Source 1
Elastic scattering due to electron impact with atoms or molecules (as shown in fig. 1.1) is a process where the energy of an incident electron is not changed, however the momentum may change.
Figure 1.1 Elastic scattering with varying impact parameters. The incident and outgoing energies $E_{inc1}$ and $E_{inc2}$ of electrons $e_1$ and $e_2$, respectively, are unchanged. The momenta $\bm{k_{inc1}}$ and $\bm{k_{inc2}}$ of the incident electrons change upon impact with the target and become $\bm{k_{out1}}$ and $\bm{k_{out2}}$ respectively. The impact parameters $p_1$ and $p_2$ are proportional to the change in momentum and scattering angle of the incident electron upon impact. The angles $\theta_1$ and $\theta_1$ are the scattering angles for electrons $e_1$ and $e_2$ respectively. source
An incident electron $e$ with energy $E_{inc}$ and momentum $\bm{k_{in}}$ collides with an atomic or molecular target, and scatters at an angle $\theta$ with respect to the direction of the incident electron with a final momentum $\bm{k_{out}}$. The momentum of an elastically scattered electron following impact with an atomic or molecular target is dependent on the interaction of the incident electron with the target’s bound and valence electrons. The angle by which the electron scatters away from the target depends on the impact parameter $p$ which is defined by the distance between the trajectory of the electron and the parallel axis running through the atomic or molecular target. The incident electrons that follow a trajectory far from the nucleus encounter a small Coulomb force and are deflected only by a small angle. On the other hand, the electrons that follow a trajectory close to the nucleus encounter a strong attractive Coulomb force and are deflected by angles of up to $180^{\circ}$. The elastic cross-section $\sigma_{elastic}$ for such scattering experiments can be measured and expressed in the following manner: $$ \sigma_{elastic}(\theta_e, E_{inc}) = \frac{d\sigma}{d\Omega} $$ where ${d\Omega}$ is the solid angle of detection of an electron energy analyzer. The cross-section $\sigma_{elastic}$ does not depend on the angle $\psi$ with respect to the detection plane.
Elastic scattering processes can lead to quantum mechanical interferences of the incident electron wave and the scattered electron following impact with the target. These quantum mechanical interferences are seen as deep and sharp dips in the cross-section measurements. If the incident electron energy is low, resonances can occur and appear in the elastic cross-section measurements. An elastic resonance occurs when an incident electron with low energy $E$ collides with a target and is captured for a time $t_{res}$ around the atom until it is ejected (as shown in fig. 1.2). During the interaction time $t_{int}$ of the electron with the target, a temporary negative ion is formed for a time $t_{res}$. The two different interaction trajectories are when the incident electron is captured then ejected creating a temporary negative ion in the process:
$$ \Ket{E_{inc}, \bm{k_{in}}} + \Ket{\psi_{ground}} \xrightarrow{indirect} \Ket{E\approx 0; \psi^{ion}_{exc}} \xrightarrow{emission}\ $$
$$ \Ket{E_{inc}, \bm{k_{out}}} + \Ket{\psi_{ground}} $$
or when the incident electron is deflected upon impact with the target:
$$ \Ket{E_{inc}, \bm{k_{in}}} + \Ket{\psi_{ground}} \xrightarrow{direct} \Ket{E_{inc}, \bm{k_{out}}} + \Ket{\psi_{ground}} $$
where $\Ket{\psi_{ground}}$ is the target ground state wavefunction. These two different interaction trajectories lead to an interference in the elastic cross-section measurements which appears as a sharp dip with a width given by Heisenberg’s uncertainty principle where $\Delta E = \frac{\hbar}{2t_{res}}$. An example of this is the $19.34,\mathrm{eV}$ elastic resonance observed from electron collisions with Helium. 2
Figure 1.2 Elastic resonance occurs by capture of the incident electron with incident energy $E_{inc}$ and momentum $\bm{k_{in}}$ for a time $t_{res}$. The electron is then ejected with the same energy as $E_{inc}$ and momentum $\bm{k_{out}}$. A temporary negative ion is formed for the time $t_{res}$ during which elastic resonance occurs. source