Editing Fourier series (section)

=== Fourier series of a Bravais-lattice-periodic function ===
<!--Linked from Reciprocal Lattice to anchor Multidimensional-->
A three-dimensional [[Bravais lattice]] is defined as the set of vectors of the form
<math display="block">\mathbf{R} = n_1\mathbf{a}_1 + n_2\mathbf{a}_2 + n_3\mathbf{a}_3</math>
where <math>n_i</math> are integers and <math>\mathbf{a}_i</math> are three linearly independent but not necessarily orthogonal vectors.  Let us consider some function <math>f(\mathbf{r})</math> with the same periodicity as the Bravais lattice, ''i.e.'' <math>f(\mathbf{r}) = f(\mathbf{R}+\mathbf{r})</math> for any lattice vector <math>\mathbf{R}</math>.  This situation frequently occurs in [[solid-state physics]] where <math>f(\mathbf{r})</math> might, for example, represent the effective potential that an electron "feels" inside a periodic crystal.  In presence of such a periodic potential, the quantum-mechanical description of the electron results in a periodically modulated plane-wave commonly known as [[Bloch's theorem|Bloch state]].

In order to develop <math>f(\mathbf{r})</math> in a Fourier series, it is convenient to introduce an auxiliary function <math display="block">g(x_1,x_2,x_3) \triangleq f(\mathbf{r}) = f \left (x_1\frac{\mathbf{a}_{1}}{a_1}+x_2\frac{\mathbf{a}_{2}}{a_2}+x_3\frac{\mathbf{a}_{3}}{a_3} \right ).</math> Both <math>f(\mathbf{r})</math> and <math>g(x_1,x_2,x_3)</math> contain essentially the same information.  However, instead of the position vector <math>\mathbf{r}</math>, the arguments of <math>g</math> are coordinates <math>x_{1,2,3}
</math> along the unit vectors <math>\mathbf{a}_{i}/{a_i}</math> of the Bravais lattice, such that <math>g</math> is an ordinary periodic function in these variables,<math display="block">g(x_1,x_2,x_3) = g(x_1+a_1,x_2,x_3) = g(x_1,x_2+a_2,x_3) = g(x_1,x_2,x_3+a_3)\quad\forall\;x_1,x_2,x_3.</math> This trick allows us to develop <math>g</math> as a multi-dimensional Fourier series, in complete analogy with the square-periodic function discussed in the previous section.  Its Fourier coefficients are<math display="block">\begin{align}
c(m_1, m_2, m_3) 
 = \frac{1}{a_3}\int_0^{a_3} dx_3 \frac{1}{a_2}\int_0^{a_2} dx_2 \frac{1}{a_1}\int_0^{a_1} dx_1\, g(x_1, x_2, x_3)\, e^{-i 2\pi \left(\tfrac{m_1}{a_1} x_1+\tfrac{m_2}{a_2} x_2 + \tfrac{m_3}{a_3} x_3\right)}
\end{align},</math> where <math>m_1,m_2,m_3</math> are all integers.  <math>c(m_1,m_2,m_3)</math> plays the same role as the coefficients <math>c_{j,k}</math> in the previous section but in order to avoid double subscripts we note them as a function.

Once we have these coefficients, the function <math>g</math> can be recovered via the Fourier series <math display="block">g(x_1, x_2, x_3)=\sum_{m_1, m_2, m_3 \in \Z } \,c(m_1, m_2, m_3) \, e^{i 2\pi \left( \tfrac{m_1}{a_1} x_1+ \tfrac{m_2}{a_2} x_2 + \tfrac{m_3}{a_3} x_3\right)}.</math> We would now like to abandon the auxiliary coordinates <math>x_{1,2,3}
</math> and to return to the original position vector <math>\mathbf{r}</math>.  This can be achieved by means of the [[reciprocal lattice]] whose vectors <math>\mathbf{b}_{1,2,3}</math> are defined such that they are orthonormal (up to a factor <math>2\pi</math>) to the original Bravais vectors <math>\mathbf{a}_{1,2,3}</math>, <math display="block">\mathbf{a}_i\cdot\mathbf{b_j}=2\pi\delta_{ij}, </math>with <math>\delta_{ij}
</math> the [[Kronecker delta]]. With this, the scalar product between a reciprocal lattice vector <math>\mathbf{Q}</math> and an arbitrary position vector <math>\mathbf{r}</math> written in the Bravais lattice basis becomes 
<math display="block">\mathbf{Q} \cdot \mathbf{r} = \left ( m_1\mathbf{b}_1 + m_2\mathbf{b}_2 + m_3\mathbf{b}_3 \right ) \cdot \left (x_1\frac{\mathbf{a}_1}{a_1}+ x_2\frac{\mathbf{a}_2}{a_2} +x_3\frac{\mathbf{a}_3}{a_3} \right ) = 2\pi \left( x_1\frac{m_1}{a_1}+x_2\frac{m_2}{a_2}+x_3\frac{m_3}{a_3} \right ),</math>which is exactly the expression occurring in the Fourier exponents. The Fourier series for <math>f(\mathbf{r}) =g(x_1,x_2,x_3)</math> can therefore be rewritten as a sum over the all reciprocal lattice vectors <math>\mathbf{Q}= m_1\mathbf{b}_1+m_2\mathbf{b}_2+m_3\mathbf{b}_3 </math>,<math display="block">f(\mathbf{r})=\sum_{\mathbf{Q}} c(\mathbf{Q})\, e^{i \mathbf{Q} \cdot \mathbf{r}},</math> and the coefficients are<math display="block">c(\mathbf{Q}) = \frac{1}{a_3} \int_0^{a_3} dx_3 \, \frac{1}{a_2}\int_0^{a_2} dx_2 \, \frac{1}{a_1}\int_0^{a_1} dx_1 \, f\left(x_1\frac{\mathbf{a}_1}{a_1} + x_2\frac{\mathbf{a}_2}{a_2} + x_3\frac{\mathbf{a}_3}{a_3} \right) e^{-i \mathbf{Q} \cdot \mathbf{r}}.</math> The remaining task will be to convert this integral over lattice coordinates back into a volume integral. The relation between the lattice coordinates <math>x_{1,2,3}</math> and the original cartesian coordinates <math>\mathbf{r} = (x,y,z)</math>  is a linear system of equations,
<math display="block">\mathbf{r} = x_1\frac{\mathbf{a}_1}{a_1}+x_2\frac{\mathbf{a}_2}{a_2}+x_3\frac{\mathbf{a}_3}{a_3},</math>which, when written in matrix form, <math display="block">\begin{bmatrix}x\\y\\z\end{bmatrix} =\mathbf{J}\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix} =\begin{bmatrix}\frac{\mathbf{a}_1}{a_1},\frac{\mathbf{a}_2}{a_2},\frac{\mathbf{a}_3}{a_3}\end{bmatrix}\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}\,,</math>involves a constant matrix <math>\mathbf{J}</math> whose columns are the unit vectors <math>\mathbf{a}_j/a_j
</math> of the Bravais lattice.  When changing variables from <math>\mathbf{r}</math> to <math>(x_1,x_2,x_3)</math> in an integral, the same matrix <math>\mathbf{J}</math> appears as a [[Jacobian matrix and determinant|Jacobian matrix]]<math display="block">\mathbf{J}=\begin{bmatrix}
\dfrac{\partial x}{\partial x_1} & \dfrac{\partial x}{\partial x_2} & \dfrac{\partial x}{\partial x_3
} \\[12pt]
\dfrac{\partial y}{\partial x_1} & \dfrac{\partial y}{\partial x_2} & \dfrac{\partial y}{\partial x_3} \\[12pt]
\dfrac{\partial z}{\partial x_1} & \dfrac{\partial z}{\partial x_2} & \dfrac{\partial z}{\partial x_3}
\end{bmatrix}\,.</math>

Its determinant <math>J
</math> is therefore also constant and can be inferred from any integral over any domain; here we choose to calculate the volume of the primitive unit cell <math>\Gamma
</math> in both coordinate systems:
<math display="block">V_{\Gamma} = \int_{\Gamma} d^3 r = J \int_{0}^{a_1} dx_1 \int_{0}^{a_2} dx_2 \int_{0}^{a_3} dx_3=J\, a_1 a_2 a_3 </math> The unit cell being a [[parallelepiped]], we have <math>V_{\Gamma}=\mathbf{a}_1\cdot(\mathbf{a}_2 \times \mathbf{a}_3)</math> and thus <math display="block">d^3r=J dx_1 dx_2 dx_3 =\frac{\mathbf{a}_1\cdot(\mathbf{a}_2 \times \mathbf{a}_3)}{a_1 a_2 a_3} dx_1 dx_2 dx_3.</math> This allows us to write <math>c (\mathbf{Q})</math> as the desired volume integral over the primitive unit cell <math>\Gamma
</math> in ordinary cartesian coordinates:
<math display="block">c(\mathbf{Q}) = \frac{1}{\mathbf{a}_1\cdot(\mathbf{a}_2 \times \mathbf{a}_3)}\int_{\Gamma} d^3 r\, f(\mathbf{r})\cdot e^{-i \mathbf{Q} \cdot \mathbf{r}}\,. </math>