I'm a chemist stuck in a mathematical problem. Please bear with me as I'm trying to express myself in Math language.

Let me explain in short terms the experimental method I'm using: X-ray diffraction. For this method, atoms in a solid can be viewed as ordered balls in space. X-rays, having similar wavelength as the distance between atoms, are diffracted by those balls and can be detected as diffraction fringes. The distance between the diffracting atoms (or planes of atoms, aka lattice distance) can then be calculated using Bragg's law We use the method to measure the symmetry of the atom ordering, their distances and placement in space.

As chemists, we like to deduce so-called lattice parameters (the lengths of the smallest repeating unit of atoms in all three cartesian coordinates) from three or more characteristic atomic plane distances. In my case the repeating unit has the form of a cuboid. The measured diffraction fringes are equivalent to the points of the fourier transform of the spacing of the atoms in real space and describe the repetition of those atoms. They are linked to the cuboid edge lengths via:

1/d^{2} = h^{2}/a^{2} + k^{2}/b^{2} + l^{2}/c^{2}

h, k and l are discrete values, characteristic to the specific diffraction fringe, d is the measured lattice distance.

Unfortunately, the cuboid is not stiff, but a, b and c have some distribution. As such I have measured an asymmetric distribution of the lattice distances d. I would like to find out how the cuboid edge lengths contribute to the measured distribution.

I fitted a normal distribution to my experimental data for 1/d^{2}, for which it is easy to deconvolute to get distributions of 1/a^{2} and so on. This doesn't reflect my data correctly, though, since I observe an asymmetric distribution of 1/d^{2}.

Do you have an idea how to extract information of the distribution of a, b and c from those three distributions of lattice spacings? In a first approximation, these three edge lengths are independent variables.