7.3 Microdomains

In the context of this manual the term microdomain refers to any small region intergrown in the lattice of the host crystal. The authors are well aware of the fact that this does not match the exact crystallographic definition of a domain. A microdomain may consist of just a single vacancy with relaxed next neighbours or a completely different structure extending over several unit cells of the host lattice. A theory for diffuse scattering by correlated microdomains was introduced by [14]. Subsequently this theory was applied to the diffuse scattering of cubic stabilized zirconia [13,20,19]. DISCUS follows another path and offers the possibility to define different microdomain types and distribute these types throughout the crystal. The diffuse scattering is then calculated from the simulated crystal containing the microdomain distribution. Depending on the type of microdomain, the host atoms are removed and replaced by the microdomain atoms. The options are laid out in a very general fashion in order to facilitate generation of many different types of defects. With this technique, small defects, extended defects, anti phase domains, finite wave trains or an incoherent intergrowth of two phases can be generated.

7.3.1 Properties of microdomains

DISCUS uses several properties to characterize the individual microdomains and the distribution of microdomains throughout the crystal. The properties characterizing an individual microdomain are briefly listed below and discussed in somewhat more detail at the end of this section.

Four steps are necessary to modify a crystal by microdomains. First, a list of microdomain types must be set up. Secondly, the distribution of these microdomain types must be characterized. Once the microdomains and their distribution have been characterized, a list of microdomain origins covering the host crystal is set up in the third step. This list can still be edited by the user before, in the fourth step, replacing the structure inside the microdomains by the new structure. These steps can be repeated and after each step the previous steps could be repeated before continuing with the next step. For example a list of microdomain types might be created and the corresponding list of microdomain origins be created. A new list of microdomain types can now be defined as well as a new distribution type. After creating a second list of origins, the crystal structure is finally modified, using both lists of microdomain origins. The program does not delete the previous microdomain types. Instead the new types are added to the list. The first set, however, is not used again. This ensures that a unique microdomain type number exists.

7.3.1.0.1 Microdomain distributions :

The microdomains can be distributed throughout the crystal by three different distribution types. They may be distributed at random, on a perfect lattice or on a paracrystalline lattice. For the random distribution, an average density of microdomains per host unit cell is to be supplied. Microdomains are created within the crystal at random position until this density is reached. DISCUS checks for a possible overlap of microdomains. The procedure stops once the estimated number of microdomains has been created, or if the introduction of a new microdomain fails for ten consecutive trials. Note, that if the crystal is not limited by a block of unit cells the algorithm will give erroneous densities.

For the lattice types, the user has to supply the base vectors in units of the direct or reciprocal sublattice. A microdomain is introduced at every grid point of the super lattice. No need exists for any commensurate relationship between the two lattices. This type of distribution will create a perfectly periodic distribution of microdomains and subsequently sharp superlattice reflexions will appear.

In case of the paracrystalline distribution, the base vectors represent the average separation between neighboring microdomains. The actual separation is modified by a three dimensional Gaussian distribution with definable sigmas. At any grid point of the paracrystalline superlattice a microdomain is introduced. The position of a new microdomain is calculated from the positions of three previous microdomains. The first microdomain used is the microdomain separated by one time the superlattice base vector ${\bf a}$ in negative direction. The other two are separated by $-{\bf b}$ and $-{\bf c}$ respectively. The base vector ${\bf a}$ is added to the position of the first microdomain. A Gaussian distributed value is added to each of the three components of the vector. The same procedure is used for the second microdomain with respect to superlattice vector ${\bf b}$ and the third microdomain respectively. The average of the three positions is used as the position for the new microdomain. The distribution of microdomains grows from the negative most corner of the crystal to the positive most corner of the crystal. A paracrystalline distribution results that is characterized by an average separation between microdomains but no long range order. The method described, does not introduce defects into the paracrystalline distribution. Each unit cell of the distribution, though of oblique shape, consists of eight microdomains at the corners. No irregularly shaped unit cells are created, nor do dislocations appear.

DISCUS also $\star$ ] $\star$ allows a microdomain distribution to be read from a file. The file type is identical to the normal structure files and the atom type number is taken as type of the microdomain. This enables the user to e.g. generate a microdomain distribution using MC simulations (see section 9).

By default, the origin of the microdomain distribution coincides with the position 0,0,0 within the crystal. The user can define a vector that is added to each microdomain origin before it is inserted into the list. No constrains apply to this vector. Two or more microdomain distributions can be created with different shift and otherwise identical distribution.

The fit of the microdomain origins to the host lattice defines whether the microdomain origin can be at any position within the sublattice or whether it is constrained to integer (and centering) vectors of the sublattice. If constrained, the origin of each microdomain is shifted to the next nearest sublattice point. You can set the type of fit to both values for each distribution type, but the effects on the microdomain distributions will be different for each distribution type. In case of a random distribution, the microdomain origin shifts to the next nearest sublattice vector. Tests for overlap of microdomains are performed after the shift. In case of a lattice distribution, the initial position of each microdomain origin is calculated as an integer multiple of the superlattice base vectors. Each individual microdomain is then shifted to the next nearest sublattice vector without effecting the other microdomains. In case of the paracrystalline distribution the position of each microdomain is calculated from the actual position of three previous microdomains. If the fit type is set to coincide with the sublattice grid, the necessary shift of any one microdomain will effect the next microdomain as well.

A correlation matrix can be set up defining probabilities for two microdomain types to be next neighbors in case of the lattice or paracrystalline distribution. The correlation matrix is currently used as an isotropic correlation. For both distribution types, the type of a current microdomain is determined from the types of the three microdomains separated by $-{\bf a}$ , $-{\bf b}$ and $-{\bf c}$ respectively. The correlations 'grow' from the negative most corner of the crystal to the positive most corner. The correlation matrix C is internally normalized such that the sum of all elements is one. The element c_ij gives the probability of microdomain type j to be next neighbor to microdomain type i. The sum of a column c_i* gives the a priory probability for the first microdomain to be of type i. The three rows of the correlation matrix representing the three neighbors are averaged and the type of microdomain determined by random number generation weighted with the respective probabilities.

7.3.1.0.2 Microdomain properties :

For each microdomain type an origin is assumed. The positions of atoms are referenced with respect to this origin, and the origins of the actual microdomains are distributed throughout the crystal. The shape property, defined by the 'boundary' command defines the boundary surface between the microdomain and the surrounding host structure. Currently this boundary surface can be a sphere, a set of planes, a rectangular block of atoms or an irregular conglomerate of atoms limited by a fuzzy boundary. The boundary type 'sphere' creates a sphere of definable radius centered at the microdomain origin. Alternatively a microdomain type can be limited by a set of planes, type 'face'. The user must supply all faces, even symmetrically equivalent faces. The distance of each face from the origin can be set individually, which allows for a very flexible adjustment to different shapes. Since the distance is set with a separate command, the two set of faces (1 0 0) and (2 0 0) represent the identical surface, unless the distance is modified. DISCUS tries to check whether the form created by the faces is closed. The origin must be inside the microdomain, while the six points at 10⁹ along each of the three sublattice base vectors must be outside. This test is certainly not a foolproof algorithm, yet a reasonable compromise. An open face will include all atoms up to the limits of the crystal. The third boundary type, 'block', is a special case of the second, and identical to the only type offered in DISCUS version 1.0. The microdomain is limited by the six {100} faces. The distances to the origin are determined from the extend of the structure read. The block defined by these six faces is cut from the host crystal and replaced by the atoms read from the structure file. It is necessary that the content type of these microdomains is a structure. DISCUS has no way of finding out, whether the new structure that was read, completely fills up the block, or whether any voids are left. For this reason it is better to use the boundary type 'fuzzy'. The fourth type of microdomain boundaries, 'fuzzy', does not set up a definable surface. Instead the extend of the microdomain is solely defined through the atoms that are read from the structure file. As for shape type 'block' the content must be set to 'structure'. Contrary to type 'block', no voids are created. For all atoms of the host crystal the distance to all microdomain atoms is calculated. If any of these distances is less than a user definable value, the host atom is removed. In effect the boundary between host and microdomain becomes fuzzy and a microdomain of boundary type 'fuzzy' does not even have to be contiguous.

Closely associated with the boundary property is the radius of the boundary surface. This property applies only to boundary types 'sphere' and 'face'. For boundary type 'block' it is determined automatically and for boundary type 'fuzzy' it does not apply. For boundary type 'sphere', the radius gives the radius of the microdomain surface. For boundary type 'face', the radius gives the distance between the microdomain origin and the face. Internally, the hkl of each face are converted to new hkl that represent the distance from the microdomain origin. The average radius of the boundary surface is the same for all microdomains throughout the crystal. The individual radius can be subject to a Gaussian distribution with user definable sigma. All atoms within this boundary surface are considered to be part of the microdomain structure and are replaced accordingly.

The content of the microdomain can be a completely new structure or a modification of the present structure. If the content is defined as structure, a new structure is read from a file identical in format to the crystal structure file. The space group and lattice constants are currently ignored. All fractional coordinates are interpreted in multiples of the host lattice. This limitation will hopefully be corrected soon. The positions of the atoms are the sum of the microdomain origin and the fractional coordinates read from the file. If applicable, a symmetry operation is applied to the fractional coordinates before adding them to the microdomain origin. The program tests whether the positions are inside the host crystal and inserted if inside. Since DISCUS cannot test for left over VOIDs, the user has to make sure that the structure is large enough to cover the whole microdomain.

Each type of microdomains can further be characterized by the orientation of the microdomain structure. Symmetry matrices can be read that will transform the structure and the corresponding boundary surface. The symmetry operation includes both a rotational and a translational part, as described in section 5.2. This symmetry operation is applied to each microdomain atom prior to inserting it into the crystal. The rotation is equally well applied to the boundary of the microdomain, while the translation is not. After the symmetry operation has been applied, DISCUS tests whether the atom is inside the microdomain or not. A large translation can therefore move atoms out of the microdomain, and these atoms will not be inserted into the microdomain. If a shift of the microdomain is needed, move the origin of the microdomain with the shift property of the distribution.

7.3.2 Related variables

The crystallographic function 'md_test' and a number of variables listed in table 7.1 are associated with microdomains. Variables marked with '(ro)' are read only variables and can not be altered by the user.

Table 7.1: Variables related to microdomains

Variable	Description
md_num[1]	Number of different microdomains (ro).
md_cre[1]	Number of different microdomains already created with the 'create' command but not yet used to modify the crystal (ro).
mc_num[1]	Number of microdomains whose origins have been distributed throughout the crystal with the 'create' command (ro).
md_rad[<i>,<j>]	Radius of the microdomain type <i>. For boundary types 'block' and 'fuzzy' <j> must be 0, for type 'sphere' <j> must be 1 and for 'face' <j> is the number of the face the radius is returned for (ro).
mc_run[1]	Number of microdomains used to modify the crystal with the 'run' command (ro).
mc_type[<i>]	Returns the microdomain type for microdomain <i> (ro).
mc_orig[i,<j>]	Coordinates for microdomain <j> (i=1,2,3 for x,y,z). This variable can only be modified for microdomains <j> which have been created but not used to modify the crystal via the command 'run'.
mc_rad[<i>]	Individual radius for microdomain <i>. This variable can only be altered for microdomains <i> not used with the 'run' command before.
md_next[<j>]	Number of the microdomain nearest to atom <i>. Negative numbers indicate that the atom is within the microdomain, positive numbers are for atoms outside.
md_dist[<i>]	Distance of the atom <i> to the surface of the nearest microdomain. If the number is negative, the atom is inside the microdomain, else outside.

The function 'md_test' returns an integer value which corresponds to the index of the microdomain nearest to the point in real space given by the three arguments x,y,z of the function. As for the variables 'md_next' and 'md_dist', a negative resulting number indicates that the given position is inside the microdomain. In contrast a positive number shows that the given point is outside the returned microdomain.

The use of these variables allows a flexible use of microdomains within the program DISCUS. However, the interpreter might be quite slow when working with large model crystals. As an alternative, a microdomain distribution can be generated externally and read from a file in the 'micro' sublevel of DISCUS.

7.3.3 Working with microdomains

Most sublevels of DISCUS allow not only the selection of atoms to be used for the particular function, the user can also specify if only atoms within or outside of microdomains shall be used by selecting the corresponding microdomain status with the command 'sele'. The following choices can be made:

Once a microdomain distribution is established within the crystal, this selection mechanism allows the user to modify only atoms outside or inside those microdomains, e.g. to generate a modulation wave. A short example shall illustrate the use of the microdomain section of DISCUS. The resulting structure containing microdomains in form of a triangle are shown in figure 7.4.

The corresponding macro file is listed below. As in previous examples, the line numbers shown allow easy reference to the different parts of the macro file but are not part of the actual file.

**Figure 7.4:** Example of microdomain distribution within a crystal
$\includegraphics[scale=0.50, angle=0]{micro.1.eps}$

After the microdomain level is entered (line 1), the sublevel is initialized because we start a new microdomain distribution. A random distribution with a density of 10% is selected (lines 3-4). The generated origins are forced to coincide with the host lattice (line 5) and the size of the microdomains is set to 5.1 Å. This is to prevent microdomains to overlap. The shape of the microdomain is set to 'fuzzy' (line 7) which means that the actual shape of the microdomain defined by its contents is used. All atoms within the host structure closer than 1.0 Å to the microdomains will be removed (line 8). Next the structure file 'md.inp' containing the microdomain structure itself is read (line 10). The original structure is inserted as orientation 1 (lines 11-12). A second orientation is obtained by simple inversion given by the matrix defined in lines 14 to 16. This 'upside down' microdomain is inserted as second orientation (lines 17-18). Now the origin distribution is generated (line 20). So far the crystal itself has not been modified. This is done in line 21 when the 'run' command generates the crystal with the given microdomain distribution as seen in figure 7.4.