Fast Ion Speed Diffusion Effect on Distributions of Fusion Neutrons

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Numerical modelling of yields and distributions of nuclear fusion products in magnetically confined plasma.

Abstract

Velocity distributions of fuel nuclei enter the formulae for distributions of products of fusion reactions in plasma. The formulae contain multiple integration, which is a computationally heavy task. Therefore, simplifications of the integrand are advantageous. One of possible simplifications is the use of closed-form analytical distributions of fast deuterons and tritons, accounting for slowing down and pitch-angle scattering and neglecting the speed diffusion. The plausibility of such a model has been studied from the viewpoint of its influence on the calculated spectra of fusion neutrons. Calculations have shown that the speed diffusion effect on suprathermal ion distribution tails does not significantly alter the qualitative behaviour of energy and angle distributions of fusion products in a beam-heated plasma.

1. Introduction

Research and development activities on fusion neutron sources for fundamental science and technological applications are being pursued by universities, laboratories, and other organizations in various countries of the world, as indicated in [1,2,3,4,5,6] and the references therein. An analysis of possibilities of constructing fusion neutron sources based on tokamaks, stellarators, and laser systems was recently published in [1]. In [2,3], the long-standing concept of using controlled nuclear fusion systems as neutron sources was considered in light of the developments of the past decade. In [4,5,6,7,8,9,10,11], the main attention was paid to tokamaks. In particular, in [4] and [5], neutron sources based on spherical and classical tokamaks, respectively, were investigated. Article [6] is dedicated to the analysis of the economic feasibility of tokamaks with conventional and superconducting windings as neutron sources applied to the transmutation of transuranium elements. Fusion neutron sources using deuterium–deuterium and deuterium–tritium reactions in a tokamak were considered in [7] and [8], respectively. In [9], an integral approach was applied to the modelling of a plasma neutron source based on a classical tokamak. Article [10] is devoted to a similar range of issues for a spherical tokamak. A three-dimensional model developed as part of the demonstration fusion neutron source DEMO-FNS project for neutronics calculations by Monte Carlo methods was presented in [11].

The yield and energy spectrum of neutrons belong to the main characteristics of the source. Along with conceptual and engineering design studies of fusion neutron sources employing magnetically confined plasma, calculations of energetic and angular distributions of fusion neutrons are essential for neutron spectroscopy [12], and for fast particle physics in plasma [13,14], as well as for estimating loads on the first walls of reactors due to fluxes of neutrons and other fast particles [15]. Modelling of tritium breeding, subcritical fission, transmutation, and other purpose blankets also requires the knowledge of spectra of incident fusion neutrons [15,16,17,18].

Reviews [19,20] consider the stages of development and prospects of fusion–fission hybrid systems and emphasize the important role of the coefficient that takes into account the increased production of neutrons in subcritical systems in the initial generations of fission chain reactions. The value of this coefficient in the formula for the number of fission neutrons per one fusion neutron can exceed two. Energies of neutrons produced in the deuterium–tritium fusion reaction are much higher than the average fission neutron energy. In this regard, the energy spectrum of a fusion neutron source strongly manifests itself in the initial generations of subcritical fission.

As a rule, in research and development works on controlled nuclear fusion in toroidal magnetic configurations, calculations of distributions of a plasma neutron source are combined with numerical modelling of the penetration of fast neutral beams into the plasma, with calculations of distributions of plasma ion velocities, simulations of neutron transport processes, and need to be included in more extensive integrated modelling. Studies [21] and [22] are devoted to the analysis of possible ranges of parameters of tokamak neutron sources with neutral beam-heated plasmas, and the examination of neutral beam-driven plasma operation scenarios of fusion neutron sources.

In magnetic plasma confinement devices, the anisotropy of fuel nuclei velocity distributions due to beam and wave heating leads to anisotropy of neutron energy spectra. An approach to calculating energy spectra of fusion neutrons, based on the use of Monte Carlo methods, exists, for example in [23,24], and an approach based on the explicit analytical formulae obtained in [25,26,27].

From the point of view of the development of the physics basis, as well as the reliability and the computational speed of integrated modelling, the use of analytical results is beneficial. The formulae for distributions of fusion products found in [25] themselves are general, i.e., suitable for arbitrary distributions of velocities of fuel nuclei. Either distributions obtained experimentally can be plugged in, or those calculated numerically. In [25], a compact simplified analytical model of anisotropic fast ion distributions neglecting the speed diffusion effect on high-energy tails was applied as a sample case, which is convenient, being expressed in a closed form. The purpose of this work was to figure out to what extent such a simplification is plausible. The overall population of energetic particles strongly influences the neutron production [22]. Both in the DEMO-FNS classical tokamak [5] and in the FNS-ST spherical tokamak [4], contributions of beam–plasma interactions to the total fusion rates are either on par with or exceed thermonuclear rates, i.e., contributions of Maxwellian populations of fuel nuclei. It is therefore preferable to evaluate the sensitivity of neutron spectra to modifications of high energy ion distribution tails caused by speed diffusion. Section 2 describes the mathematical models used for this purpose, and Section 3 presents the results, followed by a summary given in Section 4.

2. Modelling Techniques

At an early stage of nuclear fusion research, possible approaches to calculating the distributions of fusion neutrons were discussed in [28], where deuterium–deuterium fusion reactions were considered and the formula for the fusion neutron spectrum was given for the case when the interacting fuel nuclei are identical and their distributions are Maxwellian, without taking into account the anisotropy of the differential cross section. Some other special cases were also considered, namely, for the generalized Maxwellian distribution with different values of perpendicular and parallel temperatures and for monoenergetic distributions. According to the available data for the ³H(²H,n)⁴He reaction, the differential cross-section exhibits a rather weak angular dependence over a wide energy range, whereas for the ²H(²H,n)³He reaction, the anisotropy of the differential cross-section is quite distinct. For some particular cases, a simplified angular dependence of the differential cross-section of the deuterium–deuterium fusion reaction was assumed in [28].

Later, in [29], a method to calculate the energy spectra of neutrons was proposed, based on chain rule differentiation of the total cross-section with respect to the emission angle cosine, which, in turn, is related to the kinetic energy of the neutron owing to energy and momentum conservation. Estimations of fusion neutron spectra in a beam-Maxwellian plasma reported in [30] were based on [28,29]. Suprathermal ions, other than those injected by heating beams, may influence the neutron spectra, e.g., so-called knocked-on deuterons and tritons formed due to close elastic collisions with fusion-born α-particles, as discussed in [31,32]. Neutron spectra obtained using Monte Carlo modelling techniques were described in [33], and afterwards in other studies of magnetic and inertial confinement fusion, such as [34,35]. For a number of particular cases, analytical expressions were considered, such as [36,37]. However, the mentioned publications [28,29,30,31,32,33,34,35,36,37] do not contain general formulae.

For the purpose of clarifying the effect of the speed diffusion of fast ions on neutron spectra, straightforward analytical results of a general form for double differential fusion reactivities $\frac{d^2{R}_{12}}{d{E}_{3}d{\Omega }_3}$ with respect to neutron energy and laboratory emission angle obtained in [25] have been used in this work. For reactions between two colliding fuel nuclei (species “1” and “2”) and two product particles (species “3” and “4”), the general methods in [25] are called the S- and the L-algorithms, enabling calculations of distributions of fusion products for arbitrary anisotropic distributions of fuel nuclei velocities, also taking into account the angular anisotropy of differential cross-sections of fusion reactions. Either of these equivalent methods are suitable. Species “3” means neutrons herein. Although the S- and the L-algorithm differ mathematically, both are 5-fold integrals essentially based on the use of the relation

$${\left(v_3-V\right)}^2=\frac{2m_4}{M{m}_3}\left(\frac{\mu {\upsilon }^2}{2}+q_f\right)$$

(1)

following from the energy and momentum conservation laws with $M$ and $\mu$ being the sum of the fuel particle masses $m_1$ and $m_2$ and their reduced mass, respectively, $v_3$ being the laboratory velocity of product “3”, $V$ being the centre of mass velocity, $\upsilon$ designating the relative velocity of the fuel particles, and $q_f$ being the energy released in an elementary fusion reaction due to the mass defect. Up-to-date values of particle masses can be found in [38].

Anisotropic distributions of fuel nuclei velocities in fusion plasma subject to neutral beam injection or radiofrequency heating were described, for example, in [39] and the references therein. This anisotropy, in turn, leads to an anisotropy of the distributions of nuclear fusion products. Modified S- and L-algorithms were published in [27] on the basis of the same geometric technique as in [26]. This method still retains the generality of [25]. Detailed descriptions of the computations of neutron spectra are explained in the corresponding references.

The compact analytical anisotropic model, used in [25] as a sample case of fuel nuclei velocity distributions, that does not take into account the speed diffusion effect, is as follows:

$$f_{1,2}(\mathrm{v},\vartheta )=\left(1-A_{1,2}\right)f_{1,2}^{(\mathrm{M})}(\mathrm{v})+A_{1,2}{\displaystyle \sum _{n=0}^{\infty }{\varphi }_n^{(1,2)}\left(\mathrm{v}\right)P_n\left(\cos \vartheta \right)}$$

(2)

where the second term describes populations of suprathermal particles of species “1” and “2” with energetic tail fractions $0\le A_{1,2}\le 1$, the term

$$f_{1,2}^{(\mathrm{M})}({\mathrm{v}}_{1,2})=\frac{1}{{\pi }^{3/2}}{\left(\frac{m_{1,2}}{2T_{1,2}}\right)}^{3/2}\exp \left(-\frac{m_{1,2}{\mathrm{v}}_{1,2}^2}{2T_{1,2}}\right)$$

(3)

is the Maxwellian distribution of bulk thermalized particles, $P_n\left(\cos \vartheta \right)$ are Legendre polynomials,

$${\varphi }_n^{(1,2)}\left(\mathrm{v}\right)=\frac{{\kappa }_{1,2}{Z}_{1,2}^{(\mathrm{b})}}{V_c^3}\left(2n+1\right){\mathcal{Z}}_n^{(1,2)}\frac{H({\mathrm{v}}_{in{j}_{}1,2}-\mathrm{v})}{b_{1,2}(\mathrm{v})}{e}^{{-}_{}\frac{n\left(n+1\right)}{{\epsilon }_{1,2}{v}_{Te}}{\displaystyle \underset{\mathrm{v}}{\overset{{\mathrm{v}}_{in{j}_{}1,2}}{\int }}\frac{c_{1,2}(\stackrel{-}{\mathrm{v}})}{b_{1,2}(\stackrel{-}{\mathrm{v}})}d\stackrel{-}{\mathrm{v}}}}$$

(4)

Dimensionless functions $b_{1,2}(\mathrm{v})$ and $c_{1,2}(\mathrm{v})$ in (4) are given by

$$b_{1,2}(\mathrm{v})=Z_{1,2}^{(\mathrm{b})}\left(1+\frac{{\mathrm{v}}^3}{V_{c_{}1,2}^3}\right)$$

(5)

$$c_{1,2}(\mathrm{v})=\frac{{\epsilon }_{1,2}{Z}^{(\mathrm{eff})}}{2}\frac{v_{Te}}{\mathrm{v}}+\frac{2{\epsilon }_{1,2}}{3\sqrt{\pi }}$$

(6)

where

$$Z^{(\mathrm{eff})}=\frac{1}{n_e}{\displaystyle \sum _{i=1}^{N_i}{n}_i{Z}_i^2}$$

(7)

The value

$$V_{c_{}1,2}={\left(\frac{3\sqrt{\pi }}{4}{Z}_{1,2}^{(\mathrm{b})}\right)}^{1/3}{\epsilon }_{1,2}{\mathrm{v}}_{Te}$$

(8)

referred to as critical velocity is proportional to the electron thermal velocity ${\mathrm{v}}_{Te}=\sqrt{2T_e/m_e}$ determined by the electron temperature $T_e$. The parameters in (8) are dimensionless quantities given by

$${\epsilon }_{1,2}={\left(\frac{m_e}{m_{1,2}}\right)}^{1/3}$$

(9)

$$Z_{1,2}^{(\mathrm{b})}=\frac{m_{1,2}}{n_e}{\displaystyle \sum _{i=1}^N\frac{n_i{Z}_i^2}{m_i}}$$

(10)

where $m_e$ is the electron mass, $n_e$ is the electron density, the summation is over the $N$ species of the background plasma ions with masses $m_i$, electric charge numbers $Z_i$ and densities $n_i$. The values

$${\kappa }_{1,2}={\left(\frac{4\pi }{3}\ln \left(1+\frac{{\mathrm{v}}_{in{j}_{}1,2}^3}{V_{c_{}1,2}^3}\right)\right)}^{-1}$$

(11)

are normalizing constants. Finally, coefficients

$${\mathcal{Z}}_n^{(1,2)}={\displaystyle \underset{0}{\overset{\pi }{\int }}{\mathcal{Z}}_{1,2}\left(\vartheta \right)P_n\left(\cos \vartheta \right)\sin \vartheta d\vartheta }$$

(12)

are dimensionless quantities determined by the unity-normalized angular dependence factor ${\mathcal{Z}}_{1,2}\left(\vartheta \right)$ of the monoenergetic fast particle source in the plasma, $H({\mathrm{v}}_{in{j}_{}1,2}-\mathrm{v})$ is the unit step function, and ${\mathrm{v}}_{in{j}_{}1,2}$ are the injection velocities of fast particles.

The model used here to take into account the speed diffusion effect is based on numerical solutions of the Landau–Boltzmann kinetic equation for the distribution functions of fuel nuclei $n_{1,2}\left(r\right)f_{1,2}\left(v\right)$ [cm^–6s³]

$$\frac{\partial \left(n_{1,2}{f}_{1,2}\right)}{\partial t}=C_{1,2}+S_{1,2}-\frac{n_{1,2}{f}_{1,2}}{{\tau }_{1,2}}$$

(13)

where $S_{1,2}$ is the monoenergetic fast particle source

$$S_{1,2}(\mathrm{v},\vartheta )=\frac{S_{1,2}^{inj}}{2\pi {\mathrm{v}}^2}\delta (\mathrm{v}-{\mathrm{v}}_{in{j}_{}1,2}){\mathcal{Z}}_{1,2}\left(\vartheta \right)$$

(14)

with $\delta (\mathrm{v}-{\mathrm{v}}_{in{j}_{}1,2})$ being the Dirac delta function and $S_{1,2}^{inj}$ [cm^–3s^–1] being the source rate, i.e., the number of injected particles of species “1” or “2” per unit volume per unit time. Fast particle lifetime is denoted by ${\tau }_{1,2}$ to enable a simple simulation of losses.

Maxwellian background plasma is assumed, and the Landau collision term for fast ion species “1” or “2” is

$$\begin{gathered} C_{1,2}=\frac{{\mathrm{v}}_{c_{}1,2}^3}{{\tau }_{s_{}1,2}}\frac{1}{{\mathrm{v}}^2}(\frac{\partial }{\partial \mathrm{v}}\left({\mathrm{v}}_{c_{}1,2}^2\frac{a_{1,2}(\mathrm{v})}{2\mathrm{v}}\frac{\partial \left(n_{1,2}{f}_{1,2}\right)}{\partial \mathrm{v}}+b_{1,2}(\mathrm{v})\left(n_{1,2}{f}_{1,2}\right)\right) \\ +\frac{c_{1,2}(\mathrm{v})}{{\mathrm{v}}_{c_{}1,2}}\frac{1}{\sin \vartheta }\frac{\partial }{\partial \vartheta }\left(\sin \vartheta \frac{\partial \left(n_{1,2}{f}_{1,2}\right)}{\partial \vartheta }\right)) \end{gathered}$$

(15)

where

$${\mathrm{v}}_{c_{}1,2}={\epsilon }_{1,2}{\mathrm{v}}_{T_e}$$

(16)

and

$${\tau }_{s_{}1,2}={\left(\frac{m_{1,2}}{Z_{1,2}e{\omega }_{pe}}\right)}^2\frac{{\mathrm{v}}_{c_{}1,2}^3}{\Lambda m_e}$$

(17)

with $Z_{1,2}$ being the electric charge numbers of the injected particles, $\Lambda$ being the Coulomb logarithm, and

$${\omega }_{pe}=\sqrt{\frac{4\pi n_e{e}^2}{m_e}}$$

(18)

being the electron plasma frequency with $e$ designating the elementary charge.

The terms containing $a_{1,2}(\mathrm{v})$ and $c_{1,2}(\mathrm{v})$ are associated with the diffusion tensor in velocity space. These terms are responsible for the speed diffusion and pitch angle scattering, respectively. The term with $b_{1,2}(\mathrm{v})$ is associated with the dynamic friction force and describes the slowing-down process. Expressions (5) and (6) given above are simplified. Complete formulae for $a_{1,2}(\mathrm{v})$, $b_{1,2}(\mathrm{v})$, and $c_{1,2}(\mathrm{v})$ are given in [39], as well as the details of obtaining numerical steady state solutions of (13).

It is worth mentioning that an analytical approach is also possible. In brief, for an isotropic case, such an approach can be described as follows. The steady state implies zero time derivative in Equation (13). The isotropy implies no angular dependence, i.e., zero angular derivative in the term with $c_{1,2}(\mathrm{v})$. Neglecting the speed diffusion, i.e., neglecting the term with $a_{1,2}(\mathrm{v})$ bearing in mind the small parameter ${\epsilon }_{1,2}$, and assuming an infinite ${\tau }_{1,2}$, i.e., no losses of particles for simplicity, a steady state isotropic solution of (13) can be readily obtained in the form of the so-called classical slowing-down distribution

$$n_{1,2}{f}_{1,2}(\mathrm{v})=\frac{S_{1,2}^{inj}{\tau }_{s_{}1,2}}{4\pi v_{c_{}1,2}^3}\frac{1}{b_{1,2}(\mathrm{v})}H({\mathrm{v}}_{in{j}_{}1,2}-\mathrm{v})$$

(19)

where the unit step function is obviously cutting off the distribution tail above ${\mathrm{v}}_{in{j}_{}1,2}$. The most straightforward way to obtain a more sophisticated analytical solution taking into account the speed diffusion can be demonstrated for the case when all background plasma species are in thermal equilibrium with equal temperatures T. Since the collision term nullifies in thermal equilibrium, the Maxwellian distribution function $\exp \left(-m_{1,2}{v}^2/2T\right)$ is to be a partial solution of the homogeneous differential equation corresponding to (13). Next, the second independent partial solution of the homogeneous equation can be found, and afterwards the general analytical solution of Equation (13) as explained in [39]. However, numerical solutions are used herein for the designated purposes of studying the effect on the resultant fusion neutron spectra.

3. Calculation Results

To study the influence of the speed diffusion effect on distributions of emission energies and angles of nuclear fusion products, calculations of energy spectra of neutrons produced in a candidate operating regime of FNS-ST [4] in ³H(²H,n)⁴He and ²H(²H,n)³He reactions were performed using the simple analytical anisotropic model (2) similarly to [25], as well as using steady-state numerical solutions of (13) where the speed diffusion effect is included. Parameterisations of differential cross-sections from [40] and [41] were used. Numerical solutions of the Landau–Boltzmann kinetic Equation (13) were obtained using [39]. The beam and plasma parameters of the FNS-ST candidate operating regime, used herein as a sample case, are similar to those used in simulations reported in [42].

In the modelling of the ³H(²H,n)⁴He reaction, mono-directional injection of monoenergetic deuterons and tritons into Maxwellian deuterium–tritium background plasma was assumed. The injection energy 130 keV, equal electron and ion temperatures 7 keV, and the electron density $n_e$ = 10¹⁴ cm⁻³ were taken as input values. Such are the FNS-ST core plasma parameters adopted in [42]. The injection angle value, i.e., the pitch angle of particles injected by the source, was 30°, in other words, the angular dependence factor ${\mathcal{Z}}_{1,2}\left(\vartheta \right)$ in (12) and (14) was taken in the form a delta-like peak at the injection angle. In fact, the source function of fast ions originating from neutral beam injection into a toroidal magnetically confined plasma is characterized by a certain angular distribution, rather than a particular injection angle as explained in [43]; however, a narrow distribution around the selected injection angle value was assumed here for simplicity. The fraction of suprathermal tail particles $A_{1,2}$ in (2) was 2.5%.

Figure 1a,b show surface plots of anisotropic distributions of deuteron velocities and triton velocities, correspondingly, calculated for the same conditions. The results of simplified calculations using Formula (2), not taking into account the speed diffusion, are depicted by darker colours, whereas the surfaces depicted by semi-transparent lighter colours refer to the solutions of Equation (13) with the speed diffusion effect. The presence of high-energy tails in the vicinity of the injection angle can be seen. The unit step function in Formula (4) is cutting off these tails above the injection velocities, as the darker surfaces illustrate. The lighter surfaces exhibit the tails extending to the regions above the injection velocities that are different for deuterons and tritons due to the difference of masses.

Figure 2 shows contour plots of the same distributions of deuteron velocities and triton velocities as shown in Figure 1. The axes are the parallel and perpendicular projections of velocity ${\mathrm{v}}_{\parallel }=\mathrm{v}\cos \vartheta$ and ${\mathrm{v}}_{\perp }=\mathrm{v}\sin \vartheta$. Model (2) is shown by dashed lines and model (13) is shown by solid lines. Contour lines of low-energy parts of distributions of thermalized particles are circular. The anisotropy of velocity distributions of high-energy particles can be seen as distortions of contour lines in the vicinity of the injection velocity and injection angle.

It should be noted that the speed diffusion is responsible for thermalization of injected fast ions governed by Equation (13). The Maxwellian term was introduced artificially in model (2) to describe the bulk plasma ion population. Therefore, the low-energy parts of darker and lighter surfaces in Figure 1 coincide as well as the low-energy contours shown by dashed and solid lines in Figure 2.

Modelling of the ²H(²H,n)³He reaction was performed assuming mono-directional injection of monoenergetic 130 keV deuterons into Maxwellian deuterium target plasma, with equal electron and ion temperatures 7 keV and the electron density $n_e$ = 10¹⁴ cm⁻³. The injection angle 90° was used as a sample case, and the fraction of suprathermal tail deuterons was 2.5%.

Figure 3a shows 3D plots of two anisotropic distributions of velocities of deuterium nuclei calculated for the same conditions. The dark-blue surface corresponds to the simplified model (2) without taking into account the speed diffusion. The light-blue surface corresponds to the semianalytical model based on (13), taking into account the speed diffusion effect. The presence of a high-energy tail in the vicinity of the injection angle can be seen. The tail is extending for the light-blue surface further than for the dark-blue surface. Figure 3b illustrates these same deuteron velocity distributions as 2D plots for three selected pitch angles: 30°, 60°, and 90°.

Double differential reactivities with respect to neutron energy $E_n$ and laboratory emission angle ${\vartheta }_n$ are shown as surface plots in Figure 4a,b for ³H(²H,n)⁴He reactions and for ²H(²H,n)³He reactions, correspondingly. Blue surfaces show the results obtained with the fast ion speed diffusion effect and grey surfaces show the results obtained without the fast ion speed diffusion effect. The presence of extra tails of suprathermal ions with somewhat higher energies in the former case is responsible for a slight increase in the reactivities; however, the shapes of the surfaces do not significantly differ, qualitatively.

For clarity, the difference between the neutron distributions obtained with and without the effect of speed diffusion of fast deuterium and tritium ions is shown in Figure 5a for the ³H(²H,n)⁴He reaction and in Figure 5b the ²H(²H,n)³He reaction.

Angularly resolved energy spectra of neutrons produced in ³H(²H,n)⁴He reactions and in ²H(²H,n)³He reactions are shown in Figure 6a and in Figure 6b, correspondingly, for the cases when calculations were made using the simplified model (2) not taking into account the speed diffusion effect, and using a more sophisticated model (13), taking into account the speed diffusion in velocity space of fuel nuclei. Although noticeable differences in fusion product spectra can be observed, the obtained results demonstrate similar qualitative behaviour.

Total, i.e., integral over the angles, energy spectra of neutrons produced in ³H(²H,n)⁴He reactions and in ²H(²H,n)³He reactions are shown in Figure 7a and in Figure 7b, correspondingly, for the cases when calculations were made without the speed diffusion effect (dark-blue colour), and with speed diffusion effect (green colour). Integration of the total energy spectra over the entire energy range results in the rate coefficient. The rate coefficient tends to be slightly greater when the speed diffusion is accounted for because of the presence of ions with higher energies in this case.

Squares in Figure 6 and Figure 7 depict the calculated values for the selected neutron energy grid, while the solid lines depict spline approximations.

4. Conclusions

Calculations of distributions of fusion neutrons in the presence of suprathermal deuterium and tritium nuclei originating form a monoenergetic source in Maxwellian plasma have been performed. The role of the speed diffusion effect in the fuel nuclei velocity space in the formation of energy distributions of products of ³H(²H,n)⁴He and ²H(²H,n)³He fusion reactions has been studied. High-energy tails of distributions of fast ions velocities in the regions below and above the beam injection velocity influence the energetic and angular distributions of nuclear fusion products.

The effect of the fast ion speed diffusion on the obtained neutron spectra is noticeable; however, it does not significantly modify the qualitative behavior of the spectra and such general parameters as full width at half maximum. Thus, the use of simplified analytical models for the ion distribution functions is reasonably justified when the knowledge of “fine structure” of neutron spectra is not required.

Advanced plasma diagnostics, such as high-resolution neutral particle analysis combined with high-resolution neutron spectrometry, may be used for experimental validation of the mathematical models described herein. Recent progress on the development of active charge exchange diagnostics of fast ion distributions and neutron spectroscopic diagnostics on Globus-M2 spherical tokamak [44] were reported in [45] and [46], respectively. An excellent powerful set of neutron diagnostics operating on the Large Helical Device (LHD) in combination with advanced diagnostics of energetic ions was described in [47]. Measurements of anisotropic distributions of fusion neutrons in experiments with deuterium plasma heated by neutral beam injection on LHD were reported in [48]. Neutron emission spectroscopy on Joint European Torus (JET) was overviewed in [49]. Prospects of various neutron diagnostics foreseen for ITER, including neutron flux monitors and spectrometers, were described in [50].

The results may be applied for simulations of energetic and angular distributions of neutrons and charged nuclear fusion products in magnetically confined plasma in the framework of the activities on the development of fusion energy reactors and fusion neutron sources employing beam–plasma interactions.