Space charge law

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The space charge law describes the relationship between electrical current strength and voltage of an evacuated two-electrode arrangement in space-charge-limited operation (e.g. tube diode with hot cathode ). Because of the early work of Clement Dexter Child and Irving Langmuir on discharge phenomena, the space charge law is sometimes called Langmuir-Child law . The relationship between the current density in an evacuated two-electrode arrangement and the electrical voltage is expressed in the Schottky equation .

Brief description

It applies

${\displaystyle I=KU^{\frac {3}{2}}}$,

where and denote anode current and voltage, respectively. The factor , the so-called space charge constant or perveance of the diode, is a variable that only depends on the shape of the electrode arrangement and is thus a tube constant. ${\displaystyle I}$ ${\displaystyle U}$ ${\displaystyle K}$

The space charge law applies to U  > 0 V. For U  <0 V applies the starting current law . The space charge law loses its validity if the cathode yield is too low or the anode voltage is too high.

Derivation

Consider two arbitrarily shaped electrodes in a vacuum, one of which (heated, arbitrarily productive cathode) was placed on the potential (first boundary condition) and the other (anode) on the potential (anode voltage, second boundary condition). For physical reasons, the associated discharge problem must be clearly solvable. If the solution is for the anode voltage , then, according to the laws of magnetohydrodynamics , neglecting the exit velocity and the relativistic mass increase of the electrons, applies to the remaining fields ${\displaystyle \phi =0}$ ${\displaystyle \phi =U}$ ${\displaystyle \phi _{0}(\mathbf {r} )}$ ${\displaystyle U=U_{0}}$

${\displaystyle v_{0}={\sqrt {2\eta _{0}\phi _{0}}},\quad \rho _{0}=-\varepsilon _{0}\Delta \phi _{0},\quad J_{0}=-\rho _{0}v_{0},\quad I_{0}=\int J_{0}\mathrm {d} A,}$

integrating over the entire surface of the anode (excluding connecting wire). Obviously there is a solution for with any choice of the non-negative number , and holds for the other fields ${\displaystyle \phi =a\phi _{0}}$ ${\displaystyle U=aU_{0}}$ ${\displaystyle a}$

${\displaystyle v={\sqrt {2\eta _{0}\phi }}={\sqrt {a}}v_{0},\quad \rho =-\varepsilon _{0}\Delta \phi =a\rho _{0},\quad J=-\rho v=a^{\frac {3}{2}}J_{0},\quad I=\int J\mathrm {d} A=a^{\frac {3}{2}}I_{0}=\left({\frac {U}{U_{0}}}\right)^{\frac {3}{2}}I_{0}={\frac {I_{0}}{U_{0}^{\frac {3}{2}}}}U^{\frac {3}{2}}.}$

Since unambiguous solvability was assumed, not only one but the solution of the discharge problem is given for. Because any choice can be made, all the solutions to the problem are available as soon as only one is known. Now, at a given voltage of the arrangement certainly depends on the shape, therefore, is a constant of the arrangement, and for the anode current thus applies ${\displaystyle \phi =a\phi _{0}}$ ${\displaystyle U=aU_{0}}$ ${\displaystyle a}$ ${\displaystyle I_{0}}$ ${\displaystyle U_{0}}$ ${\displaystyle I_{0}/U_{0}^{3/2}}$

${\displaystyle I=KU^{\frac {3}{2}},\quad K={\frac {I_{0}}{U_{0}^{\frac {3}{2}}}}.}$

The space charge law obviously implies an infinitely high cathode yield, because it follows from it . ${\displaystyle U\to \infty }$ ${\displaystyle I\to \infty }$

The constant K

The constant depends on the anode surface and the distance between cathode and anode and the shape of cathode and anode. Barkhausen assumes a thin cathode wire that stands in the middle of an anode tube with a length and a radius . is the speed of the electron after flying through a voltage . is the elementary charge , the electron mass and the electric field constant . ${\displaystyle K}$ ${\displaystyle F}$ ${\displaystyle r}$ ${\displaystyle l}$ ${\displaystyle r}$ ${\displaystyle v_{1}}$ ${\displaystyle U}$ ${\displaystyle e}$ ${\displaystyle m_{\mathrm {e} }}$ ${\displaystyle \varepsilon _{0}}$

${\displaystyle F=2\pi rl}$

${\displaystyle v_{1}={\sqrt {\frac {2eU}{m_{\mathrm {e} }}}}}$

${\displaystyle K={\frac {4}{9}}\varepsilon _{0}{\frac {v_{1}}{\sqrt {U}}}{\frac {F}{r^{2}}}}$

${\displaystyle K={\frac {4}{9}}\varepsilon _{0}{\sqrt {\frac {2e}{m_{\mathrm {e} }}}}{\frac {F}{r^{2}}}}$

literature

• H. Barkhausen: Textbook of electron tubes, 1st volume general basics. 11th edition. S. Hirzel Verlag, Leipzig 1965, p. 46ff.

Individual evidence

1. ↑ Clement Dexter Child: Discharge From Hot CaO . In: Physical Review (Series I) . tape 32 , no. 5 , 1911, pp. 492-511 , doi : 10.1103 / PhysRevSeriesI.32.492 ( PDF [accessed February 5, 2010]). PDF ( Memento of the original from June 23, 2007 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.  
2. ^ Irving Langmuir: The Effect of Space Charge and Residual Gases on Thermionic Currents in High Vacuum . In: Physical Review (Series II) . tape 2 , no. 6 , 1913, pp. 450-486 , doi : 10.1103 / PhysRev.2.450 .