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: [tex]2\vec{k}\cdot\vec{G}+G^2=0[/tex]

Homework Equations

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

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

The Attempt at a Solution

Starting with the diffraction condition [itex]\Delta\vec{k}=\vec{G}[/itex]

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

Using the law of cosines we now have

[tex]
\begin{align}
|\vec{k}+\vec{G}|^2 &= |\vec{k}|^2 & \nonumber \\
|\vec{k}|^2 + |\vec{G}|^2 -2|\vec{k}||\vec{G}|\cos\theta &= |\vec{k}|^2 & \text{(1)} \\
\end{align}
[/tex]

[itex]\cos\theta[/itex] is the cosine of the angle between the vectors [itex]\vec{k}[/itex] and [itex]\vec{G}[/itex] which is just

[tex]
\begin{align*}
\cos\theta=\frac{\vec{k}\cdot\vec{G}}{|\vec{k}||\vec{G}|}
\end{align*}
[/tex]

Substituting the above into equation (1)

[tex]
\begin{align*}
|\vec{k}|^2 + |\vec{G}|^2 -2\vec{k}\cdot\vec{G} &= |\vec{k}|^2 \\
|\vec{G}|^2 -2\vec{k}\cdot\vec{G} &= 0 \\
G^2 -2\vec{k}\cdot\vec{G} &= 0 \\
\end{align*}
[/tex]

However the book is showing [itex]2\vec{k}\cdot\vec{G} + G^2 = 0[/itex]

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 [itex]\theta[/itex] is the angle between [itex]\vec{k}[/itex] and [itex]\vec{G}[/itex] then the angle in the law of cosines should be [itex]180^o-\theta[/itex]