Cyclotron resonance

Cyclotron resonance describes the resonant absorption of electromagnetic waves by charged particles (e.g. by free electrons or electrons in a solid body) that are in a constant magnetic field . The name is derived from the cyclotron , a particle accelerator; there the particles move with the cyclotron frequency on a circular path.

In plasmas , the cyclotron resonance of the electrons can be used to couple energy into the plasma, i.e. to increase the kinetic energy of the electrons and thus to heat the plasma (electron cyclotron resonance, ECR). This method is used in EZR ion sources.

In attempts to technical fusion a very high temperature of the ions (different hydrogen - isotopes ) needed. An additional heating of the ions can be achieved, among other things, by ion cyclotron resonance heating (IZR).

The investigation of the cyclotron resonance of the electrons (or " holes ") of a material is also a method of solid-state physics to determine the effective mass of the charge carriers.

High-performance microwave generators ( gyrotrons and magnetrons ) work with the cyclotron resonance of free electrons.

The cyclotron resonance of charged particles in a Penning trap can be used to determine their relationship between mass and charge or, if the charge is known, its mass.

Theoretical basis

Without an electric field, only the Lorentz force acts on an electron (charge - e ) with the velocity v in the magnetic field B.

${\displaystyle \mathbf {F} =-{\frac {e}{c}}\ \mathbf {v} \times \mathbf {B} .}$

A free electron follows a circular orbit or a helix; the cyclotron frequency is the frequency of the electron's orbit.

In the solid state, the speed is determined by the dispersion relation , ie by the energy and the wave vector given ${\displaystyle E}$ ${\displaystyle \mathbf {k} }$

${\displaystyle \mathbf {F} =-{\frac {e}{c}}\ \mathbf {v} (\mathbf {k} )\times \mathbf {B} .}$

${\displaystyle \mathbf {\dot {k}} =-{\frac {e}{c\hbar ^{2}}}{\frac {\partial E(\mathbf {k} )}{\partial \mathbf {k} }}\times \mathbf {B} .}$

The electron experiences a force that is perpendicular to the magnetic field B and, in k space, perpendicular to the gradient of the E ( k ) surface. It thus moves on a surface of constant energy. This can of course also be concluded for reasons of energy conservation, since a temporally constant magnetic field does not cause any change in the energy of the deflected particle. In the solid, an electron remains on the Fermi surface during its movement .

Assuming a free electron gas, this results in the classic cyclotron frequency , at which each electron has the same orbital time. However, this is not the case in solids. In order to obtain a generally valid expression for the orbital frequency, the mass of the particle must therefore be replaced by the effective mass of the particle. This results in ${\displaystyle m^{*}}$

${\displaystyle \omega _{\mathrm {C} }={\frac {eB}{m^{*}}}}$

with
B - magnetic flux density - cyclotron frequency or rotation frequency m * - effective particle mass (here: effective electron mass) e - elementary charge
${\displaystyle \omega _{C}}$

Cyclotron resonance in solid state physics

A crystal sample that is in a static magnetic field B at low temperatures (approx. 4 Kelvin) is irradiated with radio waves. The radio waves accelerate the charge carriers, which are deflected into spiral paths by the magnetic field. The absorption of the waves becomes maximum when the frequency of the radio wave is equal to or a multiple of the cyclotron frequency :

${\displaystyle \omega _{\mathrm {em} }=p\cdot \omega _{\mathrm {C} }\qquad p\in \mathbb {N} }$

With a known magnetic field strength, the effective mass of the charge carrier can be read off. ${\displaystyle m^{*}}$

In the case of a semiconductor, the sample must also be irradiated with light, the photons of which have sufficient energy to lift the electrons into the conduction band.

See also

Cyclotron , cyclotron frequency , Penning trap , gyrotron , cyclotron resonance heating

literature

• Bernard Sapoval, Claudine Hermann: Physics of Semiconductors , Springer Verlag, 2005. ISBN 0387406301
• Konrad Kopitzki: Introduction to Solid State Physics , Teubner Verlag, 2004. ISBN 3519430835

Individual evidence

1. ^ NW Ashcroft, ND Mermin: Solid state physics (College Edition). Harcourt College Publishers 1976, 214