Crystallography and Mineral Crystal Systems

by Ritika

The six crystal systems could be well understood by getting a fair knowledge about the topic of solid geometry. For this one must have a base knowledge about crystallographic axes. In broad classification, there are six crystal systems in general. The crystallographic axes play a vital role to segregate and place all well known minerals into these six crystal systems.

A Crystal is a regular polyhedral form which is surrounded by faces which are smooth. A Crystal is also termed as a chemical compound in solid state due to inter atomic forces action. Crystallography refers to the complete study of mineralogy. The flat planes which bounds the solid are referred to as crystal faces.

Crystal Systems

The six crystal systems could be well understood by getting a fair knowledge about the topic of solid geometry. For this one must have a base knowledge about crystallographic axes. In broad classification, there are six crystal systems in general. The crystallographic axes play a vital role to segregate and place all well known minerals into these six crystal systems.

Types of Crystal Systems:

The six general types of crystal systems are:

• Cubic also referred as isometric

In this type of crystal system, the three crystallographic axes have equal length. Also, in this the three crystallographic axes intersect at 90 degrees with each other. All fifteen crystal forms are closed in the isometric crystal system. The isometric or cubic crystal form has the highest degree of symmetry among all the six crystal systems.

The seven forms of crystal which have the same symmetry are namely:

• CUBE:

Some examples to name in this are galena, pyrite, fluorite, perovskite.

• OCTAHEDRON:

Minerals like magnetite, chromite, franklinite, spinel, pyrochlore, cuprite, gold, and diamond fall in this group.

• DODECAHEDRON:

  Magnetite and sodalite are placed in this category

• TETRAHEXAHEDRON:

Fluorite, magnetite or copper and garnet are examples of this form.

• TRAPEZOHEDRON:

  Example minerals of this form are analcime and leucite.

• TRISOCTAHEDRON

• HEXOCTAHEDRON

The other eight forms are namely:

• TETRAHEDRON:

This is a four faced form and some minerals to name in this form are diamond, helvite, and sphalerite.

• TRISTETRAHEDRON:

This form generally has twelve triangular faces and mineral instances of this type are sphalerite and boracite.

• HEXTETRAHEDRON:

This is similar to hexoctahedron and the mineral diamond is generally available in this form.

• DELTOID DODECAHEDRON:

  The resulting face in this is rhombic with 12-faced form.

• GYROID:

Generally there is no natural mineral crystallizing in this form and this type is classified to have in it a pyritohedron and the diploid.

• PYRITOHEDRON:

This has twelve pentagonal faces in it and the mineral Pyrite is available in this form.

• DIPLOID:

This has a look similar to the pyritohedron with resulting face as trapezia. The only mineral in this form is Pyrite.

• TETARTOID:

The mineral named Cobaltite usually crystallizes in this form.

• Tetragonal:

This crystal system has three crystallographic axes intersecting at 90 degrees as cubic crystal system but only two of the crystallographic axes are of equal length. The third crystallographic axis has different length from the other two. In this, there are open forms consisting of order types namely of first order, second order and third order. The third prism form is referred to as the ditetragonal prism.

• Orthorhombic:

The orthorhombic crystal system like the previous two crystal systems have three crystallographic axes intersecting at 90 degrees. But in contrast, in this crystal system all the three crystallographic axes are of different length. In this system, the highest available symmetry is 2-fold.

The orthorhombic system has generally three symmetry classes. They are:

• The pinacoid also referred to as the parallelohedron:

This consists of 2 parallel faces and is available in three different crystallographic orientations.

• The rhombic prism:

This open form class has four faces with two of them intersecting each other.

• The rhombic dipyramid:

This class has in it eight triangular faces and all these intersect the three crystallographic axes.

There is yet other symmetry class in the orthorhombic crystal system which has the lowest symmetry named as rhombic disphenoid, also termed as rhombic tetrahedron.

• HEXAGONAL:

This crystal system is defined with four crystallographic axes of which three of the axes lie in the same plane and intersect among themselves at 120 degrees. The fourth crystallographic axes are of either shorter or longer length as compared to the third axes. This system is thus referred by the axial cross relationships. This has in it seven possible classes each of these having 6-fold symmetry placed in the division named as hexagonal division and five possible classes each of these having 3-fold symmetry placed in the division named trigonal division.

The possible classes of the hexagonal division are:

• Normal which is also referred to as the Dihexagonal dipyramidal class:

  This has seven forms under it named as:

1. Base or basal pinacoid:

   This has 2 faces in it and is denoted by Miller indices as (0001) and (000-1).

2. First order prism

3. Second order prism

4. Dihexagonal prism:

   This has twelve bounding faces and is therefore a 12-sided prism and beryl crystal is instance to name in this category.

5. First order pyramid

6. Second order pyramid

7. Dihexagonal dipyramid:

   Minerals like zincite, wurtzite, and greenockite fall in this class category.

• Hexagonal Trapezohedral:

  The minerals to name in this category are quartz and kalsilite.

• Hexagonal Dipyramid :

  The apatite group minerals fall in this category.

• Trigonal Dipyramid

• Hemimorphic:

  Minerals like incite, wurtzite, and greenockite are present in this class.

The possible classes of the trigonal division are:

• Hexagonal-scalenohedral:

This has two principal forms under the class namely:

• rhombohedron and
• the hexagonal scalenohedron

The mineral to name generally in this class is Calcite.

• Trigonal-trapezohedral:

The forms to name in this class are nacoid, trigonal prisms, hexagonal prism, ditrigonal prisms, and rhombohedrons. Quartz is the common mineral to name in this class.

• Rhombohedral:

Dolomite and ilmenite are the two most common minerals of this class.

• Monoclinic:

Monoclicnic crystal systems like orthorhobmic crystal system have all the three crystallographic axes are of different length. But here the crystallographic axes intersect at an oblique angle. Generally the dome referred as {hkl} is best form present in this class. There are two sides of orientations possible in dome referred as {hkl} and {-hkl).

• Triclinic:

In triclicnic crystal system, both the length and angle of intersection all the three crystallographic axes are of different length and different angles respectively. The triclinic pinacoid is also termed as parallelohedron which has in it two identical and parallel faces.

Crystal Groups Forms and Classes

by Ritika

The crystal groups are broadly classified into thirty two classes of symmetry. These broad classifications of crystal classes are got based on the exterior form of crystal. These broad classifications of thirty two crystal classes further have 230 sub space groups. These vary among themselves and are classified based on the analysis and observation of x-rays. It is also an interesting fact that even crystals made from the same mineral possess different crystal forms. This is attributed to each mineral’s crystal conditions of growth.

The crystal groups are broadly classified into thirty two classes of symmetry. These broad classifications of crystal classes are got based on the exterior form of crystal. These broad classifications of thirty two crystal classes further have 230 sub space groups. These vary among themselves and are classified based on the analysis and observation of x-rays. It is also an interesting fact that even crystals made from the same mineral possess different crystal forms. This is attributed to each mineral’s crystal conditions of growth.

Parameters for Crystal Groups and Classes

The two basic parameters used for the formation of crystal groups and classes are:

• Crystallographic axes
• Forms

In the above, if the axes are represented as a, b, c then the parameter to be maintained by the crystallographic axes for the formations of crystal groups and classes are

a=1, b=1, c=1

The method of drawing the images and construction of the crystal groups and classes use the above morphology. Each crystal class is a member of one of the six crystal systems, namely

• Isometric
• Hexagonal
• Tetragonal
• Orthorhombic
• Monoclinic and
• Triclinic Crystal systems

In the above, the hexagonal crystal system is further sub divided into hexagonal and rhombohedral systems.

About Crystal Forms and Thirty Two Crystal Classes

The technical term used to refer the outward appearance of crystal is ‘habit’. The term ‘form’ is a more casual usage by people to denote the outward appearance of crystal. The outward appearance of crystals includes in it the attributes like drusy, tabular, massive, encrusting, equidimensional, reniform and acicular.

A simple crystal may consist of only a single crystal form. A more complicated crystal may be a combination of several different forms. Fifteen different forms are possible within the isometric or cubic system. These include the hexoctahedron, gyroid, hextetrahedron, diploid, and tetartoid, among others. The crystal forms of the remaining five crystal systems are the monohedron or pedion, parallelohedron or pinacoid, dihedron, or dome and sphenoid, disphenoid, prism, pyramid, dipyramid, trapezohedron, scalenohedron, rhombohedron, and tetrahedron.

Crystal form is nothing but a term used to denote a group of crystal faces and each of the elements present in this group have the same relationship to the elements of symmetry of a given crystal system. It is also a mandatory requirement that each of the crystal faces present in this group has the internal geometrical relationships as other crystal faces present in the same group. That is, the physical and chemical properties are same for all crystal faces present in the same group. The reason for this attribute is the atomic arrangement of all atoms used for composing each of the crystal face present in the group is same.

There are two classifications with respect to crystal forms. They are:

• General form
• Special forms

General form also denoted as {hkl} denotes the name for each of the 32 classes. In each of these 32 classes, there is a form associated with each crystal class in which the faces intersect each crystallographic axes at different lengths. The rest of the forms are called special forms.

Types of Crystal Forms

The crystal forums are broadly classified as two types namely

• Open  Forms
• Closed Forms

Open  Forms:

Open forms are those groups of faces that are all related by symmetry but that do not completely enclose a volume of space. These types of open form crystal require additional faces also. There are about 18 open forms.

Some examples of open forms are namely

• Pedion: This refers to flat faces that are not parallel and are geometrically linked to any other faces.
• Pinacoid: Pinacoids have only two parallel faces that forms tabular crystal. An example of such type of crystal form is ruby.
• Dome: This type of crystal form is available in monoclinic and orthorhombic minerals. For instance, the intersection of two faces would result in mirroring effect which result in this form namely dome.
• Sphenoid: This type of crystal form has two-fold rotational axes. For example, monoclinic and orthorhombic minerals have such crystal forms.
• Pyramid :In this type of crystal form multiple facets converge on a single crystallographic axis. However isometric, monoclinic or triclinic systems do not have this crystal form.
• Prism :In case of prisms, the facets run parallel to an axis of a crystal, however they do not converge with the same. The isometric or triclinic minerals do not possess this type of crystal form. Quartz possesses this type of crystal form with two sets of three sided prisms. There are also sub types of prism, namely:
• Hexagonal Prisms
• Triangular Prisms

Hexagonal prisms have two hexagonal bases which are connected by a set of six rectangular faces. In addition an important point is these six faces run parallel to each other but do not converge with axes in the crystal. In case of triangular prisms it has two triangular bases which are well connected by a set of three rectangular faces. These also posses the basis attribute namely all faces run parallel to each other but do not converge with an axes in the crystal.

Closed Forms

The groups of faces which are all related by symmetry that are completely enclosed in a volume of space are called Closed forms. There are about 30 closed forms.These closed forms are broadly subdivided into two categories, namely

• Closed Isometric Forms
• Closed Non-Isometric Forms

Closed Isometric Forms

The main crystal forms in crystal system are:

• Hexahedron :This form has eight points, six faces, and twelve edges. They are all perpendicular to each other, forming 90 degree angles.
• Octahedron :This form type has two four sided pyramids lying base to base which are totally symmetrical with no top, or bottom and has eight faces.
• Tetrahedron :This form type has four equilateral triangular faces.
• Dodecahedron :There are twelve faces in this crystal form types and there are four sub types of dodecahedron namely:
• Symmetrical pentagonal dodecahedrons
• Asymmetrical pentagonal dodecahedrons
• Delta dodecahedrons
• Rhombic dodecahedrons
• Hexoctahedron :This is a multi-faceted dodecahedron. This has 48 triangular faces in it. The above include the important gemstones namely diamond, garnet, spinel and various other symmetrical gemstones.

Closed Non-Isometric Forms

The closed non-isometric forms have six sub-categories:

• Hexagonal (Trigonal) Closed Forms
• Tetragonal Closed Forms
• Rhombohedral Closed Forms
• Orthorhombic Closed Forms
• Monoclinic Closed Forms
• Triclinic Closed Forms

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.

Bravais Crystal Lattice

by Ritika

Bravais crystal lattice in crystallography is used to explain the geometrical symmetry of a crystal in details. The other term used to refer to bravais crystal lattice is space lattice.

There are about fourteen distinct bravais crystal lattices. A crystal structure is one of the characteristics of minerals which form the base for bravais crystal lattice in crystallography. Those are the 14 bravais lattices used to describe crystal structures. A bravais lattice is the period arrangement of points that through repeated translation of the lattice vectors will fill space. All crystals could be defined in detail by any of these fourteen bravais crystal lattices. Bravais lattice took its name after Auguste Bravais.

Lattice Type and Lattice Centering

The kinds of lattice centering used by bravais lattice system are:

• Primitive Centering: In this, lattice points lie on the corner of the cells alone.
• Body Centered: In the body centered lattice centering, there are lattice points lying on the corner of the cells along with one more lattice point placed on the centre of the cell.
• Face Centered: In the face centered style, there are lattice points lying on the corner of the cells along with a lattice point placed at centre of each of the faces of the cell.
• Centered on a Single Face: In this, as the name implies, there is one additional lattice point at the center of one of the cell faces.

The seven crystal systems are combined with the various possible lattice centering defined as above which results in bravais crystal lattice in crystallography. The two dimensional centering is simple since a parallelogram has only one face whereas the three dimensional ones have more options which leads to six different centering arrangements.

Fourteen Bravais Crystal Lattices

The bravis crystal lattice has fourteen bravais lattices which all occupy the three-dimensional space. These fourteen bravais lattices are defined based on the seven crystal systems. There are 230 space groups with 32 crystallographic point groups. Except quasi crystals, all other crystalline materials lie in one of the fourteen bravais crystal lattices. The fourteen bravais crystal lattices or space lattices are:

• Triclinic Bravais Crystal Lattice
• Simple Monoclinic Bravais Crystal Lattice
• Base Centered Monoclinic Bravais Crystal Lattice
• Simple Orthorhombic Bravais Crystal Lattice
• Base-Centered Orthorhombic Bravais Crystal Lattice
• Body-Centered Orthorhombic Bravais Crystal Lattice
• Face-Centered Orthorhombic Bravais Crystal Lattice
• Hexagonal Bravais Crystal Lattice
• Rhombohedral or Trigonal Bravais Crystal Lattice
• Simple Tetragonal Bravais Crystal Lattice
• Body-Centered Tetragonal Bravais Crystal Lattice
• Simple Cubic or Isometric Bravais Crystal Lattice
• Body-Centered Cubic Bravais Crystal Lattice
• Face-Centered Cubic Bravais Crystal Lattice

Let’s know a little about each of the above fourteen bravais crystal lattices.

Triclinic Bravais Crystal Lattice

In the triclinic bravais crystal lattice, vectors of unequal length are used for defining the crystal system. In addition, in this crystal lattice, the three vectors used are not mutually orthogonal.

Simple Monoclinic Bravais Crystal Lattice and Base Centered Monoclinic Bravais Crystal Lattice

The monoclinic bravais lattice also is defined by using vectors of unequal length. The resulting structure is a rectangular prism with base having the shape of a parallelogram. In this bravais crystal lattice, two pairs of perpendicular vectors are used with third pair having an angle other than 90 degrees.

Simple Orthorhombic Bravais Crystal Lattice , Base-Centered Orthorhombic Bravais Crystal Lattice , Body-Centered Orthorhombic Bravais Crystal Lattice and Face-Centered Orthorhombic Bravais Crystal Lattice

If a cubic lattice is stretched along two lattice vectors it results in a rectangular prism with base having the shape of rectangle and this is termed as orthorhombic lattices. In the orthorhombic lattices all the three bases intersect at 90 degrees with also the three vectors being mutually orthogonal.

Hexagonal Bravais Crystal Lattice

In the hexagonal crystal lattice, the symmetry is equal as the right prism has a hexagonal base. An example of this is graphite.

Rhombohedral or Trigonal Bravais Crystal Lattice

The rhombohedral bravais crystal lattice is also termed as Trigonal bravais crystal which is well defined by using vectors of equal length. In addition, it is also important to note that all the three vectors used to define the rhombohedral bravais crystal lattice are not mutually orthogonal. The rhombohedral bravais crystal lattice is similar to the cubic system being stretched along diagonally across the body.

Simple Tetragonal Bravais Crystal Lattice and Body-Centered Tetragonal Bravais Crystal Lattice

Tetragonal crystal lattice is obtained by using a cubic lattice and stretching the same along a lattice vector. The tetragonal crystal lattice is similar to a rectangular prism having the base as the shape of square.

Simple Cubic or Isometric Bravais Crystal Lattice

Each corner of a cube is defined with a lattice point in the case of cubic crystal system. Also, each lattice point shares equal spacing between eight adjacent cubes

Body-Centered Cubic Bravais Crystal Lattice

In the case of body-centered cubic crystal system, there are eight corner points defined with a lattice point in each of these right corner points. In addition to this one, more lattice point is used to define the center of the unit cell.

Face-Centered Cubic Bravais Crystal Lattice

In the face-centered cubic crystal system, the lattice points are placed on the faces of the cube.

Bravais lattices in Two Dimensional Space

There are five Bravais lattices in two dimensional spaces. They are:

• Oblique
• Rectangular
• Centered Rectangular
• Hexagonal and
• Square

Bravais lattices in Four Dimensional Space

There are fifty two bravais lattices in four dimensional space. Of these fifty two bravais lattices, twenty one are primitive and thirty one are centered.

The point group of the bravais lattice is the set of all point operations that leave the lattice invariant. Also, the bravais lattices associated with the same space group are considered the same type, although they are not equivalent. That is, the bravais lattices associated with the same point group as classified as the same crystal system. It is vital to note that a two-dimensional honeycomb do not form a bravais lattice. A bravais lattice is a lattice in which every lattice points have exactly the same environment. That is, the bravais lattice can be spanned by primitive vectors.

Crystallographic Axes and its Symmetry Operations

by Ritika

Symmetry operations are used to describe the crystal’s outward symmetry. Symmetry operations help to define the manner in which a crystal can repeat the facets or faces on their crystal’s surface.

Mirror Plane

The plane that is used to reflect a face from one side of the crystal to the other is termed as mirror plane. It is important to note that while being reflected using the concept of mirror plane, the face of reflection that is maintained is identical but reversed in orientation. That is, for example if the original face has any right handed characteristics, then the reflected face should represent all the same attributes as the original face with a left handed attribute.

Center Symmetry Operation

Another symmetry operation that is worth knowing is called ‘center’. The center symmetry operation refers to an operation which would invert the original face of a crystal through the center of the crystal. In fact, the resulting effect stands similar to the operation named as roto inversion axis which is explained in detail below. That is, by this operation, every point of the crystal is inverted to the other side of the crystal. The center operation is mostly applied in triclinic system which follows a single fold rotational axis by which with just a single rotation the crystal face returns to the original face of rotation.

Rotational Axis

Rotational axis is an imaginary line which acts an axis and is drawn on a crystal. By rotating the crystal along the axis, it is possible to repeat a crystal face. Thus it is possible to generate new crystal faces at consistent intervals of rotation done as explained above. It is also vital to note that the resulting face should be identical to the original face only if the orientation is reversed. The next point of consideration is the determination of interval for rotation of crystal face. The determination of interval for rotation of crystal face is done by dividing the full turn into equal segment intervals. That is, for instance 360 degrees is divided into a segment of four 90 degrees that results in four fold rotational axis. Thus the numbers of folds the rotational axes can have are one, two, three, four or six. This means that a singe fold axis of rotational axis of rotation would rotate the crystal in 360 degree intervals. The two fold interval of rotational axis of rotation would rotate the crystal in 180 degrees, three fold in 120 degrees each, four fold in 90 degrees as explained before and six fold of rotational axis of rotation would rotate the crystal in 60 degrees.

Rotoinversion Axis

Rotoinversion does the functionality of both rotational axis and inversion along with this. That is, the rotoinversion axis after performing the functionality of rotation once would invert the face of crystal along the center of crystal to the opposite side. Thus, the out coming face would be totally flipped. For instance, if the original face is up, the resulting face would be down and if the original face is right the resulting face would be left. This operation of rotoinverion axis is done until the operation returns to the original face. Apart from this, the rules, determination of interval for rotation namely folds explained above for rotational axis all holds good even for rotoinversion axis.

Crystallographic Axes

The crystallographic axes are used mainly by crystallographers. These axes are similar to the geometric axes and are used for plotting the orientations of faces and symmetry elements in crystals. It is important to know that it is not vital that the crystallographic axes should be part of symmetry of the crystals. It can also be present or not within the symmetry of the crystals. Generally, it would be present in the symmetry of crystal because crystallographers would try the orientation mostly along the planes and axes of symmetry to study the operations and orientations of crystals in depth.

The seven system of crystallography along with the folds required for the axis of rotation are given below:

• ISOMETRIC: The isometric crystal system requires four three fold axes of rotation. Instances of minerals and crystals in this system are spinel, lazurite, analcime, galena, gold, fluorite, almandine, halite, cobaltite, diamond, tetrahedrite, bixbyite and so on.
• TETRAGONAL: The tetragonal crystal system requires a single four fold axis of rotation. Instances of minerals and crystals in this system are zircon, carletonite, rutile, scapolite, anatase, vesuvianite, narsarsukite, autunite, xenotime, thorite, zeunerite and so on.
• HEXAGONAL: This crystal system need a single six fold axis of rotation. Instances of minerals and crystals in hexagonal system are aquamarine, gmelinite, pyrrhotite, apatite, ettringite, molybdenite, hanksite, thaumasite, vanadinite and so on.
• TRIGONAL: This crystal system need a single three fold axis of rotation. Instances of minerals and crystals in trigonal system are sapphire, ankerite, ruby, sturmanite, magnesite, pyrargyrite, hematite,rhodochrosite, cinnabar,elbaite and so on.
• ORTHORHOMBIC: The orthorhombic crystal system should possess three two fold axes of rotation or one two fold axis of rotation with two mirror planes along with this. Instances of minerals and crystals in this system are topaz, staurolite, barite, anhydrite, chrysoberyl, olivine, celestite and so on.
• MONOCLINIC: The monoclinic crystal system requires either a single two fold axis of rotation or a single mirror plane. Instances of minerals and crystals in this system are brazilianite, aegirine, azurite, borax, catapleiite, muscovite, huebnerite, crocoite and so on.
• TRICLINIC: This crystal system should possess either a center operation of symmetry or translational symmetry. Instances of minerals and crystals in triclinic system are babingtonite, inesite, bytownite, kyanite, turquoise, albite, rhodonite, oligoclase and so on.

Generally the substances that are non-crystalline are amorphous. They do not have any symmetry and so could not be classified under any crystallographic system.

The symmetry of the lattice is thus used for determining the angular relationships between crystal faces. The measurements of the angles between crystal face is used for calculating the relative lengths of sthe crystallographic axes or unit cell edges. The crystallographic axes help to define a coordinate system within the crystal.

Crystallography Topology

by Ritika

Crystallography refers to the analysis of atoms in crystals and topology refers to the study of distortion and invariant connectivity characteristics of mathematical objects. Thus, crystallographic topology deals with association of the two attributes together. The topology of crystallographic groups is approached using orbifolds, and that of simple crystal structures using Morse functions on orbifolds.

Orbifolds

W. D. Dunbar is considered as the father of orbifold as he started the first study of orbifolds in 1981.Crystallographic orbifold O is defined as the quotient space of either a sphere SP, or Euclidean EU, space modulo a discrete crystallographic symmetry group, G and is denoted as

O= (SP or EU)/G

An orbifold comprises a basic topological space along with a singular set embedded in it. Moreover, an interesting fact is that a properly bound fundamental domain within a space group’s unit cell is an orbifold.

Crystallographic Orbifolds Types

There are three types of groups in which general crystallography could be divided in terms of crystallographic orbifolds. They are:

Point Groups

This is used for explaining the symmetry of special positions within a space group and is also termed as Wyckoff site symmetries.

Plane Groups

The space groups projected along their primary axes of symmetry become plane groups.

Space Groups

In case of space groups, all of them have a parent point group. The main use of space groups are for classification of schemes. It is important to note that order of a space group is always infinite.

The crystallographic orbifolds associated respectively with the above groups are elliptic 2-orbifolds,Euclidean 2-orbifolds, and Euclidean 3-orbifolds. Thus, the above are referred to as point orbifolds, plane orbifolds, and space orbifolds respectively.

Critical Nets and Orbifolds

Morse functions are the basis of critical net in crystallography and this plays a vital role in crystal chemistry and crystallographic topology. Critical net is defined as mathematical mapping from Euclidean 3-space to Euclidean 1-space. This could be used to orbifold so that the Euclidean 1-space of density is deformed vertical in the page. The critical-net-on-orbifold model features the conventional crystallographic invariant lattice complexes and permits concise quotient-space topological figures to be drawn without any repetitions that are attributed to normal crystal structure figures.

Lattice Complex

Lattice Complex takes a vital role in crystallography and the history of lattice complex started many years before in the branch of crystallography. Lattice complexes refer to the configurations of points that recur at least once but generally repeatedly throughout the family of all space groups. It is important to note that points on symmetry elements have smaller total unit cell occupancy and this is called the Wyckoff site multiplicity.

The critical points are best described as representing 0-, 1-, 2-, and 3-dimensional cells in a topological Morse function. In this, generally non-degenerate crucial points are taken into consideration here because a degenerate crucial point can at all times be distorted into a series of non-degenerate ones via the morsification process. A degenerate critical point will have a singular second derivative matrix with one or more zero or nearly zero eigen values. In fact, the critical points are present where the first derivative with respect to global density will be zero. Also, a 3-3 symmetric matrix occurs as the second derivative at that particular point. Moreover, purely when the critical point is fully non-degenerate, will this have a non-zero determinant.

Color Crystallographic Groups

The group or normal-subgroup relationships could be well defined by using the concept of color crystallographic groups and this concept began in early 1984. The color crystallographic groups possess both symmetry and anti symmetry operators which is used for defining the above relationships in a well structured manner. The crystallographic bicolor group set belonging to each group helps in explaining the one index-2 group or subgroup pair for regular crystallographic groups. In the bicolor crystallographic group, each element has an associated even or odd binary parity flag. This is calculated and arrived based on the product of group generator parities that produce the element.

The concept of anti symmetry plays a vital role in color crystallographic groups. The term anti symmetry was coined by Heesch and Shubnikov. Bicolor groups also referred to as magnetic groups helps to study and explain in detail concurrently the arrangement of atoms also refered as regular symmetry along with the up or down magnetic spin vector orientations which is referred to as anti symmetry for magnetic atoms in any crystal. The use of critical net on orbifold drawing expands in more areas and to name one in this direction is that it is used for explaining the complete summary of the structure’s local and global topology if along with the critical net on orbifold the lattice complex information for each critical point site is added and also the Wyckoff site multiplicities being recorded on the same drawing.

There are totally thirty six cubic crystallographic space groups and 194 space groups. However, the thirty six cubic crystallographic space groups are unlike the 194 space groups. This is because each of them has body diagonal 3-fold axes that arise from their tetrahedral and octahedral point groups. Also, the cubic groups’ orbifolds are uncomplicated as compared to simpler space groups that are derivatives of cyclic and dihedral point groups. The different crystal shapes taken by different crystals is because of the prototypes associated, for instance, the different crystal shapes in muscovite are due to stacking sequence shifts and not due to different atomic structures. The technical term used to refer the different shapes occupied by crystal is polytype. Sometimes, it is also possible to get polytype occurred when substitution causes distortion in the shape. Generally, structural distortion takes place in compounds that crystallize at different temperatures and or pressures.

Thus, the benefits of orbifolds and critical nets on crystallographic orbifolds are that it gives a detailed and concise closed-space portrait of the topology for crystallographic groups and simple crystal structures. In fact, a well detailed crystallographic orbifold atlas, if prepared, would help in giving and projecting the complete tabulation of the topological properties of crystallographic orbifolds. This would help and would be useful to crystallographers in various ways for determining the attribute of crystals.

Techniques for Growing Crystals

by Ritika

A consistent repeated pattern of connected atoms or molecules would result in formation of a crystal. There are several techniques used for growing crystals.

Nucleation

The process of growing crystals is termed as nucleation. In the process of nucleation, the molecules and atoms used for used for forming crystals are dissolved into their individual units inside a solvent. Because of this dissolving process of solute elements, namely the atoms or molecules inside the solvent, the solvent gets associated with each other and results in forming a bond with each other. Thus, the connection of more and more subunits as above results in production of larger unit. This process gets repeated in such a manner that the size becomes larger that the resultant crystal falls out of the solution. That is, the crystallization process is repeated and resultant crystal is formed. Even after this resultant crystal formation, other solute molecules which are left unattached get attached to the surface of the crystal. This process gets repeated until a state of equilibrium or balance is achieved between the solute molecules in the crystal and those that remain in the solution.

It is also vital to note that the surface which is rough would be always best for growing crystal that is for the process of nucleation than surface which is smooth. One can view this by doing a simple experiment by trying the process of nucleation on a rough piece of string and on a smooth side of a glass. You are sure to observe that crystals get formed more easily on the rough piece of string than on the smooth side of a glass.

Technique for Growing Crystal

There are three normal steps involved in the general technique of growing crystals. The three steps followed are:

• Process of forming a saturated solution
• Process of growing seed crystal
• Repeat the process of growth

Process of Forming a Saturated Solution

The primary step for growing crystals is the formation of saturated solution. The solution enables the probability of solute particles to come together and form a nucleus which would result in formation of a crystal. Maximum care must be taken to prepare the solution as above to achieve the best resultant crystal. In other words, it is best to always use a concentrated solution with as much solute as you can dissolve, namely a saturated solution. It is also possible in some cases, that nucleation is achieved by association between solute particles in the solution without any assistance which is termed as unassisted nucleation. However, to increase the probability of forming crystals, assisted nucleation always must be provided. That is, a place for association for solute particles to aggregate and formation of crystal is done which increases the probability of formation of crystal.

First, one must take a saturated solution for the technique of growing crystal. If you prefer to take dilute solution it would take more time for the air to evaporate some liquid and make it saturated. But if you take directly saturated solution, the crystal formation could begin right from that instance. Add enough water as and when needed to the crystal solution. Also, vital care must be taken while selecting the saturated selection as wrong choice of saturated solution would result in dissolution of the crystals formed. One good guideline for making the saturated solution is to add crystal solutes like alum, sugar, salt and so on to solvents like water. One can also choose other solvents based on the solute taken. For completely dissolving the solute in the solvent, carry out the step of stirring the mixture of solvent and solute added. If you find that even after stirring there is some solute left, without dissolving try to heat the mixture which would enable the solute to dissolve completely. For the process of boiling one can use boiling water or even try heating the mixture of solute and solvent directly on the stove or in a microwave.

Process of Growing Seed Crystal

For growing large amount of crystals, one can pour the saturated solution on substrate like rocks or brick or sponge. This could be then protected from dust by covering the above with paper towels or any other covering material that would protect the setup from dust as well as allow the liquid to slowly evaporate.

For growing a single larger crystal, the technique used is to first get a seed crystal for the same. To get a seed crystal, pour some saturated solution on a plate. This setup is then allowed to evaporate which would result in the formation of crystals at the bottom of plate. The crystal formed at the bottom is called seed crystal which could be got by scarping. Another technique that could be used for growing seed crystals is to pour the saturated solution on a smooth container for instance, a glass jar. Then place a rough element like a piece of string into the saturated solution. This would enable the crystals to get formed on the rough element i.e the string. These crystals are the seed crystals.

Repeat the Process of Growth

We have seen above two techniques for the process of forming seed crystals. Let us now see how to proceed with crystal growth using the seed crystal. Consider the case of seed crystal of being formed on a string as explained before. In that case, first pour the liquid into a clean container. Then suspend the string with seed crystal in this liquid and also cover the container with some material like paper towel. Ensure not to seal it with a lid. Continue this process of growing crystals and take care to pour the liquid into a clean container as soon as you see crystals growing on the container.

If the seed crystal was formed on plate as explained before, then first tie it onto a nylon fishing line. Ensure you have your material smooth which would enable it to attract crystals. This would enable growth of seed crystals faster and more easily. Then suspend the crystal in a clean container with saturated solution and the process to be followed is the same for growing crystal as for growing the seed crystal present on the string.