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# 物理代写|固体物理代写solid physics代考|Crystal structures

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## 物理代写|固体物理代写solid physics代考|The basis

Once the lattice has been determined, we can generate the actual crystal structure of a solid material by simply assigning to each lattice point the very same basis, whose definition was provided in section 2. In this case the actual position $\mathbf{R}$ of an ion is Bravais crystal, we obviously have $\mathbf{R}_{\mathrm{b}}=0$ ). The resulting periodic crystal structure eventually corresponds to a physical object, while the lattice was just an abstract geometrical entity. We nevertheless remark that such a crystal structure is a still idealised physical system since, at variance with real solids, (i) it is infinite and (ii) it does not contain any imperfection.

In the most general case, a basis will contain two or more atoms (or molecules), whose relative positions must be the same everywhere ${ }^{4}$ in order to ensure the translational invariance. On the other hand, when the basis contains just one single atom, the resulting structure is referred to as a monoatomic Bravais crystal. This is indeed a special case of crystal structure, since it can be equivalently described by using a lattice with a basis, provided that a non-primitive unit cell has been $\operatorname{chosen}^{5}$.

## 物理代写|固体物理代写solid physics代考|Classification of the crystal structures

A large variety of solids (mostly metallic) crystallise as Bravais crystals, mainly in the bcc and fcc form 6 . More specifically (standard chemical symbols hereafter appearing are defined in the periodic table reported in appendix A) we have:

1. The bec monoatomic Bravais crystal structure: it is typically assumed by alkali metals (Li, Na, K, Rb, Cs), by some transition metals $(\mathrm{V}, \mathrm{Cr}, \mathrm{Fe}, \mathrm{Nb}$, Mo, Ta, W), by some alkaline-earth metals (Ba) or by other metals (Tl).
2. The fcc monoatomic Bravais crystal structure: it is typically assumed by noble elements (Ne, Ar, Kr, Xe), by some transition metals (Co, Ni, Cu, Rh, Pd, $\mathrm{Ag}, \mathrm{Ir}, \mathrm{Pt}, \mathrm{Au}$ ), by some alkaline-earth metals (Ca, Sr), by other metals (Al, $\mathrm{Pb})$, by some lanthanides (La, Pr, Yb) or actinides (Th).
The set of non-Bravais crystals is of course much more rich, including elemental as well as compound solids. Some important crystal structures with a basis are:
3. The sodium chloride structure: it consists of an equal number of atoms of two $A$ and $B$ chemical species, placed at alternate points of an sc lattice. The resulting crystal structure is described as an fcc lattice with a basis of two different atoms in position $(0,0,0) a$ and $(1 / 2,1 / 2,1 / 2) a$, where $a$ is the cubic lattice constant. This crystal structure is typical of $A B$ ionic solids, where $\mathrm{A}$ is any alkali metal (Li, Na, $\mathrm{K}, \mathrm{Rb}, \mathrm{Cs}$ ) and $\mathrm{B}$ any halogen atom $(\mathrm{F}, \mathrm{Br}, \mathrm{Cl}, \mathrm{I})$. Similarly, $\mathrm{A}-\mathrm{B}$ compounds crystallise in this form, where $A$ is a metal (Ag, Mg, Ca, Sr or Ba) and $B$ is a non-metal $(O, S, S e$ or $\mathrm{Te})$ element.
4. The cesium chloride structure: it consists in bcc lattice whose sites are occupied by atoms of two $A$ and $B$ chemical species so that each $A$-atom has eight nearest neighbouring B-atoms (and vice versa). The resulting crystal structure is described as an sc lattice with a basis of two different atoms in position $(0,0,0) a$ and $(1 / 2,1 / 2,1 / 2) a$, where $a$ is the cubic lattice constant. We find in this crystal structure Cs-based materials, like $\mathrm{CsCl}, \mathrm{CsBr}$, and CsI.
5. The diamond structure: it consists of two inter-penetrating fcc monoatomic Bravais lattices, displaced along the diagonal of the cubic conventional unit cell by $a / 4$, where $a$ is the cubic lattice constant. The two sub-lattices are occupied by atoms of the same chemical species. The resulting crystal structure is described as an fcc lattice with a basis of two identical atoms in position $(0,0,0) a$ and $(1 / 4,1 / 4,1 / 4) a$. Elemental semiconductors (Si, Ge, $\alpha-\mathrm{Sn})$ and diamond (C) crystallise in this structure.
6. The zincblende structure: it is similar to the diamond structure, but the two sub-lattices are occupied by atoms of two different chemical species. Compound semiconductors crystallise in this structure, in any IV-IV (SiC) or III-V (where III $=\mathrm{Al}$, Ga, In and $\mathrm{V}=\mathrm{P}, \mathrm{As}, \mathrm{Sb}$ ) or II-VI (where II = Zn, $\mathrm{Cd}, \mathrm{Hg}$ and $\mathrm{VI}=\mathrm{S}, \mathrm{Se}, \mathrm{Te})$ combination. Other crystals assuming this structure are $\mathrm{CuA}$ (where $A=\mathrm{F}, \mathrm{Cl}, \mathrm{Br}, \mathrm{I}), \mathrm{BeB}$ (where $\mathrm{B}=\mathrm{S}, \mathrm{Se}, \mathrm{Te}$ ), MnC (where $C=S$, Se).
7. The hexagonal structure: it consists of two inter-penetrating hexagonal Bravais lattices, shifted from one another by a displacement vector $(a / 3, b / 3, c / 2)$, where $a=b$ and $c$ are defined in figure $2.7$. The ‘ideal’ hexagonal structure has a $c / a=\sqrt{8 / 3}$ ratio, while actual hexagonal crystals deviate from this value. The ideal cla ratio is calculated by assuming that each lattice point is occupied by a hard sphere, a situation which is referred to as hexagonal close packing (see next section for more detail). However, atoms are not rigid spheres and, therefore, in real materials this ratio can assume different values. We find in crystal structure many elemental solids like those made by $\mathrm{Cd}, \mathrm{Mg}, \mathrm{Nd}, \mathrm{Os}$, $\mathrm{Sc}, \mathrm{Ti}, \mathrm{Zn}$, and $\mathrm{Zr}$ with a cla ratio of $1.89,1.62,1.61,1.58,1.59,1,59$, $1.86$, and $1.59$, respectively.

## 物理代写|固体物理代写SOLID PHYSICS代考|Packing

There is still a remaining criterion for classifying atomic architectures to be discussed. It is based on the assumption to treat atoms as attracting hard spheres. While this is clearly a very crude approximation, it is reasonably well satisfied by metals and this represents the phenomenological foundation for its applicability.

Since atoms are looked at as ‘hard’ spheres, they cannot overlap; however, because of their mutual attraction, they tend to assume an arrangement that minimises the total energy of the system. This implies that they tend to pack as closely as possible.

Let us start by arranging identical hard spheres on a plane: in the closest packing configuration the centres of the spheres lie on a two-dimensional triangular lattice, in positions marked by A letters in figure $2.13$. By looking at this configuration from the top, we can identify for the second layer the new triangular lattice (lying on a plane parallel to the first layer) marked by B letters. While this choice is unique, when adding a third layer we can add spheres on the triangular lattice marked by $C$ letters or, alternatively, on the triangular lattice once again marked by A letters (in both cases the third layer lies on a plane parallel to the two previous ones). The corresponding stacking sequence is $\mathrm{ABCABCABC \cdots \text {and } A B A B A B A B \cdots \text { , respectively: }}$ they are named $f c c$ structure and hexagonal close-packed (hpc) structure. The atomic layers so generated correspond to (111) planes of the fcc lattice or to the basal plane of the hexagonal lattice. In both configurations each atom has 12 nearest neighbours: any model according to which the total energy of a crystal only depends on the number of nearest neighbours must necessarily predict the very same energy for fcc and hpc structures.

The number of different ways to pack hard spheres in arrangement other than the close-packed one is actually infinite. In table $2.1$ we summarise some properties of the packing in cubic lattices. The packing fraction is the volume fraction occupied by the hard spheres: the labelling ‘close packing’ for fcc is justified by the fact that its packing fraction is maximum.

## 物理代写|固体物理代写SOLID PHYSICS代考|CLASSIFICATION OF THE CRYSTAL STRUCTURES

1. bec 单原子布拉维晶体结构：通常由碱金属假定大号一世,ñ一种,ķ,Rb,Cs, 通过一些过渡金属(在,Cr,F和,ñb, Mo, Ta, W), 通过一些碱土金属乙一种或其他金属吨l.
2. fcc 单原子 Bravais 晶体结构：通常由贵重元素假定ñ和,一种r,ķr,X和, 通过一些过渡金属C这,ñ一世,C在,RH,磷d,$一种G,一世r,磷吨,一种在$, 通过一些碱土金属C一种,小号r, 通过其他金属一种l,$磷b$，一些镧系元素大号一种,磷r,是b或锕系元素吨H.
非布拉维晶体的集合当然要丰富得多，包括元素固体和化合物固体。一些具有基础的重要晶体结构是：
3. 氯化钠结构：由相等数量的两个原子组成一种和乙化学物质，放置在 sc 晶格的交替点上。所得晶体结构被描述为具有两个不同位置原子的 fcc 晶格(0,0,0)一种和(1/2,1/2,1/2)一种， 在哪里一种是立方晶格常数。这种晶体结构是典型的一种乙离子固体，其中一种是任何碱金属大号一世,ñ一种,$ķ,Rb,Cs$和乙任何卤原子(F,乙r,Cl,一世). 相似地，一种−乙化合物以这种形式结晶，其中一种是金属一种G,米G,C一种,小号r这r乙一种和乙是非金属(这,小号,小号和或者吨和)元素。
4. 氯化铯结构：它由bcc晶格组成，其位置被两个原子占据一种和乙化学物种，使每个一种-atom 有八个最近的相邻 B 原子一种nd在一世C和在和rs一种. 所得晶体结构被描述为具有两个不同位置原子的 sc 晶格(0,0,0)一种和(1/2,1/2,1/2)一种， 在哪里一种是立方晶格常数。我们在这种晶体结构中发现基于 Cs 的材料，如CsCl,Cs乙r, 和 CsI。
5. 金刚石结构：它由两个相互穿透的 fcc 单原子 Bravais 晶格组成，沿立方常规晶胞的对角线位移一种/4， 在哪里一种是立方晶格常数。这两个子晶格被相同化学种类的原子占据。所得晶体结构被描述为具有两个相同原子位置的 fcc 晶格(0,0,0)一种和(1/4,1/4,1/4)一种. 元素半导体小号一世,G和,$一种−小号n美元和钻石C在这个结构中结晶。 6. 闪锌矿结构：与金刚石结构相似，但两个亚晶格被两种不同化学物质的原子占据。化合物半导体以这种结构结晶，在任何 IV-IV小号一世C或 III-V在H和r和一世一世一世$=一种l$,G一种,一世n一种nd$在=磷,一种s,小号b$或 II-VI在H和r和一世一世=从n,$Cd,HG$一种nd$在一世=小号,小号和,吨和C这米b一世n一种吨一世这n.这吨H和rCr是s吨一种ls一种ss在米一世nG吨H一世ss吨r在C吨在r和一种r和\mathrm{CuA}(在H和r和A = \ mathrm {F}, \ mathrm {Cl}, \ mathrm {Br}, \ mathrm {I}), \ mathrm {BeB(在H和r和\ mathrm {B} = \ mathrm {S}，\ mathrm {Se}，\ mathrm {Te),米nC(在H和r和C = S \$, Se)。
7. 六边形结构：它由两个相互贯穿的六边形 Bravais 晶格组成，通过位移矢量相互移动(一种/3,b/3,C/2)， 在哪里一种=b和C在图中定义2.7. “理想的”六边形结构具有C/一种=8/3比，而实际的六方晶体偏离该值。理想的 cla 比是通过假设每个晶格点被一个硬球占据来计算的，这种情况称为六方密堆积s和和n和X吨s和C吨一世这nF这r米这r和d和吨一种一世l. 但是，原子不是刚性球体，因此，在实际材料中，该比率可以采用不同的值。我们在晶体结构中发现许多元素固体，例如由Cd,米G,ñd,这s, 小号C,吨一世,从n， 和从r具有 cla 比率1.89,1.62,1.61,1.58,1.59,1,59,1.86， 和1.59， 分别。

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