In physics, the thermal de Broglie wavelength (, sometimes also denoted by ) is roughly the average de Broglie wavelength of particles in an ideal gas at the specified temperature. We can take the average interparticle spacing in the gas to be approximately where is the volume and 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 or a 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 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 massive, non-interacting particles, the thermal de Broglie wavelength can be derived from the calculation of the partition function. Assuming a 1-dimensional box of length , the partition function (using the energy states of the 1D particle in a box) is Since the energy levels are extremely close together, we can approximate this sum as an integral: Hence, where is the Planck constant, is the mass of a gas particle, is the Boltzmann constant, and is the temperature of the gas. This can also be expressed using the reduced Planck constant as Massless particles For massless (or highly relativistic) particles, the thermal wavelength is 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