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Condensa

Bose–Einstein Condensate Physics

\[\frac{N_0}{N} = 1 - \left(\frac{T}{T_c}\right)^{3/2}\quad(T\le T_c)\]
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Can atoms share one state? Condensa is a Bose–Einstein condensate. Below a critical temperature, a large fraction of a gas of bosons drops into the single lowest-energy state. Bose and Einstein predicted it in 1924–25, and it was first made in a lab in 1995. A 3" pixel-art sticker; equation, symbols and sources in the physics panel below.

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Size3″ × 3″

3″ × 3″ kiss-cut sticker.

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The physics

Below a critical temperature, a gas of bosons starts collecting in its lowest quantum state. The fraction in that state grows as the temperature drops, following 1 − (T/Tc)^(3/2), and the critical temperature depends on the gas's density and particle mass.

Condensa is what happens when bosons get cold enough to share one state.

Below a critical temperature T_c a macroscopic fraction of an ideal Bose gas piles into the ground state: N_0/N = 1 − (T/T_c)^{3/2}. At T = 0 everyone is downstairs; above T_c the condensate fraction is zero.

The chart on the sticker is that curve. ζ(3/2) ≈ 2.612 fixes T_c for a given density and mass. On the building, the upper floors are nearly empty and the ground floor is packed.

Satyendra Nath Bose and Albert Einstein predicted this in 1924–25; dilute atomic gases first crossed T_c in 1995.

Everyone in the ground floor.

Equations

\[\frac{N_0}{N} = 1 - \left(\frac{T}{T_c}\right)^{3/2}\quad(T\le T_c)\]
\[kT_c = \frac{2\pi\hbar^2}{m}\left(\frac{n}{\zeta(3/2)}\right)^{2/3}\]
SymbolMeaningUnit
\(N_0\) number of bosons in the ground state —
\(N\) total number of bosons —
\(T\) temperature \(\mathrm{K}\)
\(T_c\) critical temperature for condensation \(\mathrm{K}\)
\(n\) number density \(\mathrm{m^{-3}}\)
\(m\) boson mass \(\mathrm{kg}\)
\(\zeta(3/2) \approx 2.612\) Riemann zeta value —

Sources

  1. Einstein (1925) Quantentheorie des einatomigen idealen Gases, Sitzungsberichte der Preussischen Akademie der Wissenschaften
  2. Bose (1924) Plancks Gesetz und Lichtquantenhypothese, Z. Phys. 26, 178
  3. Pethick & Smith, Bose–Einstein Condensation in Dilute Gases, Ch. 1
  4. Pathria & Beale, Statistical Mechanics, Ch. on ideal Bose gas

Source Einstein, A. (1925). Quantentheorie des einatomigen idealen Gases. Zweite Abhandlung. Sitzungsberichte der Preußischen Akademie der Wissenschaften (Berlin), Physikalisch-mathematische Klasse, 3–14.

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