Analytic problems involving diffraction

1.The Fourier transform of a helix.

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

rH(θ)=R(cosθx^+sinθy^)+h2πθz^  

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 ρ up to multiplicative prefactors. Here R and h are constants.

ρ(r)=δ(rrH(θ))dθ.

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  ρ(r)=δ(|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:

CL(x)=n=LLδ(xn)

Normally the comb function has  L=. Calculate the Fourier transform of CLfor finite L.

Hint. Write CL as the product of C(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.