# Methods for Line Shapes in Plasmas in the Presence of External Electric Fields

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Formulation

**d**the dipole moment and ${\alpha}_{0},\beta $ in principle complete sets of states. The density matrix has been assumed to be trivial. This may be written as (by using $t={t}_{2}-{t}_{1}$):

**e**direction is:

**e**is:

## 3. Time-Periodic Fieds: Floquet Theory

**e**.

#### 3.1. Qualitative Remarks on the B-Matrix

#### 3.2. General Lineshape and Static Solutions

- a
- If ${\Delta}_{nk}$ is independent of ${\mathbf{F}}_{\mathbf{i}}$, we get the usual static results of $\delta $-functions for central components at the Blokhitsev positions or integer multiples of $\Omega $.
- b
- For $\omega =n\Omega $, the $\delta $-function argument is zero for ${\Delta}_{nk}\left({\mathbf{F}}_{\mathbf{i}}\right)=0$, which is typically satisfied for ${F}_{i}\approx 0$, at least for H-like species. However, then the distribution function $W\left(0\right)=0$ ensures a zero result which could show up as a dip. Hence in this view one might expect intensity drops and these should practically coincide with the Blokhintsev peaks at $n\Omega $ to the extent that $|{\Delta}_{k}|\ll \Omega $.

#### 3.3. Numerical Floquet Solution

#### 3.3.1. Direct B Eigendecomposition

- Determination of B. Since ${U}_{0}\left(0\right)=I=P\left(0\right)=P\left(T\right),{U}_{0}\left(T\right)={e}^{-\u0131BT}$, compute once and for all ${U}_{0}\left(T\right)$, i.e., the time evolution for one period in the presence of the periodic field and no plasma. Since U(T) must be unitary, accuracy is typically important and since the problem is typically stiff, a geometric integrator, preserving unitarity, should be used [18].
- Diagonalize ${U}_{0}\left(T\right)=E{e}^{\u0131\Lambda T}{E}^{-1}$ to obtain the eigenvalue decomposition of B once and for all. Since ${U}_{0}\left(T\right)$ is unitary, the best way is to compute the Schur factorization: $U\left(T\right)=XT{X}^{\u2020}$, where X is unitary, and since $U\left(T\right)$ is unitary, the upper triangular matrix T is unitary and hence diagonal.Thus the columns of X are eigenvectors of $U\left(T\right)$ and form an orthonormal basis.
- Compute, also once and for all, the satellite structure: For all distinct eigenvalues ${\lambda}_{i}$ of B, consider all sattelite positions ${\Delta}_{i}+n\Omega $, for all integer n resulting in satellite position in the region of interest, where ${\Delta}_{i}$ refers to all combinations of differences in the upper-lower level Floquet exponents. This also helps optimize the frequency grid.
- Compute interpolation tables for each entry of the $Z\left(\tau \right)$ matrices for $0\le \tau \le T$, so that when ${U}_{0}\left(t\right)$ is required, we interpolate to get $Z\left(tmodT\right)$ and multiply by ${e}^{\u0131\Lambda}{E}^{\u2020}$.
- Next, solve for the time evolution in the plasma microfield $V\left(t\right)$, dressed by ${U}_{0}\left(t\right)$.

#### 3.3.2. Spectral Methods

#### 3.3.3. Analytical Solutions

## 4. Discussion

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Appendix A. Floquet Theory

**Lemma**

**A1.**

**Lemma**

**A2.**

**Lemma**

**A3.**

## Appendix B. B-Matrix for a Periodic Interaction with No Time-Independent Term

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**Figure 1.**${z}_{n}$ coefficients as a function of n, showing the rapid drop with increasing $\left|n\right|$.

**Figure 2.**Combined Blokhinstev ( $n=-1,0,1$) and Floquet (${\Delta}_{k}$) components as labelled in Equation (26). for H-like Si $\u0141-\gamma $ (with fine structure included) under conditions similar to those in Refs. [10,11]. The components form three clusters for $n=-1,0,1$, with each cluster member corresponding to a different Floquet exponent. The intensities are relative percent intensities (they add to 100).

**Figure 3.**Real part of (${C}_{nk}\left(t\right)$) for five of the strongest components. for H-like Si $L-\gamma $ (with fine structure included) under conditions similar to those in Refs. [10,11], namely electron density 3.6 × 10${}^{22}$ e/cc, electron and ion temperatures 500 and 1 eV respectively and a linearly polarized oscillatory field ${E}_{0}cos(\Omega t)$ with ${E}_{0}=0.6$ GV/cm and $\Omega =1.07\times {10}^{16}$ s${}^{-1}$, as well as a static field ${F}_{static}$ of magnitude 2.1 GV/cm in the direction perpendicular to the oscillatory field. Fine structure is included in the calculations. The sum of the autocorrelation functions in the parallel and perpendicular directions is displayed. The calculation used 100 plasma particle configurations and an impact tail for long times was recognizable.

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Alexiou, S.
Methods for Line Shapes in Plasmas in the Presence of External Electric Fields. *Atoms* **2021**, *9*, 30.
https://doi.org/10.3390/atoms9020030

**AMA Style**

Alexiou S.
Methods for Line Shapes in Plasmas in the Presence of External Electric Fields. *Atoms*. 2021; 9(2):30.
https://doi.org/10.3390/atoms9020030

**Chicago/Turabian Style**

Alexiou, Spiros.
2021. "Methods for Line Shapes in Plasmas in the Presence of External Electric Fields" *Atoms* 9, no. 2: 30.
https://doi.org/10.3390/atoms9020030