Editing Electron cyclotron resonance (section)

{{Short description|Phenomenon observed in physics}}
{{Use dmy dates|date=July 2023}}
'''Electron cyclotron resonance''' ('''ECR''') is a phenomenon observed in [[plasma physics]], [[condensed matter physics]], and [[accelerator physics]]. It happens when the frequency of incident radiation coincides with the natural frequency of rotation of electrons in magnetic fields. A free [[electron]] in a static and uniform [[magnetic field]] will move in a circle due to the [[Lorentz force]].  The circular motion may be superimposed with a uniform axial motion, resulting in a [[helix]], or with a uniform motion perpendicular to the field (e.g., in the presence of an electrical or gravitational field) resulting in a [[cycloid]].  The [[angular frequency]] (''ω''&nbsp;= 2''π''[[Frequency|''f'']] ) of this ''[[cyclotron motion]]'' for a given magnetic field strength ''B'' is given (in [[SI]] units)<ref>In SI units, the elementary charge ''e'' has the value {{val|1.602|e=−19|ul=C}}, the mass of the electron ''m''{{sub|e}} has the value {{val|9.109|e=−31|ul=kg}}, the magnetic field ''B'' is measured in [[tesla (unit)|tesla]]s, and the angular frequency ω is measured in [[radian]]s per second.</ref> by
: <math>\omega_\text{ce} = \frac{eB}{m_\text{e}}</math>.
where <math>e</math> is the [[elementary charge]] and <math>m_\text{e}</math> is the mass of the electron. For the commonly used [[microwave]] frequency [[ISM band|2.45&nbsp;GHz]] and the bare electron charge and mass, the resonance condition is met when ''B'' = {{val|0.0875|ul=T}}.

For electrons moving at relativistic speeds ''v'', the formula needs to be adjusted according to the [[special theory of relativity]] to:
: <math>\omega_\text{ce} = \frac{eB}{\gamma m_\text{e}}</math>
where
* ''m''<sub>e</sub> is the electron rest mass
* <math>\gamma = \frac{1}{\sqrt{1 - \left(\frac{v}{c}\right)^2}}</math>.