A2 - structural significance

• intensities are plotted against the scattering wavevector, $q:$

where, $\lambda$ is the wavelength of x-rays/neutrons, and $\theta$ is the scattering angle

• physically, $\mathrm{\Delta }p=\hslash q$ is the momentum transfer after the scattering
• plotting patterns against $q$ makes it $\lambda$-independent, which would not be the case for plot against $\theta$

image: CJ Gommes, S Jakschb and S Frielinghausb (2021)

• scattering peaks can be interpreted in terms of diffraction by the periodic pore structure in the nm scale

• many materials do not exhibit periodicity, hence their SAS patterns do not exhibit any sharp peaks

• linear sales have featureless plots

• logarithmic scales reveal a plateau at small $q$ followed by an oscillatory decrease

• bragg's law is concerned with diffraction very specific to scattering from periodic structures

where, $d$ is the periodicity of structure, ${\theta }_d$ is the diffraction angle, and $n\in \mathbb{Z}>0$

• this is not a general law, but a consequence of the general principles particularised to spatially periodic structures

• assumptions:

  • the emission of secondary waves are isotropic
  • they reach the detector without being scattered twice

• only the unperturbed incoming waves are responsible for secondary waves

• the phase, $\varphi$, of a secondary wave reaching the detector is solely the function of the position of the scattering centre, $(x,y,z)$

• for any given angle, a characteristic phase map can be calculated, ${\varphi }_{\theta }(x,y,z)$

• the intensity can be obtained by comparing ${\varphi }_{\theta }$ with the actual spatial distribution of electrons/nuclei in the sample

image: CJ Gommes, S Jakschb and S Frielinghausb (2021)

• the phase map is obtained by:

• it takes constant values on geometric planes oriented at an angle of $\theta /2$ wrt the incident beam

• it increases by $2\pi$ over ${\lambda }^{\ast }=2\pi /q$
• for SAS< $\sin (\theta /2)\simeq \theta /2$, so ${\lambda }^{\ast }\simeq \lambda /\theta$