+\begin{align}
+ S(E_1) &= \der{I}{E_1} = e A \frac{d}{dE_1} \left( 1 - \sigma(E_1) \right) = e A \der{\sigma(E_1)}{E_1} \label{tcs_simple}
+\end{align}
+
+A form of \eqref{tcs_simple} is usually assumed when constructing a more detailed theory of secondary electron current formation, as the location of peaks due to $\der{\sigma(E_1)}{E_1}$ is the same in both \eqref{tcs_simple} and \eqref{tcs_full}. Figure \ref{komolov1979} illustrates an idealised total current spectrum.
+
+\begin{figure}[H]
+ \centering
+ \includegraphics[width=0.8\textwidth]{figures/tcs/komolov1979_tcs.png}
+ \caption{An idealised total current spectrum (Komolov et al 1979 \cite{komolov1979})}
+ \label{komolov1979}
+\end{figure}
+
+\subsection{Contact Potential and the Surface Peak}
+
+The case $E_1 = 0$ does not in general correspond to $U = 0$. Figure \ref{contact.pdf} describes a simplified step potential model for the surfaces of the cathode and sample. In a classical approximation, primary electrons will be elastically scattered from the sample unless they have energy greater than its vacuum level.
+
+At the cathode, electrons are thermionically excited from the conduction band to the vacuum level. The vacuum levels of the cathode and surface are not generally equal. The contact potential is the difference in work functions of the materials. As $U$ approaches the contact potential, primary electrons are able to penetrate into the sample, and a sharp peak termed the surface peak is seen in $S(U)$. The maxima of this peak occurs at an applied potential $U$ such that $E_1 = 0$; its shape is indicative of $f(E - E_1)$.
+
+\begin{figure}[H]
+ \centering
+ \includegraphics[width=0.80\textwidth]{figures/tcs/contact.pdf}
+ \caption{Illustration of the step potential model for the surfaces}
+ \label{contact.pdf}
+\end{figure}
+
+
+
+
+\subsection{Electron-Electron Interactions}
+\label{tcs_e}
+
+
+In the Sommerfield free electron gas approximation, electrons are assumed to be contained within a rectangular potential well, with energies given by $E_k = \frac{\hbar^2 k^2}{2m}$, where $\vect{k}$ is the electron wave vector. Electrons fill states up to the Fermi level, with density of states $N(E_k)$ proportional to $\sqrt{E_k}$.
+
+An incoming primary electron interacting with the free electron gas may either excite a single electron to above the Fermi level, or a collective vibration (plasmon). A plasmon may be excited if $E_1 \geq \hbar \omega_p$.
+
+The response of a free electron gas to an external electric field is characterised by $\phasor{\epsilon}$. The imaginary part of $\phasor{\epsilon}$ describes the attenuation of the electric field; it describes the process of energy transfer from the primary electron to the electron gas. Since the intensity of the electric field entering the solid decreases by a factor of $|\phasor{\epsilon}^2|$, the efficiency of energy transfer is given by:
+