Understanding Diffraction Condition in Kittle's Intro to Solid State Physics

[Mart1234]

TL;DR

Diffraction Condition derivation in Kittle's Introduction to Solid State Physics

I am going over the diffraction condition section in Kittle's Introduction to Solid State Physics physics and I am having a hard time understanding why the phase difference angle for the incident wave is positive while the phase angle difference for the diffracted wave is negative. Thank you for the help.

 

[anuttarasammyak]

TL;DR Summary: Diffraction Condition derivation in Kittle's Introduction to Solid State Physics

I am having a hard time understanding why the phase difference angle for the incident wave is positive while the phase angle difference for the diffracted wave is negativeiy

I see in the figure attached that
$$\mathbf{k}\cdot \mathbf{r}>0.....(1)$$
The phase of upside or outside wave advances until reflection.
$$\mathbf{k'}\cdot \mathbf{r}<0.....(2)$$
The phase of upside or outside wave advance after reflection also so sign must be inverted for addition to (1) to give the full phase difference by reflection.