In a hexagonal close-packed (HCP) material, let a represent the interatomic spacing within a close-packed layer...

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Mechanical Engineering

In a hexagonal close-packed (HCP) material, let a represent theinteratomic spacing within a close-packed layer (plane), and let crepresent the lattice parameter normal (perpendicular) to theselayers. There are two close-packed layers stacked along c in theconventional unit cell, as described in Lecture 3. (i) Find anexpression for a in terms of the atomic radius R. (ii) Now imaginethe same close-packed layers, made from the same atoms as before(i.e. R does not change), but now stacked to form a cubic closepacked (CCP) material. There are three close-packed layers stackedalong the body diagonal of the conventional face centered cubic(FCC) unit cell of CCP material. Find an expression that relatesthe atomic radius R to the close-packed layer spacing in this unitcell. (Hint: both R and the close-packed layer spacing can bewritten in terms of the length of any edge of the FCC unit cell.)(iii) Use your answer from (ii) to obtain an expression thatrelates c to R in the HCP structure. (iv) Use your answers from (i)and (iii) to show that c/a ~ 1.633 in perfect HCP materials. (v)Many HCP materials do not have this ideal value of c/a. What is thepractical significance / consequence of departures from the idealc/a value?

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SOLUTIONi We can find expression byIn the cubic system all the faces of the cube areequivalentthat is they have similar indicesHowever this is not the case in the hexagonal system The sixprism faces for example have indices 1 0 0 0 1 01 1 0 10 0 0 1 0 1 1 0 which are not sameIn order to address this a fourth axis a3 See Answer

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