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(A A) (A A) (A A) V (2) (2) / - (A A A) - (A A A) - (2) / V

This document discusses the volume of the primitive unit cell (VBZ) and reciprocal lattice vectors. It also discusses the structure factor S(v1v2v3) for diamond crystals with a face-centered cubic (fcc) lattice structure. Key points include: - The volume of the primitive unit cell (VBZ) is calculated in terms of the primitive lattice vectors and reciprocal lattice vectors. - The structure factor S is calculated for an fcc lattice with a diamond crystal basis. - For the diamond crystal basis, the structure factor S(basis) is only non-zero when the sum of the Miller indices v1 + v2 + v3 is equal to 4n, where n is

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Simon Savitt
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62 views2 pages

(A A) (A A) (A A) V (2) (2) / - (A A A) - (A A A) - (2) / V

This document discusses the volume of the primitive unit cell (VBZ) and reciprocal lattice vectors. It also discusses the structure factor S(v1v2v3) for diamond crystals with a face-centered cubic (fcc) lattice structure. Key points include: - The volume of the primitive unit cell (VBZ) is calculated in terms of the primitive lattice vectors and reciprocal lattice vectors. - The structure factor S is calculated for an fcc lattice with a diamond crystal basis. - For the diamond crystal basis, the structure factor S(basis) is only non-zero when the sum of the Miller indices v1 + v2 + v3 is equal to 4n, where n is

Uploaded by

Simon Savitt
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© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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HW2

2.3 By definition of the primitive reciprocal lattice vectors

(a 2  a 3 )  (a 3  a1 )  (a1  a 2 )
VBZ = (2)3 = (2)3 / | (a1  a 2  a 3 ) |
| (a1  a 2  a 3 )3 |
= (2)3 / VC .

For the vector identity, see G. A. Korn and T. M. Korn, Mathematical handbook for scientists and
engineers, McGraw-Hill, 1961, p. 147.

2.4 (a) This follows by forming

1 − exp[−iM(a  k)] 1 − exp[iM(a  k)]


|F|2 = 
1 − exp[−i(a  k)] 1 − exp[i(a  k)]
1 − cos M(a  k) sin 12 M(a  k)
2
= = .
1 − cos(a  k) sin 2 12 (a  k)

1
(b) The first zero in sin M occurs for  = 2/M. That this is the correct consideration follows from
2

1 1 1
sin M(h + ) = sin Mh cos M + cos Mh sin M.
2 zero,
2 1 2
as Mh is
an integer

−2i(x j v1 +y j v2 +z j v3 )
2.5 S (v1v2 v3 ) = f  e
j

1 1 1
Referred to an fcc lattice, the basis of diamond is 000; . Thus in the product
4 4 4

S(v1v2 v3 ) = S(fcc lattice)  S (basis) ,

we take the lattice structure factor from (48), and for the basis
1
−i  (v1 + v2 + v3 ).
S (basis) = 1 + e 2

Now S(fcc) = 0 only if all indices are even or all indices are odd. If all indices are even the structure factor
of the basis vanishes unless v1 + v2 + v3 = 4n, where n is an integer. For example, for the reflection (222)
we have S(basis) = 1 + e–i3 = 0, and this reflection is forbidden.

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