Research Article
Research Article
Lawson criterion for different scenarios of using D-3He fuel in fusion reactors*
expand article infoAlexandr I. Godes, Vladimir L. Shablov
‡ IATE MEPhI, Obninsk, Russia
Open Access


The paper is devoted to refining the Lawson criterion for three scenarios of using D-3He fuel in fusion reactors (fully catalyzed and non-catalysed D-D cycles and a D-3He cycle with 3He self-supply). To this end, a new parameterization of the D + 3He → p + 4He fusion reaction cross-section and astrophysical factor has been developed based on the effective radius approximation (Landau-Smorodinsky-Bethe approximation), which is a model-free theoretical approach to investigating near-threshold nuclear reactions, including resonant reactions. In the framework of this approximation, experimental data from studies in the NACRE II and EXFOR libraries, believed to provide the most reliable results to date, have been described within the accuracy declared in the studies in question in the energy range of 0 to 1000 keV, and the fusion reactivity averaged over the Maxwell distribution has been calculated. The results obtained are in good agreement with the calculations based on the R-matrix theory and the NACRE II fusion reactivity data. For the fully catalyzed D-D cycle and the cycle with 3He self-supply, the Lawson criterion and the triple Lawson criterion have been calculated based on solving the equations of the stationary process kinetics in a fusion reactor for three fuel ions (D, 3He, and T) taking into account the potential for external supply of 3He and p and 4He impurity ions removed from the reaction zone. The parameters of the triple Lawson criterion found are as follows: nτT = 6.42∙1016 cm-3∙s∙keV (T = 54 keV) for the fully catalyzed D-D cycle, nτT = 1.03∙1017 cm-3∙s∙keV (T = 45 keV) for the cycle with 3He self-supply, and nτT = 4.89∙1016 cm-3∙s∙keV (T = 67 keV) for the non-catalyzed D-D cycle with equimolar D-3He fuel.


thermonuclear reactions, effective radius approximation, cross-section and astrophysical factor of D + 3He → p + 4He fusion reaction, Lawson criterion for different D-3He fuel use scenarios


Since the mid-1950s, the purpose of investigations in the field of controlled nuclear fusion (CNF) has been to reach and exceed the Lawson criterion (Lawson 1957; Zhdanov et al. 2017; Wurzel and Hsu 2022) that defines the conditions for achieving a self-sustained nuclear reaction. This requires three conditions as follows to be simultaneously satisfied:

  • thermonuclear fuel needs to be heated to such temperatures with which the kinetic energy of nuclei becomes sufficient for their tunneling through the Coulomb barrier with a noticeable probability;
  • the concentration of the plasma initiated in the course of heating needs to be rather high;
  • the plasma temperature and concentration need to be maintained for a definite time, τ, referred to as the plasma confinement time.

In the simplest case, when one does not take into account different conversion efficiencies of the energy generated in the fusion reactor (FR), which is used then in part or in full to heat and confine the plasma and is neglected by the radiation losses, the Lawson criterion has the form

neτ ≥ 12kT⁄(⟨σνQ), (1)

where k is the Boltzmann constant, ⟨σν⟩ is the fusion reactivity the value of which is assumed for isothermal plasma, that is, for temperature Т (it is assumed that there is a dominant reaction as in the event of deuterium-tritium plasma), and Q is the kinetic energy of the charged particles resulting from the fusion reaction. Despite being approximate, formula (1) is of a universal nature (that is, does not practically depend on the thermonuclear facility type) and gives a reference value for the plasma confinement parameter neτ, the achievement of which leads to a self-sustained fusion reaction. Besides, formula (1) shows the important role played by the correct definition of quantity ⟨σν⟩. A change of even several percent in the reaction rate is capable to affect in a noticeable way the parameters of advanced thermonuclear facilities, specifically, the plasma confinement parameter. It is exactly the Lawson criterion that defines the smallest possible frequency of fusion reactions per second required for sustaining steadily the reaction in a material medium. Although this quantity fits the reaction cross-section averaging based on the Maxwell distribution with temperature T in the situation under consideration (while real processes are not necessarily described by the Maxwell distribution, as, for example, in the event of plasma in an strong magnetic field (Artsimovich 1964), ⟨σν⟩ is also the reference value for calculating real fusion systems, this case requiring knowing the reaction reactivity in a broad temperature range. We shall point to astrophysical investigations, another field of science, which requires knowing values ⟨σν⟩ with a good accuracy.

The purpose of this study is to determine more precisely the Lawson criterion for fusion reactors on the base of D-3He fuel viewed as the immediate competitor to D-Т fuel. The major advantage of a D-3He reactor, as compared with a D-T reactor, is its low level of the neutron flux from plasma with which the lifetime of the reactor first wall is expected to reach 30 to 40 years (Khvesyuk and Chirkov 2000; Ryzhkov and Chirkov 2017). The D-3He fuel cycle is characterized, however, by much higher Lawson criterion values. We shall remind that, as of the late 1950s, the Lawson criterion was neτ = 1014 cm-3·s (Т = 17 keV) for equimolar D-Т fuel, and neτ = 1015 cm-3·s (Т = 87 keV) for equimolar D-3He fuel (Shirokov and Yudin 1980), as more current estimates are as follows (Basko 2007): neτ = 1.6·1014 cm-3·s (Т = 26 keV) for D-Т fuel; neτ = 8.1·1014 cm-3·s (Т = 105 keV) for D-3He fuel. The differences that have occurred are associated to a large extent with determining more accurately the temperature dependence of the reaction reactivity. The paper will present the results of calculating the temperature dependence of the D + 3He → p + 4He reaction reactivity based on the parametrization of its astrophysical factor in the framework of the effective radius approximation (Bethe 1949; Barit and Sergeev 1969; Landau and Lifshitz 1977; Karnakov et al. 1991), and a comparison will be provided against three most common parametrizations of value ⟨σν⟩. Three options will be considered for the D-3He fuel cycle (Khvesyuk and Chirkov 2000; Stott 2005):

  • a fully catalyzed D-D cycle in which the generated tritium and 3He are fully used as secondary fuel;
  • a non-catalyzed D-D cycle when intermediate 3He and T are removed after they give up most of their thermonuclear kinetic energy for plasma heating but prior to further combustion with deuterium;
  • 3He self-supply mode.

Parametrization of the astrophysical factor and temperature dependence of the D + 3He → p + 4He reaction reactivity in different approaches

Since the 1950s, multiple experimental and theoretical studies have been undertaken on the fusion cross-sections and thermal reactivities (see the overview in Bosch and Hale 1992). Different parametrizations were developed for the reaction cross-sections and reactivities (Kozlov 1962; Fowler et al. 1967; Peres 1979; Caughlan and Fowler 1988; Belov and Kalitkin 2017; Godes et al. 2019), the differences in which are associated both with their methodological basis and the fact that they were created at different times and were based so on varying experimental material. As applied to the D + 3He → p + 4He reaction, the developed parametrizations, which are resonant in the low-energy region, can be divided into two groups:

Group 1

Parametrizations based on physical models and approximations, including:

  • the R-matrix Wigner theory that formed the basis for the parametrizations of cross-sections and rates for the key thermonuclear reactions in Bosch and Hale 1992;
  • the Breit-Wigner approximation used in Kozlov 1962; Fowler et al. 1967; Caughlan and Fowler 1988;
  • a model-free approach – the effective radius approximation (ERA) (Barit and Sergeev 1969; Karnakov et al. 1991);
  • the resonance coupled channel model (Godes et al. 2019).

The advantage of these parametrizations is their immediate physical meaning, correct threshold behavior, use of few adjustable parameters, and possibility for being practically employed in other fields of physics, e.g., in nuclear spectroscopy, and the disadvantage is a limited applicability area.

Group 2

Parametrizations based on mathematical methods for approximation of experimental data, including:

  • a Padé approximation of the reaction cross-section and rate with the correct threshold behavior (Peres 1979) used further in Bosch and Hale 1992;
  • the regularized double period method (Belov and Kalitkin 2017), in which regularization by the A.N. Tikhonov stabilizer is used to exclude nonphysical oscillations of an approximating trigonometric series caused by major errors in experimental points.

Both methods are suitable for solving a specialized problem, that is, to determine the temperature dependence of the reaction reactivities but are unfit for addressing other problems. Besides, the double period method requires a large number of experimental points which is problematic in the event of the reaction of interest since the consideration includes studies with an insufficiently defined methodological framework (see the discussion in (Moller and Besenbaher 1980; Krauss et al. 1987; Bosch and Hale 1992; Geist et al. 1999).

The results of this study will be compared with data from the most common of the current parametrizations (Kozlov 1962; Caughlan and Fowler 1988; Bosch and Hale 1992) and NACRE II (Xua et al. 2013).

The NACRE II (Nuclear Astrophysics Compilation of Reactions) parametrization contains data on the rates of 34 exoergic reactions caused by charged particles with a mass number of A < 16. In NACRE II, the reaction rates are presented in a tabulated format in a temperature range of 1·106 K ≤ T ≤ 1·1010 K and contains experimental data from before 2013. The tables present values of the low, high and adopted estimates for value NAσν⟩. NACRE II is based on calculations of cross-sections using different theoretical models (the distorted wave Born approximation, the Breit-Wigner approximation, etc.).

We shall note that in the event of the D + 3He → p + 4He reaction, the previous NACRE version uses the results obtained in Fowler et al. 1967, Caughlan and Fowler 1988, based on the Breit-Wigner approximation with a constant width, though it would be more reasonable to employ an approach using an approximation with an energy-dependent width (Wildermuth and Tang 1999) for the near-threshold thermonuclear resonance of the 5Li** nucleus through which this reaction proceeds in the low-energy region. In NACRE II, resonance reactions are described using the distorted waves Born approximation (for the non-resonance part of the amplitude) combined with the Breit-Wigner approximation (for the resonance part of the amplitude) taking into account, where required, the dependence of the resonance width on energy.

Parametrization of the astrophysical factor and temperature dependence on the D + 3He → p + 4He reaction rate based on the effective radius approximation

The effective radius approximation (ERA) is a model-free approach and operates on experimentally observed quantities, including scattering length, effective radius and potential shape parameter.

For low-energy scattering in a system of two charged particles, this approximation is based on the following expression for the S-matrix element S11(E) of the s-wave elastic scattering

S11(E)=e2iσ0(E)ctgδ0(k)+ictgδ0(k)-i, (2)

where σ0 (E) = argΓ (1 + ) is the Coulomb scattering phase, and η = η(k)is the Coulomb parameter

η=Z1Z2e2mr2k (3)

or η = (kac)–1; ac is the Bohr radius for a pair of synthesized nuclei with reduced mass mr:

ac = ℏ2⁄(Z1Z2e2mr). (4)

In the framework of ERA, the nuclear – сoulomb shift, δ0(k), is determined by the expression

ac −1 [D (k)ctgδ0(k) + 2h (k)] = −a0−1 + 0.5r0k2, (5)

where a0 is the scattering length, r0 is the effective radius, and D (k) is the Coulomb barrier penetrability

D (k) = 2π⁄(e2πη − 1), (6)

h (k) = Reψ () − ln(η) = Reψ (i⁄(kac)) + ln(kac), (7)

where ψ(z) is the logarithmic derivative of the gamma function. In the event of absorption, if any, the nuclear-сoulomb shift becomes a complex value, and the scattering length and the effective radius become so complex values as well (Karnakov et al. 1991).

Then the following equality takes place

ω (k) = D (k)ctgδ0(k) − iD (k) = φ (k2) − 2h (k) (8)

with the function φ(k2) of the form

φ (k2)= −aca0 + 0.5r0ack2 = a0 + a1k2i (β0 + β1ac2k2). (9)

In a general case, the terms with k4 and k6 may be taken into account in the function φ(k2). As the result, the reaction cross-section is equal to

σr (E) = g (πk2)(1 −|S11|2), (10)

where g is the spin factor, (2J + 1)/[(2S1 +1)(2S2 +1)], equal, in the event of the D + 3He → p + 4He reaction, to 2/3, or

σr (E) = 8πβ (k)D (k)⁄(3k2|ω (k)|2). (11)

We shall present the expression (11) as

σr(E)=8π3k2β(k)D(k)a(k)-2h(k)2+β(k)+D(k)2 , (12)

α (k) = α0 + α1(kac)2 + α2(kac)4, β (k) = β0 + β1(kac)2 + β2(kac)4.

From (12), the following expression is obtained for the astrophysical factor S (E)=Ee2πηαr (E):

S(E)=8π223mr(1-e-2πη)β(k)(α(k)-2h(k))2+(β(k)+D(k))2 , (13)

or (taking into account numerical factors)

S(E)=9.11(1-e-2πη)β(k)(α(k)-2h(k))2+(β(K)+D(k))2(MeV·b) . (14)

ERA applicability condition: parameter kR ≤ 1 where R is the radius of action for nuclear forces (Landau and Lifshitz 1977), approximately equal, in accordance with Xua et al. 2013, to 1 Fm, so, with E ≤ 1 MeV, the said parameter does not exceed 0.25.

Besides, ERA is not applicable with E ≤ 0.02 keV when the laboratory electron screening effects manifest themselves in the nuclear reaction cross-sections (Xua et al. 2013).

The ERA parameters, using which the astrophysical factor in Fig. 1 has been calculated and which ensure the approximation of the currently most reliable experimental data from Moller and Besenbaher 1980, Krauss et al. 1987, Geist et al. 1999 with the accuracy stated therein, are as follows:

Figure 1. 

The astrophysical S-factor of the D + 3He → p + 4He reaction.

α 0 = 0.117002, α1 = 0.191855, α2 = −0.01225, β0 = 0.00937, β1 = 0.006658, β2 = 0.000582. (15)

The presented parameters also agree with the data on the elastic D-3He scattering Balashko 1965 (Fig. 2).

Figure 2. 

The ratio of the elastic D-3He scattering cross-section to the Rutherford cross-section at a scattering angle of 90 °C in the mass center system.

Another set of parameters based on experimental data in Moller and Besenbaher 1980, Krauss et al. 1987 was obtained in Alper et al. 2021:

α 0 = 0.05431, α1 = 0.25077, α2 = −0.02825, β0 = 0.00205, β1 = 0.00707, β2 = 0.00169.

The elastic scattering description in this case is somewhat worse and is not provided herein.

The fusion reaction rate is determined as:

σν⟩ = ∫0 νσ (E)F (E)dE, (16)

Where F (E) is the function of the Maxwell distribution by energy:

F(E)=2π1(kT)3/2Ee-E/(kT) , (17)

The results of calculating the temperature dependence of the rate of the reaction under investigation are shown in Fig. 3.

Figure 3. 

The temperature dependence of the rate of the reaction.

Lawson criterion for different scenarios of using D-3He fuel in fusion reactors

In the currently most common form, the Lawson criterion for fusion reactors with magnetic plasma confinement is written as follows Wurzel and Hsu 2022, Basko 2007, Stott 2005:

= 1.5(1 + ⟨Z⟩)T⁄[(Q−1 + fc)AfAbrAee], (18)

where T is the plasma temperature, keV, n is the concentration of ions, Q is the energy gain factor, e.g. the ratio of the volume-average fusion energy to the volume-average energy delivered to the plasma from external sources, fc is the part of the thermonuclear energy absorbed in plasma, Af is the function that defines the fusion power, Abr and Ae-e are the volume-average power losses via bremsstrahlung radiation on ions and on electrons respectively, ⟨Z⟩=∑Zjnj/∑nj is the average charge of plasma ions, and ne=⟨Zn is the concentration of electrons. Af and fcAf are defined in the expression (18) by the relations

Afn 2 = 18533⟨σνDHenDnHe + 0.5(3269⟨σνDDnHe + 4033⟨σνDDpT)nD2 + 17589⟨σν⟩DTnDnT,

(1 − fc) Afn2 = 1225⟨σνDDnHe nD2 + 14028⟨σνDTnDnT, (19)

such that (1 – fc)Af n2 is the power comes out via the neutrons.

The neutron energy of 14.028 MeV from the DT → nα reaction was obtained using the alpha particle mass of 4.001506 a.m.u. (Tiesinga et al. 2021) and with regard for the relativistic effects (see the Appendix 1). Functions Abr and Ae-e were taken from Khvesyuk and Chirkov 2000. There is no term in (18) that fits the synchrotron radiation – such radiation believed to be fully absorbed in plasma.

We shall consider the fully catalyzed D-D cycle when the Т and 3He formed remain in plasma and burn together with deuterium.

The steady-state operation kinetics of a quasi-infinite FR is defined by the condition that there are constant concentrations of D, 3He and Т fuel ions and p and 4He impurity ions maintained in plasma. The plasma is assumed to be isothermal, and helium ions are assumed to be doubly ionized. Taking into account the main values in terms of energy production and reaction rates in the temperature region of interest of 50 to 150 keV, the kinetics equations for the D, 3He and Т fuel ions and the protons and alpha particles to be removed from the reaction zone are written as follows:

dnDdt=σνDHepαnDnHe0.5(σνDDpT+σνDDnHe)nD2σνDTnαnDnT+dnD_extdt (20)

dnHedt=-σνDHepαnDnHe+0.5σνDDnHenD2+dnHe_ext dt (21)

dnTdt=0.5σνDDpTnD2-σνDDnαnDnT-λnT (22)

dnαdt=σνDHepαnDnHe+σνDTnαnDnT-Γα (23)

dnpdt=σνDHepαnDnHe+0.5σνDTpTnD2-Γp. (24)

dn D_ext /dt describes the FR supply with deuterium, and dnHe_ext /dt describes the FR supply with3He; λ in (22) is the rate of the 3H beta decay; Гα and Гр describe the removal of alpha particles and protons from the FR: Γj = njj, (τj is the confinement time for particles of type j). The presented form of the kinetics equations means, specifically, that the hydrogen and carbon cycle reactions, the Т + Т, Т + 3He and 3He + 3He reactions, and the energy of the tritium beta decay are neglected. Besides, the presented system of equations does not take into account the escape of fuel ions from the reaction zone as it was done, e.g., in Stott 2005. Since dnT/dt = 0 the equilibrium concentration of tritium is equal to

nT=0.5σνDDpTσνDTnαnD+λnD . (25)

Formula (25) allows estimating the contribution from the tritium beta decay for which λnT and ⟨σνDT→nα nDnT need to be compared. We shall take into account that λ ≈ 2.9∙1⁻9 s⁻1 and in a temperature range of 1·108 to 1·109 K:

σνDTnD~(10−18 ÷ 2 ∙ 10−15)nDcm3 ∙ s−1 (26)

so with nD ≥ 1 ∙ 1012см–3, the contribution from beta decay in (25) may be neglected. Then

nT=0.5σνDDpTσνDTnα=αγTn,γT=0.5σνDDpTσνDTnα, (27)

where n = nD + nT + nHe + np + nα is the concentration of plasma ions, and α is the relative concentration of deuterium (nD = αn). Expression (27) coincides with the findings in Wurzel and Hsu 2022, Khvesyuk and Chirkov 2022. Guided by the NACRE II data with T = 105 keV (the anticipated temperature of the Lawson criterion for the fully catalyzed mode), one can estimate the tritium generation:

NAσνDDpT = 1.54 ∙ 107 cm3mol−1s−1, NAσνDT = 4.76 ∙ 108 cm3mol−1s−1 so nT/nD = 0.5⟨σνDD→pT nD / (⟨σνñDT→nαnD + λ) ≈ 1.6∙10–2.

Similarly, the equilibrium concentration of 3He is equal (in the 3He self-supply mode) to:

nHe=0.5σνDDnHeσνDHepαnD=αγHen,γHe=0.5σνDDnHeσνDHepα (28)

In the event of supply with 3He, if any, expression (28) shall be substituted for

nHe=0.5σνDDnHeσνDHepαnD+dnHeext/dtσνDHepαnD=αγHenδ , (29)

where δ > 1 in the supply mode is the external parameter that can be attributed to the volume average power of the FR.

The volume-average power of the fusion reaction Pf = Afn2 in the new notation is described as:

Pf = 18533⟨σνDHeα2γHen2 + 0.5∙4033⟨σνDDpTα2n2 + 0.5∙3269⟨σνDDnHeα2n2++17589⟨σνDDα2γTn2. (30)

The volume-average power, Pc, generated in plasma by charged particles is

Pf = 18533⟨σνDHeα2γHen2 + 0.5∙4033⟨σνDDpTα2n2 + 0.5∙819⟨σνDDnHeα2n2 ++ 3561⟨σνDTα2γTn2 = Acn2 = fcAfn2. (31)

In the 3He supply mode, parameter γHe in (30) and (31) needs to be multiplied by δ.

The following ratios take place:

(nD + nHe + nT + nP + nα)⁄n = α + αγHeδ + αγT + nPn + nαn = 1. (32)

It stems from (22) – (24) that Γα = Γp in the steady-state mode, so

nPnα=τPτα=ξ=σνDHepαγHeδ+0.5σνDDpTσνDHepαγHeδ+σνDTnαγT . (33)

The results of determining the Lawson criterion and the triple Lawson criterion nτT for the thermonuclear reaction ignition mode, (Q = ∞), and for the case with ξ = 1 are presented below.

3He self-supply mode (δ = 1)

It was found as the result of a numerical simulation that parameter nτT in the self-supply mode is determined in the deuterium concentration range of 0.56 ≤ α ≤ 0.9. The minimum for this parameter (triple Lawson criterion) fits value α = 0.89 and is equal to:

nτT = 1.03∙1017 cm–3∙s∙keV (T = 45 keV).

With value α being as that, the Lawson criterion looks as follows:

nτ = 1.48∙1015 cm–3∙s (T = 130 keV).

The presented results is qualitatively consistent with the results obtained in Khvesyuk and Chirkov 2022 for the triple Lawson criterion nτT = (1–2)∙1017 cm–3∙s∙keV (T = 50–70 keV).

The concentration of plasma ions and its charge characteristics are as follows:

n D = 0.89n, nHe = 0.094n, nT = 0.0046n, nα = 0.0055n, np = 0.0055n,

Z⟩ = 1.1, ⟨Z2⟩ = 1.3.

Fully catalyzed D-D cycle

In the event of the fully catalyzed cycle, the triple Lawson criterion fits similar concentrations of D and 3He which could be expected in advance proceeding from the energy considerations:

nτT = 6.42∙1016 cm–3∙s∙keV (T = 54 keV, nD/n = nHe/n = 0.46, δ = 10.87).

The concentration of plasma ions and its charge characteristics are as follows:

n D = 0.46n, nHe = 0.46n, nT = 0.0029n, nα = 0.0386n, np = 0.0386n,

Z⟩ = 1.5, ⟨Z2⟩ = 2.5.

The Lawson criterion that fits the above parameters nD and δ has the form:

nτ = 8.35∙1014 cm–3∙s (T = 123 keV).

For comparison, we shall present the minimum value of parameter nτT with nD/n = 0.5:

nτT = 8.02∙1016 cm–3∙s∙keV (T = 56 keV, δ = 7.3).

n D = 0.5n, nHe = 0.313n, nT = 0.0036n, nα = 0.0845n, np = 0.0845n,

Z⟩ = 1.4, ⟨Z2⟩ = 2.2.

The obtained results are close to those presented in Wurzel and Hsu 2022 both for the Lawson criterion

nτ = 6.20∙1014 cm–3∙s (T = 106 keV),

and for the triple criterion

nτT = 5.20∙1016см–3∙s∙keV (T = 68 keV).

However, the direct comparison is hard to make since no deuterium and 3He concentrations are given in Wurzel and Hsu 2022.

Non-catalyzed D-D cycle for equimolar D-3He fuel (nD = 0.5n)

nτ = 6.01∙1015 cm–3∙s (T = 103 keV), nτT = 4.89∙1016 cm–3∙s∙keV (T = 67 keV).


The key result from the above consideration is the determination (as part of a model problem) of the Lawson criterion and the triple Lawson criterion for D-3He fueled thermonuclear devices with magnetic plasma confinement based on a refined temperature dependence of the D + 3He → p + 4He fusion reaction rate found using a new parametrization of the reaction cross-section and astrophysical factor in the effective radius approximation. The calculated reaction rate values are in a good agreement with the R-matrix theory results and the data contained in the NACRE II library, but, unlike these approaches, the effective radius approximation does not require an extensive computational power. It was found that in the event of a fully catalyzed cycle, the triple Lawson criterion fits equimolar D-3He fuel, (nD/n = nHe/n = 0.46), with the following parameters: nτT = 6.42∙1016 cm–3∙s∙keV (T = 54 keV), and is characterized by the smallest possible relative concentration of impurities. It is suggested that the developed approach to investigating the performance of particular thermonuclear systems with magnetic plasma confinement be used at the next stage.


  • Alper IB, Godes AI, Shablov VL (2021) New Parametrization for the 3He(d,p) 4He Fusion Reaction Rate and Refinement of the Lawson Criterion for D-3He Thermonuclear Reactors. Journal of Physics: Conference Series 2103: 012197.
  • Artsimovich LA (1964) Controlled Thermonuclear Reactions. Gordon and Breach Science Publishers, NY, 405 pp.
  • Balashko YuG (1965) Investigations of Elastic Scattering of Charged Particles on Some Light Nuclei at Low Energies. Trudy FI AN SSSR. im. P.N. Lebedeva, 33: 66–126. [in Russian]
  • Barit IYa, Sergeev VA (1969) Analysis of Resonance Reactions H3(d,n)He4 and He3(d,p)He4 in the Interaction Effective Range Approximation. Trudy FI AN SSSR. im. P.N. Lebedeva, 44: 3–15. [in Russian]
  • Basko MM (2007) Physical Foundations of the Inertial Thermonuclear Fusion. ITEPh Publ., Moscow, 147 pp. [in Russian]
  • Belov AA, Kalitkin NN (2017) Regularization of the double period method in processing experimental curves. Zhurnal Vychislitel’noy Matematiki i Matematicheskoy Fiziki 57(11): 1171–1181. [in Russian]
  • Godes AI, Kudriavtseva AS, Shablov VL (2019) Analytical 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-3He scattering. Izvestiya vuzov. Yadernaya Energetika 2: 198–207. [in Russian]
  • Karnakov BM, Mur VD, Pozdnyakov SG, Popov VS (1991) Poles and resonances in low-energy scattering of charged particles. Yadernaya Fisika 54(2(8)): 400–403. [in Russian]
  • Khvesyuk VI, Chirkov AYu (2000) Energy production in ambipolar reactors with D-T, D-3He and D-D fuel cycles. Pis’ma v Zhurnal Eksperimental’noy i Teoreticheskoy Fiziki 26(21): 61–65. [in Russian]
  • Landau LD, Lifshitz EM (1977) Course of Theoretical Physics. Vol. 3. Quantum Mechanics. Pergamon Press, Oxford, 671 pp.
  • Ryzhkov SV, Chirkov AYu (2017) Systems of Alternative Thermonuclear Power Engineering. Fizmatlit Publ., Moscow, 200 pp. [in Russian]
  • Shirokov YuM, Yudin NP (1980) Nuclear Physics. Nauka Publ., Moscow, 729 pp. [in Russian]
  • Wildermuth K, Tang JC (1999) A Unified Theory of the Nucleus. Vieweg Publ. Braunschweig, 389 pp.
  • Wurzel SE, Hsu SC (2022) Progress toward fusion energy breakeven and gain as measured against the Lawson criterion. Physics of Plasmas 29(6): 062103.
  • Xua Y, Takahashia K, Gorielya S, Arnoulda M, Ohtac M, Utsunomiyad H (2013) NACRE II: an Update of the NACRE Compilation of Charged-Particle-Induced Thermonuclear Reaction Rates for Nuclei with Mass Number A < 16. Astronomy & Astrophysics 549: A106. [120 pp]
  • Zhdanov SK, Kurnaev VA, Romanovsky MK, Tsvetkov IV (2017) Fundamentals of Physical Processes in Plasma and Plasma Installations. MEPhI Publ., Moscow, 200 pp. [in Russian]

Appendix 1

Let us consider an exoergic reaction involving the formation of particles 1 and 2 with rest energies E01 and E02and energy yield Q. We assume that the ingoing channel was characterized by a vanishingly small total momentum as is in the event of fusion reactions. Then the momentums of particles 1 and 2 have practically the same modules. We shall use a relativistic relation between the momentum and the energy, p2c2 = E2E02 = T2 + 2TE0, T = EE0 is the kinetic energy of the particle. This leads to a system of two equations:

T 1 2 + 2T1E01 = T22 + 2T2E02 and T1 + T2 = Q. (A.1)

The solution has the form:

T1=QE02E01+E02+0.5Q2(E01-E02)(E01+E02+Q)(E01+E02) . (A.2)

The first term in (A.2) fits the nonrelativistic approximation:

T 1 nonrel = QE02/(E01 + E02) = Qm2/(m1 + m2).

If particle 1 is a neutron and particle 2 is an alpha particle, the mass of which is assumed to be equal to 4.001506 a.m.u. as recommended by the CODATA system of physical constants (Tiesinga et al. 2021), then T1nonrel = 14.048 MeV.

The correction to the first term in (A.2) is negative and constitutes a fraction of it


The neutron energy is therefore equal to T1 = 14.048(1 + η) = 14.028 MeV.

Russian text published: Izvestiya vuzov. Yadernaya Energetika (ISSN 0204-3327), 2023, n. 2, pp. 134–147.
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