20.3: Mathematical Representation of Reciprocal Lattice
- Page ID
- 8299
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.