Resonance width for a particle-core coupling model with a square-well potential
aa r X i v : . [ nu c l - t h ] D ec Prog. Theor. Exp. Phys. , 00000 (19 pages)DOI: 10.1093 / ptep/0000000000 Resonance width for a particle-core couplingmodel with a square-well potential
K. Hagino , , , H. Sagawa , , S. Kanaya , and A. Odahara Department of Physics, Tohoku University, Sendai 980-8578, Japan Research Center for Electron Photon Science, Tohoku University, 1-2-1 Mikamine, Sendai982-0826, Japan Department of Physics, Kyoto University, Kyoto 606-8502, Japan RIKEN Nishina Center, Wako 351-0198, Japan Center for Mathematics and Physics, University of Aizu, Aizu-Wakamatsu, Fukushima 965-8560,Japan Department of Physics, Osaka University, Osaka 560-0043, Japan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
We derive a compact formula for the width of a multi-channel resonance state. Tothis end, we use a deformed square-well potential and solve the coupled-channels equa-tions. We obtain the S -matrix in the Breit-Wigner form, from which partial widths canbe extracted. We apply the resultant formula to a deformed nucleus and discuss thebehavior of partial width for an s -wave channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Subject Index D13,D29
1. Introduction
Much attention has been paid recently to the study of unbound states in nuclei near theneutron and proton drip lines, stimulated by the rapid progress of radioactive ion beamexperiments [1–3]. This applies both to unbound nuclei beyond the drip lines and to statesin bound nuclei above the threshold of a particle emission. A particular interest is in single-particle resonance states, which decay to a neighboring nucleus by emitting one nucleon.Such resonances also have a large impact on the r-process and rp-process nucleosyntheses.Bohr and Mottelson have derived a simple formula for the width of a single-particle resonancestate using a spherical square-well potential [4]. The formula has been used, e.g., in Ref. [5].In this paper, we extend the formula derived by Bohr and Mottelson to multi-channelresonances. That is, we discuss resonances in a particle-core system, in which several angularmomentum components are coupled together due to excitations of the core nucleus. Suchchannel coupling effects have been known to play an important role in one nucleon decays ofunbound nuclei [6–14] (see also Ref. [15] for a two-proton decay). Single-particle resonancesin a deformed potential have also been discussed, e.g., in Refs. [16–19]. In particular, thecoupling between s -wave and d -wave components often plays a crucial role in neutron-richnuclei close to the drip-line [18, 19]. The extended formula presented in this paper will providea simple estimate of partial and total widths for such multi-channel resonance states.The paper is organized as follows. In Sec. II, we consider a spherical square-well potentialand summarize the formula of Bohr and Mottelson. In Sec. III, we extend the formula tomulti-channel cases. We apply the extended formula to a deformed nucleus by taking into c (cid:13) The Author(s) 2012. Published by Oxford University Press on behalf of the Physical Society of Japan.This is an Open Access article distributed under the terms of the Creative Commons Attribution License(http://creativecommons.org/licenses/by-nc/3.0), which permits unrestricted use,distribution, and reproduction in any medium, provided the original work is properly cited. ccount the rotational excitations of the core nucleus, which results in a mixture of s and d waves. We discuss the partial width of the s -wave channel, for which a resonance may notexist in the absence of the coupling to the d -wave component. We then summarize the paperin Sec. IV.
2. Spherical square-well potential
In this paper, we consider a two-body system with a neutron and a core nucleus. Let us firstconsider a positive energy state of the system in a spherical square-well potential given by, V ( r ) = V θ ( R − r ) , (1)where V ( <
0) and R are the depth and the range of the potential, respectively. Here, θ ( x )is the step function.We shall closely follow Ref. [4] and derive the S -matrix for this potential. Writing the wavefunction for the relative motion as Ψ( r ) = u l ( r ) r Y lm (ˆ r ) , (2)where l and m are the orbital angular momentum and its z component, respectively, and Y lm (ˆ r ) is the spherical harmonics, the Schr¨odinger equation for the radial wave function, u l ( r ), reads, (cid:18) − ~ µ d dr + V ( r ) + l ( l + 1) ~ µr − E (cid:19) u l ( r ) = 0 , (3)where µ is the reduced mass and E ( >
0) is the energy. The solution of this equation withthe spherical square-well potential reads u l ( r ) = A l Krj l ( Kr ) ( r < R ) , (4)= krh ( − ) l ( kr ) − U l ( E ) krh (+) l ( kr ) ( r ≥ R ) , (5)with K = q µ ~ ( E − V ) and k = q µ ~ E . Here, A l is a constant and U l is the S -matrix, whichare given in terms of the phase shift δ l as U l = e iδ l . h ( ± ) l ( x ) is the spherical Hankel functions,which are given by h ( ± ) l ( x ) = − n l ( x ) ± ij l ( x ) using the spherical Bessel function, j l ( x ), andthe spherical Neumann function, n l ( x ) [20].From Eq. (5), one obtains U l = L l − S l + iP l L l − S l − iP l e iφ l , (6)where L l ≡ R (cid:18) ddr u l ( r ) (cid:19) r = R u l ( R ) , (7)is the logarithmic derivative of the wave function at r = R . Notice that the logarithmicderivative of the wave function is continuous at r = R , and L l can be evaluated using either q. (4) or Eq. (5). S l and P l in Eq. (6) are defined as, S l = G l ( R ) G ′ l ( R ) + F l ( R ) F ′ l ( R ) G l ( R ) + F l ( R ) , (8) P l = G l ( R ) F ′ l ( R ) − F l ( R ) G ′ l ( R ) G l ( R ) + F l ( R ) = kRG l ( R ) + F l ( R ) , (9)respectively. Here, we have followed Ref. [4] and introduced shorthanded notations definedas, F l ( r ) ≡ kr j l ( kr ) , G l ( r ) ≡ − kr n l ( kr ) , (10) F ′ l ( r ) ≡ R dF l dr , G ′ l ( r ) ≡ R dG l dr . (11)Note that S l in Eq. (8) and P l in Eq. (9) correspond to the shift function and the penetrabilityin the R-matrix method, respectively [22–25]. The phase factor e iφ l in Eq. (6) is the hardsphere scattering term (which is a part of the background S -matrix) given by e iφ l = G l ( R ) − iF l ( R ) G l ( R ) + iF l ( R ) . (12)Notice that the same formulas can be applied to a proton case by replacing the sphericalBessel and Neumann functions in Eqs. (10) and (11) with the corresponding Coulomb wavefunctions. In this case, the potential V ( r ) is given by, V ( r ) = V θ ( R − r ) + Ze r θ ( r − R ) , (13)where Z is the charge number of the core nucleus.A resonance energy, E r , may be approximately defined as the energy at which L l − S l inthe denominator in Eq. (6) vanishes, that is, L l ( E r ) − S l ( E r ) = 0. Expanding the quantity L l − S l around this energy as, L l ( E ) − S l ( E ) ∼ L l ( E r ) − S l ( E r ) − γ l ( E − E r ) = − γ l ( E − E r ) , (14)with − γ l ≡ (cid:20) ddE ( L l − S l ) (cid:21) E = E r , (15)one obtains U l = e iδ l = − i Γ l E − E r + i Γ l ! e iφ l , (16)where the resonance width Γ l is defined asΓ l ≡ P l γ l . (17)If one neglects the background phase shift, φ l , the S -matrix is U l = − E = E r , which isequivalent to δ l = π/
2. Notice that, because of the presence of the background phase shift,in general the resonance energy, E r , deviates from the energy at which the phase shift, δ l ,passes through π/
2. This is the case especially for a broad resonance, for which the deviationis significant and thus the phase shift does not cross π/ δ l = π/ r (MeV)0246 Γ ( M e V ) GamowBohr-Mottelsonapproximation
Fig. 1
The resonance widths for d -wave scattering in the mass region of A ∼
30 as afunction of the resonance energy. Here, the depth of the square-well potential, V , is variedto obtain different resonance energies, E r . The solid line is obtained with the present method,while the dashed line is obtained by seeking the Gamow state with a complex energy. Thedotted lines denote the results of the approximate formula, Eq. (22), for the resonance widthfor a square-well potential.phase shift is negligible within the resonance width. In that case, one may alternativelylocate the resonance energy as the energy at which the energy derivative of the phase shifttakes a maximum as a function of energy [17], since the background phase shift is expectedto be a slow function of energy.A care must be taken for an s -wave scattering. In this case, one would not expect to havea resonance since there is no centrifugal barrier. For a shallow square-well potential, thecondition L ( E r ) − S ( E r ) = 0 may be satisfied at some energy E r . Even in that situation,however, the background phase shift is large and the total phase shift, δ , would not showa clear resonance behavior. Especially, the phase shift would not pass through π/
2. For adeeper potential, the phase shift may pass through π/
2, but, for s -wave scattering, it isalways downwards in energy [26]. It is then misleading to call it a physical resonance state.Notice that in the present method a pole of the S -matrix is approximately obtained byneglecting the term − iP l in the denominator in Eq. (6) to determine the resonance energy, E r .This procedure is justified when the resonance width is small. Figure 1 shows a comparisonbetween the resonance widths for d -wave scattering obtained with two different methods.To this end, we take the radius parameter in the square-well potential as R = 3 .
95 fm andvary the depth parameter, V , from −
27 MeV to −
21 MeV. The mass µ is taken to be µ = 30 m N /
31, where m N is the neutron mass, in order to simulate a nucleus in the massregion of A ∼
30. The dashed line in the figure is obtained by seeking the resonance polewithout an approximation. To this end, we impose the outgoing boundary condition to theradial wave function with a complex energy, E = E r − i Γ /
2. The solid line, on the otherhand, is obtained with the present method by seeking the resonance energy which satisfies L l ( E r ) − S l ( E r ) = 0. The resonance width is then evaluated according to Eqs. (15) and (17).To this end, the logarithmic derivative, L l , is evaluated using the wave function for r < R s, L l = 1 + KR j ′ l ( KR ) j l ( KR ) , (18)from which the energy derivative of L l is computed as [4] ∂L l ∂E (cid:12)(cid:12)(cid:12)(cid:12) E = E r = − µR ~ (cid:20) − l ( l + 1) K R + S l ( S l − K R (cid:21) . (19)The energy derivative of S l in Eq. (15) is simply computed in a numerical manner.In Fig. 1, one can see that the resonance width obtained with the present method behavesqualitatively the same as that obtained with the complex energy method, although the formerwidth tends to be smaller than the latter width. See also Appendix A for a comparisonof the resonance widths obtained with the square-well potential to those with a smoothWoods-Saxon potential.An approximate formula for the resonance width, Eq. (17), has been derived in Ref. [4]using the approximate forms of the spherical Bessel and Neumann functions [21], j l ( x ) ∼ x l (2 l + 1)!! − x l + 3 + · · · ! , (20) n l ( x ) ∼ − (2 l − x l +1 − x − l + · · · ! , (21)which are valid for x ≪
1. The resultant formula reads [4], γ l ∼ ~ µR l − l + 1 ( l = 0 , kR ≪ . (22)This formula has the same dependence on the radius R and the reduced mass µ as theso called Wigner limit for the resonance width [22]. However, the underling model wavefunction is completely different. That is, the Wigner limit is obtained with a constant wavefunction, which would be valid for a non-resonant state, while Eq. (22) is obtained with awave function at a resonance condition, which has a large spatial variation. Notice that Eq.(22) is not applicable for l = 0. Even though a different form of the approximate formula canbe derive for l = 0 [4], we do not discuss it here since an s -wave hardly forms a resonance inthe single-channel problem.The resonance widths evaluated with the simple approximate formula, Eq. (22), are shownby the dotted line in Fig. 1. One can see that the approximate formula indeed works well forsmall values of the resonance energy, i.e., for narrow resonances, while the deviation fromthe exact calculation (the solid line) becomes significantly large for broad resonances.We notice that the results of the square-well potential are not sensitive to the value of thedepth parameter, V . In order to demonstrate this, Fig. 2 shows the results obtained with afixed value of V , that is, V = −
27 MeV (the filled circles). Notice that Eq. (19) and dS l /dE are functions of energy E = E r for a given value of V . We simply vary the energy, E r , in orderto evaluate the resonance width, even though the resonance condition, L l ( E r ) − S l ( E r ) = 0,may not be satisfied. The solid and the dotted lines are the same as those in Fig. 1. One cansee that the filled circles closely follow the solid line in the region shown in the figure. Theapproximate formula, Eq. (22), is independent of the depth parameter. The figure indicatesthat the resonance width is not sensitive to the value of V even outside the region in which r (MeV)0246 Γ ( M e V ) fixed V Fig. 2
The resonance width obtained by fixing the value of the depth parameter of thesquare-well potential (the filled circles) for the same setting as in Fig. 1. The meanings ofthe solid and the dotted lines are the same as in Fig. 1.the approximate formula works (that is, E r &
3. Resonance in a particle-core coupling model
Let us now extend the formula for the resonance width discussed in the previous sectionto a multi-channel case. To this end, we consider a system with a particle coupled to thequadrupole motion of a core nucleus, whose potential is given by [18, 26], V ( r ) = V θ R + R X m α m Y ∗ m (ˆ r ) − r ! , (23) ∼ V θ ( R − r ) + V R X m α m Y ∗ m (ˆ r ) δ ( R − r ) , (24)where α m is the collective coordinate for the quadrupole motion of the core nucleus. Noticethat for simplicity we have expanded the potential up to the first order of α m .We expand the total wave function asΨ α ( r ) = X α u αα ( r ) r | α i , (25)where α denotes the entrance channel (see Eq. (33) below) and the channel wave function | α i is given by | α i ≡ | ( ljI c ) IM i = X m j ,M c h jm j I c M c | IM i| φ I c M c i|Y jlm j i . (26)Here, j = l + s is the total single-particle angular momentum, I c and M c are the angularmomentum and its z -component of the core nucleus, respectively. I and M are the total angu-lar momentum of the system and its z -component, respectively, which are both conserved. n Eq. (26), | φ I c M c i and Y jlm j (ˆ r ) are the wave function for the core nucleus and the spin-angular wave function for the valence particle, respectively. When there is no spin-dependentinteraction, such as the spin-orbit interaction, the spin quantum number is conserved, andone can alternatively use the channel wave function given by | α i ≡ | ( lI c ) IM i = X m l ,M c h lm l I c M c | IM i| φ I c M c i| Y lm l i . (27)We shall use throughout the paper the general notation | α i for the channel wave functions,which denote either Eq. (26) or (27). The coupled-channels equations for the wave functions u αα ( r ) then read, (cid:20) − ~ µ d dr + V θ ( R − r ) + l α ( l α + 1) ~ µr + ǫ α − E (cid:21) u αα ( r )= − V R δ ( r − R ) X α ′ h α | X m α m Y ∗ m (ˆ r ) | α ′ i u α ′ α ( r ) (28) ≡ − δ ( r − R ) X α ′ C αα ′ u α ′ α ( r ) , (29)with C αα ′ ≡ V R h α | X m α m Y ∗ m (ˆ r ) | α ′ i . (30)Here, ǫ α is the excitation energy of the core state, | φ I c M c i . The explicit form of the couplingterm, Eq. (30), is given in Appendices B and C both for rotational and vibrational cases.The solution of the coupled-channels equations for r < R is given as, u αα ( r ) = A αα K α rj l α ( K α r ) ( r < R ) , (31)with K α = q µ ~ ( E − V − ǫ α ). Using the diagonal matrix ˜ F ( r ) defined as˜ F αα ′ ( r ) = K α rj l α ( K α r ) δ α,α ′ , (32)Eq. (31) is also written in a matrix form as u ( r ) = ˜ F ( r ) A , where u ( r ) and A are thematrices whose components are given by u αα ′ ( r ) and A αα ′ , respectively. The solution of thecoupled-channels equations for r ≥ R , on the other hand, is given by, u αα ( r ) = k α r h ( − ) l α ( k α r ) δ α,α − r k α k α U αα k α r h (+) l α ( k α r ) ( r ≥ R ) , (33)where U αα is the S -matrix and k α is defined as k α = p µ ( E − ǫ α ) / ~ . The matchingconditions of the wave functions at r = R are given by, u ( R < ) = u ( R > ) , (34) − ~ µ ( u ′ ( R > ) − u ′ ( R < )) = − Cu ( R ) , (35)where the prime denotes the radial derivative and R > and R < are defined as R + ǫ and R − ǫ , respectively, with ǫ being an infinitesimally small number. Notice that because of thedelta function in Eq. (29) the derivative of the wave functions is not continuous at r = R .When the core nucleus is excited to the channel α , the relative energy decreases by ǫ α .Such excitation is kinematically allowed only for E > ǫ α . Those channels which satisfy this ondition are referred to as open channels. For the channels with E < ǫ α , the excitations arekinematically forbidden and those channels are called closed channels. Even in this case, the virtual excitations are still possible, which can influence the dynamics of the open channels.For a closed channel, the channel wave number, k α , becomes imaginary, iκ α , and the firstterm on the right hand side of Eq. (33) diverges asymptotically. This wave function is thusunphysical. Nevertheless, the same form of wave functions can be formally employed, if thephysical S -matrix is restricted to the n × n submatrix of S , where n is the number of openchannels [28].As in the spherical case, the wave functions for r ≥ R , Eq. (33), can be solved for thematrix ˜ U , whose components are defined as ˜ U αα ′ = p k α ′ /k α U αα ′ , and one obtains (see Eq.(6)), ˜ U = ( G ( R ) + i F ( R )) − ( L > − S − i P ) − ( L > − S + i P )( G ( R ) − i F ( R )) , (36)where G ( r ), F ( r ), S , P are diagonal matrices whose diagonal components are given by G l α ( r ), F l α ( r ), S l α , and P l α , respectively (see Eqs. (8), (9), and (10)). L > is the logarithmicderivative of the wave functions [27], L ≡ R (cid:18) ddr u ( r ) (cid:19) r = R u − ( r ) , (37)evaluated with the wave functions for r ≥ R . Because of the matching conditions, Eqs. (34)and (35), L > is related to L < (that is, the logarithmic derivative evaluated with the wavefunctions for r < R ) as, L > = 2 µR ~ C + L < . (38)Notice that L < is a diagonal matrix, whose components are given by( L < ) αα ′ = ˜ F ′ l α ( R )˜ F l α ( R ) δ α,α ′ , (39)with ˜ F ′ l ( r ) ≡ R d ( ˜ F l ( r )) /dr .Noticing that L > − S + i P = L > − S − i P + 2 i P and the fact that the matrix ( G ( R ) + i F ( R )) − ( G ( R ) − i F ( R )) is diagonal with the diagonal elements given by Eq. (12), Eq. (36)can be transformed to˜ U αα ′ = e iφ lα δ α,α ′ +2 i ( G l α ( R ) + iF l α ( R )) − ( L > − S − i P ) − αα ′ P α ′ ( G l ′ α ( R ) − iF l ′ α ( R )) . (40)If we write G l ( R ) − iF l ( R ) = p G l ( R ) + F l ( R ) e iφ l and use Eq. (9), the matrix elementsof S then read, U αα ′ = e iφ lα δ α,α ′ + 2 ie iφ lα p P α ( L > − S − i P ) − αα ′ p P α ′ e iφ lα ′ . (41)We next rewrite the matrix ( L > − S − i P ) − as( L > − S − i P ) − = (1 − i ( L > − S ) − P ) − ( L > − S ) − . (42)Note that the inverse of the matrix L > − S can be written as,( L > − S ) − = cof( L > − S )det( L > − S ) , (43) here det( A ) denotes the determinant of the matrix A while cof( A ) is the cofactor matrixtransposed, that is, det( A ) = X j A ij cof( A ) ji , (44)for any i . Suppose that the determinant det( L > − S ) is zero at E = E r . Then, around thisenergy the inverse of L > − S is approximately given by,( L > − S ) − ∼ − cof( L > − S ) | E = E r d E − E r , (45)where the coefficient d is defined as (see Eq. (15)), d ≡ − ddE det ( L > − S ) | E = E r . (46)Notice that Eq. (44) indicates that when det( A ) = 0 the elements of a matrix cof( A ) isgiven in a separable form as cof( A ) ij = c i y j , where c i is a constant and the vector y satisfies Ay = 0. The matrix L > − S is a real symmetric matrix and one can choose the normalizationof the vector y such that [28] ( L > − S ) − αα ′ ∼ − γ α γ α ′ E − E r , (47)with γ α = − [cof( L > − S ) | E = E r ] ααddE det ( L > − S ) | E = E r . (48)Using Eqs. (41), (42), and (47), one finally obtains U αα ′ = e iφ lα δ α,α ′ − i √ Γ α √ Γ α ′ E − E r + i Γ tot ! e iφ lα ′ , (49)with Γ α = 2 γ α P α , (50)and Γ tot = X α Γ α . (51)This is a well known Breit-Wigner formula for a multi-channel resonance (see, e.g., Refs.[22–24, 29, 30]). One may also regard this as a special case of the R -matrix formula.Notice that a resonance width for a deformed potential, V ( r ) = V θ ( R + Rβ Y (ˆ r ) − r )[18], can be also evaluated with exactly the same formula using the channel wave functionsof | α i = | Y lK i , K being the z -component of the angular momentum, and setting all ǫ α to bezero in Eq. (28). In this case, the matrix elements of the coupling potential, Eq. (30), aregiven as C αα ′ = V R h Y lK | Y | Y l ′ K i [18].The upper panel of Fig. 3 shows the result of a two-channel calculation. For this purpose,we consider the rotational coupling with the basis given by Eq. (27) with the total angularmomentum of I = 0 (see Appendices B and C), that is, a two channel coupling between | + ⊗ s i and | [2 + ⊗ d ] ( I =0) i . The core nucleus is assumed to be Mg, and we take ǫ to bethe excitation energy of the first 2 + state, that is, 1.48 MeV. The deformation parameter β is estimated to be β = 0 .
21 by using Eq. (B10) in Appendix B with the measured B ( E B ( E ↑ = 241(31) e fm [34] (see also Ref. [35]) and the radius of R = 5 . r (MeV)012 Γ ( M e V ) s-waved-wavetotal0 1 2 3 4E (MeV)0.811.21.41.61.822.2 δ s u m / π V = -24.5 MeVV = -23 MeV (a)(b) I = 0 ε Fig. 3 (The upper panel) The resonance widths for a two-channel rotational couplingwith the total angular momentum I = 0 as a function of the resonance energy, E r . Here, thedepth of the square-well potential, V , is varied to obtain different resonance energies, E r . Thedashed and the dot-dashed lines show the partial widths for the | + ⊗ s i and | [2 + ⊗ d ] ( I =0) i channels, respectively. The threshold energy for the d -wave channel ( ǫ = 1 .
48 MeV) is alsodenoted by the arrow. The solid line shows the total width. (The lower panel) The eigenphasesums as a function of energy, E . The solid line is obtained with V = − . V = −
23 MeV. Theresonance energy and the total width are E r = 0 .
97 MeV and Γ tot = 0 .
98 MeV, respectively,for V = − . E r = 2 .
07 MeV and Γ tot = 0 .
96 MeV for V = −
23 MeV. p / β - γ spectroscopy measurement for the Mg nucleus. In the figure, the dashed andthe dot-dashed curves show the partial width for the s and d wave channels, respectively,as a function of the resonance energy, E r . The solid line shows the total width. To evaluatethese, we vary the depth parameter of the square-well potential from V = −
24 MeV to −
19 MeV. For each value of V , we first look for the resonance energy, E r , which satisfiesdet( L > ( E r ) − S ( E r ) = 0. We find that this procedure is essential in order to obtain physicalvalues of the partial widths, in contrast to the spherical case in which one can choose V somewhat arbitrarily. After the resonance energy is found in this way, the partial widthsare then evaluated according to Eqs. (50), (D9), and (D10). For E < ǫ , the d -wave channel r (MeV)0123 Γ ( M e V ) s-waved-wavetotal0 1 2 3 4E (MeV)00.20.40.60.811.21.41.6 δ s u m / π V = -25.6 MeVV = -23 MeV (a)(b) I = 2
Fig. 4
Same as Fig. 3, but for the total angular momentum of I = 2, with channels | + ⊗ d i and | + ⊗ s i . In the lower panel, the eigenphase sums are plotted for V = − . −
23 MeV. The resonance energy and the width are E r = 0 .
99 MeV and Γ tot = 0 .
17 MeV,respectively, for the former potential, while they are E r = 2 .
46 MeV and Γ tot = 0 .
98 MeVfor the latter potential. The kinks at the threshold energy for the excited channel ( E = 1 . S l =2 is evaluated bychanging k in the following formula [4] S l =2 = −
18 + 3 k R k R + k R , (52)to iκ with κ = p µ | E − ǫ | / ~ as S l =2 = − − κ R − κ R + κ R . (53)One can see that the partial width for the d -wave channel becomes dominant as the reso-nance energy increases, and thus the resonance state gradually changes to a d -wave characteras a function of energy. It is interesting to see that the resonance state exists even whenthe d -wave channel is closed at E r < ǫ . In order to see this more clearly, the lower panelof Fig. 3 shows the eigenphase sum, δ sum , defined as a sum of eigen phase shifts δ α , thatis, δ sum = P α δ α [17, 29, 30] as a function of energy, E . Here, the eigen phase shifts are efined as λ α = e iδ α , where λ α is an eigenvalue of the S -matrix. The eigenphase sum is ageneralization of the phase shift for a single-channel case, and shows a resonance behav-ior with the resonance energy E r and the width Γ tot [29, 30]. In the figure, the solid andthe dashed lines are obtained with V = − . −
23 MeV, respectively. Here, wecalculate the eigenphase sum according to the formula e iδ sum = det( U ) with the S -matrixcomputed by Eq. (41). For the former value of the potential depth, the resonance energyand the total width are found to be E r = 0 .
97 MeV and Γ tot = 0 .
98 MeV, respectively, whilefor the latter they are E r = 2 .
07 MeV and Γ tot = 0 .
96 MeV. One can see that the eigen-phase sum increases rapidly around E ∼ E r for each case, which is expected as a resonancebehavior. An interesting thing to see is that the s -wave channel shows a resonance behavioras a consequence of the coupling to the d -wave channel, even when the d -wave channel iskinetically forbidden (that is, a closed channel). Another interesting thing is that the squarewell potential with V = − . d -wave bound state. When the energy E − ǫ α for a closed channel ( E − ǫ α <
0) coincides with the energy of a bound state, thephase shift for open channels shows a resonance behavior. This is referred to as the Feshbachresonance, and plays an important role in the physics of cold atoms [37]. Since there is nobound state in the present case, the resonance behavior shown by the solid line in the lowerpanel of Fig. 3 should have a different character from a Feshbach resonance. Notice that theresonance width is a decreasing function of E r before the d -wave channel is open, i.e., for E r < .
48 MeV. This is partly due to a strong energy dependence of S given by Eq. (53),which is much stronger than the energy dependence of S for open channels given by Eq.(52).The resonance widths for the total angular momentum I = 2 are shown in the upper panelof Fig. 4. In this case, the channels | + ⊗ d i and | + ⊗ s i are coupled together. One can seethat the resonance width is similar to the single-channel case shown in the lower panel of Fig.1. That is, the d -wave channel dominates the total width with only a small contribution fromthe s -wave channel. The eigenphase sums for V = − . V = −
23 MeV are shownin the lower panel of Fig. 4. The resonance energy and the total width are E r = 0 .
99 MeVand Γ tot = 0 .
17 MeV, respectively, for the former potential, while they are E r = 2 .
46 MeVand Γ tot = 0 .
98 MeV for the latter potential. One can clearly see the expected resonancebehavior, especially for the former potential with a smaller resonance width. In addition, onecan also see a kink at the threshold energy for the excited channel, that is, at E = ǫ = 1 . s -wave due to the absence of the centrifugal barrier while the partial width for the d -wave channel is largely suppressed. This was not seen in Fig. 4 due to the threshold energy,because of which the s -wave channel contributes only at energies above the threshold. Inorder to gain a deeper insight into the role of the s -wave channel, we repeat the samecalculation as in Fig. 4, but by setting the excitation energy of the 2 + state of the corenucleus to be zero. The results are shown in Fig. 5. As expected, the partial width for the s -wave channel is larger than that for the d -wave channel, at energies below about 0.8 MeV.At very low energies, the partial width for the d -wave channel is largely suppressed dueto the finite centrifugal barrier. A similar mechanism appears in the two-neutron decay of r (MeV)012 Γ ( M e V ) s-waved-wavetotalI = 2, E + = 0 MeV Fig. 5
Same as the upper panel of Fig. 4, but obtained by setting the excitation energyof the core nucleus to be zero. O, for which the resonance energy is extremely small and the decay dynamics is largelydetermined by the s -wave component in the wave function [40, 41].
4. Summary
We have derived a compact formula, Eq. (48), for partial decay widths for a multi-channelresonance state using a deformed square-well potential. This was an extension of the formuladerived in Ref. [4] for a single-channel case to a resonance in a particle-core coupling model.We have applied the formula to a two-channel problem with s and d wave couplings. We haveshown that, even though a pure s -wave state hardly forms a resonance due to the absenceof the centrifugal barrier, a resonance may appear as a consequence of the coupling to the d -wave channel. This is the case even when the d -wave channel is closed and/or there is nobound state in the excited channel. We have also shown that the s -wave component providesa dominant contribution at low energies, whenever it is available, while the resonance changesto a d -wave character as the resonance energy increases.The resonance formula obtained in this paper is simple and semi-analytic. In particular,one does not need to solve the coupled-channels equations numerically. The formula is thususeful, at least qualitatively, in analyzing experimental data e.g., for beta-delayed neutronemissions from neutron-rich nuclei, even though several approximations used in this papermay have to be carefully examined for quantitative discussions. We will present such analysisin a separate paper. Acknowledgments
We thank M. Ichimura, I. Hamamoto, and T. Shimoda for useful discussions. This workwas supported in part by JSPS KAKENHI Grant Numbers JP16K05367, JP17J02034, andJP18KK0084.
A. Comparison between a square-well and a Woods-Saxon potentials
In this Appendix, we compare the resonance width obtained with the spherical square-well(SW) potential with that with a smooth Woods-Saxon (WS) potential. To this end, we seek Γ ( M e V ) Woods-SaxonSquare-Well r (MeV)0123 Γ ( M e V ) (a) R SW = R WS (b) R SW = 1.29 R WS Fig. A1
The resonance widths as a function of the resonance energy for d -wave scatteringin the mass region of A ∼
30. The solid and the dashed lines show the results of a Woods-Saxon and a square-well potentials, respectively. In the upper panel, the radius of the square-well potential is set to be the same as that of the Woods-Saxon potential, while it is increasedby a factor of p / d -wave Gamow state for both the potentials (see the dashed line in Fig. 1). For theWoods-Saxon potential, we take the radius and the diffuseness parameter in the Woods-Saxon as R WS = 3 .
95 fm and a WS = 0 .
67 fm [4], respectively, and vary the depth parameter, V (WS)0 , from −
29 MeV to −
14 MeV. For the square-well potential, we consider two differentchoices of the radius parameter, R . One is to take the same value as in the Woods-Saxonpotential (the upper panel), and the other is to increase the radius by a factor of p / h r i = R d r r V ( r ) R d r V ( r ) (A1)is approximately the same between the Woods-Saxon and the square-well potentials. For R = R WS , we vary the depth parameter V from −
27 MeV to −
21 MeV, while we vary itfrom −
16 MeV to − . R = p / R WS .The solid and the dashed lines in Fig. A1 show the results of the Woods-Saxon and thesquare-well potentials, respectively. One can see that the resonance width obtained with thesquare-well potential behaves qualitatively the same as that obtained with the Woods-Saxonpotential. For the increased radius parameter (the lower panel), the square well potentialreproduces even quantitatively the result of the Woods-Saxon potential, especially for small alues of the resonance width. A similar behavior can be found also in absorption crosssections, which are relevant to astrophysical fusion reactions [42]. B. Matrix elements of the coupling potential
In this Appendix, we give explicit forms of the matrix elements of the coupling potential,Eq. (30), with the channel wave functions given by Eq. (26) [31]. Using Eq. (7.1.6) in Ref.[32], one obtains V R h ( ljI c ) IM | X m α m Y ∗ m | ( l ′ j ′ I ′ c ) IM i = V R ( − ) j ′ + I c + I ( I I c j j ′ I ′ c ) hY jl || Y ||Y j ′ l ′ i h φ I c || α || φ I ′ c i , (B1)where ( I I c j j ′ I ′ c ) denotes the Wigner’s 6- j symbol. The reduced matrix element hY jl || Y ||Y j ′ l ′ i is calculated as [32], hY jl || Y ||Y j ′ l ′ i = δ l + l ′ +2 , even ( − + j √ j ˆ j ′ √ π j j ′ / − / ! , (B2)where the bracket denotes the 3- j symbol, and ˆ j and ˆ j ′ are defined as √ j + 1 and √ j ′ + 1,respectively. In order to evaluate the reduced matrix element, h φ I c || α || φ I ′ c i , one needs aspecific model for φ I c , which is given below. Notice that the E B ( E I c → I ′ c ) = 12 I c + 1 |h φ I c || ˆ T (E2) || φ I ′ c i| , (B3)with the E T (E2) m = 3 e π Z c R α m , (B4)where Z c is the proton number of the core nucleus. B.1. Vibrational coupling
We first discuss the vibrational coupling of spherical nuclei. In this case, the surface coordi-nate α m is regarded as a coordinate for a harmonic oscillator. It is related to the phononcreation and annihilation operators as [33] α m = β √ (cid:16) a † m + ( − m a − m (cid:17) , (B5)where β / √ I c = 0 andthe one phonon state with I c = 2 are given as, | φ i = | i , (B6) | φ m i = a † m | i , (B7)respectively, where | i is the vacuum state for the harmonic oscillator. From these wavefunctions, one finds h φ || α || φ i = β , (B8) nd thus [11] V R h ( ljI c = 2) IM | X m α m Y ∗ m | ( l ′ j ′ I ′ c = 0) IM i = V R β √ π (cid:28) j ′
12 20 (cid:12)(cid:12)(cid:12)(cid:12) j (cid:29) . (B9)In the vibrational model, there is no coupling from the one phonon state to the same state(that is, the reorientation term). From Eq. (B3), the value of β can be estimated from ameasured B ( E
2) strength from the ground state to the one phonon state, B ( E ↑ , as [31] β = 4 π Z c R r B ( E ↑ e . (B10)In a similar way, the wave function for the two phonon states is given as, | φ I c M c i = 1 √ a † a † ] ( I c M c ) | i , (B11)from which one finds h φ I c || α || φ I c =2 i = r I c + 1)5 β . (B12)Similar to the 1 phonon state, there is no coupling among the 2 phonon states, as well asthe coupling between the 2 phonon states and the ground state. B.2. Rotational coupling
We next consider the rotational coupling of deformed nuclei. In this case, we first transformthe surface coordinate α m to the body-fixed coordinate as a m = X m ′ D m ′ m ( ϕ d , θ d , χ d ) α m ′ , (B13)where ϕ d , θ d , and χ d are the Euler angles which specify the body-fixed frame, and D m ′ m isthe Wigner’s D -function. For axial deformation, only the m = 0 component in a m is finite.In this case, one has α m = β r π Y m ( θ d , ϕ d ) , (B14)with a = β being a deformation parameter. For the ground state rotational band in even-even nuclei, the wave function of the rotational states reads | φ I c M c i = | Y I c M c i , (B15)and one obtains h φ I c || α || φ I ′ c i = β r π h Y I c || Y || Y I ′ c i = ( − ) I c β ˆ I c ˆ I ′ c I c I ′ c ! , (B16)with h Y l || Y || Y l ′ i = ( − l √ l ˆ l ′ √ π l l ′ ! . (B17)Notice that the relation between the deformation parameter β and the B ( E
2) value is thesame as in the vibrational case, Eq. (B10). The coupling matrix element between the groundstate and the first 2 + state is also the same as in the vibrational case and is given by Eq.(B9). A difference from the vibrational coupling comes from the reorientation term, that is,the self-coupling from the 2 + state to the same state [31]. . Matrix elements with the channel wave functions of Eq. (27) In this Appendix, we give the matrix elements with the channel wave functions given by Eq.(27) instead of Eq. (26). The equation which corresponds to Eq. (B1) in Appendix B thenreads V R h ( lI c ) IM | X m α m Y ∗ m | ( l ′ I ′ c ) IM i = V R ( − ) l ′ + I c + I ( I I c l l ′ I ′ c ) h Y l || Y || Y l ′ i h φ I c || α || φ I ′ c i . (C1)The equation that corresponds to Eq. (B9) in Appendix B reads V R h ( lI c = 2) IM | X m α m Y ∗ m | ( l ′ I ′ c = 0) IM i = V R β √ π h l ′ | l i , (C2)both for the vibrational and for the rotational couplings. For the rotational coupling, thereorientation term with l = l ′ = 2, I c = I ′ c = 2 and I = 0 is given by [31] V R h [ d ⊗ + ] (00) | X m α m Y ∗ m | [ d ⊗ + ] (00) i = V R r π β . (C3)The reorientation term with l = l ′ = 0 vanishes due to the selection rule of angularmomentum. D. Two-channel coupling
When only two-channels are involved in the coupling, the matrix L > − S in Eq. (47) is a 2 × L > − S has componentsgiven by ( L > − S ) = 2 µ ~ RC + L − S , (D1)( L > − S ) = 2 µ ~ RC + L − S , (D2)( L > − S ) = ( L > − S ) = 2 µ ~ RC , (D3)where S i ( i = 1 ,
2) are given by Eq. (8) while L i ( i = 1 ,
2) are given by L i = 1 + K i R j ′ l i ( K i R ) j l i ( K i R ) . (D4)The determinant of the matrix L > − S readsdet( L > − S ) = ( L > − S ) ( L > − S ) − ( L > − S ) . (D5)Notice that the energy derivative of L i is given as ∂L i ∂E = − µR ~ " − l i ( l i + 1) K i R + 1 K i R j ′ l i ( K i R ) j l i ( K i R ) + (cid:18) j ′ l i ( K i R ) j l i ( K i R ) (cid:19) . (D6)In the spherical case, one can use the resonance condition, L i = S i , to obtain the approxi-mate formula for ∂L i ∂E [4]. In contrast, in the multi-channel case, the resonance condition is omewhat more complicated, that is, det( L > − S ) = 0, and a simple approximate formulafor ∂L i ∂E cannot be obtained.Since the elements of the cofactor matrix of the 2 × L > − S are given bycof( L > − S ) = ( L > − S ) , (D7)cof( L > − S ) = ( L > − S ) , (D8)the γ i in Eq. (48) reads γ = − µ ~ RC + L − S ddE det ( L > − S ) | E = E r , (D9) γ = − µ ~ RC + L − S ddE det ( L > − S ) | E = E r , (D10)where the numerators are evaluated at E = E r . The partial and the total widths are thenevaluated according to Eqs. (50) and (51), respectively.In the no-coupling limit, the coupling matrix C vanishes, and thus the determinant of L > − S becomes det( L > − S ) = ( L − S )( L − S ) . (D11)Suppose that L − S = 0 at E = E r . Then, the energy derivative of the determinant reads ddE det( L > − S ) | E = E r = − γ l ( L − S ) , (D12)where γ l is given by Eq. (15). This leads to γ = γ l and γ = 0, which is consistent withthe single-channel case discussed in Sec. II. References [1] M. Pf¨utzner, M. Karny, L.V. Grigorenko, and K. Riisager, Rev. Mod. Phys. , 567 (2012).[2] P.J. Woods and C.N. Davids, Ann. Rev. Nucl. Part. Sci. , 541 (1997).[3] T. Nakamura, H. Sakurai, and H. Watanabe, Prog. in Part. and Nucl. Phys. , 53 (2017).[4] A. Bohr and B.R. Mottelson, Nuclear Structure (W.A. Benjamin, Reading, MA, 1969), Vol. I, Appendix3F-2 and p. 239.[5] D.J. Millener, C.B. Dover, and A. Gal, Prog. Theo. Phys. Suppl. , 307 (1994).[6] A.T. Kruppa, B. Barmore, W. Nazarewicz, and T. Vertse, Phys. Rev. Lett. , 4549 (2000).[7] B. Barmore, A.T. Kruppa, W. Nazarewicz, and T. Vertse, Phys. Rev. C , 054315 (2000).[8] E. Maglione and L.S. Ferreira, Phys. Rev. C , 047307 (2000).[9] C.N. Davids and H. Esbensen, Phys. Rev. C , 054302 (2000).[10] H. Esbensen and C.N. Davids, Phys. Rev. C , 014315 (2000).[11] C.N. Davids and H. Esbensen, Phys. Rev. C , 034317 (2001).[12] K. Hagino, Phys. Rev. C , 041304 (2001).[13] M. Karny et al. , Phys. Rev. Lett. , 012502 (2003).[14] K. Fossez, J. Rotureau, N. Michel, Q. Liu, and W. Nazarewicz, Phys. Rev. C , 054302 (2016).[15] S.M. Wang and W. Nazarewicz, Phys. Rev. Lett. , 212502 (2018).[16] E. Maglione, L.S. Ferreira, and R.J. Liotta, Phys. Rev. Lett. , 538 (1998); Phys. Rev. C , R589(1999).[17] K. Hagino and Nguyen Van Giai, Nucl. Phys. A735 , 55 (2004).[18] K. Yoshida and K. Hagino, Phys. Rev. C , 064311 (2005).[19] I. Hamamoto, Phys. Rev. C , 024301 (2005); Phys. Rev. C , 064308 (2006); Phys. Rev. C , 054311(2008).[20] We follow Ref. [21] to use the phase convention for the spherical Neumann function, with which theasymptotic form is given by n l ( kr ) → − cos( kr − lπ/
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