Isotropic solid

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In condensed matter physics and continuum mechanics, an isotropic solid refers to a solid material for which physical properties are independent of the orientation of the system. While the finite sizes of atoms and bonding considerations ensure that true isotropy of atomic position will not exist in the solid state, it is possible for measurements of a given property to yield isotropic results, either due to the symmetries present within a crystal system, or due to the effects of orientational averaging over a sample (e.g. in an amorphous solid or a polycrystalline metal). Isotropic solids tend to be of interest when developing models for physical behavior of materials, as they tend to allow for dramatic simplifications of theory; for example, conductivity in metals of the cubic crystal system can be described with single scalar value, rather than a tensor. [1] Additionally, cubic crystals are isotropic with respect to thermal expansion [2] and will expand equally in all directions when heated. [3]

Isotropy should not be confused with homogeneity, which characterizes a system’s properties as being independent of position, rather than orientation. Additionally, all crystal structures, including the cubic crystal system, are anisotropic with respect to certain properties, and isotropic to others (such as density). [4] The anisotropy of a crystal’s properties depends on the rank of the tensor used to describe the property, as well as the symmetries present within the crystal. The rotational symmetries within cubic crystals, for example, ensure that the dielectric constant (a 2nd rank tensor property) will be equal in all directions, whereas the symmetries in hexagonal systems dictate that the measurement will vary depending on whether the measurement is made within the basal plane. [5] Due to the relationship between the dielectric constant and the optical index of refraction, it would be expected for cubic crystals to be optically isotropic, and hexagonal crystals to be optically anisotropic; Measurements of the optical properties of cubic and hexagonal CdSe confirm this understanding. [6]

Nearly all single crystal systems are anisotropic with respect to mechanical properties, with Tungsten being a very notable exception, as it is a cubic metal with stiffness tensor coefficients that exist in the proper ratio to allow for mechanical isotropy. In general, however, cubic crystals are not mechanically isotropic. However, many materials, such as structural steel, tend to be encountered and utilized in a polycrystalline state. Due to random orientation of the grains within the material, measured mechanical properties tend to be averages of the values associated with different crystallographic directions, with the net effect of apparent isotropy. As a result, it is typical for parameters such as the Young's Modulus to be reported independent of crystallographic direction. [7] Treating solids as mechanically isotropic greatly simplifies analysis of deformation and fracture (as well as of the elastic fields produced by dislocations [8] ). However, preferential orientation of grains (called texture) can occur as a result of certain types of deformation and recrystallization processes, which will create anisotropy in mechanical properties of the solid. [7]

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<span class="mw-page-title-main">Anisotropy</span> In geometry, property of being directionally dependent

Anisotropy is the structural property of non-uniformity in different directions, as opposed to isotropy. An anisotropic object or pattern has properties that differ according to direction of measurement. For example, many materials exhibit very different physical or mechanical properties when measured along different axes, e.g. absorbance, refractive index, conductivity, and tensile strength.

<span class="mw-page-title-main">Crystal</span> Solid material with highly ordered microscopic structure

A crystal or crystalline solid is a solid material whose constituents are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macroscopic single crystals are usually identifiable by their geometrical shape, consisting of flat faces with specific, characteristic orientations. The scientific study of crystals and crystal formation is known as crystallography. The process of crystal formation via mechanisms of crystal growth is called crystallization or solidification.

<span class="mw-page-title-main">Isotropy</span> Uniformity in all orientations

In physics and geometry, isotropy is uniformity in all orientations. Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix a- or an-, hence anisotropy. Anisotropy is also used to describe situations where properties vary systematically, dependent on direction. Isotropic radiation has the same intensity regardless of the direction of measurement, and an isotropic field exerts the same action regardless of how the test particle is oriented.

<span class="mw-page-title-main">Liquid crystal</span> State of matter with properties of both conventional liquids and crystals

Liquid crystal (LC) is a state of matter whose properties are between those of conventional liquids and those of solid crystals. For example, a liquid crystal can flow like a liquid, but its molecules may be oriented in a common direction as in a solid. There are many types of LC phases, which can be distinguished by their optical properties. The contrasting textures arise due to molecules within one area of material ("domain") being oriented in the same direction but different areas having different orientations. An LC material may not always be in an LC state of matter.

<span class="mw-page-title-main">Crystal structure</span> Ordered arrangement of atoms, ions, or molecules in a crystalline material

In crystallography, crystal structure is a description of ordered arrangement of atoms, ions, or molecules in a crystalline material. Ordered structures occur from intrinsic nature of constituent particles to form symmetric patterns that repeat along the principal directions of three-dimensional space in matter.

Crystal optics is the branch of optics that describes the behaviour of light in anisotropic media, that is, media in which light behaves differently depending on which direction the light is propagating. The index of refraction depends on both composition and crystal structure and can be calculated using the Gladstone–Dale relation. Crystals are often naturally anisotropic, and in some media it is possible to induce anisotropy by applying an external electric field.

<span class="mw-page-title-main">Birefringence</span> Refractive property of materials

Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light. These optically anisotropic materials are described as birefringent or birefractive. The birefringence is often quantified as the maximum difference between refractive indices exhibited by the material. Crystals with non-cubic crystal structures are often birefringent, as are plastics under mechanical stress.

In physics, a ferromagnetic material is said to have magnetocrystalline anisotropy if it takes more energy to magnetize it in certain directions than in others. These directions are usually related to the principal axes of its crystal lattice. It is a special case of magnetic anisotropy. In other words, the excess energy required to magnetize a specimen in a particular direction over that required to magnetize it along the easy direction is called crystalline anisotropy energy.

Polarizability usually refers to the tendency of matter, when subjected to an electric field, to acquire an electric dipole moment in proportion to that applied field. It is a property of particles with an electric charge. When subject to an electric field, the negatively charged electrons and positively charged atomic nuclei are subject to opposite forces and undergo charge separation. Polarizability is responsible for a material's dielectric constant and, at high (optical) frequencies, its refractive index.

<span class="mw-page-title-main">Texture (chemistry)</span>

In physical chemistry and materials science, texture is the distribution of crystallographic orientations of a polycrystalline sample. A sample in which these orientations are fully random is said to have no distinct texture. If the crystallographic orientations are not random, but have some preferred orientation, then the sample has a weak, moderate or strong texture. The degree is dependent on the percentage of crystals having the preferred orientation.

<span class="mw-page-title-main">Photoelasticity</span> Change in optical properties of a material due to stress

In materials science, photoelasticity describes changes in the optical properties of a material under mechanical deformation. It is a property of all dielectric media and is often used to experimentally determine the stress distribution in a material.

<span class="mw-page-title-main">Dendrite (crystal)</span> Crystal that develops with a typical multi-branching form

A crystal dendrite is a crystal that develops with a typical multi-branching form, resembling a fractal. The name comes from the Ancient Greek word δένδρον (déndron), which means "tree", since the crystal's structure resembles that of a tree. These crystals can be synthesised by using a supercooled pure liquid, however they are also quite common in nature. The most common crystals in nature exhibit dendritic growth are snowflakes and frost on windows, but many minerals and metals can also be found in dendritic structures.

Seismic anisotropy is the directional dependence of the velocity of seismic waves in a medium (rock) within the Earth.

<span class="mw-page-title-main">Etching (microfabrication)</span> Technique in microfabrication used to remove material and create structures

Etching is used in microfabrication to chemically remove layers from the surface of a wafer during manufacturing. Etching is a critically important process module in fabrication, and every wafer undergoes many etching steps before it is complete.

<span class="mw-page-title-main">Mesophase</span> Intermediate phase of matter between solid and liquid

In chemistry and chemical physics, a mesophase or mesomorphic phase is a phase of matter intermediate between solid and liquid. Gelatin is a common example of a partially ordered structure in a mesophase. Further, biological structures such as the lipid bilayers of cell membranes are examples of mesophases. Mobile ions in mesophases are either orientationally or rotationally disordered while their centers are located at the ordered sites in the crystal structure. Mesophases with long-range positional order but no orientational order are plastic crystals, whereas those with long-range orientational order but only partial or no positional order are liquid crystals.

<span class="mw-page-title-main">SRAS</span>

SRAS a non-destructive acoustic microscopy microstructural-crystallographic characterization technique commonly used in the study of crystalline or polycrystalline materials. The technique can provide information about the structure and crystallographic orientation of the material. Traditionally, the information provided by SRAS has been acquired by using diffraction techniques in electron microscopy - such as EBSD. The technique was patented in 2005, EP patent 1910815.

<span class="mw-page-title-main">Gabrielite</span> Sulfosalt mineral

Gabrielite is an extremely rare thallium sulfosalt mineral with a chemical formula of Tl6Ag3Cu6(As,Sb)9S21 or Tl2AgCu2As3S7.

Brillouin spectroscopy is an empirical spectroscopy technique which allows the determination of elastic moduli of materials. The technique uses inelastic scattering of light when it encounters acoustic phonons in a crystal, a process known as Brillouin scattering, to determine phonon energies and therefore interatomic potentials of a material. The scattering occurs when an electromagnetic wave interacts with a density wave, photon-phonon scattering.

Resonant ultrasound spectroscopy (RUS) is a laboratory technique used in geology and material science to measure fundamental material properties involving elasticity. This technique relies on the fact that solid objects have natural frequencies at which they vibrate when mechanically excited. The natural frequency depends on the elasticity, size, and shape of the object—RUS exploits this property of solids to determine the elastic tensor of the material. The great advantage of this technique is that the entire elastic tensor is obtained from a single crystal sample in a single rapid measurement. At lower or more general frequencies, this method is known as acoustic resonance spectroscopy.

<span class="mw-page-title-main">Huygens principle of double refraction</span> Optical principle

Huygens principle of double refraction, named after Dutch physicist Christiaan Huygens, explains the phenomenon of double refraction observed in uniaxial anisotropic material such as calcite. When unpolarized light propagates in such materials, it splits into two different rays, known as ordinary and extraordinary rays. The principle states that every point on the wavefront of birefringent material produces two types of wavefronts or wavelets: spherical wavefronts and ellipsoidal wavefronts. These secondary wavelets, originating from different points, interact and interfere with each other. As a result, the new wavefront is formed by the superposition of these wavelets.

References

  1. ↑ Ashcroft, Neil W.; Mermin, N.David (1976-01-02). Solid State Physics . Cengage Learning. pp.  250. ISBN   9780030839931.
  2. ↑ Newnham, Robert. E. (2005-01-27). Properties of Materials: Anisotropy, Symmetry, Structure. Oxford University Press. pp. 60–64. ISBN   9780198520764.
  3. ↑ Vail, J.M. (2003-04-24). Topics in the Theory of Solid Materials. CRC Press. pp. 34–47. ISBN   9780750307291 . Retrieved 31 January 2014.
  4. ↑ Nye, J.F. (1985-07-11). Physical Properties of Crystals: Their Representation by Tensors and Matrices. Oxford University Press. pp. xv–xvi. ISBN   9780198511656.
  5. ↑ Newnham, Robert. E. (2005-01-27). Properties of Materials: Anisotropy, Symmetry, Structure. Oxford University Press. pp. 79–85. ISBN   9780198520764.
  6. ↑ Ninomiya, Susumu; Adachi, Sadao (1995-06-19). "Optical properties of cubic and hexagonal CdSe". Journal of Applied Physics. 78 (7): 4681–4689. Bibcode:1995JAP....78.4681N. doi:10.1063/1.359815.
  7. 1 2 Courtney, Thomas H. (2005-12-01). Mechanical Behavior of Materials. Waveland Press Inc. pp. 47–61. ISBN   9781577664253.
  8. ↑ Cai, Wei; Nix, William D. (2016-08-21). Imperfections in Crystalline Solids. Cambridge University Press. pp. 369–417. ISBN   9781107123137.


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