# Space charge

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Fig. 1: flashes are discharges of space charges in the clouds have built

A space charge is a spatially distributed electrical charge in a non-conductive medium . It is caused by an excess of negative or positive charge carriers .

Space charges are important in rooms in which charged particles are supposed to move in a certain way. Space charge effects occur in many electronic components , e.g. B. in electron tubes , semiconductor diodes and transistors , and have a decisive influence on their electronic properties .

Space charge effects also play an important role in electron and ion sources and in particle accelerators . The electric fields associated with the space charges are often undesirable here, as they limit the achievable quality of important beam properties such as intensity or energy sharpness.

When designing gas and glow discharge tubes , space charges must be taken into account.

In nature, the movement of water droplets and ice crystals in thunderstorm clouds can create space charges that discharge in the form of lightning bolts .

## Space charges in electron tubes

Fig. 2: Vacuum diode with electron cloud
Fig. 3: Current-voltage characteristic of the vacuum diode. Dashed line: saturation currents for three different cathode temperatures

In electron tubes, space charges are generated by hot cathodes ( Edison-Richardson effect ). In order to avoid undesired interactions of the generated electrons with gas and to protect the hot cathode, the tubes are operated in a vacuum .

The space charge effects occurring in a tube are shown in Fig. 2 using the example of a simple tube diode . The electrons emitted by the hot cathode of the tube are drawn off to the anode . The electrons themselves generate electrical fields and thus considerably distort the field distribution caused by the anode voltage .

### Space charge-limited anode current

This can go so far that no field arrives at the point of origin of the electrons (the hot cathode), since it is already intercepted beforehand by the space charges. In this case, the anode current is no longer dependent on the number of electrons emitted by the cathode, but only on the anode voltage. This area of ​​the current-voltage characteristic is called the space charge-limited current (see Fig. 3).

#### calculation

The anode current or the current density can be calculated using Langmuir's or Langmuir-Child's space charge law : ${\ displaystyle I _ {\ mathrm {a}}}$${\ displaystyle j}$

${\ displaystyle I _ {\ mathrm {a}} = jS = {\ frac {4} {9}} \ varepsilon _ {0} {\ sqrt {\ frac {2e} {m _ {\ mathrm {e}}}} } {\ frac {S {U _ {\ mathrm {a}}} ^ {3/2}} {d ^ {2}}}}$.

With

• the irradiated anode area ${\ displaystyle S}$
• the vacuum dielectric constant ${\ displaystyle \ varepsilon _ {0}}$
• the elementary charge ${\ displaystyle e}$
• the electron mass ${\ displaystyle m _ {\ mathrm {e}}}$
• the anode voltage ${\ displaystyle U _ {\ mathrm {a}}}$
• the distance between cathode and anode.${\ displaystyle d}$

The equation applies under the following (only approximately valid) assumptions:

1. The electric field is homogeneous , i.e. H. the two electrodes are planar, parallel equipotential surfaces each of infinite extent
2. The electrons have zero speed when they exit the cathode
3. There are only electrons between the electrodes
4. The current is space charge limited
5. There is a steady state ; in particular, the anode voltage has not changed within the settling time .

### Saturation current

With high anode voltages, no additional anode current can be drawn by increasing the anode voltage. This saturation current is reached when the anode voltage is so high that it cannot be compensated by the space charge. In this case all electrons generated by the cathode are sucked out. The more electrons the cathode emits, the greater the saturation current (shown schematically in Fig. 3 by three dashed saturation characteristics for different cathode temperatures).

Between the cathode and anode there is a position-dependent density distribution of the electrons , which regulates itself so that the current density is the same everywhere. So z. For example, a decrease in the current density in a certain area immediately causes space charge to accumulate here, which shields the penetration of the anode voltage on the charge in front of it, so that the current density also decreases there until a state of equilibrium has been established.