Physics/Option Modules/Turning Points in Physics/Thermionic Emission
Physics · Turning Points in Physics · A2 Level

Thermionic Emission

Cathode rays don't appear from nowhere — the cathode has to actually release free electrons before an electric field can accelerate them into a beam. Heating the cathode is how that release happens, and the same simple energy equation that describes it reappears again and again throughout this module.

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The work function: why electrons don't just leave on their own

A metal contains a large number of free (conduction) electrons, able to move relatively freely through the metal's structure — this is precisely what makes metals good electrical conductors. But being free to move within the metal is not the same as being free to leave it. At the surface, an electron trying to escape into the surrounding vacuum or air is pulled back by the attraction of the positive metal ions left behind, so escaping requires doing a certain minimum amount of work against that attraction.

This minimum energy is called the work function of the metal, usually given the symbol $\phi$. At room temperature, the conduction electrons in a typical metal simply don't have enough kinetic energy to overcome it — they rattle around inside the metal, but almost none escape. If the metal is heated, however, its conduction electrons gain extra thermal kinetic energy, and the proportion of electrons with enough energy to overcome the work function rises sharply. Once heated enough, a small but steady stream of electrons boils off the surface — a process called thermionic emission ("thermionic" simply meaning "electrons released by heat").

Producing the cathode ray beam

In a cathode ray tube, the cathode is heated either directly (by passing a current through a wire filament that acts as the cathode itself) or indirectly (via a separate heater element positioned close to, but not part of, the cathode). Once thermionic emission releases electrons from the cathode's surface, the potential difference across the tube accelerates them toward the anode — this is what forms the beam described in the Cathode Rays lesson.

Accelerating an electron: $eV = \tfrac{1}{2}mv^2$

Once an electron has been freed from the cathode by thermionic emission, it still starts out essentially at rest (its thermal speed is tiny compared to the speeds reached once it's accelerated by the tube's field). The potential difference $V$ between cathode and anode then does work on the electron as it crosses the gap, and — since no other significant forces act on it in the vacuum of the tube — all of that work converts directly into the electron's kinetic energy.

DerivationKinetic energy gained by an accelerated electron
1
The work done by an electric field on a charge $e$ moved through a potential difference $V$ is, by the definition of potential difference itself, $W = eV$.
2
The electron starts from rest, so all of this work goes into giving it kinetic energy — none is lost to friction or radiation in the vacuum of the tube, and none was already present as kinetic energy before the acceleration.
3
By the work-energy principle, the work done equals the kinetic energy gained: $W = \Delta E_k = \tfrac{1}{2}mv^2 - 0$.
4
Combining these two expressions for the same quantity of work:
$eV = \tfrac{1}{2}mv^2$
This equation converts an accelerating voltage directly into a beam speed (or vice versa), and it underlies every electron gun used throughout this module — including in the electron-diffraction and electron-microscope calculations that build on it later.
▪ derived
Worked exampleFinding an electron's speed from the accelerating voltage
1
An electron is accelerated from rest through a potential difference of $2500\,\text{V}$. Find its speed as it reaches the anode. (Electron charge $e = 1.6\times10^{-19}\,\text{C}$, electron mass $m = 9.11\times10^{-31}\,\text{kg}$.)
2
Rearrange $eV = \tfrac{1}{2}mv^2$ for $v$:
$v = \sqrt{\dfrac{2eV}{m}}$
3
Substitute the values: $v = \sqrt{\dfrac{2 \times 1.6\times10^{-19} \times 2500}{9.11\times10^{-31}}}$
4
$v = \sqrt{\dfrac{8.0\times10^{-16}}{9.11\times10^{-31}}} = \sqrt{8.78\times10^{14}}$
5
$v \approx 2.96\times10^{7}\,\text{m}\,\text{s}^{-1}$
This is about 10% of the speed of light — fast, but still well within the range where the non-relativistic equation $eV=\tfrac12mv^2$ gives an accurate answer. At much higher accelerating voltages this equation stops being accurate and a relativistic treatment is needed instead.
▪ answer

Where else thermionic emission is used

Thermionic emission is not a one-off trick specific to nineteenth-century laboratory apparatus — heating a cathode remains the standard way to produce a controllable, free beam of electrons in a vacuum, and the same principle appears throughout electron-beam technology.

  • Cathode ray tube (CRT) displays. Older televisions and oscilloscopes use a heated cathode (an "electron gun") to produce the electron beam that's steered across a fluorescent screen to build up an image.
  • X-ray tubes. A heated filament releases electrons by thermionic emission, which are then accelerated through a large potential difference and slammed into a metal target to produce X-rays — covered in detail in the Medical Physics option module.
  • Vacuum-tube electronics. Before the transistor, thermionic valves (vacuum tubes) used a heated cathode as the source of electrons for amplifying and switching electrical signals, and specialist high-power valves still use the same principle today.
Two separate ideas, easy to conflate

Thermionic emission (heating the cathode to free electrons in the first place) and the acceleration equation $eV=\tfrac12mv^2$ (speeding those free electrons up across a potential difference) are two separate stages of producing a fast electron beam, and exam questions often test both stages within the same scenario. Emission gets the electrons out of the metal; acceleration is what then gives them useful kinetic energy.

Key takeaways

  • A metal's work function $\phi$ is the minimum energy a conduction electron needs to escape the metal's surface.
  • Heating a cathode gives its conduction electrons extra thermal kinetic energy; once enough electrons exceed the work function, they escape — this is thermionic emission.
  • An electron accelerated from rest through a potential difference $V$ gains kinetic energy $eV$, giving $eV = \tfrac{1}{2}mv^2$.
  • This equation converts an accelerating voltage directly into a beam speed, and reappears throughout the module wherever an electron gun is involved.
  • Thermionic emission is used in CRT displays, X-ray tubes, and vacuum-tube electronics generally, wherever a free electron beam is needed.
  • Emission (freeing electrons from the metal) and acceleration (speeding them up in a field) are distinct physical stages, both usually needed to produce a useful electron beam.
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