Research Article 
Corresponding author: Alexandr I. Godes ( godes.ai@yandex.ru ) Academic editor: Georgy Tikhomirov
© 2023 Alexandr I. Godes, Vladimir L. Shablov.
This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Citation:
Godes AI, Shablov VL (2023) Lawson criterion for different scenarios of using D^{3}He fuel in fusion reactors. Nuclear Energy and Technology 9(4): 981990. https://doi.org/10.3897/nucet.9.114267

The paper is devoted to refining the Lawson criterion for three scenarios of using D^{3}He fuel in fusion reactors (fully catalyzed and noncatalysed DD cycles and a D^{3}He cycle with ^{3}He selfsupply). To this end, a new parameterization of the D + ^{3}He → p + ^{4}He fusion reaction crosssection and astrophysical factor has been developed based on the effective radius approximation (LandauSmorodinskyBethe approximation), which is a modelfree theoretical approach to investigating nearthreshold 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 Rmatrix theory and the NACRE II fusion reactivity data. For the fully catalyzed DD cycle and the cycle with ^{3}He selfsupply, 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, ^{3}He, and T) taking into account the potential for external supply of ^{3}He and p and ^{4}He impurity ions removed from the reaction zone. The parameters of the triple Lawson criterion found are as follows: nτT = 6.42∙10^{16} cm^{3}∙s∙keV (T = 54 keV) for the fully catalyzed DD cycle, nτT = 1.03∙10^{17} cm^{3}∙s∙keV (T = 45 keV) for the cycle with ^{3}He selfsupply, and nτT = 4.89∙10^{16} cm^{3}∙s∙keV (T = 67 keV) for the noncatalyzed DD cycle with equimolar D^{3}He fuel.
thermonuclear reactions, effective radius approximation, crosssection and astrophysical factor of D + ^{3}He → p + ^{4}He fusion reaction, Lawson criterion for different D^{3}He fuel use scenarios
Since the mid1950s, the purpose of investigations in the field of controlled nuclear fusion (CNF) has been to reach and exceed the Lawson criterion (
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_{e}τ ≥ 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 deuteriumtritium 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_{e}τ, the achievement of which leads to a selfsustained 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 crosssection 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 (
The purpose of this study is to determine more precisely the Lawson criterion for fusion reactors on the base of D^{3}He fuel viewed as the immediate competitor to DТ fuel. The major advantage of a D^{3}He reactor, as compared with a DT 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 (
Since the 1950s, multiple experimental and theoretical studies have been undertaken on the fusion crosssections and thermal reactivities (see the overview in
Parametrizations based on physical models and approximations, including:
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.
Parametrizations based on mathematical methods for approximation of experimental data, including:
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 (
The results of this study will be compared with data from the most common of the current parametrizations (
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 1·10^{6} K ≤ T ≤ 1·10^{10} K and contains experimental data from before 2013. The tables present values of the low, high and adopted estimates for value N_{A}⟨σν⟩. NACRE II is based on calculations of crosssections using different theoretical models (the distorted wave Born approximation, the BreitWigner approximation, etc.).
We shall note that in the event of the D + ^{3}He → p + ^{4}He reaction, the previous NACRE version uses the results obtained in
The effective radius approximation (ERA) is a modelfree approach and operates on experimentally observed quantities, including scattering length, effective radius and potential shape parameter.
For lowenergy scattering in a system of two charged particles, this approximation is based on the following expression for the Smatrix element S_{11}(E) of the swave 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 σ_{0} (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} = ℏ^{2}⁄(Z_{1}Z_{2}e^{2}m_{r}). (4)
In the framework of ERA, the nuclear – сoulomb shift, δ_{0}(k), is determined by the expression
a_{c} ^{−1} [D (k)ctgδ_{0}(k) + 2h (k)] = −a_{0}^{−1} + 0.5r_{0}k^{2}, (5)
where a_{0} is the scattering length, r_{0} is the effective radius, and D (k) is the Coulomb barrier penetrability
D (k) = 2π⁄(e^{2}^{πη} − 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 nuclearсoulomb shift becomes a complex value, and the scattering length and the effective radius become so complex values as well (
Then the following equality takes place
ω (k) = D (k)ctgδ_{0}(k) − iD (k) = φ (k^{2}) − 2h (k) (8)
with the function φ(k^{2}) of the form
φ (k^{2})= −a_{c}⁄a_{0} + 0.5r_{0}a_{c}k^{2} = a_{0} + a_{1}k^{2} − i (β_{0} + β_{1}a_{c}^{2}k^{2}). (9)
In a general case, the terms with k^{4} and k^{6} may be taken into account in the function φ(k^{2}). As the result, the reaction crosssection is equal to
σ_{r} (E) = g (π⁄k^{2})(1 −S_{11}^{2}), (10)
where g is the spin factor, (2J + 1)/[(2S_{1} +1)(2S_{2} +1)], equal, in the event of the D + ^{3}He → p + ^{4}He reaction, to 2/3, or
σ_{r} (E) = 8πβ (k)D (k)⁄(3k^{2}ω (k)^{2}). (11)
We shall present the expression (11) as
${\sigma}_{r}\left(E\right)=\frac{8\pi}{3{k}^{2}}\frac{\beta \left(k\right)D\left(k\right)}{{\left(a\left(k\right)2h\left(k\right)\right)}^{2}+{\left(\beta \left(k\right)+D\left(k\right)\right)}^{2}}$ , (12)
α (k) = α_{0} + α_{1}(ka_{c})^{2} + α_{2}(ka_{c})^{4}, β (k) = β_{0} + β_{1}(ka_{c})^{2} + β_{2}(ka_{c})^{4}.
From (12), the following expression is obtained for the astrophysical factor S (E)=Ee^{2πη}α_{r} (E):
$S\left(E\right)=\frac{8{\pi}^{2}{\hslash}^{2}}{3{m}_{r}(1{e}^{2\pi \eta})}\frac{\beta \left(k\right)}{\left(\alpha \right(k)2h(k){)}^{2}+(\beta \left(k\right)+D\left(k\right){)}^{2}}$ , (13)
or (taking into account numerical factors)
$S\left(E\right)=\frac{9.11}{(1{e}^{2\pi \eta})}\frac{\beta \left(k\right)}{\left(\alpha \right(k)2h(k){)}^{2}+(\beta \left(K\right)+D\left(k\right){)}^{2}}(MeV\xb7b)$ . (14)
ERA applicability condition: parameter kR ≤ 1 where R is the radius of action for nuclear forces (
Besides, ERA is not applicable with E ≤ 0.02 keV when the laboratory electron screening effects manifest themselves in the nuclear reaction crosssections (
The ERA parameters, using which the astrophysical factor in Fig.
α _{0} = 0.117002, α_{1} = 0.191855, α_{2} = −0.01225, β_{0} = 0.00937, β_{1} = 0.006658, β_{2} = 0.000582. (15)
The presented parameters also agree with the data on the elastic D^{3}He scattering
Another set of parameters based on experimental data in
α _{0} = 0.05431, α_{1} = 0.25077, α_{2} = −0.02825, β_{0} = 0.00205, β_{1} = 0.00707, β_{2} = 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.
In the currently most common form, the Lawson criterion for fusion reactors with magnetic plasma confinement is written as follows
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 volumeaverage fusion energy to the volumeaverage 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_{ee} are the volumeaverage power losses via bremsstrahlung radiation on ions and on electrons respectively, ⟨Z⟩=∑Z_{j}n_{j}/∑n_{j} is the average charge of plasma ions, and n_{e}=⟨Z⟩n is the concentration of electrons. A_{f} and f_{c}A_{f} are defined in the expression (18) by the relations
A_{f}n ^{2} = 18533⟨σν⟩_{DHe}_{→}_{pα}n_{D}n_{He} + 0.5(3269⟨σν⟩_{DD}_{→}_{nHe} + 4033⟨σν⟩_{DD}_{→}_{pT})n_{D}^{2} + 17589⟨σν⟩_{DT}_{→}_{nα}n_{D}n_{T},
(1 − f_{c}) A_{f}n^{2} = 1225⟨σν⟩_{DD}_{→}_{nHe} n_{D}^{2} + 14028⟨σν⟩_{DT}_{→}_{nα}n_{D}n_{T}, (19)
such that (1 – f_{c})A_{f} n^{2} 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. (
We shall consider the fully catalyzed DD cycle when the Т and ^{3}He formed remain in plasma and burn together with deuterium.
The steadystate operation kinetics of a quasiinfinite FR is defined by the condition that there are constant concentrations of D, ^{3}He and Т fuel ions and p and ^{4}He 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, ^{3}He and Т fuel ions and the protons and alpha particles to be removed from the reaction zone are written as follows:
$\frac{d{n}_{D}}{dt}=\u27e8\sigma \nu {\u27e9}_{DHe\to p\alpha}{n}_{D}{n}_{He}0.5(\u27e8\sigma \nu {\u27e9}_{DD\to pT}+\u27e8\sigma \nu {\u27e9}_{DD\to nHe}){n}_{D}^{2}\u27e8\sigma \nu {\u27e9}_{DT\to n\alpha}{n}_{D}{n}_{T}+\frac{d{n}_{D\text{\_ext}}}{dt}$ (20)
$\frac{d{n}_{He}}{dt}=\u27e8\sigma \nu {\u27e9}_{DHe\to p\alpha}{n}_{D}{n}_{He}+0.5\u27e8\sigma \nu {\u27e9}_{DD\to nHe}{n}_{D}^{2}+\frac{d{n}_{He\_\text{ext}}}{dt}$ (21)
$\frac{d{n}_{T}}{dt}=0.5\u27e8\sigma \nu {\u27e9}_{DD\leftarrow pT}{n}_{D}^{2}\u27e8\sigma \nu {\u27e9}_{DD\leftarrow n\alpha}{n}_{D}{n}_{T}\lambda {n}_{T}$ (22)
$\frac{d{n}_{\alpha}}{dt}=\u27e8\sigma \nu {\u27e9}_{DHe\to p\alpha}{n}_{D}{n}_{He}+\u27e8\sigma \nu {\u27e9}_{DT\to n\alpha}{n}_{D}{n}_{T}{\Gamma}_{\alpha}$ (23)
$\frac{d{n}_{p}}{dt}=\u27e8\sigma \nu {\u27e9}_{DHe\to p\alpha}{n}_{D}{n}_{He}+0.5\u27e8\sigma \nu {\u27e9}_{DT\to pT}{n}_{D}^{2}{\Gamma}_{p}$. (24)
dn
_{D_ext} /dt describes the FR supply with deuterium, and dn_{He_ext} /dt describes the FR supply with^{3}He; λ in (22) is the rate of the ^{3}H 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 + ^{3}He and ^{3}He + ^{3}He 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
${n}_{T}=\frac{0.5\u27e8\sigma \nu {\u27e9}_{DD\leftarrow pT}}{\u27e8\sigma \nu {\u27e9}_{DT\leftarrow n\alpha}{n}_{D}+\lambda}{n}_{D}$ . (25)
Formula (25) allows estimating the contribution from the tritium beta decay for which λn_{T} and ⟨σν⟩_{DT→nα} n_{D}n_{T} need to be compared. We shall take into account that λ ≈ 2.9∙1⁻^{9} s⁻^{1} and in a temperature range of 1·10^{8} to 1·10^{9} K:
⟨σν⟩_{DT}_{→}_{nα}n_{D}~(10^{−18} ÷ 2 ∙ 10^{−15})n_{D}cm^{3} ∙ s^{−1} (26)
so with n_{D} ≥ 1 ∙ 10^{12}см^{–3}, the contribution from beta decay in (25) may be neglected. Then
${n}_{T}=\frac{0.5\u27e8\sigma \nu {\u27e9}_{DD\to pT}}{\u27e8\sigma \nu {\u27e9}_{DT\to n\alpha}}=\alpha {\gamma}_{T}n,{\gamma}_{T}=\frac{0.5\u27e8\sigma \nu {\u27e9}_{DD\to pT}}{\u27e8\sigma \nu {\u27e9}_{DT\to n\alpha}}$, (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
N_{A}⟨σν⟩_{DD}_{→}_{pT} = 1.54 ∙ 10^{7} cm^{3}mol^{−1}s^{−1}, N_{A}⟨σν⟩_{DT}_{→}_{nα} = 4.76 ∙ 10^{8} cm^{3}mol^{−1}s^{−1} so n_{T}/n_{D} = 0.5⟨σν⟩_{DD→pT} n_{D} / (⟨σνñ_{DT→nα}n_{D} + λ) ≈ 1.6∙10^{–2}.
Similarly, the equilibrium concentration of ^{3}He is equal (in the ^{3}He selfsupply mode) to:
${n}_{He}=\frac{0.5\u27e8\sigma \nu {\u27e9}_{DD\to nHe}}{\u27e8\sigma \nu {\u27e9}_{DHe\to p\alpha}}{n}_{D}=\alpha {\gamma}_{He}n,{\gamma}_{He}=\frac{0.5\u27e8\sigma \nu {\u27e9}_{DD\to nHe}}{\u27e8\sigma \nu {\u27e9}_{DHe\to p\alpha}}$ (28)
In the event of supply with ^{3}He, if any, expression (28) shall be substituted for
${n}_{He}=\frac{0.5\u27e8\sigma \nu {\u27e9}_{DD\to nHe}}{\u27e8\sigma \nu {\u27e9}_{DHe\to p\alpha}}{n}_{D}+\frac{d{n}_{{He}_{\text{ext}}}/dt}{\u27e8\sigma \nu {\u27e9}_{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 volumeaverage power of the fusion reaction P_{f} = A_{f}n^{2} in the new notation is described as:
P_{f} = 18533⟨σν⟩_{DHe}_{→}_{pα}α^{2}γ_{He}n^{2} + 0.5∙4033⟨σν⟩_{DD}_{→}_{pT}α^{2}n^{2} + 0.5∙3269⟨σν⟩_{DD}_{→}_{nHe}α^{2}n^{2}++17589⟨σν⟩_{DD}_{→}_{nα}α^{2}γ_{T}n^{2}. (30)
The volumeaverage power, P_{c}, generated in plasma by charged particles is
P_{f} = 18533⟨σν⟩_{DHe}_{→}_{pα}α^{2}γ_{He}n^{2} + 0.5∙4033⟨σν⟩_{DD}_{→}_{pT}α^{2}n^{2} + 0.5∙819⟨σν⟩_{DD}_{→}_{nHe}α^{2}n^{2} ++ 3561⟨σν⟩_{DT}_{→}_{nα}α^{2}γ_{T}n^{2} = A_{c}n^{2} = f_{c}A_{f}n^{2}. (31)
In the ^{3}He 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_{He}δ + αγ_{T} + n_{P}⁄n + n_{α}⁄n = 1. (32)
It stems from (22) – (24) that Γ_{α} = Γ_{p} in the steadystate mode, so
$\frac{{n}_{P}}{{n}_{\alpha}}=\frac{{\tau}_{P}}{{\tau}_{\alpha}}=\xi =\frac{\u27e8\sigma \nu {\u27e9}_{DHe\to p\alpha}{\gamma}_{He}\delta +0.5\u27e8\sigma \nu {\u27e9}_{DD\to pT}}{\u27e8\sigma \nu {\u27e9}_{DHe\to p\alpha}{\gamma}_{He}\delta +\u27e8\sigma \nu {\u27e9}_{DT\to n\alpha}{\gamma}_{T}}$ . (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.
It was found as the result of a numerical simulation that parameter nτT in the selfsupply 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∙10^{17} cm^{–3}∙s∙keV (T = 45 keV).
With value α being as that, the Lawson criterion looks as follows:
nτ = 1.48∙10^{15} cm^{–3}∙s (T = 130 keV).
The presented results is qualitatively consistent with the results obtained in
The concentration of plasma ions and its charge characteristics are as follows:
n _{D} = 0.89n, n_{He} = 0.094n, n_{T} = 0.0046n, n_{α} = 0.0055n, n_{p} = 0.0055n,
⟨Z⟩ = 1.1, ⟨Z^{2}⟩ = 1.3.
In the event of the fully catalyzed cycle, the triple Lawson criterion fits similar concentrations of D and ^{3}He which could be expected in advance proceeding from the energy considerations:
nτT = 6.42∙10^{16} cm^{–3}∙s∙keV (T = 54 keV, n_{D}/n = n_{He}/n = 0.46, δ = 10.87).
The concentration of plasma ions and its charge characteristics are as follows:
n _{D} = 0.46n, n_{He} = 0.46n, n_{T} = 0.0029n, n_{α} = 0.0386n, n_{p} = 0.0386n,
⟨Z⟩ = 1.5, ⟨Z^{2}⟩ = 2.5.
The Lawson criterion that fits the above parameters n_{D} and δ has the form:
nτ = 8.35∙10^{14} cm^{–3}∙s (T = 123 keV).
For comparison, we shall present the minimum value of parameter nτT with n_{D}/n = 0.5:
nτT = 8.02∙10^{16} cm^{–3}∙s∙keV (T = 56 keV, δ = 7.3).
n _{D} = 0.5n, n_{He} = 0.313n, n_{T} = 0.0036n, n_{α} = 0.0845n, n_{p} = 0.0845n,
⟨Z⟩ = 1.4, ⟨Z^{2}⟩ = 2.2.
The obtained results are close to those presented in
nτ = 6.20∙10^{14} cm^{–3}∙s (T = 106 keV),
and for the triple criterion
nτT = 5.20∙10^{16}см^{–3}∙s∙keV (T = 68 keV).
However, the direct comparison is hard to make since no deuterium and ^{3}He concentrations are given in
nτ = 6.01∙10^{15} cm^{–3}∙s (T = 103 keV), nτT = 4.89∙10^{16} 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^{3}He fueled thermonuclear devices with magnetic plasma confinement based on a refined temperature dependence of the D + ^{3}He → p + ^{4}He fusion reaction rate found using a new parametrization of the reaction crosssection and astrophysical factor in the effective radius approximation. The calculated reaction rate values are in a good agreement with the Rmatrix 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^{3}He fuel, (n_{D}/n = n_{He}/n = 0.46), with the following parameters: nτT = 6.42∙10^{16} 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 E_{02}and 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, p^{2}c^{2} = E^{2} – E_{0}^{2} = T^{2} + 2TE_{0}, T = E–E_{0} is the kinetic energy of the particle. This leads to a system of two equations:
T _{1} ^{2} + 2T_{1}E_{01} = T_{2}^{2} + 2T_{2}E_{02} and T_{1} + T_{2} = Q. (A.1)
The solution has the form:
${T}_{1}=\frac{Q{E}_{02}}{{E}_{01}+{E}_{02}}+\frac{0.5{Q}^{2}({E}_{01}{E}_{02})}{({E}_{01}+{E}_{02}+Q)({E}_{01}+{E}_{02})}$ . (A.2)
The first term in (A.2) fits the nonrelativistic approximation:
T _{1} ^{nonrel} = QE_{02}/(E_{01} + E_{02}) = Qm_{2}/(m_{1} + m_{2}).
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 (
The correction to the first term in (A.2) is negative and constitutes a fraction of it
$\eta =\frac{0.5{Q}^{2}({E}_{01}{E}_{02})}{({E}_{01}+{E}_{02}+Q)({E}_{01}+{E}_{02})}\approx 1.4\xb7{10}^{3}$.
The neutron energy is therefore equal to T_{1} = 14.048(1 + η) = 14.028 MeV.