the temperature at which bose einstein condensation starts
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This transition to BEC occurs below a critical temperature, which for a uniform three-dimensionalgas consisting of non-interacting particles with no apparent internal degrees of freedom is given by:
{\displaystyle T_{c}=\left({\frac {n}{\zeta (3/2)}}\right)^{2/3}{\frac {2\pi \hbar ^{2}}{mk_{B}}}\approx 3.3125\ {\frac {\hbar ^{2}n^{2/3}}{mk_{B}}}}
where:
{\displaystyle \,T_{c}} is the critical temperature,{\displaystyle \,n} is the particle density,{\displaystyle \,m} is the mass per boson,{\displaystyle \hbar } is the reduced Planck constant,{\displaystyle \,k_{B}} is the Boltzmann constant, and{\displaystyle \,\zeta } is the Riemann zeta function; {\displaystyle \,\zeta (3/2)\approx 2.6124.}
Interactions shift the value and the corrections can be calculated by mean-field theory.
This formula is derived from finding the gas degeneracy in the bose gas using Bose–Einstein statistics.
{\displaystyle T_{c}=\left({\frac {n}{\zeta (3/2)}}\right)^{2/3}{\frac {2\pi \hbar ^{2}}{mk_{B}}}\approx 3.3125\ {\frac {\hbar ^{2}n^{2/3}}{mk_{B}}}}
where:
{\displaystyle \,T_{c}} is the critical temperature,{\displaystyle \,n} is the particle density,{\displaystyle \,m} is the mass per boson,{\displaystyle \hbar } is the reduced Planck constant,{\displaystyle \,k_{B}} is the Boltzmann constant, and{\displaystyle \,\zeta } is the Riemann zeta function; {\displaystyle \,\zeta (3/2)\approx 2.6124.}
Interactions shift the value and the corrections can be calculated by mean-field theory.
This formula is derived from finding the gas degeneracy in the bose gas using Bose–Einstein statistics.
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