Thermal Diffuse Scattering Intensity Research Articles

Implications of X-ray thermal diffuse scattering in integrated Bragg intensities of silicon and cubic boron nitride

This article describes a theoretical quantification of the thermal diffuse scattering (TDS) contribution to the integrated intensities of single-crystal diffraction from silicon and cubic boron nitride. The TDS intensity is calculated ab initio and the intensity contribution to the Bragg peak determined by integration of a volume of appropriate size. The effect of including TDS in the structure factors is assessed in a subsequent refinement, where the effect on the atomic displacement parameters and the residual Fourier maps is inspected. In both compounds changes caused solely by the TDS inclusion are observed. Particularly, adding TDS to the structure factors of silicon produces significant features in the residual Fourier maps and a change in the atomic displacement parameter. Significant effects are also present at low temperatures. Thus, it is important to address the effect of TDS when pursuing high-accuracy charge densities or if analyzing the atomic displacement parameters.

Coherence of k-space electrons: application to TDS electrons by DBI

Methods of controlled electron interference in k-space on the diffraction plane by means of an electron biprism invented during the Tonomura Electron Wavefront Project are briefly reviewed. The results presented show the partial coherence of self-interfered diffusely scattered electrons, elastically scattered and inelastically scattered, found outside and in between the Bragg diffracted beams often referred to as thermal diffuse scattering (TDS) of electrons. The interference fringes formed in the TDS intensity have been used to calculate a mean displacement of the atom of u ≈ 12 pm in Aluminum in the direction perpendicular to the Bragg planes.

Quantification of electron–phonon scattering for determination of temperature variations at high spatial resolution in the transmission electron microscope

This work investigates the quantification of electron–phonon thermal diffuse scattering (TDS) for detection of temperature variations with nanometer spatial resolution in transmission electron microscopy (TEM). Observations of TDS intensity for (100) single crystal Si and Ge show interdependences of temperature and sample thickness which can be understood through the angular distributions of electron–phonon scattering as a function of temperature. The temperature sensitivity of the integrated TDS intensity can be of the order of 10−3 K−1 for Si and Ge. This shows that measurement of the TDS intensity in the TEM is a promising means for nanoscale temperature measurement; our measurements to date have demonstrated that temperature changes as small as 5 K are detectable.

Simulations of Kikuchi patterns due to thermal diffuse scattering on MgO crystals.

Inelastic scattering of fast transmission electrons from a perfect crystal is investigated using the Bloch wave theory. A comprehensive expression for the scattering of electrons is given, which includes both elastic and inelastic multiple scatterings. This expression is an extended form of Fujimoto's expression for elastic scattering (J. Phys. Soc. Japan 14:1558 (1959)). For the approximation of single inelastic scattering, the expression becomes equivalent to the formula of Rez et al. (Phil. Mag. 35: 81 (1977)). When thermal diffuse scattering (TDS) is considered using the Einstein model or the scattering factor for TDS given by Hall and Hirsch (Proc. R. Soc. A 286: 158 (1965)), Rossouw and Bursill's expression (Acta Cryst. A 41: 320 (1985)) is derived. This expression has been used in computer simulations of TDS intensity distribution (Kikuchi pattern). It is shown that the simulations for magnesium oxide (MgO) using 357 beams agree quite well with the experimental ones.

Thermal diffuse X-ray scattering study of anharmonicity in cubic barium titanate

Two BaTiO 3 single crystals with one containing 0.3 at.% Fe 3+, substituted for Ti 4+, have been examined by in-situ X-ray thermal diffuse scattering (TDS) measurement above the Curie temperature (∼ 130°C). Results from the pure sample show two-dimensional TDS distribution on the {1 0 0}; they are attributed to the overdamped [0 1 0] TO mode and to the anisotropic on (1 0 0) TA modes. The TDS distribution and negative temperature dependence of the intensity near the zone center agree well with the previous neutron inelastic scattering results found in the literature. The TDS distribution from the doped sample showed a similar pattern to that of the pure sample. However, some differences were found at smaller wave vectors where a positive temperature dependence of the intensity is seen as well as a more isotropic distribution. The heterovalent doping is believed to have changed the lattice anharmonicity. The TDS intensity calculations have been carried out using the normal mode analysis under the harmonic approximation. The calculated results showed good agreement with the observed data at | q| > 0.25 for the two samples. The anharmonic correction was necessary, however, for the TDS intensity calculations near the zone center, i.e. | q| < 0.25 , for which the imaginary part of the response function for the [0 1 0] TO mode was used in the calculations. In curve fitting the damping constants of the pure sample were found slightly larger than those of the doped sample, implying a decrease of anharmonicity due to heterovalent doping. This effect may be understood using a hole/electron-relaxation coupling concept.

Detailed Calculations of Thermal Diffuse Scattering

Convergent beam electron diffraction (CBED) produces a diffraction pattern from a single point on the specimen using a focused probe in the scanning transmission electron microscope (STEM). Because the incident wavefunction is a focused probe, each diffraction spot is enlarged to a disk the size of the objective aperture. Thermal vibration in the specimen reduces the intensity of the diffraction disks and introduces scattering in between the disk. This normally forbidden scattering between the diffraction disks is referred to as thermal diffuse scattering (TDS). The effects are most pronounced at large scattering angles, where TDS scattering can greatly reduce the intensity of the higher order Laue zone (HOLZ) lines. Previous work on modeling the TDS intensity involved the so-called frozen phonon method and the Einstein model of lattice vibration. The Einstein model assumes that each atom in the specimen vibrates independently of every other atom in the specimen and possible correlations among adjacent atoms are ignored.

Evaluation of phonon scattering in electron diffraction using image plates and electron energy filtering

There have been few studies of thermal diffuse scattering (TDS) by electron diffraction although this scattering is easily observed. The TDS intensity distribution is as a rule strongly anisotropic which can be ascribed to scattering from individual phonon modes. Results reported so far in the literature are all based on qualitative or semi-quantitative use of the observed intensity distributions. However, at present several new methods (energy filters, imaging plate, CCD, digitized film) are available to extract data for detailed quantitative interpretation. In the present work the Zeiss 912 Omega energy filtering microscope has been used, combined with the Fuji Imaging Plate and a low temperature specimen holder. This opens new possibilities both for the study of soft modes and other TDS phenomena related to phase transitions, and also static disorder and order-disorder effects. By comparison with synchrotron work on TDS , crystals may be very much smaller and count rates are much higher. However, multiple scattering and Kikuchi lines may complicate the interpretation unless thin samples are used.

Pre-Transitional Behavior in Cubic Potassium Niobate

ABSTRACTPre-transitional behavior prior to the onset of the cubic-to-tetragonal phase transformation in KnbO3 has been studied with synchrotron x-ray scattering at NSLS, Brookhaven National Lab. Measurements of the Thermal Diffuse Scattering (TDS), Grazing Incidence X-ray Scattering (GIXS), and precursor Bragg scattering have been performed at a series of temperatures on a single crystal sample in the cubic phase (Tc = 405 °C). The TDS was measured in the (001), (011) and (111) cubic planes. The GIXS was measured above, at and below the critical scattering angle of the sample. Based on localized soft mode theory, dynamic tetragonal fluctuations may be expected prior to the onset of the phase transition initiated by defects or surfaces. The GIXS data indicate that the tetragonal phase is apparent prior to the phase transition of the bulk - - consistent with the notion that nucleation originates from potent defect sites during a martensitic-like displacive transformation. The TDS intensity distribution is disk-shaped; arising from a localized, low energy transverse optic mode in the [100] directions extending out to the Brillouin zone boundary (reduced wave vector, q =0.5). Scans made in the (011) planes show TDS intensity extending out to a reduced wave vector of q ˜ 0.2. Preliminary analysis indicates that a low energy transverse optic mode extends in the [011] direction as well. The TDS intensity decreases as the transition temperature is approached from above. Comparison of the TDS results with inelastic neutron scattering data will be made.

Modelling of Bragg intensity profiles. 2. Allowance for thermal diffuse scattering

A method for empirical determination of TDS (thermal diffuse scattering) corrections in X-ray structure analysis is developed. Thermal diffuse scattering is introduced into the model suggested earlier [Chulichkov et al. (1987). Sov. Phys. Crystallogr. 32, 649–653; Laktionov, Chulichkov, Chulichkova, Fetisov, Pyt'ev & Aslanov (1989). J. Appl. Cryst. 22, 315–320] for measuring the Bragg reflection intensity profile with a four-circle diffractometer. The improved model in combination with the mathematical reduction method enables the extraction of TDS and background intensity values from the experimental intensity profile. Thus the problem of TDS correction in crystal structure refinement is solved without knowledge of the sample's elasticity constants. The comparison of TDS corrections obtained by this method with those calculated from elasticity constants for hexamethylenetetramine, C6H2N4, and sulfur, S8, crystals shows good agreement.

First- and second-order thermal diffuse scattering (TDS) intensity in molecular crystals: influence on crystal structure parameters

First- and second-order thermal diffuse scattering (TDS) intensities are calculated in the long-wave approximation allowing for dispersion (LWD) in monoclinic phenothiazine from polarization vectors and lattice-mode frequencies obtained from lattice dynamical calculations within the harmonic approximation and the external Born-von Kármán formalism using an atom-atom potential function in the form V(r) = -A/r6 + B exp (-Cr). The influence of first- and second-order TDS intensity on electronic density maps is analysed and compared. Least-squares refinements of positional and thermal parameters are carried out in different ranges of sin θ/λ taking into account both first- and second-order TDS contributions and the results are discussed.

Lattice-dynamical calculation of second-order thermal diffuse scattering in molecular crystals

A computer procedure has been developed to calculate second-order thermal diffuse scattering (TDS) intensity for molecular crystals from lattice-dynamical calculations with an atom-atom potential in the Born-von Karman formalism. It is applied to monoclinic phenothiazine and different contributions to second-order TDS intensity, acoustic-acoustic, acoustic-optic and optic-optic, are compared. Calculations are also performed in the long-wave approximation allowing for dispersion (LWD) and correction factors of Bragg intensities due to TDS contribution in the LWD approximation are, generally but not always, lower than lattice-dynamical ones; the ratio between LWD and 'exact' factors ranges from 0.4 to 1.4 for reflections considered.

Errors in atomic parameters and in electron density distributions due to thermal diffuse scattering of X-rays

Errors in X-ray diffraction intensities due to first-order thermal diffuse scattering (TDS) have been calculated from the elastic constants of the crystal. Formulae given in the literature were generalized in such a way that they include quantum effects and zero-point vibrations. It is shown that, within the approximations used, the influence of quantum effects and zero-point vibrations is negligible above 5K if ordinary measuring conditions are applied. For C6H5COCOC6H5 and NH4HC2O4.½H2O an estimate is made of the influence of TDS intensity errors on least-squares parameters and electron density distributions at 15, 110 and 293K. A critical discussion of the long-wave method used is given.