Form Factor Amplitude Research Articles

A thermal form factor series for the longitudinal two-point function of the Heisenberg–Ising chain in the antiferromagnetic massive regime

We consider the longitudinal dynamical two-point function of the XXZ quantum spin chain in the antiferromagnetic massive regime. It has a series representation based on the form factors of the quantum transfer matrix of the model. The nth summand of the series is a multiple integral accounting for all n-particle–n-hole excitations of the quantum transfer matrix. In previous works, the expressions for the form factor amplitudes appearing under the integrals were either again represented as multiple integrals or in terms of Fredholm determinants. Here, we obtain a representation which reduces, in the zero-temperature limit, essentially to a product of two determinants of finite matrices whose entries are known special functions. This will facilitate the further analysis of the correlation function.

Efficient computation of the scattering intensity from systems of nonspherical particles

The analysis of the angle dependence of the elastic scattering of radiation from a sample is an efficient and non-invasive technique that is used in fundamental science, in medicine and in technical quality control in industry. Precise information on the shape, size, polydispersity and interactions of a colloidal sample is readily obtained provided an underlying scattering model, i.e. form and structure factors, can be computed for the sample. Here, a numerical method that can efficiently compute the form factor amplitude (and thus the scattering intensity) of nonspherical scatterers through an importance sampling algorithm of the Fourier integral of the scattering density is presented. Using the precomputed form factor amplitudes, the calculation of the scattering intensity at any particle concentration then scales linearly with the particle number and linearly with the number of q points for its evaluation. This is illustrated by an example calculation of the scattering by concentrated suspensions of ellipsoidal Janus particles and the numerical accuracy for the computed form factor amplitudes is compared with analytical benchmarks.

Three-dimensional structure of electron density fluctuations in the Hall thruster plasma: E¯×B¯ mode

Collective scattering measurements have been conducted on the plasma of a Hall thruster, in which the electron density fluctuations are fully characterized by the dynamic form factor. The dynamic form factor amplitude distribution has been measured depending on the k-vector spatial and frequency components at different locations. Fluctuations are seen as propagating waves. The largest amplitude mode propagates nearly along the cross-field direction but at a phase velocity that is much smaller than the E¯×B¯ drift velocity. Refined directional analysis of this largest amplitude mode shows a thin angular emission diagram with a mean direction that is not strictly along the E¯×B¯ direction but at small angles near it. The deviation is oriented toward the anode in the (E¯,E¯×B¯) plane and toward the exterior of the thruster channel in the (B¯,E¯×B¯) plane. The density fluctuation rate is on the order of 1%. These experimentally determined directional fluctuation characteristics are discussed with regard to the linear kinetic theory model and particle-in-cell simulation results.

Multipole decomposition of the elastic atomic photon scattering form factor and the anomalous scattering factor

The elastic atomic photon scattering form factor amplitude Af0, obtained from the high energy and low momentum transfer (K) limit of the exact (within the independent particle approximation (IPA)) Rayleigh relativistic second order S-matrix scattering amplitude Asm is decomposed into an infinite sum of its electric and magnetic field multipoles. The energy-dependent part of Af0 (the scattering angle-dependent part of Af0 is incorrect due to the low K approximation), combined with the angle-dependent part of Asm, yields a form factor amplitude Af (different versions of Af and Af0 will be described in the text) that can be used to decompose the anomalous scattering factor Aasf into its multipoles. Af0 which initially contains coherent and incoherent factors is reduced to a single sum over the multipoles of only coherent terms by assuming that the charge density is spherically symmetric. At photon energies up to about ten times the K-shell binding energy it takes up to quadrupole Aasf for light atoms of Z ≤ 20 and up to the fourth multipole for heavy atoms such as uranium for an accurate Aasf (within IPA). The multipole contributions were found to become somewhat larger as the scattering angle becomes larger (greater than 120°); however there were partial cancellations of such effects.

Elastic scattering of gamma rays and X-rays

Studies of elastic gamma ray scattering were pursued independently by the groups of Prof. Ghose and the author for several decades in spite of somewhat meagre support. Several techniques for such studies developed by the two groups and some of the results obtained in the energy range from tens of keV to about 1.5 MeV are described briefly. The theoretical background necessary for understanding these results is also outlined. The results showed the importance of Modified Relativistic Form Factor (MRFF) approximation in the explanation of atomic Rayleigh scattering cross sections in the small angle regime and the necessity for an inclusion of real Delbrück scattering amplitudes at large scattering angles. Dispersion corrections to form factor amplitudes or the so-called anomalous scattering factors are shown to be needed at photon energies close to electron binding energy thresholds. A few promising future extensions of such studies are indicated at the end.

Elastic scattering of photons

Abstract Photon scattering from bound electrons of atoms, a component of the elastic photon-atom scattering amplitude called Rayleigh scattering, is discussed. General features of the many-body scattering amplitude and its partitioning into Rayleigh and Delbruck (and nuclear) single-electron transition scattering amplitudes are examined. The use of the state-of-the-art precise second-order S-matrix calculations of Rayleigh scattering in terms of single-electron transition-amplitudes has led to significant progress in our understanding of the scattering process. The importance of relativistic, higher multipole, and bound-bound contributions in calculating anomalous scattering factor deviations from form factor amplitudes must be emphasized. Accurate interpolation of cross sections in the three-dimensional space of scattering angle, photon energy, and atomic number, utilizing the available published S-matrix data, has permitted extensive tabulation of differential scattering cross sections. S-matrix results may be compared with experiments to assess their validity; they may be compared with simpler but more approximate approaches, to identify the extent of the utility of such approaches and to develop simpler prescription schemes which can give results comparable to the S-matrix results. In spite of their many successes, the present second order S-matrix methods also have limitations and shortcomings, observed in certain recent experiments. We describe the explanation of these experiments in terms of a composite theory which also incorporates non-local exchange and correlation effects. We end with a discussion of other future issues.

Elastic scattering of 152Eu and 154Eu γ-rays

The elastic scattering of 344, 779, 964, and 1408 keV γ-rays from a source containing 152Eu and 154Eu has been measured for scattering through 15°, 20°, 30° and 45° by targets of Mo, Sn, Ta, Pb and U. These measurements extend the region of our previous investigation to momentum transfers greater than 10 Å −1 where the validity of the modified form factor approximation in calculating total atom Rayleigh scattering amplitudes is questionable, and where the influence of Delbrück scattering is apparent for γ-ray energies above 1 MeV and for the larger Z targets. The experimental cross sections have been normalized to theoretical carbon Compton cross sections at 7° and agree closely with the recent theoretical elastic cross sections of Kane et al. for Rayleigh plus nuclear Thomson plus Delbrück scattering in which exact numerical partial wave inner shell Rayleigh amplitudes are used. The results are also in reasonable agreement with cross sections in which the total atom Rayleigh scattering is represented entirely by DHFS modified form factor amplitudes.

Elastic Scattering of 152Eu ? Rays by Tungsten

Differential cross sections for the elastic scattering of 344, 444, 779, 964, 1086, 1112 and 1408 keY y rays by tungsten have been measured for six scattering angles ranging from 7� to 45� and are compared with theoretical cross sections for Rayleigh plus nuclear-Thomson plus, for the 1408 keY case, Delbriick scattering. For small momentum transfers the experimental results are in best agreement with cross sections computed using the Hartree-Fock form factor Rayleigh amplitudes, but at large momentum transfers the experimental results lie below these theoretical cross sections and also those computed using both the 'exact' Rayleigh amplitudes of Kissel and Pratt and the Dirac point-Coulomb form factor amplitudes. Inclusion of the Papatzacos and Mork Delbruck amplitudes at 1408 keY reduces the difference between experimental and theoretical cross sections for scattering angles less than 25�, but worsens the situation for larger angles.:

Neutron Number Dependence of Interference in Two-Neutron Transfer Reactions and the Nuclear Deformation

By making use of a simple model for the nuclear deformation, the number dependence of the form factors for inelastic scattering and two-neutron transfer processes are discussed for even-even deformed nuclei in the rare earth region. The form factor amplitudes are expressed in terms of the Legendre polynomials, whose argument is a func­ tion of the number. It is easy to understand, in the perturbative sense, a number dependence of the interference between the direct and two-step processes observed in (p, t) reactions leading to the 0, + and 2, + states. The interference for the 2, + state in deformed nuclei shows a striking parallelism with that for the quadrupole vibrational state in spherical nuclei. It has been shown that the two-neutron transfer reactions on deformed nuclei 1l can be well described in terms of the coupled-channel Born approximation (CCBA), 2l which includes the effect of multi-step processes via inelastic scattering. It has also been dezywnstrated that the same is ;true in the case of the two-neutron transfer reactions leading to the collective vibrational state in spherical nuclei. 3l~sl The purpose of the present paper is to explain,. in the perturbative sense, the neutron number dependence of the interference between direct and two-step processes in the (p, t) and ( t, p) reactions leading to the ground band members in even-even deformed nuclei in the rare earth region. The explanation applies equally well to other deformed nuclei. The interference observed in the (p, t) reactions, leading to the 2 1 + state in the spherical vibrational nuclei has been treated by Udagawa and coworkers. 3l· 5l· 6l They have pointed.ouel.al in the frame­ work of the quasiparticle RPA that a number dependence of the form factor amplitudes originates from the dynamical role of the ground state correlations and the gross number dependence of the occupation coefficients vv and uu, and that the number dependence of the form factor amplitudes provides .an explanation to the sign change of the interference observed in the middle of a major shell. In § 2, the transition amplitude for the A(p, t)B reaction is given in a second­ order perturbative form of the CCBA description. In this procedure, it is shown that the amplitude of the core-excitation model studied by Kozlowsky et aFl is