Debye-Waller factor

The Debye-Waller factor ( DWF , after Peter Debye and Ivar Waller ) describes the temperature dependence of the intensity of the coherently elastically scattered radiation on a crystal lattice . Only this elastic scattering is subject to the Laue conditions ; the complementary, inelastic scattering is called thermal-diffuse .

In neutron scattering , the term Debye-Waller factor is sometimes applied indiscriminately to coherent and incoherent scattering; Sometimes the more precise term Lamb-Mössbauer factor is used for the latter .

${\displaystyle I=I_{0}\mathrm {e} ^{-1/3\,\left|{\vec {G}}\right|^{2}{\overline {{u}^{2}}}}}$

Here I _{0 is} the intensity of the incident wave, which is reduced to I by the factor of the exponential function . G is a reciprocal lattice vector and u is the temperature-dependent oscillation amplitude of the atoms.

The Bragg diffraction reflections are more dampened due to the lattice vibrations, the higher the temperature and the higher their order.

When considering a harmonic oscillator with the energy:

${\displaystyle {\overline {E}}={\frac {1}{2}}M{{\omega }^{2}}{\overline {{u}^{2}}}={\frac {3}{2}}k_{b}T}$

the temperature-dependent Debye-Waller factor can be written as follows:

${\displaystyle {DWF}=\mathrm {e} ^{-k_{b}T\left|{\vec {G}}\right|^{2}/({M{\omega }^{2}})}}$

Derivation

The structure factor is a measure of the relative intensity of a by Miller indices , , specific diffraction reflex. ${\displaystyle F_{hkl}}$ ${\displaystyle h}$ ${\displaystyle k}$ ${\displaystyle l}$

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

The sum runs over all atoms of the base. A position vector that points from a fixed reference point within the unit cell to the nucleus of the -th atom is a reciprocal lattice vector and the atomic scattering factor of the -th atom: ${\displaystyle {\vec {r}}_{i}}$ ${\displaystyle i}$ ${\displaystyle {\vec {G}}=h{\vec {b}}_{1}+k{\vec {b}}_{2}+l{\vec {b}}_{3}}$ ${\displaystyle f_{i}}$ ${\displaystyle i}$

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

${\displaystyle V_{A_{i}}}$is the volume and the scattering power (e.g. electron density in X-ray diffraction, charge density in electron diffraction) of the -th atom. ${\displaystyle n_{i}({\vec {\tilde {r}}}\,)}$ ${\displaystyle i}$

If one 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. ${\displaystyle {\vec {r}}_{i}}$ ${\displaystyle {\vec {r}}_{i}}$ ${\displaystyle {\vec {r}}_{i,0}}$ ${\displaystyle {\vec {u}}_{i}(t)}$

${\displaystyle {\vec {r}}_{i}(t)={\vec {r}}_{i,0}+{\vec {u}}_{i}(t)}$

The oscillation periods are very short ( s) compared to the observation period, so that a time average is always measured. The mean value of the structure factor over time is ${\displaystyle <10^{-10}}$

${\displaystyle {\overline {F_{hkl}}}=\sum _{i}f_{i}\,{\overline {\exp \left[i\,{\vec {G}}\cdot \left({\vec {r}}_{i.0}+{\vec {u}}_{i}(t)\right)\right]}}=\sum _{i}f_{i}\,\exp \left[i\,{\vec {G}}\cdot {\vec {r}}_{i.0}\right]\,{\overline {\exp \left[i\,{\vec {G}}\cdot {\vec {u}}_{i}(t)\right]}}}$

For small deflections, the exponential function is developed up to the second order

${\displaystyle {\overline {\exp \left[i\,{\vec {G}}\cdot {\vec {u}}_{i}(t)\right]}}\approx 1+i\,{\overline {{\vec {G}}\cdot {\vec {u}}_{i}(t)}}-{\frac {1}{2}}{\overline {\left({\vec {G}}\cdot {\vec {u}}_{i}(t)\right)^{2}}}}$

The first order disappears , since the deflections take place statistically in all spatial directions (temporal mean value of is zero) and are not correlated with the direction of . The second order is ${\displaystyle {\overline {{\vec {G}}\cdot {\vec {u}}_{i}(t)}}=0}$ ${\displaystyle {\vec {u}}_{i}(t)}$ ${\displaystyle {\vec {u}}_{i}}$ ${\displaystyle {\vec {G}}}$

${\displaystyle {\overline {\left({\vec {G}}\cdot {\vec {u}}_{i}(t)\right)^{2}}}=|{\vec {G}}|^{2}\,{\overline {|{\vec {u}}_{i}(t)|^{2}}}\,{\overline {\cos ^{2}\theta }}}$

Where is the angle between and . One averages over all directions in three-dimensional space, i.e. integration over the unit sphere: ${\displaystyle \theta }$ ${\displaystyle {\vec {G}}}$ ${\displaystyle {\vec {u}}_{i}(t)}$ ${\displaystyle \cos ^{2}{\theta }}$

${\displaystyle {\overline {\cos ^{2}\theta }}={\frac {\int _{0}^{2\pi }\mathrm {d} \phi \int _{0}^{\pi }\mathrm {d} \theta \,\sin \theta \cos ^{2}\theta }{\int _{0}^{2\pi }\mathrm {d} \phi \int _{0}^{\pi }\mathrm {d} \theta \,\sin \theta }}={\frac {1}{3}}}$

Inserted into the exponential function this gives:

${\displaystyle {\overline {\exp \left[i\,{\vec {G}}\cdot {\vec {u}}_{i}(t)\right]}}\approx 1-{\frac {1}{6}}\,|{\vec {G}}|^{2}\,{\overline {|{\vec {u}}_{i}(t)|^{2}}}\approx \exp \left[-{\frac {1}{6}}\,|{\vec {G}}|^{2}\,{\overline {|{\vec {u}}_{i}(t)|^{2}}}\right]}$

The structure factor is now written:

${\displaystyle {\overline {F_{hkl}}}=\sum _{i}f_{i}\,\exp \left[i\,{\vec {G}}\cdot {\vec {r}}_{i.0}\right]\,\exp \left[-{\frac {1}{6}}\,|{\vec {G}}|^{2}\,{\overline {|{\vec {u}}_{i}(t)|^{2}}}\right]}$

For atoms of the same kind is approximately the same for all . So you can pull the second exponential factor in front of the sum: ${\displaystyle {\overline {|{\vec {u}}_{i}(t)|^{2}}}}$ ${\displaystyle i}$ ${\displaystyle {\overline {|{\vec {u}}\,|^{2}}}}$

${\displaystyle {\overline {F_{hkl}}}=\exp \left[-{\frac {1}{6}}\,|{\vec {G}}|^{2}\,{\overline {|{\vec {u}}(t)|^{2}}}\right]\,\sum _{i}f_{i}\,\exp \left[i\,{\vec {G}}\cdot {\vec {r}}_{i.0}\right]=\exp \left[-{\frac {1}{6}}\,|{\vec {G}}|^{2}\,{\overline {|{\vec {u}}(t)|^{2}}}\right]\,F_{hkl}^{\,0}}$

${\displaystyle F_{hkl}^{\,0}}$is the structure factor of the static case (rigid lattice, no movement of the atoms). The intensity is proportional to the square of the amount of the structure factor . The time averaged intensity is thus ${\displaystyle I=c|F_{hkl}|^{2}}$

${\displaystyle {\overline {I}}=c\,|{\overline {F_{hkl}}}|^{2}=c\,|F_{hkl}^{\,0}|^{2}\,\exp \left[-{\frac {1}{3}}\,|{\vec {G}}|^{2}\,{\overline {|{\vec {u}}|^{2}}}\right]=I_{0}\exp \left[-{\frac {1}{3}}\,|{\vec {G}}|^{2}\,{\overline {|{\vec {u}}|^{2}}}\right]}$

Compared to the static case , the averaged intensity is reduced by the Debye-Waller factor . ${\displaystyle I_{0}=c|F_{hkl}^{\,0}|^{2}}$ ${\displaystyle \exp \left[-{\frac {1}{3}}\,|{\vec {G}}|^{2}\,{\overline {|{\vec {u}}|^{2}}}\right]}$

The Debye-Waller factor is a maximum of 1 if the atoms do not vibrate (corresponds to the static case, approximately at K). The higher the temperature, the higher the exponential factor. The reflections are not broadened by the thermal movement of the atoms, but their intensity is reduced. However, a diffuse background appears between the reflections as a result of the conservation of energy. ${\displaystyle T=0}$ ${\displaystyle {\overline {|{\vec {u}}|^{2}}}}$

The Debye-Waller factor and thus the intensity is also smaller, the larger it is, i.e. the higher the Miller indices of the family of lattice planes at which the Bragg reflection takes place. ${\displaystyle |{\vec {G}}|^{2}}$

Individual evidence

1. ↑ Peter Debye: Interference of X-rays and thermal movement . In: Ann. d. Phys. . 348, No. 1, 1913, pp. 49-92. bibcode : 1913AnP ... 348 ... 49D . doi : 10.1002 / andp.19133480105 .
2. ^ Ivar Waller: On the question of the effect of thermal movement on the interference of X-rays . In: Springer (Ed.): Journal of Physics A . 17, Berlin / Heidelberg, 1923, pp. 398-408. bibcode : 1923ZPhy ... 17..398W . doi : 10.1007 / BF01328696 .
3. ↑ C. Kittel, Introduction to Solid State Physics , 7th Edition, Oldenbourg, 1986, ISBN 3-486-20240-5 , Appendix A, p. 680ff