# 物理代写|核物理代写nuclear physics代考|PHZ4303

#### Doug I. Jones

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

In most cases, there are two daughter nuclei (fission fragments), although in about one in 300 fission events a third nucleus is produced. This is usually a small nucleus, such as ${ }_1^3 \mathrm{H}$ (tritium) or ${ }_2^4 \mathrm{He}$ ( $\alpha$-particle).

The Liquid Drop Model favours splitting into two fragments of approximately equal atomic mass, $Z$, and atomic mass number, $A$. This is, however, not what is observed. The percentage fission yields for different atomic mass number, $A$, are shown in Fig. 9.6, for the case of the fission of ${ }{92}^{238} \mathrm{U}$. The maximum yield occurs when the atomic mass numbers, $A{+}$and $A_{-}$, of the two fission fragments are in a ratio between $1.3$ and $1.5$. The reason for this asymmetry is not known, but a hint can be obtained from the fact that the favoured atomic mass number for the heavier nuclide is around 132 and this appears to be independent of the parent nuclide which undergoes fission. This is the atomic mass number of the doubly magic nuclide ${ }_{50}^{132} \mathrm{Sn}$. This might be the origin of the peak in the yield around $A=132$, i.e. there is a preference for one of the fission fragments to have atomic number and atomic mass number close to that of this doubly magic nuclide.

As can be seen from Fig. 3.4, the number of neutrons per proton in stable isotopes increases with increasing atomic number. This means that the parent nuclide in a fission process always has too many neutrons for the fission fragment nuclides to be stable. Most fission processes are therefore accompanied by the emission of two or three neutrons, known as “prompt neutrons” as they are emitted simultaneously with the fission process.

The prompt release of energy in a fission process, $Q$, is the difference between the binding energy of the parent nuclide and the fission products. For example, for the fission of ${ }{92}^{236} \mathrm{U}$, $${ }{92}^{236} \mathrm{U} \rightarrow{ }{54}^{140} \mathrm{Xe}+{ }{38}^{94} \mathrm{Sr}+2 n,$$
the binding energies of isotopes ${ }{92}^{236} \mathrm{U},{ }{54}^{140} \mathrm{Xe}$ and ${ }_{38}^{94} \mathrm{Sr}$ are $1790.4 \mathrm{MeV}, 1160.7 \mathrm{MeV}$ and $807.8 \mathrm{MeV}$, respectively, and so the energy released by this fission is $178 \mathrm{MeV}$. Most of this energy goes into the kinetic energy of the fission products (including the prompt neutrons), but in many cases the fission fragments are produced in excited states and then decay emitting $\gamma$-rays.

Notwithstanding the prompt neutron emission, the fission fragments are still neutron rich (they have too many neutrons for stability). For example, in the fission reaction of (9.4), the heaviest stable isotope of strontium has atomic mass number 88 and the heaviest stable isotope of xenon has atomic number 134. This means that there are 12 too many neutrons for stability. These neutron-rich fission products usually undergo $\beta$-decay in a fairly long decay chain until stability is reached. For example, the fission fragment ${ }{54}^{140} \mathrm{Xe}$ decays in 4 stages to ${ }{58}^{140} \mathrm{Ce}$ (cerium), which is stable. Several of these $\beta$-decays will be to daughter nuclides in an excited state, and so there are also $\gamma$-rays emitted in conjunction with these decays. On average there are $10 \gamma$-rays emitted for each fission event. These secondary decays also release energy. The total energy produced in a fission process from beginning to end is usually about $10 \%$ higher than the prompt fission energy generated by the initial fission reaction.

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

Spontaneous fission occurs as a result of quantum tunnelling, whose probability is very small. Much more likely is induced fission (which is actually how fission was first observed [72]). In this case the parent nucleus, with atomic mass number $A$, is bombarded with a neutron. The neutron is absorbed if the binding energy of the isotope with atomic number $A+1$ exceeds the binding energy of the isotope with atomic number, $A$, and the excess energy is released in the form of vibrational energy. If this neutron absorption energy is greater than the height of the fission potential barrier, then fission can proceed promptly – no quantum tunnelling is required.
An example of this is the process
$$n+{ }{92}^{235} \mathrm{U} \rightarrow{ }{92}^{236} \mathrm{U} \rightarrow{ }{56}^{141} \mathrm{Ba}+{ }{26}^{92} \mathrm{Kr}+3 n,$$
shown schematically in Fig.9.7. The binding energy of ${ }{92}^{235} \mathrm{U}$ is $1783.9 \mathrm{MeV}$, whereas the binding energy of the more stable isotope, ${ }{92}^{236} \mathrm{U}$, is $1790.4 \mathrm{MeV}$. The height of the potential barrier (for the above fission process) is $5.6 \mathrm{MeV}$, which is smaller than the binding energy of the extra neutron in the isotope ${ }_{92}^{236} \mathrm{U}$, so that the absorption of the neutron releases sufficient energy to overcome the potential barrier. In this case the bombarding neutrons need not be energetic. Fission can be induced by thermal neutrons which have a kinetic energy of around $0.025 \mathrm{eV}$. Nuclides for which induced fission can be accomplished using thermal neutrons are called “fissile”.

On the other hand, if the binding energy of the extra neutron is insufficient to overcome the potential barrier, the incident neutron needs to have sufficient kinetic energy to overcome the shortfall, thereby inducing fission. Nuclides for which fission can be induced by bombardment with neutrons of sufficient kinetic energy are called “fissionable”. An example of a fissionable (but not fissile) nuclide is ${ }{92}^{238} \mathrm{U}$ (the most abundant isotope of uranium). This has a fission barrier height of $6.3 \mathrm{MeV}$. Its binding energy is $1801.7 \mathrm{MeV}$, whereas the binding energy of ${ }{92}^{239} \mathrm{U}$ is 1806.5 MeV. The vibrational energy generated by the absorption of a neutron by a nucleus of ${ }{92}^{238} \mathrm{U}$ is therefore $4.8 \mathrm{MeV}$, which is insufficient to overcome the barrier potential. In order to induce fission in ${ }{92}^{238} \mathrm{U}$, the incident neutrons need to have at least enough kinetic energy to make up the shortfall of $1.5 \mathrm{MeV}$. It is very often the case that isotopes with an odd number of neutrons are fissile, whereas isotopes with an even number of isotopes are not. This is due to the pairing term contribution to the nuclear binding energy – adding a neutron to an isotope with initially an unpaired neutron produces an isotope in which all the neutrons are paired, thereby significantly increasing its binding energy.

## 物理代写|核物理代写核物理代考|诱导裂变

$$n+{ }{92}^{235} \mathrm{U} \rightarrow{ }{92}^{236} \mathrm{U} \rightarrow{ }{56}^{141} \mathrm{Ba}+{ }{26}^{92} \mathrm{Kr}+3 n,$$
。${ }{92}^{235} \mathrm{U}$的结合能是$1783.9 \mathrm{MeV}$，而更稳定的同位素${ }{92}^{236} \mathrm{U}$的结合能是$1790.4 \mathrm{MeV}$。势垒的高度(对于上述裂变过程)为$5.6 \mathrm{MeV}$，小于同位素${ }_{92}^{236} \mathrm{U}$中多余中子的结合能，使中子的吸收释放出足够的能量来克服势垒。在这种情况下，轰击中子不一定是高能的。裂变可由动能约为$0.025 \mathrm{eV}$的热中子诱发。可以利用热中子进行诱导裂变的核素称为“可裂变的”

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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。

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