The FCC Microscopic Model: Links, Continuum Limit and Dispersion

01What microscopic links must explain

The final manuscript adopts twelve-neighbour FCC connections as its preferred microscopic ground-state model and derives the continuum operator and full discrete dispersion from a specified link law. Its Chapter 4 summary also states that close packing alone does not exclude other realizations, and that FCC selection and equal sector speeds require dynamical and constitutive grounds.

A continuum calculates propagation from restoring and kinetic coefficients without explaining their origin. Microscopic links must specify adjacency, updating, the long-wave variables, directional dispersion and cutoff. Chapters 4 and 5 use FCC as a concrete microscopic realization with calculable adjacency; its constituents are spatial-state degrees of freedom, not ordinary atoms located in space.

The conventional cell edge is $a_0$ and nearest-neighbour distance $L_0=a_0/\sqrt2$. Four representatives in the conventional cell describe one FCC Bravais lattice, not four constituent species. Cell length, neighbour step, core radius and phase-period length have different definitions and require dynamical matching before identification.

Twelve neighbours, cubic symmetry and tetrahedral/octahedral channels organize possible density, shear and orientation responses. They do not establish ground-state selection, correct sector speeds or a particle spectrum. Close packing does not replace interaction energy and competing-ground-state calculations.

02Choose a link law before deriving motion

Chapter 5 selects a real scalar or polarization component $q_{\mathbf R}(t)$ at lattice position $\mathbf R$. Oppositely paired, reciprocal and position-independent links have discrete action

$Z_d>0$ normalizes time response, $[\Gamma_n]=\mathrm{s^{-2}}$, and $J_{\mathbf R}$ couples to the component. For length-dimensional $q$, $Z_d$ has mass units and each bracket energy units; this does not prove the sites are material balls. The factor $1/4$ includes double counting of undirected links in the oriented whole-lattice sum.

Variation must include contributions from neighbouring links. Reciprocity gives

Adjacency and response coefficients enter separately. Complete vector or internal-orientation dynamics carries additional internal indices, so this scalar law is not automatically the dynamics of every spatial degree of freedom.

03From twelve links to a continuum operator

Nearest-neighbour vectors and their second moment are

Each plane has four independent sign combinations. For the $x$ second moment, the $xy$ and $xz$ groups each contribute $a_0^2$, with the remaining group zero. Sign pairing cancels mixed moments and gives $\sum_n\mathbf d_n=0$.

At $ka_0\ll1$, expand neighbouring values:

Equal opposite-link coefficients remove odd orders. The leading speed-squared tensor is

It has $\mathrm{m^2s^{-2}}$ units. Equal links give $C_{ij}=\Gamma_0a_0^2\delta_{ij}$ and $c_q^2=\Gamma_0a_0^2$. Matching $c_q$ to $c$ requires a parameter condition; equal longitudinal/transverse speeds require checking their actual link responses too.

Slowly varying reciprocal links require the weighted divergence equation from the same link energy, developed in continuum dynamics. Coarse graining must also retain kinetic and source normalization. A spatial second moment alone does not establish action equivalence.

04Exact dispersion and its long-wave expansion

Insert $q_{\mathbf R}\propto e^{i(\mathbf k\cdot\mathbf R-\omega t)}$ into the equal-link source-free law. Time differentiation gives $-\omega^2$ and a link $e^{i\mathbf k\cdot\mathbf d_n}-1$. Opposite links cancel imaginary parts, giving $\omega^2=\Gamma_0\sum_n[1-\cos(\mathbf k\cdot\mathbf d_n)]$. Summing the groups yields

Source-equation check. The half-angle squared-sine sum preceding (5.1.12) prints the coefficient $2c^2/a_0^2$, which is inconsistent with (5.1.11). Applying $1-\cos x=2\sin^2(x/2)$ to the three $i<j$ pairs and their two signs requires $4c^2/a_0^2$; the printed coefficient would instead give $c^2k^2/2$ at long wavelength. This page uses the cosine spectrum verified by direct twelve-neighbour summation and consistent with Chapter 4 (4.12). The coefficient discrepancy is recorded separately, not presented as an additional established relation.

Every $1-\cos$ is nonnegative, so the full dispersion is nonnegative for $\Gamma_0>0$. With $c_q^2=\Gamma_0a_0^2$,

The leading term is isotropic; fourth order retains directions. A low-order lack of visible axes can coexist with higher-order directional dispersion. The expansion needs $ka_0\ll1$; extrapolating its negative fourth-order correction to high wave number creates negative frequency squares absent from the exact expression.

Chapter 5 section 6 also discusses positive higher-gradient stabilizing energy. That is a different constitutive choice, not a term to append freely to this equal-neighbour negative dispersion truncation. General links require their own higher-order calculation.

05Anisotropy and causal boundaries

Chapter 5 considers a core moving along $x$, with $\Gamma_\parallel$ for $xy,xz$ groups and $\Gamma_\perp$ for $yz$ links. The selected scalar model gives

This displays directional response without determining the velocity dependence of either coefficient. Directionality of a moving source also differs from directionality of the free operator: an isotropic causal equation can give anisotropic fields for moving sources.

Vector, orientation or microrotation models use $C_{ai,bj}\partial_i\partial_jq_b$. Cubic symmetry need not give full rotational invariance in higher-rank tensors; scalar second moments do not replace complete branch and polarization calculations.

Finite adjacency alone does not give a strict zero-tail microscopic front for continuous-time lattice equations. The long-wave hyperbolic model has its own domain of dependence. If microscopic refresh also has a strict speed boundary $c$, the update law must satisfy that separately. Phase, group, long-wave and front speeds cannot be identified by sharing a numerical label.

06Comparing realizations against the same continuum

The candidate's calculable results are its continuum moments and dispersion after a link law is supplied. Neither metrological definitions nor coordination number imply a unique spatial lattice, and this scalar model does not derive every particle branch or stability condition.

OP-SD-004 requests variables, link tensors, time rules, coarse-graining errors, branches and readout responses. Alternative lattices, networks and nonlattice models must be compared on those same objects. FCC-dependent calculations are tested with their realization, while independent continuum results retain their own conditions.