Electron cyclotron resonance (ECR) is a phenomenon observed inplasma physics,condensed matter physics, andaccelerator physics. It happens when the frequency of incident radiation coincides with the natural frequency of rotation of electrons in magnetic fields. A freeelectron in a static and uniformmagnetic field will move in a circle due to theLorentz force. The circular motion may be superimposed with a uniform axial motion, resulting in ahelix, or with a uniform motion perpendicular to the field (e.g., in the presence of an electrical or gravitational field) resulting in acycloid. Theangular frequency (ω = 2πf ) of thiscyclotron motion for a given magnetic field strengthB is given (inSI units)^[1] by

${\displaystyle {\omega }_{\text{ce}}=\frac{eB}{m_{\text{e}}}}$.

where${\displaystyle e}$ is theelementary charge and${\displaystyle m_{\text{e}}}$ is the mass of the electron. For the commonly usedmicrowave frequency2.45 GHz and the bare electron charge and mass, the resonance condition is met whenB =0.0875 T.

For electrons moving at relativistic speedsv, the formula needs to be adjusted according to thespecial theory of relativity to:

${\displaystyle {\omega }_{\text{ce}}=\frac{eB}{\gamma m_{\text{e}}}}$

where

• m_e is the electron rest mass
• ${\displaystyle \gamma =\frac{1}{\sqrt{1-{\left(\frac{v}{c}\right)}^2}}}$.

An ionizedplasma may be efficiently produced or heated by superimposing a staticmagnetic field and a high-frequencyelectromagnetic field at the electron cyclotronresonance frequency. In the toroidal magnetic fields used inmagnetic fusion energy research, the magnetic field decreases with the major radius, so the location of the power deposition can be controlled within about a centimetre. Furthermore, the heating power can be rapidly modulated and is deposited directly into the electrons. These properties make electron cyclotron heating a very valuable research tool for energy transport studies. In addition to heating, electron cyclotron waves can be used to drive current. The inverse process ofelectron cyclotron emission can be used as adiagnostic of the radial electron temperature profile.

The use of electron cyclotron resonance for efficient plasma generation, especially to obtain large numbers of multiply charged ions,^[2][3] has been applied in diverse fields:

• advanced cancer treatment, where ECRion sources are crucial forproton therapy,^{[citation needed]}
• advancedsemiconductor manufacturing, especially for high densityDRAM memories, throughplasma etching or otherplasma processing technologies,^{[citation needed]}
• electric propulsion devices forspacecraft propulsion, where a broad range of devices (HiPEP, someion thrusters, orelectrodeless plasma thrusters),^{[citation needed]}
• forparticle accelerators, on-line mass separation and radioactive ion charge breeding,^[4]
• and, as a more mundane example, painting of plastic bumpers for cars.^{[citation needed]}

The ECR ion source makes use of the electron cyclotron resonance to ionize a plasma. Microwaves are injected into a volume at the frequency corresponding to the electron cyclotron resonance, defined by the magnetic field applied to a region inside the volume. The volume contains a low pressure gas. The alternating electric field of the microwaves is set to be synchronous with the gyration period of the free electrons of the gas, and increases their perpendicular kinetic energy. Subsequently, when the energized free electrons collide with the gas in the volume they can cause ionization if their kinetic energy is larger than the ionization energy of the atoms or molecules. The ions produced correspond to the gas type used, which may be pure, a compound, or vapour of a solid or liquid material.

ECR ion sources are able to produce singly charged ions with high intensities (e.g.H⁺ andD⁺ ions of more than 100 mA (electrical) in DC mode^[5] using a 2.45 GHz ECR ion source).

For multiply charged ions, the ECR ion source has the advantages that it is able to confine the ions for long enough for multiple collisions and multiple ionization to take place, and the low gas pressure in the source avoids recombination. The VENUS ECR ion source atLawrence Berkeley National Laboratory has produced in intensity of 0.25 mA (electrical) ofBi²⁹⁺.^[6]

Some important industrial fields would not exist without the use of this fundamental technology, which makes electron cyclotron resonance ion and plasma sources one of the enabling technologies of today's world.

Within a solid the mass in the cyclotron frequency equation above is replaced with theeffective mass tensor${\displaystyle m^{\ast }}$. Cyclotron resonance is therefore a useful technique to measureeffective mass andFermi surface cross-section in solids. In a sufficiently high magnetic field at low temperature in a relatively pure material

where${\displaystyle \tau }$ is the carrier scattering lifetime,${\displaystyle k_{\text{B}}}$ is theBoltzmann constant and${\displaystyle T}$ is temperature. When these conditions are satisfied, an electron will complete its cyclotron orbit without engaging in a collision, at which point it is said to be in a well-defined Landau level.

• Cyclotron
• ARC-ECRIS
• Ion cyclotron resonance
• Synchrotron
• Gyrotron
• De Haas–van Alphen effect

• "Personal Reminiscences of Cyclotron Resonance", G. Dresselhaus, Proceedings of ICPS-27 (2004). This paper describes the early history of cyclotron resonance in its heyday as aband structure determination technique.

• Waves in plasmas
• Condensed matter physics
• Electric and magnetic fields in matter
• Ion source
• Particle accelerators

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