+The primary electron current $I_1$ for a mean energy $E_1 = e U$ can be written as:
+\begin{align*}
+ I_1(E_1) &= e A \int_{0}^{\infty} f(E - E_1) dE
+\end{align*}
+Where $A$ is the surface area irradiated by the beam.
+
+Introducing the secondary emission coefficient $\sigma(E)$, which gives the probability for a primary electron of energy $E$ to give rise to a secondary electron, we can write the secondary electron current as:
+\begin{align*}
+ I_2(E_1) &= e A \int_{-E_1}^{\infty} \sigma(E_1)f( E - E_1) dE
+\end{align*}
+
+The total current $I$ may then be written as:
+\begin{align*}
+ I(E_1) &= e A \left[ \int_{0}^{\infty} f(E - E_1) dE - \int_{-E_1}^{\infty} \sigma(E_1)f( E - E_1) dE \right]
+\end{align*}
+
+Differentiating, using the fundamental theorem of calculus, we can determine the total current spectrum:
+\begin{align*}
+ S(E_1) = \der{I}{E_1} &= e A \left\{ [ 1 - \sigma(0)] f(-E_1) - \int_{0}^{\infty} f(E - E_1) \der{\sigma(E_1)}{E_1} dE \right\}
+\end{align*}
+
+The first term in the above expression is determined solely by the distribution of primary electrons $f$. This term will be maximised when $E_1 = 0$; meaning that $U$ is equal to the contact potential $c$ between the cathode and sample.
+
+The second term contains all dependence of $S(E_1)$ on characteristics of the sample. At the threshold for a particular process, the secondary emission efficiency $\sigma(E_1)$ is expected to undergo a sharp change. This results in a well defined maxima or minima in the derivative $\der{\sigma(E_1)}{E_1}$, which can be seen as a corresponding maxima or minima in the total current spectrum $S(E_1)$. From the convolving function $f(E - E_1)$, it can be seen that the distribution of primary electron energy determines the degree to which $\der{\sigma(E_1)}{E_1}$ may be resolved from measurement of $S(E_1)$.
+
+The total current spectrum $S(E_1) = \der{I}{E_1}$ can be obtained from measurement of $I(E_1)$ using a finite difference approximation. Often, the conventional ammeter and DC power supply in Figure \ref{tcs_simple.pdf} are replaced with a lock-in amplifier and AC power supply, as in Komolov's description \cite{komolov}. Lock-in amplifier techniques have the advantage of measuring $S(E)$ directly. The lock-in amplifier also eliminates unwanted sources of noise. For this study, the lock-in amplifier approach was inpractical due to the limitations on available equipment. For future studies, it is suggested that the lock-in amplifier approach be adopted.
+
+
+\subsubsection{The Secondary Emission Coefficient}
+
+$\sigma(E)$ can be written as the sum of two components, representing the probability for secondary electrons arrising due to elastic reflections or any mechanism involving primary electron energy loss.
+
+
+
+
+
+
+
+
+
+
+\subsection{Ellipsometry}
+
+Ellipsometry is an optical technique most commonly used to determine the thickness of multilayered thin films. Ellipsometry can also be used to determine the optical constants and properties of unknown materials.
+
+Essentially, ellipsometry measures the change in polarisation of light reflected from a surface. This change in polarisation can be related to properties of the surface if knowledge of the surface is correctly applied. For a bulk sample, the change in polarisation can be directly related to the optical constants of the material.
+
+
+\subsection{Vacuum Techniques and Sample Preparation}
+
+Both the TCS experiments and the deposition of films must be performed in a vacuum. For convenience and simplicity, a single vacuum chamber at CAMSP has been repurposed to perform both of these tasks. The chamber can be pumped by a molecular turbo pump, backed by a rotaray pump, to a base pressure of $2\times10^{-8}$ mbar, or by the rotary pump alone to a base pressure of $1\times10^{-3}$ mbar. The pressure is monitored using either a pirani or ion gauge (for pressures greater than and less than $10^{-3}$ mbar respectively).
+
+%TODO: Insert graphs of pressure in chamber
+
+Figure \ref{} shows a diagram of the vacuum chamber used both for the creation of nanostructured thin films and their study using TCS. A rotatable sample holder is positioned in the centre of the chamber. One flange of the chamber houses the electron gun used for TCS measurements, whilst the opposite flange contains feedthroughs on which tungsten filament evaporators are mounted. This setup allows for almost immediate study of evaporated films by simple rotation of the sample holder to face the gun.
+
+
+The evaporators consist of a tungsten wire filament attached between two feedthroughs. A piece of a desired metal is folded over the apex of the tungsten wire. The metal can be heated by passing a current through the filament; near the metal's melting point it begins to evaporate. To clean the metal surface and ensure uniform evaporation, this procedure is first performed at low pressure (below $10^{-6}$ mbar) with no sample in the chamber, with the current increased until the metal piece begins to melt and forms a ball on the wire. Figure \ref{} shows an image of an evaporator that has been prepared for use.
+
+This study focused primarily on depositing Au films on an Si substrate, at both high and low pressures. The substrates and sample holders were cleaned in an acetone bath immediately prior to insertion in the vacuum chamber.
+
+\pagebreak