# Structural Characteristics of Ion Holes in Plasma

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## Abstract

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## 1. Introduction

## 2. Theoretical Formalism

## 3. Results and Discussion

#### 3.1. Effects of Potential on IHs

#### 3.1.1. ${T}_{i}<{T}_{e}$

#### 3.1.2. ${T}_{i}>{T}_{e}$

#### 3.2. Effects of Temperature Ratio, ${T}_{r}$

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The width-amplitude relation for different configuration of ion and electron distribution function is shown for temperature ratio, ${T}_{r}=0.5$. The blue-shaded regions correspond to the parameter space that permits the existence of ion holes.

**Figure 2.**The width-amplitude relation for different configuration of ion and electron distribution function is shown for temperature ratio, ${T}_{r}=3$.

**Figure 3.**The dependence of trapped ion density for different $\psi $ and $\delta $ is shown. The left panel shows the case for different $\psi $, for fixed ${T}_{r}=0.5$ and $\delta =10$. The right panel shows the case for different $\delta $ keeping the ${T}_{r}=0.5$ and $\psi =0.01$ constant.

**Figure 4.**Three-dimensional structure of the trapped ion distribution function ${f}_{\mathrm{tr}}(x,v)$ versus x and v, for different values of $\psi $, while keeping ${T}_{r}=0.5$ and $\delta =10$ as fixed. The color bar indicates the density. The two-dimensional projection of ${f}_{\mathrm{tr}}(x,v)$ is displayed underneath the surface plot of the trapped ion distribution function.

**Figure 5.**The same as Figure 4, except that the potential width $\delta $ is varied, while keeping ${T}_{r}=0.5$ and $\psi =0.01$ as constants.

**Figure 6.**The characteristics of trapped ion density for different $\psi $ and $\delta $ are shown. The top panels show the case for different $\psi $ keeping the ${T}_{r}=3$ and $\delta $ constant. In the top left panel, the values of $\psi $ are taken from Region 1, and $\delta $ is assumed to be 10. In the top-right panel, the values of $\psi $ are taken from Region 2, and $\delta $ is assumed to be 60. The bottom panels show the case for different $\delta $ keeping the ${T}_{r}=3$ and $\psi $ constant. In the bottom left panel, $\psi $ is assumed to be 0.01 (from Region 1), whereas in the bottom right panel $\psi $ is assumed to be 2 (from Region 2) for different values of $\delta $.

**Figure 7.**The trapped ion distribution function ${f}_{\mathrm{tr}}(x,v)$ versus x and v, as well as 2D projection, for different values of $\psi $, and for fixed ${T}_{r}=3$ and $\delta =40$.

**Figure 8.**The same as Figure 7, except that the choice of $\psi $ and $\delta $ designate Region 2. For each panel, a small hump at the center of the phase space distribution can be observed.

**Figure 9.**The trapped ion distribution function for different $\delta $ keeping ${T}_{r}=3$ and $\psi =0.01$ as constants. The values for width and amplitude are from Region 1.

**Figure 10.**The trapped ion distribution function for different $\delta $ keeping ${T}_{r}=3$ and $\psi =3$ as constants. The input amplitude represent Region 2. The central hump characterizes the phase space distribution for each case.

**Figure 11.**The characteristics of trapped ion density ${n}_{tr}$ for different ${T}_{r}$ is shown keeping $\psi $ and $\delta $ constant. The left panel shows the case for different ${T}_{r}$ keeping $\psi =0.01$ and $\delta =10$ as fixed, i.e., from Region 1. The right panel show the case for different ${T}_{r}$ keeping $\psi =3$ and $\delta =40$ as constants, i.e., from Region 2.

**Figure 12.**The trapped ion distribution function ${f}_{\mathrm{tr}}(x,v)$ versus x and v, as well as 2D projection, for different values of ${T}_{r}$, keeping $\delta =10$ and $\psi =0.01$ constants. The values for width and amplitude are from Region 1.

**Figure 13.**The same as Figure 12, except that $\delta =40$ and $\psi =3$ are used. The input parameters are from Region 2. The central hump characterizes the phase space distribution for all the cases considered here.

**Figure 14.**The characteristics of trapped ion distribution function for different ${T}_{r}$ keeping $\delta =10$ and $\psi =0.01$ as constants. The emergence of saddle point after ${T}_{r}$ crosses the threshold, ${T}_{r}=1$, denotes the occurrence of forbidden gap in the width-amplitude plot.

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**MDPI and ACS Style**

Aravindakshan, H.; Kakad, A.; Kakad, B.; Yoon, P.H.
Structural Characteristics of Ion Holes in Plasma. *Plasma* **2021**, *4*, 435-449.
https://doi.org/10.3390/plasma4030032

**AMA Style**

Aravindakshan H, Kakad A, Kakad B, Yoon PH.
Structural Characteristics of Ion Holes in Plasma. *Plasma*. 2021; 4(3):435-449.
https://doi.org/10.3390/plasma4030032

**Chicago/Turabian Style**

Aravindakshan, Harikrishnan, Amar Kakad, Bharati Kakad, and Peter H. Yoon.
2021. "Structural Characteristics of Ion Holes in Plasma" *Plasma* 4, no. 3: 435-449.
https://doi.org/10.3390/plasma4030032