# Line Shapes in a Magnetic Field: Trajectory Modifictions II: Full Collision-Time Statistics

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Formulation

- the distribution functions, e.g., the Maxwellian velocity distribution is $not$ affected by the B-field.
- the shielding is also not affected, e.g., Debye screening

#### 2.1. Collision-Time Statistics

**R**(t) are as follows: The z-coordinate of the trajectory is

- If ${r}_{L}<{R}_{max}$ and $\rho \le {R}_{max}-{r}_{L}$, ${R}_{max}\ge \rho +{r}_{L}$ and Equation (6) is satisfied for $t={t}_{i}$ for $any$ $\theta ,\psi $ and ${\omega}_{L}t$. Therefore in this case we have no restriction and $\theta $ and $\psi $ can independently take any value.
- In all other cases, Equation (6) results in the restriction:$$arccos(\frac{{R}_{max}^{2}-{\rho}^{2}-{r}_{L}^{2}}{2\rho {r}_{L}})\le |{\omega}_{L}t+\psi -\theta |$$

- If ${r}_{L}\ge {R}_{max}$, then$$0\ge \frac{{R}_{max}^{2}-{r}_{L}^{2}-{\rho}^{2}}{2\rho {r}_{L}}\ge -1$$The left of the inequality follows because ${R}_{max}\le {r}_{L}$ alone. The right part also follows since$${r}_{L}-{R}_{max}\le \rho \le {r}_{L}+{R}_{max}\Rightarrow -{R}_{max}\le \rho -{r}_{L}\le {R}_{max}$$Hence$${R}_{max}^{2}\ge {\rho}^{2}+{r}_{L}^{2}-2\rho {r}_{L}\Rightarrow \frac{{R}_{max}^{2}-{r}_{L}^{2}-{\rho}^{2}}{2\rho {r}_{L}}\ge -1$$
- For ${r}_{L}\le {R}_{max}$, Equation (9) is also valid since ${r}_{L}-{R}_{max}\le 0\le \rho $, as is Equation (10). Since, as already discussed the case $\rho \le {R}_{max}-{r}_{L}$ imposes no restriction, we only consider here the case ${R}_{max}-{r}_{L}\le \rho \le {R}_{max}+{r}_{L}$, which implies $\rho -{r}_{L}\le {R}_{max}\le \rho +{r}_{L}$. It thus remains to show that$$\frac{{R}_{max}^{2}-{r}_{L}^{2}-{\rho}^{2}}{2\rho {r}_{L}}\le 1$$

- For ${r}_{L}\le {R}_{max},\rho \le {r}_{L}$, Equation (6) is satisfied for $t={t}_{i}$ for any $\theta ,\psi $ and ${\omega}_{L}t$ and there is no restriction on $\psi ,\theta $ and ${\omega}_{L}t$.
- In all other cases, the restriction imposed by Equation (7) applies and the argument of the inverse cosine is always absolutely $\le 1$.

#### 2.2. Collision Volume

#### 2.3. Generating Perturbers

## 3. Conclusions

## Funding

## Conflicts of Interest

## Appendix A. Calculation of the I_{3} and J_{3} Double Integrals

#### Appendix A.1. The I_{3} Integration

#### The J_{3} Integration

## References

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**Figure 1.**x-y trajectory projections for ${r}_{L}\ge {R}_{max}$. Shown is the annular region between concentric circles with the origin (the emitter position) as center and radii ${r}_{L}-{R}_{max}$ and ${r}_{L}+{R}_{max}$, respectively. For an impact parameter at a distance $\rho $ from the center in the annular region, a circle with radius ${r}_{L}$ (dashed) represents the projection of the perturber path in the x-y plane. Hence a point on that circle is $\left(\rho cos\theta +{r}_{L}cos(\omega t+\psi ),\rho sin\theta +{r}_{L}sin(\omega t+\psi )\right)$, with $\psi $ the angle on the dashed circle. This must be no more than ${R}_{max}$ away from the center, else this perturber does not contribute.

**Figure 4.**Illustration of the relation between $\theta $ and $\psi $ for the case ${r}_{L}\gg {R}_{max}$. Impact parameters $\rho $ lie between the dashed and dash-dotted circles with radii ${r}_{L}-{R}_{max}$ and ${r}_{L}+{R}_{max}$. The part of the circular trajectory projections that are within ${R}_{max}$(bold circle) of the emitter (i.e., the center) are in the opposite direction of the impact parameter vector, e.g., to the north for the southern circular trajectory projection.

${\mathit{\omega}}_{\mathit{L}}\mathit{\tau}$ | $\frac{{\mathit{r}}_{\mathit{L}}}{{\mathit{R}}_{\mathit{max}}}$ | $\frac{\mathit{\rho}}{\mathit{max}({\mathit{\rho}}_{1},{\mathit{r}}_{\mathit{L}})}$ | $\frac{\mathbf{\Delta}\mathit{\psi}(\mathit{\rho})}{2\mathit{\pi}}$ |
---|---|---|---|

$\ge 2\pi $ | 1 | ||

$<2\pi $ | $\le 1$ | $\le 1$ | 1 |

$<2\pi $ | $\le 1$ | $>1$ | $\left(1-\frac{arccos(\frac{{R}_{max}^{2}-{\rho}^{2}-{r}_{L}^{2}}{2\rho {r}_{L}})}{\pi}\right)+\frac{{\omega}_{L}\tau}{2\pi}$ |

$<2\pi $ | $>1$ | $\left(1-\frac{arccos(\frac{{R}_{max}^{2}-{\rho}^{2}-{r}_{L}^{2}}{2\rho {r}_{L}})}{\pi}\right)+\frac{{\omega}_{L}\tau}{2\pi}$ |

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**MDPI and ACS Style**

Alexiou, S. Line Shapes in a Magnetic Field: Trajectory Modifictions II: Full Collision-Time Statistics. *Atoms* **2019**, *7*, 94.
https://doi.org/10.3390/atoms7040094

**AMA Style**

Alexiou S. Line Shapes in a Magnetic Field: Trajectory Modifictions II: Full Collision-Time Statistics. *Atoms*. 2019; 7(4):94.
https://doi.org/10.3390/atoms7040094

**Chicago/Turabian Style**

Alexiou, Spiros. 2019. "Line Shapes in a Magnetic Field: Trajectory Modifictions II: Full Collision-Time Statistics" *Atoms* 7, no. 4: 94.
https://doi.org/10.3390/atoms7040094