ISSN 1087-6596, Glass Physics and Chemistry, 2012, Vol. 38, No. 1, pp. 49–54. © Pleiades Publishing, Ltd., 2012. 49 1 INTRODUCTION Three kinds of sphere-packing structure (SPS) corre- sponding to simple cubic (SC), body-centered cubic (BCC), and face-centered cubic (FCC) lattices are the most popular among all SPSs. The SPSs of above three kinds have the following properties: the packing densities are π/6 ≈ 0.523, ( )π/8 ≈ 0.680, and π/3 ≈ 0.740, respectively; the contact numbers (with the surrounding spheres) are 6, 8, and 12, respectively; and the space groups are Pm m, and Im m, Fm m, respectively. Here, the packing density is defined as the ratio of the volume occupied by the spheres to the total volume of space. All the three SPSs are present in a cubic system but belong to different space groups. A continuous defor- mation extending over these three SPSs has never been reported in the previous papers. Authors have roughly summarized the continuous deformation by oral pre- sentations on several conferences [1–5]. In this paper, we show the existence of a continuous deformation and explain the changes in the packing densities, con- tact numbers, and space groups caused by the defor- mation in detail. CONTINUOUS DEFORMATION OF SPHERE-PACKING STRUCTURES FCC-SPS is a layer stacking of packed spheres in a two-dimensional (2D) hexagonal lattice or triangular 1 The article is published in the original. 3 2 3 3 3 lattice (TL), and it is one of the densest SPSs. The configuration of each layer in the FCC-SPS depends on three types of positions: A, B, and C. The stacking of the layers for the FCC-SPS is described as an infi- nite sequence of these positions (…ABCABCAB- CABC…) and hence called ABC-stacking. Both the SC-SPS and BCC-SPS can also be regarded as ABC-stacking of the layers that are sphere- arrangements on TLs. The above facts are seldom described in textbooks but are indispensable for the study of the continuous deformation discussed in this paper. In the case of the SC- and BCC-SPS, the layers of the spheres lie on the planes parallel to the (111)- plane in the cubic setting. The layers in the FCC-SPS are also parallel to the (111)-plane. All the layers are at different heights along a threefold rotational axis [111] but can be classified into the three types of positions A, B, or C depending on their projected positions on the (111)-plane. The projection of the layers as per the A, B, and C positions results in the formation of a TL. Further, an integrated projection of the three types of layers results in a TL. Figure 1 shows a layer each of the FCC-, SC-, and BCC-SPS. In the FCC-SPS, the TL comprises mutu- ally contacting spheres. However, in the other two SPSs, the spheres are separated at regular intervals. The radius of the sphere is R. The lattice constants L of the TLs are 2R, 2 ≈ 2.828R, and 4sin[(1/2)arc- cos(–1/3)]R ≈ 3.266R for FCC-, SC-, and BCC-SPS, respectively. In Fig. 1, the TLs have variable lattice constant L and a fixed sphere radius R; the opposite 2 R Continuous Deformation of Sphere-Packing Structures Extending over Simple, Body-Centered, and Face-Centered Cubic Lattices 1 Yoshinori Teshima a and Takeo Matsumoto a, b a National Institute of Advanced Industrial Science and Technology, (AIST) 1-2 Namiki, Tsukuba, 305-8564 Japan b Kanazawa University (Emeritus Professor), 2-77 Tsuchisimizu, Kanazawa, 920-0955 Japan (private address) e-mail: yoshinori.teshima@gmail.com Received January 8, 2011 Abstract—We have shown the existence of a continuous deformation extending over three spheres-packing structures corresponding to simple cubic, body-centered cubic, and face-centered cubic lattices. Throughout the continuous deformation, each sphere makes contact with at least six spheres, and the entire structure sus- tains a packing structure. The changes in the packing densities, contact numbers, and space groups caused by the deformation processes are explained in detail. Keywords: continuous deformation, sphere-packing structure, space group, packing density, contact number DOI: 10.1134/S1087659612010154