Re: Relative Volume of Texture Components in Rolled Metal

[Ralf Hielscher]

Dear John Carpenter,

thank you for this question. The point is that you underestimate how large the orientation space is compared to a ball of radius 5 degree.

Lets start by estimating the volume of a 5 degree ball of an uniform ODF.  Since for cubic - orthorhombic symmetry the orientation space is 0 - 90 degree in each of the three Euler angles, we have 90 degree / 5 degree = 18, i.e. the ball is about 1/18 of the total volume in each Euler angle. This results in 1/18 * 1/18 * 1/18 = 0.017% of the total volume. Since we have not respected the 3 fold axis of cubic symmetry we still have to multiply the result by 3 which gives the estimate of 0.05% for the volume of the ball. Thus a 5 degree ball is very, very small in orientation space. 

Lets make this estimate more exact by using MTEX and apply the volume command to the uniform ODF

volume(uniformODF(cs,ss),oc1,5*degree)

>> 0.0034

This value of 0.34% is much larger then what we estimated before, which is because we did not respect the specific geometry of the orientation space. Furthermore, the volume you obtained, i.e., 1.28%, is about 4 times the volume for the unimodal ODF which fits quite well your ODF plot which shows a value of 4.9 mrd at the center orientation.

I hope this helps.

Ralf.