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\begin{document}
\title{Measurements of Index of Refraction of the Whistler Wave Using Appleton's Equation}
\author{UCLA Physics 180E }
\date{\today}
\maketitle
\selectlanguage{english}
\begin{abstract}
\textbf{Abstract: } Using an antenna to generate whistler waves and a magnetic field probe to measure changes in magnetic field, index of refractions, $\eta$, were measured and compared to theoretical value. A simulation was also created using exact theory to compare with measured results. The measured results were found to be at a maximum $\pm 15 \%$ off of the theoretical value. Measurements were taken on two different occasions with the first taking axial measurements at different frequencies, and the second taking a plane of data at a set frequency.
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\end{abstract}%
\section{Introduction}
Radio emission from the ionosphere can produce a whistling sound in the audio frequency that can be heard[1]. The whistling sounds are described as groups of descending tones which are called the whistler mode. When lightning hits the southern hemisphere it produces a range of radio waves, some of which can travel along the earths magnetic field lines from the southern hemisphere to the northern hemisphere[1].These waves are called extraordinary waves .The extraordinary waves emit two solutions to the wave equation named L and R waves.The L and R refer to left and right hand circularly polarized. The waves that describe the whistling sound are R waves and they will be detected in the north and the different frequencies of these waves will travel at different speeds. For $\omega<\frac{\omega_{ce}}{2}$ the phase and group velocities increase with frequency, where $\omega_{ce}=\frac{eB}{m}$ is the electron cyclotron frequency[1]. Due to this, the lower frequencies will arrive at the northern hemisphere later than the higher frequencies will, causing the descending tone in the whistler mode. These R waves waves that travel along the magnetic field lines are called whistler waves and these waves can only propagate for $\omega<\frac{\omega_ce}{2}$. This lab seeks out to measure the dispersion relation of the whistler waves and to find the wave patterns theoretically and experimentally in the inductively coupled plasma device using Appletons equation.
\section{Theory}
Appleton's equation is given by
\begin{tiny}{\[\eta^2=1-\frac{\omega_{pe}^2}{\omega^2\left((1+\frac{i\nu}{\omega})-\frac{\omega_{ce}^2\sin^2\theta}{2\omega^2\left(1-\frac{\omega_{pe}^2}{\omega^2}\right)}\pm\sqrt{\frac{(\omega_{ce}^2\sin^2\theta)^2}{4(1-\frac{\omega_{pe}^2}{\omega^2})}+\frac{\omega_{ce}^2cos^2\theta}{\omega^2}}\right)}\]}
\end{tiny}
for an infinite plasma[2]. This describes the index of refraction for a whistler wave where $\eta^2=\left(\frac{kc}{\omega}\right)^2$. $\omega_{pe}$ is the plasma frequency, $\omega_{ce}$ is the electron cyclotron frequency, and $\nu$ is the rate of collisions in the plasma.The angle $\theta$ refers to the angle the waves make with respect to the background magnetic field $B_0$.The first assumption made is that the first experiment's waves are made with $\theta=0$ for waves parallel to $B_0$ and that damping is slight so that $\nu\approx0$ [2].Making these assumptions it is found that Appleton's equation reduces to \[n^2=1-\frac{\omega_{pe}^2}{\omega(\omega-\omega_{ce})}\].Noting that $\omega<\omega_{ce}$ then this results in an equation for index of refraction \[n^2\approx\frac{\omega_{pe}^2}{\omega\omega_{ce}}\]. Appleton's equation can be derived from the plasma force equation,Maxwell's equations, and using fourier analysis to show pertubations of the magnetic field are in the form $\vec{B_1}(\vec{r},t)=Be^{i(\vec{k} \cdot \vec{r}-\omega t)}$.The plasma force equation is written assuming that the particles are cold so that there are no pressure gradients and that the quantity $\vec{v}\cdot\nabla\approx0$ in the convective derivative, then the force equation is $m\frac{\partial\vec v}{\partial t}=q(\vec{E}+\vec{v}\times\vec{B})-m\nu\vec{v}$ for electrons.The two Maxwell's equations used are $\nabla\times\vec{E}=-\frac{\partial\vec{B}}{\partial{t}}$ and $\nabla\times\vec{B}=\mu_0\vec{J}+\mu_0\epsilon_0\frac{\partial\vec{E}}{\partial t}$.Linearize equations involving $\vec{B}$ and solve the force equations for $\frac{kc}{\omega}$.
\section{Experimental Setup}
The plasma was created by an inductively coupled RF source operating at a power of 120 W with $4\times 10^{-4}$ Torr Argon as the working gas. A voltage sweep was put across a Langmuir probe 70 cm from the RF source 1.3 ms into the afterglow and over a 101 $\mu s$ range. A magnetic field was applied to the device through the use of two sets of four coaxial magnets created through the use of 25.4A and 63.0A currents laid out around the device and resulting in a radially symmetric magnetic field through the device, averaging to 60 Gauss along the length. Whistler waves were generated in the plasma using a wave form generating antenna, set to 40 MHz for the experiment with data taken in a plane and set to 9 separate varying frequencies from 40 MHz to 120 MHz, each 10 MHz apart. A B-dot probe was used to measure the oscillations in the magnetic field due to the wave over a range of 70-90cm from the RF source axially and -15 degrees to 15 degrees radially for the set taken in a plane and varying axial lengths for the linear data set such that 10 data points were taken per wavelength of the measured wave. The B-dot probe used in this experiment consists of 3 sets of loops, each with a single wire sheathed by an insulating ceramic exterior. The loops are positioned orthonormal to one another with $\hat{z}$ along the central axis, and $\hat{x}$, $\hat{y}$, representing horizontal and vertical axes respectivily.When the loops detect a change in magnetic flux, will induce a current by Faraday's law in these conducting loops.The output voltage is recorded electronically and from this a measurement for $B(t)$ is obtained [3].
\section{Data Analysis}
The index of refraction was measured from finding the wavelengths from the data taken at given frequencies and angles, and then plugged into the equation for index of refraction
\[\eta=\frac{ck}{\omega}\]
where $\eta$ is index of refraction, $k$ is the wavenumber, and $\omega$ is the angular frequency. Since $k=\frac{2\pi}{\lambda}$ and $\omega=2\pi f$, then index of refraction can be expressed as
\[\eta=\frac{c}{\lambda f}\]
The results can be seen in figures 1 through 4. Wavelengths were found by finding the distance between two maximums or two minimums and then multiplying by the scale factor used when taking the measurement, which changed based on the measured frequency. As it can be seen in figure 2, the measured index of refraction concurred with the simulated data, confirming the validity of Appleton's equation over the frequency range measured. This method was limited at the low frequency that was used on the plane data, 40 MHz, by the angle at which the wavelength was measured. At angles above 45\textsuperscript{o}, full and even half wavelengths cannot be seen, which is reflected in figure 3. Up until 25\textsuperscript{o}, the measured index of refractions agreed with the theoretical values. Index of refractions of angles larger than 25\textsuperscript{o} led to some incongruities with the largest difference being at 30\textsuperscript{o} with 18\% error from expected value. Larger error margins could have probably come about from wavelengths measured at larger angles where only half of the wavelength was measured and doubled to acquire the full wavelength, potentially having the simulated data be within the error bars.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/Measured-whistler-waves-final/Measured-whistler-waves-final}
\caption{{Measured whistler waves excited at 40 MHz with a magnetic field of 60 Gauss along the central axis. The measured wavelength along the central axis ($\theta$ = 0\textsuperscript{o}) was 9.8 cm. Waves propagate from the source, shown on the left side. Brighter or darker spots indicate more positive or more negative magnitudes respectively. Dampening can be seen as the wave propagates down the chamber. Figure 4 shows a clearer example of dampening.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/index-freq/index-freq}
\caption{{Measured and simulated index of refraction plotted against frequency. As can be seen, the measured data matches the simulated data to within the error bars, confirming Appleton's theory.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/index-angle/index-angle}
\caption{{Measured and simulated index of refraction plotted against angle off central axis. The measured data concurs with Appleton's theory until 25\textsuperscript{o} within the error bars. The index of refraction of larger angles started to diverge off the simulation data. The measured angle size was limited by the frequency and method used to determine the wavelengths.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/whistler-decay/whistler-decay}
\caption{{Time evolution of 120 MHz wave from t=426 $\mu s$ to t=556 $\mu s$. Starting from Position = 0 cm, the wave propagates and dampens as it moves into the chamber. The red line shows the maximum amplitude of the wave at a particular time.%
}}
\end{center}
\end{figure}
Simulations were made using IDL software and are as shown here. The simulations were generated using Appleton's equations. The simulation generated wavefronts at different angles and summed them to create an interference pattern. Several simulations are shown with different different frequency and magnetic field settings. Figure 5 illustrates simulated whistler waves under the conditions used during the plane data measurements. Comparing to figure 1, it can be seen that the simulated data and the measured data agree with one another. There are some dissimilarities with the rate of decay, but otherwise the measured data matches quite well with the simulated data. The assumptions that were made when creating the simulation were that the magnetic field was constant along the radial direction, as well as using a constant density ($3.5 \times 10^{11} cm^{-3}$). Due to problems when taking density measurements using the langmuir probe (not all channels were saved), that source could not be used. However, the used density was reasonable when compared to previous labs. Also that the plasma was infinite (not confined in a chamber), and that the plasma was cold, about 1.5 eV, so low ion movement. The code used to generate the simulation is attached to this report.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/40MHz60G/40MHz60G}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/60MHz80G/60MHz80G}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/100MHz60G1/100MHz60G1}
\end{center}
\end{figure}
\section{Discussion of Results}
The measured data confirms Appleton's equation as a good theory for whistler waves in a plasma, with a majority of the theoretical data falling between the error bars. The data was mainly limited by the method in which the wavelengths were acquired. If the data set taken covered a larger set of radial positions, more wavelengths at larger angles could have been measured.If allowed to do the experiment, proper density measurements would have been taken to allow for greater precision for the index of refraction.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/index-vs-freq/index-vs-freq}
\caption{{Frequency vs index of refraction for 40 G, 60 G, and 80 G magnetic fields. The effect of the magnetic field on the index of refraction can be seen with cutoff occurring at the electron cyclotron frequency,~ $f_{ce}=\frac{qB}{2\pi ~ m_e}$.%
}}
\end{center}
\end{figure}
\section{Appendix}
The electron cyclotron frequency is given by \[\omega_{ce}=\frac{eB}{m_e}\] where $e$ is the elementary charge, $B$ is the magnetic field and $m_e$ is the electron mass.
The plasma frequency is given by \[\omega_{pe}=\sqrt{\frac{ne^2}{m\epsilon_0}}\] where $n$ is the plasma density,and $\epsilon_0$ is the permittivity of free space.
The voltage measured by the bdot probe is given by \[V_{measured}=N_{loop}A\frac{\partial\vec{B}_{norm}}{\partial t}\] where $N$ is the number of loops of a side of the bdot probe, $A$ is the area of a face of the probe,and $\frac{\partial\vec{B}_{norm}}{\partial t}$ is the change in magnetic field normal to the face of the probe.
The phase velocity is \[v_{Phase}=\frac{\omega}{k}\] where $\omega$ is the wave frequency and $k$ is the wave number.
\section{Reference}
[1]-Chen, Francis F. "Chapter 4: Waves in Plasmas." Introduction to Plasma Physics. Boston, MA: Springer US, 1995. 132. Print.\\
[2]-Gekelman, Walter. Week 4 notes, 180E Lab Course, UCLA, Spring 2013\\
[3]-"PEPL: Plasma Diagnostics: B-dot Probe." Plasma Diagnostics. The University of Michigan, Department of Aerospace Engineering, 06 July 2010. Web. 18 Mar. 2014.
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