Analytic behavior of the scattering amplitude at zero energy

1961 ◽  
Vol 21 (2) ◽  
pp. 249-273 ◽  
Author(s):  
W. Zimmermann
Keyword(s):  
Scattering Amplitude ◽  
Zero Energy ◽  
Analytic Behavior

scholarly journals Magnetic scattering at low energy in two dimensions

1999 ◽  
Vol 155 ◽  
pp. 95-151 ◽  
Author(s):  
Hideo Tamura
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Scattering Amplitude ◽  
Two Dimensions ◽  
Low Energy ◽  
Magnetic Scattering ◽  
Main Body ◽  
Two Dimensional ◽  
Zero Energy ◽  
Magnetic Potentials

AbstractWe study the asymptotic behavior at low energy of scattering amplitudes in two dimensional magnetic fields with compact support. The obtained result depends on the total flux of magnetic fields. It should be noted that magnetic potentials do not necessarily fall off rapidly at infinity. The main body of argument is occupied by the resolvent analysis at low energy for magnetic Schrödinger operators with perturbations of lang-range class. We can show that the dimension of resonance spaces at zero energy does not exceed two. As a simple application, we also discuss the scattering by magnetic field with small support and the convergence to the scattering amplitude by δ-like magnetic field.

Study of charge transfer in FeTi

1983 ◽  
Vol 41 ◽  
pp. 296-297
Author(s):  
J. Taft∅
Keyword(s):  
Charge Transfer ◽  
Charge Distribution ◽  
Electron Scattering ◽  
Periodic System ◽  
Electron Distribution ◽  
Scattering Amplitude ◽  
Transition Elements ◽  
Hartree Fock ◽  
Cscl Structure ◽  
The Right

It is well known that for reflections corresponding to large interplanar spacings (i.e., sin θ/λ small), the electron scattering amplitude, f, is sensitive to the ionicity and to the charge distribution around the atoms. We have used this in order to obtain information about the charge distribution in FeTi, which is a candidate for storage of hydrogen. Our goal is to study the changes in electron distribution in the presence of hydrogen, and also the ionicity of hydrogen in metals, but so far our study has been limited to pure FeTi. FeTi has the CsCl structure and thus Fe and Ti scatter with a phase difference of π into the 100-ref lections. Because Fe (Z = 26) is higher in the periodic system than Ti (Z = 22), an immediate “guess” would be that Fe has a larger scattering amplitude than Ti. However, relativistic Hartree-Fock calculations show that the opposite is the case for the 100-reflection. An explanation for this may be sought in the stronger localization of the d-electrons of the first row transition elements when moving to the right in the periodic table. The tabulated difference between fTi (100) and ffe (100) is small, however, and based on the values of the scattering amplitude for isolated atoms, the kinematical intensity of the 100-reflection is only 5.10-4 of the intensity of the 200-reflection.

Single Atom Image Contrast in Fixed Beam Dark Field Electron Microscopy

1974 ◽  
Vol 32 ◽  
pp. 366-367
Author(s):  
Wah Chi
Keyword(s):  
Scattering Amplitude ◽  
Present Report ◽  
Dark Field ◽  
Image Contrast ◽  
Single Atom ◽  
Optical Theorem ◽  
Hartree Fock ◽  
Heavy Atoms ◽  
Beam Stop ◽  
Fixed Beam

Resolution and contrast are the important factors to determine the feasibility of imaging single heavy atoms on a thin substrate in an electron microscope. The present report compares the atom image characteristics in different modes of fixed beam dark field microscopy including the ideal beam stop (IBS), a wire beam stop (WBS), tilted illumination (Tl) and a displaced aperture (DA). Image contrast between one Hg and a column of linearly aligned carbon atoms (representing the substrate), are also discussed. The assumptions in the present calculations are perfectly coherent illumination, atom object is represented by spherically symmetric potential derived from Relativistic Hartree Fock Slater wave functions, phase grating approximation is used to evaluate the complex scattering amplitude, inelastic scattering is ignored, phase distortion is solely due to defocus and spherical abberation, and total elastic scattering cross section is evaluated by the Optical Theorem. The atom image intensities are presented in a Z-modulation display, and the details of calculation are described elsewhere.

LACBED with Quantitative Anomalous Absorption Corrections

1990 ◽  
Vol 48 (2) ◽  
pp. 528-529
Author(s):  
C.J. Rossouw ◽  
L.J. Allen ◽  
P.R. Miller
Keyword(s):  
Momentum Transfer ◽  
Scattering Amplitude ◽  
Incident Beam ◽  
Waller Factor ◽  
Beam Momentum ◽  
Anomalous Absorption ◽  
Quantitative Calculation ◽  
Ewald Sphere ◽  
Image Position ◽  
Incident Beam Energy

An Einstein model for thermal diffuse scattering (TDS) has enabled quantitative calculation of the absorptive potential V'(r). This allows anomalous absorption to be accounted for in LACBED contrast. Fourier coefficients Vg-h of the absorptive component from each atom α are calculated from integrals of the formwhere fα is the scattering amplitude and M(Q) the Debye-Waller factor. Integration over the Ewald sphere (dΩ) requires the momentum transfer q to have values up to 2ko (the incident beam momentum). Dynamical ‘dechannelling’ is accounted for by the terms g ≠ h. The crystal absorptive potential is obtained by coherently summing over these atomic absorptive potentials within the unit cell. Unlike the elastic potential, the absorptive potential is a strong function of incident beam energy Eo, since the range of momentum transfer q and associated solid angles dΩ change with the Ewald sphere radius.Fig. 1 shows a LACBED pattern of the zeroth order beam from Si aligned along a <001> zone axis.

Direct imaging of charge transfer

1996 ◽  
Vol 54 ◽  
pp. 680-681
Author(s):  
Yimei Zhu ◽  
J. Tafto
Keyword(s):  
Charge Transfer ◽  
Electron Diffraction ◽  
Scattering Amplitude ◽  
High Temperature Superconductors ◽  
Charge Carriers ◽  
Image Charge ◽  
Net Charge ◽  
Long Time ◽  
Electron Holes ◽  
First Time

The electron holes confined to the CuO2-plane are the charge carriers in high-temperature superconductors, and thus, the distribution of charge plays a key role in determining their superconducting properties. While it has been known for a long time that in principle, electron diffraction at low angles is very sensitive to charge transfer, we, for the first time, show that under a proper TEM imaging condition, it is possible to directly image charge in crystals with a large unit cell. We apply this new way of studying charge distribution to the technologically important Bi2Sr2Ca1Cu2O8+δ superconductors.Charged particles interact with the electrostatic potential, and thus, for small scattering angles, the incident particle sees a nuclei that is screened by the electron cloud. Hence, the scattering amplitude mainly is determined by the net charge of the ion. Comparing with the high Z neutral Bi atom, we note that the scattering amplitude of the hole or an electron is larger at small scattering angles. This is in stark contrast to the displacements which contribute negligibly to the electron diffraction pattern at small angles because of the short g-vectors.

Direct measurement of electron-diffraction-pattern intensities using an energy loss spectrometer

1989 ◽  
Vol 47 ◽  
pp. 668-669
Author(s):  
A. G. Jackson ◽  
M. Rowe
Keyword(s):  
Thermal Effects ◽  
Dynamic Range ◽  
Scattering Amplitude ◽  
Dynamical Theory ◽  
Site Symmetry ◽  
Proof Of Concept ◽  
Parallel Acquisition ◽  
Kinematic Approximation ◽  
Speed Up ◽  
Structure Calculations

Diffraction intensities from intermetallic compounds are, in the kinematic approximation, proportional to the scattering amplitude from the element doing the scattering. More detailed calculations have shown that site symmetry and occupation by various atom species also affects the intensity in a diffracted beam. [1] Hence, by measuring the intensities of beams, or their ratios, the occupancy can be estimated. Measurement of the intensity values also allows structure calculations to be made to determine the spatial distribution of the potentials doing the scattering. Thermal effects are also present as a background contribution. Inelastic effects such as loss or absorption/excitation complicate the intensity behavior, and dynamical theory is required to estimate the intensity value.The dynamic range of currents in diffracted beams can be 104or 105:1. Hence, detection of such information requires a means for collecting the intensity over a signal-to-noise range beyond that obtainable with a single film plate, which has a S/N of about 103:1. Although such a collection system is not available currently, a simple system consisting of instrumentation on an existing STEM can be used as a proof of concept which has a S/N of about 255:1, limited by the 8 bit pixel attributes used in the electronics. Use of 24 bit pixel attributes would easily allowthe desired noise range to be attained in the processing instrumentation. The S/N of the scintillator used by the photoelectron sensor is about 106 to 1, well beyond the S/N goal. The trade-off that must be made is the time for acquiring the signal, since the pattern can be obtained in seconds using film plates, compared to 10 to 20 minutes for a pattern to be acquired using the digital scan. Parallel acquisition would, of course, speed up this process immensely.

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