What Can You Calculate?

So, if we have the amplitudes and phases of each of the diffraction spots, we can go ahead and calculate the electron density at various points in x,y,z space.

How's it done in practice?

The first thing to do is re-formulate the equation:

First, combine the exponents:

Then, remembering that , this becomes

Note: this expression makes the electron density a complex number. For some applications that's OK, but for most of what we'll be talking about, there are conditions governing the structure factor amplitudes that will cancel out the imaginary terms.

Accordingly, the most commonly used expression for the electron density is:

Now, if you were given a list of structure factor amplitudes and phases, you could go ahead and calculate the electron density. To do it in a systematic way, you'd set up a grid of x,y,z values covering the unit cell, and for each x,y,z point, you'd calculate the sum on the right. You'd take each reflection, multiply its indices by x,y,z, add that to the phase, get the cosine, multiply it by the amplitude, and add it to some sum. After you'd done that for all the reflections, you'd divide by the volume to get the electron density for that point. Then you'd move to another x,y,z point and do it again. By repeating this over and over, you'd generate the three-dimensional electron density function that you could then contour in sections and put on plastic sheets or even better, contour it in three-space and show it on a workstation screen.

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copyright © Ron Stenkamp stenkamp@u.washington.edu Most recent update 1/7/97