Oscillations #
Collision Frequencies #
Gases have three degrees of freedom to oscillate, but plasmas have four: the electron fluid adds an additional parameter.
One way to see how the electron fluid oscillates, take a region of plasma and displace the electrons by a distance \( x \) and release. Then, watch how the electrons oscillate.
Gauss’s law gives the resulting electric field (same as a parallel-plate capacitor.
\[E = \frac{n e x}{\epsilon_0}\]The restoring force is therefore
\[F = \frac{x n e^2}{\epsilon_0}\]We know forces of that form. When released, it will undergo simple harmonic motion
\[\omega = \sqrt{\frac{K}{m}} \qquad K = \frac{ ne^2}{\epsilon_0}\]We define the plasma frequency as
\[\omega_{pe} = \sqrt{ \frac{ne^2}{m_e \epsilon_0}}\]If we take the displacement out to \( \lambda_D \) then the peak velocity is the electron thermal speed \( v_e \)
\[\omega_{pe} \lambda_D = v_e\]Taking a look back at the collision frequencies, the electron-electron collision frequency \( \nu_{ee} \) (associated with the energy transfer) is close to the electron-ion collision frequency \( \nu_{ei} \) associated with momentum transfer, and both are about
\[\nu_{ee} \approx \nu_{ei} \approx \frac{v_e}{\lambda_D \Lambda} = \frac{\omega_{pe}}{\Lambda}\]So the electron-electron collision frequency will be much less than the plasma frequency if \( \Lambda \gg 1 \)
\[\Lambda \gg 1 \rightarrow \nu_{ee} \ll \omega_{pe}\]Radiation #
Cyclotron radiation #
As an electron moves through a magnetic field, it takes a helical path. The acceleration is
\[\vec a = \frac{v^2}{r} = v_{th} \omega _c\]Only the component of \( \vec a \) perpendicular to \( r \) can cause radiation in the \( r \) direction.
\( \theta \)-pinch #
Called theta-pinch because current \( I \) is in the \( \theta \) direction. The magnetic field is constant within the plasma, and constant outside the plasma (inside the device).
\[\frac{B_0 ^2}{2 \mu_0} \]Z-pinch #
Talk about a Z-pinch that’s stationary and compressing. Assume T is a constant (burning through). The radiated power is then
\[P_{rad} \propto n^2\]The ohmic heating goes like \( j^2 \)
\[P_{ohmic} \propto j^2\]So the key parameter in the ratio is \( j/n \).
There is no \( B_{\parallel} \) so we need to balance the pressure as well.
Consider ohmically heating a pure Z-pinch like FRC or ZaP.
First, assume that the power radiated per unit volume goes like
\[n^2 f (T) = P/V\]The dependence is true for a fixed impurity fraction for line radiation in steady state, and for Bremsstrahlung.
Ignore \( \ln \Lambda \) dependency in ohmic heating so
\[P/V = \eta(T) j^2\]The pressure balance demands
\[\frac{B^2}{2 \mu_0} = p = 2 nk T_0\]For Bremsstrahlung temperature and density dependence are
\[P_{rad} = c_1 n^2 T^{1/2}\] \[P_{ohm} = c_2 j^2 T^{-3/2}\]Pressure balance
\[n T \propto B^2 \propto j^2 r^2\] \[nT = c_3 j^2 r^2\] \[T = c_e \frac{j^2 r^2}{n}\]So if we’re going to get any heating, we need the ohmic heating to be greater than the radiation
\[c_2 j^2 T^{-3/2} \geq c_1 n^2 T^{1/2}\]using
\[T = c_3 \frac{j^2 r^2}{n}\] \[\rightarrow \frac{c_2}{c_1 c_3 ^2} > j^2 r^4 \sim I^2\]Density cancels out, and we’re left with a current term (\( I^2 \)) called the Pease current
\[I \leq \approx 1 MA\]Z-pinch radiatively collapses above 1MA for Bremsstralung radiation. Counter-intuitively, the current must be less than the Pease current in order to continue heating. There is an even lower current for line radiation going up through the burn-through curve.
Skin Depths in Plasma #
Two types of skin depths in plasma: collisional and collisionless
\[\delta = \sqrt{ \frac{ 2}{\mu_0 \omega \sigma}} \quad \text{Collisional}\] \[\delta = \frac{c}{\omega_{pe}} \quad \text{Electron collisionless skin depth}\] \[\delta = \frac{c}{\omega_{pi}} \quad \text{Ion collisionless skin depth}\]Collisionless skin depth is due to electron inertia, which allows the field to penetrate even if it is very hot and a good conductor. What is magnitude? Look at E production and B shielding
\[\curl E = - \pdv{B}{t}\] \[\curl B = \mu_0 j + \mu_0 \epsilon_0 \pdv{E}{t}\]Electrons accelerate with E
\[m \pdv{v}{t} = E e\]Doing a wave analysis assuming spatial solutions that decay off as \( 1 / \delta \) and oscillate with frequency \( \omega \), we have
\[\frac{1}{\delta} \sim \grad\] \[\omega \sim \pdv{}{t}\] \[\frac{E}{\delta} = - \omega B\] \[m \omega v = - e E\] \[j = - n e v\] \[\frac{B}{\delta} = \mu_0 (- n e v) + \mu_0 \epsilon_0 \omega E = \frac{E}{\delta ^2 \omega}\] \[\frac{E}{\delta ^2 \omega} = \frac{\mu_0 n e^2}{m} \frac{E}{\omega} + \mu_0 \epsilon_0 \omega E\]That’s the dispersion relation for our wave solutions. Plugging in \( c \)
\[- \frac{c^2}{\delta ^2} \frac{1}{\omega} = \frac{n e^2}{\epsilon_0 m }\frac{1}{\omega} + \omega\]That lets us identify the electron plasma frequency