Semiclassical propagator approach for emission processes. I. Two body non-relativistic case
aa r X i v : . [ nu c l - t h ] F e b Semiclassical propagator approach for emission processes.I. Two-body non-relativistic case
S.A. Ghinescu , ∗ and D.S. Delion , , , ”Horia Hulubei” National Institute of Physics and Nuclear Engineering,30 Reactorului, POB MG-6, RO-077125, Bucharest-M˘agurele, Romˆania Department of Physics, University of Bucharest, 405 Atomi¸stilor,POB MG-11, RO-077125, Bucharest-M˘agurele, Romˆania Academy of Romanian Scientists, 3 Ilfov RO-050044, Bucharest, Romˆania Bioterra University, 81 Gˆarlei RO-013724, Bucharest, Romˆania (Dated: February 17, 2021)We compare the coupled channels procedure to the semiclassical approach to describe two-bodyemission processes, in particular α -decay, from deformed nuclei within the propagator method. Weexpress the scattering amplitudes in terms of a propagator matrix, describing the effect of thedeformed field, multiplied by the ratio between internal wave function components and irregularCoulomb waves. In the spherical case the propagator becomes diagonal and scattering amplitudesacquire the well-known form. We describe a more rigorous formulation of the 3D semiclassicalapproach, corresponding to deformed potentials, which leads to the exact results and we also comparethem with the much simpler expressions given by the Angular Wentzel-Krames-Brillouin (AWKB)and Linearized WKB (LWKB) with its approximation, known as Fr¨oman WKB (FWKB) method.We will show that LWKB approach is closer than AWKB to the exact coupled-channels formalism.An analysis of alpha-emission from ground states of even-even nuclei evidences the important roleplayed by deformation upon the channel decay widths. I. INTRODUCTION
The exact description of emission processes is pro-vided by outgoing solutions in continuum of the equa-tion of motion. When the masses of the emitted parti-cles are much larger than the energy release (Q-value)the non-relativistic Schr¨odinger equation is used, whilein the other case a relativistic approach is employedwithin the Klein-Gordon equation for boson emission andDirac equation for fermion emission. The case of non-relativistic two-body processes refers by the one protonemission, alpha and heavy cluster decays P → D + C ,while two-proton emission P → D + p + p belongs tothe field of non-relativistic three-body dynamics. Themost important relatistic three-body emission processesis given by the β + decay p → n + e + + ν , where thepositron mass has a comparable value to the Q-value butit penetrates a very large barrier, comparable to the pro-ton emission case. The exact solutions for deformed emit-ters in all these cases are provided within the coupledchannels (CC) approach with an outgoing asymptotics[1]. All these processes are basically described by thequantum penetration of a particle/cluster through an in-ternal nuclear plus an external Coulomb barrier, charac-terized by a relative small ratio between the Q-value andbarrier height. In this case semiclassical solutions providevery good approximations and our purpose is to analyzesuch solutions in the most general cases, by applying theso-called propagator method, already described for theCC approach in Ref. [1]. ∗ Corresponding author:[email protected]
We will describe in this paper the two-body non-relativistic emission, where a very good approximationis given by the semiclassical Wentzel-Kramers-Brillouin(WKB) approach [2–4]. The problem of a formulatinga general three-dimensional (3D) WKB theory for sys-tems lacking spherical symmetry has a long history. Thefirst successful attempt is due to Fr¨oman [6] who ob-tained a ”semi-analytic” expression for the wave-functionof an alpha particle inside a large barrier using geomet-rical considerations. His attempt is not, however, freefrom caveats especially due to the intuitive approach hefollowed. We present here a more rigorous formulationwhich leads to the so called Linearized WKB (LWKB),which has as a particular case the Fr¨oman method. Wealso compare them with the much simpler expressionwhich has seen extensive use by many authors [4, 5]. Wewill refer to this method as ”Angular WKB” or AWKB,in short. In the end we show that both methods agreewith the exact coupled-channels formalism for small toreasonable deformations. We will apply these considera-tions in the case of alpha decays to ground and excitedstates.
II. MATHEMATICAL FORMULATION
To give a full account of all steps we begin with thespherically symmetric problem, the reason being that thecentrifugal term in the potential appears naturally whenone builds up the deformed solution as an extension ofthe spherical one.
A. Spherical emitters
Let us consider a binary emission process P ( J i ) → D ( J f ) + C ( L ) (2.1)where J i/f denotes the initial/final spin parity of the par-ent (P)/daughter (D) nucleus and L the angular momen-tum carried by the emitted cluster (C). For simplicity weconsider the cluster with a boson structure (an alpha par-ticle or heavier cluster). We will also assume an initialground state J i = 0, leading to J f = L , i.e. a coupleddaughter-cluster dynamics with the total spin L ⊗ L = 0.The Schr¨odinger equation governs the dynamics of thebinary D+C system inside a spherically symmetric po-tential barrier V ( r ) (cid:20) − ~ µ ∆ + V ( r ) (cid:21) Ψ ( r ) = E Ψ ( r ) , (2.2)where r = ( r, θ, φ ) denotes the position vector of the clus-ter in the center of mass (CM) of the system in spheri-cal coordinates and µ = m C m D / ( m C + m D ) defines thedaughter-cluster reduced mass. The generalisation to theemission of fermions is straightforward. Notice that in-side the external Coulomb barrier the standard multipoleexpansion Ψ ( r ) = X L f L ( r ) r Y L ( θ ) , (2.3)leads at a large distance to the following equations forradial components (cid:20) − d dρ + L ( L + 1) ρ + χρ − (cid:21) f l ( r ) = 0 , (2.4)depending upon the Coulomb parameter χ = 2 Z D Z C ~ v , (2.5)and reduced radius ρ = kr, k = r µE ~ . (2.6)We employ the semiclassical ansatz by writingΨ ( r ) ≡ exp (cid:20) i ~ S ( r ) (cid:21) . (2.7)Upon inserting this expression in Eq. (2.2), we obtain − i ~ µ ∆ S ( r ) + 12 µ [ ∇ S ( r )] + V ( r ) = E , (2.8)where ∆ and ∇ denote the laplacian and gradient respec-tively in spherical coordinates.The semiclassical prescription requires the exponent S ( r ) to be expanded in powers of ~ as S ( r ) = S (0)0 ( r ) + ~ S (1)0 ( r ). We plug the expansion in Eq. (2.8) and group coefficients of equal powers of ~ to obtain the followingsystem of equations ~ : (cid:16) ∇ S (0)0 ( r ) (cid:17) = − K ( r ) ~ : − i S (0)0 ( r ) + ( ∇ S (0)0 ( r ))( ∇ S (1)0 ( r )) = 0 , (2.9)where we defined the ”radial dependent momentum” K ( r ) ≡ s µE (cid:20) V ( r ) E − (cid:21) . (2.10)We show in the Appendix that an ”outgoing” solution ofthis system is given byΨ ( r, θ ) = X L c L Y (WKB) L ( θ ) p K ,L ( r ) exp (cid:20)Z r r drK ,L ( r ) (cid:21) , (2.11)where Y ( W KB ) L is the spherical harmonic Y L in the WKBapproximation, c L are some constants, r is the externalturning point defined as the largest solution of the equa-tion V ( r ) = E and K ,L ( r ) ≡ vuut µE " V ( r ) E − (cid:0) L + (cid:1) k r . (2.12)Note that we have dropped the φ dependence, which ap-pears only in the form of a phase since the potentialis spherically symmetric. This simplifies expressions inboth the spherical and deformed cases without loss ofgenerality. B. Deformed emitters
We turn now to solving the deformed problem. Inthe laboratory system of coordinates the dynamics ofthe emission process (2.1) is described by the followingSchr¨odinger equation h b H ( R ) + b H D (Ω) + V ( R , Ω) i Φ( R , Ω) = E Φ( R , Ω) , (2.13)where b H ( R ) denotes the Hamiltonian of the daughter-cluster motion depending on the relative coordinate R =( r, b R ) and b H D (Ω) describes the internal daughter motiondepending on its coordinate Ω, which is given by Eulerangles for rotational motion. We will consider an axiallysymmetric daughter-cluster interaction which can be es-timated within the double folding procedure [7–9] by thefollowing expansion V ( R , Ω) = V ( r ) + X λ> V λ ( r ) r π λ + 1 h Y λ (Ω) ⊗ Y λ ( b R ) i = V ( r ) + X λ> V λ ( r ) Y λ ( b r ) ≡ V ( r ) + V d ( r ) , (2.14)where b r is the daughter-particle angle, defining the intrin-sic system of coordinates r = ( r, b r ), with V , the isotropiccomponent (monopole), and V d ( r ), the purely anisotropicpart. We expand solution in the intrisic system to obtainin a standard way the coupled system of equations. Byneglecting the off-diagonal Coriolis terms within the so-called adiabatic approach one obtains at large distancesa similar to (2.4) form, but with different Coulomb pa-rameters and reduced radii in each channel [1, 6] χ L = χǫ L ρ L = ρǫ L ǫ L ≡ r − E L E , (2.15)where E L denotes the excitation energy of the daughternucleus. This corresponds to the energy replacements E → E − E L in each channel.In order to analyze the specific features of the deformedWKB approach we will first neglect the excitations ener-gies of the daughter nucleus, which will be consideredlater in applications. The corresponding Schrodingerequation now reads (cid:20) − ~ µ ∆ + V ( r ) + V d ( r ) (cid:21) Ψ( r ) = E Ψ( r ) . (2.16)We propose a semiclassical ansatz similar to the one inthe spherical caseΨ( r ) = exp (cid:20) i ~ S ( r ) (cid:21) , (2.17)from which we obtain the deformed equivalent of the sys-tem in Eq. (2.9) by making again the expansion in powersof ~ as S ( r ) = S (0) ( r ) + ~ S (1) ( r ) ~ : (cid:16) ∇ S (0) ( r ) (cid:17) = − K ( r ) ~ : − i S (0) ( r ) + ( ∇ S (0) ( r ))( ∇ S (1) ( r )) = 0 , (2.18)where we have defined K ( r ) ≡ s µE (cid:20) V ( r ) E − V d ( r ) E (cid:21) . (2.19)The approach followed by Fr¨oman to solve Eqs. (2.18)is known today as the linearization of the Eikonal equa-tion which applies to S (0) in our case. This approxi-mation consists in isolating the spherical part K ( r ) de-fined by Eq. (2.10) in the first equation (2.18). We callthis approach as Linearized WKB (LWKB). This canbe achieved through the binomial approximation if V d is small compared with V (in the following we omit the spatial variables trusting no ambiguity arises) K ( r ) = s µE (cid:18) V E − (cid:19)s V d /EV /E − ≈ s µE (cid:18) V E − (cid:19) (cid:18) V d /EV /E − (cid:19) ≡ K + ∆ KK , (2.20)where we have defined∆ K ≡
12 2 µV d . (2.21)As we mentioned, Fr¨oman WKB approach (FWKB) isa particular case of LWKB and it corresponds to a pureCoulomb potential of a deformed nucleus with a sharpdensity distribution. In this case various multipoles of V d have closed analytic expressions.It is clear now that, since we have isolated the sphericalcontribution, we can use the solution from its associatedproblem. We write S (0) ( r ) as S (0) ( r ) = S (0)0 ( r ) + D ( r ) , (2.22)where S (0)0 is the solution of the spherical problem givenby Eq. (2.11), and D ( r ) is the correction arising fromthe potential deformation. Then we replace this defini-tion together with Eqs. (2.20,2.21) inside Eq. (2.18) andobtain (cid:16) ∇ S (0)0 (cid:17) + ( ∇ D ) + 2 (cid:16) ∇ S (0)0 (cid:17) ( ∇ D ) = − (cid:18) K + ∆ K K + 2∆ K (cid:19) . (2.23)The essence of the linearized eikonal approximationconsists in neglecting terms of powers higher than 1 inboth ∆ K and ∇ D . We enforce now this idea and, aftersmall simplifications, we obtain( ∇ S (0)0 )( ∇ D ) = − ∆ K . (2.24)As shown in Appendix, the partial derivatives of S (0)0 are given by ∂S (0)0 ∂θ = (cid:18) L + 12 (cid:19) ~ ∂S (0)0 ∂r = ± iK ,L ( r ) , (2.25)where L is the angular momentum quantum number.We see now that our problem reduces to solving theequation iK ,L ( r ) ∂D∂r + (cid:0) L + (cid:1) ~ r ∂D∂θ = − ∆ K . (2.26) L = θ = = θ = π / = θ = = θ = π / D L ( r , θ ) / D L ( L W K B ) ( r , θ )
15 20 25 30 r ( fm ) FIG. 1. Ratio between the solution of eq. (2.26) and LWKBapproximation versus radius for L = 0 , θ = 0 , π/ This equation does not have a closed form solutionunless the deformed potential is of the form V d ( r, θ ) = V ( θ ) /r , which is not the case for axial deformations.Consequently, the approximation used is that eventhough the potential is no longer spherically symmet-ric, the classical trajectory of the emitted cluster wouldstill be a straight line and one can integrate this systemradially by setting formally ∂D/∂θ = 0. This approxi-mation is somewhat justified also by the coefficients ofthe two partial derivatives: far away from the turningpoints K ,L ( r ) is of the order 1, while ( L + 1 / /r ∝ . r = r , where the field is spherical (hence D = 0 for all l ) and the exponent becomes i ~ D ( r, θ ) = − ~ Z rr dr ′ ∆ K ( r ′ , θ ) K ,L ( r ′ ) . (2.27)We note here that this is the correct use of the WKBapproximation since it gives the expected asymptotic be-havior, while in [6] the author performs the integrationstarting from the nuclear surface towards the turningpoint. This observation is useful, however, only if onedesires to compute the wave-function of the alpha parti-cle inside the barrier at a specific point. By contrast, ifwe wish to compute only the penetrability, both expres-sions are equally valid.The last step would be to consider the effect of thedeformation on the quantum term S (1) in Eq. (2.18), butthis proves to be quite small compared to what we havediscussed already so we omit the correction, keeping onlythe spherical part S (1) ( r ) ≈ S (1)0 ( r ). The procedure is the same, the derivative with respect to θ is neglected in the S term and the integration is carried out radially. Thisapproximation performs rather well as we will show inthe following section.We turn our focus on the AWKB method. In the firstfew paragraphs of this chapter we claimed it is more el-egant than the one of Fr¨oman and now we will providesome arguments. In order not to repeat all the equationswe refer the reader to the system from Eq. (2.18). If wedo not attempt to linearlize this equation, the only waytowards a ”semi-analytic” expression is again radial in-tegration. We set ∂S (0) /∂θ = 0, but this is not enough.We do not retrieve in this way the angular momenta enu-meration, hence we still have to separate the sphericalcontribution. We can do this by writing K ( r ) = K ( r ) + ∆ K (AWKB) ( r ) , (2.28)where we have defined∆ K (AWKB) ( r ) = 2 µV d ( r ) . (2.29)We now use the definition of S (0) from Eq. (2.22) withoutneglecting any term to write( ∇ S (0)0 ) +( ∇ D ) +2( ∇ S (0)0 )( ∇ D ) = − K − ∆ K (AWKB) . (2.30)As per Eq. (2.9), ( ∇ S (0)0 ) = − K , and if we set ∂D/∂θ = 0 again, we obtain (cid:18) ∂D∂r (cid:19) + 2 ∂S (0)0 dr ∂D∂r = − ∆ K (AWKB) . (2.31)The derivative of S (0)0 with respect to r is given inEq. (2.25) and we solve this quadratic equation for dD/dr as ∂D∂r = ∓ iK ,L ± i q K ,L + ∆ K (AWKB) ≡ ∓ iK ,L ( r ) ± iK L ( r ) , (2.32)where we have denoted K L ( r ) = vuut µE V E − (cid:0) L + (cid:1) k r + V d ( r ) E ! . (2.33)Upon integrating the last equation, we retrieve the well-known (but not proved) inclusion of the centrifugal po-tential in the 3D WKB exponent i ~ D ( r, θ ) = − i ~ S (0)0 ( r ) + 1 ~ Z r r dr ′ K L ( r ′ , θ ) , (2.34)for some radius r > r where the function is known.So it turns out that the ”mixed” representation wherethe centrifugal term is included a priori is actually lessapproximate than Fr¨oman’s method, at least in principle.Now, regarding the second term in the expansion, in thiscase it is given as an extension of the spherical case S (1) ( r, θ ) = i p K L ( r, θ ) . (2.35)This result follows if one integrates radially the origi-nal system of equations with the centrifugal potentialinserted in the exponent as described above. C. Propagator method
Now, since we have build the wave-functions at all co-ordinates ( r, θ ), we can compare the WKB results withthe exact coupled channels (CC) one. To achieve this wemust build the fundamental matrix of solutions in theWKB case. The exact CC fundamental matrix of solu-tions is defined by the following asymptotics [1] H ( CC ) LL ′ ( r ) → r →∞ H (+) L ( kr, χ ) δ LL ′ , (2.36)in terms of the outgoing Coulomb-Hankel spherical waves H (+) L ( χ, kr ) = G L ( χ, kr ) + iF L ( χ, kr ). Thus, each col-umn of the fundamental matrix of solutions is obtainedby integrating backwards the coupled system of differen-tial equations, starting with above mentioned asymptoticvalue. Notice that inside the Coulomb barrier this ma-trix has practically real values, due to the fact that hereone has G L ( χ, kr ) >> F L ( χ, kr ). Therefore in practicalcalculations one uses only the irregular Coulomb wave atlarge distance. The general solution with a given angularmomentum is built as a superposition of columns f L ( r ) = X L ′ H ( CC ) LL ′ ( r ) N L ′ → r →∞ N L H (+) L ( kr, χ ) . (2.37)This expression can be used to find scattering amplitudes N L in terms of components of the internal function atsome radius r inside the barrier by using the matchingcondition f ( int ) L ( r ) = f L ( r ) N L = 1 H (+) L X L ′ K ( CC ) LL ′ ( r ) f ( int ) L ′ ( r ) , (2.38)where we introduced the propagator matrix [1] as follows K ( CC ) LL ′ ( r ) ≡ H (+) L ( χ, kr ) h H ( CC ) LL ′ ( r ) i − ≈ G L ( χ, kr ) h H ( CC ) LL ′ ( r ) i − , (2.39)with the following property K ( CC ) LL ′ ( r ) → V d → δ LL ′ , (2.40)which takes place for a sperical interaction, or for a de-formed interaction at large distance where it becomesspherical.We observe from Eqs. (2.27) and (2.34) that the com-plete wave-function in both cases can be written asΨ( r, θ ) = X L ψ L ( r, θ ) Y L ( θ ) , (2.41) where ψ L ( r, θ ) are built up using the WKB functions ψ L ( r, θ ) = exp (cid:26) i ~ (cid:16) S (0) ( r, θ ) + ~ S (1) ( r, θ ) (cid:17)(cid:27) , (2.42)with both S (0) and S (1) depending on L , as we haveshown. Then, the radial components of the completewave-function are given byΨ L ( r ) = X L ′ Z d Ω Y L ψ L ′ ( r ) Y L ′ , (2.43)which can readily be translated to the fundamental ma-trix with the asymptotics (2.36) as H ( r ) ≡ H L,L ′ ( r ) = Z d Ω Y L ψ L ′ ( r ) Y L ′ . (2.44)We now particularize Eq. (2.44) in the AWKB andLWKB approaches. For the AWKB approximation, wehave H (AWKB) L,L ′ ( r ) = Z d Ω Y L (Ω) Y L ′ (Ω) × exp ( k Z r ,L ( θ ) r dr ′ K L ′ ( r ′ , θ ) ) , (2.45)where r ,L ( θ ) are the angle dependent external turningpoints, i.e. the largest root of the equation V ( r ) E − V d ( r, θ ) E + (cid:0) L + (cid:1) k r = 0 , (2.46)at each angle. For LWKB approach, we can isolate thespherical contribution and write H (LWKB) L,L ′ ( r ) = G L ′ ( r ) × Z d Ω Y L (Ω) Y L ′ (Ω) exp (cid:20) i ~ D ,L ( r, θ ) (cid:21) (2.47)where, according to Eq. (2.27) D ,L ( r, θ ) ≡ − k R r ,L r dr ′ ∆ K ( r ′ ,θ ) K ,L ( r ′ ) , (2.48)is the deformed part of the exponential dependence gen-erating the fundamental matrix of solutions. Here, G L ( r )is the solution of the spherical problem and given by (seeappendix) G L ( r ) = 1 p K ,L ( r ) exp (cid:20)Z r ,L r dr ′ K ,L ( r ′ ) (cid:21) , (2.49)and r ,L is the spherical external turning point, i.e. thelargest solution of the equation V ( r ) E − (cid:0) L + (cid:1) k r = 0 . (2.50)We note here that a somewhat similar treatment hasbeen made by Stewart et al in [5] although the centrifugalpotential is introduced ad hoc , unlike in the AWKB ap-proach of our paper. We also mention that in [6] the an-gular momentum dependence of the deformed correctionis more approximate, while here we account for it com-pletely. More precisely, the exponent in the deformedcorrection of Fr¨oman’s original work contains the ratio∆ K/K ( r ), but our LWKB treatment gives the rigorousangular momentum dependence of the deformed term.A close inspection reveals that the LWKB deformedterm in the fundamental matrix Eq. (2.47) can be re-garded as a matrix which becomes unity in the case of0 deformation. In order to compare the two approxima-tions (with each other and with the CC equivalent), wehave to force the spherical part in the AWKB method.This is done by defining G L,L ′ ( r ) ≡ G L ( r ) δ L,L ′ , (2.51)where G L are the solutions of the spherical problem (2.2).With this definition we can impose (in matrix form) H (AWKB) ≡ G ∆ H (AWKB) , (2.52)from which we get the deformed term ∆ H as∆ H (AWKB) L,L ′ = 1 G L H (AWKB) L,L ′ . (2.53)A similar expansion can be performed for the CC fun-damental matrix, but with G (CC) L and H (CC) , the exact spherical wave function for channel L and the exact de-formed fundamental matrix respectively∆ H (CC) L,L ′ = 1 G (CC) L H (CC) L,L ′ . (2.54)For LWKB method, where the spherical term is alreadyseparated, we have∆ H (LWKB) L,L ′ = Z d Ω Y L (Ω) Y L ′ (Ω) exp (cid:20) i ~ D ,L ( r, θ ) (cid:21) . (2.55)Notice that the propagator matrix (2.39) in all cases isgiven by the obvious relation K LL ′ = ∆ H − LL ′ . (2.56)We could also perform a more symmetric decompositionof the AWKB and CC fundamental matrices by definingthe matrix (we drop the AWKB and CC indexes in thereminder of this section) G L,L ′ ( r ) = p G ( r ) 0 0 ... p G ( r ) 0 ...... ... ... ... , (2.57)with which the deformed fundamental matrices can bewritten as H = G ∆ H G . (2.58) We invert the above equation and perform the sumsin the matrix multiplication to obtain the analogs ofEqs. (2.53,2.54) in the form∆ H L,L ′ = 1 √ G L G L ′ H L,L ′ . (2.59) III. NUMERICAL RESULTS
In this section we compare the two approximationswith the exact solution given by the CC method. Wealso perform a systematic analysis of alpha decays fromeven-even emitters within the deformed WKB approach.
A. Coupled channels approach versus WKB θ = o θ = o Q = λ = λ = λ = - - - - V λ ( M e V ) r ( fm ) FIG. 2. Realistic alpha-daughter double-folding potentialversus radius for the system
T h + α , plotted by a solidline for θ = 0 o and θ = 90 o . The multipoles are given bydotted ( λ =0), dashed ( λ =2) and dot-dashed lines ( λ =4). Thehorizontal line corresponds to the Q -value of the emissionprocess U → T h + α Q=E=4.270 MeV.
The realistic cluster-core interaction, given by thedouble-folding procedure [7–9], is plotted in Fig. 2 ver-sus radius for the binary deformed system
T h + α with a quadrupole deformation β =0.215. The two solidcurves correspond to θ = 0 o and θ = 90 o , respectivelly.The multipoles in Eq. (2.14) are given by dotted ( λ =0),dashed ( λ =2) and dot-dashed lines ( λ =4).First we compare the amplitudes at the matching ra-dius following the recipe in [5] for the U nucleuswith scattering amplitudes (normalized to unity) N = {√ . , √ . , √ . } . By using the expression of thetotal decay width [1]Γ = X L = even Γ L = ~ v X L = even | N L | , (3.1)and (3.2) one can estimate the wave-function amplitudesat any point rf L ( r ) = X L ′ H LL ′ ( r ) r Γ L ′ ~ v . (3.2)and must be normalized to unity as f L ( r ) → f L ( r ) P L ′ | f L ′ ( r ) | . (3.3)We present the resulting amplitudes at R m = 12 . TABLE I. R m = 12 . L Stewart et al
LWKB AWKB0 +0 .
83 +0 .
83 +0 . − . − . − . − . − . − . ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ β = ○ CCLWKBAWKB ( a ) × ψ L ( r , θ ) L = ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ - - L = ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ - - L = ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ β = ( b ) × ψ L ( r , θ ) ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ - - ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ - - ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ β = ( c ) × ψ L ( r , θ ) ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ - - ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ - - ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ β = ( d ) × ψ L ( r , θ ) θ ( deg ) ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ - - θ ( deg ) ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ - - θ ( deg ) FIG. 3. The L = 0 , , β = 0 . β = 0 . H . We have analyzed the accuracy of LWKB and AWKBapproximations with respect to CC values. The resultsare given in Fig 3, where we plotted the CC wave functioncomponents for L = 0 , , β =0 . β = 0 . β increases, we see that LWKB ap-proximation gives a reasonable relative error σ X ( r ) = vuuuut P LL ′ h H ( X ) LL ′ ( r ) − H ( CC ) LL ′ ( r ) i P LL ′ h H ( CC ) LL ′ ( r ) i ,X = LW KB , AW KB , (3.4)about σ LW KB ( r B ) ∼
3% for β = 0 . σ AW KB ( r B ) ∼ β = β = β = ( a ) LWKB σ
15 20 25 30 r ( fm ) ( b ) AWKB
15 20 25 30 r ( fm ) FIG. 4. (a) The relative error versus radius of the LWKBfundamental system of solutions versus its coupled channelscountert. (b) Same as in (a) but for AWKB approach.
B. Approximated interaction potential
The region between the internal turning point and thebarrier maximum can be approximated with a good ac-curacy by an inverted parabola V ( r ) − E = ( V B − E ) " − (cid:18) r B − rr B − r (cid:19) ≡ V frag (1 − x ) , (3.5)in terms of the fragmentation potential V frag = V B − E , (3.6)and dimensionless coordinate x = r B − rr B − r , (3.7)where r denotes the internal turning radius. The har-monic oscillator (ho) frequency parameter of the invertedparabola is given by ~ ω = 1 r B − r r V frag d α , (3.8)in terms of the kinetic alpha-particle parameter d α = 2 µ α ~ ≈ . M eV − f m − . (3.9)Our previous analysis has shown that ~ ω ∼ V C ( r ) = 2 Z D /r at the barrier maxi-mum r B , because the difference with respect to the exactvalue is very small V B = 0 . V C ( r B ).The scattering amplitude is given at the barrier radiusby using (2.38) N L = 1 G L ( χ, ρ B ) X L ′ K LL ′ ( r B ) f ( int ) L ′ ( r B ) , (3.10)where the WKB estimate of the internal wave-function isgiven at the barrier radius r B by the Heel-Wheller ansatz[10] f ( int ) L ( V frag ) = √ p L C N,L (cid:18) EV frag (cid:19) exp (cid:16) − S (0) N (cid:17) , (3.11)in terms of the spherical nuclear action S (0) N = πV frag ~ ω , (3.12)and nuclear centrifugal term, given by the binomial ap-proximation as follows C N,L = exp "(cid:18) L + 12 (cid:19) d d = δ k s EV frag (cid:20) δ ∆ r B + r B ∆ (cid:18) arctan δ ∆ − π (cid:19)(cid:21) δ = r B − r , ∆ = q r B − δ . (3.13)The factor p L is called alpha-formation probability, whichcan be determined by experimental channel widths.Let us point out that we can use the potential, definedby (3.5), not only for the spherical part, but also for adeformed potential with V B = V B ( θ ) being the maximumbarrier height along the angle θ , r B = r B ( θ ) its positionand r = r ( θ ) the internal turning radius, which linearlydepend upon the quadrupole deformation r a ( θ ) = r a, [1 + b a β Y , ( θ )] , a = B, . By using this ansatz we can easily estimate the deformedpart if the internal action D N ( θ ) defined by Eq. (2.48).The WKB estimate of the Coulomb spherical multipolein (3.10) is given by G L ( χ, ρ ) = C C,L (cot α ) exp (cid:16) S (0) C (cid:17) , (3.14)in terms of the spherical Coulomb action S (0) C = χ (cid:18) α −
12 sin 2 α (cid:19) , (3.15) and Coulomb angular momentum term, given in a stan-dard way by the binomial approximation C C,L = exp "(cid:18) L + 12 (cid:19) c c = tan αχ . (3.16)Here, we introduced the following parametercos α = ρχ = EV C ( r ) . (3.17)Notice that the above semiclassical estimate, valid for apure Coulomb potential, gives 3% accuracy with respectto the exact function around the barrier region.In order to estimate the fundamental and propagatormatrix we used the separable LWKB approach (2.55).A simplified form is given by the Fr¨oman approach(FWKB) [6], which neglects the centrifugal barrier in(2.48) and uses a sharp density distribution at the nu-clear surface R = R [1 + β Y ( θ )]. The result is pro-portional to the quadrupole deformation parameter andLegendre polinomial i ~ D ( F W KB ) C ( θ ) = − β B ( χ, ρ ) P (cos θ ) B ( χ, ρ ) ≡ χ √ π sin 2 α (cid:0) α (cid:1) . (3.18)Thus, the deformed part of the fundamental matrix(2.55) within Fr¨oman approach is given by∆ H ( F W KB ) LL ′ ( β , χ, ρ ) = Z − d cos θ P L (cos θ ) P L ′ (cos θ ) × exp [ − β B ( χ, ρ ) P (cos θ )] , (3.19)in terms of normalized Legendre polinomials P L (cos θ ) = r L + 1 P L (cos θ ) . (3.20)Therefore the Fr¨oman propagator matrix (2.56) is givenby K LL ′ ( β , χ, ρ ) = h ∆ H ( F W KB ) LL ′ ( β , χ, ρ ) i − = ∆ H ( F W KB ) LL ′ ( − β , χ, ρ ) . (3.21) C. Alpha decay systematics
We analyzed available experimental decay widths con-cerning alpha-transitions from 168 the ground state ofeven-even emitters with J i = 0 to final states with J f = L = 0 , , , ... . In Fig. 5 (a) are given the val-ues of the reduced radius versus the Coulomb parameterat the barrier radius r B = 1 . A / + 4 / ) by usingtransitions between ground states [10]. In the panel (b) ■ ■ ■■ ■■ ■■ ■■ ■ ■■ ■ ■■ ■ ■■ ■ ■■ ■ ■ ■■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■■■ ■ ■ ■ ■ ■ ■ ■ ■ ■■■■ ■ ■ ■■ ■ ■ ■ ■ ■ ■■■■■ ■ ■ ■ ■■ ■ ■■■■■■ ■ ■ ■ ■ ■■■■■■ ■ ■ ■ ■ ■ ■ ■■■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■■ ■■ ■ ■ ■■ ■■ ■■■■■ ( a ) ρ B ■ ■ ■■ ■■ ■■ ■■ ■ ■■ ■ ■■ ■ ■■ ■ ■■ ■ ■ ■■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■■■ ■ ■ ■ ■ ■ ■ ■ ■ ■■■■ ■ ■ ■■ ■ ■ ■ ■ ■ ■■■■■ ■ ■ ■ ■■ ■ ■■■■■■ ■ ■ ■ ■ ■■■■■■ ■ ■ ■ ■ ■ ■ ■■■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■■ ■■ ■ ■ ■■ ■■ ■■■■■ ( b ) α B ( d e g )
30 35 40 45 50 55 60 65 χ B FIG. 5. (a) Reduced radius ρ B as a function the Coulombparameter at the barrier radius for alpha-transitions betweenground states on even-even nuclei [10]. (b) Same as in (a),but for the angle α B . ■ ■ ■■ ■■ ■■ ■■ ■ ■■ ■ ■■ ■ ■■ ■ ■■ ■ ■ ■■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■■■ ■ ■ ■ ■ ■ ■ ■ ■■■■■ ■ ■ ■■■■ ■ ■ ■■■■■■ ■ ■ ■ ■■ ■■■■■■■ ■ ■ ■ ■ ■■■■■■ ■ ■ ■ ■ ■ ■ ■■■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■■ ■■ ■ ■ ■■■■ ■ ■ ■■ ■■ ■ ■ ■■ ■■ ■■■■■ - d / c
25 35 45 55 65 χ B FIG. 6. Ratio between centrifugal temrs − d/c versusCoulomb parameter. we plotted the corresponding angle α B defined by Eq.(3.17). They span the following intervals (except oneisolated point) χ B ∈ [33 , ρ B ∈ [5 , α B ∈ [50 o , o ] . (3.22)The last interval corresponds to a ratio between Q -valueand the height of the Coulomb barrier EV B ∈ [0 . , . , (3.23)proving that the WKB approximation is very good forthis kind of emission processes.We analyzed the contribution of nuclear (3.13) andCoulomb centrifugal factors (3.16). ¿From Fig. 6 wenotice that the nuclear term is much smaller that itsCoulomb counterpart − d/c ∈ [0 . , . , (3.24) ○ ○ ○○ ○○ ○○ ○○ ○ ○○ ○ ○○ ○ ○○ ○ ○○ ○ ○ ○○ ○ ○ ○ ○ ○○ ○ ○ ○ ○ ○ ○○ ○ ○○ ○ ○ ○ ○ ○ ○ ○ ○ ○○ ○ ○ ○ ○ ○ ○ ○ ○○ ○ ○ ○ ○ ○ ○○○ ○ ○ ○ ○ ○ ○ ○ ○○○○○ ○ ○ ○○○ ○ ○ ○ ○○○○○○ ○ ○ ○ ○○ ○○○○○○○ ○ ○ ○ ○ ○○○○○○ ○ ○ ○ ○ ○ ○ ○○○ ○ ○ ○ ○ ○ ○ ○ ○○ ○ ○ ○ ○ ○ ○○ ○○ ○ ○ ○○○○ ○ ○ ○○ ○○ ○ ○ ○○ ○○ ○○○○○■ ■ ■■ ■■ ■■ ■■ ■ ■■ ■ ■■ ■ ■■ ■ ■■ ■ ■ ■■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■■■ ■ ■ ■ ■ ■ ■ ■ ■ ■■■■ ■ ■ ■■■ ■ ■ ■ ■■■■■■ ■ ■ ■ ■■ ■■■■■■■ ■ ■ ■ ■ ■■■■■■ ■ ■ ■ ■ ■ ■ ■■■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■■■■ ■ ■ ■■ ■■ ■ ■ ■■ ■■ ■■■■■ ○ S C ■ S N S ( )
30 40 50 60 χ B FIG. 7. Coulomb action S (0) C (open symbols) nuclear ac-tion S (0) N (dark symbols) versus Colomb parameter for alpha-transitions between ground states of even-even nuclei. and therefore it can be neglected.We then compared the Coulomb to the internal nuclearaction terms. First of all we notice from Fig. 7 thatthe spherical Coulomb term, plotted by open symbols, ismuch larger than the nuclear one, given by dark symbols S (0) N ∼ << S (0) C ∈ [12 , . (3.25) β = β = β = β = β = S N ( θ ) S C ( θ ) S N ( θ ) , S C ( θ ) θ ( deg ) FIG. 8. Nuclear action (3.26) versus θ within LWKB ap-proach at the barrier radius r B for β = 0 (dots) and for β = 0 . Thus, in Fig. 8 the lower plot gives the nuclear action S N ( θ ) = S (0) N + D N ( θ ) , (3.26)within LWKB approach at the barrier radius r B for β = 0 by dots and for β = 0 . S C ( θ ) = S (0) C + D C ( θ ) , (3.27)Notice that the Fr¨oman FWKB estimate for the Coulombaction (3.18), plotted by a dashed line, gives close values. χ = ρ = χ = ρ = R ( β , χ , ρ ) - - - - β FIG. 9. Ratio between diagonal and all propagator matrixelements R ( β , χ, ρ ) (3.43) versus quadrupole deformation foran alpha decay with average parameters χ = 45 , ρ = 10(lower dashed line) and proton emission with χ = 20 , ρ = 2(upper solid line). Therefore it turns out that the angular dependenceis practically given by the Coulomb terms, due to thefact the the nuclear part is practically constant in thisscale S N ( θ ) ∼ S (0) N ( θ ). Thus, the largest internal func-tion computing scattering amplitudes in Eq. (3.10) ispractically monopolar | f ( int )0 | >> | f ( int ) L | , L = 2 , , ... and therefore one obtains the following estimate N L ≈ f ( int )0 ( V frag ) G L ( χ B , ρ B ) K L ( β , χ B , ρ B ) . (3.28)Indeed, by using Eq. (3.2) with realistic channel decaywidths, it turns out that | f ( int ) L | < − | f ( int )0 | , L =2 , ,
6. Until now we analyzed transitions between groundstates. Deformation effects are probed by the analysis ofthe fine-structure revealed by transitions to excited statesin the daughter nucleus. We neglected in our formalismthe contribution of the daughter dynamics. The emit-ted alpha-particle with angular momentum L is coupledwith the same angular momentum of the daughter nu-cleus to the initial spin J i = 0. Thus, in each channelthe energy is replaced by E → E − E L , where E L is the excitation energy of the daughter nucleus [1]. Aswe already mentioned, by neglecting non-diagonal Corio-lis matrix elements in the intrinsic system of coordinates,the decoupled system of equations at large distances (2.4)becomes formally the same [1, 6], but the Coulomb pa-rameter and reduced radius for each channel are given byEqs. (2.15). At the barrier radius these relations become χ L = χ B ǫ L ρ L = ρ B ǫ L ǫ L ≡ r − E L E , (3.29)where the values χ B , ρ B are the barrier values for L = 0.Thus, the total decay width (3.1) becomes a superposi-tion of channel decay widths as followsΓ = X L = even Γ L = X L = even ~ v L | N L | , (3.30)in terms of the channel velocity v L = s Eµ ǫ L , (3.31)where the scattering amplitude (3.28) is replaced by N L = f ( int )0 ( V Lfrag ) G L ( χ L , ρ L ) K L ( β , χ B , ρ B ) . (3.32)Thus, each channel decay width (3.30) for transitionsfrom the ground state with J i = 0 to final states with J f = L becomes factorizedΓ L = Γ (0) L ( χ L , ρ L ) D L ( β , χ L , ρ L ) , (3.33)into a spherical ”monopole”Γ (0) L ( χ L , ρ L ) = ~ v L " f ( int )0 ( V Lfrag ) G ( χ L , ρ L ) = ~ v L p exp h − (cid:16) S (0) C ( χ L , ρ L ) + S (0) N ( V Lfrag ) (cid:17)i , (3.34)in terms of the channel fragmentation potential V Lfrag = V B − ( E − E L ) = V frag + E L , (3.35)and centrifugal-deformation factor D L ( β , χ L , ρ L ) = exp (cid:20) − α L χ L L ( L + 1) (cid:21) × K L ( β , χ B , ρ B ) , (3.36)induced by the deformed Coulomb field. Here we usedthe exact quantum expression L ( L + 1) due to the factthat the alpha-decay fine structure involves low values ofthe angular momentum.One can also factorize the ”monopole” decay widthΓ (0) L ( χ L , ρ L ) = γ ( V Lfrag ) P ( χ L , ρ L ) , (3.37)1 ● ● ● ●●● ●● ●●●●● ● ● ●● ● ●● ● ● ● ●● ● ● ● ●●●●●● ● ● ● ● ●●●●●●●● ●●●● ●●● ●●●●● ● ● ●● ●● ● ● ● ●● ● ● ● ●●●● ● ● ● ● ● ●●●●●●●● ●●●●●●●●● ●● ●●●●● ●●● ●●●● ●●○ ○ ○ ○○○ ○○ ○○○○○ ○ ○ ○○ ○ ○○ ○ ○ ○ ○○ ○ ○ ○ ○○○○○○ ○ ○ ○ ○ ○○○○○○○○ ○○○○ ○○○ ○○○○○ ○ ○ ○○ ○○ ○ ○ ○ ○○ ○ ○ ○ ○○○○ ○ ○ ○ ○ ○ ○○○○○○○○ ○○○○○○○○○ ○○ ○○○○○ ○○○ ○○○○ ○○▲ ▲ ▲ ▲▲▲ ▲▲ ▲▲▲▲▲ ▲ ▲ ▲▲ ▲ ▲▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲▲▲▲▲▲ ▲ ▲ ▲ ▲ ▲▲▲▲▲▲▲▲ ▲▲▲▲ ▲▲ ▲ ▲▲▲ ▲▲ ▲ ▲ ▲▲ ▲▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲▲▲ ▲▲▲▲ ▲▲▲▲▲ ▲▲ ▲▲▲▲▲ ▲▲▲ ▲▲▲▲ ▲▲ ( a ) K L ( β χ , ρ ) ● ● ● ●●● ●● ●●●●● ● ● ●● ● ●● ● ● ● ●● ● ● ● ●●●●●● ● ● ● ● ●●●●●●●● ●●●● ●●● ●●●●● ● ● ●● ●● ● ● ● ●● ● ● ● ●●●● ● ● ● ● ● ●●●●●●●● ●●●●●●●●● ●● ●●●●● ●●● ●●●● ●●○ ○ ○ ○○○ ○○ ○○○○○ ○ ○ ○○ ○ ○○ ○ ○ ○ ○○ ○ ○ ○ ○○○○○○ ○ ○ ○ ○ ○○○○○○○○ ○○○○ ○○○ ○○○○○ ○ ○ ○○ ○○ ○ ○ ○ ○○ ○ ○ ○ ○○○○ ○ ○ ○ ○ ○ ○○○○○○○○ ○○○○○○○○○ ○○ ○○○○○ ○○○ ○○○○ ○○▲ ▲ ▲ ▲▲▲ ▲▲ ▲▲▲▲▲ ▲ ▲ ▲▲ ▲ ▲▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲▲▲▲▲▲ ▲ ▲ ▲ ▲ ▲▲▲▲▲▲▲▲ ▲▲▲▲ ▲▲ ▲ ▲▲▲ ▲▲ ▲ ▲ ▲▲ ▲▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲▲▲ ▲▲▲▲ ▲▲▲▲▲ ▲▲ ▲▲▲▲▲ ▲▲▲ ▲▲▲▲ ▲▲ ( b ) D L ( β χ , ρ ) - - β FIG. 10. (a) The deformation factor K L ( β , χ B , ρ B ) versusdeformation for L = 0 (dark circles), L = 2 (open circles) and L = 4 (triangles).(b) Same as in (a), but for the centrifugal-deformation factor D L ( β , χ, ρ ) (3.36). in terms of the channel reduced width and penetrability γ ( V Lfrag ) = p exp h − S (0) N ( V Lfrag ) i P ( χ L , ρ L ) = ~ v L exp h − S (0) C ( χ L , ρ L ) i , (3.38)and therefore the channel decay width can be factorizedΓ L = γ ( V Lfrag ) P L ( β , χ L , ρ L ) , (3.39)in terms of the channel reduced width and deformed pen-etrability P L ( β , χ L , ρ L ) = P ( χ L , ρ L ) D L ( β , χ L , ρ L ) . (3.40)Our estimate has shown that the following approximation D L ( β , χ L , ρ L ) ≈ D L ( β , χ B , ρ B ) , (3.41)remains valid within 5% accuracy at β = 0 . T ∼ log " G ( r B ) f ( int )0 ( r B ) ∼ h S (0) C + S (0) N i = 2 χ (cid:18) α −
12 sin 2 α (cid:19) + πV frag ~ ω ∼ a Z √ E + b , (3.42) where a and b are constants.Let us stress on the fact that the factorized representa-tion (3.33) with (3.34) and (3.36) remains valid for any ofthe above described approximations AWKB, LWKB andFWKB. Moreover, we have shown in Fig. 8 that LWKBresults are close to the Fr¨oman FWKB approach. Inorder to point out on the deformation effect of the prop-agator matrix we plotted in Fig. 9 by the lower dashedline the following ratio R ( β , χ, ρ ) = S diag S total = s P L K LL ( β , χ, ρ ) P LL ′ K LL ′ ( β , χ, ρ ) , (3.43)for χ = 45 , ρ = 10 by using FWKB. This quantity givesan overall characteristics on the coupling between chan-nels induced by the quadrupole deformation. One seesthat the overall deformation effects are rather strong, i.e. R ∼ .
73 at β ∼ .
3. As a comparison, we plotted by theupper solid line the same ratio for proton emission cor-responding to characteristic parameters χ = 20 , ρ = 2.Notice a significantly smaller effect R ∼ .
93 at the samedeformation. One can conclude that the deformation ef-fect is mainly enhanced by the increase of the Coulombparameter χ .We then analyzed the influence of the deformation oneach channel decay width by plotting in Fig. 10 (a) thedeformation factor, i.e. the propagator matrix elementsquared K L ( β , χ B , ρ B ) multiplying the spherical decaywidth, versus deformation for L = 0 (dark circles), L = 2(open circles) and L = 4 (triangles). One clearly seesthat the deformation effect induced by the Coulomb bar-rier plays a significant role on each partial decay widthfor β > .
1, especially for the quadrupole L = 2, butalso for the monopole L = 0 channel. In the panel (b) weplotted the centrifugal-deformation factor (3.36) versusdeformation. One clearly sees that the L = 0 , HF L = log γ ( V frag ) γ ( V Lfrag ) ∼ log p p L + 2 (cid:2) S N ( V Lfrag ) − S N ( V frag ) (cid:3) log e ∼ (log p − log p L ) + E L . (3.44)We investigate the experimental hindrance factor HF L ( exp ) = γ ( exp ) γ L ( exp ) , (3.45)defined in terms of the experimental reduced width γ L ( exp ) = Γ L ( exp ) P L ( β , χ L , ρ L ) . (3.46)2 ○○○○○○○○○○○○○○○○○○ ●●●●●●●●●●●●●●●●●● □□ □□□□□ □□□□□ □□□□□□ HF HF HF l o g H F L ( e x p ) E ( MeV ) FIG. 11. Experimental hindrance factor (3.45) versus theexcitation energy of the daughter nucleus for L = 2, (opencircles) L = 4 (dark circles) and L = 6 (open squares). In Fig. 11 we plotted HF L ( exp ) versus the excitationenergy of the daughter nucleus E L , for L = 2, (opencircles) L = 4 (dark circles) and L = 6 (open squares)corresponding to 18 well deformed emitters above Pbwith measured channel decay widths. One indeed seesthat the general trend follows the linear energy depen-dence of Eq. (3.44). At the same time, notice the localstrong decrease of each log HF and log HF alongwith the increase of the excitation energy. This featureis given by the strong increase of the channel probability p L with respect to the excitation energy increase alongeach L -channel. IV. CONCLUSIONS
We compared the exact coupled channels procedureto the semiclassical approach to describe two-body emis-sion processes from deformed nuclei by using the prop-agator method. We expressed within this approach thevector of scattering amplitudes in terms of a propagatormatrix multiplied by the vector of internal radial wavefunction components divided to the vector of irregularCoulomb waves. We described in a rigorous way the 3Dsemiclassical approach, corresponding to deformed po-tentials, which leads to the exact results for the propa-gator matrix. We compared them with the much sim-pler expressions given by the AWKB and LWKB withits approximation, known as Fr¨oman method. We have shown that LWKB approach is closer than AWKB to theexact coupled-channels formalism. Each channel decaywidth becomes factorized into spherical and centrifugal-deformed terms. An analysis of deformation effects foralpha-emission from ground states of even-even nucleiwas performed. We evidenced the important role playedby deformation.
ACKNOWLEDGMENTS
This work was supported by the grant of the RomanianMinistry Education and of Research PN-18090101/2019-2021 and by the grant of the Institute of Atomic Physicsfrom the National Research – Development and Innova-tion Plan III for 2015 -2020/Programme 5/Subprograme5.1 ELI-RO, project ELI-RO No 12/2020.
Appendix: WKB wave functionand quantization in 3D
What we present in this appendix is not new by anymeans, but to the best of our knowledge, there is nocomprehensive work clearly stating all considerations in-volved in finding a proper solution for the spherical WKBsystem from Eqs. (2.9). We start by re-writing system(2.9) ~ : (cid:16) ∇ S (0)0 ( r ) (cid:17) = − K ( r ) ~ : − i S (0)0 ( r ) + ( ∇ S (0)0 ( r ))( ∇ S (1)0 ( r )) = 0 (A.1)The equation for the first order in ~ can be solved bymeans of separation of variables. We, thus, write S (0)0 ( r ) = A (0) ( r ) + B (0) ( θ ) + C (0) ( φ )and obtain h ∇ S (0)0 ( r ) i == (cid:18) dA (0) ( r ) dr (cid:19) + (cid:18) r dB (0) ( θ ) dθ (cid:19) + (cid:18) r sin θ dC (0) ( φ ) dφ (cid:19) = − µ ( V ( r ) − E ) (A.2)from here on we employ the notation K ( r ) ≡ s µE (cid:20) V ( r ) E − (cid:21) (A.3)We separate φ in the above equation and solve for C ( φ ) − (cid:18) dC (0) dφ (cid:19) = − λ φ = r sin θ "(cid:18) dA (0) dr (cid:19) + 1 r (cid:18) dB (0) dθ (cid:19) + K (A.4)3where λ φ is a separation constant, which will be deter-mined through quantization. As expected, we obtain aperiodic dependence on φ in our wave function through C (0) ( φ ) = ± λ φ φ (A.5)We now separate θ − (cid:18) dB (0) dθ (cid:19) − λ φ sin θ = − λ θ = r "(cid:18) dA (0) dr (cid:19) + K (A.6)where λ θ is another separation constant. Rearrangingthe θ part gives dB (0) dθ = ± s λ θ − λ φ sin θ (A.7)The closed form of B ( θ ) is not, at this point, of interestto us so we proceed with the last variable for which (cid:18) dA (0) dr (cid:19) = − K − λ θ r (A.8)which, upon expanding all terms, takes the familiar form dA (0) dr = ± i s µE (cid:18) V ( r ) E − λ θ r (cid:19) (A.9)Now we could address the quantization procedure. Theastute reader can already guess that if one stops hereand performs the quantization only to the first order,the separation constants would become [11] (by straightforward identification) λ φ = M ~ λ θ = (cid:18) L + 12 (cid:19) ~ (A.10)where M is the magnetic quantum number and L is theusual orbital quantum number. A full account of theabove expressions and the reason for the Langer correc-tion ( p L ( L + 1) → L + 1 /
2) [12] will be given later on.However, to increase the accuracy of the WKB methodwe have to compute also the second order contributionwhich, rigorously speaking, must enter in the quantiza-tion procedure. We notice that the second equation in(A.1) is also separable given the expression we found forthe first order contribution. We can thus write S (1)0 ( r ) = A (1) ( r ) + B (1) ( θ ) + C (1) ( φ ) (A.11)which implies (by direct substitution) i (cid:18) d A (0) dr + 2 r dA (0) dr + 1 r d B (0) dθ + cot θr dB (0) dθ (cid:19) = dA (0) dr dA (1) dr + 1 r dB (0) dθ dB (1) dθ + 1 r sin θ dC (0) dφ dC (1) dφ (A.12) We separate first the φ dependence and obtain ir sin θ (cid:18) d A (0) dr + 2 r dA (0) dr + 1 r d B (0) dθ + cot θr dB (0) dθ (cid:19) − r sin θ (cid:18) dA (0) dr dA (1) dr + 1 r dB (0) dθ dB (1) dθ (cid:19) = dC (0) dφ dC (1) dφ = γ φ (A.13)where γ φ is another separation constant and, solving for C (1) we find C (1) ( φ ) = ± γ φ λ φ φ. (A.14)where γ φ is another separation constant. We address nowthe quantization of the φ motion. The generally acceptedsemiclassical quantization is the Einstein-Brillouin-Keller(EBK) condition which reads [13]12 π I P q dq = (cid:18) n q + µ i b i (cid:19) ~ (A.15)where q is the generalized variable, P q is its associ-ated generalized momentum, n q is the standard quantumnumber for that variable, µ q , b q are Maslov indexes ( µ q is the number of conventional turning points along theintegration path and b q is the number of hard-wall turn-ing points along the integration path). The integrationpath is the path traversed by the classical particle in onecomplete period. In the case of the φ motion, the integra-tion path is [0 , π ] since this corresponds to a complete φ period and the whole range is classically allowed. Thegeneralized momentum is given by P φ ≡ ∂S∂φ = (cid:18) λ φ + ~ γ φ λ φ (cid:19) (A.16)The quantization condition reads12 π I P φ dφ = 12 π Z π dφ (cid:18) λ φ + ~ γ φ λ φ (cid:19) = n φ ~ (A.17)since the integrand and we obtain λ φ + ~ γ φ λ φ = n φ ~ (A.18)We now turn back to θ and apply a similar reasoningas we did for φdB (0) dθ dB (0) dθ + γ φ sin θ − i d B (0) dθ − i cot θ dB (0) dθ = ir (cid:18) d A (0) dr + 2 r dA (0) dr (cid:19) = γ θ (A.19)4where γ θ is another separation constant. After some re-arrangements we can write dB (1) dθ = i dB (0) dθ d B (0) dθ + i θ +1 dB (0) dθ (cid:18) γ θ − γ φ sin θ (cid:19) (A.20)with γ θ another separation constant. A discussion iscalled for here regarding the last two separation con-stants. With the constraints derived up to now, theycould take any value subject, of course, to the quantiza-tion conditions. We saw that for the φ dependence, γ φ does not make any difference aside from a phase. Theproblem, however arises for the θ dependence. If we nowtry to make the analogy with the exact result, we see that γ φ and γ θ should be set to 0. Indeed starting from thesystem of equations for the Legendre associated functions P L,M ( θ ) and the azimuth function Φ d P L,M dθ + cot θ dP L,M dθ + (cid:18) L ( L + 1) − M sin θ (cid:19) P L,M = 0 d Φ dφ = − M Φand perform the semiclassical expansion on both equa-tions independently, we see that only 2 constants arise.This implies that γ φ = γ θ = 0.In the light of the above considerations we can write dB (1) dθ = i dB (0) dθ d B (0) dθ + i θ (A.21)with the solution given by B (1) ( θ ) = i (cid:12)(cid:12)(cid:12)(cid:12) dB (0) dθ (cid:12)(cid:12)(cid:12)(cid:12) + i θ ) (A.22)Now we have to determine λ θ which is done throughthe quantization condition12 π I P θ dθ = (cid:18) n θ + µ θ b θ (cid:19) ~ (A.23)with P θ ≡ dS dθ = dB (0) dθ + ~ dB (1) dθ (A.24)In this case, the classically allowed range for θ is [ π − γ, π − γ ] where γ = arccos( λ φ /λ θ ), meaning that the inte-gration contour is 2 times this range. Moreover, there aretwo classical turning points at the end of the range with no hard-walls, hence µ θ = 2 and b θ = 0 so Eq. (A.23)becomes1 π Z π + γ π − γ dθ s λ θ − λ φ sin θ ≡ I θ = (cid:18) n θ + 12 (cid:19) ~ (A.25)because the logarithms evaluated along this contour giveno contribution (no poles inside the integration domain).To solve the integral above, we follow the approach inchapter 13 of [14], but we mention that the θ allowed re-gion is the one given above. First we change the variableusing cos θ = sin γ sin η which gives after some rearrangements I θ = λ θ π Z π − π dη sin γ cos η − sin γ sin η then, with another change of variable u = tan η the integral becomes I θ = λ θ π Z ∞−∞ du sin γ (1 + u ) (1 + u cos γ ) (A.26)= λ θ π Z ∞−∞ du (cid:18)
11 + u − cos γ u cos γ (cid:19) (A.27)= λ θ π (arctan u − cos γ arctan ( u cos γ )) (cid:12)(cid:12)(cid:12) ∞−∞ (A.28)= λ θ − λ φ (A.29)Adding the result above to Eq. (A.18) and taking intoaccount that γ φ = 0 gives λ θ = (cid:18) n θ + n φ + 12 (cid:19) ~ (A.30)which, if we denote L = n θ + n φ , can be written as λ θ = L + 12 (A.31)Now, we can solve for r and we find that dA (1) dr = i dA ( dr d A (0) dr + ir which gives A (1) ( r ) = i (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) dA (0) dr (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) + i log( r ) (A.32)We now gather all results together and obtain the WKBapproximation of the wave function for the 3D motion asa superposition of ”incoming” and ”outgoing” functions5Ψ ( r ) = Y (WKB) LM ( θ, φ ) p K ,L ( r ) (cid:26) c ( out ) L exp (cid:20)Z rr dr ′ K ,L ( r ′ ) (cid:21) + c ( in ) L exp (cid:20) − Z rr dr ′ K ,L ( r ′ ) (cid:21)(cid:27) , (A.33)where Y (WKB) L,M is the WKB approximation of the spher-ical harmonic Y L,M and r is the starting point of inte-gration. We do not give the closed form of Y L,M since itis more complicated and not useful as we can use the ex-act result. However, the reader is advised to consult ref.[15] for a complete account, or [16], for the case M = 0.We also mention here the works of Robnik [17, 18] whoattempts a general quantization to all orders under someconjecture and the work of Salasnich [19]. Both authorsshow that under some special circumstances, the quan- tum eigenvalue of the angular momentum operator canbe retrieved from semiclassical calculations.Finally, since it is helpful for the studies in this work,we give here the form of the radial part of an outgoingsolution of the spherical problem G L ( r ) = 1 p K ,L ( r ) exp (cid:20)Z r ,L r dr ′ K ,L ( r ′ ) (cid:21) , (A.34)where r ,L is the external turning point. [1] D.S. Delion, Theory of Particle and Cluster Emission ,(Springer, 2010)[2] G. Gamow, Z. Phys. , 204 (1928).[3] E.U. Condon and R.W. Gurney, Nature , 439 (1928).[4] Delion, D. S. and Liotta, R. J. and R. Wyss, Phys. Rev.C , 051301 (2015).[5] T.L. Stewart, M.W. Kermode, D.J. Beachey, N. Rowley,I.S. Grant, and A.T. Kruppa, Phys. Rev. Lett. , 36(1996).[6] P.O. Fr¨oman, Mat. Fys. Scr. Dan. Vid. Selsk. no. 3(1957).[7] G. Bertsch, J. Borysowicz, H. McManus, and W.G. Love,Nucl. Phys. A , 399 (1977).[8] G.R. Satchler and W.G. Love, Phys. Rep. , 183 (1979).[9] F. Carstoiu and R.J Lombard, Ann Phys. (NY), , 279(1992). [10] D.S. Delion and A. Dumitrescu, Phys. Rev. C ,014327 (2020).[11] L.J. Curtis and D.J. Ellis, Amer. J. Phys., (2004).[12] R.E. Langer, Phys. Rev. , 669 (1937).[13] M. Brack and R. Bhaduri, Semiclassical Physics (AvalonPublishing, 2003).[14] H. Goldstein, C.P. Poole, and J.L. Safko,
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