Teaching set #6 - From local order to nanodomains
When most atoms within a solid are arranged in some ordered fashion and such arrangement repeats even only a few times, the solid is classified as 'crystalline' (https://dictionary.iucr.org/Crystal). This definition is rather flexible and somewhat vague, yet there is something we do know with certainty: 'crystalline' does not mean 'periodic'. In fact, many characteristics of crystalline materials are incompatible with periodicity*: from their finite size, to missing atoms, to the fact that in every unit cell atoms are slightly displaced from the positions they occupy on average—and the list goes on.
Just as an ordered repetition of atoms can result in nano- or larger crystals depending on how many times the repeat occurs, any kind of crystal aperiodicity can be locally ordered, creating coherent structures that can extend across many unit cells. When a structural deviation from average (atom displacement, vacancy, etc.) influences other deviations around it, we refer to this state as 'correlated disorder', since these crystal imperfections are not completely unaffected by one another, but more or less strongly linked, correlated.
The following set of images explores the local structure–crystal domain continuity: the ordering of deviations from periodicity across different length scales, which can even result in the formation of nanodomains behaving like 'crystals within crystals'.
To display this phenomenon, we analyse Monte-Carlo-simulated crystals of CaTiO3, where 20% of the Ca atoms are missing. This does not obviously represent a real material, but we use it as a toy model to display how different local defect structures can be formed, and to study their influence on the resulting diffuse scattering patterns. The intensity scales have been arbitrarily chosen to enhance the diffuse features.
* Here we adopt the definition of a 'periodic structure' as "structure that strictly repeats its parameters (atomic fractional coordinates, atom types, etc.) at regular intervals". Order, on the other hand, allows for more flexibility as it does not require strict repetition of coordinate or type of atoms. Like for 'crystallinity', 'order' defines a spectrum of levels, so that there can be more or less order, all the way down to complete randomness. Liquids like water also have an ordered structure, locally, driven by the network of reversible hydrogen bonds.
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01. Unit cell of CaTiO3 used for the simulations, and hk0 plane of its reciprocal space.
02. Fully ordered, defect-free crystal of CaTiO3 (80x80 unit cells) and hk0 plane of its reciprocal space. A magnification of the ideal periodic structure is shown on the upper-left corner.
03. Fully ordered crystal of CaTiO3 without Ca atoms (80x80 unit cells) and hk0 plane of its reciprocal space. A magnification of the ideal periodic structure is shown on the upper-left corner.
04. Randomly disordered crystal of CaTiO3 (80x80 unit cells) and the hk0 plane of its reciprocal space.
05. Locally ordered vacancies in a CaTiO3 crystal (80x80 unit cells) and the resulting hk0 reciprocal plane.
To achieve this disorder correlation, the proximity of vacancy defects is slightly favoured along the crystal directions a and b, in the nearest-neighbouring unit cells. 10 MC cycles were used.
06. Locally ordered vacancies in a CaTiO3 crystal (80x80 unit cells) and the resulting hk0 reciprocal plane.
To achieve this disorder correlation, the proximity of vacancy defects is favoured along the crystal directions a and b, in the nearest-neighbouring unit cells. 1000 MC cycles were used.
07. Locally ordered vacancies in a CaTiO3 crystal (80x80 unit cells) and the resulting hk0 reciprocal plane.
To achieve this disorder correlation, the proximity of vacancy defects with Ca atoms is slightly favoured along the crystal directions a and b, in the nearest-neighbouring unit cells. 1000 MC cycles were used.
08. Locally ordered vacancies in a CaTiO3 crystal (80x80 unit cells) and the resulting hk0 reciprocal plane.
To achieve this disorder correlation, the proximity of vacancy defects with Ca atoms is favoured along the crystal directions a and b, up to the third-neighbouring unit cells. Only 10 MC cycles were used.
09. Locally ordered vacancies in a CaTiO3 crystal (80x80 unit cells) and the resulting hk0 reciprocal plane.
To achieve this disorder correlation, the proximity of vacancy defects with Ca atoms is favoured along the crystal directions a and b, up to the third-neighbouring unit cells. Only 20 MC cycles were used.
10. Locally ordered vacancies in a CaTiO3 crystal (80x80 unit cells) and the resulting hk0 reciprocal plane.
To achieve this disorder correlation, the proximity of vacancy defects with Ca atoms is favoured along the crystal directions a and b, up to the third-neighbouring unit cells. Only 40 MC cycles were used.
11. Locally ordered vacancies in a CaTiO3 crystal (80x80 unit cells) and the resulting hk0 reciprocal plane.
To achieve this disorder correlation, the proximity of vacancy defects with Ca atoms is favoured along the crystal directions a and b, up to the third-neighbouring unit cells. 1000 MC cycles were used.
12. Locally ordered vacancies in a CaTiO3 crystal (80x80 unit cells) and the resulting hk0 reciprocal plane.
To achieve this disorder correlation, the proximity of Ca atoms with Ca atoms is favoured along the crystal directions a+b and a-b. Only 10 MC cycles were used.
13. Locally ordered vacancies in a CaTiO3 crystal (80x80 unit cells) and the resulting hk0 reciprocal plane.
To achieve this disorder correlation, the proximity of Ca atoms with Ca atoms is favoured along the crystal directions a+b and a-b. 1000 MC cycles were used.
14. Locally ordered vacancies in a CaTiO3 crystal (80x80 unit cells) and the resulting hk0 reciprocal plane.
To achieve this disorder correlation, the proximity of vacancy defects with Ca atoms is favoured along the crystal directions a+b and a-b. 1000 MC cycles were used.
15. Locally ordered vacancies in a CaTiO3 crystal (80x80 unit cells) and the resulting hk0 reciprocal plane.
To achieve this disorder correlation, the proximity of vacancy defects with Ca atoms is favoured along the crystal directions a+b and a-b, and the proximity of Ca atoms with Ca atoms is favoured along the longer diagonals 2a+2b and 2a-2b. 1000 MC cycles were used.
16. Locally ordered vacancies in a CaTiO3 crystal (80x80 unit cells) and the resulting hk0 reciprocal plane.
To achieve this disorder correlation, the proximity of vacancy defects with other vacancy defects is favoured along the crystal direction a. 10 MC cycles were used.
17. Locally ordered vacancies in a CaTiO3 crystal (80x80 unit cells) and the resulting hk0 reciprocal plane.
To achieve this disorder correlation, the proximity of vacancy defects with other vacancy defects is favoured along the crystal direction a. 1000 MC cycles were used.
18. Locally ordered vacancies in a CaTiO3 crystal (80x80 unit cells) and the resulting hk0 reciprocal plane.
To achieve this disorder correlation, the proximity of vacancy defects with Ca atoms is favoured along the crystal direction a. 1000 MC cycles were used.
19. Locally ordered vacancies in a CaTiO3 crystal (80x80 unit cells) and the resulting hk0 reciprocal plane.
To achieve this disorder correlation, the proximity of vacancy defects with Ca atoms is favoured along the crystal direction a, for 3 unit cells radius. 1000 MC cycles were used.
20. Locally ordered vacancies in a CaTiO3 crystal (80x80 unit cells) and the resulting hk0 reciprocal plane.
To achieve this disorder correlation, the proximity of vacancy defects with other vacancies is favoured along the crystal direction a and b, while being disfavoured along a+b and a-b. 1000 MC cycles were used.
21. Locally ordered vacancies in a CaTiO3 crystal (80x80 unit cells) and the resulting hk0 reciprocal plane.
To achieve this disorder correlation, the proximity of vacancy defects with other vacancies is favoured along the crystal direction a+b and a-b, while being disfavoured along a, b, 2a, 2b, and the longer diagonals 2a+b and 2a-b, a+2b and a-2b. 1000 MC cycles were used.
22. Locally ordered vacancies in a CaTiO3 crystal (80x80 unit cells) and the resulting hk0 reciprocal plane.
To achieve this disorder correlation, the proximity of vacancy defects with other vacancies is favoured along the crystal direction a, b, a+b and a-b, 2a, 2b, while being disfavoured along 3a, 3b, and the longer diagonals 2a+b and 2a-b, a+2b, a-2b, 2a+2b and 2a-2b. 1000 MC cycles were used.
If you have interesting ideas for other types of local structures feel free to drop a message and I will be happy to consider them for future additions to this collection. Thank you for your interest!
