Why is intensity decreasing for higher orders of Bragg (X-ray) diffraction?

Question

Bragg diffraction for X-rays by a crystal structure is governed by

$$ n\lambda=2dsin\theta ,$$

where $n$ is an integer and the diffraction order, $\lambda$ is the x-ray wavelength, $d$ the lattice spacing and $\theta$ the angle of incidence, taken from the horizontal.

I obtained the x-ray diffraction spectrum of NaCl, clearly showing all expected features, namely the first, second and third orders of the characteristic $K_{\alpha,\beta}$ peaks, enveloped by the bremsstrahlung spectrum.

The repeated peaks decrease in height and increase in spacing, with order - but why? I can explain the increase in spacing, but why do fewer x-rays interfere constructively at higher (steeper) angles of incidence? My suggestion is that this has something to do with the scattering mechanism (Thomson scattering) but works I have looked at are highly theoretical, and beyond my current scope. Is there a simple or heuristic way of looking at this?

Answer 1

The diffraction pattern is essentially a Fourier transform of the product of the illumination beam and the diffractive structure. The envelope of the power spectrum of the far-field diffraction pattern is determined by the width of the illumination beam. See Fraunhofer diffraction.

Answer 2

first of all take care that you obtain a pattern from diffraction (elastic scattering), not a spectrum (anelastic scattering).

the peaks decresaes in intesity at higher angle on account of the form factor. The dimensions of the atomic diameter and of the X-ray wavelength are similar (X-ray are scattered by electrons).

imagine two parallel X-rays scattered at two different positions of a same electronic cloud (that we assume a sphere). if the scattering is 0, the two X-rays will emerge from the atomic cloud without changing their original direction, the path difference will be 0 and hence no destructive interference will be present.

now consider a scattering angle different from 0. the scattering angle will produce a slight difference between their path that increases with the increase of the angle. This will produce a progressive destructive interference in the scattered intnsity and hence you see a decrease of the peaks intensities at higher angles.

in neutron diffraction this does not occur because the scatterer points are the atomic nuclei, whose size is several order of magnitude lower that that of the radiation wavelength. In neutron diffraction you will see high intensity (nuclear) peaks at high angle. Nontheless, with neutron you also have scattering by unpaired electrons forming a magnetic structure, that is magnetic peaks. In this latter case you have a similar situation than X-ray, since that your scattering centre is now in the electron cloud. At the end in neutron diffraction pattern you will have magnetic peaks whose intensity rapidly decreases with scattering angle and nuclear peaks whose intensity is not depenedent on the angle.