NF4IP

I spent much of Monday trying to figure out how to download NF4IP. However, the included instructions on there did not work. I tried to download all the files within a new conda environment. I cloned the repository and ran the setup.py file by doing

python -m pip install -e setup.py

which did not work. Instead I had to first download certain packages (like cement) from the requirements.txt file

python -m pip install -r requirements.txt

and then type

python -m pip install -e .

However, when I try to run the NF4IP generate command, I get an error saying that I do not have ‘horovod’ installed. When I try to install horovod, I get an error with failing to build the wheel and cmake.

Fuchs Model Summary

On Tuesday, I had to proctor for Jared’s students for their 1250 final exam. Here I am going to summarize the Fuchs Model and Tom’s implementation of it.

The Fuchs Function will be denoted $f_F(\lambda, I_0, \tau_{FWHM}, t, w_0)$ which is a function of wavelength, intensity, full-width half max of laser pulse, target thickness, and spot size (beam waist). First, the laser energy is calculated from the following formula

\begin{equation} E_L = \frac{\pi}{2} \frac{I_0 \omega_0^2}{2 \log(2)} \tau_{FWHM} \sqrt{\frac{\pi}{2}} \end{equation}

If the target is not at the point of focus (say a distance $z$ along the beam axis from the point of focus), the actualy intensity on target will be

\begin{equation} I = I_0 \frac{w_0^2}{w(z)^2} \end{equation}

where

\begin{equation} w(z) = w_0 \sqrt{1 + (z/z_R)^2} \end{equation}

is the beam waist at a point a distance $z$ from the point of focus and $z_R$ is the Rayleigh range which is a quantity that is related the curvature of a Gaussian Beam.

From this, the the hot electron temperature of the electrons produced in the pre-plasma is given by the laser ponderomotive potential

\begin{equation} T_p = m_e c^2 (\gamma - 1) \end{equation}

where

\begin{equation} \gamma \approx \sqrt{1 + a_0^2/2} \end{equation}

and $a_0$ is the normalized vector potential

\begin{equation} a_0 = \frac{e E_0}{m_e \omega c} \approx 0.85 (\frac{\lambda}{1 \mu m})( \frac{I}{10^{18} W cm^{-2}})^{1/2} \end{equation}

The electron density on the target front surface can be estimated by assuming the average hot electron has temperature $T_p$ and the energy of the electrons that are absorbed as fast electrons can be represented as some fraction of the total laser energy $f E_L$. Thus, dividing the average energy from the total energy in electrons would give us the number of electrons $N_e$

\begin{equation} N_e = f \frac{E_L}{T_p} \end{equation}

This fraction has been found to be related on the incident laser intensity (with a max of 0.5) as

\begin{equation} f = 1.2 \times 10^{-15} I^{0.74} (W cm^{-2}) \end{equation}

(Tom uses 2.53 instead of 0.74 because that apparently fixes the issue of under-predicting Wright-Patt Max Proton Energies)

To calculate the number density, we would need to divide by the volume of the sheath of electrons. We can calculate the cross-sectional area as the radius of a circle $r_0$ that is the initial zone over which the electron is accelerated at the target front surface and should be related to the spot size (half of the spot size, but Tom uses $\w_0 / \sqrt{2 \Log(2)}$). But also, the radius should be a little larger because the electrons will diverge from the axis by some angle $\theta$ (Fuchs uses 25 deg but Tom uses 28 deg) which will increase the radius by $d \tan(\theta)$ (where d is the thickness of the target). As a result, the cross-section of the sheath will be $S_{SH} = \pi (r_0 + d \tan(\theta))^2$. The thickness of the sheath will be the speed of light times the duration of a laser pulse $c \tau$. Putting this together, we get the number density as

\begin{equation} n_{e,0} = \frac{N_e}{c \tau_L S_{SH}} \end{equation}

Next, we need to define an effective acceleration time $t_{acc} \sim 1.3 \tau_L$ (Tom uses 0.242). The normalized acceleration time is given by

\begin{equation} t_p = \omega_{pi} t_{acc} / \sqrt{2 \exp(1)} \end{equation}

where $\omega_{pi}$ is the plasma ion frequency

\begin{equation} \omega_{pi} = \sqrt{\frac{Z_i e^2 n_{e0}}{m_i \epsilon_0}} \end{equation}

Using this, the Fuchs paper gives the maximum (cutoff) energy that can be gained by the accelerated ions based on a simple fluid model as

\begin{equation} E_{max} = 2 T_p Log(t_p + \sqrt{t_p^2 + 1})^2 \end{equation}

Update on Heating Simulations

Seems like the heating results with the Spline Shape Function and current smoothing give qualitatively similar results (although just much less heating). One question that Chris raised was that if the results would change if I picked a different number of cells but with the same resolution (e.g. 64 x 64 or 16 x 16 but keeping $\Delta x$ constant). Another thing that I could explore is the peculiarity in the 10eV simulations. The 10eV simulations do not follow the trend of increasing cell size means more heating. This seems to very much go against what the Arber formula and intuition would have us believe.