MHB Does this problem warrant a taylor expansion? (solid state physics)

[skate_nerd]

The whole problem I'm doing here is not even really relevant, so I won't go too much into it...I'm told to find an atomic form factor given some certain conditions, and I do a big gross integral and got this:
$$f=(\frac{4}{4+(a_oG)^2})^2$$
where \(a_o\) is the Bohr radius and \(G\) is the magnitude of an arbitrary reciprocal lattice vector.
After finding this, I am asked to find the atomic form factor as
\(a_o>>\lambda\) and as
\(a_o<<\lambda\).
This seems pretty simple because there is a relation between \(\lambda\) and \(G\):
$$G=\frac{4\pi}{\lambda}\sin\theta$$
Plugging that into my formula for the form factor, I quickly assumed that \(a_o>>\lambda\) will just make \(\frac{a_o}{\lambda}\) really large, so the denominator of the form factor gets really large and then is approximately zero. And as \(a_o<<\lambda\), \(\frac{a_o}{\lambda}\) is approximately zero, so the form factor ends up being 1.

The problem, however, also asks for both of the limiting cases if the form factor ends up depending on \(\theta\). I'm not sure why this would be asked for each case if one of them didn't end up with a \(\theta\) dependence...
So I'm asking for some advice, would it make sense to use a taylor expansion on either or both of these limiting cases? Why or why not?

Thanks for your attention!

 

[I like Serena]

The whole problem I'm doing here is not even really relevant, so I won't go too much into it...I'm told to find an atomic form factor given some certain conditions, and I do a big gross integral and got this:
$$f=(\frac{4}{4+(a_oG)^2})^2$$
where \(a_o\) is the Bohr radius and \(G\) is the magnitude of an arbitrary reciprocal lattice vector.
After finding this, I am asked to find the atomic form factor as
\(a_o>>\lambda\) and as
\(a_o<<\lambda\).
This seems pretty simple because there is a relation between \(\lambda\) and \(G\):
$$G=\frac{4\pi}{\lambda}\sin\theta$$
Plugging that into my formula for the form factor, I quickly assumed that \(a_o>>\lambda\) will just make \(\frac{a_o}{\lambda}\) really large, so the denominator of the form factor gets really large and then is approximately zero. And as \(a_o<<\lambda\), \(\frac{a_o}{\lambda}\) is approximately zero, so the form factor ends up being 1.

The problem, however, also asks for both of the limiting cases if the form factor ends up depending on \(\theta\). I'm not sure why this would be asked for each case if one of them didn't end up with a \(\theta\) dependence...
So I'm asking for some advice, would it make sense to use a taylor expansion on either or both of these limiting cases? Why or why not?

Thanks for your attention!

Hi skatenerd!

Can it be that something like the following is intended? (Wondering)
\begin{array}{c|c|c|}
& a_0 \gg \lambda & a_0 \ll \lambda \\
\hline
\theta=0 & f = 1 & f=1\\
\hline
\theta = \pm \frac \pi 2 & f = 0 & f = 1\\
\hline
\end{array}