^{3}He fuel in fusion reactors

Corresponding author: Alexandr I. Godes (

Academic editor: Georgy Tikhomirov

The paper is devoted to refining the Lawson criterion for three scenarios of using D-^{3}He fuel in fusion reactors (fully catalyzed and non-catalysed D-D cycles and a D-^{3}He cycle with ^{3}He self-supply). To this end, a new parameterization of the D + ^{3}He → p + ^{4}He 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 ^{3}He 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, ^{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: ^{16} cm^{-3}∙s∙keV (^{17} cm^{-3}∙s∙keV (^{3}He self-supply, and ^{16} cm^{-3}∙s∙keV (^{3}He fuel.

^{3}He → p +

^{4}He fusion reaction

^{3}He fuel use scenarios

Since the mid-1950s, the purpose of investigations in the field of controlled nuclear fusion (

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 (

_{e}

where _{e}τ, 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 ⟨

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 D-Т fuel. The major advantage of a D-^{3}He 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 (^{-3}He fuel cycle is characterized, however, by much higher Lawson criterion values. We shall remind that, as of the late 1950s, the Lawson criterion was _{e}τ = 10^{14} cm^{-3}·s (_{e}τ = 10^{15} cm^{-3}·s (^{3}He fuel (_{e}τ = 1.6·10^{14} cm^{-3}·s (_{e}τ = 8.1·10^{14} cm^{-3}·s (^{3}He 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 + ^{3}He → p + ^{4}He reaction reactivity based on the parametrization of its astrophysical factor in the framework of the effective radius approximation (^{3}He fuel cycle (

a fully catalyzed D-D cycle in which the generated tritium and
^{3}He are fully used as secondary fuel;

a non-catalyzed D-D cycle when intermediate
^{3}He and T are removed after they give up most of their thermonuclear kinetic energy for plasma heating but prior to further combustion with deuterium;

^{3}He 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 ^{3}He → p + ^{4}He reaction, the developed parametrizations, which are resonant in the low-energy region, can be divided into two groups:

Parametrizations based on physical models and approximations, including:

the

the Breit-Wigner approximation used in

a model-free approach – the effective radius approximation (

the resonance coupled channel model (

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:

a Padé approximation of the reaction cross-section and rate with the correct threshold behavior (

the regularized double period method (

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 ^{6} K ≤ ^{10} K and contains experimental data from before 2013. The tables present values of the low, high and adopted estimates for value _{A}⟨

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 ^{5}Li^{**} 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.

The effective radius approximation (

For low-energy scattering in a system of two charged particles, this approximation is based on the following expression for the _{11}(

where _{0}

or η = (_{c}^{–1}; _{c} is the Bohr radius for a pair of synthesized nuclei with reduced mass _{r}

_{c}^{2}⁄(_{1}_{2}^{2}_{r}

In the framework of _{0}(

_{c}^{−1} [_{0}(_{0}^{−1} + 0.5_{0}^{2}, (5)

where _{0} is the scattering length, _{0}

^{2}^{πη}

_{c}_{c}

where ψ(

Then the following equality takes place

_{0}(^{2}) − 2

with the function φ(^{2}) of the form

^{2})= −_{c}_{0} + 0.5_{0}_{c}k^{2} = _{0} + _{1}^{2} − _{0} + _{1}_{c}^{2}^{2}). (9)

In a general case, the terms with ^{4} and ^{6} may be taken into account in the function φ(^{2}

_{r}^{2})(1 −|_{11}|^{2}), (10)

where _{1} +1)(2_{2} +1)], equal, in the event of the D + ^{3}He → p + ^{4}He reaction, to 2/3, or

_{r}^{2}|^{2}). (11)

We shall present the expression (11) as

_{0} + _{1}(_{c}^{2} + _{2}(_{c}^{4}, _{0} + _{1}(_{c}^{2} + _{2}(_{c}^{4}.

From (12), the following expression is obtained for the astrophysical factor ^{2πη}_{r}

or (taking into account numerical factors)

Besides,

The

The astrophysical ^{3}He → p + ^{4}He reaction.

_{0} = 0.117002, _{1} = 0.191855, _{2} = −0.01225, _{0} = 0.00937, _{1} = 0.006658, _{2} = 0.000582. (15)

The presented parameters also agree with the data on the elastic D-^{3}He scattering

The ratio of the elastic D-^{3}He 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

_{0} = 0.05431, _{1} = 0.25077, _{2} = −0.02825, _{0} = 0.00205, _{1} = 0.00707, _{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:

⟨_{0}^{∞}

Where

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

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

^{−1} + _{c}_{f}_{br}_{e}_{−}_{e}

where _{c}_{f}_{br}_{e-e}_{j}n_{j}/∑n_{j}_{e}_{f}_{c}A_{f}

_{f}n^{2} = 18533⟨_{DHe}_{→}_{pα}n_{D}n_{He}_{DD}_{→}_{nHe}_{DD}_{→}_{pT}_{D}^{2} + 17589⟨σν⟩_{DT}_{→}_{nα}n_{D}n_{T}

(1 − _{c}_{f}n^{2} = 1225⟨_{DD}_{→}_{nHe} n_{D}^{2} + 14028⟨_{DT}_{→}_{nα}n_{D}n_{T}

such that (1 – _{c}_{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. (_{br}_{e-e}

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

The steady-state operation kinetics of a quasi-infinite ^{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:

_{D_ext} /_{He_ext} /^{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 _{j}_{j}_{j}_{j}^{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 _{T}/dt = 0

Formula (25) allows estimating the contribution from the tritium beta decay for which λ_{T} and ⟨_{DT→nα}_{D}_{T} need to be compared. We shall take into account that ^{9} s⁻^{1} and in a temperature range of 1·10^{8} to 1·10^{9} K:

⟨_{DT}_{→}_{nα}n_{D}^{−18} ÷ 2 ∙ 10^{−15})_{D}^{3} ∙ s^{−1} (26)

so with _{D} ≥ 1 ∙ 10^{12}см^{–3}, the contribution from beta decay in (25) may be neglected. Then

where _{D}_{T}_{He}_{p}_{α}_{D}

_{A}_{DD}_{→}_{pT}^{7} cm^{3}mol^{−1}^{−1}, _{A}_{DT}_{→}_{nα}^{8} cm^{3}mol^{−1}^{−1} so _{T}/_{D} = 0.5⟨_{DD→pT}_{D} / (⟨_{DT→nα}n_{D} + λ) ≈ 1.6∙10^{–2}.

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

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

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

The volume-average power of the fusion reaction _{f}_{f}n^{2} in the new notation is described as:

_{f}_{DHe}_{→}_{pα}α^{2}_{He}n^{2} + 0.5∙4033⟨_{DD}_{→}_{pT}α^{2}^{2} + 0.5∙3269⟨_{DD}_{→}_{nHe}α^{2}^{2}++17589⟨_{DD}_{→}_{nα}α^{2}_{T}n^{2}. (30)

The volume-average power, _{c}, generated in plasma by charged particles is

_{f}_{DHe}_{→}_{pα}α^{2}_{He}n^{2} + 0.5∙4033⟨_{DD}_{→}_{pT}α^{2}^{2} + 0.5∙819⟨_{DD}_{→}_{nHe}α^{2}^{2} ++ 3561⟨_{DT}_{→}_{nα}α^{2}_{T}n^{2} = _{c}n^{2} = _{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:

(_{D}_{He}_{T}_{P}_{α}_{He}δ_{T}_{P}_{α}

It stems from (22) – (24) that _{α}_{p}

The results of determining the Lawson criterion and the triple Lawson criterion

It was found as the result of a numerical simulation that parameter

^{17} cm^{–3}∙s∙keV (

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

^{15} cm^{–3}∙s (

The presented results is qualitatively consistent with the results obtained in ^{17} cm^{–3}∙s∙keV (

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

_{D} = 0.89_{He} = 0.094_{T} = 0.0046_{α} = 0.0055_{p} = 0.0055

⟨^{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:

^{16} cm^{–3}∙s∙keV (_{D}/_{He}/

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

_{D} = 0.46_{He} = 0.46_{T} = 0.0029_{α} = 0.0386_{p} = 0.0386

⟨^{2}⟩ = 2.5.

The Lawson criterion that fits the above parameters _{D} and δ has the form:

^{14} cm^{–3}∙s (

For comparison, we shall present the minimum value of parameter _{D}/

^{16} cm^{–3}∙s∙keV (

_{D} = 0.5_{He} = 0.313_{T} = 0.0036_{α} = 0.0845_{p} = 0.0845

⟨^{2}⟩ = 2.2.

The obtained results are close to those presented in

^{14} cm^{–3}∙s (

and for the triple criterion

^{16}см^{–3}∙s∙keV (

However, the direct comparison is hard to make since no deuterium and ^{3}He concentrations are given in

^{15} cm^{–3}∙s (^{16} cm^{–3}∙s∙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 cross-section and astrophysical factor in the effective radius approximation. The calculated reaction rate values are in a good agreement with the ^{3}He fuel, (_{D}/_{He}/^{16} cm^{–3}∙s∙keV (

^{3}He(d,p)

^{4}He Fusion Reaction Rate and Refinement of the Lawson Criterion for D-

^{3}He Thermonuclear Reactors. Journal of Physics: Conference Series 2103: 012197.

^{3}(d,n)He

^{4}and He

^{3}(d,p)He

^{4}in the Interaction Effective Range Approximation. Trudy FI AN SSSR. im. P.N.

^{3}He(d,p)

^{4}He Reaction at Low Energies. Physical Review C, 60: 054003.

^{3}He scattering. Izvestiya vuzov.

^{3}He and D-D fuel cycles.

^{3}He fusion fuel cycles with

^{3}He production.

^{3}He reactions.

^{3}He + D nuclear-reaction cross section.

^{3}He and D-D Fusion Fuels.

Let us consider an exoergic reaction involving the formation of particles 1 and 2 with rest energies E01 and E_{02}and energy yield ^{2}^{2} = ^{2} – _{0}^{2} = ^{2} + 2_{0}, _{0} is the kinetic energy of the particle. This leads to a system of two equations:

_{1}^{2} + 2_{1}_{01} = _{2}^{2} + 2_{2}_{02} and _{1} + _{2} =

The solution has the form:

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

_{1}^{nonrel} = _{02}/(_{01} + _{02}) = _{2}/(_{1} + _{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 (_{1}^{nonrel} = 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 _{1} = 14.048(1 + η) = 14.028 MeV.

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