The generalized Breit operator of the interaction between two quasimolecular electrons in the framework of the third-order effects of quantum electrodynamics

The problem of interaction of two quasimolecular electrons located at an arbitrary distance from each other and near different atoms (nuclei) is solved. The interaction is considered as a second-order effect of quantum electrodynamics in the coordinate representation. It is shown that a consistent account for the natural condition of the interaction symmetry with respect to both electrons leads to an additional contribution to the relativistic interaction of the two quasimolecular electrons compared with both the standard Breit operator and the generalized Breit operator known previously. The generalized Breit–Pauli operator and the operator of electric dipole–dipole interaction of two quasimolecular electrons located at an arbitrary distance from each other are obtained. Modern methods of accounting for the relativistic and correlative effects in the problem of ion–atom interactions are discussed.

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The infra-red spectra of crystals

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1960.0194 ◽

1960 ◽

Vol 258 (1294) ◽

pp. 377-401 ◽

Cited By ~ 72

Keyword(s):

Fourth Order ◽

Second Order ◽

Order Moment ◽

Order Effects ◽

Ionic Crystals ◽

Third Order ◽

Main Band ◽

The Third ◽

Cross Terms ◽

Infra Red

The higher-order effects in the intrinsic infra-red absorption of crystals are investigated in a systematic way. In agreement with a previous paper which dealt with the static dielectric constant, it is found that in the case of ionic crystals the third- and fourth-order potential, the second- and the third-order dipole moment, and the cross-terms between the second-order moment and the third-order potential, all contribute terms of the same order to the infra-red spectrum. In the lowest approximation, the third-order moment and the fourth-order potential only affect the absorption in the immediate neighbourhood of the maximum and hence have little effect on the shape of the spectrum. The broadening of the main band is due mainly to the third-order potential, while the side bands may be caused by the second-order moment as well as by the third-order potential and by cross-terms between the two. But due to an internal field effect, in strongly ionic crystals a large second-order moment automatically leads to a large third-order potential; thus a large second-order moment may increase the width of the main band as well as the intensity of the side bands. Although the intrinsic infra-red absorption of valency crystals, such as diamond or germaniam, is due to the second-order moment only, nevertheless, there is a strong similarity between the expressions for the infra-red absorption of valency crystals and for the side-band absorption of ionic crystals. This similarity suggests that the spectra of all ionic crystals should exhibit a number of secondary maxima. The available experimental evidence does not seem sufficient to decide whether this suggestion is correct.

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On one-loop corrections to the CPT-even Lorentz-breaking extension of QED

International Journal of Modern Physics A ◽

10.1142/s0217751x18500185 ◽

2018 ◽

Vol 33 (02) ◽

pp. 1850018 ◽

Cited By ~ 5

Author(s):

T. Mariz ◽

R. V. Maluf ◽

J. R. Nascimento ◽

A. Yu. Petrov

Keyword(s):

Standard Model ◽

Gauge Field ◽

Quantum Electrodynamics ◽

Renormalization Scheme ◽

Standard Model Extension ◽

Third Order ◽

The Third ◽

Model Extension ◽

The Standard Model ◽

Tensor Formula

In this paper, we describe the quantum electrodynamics added by Lorentz-violating CPT-even terms in the context of the standard model extension. We focus our attention on the fermion sector, represented by the CPT-even symmetric Lorentz-breaking tensor [Formula: see text]. We adopt a generic form that parametrizes the components of [Formula: see text] in terms of one four-vector, namely, [Formula: see text]. We then generate perturbatively, up to the third order in this tensor, the aether-like term for the gauge field. Finally, we discuss the renormalization scheme for the gauge propagator, by taking into account [Formula: see text] traceless [Formula: see text] and, trivially, [Formula: see text] [Formula: see text].

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Probe size and 5th-order aberration

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100125609 ◽

1987 ◽

Vol 45 ◽

pp. 134-135 ◽

Cited By ~ 1

Author(s):

Zhifeng Shao

Keyword(s):

Electron Probe ◽

Spherical Aberration ◽

Wave Length ◽

Cylindrical Symmetry ◽

Electron Wave ◽

Third Order ◽

The Third ◽

Probe Radius ◽

Small Probe ◽

Probe Size

A small electron probe has many applications in many fields and in the case of the STEM, the probe size essentially determines the ultimate resolution. However, there are many difficulties in obtaining a very small probe.Spherical aberration is one of them and all existing probe forming systems have non-zero spherical aberration. The ultimate probe radius is given byδ = 0.43Csl/4ƛ3/4where ƛ is the electron wave length and it is apparent that δ decreases only slowly with decreasing Cs. Scherzer pointed out that the third order aberration coefficient always has the same sign regardless of the field distribution, provided only that the fields have cylindrical symmetry, are independent of time and no space charge is present. To overcome this problem, he proposed a corrector consisting of octupoles and quadrupoles.

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Children’s Recall of Approximations to English

Journal of Speech and Hearing Research ◽

10.1044/jshr.1602.201 ◽

1973 ◽

Vol 16 (2) ◽

pp. 201-212 ◽

Cited By ~ 3

Author(s):

Elizabeth Carrow ◽

Michael Mauldin

Keyword(s):

Language Development ◽

Fourth Order ◽

Third Order ◽

Memory Processes ◽

The Third ◽

General Index ◽

General Linguistic

As a general index of language development, the recall of first through fourth order approximations to English was examined in four, five, six, and seven year olds and adults. Data suggested that recall improved with age, and increases in approximation to English were accompanied by increases in recall for six and seven year olds and adults. Recall improved for four and five year olds through the third order but declined at the fourth. The latter finding was attributed to deficits in semantic structures and memory processes in four and five year olds. The former finding was interpreted as an index of the development of general linguistic processes.

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