密堆积Close-packing of equal spheres

每层球按六方排列，每个球与同层 6 个球相切；上一层的球坐在下层三个球形成的凹穴中。第三层若与第一层对齐即得 hcp，若落在第一、二层都未占据的位置则得 ccp。两种结构中每个球的配位数都是 12。体心立方（bcc）不是密堆积，配位数 8，空间利用率 68%；简单立方仅 52%。

Spheres in each layer are arranged hexagonally, each touching six neighbours, and the next layer sits in the hollows formed by three spheres below. If the third layer lies over the first the result is hcp; if it occupies positions used by neither of the first two, ccp. Both give each sphere 12 nearest neighbours. Body-centred cubic is not close-packed: it has coordination 8 and fills 68% of space, and simple cubic only 52%.

密堆积中存在两类空隙：四面体空隙（由 4 个球围成，半径约为球的 0.225 倍）和八面体空隙（由 6 个球围成，半径约 0.414 倍）。n 个球形成 2n 个四面体空隙和 n 个八面体空隙。离子晶体常可看作大的阴离子作密堆积、小的阳离子填充空隙：NaCl 中 Na⁺ 填满 ccp Cl⁻ 的全部八面体空隙，闪锌矿 ZnS 中 Zn²⁺ 填一半四面体空隙。

Close packings contain two kinds of hole: tetrahedral (surrounded by four spheres, radius about 0.225 of the sphere's) and octahedral (six spheres, about 0.414). n spheres create 2n tetrahedral and n octahedral holes. Ionic crystals can often be viewed as close-packed large anions with small cations in the holes: in NaCl, Na⁺ fills all the octahedral holes of ccp Cl⁻, and in zinc blende, Zn²⁺ fills half the tetrahedral holes.

1611 年开普勒猜想 74.05% 是等径球最密堆积的上限，这一“开普勒猜想”直到 1998 年才由黑尔斯借助计算机证明（2017 年形式化验证完成）。采用 ccp 的金属有 Cu、Ag、Au、Al、Ni、Pt；采用 hcp 的有 Mg、Zn、Ti、Co；采用 bcc 的有 Na、K、Fe、Cr、W。

In 1611 Kepler conjectured that 74.05% is the maximum density for packing equal spheres; Hales proved this with computer assistance in 1998, and a formal verification was completed in 2017. Metals with ccp include Cu, Ag, Au, Al, Ni and Pt; with hcp Mg, Zn, Ti and Co; and with bcc Na, K, Fe, Cr and W.