The deceleration of a monoenergetic rarefied ion beam in nonisothermal plasma in a constant magnetic field has been studied. It is shown how ion-acoustic turbulence generated by a current along the magnetic field leads to an effective decrease in the velocity of ions moving with a speed higher than the speed of ion sound. When an ion beam is injected into plasma across the magnetic field, the trajectory of the ions has the form of a contracting spiral elongated along the magnetic field. The deceleration occurs due to the Cherenkov interaction of ions with ion-acoustic waves and stops when the ion velocity decreases to the speed of ion sound. The braking lengths and the beam velocity components, which are set at the moment of stopping braking, are found. After the end of deceleration, the ions move with constant velocity in a spiral along the magnetic field.

## INTRODUCTION

Ion beams are used in studies of plasma properties in small tokamaks.^{1–3} The interaction of beams with plasma occurs under conditions when the temperature of electrons $Te$ exceeds the temperature of ions $Ti$, and the velocity of relative motion of electrons and ions is greater than the speed of ion sound. In a nonisothermal plasma, the propagation of ion-acoustic waves is possible (see, for instance, Refs. 4–7). At high current densities, ion-acoustic instability develops in a nonisothermal plasma. In this case, the plasma state is characterized by the presence of charge density pulsations with a spatial scale comparable to or greater than Debye electron radius. Quasistationary state of pulsations–ion-acoustic turbulence (IAT) is established due to the competition of Cherenkov radiation of waves by electrons, damping of waves due to Cherenkov interaction with ions, and induced scattering of waves on plasma ions. Under these conditions, it is the scattering of the beam ions by charge density pulsations that determines deceleration if their velocity exceeds the velocity of ion sound.^{8} The theory presented in Ref. 8 does not take into account the effect of the magnetic field on the movement of ions. At the same time, in small tokamaks TUMAN-3M, TEXT-U, or J-TEXT, there is a magnetic field **B**, the strength of which is $1\u2009T$ or more.^{9–12} Such a magnetic field can significantly change the trajectory of the beam ions and, thereby, affect the braking process.

The purpose of this report is to study the effect of magnetic field on the deceleration of a rarefied ion beam in a plasma with developed IAT. At the same time, the conditions in which the magnetic field does not affect the IAT are considered. Under these conditions, the magnetic field does not affect the braking process of the beam propagating along the magnetic field, and the results of Ref. 8 can be used to describe the braking. Therefore, the deceleration of an ion beam is further considered, the initial velocity of which is directed across the magnetic field and exceeds the velocity of ion sound. Deceleration occurs due to the Cherenkov interaction of the beam ions with ion-acoustic waves. Braking stops when the beam velocity decreases to the speed of ion sound. Further deceleration, which is due to other reasons, is not considered in this report. During the deceleration, the beam ions can make several turns around the direction of the magnetic field. The number of turns depends on the strength of the magnetic field and the level of IAT, which determines the efficiency of ion scattering of the beam on charge density pulsations. The trajectory has the form of a contracting spiral elongated along the magnetic field. The movement of ions along **B** occurs due to the fact that during the Cherenkov interaction with waves, ions acquire the velocity component along **B**. After the deceleration stops, the beam ions move along **B** in a spiral of constant radius. For several values of the electric field strength determining the level of IAT, the braking lengths, the values of the velocity components, and the Larmor radii at the end of the beam ion deceleration are calculated. Numerical calculations are given for the deceleration of deuterium ion beam in hydrogen plasma, but they can be used to describe the deceleration of other ions by simply redefining the effective collision frequency.

## II. ION-ACOUSTIC WAVES NUMBER DENSITY

*k*. Due to the Cherenkov radiation of the waves by electrons, their damping due to the Cherenkov interaction with resonant ions, and induced scattering by thermal ions, a quasistationary distribution of ion-acoustic waves $N(k)=N(k,\u2009cos\u2009\theta k)$ is established, where $\theta k$ is the angle between the wave vector k and the vector $\u2212eE$, where

*e*is the electron charge. Approximately, we can present the function $N(k)$ as a product of two functions $N(k)=N(k)\Phi (cos\u2009\theta k)$. In this case, the wave number modulo distribution has the following form

^{13}[see formula (2.117)]:

*E*. In a relatively weak field for the function $\Phi (cos\u2009\theta k)$, we have

^{14,15}

^{16}and the field strength $EN$ is determined by the plasma parameters and has the following form:

## III. THE EQUATION FOR THE ION BEAM VELOCITY

*t*. That is, we consider that the thermal velocity of the beam ions is much less than the velocity of the directional motion of ions. Otherwise, there is no point in talking about a particle beam. Let us use a quasilinear equation for the ion beam distribution function,

*q*is the charge of the beam ions,

*m*is their mass,

*c*is the speed of light, and $D\alpha \beta (v)$ is the tensor of quasilinear diffusion, which describes the interaction of the beam ions with ion sound,

**v**and integrate over velocities. Then, using the continuity equation and the type of the beam ions distribution function, we obtain a system of equations for the velocity components. Taking into account the axial symmetry of the ion-acoustic waves number density, it is natural to write down the system of these equations in a cylindrical coordinate system with an axis of symmetry along a constant magnetic field. In this case, for the velocity components $u=(u\rho ,u\phi ,uz)$, we obtain the equations

*z*in Eq. (7) is carried out in the range from $\zeta min=kminrDe$ to $\zeta max=kmaxrDe$. Due to the rapid decrease in the function $N(k)$ at $krDe\u223c1$ [see Eq. (1)], with an accuracy $0.15%$, we can accept $\zeta max=1$. The value of $\zeta min$ depends on how strongly ion–ion collisions affect the damping of ion-acoustic waves. Then, we assume that $\zeta min\u223c\nu ii\omega LerDi2/\omega Li2rDe2(1+\delta )\u226a1$, where $\nu ii$ is the ion–ion collision frequency in plasma. Note that under these conditions, the Kadomtsev–Petviashvili spectrum is realized with good accuracy in the range of wave numbers from $kmin$ to $kmax$. In this case, for $\nu f$, we have

Moving from kinetic Eqs. (4)–(6) for the components of the average ion velocity, we excluded from consideration the possibility of describing the broadening of the ion velocity distribution. Nevertheless, it is still possible to determine the average values of the lengths and times of deceleration of the beam ions in the plasma with IAT. Since ion scattering occurs at small angles in a plasma with IAT, as in a rarefied laminar plasma, the approximation used is just as productive.

## IV. ION BEAM DECELERATION

**B**on the movement of ions. However, we will not take into account the influence of magnetic field on the ion-acoustic waves number density. The latter is justified if the cyclotron frequency of electrons is less than their Langmuir frequency, that is, the energy density of the magnetic field is relatively small $B2/4\pi <nemec2$ (see Ref. 18). When this condition is met, the influence of magnetic field on the Cherenkov interaction of electrons with ion-acoustic waves can be ignored. However, if this condition is met, the magnetic field pressure may be greater than the plasma pressure. In this case, an inhomogeneous magnetic field can have a strong effect on the plasma density. That is, in a strong inhomogeneous magnetic field, its effect on the IAT can manifest itself through a change in plasma parameters. However, the IAT distribution established under conditions $B2/4\pi <nemec2$ will remain unchanged if the particle density and temperature are properly determined. In connection with the possible influence of magnetic field on the IAT and deceleration of ions, we note another process that can manifest itself in the plasma of Tokamaks. Due to the anisotropy of the IAT, anisotropic turbulent heating of ions occurs (Refs. 13 and 15). The anisotropy of effective ion temperatures leads to the Weibel instability development. The fluctuations in the magnetic field strength that occur in this case can be roughly estimated by $\Delta B\u223c(4\pi ni\kappa \Delta Ti)1/2$, where $\Delta Ti\u223cTi$ is the degree of ion temperature anisotropy. If $\Delta B\u226aB$, then the effect of magnetic field fluctuations on ion deceleration can be ignored. The following discusses the conditions under which $\Delta B/B\u223c0.01$. When numerically solving Eq. (6), it is convenient to switch to a new variable $s=s(\rho ,\phi ,z)$, which sets the position of the beam ions on a curved trajectory. Along the trajectory, all velocity components depend only on

*s*, and Eq. (6) has the following form:

We present a numerical solution of these equations in the case of deceleration of deuterium ion beam, for which $q=ei$ and $m=2mi$, in hydrogen plasma placed in magnetic field $B=1\u2009T$. In this case, the cyclotron frequency of the deuterium ions is equal to $\Omega =4.8\xd7107\u2009s\u22121$. The densities of electrons $ne$, ions $ni$, and their temperatures $Te$ and $Ti$ are assumed to be equal to $Te=300\u2009eV$, $Ti=50\u2009eV$, and $ne=ni=2\xd71013\u2009cm\u22123$. In these conditions, the velocity of sound is $vs=1.7\xd7107\u2009cm/s$, the frequency $\nu f=4.1\xd7\u2009107\u2009s\u22121$, the electric field strength $EN=18\u2009V/cm$, and the parameter $\delta =6$ (Ref. 16). Note that since $(1+\delta )\u226a\omega Le/\omega Li$, the term $qE/m$ in the third equation of the system (9)–(11) leads to a change in the results of numerical calculations by not more than $14%$.

If the initial velocity of the beam is directed along the symmetry axis of the IAT, then the beam is decelerating in the same way as in the absence of magnetic field.^{8} Therefore, below we consider the case when the initial velocity of the beam is directed across the symmetry axis, that is, across the magnetic field, and is equal to $5vs$. The numerical solution of the system (9–12) for two values of the electric field strength is shown in Figs. 3 and 4. In the figures, the value of coordinates is given in units of $vs/\nu f=0.4\u2009cm$. As can be seen from these figures, deuterium ions move in a contracting spiral deep into the plasma. As they decelerate, their effective Larmor radius $u\u22a5/\Omega $ decreases, and a displacement along the magnetic field occurs. The displacement along the **B** occurs due to the fact that during the Cherenkov interaction with ion-acoustic waves, the deuterium ions receive an impulse from waves that are concentrated in the area of angles $0<\theta k<\pi /2$. The efficiency of deceleration depends on the magnitude of the ratio $E/EN$, that is, on the magnitude of the ion-acoustic waves number density. In a weak electric field, when $E/EN=10<(1+\delta )2=49$, as deuterium ions penetrate deep into the plasma, they manage to make several turns along a spiral of decreasing radius. The deceleration stops when the velocity of ions *u* turns out to be equal to the speed of sound $vs$. At the same time, $uzf$ and $u\u22a5f$ have well-defined finite values, which are given in Table I and satisfy the ratio $uzf2+u\u22a5f2=vs2$. Due to the presence of $uzf$ and $u\u22a5f$, further trajectory of deuterium ions is a spiral with a radius of $u\u22a5f/\Omega $ and period of $2\pi uzf/\Omega $ along the magnetic field (see Fig. 3). In a strong electric field, when $E/EN=100>(1+\delta )2=49$, the ion deceleration is more effective due to the high level of turbulence. Therefore, in Fig. 4, before the velocity *u* turns out to be equal to the speed of sound $vs$, the trajectory contains only half a turn around the symmetry axis. This section of the trajectory is shorter than at a lower level of turbulence. The values of the length of the trajectory section $sf$, where $u>vs$ and the Cherenkov interaction of deuterium ions with waves is possible, are given in Table I. There are also values of the Larmor radius $rLf$ at the moment of stopping braking. From Table I, it can be seen how the length of $sf$ decreases with an increase in the level of turbulence, that is, $E/EN$. Obviously, the decrease in the initial beam velocity leads to the decrease in $sf$. The quantitative decrease in $sf$ can be seen from Table I, which shows the results of calculations of $sf$, $rLf$, $uzf$, and $u\u22a5f$ for two values of the initial ion velocity: $5vs$ and $3vs$. Note that the accuracy of calculations of all values at $E/EN\u223c(1+\delta )2$ is low, which is due to the low accuracy of describing the ion-acoustic waves number density at such electric field strengths. Data in tables at $E/EN=(1+\delta )2=50$ are obtained using the angle distribution of IAT shown in Fig. 2. Using an expression of the form (2) gives a greater value of the breaking length by about 3.6 times.

. | $u\u22a5(0)=5vs$ . | $u\u22a5(0)=3vs$ . | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

$E/EN$ . | $tf\u2009(10\u22127\u2009s)$ . | $sf\u2009(cm)$ . | $rLf\u2009(cm)$ . | $u\u22a5f/vs$ . | $uzf/vs$ . | $tf\u2009(10\u22127\u2009s)$ . | $sf\u2009(cm)$ . | $rLf\u2009(cm)$ . | $u\u22a5f/vs$ . | $uzf/vs$ . |

5 | 4.65 | 31.0 | 0.153 | 0.431 | 0.902 | 1.12 | 4.80 | 0.153 | 0.431 | 0.902 |

10 | 3.72 | 24.7 | 0.174 | 0.490 | 0.872 | 0.884 | 3.74 | 0.174 | 0.491 | 0.871 |

50 | 0.691 | 4.40 | 0.288 | 0.811 | 0.585 | 0.143 | 0.566 | 0.288 | 0.811 | 0.585 |

100 | 0.489 | 3.15 | 0.288 | 0.811 | 0.585 | 0.101 | 0.401 | 0.288 | 0.811 | 0.585 |

200 | 0.346 | 2.22 | 0.288 | 0.812 | 0.584 | 0.071 | 0.283 | 0.288 | 0.811 | 0.585 |

. | $u\u22a5(0)=5vs$ . | $u\u22a5(0)=3vs$ . | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

$E/EN$ . | $tf\u2009(10\u22127\u2009s)$ . | $sf\u2009(cm)$ . | $rLf\u2009(cm)$ . | $u\u22a5f/vs$ . | $uzf/vs$ . | $tf\u2009(10\u22127\u2009s)$ . | $sf\u2009(cm)$ . | $rLf\u2009(cm)$ . | $u\u22a5f/vs$ . | $uzf/vs$ . |

5 | 4.65 | 31.0 | 0.153 | 0.431 | 0.902 | 1.12 | 4.80 | 0.153 | 0.431 | 0.902 |

10 | 3.72 | 24.7 | 0.174 | 0.490 | 0.872 | 0.884 | 3.74 | 0.174 | 0.491 | 0.871 |

50 | 0.691 | 4.40 | 0.288 | 0.811 | 0.585 | 0.143 | 0.566 | 0.288 | 0.811 | 0.585 |

100 | 0.489 | 3.15 | 0.288 | 0.811 | 0.585 | 0.101 | 0.401 | 0.288 | 0.811 | 0.585 |

200 | 0.346 | 2.22 | 0.288 | 0.812 | 0.584 | 0.071 | 0.283 | 0.288 | 0.811 | 0.585 |

We present the results of calculations of the deceleration of a deuterium ion beam for another case when the particle temperatures and density of hydrogen plasma are close to those realized in the experiment:^{10} $Te=600\u2009eV$, $Ti=100\u2009eV$, and $ne=ni=3\xd71013\u2009cm\u22123$. Under these conditions, the speed of sound, the effective collision frequency, and the characteristic spatial scale and strength of the electric field are slightly higher: $vs=2.4\xd7107\u2009cm/s$, $\nu f=5.0\xd7107\u2009s\u22121$, $vs/\nu f=0.48\u2009cm$, and $EN=31\u2009V/cm$. At the same time, the influence of the magnetic field $B=1\u2009T$ on the trajectory of deuterium ions is slightly weaker (see Figs. 5 and 6). The calculation results of $sf$, $rLf$, $uzf$, and $u\u22a5f$ for two ion velocity values $5vs$ and $3vs$ are shown in Table II.

. | $u\u22a5(0)=5vs$ . | $u\u22a5(0)=3vs$ . | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

$E/EN$ . | $tf\u2009(10\u22127\u2009s)$ . | $sf\u2009(cm)$ . | $rLf\u2009(cm)$ . | $u\u22a5f/vs$ . | $uzf/vs$ . | $tf\u2009(10\u22127\u2009s)$ . | $sf\u2009(\u2009cm)$ . | $rLf\u2009(cm)$ . | $u\u22a5f/vs$ . | $uzf/vs$ . |

5 | 3.79 | 35.9 | 0.216 | 0.431 | 0.902 | 0.915 | 5.56 | 0.216 | 0.431 | 0.902 |

10 | 3.03 | 28.6 | 0.246 | 0.490 | 0.872 | 0.721 | 4.33 | 0.246 | 0.491 | 0.871 |

50 | 0.563 | 5.10 | 0.406 | 0.811 | 0.585 | 0.116 | 0.657 | 0.406 | 0.811 | 0.585 |

100 | 0.398 | 3.65 | 0.406 | 0.811 | 0.585 | 0.083 | 0.466 | 0.406 | 0.811 | 0.585 |

200 | 0.282 | 2.58 | 0.407 | 0.812 | 0.584 | 0.058 | 0.328 | 0.406 | 0.811 | 0.585 |

. | $u\u22a5(0)=5vs$ . | $u\u22a5(0)=3vs$ . | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

$E/EN$ . | $tf\u2009(10\u22127\u2009s)$ . | $sf\u2009(cm)$ . | $rLf\u2009(cm)$ . | $u\u22a5f/vs$ . | $uzf/vs$ . | $tf\u2009(10\u22127\u2009s)$ . | $sf\u2009(\u2009cm)$ . | $rLf\u2009(cm)$ . | $u\u22a5f/vs$ . | $uzf/vs$ . |

5 | 3.79 | 35.9 | 0.216 | 0.431 | 0.902 | 0.915 | 5.56 | 0.216 | 0.431 | 0.902 |

10 | 3.03 | 28.6 | 0.246 | 0.490 | 0.872 | 0.721 | 4.33 | 0.246 | 0.491 | 0.871 |

50 | 0.563 | 5.10 | 0.406 | 0.811 | 0.585 | 0.116 | 0.657 | 0.406 | 0.811 | 0.585 |

100 | 0.398 | 3.65 | 0.406 | 0.811 | 0.585 | 0.083 | 0.466 | 0.406 | 0.811 | 0.585 |

200 | 0.282 | 2.58 | 0.407 | 0.812 | 0.584 | 0.058 | 0.328 | 0.406 | 0.811 | 0.585 |

Tables I and II show calculations for $B=1\u2009T$ in the case when the initial ion velocity is orthogonal to the magnetic field and the axis of IAT symmetry. If one changes the magnetic field magnitude, then only the magnitude of the Larmor radius will change in the tables. The spiral trajectory of the ions will contract or expand. The spiral in figures will expand or contract, and its length will remain unchanged. This effect of the magnetic field is due to two reasons. First, the magnetic field does not do any work. Second, the IAT has axial symmetry. Due to this symmetry, the appearance of azimuthal velocity component does not lead to a change in the length and time of ion deceleration. The situation changes somewhat if the initial ion velocity is directed at an oblique angle to the anisotropy axis. In this case, as can be seen from Table III, other physical characteristics also change due to the anisotropy of effective ion collision frequencies. Table III shows the calculation results for the same plasma parameters as in Table I, and only the initial velocity of the beam ions has two components: $u\u22a5(0)=3vs$, $uz(0)=4vs$, and $u(0)=5vs$. The effective frequency of ion collisions when moving along the anisotropy axis is less than when moving across. As a result, the values of the braking lengths and times in Table III differ from those shown in Table I.

$E/EN$ . | $tf\u2009(10\u22127\u2009s)$ . | $sf\u2009(cm)$ . | $u\u22a5f/vs$ . | $uzf/vs$ . |
---|---|---|---|---|

5 | 4.69 | 32.2 | 0.433 | 0.904 |

10 | 3.41 | 23.3 | 0.485 | 0.875 |

50 | 0.495 | 3.26 | 0.273 | 0.962 |

100 | 0.351 | 2.31 | 0.385 | 0.923 |

200 | 0.246 | 1.63 | 0.309 | 0.954 |

$E/EN$ . | $tf\u2009(10\u22127\u2009s)$ . | $sf\u2009(cm)$ . | $u\u22a5f/vs$ . | $uzf/vs$ . |
---|---|---|---|---|

5 | 4.69 | 32.2 | 0.433 | 0.904 |

10 | 3.41 | 23.3 | 0.485 | 0.875 |

50 | 0.495 | 3.26 | 0.273 | 0.962 |

100 | 0.351 | 2.31 | 0.385 | 0.923 |

200 | 0.246 | 1.63 | 0.309 | 0.954 |

## V. CONCLUSION

Above, the deceleration of a rarefied ion beam in a plasma with an IAT generated by a current along constant magnetic field is studied. In numerical calculations, the plasma parameters were chosen close to those realized in TUMAN-3M. It is shown that, under the influence of magnetic field and IAT, the trajectories of deuterium beam ions have a form of contracting spiral. Characteristic braking lengths were calculated. Since the theory takes into account only Cherenkov interaction of beam ions with ion-acoustic waves, the deceleration stops when the ion velocity becomes equal to the speed of ion sound. The description of the further deceleration of ions implies the inclusion in the theory of other causes of deceleration, in particular, collisions with the bulk of ions. Another limitation of the theory is the assumption of uniformity of plasma parameters. In this regard, the results obtained are of interest only in conditions when the characteristic scales of plasma inhomogeneity exceed the size of the Larmor radius of the beam ions. The latter is usually the case (see the data in Tables I and II). Note that the stated theory does not cover the entire wealth of modes implemented in the TUMAN-3M, TEXT-U, and J-TEXT installations^{19–23} and needs further development. At the same time, the obtained results may be useful, since the deceleration of the ion beam occurs at shorter distances at which the temperature and density of the plasma change and at times much shorter than the plasma confinement time.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**A. A. Shelkovoy:** Investigation (equal); Supervision (equal); Writing – original draft (equal). **S. A. Uryupin:** Investigation (equal); Supervision (equal); Writing – original draft (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available within the article.

## REFERENCES

*Principles of Plasma Electrodynamics*