e. s ( int ) l ( β, r ) = X l ′ K ll ′ ( β, r ) f ( int ) l ′ ( r ) (2.4)For spherical emitters with K ll ′ = δ ll ′ it coincides withthe wave function component f ( int ) l ( r ).The half life is defined by the inverse of the total decaywidth, i.e. T = ~ ln . (2.5) -10-50510152025 30 35 40 45 50 55 60 FIG. 1:
Logarithm of half lives for α -decays from even-evennuclei versus Coulomb parameter (2.3). Differents lines con-nect decays from nuclei with the same charge number. Inside the Coulomb barrier the complex Coulomb-Hankel function practically coincides with the real irregu-lar Coulomb function and has a very simple WKB ansatz[4] H (+) l ( χ, ρ ) ≈ exp h χ (cid:16) arccos √ x − p x (1 − x ) (cid:17)i(cid:0) x − (cid:1) / C l ≡ H (+)0 ( χ, ρ ) C l (2.6)where, with the external turning point r e = Z Z e /E and barrier energy V = Z Z e /r , there were introducedthe following notations x = ρχ = rr e = EV C l = exp (cid:20) l ( l + 1) χ r χρ − (cid:21) . (2.7) There were considered here for simplicity transitions be-tween states with the same angular momentum l , as forinstance α -decays or proton emission processes betweenground states. Thus, the logarithm of the so-called re-duced half-life, corrected by the exponential centrifugalfactor squared C l , defined by the second line of this re-lation, i.e. log T red = log TC l = log ~ ln C l − log P l − log γ l , (2.8)should be proportional with the Coulomb parameter, i.e. log T red = a χ + b , (2.9)Notice that in most of decay processes between groundstates one has boson fragments, with zero angular mo-mentum, i.e. C l = 1. The case with l = 0 is connectedwith fermions, i.e. proton emission, where C l = 1. FIG. 2:
Logarithm of half lives for heavy cluster decays and thecorresponding α -decays from the same mother nuclei versusCoulomb parameter (2.3). Differents symbols denote chargenumber of the emitted cluster. The above relation is also called Geiger-Nuttall law,discovered in 1911 for α -decays between ground states(where the angular momentum carried by the α -particleis l = 0). The explanation of this law was given by G.Gamow in 1928 [2], in terms of the quantum-mechanicalpenetration of the Coulomb barrier, i.e. the first line ofEq. (2.2). It is characterized by the Coulomb parameter,which is proportional with the ratio Z D / √ Q α .The α -decays between ground states are characterizedby a remarkable regularity, especially for transitions be-tween ground states of even-even nuclei. The fact that α -transitions along various isotopic chains lie on separatelines, as stated by the Viola-Seaborg rule [3], i.e. log T = a Z D + a √ Q α + b Z D + b , (2.10)is connected with different α -particle reduced widths,multiplying the penetrability in (2.1). From Eq. (2.8)it becomes clear that the reduced width should dependupon the charge of the daughter nucleus. This feature isshown in figure 1. Still in doing systematics along neu-tron chains there are important deviations with respectto this rule, as for instance in α -decay from odd mass nu-clei, and this feature is strongly connected with nuclearstructure details. Let us mention that different forms ofthe Viola-Seaborg relation were used in Refs. [6, 7].The Viola-Seaborg rule can be generalized for heavy-cluster decays [8], as it is shown in figure 2. Here the an-gular momenta carried by emitted fragments are zero. InRef. [9] it was proposed the following generalized Viola-Seaborg rule for the heavy cluster emission log T = a Z D Z C √ Q + a Z D Z C + b + c , (2.11)with the following set of parameters a = 1 . , a = 0 . , b = − . ,c = 0 ( even ) , = 1 .
402 ( odd ) ,where c is the blocking parameter for odd-mass nuclei.Thus, from Eq. (2.8) the reduced width in this caseshould depend upon the product between daugher andcluster charges.An interesting feature can be seen by plotting in fig-ure 3 the logarithm of the reduced half life (2.8) ver-sus the Coulomb parameter for various proton emitters[10, 11]. In this case most of emitters have non-vanishingangular momentum. The data are rougly divided intotwo regions, corresponding to Z <
68 (open circles) and
Z >
68 (dark circles). This pattern can be asimilitatedwith a generalized Viola-Seaborg rule for the two groupsof charge numbers.The situation with binary cold fission is quite different.Here there is not a simple linear dependence between thehalf-life for a given isotopic partition and the Coulombparameter, due to the fact that mass asymmetry changesduring the scission process.
3. A SIMPLE MODEL FOR THE REDUCEDWIDTH
The decay processes can be schematically described bythe following cluster-daugher spherical potential V ( r ) = ~ ω β ( r − r ) v , r ≤ r B = Z D Z C e r ≡ V C ( r ) , r > r B (3.1) -10-8-6-4-202 15 16 17 18 19 20 21 22 23 24 25 FIG. 3:
Logarithm of reduced half lives (2.8) for proton emit-ters versus Coulomb parameter (2.3). Open circles denoteemitters with
Z < , while dark circles emitters with Z > . where r = 1 . A / D is the surface radius of the daughernucleus. Indeed, microscopic calculations have shownthat the preformation factor of the α -particle has a Gaus-sian shape, centered on the nuclear surface [12]. More-over, the spherical component of the prefomation am-plitude gives more than 90% contribution in the α -decaywidth of deformed nuclei. Notice that the radial equationof the shifted harmonic oscillator (ho) potential is similarwith the equation of the one-dimensional oscillator, buthaving approximate eigenvalues given by E nl = ~ ω (cid:18) n + 12 (cid:19) + ~ l ( l + 1)2 µr . (3.2)By considering Q -value as the first eigenstate in theshifted ho well Q − v = ~ ω , together with the conti-nuity condition at the top of the barrier r B , one obtainsthe following relation ~ ω β ( r B − r ) V frag ( r B ) + 12 ~ ω , (3.3)where it was introduced the so called fragmentation (ordriving) potential, as the difference between the top ofthe Coulomb barrier and Q-value V frag ( r B ) = V C ( r B ) − Q . (3.4)The second component in Eq. (2.8) contains thelogarithm of the Coulomb-Hankel function inside theCoulomb barrier which, according to Eq. (2.6), is pro-portional with the Colomb parameter χ . The third partcontains the reduced width squared which, according toEq. (2.2), is proportional with the modulus of the inter-nal wave function squared. For a shifted ho well one hasfor the ground state | f ( int )0 ( r ) | = A e − β ( r − r ) . (3.5)By using the notation γ ≡ γ and Eq. (3.3) one obtainsthe following relationlog γ ( r B ) = − log e ~ ω V frag ( r B ) + log ~ A eµr B . (3.6) -4-3.5-3-2.5-2-1.5-1-0.50 16 18 20 22 24 -4-3.5-3-2.5-2-1.5-1-0.50 16 18 20 22 24-4-3.5-3-2.5-2-1.5-1-0.50 16 18 20 22 24 -4-3.5-3-2.5-2-1.5-1-0.50 16 18 20 22 24 FIG. 4:
The logarithm of the α -decay reduced width squaredversus the fragmentation potential (3.4) for regions of the nu-clear chart described by (4.1). In this way one obtains that indeed the logarithm ofthe half life is of the Viola-Seaborg typelog T = c ( r B ) χ + c V frag ( r B ) + c ( r B , A ) , (3.7)because the fragmentation potential contains the product Z D Z C . Its coefficient depends upon the touching radius,but this radius has a very small variation along variousisotopic chains. Notice that the slope in Eq. (3.6) hasa negative value and it is connected with the shape ofthe interaction potential (ho energy ~ ω ), while the freeterm gives information about the amplitude of the clusterwave function.Our calculation has shown that the linear relation (3.6)but with different coefficients, remains valid in the mostgeneral case of the double folding plus repulsive interac-tion between fragments, used in Refs. [13, 14].
4. DECAY RULE FOR REDUCED WIDTHS
Most of experimental data refer to the α -decay. There-fore there were analyzed reduced widths in α -decays con-necting ground states of even-even nuclei. In figure 4 it isplotted the logarithm of the experimental reduced widthsquared, by using the above relation versus the fragmen-tation potential. The data are divided into five regionsof even-even α emitters as follows1) Z < , < N < F ig. c ) , stars ; (4.1)2) Z < , < N < F ig. a ) , crosses ;3) Z > , < N < F ig. b ) , circles ;4) Z > , < N < F ig. c ) , squares ;5) Z > , N > F ig. d ) , triangles . In calculations it was used the value of the touchingradius, i.e. r B = 1 . A / D + A / C ) , (4.2)where the approximation K ll ′ ≈ δ ll ′ in Eq. (2.4) is ful-filled with 90% accuracy for the most deformed nuclei[4]. Notice that regions 1-4 contain rather long isotopicchains, while in the last region 5 one has not more thantwo isotopes/chain. This is the reason why, except for thelast region 5, the reduced width decreases with respectto the fragmentation potential, according to the theoret-ical prediction given by (3.6). Notice that in the regions1 and 4, above Sn and
Pb double magic emitters,respectivelly, the ho parameter of the α -daughter poten-tial is larger than for regions 2 and 3, corresponding tocharge numbers around the double magic nucleus Pb.On the other hand, one obtains the largest amplitudes ofthe cluster wave function in the regions 2 and 3.An interesting observation can be made from figure 5,where it is plotted the difference log T − c V frag ( r B ) − c ( r B , A ) versus the Coulomb parameter χ , by usingthe same five symbols for the above described regions.Amazingly enough there were obtained three lines, cor-responding to different amplitudes of the cluster wavefunction. The regions 1 and 4, corresponding to emittersabove double magic nuclei Sn and
Pb, respectivelly,have practically the same internal amplitudes A . Thesame is true for the regions 3 and 5.The linear dependence of log γ versus the fragmenta-tion potential (3.6) remains valid for any kind of clusteremission. This fact is nicely confirmed by heavy clus-ter emission processes in figure 6 (a), where it is plottedthe dependence between the corresponding experimentalvalues for the same decays in figure 2. Here it is also plot-ted a similar dependence for α -decays corresponding tothe same heavy cluster emitters. The straight line is thelinear fit for cluster emission processes, except α -decays log γ = − . V C − Q ) + 15 . . (4.3) FIG. 5:
The difference log T − c V frag ( r B ) − c versus theCoulomb parameter χ for five different regions described by(4.1). The straight lines are the corresponding linear fits. The above value of the slope − log e / ~ ω in Eq. (3.6)leads to ~ ω ≈ . α -decay case. The relative large scattering ofexperimental data around the straight line in figure 6 canbe explained by the simplicity of the used cluster-corepotential (3.1).Let us mention that a relation expressing the spectro-scopic factor (proportional with the reduced width) forcluster emission processes was derived in Ref. [15] S = S ( A C − / α , (4.4)where A C is the mass of the emitted light cluster and S α ∼ − . As can be seen from figure 6 (b) between A C and V frag there exists a rather good linear depen-dence and therefore the above scalling law can be easilyunderstood in terms of the fragmentation potential.Concerning the reduced widths of proton emitters inRefs. [11, 16] it was pointed out the correlation betweenthe reduced width and the quadrupole deformation. Thisfact can be seen in figure 7 (a), where the region with Z <
68 corresponds to β > . Z >
68 to β < . K ll ′ ( β, r ) in Eq. (2.4).Notice that the two dark circles with the smallest reducedwidths correspond to the heaviest emitters with Z > -20-18-16-14-12-10-8-6-4-20 10 15 20 25 30 35 40 45 50 55 600510152025303540 10 15 20 25 30 35 40 45 50 55 60
FIG. 6: (a) The logarithm of the reduced width squared versusthe fragmentation potential (3.4). Different symbols corre-spond to cluster decays in figure 2. The straight line is thelinear fit (4.3) for cluster emission processes, except α -decay.(b) Cluster mass number versus the fragmentation potential. tential, corresponding to the two regions of charge num-bers, have roughly the same slopes, but different valuesin origin. Thus, the two diferrent lines in figure 3 canbe directly connected with similar lines in figure 7 (b).They correspond to different orders of magnitude of thefragmentation potential, giving different orders to wavefunctions and therefore to reduced widths.A special case is given by the two-proton emission.This process was predicted log time ago [17], but onlyfew such emitters were recently detected until now. Letus mention that the most recent general treatment of thetwo-proton emission process, assuming a three-body dy-namics, is given in Refs. [18] (and the references therein).The experimental half-lives versus Coulomb parameterare given in figure 8 (a) by triangles. The emitter chargesare also pointed out. Here it is assumed a simplified ver-sion, where the light emitted cluster is supposed to bea di-proton with the charge Z C = 2. In this case onecan use the factorisation of the decay width (2.1). Onesees that half lives (triangles) follow the general trend(dashed line) of the usual proton emitters, given in thesame figure by the symbols in figure 3. In figure 8 (b)it is plotted the logarithm of the reduced width squaredversus the fragmentation potential by triangles. One in-deed observes that the slope of the fitting dashed line hasa negative value − log e / ~ ω , but with a much largervalue in comparison with proton emitters, given by thesame symbols as in figure 7 (b). The three lines in thisfigure, corresponding to proton and two-proton radioac- -3-2.5-2-1.5-1-0.50-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4-3-2.5-2-1.5-1-0.50 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 FIG. 7: (a) The logarithm of the reduced width squared versusthe quadrupole deformation. By open circles are given emit-ters with
Z < , while by dark circles those with Z > for proton emission. The two regression lines fit the corre-sponding data. (b) The logarithm of the reduced width squaredversus the fragmentation potential (3.4). The symbols are thesame as in (a). tivity, respectivelly, are given by log γ = − . V C − Q ) + 1 . , Z < log γ = − . V C − Q ) + 3 . , Z > log γ = − . V C − Q ) + 4 . . (4.5)The ho energy is ~ ω ≈ . α -decay) and ~ ω ≈ . R C andtherefore the fragmentation potential. In reality the di-proton is not a bound system and it changes its size dur-ing the barrier penetration. Our microscopic estimate,by using a pairing residual interaction between emittedprotons from Fe, evidenced that the size of the wavepacket increases in the relative coordinate by 1 fm overa distance of 1 fm in the region of the nuclear surface.Actually the reduced width defined by (2.1) can be gen-eralized by using the other extreme scenario, given by asequential emission, where this relation is integrated overall possible energies of emitted particles [19].An interesting observation concerns the amplitude A in Eq. (3.6), given by the value of fitting lines in ori-gin. It has similar values for both two-proton and protonemitters with Z > -25-20-15-10-50 0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25-20-18-16-14-12-10-8-6-4-20 0 2 4 6 8 10 12 14
FIG. 8: (a) The logarithm of the half life versus Coulombparameter for two-proton emitters (triangles). By circles aregiven data for proton emitters in figure 3. (b) The logarithm ofthe reduced width squared versus the fragmentation potential(3.4) for two-proton emitters (triangles). The same quantityis given by circles for proton emitters in figure 7 (b). The solidlines fit proton data, while the dashed lines fit two-proton data.
Now let us analyze α -decay processes to excited low ly-ing 2 + states. There were considered more than 70 decaysof even-even rotational, vibrational and transitional nu-clei [13, 14]. The hindrance factor (HF) is defined as theratio of reduced widths squared connecting the groundstates and ground to excited states with the angular mo-mentum l = 2, i.e. HF = γ γ . (4.6)Thus, by using (3.6), the logarithm of the HF becomesproportional with the excitation energy of the daughternucleus log HF = log e ~ ω E ( ex )2 + log A A . (4.7)It is worth mentioning that this relation is equivalentwith the Boltzman distribution for the reduced width tothe excited state γ . In Refs. [20, 21] such a dependencewas postulated in order to describe HF’s.In figure 9 it is plotted the logarithm of the HF versusthe excitation energy for rotational nuclei with E ( ex )2 < A ≈ A . Noticethat the region 4 in figure 9 (b), with Z > , < -1-0.500.511.522.53 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-1-0.500.511.522.53 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 FIG. 9:
The logarithm of the hindrance factor versus the ex-citation energy of the daughter nucleus for rotational nuclei.The symbols and numbers correspond to the regions given byEq. (4.1).
N < E ( ex )2 >
5. CONCLUSIONS
Cold emission processes in terms of the well known fac-torisation of the decay width between the penetrabilityand reduced width squared, were analyzed. Based on asimple model of the two-body dynamics, namely a shiftedharmonic oscillator potential surounded by the Coulombinteraction, it was derived an universal analytical relationexpressing the logarithm of the reduced width squared asa linear function in terms of the fragmentation potential,defined as the difference between the Coulomb barrierand the Q-value. Notice that the slope has the same order of magnitude, corresponding to an ho energy ~ ω ≈ . -10123 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-10123 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-10123 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 FIG. 10:
The same as in figure 9, but for transitional andvibrational nuclei.
This rule is a consequence of the fact that the logarithmof the wave function squared is proportional with the dif-ference between the height of the Coulomb potential ata given radius and the energy of the system. It is ful-filled with a reasonable accuracy by experimental data,describing transitions between ground states as well asfor α -transitions to excited states in the most relevantregion with Z > , < N <
6. ACKNOWLEGMENTS
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