X-ray question about form factors

[appletree]

Homework Statement

Make an approximate calculation of the x-ray atomic form factor for He by assuming that its electron density is given by a hydrogenic ground state.

Homework Equations

Need help with this.

The Attempt at a Solution

I attempted to use the hydrogenic model, but could not figure out how to do this. There are additional parts to this question, but I just need a little help on getting started so that I can get this part, and then I can do the parts that follow. Thank you.

 

[mark726]

To calculate the x-ray atomic form factor for He, we can use the following equation:

f(x) = ∫ρ(r)exp(-2πix·r) dV

where f(x) is the form factor, ρ(r) is the electron density, and x is the scattering vector.

Since we are assuming that He has a hydrogenic ground state, we can use the following expression for the electron density:

ρ(r) = (1/πa0^3)exp(-r/a0)

where a0 is the Bohr radius.

Substituting this into the equation for the form factor, we get:

f(x) = (1/πa0^3)∫exp(-r/a0)exp(-2πix·r) dV

We can rewrite this as:

f(x) = (1/πa0^3)∫exp[-(r/a0 + 2πix·r)] dV

Using the spherical coordinates, we can write dV as r^2sinθdrdθdφ.

Substituting this into the equation, we get:

f(x) = (1/πa0^3)∫r^2sinθexp[-(r/a0 + 2πix·r)] drdθdφ

The integration over φ and θ will give us constants, so we can write:

f(x) = (1/πa0^3)∫r^2exp[-(r/a0 + 2πix·r)] dr

We can now substitute r = u/a0, and the integral becomes:

f(x) = (1/πa0^3)∫u^2exp(-u - 2πix·a0u) du

This integral can be evaluated using integration by parts, and the final expression for the form factor is:

f(x) = (1/πa0^3)[(1 + 2πix·a0)^-3]

This is the approximate calculation of the x-ray atomic form factor for He.