[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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