📚 IB Physics: Radioactive Decay Exam Guide (HL Included) | IB 物理：放射性衰变考点精讲（含HL）
Radioactive decay is a core topic in the IB Physics syllabus, appearing in both SL and HL. You must be able to write decay equations, interpret decay graphs, and understand the statistical nature of the process.
放射性衰变是IB物理课程的核心主题，在SL和HL中都会出现。你必须能够写出衰变方程、解读衰变图，并理解这一过程的统计性质。
This article organises the essential definitions, equations, and exam strategies, including the extra HL concepts such as decay constant derivations and the continuous beta spectrum.
本文整理了必要的定义、方程和应试策略，并包含HL的额外概念，如衰变常数的推导和连续β能谱。
1. Nucleus Notation and Types of Decay | 核素符号与衰变类型
A nuclide is written with the mass number as a superscript and the atomic number as a subscript before the chemical symbol, for example ²³⁸₉₂U.
核素的写法是在元素符号前用上标表示质量数、下标表示质子数，例如²³⁸₉₂U。
The mass number A is the total number of protons and neutrons; the atomic number Z is the number of protons.
质量数A是质子数和中子数之和；原子序数Z是质子数。
There are four common types of decay: alpha (α), beta-minus (β⁻), beta-plus (β⁺), and gamma (γ). Naturally occurring radioactive nuclides typically emit α, β⁻, and γ, while proton-rich nuclides can undergo β⁺ decay.
有四种常见衰变类型：α衰变、β⁻衰变、β⁺衰变和γ衰变。天然放射性核素通常发射α、β⁻和γ，而富质子核素可发生β⁺衰变。
2. Alpha Decay | α衰变
An alpha particle is a helium nucleus, written as ⁴₂He or α²⁺. In alpha decay, the parent nucleus loses two protons and two neutrons, so the mass number decreases by 4 and the atomic number decreases by 2.
α粒子是氦核，写作⁴₂He或α²⁺。在α衰变中，母核失去两个质子和两个中子，因此质量数减少4，原子序数减少2。
A typical equation is:
典型方程为：
²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He
Alpha particles have low penetrating power (a few centimetres of air or a sheet of paper can stop them) but very strong ionising power.
α粒子穿透力很弱（几厘米空气或一张纸即可阻挡），但电离能力非常强。
3. Beta-Minus Decay | β⁻衰变
Beta-minus decay occurs when a neutron changes into a proton, an electron, and an antineutrino. The electron is emitted as the β⁻ particle.
β⁻衰变发生时，一个中子转变为质子，同时释放一个电子和一个反中微子。释放的电子即β⁻粒子。
The general equation for the decay of a neutron inside the nucleus is:
核内中子的衰变方程一般写作：
n → p + e⁻ + ṽₑ
For example, carbon-14 decays to nitrogen-14:
例如，碳-14衰变为氮-14：
¹⁴₆C → ¹⁴₇N + ⁰₋₁e + ṽₑ
The atomic number increases by 1, while the mass number stays the same because the total number of nucleons is unchanged.
原子序数增加1，质量数保持不变，因为核子总数没有改变。
4. Beta-Plus Decay | β⁺衰变
Beta-plus decay is the reverse process: a proton changes into a neutron, a positron, and a neutrino. The emitted positron is the antiparticle of the electron.
β⁺衰变是逆过程：一个质子转变为中子，同时释放一个正电子和一个中微子。发射的正电子是电子的反粒子。
The equation is:
方程为：
p → n + e⁺ + νₑ
An example is carbon-11 decaying to boron-11:
例如碳-11衰变为硼-11：
¹¹₆C → ¹¹₅B + e⁺ + νₑ
In beta-plus decay, the atomic number decreases by 1 and the mass number is again unchanged.
在β⁺衰变中，原子序数减少1，质量数同样不变。
5. Gamma Decay | γ衰变
Gamma decay does not change the composition of the nucleus. It only releases excess energy in the form of a high-energy photon.
γ衰变不改变原子核的组成，只以高能光子的形式释放多余能量。
It frequently follows alpha or beta decay when the daughter nucleus is left in an excited state. For example:
γ衰变通常发生在α或β衰变之后，此时子核处于激发态。例如：
⁶⁰₂₇Co* → ⁶⁰₂₇Co + γ
Gamma rays are very penetrating and require thick lead or concrete to absorb them.
γ射线穿透力极强，需要厚厚的铅块或混凝土才能吸收。
6. Conservation Laws in Decay Equations | 衰变方程中的守恒定律
Every decay equation must satisfy conservation of mass number (nucleon number), conservation of charge, and conservation of energy-momentum.
每个衰变方程都必须满足质量数守恒、电荷守恒和能量-动量守恒。
For example, in the beta-minus decay of ¹⁴₆C, the mass number is 14 on both sides and the total charge is +6 on the left and +7 − 1 = +6 on the right.
例如，在¹⁴₆C的β⁻衰变中，两边质量数均为14；左边总电荷为+6，右边为+7−1=+6。
When balancing any equation, always check the superscripts and subscripts explicitly. A common exam question is to find the missing particle.
平衡任何方程时，务必检查上标和下标。常见的考题是求缺失的粒子。
7. Half-Life and Decay Constant | 半衰期与衰变常数
The half-life t½ is the time required for half of the radioactive nuclei in a sample to decay. It is measured in seconds, minutes, years, or any suitable time unit.
半衰期t½是指样品中一半放射性原子核发生衰变所需的时间。可用秒、分钟、年等合适的时间单位来度量。
The decay constant λ is the probability per unit time that a given nucleus will decay. The relationship is:
衰变常数λ是某个原子核在单位时间内发生衰变的概率。两者关系为：
t½ = (ln 2) / λ
Therefore λ = (ln 2) / t½. In IB, this derivation is expected in both SL and HL, but HL often asks you to calculate λ from a decay graph.
因此 λ = (ln 2) / t½。在IB中，SL和HL都要求掌握这一推导，但HL更常要求你从衰变图像计算λ。
8. Exponential Decay Law and Activity | 指数衰变定律与活度
The number of radioactive nuclei remaining after time t follows an exponential decay law:
经过时间t后剩余的放射性核数目满足指数衰变定律：
N(t) = N₀ e^(−λt)
Here N₀ is the initial number of nuclei, and λ is the decay constant.
其中N₀为初始核数目，λ为衰变常数。
The activity A is the number of decays per second, measured in becquerels (Bq). It is proportional to N:
活度A是每秒衰变次数，单位为贝克勒尔（Bq）。它与N成正比：
A = λN
The activity also decays exponentially: A = A₀ e^(−λt). On a semi-log graph of N versus t, the gradient is −λ.
活度同样指数衰减：A = A₀ e^(−λt)。在半对数图中，以N对t作图，斜率为−λ。
9. Decay Diagrams and Decay Chains | 衰变图与衰变链
A decay diagram plots neutron number N on the vertical axis and proton number Z on the horizontal axis. Alpha decay moves the nuclide two steps left and two steps down; beta-minus decay moves it one step right and one step down.
衰变图以中子数N为纵轴、质子数Z为横轴。α衰变使核素向左移两步、向下移两步；β⁻衰变使其向右移一步、向下移一步。
Beta-plus decay moves one step left and one step up, while gamma decay is shown as a vertical or nearly vertical transition between energy levels of the same nuclide.
β⁺衰变使核素向左移一步、向上移一步；γ衰变则表现为同一核素能级之间垂直或近似垂直的跃迁。
Some heavy nuclides undergo a long decay chain. For example, ²³⁸₉₂U eventually decays through 14 steps to stable ²⁰⁶₈₂Pb.
某些重核素会发生很长的衰变链。例如²³⁸₉₂U经过14步最终衰变为稳定的²⁰⁶₈₂Pb。
10. Radioactive Dating | 放射性测年
Carbon-14 dating uses the beta-minus decay of ¹⁴₆C with a half-life of about 5730 years. The activity in a living organism is roughly constant because it continuously absorbs carbon-14 from the environment.
碳-14测年利用¹⁴₆C的β⁻衰变，其半衰期约为5730年。生物体在存活时从环境中不断吸收碳-14，因此其体内活度大致稳定。
After death, no new carbon-14 is absorbed, and the observed activity decreases according to the exponential law. The age t can be found from:
死亡后不再吸收碳-14，观测到的活度按指数规律减少。样品的年龄t可由下式求出：
t = (1/λ) ln(A₀/A)
Because carbon-14 has a relatively short half-life, this method is useful for objects up to about 50,000 years old. Other isotopes such as uranium-238 are used for older rocks.
由于碳-14的半衰期相对较短，该方法适用于约5万年以内的物体。更古老岩石的测年可用铀-238等其他同位素。
11. Detection and Radiation Safety | 辐射探测与防护
Common detectors include the Geiger-Müller tube, the cloud chamber, and the scintillation counter. The Geiger-Müller tube produces a pulse for each ionising particle that enters it.
常见的探测器包括盖革-米勒管、云室和闪烁计数器。盖革-米勒管每进入一个电离粒子就产生一个脉冲。
Radiation dose depends on the type of radiation and the tissue involved. In the laboratory, safe practice includes minimising time near a source, maximising distance, and using appropriate shielding.
辐射剂量取决于辐射类型和所涉及的器官。在实验室中，安全操作包括尽量缩短接触源的时间、尽量增大与源的距离，并使用合适的屏蔽。
Shielding rules: α particles are stopped by paper, β⁻ particles by a few millimetres of aluminium, and γ rays by lead or concrete. A beta source should be handled with tongs because the braking radiation (X-rays) from beta in metal can be dangerous.
屏蔽规则：α粒子可用纸挡住，β⁻粒子可用几毫米厚的铝挡住，γ射线则需用铅或混凝土屏蔽。β源应使用镊子夹取，因为β粒子在金属中产生的轫致辐射（X射线）是危险的。
12. HL: Beta Spectrum and Quantum Tunnelling | HL拓展：β能谱与量子隧穿
In beta-minus decay, the emitted electron has a continuous range of energies, not discrete values. This was initially puzzling because alpha particles have discrete energies.
在β⁻衰变中，发射出的电子能量是连续分布，而不是离散值。这一现象最初令人困惑，因为α粒子的能量是离散的。
The explanation is that the decay also produces an antineutrino, which carries away a variable amount of energy. The electron and antineutrino share the total energy release in random proportions.
解释是衰变过程还产生一个反中微子，它带走了可变比例的能量。电子和反中微子随机分配总释放能量。
For alpha decay, HL students should understand that the alpha particle is emitted by quantum tunnelling through the Coulomb barrier around the nucleus. The decay constant λ therefore depends on both the energy of the alpha particle and the height and width of the barrier.
关于α衰变，HL学生应理解α粒子是通过量子隧穿穿过核周围的库仑势垒而发射的。因此衰变常数λ既取决于α粒子的能量，也取决于势垒的高度和宽度。
These HL concepts are often tested in Paper 2 or Paper 3 data-analysis questions, especially when interpreting beta spectra or explaining the role of the neutrino.
这些HL概念常在Paper 2或Paper 3的数据分析题中出现，尤其是解释β能谱或说明中微子作用的时候。
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