双语化学辞典Chem Lexicon
概念 · 固体化学 · 晶体学 Solid-state chemistry · Crystallography

晶胞Unit cell

拼音Pinyin jīng bāo 别名Also 单胞; unit cell

定义Definition

晶胞是能完整反映晶体内部原子排列规律和对称性的最小平行六面体重复单元。整个晶体可以看作由大量相同的晶胞在三维空间无隙并置而成。

The unit cell is the smallest parallelepiped repeat unit that fully shows the arrangement and symmetry of atoms in a crystal. The whole crystal can be regarded as identical unit cells stacked side by side in three dimensions without gaps.

计数与计算Counting and calculations

计算晶胞中原子数时:顶点原子被 8 个晶胞共享(计 1/8),棱上原子被 4 个共享(计 1/4),面心原子被 2 个共享(计 1/2),体内原子完全属于该晶胞。例如简单立方含 1 个原子,体心立方 2 个,面心立方 4 个。晶体密度 ρ = Z·M / (N_A·V),Z 为晶胞内的化学式单位数。原胞只含一个阵点,而惯用晶胞为了体现对称性可能含多个阵点。

When counting atoms in a cell, a corner atom is shared by 8 cells (counts 1/8), an edge atom by 4 (1/4), a face atom by 2 (1/2), and an interior atom belongs wholly to the cell. Simple cubic thus has 1 atom per cell, body-centred cubic 2 and face-centred cubic 4. Density ρ = Z·M / (N_A·V), where Z is the number of formula units per cell. A primitive cell contains one lattice point, while the conventional cell may contain several to display the full symmetry.

典型晶胞Typical cells

NaCl 为面心立方,Z = 4;CsCl 为简单立方型(Cs⁺ 在体心),Z = 1;金刚石为面心立方加四个四面体空隙,Z = 8;铁(α-Fe)为体心立方,铜、铝、金为面心立方,镁、锌、钛为六方密堆积。

NaCl is face-centred cubic with Z = 4; CsCl is primitive cubic with Cs⁺ at the body centre, Z = 1; diamond is face-centred cubic with four tetrahedral sites filled, Z = 8; α-iron is body-centred cubic, copper, aluminium and gold are face-centred cubic, and magnesium, zinc and titanium are hexagonal close-packed.

测定Determination

晶胞参数由 X 射线、中子或电子衍射峰的位置确定,原子在晶胞中的位置由衍射强度推算。剑桥结构数据库(CSD)收录了 100 多万个有机和金属有机晶体结构,无机晶体结构数据库(ICSD)收录了数十万个无机结构。

Cell parameters come from the positions of X-ray, neutron or electron diffraction peaks, and atomic positions from their intensities. The Cambridge Structural Database holds over a million organic and metal–organic structures, and the Inorganic Crystal Structure Database hundreds of thousands of inorganic ones.

反应Reactions

ρ = Z·M / (N_A·a3)立方晶胞ρ = Z·M / (N_A·a3)(Unit cell)

a 为晶胞边长。

a is the cell edge.

Cu:面心立方 Z = 4,a = 361.5 pm ⟶ ρ = 4×63.55/(6.022×1023×(3.615×10−8)3) ≈ 8.93 g/cm3Cu: Z = 4, a = 361.5 pm ⟶ ρ = 4×63.55/(6.022×1023×(3.615×10−8)3) ≈ 8.93 g/cm3

由晶胞计算铜的密度

Copper's density from its unit cell

年表Timeline

  • 1913

    首批晶胞结构测定

    The first unit-cell structures are determined

  • 1965

    剑桥结构数据库建立

    The Cambridge Structural Database is founded

相关公式Formulas

堆积效率(空间利用率) Packing efficiency→
\[\varphi = \frac{Z\cdot\tfrac43\pi r^3}{V_\text{晶胞}}\]\[\varphi = \frac{Z\cdot\tfrac43\pi r^3}{V_\text{cell}}\]
由晶胞求晶体密度 Crystal density from the unit cell→
\[\rho = \frac{Z\,M}{N_A\,V_\text{晶胞}} = \frac{Z\,M}{N_A\,a^3}\ (\text{立方})\]\[\rho = \frac{Z\,M}{N_A\,V_\text{cell}} = \frac{Z\,M}{N_A\,a^3}\ (\text{cubic})\]
例题Worked example

NaCl:Z = 4,a = 5.640 Å:ρ = 4×58.44/(6.022×10²³×(5.640×10⁻⁸)³) = 2.165 g/cm³。

NaCl: Z = 4, a = 5.640 Å → ρ = 2.165 g/cm³.