📚 The Geiger–Marsden Experiment and the Nuclear Model of the Atom | α粒子散射实验与核式结构模型
The Geiger–Marsden experiment, also known as the Rutherford gold foil experiment, is one of the most important milestones in modern physics. It revealed that the atom consists of a tiny, dense, positively charged nucleus surrounded by mostly empty space, and it led directly to the nuclear model of the atom.
盖革–马斯登实验，又称卢瑟福金箔实验，是现代物理学最重要的里程碑之一。它揭示出原子由一个极小、致密、带正电的原子核以及核外大片的“空荡”空间构成，并直接催生了原子的核式结构模型。
1. Historical Background | 历史背景
By the early 20th century, J.J. Thomson had discovered the electron and proposed the "plum pudding" model, in which negative electrons were embedded in a diffuse sphere of positive charge. This model predicted that alpha particles passing through thin matter would be deflected only slightly.
20世纪初，J.J.汤姆孙发现了电子，并提出了“葡萄干布丁”模型：带负电的电子嵌在均匀分布的正电荷球体中。该模型预言，α粒子穿过薄物质时只会发生微小偏转。
To test this prediction, Ernest Rutherford supervised Hans Geiger and Ernest Marsden at the University of Manchester. They bombarded a very thin gold foil with a beam of alpha particles emitted from a radioactive source.
为了检验这一预言，欧内斯特·卢瑟福在曼彻斯特大学指导汉斯·盖革和欧内斯特·马斯登进行实验。他们用放射源发出的α粒子束轰击极薄的金箔。
2. Experimental Setup | 实验装置
The main components of the setup were as follows:
实验装置的主要组成部分如下：
- Alpha source: radium or polonium, emitting fast helium nuclei (α particles).
- α放射源：镭或钋，发射高速氦原子核（α粒子）。
- Gold foil: extremely thin (~10⁻⁶ m) so most particles passed through.
- 金箔：极薄（约10⁻⁶ m），使大多数粒子能够穿过。
- Zinc sulfide screen: emitted a flash of light when struck by an α particle, allowing detection.
- 硫化锌荧光屏：被α粒子击中时会发出闪光，从而检测粒子位置。
- Microscope: used to view the flashes at different angles around the foil.
- 显微镜：用于观察金箔周围不同角度处的闪光。
α source → collimator → gold foil → zinc sulfide screen
α放射源 → 准直器 → 金箔 → 硫化锌荧光屏
3. Observed Results | 观察到的结果
The results were completely unexpected according to the plum pudding model:
实验结果是“葡萄干布丁”模型完全无法预料的：
| Observation | 观察结果 | Percentage / Significance | 比例/意义 |
| Most α particles passed straight through | ~99% / atom is mostly empty space |
| Some particles were deflected by small angles | Small fraction / electrostatic interaction |
| A few particles were deflected through large angles (>90°) | ~1 in 8000 / close approach to a concentrated charge |
| A very few bounced back (almost 180°) | Very rare / very massive, small positive nucleus |
Some alpha particles were even reflected back toward the source. Rutherford reportedly said: "It was almost as incredible as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you."
有些α粒子甚至被反弹回放射源方向。据说卢瑟福曾感叹：“这简直就像你用15英寸的炮弹去轰击一张薄纸，炮弹却弹回来打中了你自己。” 由此可见，原子内部存在一个极其集中的正电荷核心。
4. Incompatibility with the Plum Pudding Model | 与“葡萄干布丁”模型的矛盾
In Thomson's model, the positive charge was spread over the whole atom (radius ≈ 10⁻¹⁰ m). The maximum electric field inside such a diffuse sphere is far too weak to reverse the momentum of a fast, doubly charged alpha particle.
在汤姆孙模型中，正电荷均匀分布在整个原子上（半径≈10⁻¹⁰ m）。这种弥散球体内最大的电场太弱，根本无法让快速飞行、带两个正电荷的α粒子反向运动。
Quantitatively, the plum pudding model predicted that all scattering angles would be small, with essentially no backward scattering. The observation of large-angle scattering therefore required a completely new model.
定量地说，布丁模型预言所有散射角都很小，几乎不存在背向散射。因此，大角度散射的发现必然要求建立一个全新的原子模型。
5. The Nuclear Model of the Atom | 原子的核式结构模型
Rutherford proposed the nuclear model in 1911. Its key features are:
卢瑟福于1911年提出了核式结构模型，其要点如下：
- The atom contains a central nucleus with most of the atom's mass and all of its positive charge.
- 原子中央存在一个原子核，它集中了原子几乎全部的质量和全部的正电荷。
- The nucleus is extremely small compared to the whole atom (nuclear radius ≈ 10⁻¹⁵ m, atomic radius ≈ 10⁻¹⁰ m).
- 原子核极其微小（核半径≈10⁻¹⁵ m，原子半径≈10⁻¹⁰ m），与整个原子相比微不足道。
- The nucleus is positively charged, with charge +Ze where Z is the atomic number.
- 原子核带正电，电荷量为+Ze，其中Z为原子序数。
- Most of the atom is empty space.
- 原子内部绝大部分是“空”的。
- Electrons orbit the nucleus at relatively large distances, held by electrostatic attraction.
- 电子在离核较远的轨道上运动，依靠静电引力束缚于原子中。
R ≈ 10⁻¹⁵ m (nucleus) vs R ≈ 10⁻¹⁰ m (atom)
原子核半径 ≈ 10⁻¹⁵ m，而原子半径 ≈ 10⁻¹⁰ m
6. Coulomb Scattering and the Closest Approach | 库仑散射与最近距离
When an alpha particle (charge +2e) moves toward a gold nucleus (charge +Ze), it experiences a repulsive Coulomb force. If it approaches head-on, it slows down until its kinetic energy is completely converted into electric potential energy.
当α粒子（电荷+2e）冲向金原子核（电荷+Ze）时，会受到库仑斥力。若粒子正面入射，它会被减速直至动能全部转化为电势能。
At the point of closest approach, d, we have:
在最近距离 d 处，满足：
Eₖ = (1/4πε₀) × (2e)(Ze) / d
Eₖ = (1/4πε₀) × (2e)(Ze) / d
Rearranging:
于是：
d = (1/4πε₀) × (2Ze² / Eₖ)
For alpha particles from a typical source (Eₖ ≈ 5 MeV) scattering off gold (Z = 79), this gives d ≈ 4.5 × 10⁻¹⁴ m. This value is much smaller than the atomic radius and gives an upper bound on the nuclear size.
对于典型放射源放出的α粒子（Eₖ≈5 MeV）在金箔（Z=79）上散射，可算出d≈4.5×10⁻¹⁴ m。该值远小于原子半径，为原子核尺寸提供了一个上限。
7. Scattering Angle and Impact Parameter | 散射角与瞄准距离
The angle through which an alpha particle is deflected depends strongly on the impact parameter b — the perpendicular distance between the initial path and the nucleus.
α粒子的偏转角大小强烈依赖于瞄准距离 b——即入射路径与原子核之间的垂直距离。
- Large b: small Coulomb force, small deflection angle.
- 大 b：库仑力小，偏转角小。
- Small b: strong Coulomb repulsion, large deflection angle.
- 小 b：库仑斥力强，偏转角大。
- b = 0: head-on collision, particle reverses direction (angle = 180°).
- b = 0：正面碰撞，粒子原路返回（偏转角=180°）。
Because most alpha particles pass far from any nucleus, most experience only tiny deflections. Only the rare head-on or near-head-on collisions produce large angles.
由于大多数α粒子都远离原子核飞过，所以它们只发生微小偏转；只有极少数正面或接近正面的碰撞才会产生大角度偏转。
8. Key Conclusions | 关键结论
The experiment supported the following physical conclusions:
该实验支持了以下重要物理结论：
- The atom is mostly empty space: most α particles pass straight through.
- 原子绝大部分是空的：多数α粒子径直穿过。
- All positive charge is concentrated in a tiny central nucleus.
- 全部正电荷集中在一个微小的中心原子核中。
- The nucleus contains almost all of the atom's mass.
- 原子核集中了原子几乎全部的质量。
- The electrons are outside the nucleus and occupy most of the atomic volume.
- 电子位于原子核之外，占据了原子的大部分体积。
9. Limitation of the Rutherford Model | 卢瑟福模型的局限
The Rutherford nuclear model could not explain the stability of the atom. According to classical electromagnetism, an orbiting electron continuously radiates energy, so its radius should shrink and it should spiral into the nucleus within ~10⁻¹¹ s.
卢瑟福核式模型无法解释原子的稳定性。根据经典电磁学，轨道电子不断辐射能量，其半径会逐渐缩小，理论计算表明电子应在约10⁻¹¹ s内坠入原子核。
This difficulty was later resolved by Niels Bohr, who introduced quantized electron orbits and energy levels in 1913. Nonetheless, the nuclear model remains the basis for all modern atomic theory.
这一困难后来由尼尔斯·玻尔解决。他在1913年引入了量子化的电子轨道和能级概念。尽管如此，核式结构模型仍然是现代原子理论的基石。
10. Exam Tips | 考点提示
Candidates are often asked to compare the predicted and observed results, state the conclusions, or calculate the closest approach.
同学们常被要求比较理论的预期结果与实验的观测结果、写出实验结论，或者计算最近距离。
- Memorize the four key observations and their corresponding conclusions.
- 牢记四条关键实验现象及其对应的结论。
- Know the order of magnitude: atomic radius ~10⁻¹⁰ m, nuclear radius ~10⁻¹⁵ m.
- 记住数量级：原子半径约10⁻¹⁰ m，核半径约10⁻¹⁵ m。
- Be able to use energy conservation to find the distance of closest approach.
- 能熟练运用能量守恒求最近距离。
- Explain that large-angle scattering requires a strong, concentrated, positive charge.
- 能解释大角度散射必然要求一个强而集中的正电荷。
Published by TutorHao | Physics Revision Series | aleveler.com
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