Symmetry | EPFL Graph Search

Symmetry () in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations, such as translation, reflection, rotation, or scaling. Although these two meanings of the word can sometimes be told apart, they are intricately related, and hence are discussed together in this article. Mathematical symmetry may be observed with respect to the passage of time; as a spatial relationship; through geometric transformations; through other kinds of functional transformations; and as an aspect of abstract objects, including theoretic models, language, and music. This article describes symmetry from three perspectives: in mathematics, including geometry, the most familiar type of symmetry for many people; in science and nature; and in the arts, covering architecture, art, and music. The opposite of symmetry is asymmetry, which refers to the absence or a violation of symmetry. Symmetry (geometry) A geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion. This means that an object is symmetric if there is a transformation that moves individual pieces of the object, but doesn't change the overall shape. The type of symmetry is determined by the way the pieces are organized, or by the type of transformation: An object has reflectional symmetry (line or mirror symmetry) if there is a line (or in 3D a plane) going through it which divides it into two pieces that are mirror images of each other. An object has rotational symmetry if the object can be rotated about a fixed point (or in 3D about a line) without changing the overall shape. An object has translational symmetry if it can be translated (moving every point of the object by the same distance) without changing its overall shape. An object has helical symmetry if it can be simultaneously translated and rotated in three-dimensional space along a line known as a screw axis.

Differential geometry applied to crystallography

Philippe Kocian

The mathematical facet of modern crystallography is essentially based on analytical geometry, linear algebra as well as group theory. This study endeavours to approach the geometry and symmetry of crystals using the tools furnished by differential geometry and the theory of Lie groups. These two branches of mathematics being little known to crystallographers, the pertinent definitions such as differentiable manifold, tangent space or metric tensor or even isometries on a manifold together with some important results are given first. The example of euclidean space, taken as riemannian manifold, is treated, in order to show that the affine aspect of this space is not at all an axiom but the consequence of the euclidean nature of the manifold. Attention is then directed to a particular subgroup of the group of euclidean isometries, namely that of translations. This has the property of a Lie group and it turns out that the action of its elements, as well as those of its Lie algebra, plays an important role in generating a lattice on a manifold and in its tangent space, too. In particular, it is pointed out that one and only one finite and free module of the Lie algebra of the group of translations can generate both, modulated and non-modulated lattices. This last classification therefore appears continuous rather than black and white and is entirely determined by the parametrisation considered. Since a lattice in a tangent space has the properties of a vector space, it always possesses the structure of a finite, free module, which shows that the assignment of aperiodicity to modulated structures is quite subjective, even unmotivated. Thanks to the concept of representation of a lattice or a crystal in a tangent space, novel definitions of the notions of symmetry operation of a space group and point symmetry operation, as well as symmetry element and intrinsic translation arise; they altogether naturally blend into the framework of differential geometry. In order to conveniently pass from one representation of a crystal in one tangent space to another or to the structure on a manifold, an equivalence relation on the tangent bundle of the manifold is introduced. This relation furthermore allows to extend the concept of symmetry operation to the tangent bundle; this extension furnishes, particularly in the euclidean case, a very practical way of representing symmetry operations of space groups completely devoid of any dependence on an origin, or, in other words, in which each and every point may be considered the origin. The investigation of the group of translations having being completed, the study of the linear parts of the isometries comes naturally. Based on the fact that the set of linear parts possesses the structure of a Lie group, several results are proven in a rigorous manner, such as the fact that a rotation angle of π/3 is incompatible with a three-dimensional cubic lattice. Procedures for determining different crystal systems in function of the type of rotation are laid out by way of the study of orthogonal matrices and their relation to the matrix associated with the type of system. Finally, the description of a crystal by its diffraction patterns is taken on. It is shown that the general aspect of such a pattern is directly linked to the action of that free and finite module of the Lie algebra of translations which generates a lattice on a manifold. In the case of modulated crystals, it is demonstrated that the appearance of supplementary spots is caused by the geometry, i.e. by the parametrisation of the manifold in which the crystal exists and not by the action of the module in the Lie algebra. Thus, there exists a neat separation: the geometrical aspect on the one hand, and the action of the group on the other. As the last topic, other ways of interpreting the diffraction pattern of a modulated structure are laid out in order to argue that mere experimental data do not warrant the uniqueness of a model. The goal of this study is by no means an attempt at overthrowing existing structural models such as the superspace-formalism or at revolutionising the methods for determining structures, but is rather aimed at sustaining that the definition of certain notions becomes thoroughly natural within the appropriate mathematical framework, and, that the term aperiodicity assigned to modulated structures no longer has a true meaning.

EPFL2009

Stability and physical properties of alkylsilane and alkylphosphonate self-assembled monolayer (SAM) films on metal oxide substrates characterized at the micro- and nanoscale

Enamul Hoque

Self-assembled monolayer (SAM) films have attracted immense attention for both fundamental and applied research. A SAM is composed of a large number of molecules with a head group that chemisorbs onto a substrate, a tail group that interacts with the outer surface of the film, and a spacer (backbone) chain group that connects the head and tail groups resulting in a coating. Interactions between spacer groups of different molecules, such as van der Waals forces and/or hydrogen bonding, hasten SAM film formation and contribute to its stability. In this dissertation, SAM and thin films have been formed onto copper and aluminum oxide surfaces by reaction with 1H,1H,2H,2H-perfluorodecyldimethylchlorosilane (PFMS), 1H,1H,2H,2H-perfluorodecyltrichlorosilane (PFTS), 1H,1H,2H,2H-perfluorodecylphosphonic acid (PFDP), octylphosphonic acid (OP), decylphosphonic acid (DP), and octadecylphosphonic acid (ODP). The properties and stability of the films were investigated employing complementary surface analysis techniques: X-ray photoelectron spectroscopy (XPS), atomic force microscopy (AFM), friction force microscopy (FFM), a derivative of AFM, contact angle measurements (CAMs), and Fourier transform infrared reflection/absorption spectroscopy (FT-IRRAS). The perfluoroalkylsilane SAM on Cu is found to be extremely hydrophobic typically having sessile drop static contact angles of more than 130° for pure water and a surface energy of 14 mJ/m2 (mN/m). FFM showed a significant reduction in the adhesive force and friction coefficient of PFMS modified Cu (PFMS/Cu) compared to unmodified Cu. Treatment by exposure to harsh conditions showed that PFMS/Cu SAM can withstand boiling nitric acid (pH=1.8), boiling water, and warm sodium hydroxide (pH=12, 60 °C) solutions for at least 30 minutes. Furthermore, no SAM degradation was observed when PFMS/Cu was exposed to warm nitric acid solution for up to 70 min at 60 °C or 50 min at 80 °C. XPS and FT-IRRAS data reveal a coordination of the PFMS silicon (Si) atom with a cuprate (CuO) molecule present on the oxidized copper substrate. The data give good evidence that the stability of the SAM film on the PFMS modified oxidized Cu surface is largely due to the formation of a siloxy-copper (-Si-O-Cu-) bond via a condensation reaction between silanol (-Si-OH) and copper hydroxide (CuOH). For a PFTS modified Cu surface (PFTS/Cu), the sessile drop static contact angle of pure water has been measured to be more than 125° and the surface energy to be typically less than 16 mJ/m2. Stability tests show that the PFTS/Cu film can survive in boiling pure water for one hour, boiling nitric acid (pH 1.5 or 1.8) for 30 minutes, sodium hydroxide solution (pH 12, 70 °C) for 30 minutes, and autoclave conditions (steam at 134 °C and 3 atmospheres) for 15 minutes. The more commonly used self-assembled monolayer (SAM) modifications of Cu surfaces, e.g. thiol compounds, are significantly less stable than PFTS/Cu. Extremely hydrophobic (low surface energy) and stable PFMS/Cu SAMs and PFTS/Cu films could be useful as corrosion inhibitors in micro/nanoelectronic devices and/or as promoters for anti-wetting, low adhesion surfaces or drop-wise condensation on heat exchange surfaces. XPS analysis confirmed the presence of perfluorinated and non-perfluorinated alkylphosphonate molecules on the PFDP, DP, and ODP SAMs deposited at the aluminum oxide coated silicon (Al/Si) surfaces. The sessile drop static contact angle of pure water on PFDP SAMs was typically more than 130° and on DP and ODP typically more than 125° indicating that the phosphonic acid SAMs reacted with Al samples were very hydrophobic. The surface roughness for PFDP/Al, DP/Al, ODP/Al, and bare Al was approximately 35 nm, as determined by AFM. The surface energy for PFDP/Al was determined to be approximately 11 mJ/m2 by the Zisman plot method compared to 21 mJ/m2 and 20 mJ/m2 for DP/Al and ODP/Al, respectively. PFDP/Al gave the lowest adhesion and friction force while bare Al gave the highest. The adhesion and friction forces for ODP/Al and DP/Al SAMs were in between. ODP, DP, and OP SAMs have been studied in detail on relatively flat aluminum oxide surfaces. The rms surface roughness for ODP/Al, DP/Al, OP/Al, and bare Al was less than 15 nm, as determined by AFM. The sessile drop static contact angle of pure water on ODP/Al and DP/Al was typically more than 115° and on OP/Al typically less than 105°. The surface energy for ODP/Al and DP/Al was determined to be approximately 21 mJ/m2 and 22 mJ/m2, respectively, compared to 26 mJ/m2 for OP/Al. ODP/Al and OP/Al were studied by FFM to better understand their micro-/nano-tribological properties. ODP/Al gave the lowest coefficient of friction values while bare Al gave the highest. The adhesion forces for ODP/Al and OP/Al were comparable. The chemical stability of ODP/Al, PFDP/Al, DP/Al, OP/Al, and PFMS/Al SAMs has been inspected by exposure to warm nitric acid (pH 1.8, 30 min, 60-95 °C). The XPS data and stability against harsh chemical conditions indicate that a type of bond forms between a phosphonic acid (PA) or silane molecule and the oxidized Al surface. Stability tests using warm nitric acid (pH 1.8, 30 min, 60-95 °C) show ODP/Al SAMs to be most stable followed by PFDP/Al, DP/Al, PFMS/Al, and OP/Al SAMs. For PFTS/Al, stability tests demonstrate that modified aluminum is able to survive exposure to warm nitric acid (pH = 1.8, 60 °C, 30 min) indicating some degree of robustness. Hydrophobic, low adhesion, and robust aluminum surfaces have useful applications for micro/nano-electromechanical systems (MEMS/NEMS), such as digital micro-mirror devices (DMDs). These studies are expected to aid in the design and selection of proper lubricants for MEMS/NEMS.

EPFL2007

Application of the superspace approach to members of the ferrite family

Ivan Orlov

The superspace concept is a new crystallographic approach for the generalisation of symmetry, structural models and structure dependent properties. Although the original motivation for the superspace was intended to describe symmetry of aperiodic crystals, the concept gained general acceptance in a wider range of applications. In particular, the method is able to combine structures into a single model, thus offering a clearer view of their hidden relationships. While the individual members of the family can have different space group symmetry and be both periodic or aperiodic, the generalization in superspace leads to a common crystallographic structure model and a common superspace group reuniting all of them. The benefits of such model are best demonstrated, if applied to an extensive and diverse family of compounds. Hexagonal ferrites is such an extensive family and a prominent example. Ferrites are widely known for their applications in motors, consumer electronics, microwave technology and, recently, in medicine. In general ferrites are complex oxides composed of various metals and oxygen. "Hexagonal" ferrites, named in opposition to cubic or spinel ferrites, form a particular group closely related to the mineral magnetoplumbite with the approximate chemical composition PbFe12O19. Their high uniaxial magnetocrystalline anisotropy renders them particularly useful for electronics. Currently the family includes more than 60 members of magnetic materials with unit cell dimension extending up to 1577 Å. The study of compounds approaching such a "biological magnitude" is necessarily complicated by different symmetries of individual structures and by the large number of possible structure models, generated by different stacking of several basic units. Fortunately, the generalization in higher dimensional space offers a mechanism for solving these difficulties. The description of the family is not only more elegant but also reveals characteristic relations which are not easily observable while dealing with individual structures. Understanding the "code" of such features permits a direct readout of crystal structures and unique structure solutions for this realm of inorganic species. In this work, the structures of the hexagonal ferrites are treated in superspace as a structural modulation of a common underlying average structure. The average layer-to-layer distance is directly related to the average structure periodicity along the layer stacking direction. In a first approximation series of equidistant layers are stacked along a specific direction. A modulation is introduced by varying the type of each layer. The fundamental atomic modulation is therefore of occupational nature and can be described by means of step-like functions, which define discontinuous atomic domains in superspace. The higher-dimensional description leads to the diffraction pattern indexed with four indices hklm. However, this indexing is not unique, providing room for various models. An appropriate choice of the model is based on common structural features of the described compounds and their embedding in superspace. In order to simplify the search for a common superspace group for a set of given space groups, especially for future superspace embeddings, a database providing analytical relations between (3+1)-dimensional superspace groups and three-dimensional space groups has been created and is now available on the World Wide Web. The first complete derivation of a subgroup-supergroup tree for (3+1)-dimensional symmetry made in the course of this thesis provided an informational ground for such database. The results obtained in the present thesis demonstrate that the superspace approach is an appropriate and powerful tool for analysing compounds sharing common structural features. Both periodic and aperiodic ferrite structures, which were investigated in the course of this work, can be better understood by the description in (3+1)-dimensional superspace.

EPFL2007