Fusion cross-sections for deuterium cycle fusion reactors (D-cycle): an analysis of geometric, Gamow- Sommerfeld and astrophysical S-factors
Abstract
Fusion reactions in the deuterium cycle (D+D, D+T and D+3He) are the main nucleus-nucleus interactions which occur in tokamaks and stellerators. These reactions are the limiting case between the Woods-Saxon potential field at nuclear distances and the Coulomb electrostatic potential (scattering) at longer distances. In this paper several fusion cross-sections, geometric, Gamow- Sommerfeld and astrophysical S-factors have been reviewed and compared with experimental data from the last ENDF/B-VIII.0 cross-section library. The XDC-fusion code has been developed to calculate fusion cross-sections, geometric, Gamow-Sommerfeld and S-factors of the deuterium-cycle (D-cycle), including resonance parameters (energy and partial width). The software estimates also fusion reaction heat (Q) and Woods-Saxon/Coulomb proximity potentials. Although relative differences between fusion cross-sections are lower than 5 %, S-factors present considerable differences between the energies and partial width (FWHM) of the single-level Breit-Wigner (SLBW) resonances. The energy at which is placed the maximum fusion cross-section is also different between cases. In conclusion, fusion reaction models for light nuclei (deuterium, tritium and helium) should be reviewed in order to apply fusion to energy production in safety conditions.
Keywords: Fusion reaction, fusion cross-section, astrophysical S-factor, Gamow-Sommerfeld factor
1. Introduction
Fusion reactions in the deuterium cycle (D+D, D+T and D+3He) are the main nucleus-nucleus interactions which occur in the deuterium cycle (D-cycle) in tokamaks and stellerators fusion reactors [Smith and Cowley 2010, Donné 2018, Zohm 2019]. Fusion reactions are the limiting case between the Woods-Saxon potential field at nuclear distances and the Coulomb electrostatic potential (scattering) at longer distances. In order to overpass the Coulomb electrostatic barrier and achieve nuclear distances to allow fusion reactions, ionized plasma is confined by a magnetic field inside tokamaks and stellerators [Knoepfel 1978, Woods 2006]
Fusion reactions are characterized by their fusion cross-sections and described by three energy dependent functions: geometric, Gamow-Sommerfeld and astrophysical S-factor. These factors retain the physical characteristics of fusion reaction: a Coulomb electrostatic potential barrier penetration, in a form of an exponential attenuation, G(E), and a strong force interaction after tunneling effect in the nuclear potential well (Woods-Saxon), at the boundary conditions, in a form of an astrophysical S(E) factor or resonance function [Miley et al 1974, Li 2002, Chen 2016].
Fusion cross section is expressed in a simplified form as
f (E) GS where ϕ is the geometric factor, a term inversely proportional with incident particle energy, S is a strong-energy correlated parameter S-factor, called also astrophysical or S-factor, which depends on the boundary conditions between nuclear well and Coulomb electrostatic field and G(E), a tunneling or penetration factor, called Gamow-Sommerfeld factor. In particular, the astrophysical S-factor is a resonance function, based on the classical formula of Breit-Wigner of single-level resonances [Breit and Wigner 1936].
Fusion cross-section of light nuclei were firstly estimated by the Naval Research Laboratory, who published a 5 parameter formula for D+D, D+T and D+3He light nuclei fusion reactions, also known as Duane 5-parameter formula, based on accelerator beam-target experiments [NRL 2019].
Corresponding author: [email protected] Further studies, such as those carried out by Bosch and Hale in 1992, included a 9 parameter formula for Gamow and S-factor, based on the R-matrix of thousand experiments, and an inverse energy dependent geometric factor. However, in the case of D(d,p)T fusion reaction, no resonance peak was included [Bosch and Hale 1992].
In the last decades, fusion cross-sections have been modeled with a complex phase-shift of the incident wavefunction at the boundaries of Coulomb electrostatic potential and Woods-Saxon potential for the nuclear well, including SLBW resonance peaks for all target nuclei (D, T or 3He) [Li et al 2006,2008, Singh et al 2019].
In order to estimate fusion cross-section, geometric, Gamow-Sommerfeld, S-factors and SLBW resonance parameters in the D-cycle, the XDC-fusion code has been developed. Fusion cross-sections equations estimated through different expression have been compared with ENDF/B-VIII.0 cross- section libraries: Duane 5-parameter formula, Bosch-Hale 9-parameter formula and a combined geometric factor based on wavenumber and a complex phase-shift S-factor. 2. The D-cycle
The deuterium cycle of fusion is composed of several interdependent nuclear reactions [Atzeni and Meyer-ter-Vehn 2001, Ball 2019]:
2 2 3 1 1 D + 1 D 1T (1.01 MeV) + 1 H ( 3.02 MeV) 2 2 3 1 1 D + 1 D 2 He (0.82 MeV) + 0 n ( 2.45 MeV) 2 3 4 1 1 D + 1T 2 He (3.52 MeV) + 0 n (14.07 MeV) 2 3 4 1 1 D + 2 He 2 He (3.67 MeV) + 1 H (14.68 MeV) which can be summarized in
2 4 1 1 6 1 D 2 He (3.52 MeV) + 1 H (3.02 MeV) + 0 n (14.07 MeV) 4 1 1 2 He (3.67 MeV) + 1 H (14.68 MeV) + 0 n (2.45 MeV) + 1.83 MeV
Furthermore, protons produced by fusion reactions can also react with low probability as
1 1 2 1 H + 1 H 1 D + e + + 0.93 MeV (99.76 %) 1 1 2 1 H + 1 H + e 1 D + + 1.95 MeV (0.24 %)
Other fusion reactions of the deuterium cycle with a low branching ratio are [Robouch et al 1993, Bystritsky et al 1998]
2 2 4 1 D + 1 D 2 He (0.08 MeV) + γ0(23.77 MeV) and with tritium as target nuclei
2 3 4 1 1 D + 1T 2 He (3.52 MeV) + 0 n (14.07 MeV) 2 3 5 * 1 D + 1T 2 He (0.05 MeV) + γ0(16.65 MeV) 2 3 5 * 1 D + 1T 2 He (3.2 MeV) + γ1(13.50 MeV)
There are other fusion reactions in which the reaction heat is positive, which have not been considered, such as
2 3 3 1 1 D + 1T 2 He + 2 0 n 2 3 3 1 1 1 D + 2 He 2 He + 1 H + 0 n
Table 1 summarizes the main characteristics of the D-cycle fusion: branching ratios (BR), reaction heat (Q) and emergent particle/gamma energies. 3. Geometric, Gamow-Sommerfeld and S-factor
3.1. Geometric factor (ϕ)
In most of the fusion cross-section expressions, geometric factor is expressed as 1/ E . In the last decades, geometric factor has been expressed as the area subtended by a de Broglie wavelength or wavenumber k, which depends on energy [Li et al 2006, 2008]
(E) 2 k 2 with
k 2E / where μ is the reduced mass of the system,
m1m2 / m1 m2
The geometric factor ϕ(E) can also be expressed with two independent terms,
2 1 (E) 2 E
The first term is a conversion factor which is included in some expressions in the Gamow-Sommerfeld or in the S-factor, with units of keV.b in the SI.
3.2. Gamow-Sommerfeld factor
3.2.1 Introduction
The Gamow-Sommerfeld factor G(E) is a penetration factor of the Coulomb barrier by the incident particle and characterize the probability of incident/target nuclei of reaching a fusion reaction. It is expressed with an exponential attenuation function as [Brennan and Coyne 1964, Humblet et al 1987]
1 G(E) exp 2 2 l
with ηl the Sommerfeld parameter, evaluated as
r 1 tp k (r) dr l l rn
with kl the wave number inside the nuclear well
1/ 2 2 k (r) E U(r) l 2
where U(r) is the potential field, rn is the classical distance of closest approach (nuclear square-well radius) and rtp is the classical turning point, that is, the distance in which the energy of the incident particle energy is E U (rtp ) [Lee and Jung 2017] 3.2.2 Case 1: Duane parametrization
Duane parametrization included an interpolation of the Gamow factor as the Mott’s form of the Coulomb barrier penetrability [Mott and Massey 1965, NRL 2019]
1 G(E) exp A1/ E 1
3.2.3 Case 2
Let us consider that the potential field U(r) can be approximated as
2ll 1 U (r) U (r) C 2 r 2
where UC is the Coulomb potential and the second term is the spin-orbit potential, with l the magnetic quantum number.
The Coulomb electrostatic potential is expressed as
Z Z e2 U (r) k 1 2 r r C e r n
with ke 1/ 4 0 the Coulomb electrostatic constant and rn is estimated as
1/3 1/3 rn r0 A1 A2
with r0 the specific radius per nucleon of fusion reaction and the classical turning point rtp as
2 rtp keZ1Z2e / E
The centrifugal term is zero at the boundaries of the nuclear potential well (l=0), then the integral is calculated as [Atzeni 2004]
1 B E E E g arccos 1 0 2 E U U U 0 0 0 where
rtp U0 U (rn ) E rn and
1/ 2 2 2 Bg g 2 keZ1Z2e / 2 is the Gamow constant.
The η0 can be also expressed in term of the lengths rn and rtp as 1 B r r r g n n n 0 arccos 1 2 E r r r tp tp tp
1/ 2 The arccosine can be expanded in powers of E /U 0 , thus
1/ 2 3/ 2 5/ 2 1 Bg E 1 E 3 E E E ... 1 0 2 E 2 U 6 U 40 U U U 0 0 0 0 0
In the low energy limit, with E U 0 , η0 reduces to the Sommerfeld parameter
1 B g 0 2 E and the Gamow factor to
G(E) exp Bg / E
2 The Sommerfeld parameter can be expressed also as 0 Z1Z2 c / v , with e / 40c the fine structure constant, and v the relative velocity of the incident particle, related with the kinetic energy as, E v2 / 2 .
3.2.4 Case 3
In some approximations, the Gamow factor is expressed in a similar way to the Sommerfeld expression as [Yoon and Wong 2000]
2 G(E) exp20 1 which is equivalent to [Li et al 2006,2008]
2 G(E) exp 2 / kac 1
with ac the Coulomb unit of length
2 ac 2 ke Z1Z2e
Other expressions have included relativistic considerations about Gamow-Sommefeld factors, which will be considered in future research [Arbuzov and Kopylova 2012]
3.3. S-factor
3.3.1 Introduction
The S-factor is a function which characterizes the width and energies of the resonance peaks of a nuclear reaction. The Breit-Wigner formula was firstly applied to the non-relativistic scattering of slow neutrons with dependence only on resonance energy position and width [Breit and Wigner 1936] In the case of a single level Breit-Wigner resonance E0, the S-factor is
if S(E) 2 2 E E0 / 2
where E0 is the energy of the single-level resonance, Γi and Γf are the partial widths of the entrance and exit channels of the fusion reaction and j is the total decay width, equal to the width j of the resonance peak at half height [Takano and Ishiguro 1976, Blatt and Weisskopf 1979, Belov et al 2019].
3.3.2 Case 1
The S-factor was interpolated by Duane through a [1/2] order Padé approximation with a constant term as [Duane 1972, Miley et al 1974, NRL 2019]
A2 S(E) 2 A5 A4 A3E 1
where the single-level Breit-Wigner (SLBW) resonance is at E0 A4 / A3 and the total resonance width is 2 / A3 (see table 2) In the case of D+D fusion reactions, in which A5=0, the S-factor equals the classical Breit-Wigner resonance formula.
3.3.3 Case 2
Further approaches increased the order of the interpolation, through a [4/4] order Padé approximation as [Bosch and Hale 1992] A E A E A E A EA S(E) 1 2 3 4 5 1 E B1 E B2 E B3 EB4
Table 3 presents the coefficients for the [4/4] order Padé approximation. In the case of fusion reactions T(d,n)4He and 3He(d,p)4He, the S-factor was interpolated in two different energy regions: [0.5-550 keV] and [550-4700 keV] for T(d,n)4He and [0.3-900 keV] and [900-4800 keV] for 3He(d,p)4He.
3.3.4 Case 3
Let us consider that the Woods-Saxon potential in the nuclear well is modeled as a square-well complex potential as
U S (r) U r iU i
where Ur and Ui are the real and imaginary part of the potential [Mohr 1957].
The complex phase shift ω(δ0) of the incident wavefunction is
2 2 2 (0 ) r ii W / Wr iWi / cot0 /
In this case, the dimensionless S-factor is defined through the function ω(δ0) as [Zhong-Li et al 2004, Li et al 2008, Jing et al 2009, Singh et al 2019] 4 S(E) i 2 2 2 r i 1/
Let us define the complex function
r ii Kr iKi rn Krn
where rn is the radius of the nuclear well, estimated as
1/3 1/3 rn r0 A1 A2
with r0 the specific radius per nucleon of the nuclear well, which depends on target/incident nuclei.
Therefore, the wavenumber K is
2 2 K Kr iKi a ib where
2 a E U 2 r 2 b U 2 i
Equaling real and imaginary parts,
K 2 K 2 a r i
2Kr Ki b
Hence,
1/ 2 a a2 b2 K r 2 b Ki 2Kr
The boundary conditions of the wavefunction Ψ in the nuclear well as
sin kr ' cos Kr (r ) r Kr n cot n n sinkr n sin Kr rrn n and in the Coulomb field as [Landau and Lifshitz 1987]
rn 1 (rn ) 2 cot0 2H ac
Therefore, equaling both conditions r cot n 2H ac then,
2 ac cot(0 ) / cot() 2H rn
The real and imaginary part of the phase function ω are obtained through
ac i Im cot() rn
ac cosr ii Im r ii rn sin r ii
ac cosr coshi isin r sinhi Im r ii rn sin r coshi i cosr sinh i a cos cosh isin sinh sin cosh i cos sinh c r i r i r i r i Im r ii rn sin r coshi i cosr sinh i sin r coshi i cosr sinhi a sin cos (cosh2 sinh 2 ) i sinh cosh (sin 2 cos2 ) c Im i r r i i i i r r r i 2 2 2 2 rn sin r cosh i cos r sinh i 2 2 2 2 r sin r cosr (cosh i sinh i ) i sinh i coshi (sin r cos r ) a i sin cos (cosh2 sinh 2 ) sinh cosh (sin 2 cos2 ) c Im i r r i i r i i r r r sin 2 cosh 2 cos2 sinh 2 n r i r i a sin cos (cosh2 sinh 2 ) sinh cosh (sin 2 cos2 ) c i r r i i r i i r r 2 2 2 2 rn sin r cosh i cos r sinh i a sin 2 sinh 2 c i r r i 2 2 2 2 2rn sin r cosh i cos r sinh i a sin 2 sinh 2 c i r r i 2 2 2 2 2rn sin r 1sinh i 1 sin r sinh i a sin 2 sinh 2 c i r r i 2 2 2rn sin r sinh i and
ac r Re cot() 2H rn 2 2 2 2 r sinr cosr (cosh i sinh i ) i sinh i cosh i (sin r cos r ) a i sin cos (cosh 2 sinh 2 ) sinh cosh (sin 2 cos2 ) c Re i r r i i r i i r r 2H r sin2 cosh 2 cos2 sinh 2 n r i r i a sin cos (cosh 2 sinh 2 ) sinh cosh (sin 2 cos2 ) c r r r i i i i i r r 2H 2 2 2 2 rn sin r cosh i cos r sinh i a sin 2 sinh 2 c r r i i 2H 2 2 2 2 2rn sin r 1sinh i 1 sin r sinh i a sin 2 sinh 2 c r r i i 2H 2 2 2rn sin r sinh i Hence,
2 ac i sin 2r r sinh 2i i Wi / 2 2 2rn sin r sinh i
2 ac r sin 2r i sinh 2i r Wr / 2 2 2H 2rn sin r sinh i
The function H can be expressed as ' 1/ ka c H ln 2kac Re 2A 1/ kac or as
H ln 2kac y kac 2A
with A the Euler constant and y(kac) related to the logarithmic derivative of the Γ function given by
' x 1 1 1 y(x) 1 2 2 2 A ln(x) x x n1 n n x which can be approximated to
1 y(ka ) ka 2 c 12 c
Table 5 presents the nuclear potential well parameters and average atomic radius for the D+D, D+T and D+3He fusion reactions. Figure 1 shows the nuclear Woods-Saxon and Coulomb electrostatic potentials for fusion reactions, whereas figures 2 and 3 presents the complex real and imaginary part of the phase function ω. From the figures, the energies at which ωr=0, corresponding with the SLBW resonances, are 1039.58 keV, 51.97 keV and 218.84 keV for D+D, D+T and D+3He, respectively. 4. Fusion cross-sections
4.1 ENDF library
ENDF/B-VIII.0 (Evaluated Nuclear Data File) is the library containing the last revision of experimental fusion cross-section reactions from Brookhaven National Laboratory, based on accelerator beam- target experiments with incident particle (proton, deuteron, triton and helium-3) into several targets (H, D ,T, Helium) [Brown et al 2018]
ENDF/B-VIII.0 libraries are available in the EXFOR database (Experimental Nuclear Reaction Data), which contains an extensive compilation of experimental nuclear reaction data [Otuka 2014].
Table 5 summarizes all the fusion cross section cases that have been used, including geometric, Gamow and S-factors. These expressions have been compared with ENDF/B-VIII.0 cross sections data for fusion reactions of the D-cycle: D(d,p)T, D(d,n)3He, T(d,n)4He and 3He(d,p)4He.
Plasma temperature has been estimated as
e T (K) E MeV kb
-23 -2 where E is the kinetic energy of the incident ions, e the electron charge and kb=1.3807 x 10 m2 kg s K-1 the Boltzmann constant.
4.2 Case 1
The fusion cross-section σf (E) was firstly characterized by Duane’s through a 5 parameter expression as
1 1 A (E) 2 A f 2 5 E exp A1 / E 1 A3E A4 1 which assumed the geometric factor as a term dependent only of energy, Gamow factor through Mott’s form of the Coulomb barrier penetrability and the astrophysical S-factor as a [2/2] order Padé approximation of the classical Breit-Wigner resonance formula.
4.3 Case 2
In this case, the S-factor is estimated by a [4/4] Padé approximation and Gamow factor by its Gamow constant,
1 1 A E(A E(A E(A EA ))) (E) 1 2 3 4 5 f E exp BG / E 1 E(B1 E(B2 E(B3 EB4 )))
4.4 Case 3
In the last case, fusion cross-section is calculated as
1 2 2 4 (E) i f 2 k 2 exp 2 / ka 1 2 2 2 c r i 1/ where 1/χ2 is the Gamow factor and β is a factor which depends on the target nuclei. 5. XSDC-Fusion code
The XSDC-Fusion code is a software to compute fusion cross sections, geometric, Gamow-Sommerfeld and S-factors in the deuterium cycle (D-cycle) and energy range from 1 keV to 5 MeV, equivalent to plasma temperatures of 11.6 x 106 to 5.80 x 1010 K. The software has been developed in Python 3.8 environment and includes ENDF fusion cross section library and estimates fusion cross-sections, energy resonances and partial widths (FWHM) of all cases above-mentioned. The XSDC-Fusion code calculates also the reaction heat (Q) of the fusion reactions and the parameters of the Woods-Saxon and Coulomb proximity potentials.
6. Results and discussion
Figures 4 to 12 show Gamow-Sommerfeld factors and astrophysical S-factors for different fusion reactions. Significant differences have been observed between Gamow and S-factors, in particular with SLBW resonance energy values and Γ (FWHM).
There are considerable differences between case 1 and the other cases considering resonance peak energy for T(d,n)4He and 3He(d,p)4He reactions. In case 1, energy resonances are placed at 78.7 MeV and 325.9 MeV, respectively, while for cases 2/3 are placed at 49.1/42 MeV and 215.3/207.8 MeV, as showed in table 6.
In case 2, the S-factor of the D(d,p)T reaction does not include the Breit-Wigner resonance peak as cases 1 and 3 in 2798.2 MeV and 931.3 keV, respectively. In addition, case 1 includes a single resonance at 2798.16 keV for reaction D(d,p)T and 3821.43 keV for reaction D(d,n) 3He, respectively, whereas case 2 has a resonance at 5012.7 keV for reaction D(d,n)3He and case 3 has a single resonance in 931.3 keV for both reactions. As a notation, this energy value is equivalent to the conversion factor keV and uma, although no physical correlation has been observed between them.
Furthermore, D+D fusion reaction for case 3 is independent of reaction products (see fig. 11), while other cases have different expressions from D(d,p)T and D(d,n)3He reactions.
Figure 13 presents ENDF/B-VIII.0 fusion cross section data obtained with accelerator-based beam- target experiments in Brookhaven National Laboratory. These fusion cross sections have been used as reference values to compare with other fusion cross-section equations (cases 1, 2 and 3).
Figures 14, 15 and 16 show fusion cross section for D+D, D+T and D+ 3He evaluated through cases 1, 2 and 3, whereas table 6 summarizes resonance energies for all cases and fusion reactions and table 7 shows cross-section maximum values and energy Emax. As observed, resonance peaks are mostly attenuated in all cases and shifted from the energy at which is placed the maximum cross-section values.
Fusion cross section differences are relatively lower between all cases under study (< 5 %) in the energy range of [0.001, 5] MeV, compared with ENDF/B-VIII.0, as observed in figures 17, 18 and 19.
In addition, the energy position of the maximum value of fusion cross-section is also shifted between cases. In cases 2 and 3, fusion reaction 3He(d,p)4He has a saw profile at energies lower than 10 keV. 7. Conclusions
During decades, fusion cross-sections have been estimated with beam-target experiments on accelerators and interpolated through several parametric equations to quantum mechanical models. Fusion cross-sections (σ), S-factors and branching ratios (BR) from these experiments should be compared with real conditions in tokamak and stellerator fusion reactors, in which target particles are not in rest and have similar kinetic and potential energies of incident particles.
S-factors characterize fusion reactions in a more detailed way than cross sections, showing SLBW resonance peaks at different energies. However, there are considerable differences between resonance energies and Γ (FWHM) as observed.
Furthermore, although there are other types of Wood-Saxon proximity potentials instead of the square-well potential (i.e. optical model), those models have been applied to fusion of heavy nuclei (Z>8) [Chepurnov 1967, Denisov 2002, Aygun 2018]
In conclusion, fusion reaction models for light nuclei (deuterium, tritium and helium) should be reviewed in order to apply fusion to energy production in safety conditions.
8. References
Arbuzov AB and Kopylova TV. On relativization of the Sommerfeld-Gamow-Sakharov factor. J High Ener Phys 9 (2012)
Atzeni S and Meyer-ter-Vehn. Nuclear fusion reactions. Oxford University Press. ISBN: 978-0-19- 85624-1 (2001)
Aygun M. Comparative analysis of proximity potentials to describe scattering of 13C, 16O, 28Si and 208Pb nuclei. Rev Mex Fis E vol 64 n.2 (2018)
Ball J. Maximizing specific energy by breeding deuterium. ArXiv:1908.00834 [physics.pop-ph] doi: 10.1088/1741-4326/ab394c
Belov AA, Kalitkin NN and Kozlitin IA. Refinement of thermonuclear reaction rates. Fusion Engineering and Design 141, 51-58 (2019)
Blatt JM and Weisskopf VF. Theoretical Nuclear Physics (Springer-Verlag, 1979)
Breit G and Wigner E. Capture of Slow Neutrons, Phys. Rev. 49, 519 (1936)
Brennan JG and Coyne JJ. Energy Dependence of the D-D reaction cross section at low Energies. J Res Natl Bur Stand A Phys Chem 68A(6): 675-682 (1964)
Bosch and Hale 1992. Improved formulas for fusion cross-sections and thermal reactivities. NUCLEAR FUSION, vol 32, N0.4 (1992)
Brown DA et al. ENDF/B-VIII.0: The 8th major release of the nuclear reaction data library with CIELO- project cross sections, new standards and thermal scattering data. Nucl Data Sheets 148(2018)1
Bystritsky VM, Grebenyuk VM, Parzhitski SS, Pen’kov FM, Sidorov VT and Stolupin VA. Experimental Investigation of dd reaction in range of ultralow energies using Z-pinch. D15-98-239. Nucl Inst Meth (1998)
Chen FF. Introduction to Plasma Physics and Controlled Fusion (vol 1) Springer (2016) ISBN 978-3-319-22309-4
Chepurnov VA and Fiz Y. 955 (1967)
Denisov VY. Interaction potential between heavy ions. Phys Letters B 526 315-321 (2002)
Donné AJH. The European roadmap towards fusion electricity. Phil. Trans. R. Soc. A 377: 20170432. (2018) doi:10.1098/rsta.2017.0432
Duane BH. Fusion Cross Section Theory, in Rept. BNWL-1685 (BattellePacific Northwest Laboratory, USA 1972)
Dudek J, Majhofer A, Skalski J, Werner T, Cwiokand S and Nazarewicz W. J. Phys.G 5,101359 (1979)
Dudek J, Szymanski Z, Werner T, Faessler A and Lima C. Phys. Rev.C261712 (1982)
Englefield MJ. Solution of Coulomb problem by Laplace Transform. J Math Anal Appl. 48, 270-275 (1974)
Ghasemi R and Sadeghi H. S-factor for radiative capture reactions for light nuclei at astrophysical energies. Results in Physics 9 151-165 (2018) Green AES. Nuclear Physics, McGraw-Hill, New York 1955.
Greiner W. Quantum Mechanics, 2nd ed. Springer-Verlag Berlin Heidelberg NY (USA) 1993.
Humblet J, Fowler WA and Zimmerman BA. Approximate penetration factors for nuclear reactions of astrophysical interest. Astron Astrophys 177, 317-325 (1987)
Jing Z, Yuan-Yong F, Shu-Hua Z, Hai-Long X, Cheng-Bo Li and Qiu-Ying M. Measurement of the astrophysical S factor for the low energy 2H(d,gamma)4He reaction. Chinese Phys C, 33, 5 (2009)
Kim Y, Mack JM, Herrmann HW, Young CS et al. Determination of the deuterium-tritium branching ratio based on inertial confinement fusion implosions. Phys Rev C85, 061601(R) (2012)
Kiptily VG. On the Core Deuterium-Tritium Fuel Ratio and Temperature Measurements in DEMO. Nuclear Fusion, 55 (2) (2015) doi: 10.1088/0029-5515/55/2/023008
Knoepfel H. Tokamak Reactors for Breakeven.A Critical Study of the Near-Term Fusion Reactor Program. Pergamon (1978)
Landau LD and Lifshitz EM. Quantum Mechanics (Pergamon, Oxford 1987) p572
Langenbrunner JL, Weller HR and Tilley DR. Two-deuteron radiative capture: Polarization observables at Ed ≤ 15 MeV. Phys Rev C 42, 1214 (1990)
Lee M and Jung Y. Quantum shielding effects on the Gamow penetration factor for nuclear fusion reaction in quantum plasmas. Phys Plasmas 24, 014502 (2017)
Li XZ. Nuclear physics for nuclear fusion. Fusion Science and Technology (2002) doi:10.13182/FST02-A201
Li XZ et al. Fusion cross sections for fusion energy. Fusion Eng and Design 81, 1517-1520 (2006)
Li XZ, Wei QM and Liu B. A new simple formula for fusion cross-sections of light nuclei. Nucl Fusion 48, 125003 (2008)
Mott NF and Massey HS. The theory of atomic collisions. Oxford University Press, London (1965)
Miley GH, Towner H and Ivich N. Fusion cross sections and reactivities. Rept COO-2218-17 (University of Illinois, Urbana, IL, USA 1974) Np (1974) doi:10.2172/4014032
Nishitani T, Shibata Y, Tobita K and Kusama Y. Fusion gamma-ray diagnostics for D-3He experiments in JT-60U. Review Scientific Instruments 72, 877 (2001) doi.org/10.1063/1.1321004
Nocente M, Källne J, Salewski M, Tardocchi M and Gorini G. Gamma-ray emission spectrum from thermonuclear fusion reactions without intrinsic broadening. Nuclear Fusion 55, 12 (2015). doi:10.1088/0029-5515/55/12/123009
NRL Plasma Formulary (2019) https://www.nrl.navy.mil/ppd/content/nrl-plasma-formulary
Ongena J. Nuclear fusion and its large potential for the future world energy supply. Nukleonika; 61(4): 425-432. (2016) doi:10.1515/nuka-2016-0070
Otuka et al. Towards a More Complete and Accurate Experimental Nuclear Reaction Data Library (EXFOR): International Collaboration Between Nuclear Reaction Data Centres (NRDC) Nucl Data Sheets 120, 272-276 (2014) Rajbongshi T and Kalita K. Systematic study of deformation effects on fusion cross-sections using various proximity potentials. Open Physics 12 (6) (2014) doi:10.2478/s11534-014-0451-1
Ran Y, Xue L, Hu S and Su RK. On the Coulomb-type potential of the one-dimensional Schrödinger equation. J Phys A: Math Gen 33 9265-9272 (2000)
Robouch BV et al. Gamma diagnostics of thermonuclear plasma using D(d,gamma) 4He reaction: Feasibility study. ETDE-IT-94-23 IAEA (1993)
Singh V, Atta D, Khan MA and Basu DN. Astrophysical S-factor for deep sub-barrier fusion reactions of light nuclei. Nuclear Physics A (986) 98-106 (2019)
Smith CL and Cowley S. The path to fusion power. Phyl Trans R Soc 368, 1091-1108 (2010) doi:10.1098/rsta.2009.0216
Takano H and Ishiguro Y. Multi-level correction to Breit-Wigner Single-Level Formula. Jour Nucl Sci and Tech 14(9), 627-639 (1977)
Woods LC. Theory of Tokamak Transport (Wiley-VCH, 2006)
Yoon JH and Wong CY. Relativistic modification of the Gamow factor. Phys Rev C 61, 044905 (2000) doi:10.1103/PhysRevC.61.044905
Zhom H. On the size of tokamak fusion power plants. Phil. Trans. R. Soc. A 377: 20170437. doi:10.1098/rsta.2017.0437
Zhong-Li X, Liu B, Chen S, Ming Wei Q and Hora H. Fusion cross-sections for inertial fusion energy. Laser and Particle Beams 22, 469-477 (2004) doi:10.1017/S026303460404011X Tables
Table 1 - Fusion reactions of the D-cycle between deuterium and other light nuclei [Langenbrunner et al 1990, Jing et al 2009, Kiptily 2014, Ongena 2016] Energy released per reaction product
Reaction BRi Q (MeV) [MeV] (a) (b) T+p ~ 0.45 -4.03 1.01 3.02 D+D 3He+n ~ 0.55 -3.27 0.82 2.45 4He+γ 10-7 -10-6 -23.85 0.08 23.77 3He+2n - +2.98 - - The D-cycle D+T 4He+n ~ 1 -17.59 3.52 14.07 5He+γ(a) 5x10-5 - 5x10-4 -16.70 0.05 / 3.2 16.65 / 13.50 3He+p+n - +2.22 - - D+3He 4He+p ~ 1 -18.35 3.67 14.68 5Li+γ (b) 5x10-5 - 5x10-4 -16.39 0.05 16.34
(a) D+T branching ratio by inertial confinement fusion (ICF) [Kim et al 2012] (b) D+3He branching ratio on JT-60U tokamak [Nishitani et al 2000]
Table 2 - Fusion cross-section parameters for Duane’s 5 parameter equation [Miley et al 1974, NRL 2019]
D(d,p)T D(d,n)3He T(d,n)4He 3He(d,p)4He Units Comments 1/2 A1 46.097 47.88 45.95 89.27 keV Gamow constant (BG) A2 372 482 5.02 x 104 2.59 x 104 keV.b - A3 4.36 x 10-4 3.08 x 10-4 1.368 x 10-2 3.98 x 10-3 keV - A4 1.22 1.177 1.076 1.297 - - A5 0 0 409 647 keV.b -
E0 2798.16 3821.43 78.65 325.88 keV Resonance energy E0=A4dd1/A3dd1
Γa 4587.16 6493.51 146.20 502.51 keV Resonance width Γa=2/A3dd1
Table 3 - S-function (Padé expansion coefficients) [Bosch and Hale 1992]
T(d,n)4He 3He(d,p)4He D(d,p)T D(d,n)3He Units [0.5-550 keV] [550-4700 keV] [0.3-900 keV] [900-4800 keV] 1/2 BG 31.3970 31.3970 34.3827 34.3827 68.7508 68.7508 keV A1 5.5576 x 104 5.3701 x 104 6.927 x 104 -1.4714 x 106 5.7501 x 106 -8.3993 x 105 keV.mb A2 2.1054 x 102 3.3027 x 102 7.454 x 108 0.0 2.5226x 103 0.0 mb A3 -3.2638 x 10-2 -1.2706 x 10-1 2.050 x 106 0.0 4.5566x 101 0.0 mb.keV-1 A4 1.4987 x 10-6 2.9327 x 10-5 5.2002 x 104 0.0 0.0 0.0 mb.keV-2 A5 1.8181 x 10-10 -2.5151 x 10-9 0.0 0.0 0.0 0.0 mb.keV-3 B1 0.0 0.0 6.38 x 101 -8.4127 x 10-3 -3.1995 x 10-3 -2.6830 x 10-3 keV-1 B2 0.0 0.0 -9.95 x 10-1 4.7983 x 10-6 -8.5530 x 10-6 1.1633 x 10-6 keV-2 B3 0.0 0.0 6.981 x 10-5 -1.0748 x 10-9 5.9014 x 10-8 -2.1332 x 10-10 keV-3 B4 0.0 0.0 1.728 x 10-4 8.5184 x 10-14 0.0 1.4250 x 10-14 keV-4
Table 4 - Averaged nuclear radius r12, Wood-Saxon nuclear potential well
US = Ur + iUi , Coulomb barrier UC and factor β [Li et al 2008, Singh et al 2018]
D(d,p)T / D(d,n)3He T(d,n)4He 3He(d,p)4He Units
r0 2.778 1.887 3.331 fm
r1 7.0 5.1 9.0 fm 2
Ur -48.52 -40.69 -11.859 MeV
Ui -263.27 -109.18 -259.02 keV U 205.71 282.42 319.97 keV C β 2 2 2 2 - Table 5 - Comparison of geometric, Gamow and S-factors of several fusion cross-section expressions
Case Φ(E) G(E) S(E) Comments Reference 1 Duane’s 5 parameter A1 A2 formula with a 1 1/ E exp 1 A5 [NRL 2019] E 2 [2/2] order Padé A4 A3E 1 approximation of S(E) 1 Bosch-Hale formula B A1 E(A2 E(A3 E(A4 EA5 ))) with a [4/4] order [Bosch and 2 1/ E exp G 1 E(B E(B E(B EB )))Padé approximation of Hale 1992] E 1 2 3 4 S(E) 1 Woods-Saxon square 2 4i complex potential [Li et al 2006, 3 / k 2 2 exp 1 2 2 2 with phase shiftω (δ0) 2008] kac r i 1/ boundary conditions
Table 6 - Comparison of resonance parameters for fusion reactions
Case 1 Case 2 Case 3 Units 2798. D(d,p)T - 2 931.3 3 3821. 5012. E0 D(d,n) He keV 4 7 T(d,n)4He 78.7 49.10 42 3He(d,p)4He 325.9 215.3 207.8 4587. D(d,p)T - 846.4 2 6493. 5517. Γ D(d,n)3He 846.4 5 9 keV (FWHM) T(d,n)4He 147.4 81.2 82.1 287.5 3He(d,p)4He 515.2 172.7 5 D(d,p)T 0.14 0.05 0.04 S(0) D(d,n)3He 0.20 0.05 0.04 MeV.b at 1 keV T(d,n)4He 23.99 11.71 4.25 3He(d,p)4He 10.34 5.77 1.23 D(d,p)T 0.37 - 0.21 D(d,n)3He 0.48 0.62 0.21 S(E) at E0 MeV.b T(d,n)4He 50.61 27.29 8.47 3He(d,p)4He 26.55 16.69 8.19
Table 7 - Comparison of fusion cross section parameters for fusion reactions
ENDF Case 1 Case 2 Case 3 Units D(d,p)T 1700 2550 7999 912 D(d,n)3He 1200 3400 1109 912 Emax keV T(d,n)4He 64.8 113.3 64.8 62.3 3He(d,p)4He 247.8 430.4 262.3 230.1 D(d,p)T 8.9 x 10-2 9.6 x 10-2 1.0 x 10-1 8.9 x 10-2 D(d,n)3He 0.104 0.109 0.105 0.089 σf at Emax barn T(d,n)4He 5.01 4.94 5.04 5.00 3He(d,p)4He 0.81 0.72 0.81 0.80 Figures
Coulomb barrier Turning point
E=UC(r) projectile Tunneling
Fusion
Figure 1 - Woods-Saxon square-well (left) and Coulomb electrostatic proximity potential (right) for fusion reactions
Figure 2 - Complex phase shift ωr(E) of fusion reactions (Re)
Figure 3 - Complex phase shift ωi(E) of fusion reactions (Im) Figure 4 - Gamow factors as a function of center-of-mass energy E from Duane’s 5 parameter equation [NRL 2019]
Figure 5 - Gamow factors as a function of center-of-mass energy E (reduced Mott’s expression)
Figure 6 - Gamow factors (1/χ2) as a function of center-of-mass energy E Figure 7 - S-factor of D+D fusion reactions as a function of center-of-mass energy E (Case 1)
Figure 8 - S-factor of D+T and D+3He fusion reactions as a function of center-of-mass energy E (Case 1)
Figure 9 - S-factor for D+D fusion reaction as a function of center-of-mass energy E (Case 2) Figure 10 - S-factor for D+T and D+3He fusion reactions as a function of center-of-mass energy E (Case 2)
Figure 11 - S-function for D+D fusion reaction as a function of center-of-mass energy E (Case 3)
Figure 12 - S-function for D+T/D+3He fusion reactions as a function of center-of-mass energy E (Case 3) Figure 13 - Fusion cross-sections for D+D/D+T/D+3He reactions as a function of center-of-mass energy E from ENDF/B-VIII.0 (Reference)
Figure 14 - Fusion cross-sections for D+D/D+T/D+3He reactions as a function of center-of-mass energy E (Case 1)
Figure 15 - Fusion cross-sections for D+D/D+T/D+3He reactions as a function of center-of-mass energy E (Case 2) Figure 16 - Fusion cross-sections for D+D/D+T/D+3He reactions as a function of center-of-mass energy E (Case 3)
Figure 17 - Relative differences of fusion cross-sections between case 1 formula and ENDF
Figure 18 - Relative differences of fusion cross-sections between case 2 and ENDF Figure 19 - Relative differences of fusion cross-sections between case 3 formula and ENDF