Structure factor

The structure factor is a measure of the scattering power of a crystal base . He gives the relative intensity of by the Laue indices , , specific diffraction reflex on. The structure factor depends on the structure of the base, the scattering power of the base atoms and their thermal motion . The direction in which the diffraction reflexes can be observed is given by the Bragg or, equivalently, the Laue condition , which is based on the pure crystal lattice (a point-like scattering center at the lattice point). ${\displaystyle F_{hkl}}$ ${\displaystyle h}$ ${\displaystyle k}$ ${\displaystyle l}$

• X-ray diffraction : The electromagnetic radiation is scattered on the electrons of the atoms . The structure factor is the Fourier transform of the electron distribution within a unit cell .
• Electron diffraction : The electrons are scattered by the Coulomb interaction on the shell electrons and the atomic nuclei . The structure factor is the Fourier transform of the charge distribution within a unit cell.
• Neutron diffraction : Neutrons interact through strong interaction with the atomic nuclei and because of their magnetic moment with the magnetic moment of the atoms. The structure factor is the Fourier transform of the nuclear distribution ( nucleon distribution) and the magnetic structure within a unit cell.

description

Principle of the Laue condition: the two beams
only interfere constructively with certain ratios of and k ' ${\displaystyle {\vec {r}},{\vec {k}}}$

One chooses a reference point within the unit cell as the origin. Two infinitesimal volume elements are considered as scattering centers, one at the reference point , one at . Let the wave vector of the incident radiation be that of the scattered one . This results in the following path difference (path difference): ${\displaystyle \mathrm {d} V}$ ${\displaystyle {\vec {0}}}$ ${\displaystyle {\vec {r}}}$ ${\displaystyle {\vec {k}}}$ ${\displaystyle {\vec {k}}'}$

${\displaystyle \Delta s({\vec {r}}\,)={\vec {r}}\cdot {\frac {{\vec {k}}'}{k'}}-{\vec {r}}\cdot {\frac {\vec {k}}{k}}}$

The phase difference is (the scattering is elastic, so ): ${\displaystyle k=k'}$

${\displaystyle \varphi ({\vec {r}}\,)=2\pi {\frac {\Delta s}{\lambda }}=k\Delta s=({\vec {k}}'-{\vec {k}}\,)\cdot {\vec {r}}}$

After the Laue condition diffraction reflections can be observed only when the change of the wave vector in the scattering process a reciprocal lattice vector corresponds to: . This results in inserted: ${\displaystyle {\vec {G}}}$ ${\displaystyle {\vec {k}}'-{\vec {k}}={\vec {G}}}$

${\displaystyle \varphi ({\vec {r}}\,)={\vec {G}}\cdot {\vec {r}}}$

Now you integrate over the volume of a unit cell and weight the phase differences with the scattering power of each volume element . The scattering power is, depending on the diffraction experiment (see above), the electron density , the charge density or the nuclear density. ${\displaystyle V_{EZ}}$ ${\displaystyle \exp \left[i\,\varphi ({\vec {r}}\,)\right]}$ ${\displaystyle n({\vec {r}}\,)}$

${\displaystyle \int _{V_{EZ}}n({\vec {r}}\,)\,\exp \left[i\,\varphi ({\vec {r}}\,)\right]\mathrm {d} ^{3}r=\int _{V_{EZ}}n({\vec {r}}\,)\,\exp \left[i\,{\vec {G}}\cdot {\vec {r}}\right]\mathrm {d} ^{3}r}$

The wave diffracted at the crystal has an amplitude that is proportional to the size just calculated.

${\displaystyle F_{hkl}=\int _{V_{EZ}}n({\vec {r}}\,)\,\exp \left[i\,{\vec {G}}\cdot {\vec {r}}\right]\mathrm {d} ^{3}r}$

${\displaystyle F_{hkl}}$is called a structure factor . This is from the Laue indices , , dependent since the reciprocal lattice vector equal is. The structure factor is thus the Fourier transform of the scattering power (e.g. the electron density) . ${\displaystyle h}$ ${\displaystyle k}$ ${\displaystyle l}$ ${\displaystyle {\vec {G}}=h{\vec {b}}_{1}+k{\vec {b}}_{2}+l{\vec {b}}_{3}}$ ${\displaystyle {\mathcal {F}}\left\{n({\vec {r}}\,)\right\}=F_{hkl}({\vec {G}})}$

The vector can be a linear combination of primitive lattice vectors write: . The scalar product in the exponent can be evaluated with the relation ( corresponds to ): ${\displaystyle {\vec {r}}}$ ${\displaystyle {\vec {a}}_{i}}$ ${\displaystyle {\vec {r}}=u_{1}{\vec {a}}_{1}+u_{2}{\vec {a}}_{2}+u_{3}{\vec {a}}_{3}}$ ${\displaystyle {\vec {a}}_{i}\cdot {\vec {b}}_{j}=2\pi \delta _{ij}}$ ${\displaystyle V_{EZ}}$ ${\displaystyle u_{i}\in [0;1]}$

${\displaystyle F_{hkl}=\int _{0}^{1}\int _{0}^{1}\int _{0}^{1}n(u_{1},u_{2},u_{3})\,\exp \left[2\pi i\left(u_{1}h+u_{2}k+u_{3}l\right)\right]\mathrm {d} u_{1}\mathrm {d} u_{2}\mathrm {d} u_{3}}$

The structure factor is a complex quantity . As a result of a diffraction experiment, one observes the intensity of the diffracted wave, which is proportional to the square of the magnitude of the structure factor : ${\displaystyle F=\rho e^{i\gamma }}$

${\displaystyle I\propto |F_{hkl}|^{2}=\rho ^{2}}$

Thus all phase information is lost. If the result of a measurement were to be available, the required quantity could be found using Fourier transformation : ${\displaystyle \gamma }$ ${\displaystyle F_{hkl}}$ ${\displaystyle n({\vec {r}}\,)}$

${\displaystyle n({\vec {r}}\,)=n(u_{1},u_{2},u_{3})=\sum _{h,k,l=-\infty }^{\infty }F_{hkl}\,\exp \left[-2\pi i\left(u_{1}h+u_{2}k+u_{3}l\right)\right]}$

Since, however, only known approximation methods such as the Patterson method must be used to solve the phase problem. ${\displaystyle |F_{hkl}|^{2}}$

Atomic scattering factor

The position vector is now broken down into a part that points from the reference point to the nucleus of the -th atom, and a vector that points from the nucleus of the -th atom to the volume element under consideration. ${\displaystyle {\vec {r}}}$ ${\displaystyle {\vec {r}}_{i}}$ ${\displaystyle i}$ ${\displaystyle {\vec {\tilde {r}}}}$ ${\displaystyle i}$

${\displaystyle {\vec {r}}={\vec {r}}_{i}+{\vec {\tilde {r}}}}$

In the equation for the structure factor, the integral over the whole unit cell is split up into a sum over smaller integration areas, namely the volumes of the individual atoms . Here is the scattering power (e.g. electron density) of the -th atom. The sum runs over all atoms of the unit cell: ${\displaystyle V_{A_{i}}}$ ${\displaystyle n_{i}({\vec {\tilde {r}}}\,)=n({\vec {r}}_{i}+{\vec {\tilde {r}}}\,)}$ ${\displaystyle i}$

${\displaystyle F_{hkl}=\sum _{i}\int _{V_{A_{i}}}n_{i}({\vec {\tilde {r}}}\,)\,\exp \left[i\,{\vec {G}}\cdot ({\vec {r}}_{i}+{\vec {\tilde {r}}}\,)\right]\mathrm {d} ^{3}{\tilde {r}}=\sum _{i}\exp \left[i\,{\vec {G}}\cdot {\vec {r}}_{i}\right]\int _{V_{A_{i}}}n_{i}({\vec {\tilde {r}}}\,)\,\exp \left[i\,{\vec {G}}\cdot {\vec {\tilde {r}}}\right]\mathrm {d} ^{3}{\tilde {r}}}$

The integral is called the atomic scattering factor (or atomic form factor ) of the -th atom: ${\displaystyle \ f_{i}}$ ${\displaystyle i}$

${\displaystyle f_{i}=\int _{V_{A_{i}}}n_{i}({\vec {\tilde {r}}}\,)\,\exp \left[i\,{\vec {G}}\cdot {\vec {\tilde {r}}}\right]\mathrm {d} ^{3}{\tilde {r}}}$

The structure factor is thus written as follows:

${\displaystyle F_{hkl}=\sum _{i}f_{i}\,\exp \left[i\,{\vec {G}}\cdot {\vec {r}}_{i}\right]}$

With the component notation introduced above:

${\displaystyle F_{hkl}=\sum _{i}f_{i}\,\exp \left[2\pi i\,\left(u_{i,1}h+u_{i,2}k+u_{i,3}l\right)\right]}$

If one also considers the thermal movement of the atoms, it is time-dependent. Now one breaks down into a mean location (equilibrium position, resting) and the deflection (time-dependent). The latter leads to the Debye-Waller factor . ${\displaystyle {\vec {r}}_{i}}$ ${\displaystyle {\vec {r}}_{i}}$ ${\displaystyle {\vec {r}}_{i,0}}$ ${\displaystyle {\vec {u}}_{i}(t)}$

example

→ Main article : Systematic extinction

As an example, the structure factor for a cesium chloride structure is calculated. The grid is a cubic primitive are 2-atomiger base, the primitive lattice vectors , , . One base atom is attached to the other . ${\displaystyle {\vec {a}}_{1}=a{\hat {e}}_{x}}$ ${\displaystyle {\vec {a}}_{2}=a{\hat {e}}_{y}}$ ${\displaystyle {\vec {a}}_{3}=a{\hat {e}}_{z}}$ ${\displaystyle {\vec {r}}_{1}={\vec {0}}}$ ${\displaystyle {\vec {r}}_{2}=(1/2)({\vec {a}}_{1}+{\vec {a}}_{2}+{\vec {a}}_{3})}$

${\displaystyle {\begin{aligned}F_{hkl}&=\sum _{i=1}^{2}f_{i}\,\exp \left[2\pi i\,\left(u_{i,1}h+u_{i,2}k+u_{i,3}l\right)\right]\\&=f_{1}\,\exp \left[2\pi i\,\left(0\cdot h+0\cdot k+0\cdot l\right)\right]+f_{2}\,\exp \left[2\pi i\,\left({\frac {1}{2}}h+{\frac {1}{2}}k+{\frac {1}{2}}l\right)\right]\\&=f_{1}+f_{2}\,\exp \left[\pi i\,\left(h+k+l\right)\right]\\&=f_{1}+f_{2}(-1)^{h+k+l}\\\end{aligned}}}$

${\displaystyle F_{hkl}={\begin{cases}f_{1}+f_{2},&{\text{wenn }}h+k+l{\text{ gerade}}\\f_{1}-f_{2},&{\text{wenn }}h+k+l{\text{ ungerade}}\end{cases}}}$

If the sum of the Laue indices is even, the diffracted X-ray beam has a high intensity; if the sum is odd, the intensity is minimal. If both base atoms have the same atomic scattering factor , the intensity is zero if the sum is odd; one speaks of complete extinction . This applies to the body-centered cubic lattice (bcc lattice) if it is described in the system of the primitive cubic lattice with two identical basic atoms: ${\displaystyle f_{1}=f_{2}=:f}$

${\displaystyle F_{hkl}={\begin{cases}2f,&{\text{wenn }}h+k+l{\text{ gerade}}\\0,&{\text{wenn }}h+k+l{\text{ ungerade}}\end{cases}}}$

literature

• Borchardt-Ott, Walter: Crystallography: an introduction for natural scientists . Springer publishing house.
• Massa, Werner: Crystal structure determination . Teubner Verlag.