Length scale

Encyclopedia

In physics

,

or distance

determined with the precision of one order (or a few orders) of magnitude. 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 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 quantum mechanics

the length scale of a given phenomenon is related to its de Broglie wavelength

where is the reduced Planck's constant and is the momentum that is being probed. In relativistic mechanics

time and length scales are related by the speed of light

. In relativistic quantum mechanics or relativistic quantum field theory, length scales are related to momentum, time and energy scales through Planck's constant and the speed of light. Often in high energy physics natural units

are used where length, time, energy and momentum scales are described in the same units (usually with units of energy such as GeV

).

Length scales are usually the operative scale (or at least one of the scales) in dimensional analysis

. For instance, in scattering theory

, the most common quantity to calculate is a cross section

which has units of length squared and is measured in barn

s. The cross section of a given process is usually the square of the length scale.

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

,

**length scale**is a particular lengthLength

In geometric measurements, length most commonly refers to the longest dimension of an object.In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire...

or distance

Distance

Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical distance...

determined with the precision of one order (or a few orders) of magnitude. 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 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

Renormalization group

In theoretical physics, the renormalization group refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales...

.

In quantum mechanics

Quantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

the length scale of a given phenomenon is related to its de Broglie wavelength

where is the reduced Planck's constant and is the momentum that is being probed. In relativistic mechanics

Relativistic Mechanics

Relativistic mechanics refers to any form of mechanics compatible with either theories of Special Relativity and General Relativity. It is a non-quantum mechanics of a system of particles or of a fluid in cases where some of the velocities are comparable to the speed of light.Special relativity...

time and length scales are related by the speed of light

Speed of light

The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

. In relativistic quantum mechanics or relativistic quantum field theory, length scales are related to momentum, time and energy scales through Planck's constant and the speed of light. Often in high energy physics natural units

Natural units

In physics, natural units are physical units of measurement based only on universal physical constants. For example the elementary charge e is a natural unit of electric charge, or the speed of light c is a natural unit of speed...

are used where length, time, energy and momentum scales are described in the same units (usually with units of energy such as GeV

GEV

GEV or GeV may stand for:*GeV or gigaelectronvolt, a unit of energy equal to billion electron volts*GEV or Grid Enabled Vehicle that is fully or partially powered by the electric grid, see plug-in electric vehicle...

).

Length scales are usually the operative scale (or at least one of the scales) in dimensional analysis

Dimensional analysis

In physics and all science, dimensional analysis is a tool to find or check relations among physical quantities by using their dimensions. The dimension of a physical quantity is the combination of the basic physical dimensions which describe it; for example, speed has the dimension length per...

. For instance, in scattering theory

Scattering theory

In mathematics and physics, scattering theory is a framework for studying and understanding the scattering of waves and particles. Prosaically, wave scattering corresponds to the collision and scattering of a wave with some material object, for instance sunlight scattered by rain drops to form a...

, the most common quantity to calculate is a cross section

Cross section (physics)

A cross section is the effective area which governs the probability of some scattering or absorption event. Together with particle density and path length, it can be used to predict the total scattering probability via the Beer-Lambert law....

which has units of length squared and is measured in barn

Barn (unit)

A barn is a unit of area. Originally used in nuclear physics for expressing the cross sectional area of nuclei and nuclear reactions, today it is used in all fields of high energy physics to express the cross sections of any scattering process, and is best understood as a measure of the...

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 radiusBohr radiusThe Bohr radius is a physical constant, approximately equal to the most probable distance between the proton and 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 atom...

(approximately 53 pm1 E-12 mTo help compare different orders of magnitude this page lists lengths between 10−12 m and 10−11 m .Distances shorter than 1 pm* 1 pm = 1 picometre = 1,000 femtometres* 1 pm = distance between atomic nuclei in a white dwarf...

)) which is set by the electron's Compton wavelengthCompton wavelengthThe Compton wavelength is a quantum mechanical property of a particle. It was introduced by Arthur Compton in his explanation of the scattering of photons by electrons...

times the fine-structure constantFine-structure constantIn physics, the fine-structure constant is a fundamental physical constant, namely the coupling constant characterizing the strength of the electromagnetic interaction. Being a dimensionless quantity, it has constant numerical value in all systems of units...

: .

- The length scale for the strong interactionStrong interactionIn particle physics, the strong interaction is one of the four fundamental interactions of nature, the others being electromagnetism, the weak interaction and gravitation. As with the other fundamental interactions, it is a non-contact force...

s (or the one derived from QCDQuantum chromodynamicsIn theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...

through dimensional transmutationDimensional transmutationIn particle physics, dimensional transmutation is a physical mechanism that transforms a dimensionless parameter into a dimensionful parameter....

) is around meters (or in natural units 100 MeV), and the "radii" of strongly interacting particles (such as the protonProtonThe proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....

) are roughly comparable. This length scale is determined by the range of the Yukawa potential. The lifetimes of strongly interacting particles, such as the rho mesonRho mesonIn particle physics, a rho meson is a short-lived hadronic particle that is an isospin triplet whose three states are denoted as , and . After the pions and kaons, the rho mesons are the lightest strongly interacting particle with a mass of roughly for all three states...

, 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 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 constantFermi's interactionIn particle physics, Fermi's interaction also known as Fermi coupling, is an old explanation of the weak force, proposed by Enrico Fermi, in which four fermions directly interact with one another at one vertex...

measured by neutronNeutronThe neutron is a subatomic hadron particle which has the symbol or , no net electric charge and a mass slightly larger than that of a proton. With the exception of hydrogen, nuclei of atoms consist of protons and neutrons, which are therefore collectively referred to as nucleons. The number of...

and muonMuonThe muon |mu]] used to represent it) is an elementary particle similar to the electron, with a unitary negative electric charge and a spin of ½. Together with the electron, the tau, and the three neutrinos, it is classified as a lepton...

decay.

- The Planck length (Planck scale) is much shorter yet - about meters ( GeV in natural units), and is derived from Newton's gravitational constantGravitational constantThe gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...

which has units of length squared.

- The Mesoscopic scale is the length at which quantum mechanical behaviours in liquids or solid can be described by macroscopicMacroscopicThe macroscopic scale is the length scale on which objects or processes are of a size which is measurable and observable by the naked eye.When applied to phenomena and abstract objects, the macroscopic scale describes existence in the world as we perceive it, often in contrast to experiences or...

concepts.

## See also

- Orders of magnitude (length)
- Extragalactic Distance Scale
- Scale heightScale heightIn various scientific contexts, a scale height is a distance over which a quantity decreases by a factor of e...