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ContentsContents

• Part 1: Material SciencePart 1: Material Science– Bonds in SolidsBonds in Solids – Crystal Geometry Crystal Geometry – Crystal Structure and DefectsCrystal Structure and Defects – Electron Theory of MetalsElectron Theory of Metals – Diffusion in SolidsDiffusion in Solids

Crystal Geometry, Crystal Geometry, Structure and DefectsStructure and Defects

Minta YuwanaMinta Yuwana

20092009

Learning Objectives

1. 1. Describe the difference in atomic/molecular structure Describe the difference in atomic/molecular structure between crystalline and noncrystalline materials.between crystalline and noncrystalline materials.

2. 2. Draw unit cells for face-centered cubic, bodycentered Draw unit cells for face-centered cubic, bodycentered cubic, and hexagonal close-packed crystal structures.cubic, and hexagonal close-packed crystal structures.

3. 3. Derive the relationships between unit cell edge length Derive the relationships between unit cell edge length and atomic radius for face-centered cubic and body-and atomic radius for face-centered cubic and body-centered cubic crystal structures.centered cubic crystal structures.

4. 4. Compute the densities for metals having facecentered Compute the densities for metals having facecentered cubic and body-centered cubic crystal structures given cubic and body-centered cubic crystal structures given their unit cell dimensions.their unit cell dimensions.

Learning Objectives

5. Given three direction index integers, sketch the direction Given three direction index integers, sketch the direction corresponding to these indices within a unit cell.corresponding to these indices within a unit cell.

6. 6. Specify the Miller indices for a plane that has been Specify the Miller indices for a plane that has been drawn within a unit cell.drawn within a unit cell.

7. 7. Describe how face-centered cubic and hexagonal close-Describe how face-centered cubic and hexagonal close-packed crystal structures may be generated by the packed crystal structures may be generated by the stacking of close-packed planes of atoms.stacking of close-packed planes of atoms.

8. 8. Distinguish between single crystals and polycrystalline Distinguish between single crystals and polycrystalline materials.materials.

9. 9. Define Define isotropy isotropy and and anisotropy anisotropy with respect to material with respect to material properties.properties.

CrystalCrystal

• solids which have a regular periodic solids which have a regular periodic arrangement in their component parties, arrangement in their component parties, bounded by flat faces, orderly arranged in bounded by flat faces, orderly arranged in reference to one another, which converge reference to one another, which converge at the edges and vertices. at the edges and vertices.

CrystalCrystal

• A crystal is symmetrical about its certain A crystal is symmetrical about its certain elements like points, lines or planes elements like points, lines or planes

• if crystal rotated about these elements, it is not possible to if crystal rotated about these elements, it is not possible to distinguish its new position from the original position. distinguish its new position from the original position.

– Symmetry is an important characteristic based on Symmetry is an important characteristic based on internal structure of crystal. internal structure of crystal.

– Symmetry helps one to classify crystals and Symmetry helps one to classify crystals and describing their behavior.describing their behavior.

• At temperatures below that of crystallization, the At temperatures below that of crystallization, the crystalline state is stable for all solids.crystalline state is stable for all solids.

Lattice Points and Space LatticeLattice Points and Space Lattice

• The atomic arrangement in crystal is The atomic arrangement in crystal is called the called the crystal structurecrystal structure. . – In perfect crystal, there is a regular In perfect crystal, there is a regular

arrangement of atoms. arrangement of atoms.

• A simple model of crystal structure, A simple model of crystal structure, – points representing the centers of ions, atoms points representing the centers of ions, atoms

or molecules. or molecules. – Such points in space are called Such points in space are called lattice pointslattice points. .


• The totality of lattice points forms a The totality of lattice points forms a crystal crystal latticelattice or or space latticespace lattice..– The space lattice is defined as an array of

imaginary points which are so arranged in space that each point has identical surroundings.


• The periodicity in the arrangement of ions, The periodicity in the arrangement of ions, atoms or molecules generally varies in different atoms or molecules generally varies in different directions.directions.

• If all the atoms, molecules or ions at the lattice If all the atoms, molecules or ions at the lattice points are identical, the lattice is called a points are identical, the lattice is called a Bravias Bravias latticelattice..

• The space lattice of a crystal is described by The space lattice of a crystal is described by means of a three-dimensional co-ordinate means of a three-dimensional co-ordinate systemsystem – the coordinate axis coincide with any three edges of the coordinate axis coincide with any three edges of

the crystal that intersect at one point and do not lie in the crystal that intersect at one point and do not lie in a single plane.a single plane.


• The space-lattice concept was introduced by The space-lattice concept was introduced by R.J. Hauy R.J. Hauy – It was postulated that an elementary unit, having all It was postulated that an elementary unit, having all

the properties of the crystal, should exist, or the properties of the crystal, should exist, or conversely that a crystal was built up by the conversely that a crystal was built up by the juxtaposition of such elementary units.juxtaposition of such elementary units.

• Define vectors representing points forming the Define vectors representing points forming the vertices of a parallelepiped vertices of a parallelepiped OABC as OABC as

OA OB OCOA OB OC a space lattice is obtained by translations a space lattice is obtained by translations

parallel to and equal to parallel to and equal to OA,OB OA,OB and and OCOC. . • The parallelepiped is called the The parallelepiped is called the unit cellunit cell..


• Most of metallic crystals have highly symmetrical Most of metallic crystals have highly symmetrical structures with closed packed atoms. structures with closed packed atoms.

• Complex lattices are frequently encountered in Complex lattices are frequently encountered in metals. metals. – It comprise of several primitive translation lattices It comprise of several primitive translation lattices

displaced in relation to each other. displaced in relation to each other.

• The most common types of space lattices are: The most common types of space lattices are: – Body centered cubic (BCC) lattices,Body centered cubic (BCC) lattices,– Face centered cubic (FCC) lattices and Face centered cubic (FCC) lattices and – Hexagonal closed packed (HCP) lattices.Hexagonal closed packed (HCP) lattices.

Lattice Points and Space Lattice:Lattice Points and Space Lattice: BasisBasis

• Crystal structure is described in terms of Crystal structure is described in terms of atoms rather than points. atoms rather than points.

• To obtain a crystal structure, an atom or a To obtain a crystal structure, an atom or a group of atoms must be placed on each group of atoms must be placed on each lattice point in a regular fashion. lattice point in a regular fashion.

• Such an atom or a group of atoms is Such an atom or a group of atoms is called the called the basisbasis – this acts as a building unit or a structural unit this acts as a building unit or a structural unit

for the complete crystal structure. for the complete crystal structure.


• A lattice combined with a basis generates A lattice combined with a basis generates the the crystal structurecrystal structure..

Space lattice + Basis → Crystal StructureSpace lattice + Basis → Crystal Structure

• A lattice is a mathematical concept, the A lattice is a mathematical concept, the crystal structure is a physical concept.crystal structure is a physical concept.

• The crystal structure is obtained by placing The crystal structure is obtained by placing the basis on each lattice point such that the basis on each lattice point such that the centre of the basis coincides with the the centre of the basis coincides with the lattice point.lattice point.


Note:Note: • the number of atoms in a basis may vary from the number of atoms in a basis may vary from

one to several thousands, one to several thousands, • the number of space lattices possible is only the number of space lattices possible is only

fourteenfourteen– one can obtain a large number of crystal structures one can obtain a large number of crystal structures

from just fourteen space lattices because of the from just fourteen space lattices because of the different types of basis available. different types of basis available.

– If the basis consists of a single atom only, a mono If the basis consists of a single atom only, a mono atomic crystal structure is obtained.atomic crystal structure is obtained.

• Copper is an example of mono atomic face-centered cubic Copper is an example of mono atomic face-centered cubic structures. structures.

• Examples of complex bases are found in biological materials.Examples of complex bases are found in biological materials.

Unit CellUnit Cell

• The smallest repetitive division of crystal The smallest repetitive division of crystal structure.structure.

• The smallest component of the space The smallest component of the space lattice.lattice.

• The basic structural unit or The basic structural unit or building blockbuilding block of the crystal structure by virtue of its of the crystal structure by virtue of its geometry and atomic positions within. geometry and atomic positions within.

Unit CellUnit Cell

• Space lattices of various substances differ Space lattices of various substances differ in the size and shape of their unit cells. in the size and shape of their unit cells.

• The distance from one atom to another The distance from one atom to another atom measured along one of the axis is atom measured along one of the axis is called the called the space space constantconstant. .

• The unit cell is formed by primitives or The unit cell is formed by primitives or intercepts a, b and c along intercepts a, b and c along XX, , Y Y and and Z Z axes respectively.axes respectively.

Unit CellUnit Cell• A unit cell can be completely described by the three A unit cell can be completely described by the three

vectors vectors a, b, a, b, and and c c ( ( OPOP, , OQ OQ and and OROR) when the length ) when the length of the vectors and the angles between them (of the vectors and the angles between them (αα, , ββ, , γγ) are ) are specified. specified.

• The three angles The three angles αα, , ββ and and γγ are called are called interfacial anglesinterfacial angles. . • Taking any lattice point as the origin, all other points on Taking any lattice point as the origin, all other points on

the lattice, can be obtained by a repeated of the lattice the lattice, can be obtained by a repeated of the lattice vectors vectors a, b a, b and and cc. .

• These lattice vectors and interfacial angles constitute the These lattice vectors and interfacial angles constitute the lattice parameter of a unit cell. lattice parameter of a unit cell.

• If the values of these intercepts and interfacial angles If the values of these intercepts and interfacial angles are known, one can easily determine the form and actual are known, one can easily determine the form and actual size of the unit cell.size of the unit cell.

Primitive CellPrimitive Cell

• Defined as a geometrical shape which, when Defined as a geometrical shape which, when repeated indefinitely in 3-dimensions, will fill all repeated indefinitely in 3-dimensions, will fill all space and is equivalent of one lattice point, i.e. space and is equivalent of one lattice point, i.e. the unit cell that contains one lattice point only at the unit cell that contains one lattice point only at the corners. the corners. – Note that in some cases the unit cell may coincide Note that in some cases the unit cell may coincide

with the primitive cell, but in general the former differs with the primitive cell, but in general the former differs from the latter in that it is not restricted to being the from the latter in that it is not restricted to being the equivalent of one lattice point.equivalent of one lattice point.

Primitive CellPrimitive Cell

• The units cells, which contain more than The units cells, which contain more than one lattice point are called non-primitive one lattice point are called non-primitive cells. cells.

• The unit cells may be primitive cells, but all The unit cells may be primitive cells, but all the primitive cells need not to be unit cells.the primitive cells need not to be unit cells.

Crystal ClassesCrystal Classes

• The atoms or molecules or ions in The atoms or molecules or ions in crystalline state are arranged in a regular, crystalline state are arranged in a regular, repetitive and symmetrical pattern,repetitive and symmetrical pattern, – but the crystal will have the external but the crystal will have the external

symmetrical shape only, if no restraint is symmetrical shape only, if no restraint is imposed during crystal growth.imposed during crystal growth.


• Crystals possess Crystals possess symmetrysymmetry– The symmetry of crystals is investigated The symmetry of crystals is investigated

by means ofby means of symmetry operationssymmetry operations, , • as a result of which the crystal coincides as a result of which the crystal coincides

with itself in various positions.with itself in various positions.


• The simplest of such operations The simplest of such operations ((rotationrotation, , reflection reflection and and translationtranslation--parallelparallel displacementdisplacement) are associated ) are associated with the with the elements of symmetryelements of symmetry.. – The simplest elements of symmetry are The simplest elements of symmetry are

the the axis and planes of symmetryaxis and planes of symmetry. .


• The shape of the crystal is said to be The shape of the crystal is said to be symmetrical if it possesses one or more symmetrical if it possesses one or more elements of symmetry. elements of symmetry.

• A group of symmetry operations, A group of symmetry operations, consisting commonly of a combination of consisting commonly of a combination of rotations, reflections and rotations with rotations, reflections and rotations with reflection, is called a reflection, is called a symmetry classsymmetry class. .


• The elements of symmetry are:The elements of symmetry are:– (i) The Symmetry Plane (i) The Symmetry Plane

– (ii) The Symmetry Axis (ii) The Symmetry Axis

– (iii) The Centre of Symmetry(iii) The Centre of Symmetry


• (i) The Symmetry Plane:(i) The Symmetry Plane: – The shape of the crystal is said to be The shape of the crystal is said to be

symmetrical about a plane if it divides symmetrical about a plane if it divides the shape into two identical halves or the shape into two identical halves or into two halves which are mirror images into two halves which are mirror images of one another.of one another. • Note that only in an ideal crystal the faces Note that only in an ideal crystal the faces

are of exactly same size.are of exactly same size.


• (ii) The Symmetry Axis:(ii) The Symmetry Axis: – If the shape can be rotated about an axis so If the shape can be rotated about an axis so

that the shape occupies the same relative that the shape occupies the same relative position in space more than once in a position in space more than once in a complete revolution, such an axis is called to complete revolution, such an axis is called to be axis of symmetry.be axis of symmetry.• Such axes may be either 2, 3, 4 or 6 fold. The axis Such axes may be either 2, 3, 4 or 6 fold. The axis

of symmetry causes the crystal to occupy more of symmetry causes the crystal to occupy more than one congruent position during rotation about than one congruent position during rotation about that axis during rotation by 360that axis during rotation by 360°°..


• (iii) The Centre of Symmetry:(iii) The Centre of Symmetry: – Within a crystal, there is some point about Within a crystal, there is some point about

which crystallographically similar faces are which crystallographically similar faces are arranged in parallel and corresponding arranged in parallel and corresponding positions, e.g., the centre of the cube is a positions, e.g., the centre of the cube is a centre of symmetry.centre of symmetry. • Note that a tetrahedron has no such centre. A Note that a tetrahedron has no such centre. A

cube has highly symmetrical shape and contains cube has highly symmetrical shape and contains many planes and axis of symmetry.many planes and axis of symmetry.


• The principal axes of a cube are four fold,The principal axes of a cube are four fold,– i.e. during each complete rotation about the i.e. during each complete rotation about the

axis, the crystal passes four through identical axis, the crystal passes four through identical positions. positions.

• The body diagonal axes are three fold and The body diagonal axes are three fold and there are six two-fold axes. there are six two-fold axes.

• The vertical axis of hexagonal prism is a The vertical axis of hexagonal prism is a six-fold axes.six-fold axes.


• A symmetry operation is one that leaves the A symmetry operation is one that leaves the crystal and its environment invariant. crystal and its environment invariant.

• Symmetry operations performedSymmetry operations performed– about a point or a lineabout a point or a line are called are called point group point group

symmetrysymmetry operations operations – by translationsby translations are called are called space group symmetry space group symmetry

operationsoperations. .

• Note that crystals exhibit both types of Note that crystals exhibit both types of symmetries independently and in compatible symmetries independently and in compatible combinations. combinations.


• The different point group symmetry The different point group symmetry elements that are exhibited by crystals: elements that are exhibited by crystals: – (i) (i) centre of symmetry or inversion centre centre of symmetry or inversion centre – (ii) (ii) reflection symmetryreflection symmetry– (iii) (iii) rotation symmetry.rotation symmetry.

Crystal System

• If all the atoms at the lattice points are If all the atoms at the lattice points are identical, the lattice is said to beidentical, the lattice is said to be Bravais Bravais lattice.lattice. – There are There are four systemsfour systems and and five possible five possible

Bravais latticesBravais lattices in two dimensions. in two dimensions.

• The four crystal systems of two The four crystal systems of two dimensional space are dimensional space are oblique, oblique, rectangular, square and hexagonalrectangular, square and hexagonal. .

Crystal System

• The rectangular crystal system has The rectangular crystal system has two two Bravais latticesBravais lattices, , – rectangular primitive and rectangular primitive and – rectangular centered.rectangular centered.

• In all, there are In all, there are five Bravais lattices.five Bravais lattices.

Crystal System

Crystal System

• Based on pure symmetry considerations, Based on pure symmetry considerations, there are only fourteen independent ways there are only fourteen independent ways of arranging points in three-dimensional of arranging points in three-dimensional space,space, – such that each arrangement is in accordance such that each arrangement is in accordance

or in confirmation with the definition of a or in confirmation with the definition of a space lattice.space lattice.

• These 14 space lattices with 32 point These 14 space lattices with 32 point groups and 230 space groups are called groups and 230 space groups are called Bravais latticesBravais lattices. .

Crystal System

• Each space lattice can be defined by Each space lattice can be defined by reference to a unit cell which, when reference to a unit cell which, when repeated in space an infinite number of repeated in space an infinite number of times, will generate the entire space times, will generate the entire space lattice. lattice.

Crystal System

• To describe basic crystal structures, To describe basic crystal structures, – the 14 types of unit cells are grouped in the 14 types of unit cells are grouped in seven seven

different classesdifferent classes of crystal lattices, of crystal lattices, • i.e. to describe basic crystal structures, i.e. to describe basic crystal structures, seven seven

different co-ordinate systemsdifferent co-ordinate systems of reference axes are of reference axes are required.required.

Crystal System

• The number of lattice points in unit cell can be The number of lattice points in unit cell can be calculated as:calculated as:– Contribution of lattice point at the corner = ⅛th of the Contribution of lattice point at the corner = ⅛th of the

pointpoint– Contribution of the lattice point at the face = ½ of the Contribution of the lattice point at the face = ½ of the

pointpoint– Contribution of the lattice point at the centre = 1 of the Contribution of the lattice point at the centre = 1 of the

pointpoint

• Every type of unit cell is characterized by the Every type of unit cell is characterized by the number of lattice points (not the atoms) in it. number of lattice points (not the atoms) in it.

Crystal System

• For example, the number of lattice points For example, the number of lattice points per unit cell for per unit cell for – simple cubic (SC) is 1simple cubic (SC) is 1– body centered cubic (BCC) is 2body centered cubic (BCC) is 2– Face centered cubic (FCC) is 4Face centered cubic (FCC) is 4

Crystal System

• Volume of a unit cell can be calculated with the Volume of a unit cell can be calculated with the help of the relationhelp of the relation

• Atomic Packing Factor (APF): – defined as the ratio of total volume of atoms in a unit

cell to the total volume of the unit cell.– also called relative density of packing (RDP).

2

• Non dense, random packing

• Dense, regular packing

Dense, regular-packed structures tend to have lower energy.

Energy

r

typical neighbor bond length

typical neighbor bond energy

Energy

r

typical neighbor bond length

typical neighbor bond energy

ENERGY AND PACKINGENERGY AND PACKING

• tend to be densely packed.

• have several reasons for dense packing:-Typically, only one element is present, so all atomic radii are the same.-Metallic bonding is not directional.-Nearest neighbor distances tend to be small in order to lower bond energy.

• have the simplest crystal structures.

METALLIC CRYSTALSMETALLIC CRYSTALS

Crystal System: Cubic

4

• Rare due to poor packing (only Po has this structure)• Close-packed directions are cube edges.

• Coordination # = 6 (# nearest neighbors)

(Courtesy P.M. Anderson)

SIMPLE CUBIC STRUCTURE (SC)SIMPLE CUBIC STRUCTURE (SC)

Crystal System: Simple Cubic

• APF for a simple cubic structure = 0.52

APF = a3

4

3(0.5a)31

atoms

unit cellatom

volume

unit cellvolume

close-packed directions

a

R=0.5a

contains 8 x 1/8 = 1 atom/unit cell

Crystal System: Face Centered Cubic

6

• Coordination # = 12

Adapted from Fig. 3.1(a), Callister 6e.


• Close packed directions are face diagonals.--Note: All atoms are identical; the face-centered atoms are shaded differently only for ease of viewing.

FACE CENTERED CUBIC STRUCTURE (FCC)FACE CENTERED CUBIC STRUCTURE (FCC)

APF = a3

4

3( 2a/4)34

atoms

unit cell atomvolume

unit cell

volume

Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell

a

7

• APF for a body-centered cubic structure = 0.74Close-packed directions: length = 4R

= 2 a

Adapted fromFig. 3.1(a),Callister 6e.

ATOMIC PACKING FACTOR: FCCATOMIC PACKING FACTOR: FCC

Crystal System: Body Centered Cubic (BCC)


8

Adapted from Fig. 3.2, Callister 6e.


• Close packed directions are cube diagonals.--Note: All atoms are identical; the center atom is shaded differently only for ease of viewing.

BODY CENTERED CUBIC STRUCTURE (BCC)BODY CENTERED CUBIC STRUCTURE (BCC)

aR

9

• APF for a body-centered cubic structure = 0.68

Close-packed directions: length = 4R

= 3 a

Unit cell contains: 1 + 8 x 1/8 = 2 atoms/unit cell

Adapted fromFig. 3.2,Callister 6e.

ATOMIC PACKING FACTOR: BCCATOMIC PACKING FACTOR: BCC

APF = a3

4

3( 3a/4)32

atoms

unit cell atomvolume

unit cell

volume

Crystal System: Tetragonal

Crystal System: Hexagonal


• ABAB... Stacking Sequence

• APF = 0.74

• 3D Projection • 2D Projection

A sites

B sites

A sites Bottom layer

Middle layer

Top layer


HEXAGONAL CLOSE-PACKED HEXAGONAL CLOSE-PACKED STRUCTURE (HCP)STRUCTURE (HCP)

Crystal System: Hexagonal

Crystal System

Crystallographic Points,Crystallographic Points,Directions, and PlanesDirections, and Planes

• POINT COORDINATESPOINT COORDINATES– The position of any point The position of any point

located within a unit cell located within a unit cell specified in terms of specified in terms of fractional multiples of the fractional multiples of the unit cell edge lengths unit cell edge lengths ((aa,,bb, and , and cc).).

POINT COORDINATESPOINT COORDINATES


SolutionSolution• From the sketch (From the sketch (aa), edge lengths for this unit ), edge lengths for this unit

cell are: cell are:

a = 0.48 nm, b = 0.46 nm, and c = 0.40 nm. a = 0.48 nm, b = 0.46 nm, and c = 0.40 nm. • Fractional lengths are: Fractional lengths are: q=1/4, r=1, s=1/2q=1/4, r=1, s=1/2• The location of point:The location of point:

– the the x x axis (to point axis (to point NN) = ¼*0.48 = 0.12 nm) = ¼*0.48 = 0.12 nm– the the yy axis (to point axis (to point OO) = 1*0.46 = 0.46 nm ) = 1*0.46 = 0.46 nm – The z axis (to point The z axis (to point PP) = ½*0.40 = 0.20 nm) = ½*0.40 = 0.20 nm

CRYSTALLOGRAPHIC DIRECTIONS

• Defined as a line between two points, or a Defined as a line between two points, or a vector.vector.

• The steps utilized in the determination of The steps utilized in the determination of the three directional indices are:the three directional indices are:1.1. A vector of convenient length is positioned A vector of convenient length is positioned

such that it passes through the origin of the such that it passes through the origin of the coordinate system. coordinate system. – Any vector may be translated throughout the Any vector may be translated throughout the

crystal lattice without alteration, if parallelism is crystal lattice without alteration, if parallelism is maintained.maintained.


2.2. The length of the vector projection on each The length of the vector projection on each of the three axes is determined; of the three axes is determined;

– these are measured in terms of the unit cell these are measured in terms of the unit cell dimensions adimensions a, , bb, , and cand c..

3.3. These three numbers are multiplied or These three numbers are multiplied or divided by a common factor to reduce them divided by a common factor to reduce them to the smallest integer values.to the smallest integer values.

4.4. The three indices, without commas, are The three indices, without commas, are enclosed in square brackets, thus: [enclosed in square brackets, thus: [uvwuvw].].

– The The uu, , vv, and , and w w integers correspond to the integers correspond to the reduced projections along the reduced projections along the xx, , yy, and , and z z axes, axes, respectively.respectively.

CRYSTALLOGRAPHIC DIRECTIONS: Hexagonal

• A problem arises, some crystallographic A problem arises, some crystallographic equivalent directions will not have the equivalent directions will not have the same set of indices. same set of indices.

• Circumvented by utilizing a four-axis, or Circumvented by utilizing a four-axis, or Miller–BravaisMiller–Bravais, , coordinate system.coordinate system.– The three aThe three a11, a, a22 and a and a33 axes are all contained axes are all contained

within a single plane (called the basal plane) within a single plane (called the basal plane) and are at 120° angles to one another. and are at 120° angles to one another.

– The The z z axis is perpendicular to this basal axis is perpendicular to this basal plane. plane.


• Directional indices will be denoted by four Directional indices will be denoted by four indices, as [indices, as [uvtwuvtw]; ]; – by convention, the first three indices pertain to by convention, the first three indices pertain to

projections along the respective aprojections along the respective a11, a, a22 and a and a33

axes in the basal plane.axes in the basal plane.

CRYSTALLOGRAPHIC PLANES

– The orientations of planes for a crystal The orientations of planes for a crystal structure are represented in a similar manner.structure are represented in a similar manner.

• The unit cell is the basis, with the three-The unit cell is the basis, with the three-axis coordinate system.axis coordinate system. – Except the hexagonal crystal system, Except the hexagonal crystal system,

crystallographic planes are specified by three crystallographic planes are specified by three MillerMiller indices indices as (as (hklhkl).).

• Any two planes parallel to each other are Any two planes parallel to each other are equivalent and have identical indices.equivalent and have identical indices.


• The procedure to determine the The procedure to determine the hh, , kk, , l l ::1.1. If the plane passes through the selected If the plane passes through the selected

origin,origin, • either another parallel plane must be constructed either another parallel plane must be constructed

within the unit cell by an appropriate translation, within the unit cell by an appropriate translation, • or a new origin must be established at the corner or a new origin must be established at the corner

of another unit cell.of another unit cell.


2.2. At this point the crystallographic plane either At this point the crystallographic plane either intersects or parallels each of the three intersects or parallels each of the three axes; axes;

• the length of the planar intercept for each axis is the length of the planar intercept for each axis is determined in terms of the lattice parameters determined in terms of the lattice parameters aa, , bb, and , and cc..

• The reciprocals of The reciprocals of a, b, ca, b, c are taken as are taken as the indexthe index. . • A plane that parallels an axis may be considered A plane that parallels an axis may be considered

to have an infinite intercept, and, therefore, a to have an infinite intercept, and, therefore, a zero indexzero index..


3.3. If necessary, these three numbers are If necessary, these three numbers are changed to the set of smallest integers by changed to the set of smallest integers by multiplication or division by a common multiplication or division by a common factor.factor.

4.4. Finally, the integer indices, Finally, the integer indices, not separated by not separated by commascommas, are enclosed within parentheses, , are enclosed within parentheses, thus: (hkl).thus: (hkl).


• An intercept on the negative side of the An intercept on the negative side of the origin is indicated by a bar or minus sign origin is indicated by a bar or minus sign positioned over the appropriate index.positioned over the appropriate index.

• Reversing the directions of all indices Reversing the directions of all indices specifies another plane parallel to, on the specifies another plane parallel to, on the opposite side of and equidistant from, the opposite side of and equidistant from, the origin.origin.


• One interesting and unique characteristic One interesting and unique characteristic of cubic crystals is of cubic crystals is – that planes and directions having the same that planes and directions having the same

indices are indices are perpendicular to one anotherperpendicular to one another; ;

• For other crystal systems there are For other crystal systems there are no no simple geometrical relationshipssimple geometrical relationships between between planes and directions having the same planes and directions having the same indices.indices.

Atomic Arrangements

• The atomic arrangement for a crystallographic The atomic arrangement for a crystallographic plane, depends on the crystal structure. plane, depends on the crystal structure. – The (110) atomic planes for FCC and BCC crystal The (110) atomic planes for FCC and BCC crystal

structures are represented in the next figures; structures are represented in the next figures; reduced-sphere unit cells are also included. reduced-sphere unit cells are also included.

– Note that the atomic packing is different for each Note that the atomic packing is different for each case. case.

– The circles represent atoms lying in the The circles represent atoms lying in the crystallographic planes as would be obtained from a crystallographic planes as would be obtained from a slice taken through the centers of the full-sized hard slice taken through the centers of the full-sized hard spheres.spheres.

Atomic Arrangements

• A “family” of planes contains all those A “family” of planes contains all those planes that are crystallographically planes that are crystallographically equivalent—that is, having the same equivalent—that is, having the same atomic packing; and a family is designated atomic packing; and a family is designated by indices that are enclosed in braces—by indices that are enclosed in braces—such as {100}. such as {100}. – For example, in cubic crystals the (111), For example, in cubic crystals the (111),

(111), (111 ), (111 ), (111 ), (111), (111 ), and (111), (111 ), (111 ), (111 ), (111), (111 ), and (111 ) planes all belong to the {111} family. (111 ) planes all belong to the {111} family.

Atomic Arrangements

• On the other hand, for tetragonal crystal On the other hand, for tetragonal crystal structures, structures, – the {100} family would contain only the (100), (100), the {100} family would contain only the (100), (100),

(010), and (010) (010), and (010) – the (001) and (001) planes are not crystallographically the (001) and (001) planes are not crystallographically

equivalent.equivalent.

• Also, in the cubic system only, planes having the Also, in the cubic system only, planes having the same indices, irrespective of order and sign, are same indices, irrespective of order and sign, are equivalent. For example, both (123) and (312) equivalent. For example, both (123) and (312) belong to the {123} family.belong to the {123} family.


Hexagonal CrystalsHexagonal Crystals

• The Miller–Bravais system of the four-The Miller–Bravais system of the four-index (index (hkilhkil) scheme is favored.) scheme is favored.

• There is some redundancy in that There is some redundancy in that i i is is determined by the sum of determined by the sum of h h and and k k throughthrough

i = -(h + k)i = -(h + k)

• Otherwise the three Otherwise the three hh, , kk, and , and l l indices are indices are identical for both indexing systems.identical for both indexing systems.

LINEAR AND PLANAR DENSITIES

• Linear density (LD) is defined as the Linear density (LD) is defined as the number of atoms per unit length whose number of atoms per unit length whose centers lie on the direction vector for a centers lie on the direction vector for a specific crystallographic direction:specific crystallographic direction:

• The units are reciprocal length nmThe units are reciprocal length nm-1-1, m, m-1-1

Linear DensityLinear Density

• For example: determine the linear density of the For example: determine the linear density of the [110] direction for the FCC crystal structure. [110] direction for the FCC crystal structure. – An FCC unit cell (reduced sphere) and the [110] An FCC unit cell (reduced sphere) and the [110]

direction therein are:direction therein are:


– Represented in the figure (Represented in the figure (b) b) are those five are those five atoms that lie on the bottom face of this unit atoms that lie on the bottom face of this unit cell; here the [110] direction vector passes cell; here the [110] direction vector passes from the center of atom X, through atom Y, from the center of atom X, through atom Y, and finally to the center of atom Z.and finally to the center of atom Z.

– With regard to the numbers of atoms, With regard to the numbers of atoms, it is it is necessary to take into account the sharing of necessary to take into account the sharing of atoms with adjacent unit cellsatoms with adjacent unit cells (as discussed in (as discussed in the atomic packing factor computations). the atomic packing factor computations).


XX

YY

ZZ

Linear DensityLinear Density• Each of the X and Z corner atoms are also shared Each of the X and Z corner atoms are also shared

with one other adjacent unit cell along this [110] with one other adjacent unit cell along this [110] direction (i.e., one-half of each of these atoms direction (i.e., one-half of each of these atoms belongs to the unit cell being considered), while belongs to the unit cell being considered), while atom Y lies entirely within the unit cell.atom Y lies entirely within the unit cell.

– Thus, there is Thus, there is an equivalencean equivalence of two atomsof two atoms along the [110] direction vector in the unit cell.along the [110] direction vector in the unit cell.

– The direction vector length is equal to 4The direction vector length is equal to 4R R (Figure (Figure bb); thus the [110] linear density for ); thus the [110] linear density for FCC isFCC is

Planar Density

• Planar density (PD) is taken as the Planar density (PD) is taken as the number of atoms per unit area that are number of atoms per unit area that are centered on a particular crystallographic centered on a particular crystallographic plane, orplane, or

• The units for planar density are nmThe units for planar density are nm-2-2, m, m-2-2..

Planar Density

• For example, consider the section of a For example, consider the section of a (110) plane within an FCC unit cell. (110) plane within an FCC unit cell.

AA BB CC

DDEE

FF

Planar Density– Although six atoms have centers that lie on Although six atoms have centers that lie on

this plane, this plane, • only one-quarter of each of atoms A, C,D, and F, only one-quarter of each of atoms A, C,D, and F, • one-half of atoms B and E, one-half of atoms B and E, • total equivalence of just 2 atoms are on that plane.total equivalence of just 2 atoms are on that plane.

– The area of this rectangular section is The area of this rectangular section is • the length (horizontal dimension) = 4the length (horizontal dimension) = 4RR, the width , the width

(vertical dimension) = 2(vertical dimension) = 2RR√√2.2.• Thus, the area of this planar region = 8Thus, the area of this planar region = 8RR22√√2 2

– The planar density is determined as follows:The planar density is determined as follows:

CLOSE-PACKED CRYSTAL STRUCTURES

• Face-centered cubic and hexagonal close-packed crystal structures have atomic packing factors of 0.74, – the most efficient packing of equal-sized spheres or

atoms. • In addition to unit cell representations, these two

crystal structures may be described in terms of close-packed planes of atoms (i.e., planes having a maximum atom or sphere-packing density);– a portion of one such plane is illustrated in Figure

3.13a.


• Both crystal structures may be generated by the Both crystal structures may be generated by the stacking of these close-packed planes on top of stacking of these close-packed planes on top of one another; one another; – the difference between the two structures lies in the the difference between the two structures lies in the

stacking sequence.stacking sequence.• Let the centers of all the atoms in one close-Let the centers of all the atoms in one close-

packed plane be labeled packed plane be labeled AA. . – Associated with this plane are two sets of equivalent Associated with this plane are two sets of equivalent

triangular depressions formed by three adjacent triangular depressions formed by three adjacent atoms, into which the next close-packed plane of atoms, into which the next close-packed plane of atoms atoms BB may rest. may rest.

– The remaining depressions are those with the down The remaining depressions are those with the down vertices, which are marked vertices, which are marked CC..


• A second close-packed plane may be A second close-packed plane may be positioned with the centers of its atoms positioned with the centers of its atoms over either over either B B or or C C sites; at this point both sites; at this point both are equivalent. are equivalent. – Suppose that the Suppose that the B B positions are arbitrarily positions are arbitrarily

chosen; the stacking sequence is termed chosen; the stacking sequence is termed AB.AB. – The centers of the third plane are situated The centers of the third plane are situated

over the over the C C sites of the first plane, This yields sites of the first plane, This yields an an ABCABCABC ABCABCABC . . .. . .

20

• ABCABC... Stacking Sequence• 2D Projection

A sites

B sites

C sitesB B

B

BB

B BC C

CA

A

• FCC Unit CellA

BC

FCC STACKING SEQUENCEFCC STACKING SEQUENCE

21

• Compounds: Often have similar close-packed structures.

• Close-packed directions --along cube edges.

• Structure of NaCl

(Courtesy P.M. Anderson) (Courtesy P.M. Anderson)

STRUCTURE OF COMPOUNDS: NaClSTRUCTURE OF COMPOUNDS: NaCl

Click on image to animate Click on image to animate


• The real distinction between FCC and HCP lies The real distinction between FCC and HCP lies in the position of the third close-packed layer.in the position of the third close-packed layer.– For HCP, the centers of this layer are aligned directly For HCP, the centers of this layer are aligned directly

above the original above the original A A positions. positions. – This stacking sequence, This stacking sequence, ABABAB . . . ABABAB . . . , is repeated , is repeated

over and over. (the over and over. (the ACACAC ACACAC . . . is equivalent). . . . is equivalent). – These close-packed planes for HCP are (0001)-type These close-packed planes for HCP are (0001)-type

planes. planes. – The correspondence between this and the unit cell The correspondence between this and the unit cell

representation is shown in following figure.representation is shown in following figure.

15

• Charge Neutrality: --Net charge in the structure should be zero.

--General form: AmXp

m, p determined by charge neutrality• Stable structures: --maximize the # of nearest oppositely charged neighbors.


- -

- -+

unstable

- -

- -+

stable

- -

- -+

stable

CaF2: Ca2+cation

F-

F-

anions+

IONIC BONDING & STRUCTUREIONIC BONDING & STRUCTURE

16

• Coordination # increases with Issue: How many anions can you arrange around a cation?

rcationranion

rcationranion

Coord #

< .155 .155-.225 .225-.414 .414-.732 .732-1.0

ZnS (zincblende)

NaCl (sodium chloride)

CsCl (cesium chloride)

2 3 4 6 8

Adapted from Table 12.2, Callister 6e.




COORDINATION # AND IONIC RADIICOORDINATION # AND IONIC RADII

17

• On the basis of ionic radii, what crystal structure would you predict for FeO?

Cation

Al3+

Fe2+

Fe3+

Ca2+ Anion

O2-

Cl-

F-

Ionic radius (nm)

0.053

0.077

0.069

0.100

0.140

0.181

0.133

• Answer:

rcationranion

0.0770.140

0.550

based on this ratio,--coord # = 6--structure = NaCl

Data from Table 12.3, Callister 6e.

EX: EX: PREDICTING STRUCTURE OF FeOPREDICTING STRUCTURE OF FeO

18

• Consider CaF2 :

rcationranion

0.1000.133

0.8

• Based on this ratio, coord # = 8 and structure = CsCl. • Result: CsCl structure w/only half the cation sites occupied.

• Only half the cation sites are occupied since #Ca2+ ions = 1/2 # F- ions.


AAmmXXpp STRUCTURES STRUCTURES

19

• Demonstrates "polymorphism" The same atoms can have more than one crystal structure.

DEMO: HEATING ANDDEMO: HEATING ANDCOOLING OF AN IRON WIRECOOLING OF AN IRON WIRE

Temperature, C

BCC Stable

FCC Stable

914

1391

1536

shorter

longer!shorter!

longer

Tc 768 magnet falls off

BCC Stable

Liquid

heat up

cool down

Solid BodiesSolid Bodies

• Solids exist in nature in two principal Solids exist in nature in two principal forms: forms: – Crystalline:Crystalline:

• SingleSingle• PolyPoly

– non-crystalline (amorphous)non-crystalline (amorphous)

• Their properties differ substantially. Their properties differ substantially.

Crystalline bodiesCrystalline bodies

• Remain solid up to a definite temperature Remain solid up to a definite temperature (melting point) at which they change from (melting point) at which they change from the solid to liquid state.the solid to liquid state.

• During cooling, the inverse process of During cooling, the inverse process of solidification takes place, again at the solidification takes place, again at the definite solidifying temperature, or point.definite solidifying temperature, or point. – In both cases, the temperature remains In both cases, the temperature remains

constant until the material is completely constant until the material is completely melted or respectively solidified.melted or respectively solidified.


• Characterized by an ordered arrangement Characterized by an ordered arrangement of their ions, atoms or molecules. of their ions, atoms or molecules.

• The properties of crystals depend on The properties of crystals depend on – the electronic structure of atoms the electronic structure of atoms – the nature of their interactions in the crystal, the nature of their interactions in the crystal, – the spatial arrangement of their ions, atoms or the spatial arrangement of their ions, atoms or

molecules, molecules, – the composition, size and shape of crystals.the composition, size and shape of crystals.


• may be either in the form of may be either in the form of – single crystalsingle crystal ( (graingrain) ) or or

– an an aggregate of many crystalsaggregate of many crystals usually usually known as known as polycrystallinepolycrystalline separated by separated by well-defined well-defined grain boundariesgrain boundaries..


• Polycrystalline material is stronger than Polycrystalline material is stronger than ordinary oneordinary one – Tiny crystals (grains) have different Tiny crystals (grains) have different

orientations with respect to each other and orientations with respect to each other and grain boundaries obstruct the movement of grain boundaries obstruct the movement of dislocations. dislocations.

• Polycrystalline material are called Polycrystalline material are called isotropicisotropic – exhibit same properties in every plane and exhibit same properties in every plane and

directiondirection

• Single crystal is called Single crystal is called anisotropicanisotropic..

Amorphous BodiesAmorphous Bodies

• Amorphous substances have no crystalline Amorphous substances have no crystalline structure in the condensed state structure in the condensed state – ordinary glass, sulphur, selenium, glycerin and most ordinary glass, sulphur, selenium, glycerin and most

of the high polymers can exist in the amorphous state. of the high polymers can exist in the amorphous state.

• Amorphous bodies, when heated, are gradually Amorphous bodies, when heated, are gradually softened in a wide temperature range and softened in a wide temperature range and become viscous and only then change to the become viscous and only then change to the liquid state. liquid state.

• In cooling, the process takes place in the In cooling, the process takes place in the opposite directionopposite direction..

24

• Single Crystals-Properties vary with direction: anisotropic.

-Example: the modulus of elasticity (E) in BCC iron:

• Polycrystals

-Properties may/may not vary with direction.-If grains are randomly oriented: isotropic. (Epoly iron = 210 GPa)-If grains are textured, anisotropic.

E (diagonal) = 273 GPa

E (edge) = 125 GPa

200 m

Data from Table 3.3, Callister 6e.(Source of data is R.W. Hertzberg, Deformation and Fracture Mechanics of Engineering Materials, 3rd ed., John Wiley and Sons, 1989.)

Adapted from Fig. 4.12(b), Callister 6e.(Fig. 4.12(b) is courtesy of L.C. Smith and C. Brady, the National Bureau of Standards, Washington, DC [now the National Institute of Standards and Technology, Gaithersburg, MD].)

SINGLE VS POLYCRYSTALSSINGLE VS POLYCRYSTALS

Amorphous BodiesAmorphous Bodies

• On repeated heating, long holding at 20-25°C or, On repeated heating, long holding at 20-25°C or, in some cases, deformation of an amorphous in some cases, deformation of an amorphous body, the instability of the amorphous state may body, the instability of the amorphous state may result in a partial or complete change to the result in a partial or complete change to the crystalline state. crystalline state. – Examples of such changes are the turbidity effect Examples of such changes are the turbidity effect

appearing in inorganic glasses on heating or in optical appearing in inorganic glasses on heating or in optical glasses after a long use, partial crystallization of glasses after a long use, partial crystallization of molten amber on heating, or additional crystallization molten amber on heating, or additional crystallization and strengthening of nylon fibers on tension.and strengthening of nylon fibers on tension.

• atoms pack in periodic, 3D arrays• typical of:

26

Crystalline materials...

-metals-many ceramics-some polymers

• atoms have no periodic packing• occurs for:

Noncrystalline materials...

-complex structures-rapid cooling

Si Oxygen

crystalline SiO2

noncrystalline SiO2"Amorphous" = NoncrystallineAdapted from Fig. 3.18(b), Callister 6e.

Adapted from Fig. 3.18(a), Callister 6e.

MATERIALS AND PACKINGMATERIALS AND PACKING

28

• Quartz is crystalline SiO2: Si4+

Na+

O2-

• Basic Unit:

Si04 tetrahedron4-

Si4+

O2-

• • Glass is Glass is amorphousamorphous• • Amorphous structureAmorphous structure occurs by adding impuritiesoccurs by adding impurities

(Na(Na++,Mg,Mg2+2+,Ca,Ca2+2+, Al, Al3+3+))• • Impurities:Impurities: interfere with formation ofinterfere with formation of crystalline structure.crystalline structure.

(soda glass)Adapted from Fig. 12.11, Callister, 6e.

GLASS STRUCTUREGLASS STRUCTURE

Single crystalSingle crystal

• No matter how large, is a single grain. No matter how large, is a single grain.

• Regular polyhedrons whose shape Regular polyhedrons whose shape depends upon their chemical composition. depends upon their chemical composition. – Single crystals of metals of many cubic Single crystals of metals of many cubic

centimeters in volume are relatively easy to centimeters in volume are relatively easy to prepare in the laboratory. prepare in the laboratory.

Single crystalSingle crystal

22

• Some engineering applications require single crystals:

• Crystal properties reveal features of atomic structure.


--Ex: Certain crystal planes in quartz fracture more easily than others.

--diamond single crystals for abrasives

--turbine bladesFig. 8.30(c), Callister 6e.(Fig. 8.30(c) courtesyof Pratt and Whitney).(Courtesy Martin

Deakins,GE Superabrasives, Worthington, OH. Used with permission.)

CRYSTALS AS BUILDING BLOCKSCRYSTALS AS BUILDING BLOCKS

WhiskersWhiskers

• Very thin filaments, hair-like single crystals Very thin filaments, hair-like single crystals of about 13 mm length and perhaps 10of about 13 mm length and perhaps 10–4 –4

cm diameter.cm diameter.

• Produced as dislocations of free crystals Produced as dislocations of free crystals and are without any structural defect. and are without any structural defect.

• Far stronger than polycrystals of same Far stronger than polycrystals of same material. material.

WhiskersWhiskers

• Used as reinforcements in materials to increase Used as reinforcements in materials to increase strength by embedding fibers of one material in strength by embedding fibers of one material in a matrix of another. a matrix of another. – The properties of fiber or whisker-reinforced The properties of fiber or whisker-reinforced

composites can often be tailored for a specific composites can often be tailored for a specific application. application.

• The increase in diameter of the whiskers The increase in diameter of the whiskers decreases its strength and increases its ductility. decreases its strength and increases its ductility.

• The cost of whiskers and the expensive The cost of whiskers and the expensive fabrication is the major disadvantage of the fabrication is the major disadvantage of the method.method.

WhiskersWhiskers

• Whiskers are the most defect-free crystalline Whiskers are the most defect-free crystalline solids available today. solids available today.

• Whiskers can bear considerably high stresses Whiskers can bear considerably high stresses both at low and relatively elevated temperatures.both at low and relatively elevated temperatures.

• The best-known composite are probably, fiber The best-known composite are probably, fiber glass, glass, – consists of glass-reinforcing fibers in a matrix of either consists of glass-reinforcing fibers in a matrix of either

an epoxy polymer or polyester an epoxy polymer or polyester • Whiskers of SiC, AlWhiskers of SiC, Al22OO33, S-Glass, graphite, , S-Glass, graphite,

boron, iron, silver, copper and tin can be boron, iron, silver, copper and tin can be produced by means of special techniques.produced by means of special techniques.

WhiskersWhiskers

• Whiskers of a wide variety of substances, Whiskers of a wide variety of substances, e.g., mercury, graphite, sodium and e.g., mercury, graphite, sodium and potassium chlorides, copper, iron, and potassium chlorides, copper, iron, and aluminium oxide, have been grown from aluminium oxide, have been grown from super saturated media. super saturated media. – Whiskers grown in this way are usually a few Whiskers grown in this way are usually a few

micrometers in diameter and up to a few micrometers in diameter and up to a few inches long. inches long.

– Some are exceptionally strong, both in bend Some are exceptionally strong, both in bend tests and in tension tests.tests and in tension tests.

WhiskersWhiskers

• In addition to exceptional strength, In addition to exceptional strength, whiskers often have unique electrical, whiskers often have unique electrical, magnetic, or surface properties.magnetic, or surface properties.– This behavior can be interpreted to mean that This behavior can be interpreted to mean that

the crystal structure of whiskers is virtually the crystal structure of whiskers is virtually perfect, particularly with respect to line perfect, particularly with respect to line defects. defects. • Actually it appears that some whiskers contain line Actually it appears that some whiskers contain line

defects whereas others do not. However, no defects whereas others do not. However, no general correlation between whisker properties general correlation between whisker properties and whisker structure have been established.and whisker structure have been established.

PolycrystallinePolycrystalline

• Most crystalline solids Most crystalline solids – made up of millions of grains made up of millions of grains – each grain constitutes microstructure each grain constitutes microstructure – commonly said as polycrystalline.commonly said as polycrystalline.

• GrainGrain– a tiny single crystals a tiny single crystals – oriented randomly with respect to each otheroriented randomly with respect to each other

23

• Most engineering materials are polycrystals.

• Nb-Hf-W plate with an electron beam weld.• Each "grain" is a single crystal.• If crystals are randomly oriented, overall component properties are not directional.• Crystal sizes typ. range from 1 nm to 2 cm (i.e., from a few to millions of atomic layers).

Adapted from Fig. K, color inset pages of Callister 6e.(Fig. K is courtesy of Paul E. Danielson, Teledyne Wah Chang Albany)

1 mm

POLYCRYSTALSPOLYCRYSTALS

POLYMORPHISM AND ALLOTROPYPOLYMORPHISM AND ALLOTROPY

• Some metals, as well as nonmetals, may Some metals, as well as nonmetals, may have more than one crystal structure, a have more than one crystal structure, a phenomenon known as phenomenon known as polymorphismpolymorphism. .

• When found in elemental solids, the When found in elemental solids, the condition is often termed condition is often termed allotropyallotropy. .

• The prevailing crystal structure depends The prevailing crystal structure depends on both the temperature and the external on both the temperature and the external pressure. pressure.

POLYMORPHISM AND POLYMORPHISM AND ALLOTROPYALLOTROPY

• One familiar example is found in carbon:One familiar example is found in carbon:– graphite is the stable polymorph at ambient graphite is the stable polymorph at ambient

conditions, conditions, – diamond is formed at extremely high pressures. diamond is formed at extremely high pressures.

• Pure iron has a Pure iron has a – BCC crystal structure at room temperature, BCC crystal structure at room temperature, – changes to FCC iron at 912°C.changes to FCC iron at 912°C.

• Most often a modification of the density and Most often a modification of the density and other physical properties accompanies a other physical properties accompanies a polymorphic transformation.polymorphic transformation.

DENSITY COMPUTATIONS

12

Element Aluminum Argon Barium Beryllium Boron Bromine Cadmium Calcium Carbon Cesium Chlorine Chromium Cobalt Copper Flourine Gallium Germanium Gold Helium Hydrogen

Symbol Al Ar Ba Be B Br Cd Ca C Cs Cl Cr Co Cu F Ga Ge Au He H

At. Weight (amu) 26.98 39.95 137.33 9.012 10.81 79.90 112.41 40.08 12.011 132.91 35.45 52.00 58.93 63.55 19.00 69.72 72.59 196.97 4.003 1.008

Atomic radius (nm) 0.143 ------ 0.217 0.114 ------ ------ 0.149 0.197 0.071 0.265 ------ 0.125 0.125 0.128 ------ 0.122 0.122 0.144 ------ ------

Density (g/cm3) 2.71 ------ 3.5 1.85 2.34 ------ 8.65 1.55 2.25 1.87 ------ 7.19 8.9 8.94 ------ 5.90 5.32 19.32 ------ ------

Crystal Structure FCC ------ BCC HCP Rhomb ------ HCP FCC Hex BCC ------ BCC HCP FCC ------ Ortho. Dia. cubic FCC ------ ------

Adapted fromTable, "Charac-teristics ofSelectedElements",inside frontcover,Callister 6e.

Characteristics of Selected Elements at 20CCharacteristics of Selected Elements at 20C

metals ceramics polymers

13

(g

/cm

3)

Graphite/ Ceramics/ Semicond

Metals/ Alloys

Composites/ fibersPolymers

1

2

20

30Based on data in Table B1, Callister *GFRE, CFRE, & AFRE are Glass,

Carbon, & Aramid Fiber-Reinforced Epoxy composites (values based on 60% volume fraction of aligned fibers

in an epoxy matrix). 10

3 4 5

0.3 0.4 0.5

Magnesium

Aluminum

Steels

Titanium

Cu,Ni

Tin, Zinc

Silver, Mo

Tantalum Gold, W Platinum

Graphite Silicon

Glass -soda Concrete

Si nitride Diamond Al oxide

Zirconia

HDPE, PS PP, LDPE

PC

PTFE

PET PVC Silicone

Wood

AFRE *

CFRE *

GFRE*

Glass fibers

Carbon fibers

Aramid fibers

Why? Metals have... • close-packing (metallic bonding) • large atomic mass Ceramics have... • less dense packing (covalent bonding) • often lighter elements Polymers have... • poor packing (often amorphous) • lighter elements (C,H,O) Composites have... • intermediate values Data from Table B1, Callister 6e.

DENSITIES OF MATERIAL CLASSESDENSITIES OF MATERIAL CLASSES

d=n/2sinc

x-ray intensity (from detector)

c25

• Incoming X-rays diffract from crystal planes.

• Measurement of: Critical angles, c, for X-rays provide atomic spacing, d.

Adapted from Fig. 3.2W, Callister 6e.

X-RAYS TO CONFIRM CRYSTAL STRUCTUREX-RAYS TO CONFIRM CRYSTAL STRUCTURE

reflections must be in phase to detect signal

spacing between planes

d

incoming

X-rays

outg

oing

X-ra

ys

detector

extra distance travelled by wave “2”

“1”

“2”

“1”

“2”

XRD patternXRD pattern

• • Atoms may assemble into Atoms may assemble into crystallinecrystalline or or amorphousamorphous structures. structures.

• • We can predict the We can predict the densitydensity of a material, of a material, provided we know the provided we know the atomic weightatomic weight, , atomicatomic radiusradius, and , and crystalcrystal geometrygeometry (e.g., FCC, (e.g., FCC, BCC, HCP).BCC, HCP).

• • Material properties generally vary with singleMaterial properties generally vary with single crystal orientation (i.e., they are crystal orientation (i.e., they are anisotropicanisotropic),), but properties are generally non-directionalbut properties are generally non-directional (i.e., they are (i.e., they are isotropicisotropic) in polycrystals with) in polycrystals with randomly oriented grains.randomly oriented grains.

27

SUMMARYSUMMARY

• Part 2: Material PropertiesPart 2: Material Properties– Mechanical PropertiesMechanical Properties– Thermal PropertiesThermal Properties– Electrical and Magnetic Properties Electrical and Magnetic Properties – SuperconductivitySuperconductivity

• Part 3: Material EngineeringPart 3: Material Engineering– Alloy Systems Alloy Systems – Phase Diagrams and Phase TransformationsPhase Diagrams and Phase Transformations – Heat TreatmentHeat Treatment – Deformation of MaterialsDeformation of Materials– CorrosionCorrosion – Organic Materials: Organic Materials:

• Polymers and ElastomersPolymers and Elastomers • WoodWood• CompositesComposites

– Nano structured MaterialsNano structured Materials

ms02

Documents

crystal lattice

cubic crystal structures

crystala crystal

perfect crystal

internal structure of

bravias lattice

space latticethe periodicity

array of imaginary points