Question

Question S. (a) What is a unit cel? Of the seven crystal systems which are the three most commonly found for metals? Sketch the unit cell for each of these three, indicating the relative dimensions/angles on each cell. [8 marks] (b) The unit cell dimension, a, for a cubic lattice is 4R/V3. If the atomic radius of iron is 0.287 nm then calculate the cell side, a. Draw a sketch and use it to explain what the equation ηλ 2d sin θ is used for. Define the terms in it and their SI units. 17 marks] (c) Sketch a typical energy versus distance graph to indicate the balance be- tween attractive and repulsive forces between atoms as they approach each other. Label the axes, the force curves and the equilibrium distance for the atomic pair 6 marks] (d) Explain the purpose of Mohs' scale and give an example of a material from near each end of the scale. What type of measurement is used nowadays to provide more quantitative data? 14 marks]

Answer 1

a) A unit cell is the basic building block of a crystal structure. The different types of unit cells give rise to different crystal structures. The three basic unit cells for metallic objects are face centered cubic, body centered cubic and hexagonal close packing structures. These are shown below (taken from Britannica Encyclopedia):

Common metallic crystal structures body-centred cubic (bcc) face-centred cubic (fcc) hexagonal close-packed (hcp) 2011 Encyolopæda Britannica, Inc.

b) Since the ionic radius is 0.287nm, the side is 4x0.287/√3 = 0.663 nm.

The formula mentioned in the question is used to calculate the crystal structure using X-ray diffraction or XRD. The equation uses a known wavelength (λ) with a known path difference (nλ, where n is any integer). The angle of diffraction of the ray (θ) is measured and the interatomic distance d is calculated. This d is constant for specific cells for specific crystals, and thus, the crystal structure of a material can be determined from this equation. The sketch below helps explain this further.

c) The graph below shows the energy distance curve for a H-H bond. When the distance is too small, repulsive forces dominate and the energy required to overcome these forces and form a bond is high. As the distance increases, the repulsion decreases and the energy required to form a bond decreases to its lowest. Beyond this, as the distance increases, the atoms are too far away to form a bond, and energy is required to bring them together and form a bond.

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d) The Moh's scale of hardness is used to determine the relative hardness. The scale consists of 10 minerals, each assigned a numeric valur from 1 to 10. Thus, talc is assigned a hardness of 1, and diamond a hardenss of 10 on the scale. A mineral softer than quartz (hardness 7 on the scale) but harder than orthoclase (hardness 6) will be assigned a value between 6 and 7. For a more quantitative test of hardness, indentation hardness tests are used, where a machine is used to repeatedly indent an mineral until am impression is formed and the effort used to impress the indentation onto the mineral provides a sense of hardness.