Dear Kirsten, the issue is not just the point at psi=0. If you map the psi space more finely you see that the sin2psi "linear" behavior is actually not linear. In your case is clearly S shaped, and you see it if you connect the points. Your film is likely to be columnar and the interaction (i.e. transmission of stress and strain) between the grains in the direction normal to the surface is completely different than the one in plane. The loading conditions are completely different (for instance you have a free surface and the stress normal to the surface is zero, you have Poisson effect loading along the normal and you can have a gradient of deformation...) and this affects the value of the average peak position sensed by the X-rays.

This non-linear behavior is observed even along the hhh direction that, according to the traditional homogenisation models (Reuss, Voigt) should be linear. Actually the linearity comes from the fact that those models consider 3D isotropy, whereas your film is only transversely isotropic.

You can find for instance some plots similar to yours and some information about a very simple interpretation in terms of grain interaction if you check e.g. the works that JD Kamminga, myself and U Welzel have pioneered at the end of the '90s with EJ Mittemeijer

Dear Kirsten, the issue is not just the point at psi=0. If you map the psi space more finely you see that the sin2psi "linear" behavior is actually not linear. In your case is clearly S shaped, and you see it if you connect the points. Your film is likely to be columnar and the interaction (i.e. transmission of stress and strain) between the grains in the direction normal to the surface is completely different than the one in plane. The loading conditions are completely different (for instance you have a free surface and the stress normal to the surface is zero, you have Poisson effect loading along the normal and you can have a gradient of deformation...) and this affects the value of the average peak position sensed by the X-rays.

This non-linear behavior is observed even along the hhh direction that, according to the traditional homogenisation models (Reuss, Voigt) should be linear. Actually the linearity comes from the fact that those models consider 3D isotropy, whereas your film is only transversely isotropic.

You can find for instance some plots similar to yours and some information about a very simple interpretation in terms of grain interaction if you check e.g. the works that JD Kamminga, myself and U Welzel have pioneered at the end of the '90s with EJ Mittemeijer

Dear Matteo,

Thanks a lot for the details answer. Could you send me a privat copy of the paper you cited? I tried to find it but without succsess.

Best regards

Kirsten

Besides what Mattteo says, a couple of pedantic points.

Bragg-Brentano geometry means theta/2theta, angle of incidence equals angle of reflection, psi=0. So to do a stress measurement, you do not have Bragg-Brentano geometry. I take it to mean that you did a Bragg-Brentano scan and followed it with asymmetric scans with a fixed detector angle and varying incident angle (so psi varied with constant scattering angle, 2theta).

Also, you really want to use a higher index plane for stress measurements, preferrably with 2theta>135. This is because the error in d for any given measurement error in 2theta becomes very small for 2theta>135.

Dear w.s.l Boyer,

You are completely right with the Bragg-Brentano definition.

Naturally higher 2theta angles would be preferable, but we do not have e.g. a Cr-tube to get higher angles. Additionally, it probably would not change the situation at Psi = 0°, since this is due to the physical structure of the sample and not due to the measurement accuracy.

Best regards

Kirsten

w.s.l. Boyer William, sorry it is something that I was used to do a lot 20 years ago, so I might be a bit rusty on this... but I don't get a match on your points.

For stress measurement if you work in Bragg-Brentano (i.e. parafocusing geometry), you get more aberrations than measurement.. Nobody savvy uses that anymore for long time... A "parallel" beam is necessary, so the divergent/convergent beam issue and the problem of the beam footprint and of the focusing on the detector is highly reduced.

Then, you work at constant scattering vector direction (not constant scattering angle) for every psi tilt: to obtain that, you have to move simultaneously omega and 2theta (where one is the motor of the secondary arm and the other the motor of the specimen (in the most common configuration with fixed source. The readers can easily figure out the same for a machine with fixed specimen). If you move only omega or only 2theta then the scattering vector tilts with respect to the surface normal during the measurement, so the peak position does not correspond to the direction defined by psi in the specimen frame, and the analysis becomes a mess.

Concerning the incident angle equal to the outgoing: this is done to measure the direction orthogonal to the surface of the specimen (at psi=0). In some cases this is not what you want (e.g. specimen with a texture pole not orthogonal to the surface).

If you don't have a psi tilt stage and a stress/texture machine with parallel (or parallel-ish) beam (good luck to do accurate stress analysis though), then you can do the stress analysis in omega tilt (instead of psi tilt) mode. In that case you still keep the scattering vector constant in direction while doing the measurement at each tilt and you work exactly by offsetting omega and doing again a theta/2theta scan. The disadvantage is that the accessible psi tilt is limited by the measurement range, whereas if you tilt in the other direction (psi tilt) you can go to +/- 90 deg. The high tilt is the most important, as you want to find the stress in the plane, not the one along the surface normal.

For the high angle measurement, I agree that going high reduces the aberrations. For a steel specimen this works perfectly. Here, however we have a reasonably strong fibre texture and an apparent large stress.. so there are not many other planes parallel to the surface (orthogonal to the scattering vector). At high angle you probably have to look for a reflection with sufficient intensity and the intensity of the peak would depend a lot on the tilt (for sure going soon to very low values if the pole is very narrow). I am sure here the measurement was done on the most intense to limit the measurement time.

Last but not least.. to tell something a bit more accurate about the local stress tensor you should at least repeat the measurement with the specimen rotated 90 deg on its surface normal, otherwise the only reasonable assumption to get the complete tensor is that the state of stress/strain in the surface is equibiaxial