Topological manifolds and vector bundles with applications to crystal physics
The space group \(\text{O}^ 7_ h\) is isomorphic to the subgroup of the group of all permutations of the set \(\mathbb{S}=\{n=(n_ 0,n_ 1,n_ 2,n_ 3)\in \mathbb{Z}^ 4|\) \(n_ 0+n_ 1+n_ 2+n_ 3\in \{0;1\}\}\) generated by \(\Lambda,\Lambda_ \sigma: \mathbb{S}\to\mathbb{S}\), \(\Lambda(n_ 0,n_ 1,n_ 2,n_ 3)= (-n_ 0+1, -n_ 1, -n_ 2, -n_ 3)\), \(\Lambda_ \sigma(n_ 0,n_ 1,n_ 2,n_ 3)= (n_{\sigma(0)},n_{\sigma(1)}, n_{\sigma(2)}, n_{\sigma(3)})\), where \(\sigma\) belongs to the group \(\sigma_ 4\) of all permutations of \(\{0,1,2,3\}\). The mapping \(\delta: \mathbb{S}\times\mathbb{S}\to\mathbb{N}\), \(\delta(n,n')=\sum^ 3_{j=0} | n_ j-n'_ j|\) is on \(\text{O}^ 7_ h\)-invariant distance and \(T_ g; \ell^ 2(\mathbb{S})\to \ell^ 2(\mathbb{S})\), \((T_ g\psi)(n)=\psi(g^{-1}(n))\), where \(g\in\text{O}^ 7_ h\), is a unitary linear representation of \(\text{O}^ 7_ h\) in the Hilbert space \(\ell^ 2(\mathbb{S})= \bigl\{\psi: \mathbb{S}\to\mathbb{C}|\) \(\sum_{n\in\mathbb{S}}|\psi(n)|^ 2<\infty \bigr\}\). The equilibrium positions of the atoms of a crystal of silicon or germanium (considered without defects and infinite) form a network of points \({\mathcal R}\) (called diamond structure) such that the first neighbours \(A_ 0\), \(A_ 1\), \(A_ 2\), \(A_ 3\) of an arbitrary point \(A\in{\mathcal R}\) form a regular tetrahedron having \(A\) as center. By using a class of natural reference frames for \({\mathcal R}\), one obtains a family \({\mathcal A}\) of bijections \(\varphi: {\mathcal R}\to\mathbb{S}\) such that \(\varphi_ 1\circ \varphi_ 2^{-1}\in \text{O}^ 7_ h\), for any \(\varphi_ 1,\varphi_ 2\in{\mathcal A}\). A family \({\mathcal A}\), regarded as an atlas of global charts, defines on \({\mathcal R}\) a structure of a discrete \(\text{O}^ 7_ h\)-topological manifold. The \(\text{O}^ 7_ h\)-invariant mathematical objects we can consider on \(\mathbb{S}\), \(\ell^ 2(\mathbb{S})\) or, more generally, on a space of sections of a vector bundle \((\mathbb{S}\times V, \text{pr}_ 1,\mathbb{S})\), where \(V\) is the space of a linear representation of \(\text{O}^ 7_ h\), can be brought on \({\mathcal R}\) by using the atlas \({\mathcal A}\), independently of the chosen chart, and may be useful in modelling.
equilibrium positions of the atoms of a crystal of silicon
scientific article; zbMATH DE number 219769
scientific article