scholarly journals Exact summation of leading logs around $$ T\overline{T} $$ deformation of O(N + 1)-symmetric 2D QFTs

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Jonas Linzen ◽  
Maxim V. Polyakov ◽  
Kirill M. Semenov-Tian-Shansky ◽  
Nika S. Sokolova
Keyword(s):  
Irrelevant Dimension ◽  
Scattering Amplitude ◽  
Exact Result ◽  
Logarithmic Approximation ◽  
Exact Summation

Abstract We consider a general (beyond $$ T\overline{T} $$ T T ¯ ) deformation of the 2D O(N + 1) σ-model by the irrelevant dimension-four operators. The theory deformed in this most general way is not integrable, and the S-matrix loses its factorization properties. We perform the all-order summation of the leading infrared logs for the 2 → 2 scattering amplitude and provide the exact result for the 2 → 2 S-matrix in the leading logarithmic approximation. These results can provide us with new insights into the properties of the theories deformed by irrelevant operators more general than the $$ T\overline{T} $$ T T ¯ deformation.

A comparison of approximations for the elastic electron–electron scattering amplitude in jellium

1982 ◽  
Vol 60 (7) ◽  
pp. 1016-1028 ◽  
Author(s):  
A. H. MacDonald ◽  
D. J. W. Geldart
Keyword(s):  
Electron Scattering ◽  
Density Functional ◽  
Scattering Amplitude ◽  
Magnetic Interaction ◽  
Exact Result ◽  
Many Body ◽  
Many Body Theory ◽  
Expansion Technique ◽  
Low Energy Electrons ◽  
Body Theory

An approximation for the electron–electron scattering amplitude, Γ, in jellium, motivated by comparing formal many-body theory and density-functional theory expressions for the jellium response function, is suggested. The approximation is compared with previously suggested approximations for Γ, using an exact result for the forward scattering limit to assess the reliability of each approximation. It is pointed out that the magnetic interaction contribution to Γ becomes dominant at low density in jellium and that this contribution is absent in most commonly adopted approximations. Mean free paths for low energy electrons in jellium-like metals have been calculated for various approximations to Γ and the magnetic-interaction effect is identified as being responsible for a previously noted discrepancy between Thomas–Fermi-like and more sophisticated approximations to Γ. A systematic method, using the parquet expansion technique, has been proposed for antisymmetrizing the density-functional based approximation to Γ when necessary.

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.

Distinctive Phonetic Features as Relevant and Irrelevant Stimulus Dimensions in Speech-Sound Discrimination Learning

1974 ◽  
Vol 17 (3) ◽  
pp. 417-425
Author(s):  
Stuart I. Ritterman ◽  
Nancy C. Freeman
Keyword(s):  
College Students ◽  
Discrimination Learning ◽  
Irrelevant Dimension ◽  
Speech Sound ◽  
Irrelevant Stimulus ◽  
Relevant Dimension ◽  
Phonetic Features ◽  
Sound Discrimination ◽  
Nonsense Syllables

Thirty-two college students were required to learn the relevant dimension in each of two randomized lists of auditorily presented stimuli. The stimuli consisted of seven pairs of CV nonsense syllables differing by two relevant dimension units and from zero to seven irrelevant dimension units. Stimulus dimensions were determined according to Saporta’s units of difference. No significant differences in performance as a function of number of the irrelevant dimensions nor characteristics of the relevant dimension were observed.

A neutron-diffraction study of potassium bromide

1973 ◽  
Vol 137 (4) ◽  
Author(s):  
G. E. Bacon ◽  
D. H. Titterton ◽  
C. R. Walker
Keyword(s):  
Neutron Diffraction ◽  
Diffraction Data ◽  
Diffuse Scattering ◽  
Scattering Amplitude ◽  
Coherent Scattering ◽  
Neutron Diffraction Study ◽  
Potassium Bromide ◽  
Neutron Diffraction Data ◽  
Thermal Diffuse Scattering ◽  
Thermal Diffuse

AbstractNeutron-diffraction data have been collected from a KBr single crystal. 380 reflections were measured, reducing to 23 when averaged over equivalents. Data were corrected for extinction and thermal diffuse scattering and refinement yielded a neutron coherent scattering amplitude

scholarly journals Rise of the DIS structure function FL at small x caused by double-logarithmic corrections

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
B. I. Ermolaev ◽  
S. I. Troyan
Keyword(s):  
Structure Function ◽  
Present Calculation ◽  
Synergic Effect ◽  
Logarithmic Approximation ◽  
Logarithmic Corrections ◽  
Small X

Abstract We present calculation of FL in the double-logarithmic approximation (DLA) and demonstrate that the synergic effect of the factor 1/x from the $$ {\alpha}_s^2 $$ α s 2 -order and the steep x-dependence of the totally resummed double logarithmic contributions of higher orders ensures the power-like rise of FL at small x and arbitrary Q2.

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