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. $\lambda_D^3n \gg 1$.

Part A. Rutherford scattering

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 $b$.

A1 Write the motion equations for both particles and show that this system is equivalent to the interaction of both particles with the effective Columb's potential with a center similar to the center of mass $CM$.

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 $CM$.

A2 How does the angular momentum of the first particle behave?

A3 Write the second Newton's law in terms of $r$, $\theta$ and their time derivatives $\ddot{r}$ and $\dot{\theta}$.

A4 Introduce the Binet substitution $u = 1/r$ and obtain the differential equation for $u$ as a function of $\theta$ in the form of a harmonic oscillator:
\[u'' + Au + B = 0,\]
where $u''$ is the second derivative of $u$ with respect to the angle $\theta$.

A5 Express the deflection angle $\chi$ in the frame of the center of mass through $b$, $v_1$, $m_{1,2}$ and $Z_{1,2}$.

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

A6 Find the $b_{90}$. Express you answer by $Z_1$, $Z_2$, $m_1$ and $v_1$.

A7 In this question we work in the lab frame where the target was originally motionless.

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 $\lambda_D / b_{90}$.

B1 Find the equation for $\Delta p$ in approximation $M \gg m$. Express the answer by $p$, $b_{90}$, $n_0$, $l$ and $\Lambda$.

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

B2 Show, that the probability $P_\text{any}$ that the electron's velocity is any is unity, i.e.
\[P_\text{any} = \int\limits_{-\infty}^{+\infty} dv_x \int\limits_{-\infty}^{+\infty} dv_y \int\limits_{-\infty}^{+\infty} dv_z f(v_x, v_y, v_z) = 1. \]

B3 Find the mean value of the velocity $\langle v \rangle$. Ignore any corrections associated with the presence of $v_d$.

B4 Estimate the mean force $\langle \vec{F} \rangle$ that acting on the volume $\Delta V$ from the ions. For simplicity, assume that $\Lambda$ is independent of the electron velocity $v$ and use the value of $\Lambda$ for $v=\langle v \rangle$. Also, ignore that $|\vec{v}|$ (not a vector, just it's absolute value) depends on the $\vec{v}_d$. Finally, you understand approximations properly if
\[ \langle \vec{F} \rangle \propto \vec{v}_d \int u^3 e^{-u^2} du.\]

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.

B5 Write the second Newton's law for $\Delta V$. Consider, that the interaction with neighboring volumes of the electron gas is canceled out.

B6 Find the current density $\vec{j}_t$ in the model under the external field $\tilde{E} e^{-i \omega t}$.

B7 With $\sigma = \mathfrak{Re} \left\lbrace \dfrac{j}{\tilde{E}e^{-i\omega t}}\right\rbrace$ find the conductivity $\sigma$ of the plasma.