Bose–Einstein Condensation Calculator

Calculator Inputs

System model

Select the ideal-gas model used for T_c.

Particle mass

Example: 87 amu for Rb-87.

Evaluation temperature

Used for λ_T, n_c(T), or N_c(T).

Uniform box inputs

Use density n

Use N and V

Number density, n

Typical ultracold densities: 10¹⁸–10²¹ m^-3.

Particle number, N

Used for trap mode, and optional box mode.

Volume, V

If using N and V, density n = N/V.

Harmonic trap inputs

Particle number, N

T_c scales as N^1/3.

Frequency unit

Mean frequency ω̄ = (ω_xω_yω_z)^1/3.

ω_x or f_x

ω_y or f_y

ω_z or f_z

Optional temperature sweep

Enable sweep table

Tmin (same unit as T)

Tmax (same unit as T)

Steps (2–300)

The sweep uses your chosen temperature unit, then converts internally to kelvin.

Reset

Formula Used

This tool uses ideal, non-interacting Bose gas thresholds. It is accurate for order-of-magnitude planning, but interactions and finite-size effects shift measured values.

Uniform 3D box

T_c = \frac{2\pi\hbar^2}{m k_B}\left(\frac{n}{\zeta(3/2)}\right)^{2/3}

n is number density (m^-3).
\zeta(3/2) \approx 2.612375.

3D harmonic trap

T_c = \frac{\hbar\bar\omega}{k_B}\left(\frac{N}{\zeta(3)}\right)^{1/3}

\bar\omega = (\omega_x\omega_y\omega_z)^{1/3} with ω in rad/s.
\zeta(3) \approx 1.202057.

Thermal de Broglie wavelength

\lambda_T = \sqrt{\frac{2\pi\hbar^2}{m k_B T}}

Condensed fraction (ideal-gas model)

Box: N₀/N = 1 - (T/T_c)^{3/2} for T < T_c.
Trap: N₀/N = 1 - (T/T_c)^{3} for T < T_c.

How to Use

Select the system model: uniform box or harmonic trap.
Enter particle mass (amu is convenient for atomic species).
For a box, provide density n, or provide N and V.
For a trap, provide N and three trapping frequencies.
Choose an evaluation temperature to compute λ_T and the condensed fraction.
Optionally enable the sweep to generate a temperature table.
Use the CSV and PDF buttons to export the tables.

Example Data Table

Illustrative values for a box model (Rb-87-like mass). Your results depend on inputs.

Mass (amu)	Density (m^-3)	Estimated T_c (K)	T (K)	Expected N₀/N
87	1.0e20	~2e-7	1.5e-7	~0.3
87	3.0e20	~6e-7	3.0e-7	~0.6

Example temperatures are order-of-magnitude guides, not calibrated measurements.

Bose–Einstein Condensation in Ideal Gases

1) What this calculator estimates

This tool estimates ideal-gas Bose–Einstein condensation thresholds for a uniform 3D box or a 3D harmonic trap. It reports the critical temperature T_c, the thermal wavelength \lambda_T, and an ideal condensed fraction N₀/N at your chosen evaluation temperature.

2) Key constants and numerical values

The transition is set by \zeta(3/2) \approx 2.612375 (box) and \zeta(3) \approx 1.202057 (trap), which come from summing the Bose occupation of excited states at saturation. The calculator also uses k_B and \hbar. In trap mode, frequencies entered in hertz are converted to angular units internally.

3) Uniform box critical temperature with density

For a homogeneous 3D gas, T_c = \frac{2\pi\hbar^2}{m k_B}\left(\frac{n}{\zeta(3/2)}\right)^{2/3}, so T_c \propto n^{2/3}/m. With m=87 amu and n=1.0\times 10^{20} m⁻³, the ideal estimate is T_c \approx 4.0\times 10^{-7} K (about 400 nK). Increasing density by a factor of 8 raises T_c by roughly a factor of 4.

4) Harmonic trap critical temperature with N

In a 3D harmonic trap, T_c = \frac{\hbar\bar\omega}{k_B}\left(\frac{N}{\zeta(3)}\right)^{1/3} with \bar\omega=(\omega_x\omega_y\omega_z)^{1/3}. For N=10^6 and (f_x,f_y,f_z)=(80,80,20) Hz, \bar\omega \approx 3.17\times 10^2 rad/s and T_c \approx 2.27\times 10^{-7} K (about 227 nK). Doubling N increases T_c by about 2^{1/3}\approx 1.26 at fixed confinement.

5) Thermal wavelength and phase-space density

The thermal de Broglie wavelength \lambda_T=\sqrt{\frac{2\pi\hbar^2}{m k_B T}} increases as temperature drops, indicating stronger wave overlap. For 87 amu at 150 nK, \lambda_T \approx 4.83\times 10^{-7} m (0.483 µm). In box mode, condensation requires n\lambda_T^3 \gtrsim \zeta(3/2), so the calculator also shows n_c(T)=\zeta(3/2)/\lambda_T^3.

6) Condensed fraction trends below T_c

Below T_c, the ideal condensed fraction follows 1-(T/T_c)^{3/2} in a box and 1-(T/T_c)^{3} in a trap. The different exponents reflect different densities of states. The optional sweep table helps you visualize how quickly N₀/N grows as temperature is lowered.

7) Interpreting outputs for design decisions

Use T_c to set a target final temperature, then read off N₀/N to estimate the condensate fraction at your operating point. In trap mode, N_c(T)=\zeta(3)(k_BT/(\hbar\bar\omega))^3 is the maximum thermal population at temperature T. In box mode, comparing n to n_c(T) gives the analogous threshold check.

8) Limits of the ideal model

Interactions, finite-size effects, and anharmonic confinement can shift measured T_c by a few percent to tens of percent, especially near threshold. Use these results as a baseline, then refine with scattering length, geometry, and calibration information when precision is required. For many planning tasks, the ideal scaling captures the dominant trends.

FAQs

1) Which model should I choose: box or trap?

Use the box model for nearly uniform gases or box traps where density is well defined. Use the trap model for harmonic confinement (optical or magnetic) where the relevant control parameters are N and the trap frequencies.

2) Why does the calculator use zeta functions?

At the transition, the excited-state population reaches a maximum given by Bose statistics. Summing those occupations produces the constants ζ(3/2) for uniform 3D gases and ζ(3) for 3D harmonic traps.

3) What does the “critical density at T” mean in box mode?

It is n_c(T)=\zeta(3/2)/\lambda_T^3, the density required to saturate excited states at the chosen temperature. If your density exceeds this value, the excess particles must accumulate in the condensate.

4) What does N_c(T) mean in trap mode?

It is the maximum number of atoms that can remain thermal at temperature T for the given trap. If your total N is larger than N_c(T), the additional atoms occupy the condensate ground state.

5) Why is the condensate fraction exponent different for box and trap?

The exponent comes from the density of states: a uniform 3D gas yields a (T/T_c)^3/2 scaling, while a harmonic trap yields (T/T_c)³. Different geometries redistribute available excited states.

6) Does this include interactions or scattering length effects?

No. It uses ideal-gas expressions. Interactions can shift T_c and modify condensed fractions by a few percent or more, especially near threshold. Use these results as a baseline before adding mean-field or beyond-mean-field corrections.

7) Can I use this for 2D or 1D gases?

Not directly. Low-dimensional systems exhibit different behavior, including quasi-condensation and Berezinskii–Kosterlitz–Thouless physics in 2D. Use a dedicated low-dimensional model when confinement is strong enough to freeze motion along one or two axes.