Some Properties of Atoms — Question 10

Question 10

A finite-size form factor can resolve a uniformly charged sphere

For a supplied scattering model of a uniform sphere of radius R, the normalized amplitude is $F(q)=3[\sin(qR)-qR\cos(qR)]/(qR)^3$. The wave-number transfer q has units of inverse length. Assume $qR\ll1$.

1. Expand F through order $(qR)^2$.

2. Find the leading intensity reduction $1-|F|^2$ and infer R if it is 0.020 at $q=2.00\times10^{10}\,\mathrm{m^{-1}}$.

Question 10 - Solution

A finite-size form factor can resolve a uniformly charged sphere

(a) Method and calculation

$$x=qR:\quad\sin x-x\cos x=x^3/3-x^5/30+O(x^7).$$$$\boxed{F(q)=1-(qR)^2/10+O((qR)^4)}.$$

(b) Interpretation and further calculation

$$|F|^2=1-(qR)^2/5+O((qR)^4)\Rightarrow1-|F|^2\simeq(qR)^2/5.$$$$\boxed{R\simeq\sqrt{5(.020)}/q=1.58\times10^{-11}\,\mathrm m}.$$

Physical check

The inferred radius is a leading-order model estimate, with qR about 0.316. The charge profile is supplied; real atomic electron densities are not uniform spheres.