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20.3: Mathematical Representation of Reciprocal Lattice

  • Page ID
    8299
  • We want reciprocal lattice vectors such that the reciprocal vector is the inverse in magnitude of the real vector and is normal to the planes separating the original vector.

    So,

    \[|\mathbf{a} *|=\frac{1}{d_{100}}=\frac{1}{|\mathbf{a}| \cos \left(\gamma-\frac{\pi}{2}\right)}\]

    and

    \[\frac{\mathbf{a} *}{|\mathbf{a} *|}=\frac{\mathbf{b} \times \mathbf{c}}{|\mathbf{b} \times \mathbf{c}|}\]

    Therefore,

    \[\mathbf{a} *=\frac{\mathbf{b} \times \mathbf{c}}{\mathbf{a} \cdot \mathbf{b} \times \mathbf{c}}\]

    and similarly:

    \[\mathbf{b} *=\frac{\mathbf{c} \times \mathbf{a}}{\mathbf{a} \cdot \mathbf{b} \times \mathbf{c}}\]

    \[\mathbf{c} *=\frac{\mathbf{a} \times \mathbf{b}}{\mathbf{a} \cdot \mathbf{b} \times \mathbf{c}}\]

    Fourier Analysis of Periodic Potential

    The periodic potential of a lattice is given by:

    \[U(\mathrm{r})=\sum_{k} U_{k} \exp (i 2 \pi \mathrm{K} \cdot \mathrm{r})\]

    where Uk is the coefficient of the potential, and r is a real position vector
    However only values of K are allowed which are reciprocal lattice vectors (S).

    Proof:

    \[U(\mathrm{r})=\sum_{S} U_{S} \exp (i 2 \pi \mathrm{S} \cdot \mathrm{r})\]

    since U(r) = U(r + R), where R is a lattice vector,

    \[\sum_{S} U_{S} \exp (i 2 \pi \mathrm{S} \cdot \mathrm{r})=\sum_{S} U_{S} \exp (i 2 \pi \mathrm{S} .(\mathrm{R}+\mathrm{r}))\]

    \[\sum_{S} U_{S}=\sum_{S} U_{S} \exp (i 2 \pi \mathrm{S} . \mathrm{R})\]

    \[\lambda=\exp (i 2 \pi \boldsymbol{S} \boldsymbol{R})\]

    \[\boldsymbol{S} \boldsymbol{R}=n\]

    where n is an integer.

    Only possible values are of the form:

    \[\boldsymbol{G}=h \boldsymbol{a}^{*}+k \boldsymbol{b}^{*}+\boldsymbol{I} \boldsymbol{C}^{*}\]

    as

    \[\boldsymbol{G} \boldsymbol{R}=h+k+I\]

    and h, k, l are integers.

    Note: This is strictly the crystallographer’s definition of reciprocal lattice vectors. In solid-state physics, the 2π factor is included as a scalar within S. The 2π factor may be omitted depending on the application.

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