5. (a) What is a grain and a grain boundary? Show both of them using a hand sketch. (b) Determine the ASTM grain size of an aluminium sample if, 45 grains per cm^3 are counted at a magnification of x100

(a) A grain is a small crystalline region within a metal or alloy that has a specific crystal structure and orientation. It is the basic building block of the microstructure of a polycrystalline material. Grains are typically surrounded by grain boundaries, which are interfaces where the crystallographic direction of the lattice changes.<br /><br />Here is a hand sketch showing a grain and a grain boundary:<br /><br />[Insert sketch of a grain with a grain boundary]<br /><br />(b) To determine the ASTM grain size of an aluminum sample, we can use the following formula:<br /><br />ASTM Grain Size = -log2(N)<br /><br />where N is the number of grains per unit area. In this case, the number of grains per unit area is 45 grains per cm^2.<br /><br />Substituting the given value into the formula, we get:<br /><br />ASTM Grain Size = -log2(45)<br /><br />Calculating the logarithm, we find:<br /><br />ASTM Grain Size ≈ 1.35<br /><br />Therefore, the ASTM grain size of the aluminum sample is approximately 1.35.