Thermal de Broglie wavelength

	Email		Print
	Bookmark		Link

Thermal de Broglie wavelength

In physics

Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
, the Thermal de Broglie wavelength is defined for a free ideal gas

Ideal gas

The ideal gas model is a model of matter in which the molecules are treated as non-interacting point particles which are engaged in a random motion that obeys conservation of energy....
of massive particles in equilibrium as:

where

The thermal de Broglie wavelength is roughly the average de Broglie wavelength of the gas particles in an ideal gas at the specified temperature. We can take the average interparticle spacing in the gas to be approximately (V/N)^1/3 where V is the volume and N is the number of particles.

Discussion

Ask a question about 'Thermal de Broglie wavelength'

Start a new discussion about 'Thermal de Broglie wavelength'

Answer questions from other users

Full Discussion Forum

Encyclopedia

In physics

Physics

Ideal gas

h is Planck's constant
m is the mass
Mass

In physical science, mass refers to the degree of acceleration a body acquires when subject to a force: bodies with greater mass are accelerated less by the same force....
of a gas particle
k is Boltzmann's constant
T is the Temperature
Temperature

In physics, temperature is a physical property of a Physical system that underlies the common notions of hot and cold; something that feels hotter generally has the greater temperature....
of the gas

The thermal de Broglie wavelength is roughly the average de Broglie wavelength of the gas particles in an ideal gas at the specified temperature. We can take the average interparticle spacing in the gas to be approximately (V/N)^1/3 where V is the volume and N is the number of particles. When the thermal de Broglie wavelength is much smaller than the interparticle distance, the gas can be considered to be a classical or Maxwell-Boltzmann gas. On the other hand, when the thermal de Broglie wavelength is on the order of, or larger than the interparticle distance, quantum effects will dominate and the gas must be treated as a Fermi gas

Fermi gas

A Fermi gas, or Free electron gas, is a collection of non-interacting fermions. It is the quantum mechanics version of an ideal gas, for the case of fermionic particles....
or a Bose gas

Bose gas

An ideal Bose gas is a quantum-mechanical version of a classical ideal gas. It is composed of bosons, which have an integer value of spin, and obey Bose-Einstein statistics....
, depending on the nature of the gas particles. The critical temperature is the transition point between these two regimes, and at this critical temperature, the thermal wavelength will be approximately equal to the interparticle distance. That is, the quantum nature of the gas will be evident for

i.e., when the interparticle distance is less than the thermal de Broglie wavelength; in this case the gas will obey Bose-Einstein statistics or Fermi-Dirac statistics

Fermi-Dirac statistics

Fermi-Dirac statistics is a part of the science of physics, that applies to a system comprised of many particles that obey the Pauli Exclusion Principle....
, whichever is appropriate. On the other hand, for

i.e., when the interparticle distance is much larger than the thermal de Broglie wavelength, the gas will obey Maxwell-Boltzmann statistics.

Massless particles

For a massless particle, the thermal wavelength may be defined as:

where c is the speed of light. As with the thermal wavelength for massive particles, this is of the order of the average wavelength of the particles in the gas and defines a critical point at which quantum effects begin to dominate. For example, when the thermal wavelength of the photons in a black body

Black body

In physics, a black body is an Physical body that absorbs all electromagnetic radiation that falls on it. No electromagnetic radiation passes through it and none is Reflection ....
radiator is of the same order as the wavelength of the radiation (or larger), the "classical" Rayleigh-Jeans law

Rayleigh-Jeans law

In physics, the Rayleigh?Jeans Law, first proposed in the early 20th century, attempts to describe the spectral radiance of electromagnetic radiation at all wavelengths from a black body at a given temperature through classical arguments....
becomes inapplicable, and the "quantum" Planck's law

Planck's law of black body radiation

For a general introduction, see black body.In physics, Planck's law describes the radiance of electromagnetic radiation at all wavelengths from a black body at temperature ....
must be used.

The massless thermal wavelength is derived from the more general definition of the thermal wavelength due to Yan (Yan 2000) described below.

General definition of the thermal wavelength

A general definition of the thermal wavelength for an ideal quantum gas in any number of dimensions and for a generalized relationship between energy and momentum (dispersion relationship) has been given by Yan (Yan 2000). It is of practical importance, since there are many experimental situations with different dimensionality and dispersion relationships. If n is the number of dimensions, and the relationship between energy (E) and momentum (p) is given by:

where a and s are constants, then the thermal wavelength is defined as:

where Γ is the Gamma function

Gamma function

In mathematics, the Gamma function is an extension of the factorial function to real number and complex number numbers. For a complex number z with positive real part the Gamma function is defined by...
. For example, in the usual case of massive particles in a 3-D gas we have n=3 , and E=p²/2m which gives the above results for massive particles. For massless particles in a 3-D gas, we have n=3 , and E=pc which gives the above results for massless particles.