Ricky’s No Laser Data

Ricky’s Simulations are located in his scratch directory at the following location on OSC:

/fs/scratch/PAS1066/oropeza/Sim_2D_SinglePulse/round_target/5_mJ/no_laser

He has simulations that vary the number of particles per cell for electrons, protons, (singly ionized) carbon, and (singly ionized) oxygen. The numbers he has are seemingly random but they are generated by a Sobol Sequence that tries to pick values that would sample the parameter space well.

The simulation set up is as follows: There is a 2800 x 2800 (28 x 28 micron) 2D grid with open boundary conditions of cell size $\Delta x = 10 $ nm. The target is angled upwards at 45 degrees in the center of the simulation volume with thickness $l_x = 460 $ nm and length $l_y = 25 \mu $ m. At the ends of the target, it is rounded off by appending a semi-circle of diameter $l_x = 460 $ nm. Below, this is drawn to scale (where each cell in this picture is 1 $\mu$ m so that each cell represents 100 x 100 simulation cells).

This Target is Liquid Ethylene Glycol with the chemical formula (CH$_2$OH)$_2$ with a number density of $1.0 \times 10^{28} \frac{1}{m^3}$. His simulations assume each of the particles are singly ionized at the start which would mean that there are the following number of particles per Ethylene Glycol: C+ (2), O+ (2) , H+ (6) e- (10). Here, H+ is simply just a proton. This would mean that the initial number densities of the particles in the target are: $n_C = 2e28, n_O = 2e28, n_p = 6e28, n_e = 1e29 \frac{1}{m^3}$. The simulation is ran from 0 to 500 fs without any laser being turned on with the initial ion temperatures set to 1eV and the initial electron temperature at either 1, 100, or 1000 eV.

Ricky outputs different quantities according to the input decks. The energy (and absorption) data are stored in the diagnostic files designated ‘diag####.sdf’ which are outputted every 1fs. The particle weights are outputted in the ‘p####.sdf’ files which are outputted every 10 fs. The carbons and oxygens are getting ionized throughout the simulation which creates new particles define in EPOCH. In the input deck, these particles are referred to as Oplus and Cplus when singly ionized and Oplusi and Cplusj where $i \in {1, 2, \ldots, 7}$ and $i \in {1, \ldots, 5}$ for higher ionizations of Oxygen and Carbon. The singly ionized state would correspond to $i = j = 0$ but epoch does not append a 0 on the name.

Analysis

Given the density and size of the target we would compute a total number of Ethylene Glycol Molecules as

\begin{equation} N = n V = (10^{28} \frac{1}{m^3})( (460 nm)(25 \mu m) + \pi (230 nm)^2 (1 m)) = 1.17 \times 10^{17} \end{equation}

so we would expect the initial number of electrons, protons, oxygens, and carbons to be $N_C = N_O = 2.33e17$, $N_p = 7.00e17$, $N_e = 1.17e18$. Below, I have a plot of the number of particles by species throughout one of the simulations which confirms these numbers.

We see the electron number go up because as the carbons and oxygens ionize, they release electrons which will increase the total electron number. One good thing to notice is that the ion and electron numbers don’t decrease towards the end of the simulation. This particular simulation was at 1keV and it shows that 500fs isn’t long enough for particles to leave the open boundaries.

For analysis, I don’t use the electron temperature because without a laser, the electron energy will transfer to the ions (if the initial electron temperature is higher than the ions) and we wouldn’t observe any heating. Instead, I take the “average” temperature in eV which is computed as follows

\begin{equation} T_{eV} = \frac{\sum_i E_i}{\sum_i N_i} \times \frac{2}{3} \times \frac{1}{1.6 \times 10^{-19}} \end{equation}

I looked at the amount of heating from the simulation and added the arber slope for comparison. The simulations with 131 and 257 electron PPC actually had the average temperature go down, so I think it would be best to ignore these simulations for the analysis. The high number of particles per cell would make the heating negligible anyway.

(The x-axis is time in fs) On the other hand, the simulations for a lower amount of electron particles per cell looked something like the following

Here, we notice that the temperature increase definitely does depend on the initial temperature.

Results

As before, I ran my simulations to see how well both the alpha and ratio models performed. I am writing the definitions of the alpha and ratio models below

ALPHA MODEL \begin{equation} \frac{dT_{eV}}{dt_{ps}} = \frac{C_1}{n_{ppc}} T_{0,eV}^{1 - \alpha/2} \Delta x_{nm}^\alpha n_{23}^{(\alpha + 1)/2} \end{equation}

where $\alpha \geq 0$, and $C_1$ is a proportionality constant. For the Arber Model, $\alpha = 2$, $C_1 = 2$.

RATIO MODEL \begin{equation} \frac{dT_{eV}}{dt_{ps}} = \frac{C_2}{n_{ppc}} T_0 \sqrt{n_{23}} \frac{1}{(R (\lambda_D / \Delta x) + 1)^2} \end{equation}

where $C_2$ is again a proportionality constant and $R$ is a multiplicative constant to modify the debye ratio.

My Previous Results Optimal Alpha Model was for $C_1 = 1.85$ and $\alpha = 0.5$ with MSE = 7.76 Optimal Ratio Model was for $C_2 = 3$ and $R \sim 9.9$ with MSE = 3.82 which are in comparison to the Arber Model with MSE = 20.2

Ricky’s Simulation Results Optimal Alpha Model was for $C_1 = 0.74$ and $\alpha = 1.0$ with MSE = 0.787 Optimal Ratio Model was for $C_2 = 2.92$ and $R \sim 34.5$ with MSE = 0.73 which are in comparison to the Arber Model with MSE = 3.99

I have fewer data points for Ricky’s Simulations so that is why there is a lower MSE. One thing to notice is the lower value of $C_1$. I think this could be due to the fact that Ricky’s Sim is of a target that is a small fraction of the total simulation volume. I have found in my previous simulations that smaller targets exhibit less numerical heating.

Additionally, Ricky’s sims have a higher value of $R$ which means that the length scale at which the heating starts becoming constant is lower at around $\frac{\lambda_D}{\Delta x} \sim \frac{1}{34}$ instead of $\frac{1}{10}$. This might also be because of the smaller target. This can also be compared to what I found earlier with my sims that the length scale at which the heating starts becoming constant is around $\frac{\lambda_D}{\Delta x} \sim 0.03 \sim \frac{1}{34}$

Things to Do

• Watch Talks from DPP next week
• Review Montgomery and Tidman E-Field Fluctuation Formula