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Every primitive basis of the Bravais lattice has the formwhere $p,q,r,s\in \mathbb{Z}$ andThe unimodular determinant condition is exactly what makes the integer change of basis invertible over $\mathbb{Z}$.

Let $b_i$ be the primitive vectors of the reciprocal lattice, withSolving these four equations givesThe six reciprocal-lattice points nearest the origin areall at distance $4\pi /\sqrt{3}$.

The Wigner-Seitz cell is bounded by the perpendicular bisectors of the segments joining the origin to those six points. It is the regular hexagon with verticesIts area equals the reciprocal primitive-cell area:This is the reciprocal lattice and Wigner-Seitz cell of the unit triangular lattice.

For a periodic potential, Bloch theorem identifies wavevectors differing by a reciprocal-lattice vector. The first Brillouin zone is the Wigner--Seitz cell just found. More generally, the $n$th Brillouin zone consists of wavevectors reached from the origin after crossing exactly $n-1$ reciprocal-lattice Bragg planes; its boundaries are the perpendicular bisectorsFor this triangular reciprocal lattice, the first zone is the central regular hexagon. The second is the sixfold-symmetric collection of regions immediately outside its six sides, bounded next by the bisectors associated with the next reciprocal points. These zones organize free-particle states into bands: Bragg coupling is strongest at their boundaries and opens energy gaps when the periodic potential is introduced.

Write $\xi =V_1$ and $\eta =\varphi$. The vector fieldgenerates a Lie point symmetry of an ordinary differential equation if its local flow maps solution graphs to solution graphs. Its $n$th prolongation of a vector field iswhereandis the total derivative operator. The infinitesimal invariance criterion is

Put $p=u^{\prime }$ and $q=u^{\prime \prime }$. Direct calculation givesandForthe invariance condition isafter imposing $q=p^2/u-u^2$. The coefficient of $p^3$ isIt must vanish identically, soSince $u>0$, integration givesfor functions $F,G$.

Now takeThenand substitution givesThus the infinitesimal invariance criterion holds.

The generated one-parameter transformation is, for $c\ne 0$,while for $c=0$ it is $\tilde{x}=x+d\epsilon$, $\tilde{u}=u$. Combining these transformations gives the two-parameter symmetry groupThis is the affine-scaling symmetry of u double prime equals u prime squared over u minus u squared.