Rutherford Scattering Experiment and Atomic Structure | 卢瑟福散射实验与原子结构

📚 Rutherford Scattering Experiment and Atomic Structure | 卢瑟福散射实验与原子结构

The Rutherford scattering experiment, conducted by Ernest Rutherford together with Hans Geiger and Ernest Marsden between 1909 and 1911, stands as one of the most decisive investigations in the history of physics. It revealed for the first time that the atom contains a tiny, massive, positively charged nucleus at its centre, completely overturning the prevailing ‘plum pudding’ model and laying the foundation for our modern understanding of atomic structure.

卢瑟福散射实验由欧内斯特·卢瑟福与汉斯·盖革、欧内斯特·马斯登于1909至1911年间共同完成，是物理学史上最具决定性的研究之一。该实验首次揭示原子中心存在一个极小、致密、带正电的原子核，彻底推翻了当时盛行的”葡萄干布丁”模型，为现代原子结构观奠定了基础。

1. The Plum Pudding Model | 葡萄干布丁模型

Before Rutherford’s experiment, the most widely accepted description of the atom was J.J. Thomson’s ‘plum pudding’ model, proposed in 1904. According to this model, the atom was a uniform sphere of positive charge, roughly 10⁻¹⁰ m in radius, with negatively charged electrons embedded throughout it like raisins in a pudding. The total positive charge exactly balanced the total negative charge, making the atom electrically neutral.

在卢瑟福实验之前，最广为接受的原子描述是J.J.汤姆逊于1904年提出的”葡萄干布丁”模型。根据这一模型，原子是一个半径约为10⁻¹⁰ m的均匀正电荷球体，带负电的电子像布丁中的葡萄干一样镶嵌其中。正电荷总量恰好与负电荷总量平衡，使原子整体呈电中性。

This model was attractive because it explained two known facts: atoms are electrically neutral, and they can emit electrons when excited. However, it made a crucial prediction about how alpha particles should behave when fired at a thin metal foil. Since the positive charge was spread uniformly over the entire atomic volume, the electric field inside the atom was weak and diffuse.

该模型的吸引力在于它能解释两个已知事实：原子呈电中性，且受激发时可以发射电子。然而，它对α粒子轰击金属薄膜时的行为做出了一个关键预言。由于正电荷均匀分布在整个原子体积内，原子内部的电场是微弱且弥散的。

2. The Alpha Particle Source | α粒子源

The experiment relied on a radioactive source, typically radium or polonium, which emitted alpha particles through nuclear decay. An alpha particle is identical to a helium-4 nucleus, consisting of two protons and two neutrons, carrying a charge of +2e. At the time, it was known that alpha particles were energetic, positively charged projectiles, but their internal structure was not yet fully understood.

实验依赖于放射性源（通常是镭或钋），通过核衰变发射α粒子。α粒子与氦-4核完全相同，由两个质子和两个中子组成，带有+2e的电荷。当时人们已知α粒子是高能、带正电的抛射体，但对其内部结构尚未完全了解。

The alpha particles used in the experiment typically had kinetic energies of approximately 5 MeV. This high energy meant that they could penetrate thin solid foils and travel in nearly straight lines through air over distances of several centimetres. Because of their relatively large mass (about 7300 times that of an electron), any significant deflection of an alpha particle would require a very strong electric force.

实验中使用的α粒子动能约为5 MeV。如此高的能量意味着它们能够穿透薄固体箔片，并在空气中沿近似直线飞行数厘米。由于α粒子的质量较大（约为电子的7300倍），要使α粒子发生显著偏转，需要极强的电场力。

3. The Experimental Setup | 实验装置

The apparatus consisted of three main components. First, a sealed lead container housed the radioactive source, with a narrow slit cut into it to produce a collimated beam of alpha particles. Second, an extremely thin gold foil, approximately 0.6 μm thick or roughly 2000 atoms deep, was placed in the path of the beam. Third, a moveable zinc sulphide (ZnS) screen surrounded the foil, which emitted a flash of visible light whenever an alpha particle struck it.

实验装置由三个主要部分组成。第一，一个密封的铅容器内放置放射源，容器上开有狭窄狭缝以产生准直的α粒子束。第二，一片极薄的金箔（厚度约为0.6 μm，相当于约2000个原子的厚度）置于束流路径上。第三，一个可移动的硫化锌（ZnS）荧光屏环绕金箔，每当α粒子击中屏幕时便发出可见闪光。

Geiger and Marsden sat in a darkened room, using a microscope to observe these tiny scintillations on the screen. By moving the microscope and screen to different angular positions around the foil, they could count the number of alpha particles scattered through various angles θ, measured from the original beam direction. This painstaking procedure required thousands of hours of observation.

盖革和马斯登坐在暗室中，用显微镜观察荧光屏上微小的闪光。通过将显微镜和屏幕移动到箔片周围不同的角度位置，他们可以统计以不同角度θ（相对于原始束流方向）散射的α粒子数目。这一艰苦的过程需要数千小时的观测。

4. The Key Observations | 关键观测结果

The experimental results surprised everyone. The overwhelming majority of alpha particles, well over 99%, passed straight through the gold foil with no measurable deflection at all. A small fraction of the particles, perhaps one in a few hundred, were deflected through small angles of less than about 10°. These observations alone were broadly consistent with the plum pudding model.

实验结果令所有人震惊。绝大多数α粒子（超过99%）径直穿过金箔，完全没有可测量的偏转。一小部分粒子（约几百分之一）被偏转了小于约10°的小角度。仅就这些观测而言，与葡萄干布丁模型大致相符。

However, the truly astonishing finding came from the larger angles. Approximately one in 8000 alpha particles was deflected through an angle greater than 90°, meaning it was sent back toward the source. A very few were even scattered through nearly 180°, bouncing almost directly backwards. Rutherford later remarked that 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.”

然而，真正令人震惊的发现来自更大角度。约每8000个α粒子中就有一个被偏转超过90°，即被反弹回源方向。极少数粒子甚至以接近180°的角度散射，几乎直接向后弹回。卢瑟福后来评论道：”这几乎就像你向一张薄纸发射15英寸炮弹，结果炮弹反弹回来击中了你一样不可思议。”

5. Why the Plum Pudding Model Fails | 葡萄干布丁模型的失败

To understand why these large-angle deflections were so unexpected, we must consider the electric forces predicted by the plum pudding model. If positive charge were spread uniformly across an atom of radius about 10⁻¹⁰ m, then a passing alpha particle would experience only a weak net electrostatic force. The maximum cumulative deflection from traversing even a few thousand atoms would be no more than a fraction of a degree.

要理解为何大角度偏转如此出人意料，我们需要考虑葡萄干布丁模型所预言的电场力。如果正电荷均匀分布在半径约为10⁻¹⁰ m的原子内，那么经过的α粒子只会感受到微弱的净静电力。即使穿越数千个原子，累积的最大偏转角也不会超过几分之一度。

Mathematically, the electric field just outside a uniformly charged sphere of radius R behaves as if all charge were concentrated at the centre, but inside the sphere the field decreases linearly with distance. For a gold atom, whose positive charge would produce a field strong enough to cause some deflection, the field strength at the surface is far too weak to reverse the motion of a 5 MeV alpha particle. Thus, any large-angle scattering was completely impossible under this model.

数学上，均匀带电球体外部的电场行为等同于所有电荷集中于球心，但球内部的电场随距离线性减小。对于金原子，其正电荷产生的电场虽足以导致一定偏转，但表面处的场强远不足以使5 MeV的α粒子反向运动。因此，在这一模型下，大角度散射是完全不可能的。

The only way to produce a deflection greater than 90° is for the alpha particle to experience a very strong electrostatic repulsion from a highly concentrated positive charge during a single close encounter. Such a force can arise only if the positive charge of the atom is confined to a volume far smaller than the atom itself.

产生超过90°偏转的唯一途径，是α粒子在一次近距离遭遇中，受到高度集中的正电荷产生的极强静电斥力。这样的力只有在原子正电荷被限制在远小于原子本身的体积内时才会出现。

6. The Distance of Closest Approach | 最接近距离的计算

Rutherford developed a quantitative analysis of the scattering process by assuming that the alpha particle and the nucleus interact purely through the Coulomb force. For a head-on collision, the alpha particle’s initial kinetic energy is entirely converted into electrostatic potential energy at the point of closest approach. This gives a direct way to estimate an upper bound for the nuclear radius.

卢瑟福通过假设α粒子与原子核之间仅通过库仑力相互作用，对散射过程进行了定量分析。对于对心碰撞，α粒子的初始动能在最接近点完全转化为静电势能。这为估算原子核半径的上限提供了直接方法。

At the distance of closest approach r, energy conservation requires that:

在最接近距离r处，能量守恒要求：

½ m v² = (1 / 4π ε₀) × (Z e × 2e) / r

where m is the alpha particle mass, v is its initial speed, Z is the atomic number of the target nucleus (Z = 79 for gold), e is the elementary charge, and ε₀ is the permittivity of free space. Solving for r:

其中m为α粒子质量，v为其初速度，Z为靶核的原子序数（金为Z = 79），e为元电荷，ε₀为真空介电常数。解出r：

r = (2 Z e²) / (4π ε₀ × ½ m v²)

Using an alpha particle kinetic energy of 5 MeV, this calculation yields r ≈ 3 × 10⁻¹⁴ m, about 1/3000 of the atomic radius. This set an upper limit on the nuclear size. The actual nuclear radius of gold is closer to 8 × 10⁻¹⁵ m (8 fm), since the alpha particle is repelled before reaching the true nuclear surface.

取α粒子动能为5 MeV，该计算给出r ≈ 3 × 10⁻¹⁴ m，约为原子半径的1/3000。这为原子核尺寸设定了一个上限。金原子核的实际半径更接近8 × 10⁻¹⁵ m（8 fm），因为α粒子在到达真实的核表面之前已被排斥。

7. The Nuclear Model of the Atom | 原子的核式模型

To explain the observed scattering pattern, Rutherford proposed in 1911 that the atom consists of a very small, dense, positively charged nucleus containing virtually all of the atom’s mass, surrounded by a much larger cloud of electrons. The nucleus has a radius on the order of 10⁻¹⁵ to 10⁻¹⁴ m, while the atom as a whole has a radius of about 10⁻¹⁰ m.

为解释观测到的散射图案，卢瑟福于1911年提出：原子由一个极小、致密、带正电且几乎包含全部原子质量的原子核构成，周围环绕着体积大得多的电子云。原子核半径约为10⁻¹⁵至10⁻¹⁴ m，而整个原子的半径约为10⁻¹⁰ m。

This model beautifully accounts for all three observations. Alpha particles that travel through the vast empty space between nuclei pass straight through undeflected. Particles that pass near a nucleus experience a relatively strong Coulomb repulsion and are deflected through moderate angles. Particles that make a nearly head-on collision approach very close to the nucleus, where the repulsive force is enormous, and are sent flying backwards.

该模型完美地解释了全部三种观测结果。穿越原子核之间广阔空间的α粒子径直通过而不发生偏转；经过原子核附近的粒子受到较强的库仑斥力而以中等角度偏转；发生近似对心碰撞的粒子则极其接近原子核，在巨大的斥力作用下被弹回。

The nuclear model also explains why the vast majority of particles pass through undeflected: the nucleus occupies such a tiny volume that the probability of a close encounter is extremely small. The ratio of nuclear volume to atomic volume is roughly:

核式模型还解释了为何绝大多数粒子不发生偏转：原子核占据的体积如此之小，近距离相遇的概率极低。原子核体积与原子体积之比约为：

Vₙᵤcₗₑᵤₛ / Vₐₜₒₘ ≈ (10⁻¹⁵ / 10⁻¹⁰)³ = 10⁻¹⁵

This means that matter, at the atomic level, is almost entirely empty space.

这意味着在原子尺度上，物质几乎是完全空旷的。

8. Comparison of the Two Models | 两种模型的对比

The following table summarises the key differences between the plum pudding model and the nuclear model, alongside their predictions for the scattering experiment:

下表总结了葡萄干布丁模型与核式模型的关键区别，以及它们对散射实验的预言：

Property	Plum Pudding Model	Nuclear Model
Positive charge distribution	Spread uniformly throughout the atom	Concentrated in a tiny central nucleus
Mass distribution	Evenly spread over atomic volume	Virtually all mass in the nucleus
Predicted scattering	Only small angles, less than 1°	Large angles possible, including back-scattering
Electron location	Embedded within the positive sphere	Outside the nucleus, orbiting at distance

9. Impact on Modern Physics | 对现代物理学的影响

Rutherford’s nuclear model revolutionised physics. It provided the essential foundation for Niels Bohr’s 1913 model of the hydrogen atom, in which electrons orbit the nucleus in quantised energy levels. Bohr’s model successfully explained the discrete spectral lines of hydrogen, something the nuclear model alone could not account for.

卢瑟福的核式模型彻底改变了物理学。它为尼尔斯·玻尔1913年的氢原子模型提供了必要基础，在该模型中，电子在量子化的能级轨道上绕核运动。玻尔模型成功解释了氢的离散光谱线，这是核式模型本身无法实现的。

The scattering technique itself became a powerful experimental tool. Modern versions are used in particle physics, materials science, and medical imaging. Furthermore, the concept of a small, dense nucleus paved the way for James Chadwick’s discovery of the neutron in 1932, the development of nuclear fission in 1938, and the entire field of nuclear and particle physics that followed.

散射技术本身成为强大的实验工具，其现代版本广泛应用于粒子物理、材料科学和医学成像。此外，小而致密原子核的概念为詹姆斯·查德威克1932年发现中子、1938年核裂变的发展以及随后的整个核物理与粒子物理领域铺平了道路。

10. Exam-Style Questions | 考试风格例题

To consolidate your understanding, consider the following typical IB-style questions:

为巩固理解，请思考以下典型的IB风格题目：

Describe the experimental arrangement used by Geiger and Marsden, and state what was observed at small and at large scattering angles.
描述盖革和马斯登使用的实验装置，并说明在小角度和大角度散射角下分别观察到了什么。

Explain why the observation of alpha particles scattered through angles greater than 90° was incompatible with the plum pudding model of the atom.
解释为何观察到α粒子以大于90°的角度散射与葡萄干布丁原子模型不相容。

An alpha particle with kinetic energy 4.8 MeV is directed head-on at a gold nucleus (Z = 79). Calculate the distance of closest approach.

r = (2 × 79 × (1.6 × 10⁻¹⁹)²) / (4π × 8.85 × 10⁻¹² × 4.8 × 10⁶ × 1.6 × 10⁻¹⁹) ≈ 4.7 × 10⁻¹⁴ m

一个动能为4.8 MeV的α粒子正对金核（Z = 79）入射，计算最接近距离。

The result is approximately 5 × 10⁻¹⁴ m, which is a few tens of femtometres and represents an upper bound for the radius of the gold nucleus. Remember to convert MeV into joules by multiplying by 1.6 × 10⁻¹³ J MeV⁻¹.

结果约为5 × 10⁻¹⁴ m，即几十飞米，代表了金原子核半径的上限。注意需要将MeV乘以1.6 × 10⁻¹³ J MeV⁻¹换算为焦耳。

In summary, the Rutherford scattering experiment is a landmark of experimental physics, demonstrating the power of careful measurement combined with bold theoretical interpretation. It replaced a comfortable but incorrect model with a startlingly accurate one, and set physics on the path toward the quantum mechanical description of matter that we use today.

总而言之，卢瑟福散射实验是实验物理学的里程碑，展示了精密测量与大胆理论解释相结合的力量。它用一个惊人准确的模型取代了一个舒适但错误的模型，并为物理学通向今天所使用的物质量子力学描述铺平了道路。

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Rutherford Scattering Experiment and Atomic Structure | 卢瑟福散射实验与原子结构