Two rays/vectors that satisfy $\Delta \mathbf{k} = \mathbf{k} - \mathbf{k}_0$ make the same angle with perpendicular plane?

Question

This question is from physics, but I think the answer is more-so fundamentally a fact of mathematics, rather than physics which is why I'm posting it here.

My textbook, Solid-State Physics, Fluidics, and Analytical Techniques in Micro- and Nanotechnology, by Madou, presents the following image and explanation in a section on x-ray diffraction and Laue equations:

Bragg’s law is equivalent to the Laue equations in one dimension as can be appreciated from an inspection of Figures 2.24 and 2.25, where we use a two-dimensional crystal for simplicity. Suppose that vector $\Delta \mathbf{k}$ in Figure 2.24 satisfies the Laue condition; because incident and scattered waves have the same magnitude (elastic scattering), it follows that incoming ($\mathbf{k}_0$) and reflected rays ($\mathbf{k}$) make the same angle $\theta$ with the plane perpendicular to $\Delta \mathbf{k}$.

So this passage seems to be saying that, if two vectors $\mathbf{k}$ and $\mathbf{k}_0$ satisfy the condition $\Delta \mathbf{k} = \mathbf{k} - \mathbf{k}_0$, where $\mathbf{k}_0$ is an incoming ray, and $\mathbf{k}$ is a reflected, outgoing ray, and if these rays have the same magnitude, then it must be that the rays make the same angle $\theta$ with the plane perpendicular to $\Delta \mathbf{k}$. Is this a mathematical fact? And if so, then does anyone have a proof of this?

I would greatly appreciate it if people would please take the time to clarify this.

score 2 · Accepted Answer · answered Jan 19 '20 at 14:28

2

Your statement is not generally true. But in the text there is one more condition that you did not use: "satisfies the Laue condition". It means that $\Delta\textbf{k}$ is perpendicular to that plane. It's easy to show that in an isosceles triangle, the perpendicular from the vertex where the two equal sides meet onto the other side is also the bisector of the angle between the two equal sides.

answered Jan 19 '20 at 14:28

Andrei

37,370

Oh, my apologies. Thanks for the correction. So, if we also have the condition that the vectors satisfy the Laue condition, then it is generally true? – The Pointer Jan 19 '20 at 14:34
1

Then yes. Say $O$ is the vertex in the plane $D$ is the intersection of the plane with $\Delta\mathbf k$, $OA=OB$ and $AB\perp OD$. Then you have two right angle triangles, with the same hypotenuse, and a common side. So the two triangles are congruent. Therefore you have the same angle. – Andrei Jan 19 '20 at 14:45
Thank you for the answer! – The Pointer Jan 19 '20 at 14:51
slightly related in that there's a $\Delta k$ involved :-) How to write this conservation of energy constraint in mathematical notation? – uhoh Apr 23 '20 at 23:41

Two rays/vectors that satisfy $\Delta \mathbf{k} = \mathbf{k} - \mathbf{k}_0$ make the same angle with perpendicular plane?

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