Thermal de Broglie wavelength

In physics, the Thermal de Broglie wavelength is defined for a free ideal gas of massive particles in equilibrium as:

$\Lambda = \sqrt{\frac{h^2}{2\pi mkT}} = \frac {h} {(2 \pi m k T)^{1/2}}$

where

h is Planck's constant
m is the mass of a gas particle
k is Boltzmann's constant
T is the Temperature of the gas

The thermal de Broglie wavelength is roughly the average Bose gas, 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

$\displaystyle \frac{V}{N\Lambda^3} \le 1 \ , {\rm or} \ \left( \frac{V}{N} \right)^{1/3} \le \Lambda$

i.e., when the interparticle distance is less than the thermal de Broglie wavelength; in this case the gas will obey Fermi-Dirac statistics, whichever is appropriate. On the other hand, for

$\displaystyle \frac{V}{N\Lambda^3} \gg 1 \ , {\rm or} \ \left( \frac{V}{N} \right)^{1/3} \gg \Lambda$

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

Derivation

For a derivation of $\displaystyle \Lambda$ , see configuration integral.

Massless particles

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

$\Lambda= \frac{ch}{2kT\pi^{1/3}}$

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 Planck's law 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:

$E=ap^s\,$

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

$\Lambda=\frac{h}{\sqrt{\pi}}\left(\frac{a}{kT}\right)^{1/s} \left[\frac{\Gamma(n/2+1)}{\Gamma(n/s+1)}\right]^{1/n}$

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=pc which gives the above results for massless particles.

References

Zijun Yan, "General thermal wavelength and its applications", Eur. J. Phys. 21 (2000) 625-631. http://www.iop.org/EJ/article/0143-0807/21/6/314/ej0614.pdf

Statistical mechanics

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