The crystal formed by three main group metal elements has the chemical formula $X_2YZ$ (M < 200), where the mass fraction of $X$ is 24.33%, and the mass fraction of $Y$ is 12.86%.  

In this crystal, $Y$ and $Z$ form a layered structure similar to graphite, with a ratio of $Y:Z = 1:1$ in each layer. The upper and lower layers cannot completely overlap and are arranged in an ABAB... sequence, where the $Y$ atoms in the AB layers are fully aligned, but the $Z$ atoms are entirely misaligned. $X$ is located above and below the center of each hexagon.  

The crystal can be divided into a hexagonal unit cell with $Y$ at the vertices. The lattice parameters are $a = 504.9\,pm$, $c = 1010\,pm$, and the distance between the $X$ atoms above and below each hexagonal ring is $343.8\,pm$.  

According to the question, select the correct option from the following choices.
由三种主族金属元素形成的晶体化学式为$X_2YZ(M<200)$，其中 $X$ 所占的质量分数为24.33%，$Y$ 所占的质量分数为12.86%。

该晶体中$Y、Z$形成类似石墨的层状结构，每层中 $Y:Z=1:1$，上下两层不能完全重合，按照 ABAB...进行排列，其中AB层 $Y$ 原子是完全重合的，但 $Z$ 完全不重合，$X$ 位于每个六边形中心上下方。

该晶体可以划分出以 $Y$ 为顶点的六方晶胞，其晶胞参数 $a=504.9\,pm,c=1010\,pm$，每个六元环上下的 $X$ 原子间的距离为 $343.8pm$。

根据题意，在下列选项中选出正确的一项。
The components of the crystal contain elements from the third main group.
The components of the crystal contain elements from the fifth main group.
The components of the crystal contain elements from the sixth main group.
In the unit cell, the position $(\dfrac{1}{3}, \dfrac{2}{3}, 0.17)$ is occupied by a $Mg$ atom.
In the unit cell, the position $(\dfrac13,\dfrac23,\dfrac12)$ is occupied by the $In$ atom.
In the unit cell, the position $(\dfrac23,\dfrac13,0)$ is occupied by the $Sb$ atom.
The crystal density is $2.44g/cm^3$
The crystal density is $2.95g/cm^3$
The crystal density is $2.82g/cm^3$
晶体的组成部分中含有第三主族元素
晶体的组成部分中含有第五主族元素
晶体的组成部分中含有第六主族元素
晶胞中 $(\dfrac13,\dfrac23,0.17)$位置是 $Mg$ 原子
晶胞中 $(\dfrac13,\dfrac23,\dfrac12)$位置是 $In$ 原子
晶胞中 $(\dfrac23,\dfrac13,0)$位置是 $Sb$ 原子
晶体密度是 $2.44g/cm^3$
晶体密度是 $2.95g/cm^3$
晶体密度是 $2.82g/cm^3$
The ratio of the relative atomic masses of the three elements is $X:Y:Z=(24.33\%/2):12.86\%:(1-24.33\%-12.86\%)=12.17:12.86:62.81$, indicating that $Z$ is the heavier metal element. From $M<200$, it is calculated that $Z<125.62$.

Assuming $Z$ is $In$, calculating $M=\dfrac{114.82}{62.81\%}=182.8$, then $Y=182.8\times12.86\%=23.51, X=182.8\times12.17\%=22.25$, with no corresponding elements.

Assuming $Z$ is $Sn$, calculating $M=\dfrac{118.71}{62.81\%}=189.0$, then $Y=189.0\times12.86\%=24.31, X=189.0\times12.17\%=23.00$, corresponding to $Mg, Na$.

Assuming $Z$ is $Sb$, calculating $M=\dfrac{121.76}{62.81\%}=193.9$, then $Y=193.9\times12.86\%=24.93, X=193.9\times12.17\%=23.60$, with no corresponding elements.

Similarly, assuming $Z$ is any other main-group metal element, no combination of elements satisfies the requirements. Thus, the crystal is $Na_2MgSn$.

<Checkpoint>[The crystal is $Na_2MgSn$](2)</Checkpoint>

The vertices of the hexagonal unit cell are occupied by $Mg$ atoms, with $Sn$ atoms in the AB layers not overlapping. Therefore, one hexagonal unit cell contains one A layer and one B layer, and $Na$ atoms are located between the layers, totaling two $Na_2MgSn$ structural units. The fractional coordinates of the $Mg$ atoms are $(0,0,0),(0,0,\dfrac12)$.

Unit cell mass $m=\dfrac{2\times189}{6.023\times10^{23}}=6.28\times10^{-22}\,g$.  
<Checkpoint>[Unit cell mass $m=6.28\times10^{-22}\,g$](0.5)</Checkpoint>

Unit cell volume $V=\dfrac{\sqrt3}2a^2c=\dfrac{\sqrt3}2\times504.9^2\times1010=2.23\times10^{-22} cm^3$.  
<Checkpoint>[Unit cell volume $V=2.23\times10^{-22} cm^3$](0.5)</Checkpoint>

Crystal density $\rho=\dfrac{m}{V}=\dfrac{6.28\times10^{-22}}{2.23\times10^{-22}}=2.82\,g/cm^3$.  
<Checkpoint>[Crystal density $\rho=2.82\,g/cm^3$](1)</Checkpoint>
三种元素的相对原子质量之比 $X:Y:Z=(24.33\%/2):12.86\%:(1-24.33\%-12.86\%)=12.17:12.86:62.81$，说明 $Z$ 是较重的金属元素。由 $M<200$ 计算得到$Z<125.62$。

假设 $Z$ 是 $In$，计算 $M=\dfrac{114.82}{62.81\%}=182.8$，则$Y=182.8\times12.86\%=23.51,X=182.8\times12.17\%=22.25$，没有对应。

假设 $Z$ 是 $Sn$，计算 $M=\dfrac{118.71}{62.81\%}=189.0$，则$Y=189.0\times12.86\%=24.31,X=189.0\times12.17\%=23.00$，对应$Mg、Na$。

假设 $Z$ 是 $Sb$，计算 $M=\dfrac{121.76}{62.81\%}=193.9$，则$Y=193.9\times12.86\%=24.93,X=193.9\times12.17\%=23.60$，没有对应。

同样地，假设 $Z$ 为其他主族金属元素，都没有对应的元素组合满足题目要求。于是晶体为 $Na_2MgSn$。

<Checkpoint>[晶体为 $Na_2MgSn$](2)</Checkpoint>

六方晶胞的顶点为 $Mg$ 原子，AB层的 $Sn$ 原子不能重合，则一个六方晶胞含有一个A层一个B层，$Na$ 原子处于层与层之间，总计包含两个 $Na_2MgSn$ 结构单元。$Mg$ 原子对应的分数坐标为 $(0,0,0),(0,0,\dfrac12)$。

晶胞质量 $m=\dfrac{2\times189}{6.023\times10^{23}}=6.28\times10^{-22}\,g$。
<Checkpoint>[晶胞质量$m=6.28\times10^{-22}\,g$](0.5)</Checkpoint>

晶胞体积 $V=\dfrac{\sqrt3}2a^2c=\dfrac{\sqrt3}2\times504.9^2\times1010=2.23\times10^{-22} cm^3$。

<Checkpoint>[晶胞体积$V=2.23\times10^{-22} cm^3$](0.5)</Checkpoint>


晶体密度$\rho=\dfrac{m}{V}=\dfrac{6.28\times10^{-22}}{2.23\times10^{-22}}=2.82\,g/cm^3$。

<Checkpoint>[晶体密度$\rho=2.82\,g/cm^3$](1)</Checkpoint>