Lattice plane

In crystallography, a lattice or network plane is a plane that is spanned by points on the crystal lattice . Their position in space is described by the Miller indices ( hkl ).

description

A crystal lattice can be described as an integer linear combination of the base vectors , and (direction of the crystal axes ). A lattice plane is defined by its points of intersection with the crystal axes. The Miller indices ( hkl ) denote the plane that goes through the three points , and . So the crystal axes of the respective crystal system intersect the planes at the reciprocal values ​​of the individual indices. An index of zero denotes a point of intersection at infinity, that is, the associated basis vector is parallel to the plane. ${\displaystyle {\vec {a}}_{1}}$ ${\displaystyle {\vec {a}}_{2}}$ ${\displaystyle {\vec {a}}_{3}}$ ${\displaystyle {\tfrac {1}{h}}{\vec {a}}_{1}}$ ${\displaystyle {\tfrac {1}{k}}{\vec {a}}_{2}}$ ${\displaystyle {\tfrac {1}{l}}{\vec {a}}_{3}}$

The reciprocal lattice vector is perpendicular to the lattice plane defined by the Miller indices ( hkl ). The vectors , and form the basis vectors of the reciprocal lattice . ${\displaystyle {\vec {G}}=h{\vec {g}}_{1}+k{\vec {g}}_{2}+l{\vec {g}}_{3}}$ ${\displaystyle {\vec {g}}_{1}}$ ${\displaystyle {\vec {g}}_{2}}$ ${\displaystyle {\vec {g}}_{3}}$

A set of grid levels consists of all parallel grid levels , each with the grid level spacing . This can be calculated from the Miller indices and the reciprocal lattice vectors: ${\displaystyle d_{\mathrm {hkl} }}$

${\displaystyle d_{\mathrm {hkl} }={\frac {2\pi }{|h\,{\vec {g}}_{1}+k\,{\vec {g}}_{2}+l\,{\vec {g}}_{3}|}}}$

For crystal systems with orthogonal axes, ie orthorhombic and higher symmetric grating ( tetragonal and cubic systems) the following formula ( , , are the lattice constants ): ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle c}$

${\displaystyle d_{\mathrm {hkl} }={\frac {1}{\sqrt {\left({\frac {h}{a}}\right)^{2}+\left({\frac {k}{b}}\right)^{2}+\left({\frac {l}{c}}\right)^{2}}}}}$

This is further simplified for cubic systems, for example, by equating : ${\displaystyle a=b=c}$

${\displaystyle d_{\mathrm {hkl} }={\frac {a}{\sqrt {h^{2}+k^{2}+l^{2}}}}}$

Derivations

A plane ( hkl ) is clearly defined by three points that are not on a straight line. This here are the intersections with the crystal axes: , and . The prefactors , , arising out of the reciprocals of the Miller indices. ${\displaystyle {\vec {P}}_{1}={\frac {1}{h}}{\vec {a}}_{1}}$ ${\displaystyle {\vec {P}}_{2}={\frac {1}{k}}{\vec {a}}_{2}}$ ${\displaystyle {\vec {P}}_{3}={\frac {1}{l}}{\vec {a}}_{3}}$ ${\displaystyle {\frac {1}{h}}}$ ${\displaystyle {\frac {1}{k}}}$ ${\displaystyle {\frac {1}{l}}}$

The points on the plane can be described by the parametric shape (with a starting point and two direction vectors that lie in the plane and are not collinear ). If two points are in the plane, their connection vector is also in the plane. The direction vectors can be constructed using this ( and ). Choose any point in the plane as the starting point (here ): ${\displaystyle {\vec {r}}={\vec {r}}_{0}+\lambda {\vec {u}}+\mu {\vec {v}}}$ ${\displaystyle {\vec {u}}={\vec {P}}_{1}-{\vec {P}}_{2}}$ ${\displaystyle {\vec {v}}={\vec {P}}_{2}-{\vec {P}}_{3}}$ ${\displaystyle {\vec {P}}_{1}}$

${\displaystyle {\vec {r}}={\frac {1}{h}}{\vec {a}}_{1}+\lambda \left({\frac {1}{h}}{\vec {a}}_{1}-{\frac {1}{k}}{\vec {a}}_{2}\right)+\mu \left({\frac {1}{k}}{\vec {a}}_{2}-{\frac {1}{l}}{\vec {a}}_{3}\right)}$

If one forms the scalar product between the reciprocal lattice vector and using the relation , then we get: ${\displaystyle {\vec {G}}=h{\vec {g}}_{1}+k{\vec {g}}_{2}+l{\vec {g}}_{3}}$ ${\displaystyle {\vec {r}}}$ ${\displaystyle {\vec {g}}_{i}\cdot {\vec {a}}_{j}=2\pi \delta _{ij}}$

${\displaystyle {\vec {G}}\cdot {\vec {r}}=\underbrace {{\frac {1}{h}}\underbrace {{\vec {G}}\cdot {\vec {a}}_{1}} _{2\pi \,h}} _{=2\pi }+\lambda \underbrace {\left({\frac {1}{h}}\underbrace {{\vec {G}}\cdot {\vec {a}}_{1}} _{2\pi \,h}-{\frac {1}{k}}\underbrace {{\vec {G}}\cdot {\vec {a}}_{2}} _{2\pi \,k}\right)} _{=0}+\mu \underbrace {\left({\frac {1}{k}}\underbrace {{\vec {G}}\cdot {\vec {a}}_{2}} _{2\pi \,k}-{\frac {1}{l}}\underbrace {{\vec {G}}\cdot {\vec {a}}_{3}} _{2\pi \,l}\right)} _{=0}=2\pi }$

For a normal vector of the plane , the scalar products with the direction vectors are equal to zero ( and ). This is exactly the case, so this is perpendicular to the plane ( hkl ). ${\displaystyle {\vec {n}}}$ ${\displaystyle {\vec {n}}\cdot {\vec {u}}=0}$ ${\displaystyle {\vec {n}}\cdot {\vec {v}}=0}$ ${\displaystyle {\vec {G}}=h{\vec {g}}_{1}+k{\vec {g}}_{2}+l{\vec {g}}_{3}}$

The grid point at the origin of the coordinates runs parallel to the plane under consideration through a plane with the indices ( hkl ). Their distance is the projection of a connection vector of both planes ( ) onto the normalized normal vector ( ). Together with the above calculation, this results in the spacing of the grid levels: ${\displaystyle P_{1}}$ ${\displaystyle {\vec {r}}-{\vec {0}}={\vec {r}}}$ ${\displaystyle {\vec {G}}/G}$

${\displaystyle {\frac {\vec {G}}{G}}\cdot {\vec {r}}={\frac {2\pi }{|h{\vec {g}}_{1}+k{\vec {g}}_{2}+l{\vec {g}}_{3}|}}\equiv d_{\mathrm {hkl} }}$

Both the lengths of the reciprocal grid vectors appear in the denominator when forming the absolute value ( ) and the projections of the grid vectors on top of each other ( with ). The latter are not equal to zero in non-orthogonal crystal systems: ${\displaystyle {\vec {g}}_{i}^{\,2}=|{\vec {g}}_{i}|^{2}}$ ${\displaystyle {\vec {g}}_{i}\cdot {\vec {g}}_{j}}$ ${\displaystyle i\neq j}$

${\displaystyle d_{\mathrm {hkl} }={\frac {2\pi }{|h{\vec {g}}_{1}+k{\vec {g}}_{2}+l{\vec {g}}_{3}|}}={\frac {2\pi }{\sqrt {h^{2}{\vec {g}}_{1}^{\,2}+k^{2}{\vec {g}}_{2}^{\,2}+l^{2}{\vec {g}}_{3}^{\,2}+2hk\,{\vec {g}}_{1}\cdot {\vec {g}}_{2}+2hl\,{\vec {g}}_{1}\cdot {\vec {g}}_{3}+2kl\,{\vec {g}}_{2}\cdot {\vec {g}}_{3}}}}}$

An orthorhombic crystal system is a right-angled crystal system with three 90 ° angles, but without axes of equal length. The lattice vectors are expressed here with respect to the canonical unit base :

${\displaystyle {\vec {a}}_{1}=a\,{\hat {e}}_{x}}$

${\displaystyle {\vec {a}}_{2}=b\,{\hat {e}}_{y}}$

${\displaystyle {\vec {a}}_{3}=c\,{\hat {e}}_{z}}$

And the corresponding reciprocal lattice vectors are also orthogonal ( for ): ${\displaystyle {\vec {g}}_{i}\cdot {\vec {g}}_{j}=0}$ ${\displaystyle i\neq j}$

${\displaystyle {\vec {g}}_{1}={\frac {2\pi }{a}}\,{\hat {e}}_{x}}$

${\displaystyle {\vec {g}}_{2}={\frac {2\pi }{b}}\,{\hat {e}}_{y}}$

${\displaystyle {\vec {g}}_{3}={\frac {2\pi }{c}}\,{\hat {e}}_{z}}$

Plug this into the above general formula for grid plane spacing:

${\displaystyle d_{\mathrm {hkl} }={\frac {2\pi }{\left|h{\frac {2\pi }{a}}\,{\hat {e}}_{x}+k{\frac {2\pi }{b}}\,{\hat {e}}_{y}+l{\frac {2\pi }{c}}\,{\hat {e}}_{z}\right|}}={\frac {1}{\sqrt {\left({\frac {h}{a}}\right)^{2}+\left({\frac {k}{b}}\right)^{2}+\left({\frac {l}{c}}\right)^{2}}}}}$

The cubic crystal system is also rectangular, but in addition the lattice constants are the same for each crystal axis and the formula is further simplified to: ${\displaystyle a=b=c}$

${\displaystyle d_{\mathrm {hkl} }={\frac {a}{\sqrt {h^{2}+k^{2}+l^{2}}}}}$

See also

• Space grid