In short, because there are only 14 unique ways of choosing nonequivalent basis vectors in 3-space and with these basis vectors, one can generate 14 unique spacial lattice types.

Mohsen, it is simple: if you want to have such high symmetry, the particles (not only atoms, but also molecules are a subject of packing) must have the symmetry corresponding with the lattice symmetry (the same or even higher). Atoms are spheres, so any symmetry is a subset of their symmetry and they can be placed at the sites with any symmetry. But if you have a particle with C1 symmetry (or 1 in HM notation), e.g. CHFClBr molecule, with no other symmetry element than identity, you cannot place it to the lattice points of the hcp/fcc lattice because of breaking such symmetry.

Mohsen, co-ordination is determined by radius ratio of the atoms involved in the bonding. When the radius ratio is equal to 1.0 (or when the atoms involved are of equal size) those show the highest co-ordination number of 12. In materials with ionic bonding co-ordination is influenced by the charges of the ions involved since + and - ions bond together to mutually neutralize the charges of each other. For example, NaCl has a fcc structure and both Na+ and Cl- are in 6-fold co-ordination, whereas in calcium fluoride CaF2 with fcc structure Ca+2 is in 8-fold co-ordination with F- and F- is in 4-fold co-ordination with Ca+2, due to differences in their charges. In some structures (lattices) a single metal atom could be in many different co-ordinations depending on the neighboring atoms involved in bonding (example Al+3 is in 4-fold and 6-fold co-ordinations in the mineral muscovite). Overall symmetry of a material depends on its chemistry (constituent atoms) and co-ordinations of the atoms/molecules/radicals show in the structures they form. The common elements in the periodic table can come together in many different combinations to make different compounds/structures (not only fcc bcc or hcp) and because of their diversity one needs 14 lattices to explain those. This indicates that although there is lot of diversity possible, many materials show very similar symmetry or many are iso-structural (structurally similar), otherwise we would have exceeded this number,or more than 14 lattices would have been necessary.

In short, because there are only 14 unique ways of choosing nonequivalent basis vectors in 3-space and with these basis vectors, one can generate 14 unique spacial lattice types.

The periodicity of the crystalline materials involves the basic repeat a basic unit called unit cell (Hauy, 1784). Thus the crystalline material is formed by the repetition in space (2-D, 3-D, 4-D, ...) cells or crystallites. In 3-D space can be defined by three cell vectors called coplanar fundamental translations. These translations have lengths and well-defined angular relationships (see attached). Therefore mathematically possible we have many groups as crystals may exist. But taking into account the equality or inequality of the moduli of the fundamental vectors of angles, the number of crystalline networks we can obtain is finite (Bravais, 1850).

"That is, only seven different kinds of cells are necessary to

cover all possible point lattices or all crystals can be classified into one of the seven

crystal systems. Nevertheless, there are other ways for fulfilling the condition that

each point has identical surroundings. In this regard, Auguste Bravais (physicist in

France) found that there are 14 possible point lattices and no more and we use Bravais lattices. Since the unit cell including two or more lattice points is chosen in the Bravais lattice for convenience, some of the Bravais lattices can be expressed by other simple lattices. For example, the face-centered cubic lattice is also described by a trigonal (rhombohedral) lattice which contains only one

lattice point."

X-Ray Diffraction Crystallography

Introduction, Examples and Solved Problems

Professor Dr. Yoshio Waseda

Professor Kozo Shinoda

ISBN 978-3-642-16634-1 e-ISBN 978-3-642-16635-8

DOI 10.1007/978-3-642-16635-8

Springer Heidelberg Dordrecht London New York

The periodicity of the crystalline materials involves the basic repetition of a basic unit called unit cell (Hauy, 1784). Thus the crystalline material is formed by the repetition in space (2-D, 3-D, 4-D, ...) of cells or crystallites. In 3-D space the cell can be defined by three non-coplanar vectors called fundamental translations. These translations have lengths and a well-defined angular relationships (see attached). Therefore mathematically we have so many possible groups as crystals may exist. But taking into account the equality or inequality of the moduli of the fundamental vectors of angles, the number of crystalline lattices we can obtain is finite (Bravais, 1850).

José! Attachment please.

[email protected]

We normally have 14 lattice which give all possible bravais lattice And goound into seven crystal systems each specified by the shape and symmetry of the unit cell.

There is no need to repeat all the more or less correct answers. However, from my point of view it is necessary to focus again on the already "wrongly" formulated questions because it already contains a typical misunderstanding.

In first approximation the Bravais lattices have nothing to do with atomic positions or crystal structures!

It is a fully geometrical concept as explained in several answers. Moreover, as first requirement they fullfill the symmetry (therefore we need sometimes centered lattices which would not be necessary in principle).

Dear Mohsen,

I am attaching a link for you. The book by BD Cullity is an excellent resurce for understanding the fundamental concepts of Crystallography and the X-ray Diffraction. If we consider that the atoms are like hard spheres then we can arrange them in seven unique ways depending on what are the relationships between the distances between their centres and the angles the lines joining these centres are making with one another. Cosult page 13 of the reference for illustration.

Seven crystal systems are generated in that way and there are only fourteen forms of these seven systems that are unquely distinct. These are also mentioned in the table on page 13 of the reference.

So there are only fourteen crystallographic lattices as you mentioned. These lattices are called Bravais Lattices.

Https://archive.org/details/elementsofxraydi030864mbp

A 3-D Bravais lattice characterizing the crystalline materials is generated by the periodic repetition of a point (node) by three non-coplanar vectors a, b and c. The angles between the three vectors are α, β and γ. Independently of the lengths of the modules a, b and c and of the values of the three angles, two lattices are different if they differ by symmetry; two lattices with the same are different if they differ in the number of first neighbours around a node.

A strict mathematical proof that the 3-D Bravais lattices are only 14 is quite troublesome, but one can convince him/herself in the following way by specializing the modules a, b, c and the angles α, β, γ and check that the generated lattices differ by symmetry (number and type of rotation axes and number and type of mirror planes)

a≠b≠c and α ≠ β ≠ γ ≠ 90° primitive triclinic lattice

a≠b≠c and α = 90° β ≠ 90° γ = 90° primitive monoclinic lattice

a≠b≠c and α = 90° β = 90° γ = 90° primitive orthorhombic lattice

a=b≠c and α = 90° β = 90° γ = 90° primitive tetragonal lattice

a=b=c and α = 90° β = 90° γ = 90° primitive cubic lattice

a=b≠c and α = 90° β = 90° γ = 120° primitive hexagonal lattice

a=b=c and α = β = γ ≠ 90° primitive rhombohedral lattice

“Primitive” means that the parallelepiped individuated by the vectors a, b and c (unit cell) has nodes only at the corners.

The next step is to consider lattices that have the symmetry of one the primitive lattices, centre at least two opposite faces and check that, while preserving the symmetry, the number of first neighbours around a node changes. Seven more Bravais lattices are obtained:

one base-centred (C) monoclinic lattices

three centred orthorhombic lattices (base-centred C, body-centred I, all face-centred F)

one body-centred I tetragonal lattice

one body-centred I and one all face-centred F cubic lattices.

On the site http://en.wikipedia.org/wiki/Bravais_lattice one can find nice figures and a more detailed explanation.

Obviously, behind the question lies a few misunderstandings, one of which has already been pointed out in the answers, namely the mixup between "lattice" and "structure". The structures of most simple metals are those of ccp, bcc and hcp. The symmetry properties of those structure are to some part described by the corresponding lattices fcc, bcc and primitive (as all hexagonal structures must have).

Please note that "bcc" is used for both a structure (containing 2 atoms with fixed atomic coordinates) and a lattice type (with an even number of atoms). The correct Bravais lattice symbols are thus F, I and P, respectively). The element structure of alpha-Mn carries a lattice denoted bcc, but its structure is not the bcc structure -- there are 58 atoms to the cell!

There is an extremely common abuse to talk about "the fcc structure" (even in the already given answers...) while in fact meaning "the ccp structure". There are lots and lots of structures that are described by a fcc lattice (common ones as NaCl, diamond, zincblende, fluorite) but NO "fcc structure", and from the note on manganese above, one easily finds how very wrong it is to let it carry "the bcc structure".

The next misunderstanding is that alloy structures are all to be described as based on bcc or close packing (mostly ccp, hcp). One reason for this idea may be that ordinary textbooks do not cover any examples further than these concepts drawn further to ordered solid solutions. In fact, even binary alloys between a metal and a non-metal are likely to have very complicated structures (I have solved quite a few containing phosphorus and arsenic) where the common concepts of filling non-metal atoms interstitially into a metal structure (not into a "lattice") cannot be used in a fruitful manner. In fact, such alloys show an extreme wealth of structures (often of low symmetries) that could not be antecipated, ...

If you then take a step further to "materials" (as in your question) -- even if they consist of solely one phase (that they not always do) -- it is imaginable that the combination of more than two elements bring about rather complex scenarios. But still, as has been pointed out strongly already, there are still only 14 different lattices that are applicable to ALL crystal structures, be they artificial or natural, inorganic or organic, i.e. millions of structures!

Despite all correct crystallographic answers there are also chemical reasons, i.e. CHEMICAL BONDING in solids!

Close packing applies for several metal and ionic bond compounds.

However, there is also covalent or mixed types of bonding that cause lowering in local and crystallographic symmetry. So, besides close packing there are further principles of structure formation. Look at rules of Mooser-Pearson (8-n) and Zintl-Klemm. Compare Al - C/Si - P/As - S/Se, Cl or look at Silicates.

Al is a metal - fcc packing

C-Diamond: 8-4 = 4 bonds per C atom, locally 4 covalent bonds, high symmetry, F-Lattice

C-graphitic: 3 * 4/3 bonds, sheets with pi-bonds, => lower symmetry, trigonal-R or Hex. stacking

Si: Group IV, 8-4 = 4 bonds, it is a semiconductor with local tetrahedral bonding that can be described by sp3-hybridisation and covalent bonds to a first approx. (yes. we have to use band structure theory, too) - the lattice is still cubic F

P: Group V, 8-5 = 3 bonds per P-atom plus free electron pair, sheets in black P, P4-units in white P, lower symmetry (orthorhombic for black P)

As: 8-5 = 3, sheets and trigonal symmetry (R)

S: 8-6 = 2 bonds per S-atom, S8-rings in orthorhombic S

Cl: 8-7 = 1 bond per Cl atom, Cl2-units in solid Cl

I would even go so far to say that for the formation of crystal structures only chemical/physical reasons exist. Crystallography is only a principle of describing crystals, not to make crystals :-). Crystallography is immediately able to say where is exactly the same bonding because of symmetry and without any knowledge about the bonding itself. However, somehow the last comment from Richard touches one interesting point; the difference between metric and symmetry. Finally only the symmetry of a property, e.g. the chemical bondings, decides about the type of Bravais lattice. The metric of a lattice can be cubic but the symmetry only triclinic what finally results in a triclinic P lattice with a surprisingly cubic metric. In data bases even for this case examples exist.

The following diagram attached below gives a general overview of the classification of space groups in three dimensions according to vol. A of the International Tables. Depending on the algebraic properties, any crystal structure can be classified according to its space group, according to its crystal class, its Bravais lattice and its crystal system. The crystal systems is both a classification of Bravais lattices AND crystal classes.

The numbers given in front of each classification cannot be predicted ab initio but can only be counted once the criteria are enumerated. Mathematicians have derived those numbers for space groups up to 6 dimensions.

It will be good if you look some books about crystal structure. For example: Kittel Ch. Introduction to solid state physics. This is the good textbook on solid state physic for students.

All of the above answers are correct but My question is that why all of the structure for be more stable don't form in fcc or hcp?

This is an awesome discussion. I'm still reading some of the responses in detail and learning. I'd also suggest that Mohsen include another topic up top, as FIVE may be included. It may also help if this topic list is changed every now and then to get more diversity in responses. There seem to be so many related topics on RG with many more participants ("followers").

I'm not yet sure of Mohsen's question even after reiteration. I'm confident I'll figure it out soon. I must have been dozing in the crystallography class. But I suppose this is an interesting topic as it has elicited many erudite responses. I'd encourage all participants to at least check off the "Interesting Question" box or otherwise up top. This will circulate the question further and help diversify the responses over a larger cross-section of the expert RG membership.

Meanwhile, here is an awesome recent lecture on crystallography in general. Friday Evening Discourse at the Royal Institution, Professor Stephen Curry - See more at: http://richannel.org/seeing-things-in-a-different-light#! Originally posted by Iuliana Cernatescu, PhD

Please join and share your wealth of wisdom with us - "X-ray Diffraction Imaging for Materials Microstructural QC" Group, LinkedIn.

LinkedIn Discussion: http://www.linkedin.com/groupItem?view=&gid=2683600&type=member&item=5803109185189064705&qid=fd90e951-bedb-42b3-8cf3-559f1f084ee8&trk=groups_most_recent-0-b-ttl&goback=%2Egde_2683600_member_266852855%2Egmr_2683600

http://richannel.org/seeing-things-in-a-different-light#!

Sorry Mohsen Saboktakin, where did you read that fcc and hcp structures are more stable tha, e.g., triclinic or monoclinic structures? It seems to me that there is a great confusion concerning the basic concepts of structure, latiice, symmetry, stability....

This is a consequence of the fact that basic courses of crystallography are rarer and rarer... Young generations of students just pick up some sparse information from general courses...

Dear Giovanni Ferraris

I think the stability of structure Is associated with coordination number and coordination number of fcc and hcp is more than the rest structure.

I totally agree with Giovanni...and also Ravi. This topic is too complex if we all mix terms which are not related to the original question (although if it pains sometime not to react).

Dear Mohsen,

you claim you have read all answers (but not learnt from them). Still your question is why not all structures are either fcc or hcp. There you again mix a lattice (fcc) with a structure type (hcp). If you want to compare two structures these are ccp and hcp, both close-packed entities. However, there is a considerable restraint for these two: The building blocks have to be identical (normally atoms of the same kind, but spherical molekyles would do, as in the case of an idealized fullerene). That means that ccp and hcp are only element structures. which is a severe constraint compared to all structures that are possible! More constraints: Non-metal element structures do not belong, except for the noble gases, and not even all metals belong to either ccp, hcp or bcc. Why? Chemical bonding, of course! Atoms are not pinballs that stack according to geometry.

Your question is thus somewhat absurd: Why on earth (or elsewhere) would you expect only structures of closepacking (or those based on this concept) be the only alternatives?

Questions need answers, but answers that can be digested by the person who formulated the question. Whenever the question itself is reflecting that the person is having some confusion, answers of highest possible standard creates more confusion or only poses more questions to him rather than answers.

Mohsen, take a NaCl structure and substitute Na+ for Ca2+ and Cl- to (C2)2- (calcium acetylide). Because of (C2)2- ion is no more spherical but linear, the cell become elongated in one direction and instead of cubic lattice you'll obtain tetragonal.

Your simplistic assumption is valid only when you have spherical objects (atoms, atomic ions) packed by ionic/metallic/vdW (again - spherically symmetric) bonding. If you have molecules with any type of bonding with directional characteristics or ions/molecules which are not spherical, this is no more true.

You cannot obtain model based on close packing of spheres if your building units aren't spheres.

One thing I know after many years of learning and teaching is that all questions are relevant until the answer is revealed. The foundation of learning is questioning! Some had to brave crucifixion & "be-heading" to question (Jesus, Galileo, Copernicus, etc.). In comparison, chides from the erudite are like gentle breeze to the open-minded! Learning should be a fun and uplifting experience.

Boy! You fellows know a lot and I'm having fun learning. Still reading. Keep the ideas flowing!

http://www.flickr.com/photos/85210325@N04/10221065324/

Mohsen, a stable crystal structure is one that is in a low state of energy. In the opposite sense, you need to add energy to a crystal to destroy its order, like heat to melt the structure.

Several different terms contribute to the energy of an atomic arrangement. There are the Coulomb attraction/repulsion between ions, covalent bonds, Van-der-Waals bonds and especially in organic materials hydrogen bonds play an important role. All these energy terms add up and particular chemical compound will be in a minimum if the sum of all energy contributions is minimum. As long as Coulomb Forces are dominant, the direction of a bond is of lesser importance and one tends to get a high symmetry structure. For more covalently bonded compounds, the bond directions i.e. bond angles become very important as well.

As there is a huge amount of different chemical compounds, you get as a consequence many different ways to arrange the atoms/molecules which will lead to a minimum in energy ans thus to a stable crystal structure.

Take Erik_Rakovsky's example. NaCl is dominated by the two Coulomb terms, the attraction between Na+ and Cl- and the repulsion between the identical species. As result, the atoms are not arranged in a closed packed structure. Still the structure can be described as a closed packed structure of Chlorine with sodium filling the gaps. For Calclium acetylide the strong covalent bond between the carbons dictates that the stable structure consists of C_C pairs.

Also look at Carbon in the forms of Diamond, Graphite and Fullerene. All are covalently bonded with C in Diamond "insisting" on four equally distributed neighbors, which gives you not a closed packed structure as the energy minimum. In Graphite, the different electronic state of Carbon forces the plane sheets which are finally stacked. Fullerene (C60) is an almost spherical molecule and this packs best in a closed packed structure of the molecules. Locally the atoms are, however, not in a closed packed environment but form the molecule.

The driving force is thus the energy minimisation, which can be achieved in many different structures other than the closed packed structures. The closed packed structures are an energy minimum only for a crystal of identical atoms/spherical molecules in which the bond between atoms is non directional. Essentially these are the crystals of nobel gases and a few metals.

For all others the bonds favour other arrangements.

Many of the roughly 400 000 known crystal structures can be grouped into classes of structures with similar building principles. Text books on Crystallography will show you the most common ones, often arranged by chemistry and/or symmetry.

1-The highest bulk coordination number is 12, found in both hexagonal close-packed (HCP) and cubic close-packed (CCP) (also known as face-centered cubic or FCC) structures. The two most common allotropes of carbon have different coordination numbers. In diamond, each carbon atom is at the center of a tetrahedron formed by four other carbon atoms, so the coordination number is four, as for methane. Graphite is made of two-dimensional layers in which each carbon is covalently bonded to three other carbons. Atoms in other layers are much further away and are not nearest neighbors, so the coordination number of a carbon atom in graphite is 3 as in ethylene, According to the above, can we tell a diamond is more stable than graphite because it's coordination number is higher than graphite?

2- which driving force cause the atoms arrange in close packed structure?

Is there a relationship between co-ordination number and phase stability? I'm unaware of any such direct correlation?

Dear Mohsen,

It is fine that you know that close packed structure have 12 neighbours irrespective of symmetry (one can imagine other stackings than those of ccp andhcp). However,

I am sorry that I must insist: There is no single structure called "the fcc structure"! That is sadly enough a very common abuse of the concepts lattice vs. structure. If you know the structure, you may easily deduce the lattice, but the opposite may be very difficult! It is thus fundamentally wrong to say "also called the fcc structure", because there are lots of structures with an fcc lattice (containing various numbers of atoms, but always quartets of them -- and may also be of different kinds) but there is only one unique ccp structure (with four atoms of the same kind). I hope I have made that clear to you.

As for coordination number vs. stability, I agree with Ravi Ananth that there is no such obvious relation. When one speaks of stability it must always be compared with another situation (phase transformation, chemical reaction etc.). As for diamond (with an fcc lattice, by the way) vs. graphite, the latter form is actually more stable thermodynamically although it has got fewer nearest neighbours. Diamond is in fact metastable at ambient pressure. Still, artificial diamond may be prepared through CVD processes due to kinetic blocking of the anticipated transformation when hydrogen is present. If diamond is heated enough it will, through a spontaneous process, transform to graphite (and burn in air). This fact hinders the use of diamond as a cutting tool material (the hardest there is...) if temperatures increase too much.

Dear Mohsen, I would only recommend as a relatively short introduction the book from Ulrich Müller - Inorganic Structural Chemistry (http://www.amazon.com/Inorganic-Structural-Chemistry-Textbook/dp/0470018658/ref=la_B001HCZHIU_1_2?s=books&ie=UTF8&qid=1383734183&sr=1-2) when you probably find your answers, because you are in many misconceptions. In fact you are assuming pouring spherical atoms of the same size into the lattice, and this is not true in general, although for some structures (noble gases, metals, some ionic crystals) this CAN be good approximation. But in general this is outvoted by directional characteristics of the most of bonding interactions even if you have atoms of the same element (e.g. structure of iodine can be derived from face centered cubic lattice, but considering deformation introduced by formation of I-I covalent bond between pairs of iodine atoms), although generally you have different atoms present in the structure or the structure cannot be described as arrangement of atoms but molecules. And molecule is generally "potatoid" and by close packing of potatoes you cannot obtain any cubic lattice :D (nor hexagonal one)

Fcc and hcp lattice built on the assumption that the atoms are perfect spheres, but it is not.

Here is an example of the imperfection of a "single crystal" of ZnSe where each lattice point is not a perfect sphere as explained earlier. We are looking at only the (224) plane of this sample, "potatoid" , we also affectionately call it the "Pringles" potato chip effect:

http://www.youtube.com/watch?v=dFCQS8oUyT0&list=PL7032E2DAF1F3941F

http://www.youtube.com/watch?v=dFCQS8oUyT0&list=PL7032E2DAF1F3941F

Mohsen, The fact that you can describe regular repeating arrays of objects in three dimensions in terms of 14 Bravias lattices which can be further amplified to 230 space groups has nothing to do with materials per se. It is simply a classification of the possibilities for packing objects together. I think there are still some space-groups for which no examples of materials/compounds crystallising in this way are known, for example.

In general, structures made up of small numbers of atoms/ions such NaCl, ZnS2, MgAl2O4 crystallise in high symmetry space groups (cubic, rhomohedral/hexagonal) where it is easy to think about the structures being dominated by the close packing (ccp or hcp) of the large anions with the metals occupying either the octahedral or tetrahedral holes created within this close-packing arrangement or possibly both (as for MgAl2O4).

If we move on to more complicated chemical compositions, then we find that these begin to favour crystallisation in crystal systems of lower systems with monoclinic or triclinic being favoured by many coordination, organometallic and organic molecules.

There are further conditions imposed by inherently chiral and inherently polar systems - the first with an implied lack of Sn symmetry (usually thought of as lack of inversion) and the second with the need for a "sense of direction" (like a left- or right-handed screw or helix). Such systems can only crystallise in small subsets of the available space groups.

I hope this helps you!

Hexagonal Close Packing (hcp) and Cubic Close Packing (alias Face Centered Cubic or Fcc) are just the two most common ways of organizing spheres in the closest possible packing in space.

They can also be viewed in terms of the successive layers of spheres that occupy all the space available on a plane by adhering to the respecive sequences of layers: AB... and ABC...

As a matter of fact, once the third layer is introduced, there are an endless (theoretically infinite) number of sequences possible, resulting in close packing modes (e.g.: ABAC..., ABCBC..., etc. etc., providing you alternate different letters). Examples of these possibilities are given by several compounds such as the carbides and several others.

So this is the simplest answer to why not all crystals could be either hcp or fcc. Incidentally, bcc does not conform to a close packing scheme.

As to the reason why there are ONY 14 crystal lattices (alias Bravais lattices), Gerry Gibbs answer (the first one) gives it in an elegant nut shell which I'll let him elaborate in more mundane terms, should he wish to do so.

However, a very important consideration should also be that the 14 Bravais lattices (which also include all three: hcp, fcc and bcc) do not necessarily represent positions occupied by atoms (at the nodes of the lattices), buth they represent "symmetry boxes" which can contain atoms in any position whtin the "box" (at coordinates x,y,z) adhering to the symmetry laws within each "box" and to their respecive bonding schemes (energies).

Romano Rinaldi has urged me to elaborate in more mundane terms my answer of why there exist 14 Bravais lattice types. At the risk of being too mundane and for what its worth: Assuming that the atoms in a ‘perfect crystal’ are arranged in a periodic fashion throughout space such that the energy of the resulting is minimized and force on each atom is zero, then the crystal may be partitioned into representative unique fundamental domains where the stoichiometry of each domain is the same as that for the crystal as a whole. Given the periodic structure of the crystal, each fundamental domain defines a region that can delineated by a parallelepiped with three basis vectors denoted D={a, b, c} that each radiate form one corner of the parallelepiped such the lattice of the structure can be defined by set L = {Ua +Vb +Wc | U,V,W Є Z}. As there are only 14 unique ways of choosing basis vectors D={a, b, c}, there can only exist 14 Bravais lattice types (See the International Tables of crystallography)..

In order to prevent misunderstandings of Geralds answer, the last sentence is a general conclusion related to any arrangement of a periodic structure. It is not related to the former sentences where he described the definition of the unit cell of a specific single structure. Even there for triclinic or moniclinic lattices he could generate more than one sets of a, b, and c and alpha, beta, gamma, except additional conditions related to the length of the basis vectors and the size of the angles are formulated.

I also want to come back to the definition Giovanni Ferraris described, and which are unfortunately incorrectly used in nearly all fundamental books about crystallography. The definition for a triclinic lattice in the International Tables for Crystallography is given by:

a,b,c and α, β, γ: NO conditions

does not mean automatically that

a≠b≠c and α ≠ β ≠ γ ≠ 90°

This only describes the general case and is highly misunderstandable. As I already pointed out in a former comment only the space group symmetry defines which symmetry the lattice has. Using Gerald Gibbs terminalogy: If the fundamental domain (or the content of the lattice point) does not show any symmetry a,b, and c can have the same size and the angles between them can be 90°, the resulting lattice is nevertheless triclinic. In this case we are talking about pseudosymmtry. The reciprocal lattive points are arranged in a cubic framework, but the related intensities do not show any symmetry.

On the other hand: If a crystal structure shows a cubic symmetry, the resulting basis vectors MUST describe the edges of a cube.

I do not mean to offend any one by muddying the water. But, as you all know, there are only three nonequivalent cubic Bravais primitive lattice types: a cubic primitive lattice type P with cell dimensions a =b=c and alpha=beta=gamma=90 deg., a cubic all face centered lattice type F with cell dimensions a=b=c and alpha=beta=gamma=60 deg and cubic body centered lattice type I with cell dimensions a=b=c, alpha=beta=gamma =109.47 deg. When dealing with cubic body centered and face centered lattice type, it is often convenient, say when indexing a powder diffraction powder pattern, to chose the basis where a=b=c and alpha=beta=gamma=90. Unlike the I and F cells defined by the primitive bases, these cell contain at least two and four fundamental domains depending on the space group symmetry type. But when optimizing a crystal structure within the framework of density functional theory, the primitive bases is chosen to reduce the space, time and cost of the calculations.

Gerald! No offense taken. We are all big boys and girls already with awesome self-images. Thanks for your uninhibited perspective.

"the primitive bases is chosen to reduce the space, time and cost of the calculations". This issue should be significantly ameliorated by the low cost of computation combined with the notably higher computation speed these days. In many of these instances it is the data acquisition rate that is dismal for diffractograms even today. Folks are still addicted to their Archaic 0D point/scintillation counters instead of the modern (high resolution, high dynamic range, high sensitivity) real time 2D detector!

http://www.flickr.com/photos/85210325@N04/8008914205/in/set-72157632729013664

http://www.flickr.com/photos/85210325@N04/8001095415/in/set-72157632729013664

I realize the expertise of RG members interested in this discussion is exceptionally high by your scores, writings and profiles. So I'm going to dare to elicit your help with the analyses of some of our real time experimental findings from BNL and AFRL recently acquired. Please feel free to jump in help me out.

1.https://www.researchgate.net/post/What_are_the_causes_for_ASYMMETRY_in_the_Bragg_X-ray_Rocking_Curve_Profile_RCP_for_a_symmetric_004_GaAs_reflection

2.https://www.researchgate.net/post/X-ray_Rocking_Curve_Analysis_of_Super-lattice_SL_Epitaxial_Structures_What_are_some_of_your_practical_experiences_with_this_method

You can knock me down a couple of notches if needed, no problem. But, I'm finding near perfect match with theory. I'm unable to find similar results elsewhere in literature yet. Am I the only one? There's got to be someone else besides me in the world that has thought of this. I'll post this in a few other discussions with different set of experts. Hopefully we could bring together a "Master Mind Group"!

[email protected]

Theory meets Experiment!

http://www.flickr.com/photos/85210325@N04/9430820747/in/set-72157635172219571

The only reason isthat because that there are only 14possible unique ways of getting nonequivalent basis vectors inthree dimensions and with these basis vectors, Only 14 unique lattice types are obtained

In the last part of the nineteenth century, Hessel, Bravais, Fedorov, Schönflies and Barlow, working independently, almost simultaneously developed the mathematics

and the physical basis of geometric crystallography, i.e. that part which deals with the internal symmetry and the ordered structure of the particles that constitute the crystalline material.

In 1830, Johann Friedrich Christian Hessel, a German physician and mineralogist, proved that, as a consequence of Haüy’s law of rational intercepts, morphological forms can combine to give exactly 32 types of crystal classes, since only two-, three-, four-, and six-fold rotation axes occur. Unfortunately, his work was recognized

by the scientific community only after his death.

Auguste Bravais, a French physicist and crystallographer, is well known for having demonstrated in 1848 the existence of the 14 three-dimensional lattices. Bravais’s work corrected the small error of the German physicist and crystallographer Moritz Ludwig Frankenheim who previously proposed 15 three-dimensional lattices.

The Bravais lattices can be either primitive or centred. In primitive lattices (P) the nodes are the only points that are equivalent, i.e. equal and with identical surrounding.

Thanks José!

Only for completion...A similar situation regarding the forgotten Hessel and the therefore "unfair" tem Bravais lattice who definitely found a very nicely readable description of the 14 lattices but did not described them first also exists for Haüy’s law of rational intercepts. Also Haüy was not the first who found it. The rational intercepts have been first described 1773 by Tobern Bergmann (1735-1784) who was working at Upsala university. Huüy submitteded his work 1780 but also knew the treatise published by Bergmann. The reason why we are using nowadays the term Bravais lattice was obviously the better promotion of Haüys work.

Something is missing in the following sentence of José Amigò: “The Bravais lattices can be either primitive or centred. In primitive lattices (P) the nodes are the only points that are equivalent, i.e. equal and with identical surrounding.” Actually also in centred lattices all the nodes are equivalent: the translational equivalence of the nodes is a basic and essential property of crystallographic lattices.

To complement the great answers above, note that if the strict periodicity is replaced by quasiperiodicity, one gets, in addition to the Bravais lattices, the so-called quasicrystals: see e.g.the Wikipedia article (http://en.wikipedia.org/wiki/Quasicrystal) or this brief presentation:

http://ccf.ee.ntu.edu.tw/~ypchiou/Photonic_Crystals/Introduction%20to%20quasicrystal.pdf

That's a very interesting question which no one seems to be answering here! Why do we have 14 Bravais systems and not only 2 (the two close packed structures)?

I guess there's a chemical restriction here. For example, in the case of water, it forms open-cage like structure upon solidification which doesn't seem close packed or energetically favorable at all. However, that's the only way water molecules can arrange themselves in 3D space due to the hydrogen bonding. You know that bonding is directional and depends on the atomic orbitals present in the play.

Another example is carbon and its allotropes; diamond and graphite. The different atomic hybridization in both allotropes (sp3 in diamond & sp2 in graphite) leads to differences in crystal structures between them.

So, it is not only about stacking the atoms in the most close-packed way, but also chemistry plays a crucial role. I hope my guess is reasonable because I don't know the answer for sure.

Dear Mohamed,

lattices have nothing primarily to do with chemistry or atoms. It is a pure mathematic conclusion of 3-dimensional arrangements of points.

The proof is more than 150 years old and only considers how many arrangements are in minimum necessary.

To @Mr. Abdelaleem: Obviously you mix up 'structure' (atoms) with 'lattice' (points). There are millions of structures to cover all the millions of crystalline phases we have to deal with. You have stuck to two element structures, but not even all element structures are close packed! @Gert Nolze is (of course) absolutely right: in 3D space no more and no less than 14 Bravais lattices uniquely exist, covering symmetry aspects as well as the need for periodicity. If the latter restriction is lost, there are more -- but never less. With the same restriction, there are only 230 space groups to cover all the millions of structures. All this is an effect of group theory, pure mathematics.

In complement to Rolf Berger and Gert Nolze we can illustrate the meaning of the Bravais lattices in 3D. The following diagram is extracted from Vol. A of the International Tables. We can see that space groups can be analysed according to different criteria relating to symmetry operations. Mathematicians use the concept of "equivalence classes" to classify space groups. Depending on the criteria, we can obtain 7 crystal systems or 32 geometrical crystal classes or 14 Bravais lattices, and so on.

First the analysis of space groups classification is purely abstract and totally independent on any specific crystal structure. Second, there is no way to predict the number of any specific class (including Bravais Lattices) in any dimension. We can only derive the full list of N-dimensional space group and count the number of specific classes for this N-dimensional space. In other words, the number of 14 Bravais Lattices in 3D is just a counting result.

Dear all,

Thanks for your answers, they are all great with an exception; they are not related to the question! I think you need to read the description of the question.

I understand why there are only 14 Bravais lattice systems and not 28 or 230. That's a topic demystified since 1850. For now, if we bring these lattice groups and add atoms at the lattice points, we obviously end with 14 unique crystal structures with 2 of these crystals are the most close packed (hcp & fcc). Now, here is the question for which I have tried to answer: Why don't all materials be hcp or fcc? That's the question!

Mohamed,

You need to carefully distinguish between the concept of a lattice and a crystal structure.

A lattice is the (somewhat purely mathematical) arrangement of identical points throughout space. With the restriction to translational symmetry, group theory can be used to deduce that there are 14 distinct lattices in 3D space.

A crystal structure is the actual arrangement of atoms onto the many possible sites within such a lattice. Which actual crystal structure will be realized depends on the type of atoms and their relative number, furthermore on external properties such as temperature, pressure, magnetic, electric field etc.

At (most) conditions, crystals are in a state of low energy, at least compared to a liquid or gas. Which arrangement of the atoms presents the lowest state of energy depends on many different aspects. There are predominantly:

- the bond(s) between atoms (Van-der-Waals, metallic, ionic, covalent, Hydrogen-bonds)

- the atomic/ionic radii

If you have atoms in which the forces between the atoms are isotropic and attractive forces except at very short distances such as noble gases or many metals such as copper or gold, the minimum state of energy corresponds indeed to a closed packed structure.

As soon as you switch to purely ionic bonds as in NaCl, KF, while you still have isotropic Coulomb forces between the atoms, things change. The dependence of these forces as function of distance differs though from the simple Van-der-Waals force in noble gases. In addition we have strong attractive and repulsive potentials. A closed packed structure would require 12 first neighbors around a central atom. If one mixes positive and negative ions in this first coordination shell, one is faced with high repulsive forces that make this arrangement a high energy state. Thus a different arrangement such as the NaCl structure is of less energy than a (hypothetical) closed packed structure. If the first coordination shell is made only of negative ions around a central positive atom, you should quickly realize that you:

-run into a contradiction, as the environment around each of these negative ions is no longer a closed packed neighborhood of identical positive ions,

- the negative ions are immediate neighbors to each other with high repulsive terms.

Next, a quick look at covalent bond such as in Carbon. Here the state of the electrons around the carbon and the characteristics of a covalent bond make a strong directional bonding the energetically most favorable state. With an sp3 hybrid state these bonds are directed into the corners of a tetrahedron. Putting carbon atoms close to each other in a closed packed fashion would require much more energy than in the diamond structure.

Once you bind several or many atoms into a molecule, the local arrangement of the atoms within a molecule is dictated by the various covalent bond within the molecule. As molecules come in all sorts of regular and irregular shapes and may bear an electric charge, packing these in the lowest energy form can be done in many different styles. Try packing sheets of paper in a closed packed structure!

So its predominantly a matter of energy that results in the hundreds of thousands of different crystal structures.

I think all have answered the question, completely.

Because physical crystals possess planes, axes and centers of symmetry, they may be allocated to one of 32 crystal classes. Further sub-divisions of these classes are obtained, however, when transnational symmetry is taken into account (by including the possibility of screw axes and glide planes). The total number of possible spatial symmetries so obtained is 230, since only 230 distinguishable patterns may be formed in three dimensions by the periodic repetition of an - asymmetric - object as a motif. Because the 230 spatial symmetries are derived from 230 three- dimensional patterns, the statement that a crystal be allocated to one of them is essentially a geometrical description - it is a statement about the geometry of the crystal when its component parts are considered to be not in motion but at rest in average positions. For many crystals such a statement is a satisfactory description of the symmetry of the crystal, that is of the mutual relation of its constituent atoms in respect of type and position.

I think....... Mohamed wants to known why are there other lattices than FCC and HCP? Am I correct? If it is so then....the relative positions occupied by the atoms in lattice depend on their relative sizes and on reaction dynamics while synthesis of the crystal. The same material can be synthesized with different crystal structures by varying synthesis conditions. It purely depends on how atoms of different relative sizes get freezed in those conditions.

...apart from the fact that FCP and HCP are no lattices...

yes you are absolutely correct Gert Nolze

and apart from the fact that it is called fcc and bcc but ccp and hcp... What is FCP?

oh...I have corrected it now. Thanks Gert Nolze

In short, because there are only 14 unique ways of choosing nonequivalent basis vectors in 3-space and with these basis vectors, one can generate 14 unique spacial lattice types.