For each spacing pair, the largest geometrical period occurs with parallel projected reciprocal vectors. The closest pair is $d_{\mathrm{Cu}}=0.208\mathrm{nm}$ and $d_{\mathrm{substrate}}=0.212\mathrm{nm}$, givingHowever, the crystal is only $10\mathrm{nm}$ across. It contains at most $10/11.024\simeq 0.91$ of this period, so the largest formal period cannot appear as a repeated fringe train inside one crystal. It can instead give a broad variation of contrast. A period of order the particle diameter is the largest directly sampled scale, and even that is poorly constrained by one particle.

The distinction matters if “see” means a complete repeat. Of the exactly parallel spacing pairs, the largest other period is approximately $2.09\mathrm{nm}$ from the $0.208$ and $0.231\mathrm{nm}$ pair. If relative rotation is continuously variable, the closest pair can instead be rotated slightly to shorten its period below $10\mathrm{nm}$, so $2.09\mathrm{nm}$ is not a universal maximum visible period for arbitrary orientations.

Let the relevant projected reciprocal lattice vectors have magnitudes $g_1=1/d_1$ and $g_2=1/d_2$, with angle $\theta$ between them. Subtracting their phases gives the slow spatial modulation $2\pi ({\mathbf{g}}_1-{\mathbf{g}}_2)\cdot \mathbf{r}$. The moiré fringe spacing formula is thereforeFor parallel planes, $\theta =0$ and $D=d_1{d}_2/|d_1-d_2|$. For equal spacings, $D=d/[2|\sin (\theta /2)|]\simeq d/|\theta |$ at a small angle measured in radians. The apparent divergence as both mismatch and rotation vanish means that there is no finite beat period; it does not predict an infinite number of visible fringes.

Moiré fringes arise when two overlapping periodic structures contribute closely spaced reciprocal lattice vectors to the transmission electron microscopy image. Their relative phase varies slowly across the image, producing a coarse modulation with a period much larger than either atomic spacing. The structures may have different lattice spacings, a small relative rotation, or both. The modulation is an interference or beat phenomenon; its spacing is not itself a new atomic-plane spacing. Its orientation is perpendicular to the difference vector of the two contributing projected reciprocal-lattice vectors.

For an ideal electrolyte with species number densities $n_i$, valences $z_i$, solvent permittivity $ϵ$ and common temperature,Increasing salt concentration supplies more mobile charge to screen a surface perturbation, so a shorter distance is needed for the potential to decay. At fixed ion composition, permittivity and temperature, ${\lambda }_D\propto c^{-1/2}$.

A fivefold concentration increase therefore givesEquivalently, the inverse screening length grows by $\sqrt{5}$. Changing ion valence changes the ionic-strength weighting as well, and highly concentrated electrolytes can depart from this dilute-solution screening model.

Choose a positively charged plate to match the question's labels $n_{+}$ for co-ions and $n_{-}$ for counterions. Let $x=0$ denote the outer edge of its compact Stern layer, and set the bulk potential to zero.

In the weak-potential Debye–Hückel approximation, the diffuse-layer potential isThe key length is the Debye–Hückel screening length. For a symmetric monovalent electrolyte, the Boltzmann distribution givesPositive co-ions are depleted near the positive plate, while negative counterions accumulate. Both approach the same bulk density as the potential decays. Attraction and repulsion by the surface compete with mixing entropy, so diffuse ions do not simply collapse into one plane. The compact Stern layer represents finite closest approach and specifically associated charge; the diffuse continuum begins outside it. For large potentials the linear exponential potential needs the nonlinear Poisson-Boltzmann equation, though the far-field screening scale remains the same in this simple electrolyte.

The microscopic pair potential falls as $-r^{-6}$, whereas integrating over two half-spaces gives an interaction energy per area $W(d)=-A_H/(12\pi d^2)$. The plate attraction therefore decays much more slowly with distance because many pairs contribute. The quantities have different dimensions, so the shared sketch uses separately normalized energies at a reference separation rather than pretending that joules and joules per square metre are directly comparable.

The attractive pressure isAt $d=100\mathrm{nm}$ and the supplied effective $A_H$,The minus sign denotes decreasing separation. A total force cannot be calculated without the interacting area: for area $A$ in square metres, $F\simeq -(13.3\mathrm{N}{\mathrm{m}}^{-2})A$. This uses the supplied nonretarded half-space model; finite film thickness, geometry and retardation at larger distances can change it.

$N_1$ and $N_2$ count the microscopic interacting units in bodies 1 and 2. Their positions are ${\mathbf{r}}_i$ and ${\mathbf{r}}_j$; the difference vector gives their pair separation. The positive coefficient $C$ specifies an attractive nonretarded pair energy $-C/r^6$ and has units $\mathrm{J}{\mathrm{m}}^6$.

Hamaker summation approximates a macroscopic interaction from Van der Waals forces by adding these microscopic pair energies. For homogeneous continua, sums are replaced by integrals over the two body volumes with number densities ${\rho }_1$ and ${\rho }_2$. Integrating the geometry changes the distance dependence and produces a Hamaker constant $A_H={\pi }^2{\rho }_1{\rho }_{2}C$ for the stated simple model.

The approximation assumes pairwise additivity, uniform material properties, a known geometry and a separation large enough to use a continuum but small enough for the chosen nonretarded interaction law. It neglects many-body electrodynamic screening, chemical bonding, atomic roughness and short-range repulsion. A medium changes the effective interaction strength; an effective measured $A_H$ across water need not equal a vacuum value inferred from isolated atom pairs.

TODO — user-deferred biology. This part belongs to the explicitly biological molecular-system question and is preserved without a solution under the requested nonbiology scope.

TODO — user-deferred biology. This part belongs to the explicitly biological molecular-system question and is preserved without a solution under the requested nonbiology scope.

TODO — user-deferred biology. This part belongs to the explicitly biological molecular-system question and is preserved without a solution under the requested nonbiology scope.

TODO — user-deferred biology. This part belongs to the explicitly biological molecular-system question and is preserved without a solution under the requested nonbiology scope.