Heating powdered metallic thorium in a nitrogen atmosphere to above $500\rm{\degree C}$ produces a reddish-brown powder. This reddish-brown powder is a thorium nitride, which exhibits good diamagnetism and electrical insulation. In the nitride crystal, both $Th$ and $N$ atoms have two chemical environments. The $Th$ atoms are in a close-packed arrangement, with a stacking sequence of $\dots ABABCBCACABABCBCAC\dots$; all tetrahedral voids or all octahedral voids between adjacent close-packed layers are occupied by $N$ atoms; the ratio of these two types of $N$ atoms is $1:1$. The unit cell parameters are $a=b=387\ \mathrm{pm}$; $c=2739\ \mathrm{pm}$; $\alpha=\beta=90\degree$; $\gamma=120\degree$. Which of the following options is correct?
将金属钍的粉末置于氮气气氛中加热至 $500\rm{\degree C}$ 以上，生成一种红褐色粉末。该红褐色粉末为钍的氮化物，其呈现出较好的抗磁性和电绝缘性。该氮化物晶体中，$\mathrm{Th}$ 原子和 $\mathrm{N}$ 原子均有两种化学环境。$\mathrm{Th}$ 原子作最密堆积，其堆积方式为$\dots ABABCBCACABABCBCAC\dots$；相邻两密堆积层之间的所有四面体空隙或所有八面体空隙被 $N$ 原子占据；这两种 $N$ 原子的数目比为 $1:1$。晶胞参数为 $a=b=387\ \mathrm{pm}$；$c=2739\ \mathrm{pm}$；$\alpha=\beta=90\degree$；$\gamma=120\degree$。下列选项中正确的是（）
$\frac{1}{2}$ of the octahedral voids are occupied, $\frac{1}{2}$ of the tetrahedral voids are occupied.
The density of the crystal is $7.030\ \mathrm{g/cm^3}$
Crystals have Th atoms with coordination number $6$ only.
There are two types of $\mathrm{Th}$ atoms with different coordination numbers in the crystal, which are $6$ and $7$, respectively. The number of atoms with the two different coordination numbers is the same.
The chemical composition of the smallest unit of the crystal that repeats with translational periodicity is $1$ formula unit.
The chemical composition of the smallest unit of this crystal that repeats with translational periodicity is $3$ chemical formulas.
There are exactly 3 correct answers among options **A** to **F**.
Only 2 of options **A** to **F** are correct.
$\frac{1}{2}$ 的八面体空隙被占据，$\frac{1}{2}$ 的四面体空隙被占据。
该晶体的密度是 $7.030\ \mathrm{g/cm^3}$
晶体中有且仅有配位数为 $6$ 的 $\mathrm{Th}$ 原子
晶体中有两种不同配位数的 $\mathrm{Th}$ 原子，分别为 $6$、$7$。两种不同配位数的原子数目相同。
该晶体的平移周期性重复的最小单元的化学组成为 $1$ 个化学式
该晶体的平移周期性重复的最小单元的化学组成为 $3$ 个化学式
选项 **A** 到 **F** 中有且仅有 3 个正确
选项 **A** 到 **F** 中有且仅有 2 个正确
Because the ratio of the number of octahedral voids to tetrahedral voids in two adjacent close-packed layers is $1:2$, the ratio of the number of nitrogen atoms filling the two types of voids is $1:1$,

There are $9$ layers of $Th$ atoms in the unit cell, corresponding to $9$ close-packed layers. Each layer can be filled with one octahedral void or two tetrahedral voids. Assume that $x$ layers are filled with octahedral voids and $(9-x)$ layers are filled with tetrahedral voids. Therefore, $1×x=2×(9-x)$, solving for $x=6$, there are $6$ octahedral void layers ($6$ octahedral voids filled), and $3$ tetrahedral void layers ($6$ tetrahedral voids filled).

Therefore, $ 2/3 $ of the octahedral void layers and $ 1/3 $ of the tetrahedral void layers are filled with nitrogen atoms. There are $ 9 $ $ Th $ atoms in the unit cell, forming $ 9 $ octahedral voids and $ 18 $ tetrahedral voids; $ N $ fills $ 6 $ octahedral voids and $ 6 $ tetrahedral voids, that is, there are $ 12 $ $ N $ atoms in the unit cell, and the chemical formula of the crystal is $Th_3N_4$.
<Checkpoint>[The ratio of the number of nitrogen atoms filling the two types of voids is $1:1$, so $ 2/3 $ of the octahedral void layers and $ 1/3 $ of the tetrahedral void layers are filled with nitrogen atoms](1)</Checkpoint>
Option **A** is incorrect.

There are $9$ $Th$ atoms in a unit cell, corresponding to $9$ close-packed layers, of which $3$ layers are $N$-filled tetrahedral void layers, and $6$ layers are $N$-filled octahedral void layers. The void layers are evenly distributed. Then $3$ $Th$ will be located between two layers of $N$-filled octahedral void layers, with a coordination number of $6$, and the other $6$ $Th$ will be located between a layer of $N$-filled octahedral void layers and a layer of $N$-filled tetrahedral void layers, with a coordination number of $7$. Therefore, there are $3$ $Th$ atoms with a coordination number of $6$ and $6$ $Th$ atoms with a coordination number of $7$.
<Checkpoint>[There are $3$ $Th$ atoms with a coordination number of $6$ and $6$ $Th$ atoms with a coordination number of $7$.](1)</Checkpoint>
Options **C**, **D** are incorrect.

The smallest unit of the crystal that is periodically repeated by translation is the lattice point, which contains a structural motif. There are $9$ layers of $Th$ atoms in a unit cell, $...ABA/BCB/CAC...$ Divided into groups of every $3$ layers of $Th$ atoms, the smallest translation repeating unit can be obtained, which is $1$ $Th_3N_4$, which is 1 chemical formula.
<Checkpoint>[The smallest translation repeating unit, which is $1$ $Th_3N_4$, which is 1 chemical formula.](1)</Checkpoint>
Option **E** is correct, option **F** is incorrect.

$\rho=\frac{ZM}{N_AV}=\frac{3\times(232.0\times3+14.01\times4)}{6.022\times10^{23}\times387^2\times2739\times\sin120\degree\times10^{-30}}=10.55\ (g/cm^3)$

<Checkpoint>[$\rho=\frac{ZM}{N_AV}=10.55\ (g/cm^3)$](1)</Checkpoint>
Option **B** is incorrect

Options **G,H** are incorrect
因为相邻两密堆积层内八面体空隙与四面体空隙的数目比为$1:2$，填入两种空隙的氮原子的数目比为$1:1$，

晶胞中有$9$层$Th$原子，对应有$9$个密堆积层，每一层可填一个八面体空隙或者两个四面体空隙，假设有$x$层填八面体，$(9-x)$层填四面体，故$1×x=2×(9-x)$，解得$x=6$，有$6$层八面体空隙层($6$个八面体空隙填充)，$3$层四面体空隙层($6$个四面体空隙填充)

所以有$ 2/3 $的八面体空隙层和$ 1/3 $的四面体空隙层被氮原子填满。晶胞中有$ 9 $个$ Th $原子，形成$ 9 $个八面体空隙和$ 18 $个四面体空隙；$ N $填入$ 6 $个八面体空隙和$ 6 $个四面体空隙，即晶胞中共有$ 12 $个$ N $原子，该晶体的化学式为$\mathrm{Th_3N_4}$。

<Checkpoint>[填入两种空隙的氮原子的数目比为$1:1$，所以有$ 2/3 $的八面体空隙层和$ 1/3 $的四面体空隙层被氮原子填满](1)</Checkpoint>

选项**A**错误。


一个晶胞内有$9$个$Th$原子，对应于$9$层密堆积层，其中$3$层为$N$填充四面体空隙层，$6$层为$N$填充八面体空隙层，空隙层平均分配。则会有$3$个$Th$位于两层填充了八面体空隙层的$N$之间，配位数为$6$，其它$6$个的$Th$位于一层填充了八面体空隙层的$N$与一层填充了四面体空隙层的$N$之间，配位数为$7$。故有$3$个配位数为$6$的$Th$原子，$6$个配位数为$7$的$Th$原子。

<Checkpoint>[有$3$个配位数为$6$的$\mathrm{Th}$原子，$6$个配位数为$7$的$\mathrm{Th}$原子。](1)</Checkpoint>

选项**C**,**D**错误。

该晶体作平移周期性重复的最小单位即为点阵点，包含一个结构基元。一个晶胞里有$9$层$Th$原子，$...ABA/BCB/CAC...$按照每$3$层$Th$原子为一组划分，可以得到最小的平移重复单元，即为$1$个$\mathrm{Th_3N_4}$，为1个化学式。

<Checkpoint>[最小的平移重复单元，即为$1$个$\mathrm{Th_3N_4}$，为1个化学式。](1)</Checkpoint>

选项**E**正确，选项**F**错误

$\rho=\frac{ZM}{N_AV}=\frac{3\times(232.0\times3+14.01\times4)}{6.022\times10^{23}\times387^2\times2739\times\sin120\degree\times10^{-30}}=10.55\ (g/cm^3)$

<Checkpoint>[$\rho=\frac{ZM}{N_AV}=10.55\ (g/cm^3)$](1)</Checkpoint>

选项**B**错误

选项**G,H**错误