Interactive Tutorial about Diffraction
Interactive example: Direct methods II

Interactive examples
Electron density
Relation H and 2H


In this interactive example you should investigate limitations that exist for the phase angles of intense reflections in centrosymmetric structures. Since the electron density must always be larger than zero, the phases of intense reflections can not be completely random. This example explores the phase relationship between two reflections H,K,L and 2H,2K,2L.

The controls in the simulator window allow you to choose a reflection H,K, its amplitude and phase angle, and the respective values for the reflection 2H,2K. Since we are dealing with a centrosymmetric structure, the phase angles are limited to either 0 or 180 degrees. All calculations are based on a unit cell with lattice constants 10 Angstroem and 90 degrees, and use the reflection H,K and its Friedel pair -H,-K as well as the reflection 0,0 with an amplitude of 1000. Since the amplitude of 0,0 is set to 1000, the amplitude of the reflections H,K and 2H,2K must be less or equal to 1000. In order to work out why they are limited to less than F(0,0) go back to the main part of the teaching tutorial. Three diagrams will be plotted. The first displays the electron density distribution within the unit cell (the upper right diagram). The second and third diagrams display the electron density along two lines through the unit cell (parallel x at y=0 and parallel y at x=0, respectively).

Start exploring and try to answer the questions below:

  • Work out the effects of all the permutations of the following combinations:
    • |F(h,k)| high versus low
    • Phase (h,k) 0 versus 180 degrees
    • |F(2h,2k)| high versus low
    • Phase (2h,2k) 0 versus 180 degrees
  • What conditions do you have to fulfill to maintain an electron density that is greater or equal to zero ?

  • Does the choice of H,K influence the occurrence of negative electron density ?

Once you are done, click here to verify your answers.
© Th. Proffen and R.B. Neder, 2003