EBSD data Analysis

 

The EBSD analysis is based in the analysis of crystalline orientation through the material, we can consider an EBSD map like a lattice of orientation points in and specific coordinates system the analysis of these points is made it by 2 methods.

 

 

Each of this ways of analysis has its advantages and disadvantages, the commercial software has the advantage that is only matter of setup in the pc and work with the different analysis that has, but one of its weakness is the fact that, if we are looking for some analysis that it doesn’t have it’s almost impossible modified the software for made this analysis. The software development, is in some way a bit slow than the analysis whit commercial software, this because it’s necessary write the program, this could be the biggest weakness in this kind of analysis but with an adequate software and mathematical develop I think that is the best way to analyses the EBSD. This kind of studies are conduced by several research

 

Basic concepts in EBSD data analysis

 

Orientation Representation

 

Orientation or orientation distribution is a fundamental concept in texture analysis and anisotropy, the first necessary action to determiner an orientation is set up a coordinate system. Two are required, one for the whole specimen and other for the crystal. The axis of the specimen are selected in order to express important surfaces or direction of the specimen one example is the rolling process then we have, a rolling direction (RD), transversal direction (TD) and a normal direction (ND), if the specimen could be defined with a simple symmetry it could be selected less direction like in specimen under uniaxial loads. Or even the system could be selected arbitrarily.

 

The second coordinate system, the crystal coordinate system is specified by directions in the crystal. The selection of directions is in principle arbitrary, but it is convenient to adapt it to the crystal symmetry.

In the figure we can see the representation of the specimen in magenta with its coordinate system in green in the directions of rolling, transversal and normal, and in black a crystal with its own coordinate system in the directions (001), (010) and (100).

 

The orientation matrix

 

With the specimen and crystal coordinate system, an orientation is define as 2 the position of the crystal coordinate system whit respect to the specimen coordinate system”

 

 

Where CC and CS are the crystal and the specimen coordinate system and g is the orientation. g can be expressed I several ways.

 

The orientation matrix is a nine element matrix, and is obtained in this way. The elements of the matrix are calculated form the angles between the axis of the crystal coordinate system and the axis of the specimen coordinate system.

 

 

The orientation matrix allows a crystal direction to be expressed in terms of the specimen direction.

 

 

The Euler angles

 

The Euler angles refer to three rotations which, when performed in the correct sequence, transform the specimen coordinate system onto the crystal coordinate system. The most commonly used convention to express the Euler angles it was formulated by Bunge, the rotations are:

 

  1. f1 about the normal direction ND transforming the transversal direction TD into TD’ and the rolling direction RD into RD’
  2. F about the axes RD’ (in its new orientation)
  3. f2 about ND’’ (in its new direction)

 

Where f1, F and f2 are the Euler angle

The elements of the matrix in terms of Euler angles are therefore given by

  

 

Euler  Colouring

 

An EBSD map is the representation in colours of the orientation of the points in Euler space, if we consider a colour scale in Red – Green – Blue of 255(this is 0 is minimum and 255 is maximum of colour), and the maximum Euler angles of f1, F, and f2  for a cubic symmetry are f1=360°, F= 90°, and f2= 90°   

 

 

 

 

 

Misorientation Angle

 

 

We could define the misorientation angle as “given two orientations (grains, crystals), the misorientation is the rotation required to rotate one set of crystal axes into coincidence with the other crystal (based on a fixed reference frame) (Rollett notes).

 

 

 

 

This mean that given to crystals with a misorientation angle of a if we apply a rotation equal a to the first crystal its orientation will be the same that the second crystal.

 

The misorientation between two crystals cold be calculated by

 

 

Where M12 is the matrix embodies the misorientation between g2 and g1, where g1 is arbitrarily chosen to be the reference orientation. The angle/axis of rotation or in this case misorientation is calculated by:

 

 

 

Misorientations and interfaces

 

 

In an Euler map the regions with different orientations are clearly placed, for the change in colour, and if we only consider an array of points plotting the misorientation we can estimate the place where a grain boundary or interface is located.

  

 

typically a grain boundary or high angle boundary is when the misorientation between the orientation points is bigger than 15° and could be represented  low angle boundaries when the misorientation is »2 to 4 degrees.

 

 Mean Orientation

 

 

It is quite often the post-processing of EBSD data for different analysis like Finite element method or the consideration of complex microstructures and textures, or the recrystallization and grain growth simulation based on the deformed state. in that case the angular precision between orientation points for data acquisition EBSD system is  »1° becomes a limiting factor in analysis of deformed or recovered microstructures.

 

Then the orientation averaging is not a minor subject. The average or mean orientation of a grain can be calculated from the lattice of orientation points in an EBSD map.  Rollett, Humbert and Glez have proposed the use of quaternion geometry for the calculus of the mean orientation.

 

Transformation of Euler Angles to Quaternions

 

The quaternions are defined as “a four component vector in relation to the axis-angle representation as follows, where [uvw] are the components of the unit vector representing the rotation axis, and q is the rotation angle” (Rollett  notes).

For the transformation of an Euler orientation to a Quaternion orientation we use the follow formula:

Following the approach of (Glez and Driver 2001)the analysis of mean orientation with quaternions is the next:

 

  1. First select a point of reference inside of the grain (q1)

q(q1,q2,q3,q4)

 

  1. Calculate the minimum misorientation of the points of the grains with respect of the selected point of the reference inside of the grain

  1. calculate the average misorientation of the points inside of the grain

  1. Then this average misorientation is calculated respect to the global frame of reference.

Then the mean orientation is m and each grain has to have its on m.