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Z. [0 1 0]. [1 0 1]. Y. X. - 2 / 3 11. z. [111]. [001]. c. y. [120]. b. 1½0. [210]. x. a. [100]. For cubic: a = b = c = a o. Miller Indices. Miller Indices. Z. Z. Z. Y. Y. Y. X. X. X. (100). (110). (111). FAMÍLIA DE PLANOS {110} É paralelo à um eixo.

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Presentation Transcript
slide2

Z

[0 1 0]

[1 0 1]

Y

X

slide3

-2/311

z

[111]

[001]

c

y

[120]

b

1½0

[210]

x

a

[100]

For cubic: a = b = c = ao

slide7

Z

Z

Z

Y

Y

Y

X

X

X

(100)

(110)

(111)

directions miller indices in hexagonal structures
Directions & Miller Indices in Hexagonal Structures

[011]

(0001)

[210]

[UVW] or [uvtw]

(hkil) or (hk·l)

diamond lattice
Diamond Lattice

(100)

(110)

slide13

Spacing of Planes

Cubic:

Cubic:

Tetragonal:

Tetragonal:

Hexagonal:

Rhombohedral:

spacing of planes
Spacing of Planes

Orthorhombic:

Monoclinic:

Triclinic:

reciprocal lattice
Reciprocal Lattice

Unit cell: b1, b2, b3

Reciprocal lattice unit cell: b1*, b2*, b3* defined by:

b3*

B

P

C

b3

b2

A

O

b1

reciprocal lattice16
Reciprocal Lattice

Like the real-space lattice, the reciprocal space lattice also has a translation vector, Kl:

Where the length of R·K is equal to:

The magnitude of the translation vector has the following relationship:

angles and inner planar spacing
Angles and Inner Planar Spacing

is  to (hkl) plane. Therefore, the angle between (h1k1l1) and (h2k2l2) planes is the angle between the Kh1k1l1 and Kh2k2l2 vectors.

Recall the dot product:

Angles between reciprocal

lattice vectors.

two dimensional lattice
Two Dimensional Lattice

Wigner-Seitz

Possible choices of primitive cell for a single 2D Bravais lattice.

first brillouin zone
First Brillouin Zone

If these lattice points now represent reciprocal lattice points, then the first Brillouin zone is just the Wigner-Seitz cell of the reciprocal lattice.

b2*

b1*

slide21

DIFRAÇÃO DE RAIOS XLEI DE BRAGG

n= 2 dhkl.sen

  • É comprimento de onda

N é um número inteiro de ondas

d é a distância interplanar

 O ângulo de incidência

Válido para sistema cúbico

dhkl= a

(h2+k2+l2)1/2

slide22

DISTÂNCIA INTERPLANAR (dhkl)

  • É uma função dos índices de Miller e do parâmetro de rede

dhkl= a

(h2+k2+l2)1/2

slide23

TÉCNICAS DE DIFRAÇÃO

  • Técnica do pó:

É bastante comum, o material a ser analisado encontra-se na forma de pó (partículas finas orientadas ao acaso) que são expostas à radiação x monocromática. O grande número de partículas com orientação diferente assegura que a lei de Bragg seja satisfeita para alguns planos cristalográficos

slide24

O DIFRATOMÊTRO DE RAIOS X

  • T= fonte de raio X
  • S= amostra
  • C= detector
  • O= eixo no qual a amostra e o detector giram

Amostra

Fonte

Detector