Three-dimensional crystals possess translational symmetry in all three directions. However, there exists a class of structures, including quasicrystals, that disrupt this property due to complex reasons. As crystallographers delved deeper, it was discovered that these structures restore translational symmetry when viewed from a higher-dimensional perspective, hence they are referred to as higher-dimensional crystals. Compound $\mathbf{A}$ is a common $(3+1)D$ crystal with the composition $\mathrm{(TTF)} \mathrm{I}_{0.7+x}(0<x<0.1)$, where $\mathrm{TTF}$ (tetrathiafulvalene) and $\mathrm{I}$ exhibit the same periodicity along the $a$ and $b$ directions, while their periodicities differ along the $c$ direction, resulting in a "vernier caliper"-like structure in this direction. If the ratio $c_{\mathrm{T}} / c_{\mathrm{TT}}$ is approximately considered an integer ratio, the combined superstructure possesses a conventional unit cell.  

The known superstructure cell parameters of $\mathbf{A}$ are $a=4.815 \mathrm{~nm}$, $b=1.605 \mathrm{~nm}$, $c \approx 2.500 \mathrm{~nm}$, $\beta=91.50^{\circ}$. If any lattice point of the two substructures is placed at the $(0,0,0)$ position in the superstructure, then $\mathrm { TTF }$ lattice points are located at $(1/3,0,0)$, $(5/6,1/2,0)$, $(0,0,1/7)$, while $\mathrm{I}$ lattice points are located at $(1/6,0,14/15)$, $(0,1/2,9/10)$, $(0,0,1/5)$ (minor variations in the coordinates of $\mathrm{TTF}$ and $\mathrm{I}$ can be neglected).  

Based on the above information, determine the lattice types of the $\mathrm{TTF}$ and $\mathrm{I}$ substructures (if only the $b$-direction is centered, it can be denoted as $B$-centered, using the superstructure coordinate system as reference).
三维晶体在 3 个方向上均有平移对称性, 然而存在包括准晶体在内的一类结构, 由于复杂的原因破坏了上述性质。随着晶体学家的深入研究, 人们发现它们在更高维的角度上看恢复了平移对称性, 于是便称其为高维晶体。化合物 $\mathbf A$ 是一种常见的$(3+1)D$ 晶体, 其组成为 $\mathrm{(TTF)} \mathrm{I}_{0.7+x}(0<x<0.1)$ , 其中 $\mathrm {TTF}$ (四硫代富瓦烯)和 $\mathrm { I}$ 在 $a, b$ 方向上均具有相同的周期, 而 $c$方向上两者周期不一致, 即在该方向上有类似 “ 游标卡尺 ” 的结构。若认为 $c_{\mathrm{T}} / c_{\mathrm{T} \mathrm{T}}$ 近似为整数比, 则两者共同的超结构具有正当晶胞。 已知 $\mathbf{A}$ 的超结构晶胞参数 $a=4.815 \mathrm{~nm} 、 b=1.605 \mathrm{~nm} 、 c \approx 2.500 \mathrm{~nm} 、 \beta=91.50^{\circ}$, 若两套子结构的任一点阵点放置在超结构中坐标 $(0,0,0)$ 的位置, 则 $(1 / 3,0,0),(5 / 6,1 / 2,0),(0,0,1/7)$处存在 $\mathrm { TTF }$ 的点阵点; $(1 / 6,0,14 / 15),(0,1 / 2,9 / 10),(0,0,1 / 5)$ 处存在 $\mathrm { I }$ 的点阵点 (可忽略$\mathrm { TTF }$ 和 $\mathrm { I }$ 坐标的微小变化)。

根据以上信息,判断 $\mathrm { TTF }$ 和 $\mathrm { I }$ 子结构各自的点阵形式(若仅有 $b$ 方向带心可以写作 $B$ 心,以超结构坐标系为准)。
TTF substructure: A-centered monoclinic;  
I substructure: A-centered monoclinic
TTF substructure: A-centered monoclinic;  
I substructure: B-centered monoclinic
TTF substructure: A-centered monoclinic;  
I substructure: C-centered monoclinic
TTF substructure: B-centered monoclinic;  
I substructure: A-centered monoclinic
TTF substructure: B-centered monoclinic;  
I substructure: B-centered monoclinic
TTF substructure: B-centered monoclinic;  
I substructure: C-centered monoclinic
TTF substructure: C-centered monoclinic;  
I substructure: A-centered monoclinic
TTF substructure: C-centered monoclinic;  
I substructure: B-centered monoclinic
TTF substructure: C-centered monoclinic;  
I substructure: C-centered monoclinic
TTF子结构：A心单斜；
I子结构：A心单斜
TTF子结构：A心单斜；
I子结构：B心单斜
TTF子结构：A心单斜；
I子结构：C心单斜
TTF子结构：B心单斜；
I子结构：A心单斜
TTF子结构：B心单斜；
I子结构：B心单斜
TTF子结构：B心单斜；
I子结构：C心单斜
TTF子结构：C心单斜；
I子结构：A心单斜
TTF子结构：C心单斜；
I子结构：B心单斜
TTF子结构：C心单斜；
I子结构：C心单斜
Analysis of the TTF Substructure

Known TTF lattice point coordinates (with the supercell as the reference frame):

The translation vectors of the TTF lattice are $(0, 0, 0) , (1/3, 0, 0) , (5/6, 1/2, 0) , (0, 0, 1/7)$

<Checkpoint>[The translation vectors of the TTF lattice are $(0, 0, 0) , (1/3, 0, 0) , (5/6, 1/2, 0) , (0, 0, 1/7)$](1)</Checkpoint>

These points are all lattice points in the TTF lattice. Therefore, the vector difference between them must also be a translation vector of the TTF lattice.

Finding translation vectors in the a, b directions:

The vector from $P₀$ to $P₁$ is $v₁ = P₁ - P₀ = (1/3, 0, 0)$. This indicates that $(1/3, 0, 0)$ is a translation vector. Then, its integer multiples are also translation vectors, such as $2 * v₁ = (2/3, 0, 0) and 3 * v₁ = (1, 0, 0)$.
The vector from $P₀$ to $P₂$ is $v₂ = P₂ - P₀ = (5/6, 1/2, 0)$.
Now we have two translation vectors $v₁$ and $v₂$ in the ab plane.
Calculate $v₂ - v₁ = (5/6, 1/2, 0) - (1/3, 0, 0) = (5/6 - 2/6, 1/2, 0) = (3/6, 1/2, 0) = (1/2, 1/2, 0)$.

<Checkpoint>[There exists a translation vector $(1/2, 1/2, 0)$ in the TTF lattice](1)</Checkpoint>

We have derived a new translation vector $t = (1/2, 1/2, 0)$. This vector is exactly the centering vector of the C-centered lattice.

<Checkpoint>[C-center corresponds to the translation vector $(1/2, 1/2, 0)$](1)</Checkpoint>

In addition to translations along the supercell basis vector directions (such as $(1,0,0)$, etc.), the TTF substructure's lattice also contains an additional translation symmetry of $(1/2, 1/2, 0)$.
This means that the TTF substructure is C-centered with respect to the supercell. Since the superstructure is monoclinic, the TTF substructure is C-centered monoclinic.

<Checkpoint>[The TTF substructure is C-centered monoclinic](1)</Checkpoint>

Analysis of the I Substructure

Known translation vectors:

$u₁ = (1/6, 0, 14/15), u₂ = (0, 1/2, 9/10), u₃ = (0, 0, 1/5)$

<Checkpoint>[The translation vectors of the I lattice are $(0, 0, 0) , (1/3, 0, 0) , (5/6, 1/2, 0) , (0, 0, 1/7)$](1)</Checkpoint>

The basis vectors of the I substructure can be written as a linear combination of the supercell basis vectors. Let the basis vectors of the I substructure be $a', b', c'$, then: $ a' = m₁₁a + m₁₂b + m₁₃c b' = m₂₁a + m₂₂b + m₂₃c c' = m₃₁a + m₃₂b + m₃₃c $. The given lattice point coordinates $(u,v,w)$ must be an integer multiple linear combination of these basis vectors. $(1/6, 0, 14/15), (0, 1/2, 9/10), (0, 0, 1/5)$ are all translation vectors. We hope to find whether there exists a translation vector $t = (p,q,r) + n₁a' + n₂b' + n₃c'$, where $(p,q,r)$ is one of $(1/2,1/2,0), (1/2,0,1/2)$, or $(0,1/2,1/2)$. Note that $u₂ = (0, 1/2, 9/10)$. $ u₂ - (0, 1/2, 1/2) = (0, 0, 9/10 - 1/2) = (0, 0, 4/10) = (0, 0, 2/5) = 2 * (0, 0, 1/5)$. Therefore, $u₂ - 2 * u₃ = (0, 1/2, 1/2)$. Since $u₂$ and $u₃$ are both translation vectors of the I lattice, their integer multiple linear combination $(u₂ - 2 * u₃)$ is also a translation vector of the I lattice.

<Checkpoint>[An integer multiple linear combination of translation vectors is also a translation vector](1)</Checkpoint>.

We have derived the translation vector $t = (0, 1/2, 1/2)$.
 
<Checkpoint>[There exists a translation vector $(0, 1/2, 1/2)$ in the I lattice](1)</Checkpoint>

This vector is exactly the centering vector of the A-centered lattice.

<Checkpoint>[A-center corresponds to the translation vector $(0, 1/2, 1/2)$](1)</Checkpoint>

The lattice of the I substructure contains a translation symmetry of $(0, 1/2, 1/2)$.
This means that the I substructure is A-centered with respect to the supercell. Since the superstructure is monoclinic, the I substructure is A-centered monoclinic.

<Checkpoint>[The I substructure is A-centered monoclinic](1)</Checkpoint>
对 $\mathrm{TTF}$ 亚结构的分析

已知的$\mathrm{TTF}$阵点坐标（以超结构晶胞为参考系）：

$\mathrm{TTF}$ 点阵的平移矢量有$ (0, 0, 0) ,  (1/3, 0, 0) ,  (5/6, 1/2, 0) ,  (0, 0, 1/7)$

<Checkpoint>[$\mathrm{TTF}$ 点阵的平移矢量有$ (0, 0, 0) ,  (1/3, 0, 0) ,  (5/6, 1/2, 0) ,  (0, 0, 1/7)$](1)</Checkpoint>

这些点都是$\mathrm{TTF}$ 点阵中的格点。因此，它们之间的矢量差也必须是$\mathrm{TTF}$ 点阵的平移矢量。

寻找 a, b 方向的平移矢量:

从$ P₀$ 到$ P₁ $的矢量为$ v₁ = P₁ - P₀ = (1/3, 0, 0)$。这说明 $(1/3, 0, 0)$ 是一个平移矢量。那么，其整数倍也是平移矢量，例如 $2 * v₁ = (2/3, 0, 0) 和 3 * v₁ = (1, 0, 0)$。
从 $P₀ $到$ P₂ $的矢量为 $v₂ = P₂ - P₀ = (5/6, 1/2, 0)$。
现在我们有两个在 ab 平面内的平移矢量$ v₁$ 和 $v₂$。
计算 $v₂ - v₁ = (5/6, 1/2, 0) - (1/3, 0, 0) = (5/6 - 2/6, 1/2, 0) = (3/6, 1/2, 0) = (1/2, 1/2, 0)。$

<Checkpoint>[$\mathrm{TTF}$ 点阵中存在平移矢量$(1/2, 1/2, 0)$](1)</Checkpoint>

我们推导出了一个新的平移矢量 $t = (1/2, 1/2, 0)$。这个矢量正好是 C 心点阵的定心矢量。

<Checkpoint>[C心对应平移向量$(1/2, 1/2, 0)$](1)</Checkpoint>

$\mathrm{TTF}$ 亚结构的点阵除了包含沿超胞基矢方向的平移（如 $(1,0,0) $等）外，还额外包含一个$ (1/2, 1/2, 0) $的平移对称性。
这意味着 $\mathrm{TTF}$ 亚结构相对于超结构晶胞是 C 心（C-centered）的。由于超结构是单斜的，所以$\mathrm{TTF}$亚结构为 C 心单斜。

<Checkpoint>[$\mathrm{TTF}$亚结构为 C 心单斜](1)</Checkpoint>

对 $\mathrm I$ 亚结构的分析

已知的平移矢量:

$u₁ = (1/6, 0, 14/15), u₂ = (0, 1/2, 9/10), u₃ = (0, 0, 1/5)$

<Checkpoint>[$\mathrm{I}$ 点阵的平移矢量有$ (0, 0, 0) ,  (1/3, 0, 0) ,  (5/6, 1/2, 0) ,  (0, 0, 1/7)$](1)</Checkpoint>

$\mathrm I$亚结构的基矢可以写成超结构基矢的线性组合。设$\mathrm I$亚结构的基矢为$ a', b', c'$，则：$ a' = m₁₁a + m₁₂b + m₁₃c b' = m₂₁a + m₂₂b + m₂₃c c' = m₃₁a + m₃₂b + m₃₃c $。给定的格点坐标 $(u,v,w)$必须是这些基矢的整数倍线性组合。$ (1/6, 0, 14/15), (0, 1/2, 9/10), (0, 0, 1/5) $都是平移矢量。 希望找到是否存在一个平移矢量 $t = (p,q,r) + n₁a' + n₂b' + n₃c'$，其中$ (p,q,r) $是 $(1/2,1/2,0), (1/2,0,1/2)$, 或$ (0,1/2,1/2) $中的一个。 注意到$ u₂ = (0, 1/2, 9/10)。 u₂ - (0, 1/2, 1/2) = (0, 0, 9/10 - 1/2) = (0, 0, 4/10) = (0, 0, 2/5) = 2 * (0, 0, 1/5)$。 因此，$u₂ - 2 * u₃ = (0, 1/2, 1/2)$。 由于 $u₂$ 和 $u₃ $都是 $\mathrm I $点阵的平移矢量，它们的整数倍线性组合$ (u₂ - 2 * u₃)$ 也是$\mathrm I $点阵的平移矢量

<Checkpoint>[平移矢量的整数倍线性组合也是平移矢量](1)</Checkpoint>。

我们推导出了平移矢量$ t = (0, 1/2, 1/2)$。
 
<Checkpoint>[$\mathrm I $点阵中存在平移矢量$(0, 1/2, 1/2)$](1)</Checkpoint>

 这个矢量正是 A 心点阵的定心矢量。

<Checkpoint>[A心对应平移向量$(0, 1/2, 1/2)$](1)</Checkpoint>

$\mathrm I $亚结构的点阵包含一个 $(0, 1/2, 1/2)$ 的平移对称性。
这意味着 $\mathrm I $ 亚结构相对于超结构晶胞是 A 心（A-centered）的。由于超结构是单斜的，所以$\mathrm I$亚结构为 A 心单斜。

<Checkpoint>[$\mathrm I$亚结构为 A 心单斜](1)</Checkpoint>