UDC 537.52

ION CURRENT TO A CYLINDRICAL PROBE OF FINITE LENGTH IN PLASMA

I. A. Tikhomirov, V. V. Tikhomirov, V. Ya. Fedyanin, V. I. Shishkovskii

(Presented by the scientific seminar of the Physics and Engineering Faculty)

The ion current to a cylindrical probe of finite length is considered for the case \(\lambda > a\). For “thin” cylindrical probes of finite length, formulas are proposed that relate the ion current to the probe to the ion concentration in the undisturbed plasma and to the dimensions of the probe.
Bibliography: 2.

Considering the motion of an ion in the space-charge layer surrounding the probe, the authors of [1] introduce two criteria:

a) a criterion for the applicability of the concept of limitation motion,

b) a criterion for the applicability of the theory of a cylindrical probe of infinite length.

The probes used in practice often do not satisfy these conditions.

In the present work, formulas are derived for the ion current to a cylindrical probe in the absence of limitation motion, which make it possible to take into account the finite length of the probes.

Let us represent the probe in the form of a prolate ellipsoid of revolution and introduce an ellipsoidal coordinate system by the formulas:

\[ X = a \cos U \,\operatorname{sh}\Theta \cos \varphi \qquad -\frac{\pi}{2} \le U \le \frac{\pi}{2}; \]

\[ Y = a \cos U \,\operatorname{sh}\Theta \sin \varphi \qquad 0 \le \Theta < \infty; \tag{1} \]

\[ Z = a \sin U \,\operatorname{ch}\Theta \qquad 0 \le \varphi < 2\pi . \]

We shall assume that the potential in the probe field depends on \(\Theta\), i.e. \(\eta = \eta(\Theta)\).

In the absence of collisions during the motion of an ion in the probe field, the laws of conservation of energy and angular momentum hold:

\[ \frac{M_i v^2}{2} = \frac{M_i v_0^2}{2} + e\eta_a, \tag{2} \]

\[ M_z = M_i v_0 a \cos U \operatorname{sh}\Theta \sin \varphi, \tag{3} \]

where \(a\) is the focal distance of the ellipsoid; \(v_0\) is the ion velocity in the undisturbed plasma; \(M_i\) is the ion mass; \(\eta_a\) is the probe potential.

In the absence of limitation motion, the collecting surface is the surface of the probe; therefore the greatest moment,

which ions that reach the probe may have \((\cos U=1,\ \sin\varphi=1)\), is equal to

\[ (M_z)_{\max}=M_i v a \operatorname{sh}\Theta_a=M_i v a, \tag{4} \]

where \(a\) is the minor semiaxis of the probe.

Taking into account the law of conservation of energy (2), we obtain

\[ (M_z)_{\max}=M_i v_0 a\sqrt{1+\frac{2e\eta_a}{M_i v_0^2}} . \tag{5} \]

Let us find the equation of the largest ellipsoid such that ions falling onto it will reach the probe,

\[ M_i v_0 a\sqrt{1+\frac{2e\eta_a}{M_i v_0^2}}=M_i v_0 a\,\operatorname{sh}\Theta'. \tag{6} \]

Hence we obtain

\[ a\,\operatorname{sh}\Theta'=a\sqrt{1+\frac{2e\eta_a}{M_i v_0^2}} . \tag{7} \]

The parameters of the ellipsoid are given by the expressions:

\[ a_1=a\,\operatorname{sh}\Theta'=a\sqrt{1+\frac{2e\eta_a}{M_i v_0^2}}, \]

\[ c_1=a\,\operatorname{ch}\Theta'=a\sqrt{1+\frac{a^2}{\alpha^2}\left(1+\frac{2e\eta_a}{M_i v_0^2}\right)} . \tag{8} \]

Here \(c\) is the major semiaxis of the probe.

The surface area is

\[ S=2\pi a_1 c_1\left(\sqrt{1-\varepsilon^2}+\frac{\arcsin\varepsilon}{\varepsilon}\right), \tag{9} \]

where

\[ \varepsilon=\frac{c_1^2-a_1^2}{c_1^2} = \frac{1-\dfrac{a^2}{\alpha^2}} {1+\dfrac{a^2}{\alpha^2}\dfrac{2e\eta_a}{M_i v_0^2}} . \]

To simplify the calculation of the ion current to the probe, we shall use the assumption of [2], i.e., we shall replace the actual velocity distribution of ions in the plasma by a monoenergetic distribution. As applied to the case under consideration, for \(f(v_0)\) we shall have

\[ f(v_0)=\frac{M_i}{2\pi}\delta\left(\frac{M_i v_0^2}{2}-E_0\right). \tag{10} \]

Then the ion current per unit length of the cylindrical probe will be equal to

\[ i=en_0\int_0^\infty S v_0^2\frac{M_i}{2\pi}\delta\left(\frac{M_i v_0^2}{2}-E_0\right)\,dv_0 . \tag{11} \]

Here \(E_0\) and \(n_0\) are the energy and concentration of ions in the undisturbed plasma.

Let us introduce the length of the cylindrical probe \(l = c \dfrac{\pi}{2}\); then, under various conditions, we obtain the following expressions for the ion current to a “thin” cylindrical probe of finite length:

1) for \(\dfrac{a^{2}}{c^{2}} \ll 1,\ \dfrac{a^{2}}{c^{2}} \cdot \dfrac{2e\eta_a}{M_i v_0^{2}} > 1\),

\[ i = 2alen_0 \sqrt{\frac{2e\eta_a}{M_i}} \sqrt{ 1 + \frac{\pi^2}{4}\frac{a^2}{l^2}\frac{e\eta_a}{E_0} } \times \left( 1 - \frac{1}{6} \frac{ 1 - \dfrac{\pi^2}{2}\dfrac{a^2}{l^2} }{ 1 + \dfrac{\pi^2}{4}\dfrac{a^2}{l^2}\dfrac{e\eta_a}{E_0} } \right), \tag{12} \]

2) for \(\dfrac{a^{2}}{c^{2}} \ll 1,\ \dfrac{a^{2}}{c^{2}} \dfrac{2e\eta_a}{M_i v_0^{2}} < 1\),

\[ i = 2alen_0 \sqrt{\frac{2e\eta_a}{M_i}} \left( 1 + \frac{3\pi^2}{8}\frac{a^2}{l^2}\frac{e\eta_a}{E_0} \right). \tag{13} \]

The formulas obtained make it possible to determine the concentration of charged particles from the ion part of the characteristic in the case of “thin” probes.

LITERATURE

1. Yu. M. Kagan, V. I. Perel. ZhTF, 39, no. 12, 2238, (1969).
2. J. Bernstein, J. Rabinowitz. Phys. Fluids, 2, 12 (1959).