Liquid X-ray scattering with a pink-spectrum undulator

X-ray scattering from a liquid using the spectrum from the undulator fundamental is examined as a function of the bandwidth of the spectrum. The synchrotron-generated X-ray spectrum from an undulator is `pink', i.e. quasi-monochromatic but having a saw-tooth-shaped spectrum with a bandwidth from 1 to 15%. It is shown that features in S(q) are slightly shifted and dampened compared with strictly monochromatic data. In return, the gain in intensity is 250-500 which makes pink beams very important for time-resolved experiments. The undulator spectrum is described by a single exponential with a low-energy tail. The tail shifts features in the scattering function towards high angles and generates a small reduction in amplitude. The theoretical conclusions are compared with experiments. The r-resolved Fourier transformed signals are discussed next. Passing from q- to r-space requires a sin-Fourier transform. The Warren convergence factor is introduced in this calculation to suppress oscillatory artifacts from the finite q_M in the data. It is shown that the deformation of r-resolved signals from the pink spectrum is small compared with that due to the Warren factor. The q-resolved and the r-resolved pink signals thus behave very differently.