Deterministic Retrieval of Complex Green’s Functions Using Hard X Rays

ibstract. The angular and energy dependence of the dynamical diffraction of gamma rays was studied experimentally in the symmetric Brag case. Mossbauer diffraction spectra were measured with high angular resolution in a pure nuclear Bragg reflection of a nearly perfect 57FeB0, single crystal. In the spectrum taken at the exact Bragg position, very broad hyperfine resonance peaks were observed. In the spectra just above and below the Bragg angle the reflection peaks were much smaller and were shifted to energy positions above and below resonance respectively. The results were interpreted by means of the dynamical theory and compared with computer calculations. The measurements represent the first direct observation of the enhancement of nuclear resonance scattering in the case of a thick crystal reflection. Furthermore the angular dependence of the resonance scattering by a perfect single crystal has been measured for the first time. Soon after the discovery of the Mossbauer effect the dynamical diffraction of gamma radiation by a regular ensemble of resonant nuclei was considered theoretically. First the diffraction of gamma rays incident on a thin perfect single-crystal film near the Bragg angle was studied (Trammell 1961). It was found that in this case the halfwidth of the resonance scattering is larger than for an isolated nucleus and increases with the thickness of the crystal film. The energy broadening of the resonance scattering was called the enhancement effect. A general treatment of the dynamical diffraction of resonant gamma rays in thick perfect single crystals was given for the hue case (Afanas’ev and Kagan 1965) and also for the Bragg case (Kagan et al1968). In the latter paper it was shown that in the case of a Bragg reflection by an infinitely thick perfect crystal, under certain conditions, an angular position can be found where the reflection becomes total in spite of the presence of a strong inelastic channel and independent of the energy distance from resonance. In the region around this angular position the reflectivity falls off rapidly. Thereby the reflectivity remains unchanged when the energy distance from resonance and the angular deviii:ion are both changed in sign. Also the Bragg reflection by thin perfect crystals was treated in a general way (Hannon and Trammell 1969). The contributions of other authors may be found summarised in review articles (Belyakov 1975, Champeney 1979). Recently the problem of filtering of

Scintillators and photoconductors used in energy integrating detectors (EIDs) have inherent variations in their imaging response to single-detected X-rays due to variations in X-ray energy deposition and secondary quanta generation and transport, which degrades DQE(f). The imaging response of X-ray scintillators to single X-rays may be recorded and studied using single X-ray imaging (SXI) experiments; however, no method currently exists for relating SXI experimental results to EID DQE(f). This work proposes a general analytical framework for computing and analyzing the DQE(f) performance of EIDs from single X-ray image ensembles using a spatial frequency-dependent pulse-height spectrum. A spatial frequency (f)-dependent gain, , is defined as the Fourier transform of the imaging response of an EID to a single-detected X-ray. A f-dependent pulse-height spectrum, , is defined as the 2D probability density function of over the complex plane. is used to define a f-dependent Swank factor, AS (f), which fully characterizes the DQE(f) degradation due to single X-ray noise. AS (f) is analyzed in terms of its degradation due to Swank noise, variations in the frequency-dependent attenuation of , and noise in which occurs due to variations in the asymmetry in each single X-ray's imaging response. Three example imaging systems are simulated to demonstrate the impact of depth-dependent variation in , remote energy deposition, and a finite number of secondary quanta, on , AS (f), MTF(f), and NPS(f)/NPS(0), which are computed from ensembles of single X-ray images. The same is also demonstrated by simulating a realistic imaging system; that is, a Gd2 O2 S-based EID. Using the latter imaging system, the convergence of AS (f) estimates is investigated as a function of the number of detected X-rays per ensemble. Depth-dependent variation resulted in AS (f) degradation exclusively due to depth-dependent optical Swank noise and the Lubberts effect. Conversely, the majority of AS (f) degradation caused by remote energy deposition and finite secondary quanta occurred due to variations in . When using input X-ray energies below the K-edge of Gd, variations in the frequency-dependent attenuation of accounted for the majority of AS (f) degradation in the GOS-based EID, and very little Swank noise and variations in were observed. Above the K-edge, however, AS (f) degradation due to Swank noise and variations in greatly increased. The convergence of AS (f) was limited by variation in ; imaging systems with more variation in required more detected X-rays per ensemble. An analytical framework is proposed that generalizes the pulse-height spectrum and Swank factor to arbitrary f. The impact of single X-ray noise sources, such as the Lubberts effect, remote energy deposition, and finite secondary quanta on detector performance, may be represented using , and quantified using AS (f). The approach may be used to compute MTF(f), NPS(f), and DQE(f) from ensembles of single X-ray images and provides an additional tool to analyze proposed EID designs.

The elastic response of a vacancy in a semi-infinite fcc copper lattice containing a free surface is calculated by using a new multiscale Green's function method. The method treats the lattice distortion near the vacancy at the atomistic level and the free surface at the macroscopic continuum level in the same formalism. The lattice is modeled using the lattice statics Green's function that fully accounts for the discrete atomistic structure of the lattice and can model a large crystallite containing a million atoms without excessive CPU effort. The method is especially useful for modeling the elastic response of nanocrystals containing point defects in which surfaces and interfaces play important roles. The method bridges the length scales seamlessly by relating the microscopic lattice distortion near a point defect to measurable macroscopic parameters of the solid such as the strain and the displacement field at a free surface. Using the interatomic potential derived by Cleri and Rosato, the lattice distortion, relaxation energy, and relaxation volume due to a vacancy are calculated in an otherwise perfect copper lattice for a million-atom model containing a free (100) surface. The calculated value of the relaxation volume is in excellent agreement with the observed value. Numerical results are also presented for the strain and the displacement fields at the free surface due to a vacancy and the interaction energy between a vacancy and the free surface in anisotropic semi-infinite copper.

The rational function fitting method has been found useful in the derivation of closed-form expressions of spatial-domain Green's functions for multilayered media. However, former implementations of the rational function fitting method lead to Green's functions expressions that are not accurate in the far field when this far field is dominated by the continuous spectrum instead of being dominated by surface waves (as it happens, for instance, in the case of lossy multilayered media). In this paper, the authors introduce a novel implementation of the rational function fitting method, which leads to Green's functions expressions that are accurate in the far field when this is dominated either by the continuous spectrum or by surface waves. In the new approach, the far-field contribution of the continuous spectrum to the Green's functions is numerically fitted in terms of functions with closed-form Hankel transforms, and this far-field contribution is explicitly added to the total least squares approximations of the Green's functions. The numerical results obtained for the Green's functions with the new approach have been compared with numerical results obtained via direct numerical integration of Sommerfeld integrals, and excellent agreement has been found despite the contribution - continuous spectrum or surface waves - dominating the far field.

Crystal thickness and extinction distance determination using energy filtered CBED pattern intensity measurement and dynamical diffraction theory fitting

A new approach which clarifies the structure of many-beam dynamical interactions of diffracted waves is developed on the basis of an analytical formula for the diffracted intensity averaged over crystal thickness. In the systematic n -beam case the orientation dependence of the averaged intensity is proved to be expressed as a linear combination of terms of resonance type ( q - a k )/[( q - a k ) 2 + b k 2 ] and b k /[( q - a k ) 2 + b k 2 ] for k =1, …, n ( n -1)/2, where - q is the tangential component of the incident wave vector to the row of reciprocal lattice points. Each term can be associated with Bragg reflections between pairs of plane wave components of Bloch waves. The parameters a k and b k indicate Bragg positions and reflection widths. Modifications of the result are briefly discussed about the critical voltage effect and the non-systematic case.

Semi-XctNet: Volumetric images reconstruction network from a single projection image via semi-supervised learning

This paper presents the closed-form Green's function for an arbitrarily biased magnetically anisotropic slab. The Green's function formulated using the transmission matrix is compared with the previously published data for two special cases: the transversely and normally biased ferrite slab. An excellent agreement is achieved. Three microwave devices are investigated using one numerical model that utilizes the derived Green's function by changing the magnetization angle. In the normal magnetization case, the ferrite substrate supports the edge-mode isolators. In the transversal magnetization case, the same structure supports the phase shifters. In the longitudinal magnetization case, the same structure supports magnetic surface wave transducers. The propagation direction in the three cases is kept in the longitudinal direction. A good agreement in all cases with previously published results is achieved. Here, we show only our analysis and results for the phase shifters and transducers, since edge isolator analysis was presented by Elshafiey et al. in 1996. In addition, the effect of the deviation of the magnetization direction from the assumed one is studied, and the improvement of the predicted results compared with experimental ones is shown.

We have calculated the Debye-Waller factor (DWF) of Cu from a model that was used successfully in earlier calculations of anharmonicity by Cowley and Shukla. The present calculation has been carried out using quasiharmonic theory, the lowest-order (λ2) anharmonic perturbation theory, and a Green's function (GF) method which sums an infinite series of the λ2−type anharmonic terms. The static approximation ω → 0 in the cubic contribution to the self-energy of the GF, introduced in the earlier work on the DWF by Shukla and Hubschle is further justified by showing that in the high-temperature limit the exact results for the λ2 anharmonic contributions (cubic and quartic) to the Helmholtz free energy are given in this approximation. Results for the DWF are also obtained for a modified version of the Morse potential with λ2 perturbation theory (PT) and the GF method. The GF results are in excellent agreement with the experimental Mossbauer and X-ray data in the entire temperature range, 300 K  T  120...

We present a new charge self-consistent scheme combining Density Functional\nand Dynamical Mean Field Theory, which uses Green's function of multiple\nscattering-type. In this implementation the many-body effects are incorporated\ninto the Kohn-Sham iterative scheme without the need for the numerically\nill-posed analytic continuation of the Green's function and of the self-energy.\nThis is achieved by producing the Kohn-Sham Hamiltonian in the sub-space of\ncorrelated partial waves and allows to formulate the Green's function directly\non the Matsubara axis. The spectral moments of the Matsubara Green's function\nenable us to put together the real space charge density, therefore the charge\nself-consistency can be achieved. Our results for the spectral functions\n(density of states) and equation of state curves for transition metal elements,\nFe, Ni and FeAl compound agree very well with those of Hamiltonian based\nLDA+DMFT implementations. The current implementation improves on numerical\naccuracy, requires a minimal effort besides the multiple scattering formulation\nand can be generalized in several ways that are interesting for applications to\nreal materials.\n

The exact reduced density-matrix functional is derived from the Luttinger-Ward functional of the single-particle Green's function. Thereby, a formal link is provided between diagrammatic many-body approaches using Green's functions on the one hand and theories based on many-body wave functions on the other. This link can be used to explicitly construct approximations for the density-matrix functional that are equivalent to standard diagrammatic re-summation techniques and to non-perturbative dynamical mean-field theory in particular. Contrary to functionals of the Green's-function, the exact density-matrix functional is convex and thus provides a true minimum principle which facilitates the calculation of the grand potential and derived equilibrium properties. The benefits of the proposed Green's-function-based density-matrix functional theory for geometrical structure optimization of strongly correlated materials are discussed.

An acceleration of continuous time quantum Monte Carlo (CTQMC) methods is a potentially interesting branch of work as they are matchless as impurity solvers of a density functional theory in combination with a dynamical mean field theory approach for the description of electronic structures of strongly correlated materials. The inversion of the $k \times k$ matrix given by the diagram expansion order $k$ in the CTQMC update and the multiplication of the $k \times k$ matrix and the non-interacting Green's function to measure the impurity Green's function are computationally time-consuming. Here, we propose the CTQMC method in combination with a machine learning technique, which would eliminate the need for multiplication of the matrix with the non-interacting Green's function. This method predicts the accurate impurity Green's function and double occupancy at low temperature, and also considers the physical properties of high Matsubara frequency in a much shorter computational time than the conventional CTQMC method.

With a superhigh-efficiency numerical algorithm, we are able to self-consistently calculate Green's function in the renormalized-ring-diagram approximation for a two-dimensional electron system with long-range Coulomb interactions. The obtained ground-state energy is found to be in excellent agreement with that of the Monte Carlo simulation. The numerical results of the self-energy, the effective mass, the distribution function, and the renormalization factor of Green's function for the coupling constants in the range $0\ensuremath{\leqslant}{r}_{s}\ensuremath{\leqslant}30$ are also presented.

We present a quantum embedding theory called dynamical configuration interaction (DCI) that combines wave function and Green's function theories. DCI captures static correlation in a correlated subspace with configuration interaction and couples to high-energy, dynamic correlation outside the subspace with many-body perturbation theory based on Green's functions. DCI takes the strengths of both theories to balance static and dynamic correlation in a single, fully ab initio embedding concept. The theory adds dynamic correlation around a fixed active space of orbitals with efficient $\mathcal{O}({N}^{5})$ scaling, while maintaining a multireference treatment of the active space. We show that treating high-energy correlation up to the $GW$ and Bethe-Salpeter equation level is sufficient even for challenging multireference problems. Our theory treats ground and excited states on equal footing, and we compute the dissociation curve of ${\mathrm{N}}_{2}$, the vertical excitation energies of small molecules, and the ionization spectrum of benzene in excellent agreement with high-level quantum chemistry methods and experiment.

In this article, we investigate the dyadic Green's function (DGF) of a perfect electromagnetic conductor sphere (PEMCS) due to electric dipoles, theoretically by employing the scattering superposition principle (SSP) and the Ohm-Rayleigh method. Then, the derived DGF is used to calculate the scattered field of a PEMCS due to an arbitrarily oriented infinitesimal electric dipole and also plane wave illumination. Numerical results are compared with those obtained by the FEKO commercial software to demonstrate the efficiency and accuracy of the proposed method. The results are in excellent agreement.