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Body-Centered Cubic Lattice

The body-centered cubic lattice, abbreviated BCC, is the cubic lattice consisting of points at the vertices and centers of the cubes in a simple cubic lattice. A realization is Z^3 union [Z^3+(1/2,1/2,1/2)]. Rescaling to covolume one gives

L_(BCC)=2^(-2/3){(m_1,m_2,m_3) in Z^3:m_1=m_2=m_3 (mod2)}.
(1)

For a Bravais lattice L of covolume one, define its renormalized Coulomb energy by

E(L)=lim_(kappa->0^+)(sum_(p in L\{0})(e^(-kappa||p||))/(||p||)-(4pi)/(kappa^2)).
(2)

Muratov and Novaga (2026) report a computer-assisted proof that L_(BCC) uniquely minimizes E(L), up to orthogonal transformations. Their comparison is among three-dimensional Bravais lattices of fixed covolume, not arbitrary point configurations or periodic crystals with multiple points per cell. The proof uses interval certificates. Muratov and Novaga (2026) report AI assistance with constructing, executing, and cross-checking the verification code.

For positive screening parameter kappa, the corresponding Yukawa energy is

E_kappa(L)=1/2sum_(v in L\{0})(e^(-kappa||v||))/(||v||).
(3)

Luo and Wei (2026) report a unique transition at kappa_* approx 1.7179218684699: among Bravais lattices in R^3 of covolume one, BCC uniquely minimizes E_kappa for 0<kappa<kappa_*, and the face-centered cubic lattice uniquely minimizes it for kappa>kappa_*, up to orthogonal transformations. Exactly these two lattice classes minimize it at kappa=kappa_*. Their proof uses interval certificates. Luo and Wei (2026) credit AI with assisting the certificate computations.

Bourne et al. (2026) proved the local optimality of BCC as a lattice quantizer for the moment int_(Vor(L))||x||^pdx among three-dimensional Bravais lattices of fixed covolume, for every 1<=p<infty. Here Vor(L) is the Voronoi cell of the origin. They also proved global optimality for p in an interval containing 2. Luo (2026) proved that for every finite p>=1 the covolume-one BCC lattice uniquely minimizes this moment among full-rank point lattices in R^3 of covolume one, up to orthogonal transformations. Luo (2026) reports AI assistance with the finite computations in the proof.

See also

Bravais Lattice, Cubic Lattice, Point Lattice

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References

Bourne, D. P.; Cagnetti, F.; and Roper, S. M. "Optimality of the BCC Lattice Among 3D Lattice Quantizers." 29 Oct 2026. https://arxiv.org/abs/2610.39705.Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lattices and Groups, 3rd ed. New York: Springer-Verlag, 1999. https://doi.org/10.1007/978-1-4757-6568-7.Luo, S. "The BCC Lattice Is the Optimal 3D Lattice Quantizer for All p>=1." 6 Nov 2026. https://arxiv.org/abs/2611.09523v1.Luo, S. and Wei, J. "On Exact Solutions to Three-Dimensional Yukawa Lattice Model and BCC to FCC Phase Transitions." 22 Oct 2026. https://arxiv.org/abs/2610.31456.Muratov, C. B. and Novaga, M. "BCC Optimality for the Three-Dimensional Coulomb Lattice Energy." 21 Oct 2026. https://arxiv.org/abs/2610.29732.

Cite this as:

Weisstein, Eric W. "Body-Centered Cubic Lattice." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Body-CenteredCubicLattice.html

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