To Thermal Diffuse Scattering Research Articles

Thermal vibrations in the inversion of dynamical electron scattering

Relativistic forward scattering of electrons at finite temperature involves the incoherent superposition of diffraction patterns formed by different snapshots of thermal atomic displacements. In experiments, thermal vibrations lead to thermal diffuse scattering (TDS), partly dominating diffraction patterns of thick specimens. This study sheds light on the effects of TDS on solutions to the inverse scattering problem using combined real- and diffraction-space information acquired in a scanning transmission electron microscope (STEM) to retrieve the object's phase. Using frozen phonon multislice within the Einstein approximation, realistic ground truth data of 20-nm-thick SrTiO3 is generated and subjected to contemporary inverse multislice schemes to retrieve the projected Coulomb potential slicewise. We first classify phase retrieval algorithms as to their assumptions on periodicity along the incident beam direction, as well as pixelwise and parametrized reconstruction methods. It is found that pixelwise object reconstructions are capable of retrieving structural details qualitatively while being prone to contain TDS-related artifacts which can result in unphysical potentials. For pixelwise reconstructions of multiple independent specimen slices, we observe that the origin of TDS, i.e., thermal atomic displacements, starts to emerge naturally. However, the quantitative assessment tends to too small mean squared thermal displacements, also when reconstructing multiple object modes. Using an atomistically parametrized inversion strategy which exploits the explicit separation of thermal vibrations and potentials, temperature and chemistry of the specimen can be retrieved quantitatively with high accuracy. Published by the American Physical Society 2024

Diffuse scattering study of aspirin forms (I) and (II)

Full three-dimensional diffuse scattering data have been recorded for both polymorphic forms [(I) and (II)] of aspirin and these data have been analysed using Monte Carlo computer modelling. The observed scattering in form (I) is well reproduced by a simple harmonic model of thermally induced displacements. The data for form (II) show, in addition to thermal diffuse scattering (TDS) similar to that in form (I), diffuse streaks originating from stacking fault-like defects as well as other effects that can be attributed to strain induced by these defects. The present study has provided strong evidence that the aspirin form (II) structure is a true polymorph with a structure quite distinct from that of form (I). The diffuse scattering evidence presented shows that crystals of form (II) are essentially composed of large single domains of the form (II) lattice with a relatively small volume fraction of intrinsic planar defects or faults comprising misoriented bilayers of molecular dimers. There is evidence of some local aggregation of these defect bilayers to form small included regions of the form (I) structure. Evidence is also presented that shows that the strain effects arise from the mismatch of molecular packing between the defect region and the surrounding form (II) lattice. This occurs at the edges of the planar defects in the b direction only.

A practical approach for STEM image simulation based on the FFT multislice method

It has been demonstrated that a high-angle annular dark-field (HAADF) STEM technique gives an image resolving atomic columns. Due to the diffusion of this technique and an improvement of its resolution, a practical procedure for image simulation becomes important for a quantitative interpretation of the HAADF image. In this report a new practical scheme for a STEM image simulation is developed based on the FFT multislice algorithm. Here, a HAADF intensity due to thermal diffuse scattering (TDS) is calculated from the absorptive potential corresponding to high-angle TDS and the wave function equivalent to the propagating probe within the sample. Contrary to the commonly used Bloch wave method, a coherent bright-field intensity and a coherent HAADF intensity are also obtained straightforwardly. The HAADF image contrast calculated for GaAs is not simply proportional to Z 2 as expected from the Rutherford scattering at high-angle, and the As/Ga contrast ratio depends on the specimen thickness. This suggests that the generation of the HAADF signal is appreciably affected by the coherent dynamical scattering. The developed procedure here will have a definitive advantage over the Bloch wave approach for simulating the HAADF images expected from a defect and interface or amorphous materials, and also the HAADF image obtained by using a Cs-corrected microscope. This is because the former requires a huge super cell, while the latter needs a large objective aperture including a large number of incident beam directions.

Surface Contribution to Inelastic Scattering of X-Rays on Thermal Phonons

The Green's function formalism is applied to the analysis of the surface contribution to the inelastic scattering of X-rays by acoustic modes of thermal excitations in a crystal. Crystals with a cubic unit cell are analysed. It is shown that the surface contribution to thermal diffuse scattering (TDS) near the surface can become comparable to and even can exceed the volume one. The same general formalism is applied to the investigation of TDS in the case of dynamical diffraction, when an X-ray standing wave is generated in the crystal.

Surface Contribution to Inelastic Scattering of X-Rays on Thermal Phonons

The Green's function formalism is applied to the analysis of the surface contribution to the inelastic scattering of X-rays by acoustic modes of thermal excitations in a crystal. Crystals with a cubic unit cell are analysed. It is shown that the surface contribution to thermal diffuse scattering (TDS) near the surface can become comparable to and even can exceed the volume one. The same general formalism is applied to the investigation of TDS in the case of dynamical diffraction, when an X-ray standing wave is generated in the crystal.

Lyotropic bicontinuous cubic phase single crystals investigated using high-resolved X-ray scattering

Single crystals of an Ia d bicontinuous direct cubic phase formed by a non-ionic surfactant in water are investigated using high-resolved X-ray diffraction. The shape of the Bragg peaks confirms the existence of a 3D long-range order inside the cubic phase. A weak diffuse scattered intensity signal is measured very near the Bragg peaks. We attribute this signal to thermal diffuse scattering (TDS) and we give an estimation of the contribution of elastic waves to this TDS.

A parametrization of inelastic scattering factors for high-energy electron diffraction

Theoretical images of model structures play a vital role in the analysis of both diffraction and phase contrast electron microscope images. In the theoretical calculations required to produce these images it is important to take absorption, due to inelastic scattering, into account. This is done by including an imaginary part in the Fourier coefficients of the lattice potential, V’g. These coefficients have been presented in a variety of ways. Radi has calculated the absorptive potentials for a number of crystals. He found that the most important contribution to these potentials is due to thermal diffuse scattering (TDS), except for the V’0 term, which does not affect image contrast. The Einstein model, in which all atoms are assumed to vibrate independently of one another, is normally used in calculating the contribution of TDS to the absorption process. This approximation allows one to calculate the absorption potential as a sum of contributions due to the different atoms in the crystal.

The effect of the surface on thermal diffuse intensities in reflection high energy electron diffraction

The contributions of bulk and surface vibrational excitations to thermal diffuse scattering (TDS) in reflection high energy electron diffraction (RHEED) patterns are investigated. It is shown that the surface phonon contribution to the differential cross-section for TDS is comparable to that from bulk phonons. Many-beam dynamical TDS RHEED calculations are presented, and both the positions and heights of diffuse peaks are shown to differ appreciably from predictions of kinematical TDS RHEED theory

Quantitative TDS absorption corrections for LACBED patterns from GaP and InAs

Large-angle convergent-beam electron diffraction (LACBED or Tanaka) patterns have been measured at 200 keV and room temperature for the 〈332〉 zone axis for the III–V semiconductors GaP and InAs. Bright-field LACBED patterns exhibit substantial differences in contrast. We demonstrate that a quantitative Einstein model for absorption due to thermal diffuse scattering (TDS) provides the most important contribution to differences in diffraction contrast rather than structure factor variations due to differences in atomic number. The absorptive potential for InAs is roughly a factor of two larger than for GaP.

Quantitative absorption corrections for electron diffraction: Correlation between theory and experiment

The absorptive potential due to thermal diffuse scattering (TDS) of fast electrons is calculated using an Einstein model. The TDS potential is substantially different from the elastic potential, being extremely peaked on atomic positions with virtually no absorption in the channels between atomic planes. This absorptive potential is used to determine imaginary (absorptive) components of the eigenvalues for each branch of the dispersion surface, using a Bloch wave description for propagating the fast electron wavefunction through a crystal. Large-angle convergent-beam electron diffraction (LACBED) contrast is calculated and compared with experiment using a 300 keV beam for various zone axes of silicon, GaAs and spinel. Changes in mean free paths for TDS under strong dynamical diffraction conditions as a function of crystal orientation are calculated, showing strong channelling/blocking characteristics due to the extremely localized TDS potential. The usual model for taking dynamical absorption effects into account, i.e. scaling the absorptive potential to the elastic potential, has severe limitations and fails to quantitatively predict LACBED contrast. We demonstrate that the Einstein model provides quantitative agreement with experiment, without the use of fitted parameters. Theory is also correlated with convergent-beam diffraction patterns, where rockin-curve contrast is used to determine the polarity of a multiply twinned epilayer of CdTe. This model is also applied to the thickness fringe method for determining the composition of AlGaAs quantum wells. Thickness fringe calculations using exact matrix methods for determining complex eigenvalues are compared with perturbative methodes.

Temperature factor of silicon by powder neutron diffraction

The temperature factor of silicon has been determined by the powder neutron diffraction technique employing a double-axis neutron diffractometer. A neutron wavelength of 1.184 Å was used in the experiment. The sample used was a fine powder of silicon of purity 99.999%. The correction to the observed intensities due to thermal diffuse scattering (TDS) was not applied as the neutron velocity of 3.34 km s-1 (corresponding to neutron wavelength of 1.184 Å) is less than the minimum velocity of sound in this crystal. The B value obtained from these experiments was found to be 0.45 (2) Å, corresponding to a mean-square vibrational amplitude of 0.017 (2) Å2 and to a Debye temperature of 531 (11) K at the sample temperature of 284 K at which the experiment was performed.

Solute-Lattice Coupling Parameters Determined from an Analysis of Diffuse Scattering of a β-Brass

The diffraction theory evolved by Matsubara, Kanzaki and Krivoglaz for solid solutions has been extended so that it may be applied to thermal diffuse scattering (TDS) from an ordered phase. The TDS intensity is calculated by using the extended theory on a β-brass (Cu–47.5 at%Zn), which shows strong diffuse streaks along the 110 rel. direction in X-ray diffraction patterns taken at room temperature. As a result, the diffuse streaks are shown to occur primarily by the soft TA2 phonon mode in the B2 type ordered lattice of the β-brass. Intensity measurements of diffuse streaks have also been made by using a single crystal of the β-brass. On subtraction of the calculated TDS intensity from the measured one, solute-lattice coupling parameters are obtained to the third neighbor of atoms. From the solute-lattice coupling parameters, three elastic constants, c11, c12 and c44, of the β-brass are estimated, and they are in good agreement with those obtained by an ultrasonic method.

Choice of scans in X-ray diffraction

In a recent paper (Werner, S. A. Acta Cryst. (1971). A27, 665) it was pointed out that the choice of scans in a neutron diffraction experiment should be based on the criterion that the diffracted beam enters the detector on its centerline for each angular setting of the crystal. The same criterion should be applied in X-ray diffraction. Since the spectral distribution of a source of X-rays and neutrons is quite different, conclusions regarding the optimum coupling between the detector and crystal motions are different in these two cases. In this paper, formulas are derived (within the framework of certain gaussian approximations) for the optimum scanning ratio g in equatorial plane X-ray diffraction experiments on single crystals. For the case when a monochromator is not used, g is independent of scattering angle 2θB for a large range of instrumental parameters and Bragg angles θB. It is found that a θ–2θ scan is essentially never advisable. An expression for g is derived for the case when a planar monochromator is used in symmetric Bragg reflection. The optimum scan is found to depend on the scattering angle, but not in such a marked way as in the neutron case. Coupling the detector and crystal motions in the manner suggested allows one to decrease the acceptance aperture to its minimum width, thus keeping the background due to thermal diffuse scattering (TDS) and incoherent scattering as low as possible.