) making interceptsin the ration 1/h: 1/k: 1/l
10
CHAPTER 1. STRUCTURE FACTOR AND LATTICE
Spacing of the
hkl
planes
We know that
e
i
G
.
r
is also the equation of a plane wave, with wavevector
G
propagating in the direct
lattice. The surfaces over which
G
.
r
is a constant are the wavefronts. The discussion in the preceeding
section implies that the
hkl
planes are precisely such wavefronts.
Therefore the spacing between two
successive
hkl
planes, for which the constant
n
0
in eqn 1.21 differs exactly by 1, must be the wavelength
associated with
G
. We thus get a very important result if
d
hkl
denotes the distance between two successive
planes, then
d
hkl
=
2
π

G

(1.22)
Density of points on an hkl plane
The density of points (per unit area) in an
hkl
plane is
n
hkl
=
d
hkl
V
(1.23)
where
V
=
a
1
.
a
2
×
a
3
is the volume of the primitive unit cell. The proof is left as an exercise.
PROBLEM : Prove eqn. 1.23
These two results will come in very handy when we discuss diffraction from a lattice.
Direction in a lattice
If a vector points along
r
=
n
1
a
1
+
n
2
a
2
+
n
3
a
3
then we denote this direction as [
n
1
,n
2
,n
3
]. Note the use of different brackets to avoid confusion with
Miller indices.
Chapter 2
Diffraction and basics of crystal
structure
Lecture notes for PH409 (JulDec2013): K. Das Gupta
References:
1. Chapter 5
Solid State Physics
, N. W. Ashcroft and N.D. Mermin
2. Chapter 1 & 2
Introduction to Solid State Physics, C. Kittel
3. Useful (free) software for visualising crystal structures:•XCrysden : •POVray : •Crystosim : sangals/crystosim/finalsolution/startpage.htmlHow does oneknowabout the structure of a solid or a liquid or a glass? There is a generic answer to this
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 Spring '19
 Physics, Crystallography, Cubic crystal system, Reciprocal lattice, Brillouin zone, Lattice points