Auto setting with Laue crystal generates inexplicable incidence/reflection angles

[ja3168]

I am modelling a Laue monochromator which deflects the beam to the left (–X in laboratory coordinates). The Si (111) crystal is asymmetric cut 14° away from the symmetric case and the incident beam is under the planes. I use the Auto Setting to calculate incidence and reflection angles for 45300 eV. By examining the coordinates at the bottom of the Info widget output, I find that the correct 2θ deflection of 5.0° in the –X direction is achieved for several combinations of parameters, e.g.:

Case 1: o.e. = 90, Planes angle = 76, Onto/under planes = –1 (this case is equivalent to o.e. = 270, Planes angle = 104, Onto/under planes = 1 ?)

Case 2: o.e. = 90, Planes angle = 104, Onto/under planes = –1 (this case is equivalent to o.e. = 270, Planes angle = 76, Onto/under planes = 1 ?)

Although the beam is propagating in the correct direction after passing the crystal, I can’t make sense of the Incidence and Reflection angles (w.r.t. the normal) reported in the GUI and the Info widget, which are:

Case 1: Incidence 11.5°, Reflection 163.5° Case 2: Incidence –16.5°, Reflection 191.5°

The difference between the incident and reflected beam angles should be 175° (180 – 2θ), if they are defined against the same reference. But the difference between the calculated angles above is only 152° (i.e. 180 – 2*14° miscut). Is it possible that a calculation somewhere in the code has mixed up θ with the miscut angle?

Also, it isn’t clear to me how the miscut “planes angle” and orientation angle are defined for the Laue case, and why the above combinations apparently give the same result. I’d appreciate it if someone could point me to any documentation where the definitions of these angles are explained.

[srio]

Hello,

there is no contradiction if you consider that in SHADOW the incident angle and reflected angle are not measured in the same direction. Like in geometric optics, incident and reflection angles are measured from the normal. The incidence angle is positive if CCW, and reflection angle is positive if CW.

I tested your case in the attached workspace and drawing.

Best regards, Manuel

20220822102638902-1.pdf laue_test_ja3168.zip

[ja3168]

Thank you for going to the effort of making the drawings and the test workspace, which have resolved my confusion. The missing piece of knowledge for me was that the incident beam can lie on the "reflected" side of the mirror normal when the asymmetry angle > 90° and hence the incident angle is negative (I had assumed that whichever side the incident beam was on would become the "incident" side by definition).

I'll summarise what I now understand to make sure I have it right:

• Bragg angle is always positive
• In the Laue case, asymmetry angle is always positive and is measured from the crystal surface on the "reflected" side
• The "onto/under" flag is +1 when onto the planes and -1 when under the planes. However, when the asymmetry angle is <90°, the onto-planes case is called "under" (-1) and the under-planes case is called "onto" (+1). (Why is this the case?)

From your diagrams it is clear that 76°, +1 is equivalent to 104°, -1 after a rotation of the crystal (at least if the crystal is planar and isotropic) as both have diffraction under the planes. However, in shadow, these two cases result in an opposite deflection in laboratory X direction when using the same OE orientation angle. I suppose this is because when the incidence angle is negative the "mirror normal" points in the opposite direction to what you would expect for a given OE orientation.