Generals 1998 11

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1 Experimental Methods

[edit] Experimental Methods

[edit] Part a

The electric field is given by:

$\mathbf{E}=-\nabla\Phi-\frac{\partial\mathbf{A}}{\partial t}$

In the initial state, we may estimate $\nabla\Phi=V_{RF}/L$ , where $V R F$ is the instantaneous voltage across the RF power supply. The vector potential $\mathbf{A}$ can be found by recalling $\mathbf{B}=\hat{z}\mu_{0}I_{RF}N/L$ . Then:

$B_{z}=\hat{z}\cdot\nabla\times\mathbf{A}=\frac{1}{r}\frac{\partial}{\partial r}\left(rA_{\phi}\right)-\frac{1}{r}\frac{\partial A_{r}}{\partial\phi}$

Since we have azimuthal symmetry, the last term is zero, and so we get $A_{\theta}=\frac{1}{2}r\mu_{0}I_{RF}N/L$ . Then the total electric field is given by:

$\mathbf{E}=-\hat{z}\frac{V_{RF}}{L}-\hat{\theta}\frac{r\mu_{0}\omega I_{RF}N}{2L}$

[edit] Part b

Using the dispersion relation:

$c^{2}k^{2}=\omega^{2}-\omega_{pe}^{2}$

We get that $k=\left(\omega^{2}-\omega_{pe}^{2}\right)^{1/2}/c$ , and the electric field has the form $e i k x$ , so that the penetration distance in the radial direction will be $d=c/\sqrt{\omega_{pe}^{2}-\omega^{2}}$ . In the axial direction, the shielding is due to Debye shielding, and so the distance is like $\lambda_{d}=\sqrt{T/\left(4\pi n_{e}e^{2}\right)}$ .

[edit] Part c

Ions will flow out of the center at rate:

$v_{s}=\sqrt{\frac{T_{e}+3T_{i}}{m_{i}}}$

So that the ion loss rate is:

$\Gamma_{l}=2\pi rLn_{i}\sqrt{\frac{T_{e}+3T_{i}}{m_{i}}}$

The ion production rate is:

$\Gamma_{ion}=n_{e}\pi r^{2}L\sqrt{\frac{T_{e}}{m_{e}}}n_{He}K\left(T_{e}\right)$

So, if $T_{e}\gg T_{i}$ and $n e = n i$ , we set the fluxes to be equal:

$K\left(T_{e}\right)=\frac{2}{rn_{He}}\sqrt{\frac{m_{e}}{m_{i}}}$

[edit] Part d

The density could be measured using a microwave interferometer. By launching microwaves through the discharge tube and then turning on the RF source, one can find the change in path length due to the plasma by measuring the phase difference through interferometry. The path length will change like $L\sim\int ndx$ , the density integrated over the path length of the microwave.