Boltzmann Distribution Function
Key idea: Archived extension note on Boltzmann population ratios, degeneracy, and the thermal suppression of higher energy levels.
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The core idea
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Learning objectives
- Use advanced reference material to reinforce mathematical, optical, material, and modern-physics models.
For two non-degenerate energy levels in thermal equilibrium, the population ratio is
Nₓ/N₀ = e^(-(Eₓ-E₀)/(kT))
where k = 1.38 × 10⁻²³ J K⁻¹ and T is in kelvin. If the levels have degeneracies gₓ and g₀, include the statistical weights:
Nₓ/N₀ = (gₓ/g₀)e^(-(Eₓ-E₀)/(kT))
At fixed degeneracy, increasing the energy gap suppresses the upper-level population, while increasing the temperature makes the populations more nearly equal. The relation assumes thermal equilibrium; it is a population model, not a rule for the energy emitted in an individual transition.
Boltzmann factor
Population ratio for equal degeneracies as the energy gap grows relative to thermal energy.
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| Energy-gap ratio (ΔE/(kT)) | e^(−x) |
|---|---|
| 0 | 1 |
| 0.5 | 0.6065 |
| 1 | 0.3679 |
| 1.5 | 0.2231 |
| 2 | 0.1353 |
| 2.5 | 0.0821 |
| 3 | 0.0498 |
| 3.5 | 0.0302 |
| 4 | 0.0183 |
| 4.5 | 0.0111 |
| 5 | 0.0067 |
This factor helps explain why high-energy oscillator states are weakly populated in Planck’s model. Return to Planck’s Hypothesis for the maintained derivation path.
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Course and syllabus information
- Course
- Advanced Physics
- Edition
- Advanced Physics