Transition temperature

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The transition temperature or critical temperature ( ) is the temperature below which a system is dominated by quantum mechanical effects. In particular, the well-known quantum mechanical statistics, the Bose-Einstein statistics and the Fermi-Dirac statistics apply in these areas . ${\displaystyle T_{\mathrm {c} }}$

Below this critical temperature, the constituents that shape the system are delocalized, i.e. a macroscopic quantum state is present. One can clearly imagine that the expansion of the individual wave packets becomes so great with decreasing temperature that they “overlap” one another and are therefore no longer distinguishable.

Such macroscopic quantum states are superconductivity and superfluid as well as the more general case of a Bose-Einstein condensate .

Examples

Transition temperatures of superfluids

There are only two types of superfluids available in the laboratory.

Superfluid	Transition temperature ${\displaystyle T_{\mathrm {c} }}$
Helium-4 ( 4 He)	2.1768 K
Helium-3 ( 3 He)	0.0026 K

The transition temperature of helium-3 is significantly lower than that of helium-4, since in this case two helium particles have to come together to form a pair ( Cooper pair ). Such a pair is unstable at higher temperatures and would be broken up by phonons .

Transition temperatures of some superconductors

At normal pressure, elements have transition temperatures of up to 9.25 K ( niobium ), in high-pressure experiments up to 20 K ( lithium , 50 GPa) have been detected. The list of transition temperatures of chemical elements provides an overview of the transition temperatures .

In compounds and alloys , the transition temperature can be up to 40 K, in high-temperature superconductors even up to 130 K.

Calculation of the transition temperature

The constituents of a system are delocalized if and only if their thermal (De Broglie) wavelength is greater than the mean distance d . ${\displaystyle \lambda _{\mathrm {deBroglie} }}$

The De Broglie wavelength of a particle with momentum p and kinetic energy is given by: ${\displaystyle E_{\mathrm {kin} }=p^{2}/(2m)}$

${\displaystyle \lambda _{\mathrm {deBroglie} }={\frac {h}{p}}={\frac {h}{\sqrt {2m\,E_{\mathrm {kin} }}}}}$

Under the simplified assumption we get: ${\displaystyle E_{\mathrm {kin} }=k\,T}$

${\displaystyle \lambda _{\mathrm {deBroglie} }={\frac {h}{\sqrt {2m\,k\,T}}}}$

The mean distance d results from the particle number density n as follows:

${\displaystyle d=n^{-1/3}}$

The transition temperature represents the critical borderline case . Equation of the two expressions and resolution according to the transition temperature yields: ${\displaystyle \lambda _{\mathrm {deBroglie} }=d}$

${\displaystyle T_{\mathrm {c} }={\frac {h^{2}\,n^{2/3}}{2\,m\,k}}}$