What Are Crystalline Solids Apex

Solids

Readings for this section.

Petrucci: Department 12-7,  to 12-9

Classification of Solids

There are several categories of classification used to group dissimilar kinds of solids.  Some focus on the style the solids are bonded together, others focus on the repeatability of the solid structure.  We will try to clarify these unlike categorizations with examples then that the educatee will be able to identify the category (ies) where any given solid fits.

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Classification based on the range of atom repeatability

The repeatability of a solid refers to the power to use atom arrangements in 1 location to predict cantlet arrangements in other locations.  In a perfectly repeatable structure, there exists a small set of atoms (chosen the unit prison cell), whose positions can be used to predict atom positions in the whole solid structure by means of repeating the unit prison cell positions co-ordinate to the item symmetry of the crystal itself.  In solids that have long-range repeatability, the atom positions exist accurately described over an extended distance, using just the unit cell construction.  In solids that practise not have long-range repeatability, the unit cell construction may at best give us an thought of the kind of bonding involved in the solid just irregularity in the arrangements of the atoms will quickly frustrate any endeavor to predict cantlet positions whatsoever altitude away from the "unit of measurement cell".

Crystalline Solids have long-range repeatability.  They contain atoms or molecules bonded together in a regular pattern.   A good example of these is quartz crystals (emperical formula: SiO_two) where each silicon is bonded to four oxygens, which in turn are bonded to 2 silicons in a continuous covalent network extending in three dimensions.  You can find these crystals in your quartz watch.  The crystal vibrates at a fixed frequency when electric charge is placed across sure directions.  Your watch uses the vibrations to keep time.

Amorphous solids (or glasses) have at best, brusque-range repeatability.  They are made up of atoms or molecules with little or no regular organization.  Quartz that has been melted into liquid and cooled (moderately) chop-chop will course glass.  Quartz glass is used for applications similar windows on lasers, and fine optics like Zeiss lenses.  Many different solids tin exist in both crystalline and baggy forms.
For a particular solid (e.grand., quartz), the glass and the crystal will take the same type of bonding, the same emperical formula and some very similar physical properties.  The glass and crystalline variances can also have some properties that are quite different.  For example,  quartz crystals do not transmit light equally in all directions considering of the crystallinity whereas the quartz glass will transmit light equally in all directions.  Quartz glass will not fracture the same style as quartz crystals.

Semi-crystalline solids have medium-range repeatability, not true long-range repeatability but some repeatability over the short range (i.due east., not totally amorphous).  Such semi-crystalline materials have different properties from both glasses and crystals.  Liquid crystals, for example, accept medium range-repeatability.

Classifications based on bond type.

Solids that are crystalline, semi-crystalline and baggy can be made from all different types of atoms that are bonded in different ways.  Thus, another way to allocate solids is to look at the type of bonds property the solid together.  These different types of bond possibilities are listed here.

Molecular solids consist of molecules that are held together by week intermolecular forces.  A prime instance of this is sulfur.  Molecules of sulfur (S₈) are held together by intermolecular forces far weaker than the covalent bonds that go on the atoms within each molecule.  This type of solid may non have a loftier melting bespeak; none are higher than 400°C.  Molecular solids may be adequately soft, i.e., they can exist hands distorted or warn abroad by physical force of some kind because of the relative weakness of the intermolecular forces that are holding the solid together.

Covalent (network) solids are made of atoms that are covalently bonded together to grade one continuous network of covalently bonded atoms.  One could almost think of this type of solids every bit macroscopic molecules (big enough to see).  Diamonds are a prime example of such solids.  This type of solid tends to have a high melting point and are unremarkably quite hard.  For example, diamond melts at 3600�C.  Network solids are not all crystalline.  Quartz, mentioned higher up, is a network solid in crystalline and in baggy forms.

Ionic solids comprise ions of opposite charge which hold together with electrostatic (Columbic) interactions.  A skilful example of this is sodium chloride (table table salt).  The crystal structure of NaCl is shown on the correct. The atoms of Na⁺ alternating with atoms of Cl^- such that each positive ion has neighboring negative ions and vice versa.  Ionic solids tend to have a melting betoken that ranges from quite depression  to moderately, depending on the strength of the ionic bond.

Metallic solids are made up of metal atoms, whose loosely held outer electrons are somewhat gratuitous of their positive cores and form a continuous dissociated sea of negative accuse binding the positive cores together.   Metal bonds are generally non-directional, which means the solid will hold together even if the material is distorted significantly.  Metals can be reshaped by hit (malleable) or drawn through small openings (ductile) the way copper is formed into wires.   Metals can have low melting points and as well tend to be soft.   The crystal structure of copper is shown in the model on the right.

Classification based on the dimensionality of the solid

In Network solids, the atoms are all held together with covalent bonds such that there are no small identifiable units (molecules or clusters) inside the structure.  The array of atoms extends continuously throughout the whole solid.  Network solids can form in different dimensionalities.

1 dimensional networks (plastic),  These tend to form very soft plastic or even waxy/tar-like solids.  One-dimensional solids by and large do not form crystals by virtue of the easy entanglement of the long "molecules", which makes long-range repeatability improbable.  Some such molecules are long enough theoretically to be measurable macroscopically individually by mass or size.

Two dimensional networks (graphite).  These have planes of atoms that tin can easily slide over each.  For example, graphite is used every bit lubricant.

Iii dimensional networks (diamond).  These tend to be very strong and hard, and can have a very loftier melting indicate.  Ceramics used to line smelters and as a heat shield on spacecraft are three-dimensional network solids.

Back to Pinnacle

Lattices

For the residuum of this section, we will restrict our word to crystalline solids.  These accept atoms or groups of atoms arranged in a regular array or lattice in 3 dimensions.  We should look at the term lattice and define information technology closely.

A lattice is a mathematical abstraction that describes the fashion the atoms or groups of atoms are repeated in space. We must non mistake the diagrams that follow for arrangements of atoms.  The following details only the mathematical points that make upwardly the lattice.  Let's first look at simple lattices in two dimensions.  We will then extend the discussion into three dimensions.

A unit of measurement jail cell is any subset of the lattice that contains enough information that the whole lattice can be rebuilt by starting with just the unit cell and translating or rotating according to the symmetry of the lattice.  The simplest unit of measurement cell is 1 that contains the least number of points and needs only translations forth the cell edges to repeat the pattern.  Unit of measurement cells comprise the "stuff" we desire to echo.  The famous Dutch artist Escher regularly created paintings that were fabricated upward of repeating units.

Can yous find the unit cell here?  The red box shows a unit of measurement cell that can be used to replicate the residue of the painting past using it every bit a virtual postage stamp to make the residue of the painting by stamping each time some unit of measurement multiples of with and elevation away from the original.  The yellow box hither shows the smallest unit cell merely not necessarily the easiest to employ.  This primitive unit cell would require an additional symmetry functioning, i.due east., that each motion by one cell length horzontally or vertically be accompanied by an inversion of the light grayness and the dark grayness shading.

two-d Lattices

The simplest of the 2d lattices is the foursquare lattice.  We can cull a unit cell (the repeating unit) such that it has equal length sides (a) and an angle of 90�.  We tin locate the cell wherever nosotros please.  One pick puts one betoken of our lattice at each corner of the cell.  We could accept alternatively, placed the unit of measurement cell such that i signal was at the middle.  Less conveniently, we could have placed the cell with ane point anywhere within its boundaries.  In all cases, there is 1 point for each unit of measurement cell (the first choice saw simply ane/four of each vertex actually inside the boundaries of the unit of measurement cell.

A hexagonal lattice contains points arranged such that a unit of measurement cell can be drawn with angles of 60� and all sides of length a.  A complete hexagon is fatigued for visual effect.

Some other type of lattice is a rectangular lattice.  The unit cell in this lattice would accept angles of xc� like the square lattice but would have different length sides a and b.

The Rhombic lattice has a unit cell with equal length sides only an angle that is neither 60� nor 90�.

Finally, the least symmetric of all two dimensional lattices is the fully rhombic (parallelogram) lattice.  Hither, the angles are neither 90�, nor 60� and the sides of the unit cell are not all the same length.

3-d Lattices

In two dimensions, at that place are only these v lattices.  In three dimensions, the lattices are called signal-group lattices or Bravais lattices. In that location are xiv Bravais lattices.  For our purposes, we will look simply at two basic types of three dimensional lattices, cubic lattices and hexagonal lattices.

Cubic lattices

There are several cubic lattices.  We will focus on 3 of them here.  These include the simple cubic lattice, the torso-centered cubic lattice and the face up-centered cubic lattice.

The simple cubic lattice has only one ane lattice point inside each unit cell.   Notice that at that place seem to exist viii spheres (points) associated with this unit jail cell (the box).  It is important to recall that but the role of the point that is inside the box is truly in the unit cell.  Since merely ane/viii of each corner betoken is actually inside the unit of measurement jail cell,  there is actually simply 8 � one/8 = 1 indicate for each unit cell.  Each lattice point has six nearest neighbors (four are shown here).  We define this as the coordination number.

  A body centered cubic (bcc) unit prison cell has two points associated with it.  The 8�ane/eight=1 corner points and the one point in the centre of the cell.  Monatomic bcc lattices plainly have a coordination number of eight

The face centered cubic (fcc) lattice, has 4 points associated with its unit cell.  The 8�1/8=one corner points and 6�one/two=3 face-centered points.  These points are only one-half inside the box and so we only get to claim one-half of the point for the unit prison cell.  An alternating proper noun for the confront centered cubic lattice is the cubic closest packing (ccp).  This is one of two closest-packing arrays where the points have packed in the most efficient (least volume) way possible.   The 3d model on the left shows the lattice points.  the diagrams on the left prove two different views of monatomic structures that use fcc packing.  The far left one shows stacking layers abc and allows us to count the nearest neighbors to a particular atom (red).  The layers in CCP are really non along the cube sides, they are perpendicular to the cube body diagonal as indicated in the right-most epitome, where the colors correspond to the colors in the left-most image.  The colored atoms in the prototype all touch on the ruby-red atom in the centre of the B layer, and so we meet that the coordination number for this type of crystal is 12.  The other diagram shows the fraction of each cantlet that is really inside the unit prison cell.

The other major class of three dimensional crystal lattice is one that gives a hexagonal unit cell.  Like the hexagonal 2d array, the hexagonal 3d array has some angles of lx� with 90� directions perpendicular to that.

The unit cell of the hexagonal closest packing (hcp) array has 2 sides of length a, separated by angles of 60� and i side of length b at an angle of ninety� to the two others.  This is the other closest packing array where the points are packed in their most efficient (least book) way possible.  You lot can see that this unit cell has one betoken fully inside and 8 apex points 4 have 8.33% within and iv accept 16.66% inside the unit cell which betwixt them add together up to one point inside.

hcp lattice with unit of measurement jail cell highlighted.

hcp lattice with closest neighbouring points highlighted.

Dorsum to Pinnacle

Metals

In metals, the valence electrons tend to be quite loosely held.  These outer electrons are easily lost or shared with little effort.  Lets consider a solid piece (crystal) of metal (for example, Sodium) where the outer orbital (due south) of the neighboring metallic atoms all overlap each other.

We meet here that the loosely held electrons can freely roam from atom to cantlet with no hindrance.  In effect, the sodium cores (+one charge) are floating in a ocean of electrons (negative charges).  This allows for several properties feature of metals.

1. malleable: if you strike the metallic, the atoms can slide over each other without breaking any actual bonds.  Gilded tin can be hammered into gold-leaf (10 atoms thick) without fracturing the metal.
2. ductile: metals can be drawn through a pocket-sized opening equally in copper is pulled through a hole in a steel plate to form wire.
3. lustrous: Because of the large number of overlapping orbitals, the free energy levels are very closely spaced such that photons of a large range of frequencies are absorbed and instantly re-emitted (alias, reflected).
4. electrically conductive: since the electrons are quite mobile, metals easily bear electricity.

Because of the non-directional bonding that occurs in metals, atoms of metal crystals tend to pack in a very efficient manor.  The ii most efficient packing methods are ones that follow the hcp and ccp lattices, which have identical packing efficiencies (most 74% of the crystal volume is really atoms).  A tertiary common packing method of metals is bcc, which is not as efficient as hcp or bcc.

Most metals tend to take monatomic crystal structures.  They have only one atom associated with each crystal lattice indicate.  This makes information technology easy to compare atomic positions with crystal lattice positions.  Nosotros can choose to exactly overlap the diminutive positions with the lattice points.  This (very common) option leads to defoliation past some who remember that the atom is the lattice point.  This is only a coincidental choice made in monatomic crystals for the sake of simplicity.

Iron tends to pack in a monatomic bcc crystal structure.  "Monatomic" means that at that place is merely one cantlet associated with each lattice point in the crystal lattice.  For convenience, we tend to put the atoms exactly on the lattice points but this is not necessary as long as all lattice points share exactly the aforementioned spatial arrangement with ane and only one atom per point.

Iron'south atoms take 8 nearest neighbors and thus have a coordination number of viii.

Copper tends to pack in a monatomic ccp arrangement, where each atom is associated with its ain lattice betoken in the ccp lattice.  Recall that this is the face up centered cubic lattice array.  Each cantlet has a coordination number of 12, i.e., there are 12 nearest neighbors for each atom of copper.

Zinc atoms tends to accommodate  themselves in a monatomic hcp crystal structure where one and but one atom of Zn is associate with each lattice betoken in the hcp crystal lattice.  Each Zn atom has 12 nearest neighbours, i.e., the coordination number is 12.

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Ionic Solids

Ionic solids do not form monatomic crystal structures but many notwithstanding form closest packing arrangements since the ionic bonds are non-directional.  To envision the crystal structure of many ionic crystals, nosotros need to look at many factors, including the relative size of the positive and negative ions and the relative number of them.   Lets await at the packing system of the atoms in a close-packing construction.  We see that there are ii types of spaces between the atoms.

1. Tetrahedral spaces exist where iv atoms come together (cantlet from layer b on tiptop of triangular pigsty from layer a).
   
2. Octahedral spaces occur where six atoms come up together (three atoms from layer b surround a hole from layer a)
   

Since the octahedral holes are larger than the tetrahedral ones, they can conform larger cations, relative to the size of the anions.

Take Sodium chloride.  The chloride ions (green in the model beneath) are significantly larger than the sodium ions (blue) and we can consider the sodium chloride to exist a closest packing fcc array of chloride ions with sodium ions filling the octahedral holes between them.  NaCl  has a diatomic fcc crystal structure, since there are two "atoms" associated with each lattice point.

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The calcium fluorite (CaF₂) structure has Calcium cations in the tetrahedral spaces between fluoride ions. (The Ca²⁺ is blueish and the F- is greenish)

Note that there are different ratios of holes to atoms for tetrahedral (two holes : i atom) versus octahedral (1:1) holes.

Dorsum to Tiptop

Cell calculations

Once we understand the structure of crystals, we can exercise many unlike types of calculations using this information.  For the sake of simplicity, nosotros will restrict ourselves to calculations involving monatomic (metallic) cubic crystal structures.

Permit'south first expect at the geometry of a cube.

The cube has all sides of length a.  Each face has a face diagonal of length b and the body diagonal has a length c.  Using Standard trigonometric relationships, we can easily derive the following relationships.

b² = 2 a^two   or    b = (2 a²)^1/2

and

c² = 3 a²   or    c = (three aⁱⁱ)^1/2.

Thus, if nosotros are dealing with a Body centred cubic construction, the torso diagonal is the only prison cell management that is a uncomplicated multiple of atom radii.

Here, nosotros see that c is equal to 4 atomic radii.  c = 4 r    or    r = c/4

By the same logic, in a face up centred cubic structure, the face diagonal would be equal to 4 times the atomic radius.

b = 4 r      or      r = b/four

Example:

The length of a unit cell of iron (monatomic bcc) is measured to be 286 pm using x-ray diffraction.  What is the size (radius) of the iron cantlet?

Since Fe is monatomic bcc, we have

\[r=\frac{c}{4}=\frac{\sqrt{3a^2}}{4}=\frac{\sqrt{3\times (286 \mathrm{pm})^ii}}{4}=124 \mathrm{pm}\]

Example:

Vanadium has a unit cell structure like atomic number 26.  10-ray diffraction shows the unit cell dimension to be 305 pm.  What is the density of Vanadium?

Of course, this kind of example tin can be carried out on any unit cell; all you lot need to know is the cell dimension and the number of atoms inside the prison cell.  We discussed a few simple cubic cells for close-packed structures simply in that location are other structures that as well fit in a cubic jail cell structure

For example,

Diamond has 8 atoms inside a cubic unit prison cell.

NaCl has a FCC structure (of 4 Cl ions) with the Na⁺ ions in the octrhedral holes (ane:1) ratio so since FCC has 4 Na⁺ ions in each unit jail cell also.

Other structures can exist handled in similar ways.

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Prof. Michael J. Mombourquette.
Copyright � 1997
Revised: September 2-April-2012.

What Are Crystalline Solids Apex,

Source: http://faculty.chem.queensu.ca/people/faculty/mombourquette/FirstYrChem/Solids/

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