Diffraction condition - Kittel's Intro to Solid State Physics 8th ed.

[DiogenesTorch]

Homework Statement

How to derive equation (22) on page 31 of Kittel's Intro to Solid State Physics 8th edition.

The equation is: $$2\vec{k}\cdot\vec{G}+G^2=0$$

Homework Equations

The diffraction condition is given by $\Delta\vec{k}=\vec{G}$ which from what I can surmise is the starting point for the derivation

Here are some other relevant equations/definitions
$$\begin{align*}<br /> \vec{k} & & \text{ incident wave vector} \\<br /> \vec{k'} & & \text{ outgoing\reflected wave vector} \\<br /> \Delta \vec{k'} = \vec{k'}-\vec{k} & & \text{ scattering vector} \\<br /> \vec{G} & & \text{ reciprocal lattice vector} \\<br /> \end{align*}$$

The Attempt at a Solution

Starting with the diffraction condition $\Delta\vec{k}=\vec{G}$

$$\begin{align*}<br /> \Delta\vec{k} &= \vec{G} & \\<br /> \vec{k'}-\vec{k} &= \vec{G} & \\<br /> \vec{k}+\vec{G} &= \vec{k'} & \\<br /> |\vec{k}+\vec{G}|^2 &= |\vec{k'}|^2 & \text{If 2 vectors are equal, their squared magnitudes are equal}\\<br /> |\vec{k}+\vec{G}|^2 &= |\vec{k}|^2 & \text{since magnitudes k and k' are equal}\\<br /> \end{align*}$$

Using the law of cosines we now have

$$\begin{align}<br /> |\vec{k}+\vec{G}|^2 &= |\vec{k}|^2 & \nonumber \\<br /> |\vec{k}|^2 + |\vec{G}|^2 -2|\vec{k}||\vec{G}|\cos\theta &= |\vec{k}|^2 & \text{(1)} \\<br /> \end{align}$$

$\cos\theta$ is the cosine of the angle between the vectors $\vec{k}$ and $\vec{G}$ which is just

$$\begin{align*}<br /> \cos\theta=\frac{\vec{k}\cdot\vec{G}}{|\vec{k}||\vec{G}|}<br /> \end{align*}$$

Substituting the above into equation (1)

$$\begin{align*}<br /> |\vec{k}|^2 + |\vec{G}|^2 -2\vec{k}\cdot\vec{G} &= |\vec{k}|^2 \\<br /> |\vec{G}|^2 -2\vec{k}\cdot\vec{G} &= 0 \\<br /> G^2 -2\vec{k}\cdot\vec{G} &= 0 \\<br /> \end{align*}$$

However the book is showing $2\vec{k}\cdot\vec{G} + G^2 = 0$

Why is Kittel losing the minus sign in front of the dot product? I have been scratching my head for hours and can't find my error. I know this is some bone-headed oversight on my part but can't seem to find my error.


Thanks in advance

 

[szynkasz]

If $\theta$ is the angle between $\vec{k}$ and $\vec{G}$ then the angle in the law of cosines should be $180^o-\theta$