matt_papers/RSA/results.tex

\section{Results}
\label{sec:results}

\subsection{Octopi}

The jamming limit coverage, $\theta_{J}$,  of the of the off lattice continuum
simulation was found to be 0.531. This value is typical for simulations of 
circles on a 2D plane.\cite{Evans93} Once the system is constrained by the
lattice, $\theta_{J}$ drops to 0.489, showing that the lattice has some small,
but mostly inconsequential, effect on the coverage at the jamming limit. 

Upon examining the radial distribution curves, g(r), in Fig. \ref{fig:octgofr},
it is observed that the lattice has no signifigant contribution to the 
distribution other than to add noise to the correlation. Also, the features
of both curves are quite simple. An initial peak at twice the radius of the
octopi corresponding to the first population shell being the closest two
circles can approach without overlapping each other. The second peak at 
four times the radius simply extends this shell out one more circle length.

\subsection{Tilted Umbrellas}

In the case of the tilted umbrellas, $\theta_{J}$ for the off lattice continuum
simulation was 0.924, and 0.918 for the simulation on the lattice. It can
be seen from this data that the lattice has even less effect on the 
tilted umbrellas as on the octopi. Also, the angular overlap allowed by the
tilted umbrellas allows for almost total surface coverage based on random
parking alone.

Because a particles sticking success is closely linked to its orientation, 
the radial distribution function and the angular distribution function show
some very interesting features (Fig. \ref{fig:tugofr}). The initial peak is 
located at aproximately one radius of the umbrella. This corresponds to the 
closest distance a second particle that is perfectly aligned with the particle
may aproach without overlapping. A quick look at the angular distribution
confirms this, showing a maximum angular correlation at this distance. The 
location of the second peak in the radial distribution corresponds to twice
the radius of the umbrella. This peak is accompanied by a dip in the angular
ditribution, showing that this peak corresponds to the case of closest approach
for two anti-aligned particles. The alignments associated with both peaks are
illustrated in Fig. \ref{fig:tuallign}.
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#	Content
1	\section{Results}
2	\label{sec:results}
3
4	\subsection{Octopi}
5
6	The jamming limit coverage, $\theta_{J}$, of the of the off lattice continuum
7	simulation was found to be 0.531. This value is typical for simulations of
8	circles on a 2D plane.\cite{Evans93} Once the system is constrained by the
9	lattice, $\theta_{J}$ drops to 0.489, showing that the lattice has some small,
10	but mostly inconsequential, effect on the coverage at the jamming limit.
11
12	Upon examining the radial distribution curves, g(r), in Fig. \ref{fig:octgofr},
13	it is observed that the lattice has no signifigant contribution to the
14	distribution other than to add noise to the correlation. Also, the features
15	of both curves are quite simple. An initial peak at twice the radius of the
16	octopi corresponding to the first population shell being the closest two
17	circles can approach without overlapping each other. The second peak at
18	four times the radius simply extends this shell out one more circle length.
19
20	\subsection{Tilted Umbrellas}
21
22	In the case of the tilted umbrellas, $\theta_{J}$ for the off lattice continuum
23	simulation was 0.924, and 0.918 for the simulation on the lattice. It can
24	be seen from this data that the lattice has even less effect on the
25	tilted umbrellas as on the octopi. Also, the angular overlap allowed by the
26	tilted umbrellas allows for almost total surface coverage based on random
27	parking alone.
28
29	Because a particles sticking success is closely linked to its orientation,
30	the radial distribution function and the angular distribution function show
31	some very interesting features (Fig. \ref{fig:tugofr}). The initial peak is
32	located at aproximately one radius of the umbrella. This corresponds to the
33	closest distance a second particle that is perfectly aligned with the particle
34	may aproach without overlapping. A quick look at the angular distribution
35	confirms this, showing a maximum angular correlation at this distance. The
36	location of the second peak in the radial distribution corresponds to twice
37	the radius of the umbrella. This peak is accompanied by a dip in the angular
38	ditribution, showing that this peak corresponds to the case of closest approach
39	for two anti-aligned particles. The alignments associated with both peaks are
40	illustrated in Fig. \ref{fig:tuallign}.