X-ray diffraction analysis of matter taking into account the second harmonic in the scattering of powerful ultrashort pulses of an electromagnetic field

Abstract

It is well known that the scattering of ultrashort pulses (USPs) of an electromagnetic field in the X-ray frequency range can be used in diffraction analysis. When such USPs are scattered by various polyatomic objects, a diffraction pattern appears from which the structure of the object can be determined. Today, there is a technical possibility of creating powerful USP sources and the analysis of the scattering spectra of such pulses is a high-precision instrument for studying the structure of matter. As a rule, such scattering occurs at a frequency close to the carrier frequency of the incident USP. In this work, it is shown that for high-power USPs, where the magnetic component of USPs cannot be neglected, scattering at the second harmonic appears. The scattering of USPs by the second harmonic has a characteristic diffraction pattern which can be used to judge the structure of the scattering object; combining the scattering spectra at the first and second harmonics therefore greatly enhances the diffraction analysis of matter. Scattering spectra at the first and second harmonics are shown for various polyatomic objects: examples considered are 2D and 3D materials such as graphene, carbon nanotubes, and hybrid structures consisting of nanotubes. The theory developed in this work can be applied to various multivolume objects and is quite simple for X-ray structural analysis, because it is based on analytical expressions.

Diagnostics of Nanosystems with the Use of Ultrashort X-Ray Pulses: Theory and Experiment (Brief Review)

“…For example, we present the calculation of scattering spectra for the Gaussian pulse at τω 0 ≫ 1. Similar to the preceding section, as shown in [103], scattering occurs at the first, ω 0 , and second, 2ω 0 , harmonics and this circumstance can be used in XRDA for a more accurate study of the structure of matter. In particular, the factor for graphene was calculated in the following analytical form [103,105] and the scattering spectra are shown in Fig.…”

“…It includes the coherent and incoherent components in the scattering spectrum, magnetic component, and dependence on the characteristics of the USP and is applicable for any number of electrons (scatterers) in the system. As shown in [103], Eq. ( 16) applied to the USP with a long duration τ (e.g., to the multicycle pulse, i.e., τω 0 ≫ 1, where ω 0 is the carrier frequency of the USP) disregarding the magnetic component of the USP is transformed to the known expression for XRDA.…”

“…In this section, we consider the scattering of X-ray USPs by complex nanosystems [99,102,103]. The wavefunction of the multiatomic system is taken in the form of Eq.…”

“…As shown in [91,102], can be represented in terms of the electron density in the atom. The resulting expression for the scattering spectrum was obtained in the form [102,103] (16)…”

“…( 16) is applicable to the multiatomic system consisting of one type of atoms. Expressions for both functions and can be obtained in the analytical form using the Dirac-Hartree-Fock model [104] for any shape of the USP, but these expressions are lengthy [102,103]. As a result, Eq.…”