📂Physics

Unit Cell

정의^{1 2}

A small unit that, by repeating the same arrangement, fills the whole space without gaps and can represent a crystal structure is called a unit cell. The set of relative positions of points inside a unit cell is called the basis.

수학적 정의

For a lattice $L = \left\{ n_{1}\mathbf{a}_{1} + n_{2}\mathbf{a}_{2} + n_{3}\mathbf{a}_{3} \mid n_{i} \in \mathbb{Z} \right\}$ given in three-dimensional space $\mathbb{R}^{3}$, let the translation vectors be $\mathbf{t} = m_{1}\mathbf{a}_{1} + m_{2}\mathbf{a}_{2} + m_{3}\mathbf{a}_{3}$ ($n_{i}$ are arbitrary integers).

For a closed subset $U \subset \mathbb{R}^{3}$ of the whole space, if there exists $\left\{ \mathbf{ t}_{k} \right\}$ satisfying the following, then $U$ is called a unit cell.

1. $\bigcup_{k} (U + \mathbf{t}_{k}) = \mathbb{R}^{3}$
2. For $k \neq \ell$, $(U + \mathbf{t}_{k})^{\circ} \cap (U + \mathbf{t}_{\ell})^{\circ} = \varnothing$

Here $+$ is the sum of sets, and $A^{\circ}$ is the interior of set $A$.

The set of relative coordinates of points in the unit cell $U$ of lattice $L$ expressed as linear combinations of the primitive lattice vectors $\mathbf{a}_{i}$ is called the basis.

$$ \mathbf{r}_{j} = x_{j} \mathbf{a}_{1} + y_{j} \mathbf{a}_{2} + z_{j} \mathbf{a}_{3}, \quad j = 1, 2, \ldots, N $$

In this case $0 \le x_{j}, y_{j}, z_{j} \le 1$.

설명

In simple terms, a unit cell is a subset that, when translated and tiled, can represent the entire space without omissions or overlaps. The conditions for a unit cell expressed in plain language are as follows.

1. It covers the entire space without gaps.
2. Different translated copies do not overlap.

Each coordinate of the basis represents the position of an atom in the actual crystal structure. Thus the size of the basis $| \left\{ \mathbf{r}_{j} \right\} | = N$ denotes the number of atoms in the unit cell.

There is an important point when indicating how many lattice points are contained in a unit cell. A lattice point located on a boundary must be counted according to the fraction of that point actually included in the cell. In the figure $(a)$, let the left gray box be the unit cell. Consider the lattice point at the center of the cell $p_{1}$. For $p_{1}$ one can choose an arbitrary neighborhood $N_{\epsilon}(p_{1})$ that is contained in the unit cell $U$. In this case $p_{1}$ counts as 'one' lattice point.

By contrast, points lying on the boundary of the unit cell, for example a point like $p_{2}$, do not admit any neighborhood that is entirely contained in the unit cell for any choice of $\epsilon$. In such a case the count of $p_{2}$ is less than $1$. Concretely, for a sufficiently small neighborhood, the ratio of the volume of the intersection between the neighborhood and the unit cell to the volume of the neighborhood is $\frac{1}{4}$, hence this lattice point contributes $\frac{1}{4}$ lattice points to the unit cell. Therefore the unit cell in $(a)$ contains a total of $1 + 4\cdot\frac{1}{4} = 2$ lattice points. Alternatively, if when filling the entire lattice by translations of the unit cell a lattice point is shared among $n$ unit cells, one can understand that the lattice point should be counted as $\frac{1}{n}$.

While the unit cell in $(a)$ contains two lattice points, in figure $(b)$ the primitive unit cell and the Wigner–Seitz unit cell contain only $1$ lattice points.

Any complex crystal structure can be described using a lattice, a unit cell, and a basis. The lattice represents the macroscopic repeating pattern of the crystal, whereas the basis describes how atoms are locally arranged. A good example to understand this is the honeycomb structure. Although the honeycomb is, contrary to intuition, not a lattice by definition, it can be represented by a lattice plus a basis. Here the set of black dots inside the hexagons forms the lattice, the dotted region is the unit cell, and the blue and yellow points are represented within the unit cell by coordinates $(\frac{1}{3}, \frac{1}{3})$ and $(\frac{2}{3}, \frac{2}{3})$ respectively; these two coordinates make up the basis.

기본 단위셀

A cell that contains only one lattice point is called a primitive unit cell. If the lattice is given as $L = \left\{ n_{1}\mathbf{a}_{1} + n_{2}\mathbf{a}_{2} + n_{3}\mathbf{a}_{3} \mid n_{i} \in \mathbb{Z} \right\}$, the set of primitive lattice vectors $\mathbf{a}_{i}$ naturally defines a primitive unit cell.

위그너-자이츠 단위셀

Choose a reference lattice point and consider the region bounded by the perpendicular bisectors between that point and its neighboring lattice points; this region is called the Wigner–Seitz unit cell. By definition it is a primitive unit cell and has no lattice points on its boundary.

관습 단위셀

A conventional unit cell is a unit cell chosen by convention for convenience. It is often chosen to be a rectangular parallelepiped (a rectangular box).