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

  1. Apr 19, 2014 #1
    1. The problem statement, all variables and given/known data

    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]


    2. Relevant 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]



    3. 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
     
  2. jcsd
  3. Apr 19, 2014 #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]
     
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