lecture 10 (10/16/2006) crystallography part 3: crystallographic axes numerical notation of crystal...
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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
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
Crystal Zone and Zone AxisCrystal Zone and Zone Axis
Zone [100]Zone [100]
Zone [001]Zone [001]
Zone [100] axis
Zone [001] axis