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

In summary: Therefore, the minus sign in front of the dot product becomes a plus sign, giving 2\vec{k}\cdot\vec{G} + G^2 = 0. In summary, to derive equation (22) on page 31 of Kittel's Intro to Solid State Physics 8th edition, start with the diffraction condition \Delta\vec{k}=\vec{G} and use the law of cosines to find the relationship between \vec{k} and \vec{G}. The angle in the law of cosines should be 180^o-\theta, leading to the equation 2\vec{k}\cdot\vec{G} + G^2 = 0.
  • #1
DiogenesTorch
11
0

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
 
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  • #2
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]
 

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

1. What is diffraction condition in solid state physics?

Diffraction condition is a phenomenon where a wave is scattered off a periodic structure, such as a crystal lattice, resulting in a specific pattern of diffracted waves. It is an important concept in solid state physics as it helps in understanding the behavior of waves in crystalline materials.

2. How is diffraction condition related to Bragg's law?

Diffraction condition is closely related to Bragg's law, which states that the diffracted waves will be in phase if the path difference between them is an integer multiple of the wavelength. This condition is necessary for constructive interference to occur and for a diffraction pattern to be observed.

3. What is the significance of diffraction condition in X-ray crystallography?

X-ray crystallography is a technique used to determine the atomic and molecular structure of crystals. Diffraction condition plays a crucial role in this technique as it allows for the diffraction of X-rays off the crystal lattice, resulting in a diffraction pattern that can be used to determine the arrangement of atoms in the crystal.

4. How is diffraction condition different from reflection?

Diffraction condition and reflection are often used interchangeably, but they are not the same. Reflection refers to the bouncing back of a wave off a surface, whereas diffraction condition specifically refers to the scattering of a wave off a periodic structure, resulting in a diffraction pattern.

5. Can diffraction condition be applied to other types of waves besides light and X-rays?

Yes, diffraction condition applies to all types of waves, including sound waves, water waves, and even matter waves such as electrons. The only requirement is that the wave must encounter a periodic structure for diffraction to occur.

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