X-rays are electromagnetic waves with wavelengths of about λ= 0.5−3.0˚A. They are produced when electrons of high energy, which were accelerated from a cath-ode towards an ancath-ode of a certain material like Mo or Cu, suddenly hit the ancath-ode.
This leads to the emission of X-rays of characteristic wavelengths. In a crystal the atoms are arranged regularly in repeating units. The smallest unit which builds up the crystal just by translation is called the unit cell. The monochromated X-rays are directed towards the crystal where they are scattered by the electrons in the molecules. As the X-rays are considered as waves they can be described with an amplitude F and a phase Φ.
F=|F|eiΦ =F eiΦ (2.1)
The amplitude of the vectorF is its modulus|F| and is denotedF. The amplitude contains information on the amount of electrons present in the unit cell, whereas the phase contains information on their relative position to each other. For convenience reasons the scattered waves are considered as reflections at so-called lattice planes.
Only for certain anglesθ between the X-ray source and the lattice planes reflections can be observed. This relation is given by the Bragg equation[2]
nλ= 2dsinθ (2.2)
withn= an integer number,λ= wavelength,d= distance between the lattice planes (resolution),θ = scattering angle. The detected reflections are described with their indices h, k and l that denote their orientation relative to the unit cell, and their intensity I. According to the kinematical theory of scattering[3] the intensity I of
the reflections is proportional to the square of the amplitude F
I ∝F2 (2.3)
Unfortunately, the phase information is lost during the measurement. This is called the phase problem in crystallography. Several methods that have been developed to solve this phase problem will be described later.
As the beam is scattered by all atoms in the crystal each atom contributes to each reflection. Thus, the structure factor Fhkl can be expressed as the summation over all atomic structure factors Fi. The individual atomic structure factors Fi are composed of the atomic scattering factor fi and the phase shift Φi caused by this atom. Fig. 2.1 shows a representation of an atomic scattering factor in the Argand diagram. In the case of atomic structure factors Fi the amplitude is the atomic scattering factor fi. For a scattering angle θ = 0° the atomic scattering factor fi is normalized to the number of the electrons of the atom and it decreases with increasing resolution.
(a) atomic structure factorFi (b) total structure factorFas a sum of allFi
Figure 2.1: Projection of the structure factorFin the Argand diagram.
Fhkl =X
i
fi{cos[2π(hxi+kyi+lzi)] +isin[2π(hxi+kyi+lzi)]} (2.4)
h, k and l are the Miller indices of the lattice planes andxi, yi and zi are the coor-dinates of atomi. The structure factor can also be calculated from the contribution
of the electron density ρ in small volume elements. The Fourier transformation is
Fhkl =|Fhkl|eiΦhkl =
Z
V
ρ(xyz)e2πi(hx+ky+lz)dV (2.5) When the Fourier backtransformation is applied the electron density can be cal-culated from the summation over all individual structure factors in the volume V with
As already mentioned the amplitudes Fhkl can be calculated from the measured intensities of the reflections, but the information on the phase Φis lost.
“Direct methods” are one of several possibilities to reconstruct the phases. They can be applied mainly for small molecules with up to about 1000 atoms and when data up to atomic resolution are (d≤1.2˚A) are available. Direct methods are based on the relation between intensities and phases and the assumption that the electron density is always positive and that its maxima are distributed evenly in the crystal.
A fundamental contribution to direct methods had the Sayre equation[4] from which the triplet phase relation can be derived[5, 6]. It is based on the relation between phases of three independent but strong reflections
Φh+h0,k+k0,l+l0 ≈Φh,k,l+ Φh0,k0,l0 (2.7)
Direct methods were developed in more detail by Karle and Hauptman[7–9]. From the initial phases an initial model can be calculated.
Other methods for the determination of phases for larger molecules like proteins are the isomorphous replacement methods like SIR or MIR, which allow to get the phases from a heavy atom derivative. Anomalous dispersion methods are based on the anomalous signal that heavy atoms show for certain wavelengths (SAD and MAD). If a structurally similar compound is available Molecular Replacement can be the method of choice.
The structure factors derived from the observed intensitiesI are denotedFobs and the calculated structure factors Fcalc. They are scaled to each other by the scale
factor k
This scale factor is always necessary when the Fcalc are set into relation to theFobs but it will not specifically be mentioned in the following equations.
With the initial model and phases an improved model can be calculated by a least-squares refinement of the model parameters, from which new and improved amplitudes and phases are obtained. This process is iteratively repeated. The model is calculated such that the squared difference in the weighted squared amplitudes minimizes if the refinement is performed against F2.
X
hkl
w(Fobs2 −Fcalc2 )2 =M in. (2.9)
The factor w weights the intensities according to their accuracy and reliability, i.e. the uncertainties in their measurement are accounted for. In the simplest case the weightsw are
w= 1
σ2(Fobs2 ) (2.10)
In the refinement program SHELXL[10, 11]this weighting scheme is extended to min-imize the differences in the squares of the errors for reflections in different intensity ranges.
The values for a and b are proposed by SHELXL during the refinement.
For the minimization of the sum of the squared differences of the squared structure factors different parameters for each atom have to be refined. For a normal small molecule Independent Atom Model (IAM) these are nine parameters per atom (three coordinates x, y, z and six anisotropic displacement parameters Uij).
For a comparison how well the model agrees with the observed dataR-factors are calculated. The conventionalR1-factor is based on a comparison of the amplitudesF
R1 =
P
hkl||Fobs| − |Fcalc||
P
hkl|Fobs| (2.13)
If the R1 is weighted the wR1 is obtained:
wR1 =
As the refinement is usually performed against the squared amplitudes F2 an R2 -factor can be calculated as
Normally, the weights are applied in the refinement, thus, the weightedwR2 is given as
An additional quality criterion is the goodness of fit GooF.
GooF =S =
with n = number of reflections and p = total number of parameters refined. The GooF not only considers the weighted error in the squared differenceFobs2 −Fcalc2 but also the degree of overdetermination of the parameters with the difference (n−p).
If the structure was refined completely and correctly and the weighting scheme was applied properly theGooF should give a value close to 1.
For unproblematic small molecule structures theR1-factor can reach values smaller than 0.05 and the wR2 values less than 0.15.
Another R-factor that is used in this work is the Rint. It is calculated from the summation over all reflections which are averaged over at least one symmetry
equiv-alent.
Rint =
P|Fobs2 − hFobs2 i|
PFobs2 (2.18)
The Rint describes the deviation of a reflection from its symmetry equivalents.
In X-ray crystallography there are several approaches to describe the electron density in a crystal. One of them is the conventional Independent Atom Model (IAM). It describes a spherical electron density around the atoms and does not account for charge transfer and bonding between atoms. Thus, quite high residual electron densities remain after an IAM refinement. The higher the resolution of the experiment the more distinct are these residuals.