AQA物理 ALevel 核物理 半衰期 结合能 裂变
从 α 粒子散射到核电站的能量核算，AQA A-level 物理把核物理单独编成一个只在 A-level 出现的单元。它的计算不难，难在概念之间的连线：一条公式连着核半径，另一条连着反应堆的工程选择。下面按官方 spec 的 3.8 条目顺序，把每一条考纲内容拆成可以直接落到卷面上的写法。
From the gold foil experiment to the energy balance of a reactor, nuclear physics is a self-contained A-level unit that rewards clear bookkeeping rather than clever algebra. The formulas are few; the marks are lost on units, on balanced equations and on the reason a chain reaction can be controlled.
核物理在三张卷里的位置
AQA A-level 物理代号 7408 是线性课程，三张卷在课程结束时一次考完，这一点写在 AQA 官网的 spec 概览页上。Paper 1 考第 1 到 5 章加 6.1 周期运动，Paper 2 考 6.2 热学、第 7 章场和第 8 章核物理，两卷都是两小时 85 分，各占 34%。Paper 3 是两小时 80 分占 32%，其中 Section A 45 分考实验与数据处理，Section B 35 分考学生自选的选修单元。核物理属于第 8 章，只在 A-level 出现，AS 阶段的学生学完 α β γ 的性质就停下了。选修单元里天体物理与医学物理会再次用上衰变的内容，选这两个单元的学生要多留一份准备。很多学生以为拿着一个 5 到 6 分的小问只要套公式，阅卷时这类题的失分大多出在核方程没配平和单位没写。
AQA A-level Physics 7408 is linear, so all three papers are sat at the end of the course; Paper 2 carries 85 marks in two hours and covers section 8 alongside thermal physics and fields, while Paper 3 adds 80 marks split into 45 marks of practical skills and data analysis plus 35 marks on one optional topic. Nuclear physics sits in section 8 and appears only at A-level, which is why AS students stop after the properties of alpha, beta and gamma radiation. In practice, examiners see the lost marks in this topic come from unbalanced decay equations and missing units rather than from the arithmetic.
α 粒子散射告诉了我们什么
卢瑟福 1911 年用 α 粒子轰击金箔，把靶原子当成一片可以穿透的区域。绝大多数 α 粒子几乎直线穿过，约八千分之一被反弹回来，这个比例只有把正电荷和几乎全部质量压缩到极小的中心才能解释。核的直径约 10^-15 m，原子直径约 10^-10 m，两者相差五个数量级，所以原子内部绝大部分是空的。最近距离的概念考的是能量转换：当 α 粒子的动能全部转成与核之间的静电势能时，它离核最近，入射能量越大最近距离越小。核式结构模型解释了为什么越靠近核偏转越厉害，也解释了为什么只有极少数粒子被弹回。常见的错误是把散射实验当成测量原子半径的手段，反弹比例只说明核极小，并不直接给出核半径的数值。阅卷时这条时间线只认模型提出的年份，题目里出现 1909 一律按实验执行的那一年处理。
Rutherford's 1911 alpha scattering experiment turned a thin gold foil into a test of the plum pudding model; roughly one alpha particle in 8000 came straight back, a result no spread-out positive charge could produce. That gives a nuclear diameter near 10^-15 m against an atomic diameter near 10^-10 m, so the atom is mostly empty space. Closest approach questions are energy questions: the distance is smallest when the particle's kinetic energy has become electrostatic potential energy, so a faster particle gets closer.
α β γ 的性质与反平方律实验
三种辐射的穿透能力可以直接对比：α 被一张纸挡住，β 需要几毫米铝片，γ 要几厘米铅才能明显衰减。电离能力顺序正好相反，α 最强 γ 最弱，所以 α 源在体外算不上危险，被吸入体内之后危害最大。AQA 的反平方律只对 γ 成立，数据手册在核物理一栏给的是 I = k / x^2。Required practical 12 就是验证这条关系：用 GM 管在不同距离测计数率，先扣掉本底辐射，再画校正计数率对 1 / x^2 的直线。距离加倍时 γ 的计数率降到四分之一，扣完本底这条关系才看得出来。本底不分来源一律要测，实验室里通常是每分钟几十个计数，并不算小。错得最多的是忘记扣本底，数据里于是多出一个常数项，直线不过原点。阅卷时这类实验题的给分点就落在扣本底和用图线验证这两句上，缺一句就少一分。
Alpha stops in a sheet of paper, beta needs a few millimetres of aluminium, and gamma needs several centimetres of lead, which is the reverse order of their ionising power. The inverse-square law applies to gamma only, written in the AQA data booklet as I = k / x^2; required practical 12 tests it by measuring count rate at several distances, subtracting the background and plotting corrected count rate against 1 / x^2. Doubling the distance should cut the corrected count rate to a quarter, and skipping the background correction is what flattens the line.
衰变常数、活度与 A = λ N
放射性衰变的随机性来自单个核每秒的衰变概率恒定，这个概率就是衰变常数 λ。对大量核来说，每秒衰变的数目等于 λ 乘以现有核数，也就是活度 A = λ N，单位贝可。数据手册写的形式是 dN/dt = -λ N 与 N = N0 e^(-λt)，活度按同一条指数衰减，所以 A = A0 e^(-λt) 可以直接用。如果题目给的是摩尔质量和阿伏加德罗常数，先算出核数 N 再乘 λ，别把质量直接代进公式。举个可用的例子：1.0 × 10^15 个碳-14 核，半衰期 5730 年，λ 约 3.8 × 10^-12 s^-1，活度约 3.8 × 10^3 Bq。学生最常把 λ 与半衰期当成可以互换的两个量，算之前先把时间统一到秒，5730 年约 1.8 × 10^11 秒。
Decay is random because each nucleus has a constant probability of decaying per second, and that probability is the decay constant. For a large population the activity is A = lambda N, and since the same exponential describes the numbers, A = A0 e^(-lambda t) applies directly; if a molar mass is given, work out N from Avogadro's constant first rather than substituting the mass. A sample of 1.0 x 10^15 carbon-14 nuclei, with a half-life of 5730 years and a decay constant near 3.8 x 10^-12 per second, has an activity of about 3.8 x 10^3 Bq.
半衰期与图线读数
半衰期由 T½ = ln 2 / λ 给出，约等于 0.693 / λ，含义是把现有核数减半所需的时间。从衰减曲线读半衰期要连续取三次减半，只读一次误差偏大。对数图更省事，把 ln A 对时间作图得到直线，斜率等于 -λ，从斜率反推出半衰期。用数字验一遍：活度从 800 Bq 降到 50 Bq 是十六倍，也就是四个半衰期；若半衰期是 6.0 小时，总共 24 小时。半衰期长的核素适合做放射性定年，碳-14 的 5730 年窗口适合几百年到几万年的样品。半衰期以小时计的核素适合医学示踪，剂量在几天之内就衰减掉。常见的错误是以为过了两个半衰期就不剩什么，实际还剩四分之一，三个半衰期后剩八分之一。
Half-life follows from T = ln 2 / lambda, and reading it off a decay curve is safer with three successive halvings than with one. A log graph is quicker: plotting ln A against time gives a straight line whose gradient is -lambda. If activity falls from 800 Bq to 50 Bq that is four half-lives, so a 6.0 hour half-life means 24 hours; treating two half-lives as complete decay, when a quarter is left, is the usual slip.
N-Z 图与四种衰变方式
稳定核素集中在 N-Z 图上的一条带里，轻核的 N 约等于 Z，重核的 N 大约是 Z 的 1.5 倍。中子过多的核素走 β- 衰变，一个中子变成一个质子并放出电子与反中微子，Z 加一，N 减一。质子过多的核素走 β+ 衰变或电子俘获，两种方式都把 Z 减一，N 加一。重核的 α 衰变使 A 减 4、Z 减 2，代表点沿带向左下移动。激发态核放出 γ 光子时 N 与 Z 都不变，所以 γ 不改变核素种类。核方程配平看 A 与 Z 两个总数守恒，铀-238 的 α 衰变写成 238 = 234 + 4 与 92 = 90 + 2 两条。这一步最容易漏的是 β- 衰变里 Z 加一，学生把电子写进产物却忘了质子数也变了。
Stable nuclei sit in a band on the N-Z graph, with N about equal to Z for light nuclei and about 1.5 times Z for heavy ones. Neutron-rich nuclei use beta-minus decay, proton-rich nuclei use beta-plus or electron capture, and heavy nuclei lose an alpha particle; gamma emission changes neither N nor Z. Balancing a decay equation means matching both A and Z, as in 238 = 234 + 4 and 92 = 90 + 2 for the alpha decay of uranium-238.
核半径与电子衍射
核半径公式是 R = R0 A^(1/3)，数据手册给出 R0 = 1.05 fm，公式本身说明核体积与核子数成正比。代入几个数字就能看出重核并不大：铝-27 得 3.15 fm，铅-208 得 6.22 fm，铀-238 得 6.51 fm。测量核半径靠电子衍射，电子的德布罗意波长远小于原子尺度，第一极小对应的角度满足 sin θ = 0.61 λ / R。电子束能量越高波长越短，衍射图样越能标定核的边界，这也是为什么用电子而不用 α 粒子做精细测量。阅卷时这条公式的给分点有一分专门给核体积与 A 成正比这句理由，只写公式不写理由通常少给分。这一步最容易把 fm 与 m 混用，1 fm = 10^-15 m，半径算完再换算回米。
Nuclear radius is R = R0 A^(1/3) with R0 = 1.05 fm in the AQA data booklet, and the formula itself says the volume of a nucleus is proportional to its nucleon number. That gives 3.15 fm for aluminium-27, 6.22 fm for lead-208 and 6.51 fm for uranium-238, all far smaller than the atom. Electron diffraction supplies the measurement, because a beam whose de Broglie wavelength is well below atomic size gives a first minimum at sin theta = 0.61 lambda / R.
质量亏损、结合能与 E = mc²
核的质量总是小于组成它的质子与中子质量之和，差额叫质量亏损 Δm。氘核是最简单的例子：质子 1.00728 u 加中子 1.00867 u 得 2.01595 u，实际氘核 2.01355 u，亏损 0.00240 u。数据手册写明 1 u 相当于 931.5 MeV，所以氘的结合能约 2.24 MeV，每核子约 1.12 MeV。E = mc² 把亏损的质量换成放出的能量，两者是同一个过程的两种记法。结合能除以核子数得到每核子结合能，铁-56 附近约 8.8 MeV，铀-235 只有约 7.6 MeV。很多学生以为质量亏损说明核里少了东西，其实那部分质量在结合时以能量的形式放掉了。计算时先用 u 求亏损再乘 931.5，比换算成千克再乘 c² 少出错。
A nucleus always weighs less than its separate protons and neutrons, and that difference is the mass defect; for deuterium the defect is 0.00240 u, which the booklet's 1 u = 931.5 MeV turns into about 2.24 MeV. Dividing binding energy by nucleon number gives about 8.8 MeV per nucleon near iron-56 against roughly 7.6 MeV for uranium-235. Working in u and multiplying by 931.5 is faster and less error-prone than converting to kilograms first.
结合能曲线决定裂变与聚变都放能
每核子结合能曲线的形状决定了核能的两个方向，峰值落在铁附近，两端都更低。重核裂变成中等质量的核以后，产物每一核子的结合能更高，差额以能量形式放出。轻核聚变成较重的核，同样是向峰的左侧到峰顶爬升。一次铀-235 裂变放出约 200 MeV，换成焦耳约 3.2 × 10^-11 J。1 kg 铀-235 完全裂变的能量约 8 × 10^13 J，量级上相当于几千吨石油。这道题的给分点通常落在峰的位置和产物结合能更高所以放能这两句上。学生最常把曲线画成单调上升，把铁-56 的峰标到曲线末端，图一错后面两句解释也跟着错。
The binding energy per nucleon curve peaks near iron-56, and both splitting heavy nuclei and fusing light ones move material towards that peak. Each uranium-235 fission releases about 200 MeV, or 3.2 x 10^-11 J, and the complete fission of 1 kg of uranium-235 yields roughly 8 x 10^13 J. The marks come from saying that the products have a higher binding energy per nucleon, not from quoting a number.
诱导裂变与反应堆里的部件
铀-235 吸收一个慢中子后变得不稳定，裂成两个中等质量核并放出两到三个中子，足够维持链式反应。反应堆用慢化剂把快中子降到热能级，石墨与重水都可以，目的是提高铀-235 的裂变概率。控制棒用硼或镉吸收中子，插入越深反应越慢，功率调节靠的是插入深度而不是温度。临界质量指能维持链式反应不衰减的最小质量，低于这个值中子逃逸太多，反应无法自持。铀-238 吸收中子后大多不裂变，天然铀里铀-235 只占约 0.7%，所以反应堆燃料需要浓缩。核废料的半衰期从几十年到上万年不等，储存方案要看具体核素而不是按总量一刀切。这道题的失分大多出在把慢化剂与控制棒的作用说反，阅卷时只认降低中子能量与吸收中子两种写法。
In induced fission a uranium-235 nucleus absorbs a slow neutron, splits into two medium-mass nuclei and releases two or three more neutrons, which is what sustains a chain reaction. A moderator such as graphite or heavy water slows neutrons to thermal energies and raises the fission probability, while boron or cadmium control rods absorb neutrons; natural uranium is only about 0.7 per cent uranium-235, so reactor fuel is enriched. Swapping the jobs of moderator and control rods is the mistake examiners record most often in this question.
核聚变与库仑势垒
聚变要把两个带正电的核推到足够近，库仑斥力先立起一道势垒。太阳靠引力约束，地球上用磁约束或惯性约束，氘-氚反应所需温度约 1.5 亿摄氏度，这是 ITER 官方给出的设计口径。氘-氚聚变放出 17.6 MeV，用质量亏损验算：反应物 2.013553 u 加 3.015501 u，产物 4.001506 u 加 1.008665 u，亏损 0.018883 u，乘 931.5 得 17.59 MeV。聚变的产物是氦与中子，不产生长寿命高放废料，这是它相对裂变的一点优势。真正的难点在维持等离子体，温度、密度与约束时间的乘积要越过点火条件。很多学生以为聚变放能是因为烧掉了质量，其实放出的能量同样来自结合能差额，只是每核子差额比裂变更大。
Fusion has to push two positive nuclei past the Coulomb barrier, and the deuterium-tritium reaction needs a plasma near 1.5 x 10^8 degrees, the figure ITER quotes for its design. The energy released is 17.6 MeV per reaction; the mass defect is 0.018883 u, which the booklet's 931.5 MeV per u turns into 17.59 MeV. Its products carry no long-lived high-level waste, and the hard part is holding temperature, density and confinement time above the ignition condition together.
计算题失分点与三周复习排期
核物理的计算集中在三处：半衰期与 λ 的互换、质量亏损与结合能、核半径公式代入。单位换算是高频失分点，5730 年要换成约 1.8 × 10^11 秒，fm 要换成 10^-15 m，MeV 与焦耳之间差 1.6 × 10^-13 倍。有效数字按题目给的数据位数取，AQA 物理卷要求非整数答案保留两到三位，中途不要过早四舍五入。三周排期可以这样分：第一周 α β γ 与散射实验加上 Required practical 12，第二周衰变方程、活度与对数图，第三周结合能、裂变聚变与两份真题的核物理部分。每周留一次错题重做，只重做核方程配平与单位换算这两类，其他错题看过一遍就够。阅卷记录里核物理小问的得分率往往低于力学同分值小问，丢分点几乎都在配平与单位上。这一节的目的很朴素，把能机械拿到的分先拿满。
The calculations cluster in three places: half-life against decay constant, mass defect against binding energy, and substitution into the nuclear radius formula. Unit conversions are the commonest loss, with 5730 years equal to about 1.8 x 10^11 s and 1 fm equal to 10^-15 m. A three-week plan works if week one covers alpha, beta, gamma and required practical 12, week two covers decay equations, activity and log graphs, and week three covers binding energy, fission and fusion against two past papers.
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