Pole figure

Pole figure

A pole figure is a graphical representation of the orientation of objects in space. For example, pole figures in the form of stereographic projections are used to represent the orientation distribution of crystallographic lattice planes in crystallography and texture analysis in materials science.

Definition

Let us consider an object with a basis attached to it. The orientation of the object in space can be determined by three rotations to transform the reference basis of space to the basis attached to the object; these are the Euler angles.

If we consider a plane of the object, the orientation of the plane can be given by its normal line. If we draw a sphere with the center on the plane, then
* the intersection of the sphere and the plane is a circle, called the "trace" ;
* the intersection of the normal line and the sphere is the pole.

A single pole is not enough to fully determine the orientation of an object: the pole stays the same if we apply a rotation around the normal line. The orientation of the object is fully determined with by the poles of two planes that are not parallel.

tereographic projection

The upper sphere is projected on a plane using the stereographic projection.

Let us consider the "x","y" the ("x","y") plane is the stereographic projection of the pole, "P"'.

It is possible to choose any .

; Definition: The pole figure is the stereographic projection of the poles used to represent the orientation of an object in space.

Geometry in the pole figure

A Wulff net is used to read a pole figure.

The stereographic projection of a trace is an arc. The Wulff net is arcs corresponding to plane which share a common axis in the ("x","y") plane.

If the pole and the trace of a plane are represented on the same diagram, then
* we turn the Wulff net so the trace corresponds to an arc of the net;
* the pole is situated on an arc, and the angular distance between this arc and the trace is 90°.

Let us consider an axis Δ, and planes belonging to the zone of this axis, i.e. Δ is in all these planes, the intersection of all the planes is Δ. If we call "P" the plane that is perpendicular to Δ, then the normals to the planes all belong to "P". Thus, the poles of the planes belonging to the same zone are on the trace of the plane perpendicular to the axis.

Application

Planes of a crystal

The structure of a crystal is often represented by the pole figure of its crystallographic plane.

A plane is chosen for as the equator, usually the (001) or (011) plane; its pole is the center of the figure. Then, the poles of the other planes are placed on the figure, with the Miller indices for each pole. The poles that belong to a zone are sometimes linked with the related trace.

Texture

"Texture" in the context of Materials Science means "crystallographic preferred orientation". If a polycrystalline material (i.e. a material composed of many different crystals or grains, like most metals, ceramics or minerals) has "texture" then that means that the crystal axes are not randomly (or, more correctly, uniformly) distributed.

To draw a "pole figure", one chooses a particular crystal direction (e.g. the normal to the (100) plane) and then plots that direction, called a pole, for each and every crystal relative to a set of directions in the material. In a rolled metal, for example, the directions in the material are the rolling direction, transverse direction and rolling plane normal.

If a large number of crystals are involved, then it is typical to make a contour plot, rather than plotting individual poles.

The full determination of the texture requires the plot of two pole figures corresponding to planes that are not parallel and that do not have the same diffraction angle (thus different interplanar distances).

Diffraction figure

Let us consider X-ray diffraction with the Laue method, or electron diffraction in a transmission electron microscope. The diffraction figure shows spots.

The is determined by the Bragg's law. It gives the orientation of the plane.

If the parameters of the optics are known (especially the distance between the crystal and the photographic film), it is figure into a pole figure.

References

* Kocks, U. F., C. Tomé, and H.-R. Wenk, Eds. (1998). Texture and Anisotropy, Cambridge University Press, Cambridge, UK, ISBN 0-521-79420-X.
* Val Randle and Olaf Engler (2000), Macrotexture, Microtexture & Orientation Mapping, Gordon & Breach, Amsterdam, Holland, ISBN 90-5699-224-4.
* Adam Morawiec, Orientations and Rotations (2003), Springer, ISBN 3-540-40734-0.
* Piotr Ozga, Pole Figures: Registration and Plot Conventions, http://www.labosoft.com.pl/pf_convention.pdf

External links

* with a step of 2° (PDF file, 1p, 272KB)
* http://www.texture.de
* http://mimp.materials.cmu.edu
* [http://code.google.com/p/mtex/ MTEX] — MATLAB toolbox for Texture Analysis
* http://labotex.com
* [http://aluminium.matter.org.uk/content/html/eng/default.asp?catid=100&pageid=1039432491 Step-by-step animated construction of a Pole Figure from aluMATTER]


Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • Pole figure — Pole figure. См. Полюсная фигура. (Источник: «Металлы и сплавы. Справочник.» Под редакцией Ю.П. Солнцева; НПО Профессионал , НПО Мир и семья ; Санкт Петербург, 2003 г.) …   Словарь металлургических терминов

  • pole figure — a pole figure is a slice across the equator of the sphere of projection with the MD (machine direction) usually defined at the top of the pole figure and either the CD (cross direction) or ND (normal direction) as the right side. Normals to… …   Mechanics glossary

  • figure — pole figure …   Mechanics glossary

  • Figure De Pôles — La figure de pôles est une manière de représenter les orientations dans l espace. Sommaire 1 Définition des pôles 2 Projection stéréographique 3 Géométrie dans la figure de pôles …   Wikipédia en Français

  • Figure de poles — Figure de pôles La figure de pôles est une manière de représenter les orientations dans l espace. Sommaire 1 Définition des pôles 2 Projection stéréographique 3 Géométrie dans la figure de pôles …   Wikipédia en Français

  • Pole splitting — is a type of frequency compensation used in an electronic amplifier in which a capacitor is introduced between the input and output sides of the amplifier with the intention of moving the pole lowest in frequency (usually an input pole) to lower… …   Wikipedia

  • Pole dance — Figure de knee hold La pole dance (anglicisme parfois traduit en danse à la barre verticale[1]) est une discipline qui mêle danse et acrobaties autour d une barre. Ce sport demande force, endurance, souplesse et sensualité. Cette danse se …   Wikipédia en Français

  • Figure De Style — Cupidon est dans ce tableau l allégorie de la liaison entre Vénus et Mars. Une figure de style (du latin figura : « dessin d’un objet », par extension sa « forme ») est un procédé d expression qui s’écarte de l’usage… …   Wikipédia en Français

  • Figure de styles — Figure de style Cupidon est dans ce tableau l allégorie de la liaison entre Vénus et Mars. Une figure de style (du latin figura : « dessin d’un objet », par extension sa « forme ») est un procédé d expression qui s’écarte …   Wikipédia en Français

  • Figure De La Terre Et Histoire Du Mètre — Certains travaux préalables à la définition et la détermination du mètre et du système métrique sont exposés dans l article Figure de la Terre et méridienne de Delambre et Méchain. Sommaire 1 L aplatissement de la Terre selon Delambre 2 Le… …   Wikipédia en Français

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”