**Intensity of reflections**

In the previous lessons we have seen how we set up a diffraction experiment and what equipment we use to obtain the diffraction patterns. The question arises, how do we get from the diffraction data to a model of the actual protein structure?

**A diffraction pattern as observed on a ****multiwire detector****.
The white spots are the reflections.**

In a diffraction experiment we measure the intensities and the
position of reflections. From the position of the reflection we
can determine its index triple (*h,k,l)* and assign the
appropriate intensity to it. This intensity is proportional to
the square of the structure (factor) amplitude, *|Fhkl|,*
and is a scalar value, i.e., a simple number. Click here to see a
typical list of indexed reflections.
There are usually a few ten-thousand reflections collected for
each crystal.

**The structure factor F****(hkl)****
**The structure factor F

(1)

This is a simple summation which extends over all
atoms *j,* with* x,y,* and* z *their
fractional coordinates. The *f (j)* is the scattering factor of atom *j *and
depends on the kind of atom and the diffraction angle of the
corresponding reflection (*h,k,l)*. For* h,k,l* =0,*
f* equals the atom's number of electrons. We immediately see
that the dimension of *F(hkl)* then must be electrons. The
exponent is *complex* (notice the *i* in there)
with *x,y,z* the fractional coordinates of each atom in
the summation, and *h, k, l* the three indices of the
corresponding reflection. Formula (1) shows that if we *know
the structure*, we can easily *calculate structure factors*.

You may find it helpful to calculate structure factors for a few simple structures to see how different atoms and positions change the values for F and how anomalous contributions to the atomic scattering factors, as used in MAD phasing, lead to the breakdown of Friedel's law.

In crystallographic studies we are in fact **dealing
with the inverse problem**, we know structure factors (or,
to be precise, **their amplitude or magnitude only**
from the measured intensites) and want to know the structure. How
to do that?

Yes, I want to calculate my very own structure factors !

**Fourier transform
**The complex exponential function
is periodic, and with the above parameters it is limited between
-1,1 for its real part and

(2)

We immediately see from looking at the dimensions
that the result of the transformation, *p(xyz)*, must be
an *electron density* : *F(hkl)* is in units of
electrons (see above) and the sum is divided by the cell volume *V*.
We are already familiar with the rest in the summation, which now
is done for each position* x,y,z *in the (normalized) unit
cell. Note also the minus sign now preceding the exponent: In
fact we are transforming an* inverse space *(the*
h,k,l'*s are actually derived from *fractional *numbers
(*1/h,1/k,1/l*) designating where corresponding lattice
planes intersect the unit cell) into a *real *or*
direct space* (the electron density at a *real *point
x,y,z in space). It is a general feature of the FT to transform
from one space into its inverse space and vice versa. So our
diffraction pattern (an image of the reciprocal space) it
transformed back into the real space of electron density. *This
transformation is accurate and in principle complete*. **If
we know the structure factors** (inverse space from
diffraction *by *electrons) **we can calculate the
actual real structure** (the density* of *the
electrons in real space)

.

**This is a picture of
electron density derived from actual data. The electron density
is calculated for each point (x,y,z) in space and points of equal
value are connected forming the characteristic wire grid
presentation.**

As formula (2) is essentially a summation over all structure factors, it is alse referred to as a Fourier synthesis or Fourier summation. You may want to explore and calculate electron density and Patterson maps from the structure factors you have created previously.

Yes, I want to calculate electron density from my own structure factors !

**Bummer
**Unfortunately, a closer look at formula (2)
reveals a small but bothersome detail : In order to perform the
FT, we need the complex structure factors

(3)

In (3) we have the actually measured *|Fhkl|*
and its corresponding unknown phase part *a(hkl)*
separated. The question arises, how can we obtain these darn
phases now to complete the transformation and get to our electron
density?

We'll see that there are a number of ways to achieve this.

** Click on K.C.'s duck to learn
more about general features of Fourier Transforms**

**Back to Introduction**

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Bernhard Rupp**

Last revised
Dezember 27, 2009 01:40