A2. Fused spheres

These are made up of a sequence of truncated spheres with radius $r$ and half height $h$ and two spherical end caps of height $r-h$. Assuming an odd total number $2l+1$ of truncated spheres we find for the truncated semi-sphere located at $z=0$

$$I_{0}(k)=\pi \int_{0}^{h} dz \, \left[ r^{2}- z^{2}\right] ... ...cos kz = \pi \left[ \frac{r^{2}}{k}\, \sin kh -f(h;k) \right]$$ (32)

and for each further ($m=1, \dots M$) truncated sphere on one side,

$$I_{m}(k)=\pi \int_{(2m-1)h}^{(2m+1)h} dz \, \left[r^{2}-\le... ... \cos 2mkh \left[ 2\frac{r^{2}}{k}\, \sin kh -2f(h;k) \right]$$ (33)

For the end caps we use the expressions given in A1, with the appropriate parameters $z_{0}$, $z_{1}$, and $r_{c}$.