Inphysics, thethermal de Broglie wavelength (${\displaystyle {\lambda }_{\text{th}}}$, sometimes also denoted by${\displaystyle \mathrm{\Lambda }}$) is a measure of the uncertainty in location of a particle of thermodynamic average momentum in an ideal gas.^[1] It is roughly the averagede Broglie wavelength of particles in an ideal gas at the specified temperature.

We can take theaverage interparticle spacing in the gas to be approximately(V/N)^1/3 whereV is the volume andN 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 orMaxwell–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 aFermi gas or aBose 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$${\displaystyle {\displaystyle \frac{V}{N{\lambda }_{\text{th}}^3}\le 1,\text{ or }{\left(\frac{V}{N}\right)}^{1/3}\le {\lambda }_{\text{th}}}}$$

i.e., when the interparticle distance is less than the thermal de Broglie wavelength; in this case the gas will obeyBose–Einstein statistics orFermi–Dirac statistics, whichever is appropriate. This is for example the case for electrons in a typical metal atT = 300K, where theelectron gas obeysFermi–Dirac statistics, or in aBose–Einstein condensate. On the other hand, for$${\displaystyle {\displaystyle \frac{V}{N{\lambda }_{\text{th}}^3}\gg 1,\text{or}{\left(\frac{V}{N}\right)}^{1/3}\gg {\lambda }_{\text{th}}}}$$

i.e., when the interparticle distance is much larger than the thermal de Broglie wavelength, the gas will obeyMaxwell–Boltzmann statistics.^[2] Such is the case for molecular or atomic gases at room temperature, and forthermal neutrons produced by aneutron source.

For massive, non-interacting particles, the thermal de Broglie wavelength can be derived from the calculation of thepartition function. Assuming a 1-dimensional box of lengthL, the partition function (using the energy states of the 1Dparticle in a box) is$${\displaystyle Z=\sum _n\exp \left(-\frac{E_n}{k_{\text{B}}T}\right)=\sum _n\exp \left(-\frac{h^2{n}^2}{8m{L}^2{k}_{\text{B}}T}\right).}$$

Since the energy levels are extremely close together, we can approximate this sum as an integral:^[3]$${\displaystyle Z={\int }_0^{\mathrm{\infty }}\exp \left(-\frac{h^2{n}^2}{8m{L}^2{k}_{\text{B}}T}\right)dn=\sqrt{\frac{2\pi m{k}_{\text{B}}T}{h^2}}L\equiv \frac{L}{{\lambda }_{\text{th}}}.}$$

Hence,$${\displaystyle {\lambda }_{\text{th}}=\frac{h}{\sqrt{2\pi m{k}_{\text{B}}T}},}$$where${\displaystyle h}$ is thePlanck constant,m is themass of a gas particle,${\displaystyle k_{\text{B}}}$ is theBoltzmann constant, andT is thetemperature of the gas.^[2]This can also be expressed using the reduced Planck constant${\displaystyle \hslash =\frac{h}{2\pi }}$ as$${\displaystyle {\lambda }_{\text{th}}=\sqrt{\frac{2\pi {\hslash }^2}{m{k}_{\text{B}}T}}.}$$

For massless (or highlyrelativistic) particles, the thermal wavelength is defined as$${\displaystyle {\lambda }_{\text{th}}=\frac{hc}{2{\pi }^{1/3}{k}_{\text{B}}T}=\frac{{\pi }^{2/3}\hslash c}{k_{\text{B}}T},}$$

wherec 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 ofblack body radiation, the classicalRayleigh–Jeans law can be applied, but when the observed wavelengths approach the thermal wavelength of the photons in the black body radiator, the quantumPlanck's law must be used.

A general definition of the thermal wavelength for an ideal gas of particles having an arbitrary power-law relationship between energy and momentum (dispersion relationship), in any number of dimensions, can be introduced.^[1] Ifn is the number of dimensions, and the relationship between energy (E) and momentum (p) is given by$${\displaystyle E=a{p}^s}$$(witha ands being constants), then the thermal wavelength is defined as$${\displaystyle {\lambda }_{\text{th}}=\frac{h}{\sqrt{\pi }}{\left(\frac{a}{k_{\text{B}}T}\right)}^{1/s}{\left[\frac{\mathrm{\Gamma }(n/2+1)}{\mathrm{\Gamma }(n/s+1)}\right]}^{1/n},}$$whereΓ is theGamma function. This definition retains the following simple form for thechemical potential in the dilute (classical ideal gas) limit:^[1]

${\displaystyle \mu =k_{\text{B}}T\ln \frac{({\lambda }_{\text{th}}{)}^{n}N}{gV},}$

for internal degeneracyg (such as spin degeneracyg = 2 for electrons), and also provides clean expressions for the thermodynamics of Fermi and Bose gases.^[1]

In particular, for a 3-D (n = 3) gas of massive or massless particles we haveE =p²/2m (a = 1/2m,s = 2) andE =pc (a =c,s = 1), respectively, yielding the expressions listed in the previous sections. Note that for massive non-relativistic particles (s = 2), the expression does not depend onn. This explains why the 1-D derivation above agrees with the 3-D case.

Some examples of the thermal de Broglie wavelength at 298 K are given below.

Species	Mass (kg)	${\displaystyle {\lambda }_{\text{th}}}$ (m)
Electron	9.1094×10−31	4.3179×10−9
Photon	0	1.6483×10−5
H2	3.3474×10−27	7.1228×10−11
O2	5.3135×10−26	1.7878×10−11

• Vu-Quoc, L.,Configuration integral (statistical mechanics), 2008. this wiki site is down; seethis article in the web archive on 2012 April 28.

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