Surface Atom Fraction in Nanoparticles

Nanoparticle Size

Atomic shells K 4

Metal (sets atomic spacing)

Auto-rotate speed 0.4

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Total atoms N

Diameter

0.29 nm

Surface atoms / fraction

1 / 100%

Average coordination number

0.0 (bulk = 12)

How it works

A close-packed metal nanocluster grows in concentric atomic shells around one central atom. Shell k (k = 1, 2, 3…) adds exactly

atoms in shell k = 10k² + 2

this is the classic Mackay/cuboctahedral "magic number" sequence (12, 42, 92, 162… giving totals 13, 55, 147, 309…). Summing shells 1..K plus the center atom gives the total atom count:

N(K) = 1 + Σ(10k² + 2) for k=1..K
     = 1 + 5K(K+1)(2K+1)/3 + 2K

Only the outermost shell sits at the surface, so the surface fraction is (10K² + 2) / N(K). Because N grows as K³ while the surface shell grows only as K², shrinking a particle makes surface atoms dominate — at K=1 (13 atoms) essentially 92% of atoms are on the surface; by K=8 (2,057 atoms) it has fallen to about 40%, and a macroscopic crystal (K → ∞) is effectively 0%.

Surface atoms have fewer nearest neighbours than interior atoms — roughly 8 bonds instead of the bulk face-centred-cubic value of 12. That reduced coordination is why surface atoms are more reactive, more mobile, and melt at a lower temperature, which is the same underlying reason small nanoparticles are better catalysts and have a depressed melting point. The simulator estimates this with:

avg. coordination ≈ (12·N_interior + 8·N_surface) / N

Positions within each shell are placed on a Fibonacci sphere for a clean, evenly-spaced 3D rendering — the atom counts per shell and the resulting surface fraction and coordination numbers are the exact combinatorial values above, independent of the rendering layout. Real particle diameter uses each metal's nearest-neighbour spacing d (≈ FCC lattice constant / √2): diameter ≈ (2K + 1)·d.

Interior atom (CN = 12) Surface atom (CN ≈ 8)

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