## Quasicrystals

*by* Ritika

Quasi crystals stand as a special case of in-commensurate crystal phase. Quasi crystals are present in a space group having more than three dimensions. For instance, structures like dihedral and icosahedral structures are categories belonging to this type. Even the two-dimensional Penrose patterns could be matched with symmetry structure of quasi crystals.

Applying the concept of crystallography to quasicrystals is an interesting topic and is based on the fact that any d-dimensional non-crystallographic point group possesses dimensional representations that are compatible with periodicity. The periodic structures could be grouped and calculated in various crystallographic methods like space groups, Bravais lattices and point groups. Quasi crystals are available in various inter metallic alloy systems. Quasicrystals are generally defined as periodic lattices.

### Higher Dimensional Space

Because of the periodicity loss, quasicrystals could not be defined in 3D-space in a method as we normally do for normal crystal structures. In other words, for normal crystal structures due to the three-dimensional translational periodicity of crystal structure, we assign Miller indices which are integer values for recording the reflections. Factors like friction, adhesion, corrosion and wear resistance could be determined and studied well in surface or interfacial regions. However, these could not give an in-depth study about structure of crystal. In case of quasi crystals, for marking the above integer indices, it needs at least five linearly independent vectors. That is polygonal quasicrystals would at least require five indices and icosahedral quasicrystals would require six indices. So quasiperiodic structures are defined as a period one in a higher dimensional space

### Types of Quasicrystals:

- Quasiperiodic in Two Dimensions: This is also referred to as polygonal or dihedral quasicrystals. It has sub elements namely octagonal, decagonal and dodecagonal. This has one periodic direction which lies perpendicular to the quasiperodic layers.
- Quasiperiodic in Three Dimensions: This type has no periodic direction and icosahedral quasicrystals fall under this type.
- New type: Icosahedral quasicrystals with broken symmetry fall in this category.

### Tiling in Quasicrystals:

Before the finding of quasicrystals, the method by which a plane was covered was using two different types of tiles which reflected a non-periodic fashion. In 1984, quasicrystals were introduced and also it was found that there exists a similarity among 3D-Penrose pattern and icosahedral quasicrystal. For instance, if one wants to record the diffraction pattern of Al-Mn quasicrystal this could be done by placing the atoms on the vertices of a 3D-Penrose pattern. These results in producing a Fourier Transform that details about the diffraction pattern of Al-Mn quasicrystal. Thus quasicrystal acts as a framework and when filled with atoms in correct fashion would result in producing quasicrystal structures.

### Symmetry and Diffraction Pattern in Quasicrystals

The diffraction pattern in quasicrystal determines the symmetry which helps in finding the type of the quasicrystal. Symmetry is expressed by the set of rotations that leave the directions of the facets unchanged. For instance, the icosahedral quasicrystal results in a Laue pattern when an x-ray beam is used along one of the five-fold axes. The complete atomic structure solution of an icosahedral quasicrystalline material was reported recently by Japanese researchers. To determine clearly about how the atoms are arranged and positioned in a quasicrystal, we can use the Patterson function that epitomizes all the information about the inter-atomic vectors contained in the diffraction pattern.

### Methods Used for Determining the Structure of Quasicrystals:

There are two methods which are sued for determining and solving the structure of quasicrystals. They are:

- 3D method
- nD structure analysis

#### 3D method

In 3D method, the input obtained from HRTEM images, well known and defined structures are used combined, which results in producing a realistic structure model.

#### nD structure analysis

In nD structure analysis method, the structure is modeled based on the elements present in nD unit cells. This is in contrast with the 3D method as this is a quantitative analysis method. Here least-squares method is used for finding and calculating the diffraction patterns and also for remodeling the same. Thus, this method makes use of various mathematical methods and tools just as used in a conventional crystal with the only difference in this method being extended to the nD space.

Quasicrystals have structures that are neither crystalline nor amorphous. However, they are intermediate structures with associated diffraction patterns. They are characterized by elements and attributes like length of adjacent lattice vectors, five-fold orientation symmetries and absence of translation symmetries. Thus, besides the crystalline and amorphous solids, the quasicrystals are a new type of space filling interesting forms of matter. Thus, the structure of quasicrystals which are not generally distributed periodically in a physical space that is, they are not periodic in 3D space are defined well in a higher dimensional space using a crystallographic approach. Quasicrystals with a hierarchical structure exhibit a self-similarity in the radial part of their direction.

The x-ray holography and quasicrystals are associated resulting in two unique experimental considerations. This showed that for the first time x-ray holography can be tested on a non crystalline solid. Also in addition, this proved that atomic positions in quasicrystals can be observed in direct space and in 3D with no prerequisite atomic model and no necessity for a sophisticated extension of the classical crystallography to a six-dimensional space. This gives structural information in direct space without presuming a prerequisite model.

For instance, icosahedral CdYb quasicrystal has been solved using X-ray diffraction data collected on the D2AM beam line. This model was further refined by using data in six dimensional analysis which included in it phase reconstruction procedure. The model also leads to a description of the hierarchical packing of the clusters. The clusters are packed together to form a ‘cluster of clusters’. This cluster of clusters in turn forms a larger cluster. The inflation property continues at infinity and is used to explain the quasicrystal’s physical properties. This method thus paves the way for further investigation into the stability and physical properties of quasicrystals. Nowadays, the growth technique of quasicrystals is perfected to a level that large ideal quasicrystals could be produced. These will contribute significantly to the understanding of the quasicrystals.