Laser-Ion Acceleration

Exercise 8 – Mora, Schreiber, and optimal TNSA pulse duration

Week 8 · discussed in the tutorial session · submit solutions by e-mail as agreed in class

This sheet accompanies Lecture 8: Crow/Mora sheath vs. test particle, Schreiber’s finite-radius model compared with Mora, and the optimal TNSA pulse duration. Hot-electron density \(n_{e,0}\) was already treated in Exercise 7, Part A.

Useful scripts in the course Exercises/ folder: e04_plasmaexpansionCrowMora.m, e05_MoraVsSchreiber.m, e05_SchreiberWithExp.m. Python is equally welcome.

1. Crow/Mora static sheath

  1. Fast electrons with density \(n_{e,0}\) exiting at a plasma–vacuum interface induce a positive surface charge. After a short transient, outgoing and returning fluxes balance: the electron density is in thermal equilibrium with the electrostatic potential, supported by a fixed step-like ion profile \(n_i=n_{i,0}\) for \(z<0\) and \(n_i=0\) for \(z\ge 0\). Formulate Poisson’s equation assuming Boltzmann hot electrons with average energy \(E_e(=k_B T_{e,h})\). Solve for the potential (tip: Crow et al. 1975; Mora 2003) and give \(E_s\) at the interface.

2. Mora expansion vs. test particle

Comparison of Mora front velocity and test-particle velocity.
Figure reference: Mora front velocity vs. a test ion in a frozen Crow potential (normalised time \(\propto\omega_{pi}t\)).
  1. Mora’s maximum ion velocity is \[ v_i \approx 2c_s\ln\!\bigl(\tau+\sqrt{1+\tau^2}\bigr)=2c_s\sinh^{-1}\tau, \] with \(c_s=\sqrt{q_i E_e/m_i}\) and \(\tau=\omega_{pi}t/\sqrt{2e}\). Compare this with the velocity a test ion of charge \(q_i e\) could gain in the fixed potential of Question 1. Produce a velocity–time plot (Mora vs. test particle), e.g. with the provided MATLAB script or your own code.

3. Approximating Schreiber and comparing to Mora

Schreiber vs Mora models with experimental data.
Figure reference: Schreiber (exact / approximate) vs. Mora for different \(R/\lambda\), with experimental data as in Schreiber et al. (2006).
  1. Schreiber’s finite-radius model reads \[ \frac{\tau_L}{\tau_0} = X\left(1+\frac12\frac{1}{1-X^2}\right)+\frac12\,\mathrm{atanh}\,X, \] where \(X=v_i/v_{i,\infty}\), \(\tau_0=R/v_{i,\infty}\), and \(v_{i,\infty}=\sqrt{2ZE_\infty/m_i}\) with \(E_\infty=(mc^2/2)\sqrt{2\eta P_L/P_R}\).
    1. Taylor-expand in \(X\) and obtain a simple approximate formula for \(v_i(\tau_L)\).
    2. Plot \(v_i/c_s\) vs. \(\tau\) for Mora and \(v_i/v_{i,\infty}\) vs. \(\tau_L/\tau_0\) for exact and approximate Schreiber. Relate the time scales via the hot-electron density (choose representative \(R/\lambda\), e.g. 100, 10, 1).

4. Optimal pulse duration in TNSA

  1. In the charged-disc / Schreiber picture the terminal ion energy scales with absorbed power, but acceleration lasts only for \(\sim\tau_L\). Using \[ \frac{\tau_L}{\tau_0} = X\left(1+\frac12\frac{1}{1-X^2}\right)+\frac12\,\tanh^{-1}X \] and \(v_{i,\infty}\propto\tau_L^{-1/4}\) at fixed energy \(E_L\), find the pulse duration \(\tau_{L,\mathrm{opt}}\) that maximises the ion energy. Evaluate for \(R=2\,\mu\mathrm{m}\) and absorbed energies \(\eta E_L=1\,\mathrm{J}\) and \(10\,\mathrm{J}\) (protons). Comment briefly on ATLAS-scale pulses (\(\sim 30\,\mathrm{fs}\), tens of joules).