An alloy $\bf M$ is composed of metal $\bf A$ and metal $\bf B$, and can form a variety of crystal structures such as $X_1$-$\bf M$ phase, $X_2$-$\bf M$ phase, $X_3$-$\bf M$ phase, and $X_4$-$\bf M$ phase. (To prevent directly searching for answers without reasoning, the real names of the structures are hidden)

$X_1$-$\bf M$ phase is a phase formed when $\bf B$ is less, and its structure can be regarded as $\bf B$ atoms randomly replacing the positions of $\bf A$ atoms in metal $\bf A$.

$X_2$-$\bf M$ phase is a cubic crystal with a unit cell parameter of $a=295~\rm pm$; $\bf A$ atoms and $\bf B$ atoms are completely ordered in the crystal.

$X_3$-$\bf M$ phase is also a cubic crystal, in which $\bf A$ atoms and $\bf B$ atoms are completely ordered, forming a four-layer nested atomic cluster of "$\bf B$ tetrahedron@$\bf A$ tetrahedron@$\bf A$ octahedron@$\bf B$ cuboctahedron"; the center of the atomic cluster is located at the vertices and body center of the unit cell (and the orientation of the atomic clusters at the vertices and body center are the same).

$X_4$-$\bf M$ phase is a hexagonal crystal, and its structure can be regarded as $\bf A$ atoms randomly replacing the positions of $\bf B$ atoms in metal $\bf B$.

Hint: (a) An ideal cuboctahedron is a highly symmetrical polyhedron surrounded by 6 square faces and 8 equilateral triangular faces; (b) The atomic radii of $\bf A, B$ are $\rm 128~pm, 133~pm$, respectively.

The following statements are made:

1. The chemical formula of $X_2$-$\bf M$ phase is $\bf A_{\rm 3} B$, and $\bf B$ and $\bf A$ are located at the vertices and face centers of the unit cell, respectively.

2. The chemical formula of $X_3$-$\bf M$ phase is $\bf AB$.

3. If the fractional coordinate of an $\bf A$ atom in the "$\bf A$ tetrahedron" layer of the $X_3$-$\bf M$ phase is $(0.328, 0.672, 0.328)$, then the theoretical value of the unit cell parameter of the $X_3$-$\bf M$ phase is $a=891~\rm pm$.

4. If the fractional coordinate of an $\bf A$ atom in the "$\bf A$ tetrahedron" layer of the $X_3$-$\bf M$ phase is $(0.328, 0.672, 0.328)$, then the theoretical value of the unit cell parameter of the $X_3$-$\bf M$ phase is $a=744~\rm pm$.

5. The lattice types of $X_2$-$\bf M$ phase and $X_3$-$\bf M$ phase are simple cubic and body-centered cubic, respectively.

What is the sum of the numbers of the correct statements?
某合金$\bf M$是由金属$\bf A$和金属$\bf B$所组成的合金，可以形成$X_1$-$\bf M$相、$X_2$-$\bf M$相、$X_3$-$\bf M$相和 $X_4$-$\bf M$相等多种多样的晶体结构。（为防止不推理直接搜索答案，隐去结构的真名）

$X_1$-$\bf M$相是$\bf B$较少时形成的相，其结构可看作$\bf B$原子无序地取代金属$\bf A$中$\bf A$原子的位置。  

$X_2$-$\bf M$相是立方晶体，晶胞参数 $a=295~\rm pm$；晶体中$\bf A$原子和$\bf B$原子完全有序排布。  

$X_3$-$\bf M$相也是立方晶体，晶体中$\bf A$原子和$\bf B$原子完全有序排布，形成”$\bf B$四面体@$\bf A$四面体@$\bf A$八面体@$\bf B$立方八面体”的四层嵌套原子簇；原子簇的中心位于晶胞的顶点和体心（且顶点与体心的原子簇的取向相同）。  

$X_4$-$\bf M$相是六方晶体，其结构可看作$\bf A$原子无序地取代金属$\bf B$中$\bf B$原子的位置。  

提示：(a)理想的立方八面体是由6个正方形面和8个正三角形面围成的高对称性多面体；(b) $\bf A,B$的原子半径分别为 $\rm 128~pm,133~pm$。

有如下说法：

1.$X_2$-$\bf M$相的化学式为 $\bf A_{\rm 3} B$，$\bf B$和$\bf A$分别在晶胞的顶点和面心。

2.$X_3$-$\bf M$相的化学式为 $\bf AB$。

3.若$X_3$-$\bf M$相的”$\bf A$四面体“层中某个$\bf A$原子的分数坐标是 $(0.328, 0.672, 0.328)$，则$X_3$-$\bf M$相晶胞参数理论值 $a=891~\rm pm$。

4.若$X_3$-$\bf M$相的”$\bf A$四面体“层中某个$\bf A$原子的分数坐标是 $(0.328, 0.672, 0.328)$，则$X_3$-$\bf M$相晶胞参数理论值 $a=744~\rm pm$。

5.$X_2$-$\bf M$相和 $X_3$-$\bf M$相的点阵型式分别是简单立方和体心立方。

正确说法编号之和为多少？
2
3
5
7
8
10
11
12
All of the above options are incorrect.
2
3
5
7
8
10
11
12
以上选项均不对
Analyze $X_2$-$\bf M$ phase: If $\bf A,B$ are in contact at the face diagonal, then $a=\rm (128+133)\sqrt 2=369~pm>295~pm$, which is inconsistent; if $\bf A,B$ are in contact at the body diagonal, then $a=\rm(128+133)\times \frac {2}{\sqrt 3}=301~pm\approx295~pm$, therefore $\bf A,B$ are in contact at the body diagonal, $\bf A,B$ are at the vertices and body center respectively, and the chemical formula is $\bf AB$, statement 1 is incorrect.

<Checkpoint>[$X_2$-$\bf M$ phase, $\bf A,B$ are in contact at the body diagonal, $\bf A,B$ are at the vertices and body center respectively, and the chemical formula is $\bf AB$](1)</Checkpoint>

Analyze $X_3$-$\bf M$ phase: The composition of $\bf B$ tetrahedron@$\bf A$ tetrahedron@$\bf A$ octahedron@$\bf B$ cuboctahedron is $\bf B_{\rm 4}\rm @\bf A_{\rm 4}\rm @\bf A_{\rm 6}\rm @\bf B_{{\rm 12}}$, the quadrilateral faces of the cuboctahedron are parallel to the faces of the cubic unit cell, and the two opposing triangular faces are parallel but oriented in opposite directions. Since the atomic clusters at the vertices and body center have the same orientation, the clusters at the vertices and body center do not share atoms, and each cluster is independent. The unit cell of the $X_3$-$\bf M$ phase contains $2\times (4+6)=20$ $\bf A$ atoms and $2\times (4+12)=32$ $\bf B$ atoms, and the chemical formula is $\bf A_{\rm 5} B_{\rm 8}$, statement 2 is incorrect.

<Checkpoint>[$X_3$-$\bf M$ phase unit cell, the clusters at the vertices and body center do not share atoms](1)</Checkpoint>

<Checkpoint>[$X_3$-$\bf M$ phase chemical formula is $\bf A_{\rm 5} B_{\rm 8}$](1)</Checkpoint>

Calculate the unit cell parameter of $X_3$-$\bf M$: Let the positional offset of the $\bf A$ atom relative to the vertex or body center be represented by the fractional coordinate $t$, then the coordinates of the $\bf A$ atoms in the "$\bf A$ tetrahedron" layer of the $X_3$-$\bf M$ phase are $(t,t,t),(t,-t,-t),(-t,t,-t),(-t,-t,t)$ and the above four each plus $(\frac12,\frac12,\frac12)$, for a total of 8. It is easy to know that $(0.328, 0.672, 0.328)=(\frac12-t,\frac12+t,\frac12-t)$, so $t=0.172$.

<Checkpoint>[$t=0.172$](1)</Checkpoint>

Assume that the $\bf B$ atoms of the "$\bf B$ tetrahedron" are in close contact, and the length of the component of the distance from the $\bf B$ atom of the "$\bf B$ tetrahedron" to the center of the cluster in the direction of the coordinate axis is $133\div\sqrt2=94.0~\rm pm$

<Checkpoint>[Length of the component of the distance from the $\bf B$ atom of the "$\bf B$ tetrahedron" to the center of the cluster in the direction of the coordinate axis is $94.0~\rm pm$](1)</Checkpoint>

Assume that the $\bf A$ atoms of the "$\bf A$ tetrahedron" and the $\bf B$ atoms of the "$\bf B$ tetrahedron" are in close contact. Let the length of the component of the distance from the $\bf A$ atom of the "$\bf A$ tetrahedron" to the center of the cluster in the direction of the coordinate axis be $w$, according to the Pythagorean theorem, $\sqrt{(w-\rm94.0~ pm)^2+
(\it w-\rm94.0~ pm)^2+(\it w+\rm94.0~ pm)^2}=128~\rm pm+133~\rm pm$,
Solving for $w$: $w=153.2~\rm pm$, unit cell parameter $a=153.2~\rm pm\div 0.172=891~\rm pm$, statement 3 is correct, statement 4 is incorrect.

<Checkpoint>[$w=153.2~\rm pm$](1)</Checkpoint>

<Checkpoint>[$X_3$-$\bf M$ phase unit cell parameter $a=891~\rm pm$](1)</Checkpoint>

The lattice types of the $X_2$-$\bf M$ phase and the $X_3$-$\bf M$ phase are simple cubic and body-centered cubic, respectively, statement 5 is correct.

<Checkpoint>[$X_2$-$\bf M$ phase and $X_3$-$\bf M$ phase lattice types are simple cubic and body-centered cubic, respectively](1)</Checkpoint>

In summary, the correct statements are 3 and 5, so choose option E.
分析$X_2$-$\bf M$相：若$\bf A,B$在面对角线接触，则 $a=\rm (128+133)\sqrt 2=369~pm>295~pm$，不符；若$\bf A,B$在体对角线接触，则 $a=\rm(128+133)\times \frac {2}{\sqrt 3}=301~pm\approx295~pm$，故$\bf A,B$在体对角线接触，$\bf A,B$分别在顶点和体心，化学式为 $\bf AB$，说法1错误。

<Checkpoint>[$X_2$-$\bf M$相中，$\bf A,B$在体对角线接触，$\bf A,B$分别在顶点和体心，化学式为 $\bf AB$](1)</Checkpoint>

分析$X_3$-$\bf M$相：$\bf B$四面体@$\bf A$四面体@$\bf A$八面体@$\bf B$立方八面体的组成是$\bf B_{\rm 4}\rm @\bf A_{\rm 4}\rm @\bf A_{\rm 6}\rm @\bf B_{{\rm 12}}$，立方八面体的四边形面和立方晶胞的面平行，处于对位的两个三角形面平行但朝向相反。由于顶点与体心的原子簇的取向相同，故顶点和体心的簇不共用原子，每个簇都是独立的。$X_3$-$\bf M$相的晶胞中有 $2\times (4+6)=20$个$\bf A$，$2\times (4+12)=32$个$\bf B$，化学式为$\bf A_{\rm 5} B_{\rm 8}$，说法2错误。

<Checkpoint>[$X_3$-$\bf M$相晶胞顶点和体心的簇不共用原子](1)</Checkpoint>

<Checkpoint>[$X_3$-$\bf M$相化学式为$\bf A_{\rm 5} B_{\rm 8}$](1)</Checkpoint>


计算$X_3$-$\bf M$的晶胞参数：设$\bf A$原子相对顶点或体心的位置偏移量以分数坐标表示为 $t$，则$X_3$-$\bf M$相的”$\bf A$四面体“层中$\bf A$原子的坐标为 $(t,t,t),(t,-t,-t),(-t,t,-t),(-t,-t,t)$以及上述四个分别加 $(\frac12,\frac12,\frac12)$，共8个。易知 $(0.328, 0.672, 0.328)=(\frac12-t,\frac12+t,\frac12-t)$，得 $t=0.172$。

<Checkpoint>[$t=0.172$](1)</Checkpoint>

设“$\bf B$四面体”的$\bf B$原子紧密接触，“$\bf B$四面体”的$\bf B$原子到团簇中心的距离在坐标轴方向的分量长度为 $133\div\sqrt2=94.0~\rm pm$

<Checkpoint>[“$\bf B$四面体”的$\bf B$原子到团簇中心的距离在坐标轴方向的分量长度为 $94.0~\rm pm$](1)</Checkpoint>

设“$\bf A$四面体”的$\bf A$原子和“$\bf B$四面体”的$\bf B$原子紧密接触。设“$\bf A$四面体”的$\bf A$原子到团簇中心的距离在坐标轴方向的分量长度为 $w$ ，根据勾股定理，$\sqrt{(w-\rm94.0~ pm)^2+
(\it w-\rm94.0~ pm)^2+(\it w+\rm94.0~ pm)^2}=128~\rm pm+133~\rm pm$，
解得：$w=153.2~\rm pm$，晶胞参数 $a=153.2~\rm pm\div 0.172=891~\rm pm$，说法3正确，说法4错误。

<Checkpoint>[$w=153.2~\rm pm$](1)</Checkpoint>

<Checkpoint>[$X_3$-$\bf M$相晶胞参数$a=891~\rm pm$](1)</Checkpoint>

$X_2$-$\bf M$相和 $X_3$-$\bf M$相的点阵型式分别是简单立方和体心立方，说法5正确。

<Checkpoint>[$X_2$-$\bf M$相和 $X_3$-$\bf M$相的点阵型式分别是简单立方和体心立方](1)</Checkpoint>

综上，正确说法有3,5，选E选项。