Lorentz Microscopy of Weak Magnetic Flux Inhomogeneities

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Abstract
The potential of Lorentz microscopy to give quantitative information about weak magnetic field inhomogeneities can be exploited fully only if the classical approximation to the electron contrast is abandoned in favor of the wave mechanical reality. The inhomogeneities cause coordinate dependent phase shifts Δφm in the coherent electron wavefunction which can be detected in principle by any electron optical method for phase contrast. The transition from a classical to a wave‐mechanical treatment of contrast is necessary if the inhomogeneity contains flux of the order of one fluxon h/2q or less. Contrast formation in the defocused mode is discussed. The image space in this mode has sections with almost classical, low contrast (due to running lateral waves) divided by the caustic mantle from other sections with wave‐mechanical, high contrast (due to standing lateral waves). The maximum contrast due to a given inhomogeneity can be predicted from its maximum magnetic phase shift Δφm = (q/ℏ) ΔΦI. The differential and total scattering cross sections of an arbitrary flux distribution for particles with charge q are calculated. The total cross section does not depend on the kinetic energy. In order to observe a flux inhomogeneity ΔΦI≪h/2q in principle, a minimum number Np min of particles must be passed through it. This number is Np min > 〈Δφ2〉−1 if the particles are recorded in pure scattered states (Fraunhofer diffraction, dark‐field microscopy). The minimum number is Np min> 〈Δφ〉−2, if the detection is achieved in some optimum arrangement of interference between the scattered and an unscattered state (around the caustic mantle in the defocused mode; interference microscopy; Zernike phase contrast). In the case of detection by interference, the minimum number of scattered particles NS min necessary for observability is NS min> 〈Δφ〉−1. This relation shows that for observability of a total flux 〈Φ〉 a minimum amount of scattered charge QS min = NS min 2q is necessary, given by the relation QS min 〈Φ〉>ℏ. A full paper will appear elsewhere.

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