In physics, the thermal de Broglie wavelength (
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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, whichever is appropriate. This is for example the case for electrons in a typical metal at T = 300 K, where the electron gas obeys Fermi–Dirac statistics, or in a Bose–Einstein condensate. 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. Such is the case for molecular or atomic gases at room temperature , and for thermal neutrons produced by a neutron source.
Massive Particles
For a free ideal gas of massive particles (with no internal degrees of freedom) in equilibrium, the thermal de Broglie wavelength can be obtained through the standard de Broglie wavelength:
In the nonrelativistic case the effective kinetic energy of free particles is
where h is the Planck constant, m is the mass of a gas particle,
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 observing the long-wavelength spectrum of black body radiation, the "classical" Rayleigh–Jeans law can be applied, but when the observed wavelengths approach the thermal wavelength of the photons in the black body radiator, the "quantum" Planck's law must be used.
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. 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 = p c which gives the above results for massless particles.
Examples
Some examples of the thermal deBroglie wavelength at 298 K are,