In the following two variants of cubic $\mathrm{ZnS}$, the metal atoms form alternating layered structures, with only one type of metal atom within the same layer, and they are all on the same plane. $\mathbf{A-F}$ represent six different elements.

In $\mathbf{ABC_2}$, $\mathbf{A}$ and $\mathbf{B}$ together occupy the $\mathrm{Zn}$ position in cubic $\mathrm{ZnS}$, and there is no three-fold rotation axis in this crystal.

In $\mathbf{DEF_2}$, $\mathbf{D}$ and $\mathbf{E}$ together occupy the $\mathrm{Zn}$ position in cubic $\mathrm{ZnS}$, and there is a three-fold rotation axis in this crystal.

How many atoms are there in a primitive unit cell of $\mathbf{ABC_2}$? How many atoms are there in a structural motif?

How many atoms are there in a primitive unit cell of $\mathbf{DEF_2}$? How many atoms are there in a structural motif?
在下面两种立方$ \mathrm {ZnS} $的变体中，金属原子形成了交替层状结构，且同层内只有一种金属原子，它们均在同一平面上，$\mathbf {A-F}$代表六种不同元素。

在$\mathbf { ABC _ { 2 }}$中，$\mathbf {A}$与$\mathbf {B}$共同占据立方$ \mathrm {ZnS} $中$\mathrm {Zn}$的位置，该晶体中无三次旋转轴。

在$\mathbf { DEF_ { 2 } }$中，$\mathbf {D}$与$\mathbf {E}$共同占据立方$ \mathrm {ZnS} $中$\mathrm {Zn}$的位置，该晶体中有三次旋转轴。

在$\mathbf { ABC _ { 2 }}$的一个正当晶胞中一共有多少个原子？一个结构基元又包括多少个原子？

而在$\mathbf { DEF_ { 2 } }$的一个正当晶胞中一共有多少个原子？一个结构基元又包括多少个原子？
4; 4; 12; 12
8; 4; 12; 12
4; 4; 12; 4
8; 4; 12; 4
12; 12; 4; 4
12; 12; 8; 4
12; 4; 4; 4
12; 4; 8; 4
All of the above options are incorrect.
4；4；12；12
8；4；12；12
4；4；12；4
8；4；12；4
12；12；4；4
12；12；8；4
12；4；4；4
12；4；8；4
以上选项均不正确
In $\mathbf { ABC _ { 2 }}$, $\mathbf {A}$ and $\mathbf {B}$ jointly occupy the position of $\mathrm {Zn}$ in the cubic $ \mathrm {ZnS} $. There is no threefold rotation axis in this crystal, so in $\mathbf { ABC _ { 2 }}$ it should be that $\mathbf {A}$ and $\mathbf {B}$ replace the two layers of $\mathrm {Zn}$ at $z=0$ and $z=0.5$ in the cubic $ \mathrm {ZnS} $ unit cell with $\mathrm {Zn}$ as the vertex, instead of replacing the close-packed layers (which would lead to a threefold rotation axis).

<Checkpoint>[The substitution in $\mathbf { ABC _ { 2 }}$ is not the substitution of close-packed layers](1)</Checkpoint>

The resulting unit cell is not a proper unit cell. Connecting the vertices and the face centers yields a proper unit cell, which should be a simple tetragonal lattice, containing 4 atoms, and the structural motif also contains 4 atoms.

<Checkpoint>[One proper unit cell of $\mathbf { ABC _ { 2 }}$ contains 4 atoms](1)</Checkpoint>

<Checkpoint>[One structural motif of $\mathbf { ABC _ { 2 }}$ contains 4 atoms](1)</Checkpoint>

In $\mathbf { DEF_ { 2 } }$, $\mathbf {D}$ and $\mathbf {E}$ jointly occupy the position of $\mathrm {Zn}$ in the cubic $ \mathrm {ZnS} $. There is a threefold rotation axis in this crystal, so it is a substitution of close-packed layers.

<Checkpoint>[The substitution in $\mathbf { DEF _ { 2 }}$ is the substitution of close-packed layers](1)</Checkpoint>

Using uppercase letters to represent the close-packed layers of $\mathrm {Zn}$ and lowercase letters to represent the close-packed layers of $\mathrm {S}$, the stacking structure of cubic $ \mathrm {ZnS} $ can be expressed as $|\mathrm{AaBbCc}|$. After $\mathbf {D}$ and $\mathbf {E}$ alternately replace Zn in layers, it will become $|\mathrm{A_DaB_EbC_DcA_EaB_DbC_Ec}|$, which is a typical R-centered hexagonal lattice. A proper unit cell contains 12 atoms, which is 3 structural motifs, i.e., one structural motif contains 4 atoms.

<Checkpoint>[One proper unit cell of $\mathbf { DEF _ { 2 }}$ contains 12 atoms](1)</Checkpoint>

<Checkpoint>[One structural motif of $\mathbf { DEF _ { 2 }}$ contains 4 atoms](1)</Checkpoint>
在$\mathbf { ABC _ { 2 }}$中，$\mathbf {A}$与$\mathbf {B}$共同占据立方$ \mathrm {ZnS} $中$\mathrm {Zn}$的位置。该晶体中无三次旋转轴，因此在$\mathbf { ABC _ { 2 }}$应为在$\mathrm {Zn}$为顶点的立方$ \mathrm {ZnS} $晶胞中，$\mathbf {A}$与$\mathbf {B}$对$z=0、z=0.5$的两层$\mathrm {Zn}$进行取代，而不是密堆积层的取代（会导致三次旋转轴）。

<Checkpoint>[$\mathbf { ABC _ { 2 }}$中的取代并不是密堆积层的取代](1)</Checkpoint>

此时得到的晶胞并非正当晶胞，连接顶点与面心，即可得到正当晶胞，应为简单四方点阵，包含4个原子，结构基元同样为4个原子。

<Checkpoint>[$\mathbf { ABC _ { 2 }}$的1个正当晶胞包含4个原子](1)</Checkpoint>

<Checkpoint>[$\mathbf { ABC _ { 2 }}$的1个结构基元包含4个原子](1)</Checkpoint>

在$\mathbf { DEF_ { 2 } }$中，$\mathbf {D}$与$\mathbf {E}$共同占据立方$ \mathrm {ZnS} $中$\mathrm {Zn}$的位置，该晶体中有三次旋转轴，因此是密堆积层的取代。

<Checkpoint>[$\mathbf { DEF _ { 2 }}$中的取代是密堆积层的取代](1)</Checkpoint>

以大写字母表示$\mathrm {Zn}$的密堆积层，小写字母表示$\mathrm {S}$的密堆积层，则立方$ \mathrm {ZnS} $的堆积结构可表示为$|\mathrm{AaBbCc}|$，$\mathbf {D}$与$\mathbf {E}$交替分层取代Zn后，则会变为$|\mathrm{A_DaB_EbC_DcA_EaB_DbC_Ec}|$，为典型的R心六方点阵，一个正当晶胞包含12个原子，为3个结构基元，即一个结构基元包含4个原子。

<Checkpoint>[$\mathbf { DEF _ { 2 }}$的1个正当晶胞包含12个原子](1)</Checkpoint>

<Checkpoint>[$\mathbf { DEF _ { 2 }}$的1个结构基元包含4个原子](1)</Checkpoint>