**Research**

Our interests are mainly in studies of materials' properties by diffraction methods. We focus on:

- Strain and defect determination through the measurement and modeling of diffraction line broadening
- Development of methods for analysis of residual strain/stress, texture, and defects in materials
- Simple programs for analysis of diffraction line broadening are also available

Materials currently of interest include:

We recently acquired Bruker D8 diffractometer with GADDS area detector. Research equipment available at the nearby National Institute of Standards and Technology (NIST) in Boulder is also used for experimental work.

We studied changes of residual stress and defects that occur upon poling of ferroelectric BaTiO_{3} polycrystals by high-resolution synchrotron-radiation diffraction. BaTiO_{3} polycrystals are studied increasingly because the particle size, which has important influence on structure and physical properties, is controlled easily. The existence of a distinct severely textured and strained surface layer was detected. Most likely, this layer has the same tetragonal structure as the main fraction. Diffraction-line-broadening analysis shows large microstructural changes, especially along the direction of spontaneous-polarization and poling-field vector. The inhomogeneous strain upon poling is about the same order of magnitude as the strain caused by electrostriction during poling and indicates a substantial increase of the dislocation density. Dislocations play an important role in bulk microstructural changes that occur upon poling. The dislocation density is on the order of 10^{9}/cm^{2} and an associated strain-energy increase of about 20 kJ/m^{3}. This implies that the application of an external poling field generates defects in the structure and increases the internal stress. The increase of internal stress influences the ferroelectric and phase-transition temperatures. Moreover, the increase of both internal stress and defect concentration may have adverse consequences on both polycrystalline and epitaxial thin films through the accelerated degradation of dielectric properties. Because some dislocation reactions lead to microcracking, this may even result in the mechanical failure.

- D. Balzar, P. W. Stephens, H. Ledbetter, E. T. Park, J. Routbort,
**Dislocation-Density Changes upon Poling of Polycrystalline BaTiO**_{3},*Physical Review B***59**(1999) 3414-3420.

We proposed an extension to the phenomenological thermodynamic Landau-Devonshire theory to include the contribution of inhomogeneous strains caused by lattice defects to the Gibbs free energy. The model yields correction terms for dielectric and ferroelectric quantities as a function of both elastic misfit strain and defectrelated strain that can be measured by x-ray-diffraction techniques. We compare the correction in Curie-Weiss temperature due to elastic and inhomogeneous strain in pristine, W and Mn 1% doped Ba0.6Sr0.4TiO3 thin films grown on the LaAlO3 substrate. If the contribution of inhomogeneous strain is included, the agreement with measurements markedly improves.

- D. Balzar, P.A. Ramakrishnan, A.M. Hermann,
**Defect-Related Lattice Strain and the Transition Temperature in Ferroelectric Thin Films**,*Physical Review B***70**(9) (2004) 092103.

We studied pristine, W and Mn 1% doped Ba0.6Sr0.4TiO3 epitaxial thin films grown on the LaAlO3 substrate that were deposited by pulsed-laser deposition (PLD). Dielectric and ferroelectric properties were determined by the capacitance measurements and X-ray diffraction was used to determine both residual elastic strains and defect-related inhomogeneous strains by analyzing diffraction line shifts and line broadening, respectively.We found that both elastic and inhomogeneous strains are affected by doping. This strain correlates with the change in Curie-Weiss temperature and can qualitatively explain changes in dielectric loss. To explain the experimental findings, we model the dielectric and ferroelectric properties of interest in the framework of the Landau-Ginzburg-Devonshire thermodynamic theory. As expected, an elastic-strain contribution due to the epilayer-substrate misfit has an important influence on the free-energy. However, additional terms that correspond to the defect-related inhomogeneous strain had to be introduced to fully explain the measurements.

- D. Balzar, P. A. Ramakrishnan, P. Spagnol, S. Mani, A.M. Hermann, M.A. Matin,
**Influence Of Strains And Defects On Ferroelectric And Dielectric Properties Of Thin-Film Barium-Strontium Titanates**,*Japanese Journal of Applied Physics***41**(2002) 6628-6632.

**Nanocomposites for Biomedical Applications**

We examined the storage and release of internal stresses in shape memory polymers reinforced with a dispersion of nanometer-scale SiC particles. A quantitative Rietveld analysis of diffraction peaks was used to measure changes in the lattice parameter of the SiC particles after permanent deformation at 25 °C, and subsequent shape recovery induced by heating to 120 °C. Under 50% compression of the composite material, the nanoparticles store a finite compressive stress, which is almost completely released during heated strain recovery. The values of the stored internal stresses in the particles are compared to values based on micromechanic calculations.

- K. Gall, M. L. Dunn, Y. Liu, G. Stefanic, D. Balzar,
**Internal Stress Storage in Shape Memory Polymer Nanocomposites**,*Applied Physics Letters***85**(2004) 290-292.

We studied the structure and electronic properties of polymer-derived silicoboron–carbonitride ceramics. Structural analysis using radial-distribution-function formalism showed that the local structure is comprised of Si tetrahedra with B, C, and N at the corners. Boron doping of SiCN leads to enhanced p-type conductivity. The conductivity variation with temperature for both SiCN and SiBCN ceramics shows Mott’s variable range hopping behavior in these materials, characteristic of a highly defective semiconductor. The SiBCN ceramic has a low, positive value of thermopower, which is probably due to a compensation mechanism.

- P.A. Ramakrishnan, Y.T. Wang, D. Balzar, Linan An, C.
Haluschka, R. Riedel, A.M. Hermann,
**Silicoboron-carbonitride ceramics: A class of high temperature, dopable electronic materials**,*Applied Physics Letters***78**(2001) 3076-3078.

**METHODS**

**Strain and defects
by analyzing diffraction-line broadening**

Many crystal-lattice imperfection cause diffraction-line broadening. Dislocations and special arrangements of point defects (vacancies, interstitials, and substitutions) manifest themselves through the lattice strain. If a crystal is broken into smaller incoherently diffracting domains by dislocation arrays, stacking faults, twins, or another extended imperfections, then domain-size broadening occurs. By analyzing angle dependence of the line broadening it is possible to distinguish and quantify these defects. However, although the understanding for these basic facts exists for a relatively long time, there is no single coherent theory which would be applicable irrespectively of crystalline symmetry, degree of diffraction-line broadening, and so on.

The following review explains more thoroughly the current status in this field:

- D. Balzar,
**Voigt-Function Model in Diffraction Line-Broadening Analysis**, in*Defect and Microstructure Analysis from Diffraction*, edited by R.L. Snyder, H.J. Bunge, and J. Fiala, International Union of Crystallography Monographs on Crystallography No. 10 (Oxford University Press, New York, 1999) pp. 94-126.

Phenomenological approaches to line-broadening analysis assume that the size-broadened and strain-broadened line profiles can be approximated by simple analytical functions. During decades of research, it became more and more obvious that neither Lorentz (Cauchy) nor Gauss functions can adequately model either size or strain effect. However, the Voigt-function (a convolution of Lorentz and Gauss functions), as a model for both size-broadened and strain-broadened profiles ("double-Voigt" method), may be more realistic and accurate. On assumption of Gaussian distribution of strains, the relationship between parameters obtained by the Warren-Averbach approximation and integral-breadth methods becomes possible. Moreover, some common occurrences in Warren-Averbach analysis, particularly the "hook" effect, functional dependence of mean-square strain on averaging distance, and ratio of volume-weighted to the are-weighted domain size, all follow from the "double-Voigt" model.

The "double-Voigt" model is covered in the following publications:

- D. Balzar, H. Ledbetter,
**Voigt-Function Modeling in Fourier Analysis of Size- and Strain-Broadened X-ray Diffraction Peaks**,*Journal of Applied Crystallography***26**(1993) 97-103.

- D. Balzar, H. Ledbetter,
**Accurate Modeling of Size and Strain Broadening in the Rietveld Refinement: the "Double-Voigt" Approach**,*Advances in X-ray Analysis***38**(1995) 397-404.

Computer program BREADTH can be used to analyze line broadening according to the "double-Voigt" method.

*Elastic
strain and stress through Rietveld refinement*

The complete texture information allows calculations of orientation-distribution function (ODF) and consequent weighting of monocrystal elastic constants. The polycrystalline elastic constants are needed to calculate accurate elastic stresses from the measured elastic strains. A simultaneous determination of complete strain tensor and texture is possible by Rietveld refinement of diffraction measurements collected at different specimen orientations. Therefore, all the strain and texture parameters can be obtained along with structural, microstructural, compositional, and other information for all the crystalline phases. The method is generally applicable and especially suitable for determination of stress and texture in multiphase materials.

The method to obtain strain tensor is explained in the following publication:

- D. Balzar, R.B. Von Dreele, K. Bennett, H. Ledbetter,
**Elastic-Strain Tensor by Rietveld Refinement of Diffraction Measurements**,*Journal of Applied Physics***84**(1998) 4822-4833.

The following publication proposes a method to determine texture-weighted strain/stress orientation distribution function and average strain/stress tensors by Rietveld refinement for arbitrary crystal symmetry:

- N.C. Popa, D. Balzar,
**Elastic strain and stress determination by Rietveld refinement: Generalized treatment for textured polycrystals for all Laue classes**,*Journal of Applied Crystallography***34**(2001) 187-195.