79urn:lsid:arphahub.com:pub:D015427E-53F6-553C-9279-2E51CD684756Nuclear Energy and TechnologyNUCET2452-3038National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)10.3897/nucet.5.3932039320Research ArticleNuclear physics Fusion and Radiation applicationAnalytical version of the resonance coupled-channel model for D + T → ^{5}He** → α + n reaction and its application for the description of low-energy D-T and D- ^{3}He scatteringGodesAlexander I.1spartakalex46@mail.ruKudriavtsevaAnna S.2ShablovVladimir L.1Obninsk Institute for Nuclear Power Engineering, NRNU MEPhI, 1 Studgorodok, Obninsk, Kaluga reg., 240040 Russian FederationObninsk Institute for Nuclear Power EngineeringObninskRussiaNational Research Nuclear University MEPhI, 31 Kashirskoe shosse, Moscow, 115409 Russian FederationNational Research Nuclear University MEPhIMoscowRussia
Corresponding author: Alexander I. Godes (spartakalex46@mail.ru)
Academic editor: Yury Korovin
20192509201953231235349BFC7D-3974-5CDB-BAE2-A9880838F6A534704971806201917082019Alexander I. Godes, Anna S. Kudriavtseva, Vladimir L. ShablovThis is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
The purpose of the present paper is the formulation of the analytical version of the resonance coupled-channel model (RCCM) originally developed for D + T → ^{5}He^{**} → α + n nuclear fusion reaction. The integral in the denominator of the Breit-Wigner type is examined in the expression for S-matrix elements of binary processes in this model. Imaginary part of this integral determines the energy-dependent decay width for the near-threshold channel. It is demonstrated that this integral can be calculated explicitly with the Binet representation for the ψ-function (the logarithmic derivation of the gamma function). As the result the explicit expression for the S-matrix elements in the form of analytical functions of the channel momenta are obtained and the equivalence of the RCCM and the effective range approximation (Landau – Smorodinsky – Bethe approximation) is established on this basis. This allows expressing the parameters of the RCCM through the model independent system characteristics: the complex scattering length and the complex effective range. Several sets of model parameters of both approaches that provide a good description of the measured data on D + T → α + n reaction and D-T elastic scattering are derived. By this means we find the location of the S – matrix poles on different Riemann sheets which corresponds to J^{π} = (3/2)^{+} state of ^{5}He and ^{5}Li nuclei. In particular, the location of the resonance (R) and shadow (S) poles is determined:
Our results agree well with previous findings. The possible generalizations of the results obtained are discussed.
Thermonuclear reactionsresonance coupled-channel modeleffective range approximationS-matrix polesresonance and shadow polesJ^{π} = (3/2)^{+} state of ^{5}He and ^{5}Li nucleiIntroduction
Reactions containing in the final state non-stable quantum systems along with other particles serve as an important source of information about characteristics of non-stable quantum systems. This is of special importance in the cases when the system under examination is difficult or impossible to observe in binary collisions, for instance, neutron excess nuclei ^{9}He, ^{10}He, ρ^{0}, η^{0}-mesons, etc. However, a different type of problem emerges in this case, associated with the fact that the process of formation and decay of non-stable system occurs under the effects of Coulomb and nuclear fields produced by accompanying reaction products. This results in the deviation of observed parameters of the resonance from their values in the case of its isolated excitation and decay. In particular, change of shape of the resonance curve, its half-width, shift of resonance maximum position and change in the correlation between the resonance decay branches along different channels are possible with the latter constituting distinguishing feature of near-threshold resonances (Komarov et al. 1987, 1992, 1996, Fazio et al. 1996, Nemets et al. 2007, Mikhailov et al. 2014, Pavlenko et al. 2010). It was demonstrated in these studies that the above-mentioned effects are mainly associated with effects of Coulomb field of accompanying particles on the resonance decay process.
Near-threshold resonances (Fazio et al. 1996, Komarov et al. 1987), the description of which is usually made with application of the so-called approximation with energy-dependent width (Komarov et al. 1996, Wildermuth and Tang 1999, Nikitiu 1983), are of particular interest. In such case many-body scattering S-matrix either has 2l +1 poles on the complex momentum plane (l being the orbital angular momentum of the resonance) when the resonance corresponds to the pair of fragments one of which is neutral (with exception of the case of s-wave when the number of poles is equal to two), or it has the infinite number of poles when a pair of charged fragments forms the resonance. In both cases one among these poles corresponds to the resonance, the second one corresponds to the so-called “shadow” pole, and it is the presence of the latter pole, which may lead to the observed physical effects (Betan et al. 2018, Miaroshi 1980). Anomalous broadening of the resonance peak corresponding to the second excited state of ^{5}He nucleus as compared with the value equal to 70 keV when this resonance is observed in elastic (n+α) collision (Arena et al. 1989), was revealed. Authors of the cited study suggested that the discovered effect is explained by the effects of the shadow pole.
Resonance coupled-channel model and its connection with effective range approximation
Let us address the issue of determination of parameters of the resonance and shadow poles corresponding to the second exited states of ^{5}He** and ^{5}Li^{**} nuclei based on the resonance coupled-channel model for D + T → α + n reaction (Bogdanova et al. 1991). Element of scattering S-matrix S_{11} corresponding to elastic D-T scattering has the following form within the framework of the resonance coupled-channel model (Bogdanova et al. 1991):
(1)
where σ_{0}(E) is the s-wave Coulomb scattering phase; Γ_{2} is the α-n channel decay width; Γ_{1}(E) is the energy-dependent width of decay along the D-T channel: Γ_{1}(E) = –2γIm I (E). The function I (E) is preset in the following form:
(2)
where K^{2} = 2μ(E + i0)/ ħ^{2}; μ is the reduced mass of the D-T system; η(k) = e^{2}μ/(ħ^{2}k) = (ka_{c})^{–1} is the Coulomb parameter of this system; a_{c} = ħ^{2}/(e^{2}μ) is the Bohr radius for the D-T pair. Values Ε_{0}, Γ_{2}, β, γ in (1), (2) are the model parameters. It transpires from (2) that the width
where C_{0}^{2}(K) = 2πη(K)(e^{2πη(}^{K}^{)} – 1)^{–1} is the Gamow multiplier, demonstrates correct threshold behavior (Wildermuth and Tang 1999, Nikitiu 1983). Cross-section of D + T → α + n reaction is equal to:
with k_{DT} = (2μE)^{1/2}/ħ. It is possible to calculate integral I (E) represented by (2) explicitly using the Binet formula for ψ-function (logarithmic derivative of gamma-function) (Bateman and Erdelyi 1953)
(5)
which will subsequently allow directly implementing analytical extension of S-matrix elements on non-physical sheet without referring to the contour deformation method as it was done in (Bogdanova et al. 1991).
For calculating I (E) we transform denominators in (2)
Thus, poles of scattering S-matrix on non-physical sheet of wave numbers are zeros of analytical function ω(K) where K = k_{1} – ik_{2}, arg K < 0.
The obtained result allows establishing connection between the resonance coupled-channel model and the effective range approximation for the system of charged particles in the presence of absorption (Karnakov et al. 1990, 1991). The following expression is written for the element of S-matrix S_{11}(E) in effective range approximation instead of (1):
where a_{0} is the scattering length; r_{0} is the effective range; D (K) = 2π/[exp(2π/(Ka_{c})) – 1] is the Coulomb barrier penetrability; h (K) = Reψ(i/(Ka_{c}))+ln (Ka_{c}).
Formula (11) is the Landau – Smorodinsky – Bethe approximation (Landau and Lifshitz 1977), and, in the presence of absorption, the scattering length and the effective range become complex values (Karnakov et al. 1990, 1991). Since D (K) = πcth(π/(Ka_{c})) – π and Imψ(ix) = 0.5/x + π/2 cth πx than the following equation is valid
Comparison of functions ω(K) and ω_{1}(K) demonstrates that if terms proportional to K^{4} and K^{6} are neglected in (9) then the connection can be established between the parameters of the models under discussion in the following form
Taking into account the first two expressions in (14) equation (4’) for D + T → α + n reaction cross-section is reduced in the approximation specified above to the following form:
which coincides with (13) within the accuracy of the term proportional to K^{4}.
Determination of model parameters and positioning of S-matrix poles of D + T → α + n and D + <sup>3</sup>He → α + p processes in the neighborhood of resonance energies of <sup>5</sup>He(3/2)<sup>+</sup> and <sup>5</sup>Li(3/2)<sup>+</sup> nuclei
Two sets of parameters of the resonance coupled-channel model well matching the cross-section of the thermonuclear fusion D + T → α + n reaction are presented in (Bogdanova et al. 1991) for the case of ^{5}He(3/2)^{+} nucleus with one of the presented sets of parameters in good agreement with measured data on the cross-section of elastic D-T scattering. Expressions (14) lead to the following two sets of effective range approximation.
Let us note that E_{0} and Γ_{2} parameters are taken from (Bogdanova et al. 1991), β parameter corresponds to the value of в E_{f}: β = (2μE_{f})^{1/2}/ ħ which is similar for both options and is equal to 2 MeV, and parameter γ is recalculated according to the values of cognominal variable presented in (Bogdanova et al. 1991).
The following set of parameters of the effective range approximation was obtained in (Karnakov et al. 1990) on the basis of analysis of thermonuclear fusion reaction:
All sets of parameters presented above were additionally analyzed to verify agreement with experimental data using parametrization of experimental data on the D + T → α + n reaction cross-section (Bosch and Hale 1992) more recent as compared with (Bogdanova et al. 1991, Karnakov et al. 1990, 1991) (Fig. 1). As it was originally expected, all six sets of parameters are in good agreement with the parametrization in question. The above fact confirms the conclusion drawn before in (Bogdanova et al. 1991) that experimental data of only one type are not sufficient for the determination of true parameters of the models. Because of this reason, the data on elastic D-T-scattering were analyzed similarly to (Bogdanova et al. 1991). Energy dependence of the ratio of elastic D-T-scattering cross-section to the Rutherford cross-section for scattering angle Ѳ = π/2 (Balashko 1965) is presented in Figure 2 in the center-of-mass system:
The energy dependence of nuclear fusion D + T → α + n reaction cross-section: solid line is the parametrization in (Bosch and Hale 1992); “dotted” line is the resonance coupled-channel model (Set 1); dashed line is the effective range approximation.
Comparison of the ratio of elastic D-T-scattering to the Rutherford cross-section with experimental data: solid line is the resonance coupled-channel model (Set 1); ”dotted line” is the effective range approximation.
https://binary.pensoft.net/fig/342013
(16)
where f_{c}(q) = –2μe^{2}(ħq)^{–2}exp(2iσ_{0}(E) – iη ln(q^{2}/(4k_{DT}^{2}))) is the Coulomb scattering amplitude; ħq is the transmitted momentum; S_{11} is the S-matrix of elastic D-T-scattering in s-wave; S^{с}_{11} = exp(2iσ_{0}(E)) is the Coulomb S-matrix in s-wave. For angle Ѳ = π/2 expression (16) acquires the following form (Bogdanova et al. 1991):
The presented values of parameters of the resonanse coupled-channel model and approximation of effective range were applied for determining the poles of scattering S-matrix on different sheets of Riemann complex momenta surfaces, i.e. for K = k_{1} – ik_{2}, arg K <0. The property of resonance denominators ω(K, Γ_{2})^{*} = ω(–K^{*}, –Γ_{2}) which is easily established on the basis of explicit form of ω(K) function (9) was applied here. Calculated values of parameter of the resonance and shadow poles of scattering S-matrix for D-T system in the neighborhood of energy of state J^{π} = (3/2)^{+} of ^{5}He nucleus are presented in Table 1.
Poles of amplitude of low-energy D-T-scattering.
Parameter set
Resonance pole
Shadow pole
Resonance coupled-channel model (Set 1)
K_{R} = (1.334 – i 0.465)/a_{c}
K_{S} = (–1.652 + i 0.034)/a_{c}
Z_{R} = 46.9 – i 37.2 _{k}eV
Z_{S} = 81.7 – i 3.5 keV
Resonance coupled-channel model (Set 2)
K_{R} = (1.334 – i 0.468)/a_{c}
K_{S} = (–1.649 + i 0.003)/ a_{c}
Z_{R} = 46.8 – i 37.5 keV
Z_{S} = 81.7 – i 0.3 keV
Effective range approximation (Karnakov et al. 1990)
K_{R} = (1.33 – i 0.45)/a_{c}
K_{S} = (–1.61 – i 0.15)/a_{c}
Z_{R} = 46 – i 36 keV
Z_{S} = 77 + i 14 keV
Let us note that the models, parameters of which describe well the behavior of nuclear fusion reaction (not only those presented above), produce close values of resonance parameters while the parameters of the shadow pole can significantly differ. It was established on the basis of comparison of different sets of parameters with experimental data that the set of parameters from (Bogdanova et al. 1991) equivalent to Set 1, is the best. Results presented in (Bogdanova et al. 1991) recently found confirmation in (Betan et al. 2018): Z_{S} ≈ 82 – i3.5 (keV). Position of Coulomb poles was determined beside that in agreement with results in (Karnakov et al. 1990).
The method developed was applied to the description of D + ^{3}He → α + p fusion reaction. It was established in this case that the best set of parameters for this reaction from the viewpoint of agreement with parametrization (Bosch and Hale 1992) is the set calculated on the basis of effective range approximation (Karnakov et al. 1991):
which agrees with results in (Karnakov et al. 1991).
Conclusion
Four-parameter representation was obtained for S-matrix elements in the resonance coupled-channel model (Bogdanova et al. 1991) allowing investigating analytical structure of S-matrix in the neighborhood of near-threshold energy on the basis of explicit expressions derived for it in the form of analytical channel momenta functions. An important consequence of the obtained expressions is the establishment of equivalence of the resonance coupled-channel model and the effective range approximation for the system of charged particles with absorption (Karnakov et al. 1990, 1991). As the result, parameters of the model can be expressed through model-free characteristics of the system – complex scattering length and complex effective range. The obtained results are planned to be used in the future for the description of near-threshold resonances for ^{7}Li, ^{8}Be nuclei and for investigating decay characteristics of these nuclei, as well as of ^{5}He and ^{5}Li in multi-particle nuclear reactions.
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* Russian text published: Izvestiya vuzov. Yadernaya Energetika (ISSN 0204-3327), 2019, n. 2, pp. 198–207.