Corrections to Magic Number Formulas for Body‑Centered Cubic Nanoclusters and New Truncated Cube Data

Abstract

We present corrections to the magic‑number formulas for body‑centered cubic (bcc) nanoclusters. The revised equations are validated by radial distribution function (RDF) calculations for several crystal structures, and we additionally report RDF‑derived data for truncated bcc and face‑centered cubic (fcc) cubes that are observed in natural systems.

Introduction

In a recent publication [1] we derived magic‑number formulas for a range of crystal nanoclusters. Crystallographers know that the bulk coordination of an ideal bcc lattice is eight. The RDF identifies nearest‑neighbor peaks relative to a central atom, and the integrated intensity of each peak yields the coordination number of those neighbors. Using the established method of Ref. [2], we calculated RDFs for a selection of crystals. For ideal bcc cubes the coordination number at the first shell is cn = 1, which requires a revised cutoff radius. Accordingly, we present results for truncated bcc and fcc clusters below.

Main Text

While reviewing the magic‑number equations presented in Ref. [1], we noted that the definition of the adjacency matrix (Eq. (1)) depends on the underlying crystal structure:

Equation (1): A(i,j) = { 1 if r_{ij} < r_c and i ≠ j; 0 otherwise }

Here, r_{ij} is the Euclidean distance between atoms i and j. For dodecahedral structures the cutoff is chosen as r_c = 1.32 × r_min to accommodate varying bond lengths. In contrast, a bcc lattice requires r_c = 2/√3 × r_min ≈ 1.15 × r_min, which alters the adjacency matrix and the resulting magic formulas. Table 1 lists the nearest‑neighbor distances and coordination numbers obtained from the RDFs. The RDF peaks are normalized to the first peak, rendering the peak positions dimensionless. While the concept of “magic numbers” is linked to shell filling, the nearest‑neighbor peaks themselves are distinct from shell boundaries.

The dodecahedron is a particularly complex case, with third‑neighbor distances at r_2 = 1.31 × r_min. Further analysis of this structure is underway. The corrected bcc results are presented in Tables 2–6. They agree with the sequence reported by van Hardeveld and Hartog [3] when the index is shifted by one (we use 0, 1, 2,… while they use 1, 2, 3,…). Perfect cubes are mathematically interesting but rarely occur naturally because of single bonds at the corners. Consequently, we have generated truncated bcc and fcc cubes with corners removed; their data appear in Tables 7 and 8. The updated magic‑number formulas for selected clusters are summarized in Table 9.

Conclusions

We have corrected the magic‑number formulas for bcc nanoclusters and extended the dataset with RDF‑derived results for truncated bcc and fcc cubes.

Availability of Data and Materials

The datasets supporting the conclusions of this article are available from the corresponding author upon request.

Abbreviations

bcc:

Body centered cubic

fcc:

Face centered cubic

RDF:

Radial distribution function