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).
![H](https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a)
![k](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40)
![l](https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac)
description
Principle of the Laue condition: the two beams
only
interfere constructively with certain ratios of and k '![{\ vec r}, {\ vec k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05fd1a6f5a6473b75d25ba6f102f2583f4a6ba4f)
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):
![\ mathrm {d} V](https://wikimedia.org/api/rest_v1/media/math/render/svg/b80507190aa9d38a279909db47b63657f2b62ba7)
![{\ vec {0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e76498919cf387316fc79d04120c59a8d430ef36)
![{\ vec {r}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6aec3c9ce13b53e9e24c98e7cce4212627884c91)
![{\ vec {k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ccd4b98d198d6538010ae815ee1199baabd3493)
![{\ vec k} '](https://wikimedia.org/api/rest_v1/media/math/render/svg/24d42a2dc714bdd2a70a8a107a45b9ee95ce05f9)
![\ Delta s ({\ vec {r}} \,) = {\ vec {r}} \ cdot {\ frac {{\ vec {k}} '} {k'}} - {\ vec {r}} \ cdot {\ frac {{\ vec {k}}} {k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f4e1193fc23226b3d04128c4ee87b5d1f63fd58)
The phase difference is (the scattering is elastic, so ):
![k = k '](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1850e16ed4ac1d1d3a126488fb6ac9511e76128)
![\ varphi ({\ vec {r}} \,) = 2 \ pi {\ frac {\ Delta s} {\ lambda}} = k \ Delta s = ({\ vec {k}} '- {\ vec { k}} \,) \ cdot {\ vec {r}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb918a0b1194a3e23b33d6ada3b3420567d92da9)
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:
![{\ vec {G}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/241bb14fd31373ea9f4a602cc59e4ec1da27b10b)
![{\ vec {k}} '- {\ vec {k}} = {\ vec {G}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8cf2d810bf1e635b533568a5990ec220ed58e45)
![\ varphi ({\ vec {r}} \,) = {\ vec {G}} \ cdot {\ vec {r}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73e810741645632e10089f3e1c7cebb6d19a3f4d)
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.
![V _ {{EZ}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5321e8517e8a1f59394f804829cafa73c8a07f5d)
![\ exp \ left [i \, \ varphi ({\ vec {r}} \,) \ right]](https://wikimedia.org/api/rest_v1/media/math/render/svg/95548c1b513beab86488ccc95075e53e67518c8b)
![n ({\ vec {r}} \,)](https://wikimedia.org/api/rest_v1/media/math/render/svg/0992aa91239c5aef633b7a5fbc927e0ecce1c448)
![\ 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](https://wikimedia.org/api/rest_v1/media/math/render/svg/5df8e0e1e972c5be35c8137039764d00be8655ef)
The wave diffracted at the crystal has an amplitude that is proportional to the size just calculated.
![F _ {{hkl}} = \ int _ {{V _ {{EZ}}}} n ({\ vec {r}} \,) \, \ exp \ left [i \, {\ vec {G}} \ cdot {\ vec {r}} \ right] {\ mathrm {d}} ^ {{3}} r](https://wikimedia.org/api/rest_v1/media/math/render/svg/01df5a2d082271966c618218691e36fcb7620e38)
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) .
![H](https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a)
![k](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40)
![l](https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac)
![{\ vec {G}} = h {\ vec {b}} _ {{1}} + k {\ vec {b}} _ {{2}} + l {\ vec {b}} _ {{3 }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7652f70490c0e22446fca5ff6dc9b98493066b26)
![{\ mathcal {F}} \ left \ {n ({\ vec {r}} \,) \ right \} = F _ {{hkl}} ({\ vec {G}})](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9af9477284735e4f031e58a45408a7fa569449c)
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 ):
![{\ vec {r}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6aec3c9ce13b53e9e24c98e7cce4212627884c91)
![{\ vec {a}} _ {{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de5ef6042eca03fa8c969e5b33adf6fe9e02476b)
![{\ vec {r}} = u _ {{1}} {\ vec {a}} _ {{1}} + u _ {{2}} {\ vec {a}} _ {{2}} + u_ { {3}} {\ vec {a}} _ {{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cf1003049ca2ad4790dc2d6e5e5c5dcfb1b1566)
![{\ vec {a}} _ {{i}} \ cdot {\ vec {b}} _ {{j}} = 2 \ pi \ delta _ {{ij}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fae73a68abbbab27d44ff98e07ee3e4aec7dd801)
![V _ {{EZ}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5321e8517e8a1f59394f804829cafa73c8a07f5d)
![u _ {{i}} \ in [0; 1]](https://wikimedia.org/api/rest_v1/media/math/render/svg/baf599d76837e24a9209540a3d7900adde3811a4)
![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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a84184b3a12bce6986fca4955ddb6716526addf)
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 :
![F = \ rho e ^ {{i \ gamma}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/002a9eeb10fe59716c056d7f2c50228826c3597d)
![I \ propto | F _ {{hkl}} | ^ {{2}} = \ rho ^ {{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53390f0c3887f91da71ad8c4586a9e59cac79c38)
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 :
![\gamma](https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a)
![F _ {{hkl}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40a256c8688cd3af0a00fb8e6dbabf5554b3bd41)
![n ({\ vec {r}} \,)](https://wikimedia.org/api/rest_v1/media/math/render/svg/0992aa91239c5aef633b7a5fbc927e0ecce1c448)
![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]](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9bde2f6570e2825b9f00b660ec30e248e0d9a7f)
Since, however, only known approximation methods such as the Patterson method must be used to solve the phase problem.
![| F _ {{hkl}} | ^ {{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c23ba21ec5ebf3a97b374cda17c4107ce41a6a9)
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.
![{\ vec {r}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6aec3c9ce13b53e9e24c98e7cce4212627884c91)
![{\ vec {r}} _ {{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de93eb4c8bca39012a94e9809c45d7fd677bf975)
![i](https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20)
![{\ vec {{\ tilde r}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16896829db223b3040f4789f558e2600c7078f92)
![i](https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20)
![{\ vec {r}} = {\ vec {r}} _ {{i}} + {\ vec {{\ tilde {r}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31aa6fbe7698e288df821821a05e6c1eb37075f7)
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:
![V _ {{A _ {{i}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0257db5a16d49f3668157a1bb2d0eb7fd47d0c7b)
![n _ {{i}} ({\ vec {{\ tilde {r}}}} \,) = n ({\ vec {r}} _ {{i}} + {\ vec {{\ tilde {r} }}} \,)](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bbb572941d8010a07599e1e4cccd904ca22dc74)
![i](https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20)
![F _ {{hkl}} = \ sum _ {{i}} \ int _ {{V _ {{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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00d8ae81b83dab3ce733d7b764b36108c223822d)
The integral is called the atomic scattering factor (or atomic form factor ) of the -th atom:
![\ f _ {{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3926cbb8e30ae6bdc7f4a6eff02bdedc9133d08)
![i](https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20)
![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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/489f0ce55a44711ef5acf412b79619eae8ec95ea)
The structure factor is thus written as follows:
![F _ {{hkl}} = \ sum _ {{i}} f _ {{i}} \, \ exp \ left [i \, {\ vec {G}} \ cdot {\ vec {r}} _ {{ i}} \ right]](https://wikimedia.org/api/rest_v1/media/math/render/svg/15dc5eab9ef9fa9de24d3cdddcda1a987dfd3a6e)
With the component notation introduced above:
![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]](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0c2213c766874843e50a6b2af65462d1db82dbd)
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 .
![{\ vec {r}} _ {{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de93eb4c8bca39012a94e9809c45d7fd677bf975)
![{\ vec {r}} _ {{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de93eb4c8bca39012a94e9809c45d7fd677bf975)
![{\ vec {r}} _ {{i, 0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5797bdc26d622aba1424bee0dcd563bc2da8b509)
![{\ vec {u}} _ {{i}} (t)](https://wikimedia.org/api/rest_v1/media/math/render/svg/d11c487f1b7e000feae83fa2c3b02702e96ef7b3)
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 .
![{\ vec {a}} _ {{1}} = a {\ hat {e}} _ {{x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9457e83134a7cfb1a7e30adc527ae1655b6cb0a)
![{\ vec {a}} _ {{2}} = a {\ hat {e}} _ {{y}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a07a3e1bb58afec5a47d07f2ed4d3c5bb696d6b3)
![{\ vec {a}} _ {{3}} = a {\ hat {e}} _ {{z}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c67c11ae77ab3fd52c8626c4cfefe5ad42619ebd)
![{\ vec {r}} _ {{1}} = {\ vec {0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b848774d619df40af11a7429fd7c1166fe636d4d)
![{\ vec {r}} _ {{2}} = (1/2) ({\ vec {a}} _ {{1}} + {\ vec {a}} _ {{2}} + {\ vec {a}} _ {{3}})](https://wikimedia.org/api/rest_v1/media/math/render/svg/0eebe7eae964e1244f08f71d2a43724232224a7b)
![{\ 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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a20165ed1f42311070082bcaa0ba8a655cffd416)
![{\ 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f22a504a17c92e87435978ea9e3fa28063fbfa68)
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:
![f _ {{1}} = f _ {{2}} =: f](https://wikimedia.org/api/rest_v1/media/math/render/svg/71c8ef16ac86e11db11ffcb8062167105c7ccf51)
![{\ displaystyle F_ {hkl} = {\ begin {cases} 2f, & {\ text {if}} h + k + l {\ text {even}} \\ 0, & {\ text {if}} h + k + l {\ text {odd}} \ end {cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09c95a151d1b6cbe7375ff4b97237dc83eb3ff4e)
literature
- Borchardt-Ott, Walter: Crystallography: an introduction for natural scientists . Springer publishing house.
- Massa, Werner: Crystal structure determination . Teubner Verlag.