## 物理代写|核物理代写nuclear physics代考|PHY471

2022年9月24日

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## 物理代写|核物理代写nuclear physics代考|Discovery of Fission

In 1934 Enrico Fermi, performed experiments in which he bombarded uranium with neutrons [70], thereby provoking transmutation of the nucleus. Uranium, with atomic number 92 , is the heaviest of the naturally occurring elements. Fermi speculated that in some cases the neutron-rich uranium underwent $\beta$-decay to create neptunium, the first transuranic element. ${ }^1$ However, Ida Noddack suggested that rather than the production of neptunium, the neutron bombardment had caused the uranium nucleus to split into two smaller nuclei [71]. The process of such splitting (induced or spontaneous) is called nuclear “fission”. We can see from Fig.3.2, which shows the distribution of binding energies per nucleon, that this process is energetically possible. For the heavier elements, such as uranium, the binding energy per nucleon is considerably lower than for elements in the middle of the periodic table, such as iron or nickel. This means that it is energetically favourable for a heavy nucleus (with atomic mass number greater than about 150) to split into two fragments of smaller nuclei, thereby releasing energy which goes into the kinetic energy of the fragments. A typical fission process releases around $200 \mathrm{MeV}$.
This hypothesis was verified in 1939 by Otto Hahn and Fritz Strassmann, who identified barium as one of the elements produced during the neutron bombardment of uranium [72].

The mechanism by which fission occurs was expounded by Lisa Meitner and her nephew Otto Frisch.

The classical picture is displayed in Fig. 9.1. The nucleus starts off (almost) spherical and then becomes distorted into an (azimuthally symmetric) ellipsoid, (b). Further distortion causes the nucleus to develop a “neck” (c). This is known as the “saddle point”. After further deformation, the two sides of the neck separate (d) into two different nuclei with smaller atomic number and atomic mass number. The point of separation is called the “scission point”.

## 物理代写|核物理代写nuclear physics代考|Spontaneous Fission

For the fission stable nuclides, the energy initially increases as the nucleus becomes less spherical (N.B. a decrease in binding energy means an increase in the rest energy of the nucleus), as shown in Fig. 9.2.

The energy increases for the ellipsoidal configuration (b) of Fig. 9.1, reaching a maximum at (c) where the neck is formed and then the energy decreases and the two fission fragments separate at (d). Although such a potential makes the nucleus classically stable against fission, spontaneous fission can nevertheless occur via quantum tunnelling in a similar way to the quantum tunnelling, which leads to $\alpha$ decay. Spontaneous fission is far less likely than $\alpha$-decay, but it does occur, albeit with half-lives which are much longer than the half-life for $\alpha$-decay. Spontaneous fission of ${ }{92}^{238} \mathrm{U}$ was first observed in 1940 by Konstantin Petrzhak and Georgy Flerov [75]. The fission half-life of ${ }{92}^{238} \mathrm{U}$ is around $10^{16}$ years – compared with the $\alpha$-decay half-life of $4.5 \times 10^9$ years.

We can make an order-of-magnitude estimate of the height of the fission potential barrier. Suppose the two fission fragments have atomic numbers $Z_1$ and $Z_2$, respectively, and atomic mass numbers $A_1$ and $A_2$. The nuclear radii of the fragments are given by $r_1=r_0 A_1^{1 / 3}$ and $r_2=r_0 A_2^{1 / 3}$. The potential maximum, $V_{\max }$, is reached just at the point of separation of two fission fragments (the scission point) and is equal to the Coulomb potential for two electrically charged spheres with charges $Z_1 e$ and $Z_2 e$ whose centres are separated by $r_1+r_2$, as shown in Fig. 9.3.
$$V_{\max }=\frac{Z_1 Z_2 \alpha \hbar c}{r_0\left(A_1^{1 / 3}+A_2^{1 / 3}\right)} .$$

The fission energy released, $Q$, is the difference between initial potential energy and the final potential energy (when the fission products are widely separated), which is equal to the difference between the sum of the binding energy of the fission fragments and the binding energy of the parent.
$$Q=B\left(A_1, Z_1\right)+B\left(A_2, Z_2\right)-B\left(A, Z_1+Z_2\right) .$$
The height, $V_{\text {height, }}$, of the potential barrier is the difference between the potential energy at scission the point (9.1) and the fission energy release, as shown in Fig. $9.4$
$$V_{\text {height }}=V_{\max }-Q \text {, }$$
with $V_{\max }$ and $Q$ given by (9.1) and (9.2), respectively.
The estimate obtained from (9.3) is not very good, because it involves the small difference between almost equal quantities ( $Q$ is of order $200 \mathrm{MeV}$, whereas the barrier height is of order $10 \mathrm{MeV}$ ) so that any fractional error in the estimate of either $Q$ or $V_{\max }$ is amplified in the determination of the barrier height. Furthermore, (9.1) overestimates $V_{\max }$ since it assumes that when the fragments separate they can be considereed as spherical charge distributions, which is not the casee.

A more careful estimate of fission barrier heights [76] is shown in Fig. 9.5, which also shows corrections to these estimates from the effects of the Shell Model. Note the substantial increase in barrier heights where either the number of protons or the number of neutrons is equal to a magic number.

## 物理代写|核物理代写核物理代考|裂变的发现

1934年，恩里科·费米(Enrico Fermi)进行了用中子轰击铀的实验[70]，从而引发了原子核的嬗变。铀的原子序数为92，是自然存在的元素中最重的。费米推测，在某些情况下，富含中子的铀发生了变化 $\beta$衰变产生镎，这是第一种超铀元素。 ${ }^1$ 然而，Ida Noddack认为，与其说是产生了镎，不如说是中子轰击导致铀核分裂成两个更小的核[71]。这种分裂的过程(诱发的或自发的)被称为核裂变。从图3.2中可以看出，这个过程在能量上是可能的，图3.2显示了每个核子结合能的分布。对于较重的元素，如铀，每核子的结合能要比元素周期表中处于中间位置的元素，如铁或镍，低得多。这意味着在能量上有利于一个重的原子核(原子质量数大于150左右)分裂成两个小原子核碎片，从而释放出能量，成为碎片的动能。典型的裂变过程在周围释放 $200 \mathrm{MeV}$这一假设在1939年被Otto Hahn和Fritz Strassmann证实，他们确定钡是铀在中子轰击过程中产生的元素之一[72]

## 物理代写|核物理代写核物理代考|自发裂变

$$V_{\max }=\frac{Z_1 Z_2 \alpha \hbar c}{r_0\left(A_1^{1 / 3}+A_2^{1 / 3}\right)} .$$

$$Q=B\left(A_1, Z_1\right)+B\left(A_2, Z_2\right)-B\left(A, Z_1+Z_2\right) .$$
$V_{\text {height, }}$势垒的值为分裂点(9.1)的势能与裂变能释放的差值，如图所示。 $9.4$
$$V_{\text {height }}=V_{\max }-Q \text {, }$$
with $V_{\max }$ 和 $Q$ 分别由(9.1)和(9.2)给出。从(9.3)得到的估计不是很好，因为它涉及到几乎相等的量( $Q$ 有秩序 $200 \mathrm{MeV}$，而障壁高度则符合规定 $10 \mathrm{MeV}$ )，因此，任何估算中的微小误差 $Q$ 或 $V_{\max }$ 在确定势垒高度时被放大。此外，(9.1)高估了 $V_{\max }$ 因为它假设当碎片分离时，它们可以被认为是球形电荷分布，但事实并非如此

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。