Synthesis of zinc oxide (ZnO) nanoflowers and nanopencils is normally carried out with the help of structure-directing chemicals and templates. Besides releasing environmentally toxic substances, the use of these chemicals restricts the practical applications of nanostructures. In this paper, we report a typical template-free and environmentally benign synthesis of ZnO nanoflowers with nanopencils-like petals through a facile hydrothermal method. The as-synthesized ZnO nanoflowers were characterized using X-ray diffraction, field emission scanning electron microscopy, tunneling electron microscopy (TEM), high resolution TEM, and energy-dispersive X-ray spectroscopy techniques. The results reveal high quality and impurity-free hexagonal wurtzite phase ZnO nanoflowers of nanopencils-like petals having a length of 100-150 nm and a diameter of 45-60 nm.
In this paper, we will analyse the scalar harmonics on a four sphere using a associated Legendre function. Then, we will use these modes to construct two types of vector harmonics on a four sphere. Finally, we will also construct three types of tensor harmonics on a four sphere. As there is a relation between de Sitter spacetime and four sphere, these modes are related to the modes on de Sitter spacetime.
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