Density matrix of inelastically scattered fast electrons

Schattschneider, P.; Nelhiebel, Michael; Jouffrey, B.

doi:10.1103/physrevb.59.10959

Cited by 85 publications

(85 citation statements)

References 17 publications

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“…For several transition channels at the same energy, the outgoing probe electron is in a mixed state, described by the reduced density matrix [34,35] and the path of rays cannot be visualized in such an easy way. Note that the total intensity is the trace of the matrix, i.e.…”

Section: Principlementioning

confidence: 99%

EMCD with an electron vortex filter: Limitations and possibilities

et al. 2017

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We discuss the feasibility of detecting spin polarized electronic transitions with a vortex filter. This approach does not rely on the principal condition of the standard energy loss magnetic chiral dichroism (EMCD) technique, the precise alignment of the crystal, and thus paves the way for the application of EMCD to new classes of materials and problems. The dichroic signal strength in the L 2,3 -edge of ferromagnetic cobalt is estimated on theoretical grounds. It is shown that magnetic dichroism can, in principle, be detected. However, as an experimental test shows, count rates are currently too low under standard conditions.

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Section: Principlementioning

confidence: 99%

EMCD with an electron vortex filter: Limitations and possibilities

et al. 2017

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“…which can be written as (14) with , and, after usual exposure times in the order of seconds, interference contrast in an image-plane hologram would be completely wiped out.…”

Section: Then We Superimposementioning

confidence: 99%

“…This coherence corresponds to ensemble coherence of the beam electrons The reduced density matrix for the fast electron after an inelastic scattering event with energy transfer E  can be written as [14] (26) with  meaning convolution.…”

Section: Density Matrix Descriptionmentioning

confidence: 99%

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Inelastic electron holography – first results with surface plasmons

Röder

Lichte

2011

Eur. Phys. J. Appl. Phys.

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Inelastic interaction and wave optics seem to be incompatible in that inelastic processes destroy coherence, which is the fundamental requirement for holography. In special experiments it is shown that energy transfer larger than some eV 15 10  undoubtedly destroys coherence of the inelastic electron with the elastic remainder. Consequently, the usual inelastic processes, such as phonon-, plasmon-or inner shell-excitations with energy transfer of several meV out to several eV 10 , certainly produce incoherence with the elastic ones. However, it turned out that within the inelastic wave, "newborn" by the inelastic process, there is a sufficiently wide area of coherence for generating "inelastic holograms". This is exploited to create holograms with electrons scattered at surface-plasmons, which opens up quantum mechanical investigation of these inelastic processes.

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“…The contrast of the resulting fringe pattern depends on the coherence between the interfering beams. This coherence is determined by the scattering process with the object [5,6], but also depends on the partial coherence of the beam electron ensemble [7]. Fresnel diffraction at the biprism rim [8] as well as aberrations of the objective lens have a share in the formation of the detected fringe pattern.…”

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Density matrix reconstruction by off-axis electron holography

Röder¹,

Lichte²

2013

Microsc Microanal

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The phase problem in transmission electron microscopy emerges through the conventional way of detecting intensities. Consequently, it is not possible to deduce the expected intensity at an arbitrary electron optical plane below the object from the measured intensity at the detector. In case of an elastically scattered electron, additional access to its phase suffices to construct a wave in the sense of a pure state, which can be propagated numerically between different electron optical planes. The case of inelastic scattering complicates the situation much more, because both scattering and detection process lead inevitably to decoherence (see e.g. [1]). Since a wave is fully coherent per definition, it is insufficient to describe the inelastically scattered electron by a wave. Here, the beam electron transfers from a pure state into a mixed state, which has to be described e.g. by a reduced density matrix. If this density matrix is known at the detector plane, it is known for all other planes (below the object). The diagonal elements of this matrix directly correspond to the measured intensity and are accessible by means of conventional TEM. The off-diagonal elements describe the coherence of the scattered electron hence require an interferometric approach for their experimental determination. Finally, the significance of the off-diagonal elements of the density matrix for inelastically scattered electrons is comparable to that of the phases for the case of pure elastic scattering. Off-axis electron holography in combination with energy filtering [2, 3] realizes experimental access to the off-diagonal elements by superposition of different partial beams in dependence on their spatial separation (shear d, see fig. 1a and 1b) [4]. The contrast of the resulting fringe pattern depends on the coherence between the interfering beams. This coherence is determined by the scattering process with the object [5,6], but also depends on the partial coherence of the beam electron ensemble [7]. Fresnel diffraction at the biprism rim [8] as well as aberrations of the objective lens have a share in the formation of the detected fringe pattern. Thus in general, a direct interpretation in terms of reduced density matrix is complicated. To identify special interpretable cases, we derive and analyze a transfer theory for this holographic measurement setup based on a generalization of the transmission cross coefficient. It turns out that for samples homogeneous perpendicular to the biprism and for small scattering angles, the effect of diffraction at the biprism and partial coherence of the electron beam ensemble can be separated from object influences. That allows a direct and model-independent procedure to determine the density matrix of a scattered electron ( fig. 1c and 1d). Limitations are given by the finite biprism diameter providing lower bounds for the shear and, if applicable, by aberrations of the energy filter. We apply this method for the investigation of the influence of surface plasmons on the coherence of the scattere...

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Density matrix of inelastically scattered fast electrons

Cited by 85 publications

References 17 publications

EMCD with an electron vortex filter: Limitations and possibilities

EMCD with an electron vortex filter: Limitations and possibilities

Inelastic electron holography – first results with surface plasmons

Density matrix reconstruction by off-axis electron holography

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