Coordination sequences and layer-by-layer growth of periodic structures

2019 ◽  
Vol 234 (5) ◽  
pp. 291-299
Author(s):  
Anton Shutov ◽  
Andrey Maleev

Abstract A new approach to the problem of coordination sequences of periodic structures is proposed. It is based on the concept of layer-by-layer growth and on the study of geodesics in periodic graphs. We represent coordination numbers as sums of so called sector coordination numbers arising from the growth polygon of the graph. In each sector we obtain a canonical form of the geodesic chains and reduce the calculation of the sector coordination numbers to solution of the linear Diophantine equations. The approach is illustrated by the example of the 2-homogeneous kra graph. We obtain three alternative descriptions of the coordination sequences: explicit formulas, generating functions and recurrent relations.


2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Philippe Jacquet ◽  
Charles Knessl ◽  
Wojciech Szpankowski

International audience The method of types is one of the most popular techniques in information theory and combinatorics. Two sequences of equal length have the same type if they have identical empirical distributions. In this paper, we focus on Markov types, that is, sequences generated by a Markov source (of order one). We note that sequences having the same Markov type share the same so called $\textit{balanced frequency matrix}$ that counts the number of distinct pairs of symbols. We enumerate the number of Markov types for sequences of length $n$ over an alphabet of size $m$. This turns out to coincide with the number of the balanced frequency matrices as well as with the number of special $\textit{linear diophantine equations}$, and also balanced directed multigraphs. For fixed $m$ we prove that the number of Markov types is asymptotically equal to $d(m) \frac{n^{m^{2-m}}}{(m^2-m)!}$, where $d(m)$ is a constant for which we give an integral representation. For $m \to \infty$ we conclude that asymptotically the number of types is equivalent to $\frac{\sqrt{2}m^{3m/2} e^{m^2}}{m^{2m^2} 2^m \pi^{m/2}} n^{m^2-m}$ provided that $m=o(n^{1/4})$ (however, our techniques work for $m=o(\sqrt{n})$). These findings are derived by analytical techniques ranging from multidimensional generating functions to the saddle point method.



2020 ◽  
Vol 235 (4-5) ◽  
pp. 157-166
Author(s):  
Anton Shutov ◽  
Andrey Maleev

AbstractExplicit formulas for coordination sequences of all 20 plane 2-uniform graphs are proved. The proof is based on the concept of layer-by-layer growth and on the canonical representation of geodesic chains in terms of special chains called as rays. The method works for a wide class of plane periodic graphs satisfying the following condition: for each sector of layer-by-layer growth there exists a graph vertex that is initial for two rays determining the sector. This generaizes the previous results where it is required that all vertices are initial for all rays.



2003 ◽  
Vol 780 ◽  
Author(s):  
P. Thomas ◽  
E. Nabighian ◽  
M.C. Bartelt ◽  
C.Y. Fong ◽  
X.D. Zhu

AbstractWe studied adsorption, growth and desorption of Xe on Nb(110) using an in-situ obliqueincidence reflectivity difference (OI-RD) technique and low energy electron diffraction (LEED) from 32 K to 100 K. The results show that Xe grows a (111)-oriented film after a transition layer is formed on Nb(110). The transition layer consists of three layers. The first two layers are disordered with Xe-Xe separation significantly larger than the bulk value. The third monolayer forms a close packed (111) structure on top of the tensile-strained double layer and serves as a template for subsequent homoepitaxy. The adsorption of the first and the second layers are zeroth order with sticking coefficient close to one. Growth of the Xe(111) film on the transition layer proceeds in a step flow mode from 54K to 40K. At 40K, an incomplete layer-by-layer growth is observed while below 35K the growth proceeds in a multilayer mode.



Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 309-320 ◽  
Author(s):  
B.S. El-Desouky ◽  
Nenad Cakic ◽  
F.A. Shiha

In this paper we give a new family of numbers, called ??-Whitney numbers, which gives generalization of many types of Whitney numbers and Stirling numbers. Some basic properties of these numbers such as recurrence relations, explicit formulas and generating functions are given. Finally many interesting special cases are derived.



Nanomaterials ◽  
2021 ◽  
Vol 11 (7) ◽  
pp. 1631
Author(s):  
Qiang Zhang ◽  
Yohanes Pramudya ◽  
Wolfgang Wenzel ◽  
Christof Wöll

Metal organic frameworks have emerged as an important new class of materials with many applications, such as sensing, gas separation, drug delivery. In many cases, their performance is limited by structural defects, including vacancies and domain boundaries. In the case of MOF thin films, surface roughness can also have a pronounced influence on MOF-based device properties. Presently, there is little systematic knowledge about optimal growth conditions with regard to optimal morphologies for specific applications. In this work, we simulate the layer-by-layer (LbL) growth of the HKUST-1 MOF as a function of temperature and reactant concentration using a coarse-grained model that permits detailed insights into the growth mechanism. This model helps to understand the morphological features of HKUST-1 grown under different conditions and can be used to predict and optimize the temperature for the purpose of controlling the crystal quality and yield. It was found that reactant concentration affects the mass deposition rate, while its effect on the crystallinity of the generated HKUST-1 film is less pronounced. In addition, the effect of temperature on the surface roughness of the film can be divided into three regimes. Temperatures in the range from 10 to 129 °C allow better control of surface roughness and film thickness, while film growth in the range of 129 to 182 °C is characterized by a lower mass deposition rate per cycle and rougher surfaces. Finally, for T larger than 182 °C, the film grows slower, but in a smooth fashion. Furthermore, the potential effect of temperature on the crystallinity of LbL-grown HKUST-1 was quantified. To obtain high crystallinity, the operating temperature should preferably not exceed 57 °C, with an optimum around 28 °C, which agrees with experimental observations.



2020 ◽  
Vol 11 (24) ◽  
pp. 10548-10551
Author(s):  
Aswani Sathish Lathika ◽  
Shammi Rana ◽  
Anupam Prasoon ◽  
Pooja Sindhu ◽  
Debashree Roy ◽  
...  


2003 ◽  
Vol 42 (Part 2, No. 5A) ◽  
pp. L445-L447 ◽  
Author(s):  
Norio Onojima ◽  
Jun Suda ◽  
Hiroyuki Matsunami


2010 ◽  
Vol 12 (19) ◽  
pp. 5053 ◽  
Author(s):  
Yi Cui ◽  
Qiang Fu ◽  
Xinhe Bao


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