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

n_eτ ≥ 12kT⁄(⟨σν⟩Q), (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 n_eτ, 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 n_eτ = 10¹⁴ cm^-3·s (Т = 17 keV) for equimolar D-Т fuel, and n_eτ = 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): n_eτ = 1.6·10¹⁴ cm^-3·s (Т = 26 keV) for D-Т fuel; n_eτ = 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.

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:

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

$S_{11}\left(E\right)=e^{2i{\sigma }_0\left(E\right)}\frac{ctg{\delta }_0\left(k\right)+i}{ctg{\delta }_0\left(k\right)-i}$, (2)

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

$\eta =\frac{Z_1{Z}_2{e}^2{m}_r}{{\hslash }^{2}k}$ (3)

or η = (ka_c)^–1; a_c is the Bohr radius for a pair of synthesized nuclei with reduced mass m_r:

a_c = ℏ²⁄(Z₁Z₂e²m_r). (4)

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

a_c ⁻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⁄(e^2πη − 1), (6)

h (k) = Reψ (iη) − ln(η) = Reψ (i⁄(ka_c)) + ln(ka_c), (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)= −a_c⁄a₀ + 0.5r₀a_ck² = a₀ + a₁k² − i (β₀ + β₁a_c²k²). (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

σ_r (E) = g (π⁄k²)(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

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

We shall present the expression (11) as

${\sigma }_r\left(E\right)=\frac{8\pi }{3k^2}\frac{\beta \left(k\right)D\left(k\right)}{{\left(a\left(k\right)-2h\left(k\right)\right)}^2+{\left(\beta \left(k\right)+D\left(k\right)\right)}^2}$ , (12)

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

From (12), the following expression is obtained for the astrophysical factor S (E)=Ee^2πηα_r (E):

$S\left(E\right)=\frac{8{\pi }^2{\hslash }^2}{3m_r(1-e^{-2\pi \eta })}\frac{\beta \left(k\right)}{\left(\alpha \right(k)-2h(k){)}^2+(\beta \left(k\right)+D\left(k\right){)}^2}$ , (13)

or (taking into account numerical factors)

$S\left(E\right)=\frac{9.11}{(1-e^{-2\pi \eta })}\frac{\beta \left(k\right)}{\left(\alpha \right(k)-2h(k){)}^2+(\beta \left(K\right)+D\left(k\right){)}^2}(MeV·b)$ . (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:

$F\left(E\right)=\frac{2}{\sqrt{\pi }}\frac{1}{(kT{)}^{3/2}}\sqrt{E}{e}^{-E/\left(kT\right)}$ , (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.

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:

nτ = 1.5(1 + ⟨Z⟩)T⁄[(Q⁻¹ + f_c)A_f − A_br − A_e−e], (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, f_c is the part of the thermonuclear energy absorbed in plasma, A_f is the function that defines the fusion power, A_br and A_e-e are the volume-average power losses via bremsstrahlung radiation on ions and on electrons respectively, ⟨Z⟩=∑Z_jn_j/∑n_j is the average charge of plasma ions, and n_e=⟨Z⟩n is the concentration of electrons. A_f and f_cA_f are defined in the expression (18) by the relations

A_fn ² = 18533⟨σν⟩_DHe→pαn_Dn_He + 0.5(3269⟨σν⟩_DD→nHe + 4033⟨σν⟩_DD→pT)n_D² + 17589⟨σν⟩_DT→nαn_Dn_T,

(1 − f_c) A_fn² = 1225⟨σν⟩_DD→nHe n_D² + 14028⟨σν⟩_DT→nαn_Dn_T, (19)

such that (1 – f_c)A_f 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 A_br and A_e-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:

$\frac{d{n}_D}{dt}=-⟨\sigma \nu {⟩}_{DHe\to p\alpha }{n}_D{n}_{He}-0.5(⟨\sigma \nu {⟩}_{DD\to pT}+⟨\sigma \nu {⟩}_{DD\to nHe})n_D^2-⟨\sigma \nu {⟩}_{DT\to n\alpha }{n}_D{n}_T+\frac{d{n}_{D\text{\_ext}}}{dt}$ (20)

$\frac{d{n}_{He}}{dt}=-⟨\sigma \nu {⟩}_{DHe\to p\alpha }{n}_D{n}_{He}+0.5⟨\sigma \nu {⟩}_{DD\to nHe}{n}_D^2+\frac{d{n}_{He\_\text{ext }}}{dt}$ (21)

$\frac{d{n}_T}{dt}=0.5⟨\sigma \nu {⟩}_{DD\leftarrow pT}{n}_D^2-⟨\sigma \nu {⟩}_{DD\leftarrow n\alpha }{n}_D{n}_T-\lambda n_T$ (22)

$\frac{d{n}_{\alpha }}{dt}=⟨\sigma \nu {⟩}_{DHe\to p\alpha }{n}_D{n}_{He}+⟨\sigma \nu {⟩}_{DT\to n\alpha }{n}_D{n}_T-{\Gamma }_{\alpha }$ (23)

$\frac{d{n}_p}{dt}=⟨\sigma \nu {⟩}_{DHe\to p\alpha }{n}_D{n}_{He}+0.5⟨\sigma \nu {⟩}_{DT\to pT}{n}_D^2-{\Gamma }_p$. (24)

dn _{D_ext} /dt describes the FR supply with deuterium, and dn_{He_ext} /dt describes the FR supply with³He; λ in (22) is the rate of the ³H beta decay; Г_α and Г_р describe the removal of alpha particles and protons from the FR: Γ_j = n_j /τ_j, (τ_j 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 dn_T/dt = 0 the equilibrium concentration of tritium is equal to

$n_T=\frac{0.5⟨\sigma \nu {⟩}_{DD\leftarrow pT}}{⟨\sigma \nu {⟩}_{DT\leftarrow n\alpha }{n}_D+\lambda }{n}_D$ . (25)

Formula (25) allows estimating the contribution from the tritium beta decay for which λn_T and ⟨σν⟩_DT→nα n_Dn_T 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:

⟨σν⟩_DT→nαn_D~(10⁻¹⁸ ÷ 2 ∙ 10⁻¹⁵)n_Dcm³ ∙ s⁻¹ (26)

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

$n_T=\frac{0.5⟨\sigma \nu {⟩}_{DD\to pT}}{⟨\sigma \nu {⟩}_{DT\to n\alpha }}=\alpha {\gamma }_{T}n,{\gamma }_T=\frac{0.5⟨\sigma \nu {⟩}_{DD\to pT}}{⟨\sigma \nu {⟩}_{DT\to n\alpha }}$, (27)

where n = n_D + n_T + n_He + n_p + n_α is the concentration of plasma ions, and α is the relative concentration of deuterium (n_D = 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:

N_A⟨σν⟩_DD→pT = 1.54 ∙ 10⁷ cm³mol⁻¹s⁻¹, N_A⟨σν⟩_DT→nα = 4.76 ∙ 10⁸ cm³mol⁻¹s⁻¹ so n_T/n_D = 0.5⟨σν⟩_DD→pT n_D / (⟨σνñ_DT→nαn_D + λ) ≈ 1.6∙10^–2.

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

$n_{He}=\frac{0.5⟨\sigma \nu {⟩}_{DD\to nHe}}{⟨\sigma \nu {⟩}_{DHe\to p\alpha }}{n}_D=\alpha {\gamma }_{He}n,{\gamma }_{He}=\frac{0.5⟨\sigma \nu {⟩}_{DD\to nHe}}{⟨\sigma \nu {⟩}_{DHe\to p\alpha }}$ (28)

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

$n_{He}=\frac{0.5⟨\sigma \nu {⟩}_{DD\to nHe}}{⟨\sigma \nu {⟩}_{DHe\to p\alpha }}{n}_D+\frac{d{n}_{{He}_{\text{ext}}}/dt}{⟨\sigma \nu {⟩}_{DHe\to p\alpha }{n}_D}=\alpha {\gamma }_{He}n\delta$ , (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 P_f = A_fn² in the new notation is described as:

P_f = 18533⟨σν⟩_DHe→pαα²γ_Hen² + 0.5∙4033⟨σν⟩_DD→pTα²n² + 0.5∙3269⟨σν⟩_DD→nHeα²n²++17589⟨σν⟩_DD→nαα²γ_Tn². (30)

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

P_f = 18533⟨σν⟩_DHe→pαα²γ_Hen² + 0.5∙4033⟨σν⟩_DD→pTα²n² + 0.5∙819⟨σν⟩_DD→nHeα²n² ++ 3561⟨σν⟩_DT→nαα²γ_Tn² = A_cn² = f_cA_fn². (31)

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

The following ratios take place:

(n_D + n_He + n_T + n_P + n_α)⁄n = α + αγ_Heδ + αγ_T + n_P⁄n + n_α⁄n = 1. (32)

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

$\frac{n_P}{n_{\alpha }}=\frac{{\tau }_P}{{\tau }_{\alpha }}=\xi =\frac{⟨\sigma \nu {⟩}_{DHe\to p\alpha }{\gamma }_{He}\delta +0.5⟨\sigma \nu {⟩}_{DD\to pT}}{⟨\sigma \nu {⟩}_{DHe\to p\alpha }{\gamma }_{He}\delta +⟨\sigma \nu {⟩}_{DT\to n\alpha }{\gamma }_T}$ . (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.

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, (n_D/n = n_He/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.

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 = E² – E₀² = T² + 2TE₀, T = E–E₀ 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:

$T_1=\frac{Q{E}_{02}}{E_{01}+E_{02}}+\frac{0.5Q^2(E_{01}-E_{02})}{(E_{01}+E_{02}+Q)(E_{01}+E_{02})}$ . (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 T₁^nonrel = 14.048 MeV.

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

$\eta =\frac{0.5Q^2(E_{01}-E_{02})}{(E_{01}+E_{02}+Q)(E_{01}+E_{02})}\approx -1.4·{10}^{-3}$.

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