The Perfect Packing Models of Isometric Spheres
According to Manegold and others (1931), there are five varieties in the systematic packing forms of isometric spheres. These five forms are distinguished by their coordination number, N, which can have values of 4,6,8,10 and 12.
The form of N=6
For Ν = 6, the isometric spheres are packed in the form of a cubic lattice (Fig. 1). The three coordinate axes are set as shown in the figure, and each layer of spheres is numbered. The radius of each sphere is set to R(cm).
Fig. 1. A perfect packing form of isometric spheres (the cubic lattice, Ν = 6).
The form of N = 8
From the case of Ν = 6, we move the even-numbered layer of spheres perpendicular to the Z-axis along the X-axis by a distance R.
The form of Ν = 10
As for the form of Ν = 8, we move the even numbered layer of spheres vertical to the Y-axis by a distance R. Then, we can make the form of N = 10.
The form of
N = 12
In the case of N = 10, the even numbered layer of spheres vertical to the Y-axis is moved in the direction of the Z-axis by a distance R. This form is the densest packing form.
The form of N = 4
This form is the Wurzite Structure: the centers of the spheres are located on the center of gravity and each apex of a regular tetrahedron.
Introduction of the net-like model
The net-like model is derived from the forms of N = 6,8,10 and 12 by the following procedure. First, the radius of each sphere shrinks from R to pR(0 < q ≤ 1), and then a line connecting each original point of contact becomes the central axis of a column having a radius qR(0 ≤ p ≤ l). A part of this transformation is shown in Figure 2, where the stippled part is the net-like model. The spheres are connected by a column that may simply be called a “branch”.
Fig. 2. A part of the net-like model (stippled part).
The crossing state of the branches
As the value of q increases, the branches become thicker and ultimately intersect. Let the angle made by two adjacent branches be θ (radian) as shown by Figure 3. The condition for the branches not to intersect or overlap is sin(θ/2) ≤ q/p. θ is π/2 in the case of N = 6, and is equal to π/3 in the case of N = 8,10 and 12. Then, this equation is expressed by the following forms,
Fig. 3. The state of two branches in contact (θ is the angle made by two branches).
The structure of the model made by uncrossed branches
The net-like model is introduced by shrinking of the spheres of the original form, and therefore the material volume is less than the original. In the case where the model is composed of uncrossed branches, we define the material volume limited to that inside the original sphere as V
u (cm3). For calculation of the volume, each branch in V
u is divided by a plane tangent to the original sphere. V
u is then given by the next equation (Watanabe, 1980).
Define n as the number per unit length of the original sphere along a line. Then n is equal to 1/(2R). The number per unit volume of the shrunken spheres, nʹ, is expressed for each coordination number as
If the model is made of a material whose density is ρ0 (g cm−3), the density of the model, ρ, is expressed
In order to bring the model close to snow, the actual value of ρ0 is fixed at 0.917 g cm−3.
The border model (in the case of N=6 and p = 1)
According to Watanabe (1989), the solid angle made by an ice bond in snow increases with the increase in snow density. Therefore, N of the high-density model must be 6.
In the case of N = 6, if p = 1 and q = 1/√2, the model has the maximum density possible with uncrossed branches, and we name this model the border model. V
u of the border model is expressed by substituting the above mentioned values into Equation (2); hence,
The density, ρ, is calculated by substituting this V
u and nʹ of Equation (3) into Equation (4)
ρ = 0.917 × (8 R
3)−1 × 5.490 R
3 = 0.629 (g cm−3).
The model made by crossed branches
Where the model density is greater than 0.629 g cm−3 it is composed of crossing branches and the following relations are valid: Ν = 6, p = 1 and (1/√2) < q. A sectional view of this model is shown in Figure 4; this section is perpendicular to the Z-axis and point 0 is the center of a sphere. The stippled part is the branch of the border model, and V
u, of this model must add the volume of parts A and Β to the volume of the border model.
Fig. 4. A sectional view of the model which is composed of crossing branches; the stippled part is the branch of the border model.
A and V
B stand for these additional volumes per branch. V
A. and V
B can be found from Figure 5 which is an expanded view of Figure 4, where a(cm), b(cm) and θʹ are defined as a = qR, b = R/√2 and cos−1
θʹ = b/a, respectively. The next equations are obtained by using V
w and are
where the part of Β on a branch has four overlapping parts, one of which is shown in Figure 5 by stippling. The volume of this part is expressed by V
w. The whole volume of part Β is equal to π(a
2 − b
2(a − b), and this volume multiplied by (2θʹ/2π)
is the volume which is shown by the part of PPʹQʹMQP in Figure 5. Then we assume that the next approximate equation is valid, hence
The area of the segment QQʹM is a
θʹ − b(a
2 − b
2)1/2 and the area of the figure PPʹQʹMQP is equal to (a
2 − b
2)θʹ thus V
w is shown by
Fig. 5. The expanded view of Fig. 4 in the direction of X-axis.
u of the crossed-branches model is equal to the volume that adds 6(V
A + V
B) to V
u of the border model, and is expressed as
3 is the volume of the border model as shown in Equation (5). Substituting a = qR and b = R/√2 into Equation (8), it follows that
Substituting Equation (9) for V
u into Equation (4), the density of the model is calculated from the value of q as shown in Table 1.
In this table, the first line is the value of q and ρ for the border model. In the case of q = 1, the model corresponds with snow and ice in which the air is enclosed.
Table 1 q and density, ρ, of the model