= Shielded toroidal-flux electron interference
In 1986, <Akira Tonomura> and collaborators coated a <ferromagnet> shaped as a <torus> with superconducting <niobium> and <copper>. A cooled stage held it in a 150 kV field-emission <electron microscope>. An <electron biprism> overlapped object and reference waves; film <holograms> were reconstructed optically. Room-temperature leakage screening selected samples; varying <copper> thickness tested electron penetration.<Evidence for Aharonov-Bohm effect with magnetic field completely shielded from electron wave by Akira Tonomura and others>[pp. 792–794, Figs. 1–4]{ref}
Below the superconducting transition, through-hole and outside <Electron> waves differed in <wave phase> by zero or $\pi$ modulo $2\pi$. Warming through it changed the <interference> reversibly. Leakage and coherent-penetration bounds do not prove zero <magnetic field>.<Evidence for Aharonov-Bohm effect with magnetic field completely shielded from electron wave by Akira Tonomura and others>[p. 794, Figs. 6–7: experimental limits]{ref}
Why these two phases? The shield's <superconducting flux quantum> is $h/(2e)$, whereas the probing <Electron> has <electric charge> $-e$. Substituting $\Phi=Nh/(2e)$ gives
$$
e^{i\delta}=e^{-iN\pi}=(-1)^N.
$$
An odd $N$ therefore moves a carrier fringe by half its spacing; an even $N$ leaves it unchanged modulo a full spacing. The phase quantization provides an internal check on the superconducting enclosure. It does not mean that zero <wave phase> proves zero trapped <magnetic flux>.<The Aharonov-Bohm effect and its applications to electron phase microscopy by Akira Tonomura>[2006, p. 49, Confirmation experiments on the AB effect; Fig. 2 on p. 48]{ref}
The readout is <electron holography> in the <off-axis holography> geometry: a known tilted reference converts an unknown relative <wave phase> into a spatial carrier displacement. Reconstructing a magnetic <wave phase> assumes stable reference phase, adequate coherence and a calibrated carrier; electrostatic charging and material traversal can also shift <wave phase>. Optical <wave phase> amplification rescales the displayed fringe displacement. The experiment's shielding and comparison tests support the magnetic inference; an interferogram by itself measures the total relative <wave phase>, rather than separately certifying every contribution. The ideal field-free model describes the limiting geometry, while finite leakage and penetration remain questions of experimental bounds.
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