Radiative Pusher

LL equation vs sokolov correction

In the sokolov paper, he had reformulated the Landau-Lifshitz equation in a more elegant mannar to meet the mass on shell requirement.

$$\frac{dp^\alpha}{d\tau} = \frac{e}{mc^2}F^{\alpha \beta}\frac{dx_\beta}{d\tau} - \frac{I_{cl}p^\alpha}{mc^2}$$

$$\frac{dx^\alpha}{d\tau} = cp^\alpha + \frac{\tau_0 f^{\alpha}_{Le}}{m}$$

$$\tau_0=2e^2/3mc^3$$

Convert into 3-vector form and using the plasma normalization, these equation becomes:

$$dp_{e,p}/dt = f_{Le,p} \mp [\bar u_{e,p} \times B] - u_{e, p} \varepsilon_{e, p}^2 (f_{Le,p} \cdot \bar{u}_{e,p})$$

$$dx_{e,p}/dt = u_{e,p} + \bar{u}_{e,p},$$

$$\varepsilon_{e,p} = \sqrt{1 + p_{e,p}^2},$$

$$u_{e, p} = p_{e, p} / \varepsilon_{e,p}$$

where

$$f_{Le,p} = \mp(E+u_{e,p}\times B), \varepsilon=\tau_0 \omega$$

$$\bar{u}_{e,p} = \varepsilon \frac{f_{Le,p}-u_{e,p}(u_{e,p} \cdot f_{Le,p})}{1+\varepsilon (u_{e,p} \cdot f_{Le,p})}$$

with $\varepsilon = \tau_0 \omega$.

$$\varepsilon = \tau_0 \omega = \frac{2}{3} \alpha^3\omega_{au}$$

and

$$\omega_{au} = \frac{\hbar\omega}{2Ry} = \frac{\hbar\omega}{\alpha^2 m_e c^2}$$

thus

$$\varepsilon = \frac{2}{3} \alpha \frac{\hbar \omega}{m_e c^2} = \frac{2}{3} \alpha \xi$$

This is the only point we need laser photon energy.

$$\mathbf{F}_{\mathrm{RR,classical}} =\frac{2}{3}\alpha_f \xi_L \bigg \lbrace \gamma \bigg[ \bigg (\frac{\partial}{\partial t}+\frac{\mathbf{p}}{\gamma}\cdot\nabla \bigg )\mathbf{E}+\frac{\mathbf{p}}{\gamma} \times \bigg (\frac{\partial}{\partial t}+\frac{\mathbf{p}}{\gamma}\cdot\nabla \bigg )\mathbf{B} \bigg]$$

$$+ \bigg[\mathbf{E}\times \mathbf{B}+\frac{1}{\gamma}\mathbf{B}\times (\mathbf{B}\times \mathbf{p})+\mathbf{E}(\mathbf{p \cdot E}) \bigg] -\gamma\mathbf{p} \bigg [ \bigg(\mathbf{E}+\frac{\mathbf{p}}{\gamma}\times \mathbf{B} \bigg)^2-\frac{1}{\gamma^2}(\mathbf{E \cdot p})^2 \bigg ] \bigg\rbrace,$$

sokolov quantum correction

when we use the quantum correction, the $\tau_0$ becomes

$$\tau_0 \rightarrow \tau_0 \frac{I_{QED}}{I_{cl}} = \tau_0 q(\chi)$$

and

$$\frac{I_{QED}}{I_{cl}} =q(\chi)=\frac{9\sqrt{3}}{8\pi} \int_0^\infty dr_0 r_0 \bigg[ \int_{r_\chi}^\infty K_{5/3}(x)dx+r_0r_{\chi}\chi^2 K_{2/3}(r_\chi)\bigg]$$

Luckily a very good approximate function has been found in Bulanov's paper:

$$q(\eta) = (1.0 + 4.8(1.0 + \eta) \log(1.0 + 1.7 \eta) + 2.44\eta^2)^{-2/3}$$

Here the definition of $\eta$ is slightly different with that in sokolov with a $2/3$ factor:

$\chi = 3/2 \eta$.

qeta = @(eta) (1.0 + 4.8 * (1.0 + eta) .* log(1.0 + 1.7 * eta) + 2.44 * eta .* eta).^-0.666;
qchi = @(chi) (1.0 + 1.04 * chi).^-1.333;
chi = 10.^linspace(-2, 2, 50);
eta = chi / 1.5;
plot(chi.^2, chi.^2 .* qchi(chi), 'linewidth', 3);
hold on
plot(eta.^2, eta.^2 .* qeta(eta), 'linewidth', 3);

xlim([1e-4, 1e4])
ylim([1e-4, 1e2])


set(gca, 'xscale', 'log', 'yscale', 'log')

set(gca, 'fontsize', 18, 'box', 'on', 'linewidth', 1.5)

legend('qchi', 'qeta')

How do we calculate electron spectra

1. we assume only high energy electrons contribute to the spectra, low energy electron only radiated through the near field manner.
2. we assume these high energy electrons radiation in the direction of their momentum.
3. we use the angle integrated spectra formula to generate a vector of spectra, which only depend on the radiation frequency.

The modified radiation spectra is :

$$\frac{d\mathcal{E}^m}{d\omega'} = \int_0^t \bigg[\sum_e \frac{I_{cl}}{\omega_c} \delta^2 (\vec n - \frac{\vec p}{p}) \delta (\log(\bar \omega) - \log \omega_c) \bigg] dt$$

and with the conversion:

$$\frac{d\mathcal{E}_{\text{rad}}(\omega', \vec{n})}{d \vec{n} d\omega'} = \int Q_{cl}\bigg(\frac{\omega'}{\bar \omega}\bigg) \frac{d\mathcal{E}^m}{d\omega'} d\log(\bar \omega)$$

with $\omega_c = \mathcal{E}\chi$

1. in each step we only calculate the modified spectra, and store it, until dump

in this step, the summation core is $I_{cl} / \omega_c$. And

$$I_{cl}/\omega_c = (2/3)^3 \frac{\chi \alpha_f}{\gamma}$$

thus, for each particle at each step we only find the $\omega_c$, $\phi$, $\theta$, and count on its weight times this quantity.

In our program, we only count $\chi Q \Delta t / \gamma$, here $Q$ is the particle weight.

2. in the post processing python file, the 2nd formula will be used with our dropped $(2/3)^3 \alpha_f$.

When we use the method from sokolov:

1. $\omega_c = \mathcal{E}_e \chi$. and find the discrete value $\log(\bar \omega)_i$ ~ $\log \omega_c$.
2. according to the direction of $\vec p$ find $\theta_j$, $\phi_k$.
3. add spectra with

$$\mathcal{E}_{i, j, k} \rightarrow \mathcal{E}_{i, j, k} + \frac{\mathcal{E}_e^2 (\vec{f}_{l} \cdot \vec{\bar{u}})\Delta t}{\omega_c \Delta\log(\bar{\omega}) \Delta \vec{n}_{j, k}}$$

where $\Delta \vec n_{j, k} = \sin(\theta_j) \Delta \theta \Delta \phi$.

in the sokolov push we only calculate :

$$\varepsilon = \int\frac{dI}{d\omega} dt d\omega = \frac{I_{cl}}{\omega_c}q(\chi) \\ =(2/3)^3 \chi \alpha q(\chi)/\gamma$$