放射性碳测年 Radiocarbon dating — 公式推导、符号与局限

\[t = \frac{t_{1/2}}{\ln2}\ln\frac{A_0}{A} = 8267\ \mathrm{a}\times\ln\frac{A_0}{A}\quad(t_{1/2} = 5730\ \mathrm a)\]

一般的母子体测年（如 U–Pb、K–Ar、Rb–Sr）general parent–daughter dating (U–Pb, K–Ar, Rb–Sr)

\[t = \frac{1}{\lambda}\ln\left(1 + \frac{N_\text{子}}{N_\text{母}}\right)\]\[t = \frac{1}{\lambda}\ln\left(1 + \frac{N_\text{daughter}}{N_\text{parent}}\right)\]

含义Meaning

活生物体与大气交换碳，¹⁴C/¹²C 保持恒定；死亡后 ¹⁴C 不再补充、按 5730 年半衰期衰减。测量残留比活度即可推算年代（利比，1949）。

Living organisms exchange carbon with the atmosphere, keeping ¹⁴C/¹²C constant; after death ¹⁴C decays with t½ = 5730 a. Measuring the remaining activity gives the age (Libby, 1949).

符号说明Symbols

符号Symbol	含义 Meaning	单位 Unit
\(A_0\)	现代（活体）碳的比活度 ≈ 0.227 Bq/g Cspecific activity of modern carbon ≈ 0.227 Bq per g C	Bq·g⁻¹
\(A\)	样品比活度sample specific activity	Bq·g⁻¹
\(N_\text{子}, N_\text{母}\)	子体、母体核数daughter and parent nuclei	1

推导Derivation

宇宙射线中子在高空产生 ¹⁴C：¹⁴N(n, p)¹⁴C，平衡时大气 ¹⁴C/¹²C ≈ 1.2×10⁻¹²。
Cosmic-ray neutrons make ¹⁴C by ¹⁴N(n,p)¹⁴C; at steady state ¹⁴C/¹²C ≈ 1.2×10⁻¹².
死亡后 \(A = A_0e^{-\lambda t}\)，解出 t。
After death \(A = A_0e^{-\lambda t}\); solve for t.
母子体法：子体积累 \(N_d = N_p(e^{\lambda t} - 1)\)，与初始子体无关时可直接求 t。
Parent–daughter: \(N_d = N_p(e^{\lambda t}-1)\).

适用条件与局限Assumptions & limitations

大气 ¹⁴C 水平历史上有波动（地磁、太阳活动；化石燃料“苏斯效应”；1950–60 年代核试验“炸弹碳”），需用树轮等校准曲线（IntCal）换算为日历年。
Atmospheric ¹⁴C has varied (geomagnetism, solar activity, fossil-fuel Suess effect, bomb carbon), so ages are calibrated with tree rings (IntCal).
适用范围约 300–50 000 年；加速器质谱（AMS）只需毫克级样品。
Range ≈ 300–50 000 years; AMS needs only milligrams.

例题Worked example

木炭 ¹⁴C 活度为现代的 25%：t = 8267×ln4 = 11 460 年（两个半衰期）。

Charcoal at 25 % of modern activity: t = 11 460 years (two half-lives).