Subthreshold resonances and resonances in the R-matrix method for binary reactions and in the Trojan Horse method

# Subthreshold resonances and resonances in the $R$-matrix method for binary reactions and in the Trojan Horse method

## Abstract

In this paper we discuss the -matrix approach to treat the subthreshold resonances for the single-level and one channel, and for the single-level and two channel cases. In particular, the expression relating the ANC with the observable reduced width, when the subthreshold bound state is the only channel or coupled with an open channel, which is a resonance, is formulated. Since the ANC plays a very important role in nuclear astrophysics, these relations significantly enhance the power of the derived equations. We present the relationship between the resonance width and the ANC for the general case and consider two limiting cases: wide and narrow resonances. Different equations for the astrophysical S-factors in the -matrix approach are presented. After that we discuss the Trojan Horse Method (THM) formalism. The developed equations are obtained using the surface-integral formalism and the generalized -matrix approach for the three-body resonant reactions. It is shown how the Trojan Horse (TH) double differential cross section can be expressed in terms of the on-the-energy-shell astrophysical S-factor for the binary sub-reaction. Finally, we demonstrate how the THM can be used to calculate the astrophysical S-factor for the neutron generator in low-mass AGB stars. At astrophysically relevant energies this astrophysical S-factor is controlled by the threshold level keV. Here, we reanalyzed recent TH data taking into account more accurately the three-body effects and using both assumptions that the threshold level is a subthreshold bound state or it is a resonance state.

###### pacs:
26., 25.70.Ef, 24.30.-v, 24.10.-i

## I Introduction

A subthreshold bound state (which is close to threshold) reveals itself as a subthreshold resonance in low-energy scattering or reactions. Subthreshold resonances play an important role in low energy processes, in particular, in astrophysical reactions. In this paper we present new -matrix equations for the reaction amplitudes and astrophysical -factors for analysis of reactions proceeding through subthreshold resonances. We consider elastic scattering and resonant reaction for subthreshold resonance coupled with open resonance channels for single and two-level cases. All the equations are expressed in terms of the formal and observable reduced widths. The observable reduced width is expressed in terms of the asymptotic normalization coefficient (ANC). As a new result, we obtain a new equation for the connection of the ANC with the observable reduced width of the subthreshold resonance, which is coupled with a resonance channel. This equation is extremely important taking into account a crucial role of the ANC in nuclear astrophysics. We also derive equations for the Trojan Horse Method (THM) reaction amplitude, triple and double differential cross sections in the presence of the subthreshold state using a generalized -matrix method and the surface-integral method. The THM is a powerful indirect technique to treat resonances. We show that the THM can equally well incorporate the subthreshold and real resonances.

The Trojan Horse (TH) double differential cross section is expressed in terms of the on-the-energy-shell (OES) astrophysical factor, which can be contributed by both the subthreshold and resonance states. It is also demonstrated how the developed theory can be used to calculate the astrophysical S-factor of the reaction , which is a neutron generator for -processes in low mass AGB stars. A special attention is given to the subthreshold keV bound state in , which behaves as a subthreshold resonance. There are few papers where S-factors have been calculated for , but the equations were not described. Our theory, especially the new Trojan Horse (TH) equations, provide a very useful tool for experimentalists to treat resonances.

The paper is organized as follows. In Section II first we consider the single-channel, single-level elastic scattering just to demonstrate how to relate the ANC and the reduced width for the subthreshold resonance. After that the two-channel case is introduced, in which the subthreshold bound state is coupled with an open channel. The connection between the ANC and the reduced width obtained for the single-channel case is generalized for the two-channel case, when one of the channels is closed and the second one is open. In Section III we consider the resonant reaction proceeding through the subthreshold state. Generalization of the standard -matrix equations for the reaction amplitudes is presented. We also give the explicit -matrix equations for astrophysical factors obtained for different cases, which we need to analyze the reaction. To follow the results obtained in Sections II and III the reader is supposed to be familiar with the classical -matrix review lanethomas58 (). In Section IV we derive the reaction amplitude, triple and double differential cross section for the indirect TH reactions proceeding through the intermediate resonances within the framework of the generalized -matrix approach. Finally, in Section V we present the analysis of the astrophysical factor for the reaction, which is the neutron generator in the AGB stars. Throughout the paper the system of units in which is used.

## Ii Elastic scattering

### ii.1 Single-channel, single-level

First we consider the single-level, single-channel -matrix approach in the presence of the subthreshold bound state (also called subthreshold resonance). The resonant elastic scattering amplitude in the channel with the partial wave can be written in the standard -matrix form lanethomas58 ():

 Tii=−2ie−2iδhsiPi(γ(s)i)2E1−Ei−[Si(Ei)−Bi+iPi](γ(s)i)2.

Here, is the reduced width amplitude of the subthreshold bound state with the binding energy , is the mass of the particle , is the relative kinetic energy, is the -matrix energy level, is the -matrix shift function in channel , is outgoing spherical wave, is the radius connecting centers-off-mass of the particles in the channel and is the relative momentum. is the energy-independent -matrix boundary condition constant, and are the penetrability factor and the channel radius in the channel , is the hard-sphere scattering phase shift in the channel .

If we choose the boundary condition parameter , in the low-energy region where the linear approximation is valid

 Si(Ei)−Si(−ε(s)i)≈dSi(Ei)dEi∣∣Ei=−ε(s)i(Ei+ε(s)i). (2)

Then at small

 Tii=2ie−2iδhsiPi(~γ(s)i)2ε(s)i+Ei+iPi(~γ(s)i)2, (3)

which has a pole at because vanishes for . Here, is the observable reduced width of the subthreshold resonance. The observable reduced width is related to the formal -matrix reduced width as

 (~γ(s)i)2=(γ(s)i)21+(γ(s)i)2[dSi(Ei)/dEi]∣∣Ei=−ε(s)i. (4)

Determining the asymptotic normalization coefficient (ANC) as the residue in the pole of the scattering amplitude corresponding to the bound-state pole muk99 (), we get for the ANC of the subthreshold state,

 [C(s)i]2=2μiRi(~γ(s)i)2W2−η(s)i,li+1/2(2κ(s)iRi), (5)

where is the Whittaker function, and are the Coulomb parameter and the bound-state wave number of the subthreshold state , is the reduced mass of and , is the charge of nucleus . We established the relationship between the ANC and the observable reduced width earlier in muk99 (), but in the next section Eq. (5) will be generalized for two-channel case.

The observable partial resonance width of the subthreshold resonance is given by

 ~Γ(s)i(Ei) =2Pi(~γ(s)i)2 =Pi(C(s)iW−η(s)ili+1/2(2κ(s)iRi))2μiRi. (6)

Eq. (6) is of a fundamental importance. It shows that the subthreshold bound state at behaves as a resonance with the resonance width expressed in terms of ANC and the Whittaker function of this bound state taken on the border . We have considered the connection between the ANC and the reduced width amplitude for the bound states. In the next section we consider the connection between the ANC and the observable resonance width for real resonances.

### ii.2 Two-channel, single-level

Now we consider the elastic scattering in the presence of the subthreshold bound state in the channel which is coupled with the second channel . The relative kinetic energies in the channels , , and channel , , are related by

 Ef≡EbB=Ei+Q,Q=mx+mA−mb−mB>0. (7)

We assume that , that is, the channel is open for . The resonance part of the elastic scattering amplitude in the channel in the single-level, two-channel matrix approach, is

 Tii = −2ie−2iδhsi (8) × Pi(γ(s)i)2E1−Ei−∑n=i,f[Sn(En)−Bn+iPn]γ2n,

where is the formal reduced width in the channel . Note that . and are the penetrability factor and the channel radius in the channel . There are two fitting parameters, and , in the single-level, two-channel -matrix fit at fixed channel radii.

Again, we use the boundary condition and . The energy in the channel and corresponds to in the channel . Assuming a linear energy dependence of at small , we get

 Tii≈2ie−2iδhsiPi(~γ(s)i)2ε(s)i+Ei+i∑n=i,fPn~γ2n, (9)

where the observable reduced width in the channel is

 ~γ2n=γ2n1+∑t=i,fγ2t[dSt(Et)/dEt]∣∣Et=E(s)t, (10)

again noticing that and . Correspondingly, the observable partial resonance width in the channel is

 ~Γn(En)=2Pn~γ2n, (11)

with the total width .

The presence of the open channel coupled to the elastic scattering channel generates an additional term in the denominators of Eqs. (8), (9) and (10). The resonant elastic scattering amplitude in channel in the presence of the subthreshold bound state channel can be obtained from Eq. (8) by replacing .

Although the scattering amplitude vanishes at it can be extrapolated to the bound-state pole bypassing using

 Tii \lx@stackrelEi→−ε(s)i≈ 2κ(s)i(−1)li+1eiπη(s)i × RiW2−ηi,li+1/2(2κ(s)iRi)(~γ(s)i)2ε(s)i+Ei+iPf~γ2f,

where is the Coulomb parameter in the channel . Again, the ANC, as a residue in the pole of the scattering amplitude, is related to the observable reduced width of the subthreshold state by Eq. (5), in which now is determined by

 (~γ(s)i)2=(γ(s)i)21+∑t=i,fγ2t[dSt(E)/dE]∣∣E=E(s). (13)

The derivation of the connection between the ANC and the reduced width of the subthreshold resonance in the presence of an open channel is a generalization of Eq. (5) and is one of the main results obtained in this paper.

It follows from Eq. (LABEL:Tiipoleextrap2ch2) that in the presence of the open channel coupled with the channel , the elastic scattering amplitude has the bound-state pole shifted into the complex plane, i.e., .

We have established a connection between the ANC, the observable reduced width and the observable resonance width (at ) of the subthreshold resonance state. But, besides the subthreshold bound state in the channel , in Eq. (9) we have also a real resonance in the channel . In muk99 (); izvAN (); ANC () the definition of the ANC was also extended for real resonances. Here we remind the connection between the resonance width, the ANC, and the reduced width for the real resonance in the channel whose real part of the complex resonance energy is located at . The ANC for the resonance state is determined as the amplitude of the outgoing resonance wave ANC (), which is the generalization of the ANC definition for the bound state. Then the general expression connecting the observable resonance width and the ANC is izvAN ()

 C2f=2(−1)lfk(0)fρei[2δlf(k(0)f)−arctan(ρ)/2](1+ρ2)1/4+(1+ρ2)−1/4, (14)

where ,

 k(0)f =√2μf(E(R)f−i~Γf(E(R)f)/2 =k(R)f−ik(I)f (15)

is the complex resonant-state momentum in the channel , is the observable resonance width at the resonance energy. is the potential (non-resonant) scattering phase shift in the channel taken at the complex resonant momentum . Thus, if the resonance is not Breit-Wigner type (, then to calculate the ANC from the resonance width one needs to calculate the non-resonant scattering phase shift at the complex energy, which is quite far from the real -axis. It makes the quantity and, correspondingly, the ANC extremely model-dependent. Hence, it does not make sense to use the ANC for non-Breit-Wigner, i.e., for broad resonances. However, for narrow resonances from Eq. (14) we get muk99 (); izvAN ()

 C2f≈μf~Γf(E(R)f)k(R)fei[2δlf(k(R)f)−iπlf]. (16)

## Iii Resonant reactions

We present in this section equations for the reaction amplitudes proceeding through the subthreshold resonance with the standard -matrix equations generalized for the subthreshold state. Based on these amplitudes we obtain the corresponding astrophysical S-factors which can be used to analyze the experimental data obtained from direct and indirect measurements. Note that here the expressions for the astrophysical factors are written in the convenient -matrix form and can be used by experimentalists for the analysis of similar reactions proceeding through the subthreshold resonance.

Let us consider the resonant reaction

 x+A→F(s)→b+B (17)

with , proceeding through an intermediate resonance, which is a resonance in the exit channel and the subthreshold bound state in the initial channel . We assume also that , that is, the channel is open at the subthreshold bound-state pole is in the channel .

The single-level, two-channel -matrix amplitude describing the resonant reaction in which in the initial state the colliding particles and have a subthreshold bound state and the resonance in the final channel , can be obtained by generalizing the corresponding equations from Refs. lanethomas58 (); barker2000 ():

 Tfi = 2ie−i(δhsi+δhsf) × √Pfγf√Piγ(s)iε(s)i+Ei+∑n=i,f[Sn(En)−Sn(E(s)n)+iPn]γ2n.

Here we remind that and . The astrophysical factor is given by

 (19)

where is the spin of the subthreshold state in the channel , which is also the spin of the resonance in the channel , is the spin of the particle , , fm is the nucleon Compton wavelength, MeV is the atomic unit mass. All the reduced width amplitudes are expressed in MeV.

Assume now that the low-energy binary reaction (17) is contributed by a few non-interfering levels. The subthreshold resonance in the channel , which is coupled to the open channel is attributed to the first level, , while other levels with are attributed to two open coupled channels and of higher energy levels and spins . The astrophysical factor is given by

 S(Ei)(keV.b)=N∑λ=1Sλ(Ei)(keV.b), (20)
 Sλ(Ei)(keV.b)=ν2NE2N20πμie2πηi^JF(λ)^Jx^JAPfλPiλγ2fλγ2iλ(Eλ−Ei−∑n=i,f[Snλ(En)−Bnλ)]γ2nλ)2+[∑n=i,fPnλγ2nλ]2.

Here, all the quantities with the subscripts correspond to the channel and level , and are the reduced width amplitudes of the resonance in the initial and final channels, , is the energy level in the channel .

Now we consider two interfering levels, and , and two channels in each level. All the quantities related to the levels and have additional subscripts or , correspondingly. We assume that the level corresponds to the subthreshold state in the channel , which decays to a resonant state corresponding to the level in the channel . The level describes the resonance in the channel , which decays into the resonant state in the channel . The level lies higher than the level but both levels do interfere. The reaction amplitude is given by

 Tfi=−2ie−i(δhsf+δhsi)√Pf√Pi∑λτγfλAλτγiτ. (22)

Here, is the level matrix in the -matrix method,

 (A−1)λτ = (−ε(s)i−Ei)δλτ −

The corresponding astrophysical factor is

 S(ExA)(keV.b) = 20πν2NE2N^JF(s)^Jx^JA1μie2πηi (24) × PfPi∣∣∑λτγfλAλτγiτ∣∣2.

## Iv Trojan Horse

One of the most striking methods of treating low-energy resonances and subthreshold resonances is the Trojan Horse Method (THM), which is a powerful indirect technique allowing one to determine the astrophysical factor for rearrangement reactions reviewpaper (). While in direct measurements, owing to the presence of the Coulomb barrier, it is difficult or practically impossible to reach the region where the peak in the astrophysical factor is generated by the low-energy resonances or subthreshold resonance reveals itself, the THM is the only method, which allows one not only to observe the peak from these resonances at but even to trace this peak down to the subthreshold bound state at . The THM involves obtaining the cross section of the binary process (17) at astrophysical energies by measuring the Trojan Horse (TH) reaction [the two-body to three-body process ( particles)] in the quasi-free (QF) kinematics:

 a+A→s+F∗→s+b+B. (25)

The Trojan Horse particle, , which has a dominant -wave cluster structure, is accelerated at energies above the Coulomb barrier. After penetrating the barrier, the TH-nucleus undergoes breakup leaving particle to interact with target while projectile , also called a spectator, flies away. From the measured cross section of the TH reaction, the energy dependence of the astrophysical factor of the binary sub-process (17) is determined. Since the transferred particle in the TH reaction is virtual, its energy and momentum are not related by the On-the-Energy-Shell (OES) equation, that is, . The main advantage of the THM is that the penetrability factor in the entrance channel of the binary reaction (17) is not present in the expression for the TH cross section Bau86 (). It allows one to study resonant reactions (17) at astrophysically relevant energies for which direct measurements are impossible or extremely difficult to perform because of the very small value of . The second advantage of the THM is that it provides a possibility to measure the cross section of the binary reaction (17) at negative owing to the off-shell character of the transferred particle in the TH reaction.

### iv.1 Trojan Horse reaction amplitude

The details of the THM are addressed in the review paper reviewpaper (). The expression for the amplitude of the reaction (25) ( for ), which is described by the diagram of Fig. 1, in the surface integral approach and distorted wave Born approximation (DWBA) has been derived in muk2011 (). In the THM the absolute cross section of the reaction (25) is not measured and is determined by normalizing the THM cross section to available direct experimental data at higher energies. That is why it makes sense to use the plane wave approximation to get the THM amplitude. The expression for the prior form of the THM resonant reaction amplitude in the plane wave approximation was derived in reviewpaper () using the generalized -matrix approach for the three-body reactions. The derived expression is valid for the reactions proceeding through the real and subthreshold resonances and takes the form (with fixed projections of the spins of the initial and final particles)

 MMsMbMBMaMA=i2π2√1μfkfφa(psx)∑JF(s)lfliilf+li∑MxMF(s)mjimjfmlfmlie−iδhsfYlfmlf(−^kf)√Riμie−iδ(hs)i ×Y∗limli(^pi)⟨jfmjflfmlf|JF(s)MF(s)⟩⟨jimjilimli|JF(s)MF(s)⟩⟨JbMbJBMB|jfmjf⟩⟨JxMxJAMA|jimji⟩ ×⟨JsMsJxMx|JaMa⟩P−1/2liTlfli~Mli. (26)

Here

 ~Mli =jli(piRi)[[Bi−1]−Di] +2ZxZAμxA137∞∫RidriOli(ki,ri)Oli(ki,Ri)jli(pi,ri), (27)

with

 Di≡Dli(pi,Ri)=Ri∂jli(pi,ri)/∂ri∣∣ri=Rijli(pi,Ri), Bi≡Bli(ki,Ri)=Ri∂Oli(ki,ri)/∂ri∣∣ri=RiOli(ki,Ri). (28)

Note that and determine the entry and exit channels of the subreaction (17) rather than the entry and exit channels of the TH reaction (25). In Eq. (26) is the Clebsch-Gordan coefficient, is the spin (its projection) of particle , is the channel spin (its projection) in the channel and is the projection of ; the sum over projection of the spin is added formally and can be dropped because the projections of the spins of the final particles and are fixed. Also, is the Fourier transform of the -wave bound-state wave function of nucleus ; because we included the sum over we replaced by ; is the outgoing spherical wave, is the spherical Bessel function. We also remind that is the radius connecting and , is the conjugated OES relative momentum, that is, , is the reduced mass of particles and , . and are the off-shell and relative momenta, correspondingly.

We see that the THM reaction amplitude is expressed in terms of the binary sub-reaction OES amplitude in the -matrix form. We slightly modified the notation of the binary sub-reaction amplitude adding the subscripts and to underscore explicitly a dependence of the binary reaction amplitude on the and relative orbital angular momenta in the entry channel and the exit channels, correspondingly, of the reaction (17). is given by any of the resonance amplitudes derived above.

The possibility to express the TH reaction amplitude in terms of the -matrix OES amplitude is the result of using the surface integral formalism and the generalized -matrix approach for the three-body reactions muk2011 (); reviewpaper (). The most important feature of the THM providing the success of the THM as an indirect technique for the analysis of the low-energy resonances, which are not reachable by direct measurements, is the absence of the penetrability factor in the entry channels of the binary sub-reaction (17). The absence of is evident because containing in is compensated by the factor in front of .

However, we pay a price for using the three-body reaction to obtain an information about the two-body sub-reaction. Two additional factor appears in the TH reaction amplitude. The first factor is . This factor contains the logarithmic derivatives of the spherical Bessel function and the outgoing spherical wave taken at the channel radius . These logarithmic derivatives are a result of the generalized -matrix approach. In addition, contains the radial integral whose integrand behaves asymptotically (at ) as . Such an integral is not converging in the usual sense and requires a regularization to provide convergence at large . This integral was not taken into account in previous THM publications. Another important factor appearing from the consideration of the three-body TH reaction is the Fourier transform of the bound-state wave function of the TH particle , which plays an important role in the determination of TH reaction kinematics. As we have underscored, in the THM the selected loosely-bound TH particle has a dominant -wave cluster structure. It is necessary for the following reasons:
(i) The Fourier transform of the -wave bound-state wave function has a peak at . The reaction kinematics for which is called quasi-free (QF) one. In the practical application to cover some energy interval at fixed one needs to select the THM reaction events with different . The larger the smaller and, hence, the smaller the TH cross section. But for a loosely-bound TH particle the decay of with increase of is much smaller than for strongly-bound particles. Typically in the THM a variation of is carried out in the interval where is the TH particle wave number, is the binding energy of with respect to the virtual decay and is the reduced mass of and . Hence, in the THM with loosely bound TH particles one has the priviledge to deviate from the QF kinematics within the interval without loosing significantly the value of the TH cross section.
(ii) There is another important reason of choosing a loosely-bound TH particle. At the probable distances between and are . The smaller the larger . At large one can treat the outgoing particle as a spectator which causes a minimal disturbance to the binary sub-reaction (17).

The factor depends on the relative momentum . From momentum conservation we get (see diagram in Fig. 1)

 pxA=mApx−mxkAmxA, (29)

where is the momentum of the virtual particle and is the momentum of the real particle , . It is convenient to consider the system in which the TH particle is at rest, that is, . Then

 pxA=−mAks+mxkAmxA (30)

and .

There is another THM important relation: the particle is virtual. Hence, . From the momentum-energy conservation in the three-ray vertices and in the diagram in Fig. 1 we get

 ExA=p2xA2μxA−p2sx2μsx−εsx. (31)

From this equation we can conclude that always . Moreover, the binding energy of the TH particle plays an important role in decreasing allowing one to measure the TH cross section at lower energies even if the beam energy is higher than the Coulomb barrier in the initial channel of the TH reaction (25) Bau86 (); reviewpaper (). It is convenient to rewrite Eq. (31) in the system where . In this system Eq. (31) can be reduced to

 ExA=mxmxAEA+ks⋅kAmxA−k2s2μsx−εsx. (32)

### iv.2 Triple and double differential cross sections of Trojan Horse reaction

The TH triple differential cross section is given by

 d3σdΩkbBdΩksFdExA =μaAμsFμbB(2π)5ksFkbBkaA1^Ja^JA ×∑MaMAMsMbMB∣∣MMsMbMBMaMA∣∣2. (33)

We remind that and are related by Eq. (7). That is why we replaced by . For practical applications it is more convenient to use the TH double differential cross section, which is obtained by integration of the the triple differential cross section over . Using the orthogonality of the Clebsch-Gordan coefficients and integrating over we get for the TH double differential cross section

 d2σdΩksFdExA =e−2πηxAφ2a(psx)160π3ν2NE2NRxAμaAμsFksFkaA (34)

We remind that . Thus the TH double differential cross section is expressed in terms of the OES astrophysical factors . By measuring the energy dependence of the TH double differential cross section we actually measure the energy dependence of the OES astrophysical factor. We underscore again that in the THM only the energy dependence of the double differential cross section on is measured.

To extract the astrophysical factor from the TH experiment we assume, for simplicity, that in the region where direct data are available only one gives a dominant contribution. Then expressing the astrophysical factor in terms of the TH double differential cross section and introducing normalization factor of the TH astrophysical factor to the available experimental data at higher energies, at which penetrability factor is not an issue and direct measurements are available, we get the TH astrophysical factor

 Sli(ExA) =NFe2πηxAkaAksF160π3ν2NE2NRxAμaAμsF ×Pli1φ2a(psx)∣∣~Mli∣∣2d2σdΩksFdExA. (35)

Here, is the energy-independent TH normalization factor. After the NF has been determined at higher energy one can determine the astrophysical factor at lower energies using the experimental TH double differential cross section.

Assume that at low energies only one does contribute then Eq. (35) can be used to determine at lower energies. If there are two or more interfering resonances then all of them have the same . If, for example, two resonances contribute with different then one can find a region where one of these resonances dominate. Once the astrophysical factor for one of the resonances is determined, the astrophysical factor for the second one can be also determined.

## V Astrophysical factor of 13C(α,n)16O reaction

The reaction is considered to be the main neutron supply to build up heavy elements from iron-peak seed nuclei in AGB stars. At temperature K, the energy range where the reaction is most effective, the so-called Gamow window illiadis (); BK16 () is within keV with the most effective energy at keV. This reaction was studied using both direct and indirect (TH) methods. Direct data, owing to the small penetrability factor, were measured with reasonable accuracy down to keV. Data in the interval keV were obtained with much larger uncertainty davids68 (); bairhaas (); drotleff (); brune (); Harissopulos (). In the paper Harissopulos () the unprecedented accuracy of was achieved at energies keV. The dominant contribution to the