Laue condition

The Laue condition , according to Max von Laue , is a description of diffraction effects on crystals that is equivalent to the Bragg condition . It provides information on the occurrence of diffraction reflections in the event of elastic scattering of X-rays , electrons or neutrons on crystals.

There are two equivalent approaches to explaining X-ray diffraction. In both, crystals are viewed as rigid periodic structures of microscopic objects. In the Bragg theory , the atoms in the crystal are arranged in parallel lattice planes with constant spacing. At these levels there is a specular reflection of the radiation. The Von Laue theory makes other assumptions:

• Describe the crystal as a Bravais lattice
• Identical microscopic objects that scatter the incident radiation sit at the grid positions
• Reflections only in directions for which the radiation scattered by the grid points constructively interferes

The Laue condition reads: Constructive interference is obtained precisely when the change in the wave vector during the scattering process corresponds to a reciprocal grid vector .

The Laue condition is based on the pure crystal lattice (a point-like scattering center at the lattice point) and specifies the direction in which diffraction reflections can be observed. The relative intensity of the reflections depends on the structure of the base, the scattering power of the base atoms and the thermal movement of the atoms. This is described by the structure factor.

Derivation of the Laue condition

Laue condition

The distance between two scattering centers (grid points) is a grid vector . Let the wave vector of the incident radiation be that of the scattered one . This results in the following path difference (path difference): ${\displaystyle {\vec {R}}}$ ${\displaystyle {\vec {k}}}$ ${\displaystyle {\vec {k}}'}$

${\displaystyle \Delta x={\vec {R}}\cdot {\frac {\vec {k}}{k}}-{\vec {R}}\cdot {\frac {{\vec {k}}'}{k'}}}$

For constructive interference , the path difference must be an integral multiple of the wavelength : ${\displaystyle \lambda }$

${\displaystyle \Delta x=m\lambda ,\quad m\in \mathbb {Z} }$

Equating gives:

${\displaystyle {\vec {R}}\cdot \left({\frac {\vec {k}}{k}}-{\frac {{\vec {k}}'}{k'}}\right)=m\lambda }$

Judging from elastic scattering of the wave number of the incident and the reflected beam is equal to: . The following must apply to all grid vectors : ${\displaystyle k=k'={\frac {2\pi }{\lambda }}}$ ${\displaystyle {\vec {R}}}$

${\displaystyle {\vec {R}}\cdot \left({\vec {k}}-{\vec {k}}'\right)=2\pi m}$   or equivalent   ${\displaystyle e^{i\,{\vec {R}}\cdot ({\vec {k}}-{\vec {k}}')}=1}$

This corresponds exactly to the determining equation for reciprocal lattice vectors : ${\displaystyle {\vec {K}}}$

${\displaystyle e^{i\,{\vec {R}}\cdot {\vec {K}}}=1}$

The Laue condition is thus: Constructive interference is obtained precisely when the change in the wave vector during the scattering process corresponds to a reciprocal grid vector.

${\displaystyle {\vec {k}}-{\vec {k}}'={\vec {K}}}$   or.   ${\displaystyle \Delta {\vec {k}}={\vec {K}}}$

For an illustration of the Laue condition see Ewaldkugel .

Laue equations and Laue indices

Reciprocal grid vectors can be expressed as a linear combination of the primitive grid vectors of the reciprocal grid , where the Laue indices (see below) are: ${\displaystyle {\vec {b}}_{i}}$ ${\displaystyle h_{i}\in \mathbb {Z} }$

${\displaystyle {\vec {K}}=h_{1}{\vec {b}}_{1}+h_{2}{\vec {b}}_{2}+h_{3}{\vec {b}}_{3}}$

The grid vectors can also be represented as a linear combination of the primitive grid vectors with : ${\displaystyle {\vec {a}}_{i}}$ ${\displaystyle n_{i}\in \mathbb {Z} }$

${\displaystyle {\vec {R}}=n_{1}{\vec {a}}_{1}+n_{2}{\vec {a}}_{2}+n_{3}{\vec {a}}_{3}}$

The scalar product of primitive lattice vectors of space and reciprocal space is: ${\displaystyle {\vec {a}}_{j}}$ ${\displaystyle {\vec {b}}_{i}}$

${\displaystyle {\vec {b}}_{i}\cdot {\vec {a}}_{j}=2\pi \delta _{ij}\ ,}$

where is the Kronecker symbol . ${\displaystyle \delta _{ij}}$

If one forms the scalar product of the Laue condition above with the primitive position vectors, one obtains the three Laue equations : ${\displaystyle \Delta {\vec {k}}={\vec {K}}}$

${\displaystyle {\vec {a}}_{1}\cdot \Delta {\vec {k}}=2\pi h_{1}}$

${\displaystyle {\vec {a}}_{2}\cdot \Delta {\vec {k}}=2\pi h_{2}}$

${\displaystyle {\vec {a}}_{3}\cdot \Delta {\vec {k}}=2\pi h_{3}}$

The three whole numbers (normally , but here there is a risk of confusion with the wave number , therefore ) are called the Laue indices . The three equations each define a cone ( degenerate into a plane for h = 0 ). So that all conditions are met, three conical surfaces must meet in this spatial direction. This explains and indicates the punctiform interference patterns of X-ray diffraction on crystal lattices. At the same time as Laue , WH Bragg and WL Bragg set up the Bragg condition for reflection on parallel surfaces at a distance d . Even if the approaches of Laue ( diffraction in all spatial directions) and Bragg ( reflection ) are different, the two effects are equivalent: Does the family of lattice planes, which in the Bragg condition have the distance d, have Miller indices ( hkl) , the interference point has the Laue indices nh nk nl , so the Laue indices are just n times the Miller indices. Due to the connection with the Bragg reflection, the Laue indices are sometimes also referred to as Bragg indices . ${\displaystyle h_{1}h_{2}h_{3}}$ ${\displaystyle h,k,l}$ ${\displaystyle k}$ ${\displaystyle h_{1},h_{2},h_{3}}$ ${\displaystyle n\lambda =2d\sin \theta }$

Alternative formulation of the Laue condition

You can write the Laue condition in an alternative form. Square the Laue condition and use (elastic diffraction): ${\displaystyle {\vec {k}}'={\vec {k}}-{\vec {K}}}$ ${\displaystyle k=k'}$

${\displaystyle k^{2}=k^{2}-2{\vec {k}}\cdot {\vec {K}}+K^{2}}$   so   ${\displaystyle {\vec {k}}\cdot {\vec {K}}={\frac {1}{2}}K^{2}}$

Share by : ${\displaystyle K}$

${\displaystyle {\vec {k}}\cdot {\frac {\vec {K}}{K}}={\frac {1}{2}}K}$

For a given one this is a plane equation in the Hesse normal form . The projection from onto the direction from is constant . A wave vector of the incident radiation fulfills the Laue condition if its tip lies in a Bragg plane . A Bragg plane is the mid-perpendicular plane on the connecting line between the origin in reciprocal space and a point . For neighboring points in the reciprocal space, this plane equation corresponds to the construction specification of the Wigner-Seitz cell of the reciprocal lattice (first Brillouin zone ). ${\displaystyle {\vec {K}}}$ ${\displaystyle {\vec {k}}}$ ${\displaystyle {\vec {K}}/K}$ ${\displaystyle K/2}$ ${\displaystyle {\vec {k}}}$ ${\displaystyle {\vec {K}}}$

From this follows the alternative formulation of the Laue condition : Constructive interference is obtained precisely when the tip of the incident wave vector lies on the edge of a Brillouin zone.

Equivalence of Laue and Bragg conditions

Laue and Bragg conditions

If you go from and to, the result is: ${\displaystyle {\vec {k}}\cdot {\vec {K}}=K^{2}/2}$ ${\displaystyle {\vec {K}}={\vec {k}}-{\vec {k}}'}$

${\displaystyle {\vec {k}}\cdot ({\vec {k}}-{\vec {k}}')={\frac {1}{2}}K^{2}}$

The angle between and is : ${\displaystyle {\vec {k}}}$ ${\displaystyle {\vec {k}}'}$ ${\displaystyle 2\Theta \,}$

${\displaystyle k^{2}\underbrace {\left(1-\cos(2\Theta )\right)} _{2\sin ^{2}(\Theta )}={\frac {1}{2}}K^{2}}$

with and cosine law ${\displaystyle {\vec {k}}^{2}={\vec {k}}'^{2}}$

Square root yields:

${\displaystyle 2k\sin(\Theta )=K\,}$

The scalar product between a reciprocal lattice vector and a lattice vector gives:

${\displaystyle 2\pi n={\vec {K}}\cdot {\vec {R}}=K\underbrace {R\cos(\alpha )} _{d},\quad n\in \mathbb {Z} }$   Consequently   ${\displaystyle K={\frac {2\pi n}{d}}}$

For a given one , this is a plane equation for a grid plane, with this plane being perpendicular. If the following linear combination is written, the vector is perpendicular to the plane of the grid . The grid plane spacing is ${\displaystyle {\vec {K}}}$ ${\displaystyle {\vec {K}}}$ ${\displaystyle {\vec {K}}}$ ${\displaystyle {\vec {K}}=h_{1}{\vec {b}}_{1}+h_{2}{\vec {b}}_{2}+h_{3}{\vec {b}}_{3}}$ ${\displaystyle (h_{1},h_{2},h_{3})}$ ${\displaystyle d}$

${\displaystyle d={\frac {2\pi }{K}}={\frac {2\pi }{|h_{1}{\vec {b}}_{1}+h_{2}{\vec {b}}_{2}+h_{3}{\vec {b}}_{3}|}}}$.

With and we get the Bragg condition (n corresponds to the order of the diffraction reflex): ${\displaystyle k=2\pi /\lambda }$ ${\displaystyle K=2\pi n/d}$ ${\displaystyle 2k\sin(\Theta )=K}$

${\displaystyle 2d\sin(\Theta )=n\lambda \,}$

Diffraction reflex

• According to Laue: Change of the wave vector by reciprocal lattice vector ${\displaystyle {\vec {K}}}$
• According to Bragg: Reflection on set of lattice planes of the crystal lattice, which is perpendicular to and the distance between them . ${\displaystyle {\vec {K}}}$ ${\displaystyle d={\frac {2\pi }{K}}}$

literature

• Martin J. Buerger : Crystallography . 1st edition. Walter de Gruyter, Berlin 1977, ISBN 3-11-004286-X .
• Neil W. Ashcroft, N. David Mermin: Solid State Physics . 2nd Edition. Oldenbourg, Munich 2005, ISBN 3-486-57720-4 .

Individual evidence

1. ↑ Neil W. Ashcroft, David N. Mermin: Solid State Physics . 2007, ISBN 978-3-486-58273-4 ( page 124 in the Google book search).
2. ^ André Authier: Early Days of X-ray Crystallography . Oxford University Press, 2013, ISBN 978-0-19-965984-5 ( limited preview in Google Book Search).
3. ↑ Gerd Koppelmann, Gert Sinn: For interference on space grids. (PDF) Interpretation and light-optical model tests. In: Paths in physics didactics. Werner B. Schneider, 1991, accessed on January 2, 2015 ( currently out of print as a book by Verlag Palm & Enke under ISBN 3-7896-0100-4 ). 
4. ↑ S. König, R. Erlebach: Crystallography and X-ray examination of crystals. (PDF) Retrieved on January 2, 2015 (undated lecture at the University of Jena, in particular page 7). 
5. ↑ Paul Katolla, Tobias Krähling: crystal studies using Debye-Scherrer photographs. (PDF) August 7, 2009, accessed on January 2, 2015 (internship protocol at the Ruhr-Uni Bochum, see page 6 in particular). 
6. ↑ JL Atwood, JW Steed: Encyclopedia of Supramolecular Chemistry , CRC Press, 2004, ISBN 978-0-8247-4724-4 .