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Energy Released in Radioactive Decay | 放射性衰变释放的能量

2026-08-29 13:01 · 本文部分内容由AI辅助创作

📚 Energy Released in Radioactive Decay | 放射性衰变释放的能量

Radioactive decay releases energy when an unstable nucleus transforms into a more stable configuration. This energy appears as the kinetic energy of emitted particles and gamma photons. The source of the energy is a small loss of rest mass, described by Einstein's mass-energy equivalence. This article explains the mass defect, binding energy, the Q-value of decay, and how to calculate the energy released in alpha, beta and gamma processes for A-Level physics.

放射性衰变在不稳定原子核转变为更稳定组态时释放能量。这些能量表现为发射粒子和γ光子的动能。能量来源于微小的静质量损失，由爱因斯坦的质能等价关系描述。本文将解释质量亏损、结合能、衰变 Q 值，以及 A-Level 物理中 α、β、γ 过程释放能量的计算方法。

1. Mass-Energy Equivalence and the Atomic Mass Unit | 质能等价与原子质量单位

Einstein' gy ΔE = Δm c². In nuclear physics, masses are conveniently expressed in atomic mass units, u. One atomic mass unit is defined as 1/12 of the mass of a carbon-12 atom.

爱因斯坦关系 E = mc² 表明质量和能量是同一量的两种形式。如果系统损失静质量 Δm，就会释放能量 ΔE = Δm c²。在核物理中，质量通常用原子质量单位 u 表示。一个原子质量单位定义为碳-12 原子质量的 1/12。

E = m c²

1 u = 1.6605 × 10⁻²⁷ kg

1 u c² = 931.5 MeV

pletely into energy gives 1.6605 × 10⁻²⁷ × (2.9979 × 10⁸)² = 1.492 × 10⁻¹⁰ J, which is 931.5 MeV. This conversion factor is the key to all decay-energy calculations: a mass loss of 1 u releases 931.5 MeV of energy.

换算因子 931.5 MeV/u 来自电子伏特定义。因为 1 eV = 1.602 × 10⁻¹⁹ J，一个原子质量单位完全转化为能量时给出 1.6605 × 10⁻²⁷ × (2.9979 × 10⁸)² = 1.492 × 10⁻¹⁰ J，即 931.5 MeV。这个换算因子是所有衰变能计算的关键：质量亏损 1 u 释放 931.5 MeV 能量。

2. Binding Energy and Nuclear Stability | 结合能与核稳定性

The binding energy of a nucleus is the energy required to separate it into its individual protons and neutrons. The mass of a nucleus is always less than the sum of the masses of its separated nucleons; this difference is the mass defect. Multiplying the mass defect by c² gives the binding energy.

原子核的结合能是将它拆散为单个质子和中子所需的能量。原子核的质量总是小于其分离核子质量之和；这个差值就是质量亏损。将质量亏损乘以 c² 即得结合能。

Δm = Z mp + N mn − m_nucleus

BE = Δm c²

The binding energy per nucleon increases from light nuclei up to iron-56 and then decreases. A decay is energetically allowed if the products have greater binding energy per nucleon than the parent, so energy is released as the system becomes more tightly bound. The table below gives typical values.

每个核子的结合能从轻核到铁-56 逐渐增大，之后又下降。如果产物的每个核子结合能大于母核，则衰变在能量上是允许的，系统变得更紧密时释放能量。下表给出典型数据。

Nuclide | 核素Binding energy per nucleon / MeV | 每核子结合能 / MeV
He-4 | 氦-47.07
C-12 | 碳-127.68
Fe-56 | 铁-568.79
U-238 | 铀-2387.57

3. Mass Defect in Decay Processes | 衰变过程中的质量亏损

In any spontaneous decay, the total rest mass of the products is less than the rest mass of the parent. This mass difference is the mass defect of the decay, and it is converted into the kinetic energy of the products and the energy of any emitted gamma photons.

在任何自发衰变中，产物的总静质量小于母核的静质量。这个质量差就是衰变的质量亏损，它转化为产物的动能以及任何发射γ光子的能量。

Δm = mparent − Σ m_products

Q = Δm c² = (mparent − Σ m_products) c²

The quantity Q is called the Q-value of the decay. If Q is positive, the decay is energetically possible; if Q is negative, the decay cannot occur spontaneously. Charge and nucleon number are still conserved in the process, even though mass is not conserved because some rest mass becomes energy.

量 Q 称为衰变的 Q 值。如果 Q 为正，衰变在能量上可能发生；如果 Q 为负，衰变不能自发发生。过程中电荷数和核子数仍然守恒，虽然质量不守恒，因为部分静质量变成了能量。

4. Energy Released in Alpha Decay | α衰变释放的能量

In alpha decay, a nucleus emits a helium-4 nucleus, which is called an alpha particle. A general example is radium-226 decaying into radon-222:

在 α 衰变中，原子核发射一个氦-4 核，即 α 粒子。一个典型例子是镭-226 衰变为氡-222：

²²⁶₈₈Ra → ²²²₈₆Rn + ⁴₂He + Q

e sum of the product masses is 226.020181 u, so Δm = 226.025410 − 226.020181 = 0.005229 u.

使用原子质量：m(Ra-226) = 226.025410 u，m(Rn-222) = 222.017578 u，m(He-4) = 4.002603 u。产物质量之和为 226.020181 u，因此 Δm = 226.025410 − 226.020181 = 0.005229 u。

Q = 0.005229 × 931.5 MeV = 4.87 MeV

Most of this energy is carried away as kinetic energy of the alpha particle and the recoiling radon nucleus. In A-Level calculations, you should always subtract the total product mass from the parent mass, not the other way round.

这些能量的大部分以 α 粒子和反冲氡核的动能形式带走。在 A-Level 计算中，你应当始终用母核质量减去产物总质量，而不是反过来。

5. Energy Released in Beta Decay | β衰变释放的能量

Beta-minus decay converts a neutron into a proton, emitting an electron and an antineutrino. Beta-plus decay converts a proton into a neutron, emitting a positron and a neutrino. Electron capture is a third process in which a nucleus captures an inner electron.

β⁻ 衰变将中子转化为质子，发射电子和反中微子。β⁺ 衰变将质子转化为中子，发射正电子和中微子。电子俘获是第三种过程，原子核俘获一个内层电子。

¹⁴₆C → ¹⁴₇N + e⁻ + ν̄ₑ + Q

β⁻: Q = [mparent(atom) − m_daughter(atom)] c²

β⁺: Q = [mparent(atom) − m_daughter(atom) − 2m_e] c²

r beta-plus decay, two electron masses, 2 × 0.000549 u = 0.001098 u, must be subtracted, equivalent to a threshold of 1.022 MeV. In carbon-14 decay, the mass difference is about 0.000168 u, so Q = 0.156 MeV, which is shared between the electron and the antineutrino.

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对于 β⁻ 衰变，使用原子质量很方便，因为发射电子的质量与子原子中多出的一个电子质量相消。对于 β⁺ 衰变，必须减去两个电子质量，即 2 × 0.000549 u = 0.001098 u，相当于 1.022 MeV 的阈值。在碳-14 衰变中，质量差约为 0.000168 u，所以 Q = 0.156 MeV，由电子和反中微子共享。

6. Gamma Emission and Excited Nuclear States | γ发射与核激发态

Gamma emission occurs when a nucleus in an excited state drops to a lower energy state and emits a high-energy photon. The nucleon number A and proton number Z do not change, so gamma emission is not a transmutation. The photon energy is equal to the energy difference between the two nuclear levels.

γ 发射发生在处于激发态的原子核跃迁到较低能态并发射高能光子时。核子数 A 和质子数 Z 不变，因此 γ 发射不是一种嬗变。光子能量等于两个核能级之间的能量差。

hf = E_initial − Efinal

Gamma emission often follows alpha or beta decay because the daughter nucleus may be left in an excited state. For example, technetium-99m emits a 140 keV gamma photon, and caesium-137 decays by beta emission to excited barium-137m, which then emits a 662 keV gamma photon. The energy of these photons can be measured to identify the nuclide.

γ 发射通常紧随 α 或 β 衰变，因为子核可能处于激发态。例如，锝-99m 发射 140 keV 的 γ 光子，铯-137 经 β 衰变到激发的钡-137m，然后发射 662 keV 的 γ 光子。这些光子的能量可用于识别核素。

7. Q-value, Kinetic Energy and Momentum Conservation | Q值与动能、动量守恒

ry, then Q = KE_products − KE_parent = KE_products. Momentum must also be conserved, so the products move in opposite directions with equal magnitudes of momentum.

衰变的 Q 值等于母核衰变前静止时产物获得的总动能。如果母核静止，则 Q = 产物动能 − 母核动能 = 产物动能。动量也必须守恒，因此产物沿相反方向运动，动量大小相等。

Q = KE₁ + KE₂ + KE₃ + …

For alpha decay, the alpha particle and the daughter nucleus have equal magnitudes of momentum. Since kinetic energy is p²/2m, the lighter alpha particle receives the larger share of the energy. The alpha particle carries a fraction (A − 4)/A of the Q-value, and the daughter nucleus carries 4/A.

对于 α 衰变，α 粒子和子核的动量大小相等。由于动能是 p²/2m，较轻的 α 粒子获得较大的能量份额。α 粒子携带 Q 值的 (A − 4)/A 部分，子核携带 4/A 部分。

ic energy is 4.87 × 222/226 = 4.78 MeV, and the radon nucleus recoils with about 0.09 MeV.

对于 Q = 4.87 MeV 的镭-226 例子，α 粒子动能为 4.87 × 222/226 = 4.78 MeV，氡核反冲动能约为 0.09 MeV。

8. Worked Example: Alpha Decay of Uranium-238 | 计算示例：铀-238的α衰变

and m(He-4) = 4.002603 u. The total product mass is 234.043601 + 4.002603 = 238.046204 u.

考虑衰变 ²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He。原子质量为 m(U-238) = 238.050788 u，m(Th-234) = 234.043601 u，m(He-4) = 4.002603 u。产物总质量为 234.043601 + 4.002603 = 238.046204 u。

Δm = 238.050788 − 238.046204 = 0.004584 u

Q = 0.004584 × 931.5 = 4.27 MeV

Q = 4.27 × 1.602 × 10⁻¹³ J = 6.84 × 10⁻¹³ J

eus receives about 0.07 MeV. This is a typical alpha-decay energy, and it explains why alpha particles from a given nuclide have discrete, characteristic energies.

α 粒子获得 4.27 × 234/238 = 4.20 MeV，钍核获得约 0.07 MeV。这是典型的 α 衰变能量，也解释了为什么给定核素发射的 α 粒子具有分立且特征性的能量。

9. Using MeV per u and MeV per Nucleon | 使用 MeV/u 与 MeV/核子

The fastest way to calculate the energy released in a decay is to find the mass defect in u and multiply by 931.5 MeV/u. This avoids converting every mass into kilograms and back again. The same method applies to fusion and fission calculations.

计算衰变释放能量最快的方法是先求出以 u 为单位的质量亏损，再乘以 931.5 MeV/u。这样可以避免把每个质量都换算成千克再换算回来。同样的方法也适用于聚变和裂变计算。

Q / MeV = Δm / u × 931.5

nergy per nucleon, divide the total energy by the total number of nucleons in the products.

如果题目要求以焦耳表示能量，就用千克质量代入 E = Δm c²，或把 MeV 结果按 1 MeV = 1.602 × 10⁻¹³ J 换算。如果题目要求每个核子的能量，就把总能量除以产物中的总核子数。

10. Energetic Feasibility and the Binding-Energy Curve | 能量阈值与结合能曲线

A decay can occur spontaneously only if Q is positive, which means the products are more tightly bound than the parent. On the binding-energy-per-nucleon curve, this corresponds to moving toward the maximum near iron-56. Light nuclei release energy by fusion, very heavy nuclei release energy by fission and alpha decay, and nuclei far from the stability line release energy by beta decay.

只有当 Q 为正，即产物比母核结合得更紧时，衰变才能自发发生。在每核子结合能曲线上，这对应于向铁-56 附近的最大值移动。轻核通过聚变释放能量，很重的核通过裂变和 α 衰变释放能量，远离稳定线的核通过 β 衰变释放能量。

The decay energy is also related to the difference in total binding energy between parent and products:

衰变能量也与母核和产物总结合能之差有关：

Q = BE_products − BE_parent

². It is useful because it shows directly that a decay releases energy when the final nucleus is more stable, that is, more tightly bound per nucleon.

这个方程等价于 Q = Δm c²。它很有用，因为它直接表明：当最终原子核更稳定，即每个核子结合得更紧时，衰变释放能量。

11. Exam Techniques and Common Mistakes | 考试技巧与常见错误

Most marks in decay-energy questions are lost through small errors rather than through lack of understanding. Check the following points carefully in every calculation.

衰变能题目中丢失的分数大多来自小错误，而不是理解不足。每次计算都要仔细检查以下几点。