Source 1
Inelastic scattering processes occur when the incident electron of energy $E_{inc}$ and a well defined momentum $\bm{k_{in}}$, excites the target upon impact to an excited state $\Ket{\psi_{exc}}$ of a discrete energy $E_{exc}$ and so scatters inelastically. This process is useful as varying incident electron energies can allow for the excitation of all states of a target; states that are not accessible via photon excitation. The inelastic scattering reactions by electron impact are simply:
$$ E_{inc}, \bm{k_{in}}, \Ket{\psi_{ground}} \xrightarrow{interaction} E_{inc} - E_{exc}, \bm{k_{out}}, \Ket{\psi_{exc}} $$
with the experimental inelastic cross-section measurements expressed as follows:
$$ \sigma_{inelastic}(\theta_e, E_{inc}, E_{exc}) = \frac{d\sigma}{d\Omega} $$
where $\theta_e$ is the inelastic scattering angle with respect to the direction of the incident electron. The solid angle of the electron energy analyzer is given by $d\Omega$, which is tuned to detect electrons of energies $E_{inc} - E_{exc}$. It is important to note that the different substate contributions cannot be identified in inelastic differential cross-section measurements $\sigma_{inelastic}$ because the states are degenerate in energy. These measurements are usually carried out with electron energy analyzers with high energy resolution. The measurements produce electron energy loss spectra in which it shows the probability of excitation (or vibration) of a state as a function of the incident electron energy $E_{inc}$ and scattering angle $\theta_e$. An example of this are studies such as the one carried out by Khakoo et al 2, in which cross-section measurements were obtained for both elastic and vibrationally inelastic scattering of low energy electrons from the ethylene ($C_2H_4$) molecule. These measurements were obtained at incident energies ranging from $0.5$ to $100,\text{ eV}$ and scattering angles ranging from $5^{\circ}$ to $130^{\circ}$.
Another example of an inelastic scattering process is seen in Compton’s experiment which was carried out by Arthur Holly Compton in 1923 3. This experiment involved the scattering of photons (X-rays or gamma rays) from electrons in a graphite target, which found an increase in the wavelength (Compton effect) of the scattered electron relative to that of the incident photon, and a transfer of energy from the incident photon to the recoiling electron.