1. Solid Crystallography
Engineering 45
Solid Crystallography
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer

2. Crystal Navigation
As Discussed Earlier A Unit Cell is completely Described by Six Parameters
Lattice Dimensions: a, b ,c
Lattice (InterAxial) Angles: ,  ,
Navigation within a Crystal is Performed in Fractional Units of the Lattice Dimensions a, b, c

3. Point COORDINATES
Cartesian CoOrds (x,y,z) within a Xtal are written in Standard Paren & Comma notation, but in terms of Lattice Fractions.
Example
Given TriClinic unit Cell at Right
Sketch the Location of the Point with Xtal CoOrds of: (1/2, 2/5, 3/4)

4. Point CoOrdinate Example
From The CoOrd Spec, Convert measurement to Lattice Constant Fractions
x → 0.5a
y → 0.4b
z → 0.75c
To Locate Point Mark-Off Dists on the Axes
Located Point (1/2, 2/5, 3/4)

5. Crystallographic DIRECTIONS
Convention to specify crystallographic directions: 3 indices, [uvw] - reduced projections along x,y,z axes
Procedure to Determine Directions
vector through origin, or translated if parallelism is maintained
length of vector- PROJECTION on each axes is determined in terms of unit cell dimensions (a, b, c); negative index in opposite direction
reduce indices to smallest INTEGER values
enclose indices in brackets w/o commas x z y x z y
[122]
[111]
[001] _
[110]
[010]

6. Example  Xtal Directions
Write the Xtal Direction, [uvw] for the vector Shown Below
Step-1: Translate Vector to The Origin in Two SubSteps

7. Example  Xtal Directions
After −x Translation, Make −z Translation
Step-2: Project Correctly Positioned Vector onto Axes

8. Example  Xtal Directions
Step-3: Convert Fractional Values to Integers using LCD for 1/2 & 1/3 → 1/6
x: (−a/2)•(6/a) = −3
y: a•(6/a) = 6
z: (−2a/3)•(6/a) = −4
Step-4: Reduce to Standard Notation:

9. Crystallographic PLANES
Planes within Crystals Are Designated by the MILLER Indices
The indices are simply the RECIPROCALS of the Axes Intersection Points of the Plane, with All numbers INTEGERS
e.g.: A Plane Intersects the Axes at (x,y,z) of (−4/5,3,1/2) Then The Miller indices:

10. Miller Indices  Step by Step
MILLER INDICES specify crystallographic planes: (hkl)
Procedure to Determine Indices
If plane passes through origin, move the origin (use parallel plane)
Write the INTERCEPT for each axis in terms of lattice parameters (relative to origin)
RECIPROCALS are taken: plane parallel to axis is zero (no intercept → 1/ = 0)
Reduce indices by common factor for smallest integers
Enclose indices in Parens w/o commas
The last one is actuall (1 1 -1)

11. Example  Miller Indices
Find The Miller Indices for the Cubic-Xtal Plane Shown Below

12. The Miller Indices Example
In Tabular Form

13. More Miller Indices Examples
Consider the (001) Plane x z y
x y z
Intercepts
Reciprocals
Reductions
Enclosure 1
0 0 1
(none needed)
(001)
Some Others
(3/4, ½, -1/4) => invert => (4/3, 2, -4) => reduce by mult by 3/2 => (2 3 -6)

14. FAMILIES of DIRECTIONS
Crystallographically EQUIVALENT DIRECTIONS → < V-brackets > notation
e.g., in a cubic system,
Family of <111> directions: SAME Atomic ARRANGEMENTS along those directions

15. FAMILIES of PLANES
Crystallographically EQUIVALENT PLANES → {Curly Braces} notation
e.g., in a cubic system,
Family of {110} planes: SAME ATOMIC ARRANGEMENTS within all those planes

16. Hexagonal Structures
Consider the Hex Structure at Right with 3-Axis CoOrds
Plane-C
The Miller Indices
Plane-A → (100)
Plane-B → (010)
Plane-C → (110)
Plane-B
BUT
Planes A, B, & C are Crystallographically IDENTICAL
The Hex Structure has 6-Fold Symmetry
Direction [100] is NOT normal to (100) Plane
Plane-A

17. 4-Axis, 4-Index System
To Clear Up this Confusion add an Axis in the BASAL, or base, Plane
Plane-C
The Miller Indices now take the form of (hkil)
Plane-A →
Plane-B →
Plane-C →
Plane-B
Plane-A

18. 4-Axis Directions
Find Direction Notation for the a1 axis-directed unit vector
Noting the Right-Angle Projections find

19. More 4-Axis Directions
[ ] directions is at 90deg to a3; i.e., it splits the 60deg angle

20. 4-Axis Miller-Bravais Indices
Construct Miller-Bravais (Plane) Index-Sets by the Intercept Method
Plane
Plane

21. 4-Axis Miller-Bravais Indices
Construct More Miller-Bravais Indices by the Intercept Method
Plane
Plane

22. 3axis↔4axis Translation
The 3axis Indices
Where n  LCD/GCF needed to produce integers-only
Example [100]
The 4axis Version
Conversion Eqns
Thus with n = 1

23. 4axis Indices CheckSum Given 4axis indices
Directions → [uvtw]
Planes → (hkil)
Then due to Reln between a1, a2, a3

24. Linear & Areal Atom Densities
Linear Density, LD  Number of Atoms per Unit Length On a Straight LINE
Planar Density, PD  Number of Atoms per Unit Area on a Flat PLANE
PD is also called The Areal Density
In General, LD and PD are different for Different
Crystallographic Directions
Crystallographic Planes

25. Silicon Crystallography
Structure = DIAMOND; not ClosePacked
Diamond APF = (8*pi*r^3/3)(8*r/sqrt(3))^3 = 34%

26. LD & PD for Silicon Si

27. LD and PD For Silicon For 100 Silicon For {111} Silicon
LD on Unit Cell EDGE
For {111} Silicon
PD on (111) Plane
Use the (111) Unit Cell Plane
1-cos3o => 13.4% lower atom density on (100) plane

28. X-Ray Diffraction → Xtal Struct.
As Noted Earlier X-Ray Diffraction (XRD) is used to determine Lattice Constants
Concept of XRD → Constructive Wave Scattering
Consider a Scattering event on 2-Waves
The Dectector collects a vast number of wave; thus in-phase wave add to the magnitude of the signal
Amplitude 100% Added
Amplitude 100% Subtracted
Constructive Scattering
Destructive Scattering

29. Path-Length Difference
XRD Quantified
X-Rays Have WaveLengths, , That are Comparable to Atomic Dimensions
Thus an Atom’s Electrons or Ion-Core Can Scatter these X-rays per The Diagram Below
Path-Length Difference

30. XRD Constructive Interference1 1’ 2’
The Path Length Difference is Line Segment SQT 2
Waves 1 & 2 will be IN-Phase if the Distance SQT is an INTEGRAL Number of X-ray WaveLengths
Quantitatively
Now by Constructive Criteria Requirement
Thus the Bragg Law

31. XRD Charateristics
The InterPlanar Spacing, d, as a Function of Lattice Parameters (abc) & Miller Indices (hkl)
By Geometry for OrthoRhombic Xtals
For Cubic Xtals a = b = c, so d

32. XRD Implementation X-Ray Diffractometer Schematic Typical SPECTRUM
T  X-ray Transmitter
S  Sample/Specimen
C  Collector/Detector
Typical SPECTRUM
Spectrum  Intensity/Amplitude vs. Indep-Index Pb

33. X-Ray Diffraction Patternz x y a b c z x y a b c z x y a b c
(110)
(211)
Intensity (relative)
(200)
Diffraction angle 2θ
Diffraction pattern for polycrystalline α-iron (BCC)

34. XRD Example  Nb Given Niobium, Nb with FIND
BCC Niobium
XRD Example  Nb
Given Niobium, Nb with
Structure = BCC
X-ray = Å
(211) Plane Diffraction Angle, 2∙θ = 75.99°
n = 1 (primary diff)
FIND
ratom
d211
Find InterPlanar Spacing by Bragg’s Law

35. Nb XRD cont To Determine ratom need The Cubic Lattice Parameter, a
Use the Plane-Spacing Equation
For the BCC Geometry by Pythagorus

36. PolyCrystals → Grains Most engineering materials are POLYcrystals 1 mm
Nb-Hf-W plate with an electron beam weld
1 mm
Each "grain" is a single crystal.
If crystals are randomly oriented, then overall component properties are not directional.
Crystal sizes typically range from 1 nm to 20 mm
(i.e., from a few to millions of atomic layers).

37. Single vs PolyCrystals
• Single Crystals
-Properties vary with
direction: anisotropic.
-Example: the modulus
of elasticity (E) in BCC iron:
• Polycrystals
-Properties may/mayNot
vary with direction.
-If grains are randomly
oriented: isotropic.
(Epoly iron = 210 GPa)
-If grains are textured,
anisotropic.
200 mm 19

38. WhiteBoard Work Planar-Projection (Similar to P3.48)
Given Three Plane-Views, Determine Xtal Structure
Also:
3.47 in 7e
Find Aw

41. xTal Planes in Simple Cubic Unit Cell
All Done for Today
xTal Planes in Simple Cubic Unit Cell

42. Planar Projection
101
101
Structure is Face Centered ORTHORHOMBIC

43. Planar Projection