Calculator
Formula Used
Critical temperature:
Tc = [2πℏ² / (m kB)] × [n / (g ζ(3/2))]^(2/3)
Thermal de Broglie wavelength:
λ = √[2πℏ² / (m kB T)]
Phase space density:
PSD = (n / g) × λ³
Here, n is number density. The mass is m. The degeneracy is g. The Bose value ζ(3/2) is about 2.612375.
How to Use This Calculator
- Select whether density comes from particle count and volume or direct density.
- Enter the particle count and volume, or enter number density.
- Choose a particle preset or enter a custom particle mass.
- Enter spin degeneracy for the available internal states.
- Add a reference temperature when you need phase space checks.
- Press Calculate to show the result above the form.
- Use CSV or PDF buttons to save the calculation report.
Example Data Table
| Case | Particle | Density | Degeneracy | Expected Scale |
|---|---|---|---|---|
| Cold atom gas | Rubidium-87 | 1e20 m⁻³ | 1 | Hundreds of nanokelvin |
| Dense helium sample | Helium-4 | 2e28 m⁻³ | 1 | Kelvin range estimate |
| Light boson model | Hydrogen-1 mass | 1e22 m⁻³ | 1 | Higher than heavy atoms |
| Multiple internal states | Sodium-23 | 5e19 m⁻³ | 3 | Lower than g equals one |
Understanding the Condensation Temperature
Bose-Einstein condensation appears when many bosons share one lowest quantum state. The transition needs very low temperature, high particle density, and particles that act as identical bosons. This calculator estimates the critical condensation temperature for an ideal uniform gas. It uses particle number density, particle mass, and spin degeneracy. The result helps compare gases, traps, and classroom examples before a detailed simulation is made.
Why Density Matters
The number density tells how many particles fit inside each cubic meter. A denser gas has smaller average spacing between particles. Quantum wave packets then overlap at a higher temperature. That overlap is the practical signal that condensation can begin. When density is low, the same particles must be cooled further. This is why ultracold atomic experiments need careful compression, evaporation, and trapping control.
Why Mass Matters
Mass has the opposite effect. Light bosons have larger thermal wavelengths at the same temperature. They can overlap more easily, so their critical temperature is higher. Heavy atoms need lower temperatures for the same density. The formula also includes spin degeneracy. Extra internal states spread particles across available quantum states. A larger degeneracy lowers the temperature needed for condensation.
How To Read Results
The main answer is critical temperature in kelvin. The tool also shows microkelvin and nanokelvin forms because laboratory values are often tiny. The density, effective spacing, and thermal de Broglie wavelength are listed for checking. Near the critical point, the phase space density reaches the Bose threshold. Values above the threshold suggest a condensed fraction may be possible, while values below it suggest a normal gas.
Useful Limits
This calculator is best for ideal, weakly interacting, uniform Bose gases. Real experiments may use harmonic traps, finite atom counts, magnetic fields, optical lattices, and interactions. Those effects can shift the observed transition. Still, the estimate is useful. It gives a clean first value, supports lesson planning, and helps test whether input data is physically reasonable.
Exporting Results
CSV output is useful for spreadsheets and repeated runs. PDF output is better for reports and homework records. Keep the selected units visible when sharing results. Small unit mistakes can move answers by many powers of ten. Always review assumptions before reporting values.
FAQs
What is Bose-Einstein condensation temperature?
It is the critical temperature where an ideal boson gas begins to place many particles into the lowest quantum state. Below this value, a macroscopic condensed fraction can appear.
Which inputs affect the result most?
Number density and particle mass have strong effects. Higher density raises the critical temperature. Higher particle mass lowers it. Spin degeneracy also lowers the result when more internal states are available.
Can I use particle count instead of density?
Yes. Select particle count and volume. The calculator converts them into number density by dividing total particles by volume in cubic meters.
What does spin degeneracy mean?
Spin degeneracy counts available internal spin states. If particles are spread across more states, each state has fewer particles. This lowers the ideal condensation temperature.
Why are results often in nanokelvin?
Many dilute atomic gases need extremely low temperatures before quantum wave packets overlap enough for condensation. Nanokelvin units make those small values easier to read.
Does this handle interacting gases?
It gives an ideal gas estimate. Interactions, traps, finite particle counts, and external fields can shift the measured transition temperature in real experiments.
What is phase space density?
Phase space density compares density with thermal wavelength. For this ideal model, condensation begins when the per-state value reaches about 2.612.
Can I export the calculation?
Yes. After calculation, use the CSV button for spreadsheet work. Use the PDF button for reports, notes, and printable records.