Packing of Spheres

Grocers and the people who stack cannon balls have probably long known that there is more than one way to pile spheres neatly. Starting with a triangular layer of close-packed spheres one can proceed to add a second layer in two ways - by placing the first sphere of the second layer either in a hole closest to the edge or in a hole second closest to the edge, and adding the rest of the spheres in close packing. For the third layer there are again two choices, the hole closest to the edge or the hole second closest. If the closest hole was chosen for both the second and third layer, and if subsequent layers are added in the same way, a regular tetrahedron will result. If the second closest hole was chosen for each successive layer a more squat pyramid will result, but both of these are crystallographically identical, and indeed, one can be transformed into the other by removing spheres from the three ascending edges until a new (but smaller) pyramid of the other type results. Both of these are in cubic closest packing and differ merely in that the individual tetrahedra of four spheres have a face parallel to the face of the pyramid in one case, and an edge in the other.

There are also two pyramids possible for hexagonal closest packing, depending on whether the second layer was formed by starting with the holes closest or second closest to the edge of the first layer. In order to have the third layer directly above the first (which results in hexagonal closest packing) it will be necessary to choose a hole closest to the edge if a second closest hole had been chosen for the second layer, and vice versa. Thus the resulting pyramid will be a step pyramid with steepness midway between the two pyramids with the cubic closest-packed arrangement. The two pyramids with h.c.p. are very similar, and one can be transformed into the other by merely removing the bottom layer (or adding a layer beneath the first layer).

It is interesting to consider the efficiency of packing of one layer of objects with circular cross section in a square box. It is customary to pack vials, bottles, circular pill boxes, etc. in simple square arrays as in figure 1. However, more objects would fit into the same box if close packing were used provided there were eight or more (7.5 to be more exact) objects in a row. We see in fig. 2 that every second row in close packing has one fewer circle, but we also see that we have one more row (nine instead of eight). Thus there are 68 circles in figure 2 as against 64 in fig. 1, and in the same sized square. The actual saving in thickness per two rows is 2(2 - √3)R, where R is the radius of the circles. Then n, the number of rows necessary to produce one extra horizontal row, is

n=√3/(2-√3) + 1 = ca. 7.5

In an infinite array the % void for square packing is 21.5%. while for close packing it is 9.3% so 15.5% more circles could be packed into an infinite square. In the above example of eight circles in the first row we found that four more circles could be packed, which is 6.25% more than in square packing; 40% of the limiting case.

Let us now examine the three-dimensional case where
spheres are packed into a square box. For the simple cubic arrangement in a box, with edges an integral multiple of the diameter of the spheres, the density is 0.5238, that is, there is 47.64% void, and there are no "boundary effects."

Starting with the same square first layer, but putting the second layer in the holes of the first closest to the side of the box and so on, leads to a body-centered tetragonal structure with axial ratio 3½ and limiting density 0.605 and % void 39.5%. While this limiting density is better than that for simple cubic, boundary effects now appear, causing every second layer to have one less sphere in each row of spheres in each direction, so that if the first layer had n2 spheres, the second will have (n - 1)^{2}, as will all even-numbered layers. For a box with 64 spheres in the first layer there will be only 49 in the second, an average of 56.5 spheres per layer. However, the layers are packed closer together in the vertical direction, with an axial ratio of 3½, meaning that there will be layers of b.c. than of simple cubic, and there will be one extra b.c. layer for every 7.47 simple cubic layers, or approximately 17 b.c. layers to 15 s.c. layers. If there are n^{2} spheres in the first layer, then there will be zn^{2} spheres in z layers of s.c. packing. For the tetragonal body centered packing there will be the average of n^{2} and (n - 1)^{2} spheres per layer, which is equal to n^{2} - n + 1/2 , and hence there will be (17/15)z(n^{2} - n + 1/2) spheres in the b. c. box of the same size. The difference of these expressions will be the difference in number of spheres. A box with 64 spheres in the first layer, and 15 s.c. layers deep would contain 15(64) = 960 spheres. The same box would contain 17 b.c. layers, and as 9 of these could contain 64 spheres and 8 containing 49 spheres, the total is 968, eight more than the s.c. case. However, if one more layer were added, the simple arrangement would gain 64 spheres while the b.c. would gain only 49, so that the simple cubic box would then have 7 more spheres than the b.c. box. So it is nip and tuck for a box with an 8 by 8 base. But for smaller bases the s.c. is more efficient and for larger bases the b.c. Thus, for a 20 x 20 base, 16 s.c. layers deep, there are 6400 spheres, while for the 18 b.c. layers there would be 6849 spheres, 7% more, almost half of the limiting value of 15.5%

Close-Packed Layers

We have seen that a square with eight circles along an edge will have nine rows of close packed circles perpendicular to this edge, and that 6.25% more circles could be packed this way than in simple square packing. Using this square as the base of a rectangular box, and substituting spheres for circles, we now proceed to add a second layer in close packing with the first. There would appear to be two ways to add second layer - as with the four types of close packed pyramids, by adding a sphere in either the hole closest to the edge (with eight spheres) or in the second closest hole. However, if we start with the closest hole on the one side, we end up with the second closest hole on the other side, so the two ways are identical for our purpose. The second layer will contain eight rows of spheres in each direction, rather than 9 by 8 as in the first layer, so there will be a loss of 8 spheres over the arrangement resulting from simply piling the spheres directly over each other. There is, however a saving of space in the vertical direction. If we proceed in h.c.p., with the third layer directly over the first, the axial ratio is 1.633, and there will be 0.367 layer thickness saved in going from the first to the second layer. i.e., 0.1835 of a layer. Then 1.633/.2(.1835) = 4.46 layers after the first to make room for one more c.p. layer; i.e., 7.46 layers of c.p. will = 6.46 simple layers.

In the simple packing of spheres directly over the first layer (close packed), etc, there will be 8 x 9 = 72 spheres per layer, or 432 in 6.46 layers. For h.c.p. there will be alternately 72 and 64, or average 68 spheres per layer, or 476 in 7.46 layers, 10% more than in simple packing over a c.p. base. If we had used c.c. packing, that is, if the third layer were not above the first as in h.c. packing,