[플라즈마 물리][Plasma Physics]CH1 Introduction - Temperature

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Temperature

The one-dimensional Maxwellian distribution is given by

\[\begin{aligned} f(v)=A e^{-\frac{mv^2}{2K_BT}}\end{aligned}\]

Unlike normal distribution gaussian equation can have a form of

\[\begin{aligned} \label{eq_4} f(x) = \frac{n}{\sqrt{2\pi}\sigma}e^{-\frac{(v-v_{\mu})^2}{2\sigma^2}}\end{aligned}\]

Where \(n\) is the number density. \(fdv\) is the number of particles per [\(m^3\)] with velocity between \(v\) and \(v+dv\), \(\frac{1}{2}mv^2\) is the kinetic energy, and \(K_B\) is the Boltzmann’s constant. The density \(n\), or number of particles per [\(m^3\)], is given by

\[\begin{aligned} n=\int_{-\infty}^{\infty}f(v)dv\end{aligned}\]

so that the constant \(A\) is found to be

\[\begin{aligned} A=n\sqrt{\frac{m}{2\pi K_BT}}\end{aligned}\]

Where \[\begin{aligned} \int_{-\infty}^{\infty}e^{-ax^2}dx = \sqrt{\frac{\pi}{a}} \end{aligned}\] is used.

\(\bullet\) meaning of T = Distribution of the particles

The width of the distribution is characterized by the constant \(T\). By computing the average kinetic energy of particles in the distribution, we can see the exact meaning of \(T\).
\[\begin{aligned} \label{eq_5} E_{av}=\frac{\int_{-\infty}^{\infty}\frac{1}{2}mu^2 f(u) du}{\int_{-\infty}^{\infty} f(u) du}\end{aligned}\]

Defining \(v_{th}=\sqrt{\frac{2K_BT}{m}}\) and \(y=\frac{u}{v_{th}}\), 1-D Maxwellian distribution can be written as

\[\begin{aligned} f(u)=Ae^{-\frac{u^2}{v_{th}^2}}\end{aligned}\]

By substitution average kinetic energy becomes

\[\begin{aligned} E_{av}&=\frac{\frac{1}{2}mAv_{th}^3 \int_{-\infty}^{\infty} e^{-y^2}y^2 dy}{Av_{th}\int_{-\infty}^{\infty} e^{-y^2}dy}\\ &=\frac{\frac{1}{2}mAv_{th}^3 \frac{1}{2}}{A v_{th}}=\frac{1}{4}mv_{th}^2=\frac{1}{2}K_BT\end{aligned}\]

Thus the average kinetic energy is \(\frac{1}{2}K_BT\).
In three dimensions,

\[\begin{aligned} f(u,v,w)=n\left( \frac{m}{2\pi K_BT } \right)^{\frac{3}{2}}e^{-\frac{\frac{1}{2}m\left( u^2 + v^2 + w^2 \right)}{K_BT}}\end{aligned}\]

Using similar calculation we get

\[\begin{aligned} E_{av}=\frac{3}{2}KT\end{aligned}\]

The general result is that \(E_{av}\) equals \(\frac{1}{2}K_BT\) per degree of freedom.

\(\bullet\) Since \(T\) and \(E_{av}\) are so closely related, it is customary in plasma physics to give temperatures in units of energy.

\(\bullet\) To avoid confusion, it is not \(E_{av}\) but the energy corresponding to \(KT\) that is used to denote the temperature.

\(\bullet\) For \(KT=1eV= 1.6\times 10^{-19}[J]\)

\[\begin{aligned} T=\frac{1.6 \times 10^{-19}}{1.38 \times 10^{-23}}=11600\end{aligned}\]

Thus the conversion factor is

\[\begin{aligned} 1eV = 11,600 ^{\circ}K \end{aligned}\]

\(\bullet\) By a \(2eV\) plasma we mean that \(KT=2eV\), or \(E_{av}=3eV\) in three dimensions.

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