Lecture 10
3-4 The diffraction condition
Consider Fig.23, the change ∆𝛫 in wave vector 𝛫 is
in the direction perpendicular to the (ℎ𝑘𝑙) planes.
Provided that scattering is elastic (I𝜥I=I𝜥I=
I∆𝜥I= 2𝑠𝑖𝑛𝜃I𝑲I
∴ I∆𝜥I =
4𝜋 sin 𝜃
𝜆
4𝜋 sin 𝜃
∆𝜥=
𝜆
𝒏
where𝒏=𝑮hkl/ I𝑮hklI unit vector along 𝑮hkl
4𝜋 sin 𝜃 𝑮
∴ ∆𝜥=
∴ ∆𝜥 =
𝜆
4𝜋 sin 𝜃
𝜆(2𝜋/𝑑ℎ𝑘𝑙 )
2𝑑ℎ𝑘𝑙 𝑠𝑖𝑛 𝜃
2𝑑ℎ𝑘𝑙 sin 𝜃
𝜆
𝑰𝑮𝑰
=
𝜆
=1
𝑮hkl
. 𝑮hkl
Bragg’s law for 𝑛 = 1
∴ ∆𝜥 = 𝑮hkl(Bragg condition)
𝚱 = 𝑮hkl+ 𝚱
2𝜋
𝜆
)
𝐾Κ
𝜽∆𝜥
Fig.23(ℎ𝑘𝑙)
Κ𝜽
When each side of this equation is squared and the
quantity 𝚱=𝚱 , the Bragg condition appears in the
form:
I𝑮I2+2𝒌. 𝑮=0
This is the central result of the theory of elastic
scattering of waves in a periodic lattice.
If 𝑮 is a reciprocal lattice vector, so is - 𝑮, then
2𝒌. 𝑮=I𝑮I2
This Eq. is another statement of the Bragg
conditionor called Bragg equation in reciprocal
lattice
Note:Eq.
𝒂*. 𝒂 = 𝒃*. 𝒃= 𝒄*. 𝒄= 𝟐𝝅,and
Eq.𝑮hkl=𝒉𝒂*+ 𝒌𝒃*+ 𝒍𝒄*require that
𝒂. ∆𝜥= 𝟐𝝅𝒉
𝒃. ∆𝜥= 𝟐𝝅𝒌This is laue equations
𝒄. ∆𝜥= 𝟐𝝅𝒍
3-5 Brillouin zone
A Brillouin zone is defined as a Wigner-Seitz
primitive cell in the reciprocal lattice.
To find this ( use the same algorithm as for finding
the Wigner-Seitz primitive cell in real space), draw
thereciprocal lattice. Thendraw vectors to all the
nearestreciprocal lattice points,thenbisect them. The
resulting Figure is your cell (see Fig. 24).
Fig. 24
Exercise:
Draw the Brillouin zone of the square lattice and
hexagonal lattice .
Solution:
Fig.25 The reciprocal lattices (dots) and
corresponding first Brillouin zones of (a) square
lattice and (b) hexagonal lattice.
Problems:
1- Show that the Bragg equation in reciprocal lattice
is equilibrium the Bragg equation in real lattice.
2- Show that the boundary of the B.Z for reciprocal
lattice satisfies the diffraction condition.
3- Use the ideas of reciprocal lattice to find a
relationship between dhkl and lattice constantsa,b,c