Arsenic can interact with most metals to form compounds with diverse compositions. The electronegativity of arsenic differs significantly from that of alkali and alkaline earth metals, resulting in intermetallic compounds between them exhibiting salt-like characteristics. The structure of the typical compound $X_3As$ bears some resemblance to that of $\alpha-Y_3Z$. 

The structure of $\alpha-Y_3Z$ consists of alternating layers of $Y_2Z$ planes and non-close-packed $Y$ layers. In the $Y_2Z$ planes, the $Y$ atoms are arranged similarly to carbon in graphite layers, with $Z$ located at the center of hexagonal rings. The $Y$ atoms in the non-close-packed layers are linearly connected to the $Z$ atoms in the adjacent layers above and below. In $X_3As$, $X$ and $As$ form a hexagonal network structure resembling graphite, with the $X$ atoms in the upper and lower layers overlapping, while the remaining $X$ atoms are positioned on either side of the hexagonal pores.

In the known unit cell of $X_3As$, the shortest X-X distance is 308.8 pm, and the lattice parameters are a = b = 509.8 pm. The parameter c is currently unknown but does not exceed 1000 pm. If the X-As distance along the c-axis direction is 299.4 pm, and assuming the three closest X atoms to As are present in quantities i, j, and k with distances r, s, and t, respectively, calculate $(ir + js + kt)/(i + j + k)$.
砷可与大多数金属相互作用，形成组成多样的化合物，砷的电负性与碱金属和碱土金属的电负性相差很大，因此它们之间的金属互化物有类盐的特征。典型的化合物$X_3As$的结构与$\alpha-Y_3Z$有一定的相似性。

$\alpha-Y_3Z$的结构是由$Y_2Z$平面层和非密置层的$Y$交替形成的，$Y_2Z$平面层的$Y$如同石墨层中的C，$Z$处于六元环的中心，非密置层的$Y$与上下两层的$Z$呈直线相连。在$X_3As$中，$X$与$As$形成类似石墨那样的六边形网状结构，上下两层$X$重合，其余的$X$排列在网孔的两边。

已知$X_3As$的晶胞中，X-X的最短距离为308.8pm，晶胞参数a=b=509.8pm，c暂时未知，但不超过1000pm，若平行于c轴方向X-As间距离为299.4pm，假设距离As最近的三种X原子分别有i、j、k个，距离分别是r、s、t，计算(ir+js+kt)/(i+j+k)。
303.1pm
309.8pm
316.0pm
330.7pm
406.0pm
303.1pm
309.8pm
316.0pm
330.7pm
406.0pm
The four relatively close X-X distances in the unit cell can be expressed as:

$$d_1=\sqrt{299.4^2+(509.8/\sqrt{3})^2}=419.8pm$$

<Checkpoint>[$$d_1=\sqrt{299.4^2+(509.8/\sqrt{3})^2}=419.8pm$$](0.5)</Checkpoint>

$$d_2=\sqrt{(c/2-299.4)^2+(509.8/\sqrt{3})^2}$$

<Checkpoint>[$d_2=\sqrt{(c/2-299.4)^2+(509.8/\sqrt{3})^2}$](0.5)</Checkpoint>

$$d_3=c-2*299.4$$

<Checkpoint>[$d_3=c-2*299.4$](0.5)</Checkpoint>

$$d_4=\sqrt{(299.4*2-c/2)^2+(509.8/\sqrt{3})^2}$$

<Checkpoint>[$d_4=\sqrt{(299.4*2-c/2)^2+(509.8/\sqrt{3})^2}$](0.5)</Checkpoint>

Solving $c_1=785.6pm$ from $d_2=308.8pm$ and substituting yields $d_3=186.8pm$, which contradicts the shortest X-X distance of 308.8pm.

Solving $c_2=907.6pm$ from $d_3=308.8pm$ and substituting gives $d_2=332.4pm$ and $d_4=328.1pm$, both greater than 308.8pm.

Solving $d_4=308.8pm$ yields $c_3=1010.8pm$, which exceeds $1000pm$.

Therefore, the unit cell parameter c is $907.6pm$.

<Checkpoint>[$c=907.6pm$](1)</Checkpoint>

The X atoms adjacent to As include three in-plane hexagonal neighbors at a distance of $509.8pm/\sqrt{3}=294.3pm$;

<Checkpoint>[The nearest X atoms to As are 3, with a distance of $509.8pm/\sqrt{3}=294.3pm$](0.5)</Checkpoint>

Two interlayer X atoms adjacent to As are at a distance of $299.4pm$;

<Checkpoint>[The second-nearest X atoms to As are 2, with a distance of $299.4pm$](0.5)</Checkpoint>

The third-nearest X atoms to As could be either the second-nearest in-plane X atoms or the interlayer X atoms not directly opposite, with distances of $509.8pm/\sqrt{3}*2=588.7pm$ and $\sqrt{(509.8/\sqrt{3})^2+(907.6/2-299.4)^2}=332.3pm$, respectively. Thus, the third-nearest X atoms are the interlayer non-opposite ones, totaling 6.

<Checkpoint>[The third-nearest X atoms to As are 6, with a distance of $332.3pm$](1)</Checkpoint>

Calculating (ir+js+kt)/(i+j+k) yields $316.0pm$.

<Checkpoint>[$(ir+js+kt)/(i+j+k)=316.0pm$](1)</Checkpoint>

<MultiModal>![The image shows the unit cell of $X_3As$, which is a hexagonal unit cell. The X coordinates are (0,0,0), (0,0,1/2), (1/3,2/3,1/3), (1/3,2/3,2/3), (2/3,1/3,1/6), (2/3,1/3,5/6); the As coordinates are (1/3,2/3,0), (2/3,1/3,1/2).

The distance between X at (0,0,0) and (1/3,2/3,2/3) is $d_1$, between (0,0,0) and (2/3,1/3,5/6) is $d_2$, between (1/3,2/3,2/3) and (1/3,2/3,1/3) is $d_3$, and between (1/3,2/3,2/3) and (2/3,1/3,5/6) is $d_4$.

The distance between X at (0,0,0) and As at (1/3,2/3,0) is $294.3pm$, between X at (1/3,2/3,2/3) and As at (1/3,2/3,0) is $299.4pm$, between X at (2/3,1/3,5/6) and As at (1/3,2/3,0) is $332.3pm$, and the distance between As and the X atom diagonally opposite in the hexagon is $588.7pm$.](/media/uploads/98e8ad22-c9e2-4b17-811f-9cea53cb62a0.png)</MultiModal>
晶胞中距离较近的4种X-X间距可以表示为：

$$d_1=\sqrt{299.4^2+(509.8/\sqrt{3})^2}=419.8pm$$

<Checkpoint>[$$d_1=\sqrt{299.4^2+(509.8/\sqrt{3})^2}=419.8pm$$](0.5)</Checkpoint>

$$d_2=\sqrt{(c/2-299.4)^2+(509.8/\sqrt{3})^2}$$

<Checkpoint>[$d_2=\sqrt{(c/2-299.4)^2+(509.8/\sqrt{3})^2}$](0.5)</Checkpoint>

$$d_3=c-2*299.4$$

<Checkpoint>[$d_3=c-2*299.4$](0.5)</Checkpoint>

$$d_4=\sqrt{(299.4*2-c/2)^2+(509.8/\sqrt{3})^2}$$

<Checkpoint>[$d_4=\sqrt{(299.4*2-c/2)^2+(509.8/\sqrt{3})^2}$](0.5)</Checkpoint>

由$d_2=308.8pm$解出的$c_1=785.6pm$，代入得到$d_3=186.8pm$，与X-X最短距离是308.8pm不符。

由$d_3=308.8pm$解出的$c_2=907.6pm$，代入得到$d_2=332.4pm$，$d_4=328.1pm$，均大于308.8pm。

由$d_4=308.8pm$解出的$c_3=1010.8pm$，大于$1000pm$。

因此晶胞参数c为$907.6pm$。

<Checkpoint>[$c=907.6pm$](1)</Checkpoint>

与As邻近的X原子有层内六边形上相邻的X，有3个，距离为$509.8pm/\sqrt{3}=294.3pm$；

<Checkpoint>[距离As最近的X有3个，距离为$509.8pm/\sqrt{3}=294.3pm$](0.5)</Checkpoint>

层间与As邻近的X原子有2个，距离为$299.4pm$；

<Checkpoint>[距离As次近的X原子有2个，距离为$299.4pm$](0.5)</Checkpoint>

距离As第三近的X原子可能是层内次近的X原子，也可能是层间不直接相对的X原子，这两种原子距离分别为$509.8pm/\sqrt{3}*2=588.7pm$和$\sqrt{(509.8/\sqrt{3})^2+(907.6/2-299.4)^2}=332.3pm$，因此第三近的X原子是层间不直接相对的，有6个。

<Checkpoint>[距离As第三近的X原子有6个，距离为$332.3pm$](1)</Checkpoint>

计算(ir+js+kt)/(i+j+k)得$316.0pm$。

<Checkpoint>[$(ir+js+kt)/(i+j+k)=316.0pm$](1)</Checkpoint>

<MultiModal>![图片是$X_3As$的晶胞，这是一个六方晶胞，X的坐标为(0,0,0), (0,0,1/2),(1/3,2/3,1/3), (1/3,2/3,2/3), (2/3,1/3,1/6), (2/3,1/3,5/6)；As的坐标为(1/3,2/3,0), (2/3,1/3,1/2)。

其中坐标为(0,0,0)和(1/3,2/3,2/3)的X之间间距为$d_1$，坐标为(0,0,0)和(2/3,1/3,5/6)的X之间间距为$d_2$，坐标为(1/3,2/3,2/3)和(1/3,2/3,1/3)的X之间间距为$d_3$，坐标为(1/3,2/3,2/3)和(2/3,1/3,5/6)的X之间间距为$d_4$。

坐标为(0,0,0)的X和坐标为(1/3,2/3,0)的As之间间距为$294.3pm$，坐标为(1/3,2/3,2/3)的X和坐标为(1/3,2/3,0)的As之间间距为$299.4pm$，坐标为(2/3,1/3,5/6)的X和坐标为(1/3,2/3,0)的As之间间距为$332.3pm$，As和与As处在六边形对角线的X之间间距为$588.7pm$。](/media/uploads/98e8ad22-c9e2-4b17-811f-9cea53cb62a0.png)</MultiModal>