The E₈ lattice is constructed in terms of icosians by matching two sets of F₄ lattices described by quaternions. Embedding the noncrystallographic group H₄ into the Weyl group W(E₈) has been described using matrix generators with an emphasis on the relevant Coxeter elements. The conjugacy classes of H₄ in terms of quaternions and the characters of the two four-dimensional irreducible representations are explicitly calculated.

The isotropic-nematic interface of a simple model of liquid-crystal molecules has been investigated using computer simulation, and by numerical minimization of the Onsager free-energy functional. The molecules are represented by long spherocylindrical particles interacting via the Kihara potential. The agreement between simulation and theory is excellent, apart from the bulk coexistence densities which are over estimated by the theory. Planar alignment of the molecules at the interface is preferred in all cases. The number density profile is found to vary monotonically, both in simulation and in theory. Biaxiality of the molecular orientational distribution near the interface is demonstrated.

Cayley-Dickson doubling procedure is used to construct the root systems of some celebrated Lie algebras in terms of the integer elements of the division algebras of real numbers, complex numbers, quaternions, and octonions. Starting with the roots and weights of SU(2) expressed as the real numbers one can construct the root systems of the Lie algebras of SO(4),SP(2)≈SO(5),SO(8),SO(9),F4 and E8 in terms of the discrete elements of the division algebras. The roots themselves display the groups structures besides the octonionic roots of E8 which form a closed octonion algebra. The automorphism group Aut(F4) of the Dynkin diagram of F4 of order 2304, the largest crystallographic group in four-dimensional Euclidean space, is realized as the direct product of two binary octahedral group of quaternions preserving the quaternionic root system of F4. The Weyl groups of many Lie algebras, such as, G2,SO(7),SO(8),SO(9),SU(3)XSU(3), and SP(3)×SU(2) have been constructed as the subgroups of Aut(F4). We have also classified the other non-parabolic subgroups of Aut(F4) which are not Weyl groups. Two subgroups of orders 192 with different conjugacy classes occur as maximal subgroups in the finite subgroups of the Lie group G2 of orders 12096 and 1344 and proves to be useful in their constructions. The triality of SO(8) manifesting itself as the cyclic symmetry of the quaternionic imaginary units e1,e2,e3 is used to show that SO(7) and SO(9) can be embedded, triply symmetric way in SO(8) and F4 in respectively.

Roughness and defects induced on few-layer graphene (FLG) irradiated by Ar+ ions at different energies were investigated using X-ray photoemission spectroscopy (XPS) and atomic force microscopy techniques. The results provide direct experimental evidence of ripple formation, sp 2 to sp 3 hybridized carbon transformation, electronic damage, Ar+ implantation, unusual defects and edge reconstructions in FLG, which depend on the irradiation energy. In addition, shadowing effects similar to those found in oblique-angle growth of thin films were seen. Reliable quantification of the transition from the sp 2-bonding to sp 3-hybridized state as a result of Ar+ ion irradiation is achieved from the deconvolution of the XPS C (1s) peak. Although the ion irradiation effect is demonstrated through the shape of the derivative of the Auger transition C KVV spectra, we show that the D parameter values obtained from these spectra which are normally used in the literature fail to account for the sp 2 to sp 3 hybridization transition. In contrast to what is known, it is revealed that using ion irradiation at large FLG sample tilt angles can lead to edge reconstructions. Furthermore, FLG irradiation by low energy of 0.25 keV can be a plausible way of peeling graphene layers without the need of Joule heating reported previously.