1. INTERACTIVE BONDING AND CRYSTAL STRUCTURE OF SOLIDS
NILESH PANCHOLI,
B.E. (Mech.), M. E. (CAD/CAM)
Assistant Professor
Department of Mechanical Engineering
L. D. College of Engineering
Ahmedabad

2. QUESTIONS
Calculate the volume of an FCC unit cell in terms of the atomic radius R. Show that the atomic packing factor of FCC unit cell is more than that of BCC
Define: (i) Metallic Bonding (ii) Covalent Bonding (iii) Orthorhombic and Tetragonal crystal structure
Cu has a FCC crystal structure and a unit cell with lattice constant of nm. What is the interplaner spacing of d111 planes? Show one prism plane (1010) and (2110) direction of HCP lattice.
Find the atomic packing factor in case of FCC crystal
If the lattice parameter of alpha iron is 286 pm what is its atomic radius?
Calculate the no. of atoms per zinc crystal structure unit cell.
Calculate and compare the atomic packing factor of the FCC and BCC cell units. What role does the atomic packing factor have on creep strengths of FCC and BCC metals at same homologous temperature?
Al is FCC, and has an atomic radius of nm. Calculate its lattice parameter.
The lattice parameter of a material having FCC structure, is nm. Determine the length of the burger’s vector along[110] direction.
Justify: FCC metals are often recommended for use at low temperature.
Justify: Four octahedral sites are associated with one FCC unit cell.

3. Why Study Atomic Structure and Interactive Bonding?
An important reason to have an understanding of inter-atomic bonding in solids is that, in some instances, the type of bond allows us to explain a material’s properties.
For example, consider carbon, which may exist as both graphite and diamond. Whereas graphite is relatively soft and has a greasy feel to it, diamond is the hardest known material. This dramatic disparity in properties is directly attributable to a type of inter atomic bonding found in graphite that does not exist in diamond.

4. Bonding Forces and Energies
Consider two isolated atoms:
When the atoms are at large inter-atomic
separation distance, the atoms do not exert any force on each other.
When the distance is decreased, an attractive force FA starts to act pulling atoms closer.
FA increases as the atoms gets closer.
But as the atoms get closer a repulsive force FR begin to act.
The net force FN between the two atoms is given by:
FN = FA + FR
At some inter-atomic distance ro, FR exactly equals FA and FN becomes Zero
FN = 0 = FA + FR
ro is called the equilibrium inter-atomic separation distance at which atoms enter into bonding
ro ≈ 0.3 nm

5. Force vs. Separation Distance
Energy vs. Separation Distance

6. Bonding Forces and Energies (Contd.)
A number of material properties depend on E0, the curve shape, and bonding type. For example,
Materials having large bonding energies typically also have high melting temperature.
At room temperature, solids formed for large bonding energies, whereas for small energies the gaseous state is favored, liquids prevail when energies are of intermediate magnitude.
The mechanical stiffness (modulus of elasticity) is dependent on the shape of the force-versus-interatomic separation curve.
The coefficient of thermal expansion (how much expands or contracts per degree change in temperature) is related to the shape of its E0-versus-r0 curve.

7. BONDING IN SOLIDS Ionic Bonding Covalent Bonding Metallic Bonding
van der Waals Bonding
Hydrogen Bonding

8. PRIMARY AND SECONDARY BONDS Primary Bonds: Chemical (strong) bonding, involves the transfer or sharing of electrons: ionic, covalent, or metallic bonds. Secondary Bonds: Physical (weak) bonding, does not involve the transfer or sharing of electrons: hydrogen and van der Waals bonds.

9. DIRECTIONAL AND NON-DIRECTIONAL BONDS Directional Bonds: Single or multiple bonds, which are localized and occur at fixed angles with respect to each other. Non-Directional Bonds: Bonding is equally probable at all angles. The bond is not localized to a specific direction.

10. THE THREE PRIMARY OR STRONG BONDS Metal to Non-Metal: Ionic
Non-Metal to Non-Metal: Covalent
Metal to Metal: Metallic

11. THE SECONDARY OR WEAK BONDS Van der Waals: Fluctuating dipoles
Hydrogen: Develops between “electropositive” and “electronegative” elements. Permanent dipoles

12. THE IONIC BOND
Ionic Bond: Arises from the electrostatic attraction between cations and anions. Because the cations are everywhere positive and the anions are everywhere negative, the bond is non-directional.

13. IONIC BOND

14. THE COVALENT BOND
Covalent Bond: Arises from the electrostatic attraction between cations/cation cores and shared electron pairs. The electrons are said to be localized, because they are confined, primarily between adjacent cations.
Hence, the covalent bond is directional.

15. COVALENT BOND

16. THE METALLIC BOND
Metallic Bond: Arises from the electrostatic attraction between cation cores and an electron cloud. The electrons are said to be delocalized, because they are not confined to any cation core, but are “free” to move between the cation cores.
Hence, the metallic bond is non-directional.

17. METALLIC BOND

18. THE VAN DER WAALS BOND
Weak, secondary bond formed by the attraction of fluctuating dipoles between, for example, atoms of the noble gases and between molecules. Van der Waals bonds are non-directional.

19. Fluctuating Dipole Bonds
Bonding Between Argon electric dipoles - +
Argon atom
Argon Electric Diploe ≈
Van der waals bond

20. Permanent Dipole Bonds
Van der waal bonds also occur between permanent polar molecules.
The bonding energies are higher than the fluctuating induced or polar molecule induced bonds.
The strongest Secondary Bonding is Hydrogen Bond. (Directional)
Examples of Hydrogen Bonding:
HF, H2O, NH3

22. CRYSTAL STRUCTURE OF SOLIDS
WHY STUDY?
• How do atoms assemble into solid structures?
• How does the density of a material depend on
its structure?
• When do material properties vary with the
sample orientation?

23. CRYSTAL SYSTEMS
Based on shape of unit cell ignoring actual atomic locations
Unit cell = 3-dimensional unit that repeats in space
Unit cell geometry completely specified by a, b, c & a, b, g (lattice parameters or lattice constants)
Seven possible combinations of a, b, c & a, b, g, resulting in seven crystal systems

24. Structure Review Defining components of a general crystal system
lengths of the axes (lattice constants) are a, b, and c
angles between the atom planes are  (alpha),  (beta), and  (gamma)
Components of the crystalline structure z x y b a c   

25. Structure Review
In solid state, particles are bonded together in rigid, crystalline structure. Seven basic crystalline structures are:
cubic: BCC, and FCC
tetragonal: Simple T and BCT
orthorhombic: simple, Body CO, Base CO, and FCO
rhombohedral
hexagonal: hexagonal close packed, FCC closed packed, complex cubic structure (diamond structure)
monoclinic: simple and base-centered
triclinic
Each crystal system has a different axis length and angles that separate the atoms

26. CRYSTAL SYSTEMS

27. SOME DEFINITIONS … Lattice: 3D array of regularly spaced points
Crystalline material: atoms situated in a repeating 3D periodic array over large atomic distances
Amorphous material: material with no such order
Hard sphere representation: atoms denoted by hard, touching spheres
Reduced sphere representation
Unit cell: basic building block unit (such as a flooring tile) that repeats in space to create the crystal structure; it is usually a parallelepiped or prism

28. METALLIC CRYSTALS • tend to be densely packed.
• have several reasons for dense packing:
-Typically, made of heavy element.
-Metallic bonding is not directional; i.e., no
restrictions as to the number and position of
nearest-neighbor atoms
-Nearest neighbor distances tend to be small in
order to lower bond energy.
• have the simplest crystal structures.
We will look at four such structures...

29. SIMPLE CUBIC STRUCTURE (SC)
• Cubic unit cell is 3D repeat unit
Rare (only Po has this structure)
• Close-packed directions (directions along which atoms touch each other)
are cube edges.
• Coordination # = 6
(# nearest neighbors)

30. ATOMIC PACKING FACTOR a R=0.5a
Adapted from Fig. 3.19,
Callister 6e.
Lattice constant
close-packed directions a
R=0.5a
contains 8 x 1/8 = 1
atom/unit cell
• APF for a simple cubic structure = 0.52

32. BODY CENTERED CUBIC STRUCTURE (BCC)
• Coordination # = 8
Adapted from Fig. 3.2,
Callister 6e.
(Courtesy P.M. Anderson)
• Close packed directions are cube diagonals.
--Note: All atoms are identical; the center atom is shaded
differently only for ease of viewing.

33. ATOMIC PACKING FACTOR: BCC
(Body Diagonal)2
= (R+2R+R)2
= (a2 + a2 + a2)
• APF for a body-centered cubic structure = p3/8 = 0.68

34. FACE CENTERED CUBIC STRUCTURE (FCC)

35. FACE CENTERED CUBIC STRUCTURE (FCC)
• Coordination # = 12
• Close packed directions are face diagonals.
--Note: All atoms are identical; the face-centered atoms are shaded
differently only for ease of viewing.

36. ATOMIC PACKING FACTOR: FCC
Adapted from
Fig. 3.1(a),
Callister 6e.
• APF for a body-centered cubic structure = p/(32) = 0.74
(best possible packing of identical spheres)

37. FCC STACKING SEQUENCE • FCC Unit Cell • ABCABC... Stacking Sequence
• 2D Projection

40. HEXAGONAL CLOSE-PACKED STRUCTURE (HCP)
Ideally, c/a = for close packing
However, in most metals, c/a ratio deviates from this value

41. HEXAGONAL CLOSE-PACKED STRUCTURE (HCP)
• 3D Projection
• 2D Projection
Adapted from Fig. 3.3,
Callister 6e.
Total No. of Atoms in HCP Crystal
= 1 (6 x 1/6) +1 (6 x 1/6) + 1 (2 x ½) + 3
= 6
• Coordination # = 12
• APF = 0.74, for ideal c/a ratio of 1.633

42. Close packed crystals A plane B plane C plane A plane
…ABCABCABC… packing
[Face Centered Cubic (FCC)]
…ABABAB… packing
[Hexagonal Close Packing (HCP)]

43. COMPARISON OF CRYSTAL STRUCTURES
Crystal structure coordination # packing factor close packed directions
Simple Cubic (SC) cube edges
Body Centered Cubic (BCC) body diagonal
Face Centered Cubic (FCC) face diagonal
Hexagonal Close Pack (HCP) hexagonal side

44. Characteristics of Selected Elements at 20C
Adapted from
Table, "Charac-
teristics of
Selected
Elements",
inside front
cover,
Callister 6e.

45. STRUCTURE OF OTHER SYSTEMS
• Structure of NaCl
(Courtesy P.M. Anderson)
• Structure of Carbon
Graphite
Diamond

46. THEORETICAL DENSITY, r Example: Copper
Data from Table inside front cover of Callister (see previous slide):
• crystal structure = FCC: 4 atoms/unit cell
• atomic weight = g/mol (1 amu = 1 g/mol)
• atomic radius R = nm (1 nm = 10 cm) -7

47. DENSITIES OF MATERIAL CLASSES
metals
>
ceramic s
polymer
Why?
Metals have...
• close-packing
(metallic bonding)
• large atomic mass
Ceramics have...
• less dense packing
(covalent bonding)
• often lighter elements
Polymers have...
• poor packing
(often amorphous)
• lighter elements (C,H,O)
Composites have...
• intermediate values
Data from Table B1, Callister 6e.

48. COMMON DIRECTIONS

49. EXAMPLES: DIRECTIONS
Draw a [1,-1,0] direction within a cubic unit cell
Determine the indices for this direction
Answer: [120]

50. CRYSTALLOGRAPHIC PLANES
Crystallographic planes specified by 3 Miller indices as (hkl)

51. Miller Index Defining particular plane of the atom with Miller Index
3 steps to determine the Miller Index
Find the intercepts x = 2, y = 3, and z = 2 from figure
Take the reciprocal of the axis length, 1/2, 1/3, and 1/2
Find the lowest common multiplier and then multiply the reciprocal
Then, the Miler Index is (3, 2, 3)…… 6*1/2, 6*1/3, 6*1/2.. Note ( )
Example, z x y
b=3
a=2
c=2

52. THREE IMPORTANT CRYSTAL PLANES

53. THREE IMPORTANT CRYSTAL PLANES
Parallel planes are equivalent

54. THANK YOU