70
4 Molecular Crystals
= f
◦ exp
−i
(N − 1)qa
2
sin
Nqa
2
sin
qa
2
,
(4.8)
where the last equality utilizes Eq. 4.7 while adopting k = qa. The last formula
is precisely the same as the amplitude of the ideal array (Eq. 4.5). Thus, crystals
possessing structural disorder again give a set of sharp diffractions.
Next, we consider the case that positions of atoms are modulated by a displacement
wave of which the period is incommensurate (irrational) to the original period:
r j = ja + Δr 0 sin jka.
(4.9)
The irrational modulation, such as k = 1/
√
2, means that there are no atoms, of
which the displacements from the ideal position, ja, are the same to each other in
this atomic array. In this sense, the array has no translational symmetry. This type
of crystalline phase is termed as an incommensurate phase because the modulated
array gives a set of sharp diffractions as follows, as far as |qΔ| | 1. The strength of
the diffracted wave by the modulated array is calculated as
A(q) = f
◦
N
j=0
exp[iq( ja + Δr 0 sin jka)]
= f
◦
N
j=0
exp(i jqa) exp(iqΔr 0 sin jka)
≈ f
◦
N
j=0
exp(i jqa) (1 + iqΔr 0 sin jka)
= f
◦
N
j=0
exp(i jqa)
+ f
◦ qΔr 0
2
N
j=0
exp(i jqa)[exp(i jka) − exp(−i jka)]
= f
◦
N
j=0
exp(i jqa)
+ f
◦ qΔr 0
2
⎧
⎨
⎩
N
j=0
exp[i j (q + k)a] −
N
j=0
exp[i j (q − k)a]
⎫
⎬
⎭
(4.10)
The sums in the last line have the same form as Eq. 4.5 with the substitution q →
(q ± k). Thus, not only qa = nπ (with n ∈ Z) but also (q ± k)a = nπ is the condition of diffractions to appear. Namely, sharp diffractions appear at q = n(π/a) ± mk
with m = −1, 0, 1. Thus, this modulated atomic array without any translational peri-