scholarly journals Generalization of the Levinson Theorem on the Asymptotic Value of the Scattering-Amplitude Phase Shift

2021 ◽  
Vol 84 (1) ◽  
pp. 29-33
Author(s):  
M. I. Krivoruchenko ◽  
K. S. Tyrin
Keyword(s):  
Phase Shift ◽  
Scattering Amplitude ◽  
Asymptotic Value ◽  
Levinson Theorem

Oscillatory Screening Effects on Elastic Collisions in Dense Electron-Ion Quantum Plasmas

2009 ◽  
Vol 64 (3-4) ◽  
pp. 237-241 ◽  
Author(s):  
Yo-Han Koo ◽  
Young-Dae Jung
Keyword(s):  
Phase Shift ◽  
Total Cross Section ◽  
Cross Section ◽  
Scattering Amplitude ◽  
Wave Number ◽  
Optical Theorem ◽  
Screening Effect ◽  
Quantum Plasmas ◽  
Quantum Wave ◽  
Scattering Phase

Abstract The oscillatory screening effects on elastic electron-ion collisions are investigated in dense quantum plasmas. The eikonal method with the modified Debye-Hückel potential is employed to obtain the scattering phase shift and scattering amplitude. In addition, the total elastic collision cross section is obtained by the optical theorem with the forward scattering amplitude in quantum plasmas. It is shown that the modified Debye-Hückel screening in quantum plasmas produces the oscillatory behaviour of the scattering phase shift. In addition, the minimum position of the phase shift is receded from the target ion with decreasing the quantum wave number. It is also found that the oscillatory screening effect suppresses the differential cross section. The total cross section is also found to be decreased due to the oscillatory screening effect. In addition, it is shown that the total cross section decreases with an increase of the quantum wave number

Impact Parameter Representation of the Scattering Amplitude

1971 ◽  
Vol 49 (14) ◽  
pp. 1885-1898 ◽  
Author(s):  
M. Razavy
Keyword(s):  
Elastic Scattering ◽  
Phase Shift ◽  
Differential Cross Section ◽  
Cross Section ◽  
Impact Parameter ◽  
Differential Cross ◽  
Scattering Amplitude ◽  
Potential Methods ◽  
Energy Dependent ◽  
The Impact

From the Lippmann–Schwinger equation, the exact and different approximate relations for the impact parameter form of the total scattering amplitude on- and off-the-energy shell are derived. The relation between the impact parameter phase shift and the range of potential is studied, and several methods of determining the potential from the impact parameter phase shift for local, nonlocal, and energy dependent interactions are obtained in Blankenbecler and Goldberger's approximation. By considering solvable examples it is shown that the Glauber's approximation, in certain cases, may be valid for all scattering angles. Finally for completely elastic scattering or for a purely absorptive potential, methods of finding the impact parameter phase shift from the differential cross section for scattering are given.

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.

Object Wave Determination by Single-Sideband Methods

1973 ◽  
Vol 31 ◽  
pp. 266-267
Author(s):  
Kenneth H. Downing ◽  
Benjamin M. Siegel
Keyword(s):  
Phase Shift ◽  
Carbon Films ◽  
Object Wave ◽  
Electron Wave ◽  
Phase Object ◽  
Heavy Atoms ◽  
Single Sideband ◽  
Thin Carbon ◽  
Additional Phase Shift ◽  
Additional Phase

Under the “weak phase object” approximation, the component of the electron wave scattered by an object is phase shifted by π/2 with respect to the unscattered component. This phase shift has been confirmed for thin carbon films by many experiments dealing with image contrast and the contrast transfer theory. There is also an additional phase shift which is a function of the atomic number of the scattering atom. This shift is negligible for light atoms such as carbon, but becomes significant for heavy atoms as used for stains for biological specimens. The light elements are imaged as phase objects, while those atoms scattering with a larger phase shift may be imaged as amplitude objects. There is a great deal of interest in determining the complete object wave, i.e., both the phase and amplitude components of the electron wave leaving the object.

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.

A Many-Beam Technique for Enhanced Resolution of Partial Dislocations

1972 ◽  
Vol 30 ◽  
pp. 646-647
Author(s):  
J. M. Oblak ◽  
B. H. Kear
Keyword(s):  
Phase Shift ◽  
Field Method ◽  
Dark Field ◽  
Bright Field ◽  
Field Technique ◽  
Weak Beam ◽  
Partial Dislocations ◽  
Enhanced Resolution ◽  
Nickel Base ◽  
Beam Technique

The “weak-beam” and systematic many-beam techniques are the currently available methods for resolution of closely spaced dislocations or other inhomogeneities imaged through strain contrast. The former is a dark field technique and image intensities are usually very weak. The latter is a bright field technique, but generally use of a high voltage instrument is required. In what follows a bright field method for obtaining enhanced resolution of partial dislocations at 100 KV accelerating potential will be described.A brief discussion of an application will first be given. A study of intermediate temperature creep processes in commercial nickel-base alloys strengthened by the Ll2 Ni3 Al γ precipitate has suggested that partial dislocations such as those labelled 1 and 2 in Fig. 1(a) are in reality composed of two closely spaced a/6 <112> Shockley partials. Stacking fault contrast, when present, tends to obscure resolution of the partials; thus, conditions for resolution must be chosen such that the phase shift at the fault is 0 or a multiple of 2π.

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.

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