# Evolution of Heavy Ion Beam Probing from the Origins to Study of Symmetric Structures in Fusion Plasmas

^{1}

^{2}

## Abstract

**:**

## 1. Preface

## 2. Introduction

_{r}and toroidal magnetic B

_{t}fields is generally accepted in magnetic confinement studies [6]. However, this hypothesis has both agreements and disagreements with experiment, so, the effect of E

_{r}to plasma confinement still remains unclear.

#### 2.1. Heavy Ion Beam Probe—A Tool for Measuring Electrical Potential and Plasma Turbulence

#### 2.2. Physical Principles of HIBP Measurements

_{e}, and magnetic potential A (or poloidal magnetic field B

_{p}or current density j). The position of the sample volume can be moved along the plasma cross section by varying the energy of the probing beam E

_{b}or the injection angle of the beam into the plasma α. Thus, the set of points observed by the detector forms a two-coordinate (E

_{b}, α) grid called the detector grid.

_{i}, y

_{i}and detection points x

_{D}, y

_{D}, and the injection angle α (ii) the physical parameters: the toroidal magnetic field B

_{t}, the energy E

_{b}, the mass m, and the electric charge q of the probing particles.

_{L}itself, but the values of the parameters on the right hand side of Equation (1), which are subject to independent restrictions. The equality R

_{L}(q, m, E

_{b}, B

_{t}) = const reduces the number of physical parameters from 4 to 3. So, for 2D optimization in the meridional plane, there are eight parameters: x

_{I}, y

_{I}, x

_{D}, y

_{D}, α, B

_{t}, E

_{b}, q. The need to pass the trajectories of particles through the ports of the vacuum chamber and the structural elements of the machine impose serious constraints on these parameters. These constraints are related to each other because the allowable tolerances for each parameter depend on the others. Such relations defy analytical description due to the complexity of the trajectory behavior. The configuration of the vacuum chamber in the meridional plane approximately determines their tolerances. The problem of optimizing experimental conditions includes the choice of optimal probing parameter values in the sense of the following goal functions:

- Pass the beam through the existing vacuum vessel ports;
- Find the detector line from the core to the edge of the plasma;
- Find a detector grid covering the maximum part of the plasma cross section;
- Optimize the range of beam energies.

_{p}, which shifts the orbits of the probing particles out of the meridional plane and transforms them into spatial curves [26]. In this case, additional parameters appear that affect the sample volume. These are the toroidal coordinates of the injector and detector, Z

_{I}and Z

_{D}, and the injection angle between the initial velocity vector and the meridional plane β. This gives us a 12-dimensional optimization problem with implicit connections between the limits of parameters and with non-formalized goal functions. Figure 2 shows an example of solving this problem by ‘the shooting’ method in T-10. Examples of solving the optimization problem in other machines are given in Section 7 and Section 8. The first goal function was achieved in all considered machines, as the chosen probing schemes ensured the passage of trajectories through the vacuum vessel ports and allowed measurements. Maximization of the second and third goal functions provides the maximum possible radial and angular sizes of the observation area in plasma. The fourth function is not always so essential, because the energy range of modern accelerators is quite large. However, for large machines, such as the W-7X stellarator and the constructed ITER tokamak reactor, this task is of great importance. As will be shown in Section 7, the chosen injection schemes allow one to connect the core and the edge of the plasma with a detector line using the acceptable beam energy.

## 3. Mathematical Problems of Determining Plasma Parameters Using HIBP

#### 3.1. Determination the Spatial Distribution of Electric Potential

_{b}. At the ionization point, which is the plasma region of interest (SV), a particle of the primary beam loses an electron with potential energy $-e{\phi}_{pl}^{SV}$ which is acquired by the secondary ionized probe ion. The total energy of secondary ions leaving the plasma is ${E}_{d}={E}_{b}+e{\phi}_{pl}^{SV}$. Therefore, the local potential in SV is equal to the energy difference:

^{+}, Tl

^{+}, Au

^{+}) and energies (on a scale of several hundred keV—several MeV). On the other hand, the measured plasma potential has the scale of the electron temperature. In the plasma core, it ranges from several hundred eV to keV. At the plasma edge, it has a scale from several eVs to hundreds of eV. Thus, according to (2), the measurement of the potential claims the computation of a small difference of two large quantities. Such measurements require high accuracy of each quantity or highly stabilized high-voltage sources and the use of precision instruments to measure the energy of the secondary beam.

#### 3.2. Determination of the Spatial Distribution of Plasma Density

_{1}and i

_{2}are the intensities of the primary and secondary beams in the SV; $\sigma =\langle \sigma {\upsilon}_{e}\rangle /{\upsilon}_{b}$ is the effective ionization cross section by electron impact, <συ

_{e}> is the ionization rate averaged over the Maxwellian distribution of the electron velocity υ

_{e}, itself a function of the electron temperature T

_{e}; υ

_{b}is the initial velocity of the probing beam, υ

_{e}> υ

_{b}; l = l(x, y) is the coordinate on the detector line; and λ(l(E)) is the longitudinal dimension of SV, the length of the arc along the primary trajectory, from which the secondary ions reach the detector. The electron density n(l) is averaged over the length λ.

_{1}and i

_{2}are the intensities of the primary beam at the exit from the ion injector and the secondary beam at the detector. In this case, Relation (3) can be directly used to determine the local values of the plasma density. However, in modern fusion machines installations, both the primary and secondary beams are significantly attenuated, when passing through the plasma. Methods for determining plasma density using HIBP in the case of strong attenuation are described in [27,28], and they were used for interpretation of experimental data from the TM-4 tokamak.

_{t}, taking into account the attenuation of along the beam trajectory, caused by elementary processes, including collisions of beam particles with plasma particles, can be written as follows:

_{b}is the ion beam current leaving the injector, n is the electron density, σ

^{12}and σ

^{23}are the known effective cross sections for collisional ionization of the probing beam (e.g., Cs in the cases of TM-4 and TJ-II) by plasma electrons.

_{t}(l) is determined experimentally; I

_{b}is a known constant. The integration lines Il and lD, and the quantities λ(l) are found by calculating the trajectory and the detector line.

#### 3.3. Determination of the Plasma Magnetic Potential (Field of Plasma Current)

_{φ}, the toroidal component of the magnetic potential:

_{0}is the injection point, s

_{i}is the ionization point, and s

_{d}is the detection point, q and q + k are dimensionless charge numbers of primary and secondary beam particles. Equation (6) connects the unknown function A

_{φ}with the angular displacement of the beam in the detector respect to the one of injector φ

_{d}− φ

_{0}, that is, with the measured quantity. Equation (6) is a nonlinear Volterra-type integral equation of the second kind. As the integration limits are variable, they depend on the coordinates of the ionization point in the plasma (points on the detector line). In contrast with the equation of the first kind, solving such equations is a well-posed problem. The function A

_{φ}is unknown only inside the plasma column, for r < a. Outside the plasma, where r > a, A

_{φ}

^{ext}is determined by its values inside the plasma as follows:

_{φ}

^{ext}is computed. Note that the integrals in (6–7) are curvilinear integrals over 3D trajectories. The arc element ds along the particle trajectory depends on the unknown function A

_{φ}: ds = ds(A

_{φ}), therefore, Equation (7) is a nonlinear integral equation.

## 4. Application of the HIBP to Fluctuation Measurements

_{e}from the beam current I

_{t}, and the poloidal magnetic field B

_{p}from the toroidal angular displacement of beam φ

_{d}or the linear beam shift ζ = φ

_{d}R, where R is the distance from the detection point to the origin of the cylindrical coordinate system [33].

_{an}is the voltage applied to the analyzer, and F is its dynamic coefficient. Thus, there is a simple linear relationship between the desired value $\delta {\phi}_{pl}^{SV}$ and the measured value $\delta i(t)$.

_{e}over the beam trajectory. In this case:

_{e}, for which the radial correlation length is large and comparable to the plasma radius, the local data on oscillations in the sample volume can be contaminated by attenuation effects accumulating along the entire trajectory. For the experimental data, considered in this paper the large-scale modes are not typical, so the density oscillations are measured locally [34].

_{d}, is described by the following expression [35]:

_{φ}/R, P

_{ζ}

_{inj}is the toroidal magnetic moment at the injection point, known constant. Note that the integrals in (10) are taken along the total paths L

_{1}and L

_{2}from the injector to the position of the sample volume (SV), and from SV to the detector. Equation (10) shows that the local term δψ

_{SV}input the main contribution to the oscillations of beam shift. The contribution of integral terms is much smaller for oscillations with shorter correlation length (or radial wavelength) due to the path integration. It was shown in Reference [36] that the integral terms do not strongly affect the local terms from MHD oscillations, therefore HIBP provides practically local measurements of magnetic fluctuations. However, a nonlocal contribution to ζ

_{d}from oscillations with a large correlation length, comparable to the length of the beam trajectory in plasma, is also possible.

## 5. Mathematical Problems of Experimental Data Processing for Fluctuations of Plasma Parameters

#### 5.1. Spectral Fourier Analysis of Oscillations

_{x}(f) and S

_{x}*(f) as its complex conjugate. Thus the power spectral density (PSD) of oscillations is:

#### 5.2. Fourier Analysis of Coherency and Cross-Phase of Oscillations

_{xy}and cross-phase θ

_{xy}, as following:

#### 5.3. Bispectral Fourier Analysis of Plasma Oscillations

_{1}, f

_{2}and f

_{3}= f

_{1}+ f

_{2}. An example of calculating bicoherence and biphase is shown in Figure 10. The excess of the bicoherence coefficient over the background at the GAM frequency indicates a significant three-wave interaction between the GAM and the broadband turbulence of potential [46]. Note that, at the GAM frequencies the biphase has finite values, while outside them it has a stochastic character. Thus, analysis of HIBP data shows that GAM in T-10 linked with broadband turbulence [47,48], and is not excited by other mechanisms, such as, e.g., the effect of energetic particles.

## 6. Measurements by Several Spatial Channels

#### 6.1. Measurement of Turbulent Particle Flux on the TJ-II Stellarator

_{p}by the potential difference, E

_{p}= (φ

_{1}− φ

_{2})/∆x. The finite distance between the investigated volumes limits the poloidal wave vector of oscillations k

_{p}= k

_{θ}< 3 cm

^{−1}. Direct measurement of E

_{p}allows us to define the radial velocity of E × B drift V

_{r}= E

_{p}/B

_{t}.

_{nE}is the Fourier cross-spectrum between density and poloidal electric field E

_{p}oscillations.

_{p}and the density oscillations [50]. Usually, AEs contribute to the turbulent particle flux ${\Gamma}_{E\times B}^{AE}$

_{,}making up an appreciable fraction of the total flux ${\Gamma}_{E\times B}$. Figure 14 shows that ${\Gamma}_{E\times B}^{AE}$ can be much larger than the broadband turbulent flux ${\Gamma}_{E\times B}^{BB}$ in the same frequency range [51]. It was shown that the value of the turbulent flux is comparable to the value of the total particle flux; therefore,

**Γ**

_{E×B}plays an essential role in the particle balance. It was shown that the transition to the improved confinement mode is characterized by suppression of broadband fluctuations of the plasma potential and density, as well as by a decrease

**Γ**

_{E×B}both at the plasma edge and in the hot core [52].

#### 6.2. Measurement of Turbulent Particle Flux in the T-10 Tokamak

_{t}= 2.4 T, and energy E

_{b}= 220 keV is shown in Figure 15. We can see that the potential difference φ

_{k}− φ

_{j}corresponds to the electric field E, directed along the segment k

_{j}. To measure the poloidal field, it is necessary to align the segment in the poloidal direction. This is achieved in the point of deepest penetration for the detector line of equal energy. [53]. Thus, we can measure the flux for each energy in one spatial point.

#### 6.3. Measurement of the GAM Poloidal Symmetry in the T-10 Tokamak

_{3}(Figure 19b) and edge φ

_{5}(Figure 19c) slits. Figure 19d shows that the coherence coefficient between φ

_{3}and φ

_{5}in the GAM frequency domain reaches unity. Figure 19e shows that the phase shift between these poloidally shifted signals is zero, indicating that GAM is poloidal symmetric, i.e., m = 0.

_{1}and S

_{5}on average to ±0.5, except for the edge, where it remained at a level of ±2 [58]. Thus, phase measurements of the potential confirmed the theoretically predicted poloidal symmetry of potential oscillations for GAM.

## 7. HIBP Diagnostics and Their Outcome for Plasma Symmetry Study

#### 7.1. TM-4 HIBP

#### 7.2. T-10 HIBP

_{t}= 1.5 T in the range 0.6 < ρ

_{SV}< 1 [63,64,65].

- The Cs ions were replaced by heavier Tl ions. The energy range was consequently increased to 220, then to 280, 300, and finally to 330 kV. These changes allow to study discharges with microwave heating (ECRH) at 2.08 < B
_{t}< 2.5 T. In discharges with B_{t}< 2.17 T, the radial region 0.2 < ρ_{SV}< 1 was investigated; - The ion beamlines were modernized in the following way:
- Two serial pairs of steering plates were installed into the primary beamline. The first plates deflect the beam “counter-current” to the extreme toroidal position near the second plates. Thus, the beam approaches the second plates not axially, but maximally shifted. The second plates deflect the beam “co-current”. As a result, a zigzag trajectory is formed, and it became possible to use the full toroidal width of the port, not half of it, as in the axial injection of the beam, and maximize the toroidal correction angle β.
- The secondary beamline was significantly expanded. The toroidal correction plates were placed inside the vacuum chamber to increase the angle of toroidal correction of the secondary trajectories β
_{3}. Correcting plates were equipped with a continuous baking system that maintains their operating temperature at the level of 220–250 °C. This allows us to completely avoid the deposition of hydrocarbon films on the surface of the plates, and to exclude the appearance of high-voltage breakdowns between the plates or from the plate to the grounded T-10 wall. Immediately before the working shot of the tokamak, the baking was turned off and the baking circuit was disconnected in order to exclude the possibility of the interaction of the closed loop with the current and the confining magnetic fields of the machine. The electromagnetic forces resulting from this interaction could deform the plates, making it impossible for the secondary beam to pass into the analyzer. The described upgrades made it possible to expand the operational limits of measurements up to the operational limits of the tokamak 140 kA < I_{pl}< 330 kA.

- The technique of scanning the entry angle of the particle beam into the plasma was implemented. Additionally, the vertical correction plates for secondary trajectories were installed to optimize the particle entry angle into the analyzer. It allowed us to obtain fragments of the radial profile per discharge;
- A control system for the beam was created, which allows us to select a complete set of control voltages in the primary and secondary ion beamlines in each subsequent tokamak shot based on the analysis of the beam position in the previous shot (injection and sweeping angles in primary and secondary beamlines α
_{1}, α_{2}, α_{3}, β_{1}, β_{2}, β_{3}). Its implementation radically simplified and accelerated the process of selecting control voltages, to significantly increase the accuracy of beam positioning, thereby providing systematic measurements both with a fixed SV position or with radial scan.Besides these two traditional HIBP operating modes, the developed system made it possible to operate in new non-standard modes. The most popular among them:- Periodic variation of the SV between two spatial positions (“colon”),
- Periodic change of the SV between several positions (“multiple points”),
- Alternation of the scan and at point measurements during one shot (“scan + point”).

- A feedback for toroidal displacement was created, which adjust the toroidal correction voltage in the secondary beamline (toroidal angle β
_{3}), depending on the measured toroidal displacement in the detector ζ_{d}. This allows us to automate the selection of the toroidal correction voltage from shot to shot, as well as during one shot, especially in shots with changing plasma current (ramp-up, ramp-down) as well as the start of the plasma discharge with sharp increase of the plasma current. The feedback system for toroidal displacement is described in detail in the Reference [67]. - The initial single-channel energy analyzer was first replaced by a two-channel and then a five-channel one. As a result, we can carry out correlation measurements over several spatial channels. In particular, in the region of maximal beam penetration into the plasma, it became possible to measure the poloidal potential correlations or E
_{p}, a turbulent particle flux**Γ**_{E×B}, as well as the poloidal density correlations or rotation of turbulence. Closer to the plasma edge, it is possible to measure the velocity of radial correlations or the radial propagation of potential and density perturbations. - An emitter-extractor unit has been developed, which makes it possible to significantly increase the primary beam current from 2–20 μA to 100–130 μA. Its use has expanded the allowable density limit for HIBP towards both high densities (plasma core) and also ultra-low densities at the plasma edge and in the scrape-off layer (SOL) [68].

#### 7.3. TJ-I HIBP

^{+}ion beam, accelerated up to energy E

_{b}= 60 keV. The scheme of the complex is shown in Figure 24. Continuing the T-10 experiment, the TJ-I used the correcting toroidal plates for the secondary beam. As well as in T-10, the analyzer of the ‘flat mirror’ type was used on the TJ-I. The TJ-I tokamak had rectangular toroidal field coils. The TJ-I was the first, where the radial scanning method was used. The results of its application for E

_{b}= 20 keV are shown in Figure 25. We see that the plasma density has a bell-shaped profile with a pronounced maximum in the center, which shifts outward along the major radius during the discharge. The distribution of the plasma potential along the radius is inhomogeneous and also evolves with the density: the density decrease is accompanied by potential evolution towards the positive values.

#### 7.4. TUMAN-3M HIBP

^{+}ions accelerated up to the energy E

_{b}= 60 keV. The scheme of the HIBP complex is shown in Figure 26. Figure shows that the detector line connects the center and edge of the plasma. It is suitable for determination of radial profiles over a total radial interval.

#### 7.5. WEGA HIBP

^{+}ions accelerated up to the energy E

_{b}= 50 keV. The scheme of the HIBP complex is shown in Figure 28, and the detector grid is shown in Figure 29.

#### 7.6. URAGAN-2M HIBP

^{+}ion beam accelerated to energies up to E

_{b}= 70 keV. The scheme of the HIBP complex is shown in Figure 30a, and the equatorial projection of trajectories is shown in Figure 30b. URAGAN-2M was recommissioned in 2006 after a long pause caused by the reconstruction. It is currently undergoing the stage of adjustment and reaching the design parameters.

#### 7.7. TJ-II HIBP—The Most Advanced Diagnostics to Study Plasma Symmetric Structures

^{+}ion beam accelerated up to the energy E

_{b}= 150 keV.

^{19}m

^{−3}. It was shown that plasma potential evolves dramatically with the density raise, changing the shape and the sign [84]. In the whole observed domain of plasma parameters (density, temperature, collisionality, way of heating: ECRH, NBI) the evolving plasma potential is symmetric in terms of LFS–HFS. Later on, the symmetric structure of plasma potential profiles was proven for more extended domain of plasma parameters [85,86].

_{b}variation [89,90].

## 8. Future Prospects to Study of Symmetric Structures in Toroidal Plasmas—Conceptual Design of the HIBP Diagnostics for Various Toroidal Devices

- (a)
- Simultaneous measurements of the all three signals for potential, density, magnetic oscillations for comprehensive analysis of the plasma phenomena, including turbulence;
- (b)
- Multichannel measurements for correlation studies, including plasma turbulence poloidal rotation and radial propagation, and turbulent particle flux;
- (c)
- Maximal extension of the detector grid for 2D mapping and study of the poloidal symmetry/symmetry braking;
- (d)
- Creation of the dual HIBP system for study of toroidal/helical symmetry.

#### 8.1. HIBP Design for the TCABR Tokamak

_{b}= 105 keV. The TCABR tokamak has rectangular toroidal field coils. The scheme of the HIBP complex is shown in Figure 39a, and the detector grid is shown in Figure 39b. At the TCABR tokamak, the polarization of the plasma edge (biasing) and heating by Alfvén waves [109] are studied. For both experimental programs, 2D data on the plasma potential and its fluctuations are required.

#### 8.2. HIBP Design for the Globus-M2 Tokamak

_{b}≤ 45 keV. Figure 40 shows the scheme of the HIBP complex. Figure 40a shows both the vertical and the equatorial projections of the calculated trajectories of the probing particles. Figure 40b shows the detector grid aiming for the study of the 2D structures at the outer half of plasma radius in the LFS.

#### 8.3. HIBP Design for the COMPASS Tokamak

_{p}< 360 kA, B

_{t}≤ 2.1, P

_{NBI}(at 40 keV) = 2 × 0.3 MW) with a, was created on the initiative of Prof. J. Stökel. COMPASS has a single-null divertor plasma configuration with elongation k = 1.8, triangularity ∆ = 0.2, and horizontal plasma size of 16 cm. The detector line, calculated in the classical probing scheme for and rectangular toroidal field coils, does not pass through the plasma center. Therefore, we consider another possible combination of probing ports, with the injection of primary particles through the lower inclined port and registration of secondary particles through the upper inclined port. The calculation and design results are shown in Figure 41. We see that the detector line connects the center and the plasma edge. Thus, this probing scheme is preferable. However, for realization of this scheme, the direction of the magnetic field B

_{t}should be reversed.

#### 8.4. HIBP Design for the TCV Tokamak

_{b}= 105 keV. The TCV has rectangular toroidal field coils. The scheme of the HIBP complex is shown in Figure 43a, and the detector grid is shown in Figure 43b.

#### 8.5. HIBP Design for the MAST Tokamak

_{b}= 105 keV. The MAST tokamak has parameters R = 0.85 m, a = 0.65 m, B

_{t}= 0.6 T, I

_{pl}= 1 MA, an extremely small aspect ratio and rectangular toroidal field coils. The vertical section of the tokamak vacuum chamber is also rectangular. The scheme of HIBP is shown in Figure 44a [115], and one of the probing options and the detector grid is shown in Figure 44b. Note that several trajectories (blue) have intersections that complicate interpretation of HIBP measurements, while other trajectories (red) have not intersections. The similar situation was in COMPASS (Figure 41).

#### 8.6. HIBP Design for the T-15MD Tokamak

_{t}≤ 2 T) combined with a small aspect ratio (R/a = 1.48 m/0.67 m). It is planned to use ohmic, ECRH, ICRH and NBI plasma heating and current drive (CD). Particular attention will be paid to the study of the role of the electric field in L-H transitions, Alfvén eigenmodes, and GAMs. These experimental programs require 2D data on the plasma potential, density, magnetic field and its fluctuations.

_{b}= 400 keV. T-15 MD magnetic system was taken into account in the detailed trajectory calculations [117]. The scheme of the HIBP complex is shown in Figure 45a [118], and the detector grid is shown in Figure 45b.

#### 8.7. HIBP Design for the W7-X Stellarator

_{t}= 3 T, R = 5.5 m, a = 0.53 m. W7-X was commissioned at the end of 2015 in the Max Planck Institute for Plasma Physics in Greifswald, Germany. W7-X intended to study the plasma confinement in the optimized 3D magnetic configuration. It uses microwave and NBI for plasma heating. Particular attention is focused on study the role of the electric field at L-H transitions and Alfvén eigenmodes. These experimental programs require data on the plasma potential and its fluctuations. A preliminary calculation of trajectories for the proposed W7-X plasma probing scheme is shown in Figure 46 [125,126].

#### 8.8. HIBP Design for the International Experimental Tokamak Reactor ITER

_{b}= 3.5 MeV [127]. This is a standard energy range for large-scale machines in operation [19]. ITER has ovoid-shaped toroidal field coils. The scheme of the HIBP complex is shown in Figure 47, and the detector grid is shown in Figure 48a. In the standard ITER mode, the nominal plasma current will lead to a noticeable toroidal shift of the trajectories of the probing particles, shown in Figure 48b. We see that with the initial toroidal shift z

_{I}= 0 at the injection point I and the value of the toroidal angle β = 7°, the toroidal coordinate of the detection point D, z

_{D}= 10 cm, and easily fits into the dimensions of the horizontal port. Because in ITER the discharge will transit to the H-mode that is always accompanied by change in the electric field [52,128,129], the measurements of the electric potential and turbulence will be an important issue for the physics of ITER. Recently, it was first shown by HIBP in the T-10 tokamak that in the discharges with low collisionality—which mimic the ITER plasma condition—the core plasma potential has positive values and an enhanced level of turbulence, so the prediction for the ITER plasma was made [130]. Verification of this prediction could be a very important and challenging task for physics studies in ITER.

## 9. Summary

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) HIBP experimental set-up at T-10 [15]: (1) accelerator, (2) primary beamline, (3) secondary beamline, (4) analyzer, (5) trajectory of primary Tl

^{+}ions (red curve), (6) trajectory of secondary Tl

^{++}ions (yellow curve), (7) ionization point or sample volume (SV). (

**b**) Photo of the HIBP diagnostic hardware. Reproduced courtesy of IAEA. Copyright 2017 IAEA.

**Figure 2.**Detector grid for T-10 with the field B

_{t}= 1.55 T: lines of equal beam energy (E

_{b}) are marked in green, lines of equal injection angle α, labeled with scanning voltage U

_{scan}are blue. The asterisks denote the nodes of the detector grid available for observation with all slits of multi-slit detector.

**Figure 4.**Plasma density profiles in TM-4: (

**a**) Ohmic heated (OH) plasma, ${\overline{n}}_{e}$ = 1.5 × 10

^{19}m

^{−3}; dash line and × is a solution of Equation (3), without attenuation; solid line and ○ is solution of Equation (4) with a quasi-solution method. (

**b**) OH (${\overline{n}}_{e}$ = 1.3 × 10

^{19}m

^{−3}) and ECR heated plasmas; ○ is HIBP, + is interferometer; □ are probes and Δ is Thomson scattering data.

**Figure 5.**(

**a**) Toroidal shift of trajectories as function of the path traversed by the probing particle in TUMAN-3M (B

_{t}= 0.5 T; I

_{pl}= 100 kA; R = 54 cm; a = 24 cm), with parabolic profile of current; s

_{α1}and s

_{α}

_{2}are distances from the injector to the plasma boundary; s

_{d}is the distance to the detector point; curves with different colors shows trajectories, born in different sample volumes, marked with black rectangles. (

**b**) Shift of beam particles in the detector as a function of energy at a total plasma current I

_{pl}= 100 kA with different profiles of current density, marked with numbers. The width of the beam track for each current profile corresponds to error in definition of the beam position relatively the meridional plane of 1 cm.

**Figure 6.**(

**a**) PSD of the potential and plasma density fluctuations measured with a heavy ion beam, and the PSD of magnetic oscillations measured with a magnetic probe in the typical T-10 ohmic discharge [15]. (

**b**) Spectrogram of plasma potential oscillations measured in the T-10 discharge with additional ECR heating from 610 to 800 ms. Reproduced courtesy of IAEA. Copyright 2017 IAEA.

**Figure 7.**Conceptual scheme of a zonal flow, a radially localized, poloidally and azimuthally symmetric (m = 0, n = 0) electrostatic potential structure, along with its resulting radial electric field E

_{r}, and turbulence stabilizing poloidal E × B flow.

**Figure 8.**Spectrogram of the coherency between the total HIBP current (plasma density) oscillations, measured in the plasma core at r/a = 0.16 and a MP signal [45]. Reproduced courtesy of IAEA. Copyright 2010 IAEA.

**Figure 9.**Spectrogram of the cross-phase between signals I

_{t}(~n

_{e}) and poloidal magnetic field B

_{p}(~ζ

_{d}) measured in the same SV in discharge with AEs. Insets show the histograms of the cross-phase computed over the areas marked by rectangles; the insets show the histograms for the cross-phases [45]. Reproduced courtesy of IAEA. Copyright 2010 IAEA.

**Figure 10.**Auto-bicoherency (

**a**) and biphase (

**b**) for the plasma potential in the ohmic discharge of the T-10 tokamak [46]. Reproduced courtesy of IAEA. Copyright 2017 IAEA.

**Figure 11.**Multi-slit energy analyzer for direct measurements of particle flux: 5-S—five entrance slits; D—detector; G—grid; GP—ground plate; HVP—high-voltage plate; W—adjustment window.

**Figure 12.**Detector line for the two-slit energy analyzer [15]. The upper slit is marked in blue, the lower one—in red. Reproduced courtesy of IAEA. Copyright 2017 IAEA.

**Figure 13.**Scheme of the poloidal electric field measurements in the plasma using the double-slit analyzer. The potential φ

_{1}and φ

_{2}are measured by the first and second slit correspondingly; ∆x is distance between the points of measurements; E

_{p}is poloidal electric field; B

_{t}is toroidal magnetic field, V

_{r}is the velocity of radial drift in crossed E

_{p}× B

_{t}fields.

**Figure 14.**The time evolution of the NBI-sustained discharge with AEs in TJ-II. (

**a**) Frequency resolved particle flux

**Γ**

_{E×B}(in a. u.) measured by HIBP at ρ = −0.5, k

_{θ}< 2 cm

^{−1}; red color means positive outward flux, and blue means negative inward flux. (

**b**) Cross-phase θ

_{Ep-ne}between E

_{p}and n

_{e}oscillations. Only the points with high Coh

_{ne- Ep}> 0.3 are shown. The color bar is in radians. Three chosen branches of the AE are marked by color ovals. (

**c**) The histograms of the cross-phase for each branch is marked with corresponding colors, indicating the flux direction; left box (red): θ

_{Ep-ne}= −3/4π, corresponds to the outward flux; central box (green): θ

_{Ep-ne}= −π/2, corresponds to zero flux; right box (blue): θ

_{Ep-ne}~0 corresponds to the inward flux. (

**d**) PSD of the turbulent particle flux, taken at some typical time instant, averaged over 1 ms. Three frequency peaks related to the AE branches identified above are marked with corresponding colors. Adapted from [51] Copyright 2012 IAEA.

**Figure 15.**Scheme for measuring the electric field in plasma using the 5-slit analyzer. The potential φ

_{k}is measured in the point S

_{k}by slit with number k = 1–5; E

_{p}is the poloidal component of the electric field required to calculate V

_{r}; the radial E

_{p}× B

_{t}drift velocity in the crossed fields, and E

_{r}is its radial component.

**Figure 16.**Time evolution of the T-10 discharge with additional ECR heating. (

**a**) Flux spectral function Γ

_{E×B}(rel. un.) measured at ρ = 0.78, k

_{θ}< 2; the red color corresponds to the outward flux, positive, blue—to the inward flux, negative. The time-frequency regions of the quasi-coherent mode (QC) and geodesic acoustic mode (GAM) are marked; (

**b**) the total particle fluxes averaged over GAM (${\Gamma}_{E\times B}^{GAM}$) and QC (${\Gamma}_{E\times B}^{QC}$ ) frequency ranges, and the line-average density ${\overline{n}}_{e}$ [54]. Reproduced with permission IoP.

**Figure 17.**Time evolution of Ohmic T-10 discharge with slow density ramp-up. (

**a**) Power spectral density (PSD) of potential oscillations measured by the central slit φ

_{3}, SV is located at r/a = 0.57; (

**b**) PSD for difference of potentials between the central and edge slits φ

_{1}− φ

_{3}= E

_{p}Δx; (

**c**) the frequency resolved spectral function of flux Γ

_{E×B}. The frequency domain of GAM is marked by rectangle [54]. Reproduced with permission IoP. Copyright IoP.

**Figure 18.**Two-point technique of correlation measurements of the plasma electric potential using the 5-slit analyzer.

**Figure 19.**Correlational measurements of electric potential with 5-slits analyzer. Traces of plasma current I

_{p}and line-averaged density ${\overline{n}}_{e}$ in discharge with B

_{t}= 2.3 T (

**a**); power spectral densities of potential oscillations measured in two most distant SVs by upper slit φ

_{1}(

**b**); and by lower slit φ

_{5}(

**c**); their coherency (

**d**); and cross-phase (

**e**) at r = 21 cm. GAM dominates in all PSDs. Adapted from [58] Copyright IAEA 2015.

**Figure 20.**The phase shift between the potential oscillations and the poloidal mode number m measured by intermediate slits of the analyzer in Ohmic discharge with B

_{t}= 2.3 T, n

_{e}= (0.9–2.4) × 10

^{19}m

^{−3}, I

_{p}= 220 kA.

**Figure 22.**Scheme of the TM-4 HIBP complex: 1—source power unit; 2—high-voltage source for accelerator; 3—ion source; 4—accelerating tube; 5—high-voltage divider; 6—ion beamline; 7—vacuum valves; 8—primary beam pick-up; 9—plasma; 10—tokamak chamber; 11—grid analyzer for secondary ions; 12—secondary-electron multiplier. Adapted from [13] Copyright IAEA 1985.

**Figure 24.**Scheme of the heavy ion beam probe on the TJ-I tokamak. (

**a**) Machine and diagnostics; (

**b**) detector grid with lines of equal injection angle from −12° to +12° (dashed), and the equal beam energy E

_{b}from 6 to 60 keV (solid lines).

**Figure 25.**The profiles of the plasma potential (

**a**) and density (

**b**) for three spatial scans in TJ-I.

**Figure 26.**Scheme of the HIBP on the TUMAN-3M tokamak, the major axis of the torus is on the left. Thin curves mark the trajectories of probing particles, I—injector, D—detector. Asterisks indicate points of the detector line available for observation. The calculation was carried out for a beam of Cs

^{+}ions with the beam energies E

_{b}= 4–13 keV, the point with E

_{b}= 10 keV is located in the plasma center.

**Figure 27.**Numerical study of the sensitivity of the toroidal shift of the trajectories to the plasma current and the initial toroidal angle β of the TUMAN-3M. Projections of the trajectories onto the equatorial plane of the torus (x, z) for different values of the plasma current I and the initial toroidal angle β (a–d); X

_{I}and X

_{D}injector and detector coordinates; asterisks mark the points of the detector line available for observation for energies E

_{b}= 4, 5, 6, 8, and 10 keV.

**Figure 29.**Detector grid for Na

^{+}ions in the WEGA stellarator. The beam energy E

_{b}and the injection angle (expressed in scanning voltage U

_{scan}) are indicated.

**Figure 30.**Scheme of the HIBP on the URAGAN-2M stellarator. The optimal detector line is marked by blue asterisks, the green solid line is the trajectory; I is injector, D is detector, and P is plasma; (

**a**) vertical plane, (

**b**) equatorial plane.

**Figure 32.**Scheme of the HIBP in TJ-II. Inset shows plasma cross-section and detector line for E

_{b}= 127 keV, scanning voltage limits are indicated: −6 kV at HFS (ρ = −1) and +4 kV at LFS (ρ = +1).

**Figure 33.**Plasma potential profile evolution in TJ-II. The first proof of the LFS–HFS symmetry [85]. Reproduced courtesy of IAEA. Copyright 2011 IAEA.

**Figure 34.**3D structure of HAE mode m/n = 4/7; (

**a**) top view; (

**b**) poloidal cut [51]. Reproduced courtesy of IAEA. Copyright 2012 IAEA.

**Figure 35.**The spatial distribution of the plasma potential (

**a**) and its oscillations (

**b**) over the vertical cross-section in TJ-II. (

**a**) Potential contours cover about one half of the plasma poloidal cross-section: the maximum of 2D potential distribution (~1000 V) is located at the magnetic axis, equipotentials are consistent with vacuum magnetic flux surfaces, and 2D potential distribution is symmetric (LFS–HFS and up–down). (

**b**) The maxima of 2D RMS distribution are located at the magnetic axis (~30 V) and at the edge (~35–40 V), equipotentials are basically consistent with vacuum magnetic flux surfaces, 2D potential distribution is not fully symmetric: at the mid-radius ~15 V at LFS vs. ~20 V at HFS, i.e., 1.3 times higher. [91].

**Figure 36.**Dual HIBP on TJ-II—the tool to study plasma symmetric structures. (

**a**) Top view of the TJ-II diagnostics layout. (

**b**) Side view of HIBP-2.

**Figure 38.**Long-range toroidal correlations in plasma potential oscillations in TJ-II [97]. Plasma potential long-range-correlations radial distribution in the ECRH discharge #39894, P

_{ECRH}= 2 × 240 kW. (

**a**) Potential coherency spectrogram coh(φ

_{1}(ρ

_{SV1}), φ

_{2}(ρ

_{SV2})) vs. radius ρ

_{SV1}by HIBP-1; ρ

_{SV1}scans from −1 at HFS to +1 at LFS, while HIBP-2 measures at fixed position ρ

_{SV2}= −0.63; (

**b**) spectrogram of the cross-phase between φ

_{1}(ρ

_{SV1}) and φ

_{2}(ρ

_{SV2}), calculated for the coherency, exceeding noise level. (

**c**) Radial distribution of potential coherency averaged by 0 < f < 20 kHz. Reproduced courtesy of IoP. Copyright 2018 IoP.

**Figure 39.**(

**a**) Scheme of the HIBP at TCABR tokamak; (

**b**) Detector grid of the TCABR tokamak for Tl ions with beam energy E

_{b}= 55–106 keV and the injection angle 65–76°. The grid calculated with a step of 5 keV and 1°.

**Figure 40.**(

**a**) Scheme of the HIBP at Globus-M2 tokamak. HIBP primary (black) and secondary (red) trajectories for a detector line of the probing scheme using 78° input port, E

_{b}= 40 keV, with the secondary beamline. Primary beamline angles are α = 45°, β = −5°. Secondary beamline angles are α = 0°, β = 20°. Top: side view, bottom: bottom view. Green star denotes the detection point, blue diamond denotes plasma center. (

**b**) Detector grid for the probing scheme shown in (

**a**).

**Figure 42.**Detector grid for the Tl ions at B

_{t}= 2.1 T and beam energies E

_{b}= 130–250 keV. The grid is calculated with a step of 10 keV. The line E

_{b}= 210 keV allows us to connect the center and edge of the plasma, 0 < r/a < 1.

**Figure 43.**(

**a**) Schematic of the HIBP complex on the TCV tokamak: B

_{t}= 1.07 T, I

_{pl}= 135 kA; (

**b**) detector grid for Tl ions, E

_{b}= 120–250 keV and the injection angle is 65–80°. The grid is drawn with a step of 10 keV and 1°.

**Figure 44.**(

**a**) Schematic of the HIBP complex on the MAST spherical tokamak: B

_{t}= 0.6 T, I

_{pl}= 1 MA; I—injector; A, B and C in inset are various detector positions. Inset demonstrates intersection of secondary trajectories; (

**b**) detector grid for Cs ions. Areas without intersection the secondary trajectories are marked in red, with the crossing—in blue. Lines of equal angle are solid, lines of equal energy are dashed.

**Figure 45.**(

**a**) Schematic of the HIBP trajectories on the T-15 MD tokamak: B

_{t}= 2 T, I

_{pl}= 2 MA; (

**b**) part of the poloidal section of T-15 MD chamber and the detector grid. The calculations were performed for Tl ion beam in reduced magnetic field B

_{t}= 1 T. Lines of equal energy E

_{b}= 80–220 keV are green.

**Figure 46.**Scheme of HIP for the W7-X stellarator. (1)—primary trajectories injected through the port A11 at the bottom of the figure; (2)—fan of secondary trajectories directed to the analyzer through the port N11; (3)—total fan of secondary trajectories.

**Figure 47.**Schematic of the HIBP complex on the fusion reactor ITER: B

_{t}= 6 T, I

_{pl}= 15 MA. The inset shows the detector grid and fragments of ion beam lines. The asterisks mark the detector line, which allows one to study the radial profiles of the plasma potential; I—injection point, D—detection point, dimensions in cm.

**Figure 48.**(

**a**) The detector grid for ITER. Calculated for E

_{b}= 2.6–3.4 MeV with steps of 0.1 MeV and angle 0.5°. The line for α = 13° allows one to probe a fragment of the potential profile at the plasma periphery, 0.76 < r/a < 0.98, by changing the energy from shot to shot. The line E

_{b}= 3.3 MeV allows one to obtain a fragment of the potential profile 0.76 < r/a < 1 by changing the injection angle β. (

**b**) Toroidal shift in the equatorial projection of the detector line with α = 14°.

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Melnikov, A.
Evolution of Heavy Ion Beam Probing from the Origins to Study of Symmetric Structures in Fusion Plasmas. *Symmetry* **2021**, *13*, 1367.
https://doi.org/10.3390/sym13081367

**AMA Style**

Melnikov A.
Evolution of Heavy Ion Beam Probing from the Origins to Study of Symmetric Structures in Fusion Plasmas. *Symmetry*. 2021; 13(8):1367.
https://doi.org/10.3390/sym13081367

**Chicago/Turabian Style**

Melnikov, Alexander.
2021. "Evolution of Heavy Ion Beam Probing from the Origins to Study of Symmetric Structures in Fusion Plasmas" *Symmetry* 13, no. 8: 1367.
https://doi.org/10.3390/sym13081367