Answer All Questions
1. (a) Show that the reciprocal lattice of a Simple Cubic Bravais lattice, with cubic primitive cell of side
a, is a simple cubic lattice with cubic primitive cell of side 2/a.
(b) Show that the reciprocal of a Simple Hexagonal Bravais lattice with lattice constants c and a is
2
1
¿
a¿ =
another Simple Hexagonal lattice with lattice constants c =
and
, where a* is
c
a √3
rotated by 30Ëš with respect to a, b* is perpendicular to a and the angle between a* and b* is 60Ëš.
(c) Depict a, b, a*, and b* on an (0001) stereographic projection. Using 4-index notation, write down
expressions for these real and reciprocal lattice vectors.
(d) The volume of the unit cell in reciprocal space can be defined as
1
¿
V =
.
V V ¿ =a¿ ∙ b¿ × c ¿ . Prove that (e) Given that the primitive unit cell of the f.c.c. lattice can be constructed by connecting the origin to
the three adjacent face centers, show that this primitive cell is rhombohedral with the rhombohedral
angle being equal to 60º (Note: a primitive unit cell contains only 1 lattice point).
(f) Using this primitive unit cell, show that the reciprocal lattice of the f.c.c. lattice is b.c.c.
(g) Using similar methodology, show that the reciprocal lattice of the b.c.c. lattice is f.c.c.
2. (a) Show by means of a (1 1´ 0) sectional drawing that [ 111 ]
cubic system, but not, in general, in the tetragonal system. is perpendicular to ( 111 ) in the (b) In a drawing of a hexagonal prism, indicate the following planes and directions:
( 1 2´ 10 ) , ( 10 ´12 ) , ( 1´ 011 ) , [ 110 ] , [ 11 1´ ] , [ 021 ] .Show cell axes.
(c) For the cubic structure show that the planes ( 1 1´ 0 ) , ( 1 ´21 ) and ( 3´ 12) belong to the zone
[111] . Confirm this by plotting these poles and this zone axis on an (001) cubic projection.
´ ) , ( 3´ 11 ) ,( 1´ 3´ 2) ? If so, what is the
(d) Do the following planes all belong to the same zone: ( 110
zone axis? Give the indices of any other plane belonging to this zone. Plot all these on the same
projection you drew in (c).
(e) Lutetium has a hexagonal structure with lattice parameters a=3.516 Ã… and c = 5.570 Ã…. Plot the
h0l net of the reciprocal lattice of this material.
(f) Draw a standard (0001) projection of beryllium (hexagonal, c/a = 1.57), showing all poles of the
´ } and the important zone circles between them.
form { 2 ´1 1´ 0 } , { 10 ´10 } , {2 1´ ´11 } and {10 11

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