Homework assignment 3

ENGR 270 Homework #3 Crystal Structure of Metals and Crystallography

Introduction to Crystal Structure:

1. For each material class or subclass listed below, write the letter from the list on the right that best describes the
type of atomic arrangement most common for that group of materials.

Arrangement

(A, B, C, D) Material Class
Atomic Arrangement

Metals A Semi-Crystalline

Ceramics B
Single crystal or

Polycrystalline

Glasses C Amorphous

Polymers D Polycrystalline

Unit Cell:

2. Define the term unit cell:

Bravais Lattices questions:

3. Define the term crystal lattice:

4. Define the term lattice parameter:

5. List the 7 different lattice systems:

6. Which are the 3 different unit cell types commonly found in metals?

7. Define the term polymorphism, and give several examples of polymorphic metals:

Cubic Unit Cells and Their Origins questions:

8. For each of the following cubic unit cells, list the number of atoms per unit cell, coordination number, and close-
packed direction (cube edge, body diagonal, or face diagonal) along which atoms touch each other.

Cubic Cell Type

Atoms

per cell

Coordination

Number Close-packed direction

Simple (SC)

Body-Centered (BCC)

Face-Centered (FCC)

9. Which of the 3 cubic unit cells above has the most efficient (densest) packing? __________

Atomic Packing Factor questions:

10. Use the close-packed direction in a BCC unit cell to derive a geometrical relationship between the atomic radius
R and lattice parameter a for a BCC material.

11. Calculate the Atomic Packing Factor for a BCC material.

Close Packing Crystal Structures questions:

12. What are the two types of close packed crystal structures, and which
stacking sequence is found in each type of structure?

Theoretical Density questions:

13. Calculate the theoretical density of copper in g/cm3, based on the fact that it has an FCC crystal structure,
atomic mass of 63.55 g/mol, and an atomic radius of 0.128 nm. Compare your result to the actual density of 8.96
g/cm3 for copper.

Close-packed

structures

Stacking sequence

14. On average, how do the material classes of metals, ceramics, and polymers rank relative to each other in terms

of density, and what are the reasons for this ranking ?

Introduction to Crystallography

15. What experimental technique, used to determine the atomic-level structure of not only inorganic materials such

as diamond and graphite but also organic molecules such as DNA, has been touted by many as the most

important scientific breakthrough of the 20th century?

Single Crystal vs Polycrystalline vs Amorphous

16. Identify each of the material samples below as (a) having a Single Crystal, Polycrystalline, or Amorphous
structure, and (b) as having properties that are Isotropic or Anisotropic.

a diamond

annealed

copper

cold-rolled

copper glass

Structure

Properties

Crystallographic Points :

17. Write the letter (A, B, C, D) of the point pictured in the figure below next to the corresponding set of

crystallographic indices for the point in the list at right.

Letter

½ ½ ½

½ 1 ½

1 ½ 0

1 0 1

z

x

y

A

B

C
D

Crystallographic Directions:

18. Write the letter (A, B, C, D) of the direction pictured in the figure below next to the corresponding set of

direction indices in the list at right.

19. Determine the indices for the following directions (A, B, C, D) of a cubic unit cell

Letter

[23̅6]

[1̅1̅0]

[001]

[012]

z

x

y

½

⅓

A

B

C D

20. Determine the indices for the following directions in the Miller-Bravais coordinate system for a Hexagonally Closed

Packed (HCP) unit cell

Crystallographic Planes:

21. Write the letter (A, B, C, D) of each shaded plane pictured below next to the corresponding set of plane indices

in the list at right.

Letter

(111)

(001̅)

(120)

(11̅0)

z

x

y

½

A

z

x

y B

z

x

y C

z

x

y
D

22. Write the Miller Indices for the following planes.

Families of Directions/Planes:

23. In the table at right, write the direction indices for each face diagonal direction pictured below. In the adjacent

column, write the indices for the direction that points exactly opposite to the one pictured.

24. Note that the directions you listed above comprise the entire <110> family in the cubic system, corresponding to

the face diagonal directions. Based on the pattern you notice in the sets of indices, write below the entire set of

direction indices for the <100> family:

z

x

y

A B

C D

E

F

25. Does this <100> family of directions correspond to edge directions or body diagonals? (circle)

26. Which family of planes represents faces of a cubic unit cell? {111} {110} {100} (circle one)

27. What is the geometrical relationship between the [100] direction and (100) plane? How about [110] direction

and (110) plane? Between any [hkl] direction and (hkl) plane in cubic systems?

Atomic Densities and Packing Factors

28. Which direction in FCC crystals has the highest linear density (circle)? <111> <110> <100>

29. Which direction in BCC crystals has the highest linear density (circle)? <111> <110> <100>

Powder X-ray Diffraction

30. What is the interplanar spacing d between adjacent planes in a crystal that result in constructive interference of

X-rays with wavelength 0.154 nm, when the Bragg angle is 19.22 degrees? (assume n=1)

31. Suppose that the interference peak in the question above was produced by a set of (110) planes in a BCC crystal.

What would be the lattice parameter a for this material?

32. Only some families of {hkl} planes produce diffraction peaks in certain crystal structures. The observed

diffractions in cubic crystals have been summarized by a set of reflection rules:

 For SC crystal structures, all planes produce diffractions.

 For BCC, diffractions occur if h+k+l gives an even result.

 For FCC, diffractions occur when h, k, and l are individually either all even or all odd.

(a) In the table below, X out the boxes where diffraction does NOT occur for each structure.

(b) In cases where diffraction is allowed, calculate the value of h2+k2+l2 and write it in the box.

h2+k2+l2 for allowable diffractions

(hkl) SC BCC FCC

(100)

(110)

(111)

(200)

(210)

(211)

(220)

33. Suppose that a sample of aluminum is subjected to an X-ray diffraction analysis.

(a) What is the lattice parameter a for aluminum, given that it has atomic radius 0.143 nm and FCC crystal

structure?

(b) What would be the expected diffractometer angles (degrees) for the first 3 peaks observed, if the analysis

used X-rays with wavelength 0.154 nm?

Use the table below as a guide for your analysis. Assume n=1 for all peaks.

(hkl) d (nm) Bragg angle 
Diffractometer

angle 2