= Flux-lattice spacing
{title2=$a=\sqrt{2\Phi_0/(\sqrt3\,\overline B)}$}
Assume a triangular array of straight, singly quantized <Abrikosov vortices> with nearest-neighbor distance $a$. Here $\overline B>0$ is the magnitude of the \b[spatially averaged internal] <magnetic field>, and $\Phi_0=h/(2e)$ is the <superconducting flux quantum>; $e>0$ is the <elementary charge>. Two lattice vectors of <length> $a$ meeting at $60^\circ$ span <area> $A_c=\sqrt3a^2/2$. One flux quantum per cell gives
$$
\overline B A_c=\Phi_0,\qquad
n_v=\frac{\overline B}{\Phi_0},\qquad
a=\sqrt{\frac{2\Phi_0}{\sqrt3\,\overline B}}.
$$
Thus multiplying $\overline B$ by five decreases $a$ by $\sqrt5$; it does not multiply the flux of each vortex by five. Using the exact SI <Planck constant> and <elementary charge>, $\Phi_0\simeq2.06783\times10^{-15}\,\mathrm{Wb}$.<Fundamental Physical Constants — Complete Listing by National Institute of Standards and Technology and others>[2022 CODATA table, exact Planck constant and elementary charge]{ref}
For chosen internal fields $0.20\,\mathrm T$ and $1.0\,\mathrm T$, the spacings are approximately \b[$109\,\mathrm{nm}$ and $48.9\,\mathrm{nm}$]. These are geometric model calculations, not measurements from a particular sample. The local <magnetic field> varies within a cell; replacing $\overline B$ by the field at a core would not give this counting relation.
\Image[/superconducting-vortex-lattice-spacing.png]
{title=More internal magnetic flux means more vortices per unit area}
{description=Original triangular-array geometry for stipulated $\xi=10\,\mathrm{nm}$ and $\lambda=100\,\mathrm{nm}$. The two core-amplitude maps use a chosen nearest-core hyperbolic-tangent profile, not a solution of the Ginzburg-Landau equations or an experimental image. The spacing curve counts one flux quantum per cell; its endpoint at $B_{c2}$ does not establish that field by geometry alone.}
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