Lecture 10 (10/16/2006) Crystallography Part 3: Crystallographic Axes Numerical Notation of Crystal Faces and Atomic Planes – Miller Indices.

Lecture 10 (10/16/2006)Lecture 10 (10/16/2006)

CrystallographyCrystallographyPart 3: Part 3:

Crystallographic AxesCrystallographic Axes

Numerical Notation of Crystal Faces Numerical Notation of Crystal Faces and Atomic Planes – Miller Indicesand Atomic Planes – Miller Indices

Point Groups of Crystal SystemsPoint Groups of Crystal Systems

Crystallographic AxesCrystallographic Axes

Isometric Crystal SystemIsometric Crystal System Axes equal length, 90ºAxes equal length, 90º

Aligned with AAligned with A44 (432, 43m, 4/m32/m) (432, 43m, 4/m32/m)

or with Aor with A22 (23, 2/m3) (23, 2/m3)

Hexagonal Crystal SystemHexagonal Crystal System 1 long or short axis (1 long or short axis (cc), 90º to 3 axes of equal ), 90º to 3 axes of equal

length (length (aa11, a, a22, a, a33), 120º to each other ), 120º to each other

cc aligned with A aligned with A6 6 or Aor A33; ; aa11, a, a22, a, a3 3 are aligned parallel are aligned parallel

to to AA2 2 or m planes or arbitrary (3, 3, 6, 6)or m planes or arbitrary (3, 3, 6, 6)

Tetragonal Crystal SystemTetragonal Crystal System 1 long or short axis (1 long or short axis (cc), two axes of equal length ), two axes of equal length

((aa11, a, a22); all axes 90º to each other ); all axes 90º to each other

cc aligned with A aligned with A44; ; aa11 and a and a22 are aligned parallel to are aligned parallel to

AA2 2 or m planes or arbitrary (4, 4)or m planes or arbitrary (4, 4)

Orthorhombic Crystal SystemOrthorhombic Crystal System three axes of unequal length (three axes of unequal length (a, b, ca, b, c); all axes 90º ); all axes 90º

to each other to each other

axes aligned parallel to Aaxes aligned parallel to A22; for mm2, c aligned ; for mm2, c aligned

parallel with Aparallel with A22, , a & ba & b normal to m planes normal to m planes

Monoclinic Crystal SystemMonoclinic Crystal System three axes of unequal length (three axes of unequal length (a, b, ca, b, c); ); ββ ≠ 90º, ≠ 90º,

αα==γγ= 90º, = 90º, bb axis is parallel to A axis is parallel to A2 2 (2, 2/m) or normal to mirror (2, 2/m) or normal to mirror

(m); (m); aa and and cc axes normal to axes normal to bb and normal to and normal to crystal faces or edgescrystal faces or edges

Triclinic Crystal SystemTriclinic Crystal System three axes of unequal length (three axes of unequal length (a, b, ca, b, c); ); α α ≠≠ β β ≠ ≠ γ γ

≠ 90º≠ 90º a, b, a, b, && c c axes normal to crystal faces or edges axes normal to crystal faces or edges

Numerical Notation for Crystal Numerical Notation for Crystal Planes and FacesPlanes and Faces

UnitUnitFaceFace

•relative values, not absolute distancesrelative values, not absolute distances•units defined by largest face cutting all three axes – Unit Faceunits defined by largest face cutting all three axes – Unit Face

Crystal Face InterceptsCrystal Face Intercepts

1c

1b 2b

1/3c

UNITUNITFACEFACE

or 1a, 1b, 1/3cor 1a, 1b, 1/3cor 3a, 3b, 1cor 3a, 3b, 1c

2/3c

3b

Miller IndicesMiller IndicesInvert intercept values and clear fractionsInvert intercept values and clear fractions

Negative Negative IndicesIndices

Law of Rational Indices –Law of Rational Indices –common faces have simple common faces have simple whole numbers for Miller Indices whole numbers for Miller Indices

Crystal Face InterceptsCrystal Face Intercepts

1c

1b 2b

1/3c

UNITUNITFACEFACE

or 1a, 1b, 1/3cor 1a, 1b, 1/3cor 3a, 3b, 1cor 3a, 3b, 1c

2/3c

3bMiller Indices(1/1,1/1,1/1) = 111(1/2,1/2,3/2) = 113(1/2,1/2,3/2) = 113(1/1,1/1,3/1) = 113(1/1,1/1,3/1) = 113(1/3,1/3,1/1) = 113(1/3,1/3,1/1) = 113

Miller Miller Indices in Indices in Hexagonal Hexagonal

SystemSystem

Crystal Zone and Zone AxisCrystal Zone and Zone Axis

Zone [100]Zone [100]

Zone [001]Zone [001]

Zone [100] axis

Zone [001] axis

Next LectureNext Lecture

Crystal Symmetry (Continued)Crystal Symmetry (Continued) Crystal FormsCrystal Forms TwinningTwinning

Read: Chapter 5, p. 201-213Read: Chapter 5, p. 201-213Read: Chapter 6, p. 240-251 for LabRead: Chapter 6, p. 240-251 for Lab

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