物理代写|固体物理代写Solid-state physics代考|PHYSICS3544

2023年3月29日

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物理代写|固体物理代写Solid-state physics代考|Attractive and Repulsive Forces

Solids are stable structure; for example, a crystal of $\mathrm{KCl}$ is more stable than a collection of free $\mathrm{K}$ and $\mathrm{Cl}$ atoms. Similarly $\mathrm{Cu}$ metal is more stable than a collection of free $\mathrm{Cu}$ atoms. This implies that $\mathrm{K}$ and $\mathrm{Cl}$ atoms (or $\mathrm{Cu}$ atoms) attract each other when they approach each other; that is, an attractive interatomic force exists which holds the atoms together. It means that energy of the solid is lower than that of free atoms. The difference in energy is needed to pull apart solid into neutral free atoms at rest at infinite separation. This is called binding energy (or cohesive energy) of the crystal.

To form a solid, two different forces must exists. An attractive force which is necessary for any bonding, a repulsive force on the other hand, is required in order to keep the atoms from getting collapsed. Magnetic forces have only a weak effect in bonding and gravitational forces are negligible. An expression for an interatomic potential can be written as
$$V(r)=\frac{A}{r^n}-\frac{B}{r^m}$$
where $A$ and $B$ are constants and $n>m$, that is, the repulsive part has to prevail for short distances. Such a potential and resulting force is shown in Fig. 2.1a, b, respectively. Figure 2.1 has a minimum at some distance $r_0$. For $r>r_0$ the potential increases gradually, approach 0 as $r \rightarrow \infty$, while $r<r_0$, the potential increases very rapidly, approaches $\infty$ at small value of $r$. The system is most stable at the minimum point which represents the equilibrium position. At equilibrium position the attractive and repulsive forces cancel each other. The binding energy $E_0$ corresponds to the energy at this minimum point. A number of solid properties depend on $E_0$, the curve shape bonding type.For example solids having large bonding energies typically also have high melting temperature, at room temperature, solid substances are formed for large bonding energies whereas for small energies gaseous state is favoured, liquids prevail when the energies are of intermediate magnitude. The mechanical stiffness (modulus of elasticity) of a material is dependent on the shape of its force versus interatomic separation curve. For relatively stiff material the slope for such a curve at $r=r_0$ position will be quite steep; slopes are shallower for more flexible material. For $V$ versus $r$ curve, a deep and narrow trough which typically occurs for materials having large bonding energies normally correlate with a low coefficient of thermal expansion.

物理代写|固体物理代写Solid-state physics代考|Ionic Bonding

Ionic bonding is always found in compounds that are composed of both metallic and nonmetallic elements, elements that are situated at the horizontal extremities of the periodic table. Atoms of metallic element have low ionization energies and hence can loose electrons readily interact with atoms of non-metallic element, that have high electron affinity.

The former atom which loses electron(s) becomes positive ion and the latter atom which gains electron(s) becomes negative ion. In the process all the atoms acquire stable or inert gas configuration and in addition an electric charge; that is they becomes ions.

Consider the formation of ionic bond in $\mathrm{NaCl}$. The electron configuration of $\mathrm{Na}$ atom is (At. No. 11) 2, 8, 1 with having one valence electron. The electronic configuration of $\mathrm{Cl}$ atom (At. No. 17) is 2,8 , and 7 having 7 electrons in the outermost orbit. $\mathrm{Cl}$ requires one more electron to attain a stable configuration. In $\mathrm{NaCl}, \mathrm{Na}$ atom loses electron and becomes positive ion while $\mathrm{Cl}$ atom gain electron and becomes negative ion. The Coulomb attraction of the oppositely charged ions produces a stable union. In the case of $\mathrm{NaCl}$, sodium and chlorine ions have acquired the configuration of nearest inert gas. The force $F_{\text {a }}$ of attraction is
$$F_{\mathrm{a}}=-\frac{e^2}{r^2}$$
where $e$ is the electronic charge and $r$ is the distance between the ions. Besides the force of attraction, a force of repulsion will also come into play when the ions are brought nearer, due to the Pauli exclusion principle. The force of repulsion $F_{\mathrm{r}}$ is
$$F_{\mathrm{r}}=\frac{m b}{r^{m+1}}$$
where $b$ is a constant and $m$ is a number. Figure 2.2 shows the $\mathrm{NaCl}$ structure. The bond energy $U$ is
$$\begin{gathered} U=-\int_{-\infty}^r\left[-\frac{e^2}{r^2}-\frac{m b}{r^{m+1}}\right] \mathrm{d} r \ U_{\text {attraction }}=-\frac{6 e^2}{r}+\frac{12 e^2}{\sqrt{2} r}-\frac{8 e^2}{\sqrt{3} r}+\frac{6 e^2}{2 r}-\frac{24 e^2}{\sqrt{5} r}+\cdots \ U_{\text {attraction }}=-\frac{e^2}{r}\left(6-\frac{12}{\sqrt{2}}+\frac{8}{\sqrt{3}}-\frac{6}{2}+\frac{24}{\sqrt{5}}-\cdots\right) \end{gathered}$$

固体物理代写

物理代写|固体物理代写Solid-state physics代考|Attractive and Repulsive Forces

$$V(r)=\frac{A}{r^n}-\frac{B}{r^m}$$

物理代写|固体物理代写Solid-state physics代考|Ionic Bonding

$$F_{\mathrm{a}}=-\frac{e^2}{r^2}$$在哪里 $e$ 是电子电荷和 $r$ 是离子之间的距离。由于泡利不 相容原理，当离子靠得更近时，除了吸引力之外，排斥 力也会发挥作用。排斥力 $F_{\mathrm{r}}$ 是
$$F_{\mathrm{r}}=\frac{m b}{r^{m+1}}$$

$$U=-\int_{-\infty}^r\left[-\frac{e^2}{r^2}-\frac{m b}{r^{m+1}}\right] \mathrm{d} r U_{\text {attraction }}=-\frac{6 e^2}{r}$$

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MATLAB代写

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