Electron optics
Encyclopedia
Electron optics deals with the focusing and deflection of electron
s using magnetic
and/or electrostatic fields.
of + 1/2. While in motion an electron possesses kinetic energy
, regardless of any imposed charge field—this could be achieved by accelerating electrons via a voltage differential into a screened "field-free" region, which initially imparts the energy required to accelerate the electron. Given sufficient voltage, the electron can be accelerated sufficiently fast to exhibit measurable relativistic effects, and the velocity must be accounted for relativistically. According to the wave particle duality, electrons can also be considered as wave propagations and therefore have associated wave properties such as wavelength, phase and amplitude.
With respect to electron optics, the nature of the electron as a charged particle causes electrons to interact with imposed electron fields, and their spin causes magnetic field interactions as well. These interactions form the fundamentals of electron optical theory.
of the magnetic field and the electron propagation vector, such that, in an infinite uniform field moving electrons take a circular motion
at a constant radius dependent upon electron velocity and field strength according to the following equation, which can be derived from circular motion:
where r is the orbit radius, m is the mass of an electron, v is the electron velocity, e is the charge on the electron and H is the applied magnetic field. Electrons that have a velocity component that is parallel to the magnetic field will rather "stretch out" the circle and form helical
paths, the pitch of which is subject to the rotation period and the parallel velocity component.
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—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
to propagate any reasonable distance, such as would be desirable in electron optic systems. Penetration in vacuum is dictated by mean free path
, a measure of the probability of collision between electrons and matter, approximate values for which can be derived from Poisson statistics.
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s using magnetic
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
and/or electrostatic fields.
Electron properties
Electrons are charged particles (point charges with rest mass). The electron also has an associated spinSpin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
of + 1/2. While in motion an electron possesses kinetic energy
Kinetic energy
The kinetic energy of an object is the energy which 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 acceleration, the body maintains this kinetic energy unless its speed changes...
, regardless of any imposed charge field—this could be achieved by accelerating electrons via a voltage differential into a screened "field-free" region, which initially imparts the energy required to accelerate the electron. Given sufficient voltage, the electron can be accelerated sufficiently fast to exhibit measurable relativistic effects, and the velocity must be accounted for relativistically. According to the wave particle duality, electrons can also be considered as wave propagations and therefore have associated wave properties such as wavelength, phase and amplitude.
With respect to electron optics, the nature of the electron as a charged particle causes electrons to interact with imposed electron fields, and their spin causes magnetic field interactions as well. These interactions form the fundamentals of electron optical theory.
Particle interpretation
Magnetic fields interact with an electron in a manner that alters velocity independently of kinetic energy (velocity magnitude). Electrons move according to the cross productCross product
In mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...
of the magnetic field and the electron propagation vector, such that, in an infinite uniform field moving electrons take a circular motion
Circular motion
In physics, circular motion is rotation along a circular path or a circular orbit. It can be uniform, that is, with constant angular rate of rotation , or non-uniform, that is, with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves circular motion of...
at a constant radius dependent upon electron velocity and field strength according to the following equation, which can be derived from circular motion:
where r is the orbit radius, m is the mass of an electron, v is the electron velocity, e is the charge on the electron and H is the applied magnetic field. Electrons that have a velocity component that is parallel to the magnetic field will rather "stretch out" the circle and form helical
Helix
A helix is a type of smooth space curve, i.e. a curve in three-dimensional space. It has the property that the tangent line at any point makes a constant angle with a fixed line called the axis. Examples of helixes are coil springs and the handrails of spiral staircases. A "filled-in" helix – for...
paths, the pitch of which is subject to the rotation period and the parallel velocity component.
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
Diffraction
Diffraction refers to various phenomena which occur when a wave encounters an obstacle. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1665...
, a full analysis of electron paths can be obtained by solving Maxwell's equation—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
Vacuum
In everyday usage, vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty". A perfect vacuum would be one with no particles in it at all, which is impossible to achieve in...
to propagate any reasonable distance, such as would be desirable in electron optic systems. Penetration in vacuum is dictated by mean free path
Mean free path
In physics, the mean free path is the average distance covered by a moving particle between successive impacts which modify its direction or energy or other particle properties.-Derivation:...
, a measure of the probability of collision between electrons and matter, approximate values for which can be derived from Poisson statistics.
See also
- Charged particle beamCharged particle beamA charged particle beam is a spatially localized group of electrically charged particles that have approximately the same velocity . The kinetic energies of the particles are typically measured in keV or MeV, much larger than the energies of particles at ambient temperature...
- Electron beam technologyElectron beam technologyFree electrons in vacuum can be influenced by electric and magnetic fields as to form a fine beam. At the spot of collision of the beam with the particles of the solid-state matter, most portion of the kinetic energy of electrons is transferred into heat. The main advantage of this method is the...
- Electron microscopeElectron microscopeAn electron microscope is a type of microscope that uses a beam of electrons to illuminate the specimen and produce a magnified image. Electron microscopes have a greater resolving power than a light-powered optical microscope, because electrons have wavelengths about 100,000 times shorter than...
- Ernst RuskaErnst RuskaErnst August Friedrich Ruska was a German physicist who won the Nobel Prize in Physics in 1986 for his work in electron optics, including the design of the first electron microscope.Ruska was born in Heidelberg...