Extended quantum diffusion approach to reactions of astrophysical interests
aa r X i v : . [ nu c l - t h ] J u l Extended quantum diffusion approach to reactions of astrophysical interests
V.V.Sargsyan , , G.G.Adamian , N.V.Antonenko , and H. Lenske Joint Institute for Nuclear Research, 141980 Dubna, Russia Institut f¨ur Theoretische Physik der Justus–Liebig–Universit¨at, D–35392 Giessen, Germany (Dated: July 13, 2020)The quantum diffusion approach is extended to low energy fusion (capture) reactions of light-and medium-mass nuclei. The dependence of the friction parameter on bombarding energy is takeninto account. A simple analytic expression is obtained for the capture probability at extreme sub-barrier energies. The calculated cross-sections are in a good agreement with the experimental data.The fusion excitation functions calculated within the quantum diffusion and WKB approaches arecompared and presented in the astrophysical S -factor representation. PACS numbers: 25.70.Ji, 24.10.Eq, 03.65.-wKey words: capture, sub-barrier fusion; dissipative dynamics
I. INTRODUCTION
Fusion reactions at energies near and below theCoulomb barrier have been an object of extensive ex-perimental and theoretical studies in the past decades[1–4]. Indeed, the heavy-ion fusion allows us to extendthe periodic table beyond the elements that can not besynthesized using neutrons and light charged particles.The fusion of light- and medium-mass nuclei plays an im-portant role in the evolution of massive stars where thebehavior of fusion excitation function at extreme sub-barrier energies determines the reaction rates. For ex-ample, towards the end of stellar life-cycle the elementsup to the iron can be synthesized. These reactions drivesthe nucleosynthesis and generates the energy in novae,supernovae, and close binary stars [5]. In Refs. [6–27],the fusion reactions involving light nuclei at low energieswere investigated both experimentally and theoretically.The recent developments in fusion reactions both in ex-periment and theory are presented in Refs. [2–4] andreferences therein.For light and medium-mass nuclei the fusion is gov-erned by the penetrability of colliding nuclei through theCoulomb and centrifugal barrier (so called capture). Ifthe collision occurs at energies permitting very large an-gular momentum, there is a possibility that the formeddinuclear system decays after the capture stage. How-ever, at energies near and below the Coulomb barrier,the contribution of large angular momenta to fusion canbe disregarded. Therefore, the description of fusion ofthese nuclei is reduced to the description of the captureof projectile by target-nucleus.To study the capture (fusion) process in heavy-ion re-actions, the quantum diffusion approach, based on thequantum master-equation for the reduced density ma-trix, has been suggested in Refs. [28–30]. In this ap-proach the collisions of nuclei are treated in terms ofa single collective variable: the relative distance R be-tween the colliding nuclei. The coupling of the relativemotion to the excitation of various channels, such as non-collective single-particle excitations, low-lying collectivemodes (dynamical quadrupole and octupole excitations of the target and projectile) lead to the fluctuation anddissipation effects. Hence, many quantum-mechanicaland non-Markovian effects, accompanying the passagethrough the potential barrier, are considered in our for-malism. The nuclear deformation effects are taken intoaccount through the dependence of the nucleus-nucleuspotential on the deformations and mutual orientations ofthe colliding nuclei [31, 32].As shown in Refs. [29–31], our model successfully de-scribes the capture (fusion) cross section in heavy-ion col-lisions at energies near and below the Coulomb barrier.In the present work we extend our approach to describethe capture (fusion) of light and medium-mass nuclei atenergies well below the Coulomb barrier. Our aim is tocalculate the fusion cross sections for nuclei of interestand importance for stellar burning. So, the approach isapplied to low-energy fusion reactions with carbon, oxy-gen, and silicon nuclei. II. FORMALISM OF THE QUANTUMDIFFUSION APPROACH
The capture cross section is the sum of the partial cap-ture cross sections [29–32] σ cap ( E c . m . ) = X J σ cap ( E c . m . , J )= πλ X J (2 J + 1) Z π/ dθ sin( θ ) × Z π/ dθ sin( θ ) P cap ( E c . m . , J, θ , θ ) , (1)where λ = ~ / (2 µE c . m . ) is the reduced de Broglie wave-length, µ = m A A / ( A + A ) is the reduced mass ( m is the nucleon mass), and the summation occurs overpossible values of angular momentum J at a given bom-barding energy E c . m . . Knowing the potential of the in-teracting nuclei for each orientation defined by the angles θ i ( i = 1 , P cap ( E c . m . , J, θ , θ ) which is the probability topenetrate throw the potential barrier in the relative co-ordinate R at a given J . P cap is obtained by integratingthe propagator G from the initial state ( R , P ) at time t = 0 to the final state ( R, P ) at time t ( R is defined withrespect to the position R b of the Coulomb barrier and P is the conjugate momentum): P cap = lim t →∞ Z R in −∞ dR Z ∞−∞ dP G ( R, P, t | R , P , t →∞
12 erfc " − R in + R ( t ) p Σ RR ( t ) . (2)Here, we use the propagator G = π − | det Σ − | / exp( − q T Σ − q ) , (3)where q T = [ q R , q P ], q R ( t ) = R − R ( t ), q P ( t ) = P − P ( t ), R ( t = 0) = R , P ( t = 0) = P , Σ kk ′ ( t ) = 2 q k ( t ) q k ′ ( t ),Σ kk ′ ( t = 0) = 0 and k, k ′ = R, P , obtained in Ref.[33] for a local inverted oscillator which replaces the realnucleus-nucleus potential in the variable R . The fre-quency ω of this local inverted oscillator with an internalturning point R in is defined from the condition of equalityof the classical actions of approximated and real potentialbarriers of the same height at given E c . m . and J . Notethat this procedure leads to the frequency depending on E c . m . and J . This local replacement of the real poten-tial by the inverted oscillator with energy-dependent fre-quency is well justified for heavy-ion reactions at energiesnear and below the Coulomb barrier [29–32, 34, 35].As at t → ∞ the internal turning point R in ≪ R ( t ),the capture cross section is defined by the ratio of themean value of the collective coordinate R ( t ) and its vari-ance Σ RR ( t ). For the explicit expressions for R ( t ) andΣ RR ( t ) we refer to our previous studies in Refs.[29–32, 35]. Using the Hamiltonian of the system, which in-cludes the collective subsystem, the environment (whichmimics the internal excitations) and the coupling be-tween the collective subsystem and the environment, asystem of non-Markovian Langevin equations for the col-lective coordinates was derived. These equations of mo-tion for the collective subsystem satisfy the quantum fluc-tuation - dissipation relations and contain the influenceof quantum, dissipative and non-Markovian effects on thecollective motion [28, 35]. The expressions for the R ( t ) and Σ RR ( t ) are R ( t ) = A t R + B t P , Σ RR ( t ) = 4 ~ ˜ λǫµγ π t Z dτ ′ B τ ′ t Z dτ ′′ B τ ′′ ∞ Z d Ω ΩΩ + γ × coth (cid:20) ~ Ω2 T (cid:21) cos[Ω( τ ′ − τ ′′ )] ,B t = 1 µ X i =1 β i ( s i + γ ) e s i t ,A t = X i =1 β i [ s i ( s i + γ ) + 2 ~ ˜ λǫγ ] e s i t . (4)Here, Σ RR (0) = 0, A = 1, and B = 0. In Eqs.(4), β = [( s − s )( s − s )] − , β = [( s − s )( s − s )] − and β = [( s − s )( s − s )] − , and s i are the real roots( s ≥ > s ≥ s ) of the following equation( s + γ )( s − ǫ ) + 2 ~ ˜ λǫγs = 0 . (5)The parameters γ , ǫ and ˜ λ determine the characteristicsof the system. The values of γ − is the memory time ofdissipation of relative motion energy by the internal sub-system or is the inverse bandwidth of the internal sub-system excitations. The non-Markovian effects appearin the calculations through γ . The instantaneous dissi-pation corresponds to taking γ → ∞ . The parameter ǫ defines the initial frequency of the collective subsys-tem and ˜ λ determines the average coupling strength ofthe collective subsystem with internal excitations. To setthese parameters [28, 35], we use the asymptotic valuesof the friction coefficient λ = − ( s + s ) (6)and potential frequency ω = ǫ (cid:18) ( s + γ )( s + γ )( s + γ )( s + γ ) − ~ ˜ λγǫ (cid:19) / . (7)Note, that ω takes into account the renormalization ofthe initial frequency due to the coupling to the internalexcitations. So, in the asymptotic limit t → ∞ , the fric-tion λ and frequency ω are related to the parameters γ , ǫ , ˜ λ , and the roots s , of Eq. (5). Setting the values of λ , ω , and γ , we determine the dynamics of the system.The use of asymptotic values of λ and ω is justified, sincethe characteristic time of reaching them is much shorterthan the characteristic time of capture.Equations (2), (4), (6), and (7) lead to the analyticexpression for the capture probability: P cap = 12 erfc "(cid:18) πs ( γ − s )2 ~ µ ( ǫ − s ) (cid:19) / µǫ R /s + P [ γ ln( γ/s )] / . (8)In the derivation of Eq. (8) the limit of low tempera-tures ( T →
0) was used, which is suitable for sub-barrierfusion. Note, that the friction λ and internal excitationwidth γ are related. If the coupling with internal degreesof freedom is disregarded, λ →
0, then the limit γ → ∞ results in the Markovian dynamics. In the case of λω ln( γ ) → const (9)at λ →
0, the well-known quantum-mechanical barriertransmission probability is obtained P cap ∼ exp[ − π ( V b − E c . m . ) / ~ ω ] . III. NUCLEUS-NUCLEUS POTENTIAL
In the case of collision of deformed nuclei the effectivenucleus-nucleus potential reads as: V = V N + V C + ~ J ( J + 1)2 µR , (10)where V N , V C , and the last summand stand for the nu-clear, Coulomb, and centrifugal potentials, respectively[37]. The potential depends on the relative distance R be-tween the center of mass of two interacting nuclei, masses A i , charges Z i and radii R i of the nuclei ( i = 1 , θ i of the deformed (with the quadrupoledeformation parameters β ( i )2 ) nuclei and angular momen-tum J . For deformed nuclei, the static quadrupole de-formation parameters are taken from Ref. [36]. For thenuclear part of potential, V N = Z ρ ( r ) ρ ( R − r ) F ( r − r ) d r d r , (11)the double-folding formalism is used, where F ( r − r ) = C [ F in ρ ( r ) ρ + F ex (1 − ρ ( r ) ρ )] δ ( r − r ) is thedensity-depending effective nucleon-nucleon interactionand ρ ( r ) = ρ ( r ) + ρ ( R − r ), F in , ex = f in , ex + f ′ in , ex ( N − Z )( N − Z )( N + Z )( N + Z ) . Here, ρ i ( r i ) and N i are the nucleondensities and neutron numbers of the light and the heavynuclei of the dinuclear system. Our calculations are per-formed with the following set of parameters: C = 300MeV fm , f in = 0.09, f ex = -2.59, f ′ in = 0.42, f ′ ex = 0.54and ρ = 0.17 fm − [37]. The densities of the nuclei aretaken in the two-parameter symmetrized Woods-Saxonform with the nuclear radius parameter r =1–1.15 fmand the diffuseness parameter a =0.47–0.56 fm depend-ing on the charge and mass numbers of the nucleus [37].The Coulomb interaction of two quadrupole deformednuclei reads as V C = Z Z e R + (cid:18) π (cid:19) / Z Z e R X i =1 , R i β ( i )2 " (cid:18) π (cid:19) / β ( i )2 × P (cos θ i ) , (12) where P (cos θ i ) is the Legendre polynomial.The calculated potentials with respect to their barriers V b are presented in Fig. 1 for two reactions with spher-ical nuclei O+ Pb and O+ O at J = 0 . Withincreasing angular momentum, the positions of the po-tential barrier R b and the minimum R m merges, and atcertain J the potential pocket disappears. This is a natu-ral limitation of J that contribute to the capture (fusion)., The large Coulomb repulsion in the case of O+ Pb -4 -3 -2 -1 0 1 2 3 4 5 6-20-15-10-505 V - V b ( M e V ) R (fm)
FIG. 1: The nucleus-nucleus potentials calculated at J = 0for the reactions O+ Pb (solid line) and O+ O (dashedline). The coordinate R is defined relative to the position R b of the Coulomb barrier. leads to a steep decline of the potential, compared to thatin the case of O+ O. So, at the fixed E c . m . − V b < R ext in the case of heavier system. IV. EXTENSION OF THE APPROACHA. Energy-dependent friction and internalexcitation bandwidth
The formalism, introduced in Sect. II, implies that thefriction λ does not depend on E c . m . . The use of the con-stant friction seems to be valid in case of fusion of ratherheavy nuclei at energies near and below (up to 5-6 MeV)the Coulomb barrier. However, in the reactions withmedium-mass and light nuclei, and/or at extreme sub-barrier energies, the dependence of the friction on E c . m . can not be ignored. This remark can be easily understoodfrom Fig. 2, where the comparison of the dependenciesof the external turning point R ext on energy is shownfor the reactions O+ Pb and O+ O. The value of R ext at given E c . m . indicates the degree of the overlapof nuclear density profiles, which is responsible for thenuclear friction. For the O+ O reaction, the valueof R ext drastically increases with decreasing E c . m . − V b R e x t ( f m ) E c.m. -V b (MeV) FIG. 2: The calculated dependencies of the external turningpoint R ext on ( E c . m . − V b ) for the reactions O+ Pb (solidline) and O+ O (dashed line). The value of R ext is definedrelative to the position R b of the Coulomb barrier. which leads to a strong reduction of the friction with re-spect to the O+ Pb reaction. At fixed E c . m . − V b ,the value of R ext is much closer to the position of thecorresponding Coulomb barrier for heavy system.To include the bombarding energy dependence of fric-tion in our model, we refer to the studies of Refs. [38, 39],where the friction, λ ( R ) = λ b (cid:18) ∇ V N ( R ) ∇ V N ( R b ) (cid:19) , (13)proportional to the square of nuclear force, was suggestedfor fusion and deep inelastic reactions. This form of λ ( R ) takes into account the overlap of nuclear surfaceson which the friction strength depends. To determine thenormalization parameter λ b , we use our previous studies[29–32], where the fusion cross section of heavy nuclei atenergies near and below (up to 4-5 MeV) the Coulombbarrier was well described with constant friction coeffi-cient ~ λ b = ~ λ ( R = R b ) = 2 MeV. The calculated de-pendencies of the friction on R are shown in Fig. 3 forthe reactions O+ Pb and O+ O. One can see therapid decrease of the friction with increasing R . Notethat the calculated capture cross sections are rather in-sensitive to the value of λ b . For example, the variationof this parameter by 2 times leads to the change of theresults of the calculations by less then 5%.In accordance with Eq. (9) the internal excitationbandwidth γ is related to the friction. We take the samerelation also in the case of coordinate-dependent frictioncoefficient λ ( R ): γ ( R ) = γ exp (cid:20) k ω ( R ) λ ( R ) (cid:21) . (14)In the case of constant friction ~ λ = 2 MeV, the bestagreement with the experimental data is archived at con- -6 -5 -4 -3 -2 -1 ( M e V ) R (fm)
FIG. 3: The calculated dependencies of the friction coef-ficients on R for the reactions O+ Pb (solid line) and O+ O (dashed line). The coordinate R is defined relativeto the position R b of the Coulomb barrier. stant internal excitation width ~ γ = 32 MeV for the re-actions with heavy nuclei [29–32]. Thus, we choose γ to have ~ γ ( R = R b ) = 32 MeV. Note that at deep sub-barrier energies the results of calculations are almost un-sensitive to γ (see subsection IV.D). In the limit λ → λω ln( γ ) → k as in Eq. (9).The value of k in Eq. (14) is a parameter to be ad-justed, and may vary for different reactions. However,our calculations show a certain universality of this pa-rameter for all considered reactions. The perfect agree-ment with the experimental cross sections is archived ifthe values of γ , ω , and λ are calculated at R = R ext andthe value of k is defined as k = α p µω b , (15)where α = π MeV / fm − and ω b = ω ( R = R b ) is thefrequency at the barrier position R b .So, in our extended model we use the values of frictionand internal excitation bandwidth which are calculatedat R = R ext : λ ( R ext ) and γ ( R ext ). Thus, the bombardingenergy dependence of γ and λ are included through theirdependence on R ext . B. Energy-dependent frequency
We use the local inverted oscillator approximationwhich means that the nucleus-nucleus interaction poten-tial at each E c . m . is locally replaced by the inverted os-cillator with own frequency. At different E c . m . , there aredifferent local inverted oscillators. As mentioned in Sect.II, for the reactions with heavy nuclei at sub-barrier ener-gies, we determine the frequency ω of the approximatedoscillator from the condition of equality of the classicalactions under the barrier of the real and approximatedpotentials. This approximation leads to the close valuesof R ext for the real and approximated potentials. For thereactions with light- and medium-mass nuclei, the sameprocedure leads to completely different values of R ext inthe cases of real and approximated potentials. Becausethe friction strongly depends on R ext , this approximationbecomes irrelevant. For the light- and medium-mass nu-clei, we suggest to match the height and position of thebarrier of the real potential with the height and positionof inverted oscillator. To determine the frequency ω atsub-barrier energies, we use the following expression V b − E c . m . = µω ( R ext − R b ) ω on E c . m . that is on R ext . C. Initial conditions and parameters
Employing Eq. (8) and the initial coordinate R andmomentum P , we calculate the capture probability P cap .Let us consider the initial conditions and parameters usedin our calculations.If the collision of nuclei occurs at sub-barrier energies E c . m . < V b , the dissipation of the kinetic energy of rel-ative motion before R ext is neglected. Hence, the R coincides with the external turning point, R = R ext ,and P = 0. Here, the values of λ , ω , γ are calculatedat R = R = R ext , λ ( R ext ), ω ( R ext ), γ ( R ext ), and corre-spondingly they depend on E c . m . .If the capture occurs at energies E c . m . abovethe Coulomb barrier V b , R = R b and P = p µE c . m . exp( − λ b t int ). Here, the dissipation ∆ E = E c . m . [1 − exp( − λ b t int )] of the kinetic energy of relativemotion is taken effectively into account by using the av-erage friction coefficient λ b and energy-dependent inter-action time estimated as t int = 1 / √ E c . m . s. For the cal-culations of σ cap ( E c . m . ) at energies above the Coulombbarrier, we use the values of λ , ω , and γ calculated atthe barrier position: ~ λ b = ~ λ ( R = R b ) = 2 MeV, ~ γ b = ~ γ ( R = R b ) = 32 MeV and ω b = ω ( R = R b ) = q µ d VdR | R = R b . D. Analytical expression for the capture atextreme sub-barrier energies
At extreme sub-barrier energies, we have the follow-ing initial conditions: P = 0 and R = R ext = Z Z e /E c . m . . Using this R and Eq. (16), we obtainthe analytical expression ω = E c . m . Z Z e − R b E c . m . (cid:18) V b − E c . m . ) µ (cid:19) / for frequency. Because at extreme sub-barrier energiesthe value of friction is small and γ ≫ ω , ω/λ ≫ ln( γ ),we derive s ≃ ω andln (cid:18) γs (cid:19) ≃ ln( γ ) ≃ k ωλ . (17)Substituting these expressions and initial conditions intoEq. (8), we finally obtain P cap = 12 erfc s π ( V b − E c . m . ) k ~ ω . (18)Note that Eq. (18) is similar to the well known quantum-mechanical barrier transmission probability but with thereplacement of the usual frequency by the effective one. V. RESULTS OF CALCULATIONS
Using the procedure described, we apply Eqs. (1),(6)–(8), (13), (14), and (16) to calculate the capturecross-section σ cap ( E c . m . ) for low-energy reactions withlight- and medium-mass nuclei. As emphasized in[6, 8, 9, 11, 13, 14, 22], the fusion reactions between car-bon and oxygen isotopes are playing a crucial role in awide variety of stellar burning scenarios. As the first stepin that direction, we compare our calculated results withthe available data. -9 -7 -5 -3 -1 c ap ( m b ) E c.m. (MeV) C+ C FIG. 4: The calculated capture cross section (line) vs E c . m . for the C+ C reaction compared with the available ex-perimental data. The experimental data marked by squares,circles, stars and triangles are taken from Refs. [6, 13, 14, 22],respectively
The results of the calculated capture cross sections andthe experimental data are shown in Figs. 4–10. In allconsidered reactions we obtain a good agreement with theexperiments. Note, that for C+ C reaction the earlymeasured data [6, 13] differ from the later ones [14, 22].Here, the mechanism that causes the oscillations of the -9 -7 -5 -3 -1 c ap ( m b ) E c.m. (MeV) C+ O FIG. 5: The same as in Fig. 4, but for the C+ O reaction.The experimental data marked by squares, circles, and starsare taken from Refs. [7, 8, 21], respectively. -7 -5 -3 -1 c ap ( m b ) E c.m. (MeV) O+ O FIG. 6: The same as in Fig. 4, but for the O+ O reaction.The experimental data marked by squares, circles, trianglesand stars are taken from Refs. [9–12], respectively. cross section in the C+ C reaction is not considered[25].Our calculated results at sub-barrier energies arerather sensitive to the coefficient k [Eq.(15)]. However,it is uniformly determined for all reactions considered.Thus, we conclude that Eq. (15) is useful for the reac-tions of astrophysical interest.At energies below the Coulomb barrier, where the crosssection drops rapidly with decreasing energy, it is moreconvenient to discuss the astrophysical S -factor, S ( E c . m . ) = E c . m . σ fus ( E c . m . ) exp[2 π ( η − η )] , (19)rather than the fusion excitation function. Here, η ( E c . m . ) = Z Z e p µ/ (2 ~ E c . m . ) is the Sommerfeld pa-rameter and η = η ( E c . m . = V b ), where V b is the Coulomb -8 -6 -4 -2 c ap ( m b ) E c.m. (MeV) C+ Si FIG. 7: The same as in Fig. 4, but for the C+ Si reac-tion. The experimental data marked by circles are taken fromRef. [23].
22 24 26 28 30 32 34 36 3810 -6 -4 -2 c ap ( m b ) E c.m. (MeV) Si+ Si FIG. 8: The same as in Fig. 4, but for the Si+ Si reac-tion. The experimental data marked by circles are taken fromRef. [19]. barrier height for the spherical interacting nuclei. Assum-ing that the capture cross section is equal to the fusioncross section, we calculate the astrophysical S -factor. InFigs. 11 and 12 the calculated S -factors versus E c . m . areshown for the reactions C+ C, C+ O, C+ Si, O+ O, and Si+ Si. A good agreement of the cal-culated excitation function with the experimental dataleads to a good description of S -factor as well. For thereactions under study, the S -factor has a maximum at E c . m . ≈ V b , where V b is the Coulomb barrier height forthe spherical interacting nuclei. The origin of the max-imum of the S -factor is the turning-off of the nuclearforces between the colliding nuclei with decreasing E c . m . .While the theory shows clear maximum, their presencein the experimental data is tenuous up to now. In therecent paper [22] on a new measurement of the C+ C
33 36 39 42 45 48 51 5410 -8 -6 -4 -2 c ap ( m b ) E c.m. (MeV) S+ Ca FIG. 9: The same as in Fig. 4, but for the S+ Ca reac-tion. The experimental data marked by circles are taken fromRef. [40].
35 40 45 50 55 6010 -7 -5 -3 -1 c ap ( m b ) E c.m. (MeV) S+ Ca FIG. 10: The same as in Fig. 4, but for the S+ Careaction. The experimental data marked by circles are takenfrom Ref. [41]. fusion cross sections, it was found that the astrophysicalS-factor exhibits a maximum around E c . m . =3.5–4 MeV.The additional measurements of different systems at low-est bombarding energies are necessary to establish theexistence of S -factor maximum. In Figs. 11 and 12,after this maximum S -factor decreases strongly with de-creasing bombarding energy, which leads to a reductionof the previously predicted astrophysical reaction rates.Note also that such a strong dependence on E c . m . , infact, contradicts the philosophy of representing the crosssection through the S -factor.Figure 11 shows a comparison between our and WKB( P cap is determined within both the WKB model andthe interaction potential of Eqs. (10)–(12)) S -factors forthe reactions C+ C and O+ O. As seen, the fluc-tuation and dissipation effects taken into account in our -1 E c . m . c ap e (- ) ( M e V m b ) E c.m. (MeV) C+ C -1 E c . m . c ap e (- ) ( M e V m b ) E c.m. (MeV) O+ O FIG. 11: The calculated astrophysical S -factor vs E c . m . forthe reactions C+ C ( η = η ( E c . m . = V b ) = 5 .
58) and O+ O ( η = 9 .
01) (solid lines). Comparison of S -factorsfrom the WKB model (dashed lines). The experimental data(symbols) are from Refs. [6, 9–14, 22]. model increase fusion (capture) probability at sub-barrierenergies and decrease at above barrier energies. VI. SUMMARY
In the collisions of light- and medium-mass nuclei atlow sub-barrier energies, the external turning point is lo-cated far from the Coulomb barrier position. This meansa weak overlap of nuclear surfaces and, correspondingly,small friction. To this end, we extended our quantumdiffusion approach and considered the friction dependingon the bombarding energy. Using the extended approach,we compared the calculated capture cross-sections withthe available experimental data. In all cases we obtaineda good description of the experiments. Comparing thefusion excitation functions calculated within the quan-tum diffusion and WKB approaches, we found that the -3 -2 -1 E c . m . c ap e (- ) ( M e V m b ) E c.m. (MeV) C+ O -3 -2 -1 E c . m . c ap e (- ) ( M e V m b ) E c.m. (MeV) C+ Si
14 16 18 20 22 24 26 28 30 32 3410 -1 E c . m . c ap e (- ) ( M e V m b ) E c.m. (MeV) Si+ Si FIG. 12: The calculated astrophysical S -factor vs E c . m . forthe reactions C+ O ( η = 7 . C+ Si ( η = 10 . Si+ Si ( η = 22 . the fluctuation and dissipation increase fusion cross sec-tion at sub-barrier energies. For the reactions C+ C, C+ O, C+ Si, O+ O, and Si+ Si, the max-imum of astrophysical S -factor at E c . m . ≈ V b was pre-dicted. However, more experimental data at low energiesis needed to confirm our predictions. Another interestingbehavior of the obtained S-factor is that its dependenceon E c . m . is quite strong at the collision energies belowthe maximum.In the limit of weak friction, which corresponds toextreme sub-barrier energies, the analytic expression(18) for the capture probability is obtained. This simpleexpression can be applied to the reactions of astro-physical interest. It determines the reaction rates fromwhich, in turn, the astrophysical S -factors are derived.The strong decline of fusion cross sections at sub-barrierenergies considerably reduces the stellar burning ratesand, moreover, leads to severe experimental problems,inhibiting the measurements in many cases. Thisdemands for the reliable theoretical methods, allowingus to extrapolate σ cap ( E c . m . ) into the experimentallyinaccessible regions at extreme sub-barrier energies.V.V.S. acknowledges the Alexander von Humboldt-Stiftung (Bonn). This work was partially supportedby Russian Foundation for Basic Research (Moscow,grant number 17-52-12015) and DFG (Bonn, contractLe439/16). [1] S. Hofmann, Lec. Notes Phys. , 203 (2009); Yu.Ts.Oganessian and V.K. Utyonkov, Nucl. Phys. A , 62 (2015). [2] B. B. Back, H. Esbensen, C. L. Jiang, and K. E. Rehm,Rev. Mod. Phys. , 317 (2014).[3] L. F. Canto, P. R. S. Gomes, R. Donangelo, J. Lubian,and M. S. Hussein, ibid. , 1 (2015).[4] C. Beck, arXiv:1812.08013v1 [nucl-ex].[5] V. V. Sargsyan, H. Lenske, G. G. Adamian, and N. V.Antonenko, Int. J. Mod. Phys. E , 1850063 (2018); ,1850093 (2018).[6] Michael G. Mazarakis and William E. Stephens, Phys.Rev. C , 1280 (1973).[7] B. Cujec and C. A. Barnes, Nucl. Phys. A266 , 461(1976).[8] P. R. Christensen, Z. E. Switkowski and R. A. Dayras,Nucl. Phys.
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