\contentsline {subsection}{\numberline {6.1}\color {red}Definitions}{2}
\contentsline {subsubsection}{\numberline {6.1.1}\color {blue}WARNING}{2}
\contentsline {subsubsection}{\numberline {6.1.2}\color {blue}emittance}{2}
\contentsline {subsubsection}{\numberline {6.1.3}\color {blue}Partition Functions}{2}
\contentsline {subsubsection}{\numberline {6.1.4}\color {blue}Longitudinal Heating Terms}{3}
\contentsline {subsubsection}{\numberline {6.1.5}\color {blue}Emittance changes}{3}
\contentsline {subsection}{\numberline {6.2}\color {red}Transverse Cooling}{3}
\contentsline {subsubsection}{\numberline {6.2.1}\color {blue}Partition Functions}{3}
\contentsline {subsubsection}{\numberline {6.2.2}\color {blue}Cooling rate vs. Energy}{4}
\contentsline {subsubsection}{\numberline {6.2.3}\color {blue}Heating Terms}{5}
\contentsline {subsubsection}{\numberline {6.2.4}\color {blue}Beam Divergence Angles}{7}
\contentsline {subsection}{\numberline {6.3}\color {red}Focusing Systems}{8}
\contentsline {subsubsection}{\numberline {6.3.1}\color {blue}Solenoid}{8}
\contentsline {subsubsection}{\numberline {6.3.2}\color {blue}Current Carrying Rod}{8}
\contentsline {subsubsection}{\numberline {6.3.3}\color {blue}Minimum Transverse Emitances}{9}
\contentsline {subsection}{\numberline {6.4}\color {red}Longitudinal and 6D Cooling}{10}
\contentsline {subsubsection}{\numberline {6.4.1}\color {blue}Longitudinal Cooling}{10}
\contentsline {subsubsection}{\numberline {6.4.2}\color {blue}Energy Loss}{11}
\contentsline {subsubsection}{\numberline {6.4.3}\color {blue}Longitudinal "Cooling"}{11}
\contentsline {subsubsection}{\numberline {6.4.4}\color {blue}Partition Functions vs. Energy}{12}
\contentsline {subsubsection}{\numberline {6.4.5}\color {blue}6D Cooling Rate vs. Energy}{13}
\contentsline {subsubsection}{\numberline {6.4.6}\color {blue}Heating Terms}{13}
\contentsline {subsubsection}{\numberline {6.4.7}\color {blue}rf and bunch length}{14}
\contentsline {subsection}{\numberline {6.5}\color {red}minimum 6D emittances vs. Energy}{15}
\contentsline {subsubsection}{\numberline {6.5.1}\color {blue}Comparisons}{18}
\contentsline {subsection}{\numberline {6.6}\color {red}Conclusion}{19}
\contentsline {subsubsection}{\numberline {6.6.1}\color {blue}Complications}{19}
