# The Hendriks-Teller Model

So I am working on understanding the Hendriks-Teller model of 1D disorder. So the way I understand it is the following. You have a random smattering of particles. Each mass is separated by some unit distance with probability p or some larger distance 1 + (rho) with probability 1- p. I want to derive the structure factor for said model. Further information can be found in Principles of Condensed Matter Physics By P. M. Chaikin, T. C. Lubensky (question 2.1) and Hendriks and Teller's original paper in the Journal of Chemical Physics, 1942.

The final structure factor should be:

$$S(q) = \frac{p(1-p)[1-\cos{\rho q}]}{1-p(1-p)-p\cos{q}-(1-p)\cos{[(1+\rho)q]}+p(1-p)\cos{\rho q}}$$

I figure I should start with something like this, the structure factor for a discrete system:

$$S(q)=\frac{1}{N}\langle \sum_{jk} e^{i \vec{q} (\vec{R_{j}}-\vec{R_{k}}) } \rangle$$

Thanks all.

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I have been reading and found that this may be helpful to this cause: $$S(q)= \sum_{j=1}^{n} f_{i}\cos{[\frac{s\pi hx}{a}]}+i\sum_{j=1}^{n} f_{i} \sin{[\frac{2\pi hx}{a}]}$$ – Dylan Sabulsky Oct 7 '12 at 15:15
On pg 242 of this document, the author talks about it a little bit. Check this out – Dylan Sabulsky Oct 7 '12 at 15:22
The comment above is equivalent by definition to the equation you wrote in the answer. Both are the Fourier transform of a sequence of delta-functions at each of the Hendricks-Teller positions. You are asked to fourier transform a quasi-periodic function without long-range order. – Ron Maimon Oct 7 '12 at 15:59
Duly noted. I'm currently struggling with how to include the probability ideas within this frame work. – Dylan Sabulsky Oct 7 '12 at 16:07

Let’s start with the definition of the structure factor from C&L, by com- bining Eq. 2.1.9 and 2.1.10. We have

$$S(q)= \frac{1}{N} \langle \sum_{\alpha,\alpha'} e^{i q (x_{\alpha}-x_{\alpha '})} \rangle$$

where the x’s refer to the mass points and the summation is understood as a double sum. Let’s break this up into parts and we’ll notice that things get simplified a bit.

$$S(q)=\frac{1}{N}[\langle \sum_{\alpha=\alpha'} e^{i q (x_{\alpha}-x_{\alpha '})} \rangle + \langle \sum_{\alpha>\alpha'} e^{i q (x_{\alpha}-x_{\alpha '})} \rangle + \langle \sum_{\alpha<\alpha'} e^{i q (x_{\alpha}-x_{\alpha '})} \rangle]$$

Now the first term in brackets is just N. The third term is just the complex conjugate of the second one. So we really only need to calculate the second (or the third) term. Let’s proceed

$$S(q)= \langle \sum_{\alpha>\alpha'} e^{i q (x_{\alpha}-x_{\alpha '})} \rangle = \sum_{\alpha>\alpha'}[p e^{iq1}+(1-p)e^{iq(1+\rho)}]^{\alpha-\alpha'}$$

The introduction of the probabilities comes from the definition of an expected value. The power of $$(\alpha-\alpha')$$ comes from the fact that you have to sum over all of the possible combinations of $$x_{\alpha}-x_{\alpha'}$$. Since this is in the exponent, this sum of exponents can be rewritten as a product of exponentials. Letting $$(\alpha-\alpha')=k$$ and evaluating the sum, we obtain

$$\sum_{k=1}^{N-1}(N-k)[pe^{iq}+(1-p)e^{iq(1+\rho)}]^{k}$$

If you stare at this long enough you come to see this is a geometric sum. We can wave our hands and end up with (Look at the bounds on the sum, N is large, think of what terms dominate. etc)

$$S(q) \approx\frac{1}{N}(1+\frac{pe^{iq}+(1-p)e^{iq(1+\rho)}}{1-[pe^{iq}+(1-p)e^{iq(1+\rho)}]}+c.c.)=\frac{1-|pe^{iq}+(1-p)e^{iq(1+\rho)}|^{2}}{|1-[pe^{iq}+(1-p)e^{iq(1+\rho)}]|^{2}}$$

The rest is just algebra from here.

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