# 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}$

## 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

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 }}}
${\ displaystyle F_ {hkl} = {\ begin {cases} f_ {1} + f_ {2}, & {\ text {if}} h + k + l {\ text {even}} \\ f_ {1} -f_ {2}, & {\ text {if}} h + k + l {\ text {odd}} \ end {cases}}}$

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: ${\ displaystyle f_ {1} = f_ {2} =: f}$

${\ displaystyle F_ {hkl} = {\ begin {cases} 2f, & {\ text {if}} h + k + l {\ text {even}} \\ 0, & {\ text {if}} h + k + l {\ text {odd}} \ end {cases}}}$

## literature

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