### 物理代写|核物理代写nuclear physics代考|ERICE2022

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|核物理代写nuclear physics代考|Inconsistency of the “Plum Pudding” Model

Let us consider what would be expected if the “Plum Pudding” model were indeed correct.

We know from Gauss’ law that at a distance $r$ from the centre of the atom, the electric field is determined by the charge enclosed in a sphere of radius $r$ surrounding the centre of the atom.

The volume of a sphere of radius $r$ is proportional to $r^{3}$. Therefore for $r$ smaller than the radius, $R$, of the atom, the electric charge enclosed with a sphere of radius $r$ is a fraction $r^{3} / R^{3}$ of the total electric charge (assuming a uniform distribution of electric charge throughout the “dough”), so that the magnitude of the electric field at a distance $r$ from the centre of the atom is given by
$$\left(\frac{r^{3}}{R^{3}}\right) \frac{Z e}{4 \pi \varepsilon_{0} r^{2}},(r \leq R) .$$
This is a maximum for $r=R$. This means that the scattering angle cannot be larger than the scattering angle corresponding to impact parameter $b=R$. For values of impact parameter $b<R$, the scattering angle decreases as $b$ decreases.

We have seen above that for $\alpha$-particles with typical kinetic energy of $5 \mathrm{MeV}$, this corresponds to a maximum scattering angle of around $3 \times 10^{-4}$ radians $\left(\approx 0.017^{\circ}\right)$. Such an angle would have been far too small to be observed in any of the GeigerMarsden experiments and they certainly would not have observed any scattering exceeding $90^{\circ}$.

## 物理代写|核物理代写nuclear physics代考|Confirmation of Rutherford Scattering Cross Section

In 1913, Geiger and Marsden [18] performed a far more accurate experiment to check the details of Rutherford’s formula (1.12). They checked the dependence of the rate on the scattering angle and found consistency with the prediction
$$N(\theta) \propto \frac{1}{\sin ^{4}(\theta / 2)}$$
Their results, shown in Fig. 1.5, agree remarkably well.
By using foils of different thickness, they showed that the number of particles scattered through a given angle was proportional to the thickness of the foil, and by using foils made from different metals (tin, silver, copper and aluminium) they were able to show that this number was proportional to the square of the atomic number, $Z$, of the material of the foil.

They were able to slow down the incident $\alpha$-particles, by placing thin sheets of mica immediately in front of the radioactive source. From this they were able to verify that the number of scattered particles was inversely proportional to the fourth power of their velocity, as indicated in (1.12).

## 物理代写|核物理代写nuclear physics代考|Inconsistency of the “Plum Pudding” Model

$$\left(\frac{r^{3}}{R^{3}}\right) \frac{Z e}{4 \pi \varepsilon_{0} r^{2}},(r \leq R)$$

## 物理代写|核物理代写nuclear physics代考|Confirmation of Rutherford Scattering Cross Section

1913 年，Geiger 和 Marsden [18] 进行了更准确的实验来检查卢瑟福公式 (1.12) 的细节。他们检查了速率对散 射角的依赖性，并发现与预测的一致性
$$N(\theta) \propto \frac{1}{\sin ^{4}(\theta / 2)}$$

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