International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 1.1, pp. 4-5

## Section 1.1.3. Angles in direct and reciprocal space

E. Kocha

aInstitut für Mineralogie, Petrologie und Kristallographie, Universität Marburg, Hans-Meerwein-Strasse, D-35032 Marburg, Germany

### 1.1.3. Angles in direct and reciprocal space

| top | pdf |

The angles between the normal of a crystal face and the basis vectors a, b, c are called the direction angles of that face. They may be calculated as angles between the corresponding reciprocal-lattice vector r* and the basis vectors , and : The three equations can be combined to give The first formula gives the ratios between a, b, and c, if for any face of the crystal the indices (hkl) and the direction angles λ, μ, and ν are known. Once the axial ratios are known, the indices of any other face can be obtained from its direction angles by using the second formula.

Similarly, the angles between a direct-lattice vector t and the reciprocal basis vectors , and are given by The angle between two direct-lattice vectors and or between two corresponding point rows and may be derived from the scalar product as Analogously, the angle between two reciprocal-lattice vectors and or between two corresponding point rows and or between the normals of two corresponding crystal faces and may be calculated as with

Finally, the angle between a first direction [uvw] of the direct lattice and a second direction [hkl] of the reciprocal lattice may also be derived from the scalar product of the corresponding vectors t and r*.