Ionization

Field Ionization

Under the influence of a sufficiently strong external electric field atoms become ionized. Particularly the dynamics of interactions between ultra-high intensity laser pulses and matter, e.g., Laser-Plasma Acceleration (LPA) with ionization injection, or Laser-Plasma Interactions with solid density targets (LPI) can depend on field ionization dynamics as well.

WarpX models field ionization based on a description of the Ammosov-Delone-Krainov model:cite:p:mpion-Ammosov1986 following Chen et al. [1].

Implementation Details and Assumptions

Note

The current implementation makes the following assumptions

  • Energy for ionization processes is not removed from the electromagnetic fields

  • Only one single-level ionization process can occur per macroparticle and time step

  • Ionization happens at the beginning of the PIC loop before the field solve

  • Angular momentum quantum number \(l = 0\) and magnetic quantum number \(m = 0\)

The model implements the following equations (assumptions to \(l\) and \(m\) have already been applied).

The electric field amplitude is calculated in the particle’s frame of reference.

\[\begin{split}\begin{aligned} \vec{E}_\mathrm{dc} &= \sqrt{ - \frac{1}{\mathrm{c}^2} \left( \vec{u} \cdot \vec{E} \right)^2 + \left( \gamma \vec{E} + \vec{u} \times \vec{B} \right)^2 } \\ \gamma &= \sqrt{1 + \frac{\vec{u}^2}{\mathrm{c}^2}} \end{aligned}\end{split}\]

Here, \(\vec{u} = (u_x, u_y, u_z)\) is the momentum normalized to the particle mass, \(u_i = (\beta \gamma)_i \mathrm{c}\). \(E_\mathrm{dc} = |\vec{E}_\mathrm{dc}|\) is the DC-field in the frame of the particle.

\[\begin{split}\begin{aligned} P &= 1 - \mathrm{e}^{-W\mathrm{d}\tau/\gamma} \\ W &= \omega_\mathrm{a} \mathcal{C}^2_{n^* l^*} \frac{U_\mathrm{ion}}{2 U_H} \left[ 2 \frac{E_\mathrm{a}}{E_\mathrm{dc}} \left( \frac{U_\mathrm{ion}}{U_\mathrm{H}} \right)^{3/2} \right]^{2n^*-1} \times \exp\left[ - \frac{2}{3} \frac{E_\mathrm{a}}{E_\mathrm{dc}} \left( \frac{U_\mathrm{ion}}{U_\mathrm{H}} \right)^{3/2} \right] \\ \mathcal{C}^2_{n^* l^*} &= \frac{2^{2n^*}}{n^* \Gamma(n^* + l^* + 1) \Gamma(n^* - l^*)} \end{aligned}\end{split}\]

where \(\mathrm{d}\tau\) is the simulation timestep, which is divided by the particle \(\gamma\) to account for time dilation. The quantities are: \(\omega_\mathrm{a}\), the atomic unit frequency, \(U_\mathrm{ion}\), the ionization potential, \(U_\mathrm{H}\), Hydrogen ground state ionization potential, \(E_\mathrm{a}\), the atomic unit electric field, \(n^* = Z \sqrt{U_\mathrm{H}/U_\mathrm{ion}}\), the effective principal quantum number (Attention! \(Z\) is the ionization state after ionization.) , \(l^* = n_0^* - 1\), the effective orbital quantum number.

Empirical Extension to Over-the-Barrier Regime for Hydrogen

For hydrogen, WarpX offers the modified empirical ADK extension to the Over-the-Barrier (OTB) published in Zhang et al. [2] Eq. (8) (note there is a typo in the paper and there should not be a minus sign in Eq. 8).

\[W_\mathrm{M} = \exp\left[ a_1 \frac{E^2}{E_\mathrm{b}} + a_2 \frac{E}{E_\mathrm{b}} + a_3 \right] W_\mathrm{ADK}\]

The parameters \(a_1\) through \(a_3\) are independent of \(E\) and can be found in the same reference. \(E_\mathrm{b}\) is the classical Barrier Suppresion Ionization (BSI) field strength \(E_\mathrm{b} = U_\mathrm{ion}^2 / (4 Z)\) given here in atomic units (AU). For a detailed description of conversion between unit systems consider the book by Mulser and Bauer [3].

Testing

[1]

M. Chen, E. Esarey, C. G. R. Geddes, C. B. Schroeder, G. R. Plateau, S. S. Bulanov, S. Rykovanov, and W. P. Leemans. Modeling classical and quantum radiation from laser-plasma accelerators. PHYSICAL REVIEW SPECIAL TOPICS-ACCELERATORS AND BEAMS, Mar 2013. doi:10.1103/PhysRevSTAB.16.030701.

[2]

Q. Zhang, P. Lan, and P. Lu. Empirical formula for over-barrier strong-field ionization. Physical Review A, 90(4):043410, October 2014. doi:10.1103/PhysRevA.90.043410.

[3]

P. Mulser and D. Bauer. High Power Laser-Matter Interaction. Volume 238. Springer Berlin Heidelberg, 2010. ISBN 978-3-540-50669-0. Series Title: Springer Tracts in Modern Physics. doi:10.1007/978-3-540-46065-7.