A-Level核物理核心概念与公式精讲
一、原子核的结构：质子和中子 | Structure of the Nucleus: Protons and Neutrons
原子核由质子和中子组成，两者统称为核子。质子带一个单位的正电荷，质量约为1.67×10⁻²⁷kg；中子不带电，质量与质子几乎相同。核外电子带负电，质量大约只有核子的1/1836。原子核的直径只有10⁻¹⁵m量级，却集中了原子几乎全部的质量，密度高达10¹⁷kg/m³。质子的数量决定元素的化学性质，也就是原子序数Z；质子与中子的总数称为质量数A。核内的核子依靠强核力束缚在一起，这种力在极短距离内远强于质子之间的静电排斥力，是原子核能够稳定存在的原因。
The nucleus of an atom contains protons and neutrons, which together are called nucleons. Each proton carries one unit of positive charge and has a mass of about 1.67 x 10^-27 kg, while the neutron is electrically neutral and almost identical in mass. Orbiting electrons are negatively charged and have roughly 1/1836 of the mass of a nucleon. A nucleus has a diameter of the order of 10^-15 m yet contains almost all of the atom's mass, which gives it a density near 10^17 kg per cubic metre. The number of protons defines the element and is called the proton number Z, while the total number of protons and neutrons is the mass number A. Nucleons are held together by the strong nuclear force, which easily overcomes the electrostatic repulsion between protons over very short distances and is the reason stable nuclei can exist at all.
二、同位素与核素记号 | Isotopes and Nuclide Notation
核素可以用符号表示：左上角写质量数A，左下角写质子数Z，旁边是元素符号X。例如碳-12写作¹²₆C，碳-14写作¹⁴₆C。质子数相同而中子数不同的原子互称同位素，它们的化学性质几乎相同，但质量和核稳定性不同。碳-12稳定而碳-14具有放射性，两者常出现在放射性测年法的题目中。铀-235与铀-238也是同位素，质子数都是92，差别在于中子数，质量数分别是235和238。
A nuclide is written with the chemical symbol X, the mass number A placed at the top left and the proton number Z at the bottom left. Carbon-12 is written as 12-C with six protons and six neutrons, while carbon-14 has two extra neutrons. Isotopes are atoms of the same element that contain the same number of protons but different numbers of neutrons, so they share almost identical chemical behaviour yet differ in mass and in nuclear stability. Carbon-12 is stable whereas carbon-14 is radioactive, and the pair frequently appears in radiocarbon dating questions. Uranium-235 and uranium-238 are another well-known isotope pair: both have 92 protons, but their mass numbers differ because uranium-238 contains three additional neutrons.
三、质量亏损与结合能 | Mass Defect and Binding Energy
把核子单独分开时的总质量，总是大于它们结合成原子核后的质量，两者之差称为质量亏损。质量并没有凭空消失，而是按照爱因斯坦的质能方程E=mc²转化成了能量。把原子核拆成单个核子所需的能量，也就是把核子结合成原子核时释放的能量，称为结合能。结合能越大，原子核结合得越牢固、越稳定。计算时先求出质量亏损Δm，再乘以光速的平方c²即可得到结合能；质量亏损1u大约对应931MeV的能量。
The total mass of the separated nucleons is always slightly greater than the mass of the assembled nucleus, and this difference is called the mass defect. No mass is actually lost; according to Einstein's equation E = mc^2 the missing mass has been converted into energy. The binding energy of a nucleus is the energy required to split it into its individual nucleons, or equivalently the energy released when those nucleons come together. A larger binding energy means a more tightly bound and more stable nucleus. To calculate it, find the mass defect Δm and multiply by the square of the speed of light c^2; a mass defect of exactly 1 u corresponds to about 931 MeV of energy.
四、结合能曲线与核稳定性 | The Binding Energy Curve and Nuclear Stability
以核子数为横轴、每个核子的平均结合能为纵轴作图，就得到结合能曲线。曲线中段最高，峰值出现在质量数约56的铁-56附近，每个核子的结合能约为8.75MeV；铁之后的重元素曲线缓慢下降。曲线形状解释了核能的来源：轻核聚变时，小原子核合并成中等质量的核，平均结合能上升，释放能量；重核裂变时，铀等大核裂成两个中等核，同样向曲线峰值靠近，也释放能量。因此曲线两端都蕴藏着可利用的核能。
Plotting the binding energy per nucleon against the nucleon number A gives the binding energy curve. The curve rises steeply for light nuclei, peaks in the middle at iron-56 with about 8.75 MeV per nucleon, and then falls slowly for heavier elements. This shape explains where nuclear energy comes from. During fusion, very light nuclei merge into a nucleus of intermediate mass, moving upward along the curve and releasing energy. During fission, a heavy nucleus such as uranium splits into two medium-sized nuclei, also moving toward the peak and releasing energy. Both ends of the curve therefore hold energy that can be exploited.
五、阿尔法、贝塔与伽马衰变 | Alpha, Beta and Gamma Decay
不稳定的原子核会自发衰变并放出射线。α衰变释放出由两个质子和两个中子组成的氦核，核的质量数减少4、质子数减少2，穿透力最弱，一张纸即可挡住。β⁻衰变中，一个中子转化为质子并放出一个电子，质量数不变、质子数增加1；β⁺衰变则相反，质子转化为中子并放出正电子。γ辐射是高能电磁波，常伴随其他衰变释放，穿透力最强，需要厚铅层才能阻挡。书写衰变方程时，必须保证方程两边的质量数与质子数之和守恒。
Unstable nuclei decay spontaneously and emit radiation. Alpha decay releases a helium nucleus containing two protons and two neutrons, so the parent nucleus loses 4 from its mass number and 2 from its proton number; alpha particles are the least penetrating and are stopped by a sheet of paper. In beta-minus decay a neutron changes into a proton and an electron is emitted, so the mass number stays the same while the proton number rises by 1. Beta-plus decay is the reverse process: a proton becomes a neutron and a positron is released. Gamma radiation is high-energy electromagnetic radiation that often accompanies other decays and is the most penetrating, requiring thick lead to absorb. When writing any decay equation, the total mass number and the total proton number must balance on both sides.
六、指数衰变定律与半衰期 | The Exponential Decay Law and Half-Life
放射性衰变是随机过程，无法预测具体哪个原子核在什么时刻衰变，但对大量原子核可以用统计规律描述。单位时间内衰变的原子核数目与当前剩余的原子核数目成正比，这就是衰变定律，数学上写成N=N₀e^(-λt)。半衰期T½是原子核数目减少到原来一半所需的时间，与初始数量无关。经过一个半衰期剩余一半，两个半衰期剩余四分之一，n个半衰期后剩余(1/2)ⁿ。考试常给出质量随时间变化的数据或图表，要求读取半衰期。
Radioactive decay is a random process: we cannot predict which individual nucleus will decay or exactly when, but a large population of nuclei obeys a statistical law. The number of decays per second is proportional to the number of nuclei still present, which leads to the exponential law N = N0 e^(-λt). The half-life T½ is the time taken for half of the nuclei to decay, and it is independent of how many nuclei are present initially. After one half-life half of the sample remains, after two half-lives a quarter remains, and after n half-lives the surviving fraction is (1/2)^n. Exam questions often provide mass-versus-time data or a graph and ask you to read off the half-life.
七、活度与衰变常数的关系 | Activity and the Decay Constant
活度A定义为单位时间内发生衰变的原子核数目，单位是贝克勒尔（Bq），1Bq等于每秒一次衰变。活度与衰变常数λ及当前原子核数目N的关系是A=λN。衰变常数λ表示每个原子核单位时间内的衰变概率，它与半衰期的关系为λ=ln2/T½，约等于0.693/T½。活度随时间同样按指数规律衰减，即A=A₀e^(-λt)。换算时注意单位：半衰期用秒作单位时，λ的单位就是s⁻¹。这类题目多为公式直接代入，但务必先把时间单位统一。
The activity A is the number of decays per second and is measured in becquerels (Bq), where 1 Bq means one decay per second. Activity is related to the decay constant λ and the number of nuclei N by A = λN. The decay constant represents the probability per unit time that a given nucleus will decay, and it is linked to the half-life through λ = ln 2 / T½, which is approximately 0.693 / T½. The activity itself also decays exponentially with time, following A = A0 e^(-λt). Watch the units carefully: when the half-life is expressed in seconds, λ comes out in units of s^-1. Most of these questions are simple substitution, but always convert the time units before plugging numbers in.
八、核裂变与链式反应 | Nuclear Fission and Chain Reactions
核裂变指重核吸收一个中子后分裂成两个中等质量的核，同时释放中子和巨大能量。典型反应为：²³⁵₉₂U + ¹₀n → ¹⁴¹₅₆Ba + ⁹²₃₆Kr + 3¹₀n，一次裂变释放约200MeV的能量。裂变产生的中子又可以引发其他铀-235核裂变，形成链式反应。核反应堆用控制棒（如镉或硼）吸收多余中子来调节反应速率，用慢化剂（如石墨或重水）把快中子减速，提高铀-235捕获中子的概率。裂变释放的热量产生蒸汽，推动汽轮机发电。
Nuclear fission occurs when a heavy nucleus absorbs a neutron and splits into two medium-mass nuclei, releasing neutrons and a large amount of energy. In a typical reaction, uranium-235 absorbs a slow neutron and splits into barium-141 and krypton-92, releasing three neutrons and about 200 MeV of energy. Those released neutrons can trigger further fissions in other uranium-235 nuclei, producing a chain reaction. In a nuclear reactor, control rods made of cadmium or boron absorb surplus neutrons to regulate the reaction rate, while a moderator such as graphite or heavy water slows the fast neutrons down so that they are more easily captured by uranium-235. The heat released turns water into steam that drives turbines and generates electricity.
九、核聚变与恒星能量来源 | Nuclear Fusion and Stellar Energy
核聚变是两个轻核结合成一个较重核的过程。太阳等恒星的能量来自氢核聚变成氦核的反应：四个氢核最终合成一个氦核，质量亏损转化为巨大的能量。聚变要求原子核靠近到强核力起作用的距离，必须先克服质子之间的静电排斥，因此需要极高的温度和压强，太阳核心的温度约为1500万摄氏度。可控聚变（如托卡马克装置）是清洁能源的重要研究方向，但目前仍难以实现持续稳定的反应。考试中要分清：裂变是重核分裂，聚变是轻核合并。
Nuclear fusion is the process in which two light nuclei join together to form a heavier nucleus. The energy of the Sun and other stars comes from hydrogen nuclei fusing into helium: four protons eventually combine to produce one helium nucleus, and the mass defect is released as an enormous amount of energy. Fusion requires the nuclei to approach within the range of the strong nuclear force, which means overcoming the electrostatic repulsion between the positively charged protons first, so extremely high temperatures and pressures are needed, with the solar core reaching about 15 million degrees Celsius. Controllable fusion in devices such as tokamaks is a major research goal for clean energy, but sustained reactions remain very difficult to achieve. In the exam, keep the distinction clear: fission splits heavy nuclei, fusion merges light ones.
十、用E=mc²计算核反应能量 | Calculating Reaction Energy with E=mc²
计算核反应释放的能量一般分三步：先分别求出反应前后所有粒子的总质量；再作差得到质量亏损Δm；最后用E=Δmc²把质量亏损换算成能量。若质量以原子质量单位u给出，1u对应931MeV，例如质量亏损0.2u时释放能量约为0.2×931≈186MeV。注意表述的准确性：核反应中总能量守恒，但静质量不守恒，亏损的质量转化为粒子的动能和辐射能。作答时还要写清单位换算过程，MeV、u与焦耳之间的转换要完整。
To calculate the energy released in a nuclear reaction, follow three steps. First, find the total mass of all the particles before the reaction and the total mass after the reaction. Second, subtract to obtain the mass defect Δm. Third, convert this mass defect into energy using E = Δm c^2. When masses are given in atomic mass units, remember that 1 u corresponds to 931 MeV, so a mass defect of 0.2 u releases roughly 186 MeV. Be precise with your wording: in nuclear reactions total energy is conserved but rest mass is not, because the missing mass appears as kinetic energy and radiation. Always show the unit conversions clearly when working between u, MeV and joules.
十一、常见错误与考试得分技巧 | Common Mistakes and Exam Techniques
核物理题目最常见的错误有三个：一是衰变方程配平错误，写完方程没有检查两边质量数之和与质子数之和是否相等；二是半衰期计算把剩余量当成衰变量，题目问已衰变比例时忘记用1减去剩余比例；三是混淆α、β、γ三种射线的穿透力与电离能力排序。得分技巧：先列出已知量并统一单位；半衰期相关的题优先使用λ=0.693/T½；涉及结合能的题先把质量单位换算成u或kg再代入E=mc²；选择题可用排除法快速判断衰变类型。计算题务必保留正确的有效数字，并写出必要的文字说明。
Three mistakes appear again and again in nuclear physics questions. The first is unbalanced decay equations: after writing an equation, always check that the total mass number and the total proton number are equal on both sides. The second is confusing the remaining fraction with the decayed fraction: when a question asks for the percentage that has decayed, subtract the surviving fraction from 1. The third is mixing up the penetrating and ionising power of alpha, beta and gamma radiation. Useful exam tactics: list the known quantities and convert units before substituting; use λ = 0.693 / T½ for half-life problems; convert masses into u or kg before applying E = mc^2; and in multiple-choice questions use elimination to identify the decay type quickly. Keep the correct number of significant figures and write a short explanation for each step in calculation questions.
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