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I was reading chapter 2 of Chaikin and Lubensky, where I got stuck at this derivation of structure function. While talking about Smectics-A liquid crystal, it was mentioned that the molecules are aligned perpendicular to the layer. The introduction of the layering indicates the presence of a mass density wave perpendicular to the layers. So, the positional correlation in the system can be described as a sinusoidal modulation of the average molecular number-density: $$\langle n(\vec{x}) = n_0 + 2n_{q_0}cos(q_0z) \rangle,$$ where $q_0 = 2\pi/l$ and the z-axis is along the normals and parallel to $\vec{n}$. The Fourier Transform of this equation leads to two Bragg peaks away from $\vec{q}=0$ in the structure function: $$S(\vec{q}) = \left| \langle n_{q_0} \rangle \right|^2 (2pi)^3 [\delta(\vec{q_z} - q_0\widehat{e_z}) + \delta(\vec{q_z} + q_0\widehat{e_z})]$$

I just want to know if they have written it approximately by ignoring the first term $n_0$ and a multiple of $2$ from the second term.

Soham Parmar
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1 Answers1

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The Fourier transform (FT) of the cosine is simple, if we use Euler's formula to write $$ \cos(x) = \frac{e^{ix}+ e^{-ix}}{2} $$ There are different "definitions" of the Fourier transform. Hence, the pre-factor of the FT depends on this definition. Hence, it's impossible to answer your second questions, unless you state the definition of the FT. Nevertheless, using Euler's formula, it is straight forward to show that the FT of the cos-term always yields the following functional form $$ FT[\cos(q_0 z)] \propto \delta(q_0) + \delta(-q_0) $$ in one-dimension. In three-dimension, we have to account for the different directions -- as you did in the question.

The FT of a constant is a $\delta$-Function. Hence, the constant term is omitted in the definition of the structure factor. From the wording this makes sense to me, because a constant density is opposite of a structured density.

Hope this helps.

NotMe
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