Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 40 (1999), No. 1, 125-140.

Periodic Sphere Packings and the Wulff-Shape

Uwe Schnell

Mathematisches Institut, Universität Siegen D-57068 Siegen, Germany e-mail: schnell@mathematik.uni-siegen.de

Abstract. The asymptotic shape of large densest sphere packings in a lattice L ⊂ E^{d}(d ≥ 2), measured by parametric density, is a polytope, the so-called Wulffshape of ideal crystals in crystallography. This was shown in [15] and [16] by density deviation derived from parametric density. Our main goal is a generalization of this approach from lattice packings to periodic packings of spheres. It turns out that the asymptotic shape of large densest periodic packings is a generalized Wulff-shape which is more complicate than in the lattice case. As an application we determine this Wulff-shape for a type of crystals with two different atoms (NaCl-type) and for a non-lattice crystal (diamond) and for a combination of both (ZnS-type). In all cases the result coincides for suitable parameter with the natural shapes. In particular the asymptotic shape in the case of the ZnS-type is a tetrahedron, i.e. not centrally symmetric. Although the intention of these generalizations is application to crystallography, the results can also be considered as contributions to finite and infinite packings of spheres.

1. Introduction

Let B^{d} denote the unit ball in Euclidean d-space E^{d}(d ≥ 2). Let V denote the volume and V (B^{d}) = κ_{d}. For x, y ∈ E^{d} let xy denote their scalar product, so that xx = ∥x∥^{2}. For X ⊂ E^{d} let lin (X), aff (X), pos (X) and conv (X) be its linear, affine, positive and convex hull, respectively. In the following let L^{d} denote the set of lattices L ⊂ E^{d} with det L ≠ 0. For L ∈ L^{d} let λ_{i}(L) = λ_{i}(B^{d}, L), i = 1, …, d, and μ(L) = μ(B^{d}, L) be its successive minima

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 1999 Heldermann Verlag

and its inhomogenous minimum (or covering radius), respectively (see e.g. [10]). In particular λ_{1}(L) is the length of the shortest nonzero vector in L. Let P(L) be the set of polytopes P with nonempty interior and facets orthogonal to suitable vectors of the polar lattice L^{∗}. In particular, P(L) contains the set of L-polytopes, i.e. convex hulls of finite subsets of L.

If C_{n} = {c_{1}, …, c_{n}} ⊂ E^{d} is a discrete set such that C_{n} + B^{d} is a packing, then its parametric density is defined (see [14]) by

δ_{ϱ}(B^{d}, C_{n}) = nκ_{d}/V ( conv (C_{n} + ϱB^{d})),  (1)

where ϱ > 0 is a parameter. For a convex body K and a packing lattice L ∈ L^{d} for K we use δ_{ϱ}(B^{d}, K) = δ_{ϱ}(B^{d}, L, K) instead of δ_{ϱ}(B^{d}, K ∩ L). In the case of lattice packings for any sequence P_{n} ∈ P(L), whose inradii tend to infinity, we have (cf. e.g. [10], p.219)

lim sup_{n→∞} δ_{ϱ}(B^{d}, P_{n}) = κ_{d}/ det L =: δ_{L},  (2)

where δ_{L} is the lattice packing density for B^{d} and L. So as a refinement the density deviation for P ∈ P(L) has been defined ([15]) as

Δ_{L}(P, ϱ) = lim_{λ→∞}(V (λP))^{1/d}{δ_{L} − δ_{ϱ}(B^{d}, λP)}.  (3)

In [15] it was proved that for P, Q ∈ P(L) with Δ_{L}(P, ϱ) < Δ_{L}(Q, ϱ), P is, in a natural sense, asymptotically ϱ-denser than Q, and the minimum of Δ_{L}(P, ϱ) is attained for the Wulff-shape

W (L, ϱ) = \{x ∈ E^{d} : ux ≤ ϱ∥u∥ − \frac{1}{2}, u ∈ L^{∗} primitive\} .  (4)

W (L, ϱ) is a polytope and it depends only on L and ϱ. Only ϱ with

ϱ > ϱ_{L} = max{(2∥u∥)^{−1} : u ∈ L^{∗}\{0}}

are considered. This guarantees that W (L, ϱ) is nonempty and no (d − 1)-dimensional P is ϱ-denser than all P ∈ P(L).

In Section 2 we give an alternative proof that W (L, ϱ) is a polytope, without using diophantine approximation and give bounds for the number of facets depending on ϱ.

It is remarkable that the principle of minimizing the volume (maximizing the density) is equivalent to the principle of minimizing the free surface energy, which is known as the Gibbs-Curie-Wulff principle (cf. e.g. [11], [5]). Another approach is in [1], where the asymptotic shape of the densest packing configurations C_{n} ⊂ L, i.e. with maximal δ_{ϱ}(B^{d}, C_{n}) for given n, is considered.

Our main goal is a generalization of Wills’ approach from lattice packings to periodic packings of spheres. Only few crystals are 1-atomic lattice packings (e.g. metals and rare gases). So a generalization to periodic packings of spheres with different radii is reasonable. For this we need some more notations.

Let K_{1}, …, K_{n} be convex bodies with volumes v_{1}, …, v_{n} and v = v_{1} + … + v_{n}. Further let L ∈ L^{d} and for t_{1}, …, t_{n} ∈ E^{d} let L_{j} = L + t_{j}, j = 1, …, n (we will always assume that t_{1} = 0). If (L_{1} + K_{1}) ∪ … ∪ (L_{n} + K_{n}) is a packing then M = L_{1} ∪ … ∪ L_{n} is a periodic packing for K_{1}, …, K_{n} (cf. [12]). In the following we use M for the point set together with the K_{i} and the v_{i}. All ideal crystals can be presented in this way and in general M is not a lattice (although chemists speak of lattices). In this application K_{1}, …, K_{n} are spheres (atoms) of different size.

For a convex body K we define L_{j}(K) = card (K ∩ L_{j}) and M(K) = card (K ∩ M) analogously to the lattice point enumerator L(K) = G(K, L) = card (K ∩ L). Further let

δ_{ϱ}(M, K) = δ_{ϱ}(B^{d}, M, K) = \frac{∑_{j=1}^{n} v_{j}L_{j}(K)}{V (K + ϱB^{d})}  (5)

be the parametric density with respect to M and ϱ. If K is the convex hull of M-points, then (5) is a generalization of (1). Further the definition is natural, if K_{1}, …, K_{n} are balls with different radii. In general the K_{i} have different shapes and the question is how to choose a gauge body C instead of B^{d} in the denominator. For the choice of C one can take chemical and physical aspects into consideration, but in this paper we choose C = B^{d} also for the general case. Note that the density in (5) may be greater than 1 for ϱ = 1. To avoid this one can use C = rB^{d} with a suitable r > 0.

The density δ_{M} of an infinite periodic packing is (cf. [12], p.25)

δ_{M} = \frac{v}{det L}.  (6)

To generalize (3), it is not sufficient to consider only dilatates to a given P ∈ P(L) because there might be many points of M close to a facet and it is more efficient to include them. So we choose a sequence from the set of polytopes P(P) ⊂ P(L), which have the same outer normals as P. Now we introduce the density deviation for periodic packings.

Definition 1. For P ∈ P(L) let

Δ_{M}(P, ϱ) = inf lim inf_{λ→∞} (V (λR_{λ}))^{1/d} {δ_{M} − δ_{ϱ}(M, λR_{λ})} ,

where the infimum is taken over all sequences {R_{λ}} with R_{λ} ∈ P(P) and R_{λ} → P.

To justify the definition of the density deviation we will prove in Section 3 that if Δ_{M}(P, ϱ) < Δ_{M}(Q, ϱ) for P, Q ∈ P(L), then P is asymptotically ϱ-denser than Q ∈ P(L) in the following natural sense.

Definition 2. Let P, Q ∈ P(L). Then P is called asymptotically ϱ-denser than Q if there is a sequence R_{λ} ∈ P(P) with R_{λ} → P such that for each sequence S_{ν} ∈ P(Q) with S_{ν} → Q and each ν ∈ ℕ a λ(ν) can be chosen such that for ν ≥ ν_{0}

M(λ(ν)R_{λ(ν)}) ≤ M(νS_{ν}) ≤ M((λ(ν) + 1)R_{λ(ν)+1})

and

min {δ_{ϱ}(M, λ(ν)R_{λ(ν)}), δ_{ϱ}(M, (λ(ν) + 1)R_{λ(ν)+1})} > δ_{ϱ}(M, νS_{ν}).

For x ∈ ℝ let {x} = x − [x] be the fractional part of x. For primitive u ∈ L^{∗} and j = 1, …, n we define

ϵ_{j}(u) = {u · t_{j}}.

Then ϵ_{j}(u) ∈ [0, 1) and ϵ_{j}(u) det L/Δ(u) represents the distance of the hyperplane containing points of L_{j} between {x : ux = 0} and {x : ux = 1} to the origin, where Δ(u) = det L∥u∥ is the determinant of the (d − 1)-sublattice of L orthogonal to u. For nonprimitive u ∈ L^{∗} let ϵ(u) = ϵ(u′), where u′ = u/m primitive. Note that ϵ_{j}(−u) = 1 − ϵ_{j}(u), for ϵ_{j}(u) ≠ 0.

Let P ∈ P(L) such that the outer normals u_{i}, i = 1, …, k, of its facets F_{i} are primitive vectors of L^{∗}. Let f_{1}, …, f_{k} denote their (d − 1)-dimensional volumes and Δ_{1}, …, Δ_{k} the (d − 1)-dimensional determinants of the corresponding sublattices of L. Note that Δ_{i} = det L∥u_{i}∥. For fixed u_{i} each L_{m}, m = 1, …, n, determines a family H_{m}(u_{i}) of parallel hyperplanes orthogonal to u_{i}. For the approximating sequence R_{λ} in Definition 1 we have to choose for each u_{i} the position of the corresponding supporting hyperplanes. We will prove that R_{λ} leads to an optimal density, if these hyperplanes belong to one certain family H_{m}(u_{i}). The following definition is motivated by the choice of this optimal family.

Definition 3. For u ∈ L^{∗} and m = 1, …, n let

τ_{m}(u) = \frac{1}{2} − ∑_{j=1}^{n} \frac{v_{j}}{v} · \{ϵ_{m}(u) − ϵ_{j}(u)\}  (7)

and

τ(u) = max_{1≤m≤n} τ_{m}(u).  (8)

We call τ(u)/∥u∥ the optimal facet density in direction of u.

The geometric meaning of τ(u) will be clear in the proof of the main result in Section 3

Δ_{M}(P, ϱ) = δ_{M}V (P)^{1/d−1} ∑_{i=1}^{k} f_{i} (ϱ − \frac{τ(u_{i})}{∥u_{i}∥}) ,  (9)

and there is a P ∈ P(L) with minimal Δ_{M}(P, ϱ), which is given by the following

Definition 4. Let

W_{M} = W (M, ϱ) = {x : ux ≤ ϱ∥u∥ − τ(u), u ∈ L^{∗} primitive}  (10)

be the generalized Wulff-shape.

For n = 1 we have τ(u) = 1/2 and hence W_{M} is a generalization of W (L_{,}ϱ) in (4). We only consider such ϱ with

ϱ > ϱ_{M} := max \{\frac{τ(u)}{∥u∥} : u ∈ L^{∗}\\{0\}\} .  (11)

This guarantees that W_{M} is nonempty and no (d − 1)-dimensional P is ϱ-denser than all P ∈ P(L).

All ideal crystals can be modeled as periodic packings of spheres in E^{3}. In Section 4 we choose specific examples with n = 2 to show that finite packings equipped with the parametric density is a useful approach to explain crystal growth based on atomic structure only ([6], p.1). We consider crystals of the NaCl-type where K_{1} and K_{2} are spheres with different radii. For this type M is a lattice again. This does not hold for the ZnS-type, named after the zinc blend. A special case of this type is the diamond which consists of only one type of atoms (carbon). For all examples there are parameters ϱ such that the asymptotic shape of the densest packing coincides with the natural shape, which is the cube for NaCl, the octahedron for the diamond and for the ZnS-type the shape may be a tetrahedron, in particular not centrally symmetric.

2. Bounds for the number of facets

To estimate the number of facets of W (L, ϱ), we first show that only sufficiently small vectors of the polar lattice contribute to the Wulff-shape. In fact we prove this for periodic packings.

Lemma 1. For L ∈ L^{d} let f : L^{∗} → ℝ with f(nu) = f(u), for n ∈ ℕ, u ∈ L^{∗} and 0 < a ≤ f(u) ≤ b, for all u. Further let W (L, f, ϱ) = ⋂_{u∈L^{∗} primitive} {y : uy ≤ ϱ∥u∥ − f(u)} and B = {u ∈ L^{∗} : f(u) ≥ b − a/2} and let there be an R > 0 such that B + RB^{d} = E^{d}. Then for ϱ > max_{u∈L^{∗} primitive} \{\frac{f(u)}{∥u∥}\} and S = max\{2R, 4R^{2}ϱ/a\}

W (L, f, ϱ) = ⋂_{{u∈L^{∗} primitive \atop ∥u∥≤S}} \{y : uy ≤ ϱ∥u∥ − f(u)\} .

In particular W (L, f, ϱ) is a polytope.

In the case that even {u ∈ L^{∗} : f(u) = b} + RB^{d} is a covering, we may choose S = max{2R, 2R^{2}ϱ/a}.

Proof. Let W′ be the set on the right side. Then it suffices to prove that W′ ⊂ W := W (L, f, ϱ). Assume x ∈ W′\W. Then there is a u ∈ L^{∗} with ∥u∥ > S and ux > ϱ∥u∥ − f(u) and without restriction we may assume that ∥u∥ is minimal. By assumption there is a v ∈ B with ∥v − \frac{u}{2}∥ ≤ R. With w := u − v we have v + w = u, ∥w − \frac{u}{2}∥ ≤ R and hence ∥v − w∥ ≤ 2R. From ∥v∥, ∥w∥ ≤ ∥u/2∥ + R < ∥u∥ it follows by the minimal property of u that vx ≤ ϱ∥v∥ − f(v) and wx ≤ ϱ∥w∥ − f(w) and so

ux ≤ ϱ(∥v∥ + ∥w∥) − b + a/2 − f(w).

The parallelogram identity leads to

(∥v∥ + ∥w∥)^{2} = 2∥v∥^{2} + 2∥w∥^{2} − (∥v∥ − ∥w∥)^{2}
≤ 2∥v∥^{2} + 2∥w∥^{2} = ∥v + w∥^{2} + ∥v − w∥^{2} ≤ ∥u∥^{2} + 4R^{2}

and hence

∥v∥ + ∥w∥ ≤ ∥u∥ + (\sqrt{∥u∥^{2} + 4R^{2}} − ∥u∥) = ∥u∥ + \frac{4R^{2}}{∥u∥ + \sqrt{∥u∥^{2} + 4R^{2}}} ≤ ∥u∥ + \frac{2R^{2}}{∥u∥} .

With ∥u∥ > 4R^{2}ϱ/a and f(u) ≤ b we get

ux ≤ ϱ∥u∥ + \frac{2ϱR^{2}}{∥u∥} − b + a/2 − f(w) ≤ ϱ∥u∥ + a − f(w) − f(u) ≤ ϱ∥u∥ − f(u),

which contradicts our assumption. The additional remark follows from f(v) = b. □

Remark. For the Wulff-shape in the lattice case (4) we have a = b = 1/2 and one can choose R equal to the covering radius μ(L^{∗}). If N(L, ϱ) denotes the number of facets of W (L, ϱ) then by Lemma 1 and μ(L^{∗})ϱ_{L} = μ(L^{∗})/(2λ_{1}(L^{∗})) ≥ 1/4

N(L, ϱ) ≤ G(8μ(L^{∗})^{2}ϱ · B^{d}, L^{∗}), for ϱ > ϱ_{L}.  (12)

We also get lower bounds and we so obtain

Theorem 1. Let ϱ ≥ 2ϱ_{L}. Then

c_{1}(L)ϱ^{d(d−1)/(d+1)} ≤ N(L, ϱ) ≤ c_{2}(L)ϱ^{d},

where c_{1}(L) = (\frac{dκ_{d}}{2^{d}κ_{d−1}} )^{2d/(d+1)} (\frac{λ_{1}(L^{∗})}{3})^{d(d−1)/(d+1)} and c_{2}(L) = κ_{d} det L12^{d}μ(L^{∗})^{2d} depend only on L.

Remarks. (1) Actually the proof shows that for ϱ > ϱ_{L}

c_{1}(L) (\frac{4(ϱ − ϱ_{L})^{2}}{ϱ} )^{d(d−1)/(d+1)} ≤ N(L, ϱ) ≤ c_{2}(L)ϱ^{d}.

(2) Theorem 1 can be interpreted as a stability result on the Wulff-shape depending on ϱ (cf. [9]).

Proof. There are two simple but useful upper bounds for the lattice point enumerator (cf. [8]):

det Λ · G(K, Λ) ≤ V (K + μ(Λ)B^{d})  (13)

and, if Λ is a packing lattice for B^{d},

κ_{d} · G(K, Λ) ≤ V (K + B^{d}).  (14)

For K = rB^{d} in (13) and (14) and with G(rB^{d}, Λ) = G(2r/λ_{1}(Λ) · B^{d}, 2/λ_{1}(Λ) · Λ) in (14) we get

G(rB^{d}, Λ) ≤ \frac{κ_{d}}{det Λ}(r + μ(Λ))^{d}  (15)

and

G(rB^{d}, Λ) ≤ (\frac{2r}{λ_{1}(Λ)} + 1)^{d} .  (16)

Now the upper bound is a consequence of (12), (15) and 1 < 4μ(L^{∗})ϱ, for ϱ > ϱ_{L}.

For the lower bound let L^{∗}\{0} = (u^{1}, u^{2}, …) with ∥u^{i}∥ ≤ ∥u^{i+1}∥, i = 1, 2, …. Further let r_{i} = ∥u^{i}∥. From (16) and r_{i} ≥ λ_{1}(L^{∗}) there is a constant η_{d} = (\frac{3}{λ_{1}(L^{∗})})^{d}, such that G(r_{i}B^{d}, L^{∗}) ≤ η_{d} · r_{i}^{d}. Consequently

r_{i} ≥ η_{d}^{−1/d} · G(r_{i}B^{d}, L^{∗})^{1/d} ≥ η_{d}^{−1/d}i^{1/d}.

Now let v^{1}, …, v^{N} ∈ L^{∗} be the lattice vectors corresponding to the N = N(L, ϱ) facets of W (L, ϱ) with ∥v^{i}∥ ≤ ∥v^{i+1}∥, i = 1, …, N − 1. It follows ∥v^{i}∥ ≥ r_{i} ≥ η_{d}^{−1/d}i^{1/d}, so the i-th

facet has distance ϱ − \frac{1}{2∥v^{i}∥} ≥ ϱ − \frac{η_{d}^{1/d}}{2i^{1/d}} to the origin. Hence its (d − 1)-volume f_{i} is bounded above by

f_{i} ≤ κ_{d−1} · (ϱ^{2} − ϱ^{2} − \frac{η_{d}^{2/d}}{4i^{2/d}} + \frac{ϱη_{d}^{1/d}}{i^{1/d}} )^{(d−1)/2} ≤ κ_{d−1} (\frac{η_{d}}{i} )^{(d−1)/(2d)} ϱ^{(d−1)/2}.

Consequently, we obtain for the surface area F of W (L, ϱ) the inequality F ≤ κ_{d−1}η_{d}^{(d−1)/(2d)} ϱ^{(d−1)/2} ∑_{i=1}^{N} \frac{1}{i^{(d−1)/(2d)}} .

Since ∑_{i=1}^{N} i^{(1−d)/(2d)} ≤ 1 + ∫_{1}^{N} x^{(1−d)/(2d)} dx = 1 − 2d/(d + 1) + 2d/(d + 1)N^{(d+1)/(2d)} ≤ 2N^{(d+1)/(2d)} we have

F ≤ 2κ_{d−1}η_{d}^{(d−1)/(2d)}ϱ^{(d−1)/2}N^{(d+1)/(2d)}.  (17)

For the inradius r of W (L, ϱ) holds r ≥ ϱ − 1/(2λ_{1}(L^{∗})) = ϱ − ϱ_{L} and with ϱ ≥ 2ϱ_{L} it follows

F ≥ dκ_{d}(ϱ − ϱ_{L})^{d−1} ≥ \frac{dκ_{d}}{2^{d−1}} ϱ^{d−1}.  (18)

From (17) and (18) we obtain

N ≥ (\frac{dκ_{d}}{2^{d}κ_{d−1}} )^{2d/(d+1)} η_{d}^{−(d−1)/(d+1)} · ϱ^{d(d−1)/(d+1)}

and Theorem 1 is proved. □

Example. For ℤ^{d} the constants in Theorem 1 are c_{1} = (\frac{3d^{2}κ_{d}^{2}}{12^{d}κ_{d−1}^{2}} )^{d/(d+1)} ∼ (\frac{6πd}{12^{d}})^{d/(d+1)} and

c_{2} = κ_{d}(3d)^{d} ∼ (18πed)^{d/2}.

For applications to crystals we can assume that M is rational with respect to L, i.e. there is a lattice Λ such that M ⊂ Λ. So we can apply Lemma 1 with R = μ(Λ^{∗}) and we get an upper bound for the number of facets N(M, ϱ) of W (M, ϱ) as in Theorem 1. Since the proof for the lower bound works for all polytopes P ∈ P(L) with W (L, ϱ) ⊂ P ⊂ ϱB^{d}, it follows with 0 ≤ τ(u) ≤ 1/2 (compare Lemma 2 in Section 3)

Corollary 1. If M ⊂ Λ, for a lattice Λ, then there are constants c_{1}(M), c_{2}(M) such that for ϱ ≥ 2ϱ_{L}

c_{1}(M)ϱ^{d(d−1)/(d+1)} ≤ N(M, ϱ) ≤ c_{2}(M)ϱ^{d}.

3. General periodic packings

The following explicit representation will later be used to minimize Δ_{M}(P, ϱ).

Theorem 2. Let P ∈ P(L) be with outer facet normals u_{i} ∈ L^{∗}, i = 1, …, k and let f_{1}, …, f_{k} denote the (d − 1)-dimensional volumes of the corresponding facets F_{1}, …, F_{k}. Then

Δ_{M}(P, ϱ) = δ_{M}V (P)^{1/d−1} ∑_{i=1}^{k} f_{i} (ϱ − \frac{τ(u_{i})}{∥u_{i}∥}) .

Proof. Let Δ_{1}, …, Δ_{k} be the (d − 1)-dimensional determinants of the sublattices of L orthogonal to u_{1}, …, u_{k}. Further let {R_{λ}} be a sequence as in Definition 1. In particular, R_{λ} has the same outer facet normals as P. Further we can assume that 0 is contained in int (P) and int (R_{λ}). For λ ≥ 1 and j = 1, …, n we define P_{j}(λ) by

P_{j}(λ) = conv (λR_{λ} ∩ L_{j}).  (19)

To determine L_{j}(λR_{λ}) = L(Q_{j}(λ)), where Q_{j}(λ) = P_{j}(λ) − t_{j}, we consider the contribution of each lattice point u ∈ L to L(Q_{j}(λ)) − V (Q_{j}(λ))/ det L. If D(u) denotes the Dirichlet-Voronoi-cell (DV-cell) corresponding to u ∈ L, then we distinguish five cases:

(1) D(u) ⊂ Q_{j}(λ),

(2) D(u) ⊂ E^{d}\Q_{j}(λ),

(3) u is contained in a facet of Q_{j}(λ),

(4) u ∈ intQ_{j}(λ) and D(u) ⊄ Q_{j}(λ),

(5) u ∉ Q_{j}(λ) and D(u) ∩ Q_{j}(λ) ≠ ∅. We consider the lattice points in four steps.

(a) The contribution of the u ∈ L of types (1) and (2) is 0. From R_{λ} → P it is clear that the facets G_{λ,i} of R_{λ} orthogonal to u_{i} have (d − 1)-volume f_{i}λ^{d−1} + o(λ^{d−1}). The convex hull of the points of L_{j} in the parallel L_{j}-hyperplane close to G_{λ,i} defines facets F_{λ,i} of Q_{j}(λ) orthogonal to u_{i} with (d − 1)-volume V_{d−1}(G_{λ,i}) + O(λ^{d−2}) = f_{i}λ^{d−1} + o(λ^{d−1}). Let H_{λ,i} = aff (F_{λ,i}), Q_{j}(λ) contained in the halfspace H_{λ,i}^{+} and Λ_{λ,i} = L ∩ H_{λ,i}. For u ∈ Λ_{λ,i} let D_{d−1}(u) be the corresponding (d−1)-dimensional DV-cell. Considering the lattice points in the inner parallel body of F_{λ,i} in H_{λ,i} with a distance, which depends only on L and P, it follows that there are \frac{f_{i}}{Δ_{i}} λ^{d−1} + o(λ^{d−1}) DV-cells D_{d−1}(u) of Λ_{λ,i} with ((D_{d−1}(u) + 2D(0)) ∩ H_{λ,i}^{+}) ⊂ Q_{j}(λ).

(b) All these lattice points u ∈ Λ_{λ,i} (of type (3)) have a contribution of 1/2.

(c) Next we show that the (negative) contribution of lattice points of type (5) whose DV-cells intersect one of these (d − 1)-cells is canceled out by those of type (4). It suffices to consider one such u ∈ Λ_{λ,i}. Let v ∈ L of type (5) be with D(v) ∩ D_{d−1}(u) ≠ ∅. By the choice of u we have (D(2u − v) ∩ H_{i}^{+}) ⊂ Q_{j}(λ) and the sum of the contribution of v and its mirror point 2u − v is 0.

(d) For the surface area of Q_{j}(λ) it holds F (Q_{j}(λ)) = ∑_{i=1}^{k} f_{i}λ^{d−1} + o(λ^{d−1}) and hence the remaining parts of the boundary of Q_{j}(λ) have a total (d − 1)-volume of o(λ^{d−1}) and so the total contribution of DV-cells intersecting those parts is o(λ^{d−1}).

Combining (a), (b), (c) and (d) we obtain (analogously to Ehrhart’s formula for the lattice point enumerator (cf. e.g. [10], [8]))

L_{j}(λR_{λ}) = L(Q_{j}(λ)) = \frac{V (P_{j}(λ))}{det L} + 1/2λ^{d−1} ∑_{i=1}^{k} \frac{f_{i}}{Δ_{i}} + o(λ^{d−1}),  (20)

for j = 1, …, n. Now we compare λR_{λ} with P_{j}(λ). We dissect λR_{λ} into the cones C_{i} = pos (F_{i}). Let c_{i}(λ) = max \{\frac{u_{i}}{∥u_{i}∥}x : x ∈ λR_{λ} ∩ M\}, c_{ij}(λ) = max \{\frac{u_{i}}{∥u_{i}∥}x : x ∈ λR_{λ} ∩ L_{j}\} and α_{ij}(λ) = c_{i}(λ)−c_{ij}(λ) ≥ 0. The contribution to V (λR_{λ})−V (P_{j}(λ)) is α_{ij}(λ)f_{i}λ^{d−1} +o(λ^{d−1}). Summing up over all facets yields

V (P_{j}(λ)) = V (λR_{λ}) − λ^{d−1} ∑_{i=1}^{k} α_{ij}(λ)f_{i} + o(λ^{d−1}).

Together with (20) and Ψ(R_{λ}, λ) = ∑_{j=1}^{n} v_{j}L_{j}(λR_{λ}) it follows

Ψ(R_{λ}, λ) = V (λR_{λ})\frac{v}{det L} + λ^{d−1} ∑_{i=1}^{k} ∑_{j=1}^{n} f_{i}v_{j} (\frac{1}{2Δ_{i}} − \frac{α_{ij}(λ)}{det L}) + o(λ^{d−1})

= δ_{M}V (λR_{λ}) + δ_{M}λ^{d−1} ∑_{i=1}^{k} f_{i} \frac{det L}{Δ_{i}} (\frac{1}{2} − ∑_{j=1}^{n} \frac{Δ_{i}}{det L}α_{ij}(λ)\frac{v_{j}}{v} ) + o(λ^{d−1}).

Without restriction we can assume that there is an index m = m(λ) with α_{im}(λ) = 0 and hence \frac{Δ_{i}}{det L}α_{ij}(λ) = ϵ_{m}(u_{i}) − ϵ_{j}(u_{i}). From the definition of τ_{m}(u_{i}) and τ(u_{i}) together with Δ_{i} = det L∥u_{i}∥ we obtain

Ψ(R_{λ}, λ) ≤ δ_{M}V (λR_{λ}) + δ_{M}λ^{d−1} ∑_{i=1}^{k} f_{i} \frac{τ(u_{i})}{∥u_{i}∥} + o(λ^{d−1}).

Now by Definition 1 and R_{λ} → P

Δ_{M}(P, ϱ) ≥ δ_{M} lim_{λ→∞} V (λR_{λ})^{1/d}
(\frac{V (λR_{λ}) + ϱF (λR_{λ}) − V (λR_{λ}) − λ^{d−1} ∑_{i=1}^{k} f_{i} \frac{τ(u_{i})}{∥u_{i}∥} + o(λ^{d−1})}{V (λR_{λ}) + o(λ^{d})})

= δ_{M} lim_{λ→∞} V (R_{λ})^{1/d−1} (ϱF (R_{λ}) − ∑_{i=1}^{k} f_{i} \frac{τ(u_{i})}{∥u_{i}∥})

= δ_{M}V (P)^{1/d−1} ∑_{i=1}^{k} f_{i} (ϱ − \frac{τ(u_{i})}{∥u_{i}∥})

On the other hand we can choose R_{λ} such that τ(u_{i}) is attained for i = 1, …, k and equality follows. □

Remark. If in the definition of the parametric density (5) B^{d} is replaced by a convex body C, then a density deviation Δ_{M}(P, C, ϱ) with respect to C can be defined. The same proof as in Theorem 2 yields

Δ_{M}(P, C, ϱ) = δ_{M}V (P)^{1/d−1} ∑_{i=1}^{k} \frac{f_{i}}{∥u_{i}∥} (ϱh_{C}(u_{i}) − τ(u_{i})) ,  □

where h_{C} is the support function of C.

Proposition 1. Let P, Q ∈ P(L) and ϱ > ϱ_{M} with Δ_{M}(P, ϱ) < Δ_{M}(Q, ϱ). Then P is asymptotically ϱ-denser than Q.

Proof. The proof of Theorem 2 shows that there is a sequence R_{λ} ∈ P(P) with R_{λ} → P and a constant c with

V (λR_{λ}) ≤ V ((λ + 1)R_{λ+1}) ≤ V (λR_{λ}) + cλ^{d−1}

and

δ_{ϱ}(M, λR_{λ}) = δ_{M} − \frac{Δ_{M}(P, ϱ)}{V (λR_{λ})^{1/d}} + o(1/λ).

On the other hand, by the definition of Δ_{M} for any sequence S_{ν} ∈ P(Q) with S_{ν} → Q holds

δ_{ϱ}(M, νS_{ν}) ≤ δ_{M} − \frac{Δ_{M}(Q, ϱ)}{V (νS_{ν})^{1/d}} + o(1/ν).

Now we define λ(ν) by

M(λ(ν)R_{λ(ν)}) ≤ M(νS_{ν}) ≤ M((λ(ν) + 1)R_{λ(ν)+1}).

From M(νS_{ν}) = nV (νS_{ν})/ det L+o(ν^{d}) and M(λ(ν)R_{λ(ν)}) = nV (λ(ν)R_{λ(ν)})/ det L+o(λ(ν)^{d}) it follows V (λ(ν)R_{λ(ν)}) = V (νS_{ν}) + o(ν^{d}) and λ(ν) = Ω(ν). Hence

δ_{ϱ}(M, λ(ν)R_{λ(ν)}) − δ_{ϱ}(M, νS_{ν}) ≥ \frac{Δ_{M}(Q, ϱ) − Δ_{M}(P, ϱ)}{V (νS_{ν})^{1/d}} + o(1/ν)
= 1/ν · (\frac{Δ_{M}(Q, ϱ) − Δ_{M}(P, ϱ)}{V (Q)^{1/d}}) + o(1/ν)

and the assertion is a consequence of δ_{ϱ}(M, λR_{λ}) − δ_{ϱ}(M, (λ + 1)R_{λ+1}) = o(1/λ). □

To prove that W_{M} is a polytope we first have to check that the assumptions of Lemma 1 are satisfied.

Lemma 2. For L ∈ L^{d} and u ∈ L^{∗}

\frac{1}{2n} ≤ τ(u) ≤ \frac{1}{2}.

Proof. The right inequality is clear and equality holds if M is a lattice. For the lower bound let λ_{j} = v_{j}/v, j = 1, …, n. Then ∑_{j=1}^{n} λ_{j} = 1 and hence ∑_{j=1}^{n} λ_{j}^{2} ≥ 1/n. With {ϵ_{m} − ϵ_{j}} + {ϵ_{j} − ϵ_{m}} ≤ 1 (we omit the (u)) we get

∑_{m=1}^{n} λ_{m} ∑_{j=1}^{n} λ{ϵ_{m} − ϵ_{j}} = ∑_{m≠j} λ_{j}λ_{m}{ϵ_{m} − ϵ_{j}}

≤ ∑_{j<m} λ_{j}λ_{m} = \frac{1}{2} (1 − ∑_{j=1}^{n} λ_{j}^{2}) ≤ \frac{1}{2} − \frac{1}{2n}.

Hence there is an m ∈ {1, … n} with ∑_{j=1}^{n} λ_{j}{ϵ_{m} − ϵ_{j}} ≤ 1/2 − 1/(2n) and so τ(u) ≥ 1/(2n). Equality holds for λ_{j} = 1/n and ϵ_{j} = (j − 1)/n, j = 1, …, n. □

From Corollary 1 we know that W_{M} is a polytope, if M is rational with respect to L. This is sufficient for application to crystals but for completeness we also consider the irrational case.

Theorem 3. If ϱ > ϱ_{M}, then W_{M} is a polytope with nonempty interior, i.e. W_{M} ∈ P(L).

Proof. From the definitions of W_{M} and ϱ_{m} we have 0 ∈ int (W_{M}). In Lemma 1 we have f(u) = τ(u), a = 1/(2n), b = 1/2 and B = {u ∈ L^{∗} : τ(u) ≥ 1/2 − 1/(4n)}. We have to prove the existence of an R > 0 with RB^{d} + B = E^{d}. For this let L = Aℤ^{d}, m ∈ L^{∗}, α_{j} = t_{j}m, j = 1, …, n. We define homogenous linear forms L_{1}, …, L_{n} by L_{j}(x) = t_{j}A^{∗}x, where A^{∗} = (A^{−1})^{t}. If ∑_{j=1}^{n} w_{j}L_{j}(x) has integer coefficients, then ∑_{j=1}^{n} w_{j}t_{j} ∈ L and ∑_{j=1}^{n} w_{j}α_{j} = (∑_{j=1}^{n} w_{j}t_{j}) m ∈ ℤ. It follows by Kronecker’s approximation theorem (see e.g. [3], p.58), that there is a v ∈ ℤ^{d} and an X > 0 such that

∥L_{j}(v) − α_{j}∥_{ℤ} < \frac{1}{8n}, j = 1, …, n

and

|v_{i}| ≤ X, i = 1, …, d,

where ∥ · ∥_{ℤ} denotes the distance to ℤ. With u = A^{∗}v − m ∈ L^{∗} we obtain ∥ut_{j}∥_{ℤ} < 1/(8n) and ∥u + m∥ ≤ ∥A^{∗}∥\sqrt{d}X. Hence τ(u) > 1/2 − 1/(4n) and the assertion is a consequence of R = ∥A^{∗}∥\sqrt{d}X + μ(L^{∗}). □

We will prove that W_{M} minimizes Δ_{M}(P, ϱ) for all P ∈ P(L) (note that Δ_{M}(P, ϱ) is invariant for dilatations of P). In fact we prove a more general result.

Lemma 3. Let u_{1}, u_{2}, … ∈ E^{d} and α_{1}, α_{2}, … ∈ ℝ, such that

W = ⋂_{i=1}^{∞} {y : u_{i}/∥u_{i}∥y ≤ α_{i}}

is a polytope with int (W) ≠ ∅. Then holds for all polytopes Q with facet normals from {u_{1}, u_{2}, …}

V (Q)^{1/d−1} ∑ α_{i}f_{i}(Q) ≥ dV (W)^{1/d},

with equality if and only if Q is homothetic to W.

Remark. Actually Lemma 3 is Wulff’s theorem (compare [5], [7], [11], [17]).

Proof. Let h be the support function of W. Then it follows from the definition of W that

h(u_{i}/∥u_{i}∥) ≤ α_{i}, for all i.

We can estimate ∑ α_{i}f_{i}(Q) with the help of a mixed volume (see [13]) and Minkowski’s inequality (see [13]) yields that

V (Q)^{1/d−1} ∑ α_{i}f_{i}(Q) ≥ V (Q)^{1/d−1} ∑ f_{i}(Q)h(u_{i}/∥u_{i}∥) = V (Q)^{1/d−1}dV_{d−1}(W, Q)
≥ V (Q)^{1/d−1}dV (W)^{1/d}V (Q)^{(d−1)/d} = dV (W)^{1/d}

with equality only if Q and W are homothetic. On the other hand, for Q = W equality holds. □

An immediate consequence is

Corollary 2. If V (Q) = V (W_{M}), for any Q ∈ P(L), then Δ_{M}(Q, ϱ) ≥ Δ_{M}(W_{M}, ϱ), with equality only if Q is a translate of W_{M}.

Remark. With C instead of B^{d} in (5) we obtain the Wulff-shape

W (M, ϱ, C) = {x : ux ≤ ϱh_{C}(u) − τ(u), u ∈ L^{∗} primitive} .

4. Application to crystal types

We consider some examples of crystal-types, which can be described as two-lattice packings. For n = 2 we assume v_{1} ≥ v_{2}. Let σ = \frac{v_{1}}{v_{1}+v_{2}} ∈ [1/2, 1]. Actually, σ is not well-defined in reality. But we can consider it as an average value. For shortness let t = t_{2} and ϵ(u) = ϵ_{2}(u), for u ∈ L^{∗}. Consequently τ(u) = 1/2 − min{ϵ(u)σ, (1 − ϵ(u))(1 − σ)}.

First we consider crystals of the NaCl-type, for which M = ℤ^{3} is a lattice again. The bigger atomic ions (Na) are placed in the even points (i.e. with even sum of coordinates) and the smaller ones (Cl) in the odd points. It is well-known that the odd and the even points in ℤ^{3} are copies of the face centered cubic (fcc) lattice with basis {(1, 1, 0)^{t}, (1, 0, 1)^{t}, (0, 1, 1)^{t}}. The translation vector is t = (1, 0, 0)^{t}. We have det L = 2, and L^{∗} has a basis {1/2(1, 1, −1)^{t}, 1/2(1, −1, 1)^{t}, 1/2(−1, 1, 1)^{t}}. Further ϵ(u) ∈ {0, 1/2} for u ∈ L^{∗}, and hence ϵ(−u) = ϵ(u) and ϱ_{M} = max\{1/2, σ/\sqrt{3}\}.

We determine W (M, ϱ) depending on ϱ and σ. For large ϱ the shape will tend to a ball and exact computation is hard. But for small ϱ we obtain the natural shapes.

Theorem 4. Let ϱ > ϱ_{M} = max\{1/2, σ/\sqrt{3}\}. Then W (M, ϱ) is

(a) a cube of edgelength (2ϱ − 1), if 1/2 ≤ ϱ ≤ 1/12(3 + \sqrt{3})(3 − 2σ).

(b) an octahedron of height 2(ϱ\sqrt{3} − σ), if σ/\sqrt{3} ≤ ϱ ≤ 1/2(1 + \sqrt{3})(σ − 1/2).

(c) a truncated octahedron, if max\{(2 + \sqrt{3})(1 − σ), 1/2(1 + \sqrt{3})(σ − 1/2)\} ≤ ϱ ≤ (\sqrt{2} + \sqrt{3})(σ − 1/2).

(d) a truncated cube, if 1/12(3 + \sqrt{3})(3 − 2σ) ≤ ϱ ≤ min\{(2 + \sqrt{3})(1 − σ), 1/4(2 + \sqrt{2})\}.

Remark. We illustrate the theorem in the following diagram (see below).

For example r(Na^{+}) = 116pm, r(Cl^{−}) = 167pm (see e.g. [4]) and σ = 0.748…. So the Wulff-shape for NaCl is a cube for ϱ ∈ (0.5, 0.592 …).

Proof. (a) We have ±(1, 0, 0)^{t}, ±(0, 1, 0)^{t}, ±(0, 0, 1)^{t} ∈ L^{∗} with ϵ = 0. The intersection of the corresponding halfspaces yields the cube C. By symmetry it suffices to prove that x_{0} = (ϱ − 1/2)(1, 1, 1)^{t} satisfies the conditions for all u ∈ L^{∗}. Therefore by assumption on ϱ

ux_{0} ≤ ∥u∥∥x_{0}∥ = (ϱ − 1/2)\sqrt{3}∥u∥ = ϱ∥u∥ + (ϱ(\sqrt{3} − 1) − \sqrt{3}/2)∥u∥
≤ ϱ∥u∥ + (1/12(3 + \sqrt{3})(3 − 2σ)(\sqrt{3} − 1) − \sqrt{3}/2)∥u∥
≤ ϱ∥u∥ − σ/\sqrt{3}∥u∥.

Uwe Schnell: Periodic Sphere Packings and the Wulff-Shape 137

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.; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; 1/2; 1; \sqrt{3}/2; 1/2; 1; \sqrt{3}/3; (\sqrt{3} + 1)/4; (\sqrt{3} + \sqrt{2})/2; (3 + \sqrt{3})/6; (2 + \sqrt{2})/4; σ; ϱ; C; O; T C; T O]

For those u with ϵ(u) = 1/2 the desired inequality is ux_{0} ≤ ϱ∥u∥ − σ/2 and is satisfied since ∥u∥ ≥ \sqrt{3}/2, for u ∈ L^{∗}. For ϵ(u) = 0, ∥u∥ ≥ \sqrt{3} ≥ \sqrt{3}/(2σ) is sufficient. So we still have to check the twelve u ∈ L^{∗} with ∥u∥ = \sqrt{2} and obviously it suffices to consider u = (1, 1, 0)^{t}. By assumption

ux_{0} = 2ϱ − 1 ≤ ϱ\sqrt{2} + (2 − \sqrt{2})1/12(3 + \sqrt{3})(3 − 2 · 1/2) − 1 < ϱ\sqrt{2} − 1/2.

(b) The 8 vectors in L^{∗} with length \sqrt{3}/2 determine the octahedron O. Again it suffices to prove that x_{0} = (ϱ\sqrt{3} − σ)(1, 0, 0)^{t} satisfies the conditions for all u ∈ L^{∗}. All further u ∈ L^{∗} have length ≥ 1 and by our assumption on ϱ

ux_{0} ≤ (ϱ\sqrt{3} − σ)∥u∥ ≤ ϱ∥u∥ + (1/2(1 + \sqrt{3})(σ − 1/2)(\sqrt{3} − 1) − σ) ∥u∥
= ϱ∥u∥ − ∥u∥/2 ≤ ϱ∥u∥ − 1/2.

(c) Let a = ϱ\sqrt{3} − σ and b = ϱ − 1/2. Then a, b ≥ 0. The six vectors in L^{∗} with length 1 and ϵ = 0 determine a cube C with height 2b and the eight vectors in L^{∗} with length \sqrt{3}/2 and ϵ = 1/2 determine an octahedron O with height 2a. By the assumptions on ϱ and σ we have b ≤ a ≤ 2b and the intersection of O and C is a truncated octahedron with 24 vertices of the type v = (0, a − b, b)^{t}. The scalar product of v with one of the six u ∈ L^{∗} with length \sqrt{2} is at most a and the third assumption on ϱ is equivalent to a ≤ ϱ\sqrt{2} − 1/2, which is the desired inequality for u. All further primitive u ∈ L^{∗} have length ≥ \sqrt{11}/2 and by our assumption σ ≥ 1/2 + ϱ(\sqrt{3} − \sqrt{2}) and hence a − b ≤ ϱ(\sqrt{2} − 1). Together with a − b ≤ b = ϱ − 1/2 one obtains by elementary calculation that ∥v∥ ≤ ϱ − 1/\sqrt{11} and hence uv ≤ ϱ∥u∥ − ∥u∥/\sqrt{11} ≤ ϱ∥u∥ − 1/2.

(d) Let a, b be as in (c). By the assumptions on ϱ and σ we have 2b ≤ a ≤ 3b and here the intersection of O and C is a truncated cube with 24 vertices of the type v = (b, b, a − 2b)^{t}. The scalar product of v with one of the six u ∈ L^{∗} with length \sqrt{2} is at most 2b and the third assumption on ϱ is equivalent to 2b ≤ ϱ\sqrt{2} − 1/2, which is the desired inequality for u. The scalar product of v with one of the 24 u ∈ L^{∗} with length \sqrt{11}/2, e.g. 1/2(3, 1, 1)^{t} is at most 5b/2 = 5/2ϱ − 5/4 ≤ ϱ\sqrt{11}/2 − 1/2. All further primitive u ∈ L^{∗} have length ≥ \sqrt{19}/2. Further, ∥v∥ ≤ b\sqrt{3} and hence

uv ≤ ϱ∥u∥ + (ϱ(\sqrt{3} − 1) − \sqrt{3}/2 − ϱ) ∥u∥ ≤ ϱ∥u∥ − 1/2.  □

Now we turn to crystals of the ZnS-type, named after the zinc blend. One special case of this type is the diamond, which consists of only one type of atoms (carbon) and consequently σ = 1/2. Again L is a face centered cubic lattice (with basis {(2, 2, 0)^{t}, (2, 0, 2)^{t}, (0, 2, 2)^{t}}) and the translation vector is t = (1, −1, 1)^{t}. We have det L = 16, and L^{∗} has a basis {1/4(1, 1, −1)^{t}, 1/4(1, −1, 1)^{t}, 1/4(−1, 1, 1)^{t}}. Further ϵ(u) ∈ {0, 1/4, 1/2, 3/4}, for u ∈ L^{∗} and ϱ_{M} = (σ + 1)/\sqrt{3}.

Theorem 5. (a) If \sqrt{3}/2 ≤ ϱ ≤ 1/2(\sqrt{3}+1) and σ = 1/2, then W (M, ϱ) is an octahedron

O with height 2ϱ\sqrt{3} − 3.

(b) If \frac{σ+1}{\sqrt{3}} < ϱ < min\{\frac{4σ+1}{2\sqrt{3}}, \frac{2}{\sqrt{3}}\}, then W (M, ϱ) is a tetrahedron with outer normal vectors (−1, −1, −1)^{t}, (1, 1, −1)^{t}, (1, −1, 1)^{t}, (−1, 1, 1)^{t}.

Remark. We illustrate the theorem in the following diagram.

.................................................................................................................................................................................. .......................................................................................................................................................................................................................................................................................................................................................................... ........................................................................................................................................................................................................................................................................................................................................................... ........... ... ... .... .... ..... ..... ..... ...... ...... ....... ........ ........ ......... ......... ......... .......... .......... .......... ........... ........... ........... ............ ............ ............. .............. .............. ............... ............... ............... ................ ................ ................. ................. ................. .................. .................. .................. ................... ................... .................... 1/2  3/4  1 \sqrt{3}/2 2/\sqrt{3}^{\frac{\sqrt{3}+1}{2}}  σ ϱ

For example r(Zn^{2+}) = 88pm, r(S^{2−}) = 170pm (see e.g. [4]) and σ = 0.878 . . .. So the Wulff-shape for ZnS is a tetrahedron for ϱ ∈ (1.084 …, 1.154 …).

Remark. Note that in the case of the diamond (a) the planes determining the octahedron are not those with the maximal density (these are the planes with normals (1, 1, 0)^{t} etc., which contain both types of atoms). The result is based on the fact that in directions (1, 1, 1) etc. there are pairs of planes which are very close together.

Proof. (a) For σ = 1/2 it follows τ(u) = 1/2 max{1 − ϵ(u), ϵ(u)} = τ(−u). The eight vectors in L^{∗} with length \sqrt{3}/4 and τ = 3/8 determine the octahedron O. It suffices to prove that x_{0} = (ϱ\sqrt{3} − 3/2)(1, 0, 0)^{t} satisfies the conditions for all further u ∈ L^{∗}. By our assumption on ϱ

ux_{0} ≤ ∥u∥(ϱ\sqrt{3} − 3/2) ≤ ϱ∥u∥ + (1/2(1 + \sqrt{3})(\sqrt{3} − 1) − 3/2) ∥u∥
= ϱ∥u∥ − ∥u∥/2.

If τ(u) = 1/4, then ∥u∥ ≥ 1/2 and hence ux_{0} ≤ ϱ∥u∥ − 1/4. If τ(u) = 3/8, then (up to the vectors which determine O) ∥u∥ ≥ \sqrt{11}/4 and so ux_{0} ≤ ϱ∥u∥ − \sqrt{11}/8 < ϱ∥u∥ − 3/8. If τ(u) = 1/2, then for ∥u∥ ≥ 1 holds ux_{0} ≤ ϱ∥u∥ − 1/2 and we still have to check the u with length \sqrt{2}/2. It suffices to consider u_{0} = (1/2, 1/2, 0)^{t}:

u_{0}x_{0} = ϱ\sqrt{3}/2 − 3/4 ≤ ϱ\sqrt{2}/2 + 1/2(\sqrt{3} + 1)1/2(\sqrt{3} − \sqrt{2}) − 3/4 < ϱ\sqrt{2}/2 − 1/2.

(b) There are four vectors u = 1/4(−1, −1, −1)^{t}, 1/4(−1, 1, 1)^{t}, 1/4(1, −1, 1)^{t}, 1/4(1, 1, −1)^{t} with ϵ(u) = 3/4 and τ(u) = (1 + σ)/4. The corresponding halfspaces are defined by (4u)x ≤ q := ϱ\sqrt{3}−(σ +1) and the intersection is a tetrahedron T with vertices (q, q, q)^{t}, (−q, q, −q)^{t}, (q, −q, −q)^{t}, (−q, −q, q)^{t}. We still have to prove that these vertices satisfy the conditions for all the other u ∈ L^{∗}.

For the negatives to the four vectors defining T holds ϵ(u) = 1/4 and τ(u) = 1/4 max{2− σ, 3σ − 1)\} and the corresponding halfspaces are defined by (4u)x ≤ p := ϱ\sqrt{3} − max\{3σ − 1, 2 − σ\}. With p − 3q = −2ϱ\sqrt{3} + min\{4, 1 + 4σ\} it follows by the assumptions on ϱ that 3q ≤ p and so the vertices of T satisfy these conditions.

If v is a vertex of T and u ∈ L^{∗} then uv ≤ q\sqrt{3}∥u∥ = ϱ∥u∥ + (2ϱ − \sqrt{3}(σ + 1))∥u∥. The assumptions on ϱ determine a triangle with vertices (σ, ϱ) = (1/2, \sqrt{3}/2), (1, 2/\sqrt{3}), (3/4, \sqrt{3}). The linear function (2ϱ − \sqrt{3}(σ + 1)) attains its maximum in a vertex and hence uv ≤ ϱ∥u∥ − 5\sqrt{3}/12∥u∥.

For ∥u∥ ≥ 6/(5\sqrt{3}) = 0.69 … we have uv ≤ ϱ∥u∥ − 1/2. So we still have to check the six vectors u with length 1/2 and ϵ(u) = 1/2 and τ(u) = σ/2, e.g. (1/2, 0, 0)^{t}. From ϱ ≤ 2/\sqrt{3} we obtain uv ≤ q/2 ≤ ϱ∥u∥ − σ/2. □

Acknowledgement. I like to thank U. Betke and the chemist W. H. E. Schwarz for helpful comments.

References

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Received April 6, 1998