Boltzmann Population Ratio

Enter the two-level energy gap ΔE (eV) and absolute temperature T; compute the high-to-low level particle ratio N₂/N₁ = exp(−ΔE/kT to understand how higher temperature increases upper-level occupancy.

📐 计算公式

玻尔兹曼布居比 N₂/N₁ = e^(−ΔE ÷ (k_B×T))，ΔE 由 eV 乘 1.602176634×10⁻¹⁹ 化为焦耳（k_B = 1.380649×10⁻²³ J/K）；上能级占比 = 比值 ÷ (1+比值) × 100%，下能级占比 = 1 ÷ (1+比值) × 100%。校验：ΔE、T 两项非空数值守卫。判据：ΔE 越大或 T 越低，上能级占比按指数衰减。

输入能级差 ΔE（eV）与温度 T，求两能级粒子数比。

能级差 ΔE (eV)

温度 T (K)

Formula & Standard References

Item	Formula / Note	Source / Standard
Boltzmann distribution	N₂/N₁ = (g₂/g₁)e^(−ΔE/kT)	Boltzmann statistics
Degeneracy	g is level degeneracy	Statistical physics
Population inversion	Requires pumping for N₂ > N₁	Laser principles
Partition function	Z = Σgᵢe^(−Eᵢ/kT)	Statistical mechanics
Temperature	k_B = 1.380649×10⁻²³ J/K	SI 2019
Applications	Spectra, lasers, chemistry	Molecular spectroscopy

📌 计算说明

N₂/N₁ = e^(−ΔE/kT)。
ΔE=1eV,T=300K → 1.6×10⁻¹⁷。

📐 Physical Constants

Constants used on this page and their authoritative sources, for verification and citation

Constant	Value	Unit	Source
Boltzmann constant k (k_B)	1.380649 × 10⁻²³	J·K⁻¹	SI defining constant (exact)
Elementary charge e	1.602176634 × 10⁻¹⁹	C	SI defining constant (exact)

Note: k and e are exact SI defining constants; when ΔE is given in eV it must be multiplied by e to convert to joules in ΔE/kT.

📚 深度解析：玻尔兹曼布居比

💡 常见使用场景

两能级粒子数之比。
激光/热平衡上下能级反转判断。
温度对激发占比的影响。

能级差 0.1eV，300K

N_高/N_低=e^(−ΔE/kT)=e^(−0.1/(8.617e-5×300))=e^(−3.86)=0.021，高态仅占约 2%。

1eV 能级差

kT≈0.0259eV，比值=e^(−38.6)≈1.6e-17，常温下几乎全在基态。

❓ 常见问题（FAQ）

kT 室温多大？: 约 0.0259 eV；ΔE≫kT 时上能级几乎不被占据。
简并度影响吗？: 严格为 g_高/g_低·e^(−ΔE/kT)，本工具按简并度 1 计。

玻尔兹曼布居比计算器