Electron optics is a mathematical framework for the calculation of electron trajectories along

electromagnetic fields An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical ...

. The term ''optics'' is used because

magnetic Magnetism is the class of physical attributes that are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particles ...

and

electrostatic Electrostatics is a branch of physics that studies electric charges at rest (static electricity). Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for am ...

lenses act upon a

charged particle beam A charged particle beam is a spatially localized group of electrically charged particles that have approximately the same position, kinetic energy (resulting in the same velocity), and direction. The kinetic energies of the particles are much lar ...

similarly to

optical lens A lens is a transmissive optical device which focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a compound lens consists of several simple lenses (''elements''), ...

es upon a light beam. Electron optics calculations are crucial for the design of

electron microscopes An electron microscope is a microscope that uses a beam of accelerated electrons as a source of illumination. As the wavelength of an electron can be up to 100,000 times shorter than that of visible light photons, electron microscopes have a hi ...

and

particle accelerators A particle accelerator is a machine that uses electromagnetic fields to propel charged particles to very high speeds and energies, and to contain them in well-defined beams. Large accelerators are used for fundamental research in particle ...

. In the

paraxial approximation In geometric optics, the paraxial approximation is a small-angle approximation used in Gaussian optics and ray tracing of light through an optical system (such as a lens). A paraxial ray is a ray which makes a small angle (''θ'') to the opti ...

, trajectory calculations can be carried out using ray transfer matrix analysis.

Electron properties

Electrons are charged particles (

point charge A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take u ...

s with

rest mass The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, ...

) with spin 1/2 (hence they are

fermions In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and le ...

). Electrons can be accelerated by suitable

electric Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described b ...

(or

) fields, thereby acquiring

kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its a ...

. Given sufficient voltage, the electron can be accelerated sufficiently fast to exhibit measurable relativistic effects. According to

wave particle duality In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (r ...

, electrons can also be considered as

matter waves Matter waves are a central part of the theory of quantum mechanics, being an example of wave–particle duality. All matter exhibits wave-like behavior. For example, a beam of electrons can be diffracted just like a beam of light or a water wav ...

with properties such as

wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...

, phase and

amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of a ...

Geometric electron optics

Magnetic fields

Electrons interact with magnetic fields according to the second term of the Lorentz force: a

cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and i ...

between the magnetic field and the electron velocity. In an infinite uniform field this results in a circular motion of the electron around the field direction with a radius given by: :

r =  \frac

where ''r'' is the orbit radius, ''m'' is the mass of an electron,

v_\perp

is the component of the electron velocity perpendicular to the field, ''e'' is the electron charge and ''B'' is the magnitude of the applied magnetic field. Electrons that have a velocity component parallel to the magnetic field will proceed along

helical Helical may refer to: * Helix A helix () is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is ...

trajectories.

Electric fields

In the case of an applied electrostatic field, an electron will deflect towards the positive gradient of the field. Notably, this crossing of electrostatic field lines means that electrons, as they move through electrostatic fields change the magnitude of their velocity, whereas in magnetic fields, only the velocity direction is modified. As electrons can exhibit non-particle (wave-like) effects such as diffraction, a full analysis of electron paths can be obtained by solving

Maxwell's equation Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. T ...

—however in many situations, the particle interpretation may provide a sufficient approximation with great reduction in complexity. One further property of electrons is that they interact strongly with matter as they are sensitive to not only the nucleus, but also the matter's electron charge cloud. Therefore, electrons require

vacuum A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or " void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often di ...

to propagate any reasonable distance, such as would be desirable in electron optic system. Penetration in vacuum is dictated by

mean free path In physics, mean free path is the average distance over which a moving particle (such as an atom, a molecule, or a photon) travels before substantially changing its direction or energy (or, in a specific context, other properties), typically as a ...

, a measure of the probability of collision between electrons and matter, approximate values for which can be derived from

Poisson statistics In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known c ...

Relativistic theory

Although not very common, it is also possible to derive effects of magnetic structures to charged particles starting from the

Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac pa ...

Diffractive electron optics

A sub-relativistic free electron propagating in

can be accurately described as a de Broglie

matter wave Matter waves are a central part of the theory of quantum mechanics, being an example of wave–particle duality. All matter exhibits wave-like behavior. For example, a beam of electrons can be diffracted just like a beam of light or a water wav ...

with a wavelength inversely proportional to its longitudinal momentum. As a result of the charge carried by the electron, electric fields, magnetic fields, or the electrostatic mean inner potential of thin, weakly interacting materials can impart a phase shift to the wavefront of an electron. Thickness-modulated

silicon nitride Silicon nitride is a chemical compound of the elements silicon and nitrogen. is the most thermodynamically stable and commercially important of the silicon nitrides, and the term "silicon nitride" commonly refers to this specific composition. It ...

membranes and programmable phase shift devices have exploited these properties to apply spatially varying phase shifts to control the far-field spatial intensity and phase of the electron wave. Devices like these have been applied to arbitrarily shape the electron wavefront, correct the aberrations inherent to

, resolve the orbital angular momentum of a free electron, and to measure

dichroism In optics, a dichroic material is either one which causes visible light to be split up into distinct beams of different wavelengths (colours) (not to be confused with dispersion), or one in which light rays having different polarizations are a ...