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\title{Optical/UV Band
Reverberation Mapping of NGC 5548 with Frequency-Resolved Techniques}

\date{August 20, 2016}

\maketitle

\begin{abstract}

The alpha radiation spectrum of Americium$^42$ 

\end{abstract}


\section{Introduction}
\label{sec:intro}

Alpha-partcicle spectroscopy is a method for testing and measuring the properties of any alpha emitter, which is a class of radioactive particles that emit alpha particles, a bound collection of two protons and two neutrons. This is one of the primary classes of radiation, along with beta and gamma radiation, and of these the only hadronic form of radiation. An alpha particle is emitted during alpha decay of a nucleus, when the nucleus gives off 2 protons and 2 neutrons. The isotope would thus lose 2 from its atomic number, decaying into an isotope of a new element. The spectrum measured from the radium sample exemplifies this, as the radium decays into daughter elements further down the periodic table. Alpha particles created in this process often have kinetic energy near 5 MeV and are highly ionizing, but with a small penetration depth. They are therefore not considered a dangerous form of radiation unless ingested.





\section{Reverberation Mapping}
\label{sec:reverbmap}

    \subsection{Continuum Reverberation}
    \label{sec:cont_reverb}
    
    \begin{equation}
        \label{eq:timelag}
        \tau\left(\lambda\right) = 
            \left(3.9 \textrm{d}\right) 
            \left(\frac{T_0}{10^4\mathrm{K}}\right)^{4/3}
            \left(\frac{\lambda}{10^4\mathrm{\AA}}\right)^{4/3}
            \left(\frac{X}{4}\right)^{4/3}
    \end{equation

    \begin{figure}
        \centering
        \includegraphics[width=2.5in]{../img/basic_geometry.png}
        \caption{Simple geometry of reverberation in the accretion disk. Some continuum emission is reprocessed before escaping toward the observer.}
        \label{fig:disk_reverb}
    \end{figure}


    \begin{figure}
        \centering
        \begin{minipage}{.475\textwidth}
            \centering
            \includegraphics[width=1\linewidth]{../img/tophat_timedomain.pdf}
            \captionof{figure}{Tophat transfer functions in the time domain show an average time lag of the reverberating curve and a constant distribution in time over an interval. An area of unity indicates no loss of signal in the response.}
            \label{fig:th_time}
        \end{minipage}
        \hfill
        \begin{minipage}{.475\textwidth}
            \centering
            \includegraphics[width=1\linewidth]{../img/tophat_freqdomain.pdf}
            \captionof{figure}{The time lags associated with each tophat function. Distinct features related to the average time lag are present (maximum, value of $\nu$ at steepest change), and complicated relationships with higher frequency waves can be noted.}
            \label{fig:th_freq}
        \end{minipage}
    \end{figure}



\section{Discussion}

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