Newtonian dynamics of imaginary time-dependent mean field theory

A Time Dependent Hartree-Fock (TDHF) based classical model is applied to sub-barrier fusion reactions using the Feynman Path Integral Method (FPIM). The fusion cross-sections and modified astrophysical S*-factors are calculated for the C+C reactions and compared to direct and indirect experimental results. Different channels cross-sections are estimated from the statistical decay of the compound nucleus. A good agreement with the direct data is found. We suggest a complementary observable given by the (imaginary) action A easily derived from theory and experiments. When properly normalized by the action in the Gamow limit it has an upper value of 1 at zero beam energies. It becomes negative at the Coulomb barrier which is Vcb=5.05±0.05MeV from direct data and Vcb=5.5MeV from model calculations. Careful measurements of the fusion cross-sections for 12 C+ 12 C reactions are crucial to our understanding of massive stars and super-bursts from accreting neutron stars [1-4]. These measurements are especially compelling at energies close or below the Gamow peak [4], i.e., below 2 MeV. Direct data measurements [5-10] start from Ecm=2.1 MeV center of mass energy showing many resonances especially when expressed in terms of the modified astrophysical S* factor [13]: S*(Ecm)=Ecm (Ecm)×exp(87.12Ecm -1/2 +0.46Ecm)= S(Ecm) ×exp(0.46Ecm). To overcome the difficulties of vanishing fusion crosssection at very low energies, indirect methods have been proposed [14,15]. Recently, indirect measurements based on the THM have been published [11] displaying a wealth of resonances in S* at center of mass energies as low as 0.8MeV. A critical reanalysis of the same experiment to include Coulomb effects in the 3-body final channel and the Distorted Wave Born Approximation (DWBA) instead of the Plane Wave Approximation (PWA) resulted in a much lower S* up to three orders of magnitude. These contrasting results reveal a strong model dependence of the THM approach and cast doubts about its general validity. Macroscopic and microscopic models [16-19] offer a reasonable reproduction of the direct data and they are sometime considered as an upper limit of the fusion cross-section [20,21]. Predictions of these models cannot reproduce the low energy THM results of ref.[11] and are way above the Coulomb renormalization analysis of [12]. In ref. [22], we proposed a microscopic model based on the Vlasov equation and the FPIM reproducing rather well the available direct data at that time for the 12 C+ 12 C system. The approach is quite successful when applied to heavier nuclei [23] and spontaneous fission [24]. Before getting involved into complex numerical calculations, which may or may not be feasible at such low energies, it is instructive to simplify the heavy ion dynamics using a robust macroscopic model. In ref. [25], we proposed a TDHF based model, dubbed Neck Model (NM), and successfully reproduced above the barrier fusion cross section as well as deep inelastic [26] and fission [27] dynamics. In this work we generalize the NM to sub barrier energies in the FPIM framework. Newton force equations are solved assuming the collective variables given by the center of mass distance R of the two colliding nuclei and their relative momenta P. The forces acting on the nuclei are given by the Coulomb and the nuclear Bass potentials before the two nuclei touch [28]. After touching, the two nuclei are described as sections of spheres joined by a cylinder of radius rN, the neck radius. In the rebounding phase, the nuclear geometry is given by two half spheres joined by sections of cones of radius rN. Volume conservation is enforced which gives suitable relations between the neck radius and the relative distance of the two nuclei and it is in good agreement to TDHF calculations [25]. In such a configuration the force is given by the surface tension times the perimeter of the neck, as for a liquid drop. During this stage nucleons are transferred through the neck, giving rise to one body-dissipation described by the Randrup-window formula [29]. To describe the subbarrier fusion dynamics, we go into imaginary times (FPIM) at the first (external) turning point [22-24] when the collective momentum P=0fm -1 . In imaginary times, the second derivative of R respect to (imaginary) time gets a i 2 =-1 contribution and the Coulomb force becomes attractive while the nuclear part repulsive. This produces an acceleration of the two nuclei towards each other until they come to a halt (because of the short-range nuclear EPJ Web of Conferences 252, 05001 (2021) HINPw6 https://doi.org/10.1051/epjconf/202125205001 © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). repulsion in imaginary times) at the second (internal) turning point. At this stage we switch from imaginary to real times again. If the nuclear force is strong enough the two nuclei fuse and this happens always for light nuclei like 12 C. The probability of fusion for the lth-partial wave is given by [22]: = (1 + exp {2 }) (1). The action (in units of ħ) is given by = ∫ . The cross section is given by: ( ) = ħ ∑ (2 + 1) (2). Pl gives the probability of decay of the compound nucleus into different channels, is the nuclear reduced mass. Following [22], the calculations will be performed at zero impact parameter only. In order to take into account the l-dependence of the transmission probability we shift the beam energy for each l as [30]: ≈ ( − ( + 1)ħ 2 < >) <R> is an effective moment of inertia of two touching spheres at the internal turning point <R> and it is slowly varying with energy. Recall that for the 12 C+ 12 C system, because of angular momentum and parity conservation laws, only even l-values are allowed. In figure 1, we plot, for different l-values as indicated, the collective momentum versus collective distance of the two nuclei at Ecm=2.5 MeV. Starting from infinite distance, the two nuclei approach each other under the influence of the Coulomb repulsion until they reach the first turning point (P=0 fm -1 ). Clearly the first turning point is farther for higher l-values or lower beam energies. At this stage, the system enters imaginary time propagation, and the Coulomb force becomes attractive resulting in a strong acceleration. At shorter distances the (repulsive in imaginary times) nuclear force given by the Bass potential violently decelerates the two nuclei until P=0 fm -1 again, the second turning point. For light nuclei the imaginary dynamics is completely determined by the Coulomb and Bass potential. When returning to real times, the two nuclei accelerate again towards each other, a neck is formed, and fusion occurs. The area inside the first and second turning points gives the action A. Notice that A increases quickly with increasing l, thus the probability of fusion decreases, eq.(1), and the cross section is dominated by low l values, eq.(2). A striking difference with the microscopic Vlasov approach, see figure 2 in ref.[22], is the oscillations seen in imaginary times. These are due to collective excitation of the single particle degrees of freedom included in the microscopic model but not in the NM. As a result, oscillations are seen in the fusion cross section of figure 3 in ref.[22] and we will discuss them further in the fig.3 below. Figure 1. Collective momentum vs distance of the two nuclei for the values of Ecm and l indicated. The areas inside the first and second turning points give the action A. After fusing, the excitation energy of the compound nucleus (CN) can be determined from the Ecm, l and the Q-values for the CN and the open decay channels. In our case the most important channels are: 12 C( 12 C, )Mg, Q=13.9 MeV; C(C, ) Ne Q=4.62 MeV; 12 C( 12 C,p) 23 Na, Q=2.24 MeV; 12 C( 12 C,n) 23 Mg, Q=-2.6 MeV; 12 C( 12 C, 8 Be) 16 O, Q=-0.2MeV. We can calculate the decay probability in certain channels within the framework of the statistical model [32]: Π = ! ( "# + 2$ "#) %#( &' ∗ ,-#) %( &' ∗ ) ./0 (3). The probability entering eq. (2) can be expressed as = Π / ∑ Π for each l-value and excitation energy 45 ∗ = − ( 6 )ħ 78 9 + :. ! = 2; + 1 and si is the spin of the emitted particles. The absorption cross section for the inverse process is given by: ./0 = ? 5 (1 − @4( 5)/ "#), where RN is the sum of the radii of the nuclei in the exit channel, VC(RN) the Coulomb barrier and Tki the kinetic energy of the emitted particle. For gamma absorption: A./0 = 75 45 (,-D . F) G ,-D G ($H) [32], ACN the mass number of the CN. The level densities contribution can be written as: %#( &' ∗ ,-#) %( &' ∗ ) = exp (− ,-# , ); the temperature is given by = I 45 ∗ /J and a= ACN /8.0 (MeV). EPJ Web of Conferences 252, 05001 (2021) HINPw6 https://doi.org/10.1051/epjconf/202125205001

Careful measurements of the fusion cross-sections for 12 C+ 12 C reactions are crucial to our understanding of massive stars and super-bursts from accreting neutron stars [1][2][3][4]. These measurements are especially compelling at energies close or below the Gamow peak [4], i.e., below 2 MeV. Direct data measurements [5][6][7][8][9][10] start from E cm =2.1 MeV center of mass energy showing many resonances especially when expressed in terms of the modified astrophysical S* factor [13]: S*(E cm )=E cm V(E cm )×exp(87.12E cm -1/2 +0.46E cm )= S(E cm ) ×exp(0.46E cm ). To overcome the difficulties of vanishing fusion crosssection at very low energies, indirect methods have been proposed [14,15]. Recently, indirect measurements based on the THM have been published [11] displaying a wealth of resonances in S* at center of mass energies as low as 0.8MeV. A critical reanalysis of the same experiment to include Coulomb effects in the 3-body final channel and the Distorted Wave Born Approximation (DWBA) instead of the Plane Wave Approximation (PWA) resulted in a much lower S* up to three orders of magnitude. These contrasting results reveal a strong model dependence of the THM approach and cast doubts about its general validity. Macroscopic and microscopic models [16][17][18][19] offer a reasonable reproduction of the direct data and they are sometime considered as an upper limit of the fusion cross-section [20,21]. Predictions of these models cannot reproduce the low energy THM results of ref. [11] and are way above the Coulomb renormalization analysis of [12]. In ref. [22], we proposed a microscopic model based on the Vlasov equation and the FPIM reproducing rather well the available direct data at that time for the 12 C+ 12 C system. The approach is quite successful when applied to heavier nuclei [23] and spontaneous fission [24]. Before getting involved into complex numerical calculations, which may or may not be feasible at such low energies, it is instructive to simplify the heavy ion dynamics using a robust macroscopic model. In ref. [25], we proposed a TDHF based model, dubbed Neck Model (NM), and successfully reproduced above the barrier fusion cross section as well as deep inelastic [26] and fission [27] dynamics. In this work we generalize the NM to sub barrier energies in the FPIM framework. Newton force equations are solved assuming the collective variables given by the center of mass distance R of the two colliding nuclei and their relative momenta P. The forces acting on the nuclei are given by the Coulomb and the nuclear Bass potentials before the two nuclei touch [28]. After touching, the two nuclei are described as sections of spheres joined by a cylinder of radius r N , the neck radius. In the rebounding phase, the nuclear geometry is given by two half spheres joined by sections of cones of radius r N . Volume conservation is enforced which gives suitable relations between the neck radius and the relative distance of the two nuclei and it is in good agreement to TDHF calculations [25]. In such a configuration the force is given by the surface tension times the perimeter of the neck, as for a liquid drop. During this stage nucleons are transferred through the neck, giving rise to one body-dissipation described by the Randrup-window formula [29]. To describe the subbarrier fusion dynamics, we go into imaginary times (FPIM) at the first (external) turning point [22][23][24] when the collective momentum P=0fm -1 . In imaginary times, the second derivative of R respect to (imaginary) time gets a i 2 =-1 contribution and the Coulomb force becomes attractive while the nuclear part repulsive. This produces an acceleration of the two nuclei towards each other until they come to a halt (because of the short-range nuclear repulsion in imaginary times) at the second (internal) turning point. At this stage we switch from imaginary to real times again. If the nuclear force is strong enough the two nuclei fuse and this happens always for light nuclei like 12 C. The probability of fusion for the lth-partial wave is given by [22]: The cross section is given by: (2). P l gives the probability of decay of the compound nucleus into different channels, is the nuclear reduced mass. Following [22], the calculations will be performed at zero impact parameter only. In order to take into account the l-dependence of the transmission probability we shift the beam energy for each l as [30]: P<R 2 > is an effective moment of inertia of two touching spheres at the internal turning point <R> and it is slowly varying with energy. Recall that for the 12 C+ 12 C system, because of angular momentum and parity conservation laws, only even l-values are allowed.
In figure 1, we plot, for different l-values as indicated, the collective momentum versus collective distance of the two nuclei at E cm =2.5 MeV. Starting from infinite distance, the two nuclei approach each other under the influence of the Coulomb repulsion until they reach the first turning point (P=0 fm -1 ). Clearly the first turning point is farther for higher l-values or lower beam energies. At this stage, the system enters imaginary time propagation, and the Coulomb force becomes attractive resulting in a strong acceleration. At shorter distances the (repulsive in imaginary times) nuclear force given by the Bass potential violently decelerates the two nuclei until P=0 fm -1 again, the second turning point. For light nuclei the imaginary dynamics is completely determined by the Coulomb and Bass potential. When returning to real times, the two nuclei accelerate again towards each other, a neck is formed, and fusion occurs. The area inside the first and second turning points gives the action A. Notice that A increases quickly with increasing l, thus the probability of fusion decreases, eq.(1), and the cross section is dominated by low l values, eq.(2). A striking difference with the microscopic Vlasov approach, see figure 2 in ref. [22], is the oscillations seen in imaginary times. These are due to collective excitation of the single particle degrees of freedom included in the microscopic model but not in the NM. As a result, oscillations are seen in the fusion cross section of figure 3 in ref. [22] and we will discuss them further in the fig.3 below. symbols) compared to the direct data (full symb total cross section is given by the triangles (cy by the diamonds (orange), n-channel by the circ p-channel (blue) by the squares. The model ag from ref. [8] and it overestimate ref. [7]. No increase of the n-channel (negative Q-value) c p-channel. The model reproduces the trend bu the data of ref. [7].
In figure 2, we compare the model result data [7,8] for the total cross section a important decay channels (D,p and n). agreement is quite good especially at energies. An over (under) estimate of the is noticeable. At lower excitation energies levels of the CN and the exit channels n important for each l-values, these levels ar in the present calculations, and we ex produce oscillations as in the data if proper Notice that any statistical model refineme the P l values entering eq. (1) but such a qua one, thus we conclude that the calculated represent an upper limit [20,21]. Our resul agreement with other model calculations [1 n of E cm (open bols) [7,8]. The yan), -channel cles (green) and grees to the data otice the faster compared to the ut overestimates s to the direct and the most The overall t the highest n (p)-channel the individual nuclei become e not included xpect them to rly considered. ent will affect antity is below cross-sections lts are in good [16][17][18][19][20].  [7,8,33,34]. given by the full (orange) circles [1 renormalized one by the full (red) trian model calculations, including 24 Mg reso the dashed (green) line [35]. The Vlaso times are given by the full (red) line [22].
To avoid the quickly decreasing fus sub-barrier energies, it is customa modified astrophysical factor S* as figure 3. The model calculations giv b at the lowest energy, slowly increa energy region. Direct data [7,8] are o calculation at higher energies an 'bumps'. The indirect data are norm data at 2.6 MeV [11]. At much lower diverges from the calculation up magnitude higher, while the Cou THM analysis is up to 3 orders bel These discrepancies need to investigated maybe within the same for a system, which does not present problem for one of the outgoing pa reaction could be 13 C( 12 C,n)…where as a spectator [12]. In ref. [35] we present model to include effects of k 26 Mg. We found a reasonable agre data [11]. In ref. [36], we have adde the Coulomb field resulting in a 15 closer to the direct data. The Vlaso are in rather good agreement to t display oscillations or 'bumps' sim repetition of those calculations e energies and different interaction interesting. Calculated and measured values of S from system to system and as a fu The original purpose in defining the to avoid the large variation of the fus factoring out the Coulomb barrier pen [4] thus leaving nuclear effects emb Whether this goal has been reached figure 3 and the literature [1-13,16-2 we would like to propose a comple which can be derived from eq. (2). figure 1 that at the lowest energie dominant, thus neglecting higher l-va T 0, the action becomes: Of course, if data could be provide value, then eq.(4) could be modified eq. (2). Notice that when dealing w cross-section P l =1. The value of t derived analytically for the case of t only (Gamow limit) [  In figure 4, we plot the Gamow normalized for the model not including resonance symbols) and the data (full symbols). reasonable agreement to the direct data, w is below [11] or above the calculations for renormalized case [12], i.e., similar to Notice that the action A has an opposite and S*, i.e., the higher is A the smaller section. Because of the clear physical int A/A G we can parameterize its behavior as: The parameters a,b,c is fitted to data or t results can be seen in figure 3. The action at the effective Coulomb barrier V cb =5.0 from the direct data [7,8] to compare V cb =5.5 MeV. The latter is the highest ener simulation and its value critically depends other nuclear potentials. The fits to the ind V cb =4.3 MeV for the Coulomb renormaliza and no solution for the original data [11]. O last result depends on the THM normalizat direct data [8]. There seem to be som between the direct data of ref. [7] and [8] w better reproduced by the model.
In conclusion, in this paper we have discu inspired macroscopic model and extend barrier energies using the Feynman Path In i.e., solving Newton dynamics below t imaginary times. The model has no free p reasonably reproduces the direct data. The data are well above or below the model de the Coulomb corrections are applied [12 Inclusion of resonances in the model [35 results of ref. [11]. Coulomb field modifi e+e-corrections gives results in better agr direct data [36]. We have introduced a ne the action A, that when properly norm bols as in figure eq.(5), the full dashed line to d action vs E cm es [35] (open We notice a while the THM r the Coulomb the figure 3. behavior to S r is the cross terpretation of (5).
theory and the becomes zero 05±0.05 MeV to the model rgy we run the on the Bass or direct data give ation case [12] Of course, this tion [11] to the me discrepancy with the former ussed a TDHF ded it to subntegral Method the barrier in parameters and THM indirect epending on if ] or not [11]. ] supports the cations due to reement to the ew observable, malized by the action in the Gamow limit gives a q and close to unity and we expect all fall in the same region.