In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, length scale is a particular
length
Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...
or
distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
determined with the precision of at most a few
orders of magnitude
An order of magnitude is an approximation of the logarithm of a value relative to some contextually understood reference value, usually 10, interpreted as the base of the logarithm and the representative of values of magnitude one. Logarithmic dis ...
. The concept of length scale is particularly important because physical phenomena of different length scales cannot affect each other and are said to
decouple. The decoupling of different length scales makes it possible to have a self-consistent theory that only describes the relevant length scales for a given problem.
Scientific reductionism
Reductionism is any of several related philosophical ideas regarding the associations between phenomena which can be described in terms of other simpler or more fundamental phenomena. It is also described as an intellectual and philosophical pos ...
says that the physical laws on the shortest length scales can be used to derive the effective description at larger length scales.
The idea that one can derive descriptions of physics at different length scales from one another can be quantified with the
renormalization group
In theoretical physics, the term renormalization group (RG) refers to a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in the ...
.
In
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
the length scale of a given phenomenon is related to its
de Broglie wavelength
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 wave ...
where
is the reduced
Planck's constant and
is the momentum that is being probed. In
relativistic mechanics
In physics, relativistic mechanics refers to mechanics compatible with special relativity (SR) and general relativity (GR). It provides a non-quantum mechanical description of a system of particles, or of a fluid, in cases where the velocities of ...
time and length scales are related by the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
. In
relativistic quantum mechanics
In physics, relativistic quantum mechanics (RQM) is any Poincaré covariant formulation of quantum mechanics (QM). This theory is applicable to massive particles propagating at all velocities up to those comparable to the speed of light ''c ...
or
relativistic quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles ...
, length scales are related to momentum, time and energy scales through Planck's constant and the speed of light. Often in
high energy physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) and b ...
natural units
In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a Coherence (units of measurement), coherent unit of a quantity. For e ...
are used where length, time, energy and momentum scales are described in the same units (usually with units of energy such as
GeV GEV may refer to:
* ''G.E.V.'' (board game), a tabletop game by Steve Jackson Games
* Ashe County Airport, in North Carolina, United States
* Gällivare Lapland Airport, in Sweden
* Generalized extreme value distribution
* Gev Sella, Israeli-Sou ...
).
Length scales are usually the operative scale (or at least one of the scales) in
dimensional analysis
In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measure (such as m ...
. For instance, in
scattering theory
In mathematics and physics, scattering theory is a framework for studying and understanding the scattering of waves and particles. Wave scattering corresponds to the collision and scattering of a wave with some material object, for instance sunli ...
, the most common quantity to calculate is a
cross section
Cross section may refer to:
* Cross section (geometry)
** Cross-sectional views in architecture & engineering 3D
*Cross section (geology)
* Cross section (electronics)
* Radar cross section, measure of detectability
* Cross section (physics)
**Abs ...
which has units of length squared and is measured in
barn
A barn is an agricultural building usually on farms and used for various purposes. In North America, a barn refers to structures that house livestock, including cattle and horses, as well as equipment and fodder, and often grain.Allen G. ...
s. The cross section of a given process is usually the square of the length scale.
Examples
*The atomic length scale is
meters and is given by the size of hydrogen atom (''i.e.'', the
Bohr radius
The Bohr radius (''a''0) is a physical constant, approximately equal to the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state. It is named after Niels Bohr, due to its role in the Bohr model of an ...
(approximately 53
pm)) which is set by the electron's
Compton wavelength
The Compton wavelength is a quantum mechanical property of a particle. The Compton wavelength of a particle is equal to the wavelength of a photon whose energy is the same as the rest energy of that particle (see mass–energy equivalence). It was ...
times the
fine-structure constant
In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by (the Greek letter ''alpha''), is a fundamental physical constant which quantifies the strength of the electromagnetic interaction between ele ...
:
.
*The length scale for the
strong interaction
The strong interaction or strong force is a fundamental interaction that confines quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is called the n ...
s (or the one derived from
QCD
In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a ty ...
through
dimensional transmutation
In particle physics, dimensional transmutation is a physical mechanism providing a linkage between a dimensionless parameter and a dimensionful parameter.
In classical field theory, such as gauge theory in four-dimensional spacetime, the couplin ...
) is around
meters (or in natural units 1000 MeV or 1 GeV), and the "radii" of strongly interacting particles (such as the
proton
A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge of +1 ''e'' elementary charge. Its mass is slightly less than that of a neutron and 1,836 times the mass of an electron (the proton–electron mass ...
) are roughly comparable. This length scale is determined by the range of the
Yukawa potential
In particle, atomic and condensed matter physics, a Yukawa potential (also called a screened Coulomb potential) is a potential named after the Japanese physicist Hideki Yukawa. The potential is of the form:
:V_\text(r)= -g^2\frac,
where is a m ...
. The lifetimes of strongly interacting particles, such as the
rho meson
Rho (uppercase Ρ, lowercase ρ or ; el, ρο or el, ρω, label=none) is the 17th letter of the Greek alphabet. In the system of Greek numerals it has a value of 100. It is derived from Phoenician letter res . Its uppercase form uses the sa ...
, are given by this length scale divided by the speed of light:
seconds. The masses of strongly interacting particles are several times the associated energy scale (500 MeV to 3000 MeV).
*The
electroweak
In particle physics, the electroweak interaction or electroweak force is the unified field theory, unified description of two of the four known fundamental interactions of nature: electromagnetism and the weak interaction. Although these two force ...
length scale is shorter, roughly
meters and is set by the rest mass of the
weak vector bosons which is roughly 100 GeV. This length scale would be the distance where a Yukawa force is mediated by the weak vector bosons. The magnitude of weak length scale was initially inferred by the
Fermi constant
In particle physics, Fermi's interaction (also the Fermi theory of beta decay or the Fermi four-fermion interaction) is an explanation of the beta decay, proposed by Enrico Fermi in 1933. The theory posits four fermions directly interacting ...
measured by
neutron
The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons beh ...
and
muon
A muon ( ; from the Greek letter mu (μ) used to represent it) is an elementary particle similar to the electron, with an electric charge of −1 '' e'' and a spin of , but with a much greater mass. It is classified as a lepton. As wi ...
decay.
*The
Planck length (Planck scale) is much shorter yet - about
meters (
GeV
in natural units), and is derived from Newton's
gravitational constant
The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in ...
which has units of length squared.
*The
Mesoscopic scale
Mesoscopic physics is a subdiscipline of condensed matter physics that deals with materials of an intermediate size. These materials range in size between the nanoscale for a quantity of atoms (such as a molecule) and of materials measuring mic ...
is the length at which quantum mechanical behaviours in liquids or solid can be described by
macroscopic
The macroscopic scale is the length scale on which objects or phenomena are large enough to be visible with the naked eye, without magnifying optical instruments. It is the opposite of microscopic.
Overview
When applied to physical phenomena an ...
concepts.
See also
*
Orders of magnitude (length)
The following are examples of orders of magnitude for different lengths.
__TOC__
Overview
Detailed list
To help compare different orders of magnitude, the following list describes various lengths between 1.6 \times 10^ metres and 10 ...
*
Extragalactic Distance Scale
The cosmic distance ladder (also known as the extragalactic distance scale) is the succession of methods by which astronomers determine the distances to celestial objects. A ''direct'' distance measurement of an astronomical object is possible o ...
*
Scale height
In atmospheric, earth, and planetary sciences, a scale height, usually denoted by the capital letter ''H'', is a distance (vertical or radial) over which a physical quantity decreases by a factor of e (the base of natural logarithms, approximate ...
References
{{DEFAULTSORT:Length Scale
Orders of magnitude (length)