Analytic problems involving diffraction

1.The Fourier transform of a helix.

(a) First parametrize it by an angle theta, so that

${\bf r_H}(\theta) = R(\cos\theta {\hat x} + \sin\theta {\hat y}) + \frac{h}{2\pi}\theta {\hat z}$ ${\bf r_H}(\theta) = R(\cos\theta {\hat x} + \sin\theta {\hat y}) + \frac{h}{2\pi}\theta {\hat z}$

$R$ and $h$ are parameters with meanings you should try to understand. The density of the helix is

Calculate the three dimensional Fourier transform of $\rho$ $\rho$ up to multiplicative prefactors. Here $R$ and $h$ are constants.

$\rho({\bf r}) = \int_{-\infty}^{\infty} \delta({\bf r} - {\bf r_H}(\theta))d\theta .$ $\rho({\bf r}) = \int_{-\infty}^{\infty} \delta({\bf r} - {\bf r_H}(\theta))d\theta .$

Hint: Look at http://physics.ucsc.edu/~josh/116C.07/index.html for a handout with relevant equations in it.

(b) Sketch what the diffraction pattern should look like for a helix.

2. Fourier for a spherical shell of radius.

Take $\rho({\bf r}) = \delta (|{\bf r}| -R)$ $\rho({\bf r}) = \delta (|{\bf r}| -R)$ . Calculate its three dimensional Fourier transform, up to multiplicative prefactors.

3. Now consider a cubic crystal of spherical shells, with lattice spacing $a$ that is greater than $2R$ . What is the Fourier Transform in this case?

4. Consider a comb function but without an infinite number of delta functions:

$C_{L} (x) = \sum_{n = - L}^{L} δ (x - n)$ $C_L(x) = \sum_{n=-L}^{L} \delta(x-n)$

Normally the comb function has $L = ∞$ $L = \infty$ . Calculate the Fourier transform of $C_{L}$ $C_L$ for finite $L$ $L$ .

Hint. Write $C_{L}$ $C_L$ as the product of $C_{∞}$ $C_\infty$ (which you know how to Fourier transform) and a top hat function. Then use the convolution theorem.

5. What does the previous problem tell you about how a diffraction pattern changes for crystals of finite size? How about finite length DNA molecules?

6. Give three examples where helices are important in biological systems. Find an example of a structure such as in problem 2.

7. How would your answer for 1 change for a double helix? Hint: use the shift theorem for fourier transforms.