\[a_{\mathrm A_{1-x}\mathrm B_x} = (1 - x)\,a_\mathrm A + x\,a_\mathrm B\]
带隙的弯曲（bowing）关系band-gap bowing relation
\[E_g(x) = (1-x)E_{g,\mathrm A} + xE_{g,\mathrm B} - bx(1-x)\]含义Meaning
固溶体（合金、混晶）的晶格常数近似为组分晶格常数按组成的线性加权（1921）。用 XRD 测晶格常数即可求组成。
The lattice constant of a solid solution is approximately the composition-weighted average of the end members (1921), so XRD lattice constants give composition.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(x\) | B 组分的摩尔分数mole fraction of B | 1 |
| \(a\) | 晶格常数lattice constant | Å |
| \(b\) | 弯曲参数bowing parameter | eV |
推导Derivation
若两种原子随机占据同一晶格位置、平均键长按组成线性变化，则晶胞参数线性插值。
If atoms randomly share sites and the mean bond length varies linearly, the cell constant interpolates linearly.
带隙一般偏离线性，用二次弯曲项描述。
Band gaps usually bow, needing a quadratic term.
适用条件与局限Assumptions & limitations
经验规律，实际常有正或负偏差；不适用于有序相或两相区。
Empirical, with frequent deviations; not for ordered phases or two-phase regions.
例题Worked example
Al_xGa₁₋ₓAs 晶格常数几乎不随 x 变化（GaAs 5.653 Å，AlAs 5.661 Å），所以可在 GaAs 上外延无应变异质结。
AlₓGa₁₋ₓAs is almost lattice-matched to GaAs (5.653 vs 5.661 Å), enabling strain-free heterostructures.