Geometry and X-ray diffraction characteristics of carbon nanotubes

Fullerene-like carbon nanostructures with tetrahedral and octahedral symmetries

“…The coordinates of the atoms of a flat carbon layer, as is well known, are determined from formulas reported previ ously [3].…”

“…On the generation of the coordinates y and z of the atoms of a flat carbon layer [3], atoms that fall into this region will finally occur on the faces of the sin cos . In order to obtain the coordinates of the points of these faces, let us act as follows: First, we translate the triangular face or the hexagonal face located in the plane to the vector H k along the x axis; next, we turn it to the angle θ around the y axis counterclockwise and then turn it to the angles , k = 1, 2, 3) around the z axis also counterclockwise.…”

“…An important study was reported by Voytekhovsky and Stepenshchikov [2], who obtained all of the combinatory types of fullerenes (in total 5770 fullerenes) with the numbers of atoms in them from 20 to 60 and established their symmetry groups. Pleshakov [3,4] considered the geometric structures of nanotubes, nanocones, and rolls and helicoidal structures and their polyhedral models and discussed the problems of the growth of nanoobjects. More recently, Pleshakov reported fullerene like icosahedral structures [5].…”

Computer Models of Helical Nanostructures

“…The current notion of nanofibers as objects with cones inserted into each other doesn't exflat experimentally observed cone's opening angles. As shown in [1,11] there are only five types of nanocones with opening angles δ = 19.19°, 38.94°, 60°, 83.62° and 112.88°, and most of angles observed with electronic microscope do not match these five standard ones; and only in a helicoid the δ angle can vary widely.…”

“…These ideas and differential geometry methods often used in material cutting [17] and Modeling of Tent Fabric Structures [18] have not been used to build complex nanoobject models, such as helical, as far as we're concerned. Explicitly or implicitly they were used for simple surfaces only, such as cylindrical, conical and scrolls [11] that can always be unfolded to a plane.…”