Electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no kn ...
s in free space can carry
quantized orbital
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
(OAM) projected along the direction of propagation. This orbital angular momentum corresponds to
helical
Helical may refer to:
* Helix, the mathematical concept for the shape
* Helical engine, a proposed spacecraft propulsion drive
* Helical spring, a coilspring
* Helical plc, a British property company, once a maker of steel bar stock
* Helicoil
A t ...
wavefronts, or, equivalently, a
phase
Phase or phases may refer to:
Science
*State of matter, or phase, one of the distinct forms in which matter can exist
*Phase (matter), a region of space throughout which all physical properties are essentially uniform
* Phase space, a mathematic ...
proportional to the
azimuth
An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north.
Mathematicall ...
al angle.
[
] Electron beams with quantized orbital angular momentum are also called electron vortex beams.
Theory
An electron in free space travelling at non-
relativistic speeds, follows the
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
for a
free particle
In physics, a free particle is a particle that, in some sense, is not bound by an external force, or equivalently not in a region where its potential energy varies. In classical physics, this means the particle is present in a "field-free" space. I ...
, that is
where
is the reduced
Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
,
is the single-electron
wave function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...
,
its mass,
the position vector, and
is time.
This equation is a type of
wave equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...
and when written in the
Cartesian coordinate system
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
(
,
,
), the solutions are given by a
linear combination of
plane wave
In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space.
For any position \vec x in space and any time t, th ...
s, in the form of
where
is the
linear momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass and ...
and
is the electron energy, given by the usual
dispersion relation
In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given the d ...
. By measuring the momentum of the electron, its
wave function must collapse and give a particular value. If the energy of the electron beam is selected beforehand, the total momentum (not its directional components) of the electrons is fixed to a certain degree of precision.
When the Schrödinger equation is written in the
cylindrical coordinate system
A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference di ...
(
,
,
), the solutions are no longer plane waves, but instead are given by
Bessel beams,
solutions that are a linear combination of
that is, the product of three types of functions: a plane wave with momentum
in the
-direction, a radial component written as a
Bessel function of the first kind
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary ...
, where
is the linear momentum in the radial direction, and finally an azimuthal component written as
where
(sometimes written
) is the
magnetic quantum number
In atomic physics, the magnetic quantum number () is one of the four quantum numbers (the other three being the principal, azimuthal, and spin) which describe the unique quantum state of an electron. The magnetic quantum number distinguishes th ...
related to the angular momentum
in the
-direction. Thus, the dispersion relation reads
. By azimuthal symmetry, the wave function has the property that
is necessarily an
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
, thus
is quantized. If a measurement of
is performed on an electron with selected energy, as
does not depend on
, it can give any integer value. It is possible to experimentally
prepare
Prepare or ''variation'', may refer to:
* PREPARE (Preparedness Against (Re-)emerging Epidemics) of the European Union (EU)
* Promoting Resilience and Efficiency in Preparing for Attacks and Responding to Emergencies (PREPARE) Act of 2017 (U.S.)
* ...
states with non-zero
by adding an azimuthal phase to an initial state with
; experimental techniques designed to
measure
Measure may refer to:
* Measurement, the assignment of a number to a characteristic of an object or event
Law
* Ballot measure, proposed legislation in the United States
* Church of England Measure, legislation of the Church of England
* Mea ...
the orbital angular momentum of a single electron are under development. Simultaneous
measurement
Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events.
In other words, measurement is a process of determining how large or small a physical quantity is as compared ...
of electron energy and orbital angular momentum is allowed because the
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
commutes with the
angular momentum operator
In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic and molecular physics and other quantum prob ...
related to
.
Note that the equations above follow for any free quantum particle with mass, not necessarily electrons. The quantization of
can also be shown in the
spherical coordinate system
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
, where the wave function reduces to a product of
spherical Bessel function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary ...
s and
spherical harmonics
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.
Since the spherical harmonics form a ...
.
Preparation
There are a variety of methods to prepare an electron in an orbital angular momentum state. All methods involve an interaction with an
optical element such that the electron acquires an azimuthal phase. The optical element can be material,
[
] magnetostatic, or electrostatic. It is possible to either directly imprint an azimuthal phase, or to imprint an azimuthal phase with a holographic diffraction grating, where grating pattern is defined by the interference of the azimuthal phase and a planar or spherical carrier wave.
Applications
Electron vortex beams have a variety of proposed and demonstrated applications, including for
mapping magnetization,
studying chiral molecules and chiral plasmon resonances,
[
] and identification of crystal chirality.
Measurement
Interferometric
Interferometry is a technique which uses the ''interference'' of superimposed waves to extract information. Interferometry typically uses electromagnetic waves and is an important investigative technique in the fields of astronomy, fiber op ...
methods borrowed from
light optics also work to determine the orbital angular momentum of free electrons in pure states. Interference with a planar reference wave,
diffractive filtering and self-interference
[
] can serve to characterize a prepared electron orbital angular momentum state. In order to measure the orbital angular momentum of a superposition or of the mixed state that results from interaction with an atom or material, a non-interferometric method is necessary. Wavefront flattening,
transformation of an orbital angular momentum state into a planar wave, or cylindrically symmetric Stern-Gerlach-like measurement
[
] is necessary to measure the orbital angular momentum mixed or superposition state.
References
{{reflist
Angular momentum
Electron beam
Electron microscopy
Orbital angular momentum of waves