9780030839931, Chapter 4, Problem 7P

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Chapter 4, Problem 7P Problem Let Nn be the number of nth nearest neighbors of given Bravais lattice point (e.g. in a simple cubic Bravais lattice N1 N2 12, etc.) Let m be the distance to the nth nearest neighbor expressed as a multiple of the nearest neighbor distance (e.g., in simple cubic Bravais lattice 1.414). Make a table of Nn and rn for 1, 6 for the fee, bcc, and SC Bravais lattices Step-by-step solution Step The total scattering amplitude for the reflection (hkl) is defined as, F(hkl)=fS Here, is called the geometrical structure The geometrical structure depends upon the geometrical arrangement of atoms within the unit Again, are the miller indices of the crystal and represent the coordinates of atom. Step Simple cubic crystals (sc) The effective number of atoms in a unit cell of simple cubic structure is Assuming that it lies at the origin, the structure factor given by the equation comes out to be unity. So the diffraction amplitude from equation F(hkl)= S becomes, F(hkl)=f Thus, all the diffraction lines predicted by Bragg's law would appear in the diffraction pattern. Step Body-centered cubic crystals (bcc) The effective number of atoms in a bcc unit cell is The diffraction amplitude for bcc crystal would The expression within the square bracket represents the structure factor for bcc crystal The structure factor becomes zero for odd values of as equals -1 if n is For even values of equals 2f Step Face-centered cubic crystal (fcc) The effective number of atoms in an fcc unit cell is The diffraction amplitude for fcc crystal would The expression within the square bracket represents the structure factor for fcc crystal. The structure factor is non-zero only k and are all even or all odd and have value equal Step of 5 The number of vectors N is equal to the number of different arrangements (permutations), which includes altering the sign of the indices The following table shows the arrangement Vector (miller indices N SC bcc fcc 0,0,0 1 X X X 1,0,0 6 X 1,1,0 12 X X 1,1,1 8 X X 2,0,0 6 X X X 2,1,0 24 X 2,1,1 24 X X 2,2,0 12 X X X 2,2,1 24 X 3,0,0 6 X 3,1,0 24 X X 3,1,1 24 X X 2,2,2 8 X X X 3,2,0 24 X 3,2,1 48 X X 4,0,0 6 X X X From the above we easily get (note we give R-squared) The following table shows and rn for =1,2,3,4,5,6 for the fcc, bcc, and SC Bravais lattices N SC R2 SC N bcc R2 bcc N fcc R2 fcc 16 1 8 3 12 2 2 12 2 6 4 6 4 3 8 3 12 8 24 6 4 6 4 24 11 12 8 5 24 5 8 12 24 10 6 24 6 6 16 8 12 Comments (1) Anonymous This is the only relevant part to the question. Structure factor/atomic form factor are not even relevant or described in this chapter