Thesis - Oh deary me

Progress? HAHAHAH I laugh in the face of progress

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+++ b/thesis/thesis.tex
@@ -77,39 +77,74 @@
 
 \section*{Acknowledgements}
 
 
 \section*{Acknowledgements}
 

-I am extremely grateful for the support offered to me by many individuals during this project. Thanks to my supervisors Prof Sergey Samarin and W/Prof Jim Williams for envisioning the project, and their invaluable support throughout the year. I would also like to thank staff members at CAMSP for assisting with the supervision of this project. In particular I am extremely grateful for the help and advice given by Dr Paul Gualiardo during the construction and testing of the Total Current Spectroscopy experiment.
+I am extremely grateful for the support offered to me by many individuals during this project. There aren't many synonyms for ``Thanks'', so I'm afraid this section may be a little repetitive.

 
 

-Thanks to Alexie ??? from CMCA for producing the SEM images which proved a valuable aid for understanding the structure of metallic-black films.
+Thanks to my supervisors Prof Sergey Samarin and W/Prof Jim Williams for envisioning the project, and their invaluable support throughout the year. I would also like to thank staff members at CAMSP for assisting with the supervision of this project. In particular I am extremely grateful for the help and advice given by Dr Paul Gualiardo during the construction and testing of the Total Current Spectroscopy experiment.

 
 

-I would also like to endorse the team at J.A Woolam, who provided replacement pins for the ellipsometer alignment detector at no charge after one of the original pins became mysteriously damaged.
+Thanks to Alexie ??? from CMCA for producing the SEM images which proved a invaluable aid for discussing the structure of the metallic-black films. Thanks to Nikita Kostylev for helping me learn the art of operating the ellipsometer. Thanks to both Prof Mikhail Kostylev and Jeremy Hughes for lending me some samples for ellipsometric analysis. I would also like to endorse the team at J.A Woolam, who provided replacement pins for the ellipsometer alignment detector at no charge after one of the original pins became mysteriously damaged. 

 
 

-Congratulations and thanks to Jeremy Hughes who successfully predicted that the emission current of the electron gun was varying sinusoidally due to the changing temperature of the room. Commisserations to Alexander Mazur, whose suspicians that the vacuum chamber contained a pulsar proved wrong.
+Congratulations to Jeremy Hughes who successfully predicted that the emission current of the electron gun was varying periodically less than a quarter of the way through the first period. Condolences to Alexander Mazur, whose theory that the vacuum chamber contained a pulsar proved unfounded. 

 
 

-Finally, thanks to all my family and friends for their support and for continuing to put up with my slow descent into madness during the last 12 months. 
+Thanks to all my family and friends for their support and for continuing to put up with my slow descent into madness during the last 12 months. 
+
+Finally, perhaps as a result of the aforementioned madness,  I would also like to thank the various pieces of equipment and inanimate objects which have been crucial to the success of this project. This includes the ellipsometer, the ADC/DAC box, my laptop computer ``Cerberus'', and the two ammeters upon which I relied upon so heavily. Rest in peace Keithly 610B. Your death was not in vain.
+
+\pagebreak

 
 \section{Introduction}
 
 The report will be organised as follows; first we will discuss literature relating to surface science and nanostructured thin films in particular. We will then give an overview of the primary experimental techniques employed during this project, before presenting experimental results. Finally, we will discuss stuff.
 
 
 \section{Introduction}
 
 The report will be organised as follows; first we will discuss literature relating to surface science and nanostructured thin films in particular. We will then give an overview of the primary experimental techniques employed during this project, before presenting experimental results. Finally, we will discuss stuff.
 

+\pagebreak
+

 \section{Overview}
 
 In this section we provide an overview of theoretical and experimental literature related to the properties of nanostructured thin films. We also include a summary of the past research which focuses on metallic-black films.
 
 \section{Overview}
 
 In this section we provide an overview of theoretical and experimental literature related to the properties of nanostructured thin films. We also include a summary of the past research which focuses on metallic-black films.
 

-\subsection{Electron Structure of Bulk and Surface}
+\subsection{Secondary Electron Emission}
+
+% What is it?
+Secondary electron emission refers to the emission of electrons from a target surface caused by interactions with an incident electron, or more commonly a beam of electrons. Electrons from the incident beam are referred to as primary electons, whilst the term ``secondary electron'' applies to all electrons reflected or emitted from the surface under bombardment. 
+
+% First mentions and early research

 
 
 
 

+% General description of secondary electron spectrum
+
+Figure \ref{se_dist.pdf} shows the general shape of an energy distribution of secondary electrons. for a primary electron energy of $E_p$. The narrow peak centred at $E = E_p$ is due to elastically scattered electrons; the width of this peak is determined by the distribution of primary electron energies, as well as the resolution of the detector. The broad peak in the low energy part of the spectrum is due to inelastically scattered electrons.
+
+
+\begin{center}
+       \includegraphics[scale=0.40]{figures/se_dist.pdf}
+       \captionof{figure}{{\bf Model of Secondary Electron Distribution}}
+       \label{se_dist.pdf}
+\end{center}
+
+% How real secondary electron spectra depend upon the surface
+
+Real secondary electron distributions also show fine structure imposed upon the inelastic part of simplified spectrum described above. This fine structure is characteristic of the electron spectra of the target surface. Near to the elastic peak, fine structure is caused by energy loss to interband transitions and plasma vibration excitation. The central part of the distribution contains fine structure due to Auger electron emission, and energy losses due to excitation of inner electrons. Fine structure at low energies is due to the structure of empty states of the solid.
+
+% Research of Komolov
+

 \subsection{Plasmons}
 
 \subsection{Plasmons}
 

+% What are they?
+A plasmon is a quasi-particle arising due to charge density oscillations in a solid. 
+
+% Early research

 
 \subsection{Metallic-Black Films}
 
 
 \subsection{Metallic-Black Films}
 

-So called ``Metallic-Black'' films are the result of deposition of metal elements at high pressure (of the order of $10^{-2}$ mbar). The films are named due to their high absorbance at visible wavelengths; they appear ``black''. There is a remarkable contrast between such films and films deposited under ultra-high vacuum, which are typically highly reflective, exhibiting similar optical properties to a bulk sample of the metal.
+% What are they?
+So called metallic-black films are the result of deposition of metal elements at a relatively high pressure (of the order of $10^{-2}$ mbar). The films are named due to their high absorbance at visible wavelengths; they appear black to the naked eye. There is a remarkable contrast between such films and films deposited under low pressure (less than $10^{-6}$mbar), which are typically highly reflective and brightly coloured.

 
 

-This phenomenom has been known since the early 20th century, with the first papers on the subject published by Pfund in the 1930s \cite{pfund}. Pfund's research has been mostly qualitative, and estabilishes the conditions for formation of metallic black films for a number of different metal elements. More extensive research has been carried out on the structural and optical properties of these films by Louis Harris and others from the 1950s until the 1970s \cite{}. Despite advances in computer technology revolutionising the field of surface science, little work has been done since Harris' contributions. %TODO: Check for any recent work
+% First mentions and early research; Pfund
+This phenomenom has been known since the early 20th century, with the first papers on the subject published by Pfund in the 1930s \cite{pfund1930}, \cite{pfund1933}. Pfund established the conditions for formation of metallic-blacks \cite{pfund1930}, and showed that the transmission spectrum of metallic black films is almost zero in visible wavelengths, but increases to a plateau in the far infrared \cite{pfund1933}. More extensive research on the structural and optical properties of these films by Louis Harris and others during the 1940s and 1950s \cite{harris1948}, \cite{harris1952}, \cite{harris1953}. It has been established that metallic blacks may be prepared in either air or inert gases

 
 

-Secondary Electron Microscope (SEM) images of Au deposited on Si at high and low pressures were produced at the Centre for Microscopy Characterisation and Analysis (CMCA), UWA. These images are similar to previous images produced by Harris \cite{}. The film imaged on the left (high pressure) appears black at visible wavelengths, whilst the film on the right (low pressure) appears golden yellow.

 
 

-%TODO: Insert images
+% Pretty pictures for purpose of this discussion
+% Not really "results", since I didn't make the images
+Secondary Electron Microscope (SEM) images of Au deposited on Si at high and low pressures (in air) were produced at the Centre for Microscopy Characterisation and Analysis (CMCA), UWA. The film imaged on the left (high pressure) appears black at visible wavelengths, whilst the film on the right (low pressure) appears golden yellow.

 
 \begin{center}
 
 
 \begin{center}
 


@@ -118,24 +153,40 @@ Secondary Electron Microscope (SEM) images of Au deposited on Si at high and low
        \includegraphics[scale=0.2]{figures/Au_BLACK_200nm.png} & %\captionof{figure}{Au-Black SEM Image} \label{Au_BLACK_200nm.png} &
        \includegraphics[scale=0.2]{figures/Au_semi-shiny_1_SEM.png} %\captionof{figure}{Au SEM Image} \label{Au_semi-shiny_1_SEM.png}          
        
        \includegraphics[scale=0.2]{figures/Au_BLACK_200nm.png} & %\captionof{figure}{Au-Black SEM Image} \label{Au_BLACK_200nm.png} &
        \includegraphics[scale=0.2]{figures/Au_semi-shiny_1_SEM.png} %\captionof{figure}{Au SEM Image} \label{Au_semi-shiny_1_SEM.png}          
        

-
+       \label{SEM_images}

 \end{tabular}
 
        \captionof{figure}{{\bf SEM images of Au deposited on Si at $2\times10^{-2}$mbar (left) and $1\times10^{-6}$mbar.} Note that the scales are very similar for both images.}              
 
 \end{center}
 
 \end{tabular}
 
        \captionof{figure}{{\bf SEM images of Au deposited on Si at $2\times10^{-2}$mbar (left) and $1\times10^{-6}$mbar.} Note that the scales are very similar for both images.}              
 
 \end{center}
 

-The structural difference between the two films is striking, and yet the exact mechanism behind the formation of the metallic-black film remains unclear. 
-% TODO: Attempt to find explanation of structure
-
-Nanostructured metal films prepared at low pressure are often approximated by an isotropic layer of spherical blobs upon the substrate, or even as a uniform layer with an ``effective'' thickness \cite{}. As the right image in Figure \ref{} shows, this is largely representative of the structure of such a film. In contrast, the metallic-black film appears non-uniform; as a result more complicated models are required to theoretically describe the properties of metallic-black films.
+% Explanation of structure?
+The structural difference between the two films is striking, and yet the exact mechanism behind the formation of the metallic-black film is not well understood. The most widely accepted explanation is that the evaporated metal particles reaching the target surface have insufficient energy to form a regular crystal lattice due to cooling through collisions with the atmosphere \cite{}. As of yet, there is no detailed theoretical description of this behaviour. 

 
 

+% Research by Harris concluding ``condensor'' like structure

 Harris et al. have produced experimental results of the transmission of metallic-black films from visible wavelengths to the far-infrared \cite{}. By modelling the film as a layer of metallic strands, acting as ``condensors'', Harris et al. arrived at an expression for the electron relaxation time of [element]-black \cite{}, leading to a a transmission spectrum in good agreement with experimental results.
 
 Harris et al. have produced experimental results of the transmission of metallic-black films from visible wavelengths to the far-infrared \cite{}. By modelling the film as a layer of metallic strands, acting as ``condensors'', Harris et al. arrived at an expression for the electron relaxation time of [element]-black \cite{}, leading to a a transmission spectrum in good agreement with experimental results.
 

+% Mckenzie
+Mckenzie has established that the presence of oxygen effects the optical and electrical properties of metallic-blacks \cite{mckenzie2006}.
+
+% Model of structure
+Nanostructured metal films prepared at low pressure are often approximated by an isotropic layer of spherical blobs upon the substrate, or even as a uniform layer with an ``effective'' thickness \cite{}. As the right image in Figure \ref{SEM_images} shows, this is a good representation of the structure of such a film. In contrast, the metallic-black film is highly non-uniform; as a result, detailed characterisation of the properties of such a film is difficult.
+
+

 
 

-More recently, it has been suggested that metallic-black films may exhibit plasmonic properties that could be exploited to increase the efficiency of thin film solar cells \cite{pandjwani}. In this work, the author shows that 

 
 

-However, currently there is no direct evidence of plasmon excitations in metallic-black films. %TODO: Check for evidence of plasmonic behaviour in metallic-black films
+% More recent research
+More recently, it was shown that Au-black coatings increased the efficiency of thin film solar cells \cite{}. In this study, a simulation approximating an Au-black film as a layer of semi-spherical structures showed plasmonic behaviour which lead to an increase in electric field behind the film. 
+
+
+% Artificially ``blackened'' thin films
+Metallic-black films have proven useful in applications requiring efficient absorption of light, including the. Recently there has been interest in artificial ``blackening'' of metal surfaces in ways which simplify the characterisation of the surfaces for practical applications. 
+
+Sondergaard et al. have produced metallic-black surfaces capable of suporting surface plasmon modes \cite{sondergaard2012}. These films exhibit similar optical properties to the previously considered ``evaporated'' metallic-black surfaces. 
+
+% What I will be doing with metallic-black films
+This project will employ Total Current Specroscopy, Ellipsometry and Optical Spectroscopy methods to investigate the difference between metallic films deposited at low pressure, and high pressure (metallic-blacks). The production and study of artificially blackened films is beyond the scope of this research.
+

 
 \begin{center}
        \includegraphics[scale=0.2]{figures/300V-01.jpg}
 
 \begin{center}
        \includegraphics[scale=0.2]{figures/300V-01.jpg}


@@ -149,22 +200,11 @@ However, currently there is no direct evidence of plasmon excitations in metalli
 
 \subsection{Secondary Electron Spectroscopy}
 
 
 \subsection{Secondary Electron Spectroscopy}
 

-Secondary Electron Spectroscopy encompasses a large group of techniques used for studying the electron spectra of surfaces and solids. In these methods a beam of primary electrons is directed at a surface. The interactions between primary electrons and the surface give rise to an energy distribution of electrons elastically and inelastically scattered from the surface. Analysis of the distribution of the scattered ``secondary'' electrons gives information about the electron energy spectrum of the target surface.
-
-
-\begin{center}
-       \includegraphics[scale=0.40]{figures/se_dist.pdf}
-       \captionof{figure}{{\bf Model of Secondary Electron Distribution}}
-       \label{se_dist.pdf}
-\end{center}
-
-Figure \ref{se_dist.pdf} shows the general shape of the secondary electron distribution for a primary electron energy of $E_p$. The narrow peak centred at $E = E_p$ is due to elastically scattered electrons; the width of this peak is determined by the distribution of primary electron energies, as well as the resolution of the detector. The broad peak in the low energy part of the spectrum is due to inelastically scattered electrons. The thick curve shows 
-
-Real secondary electron distributions also show fine structure imposed upon the inelastic part of simplified spectrum described above. This fine structure is characteristic of the target surface. Near to the elastic peak, fine structure is caused by energy loss to interband transitions and plasma vibration excitation. The central part of the distribution contains fine structure due to Auger electron emission, and energy losses due to excitation of inner electrons. Fine structure at low energies is due to the structure of empty states of the solid.
+Secondary Electron Spectroscopy encompasses a large group of techniques which exploit secondary electron emission for studying the electron spectra of surfaces and solids. In these methods a beam of primary electrons is directed at a surface. The interactions between primary electrons and the surface give rise to an energy distribution of electrons elastically and inelastically scattered from the surface. Analysis of the distribution of the scattered ``secondary'' electrons gives information about the electron energy spectrum of the target surface.

 
 

-Techniques of Secondary Electron Spectroscopy can be divided into two classes. Energy-resolved methods are based upon observation of the secondary electron distribution at a fixed primary electron energy. These methods aim to examine specific secondary emission processes which occur within a selected energy interval. In contrast to Energy-resolved methods, Total Current (or Yield) methods measure the total current of secondary electrons as a function of primary electron energy. 
+Techniques of Secondary Electron Spectroscopy can be divided into two classes. Energy-resolved methods are based upon observation of the secondary electron distribution at a fixed primary electron energy. These methods aim to examine specific secondary emission processes which occur within a selected energy interval. The angular distribution of emitted electrons is often also recorded.

 
 

-The focus of this project has been low energy Total Current Spectroscopy (TCS). While Total Current methods provide less detailed information about secondary emission processes within a solid, they are useful for characterisation of the electron structure. Total Current methods are also simpler to realise experimentally, as they do not require energy analysers, and current measurement may be performed external to the vacuum chamber.
+In contrast to Energy-resolved methods, Total Current (or Yield) methods measure the total current of secondary electrons as a function of primary electron energy. The focus of this project has been low energy Total Current Spectroscopy (TCS). While Total Current methods provide less detailed information about secondary emission processes within a solid, they are generally simpler to realise experimentally as they do not require energy analysers, and current measurement may be performed external to the vacuum chamber, using a conventional ammeter.

 
 
 \subsection{Total Current Spectroscopy}
 
 
 \subsection{Total Current Spectroscopy}


@@ -172,7 +212,6 @@ The focus of this project has been low energy Total Current Spectroscopy (TCS).
 Figure \ref{} shows a simplified schematic for the Total Current Spectroscopy experiments conducted during this study. An electron gun is used to produce the beam of primary electrons. Electrons are emitted from a cathode held at negative potential relative to the target. These electrons are focused into a beam and accelerated onto the target through the electric field produced by a series of electrodes. A detector is used to measure the total current passing through the target.
 
 
 Figure \ref{} shows a simplified schematic for the Total Current Spectroscopy experiments conducted during this study. An electron gun is used to produce the beam of primary electrons. Electrons are emitted from a cathode held at negative potential relative to the target. These electrons are focused into a beam and accelerated onto the target through the electric field produced by a series of electrodes. A detector is used to measure the total current passing through the target.
 
 

-A major advantage of Total Current Spectroscopy methods is the relative simplicity of the experimental setup. Because energy resolution of secondary electrons is not required, current measurement can be performed external to the vacuum chamber, using a conventional ammeter.

 
 If the current incident upon the sample is $I_{\text{total}}$, and the current of secondary electrons scattered from the surface is $I_r$, then the transmitted current $I_t$ is given by:
 
 
 If the current incident upon the sample is $I_{\text{total}}$, and the current of secondary electrons scattered from the surface is $I_r$, then the transmitted current $I_t$ is given by:
 


@@ -187,13 +226,18 @@ In this case, differentiating with respect to $E$:
        \der{I_t}{E} &= - \der{I_r}{E}
 \end{align*}
 
        \der{I_t}{E} &= - \der{I_r}{E}
 \end{align*}
 

-Figure \ref{} is a block diagram of the experimental setup including measurement and control systems external to the vacuum chamber.
+Figure \ref{block_diagram.pdf} is a block diagram of the experimental setup including measurement and control systems external to the vacuum chamber.
+Note that the emission current ammeter is optional, and was used for testing purposes.
+
+\begin{center}
+       \includegraphics[scale=0.60]{figures/block_diagram.pdf} \label{block_diagram.pdf}
+\end{center}

 
 \subsubsection{Electron Optics}
 
 The electron gun used for this experiment was repurposed from an old Cathode-Ray Oscilloscope (CRO). Figure \ref{} shows a simplified diagram of the electron gun, whilst Figure \ref{} shows a photograph of the gun. 
 
 
 \subsubsection{Electron Optics}
 
 The electron gun used for this experiment was repurposed from an old Cathode-Ray Oscilloscope (CRO). Figure \ref{} shows a simplified diagram of the electron gun, whilst Figure \ref{} shows a photograph of the gun. 
 

-The full circuit diagram for the electron gun control circuit is shown in Appendix A. \cite{}
+The full circuit diagram for the electron gun control circuit is shown in Appendix A. %\cite{}

 
 \subsubsection{Automatic Data Acquisition}
 
 
 \subsubsection{Automatic Data Acquisition}
 


@@ -210,8 +254,7 @@ Ellipsometry is an optical technique most commonly used to determine the thickne
 
 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.
 
 
 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.
 

-\subsubsection{Description of the Polarisation state of Light}
-
+A

 
 
 
 
 
 


@@ -223,6 +266,9 @@ A Variable Angle Spectroscopic Ellipsometer at CAMSP has been used to perform a
 
 The VASE
 
 
 The VASE
 

+It is also possible to conduct reflection and transmission spectroscopy experiments using the VASE.
+
+

 \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).
 \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).


@@ -244,6 +290,12 @@ This study focused primarily on depositing Au films on an Si substrate, at both
 
 \subsection{Ellipsometric Measurements}
 
 
 \subsection{Ellipsometric Measurements}
 

+\subsection{Transmission Spectra of Metal Films}
+
+Using the VASE and a commercial spectrometer (OceanOptics) in independent experiments, we obtained transmission spectra for metallic-black and some other metallic films.
+
+
+

 
 \section{Achievements}
 
 
 \section{Achievements}
 


@@ -254,7 +306,7 @@ This study focused primarily on depositing Au films on an Si substrate, at both
 Figure \ref{} shows the complete electron gun control circuit. The circuit was designed and constructed as part of this project. The design is based upon examples found in \cite{komolov} and \cite{Moore}.
 
 \subsection*{Electron Optics}
 Figure \ref{} shows the complete electron gun control circuit. The circuit was designed and constructed as part of this project. The design is based upon examples found in \cite{komolov} and \cite{Moore}.
 
 \subsection*{Electron Optics}

-The electron gun has been recycled from a Cathode Ray Oscilloscope. Figure \ref{} shows a diagram and photograph of the gun. The gun contains a total of 9 electrodes; several electrodes are held at the same potential, as shown in the figure. From figure \ref{} it can be seen that the electrode potentials are referenced to the initial energy, rather than ground. Because of the relatively large distance between the gun and sample (held at ground), this ensures that changes in initial energy do not significantly effect the focusing properties of the gun.
+The electron gun has been recycled from a Cathode Ray Oscilloscope. Figure \ref{} shows a diagram and photograph of the gun. The gun contains a total of 9 electrodes; several electrodes are held at the same potential, as shown in the figure. As shown in figure \ref{}, the electrode potentials are referenced to the cathode, not signal ground. Because of the relatively large distance between the gun and sample (held at ground), this ensures that changes in initial energy do not significantly effect the focusing properties of the gun.

 
 The optimum potentials of the gun electrodes were determined manually by focusing the gun on an Au film (deposited on Si). I-V curves obtained by measuring current through the sample as a function of initial energy were obtained. The electrode potentials were systematically altered to ensure the curves were as close as possible to the ideal model.
 
 
 The optimum potentials of the gun electrodes were determined manually by focusing the gun on an Au film (deposited on Si). I-V curves obtained by measuring current through the sample as a function of initial energy were obtained. The electrode potentials were systematically altered to ensure the curves were as close as possible to the ideal model.
 


@@ -269,7 +321,7 @@ The filament is surrounded by a conducting cylindrical electrode commonly called
 
 \subsection*{Einzel Lens}
 
 
 \subsection*{Einzel Lens}
 

-Electrodes passing through the aperture of the venault are accelerated towards the target through six electrodes. The first and last pair of electrodes are at the same potential; the central pair are held at a different potential. Such an arrangement can be considered as an ``Einzel'' or ``zoom'' lens. It can be shown \ref{Moore} that the 
+%Electrodes passing through the aperture of the venault are accelerated towards the target through six electrodes. The first and last pair of electrodes are at the same potential; the central pair are held at a different potential. Such an arrangement can be considered as an ``Einzel'' or ``zoom'' lens. It can be shown \ref{Moore} that the 

 
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