1.3.1- Planos y Direcciones Critalinas

Direcciones y PlanosCristalinos

Direcciones y PlanosDirecciones y PlanosCristalinosCristalinos

Point Coordinates in an Orthogonal Coordinate System Simple Cubic

Point Coordinates in an Orthogonal Point Coordinates in an Orthogonal Coordinate System Simple CubicCoordinate System Simple Cubic

Crystallographic Directions in Cubic SystemCrystallographic Directions in Cubic SystemCrystallographic Directions in Cubic System

• Determination of the directional indices in cubic system:Four Step Procedure (Text Book Method)

1. Draw a vector representing the direction within the unit cell such that it passes through the origin of the xyz coordinate axes.

2. Determine the projections of the vector on xyz axes.3. Multiply or divide by common factor to obtain the three

smallest integer values.4. Enclose the three integers in square brackets [ ].

e.g. [uvw]

u, v, and w are the integers

Crystallographic Directions in Cubic System

Crystallographic Directions in Crystallographic Directions in Cubic SystemCubic System

[120]

[110]

[111]

Head and Tail Procedure for determining Miller Indices for Crystallographic

Directions

Head and Tail Procedure for determining Head and Tail Procedure for determining Miller Indices for Crystallographic Miller Indices for Crystallographic

DirectionsDirections

1. Find the coordinate points of head and tail points.

2. Subtract the coordinate points of the tail from the coordinate points of the head.

3. Remove fractions.4. Enclose in [ ]

Indecies of Crystallographic Directions in Cubic System

Indecies of Crystallographic Directions in Indecies of Crystallographic Directions in Cubic SystemCubic System

Direction A

Head point – tail point

(1, 1, 1/3) – (0,0,2/3)

1, 1, -1/3

Multiply by 3 to get smallest integers

3, 3, -1

A = [33Ī]

_ _ B = [403]

Direction BHead point – tail point(0, 1, 1/2) – (2/3,1,1)-2/3, 0, -1/2Multiply by 6 to get smallest integers

C = [???] D = [???]

Indices of Crystallographic Directions in Cubic System

Indices of Crystallographic Directions in Indices of Crystallographic Directions in Cubic SystemCubic System

A = [???]

_ _

C = [0I2]

B= [???]_

D = [I2I]

Direction C

Head Point – Tail Point

(1, 0, 0) – (1, ½, 1)

0, -1/2, -1

Multiply by 2 to get the smallest integers

Direction DHead Point – Tail Point(1, 0, 1/2) – (1/2, 1, 0)1/2, -1, 1/2Multiply by 2 to get the smallest integers

Crystallographic Directions in Cubic System

Crystallographic Directions in Crystallographic Directions in Cubic SystemCubic System

[210]

Indices of a Family or FormIndices of a Family or FormIndices of a Family or Form

00]1[],1[000],1[0[001],[010],[100], >100 ≡<

]11[1],111[1],11[],111[11],1[1],1[1],1[11[111],111 >< ≡

01]1[1],1[010],1[],101[],11[00],11[

]1[10],1[010],1[1[101],[011],[110],110 >< ≡

MILLER INDICES FOR MILLER INDICES FOR CRYSTALLOGRAPHIC PLANESCRYSTALLOGRAPHIC PLANES

• Miller Indices for crystallographic planes are the reciprocals of the fractional intercepts (with fractions cleared) which the plane makes with the crystallographic x,y,z axes of the three nonparallel edges of the cubic unit cell.

• 4-Step Procedure:1. Find the intercepts that the plane makes with the three

axes x,y,z. If the plane passes through origin change the origin or draw a parallel plane elsewhere (e.g. in adjacent unit cell)

2. Take the reciprocal of the intercepts3. Remove fractions4. Enclose in ( )

Miller Indecies of Planes in Crystallogarphic Planes in Cubic System

Miller Indecies of Planes in Miller Indecies of Planes in Crystallogarphic Planes in Cubic SystemCrystallogarphic Planes in Cubic System

Drawing Plane of known Miller Indices in a cubic unit cell

Drawing Plane of known Miller Indices in Drawing Plane of known Miller Indices in a cubic unit cella cubic unit cell

Draw ( ) plane110

Miller Indecies of Planes in Crystallogarphic Planes in Cubic System

Miller Indecies of Planes in Miller Indecies of Planes in Crystallogarphic Planes in Cubic SystemCrystallogarphic Planes in Cubic System

Origin for AOrigin for B

Origin for A

A = (IĪ0) B = (I22) A = (2IĪ) B = (02Ī)

CRYSTALLOGRAPHIC PLANES AND DIRECTIONS IN HEXAGONAL UNIT CELLS

Miller-Bravais indices -- same as Miller indices for cubic crystals except that there are 3 basal plane axes and 1 vertical axis.

Basal plane -- close packed plane similar to the (1 1 1) FCC plane.

contains 3 axes 120o apart.

Direction Indices in HCP Unit CellsDirection Indices in HCP Unit Cells –

[uvtw] where t=-(u+v)Conversion from 3-index system to 4-index system:

'

''

''

)(

)2(31

)2(31

wwvut

uvv

vuu

=

+−=

−=

−=

][][ ''' uvtwwvu →

MillerMiller--Bravais Indices for Bravais Indices for crystallographic directions in HCPcrystallographic directions in HCP

[011] = ]3121[__

[1-1-1] = ]2311[__

[-101] = ]1132[_

MillerMiller--Bravais Indices for Bravais Indices for crystallographic planes in HCPcrystallographic planes in HCP

MillerMiller--BravaisBravais indices areindices are h,k,i,lh,k,i,lwith i = with i = --((h+kh+k).).

Basal plane indices (0 0 0 1)Basal plane indices (0 0 0 1)

Intersecciones a1= Intersecciones a1= --1 a2 = 0 c = 01 a2 = 0 c = 0Recíprocos Recíprocos --1 0 01 0 0Índices de Índices de MillerMiller ((--100)100)Índices de Índices de MillerMiller--BravaisBravais ((--1010)1010) donde i = donde i = --(h + k)(h + k)

_(1211)

Intersecciones a1= Intersecciones a1= --1 a2 = 1/2 c = 11 a2 = 1/2 c = 1Recíprocos Recíprocos --1 2 11 2 1Índices de Índices de MillerMiller ((--121)121)Índices de Índices de MIllerMIller--BravaisBravais ((--1212--11)11) donde i = donde i = --(h + k)

Intersecciones a1= 1 a2 = 0 c = ½Intersecciones a1= 1 a2 = 0 c = ½Recíprocos 1 0 2Recíprocos 1 0 2Índices de Índices de MillerMiller (102)(102)IndicesIndices de de MIllerMIller--BravaisBravais ((1010--12)12) donde i = donde i = --(h + k)(h + k) (h + k)

Atomic Arrangement on (110) plane in FCC

Atomic Arrangement on (110) Atomic Arrangement on (110) plane in FCC plane in FCC

Atomic Arrangement on (110) plane in BCC

Atomic Arrangement on Atomic Arrangement on (110) plane in BCC(110) plane in BCC

Atomic arrangement on [110] direction in FCC

Atomic arrangement on [110] Atomic arrangement on [110] direction in FCCdirection in FCC

Linear and Planar Atomic DensitiesLinear and Planar Atomic DensitiesLinear and Planar Atomic Densities

• Linear Density “LD”is defined as the number of atoms per unit length whose centers lie on the direction vector of a given crystallographic direction.

vectordirection ofLength directionon centered atoms of No.

=LD

Linear DensityLinear DensityLD for [110] in BCC.No. of atom centered on the direction

vector [110] = 1/2 +1/2 = 1

Length of direction vector [110] = √2 a

a = 4R/ √ 3

243

)3/4(21

21

RRaLD ===

[110]

√√ 2a

Linear DensityLinear Density

• LD of [110] in FCC# of atom centered on the direction

vector [110] = 2 atoms

Length of direction vector [110] = 4R

LD = 2 /4R

LD = 1/2RLinear density can be defined as reciprocal of the repeat distance ‘r’

LD = 1/r

Planar DensityPlanar Density• Planar Density “PD”is defined as the number of atoms per unit

area that are centered on a given crystallographic plane.

No of atoms centered on the planePD = —————————————

Area of the plane

Planar Density of (110) plane in FCCPlanar Density of (110) plane in FCC# of atoms centered on the

plane (110) = 4(1/4) + 2(1/2) = 22 atoms

Area of the plane= (4R)(2R √ 2) = 8R22

(111) Plane in FCC

241

282

22110 RRatomsPD ==

a = 2R √√ 2

4R

Closed Packed Crystal StructuresClosed Packed Crystal Structures

• FCC and HCP both have:CN = 12 and APF = 0.74

APF= 0.74 is the most efficient packing.

Both FCC and HCP have Closed Packed PlanesFCC ----(111) plane is the Closed Packed PlaneHCP ----(0001) plane is the Closed Packed PlaneThe atomic staking sequence in the above twostructures is different from each other

Closed Packed Structures

Closed Packed Plane Stacking in HCPClosed Packed Plane Stacking in HCP

Closed Packed Plane Stacking in FCCClosed Packed Plane Stacking in FCC

Crystalline and Crystalline and NoncrystallineNoncrystalline MaterialsMaterialsSingle CrystalsSingle Crystals

• For a crystalline solid, when the periodic and repeated arrangement of atoms is perfect or extends throughout the entirety of the specimen without interruption, the result is a single crystal.

• All unit cells interlock in the same way and have the same orientation.

• Single crystals exist in nature, but may also be produced artificially.

• They are ordinarily difficult to grow, because the environment must be carefully controlled.

• Example: Electronic microcircuits, which employ single crystals of silicon and other semiconductors.

Polycrystalline MaterialsPolycrystalline MaterialsPolycrytallinePolycrytalline MaterialsMaterials

PolycrystallinePolycrystalline crystalline solids crystalline solids composed of many small composed of many small crystals or grains.crystals or grains.

Various stages in the solidification :Various stages in the solidification :a)a) Small crystallite nuclei Growth Small crystallite nuclei Growth

of the crystallites.of the crystallites.

b)b) Obstruction of some grains that Obstruction of some grains that are adjacent to one another is are adjacent to one another is also shown.also shown.

c)c) Upon completion of Upon completion of solidification, grains that are solidification, grains that are adjacent to one another is also adjacent to one another is also shown.shown.

d)d) Grain structure as it would Grain structure as it would appear under the microscope.appear under the microscope.

AnisotropyAnisotropy• The physical properties of single crystals of some

substances depend on the crystallographic direction in which the measurements are taken.

• For example, modulus of elasticity, electrical conductivity, and the index of refraction may have different values in the [100] and [111] directions.

• This directionality of properties is termed anisotropy.

• Substances in which measured properties are independent of the direction of measurement are isotropic.

http://www.youtube.com/watch?v=FSBnkt6P2lA&feature=related

http://www.youtube.com/watch?v=Rm-i1c7zr6Q

1.3.1- Planos y Direcciones Critalinas

Documents

Transcript of 1.3.1- Planos y Direcciones Critalinas