l02-Crystallographic Planes

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L-02

ENGINEERINGMATERIALS

CRYSTALLOGRAPHIC PLANES


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The Structure of Crystalline Solids

why study The Structure of Crystalline Solids?

The properties of some materials are directly relatedto their crystal structures, e.g., pure and undeformed

magnesium and beryllium, having one crystal

structure, are much more brittle (i.e., fracture at lowerdegrees of deformation) than are pure andundeformed metals such as gold and silver that haveyet another crystal structure.

Furthermore, significant property differences existbetween crystalline and noncrystalline materialshaving the same composition.

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For example, noncrystalline ceramics and polymersnormally are optically transparent; the same

materials in crystalline (or semi crystalline) formtend to be opaque or, at best, translucent.

1.INTRODUCTION: structure of materials,specifically, to some of the arrangements that may

be assumed by atoms in the solid state. Crystalline, noncrystalline solids, three common

crystal structures, scheme by whichcrystallographic points, directions, and planes are

expressed. Single crystals, polycrystalline, and noncrystalline

materials are considered.

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Crystal Structures

2 FUNDAMENTAL CONCEPTS

Crystalline: Solid materials may be classified according to the regularity

with which atoms or ions are arranged with respect to oneanother. A crystalline material is one in whichthe atoms are

situated in a repeating or periodic array over large atomicdistances; that is, long-range order exists, such that uponsolidification, the atoms will position themselves in arepetitive three-dimensional pattern, in which each atom isbonded to its nearest-neighbor atoms.

All metals, many ceramic materials, and certain polymersform crystalline structures under normal solidificationconditions.

For those that do not crystallize, this long-range atomic orderis absent.

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crystal structure Some of the properties of crystalline solids depend on

the crystal structure ofthe material, the manner in

which atoms, ions, or molecules are spatially arranged.

There is an extremely large number of different crystal

structures all having long range atomic order; these varyfrom relatively simple structures for metals to

exceedingly complex ones, as displayed by some of the

ceramic and polymeric materials.

The present discussion deals with several common

metallic crystal structures.5


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(c) an aggregate of

many atoms

Figure 3.1 For the face centered

cubic crystal structure

(a) a hardsphere unit

cell representationA reduced-sphereunit cell


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lattice When describing crystalline structures, atoms (or ions)

are thought of as being solid spheres having well-defined diameters.

This is termed the atomic hard sphere model in whichspheres representing nearest-neighbor atoms touchone another.

An example of the hard sphere model for the atomicarrangement found in some of the common elementalmetals is displayed in Figure 3.1c.

In this particular case all the atoms are identical.Sometimes the term lattice is used in the context ofcrystal structures; in this sense lattice means a three-dimensional array of points coinciding with atompositions (or sphere centers).

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3.UNIT CELLS The atomic order in crystalline solids indicates that

small groups of atoms form a repetitive pattern. Thus, in describing crystal structures, it is often

convenient to subdivide the structure into smallrepeat entities called unit cells.

Unit cells for most crystal structures areparallelepipeds or prisms having three sets of parallelfaces; one is drawn within the aggregate of spheres(Figure 3.1c), which in this case happens to be a cube.

A unit cell is chosen to represent the symmetry of thecrystal structure, wherein all the atom positions in thecrystal may be generated by translations of the unitcell integral distances along each of its edges.

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Thus, the unit cell is the basic structural unit

or building block of the crystal structure and

defines the crystal structure by virtue of itsgeometry and the atom positions within.

Convenience usually dictates that

parallelepiped corners coincide with centers

of the hard sphere atoms.

Furthermore, more than a single unit cell may

be chosen for a particular crystal structure;

however, we generally use the unit cell having

the highest level of geometrical symmetry.


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4 METALLIC CRYSTAL STRUCTURES The atomic bonding in this group of materials is metallic

and thus non directional in nature.

Consequently, there are minimal restrictions as to the

number and position of nearest-neighbor atoms; this

leads to relatively large numbers of nearest neighbors

and dense atomic packings for most metallic crystalstructures.

Also, for metals, using the hard sphere model for the

crystal structure, each sphere represents an ion core.

Table 3.1 presents the atomic radii for a number ofmetals.

Three relatively simple crystal structures are found for

most of the common metals: FCC, BCC, HCP10


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The Face-Centered Cubic Crystal Structure

a unit cell of cubic geometry

a is hard sphere model for the FCC unit cell.

In b atom centers are represented by smallcircles to provide a better perspective of atompositions.

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Fig a Fig bFig 3.1


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FCC this crystal structure is found in copper,

aluminum, silver, and gold

Total of four whole atoms contained in FCC

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BODY CENTERED CRYSTAL STRUCTURE (BCC)

Fig 3.2


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BCC

Chromium, iron, tungsten, metals exhibit a BCC structure

atomic packing

factor for BCC is 0.68

Fig 3.2


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The Hexagonal Close-Packed Crystal Structure

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For the hexagonal close-packed crystal structure, (a) a reduced-sphere unitcell (a

and c represent the short and long edge lengths, respectively), and (b) an

aggregate of many atoms.

Figure 3.3


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The c/a ratio should be 1.633; however,

for some HCP metals this ratio deviatesfrom the ideal value.

The coordination number and the atomic

packing factor for the HCP crystalstructure are the same as for FCC: 12 and

0.74, respectively.

The HCP metals include cadmium,magnesium, titanium, and zinc; some of

these are listed in Table 3.1.

HCP


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5 DENSITY COMPUTATIONS A knowledge of the crystal structure of a metallic solid

permits computation of its theoretical density throughthe relationship.

Where

n number of atoms associated with each unit cell

A atomic weight

VC volume of the unit cell

NA Avogadros number16.023X1023 atoms/mol

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6-POLYMORPHISM AND ALLOTROPY Some metals, as well as nonmetals, may have more than one

crystal structure, a phenomenon known as polymorphism.

When found in elemental solids, the condition is often

termed allotropy.

The prevailing crystal structure depends on both the

temperature and the external pressure.

One familiar example is found in carbon: graphite is the stable

polymorph at ambient conditions, whereas diamond is

formed at extremely high pressures.

Also, pure iron has a BCC crystal structure at room

temperature, which changes to FCC iron at 912 deg C . Most

often a modification

of the density and other physical properties accompanies a

polymorphic transformation.21


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7.CRYSTAL SYSTEMS lattice parameters of a crystal structure are

three edge lengths a, b, and c, and the threeinter axial angles.

there are seven different possible combinations

ofa, b, and c and,, and

Cubic

A=b=c = = = 90

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UNIT CELL

Figure 3.4


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Specification of Point Coordinates

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Location of Point Having Specified CoordinatesFor the unit cell shown in the accompanying sketch (a), locate the point having

coordinates , 1,


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9.CRYSTALLOGRAPHIC DIRECTIONS Figure 3.6 The [100], [110], and [111]

directions within a unit cell..

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Fig 3.6


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Determination of Directional Indices

Determine the indices for the direction shown in the accompanying figure.

Pl recall the concept of vector analysis here


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Construction of Specified Crystallographic Direction Draw a [I0] direction within a cubic unit cell.

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Hexagonal Crystals

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Figure 3.7 Coordinate axis system for a hexagonal unit cell (MillerBravais scheme).


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3.10 CRYSTALLOGRAPHIC PLANES The orientations of planes for a crystal

structure are represented in a similar manner. Again, the unit cell is the basis, with the three-

axis coordinate system as represented in

Figure 3.4. In all but the hexagonal crystal system,

crystallographic planes are specified by threeMiller indices as (hkl).

.Any two planes parallel to each other areequivalent and have identical indices.

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The procedure employed in determination of

the h, k, and l index numbers is as follows:

1. If the plane passes through the selected origin, either another parallel plane

must be constructed within the unit cell by an appropriate translation, or a

new origin must be established at the corner of another unit cell.2. At this point the crystallographic plane either intersects or parallels each of

the three axes; the length of the planar intercept for each axis is determined

in terms of the lattice parameters a, b, and c.

3. The reciprocals of these numbers are taken. A plane that parallels an axis

may be considered to have an infinite intercept, and, therefore, a zero index.4. If necessary, these three numbers are changed to the set of smallest integers

by multiplication or division by a common factor.3

5. Finally, the integer indices, not separated by commas, are enclosed within

parentheses, thus: (hkl).


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CRYSTALLOGRAPHIC PLANES (CONTD) An intercept on the negative side of the origin is

indicated by a bar or minus sign positioned over theappropriate index.

Furthermore, reversing the directions of all indicesspecifies another plane parallel to, on the opposite sideof and equidistant from, the origin.

Several low-index planes are represented in Figure 3.9.

One interesting and unique characteristic of cubiccrystals is that planes and directions having the sameindices are perpendicular to one another; however, for

other crystal systems there are no simple geometricalrelationships between planes and directions having thesame indices.

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Figure 3.9 Representations of a series each of (a) (001), (b) (110), and (c) (111) crystallographicplanes.

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Determination

of Planar

(Miller) Indices:Determine the Miller indices for the plane

shown in the accompanying sketch (a).

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l02-Crystallographic Planes