The pecularities of structure and diffraction by misfit mixed-layer nanotubes

The analysis of diffraction by separate mixed-layer goffered nanotube’s lattice is offered. Two extreme cases of the large and small size of coherent scattering regions (CSR) in a radial direction are considered. The qualitative explanation of observed diffraction effects is given


INTRODUCTION
The misfit mixed-layer nanotubes SnS 2 /SnS with a various degree of ordering in cylindrical layers SnS 2 and SnS alternation were synthesized in the Weizmann Institute of Science (Israel) by R. Tenne group [1]. Nanotubes were synthesized on the basis of flat layered crystals SnS 2 by extraction of a part of S atoms and forming SnS layers. The misfit of layers SnS 2 and SnS, having not only various parameters of lattices, but also symmetry of layer, results in a curvature of flat packs and forming nanotubes.
During an experimental research the nanotubes with rather original structure were found, on the TEM-images of which the periodically alternating in a longitudinal direction of nanotube light and dark radial (or nearly so radial) strips were observed ( fig. 1 and 2, violet line -nanotube axis, light-blue -zero layer line). More detailed analysis of the TEM-images has shown, that within the strips the layers have wavy character, and the radius of layers curvature is approximately constant. Hence, to coordinate the lattices the nanotubes layers are bent not only in a direction of the cylinder's circle, but also in direction of its axis, forming, thus, goffered nanotube. Originally such nanotubes were named "strained", however term "goffered" more corresponds to character of structure.
It is obvious, that such way of coordination is possible only for the layers with close longitudinal (along a nanotube's axis) parameters of lattices a [2]. The analysis of microdiffraction patterns has shown, that two types of layers SnS 2 always take place in SnS 2 /SnS nanotubes: with a = 0,36 nm and developed on π/2 with a = 0,63 nm, while the parameter a of SnS layer is equal 0,58 nm. Apparently, the additional bend in a direction of nanotube's axis arises in pair of layers SnS 2 and SnS, where the layer SnS 2 has a = 0,63 nm, and this layer is positioned on the external side of the bend. Really, the goffering takes place only in mixed-layer nanotubes OT (O -SnS layer, T -SnS 2 layer) and is absent in structures OTT or OTOTT, where the presence of an additional SnS 2 layer interferes with a bend.
The "additional" layer lines with reflexes are distinctly observed in the microdiffraction patterns of goffered nanotubes ( fig. 1 and 2). Similar additional layer lines, located close to basic ones, are known in superlattices diffraction researches. It allows to offer the model of nanotube structure, based on a wavy superlattice in a longitudinal direction, and to apply the known approaches to interpretation its diffraction pattern. The measurements have shown, that, as in usual superlattices, inverse value of distance Δ s from the basic layer line up to additional one ( fig. 1) well corresponds to longitudinal periodicity in the TEM-image, equal ≈ 5,4 nm.
The microdiffraction patterns on fig. 1 and 2 are rather similar, however there is an essential difference. The distribution of intensity on an additional layer line near to basic layer line 20l on fig. 1 is similar to a profile of the basic line. The analogous distributions on microdiffraction pattern of goffered nanotube on fig. 2 have obvious displacements in a direction of nanotube's axis. Interpretation of this effect requires theoretical research of a problem.  As a first step let's consider positions of the lattice sites of circular orthogonal [2] goffered nanotube, shown on a fig. 3 with the basic designations, and basic features of diffraction by it.

THE LATTICE OF GOFFERED NANOTUBE
Let nanotube, oriented along axis z, consists of the ordered alternating "goffered" pairs of layers type of A and B, having the superperiod λ and radius of the goffer bend r g . Let both layers of every m-th pair has the same centre of goffer curvature, located on a circle of radius O m ( fig. 3). Let the radius of internal layer's point, most remote from nanotube's axis, is equal ρ 0 , d A and d B -thickness of layers A and B, accordingly, and The discussed way of the coordination means that Δε -an angle, under which the coordinated cell of pair layers is visible from the centre of curvature of pair, becomes the crystallographic constant. Longitudinal parameter a of a lattice of coordinated layers pair has no definite value and for convenience can be chosen on an internal surface of pair.
Let's number the sites of lattice, formed by pairs, within the limits of superlattice's wave by integer variable t, and waves -by n. Then the angular position ε t of any pair's sites concerning their centre of curvature is possible to write down as: and z-coordinates of these sites and number of cells on the length of wave -as: under obvious condition Radius of circles, on which the centres of curvature of an internal layer's waves are located, is equal to ρ 0 -r g . Hence, it is possible to write down the polar radiuses of sites of considered lattice as: From here we can find an angle, under which the circular parameter b at ε t = 0 is seen from the nanotube's axis, and by it -the angular positions of sites: where v -number of site on the circle, p m -number of sites on the m-th pair's circles (integer by definition), ε m -initial azimuthal angular phase of the appropriate layer. It is obvious, that the circular parameter b has some variation within the limits of superlattice wavelength λ in considered model. However it is known [3], that this parameter does not influence on circular nanotube's strong (k = 0) reflexes, which consideration is the purpose of research. International Letters of Chemistry, Physics and Astronomy Vol. 2 9

THE STRONG REFLEXES AMPLITUDE
Let's write down an amplitude of diffraction by lattice, defining by expressions (1), (3) and (4), in cylindrical coordinates system {R, φ*, z*} in reciprocal space: M and N -number of layer pairs in nanotube and its length (in units λ), accordingly. Sum over goffer periods (over n) is easily calculated and has sharp maxima, equal to N, at The expression (7) defines the system of layer planes in reciprocal space, on which all sites of reciprocal lattice take place. In a plane {R, z*}, that is in the section of these planes by Evald sphere, that corresponds to the usual electron microdiffraction experiment, it gives the system of close located to each other layer lines (7), which numbering is the same with values of index h 1 . Let's be limit by half plane z* ≥ 0, that means h 1 ≥ 0. With taking into account (7) amplitude transforms in: Let's expand the second exponent in (8) into a series of cylindrical waves according to where, in view of triviality of sum over v, the first addendum looks like and gives the amplitude of so-called "strong " reflexes [3].
With the purpose of estimation of strong reflexes intensity distribution let's approximate the Bessel function in (10) by cosine:   Let's consider the sum over t, having presented cosine in an exponential form: (11) With taking into account (3) and (6), the sum S 1 : Let's expand two last exponents into a series of cylindrical waves according to which has a sharp maximum at x, equal to an integer of 2π, the height of a maximum is equal to M, and its width is inverse proportional to this value. For example, in the case of S 0 : x = q'Δε, M = T, the maximum of function is realized at However addendum S 1 of amplitude contains also the multiplier, depending on the index of summation over nanotube's layers (over m). This summation, after rejection of factors, insignificant for this analysis, gives the amplitude's multiplier G(2πRd), which also are looking like (16). Hence, the amplitude of strong reflexes represents a number of addendums, each of International Letters of Chemistry, Physics and Astronomy Vol. 2