1. STRUCTURE OF METALS
Materials Science

2. Chapter 1: The Structure of Metals
Figure 1.1 An outline of the topics described in Chapter 1.

3. Why Study Crystal Structure of Materials?
The properties of materials are directly related to their crystal structures
Significant property differences exist between crystalline and non-crystalline materials having the same composition, diamond Vs graphite
Significance of structure of crystal:
Pure and undeformed magnesium and beryllium, having one crystal structure, are much more brittle (i.e., fracture at lower degrees of deformation) than are pure and undeformed metals such as gold and silver that have yet another crystal structure
Opposite of “Crystalline” is Amorphous. It means lacking definite form; having no specific shape; formless. Diamond is crystalline while graphite is amorphous. Silica may opaque or may be transparent

4. Crystalline and Crystal Structure
A crystalline material is one in which the atoms are situated in a specific order (repeating or periodic array over large atomic distances)
All metals, many ceramics, and some polymers make crystalline structure
Some of the properties of crystalline solids depend on the crystal structure (manner in which atoms, ions or molecules are spatially arranged) of the material
1. Solid materials may be classified according to their crystal structure
2. Ice, NaCl are crystalline solids; the glass for drinking water is an amorphous solid. Heat-treatment makes glass partly crystalline so that it becomes transparent. Nylon, polypropylene are crystalline polymers.
3. Spatially means: when solidification of metal occurs, atoms will position themselves in specific 3D pattern.
4. There is an extremely large number of different crystal structures all having long range atomic order; these vary from relatively simple structures for metals to exceedingly complex ones, as displayed by some of the ceramic and polymeric materials.

5. Lattice
When describing crystalline structures, atoms are considered as being solid spheres having well-defined diameters
Atomic hard sphere model -> in which spheres representing nearest-neighbor atoms touch one another
Lattice is a regularly spaced 3D array of points coinciding with atom positions (or centers)
From top-bottom Figs represent FCC crystal structure as: Hard sphere unit cell ; Reduced sphere unit cell; an aggregate of many atoms in FCC crystal.

6. Unit Cells
Unit Cell is the smallest group of atoms whose repetition makes a crystalline solid.
Unit cells are parallelepipeds or prisms having three sets of parallel faces
The unit cell is the basic structural unit or building block of the crystal structure
A unit cell is chosen to represent the symmetry of the crystal structure
1. The atomic order in crystalline solids indicates that small groups of atoms form a repetitive pattern
2. parallelepiped -> a three-dimensional figure formed by six parallelograms; Prism -> a polyhedron with two congruent and parallel faces (the bases) and whose lateral faces are parallelograms. In first figure of previous slide it happens to be a cube.
3. The unit cell is the basic structural unit or building block of the crystal structure and defines the crystal structure by virtue of its geometry and the atom positions within

7. Crystal Structure of Metals
Common crystal structures for metals:
Body-centered cubic (BCC) - alpha iron, chromium, molybdenum, tantalum, tungsten, and vanadium.
Face-centered cubic (FCC) - gamma iron, aluminum, copper, nickel, lead, silver, gold and platinum.
Hexagonal close-packed - beryllium, cadmium, cobalt, magnesium, alpha titanium, zinc and zirconium.

8. Crystal Structure of Metals
The Face-Centered Cubic Crystal Structure
The Body-Centered Cubic Crystal Structure
The Hexagonal Close-Packed Crystal Structure
0. The atomic bonding in this group of materials is metallic and thus nondirectional in nature. Consequently, there are minimal restrictions as to the number and position of nearest-neighbor atoms; this leads to relatively large numbers of nearest neighbors and dense atomic packings for most metallic crystal structures. The atoms tend to arrange themselves in a position that guarantees minimum energy of the system, thus the atoms in a unit cell of metals tend to as close as possible. This gives rise to dense crystalline structures

9. Face-Centered Cubic Structure (FCC)
FCC -> a unit cell of cubic geometry, with atoms located at each of the corners and the centers of all the cube faces
For the fcc crystal structure, each corner atom is shared among eight unit cells, whereas a face-centered atom belongs to only two
How many atoms in a unit cell? one-eighth of each of the eight corner atoms and one-half of each of the six face atoms, or a total of four whole atoms, may be assigned to a given unit cell
copper, aluminum, silver, and gold have fcc
Cell volume= cube volume (generated from the center of atom at corner)
2. Bottom figure shows a hard sphere model for the FCC unit cell. …

10. Face-Centered Cubic Crystal Structure
The face-centered cubic (fcc) crystal structure: (a) hard-ball model; (b) unit cell; and (c) single crystal with many unit cells
Example: Aluminum (Al)
Moderate strength
Good ductivity

11. FCC – Exercise- 1 Derive: ;
Where a = side length of the unit cell cube
And R = Radius of the atom sphere
From right angle triangle:
Solving for a:
The FCC unit cell volume:

12. FCC – Coordination Number and APF
For metals, each atom has the same number of touching atoms, which is the coordination number
For fcc, coordination number is 12
The APF (Atomic Packing Factor) is the sum of the sphere volumes of all atoms within a unit cell divided by the unit cell volume
1. the coordination number of a central atom in a molecule or crystal is the number of its nearest neighbors
2. Take the front-most yellow atom in the figure and determine its coordination number (the front face atom has four corner nearest-neighbor atoms surrounding it, four face atoms that are in contact from behind, and four other equivalent face atoms residing in the next unit cell to the front)
3. … (assuming the atomic hard sphere model)
4. … which is the maximum packing possible for spheres all having the same diameter (calculated on a piece of paper). …. Metals typically have relatively large atomic packing factors to maximize the shielding provided by the free electron cloud (a deep solid state physics concept). This is y fcc metals are very good conductors (electrons repelled from electrons of nearest atoms and become free)

13. Example 3.2—APF for FCC
Considering atom as a sphere: Volume of cell: Finally, 1

14. Body-Centered Cubic Structure (BCC)
BCC -> a cubic unit cell with atoms located at all eight corners and a single atom at the cube center
Center and corner atoms touch one another along cube diagonals

15. Body-Centered Cubic Crystal Structure
The body-centered cubic (bcc) crystal structure: (a) hard-ball model; (b) unit cell; and (c) single crystal with many unit cells
Example: Iron (Fe)
Good strength
Moderate ductivity

16. Assignment
For BCC unit cell, derive that the side length a obeys the relation:
Where a = side length of the unit cell cube
And R = Radius of the atom sphere
First take diagonal of the cube and apply Pythagoras theorem and then take diagonal of the face of the cube and then apply Pythagoras theorem and then solve the two equations
Hint. Cube diagonal. 4R= a.sqrt(3)

17. BCC- Cont Chromium, iron, tungsten exhibit bcc structure
No of atoms in BCC unit cell: 2
The coordination number for the BCC is 8
the atomic packing factor for BCC lower—0.68 versus 0.74 (FCC)
2. … :the equivalent of one atom from the eight corners, each of which is shared among eight unit cells, and the single center atom, which is wholly contained within its cell
3. … each center atom has as nearest neighbors its eight corner atoms

18. Packing Factor – FCC vs BCC
Left -> FCC (more packed and dense, 0.74); Right -> BCC (0.68)

19. Prove that APF for BCC unit cell is 0.68

20. Hexagonal Close-Packed Crystal (HCP)
Each of top and bottom faces of the unit cell consist of 6 atoms that form regular hexagons and surround a single atom in the center
Another plane that provides 3 additional atoms to the unit cell is situated between the top and bottom planes
The atoms in this mid-plane have as nearest neighbors atoms in both of the adjacent two planes

21. Hexagonal Close-Packed Crystal Structure
The hexagonal close-packed (hcp) crystal structure: (a) unit cell; and (b) single crystal with many unit cells
Example: Beryllium, Zinc
Low strength
Low ductivity

22. Hexagonal Close-Packed Crystal (HCP)
The equivalent of six atoms is contained in each unit cell
If a and c represent, respectively, the short and long unit cell dimensions the c/a ratio should be 1.633
The coordination number and the APF for the HCP are the same as for FCC: 12 and 0.74, respectively
The HCP metals include cadmium, magnesium, titanium, and zinc, etc
… : one-sixth of each of the 12 top and bottom face corner atoms, one-half of each of the 2 center face atoms, and all 3 midplane interior atoms
(c/a calculation is bit complicated) … however, for some HCP metals this ratio deviates from the ideal value.
a= 2R, c= 4R sqrt (2/3)

23. Density Computations
Density of a material can be computed from its crystalline structure
n = number of atoms associated with each unit cell
A = atomic weight
VC = volume of the unit cell
NA = Avogadro’s number (6.023 × 1023 atoms/mol)
The mole is defined as the amount of substance that contains as many elementary entities (e.g., atoms, molecules, ions, electrons) as there are atoms in 0.012 kg of the isotope carbon-12 (12C).[1] Thus, by definition, one mole of pure 12C has a mass of exactly 12 g. The amount of substance n (in mol) and the number of atoms or molecules contained in it, N, are proportional, and the proportionality constant N/n is known as Avogadro constant. By definition it has the dimension of the inverse of amount of substance (unit mol−1) and its experimentally determined value is (30)×1023 mol−1

24. EXAMPLE PROBLEM 3.3
Copper has an atomic radius of nm, an FCC crystal structure, and an atomic weight of 63.5g/mol. Compute its theoretical density and compare the answer with its measured density. Solution: The crystal structure is FCC, n (atoms) = 4 ACu = 63.5g/mol VC = a3 = [2R(2)1/2]3 (For FCC)= 16R3(2)1/2 ; R (atomic radius) = 0.128nm Using the equation:
Vc = a^3 = [2R(2)^0.5]^3

25. EXAMPLE PROBLEM 3.
The literature value for density for Cu is 8.94g/cm3
4. … which is in very close agreement that we have calculated

26. Significance of Crystal Structure: Polymorphism and Allotropy
Polymorphism: Some metals and non-metals may have more than one crystal structure . This phenomenon is known as Polymorphism, e.g., Iron has BCC structure at room temperature , but FCC structure at 912C.
Allotropy: When the above phenomenon is found in elemental solids, called Allotropy, e.g., C has two allotropes: Graphite is formed at room temperature and Diamond is formed at high temperature and pressure.
Note the difference in properties due to change in crystal structure of same metal/element

27. Crystalline and Non Crystalline Materials
Single Crystal
Polycrystalline Materials
Anisotropy

28. 1. Single Crystal
For a crystalline solid, when the repeated arrangement of atoms is perfect or extends throughout the entirety of the specimen without interruption, the result is a single crystal
If the extremities of a single crystal are permitted to grow without any external constraint, the crystal will assume a regular geometric shape having flat faces
Within the past few years, single crystals have become extremely important in many of our modern technologies.
Garnet
1. Single crystals exist in nature, but they may also be produced artificially. They are ordinarily difficult to grow, because the environment must be carefully controlled
A photograph of a garnet single crystal is shown
3. Applications: in particular electronic microcircuits, which employ single crystals of silicon and other semiconductors.

29. 2. Polycrystalline Materials
Most crystalline solids are composed of a collection of many small crystals or grains; such materials are termed polycrystalline
Initially, small crystals or nuclei form at various positions. These have random crystallographic orientations
2. Various stages in the solidification of a polycrystalline specimen are represented schematically … … as indicated by the square grids

30. 2. Polycrystalline Materials
Growth of the crystallites; the obstruction of some grains that are adjacent to one another
Upon completion of solidification, grains having irregular shapes have formed
The grain structure as it would appear under the microscope; dark lines are the grain boundaries
there exists some atomic mismatch within the region where two grains meet; this area, called a grain boundary
1. The small grains grow by the successive addition from the surrounding liquid of atoms to the structure of each … … The extremities
of adjacent grains intrude into one another as the solidification process approaches completion
2. … The crystallographic orientation varies from grain to grain.

31. 3. Anisotropy
The physical properties of single crystals of some substances depend on the crystallographic direction in which measurements are taken
This directionality of properties is termed anisotropy
The extent and magnitude of anisotropic effects in crystalline materials are functions of the symmetry of the crystal structure
… For example, the elastic modulus, the electrical conductivity, and the index of refraction may have different values in the [100] and [111] directions
… and it is associated with the variance of atomic or ionic spacing with crystallographic direction. Substances in which measured properties are independent of the direction of measurement are isotropic
the degree of anisotropy increases with decreasing structural symmetry—triclinic structures (having three unequal crystal axes intersecting at oblique angles) normally are highly anisotropic
Table -> The modulus of elasticity values at [100], [110], and [111] orientations for several materials are presented in the table

32. 3. Anisotropy
For many polycrystalline materials, the crystallographic orientations of the individual grains are totally random.
Under these circumstances, even though each grain may be anisotropic, a specimen composed of the grain aggregate behaves isotropically
Also, the magnitude of a measured property represents some average of the directional values

33. 3. Anisotropy
The magnetic properties of some iron alloys used in transformer cores are anisotropic—that is, grains (or single crystals) magnetize in a <100>-type direction easier than any other crystallographic direction

34. Non-crystalline Solids
Non-crystalline solids lack systematic and regular arrangement of atoms over relatively large atomic distances. Also called amorphous (without form)
Compare SiO2 . Crystalline structure has ordered atoms while non-crystalline structure has disorder atoms.
Each Si atom is bonded to 3 O atoms in crystalline as well as non-crystalline structures.
Metals and some ceramics form crystalline structures
Polymers are mostly non-crystalline. However, some are semi-crystalline
Non-Crystalline
Crystalline

35. Numerical Problems Problems 3.2 to 3.19, 3.23 to 3.25, 3.27 to 3.32,
and 3.37 to 3.41
Research Assistant will help you out to solve some of problems during tutorial lecture.
1, 3.13, 3.14= Asses structure from n (no of atoms)