An FCC crystal may be described by a \"conventional\" (i.e. non - primitive bin
ID: 1893649 • Letter: A
Question
An FCC crystal may be described by a "conventional" (i.e. non - primitive bin convenient) unit cell which contains four lattice points. More properly, this unit cell is described as simple cubic with a basis, of four atoms located at (0,0,0), a(1/2, 1/2, 0),a(1/2, 0, 1/2), a(0, 1/2, 1/2). Show that the structure factor of this basis is Scans = f{1 + e - ix(h + k) + e - ix(k + l) + e - ix(l + k)} and hence derive the rule for FCC crystals indexed on a conventional unit cell. that the allowed Bragg reflections must satisfy h, k, l all even or all odd.Explanation / Answer
The structure factor of the basis is defined as G = jfj exp(-i.G).......................(1)
the usual form of this result follows on writing rj=xja+yjb+zjc and G=hA+kB+lC, then for reflection hkl,
(hkl)=jfj exp-i2(xjh+yjk+zjl)....................................(2)
The basis of the fcc structure referred to the cubic cell has identical atoms at (000); (01/21/2);(1/201/2);
(1/21/20).Thus eq. (2) becomes
(hkl) = f[1+e-i(k+l)+e-i(h+l)+e-i(h+k)]
If all indices are even integral , =4f; similarly if all indices are odd integrals.
But if only one of the integers is even, two of the exponents will be odd multiples of -i and will vanish. If only one of the integers is odd, the same argument applies and will vanish. Thus in the fcc lattice no reflection can occur for which the indices partly even and partly odd.
i.e., allowed reflections for cubic crystals of FCC (extinctions rules) is :when h,k, and l are all odd or all even