In the simplest model of the plasma we take into account only the interaction of the electrons with the external field and the shielding effect, so we write their equation of motion as

\[m\ddot{\vec{r}} = \tilde{E}e^{-i \omega t}.\]

In this model the conductivity of the plasma $\sigma$ is zero. Let's add the interaction between charged particles, in particular their scattering on each other. We assume that the plasma consists of electrons with the charge $-e=-1.60 \cdot 10^{-19}~\text{C}$ and the mass $m=9.10 \cdot 10^{-31}~\text{kg}$ and ions with the charge $Ze$ and the mass $M\gg m$. The averaged over the Debye length concentration of ions in the plasma is $n_0$ and the concentration of electrons is $n$. Also, the plasma is "hot" i.e. $\lambda_D^3n \gg 1$.

Let's consider the particle with the charge $Z_1 e$ and the mass $m_1$ which is the projectile, and the particle with the charge $Z_2 e$ and the mass $m_2$ which is the target. The velocity of the first particle is $v_1$ when it's far from the target and the impact parameter is $b$.

Let's consider the dynamics in the center of mass frame: in this case the dynamic equations of the particles are separated. For further discussion we will use the polar coordinate system with the reference point $CM$.

\[u'' + Au + B = 0,\]

where $u''$ is the second derivative of $u$ with respect to the angle $\theta$.

Let $b_{90}$ the impact parameter $b$ for which the defelction angle $\chi$ is $90^\circ$.

What is the difference $\Delta p(b)$ of the momuntum of the first particle projection onto the direction of the original motion? Work with the small angle approximation $b \gg b_{90}$. Express your answer by the original momentum of the particle $p$, $m_{1,2}$, $b$ and $b_{90}$.

Part B. Scattering in matter

Let's look at how scattering changes the dynamics of electrons in plasma. First of all, we have to take into account that in matter we have to average of $\Delta p(v)$ over several scattering events. The formula for the average $\Delta p$ when the projectile has traveled the distance $l$ in the matter consisting of motionless ions with the concentration $n_0$ is

\[\Delta p = \int\limits_0^\infty \Delta p(b) \cdot n_0 2\pi b l \, db.\]

This intergral diverges at $b=0$ and also at $b=\infty$, so we have to cut-off it. Firstly, we use simplified formula for $p(b)$ in approximation $b \gg b_{90}$, so let's use the $b_{90}$ as the bottom limit. Secondly, according the shielding effect we have to use $\lambda_D$ as the top limit. Let $\Lambda$ be $\lambda_D / b_{90}$.

Thermal motion of electrons is much faster than drifting under the external field. Let's consider the small volume $\Delta V$ of the electron gas with the size which approximately equal to $\lambda_D$. Let this volume drift with the velocity $\vec{v}_d$ in the matter consisting of the motionless ions. In this volume there is an electron velocity distribution which we will consider to be Mawellian:

\[ f (\vec{v}) = \left( \frac{m}{2\pi k_\text{B}T} \right)^{3/2} \exp \left( -\frac{m(\vec{v}-\vec{v}_d)^2}{2k_\text{B}T} \right),\]

where $f (\vec{v})$ is the probability density function, i.e. the probability $dP$ that the velocity is in the region $v_x \in (v_x, v_x + dv_x)$, $v_y \in (v_y, v_y + dv_y)$, $(v_z, v_z + dv_z)$ is

\[dP = f(\vec{v}) \, dv_x \, dv_y \, dv_z.\]

The Gauss integral can be useful for further calculations:

\[\int\limits_{-\infty}^{+\infty} e^{-x^2}dx = \sqrt{\pi}.\]

\[P_\text{any} = \int\limits_{-\infty}^{+\infty} dv_x \int\limits_{-\infty}^{+\infty} dv_y \int\limits_{-\infty}^{+\infty} dv_z f(v_x, v_y, v_z) = 1. \]

\[ \langle \vec{F} \rangle \propto \vec{v}_d \int u^3 e^{-u^2} du.\]

Note, that all of these approximations don't change the crux of the phenomena, the only thing that might be affected is a different numerical coefficient before.