# 物理代写|粒子物理代写Particle Physics代考|PHYS422

## 物理代写|粒子物理代写Particle Physics代考|The Collective Model

The Shell Model has its shortcomings. In spite of its great success, the usefulness of the Shell Model should not be overstated. It has a limited range of validity – it can explain phenomena mainly relevant to the light spherical nuclei, but even in this case one observes discrepancies between the predictions of the model and experiment. These discrepancies are even larger for heavier nuclei. We have already seen that the Shell Model does not predict magnetic dipole moments or the spectra of excited states very well.

One further failing of the Shell Model is the prediction of electric quadrupole moments. The Shell Model predicts very small values for these. However, for heavier nuclei with $A$ in the range of 150-190 and for $A>220$, these electric quadrupole moments are found to be rather large. The failure of the Shell Model to correctly predict electric quadrupole moments arises from the assumption that the nucleons move in a spherically symmetric potential.

A model that generalizes the Shell Model is the Collective Model, which considers the effect of a non-spherically symmetric potential (leading to substantial deformations for large nuclei and consequently large electric quadrupole moments) and takes into account interactions between nucleons. One of the most striking consequences of the Collective Model is the explanation of low-lying excited states of heavy nuclei. These excitations are of two types:

• Rotational States. A nucleus whose nucleon density distributions are spherically symmetric (zero quadrupole moment) cannot have rotational excitations (this is analogous to the application of the principle of equipartition of energy to monatomic molecules for which there are no degrees of freedom associated with rotation). On the other hand, a nucleus with a non-zero quadrupole moment can have excited levels due to rotation perpendicular to the (rotational) axis of symmetry.

Some nuclides have a higher binding energy than some of their neighbours. When this is the case, it is often energetically favourable for a nuclide with a lower binding energy (“parent nuclide”) to decay into one with a higher binding energy (“daughter nuclide”) plus another particle or particles associated with different types of radioactive decay. In many cases, a given element will have two or more naturally occurring radioactive isotopes in addition to two or more stable isotopes.
There are three main types of radioactivity corresponding to three different types of emitted particles. The sum of the kinetic energies of the final-state particles is the difference in the binding energy between the parent and daughter nuclides and is usually a few $\mathrm{MeV}$.

• ” $\alpha$-decay” – The emission of an $\alpha$-particle, which is a ${ }2^4 \mathrm{He}$ (helium) nucleus. In such decays the daughter nuclide has an atomic number which is two less than that of the parent and an atomic mass number four less than that of the parent. The rest energy of the $\alpha$-particle, $m\alpha c^2$, is much larger than its kinetic energy, so that it travels at a speed much lower than the speed of light. It is also doubly charged and therefore easily ionizes media through which it travels. It can be shown that if the electrons in an absorbing material are treated as free particles
• (a reasonable approximation as their binding energy is certainly small compared with the kinetic energy of the emitted $\alpha$-particles), then the energy loss per unit distance travelled through the material is inversely proportional to the square of the velocity. This means that as the $\alpha$-particles have a small velocity when they enter the absorbing material, they lose most of their initial energy very rapidly and can only penetrate a thin sheet of paper and are absorbed by a single layer of human skin.
• ” $\beta$-decay” – The emission of an electron or a positron.
Positron emission is accompanied by the emission of a very low-mass particle called a “neutrino”. Electron emission is accompanied by the antiparticle of the neutrino (the antineutrino). The daughter nuclide has atomic number one more (in the case of electron emission) or one less (in the case of positron emission) than that of the parent but has the same atomic mass number. The difference in the binding energies is equal to the total energy of the decay products. The electron rest energy is usually small in comparison with the energy with which they are emitted in $\beta$-decay so that they travel ultra-relativistically, i.e. with almost the speed of light. As such they have much higher penetrating power than $\alpha$-particles and can penetrate as much as a few millimetres of aluminium or a few centimetres of human tissue.

## 物理代写|粒子物理代写粒子物理学代考|集体模式

Shell模型有它的缺点。尽管贝壳模型取得了巨大的成功，但它的作用不应被夸大。它的有效性范围有限——它可以解释主要与轻球形核有关的现象，但即使在这种情况下，人们也可以观察到模型的预测和实验之间的差异。对于较重的原子核，这种差异甚至更大。我们已经知道，壳模型不能很好地预测磁偶极矩和激发态谱

Shell模型的另一个缺点是对电四极矩的预测。壳模型预测的这些值非常小。然而，对于较重的原子核，$A$在150-190范围内，对于$A>220$，这些电四极矩被发现相当大。Shell模型不能正确预测电四极矩的原因是假设核子以球对称势运动

• 转动状态。一个原子核的核子密度分布是球对称的(零四极矩)，它不可能有旋转激励(这类似于将能量均分原理应用到没有旋转自由度的单原子分子上)。另一方面，具有非零四极矩的原子核可以由于垂直于(旋转)对称轴的旋转而有激发能级。

## 物理代写|粒子物理代写粒子物理学代考|放射性类型

.

• ” $\alpha$ -decay” – $\alpha$粒子的发射，这是一个${ }2^4 \mathrm{He}$(氦)核。在这种衰变中，子核素的原子序数比父核素的原子序数小2，原子质量数比父核素的原子质量数小4。粒子$\alpha$$m\alpha c^2$的剩余能量比它的动能大得多，因此它的运动速度比光速低得多。它还带有双重电荷，因此很容易使它所经过的介质电离。可以表明，如果将吸波材料中的电子视为自由粒子
• (这是一个合理的近似，因为它们的结合能肯定小于发射的$\alpha$ -粒子的动能)，那么通过该材料的单位距离的能量损失与速度的平方成反比。这意味着，由于$\alpha$ -粒子在进入吸收材料时速度很小，它们很快就失去了大部分初始能量，只能穿透一张薄纸，被人的一层皮肤吸收
• ” $\beta$ -decay” -电子或正电子的发射。正电子的发射伴随着一种称为“中微子”的极低质量粒子的发射。电子发射伴随着中微子的反粒子(反中微子)。子核素的原子序数比父核素多1(在电子发射的情况下)或少1(在正电子发射的情况下)，但原子质量数相同。结合能的差等于衰变产物的总能量。电子静止能量通常与它们在$\beta$衰变中发射的能量相比较小，因此它们以超相对论的速度运动，即几乎以光速运动。因此，它们的穿透力比$\alpha$ -颗粒高得多，可以穿透几毫米厚的铝或几厘米厚的人体组织

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