At the outset, surface diffusion is generally assumed to be simple. On the contrary to the belief, it is rather difficult to precisely estimate. This problem arises from the difficulty in describing the surface as “atomically flat,” as surfaces also have grooves, scratches, steps, and so on. Moreover, surfaces are easily susceptible to contamination. The general view is that the activation energy for surface diffusion is considerably less than that for the bulk diffusion as a surface atom has almost half the nearest-neighbor atoms compared to an atom in the bulk. So, it is relatively easier for a surface atom to hop from one lattice position to another equivalent atom position.

2.3.7

Summary

This section discussed a very important kinetic phenomenon in materials known as diffusion. Diffusion has a vital role to play in several nuclear-specific materials, which will be elucidated in later chapters. Both phenomenological and atomistic theories of diffusion are discussed. Microstructural paths of diffusion have an important bearing on the activation energy and thus on diffusion rates; activation energy values in a material would be sequenced from smaller to higher magnitudes as shown in the following: surface (Qf) < grain boundary (Qgb) < dislocation core (Qc) < lattice self-diffusion (Ql).

Problems

2.1 Given the following information on tantalum (Ta): atomic number = 73, atomic mass = 180.95 amu, atomic radius = 0.1429 nm, density = 16.6 g/cc m~3. (a) Find out the number of atoms per mm3. (b) What is the atomic pack­ing factor? (c) If it is cubic, what is its Bravais lattice?

2.2 What is polymorphism? Give four examples.

2.3 The unit cell of uranium has orthorhombic symmetry with a, b, and c lattice parameters of 0.286, 0.587, and 0.495 nm, respectively. If its density, atomic weight, and atomic radius are 19.05 gcm~3, 238.03 gmol-1, and 0.1385 nm, respectively. Calculate the atomic packing factor.

2.4 Calculate the density of uranium carbide (UC) given the atomic radii of U and C species.

2.5 Copper has an FCC structure with an atomic radius of 0.128 nm and atomic weight of 63.5 g mol-1.

a) Calculate its density in g cm-3?

b) Draw a neat sketch of a unit cell with axes appropriately shown and depict the planes and directions: (100), (110) and (111); [110] and [111].

c) Calculate the planar density (per cm2) of atoms on the above planes?

d) Calculate the linear density (per cm) of atoms along these directions?

2.6 What is the angle between the planes (011) and (001) in a cubic crystal?

2.7 a) What are the crystal structures of UC and UO2 (draw a neat sketch of a

unit cell and show the ionic positions) and what are the advantages of UO2 over UC as a nuclear fuel?

b) In UO2, what are the coordination numbers of U and O ions?

c) Given that the ionic radii of U and O are 0.97 and 1.32 A, respectively, cal­culate the lattice constant of UO2.

2.8 Draw a neat sketch of a unit cell of Fe (BCC) and depict a close-packed plane and close-packed direction (i. e., the densest plane and direction) showing clearly the choice of coordinates.

a) What are the Miller indices of the specific plane and direction you chose?

b) Calculate the planar and linear atomic densities for the plane and direction above?

2.9 a) Compute and compare the linear densities of the [110] and [111] directions

for BCC.

b) Calculate and compare the planar densities of the (100) and (111) planes for FCC.

2.10 Show that the ideal c/a ratio for HCP crystals is 1.633?

2.11 On a neat sketch of a HCP crystal, show the following (clearly depict the axes):

a) Prism plane (1010) and a close-packed direction in that plane — What are the Miller indices of the direction chosen and what is the interatomic distance?

b) Pyramidal plane (1122) and a close-packed direction in that plane — What are the Miller indices of the direction chosen?

c) A basal plane and a close-packed direction in that plane — What are the Miller indices of the plane and direction chosen?

2.12 a) If the activation energy for vacancy formation in Fe is 35 kcal mol-1, what

is the vacancy concentration at (i) 900 °C and (ii) 400 °C? b) Calculate the number of vacancies per unit cell at 900 °C in the above problem.

2.13 In alkali halides, such as NaCl, one finds intrinsic and extrinsic vacancies (cat­ion vacancies) due to divalent impurity such as Ca. On an Arrhenius plot show (schematically) the temperature variation of vacancy (cation) concentra­tion indicating the extrinsic and intrinsic regions. Note down the equations relevant to these regions.

2.14 Compare and contrast the edge and screw dislocations.

2.15 Show how the dislocation model of low-angle grain boundaries can explain the creation of high-angle grain boundaries.

2.16 Show that Hume-Rothery’s rules apply to copper-nickel alloy system. Why is the solubility of carbon in FCC iron more than in BCC iron?

2.17 Steel surfaces can be hardened by carburization, the diffusion of carbon into the steel from a carbon-rich atmosphere. During one such treatment at 1000 °C, there is a drop in carbon concentration from 5.0% to 4.0% carbon between 1 and 2 mm from the surface of the steel. Estimate the flux of carbon into the steel in this near-surface region to be 2.45 x 1019 atoms per m2 s (The density of y-Fe at 1000 °C is 7.63 gcm-3.).

2.18 A steel with 0.2% C is to be carburized in a carburizing atmosphere to reach a carbon concentration of 1.1% at the surface. After 10 h at 890 °C, at what depth below the surface one would find 0.4% C concentration? (For diffusion of C in austenite, D0 = 2.0 x 10-5 s-1, and Q = 140 kJ mol-1)

2.19 a) Given the diffusion data (D in cm2 s-1) for yttrium in chromium oxide at

different temperatures, find the activation energy and diffusion coefficient

(D0): 1.2 x 10-13 @800 °C; 5.4 x 10-13 @850 °C; 6.7 x 10-13 @900 °C;

1.8

x 10-12 @950 °C; and 4.6 x 10-12 @1000 °C. b) Find D at 925 °C.