Polarizability

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Polarizability is the ability to form instantaneous dipoles. It is a property of matter. Polarizabilities determine the dynamical response of a bound system to external fields, and provide insight into a molecule's internal structure. [1] In a solid, polarizability is defined as dipole moment per unit volume of the crystal cell. [2]

In electromagnetism, there are two kinds of dipoles:

The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. The SI units for electric dipole moment are coulomb-meter (C⋅m); however, the most commonly used unit in atomic physics and chemistry is the debye (D).

Electric polarizability

Definition

Electric polarizability is the relative tendency of a charge distribution, like the electron cloud of an atom or molecule, to be distorted from its normal shape by an external electric field.

An atom is the smallest constituent unit of ordinary matter that has the properties of a chemical element. Every solid, liquid, gas, and plasma is composed of neutral or ionized atoms. Atoms are extremely small; typical sizes are around 100 picometers. They are so small that accurately predicting their behavior using classical physics – as if they were billiard balls, for example – is not possible. This is due to quantum effects. Current atomic models now use quantum principles to better explain and predict this behavior.

A molecule is an electrically neutral group of two or more atoms held together by chemical bonds. Molecules are distinguished from ions by their lack of electrical charge. However, in quantum physics, organic chemistry, and biochemistry, the term molecule is often used less strictly, also being applied to polyatomic ions.

An electric field surrounds an electric charge, and exerts force on other charges in the field, attracting or repelling them. Electric field is sometimes abbreviated as E-field. The electric field is defined mathematically as a vector field that associates to each point in space the force per unit of charge exerted on an infinitesimal positive test charge at rest at that point. The SI unit for electric field strength is volt per meter (V/m). Newtons per coulomb (N/C) is also used as a unit of electric field strength. Electric fields are created by electric charges, or by time-varying magnetic fields. Electric fields are important in many areas of physics, and are exploited practically in electrical technology. On an atomic scale, the electric field is responsible for the attractive force between the atomic nucleus and electrons that holds atoms together, and the forces between atoms that cause chemical bonding. Electric fields and magnetic fields are both manifestations of the electromagnetic force, one of the four fundamental forces of nature.

The polarizability ${\displaystyle \alpha }$ in isotropic media is defined as the ratio of the induced dipole moment ${\displaystyle {\boldsymbol {p}}}$ of an atom to the electric field ${\displaystyle {\boldsymbol {E}}}$ that produces this dipole moment. [3]

Isotropy is uniformity in all orientations; it is derived from the Greek isos and tropos. Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix 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.

${\displaystyle \alpha ={\frac {\boldsymbol {p}}{\boldsymbol {E}}}}$

Polarizability has the SI units of C·m2·V−1 = A2·s4·kg−1 while its cgs unit is cm3. Usually it is expressed in cgs units as a so-called polarizability volume, sometimes expressed in Å 3 = 10−24 cm3. One can convert from SI units to cgs units as follows:

The International System of Units is the modern form of the metric system, and is the most widely used system of measurement. It comprises a coherent system of units of measurement built on seven base units, which are the second, metre, kilogram, ampere, kelvin, mole, candela, and a set of twenty prefixes to the unit names and unit symbols that may be used when specifying multiples and fractions of the units. The system also specifies names for 22 derived units, such as lumen and watt, for other common physical quantities.

The angstrom or ångström is a unit of length equal to 10−10 m; that is, one ten-billionth of a metre, 0.1 nanometre, or 100 picometres. Its symbol is Å, a letter of the Swedish alphabet.

${\displaystyle \alpha (\mathrm {cm} ^{3})={\frac {10^{6}}{4\pi \varepsilon _{0}}}\alpha (\mathrm {C} \cdot \mathrm {m} ^{2}\cdot \mathrm {V} ^{-1})={\frac {10^{6}}{4\pi \varepsilon _{0}}}\alpha (\mathrm {F} \cdot \mathrm {m} ^{2})}$ ≃ 8.988×1015 × ${\displaystyle \alpha (\mathrm {F} \cdot \mathrm {m} ^{2})}$

where ${\displaystyle \varepsilon _{0}}$, the vacuum permittivity, is ~8.854 × 10−12 (F/m). If the polarizability volume is denoted ${\displaystyle \alpha '}$ the relation can also be expressed generally [4] (in SI) as ${\displaystyle 4\pi \varepsilon _{0}\alpha '=\alpha }$.

In electromagnetism, absolute permittivity, often simply called permittivity, usually denoted by the Greek letter ε (epsilon), is the measure of capacitance that is encountered when forming an electric field in a particular medium. More specifically, permittivity describes the amount of charge needed to generate one unit of electric flux in a given medium. A charge will yield more electric flux in a medium with low permittivity than in a medium with high permittivity. Permittivity is the measure of a material's ability to store an electric field in the polarization of the medium.

The polarizability of individual particles is related to the average electric susceptibility of the medium by the Clausius-Mossotti relation.

In electricity (electromagnetism), the electric susceptibility is a dimensionless proportionality constant that indicates the degree of polarization of a dielectric material in response to an applied electric field. The greater the electric susceptibility, the greater the ability of a material to polarize in response to the field, and thereby reduce the total electric field inside the material. It is in this way that the electric susceptibility influences the electric permittivity of the material and thus influences many other phenomena in that medium, from the capacitance of capacitors to the speed of light.

Polarizability for anisotropic or non-spherical media cannot in general be represented as a scalar quantity. Defining ${\displaystyle \alpha }$ as a scalar implies both that applied electric fields can only induce polarization components parallel to the field and that the ${\displaystyle x,y}$ and ${\displaystyle z}$ directions respond in the same way to the applied electric field. For example, an electric field in the ${\displaystyle x}$-direction can only produce an ${\displaystyle x}$ component in ${\displaystyle {\boldsymbol {p}}}$ and if that same electric field were applied in the ${\displaystyle y}$-direction the induced polarization would be the same in magnitude but appear in the ${\displaystyle y}$ component of ${\displaystyle {\boldsymbol {p}}}$. Many crystalline materials have directions that are easier to polarize than others and some even become polarized in directions perpendicular to the applied electric field, and the same thing happens with non-spherical bodies. Some molecules and materials with this sort of anisotropic behavior are often optically active, exhibiting effects such as birefringence of light.

To describe anisotropic media a polarizability rank two tensor or ${\displaystyle 3\times 3}$ matrix ${\displaystyle \alpha }$ is defined,

${\displaystyle \mathbb {\alpha } ={\begin{bmatrix}\alpha _{xx}&\alpha _{xy}&\alpha _{xz}\\\alpha _{yx}&\alpha _{yy}&\alpha _{yz}\\\alpha _{zx}&\alpha _{zy}&\alpha _{zz}\\\end{bmatrix}}}$

The elements describing the response parallel to the applied electric field are those along the diagonal. A large value of ${\displaystyle \alpha _{yx}}$ here means that an electric-field applied in the ${\displaystyle x}$-direction would strongly polarize the material in the ${\displaystyle y}$-direction. Explicit expressions for ${\displaystyle \alpha }$ have been given for homogeneous anisotropic ellipsoidal bodies. [5] [6]

Application in crystallography

The matrix above can be used with the molar refractivity equation and other data to produce density data for crystallography. Each polarizability measurement along with the refractive index associated with its direction will yield a direction specific density that can be used to develop an accurate three dimensional assessment of molecular stacking in the crystal. This relationship was first observed by Linus Pauling. [2]

${\displaystyle R={\displaystyle \left({\frac {4\pi }{3}}\right)N_{a}a_{c}=\left({\frac {M}{p}}\right)\left({\frac {n^{2}-1}{n^{2}+2}}\right)}}$

R = Molar refractivity , ${\displaystyle N_{a}}$ = Avogadro's number, ${\displaystyle a_{c}}$ = electronic polarization, p = density, M = Molar mass, n = refractive index

Tendencies

Generally, polarizability increases as the volume occupied by electrons increases. [7] In atoms, this occurs because larger atoms have more loosely held electrons in contrast to smaller atoms with tightly bound electrons. [7] [8] On rows of the periodic table, polarizability therefore decreases from left to right. [7] Polarizability increases down on columns of the periodic table. [7] Likewise, larger molecules are generally more polarizable than smaller ones.

Water is a very polar molecule, but alkanes and other hydrophobic molecules are more polarizable. Water with its permanent dipole is less likely to change shape due to an external electric field. Alkanes are the most polarizable molecules. [7] Although alkenes and arenes are expected to have larger polarizability than alkanes because of their higher reactivity compared to alkanes, alkanes are in fact more polarizable. [7] This results because of alkene's and arene's more electronegative sp2 carbons to the alkane's less electronegative sp3 carbons. [7]

Ground state electron configuration models are often inadequate in studying the polarizability of bonds because dramatic changes in molecular structure occur in a reaction. [7]

Magnetic polarizability

Magnetic polarizability defined by spin interactions of nucleons is an important parameter of deuterons and hadrons. In particular, measurement of tensor polarizabilities of nucleons yields important information about spin-dependent nuclear forces. [9]

The method of spin amplitudes uses quantum mechanics formalism to more easily describe spin dynamics. Vector and tensor polarization of particle/nuclei with spin S ≥ 1 are specified by the unit polarization vector ${\displaystyle {\boldsymbol {p}}}$ and the polarization tensor P`. Additional tensors composed of products of three or more spin matrices are needed only for the exhaustive description of polarization of particles/nuclei with spin S32 . [9]

Related Research Articles

In electrodynamics, circular polarization of an electromagnetic wave is a polarization state in which, at each point, the electric field of the wave has a constant magnitude but its direction rotates with time at a steady rate in a plane perpendicular to the direction of the wave.

Physisorption, also called physical adsorption, is a process in which the electronic structure of the atom or molecule is barely perturbed upon adsorption.

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.

In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller than any relevant dimension of the body; so that its geometry and the constitutive properties of the material at each point of space can be assumed to be unchanged by the deformation.

In the calculus of variations, a field of mathematical analysis, the functional derivative relates a change in a functional to a change in a function on which the functional depends.

In atomic physics, hyperfine structure refers to small shifts and splittings in the energy levels of atoms, molecules, and ions, due to interaction between the state of the nucleus and the state of the electron clouds.

The Stark effect is the shifting and splitting of spectral lines of atoms and molecules due to the presence of an external electric field. It is the electric-field analogue of the Zeeman effect, where a spectral line is split into several components due to the presence of the magnetic field. Although initially coined for the static case, it is also used in the wider context to describe the effect of time-dependent electric fields. In particular, the Stark effect is responsible for the pressure broadening of spectral lines by charged particles in plasmas. For most spectral lines, the Stark effect is either linear or quadratic with a high accuracy.

In classical electromagnetism, polarization density is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is placed in an external electric field, its molecules gain electric dipole moment and the dielectric is said to be polarized. The electric dipole moment induced per unit volume of the dielectric material is called the electric polarization of the dielectric.

A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles on a sphere. These series are useful because they can often be truncated, meaning that only the first few terms need to be retained for a good approximation to the original function. The function being expanded may be complex in general. Multipole expansions are very frequently used in the study of electromagnetic and gravitational fields, where the fields at distant points are given in terms of sources in a small region. The multipole expansion with angles is often combined with an expansion in radius. Such a combination gives an expansion describing a function throughout three-dimensional space.

Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. An individual photon can be described as having right or left circular polarization, or a superposition of the two. Equivalently, a photon can be described as having horizontal or vertical linear polarization, or a superposition of the two.

In atomic, molecular, and optical physics and quantum chemistry, the molecular Hamiltonian is the Hamiltonian operator representing the energy of the electrons and nuclei in a molecule. This operator and the associated Schrödinger equation play a central role in computational chemistry and physics for computing properties of molecules and aggregates of molecules, such as thermal conductivity, specific heat, electrical conductivity, optical, and magnetic properties, and reactivity.

There are various mathematical descriptions of the electromagnetic field that are used in the study of electromagnetism, one of the four fundamental interactions of nature. In this article, several approaches are discussed, although the equations are in terms of electric and magnetic fields, potentials, and charges with currents, generally speaking.

Deformation in continuum mechanics is the transformation of a body from a reference configuration to a current configuration. A configuration is a set containing the positions of all particles of the body.

In physics, magnetization dynamics is the branch of solid-state physics that describes the evolution of the magnetization of a material.

In the natural sciences, an intermolecular force is an attraction between two molecules or atoms. They occur from either momentary interactions between molecules or permanent electrostatic attractions between dipoles. They can be explained using a simple phenomenological approach, or using a quantum mechanical approach.

The Rashba effect, also called Bychkov-Rashba effect, is a momentum-dependent splitting of spin bands in bulk crystals and low-dimensional condensed matter systems similar to the splitting of particles and anti-particles in the Dirac Hamiltonian. The splitting is a combined effect of spin–orbit interaction and asymmetry of the crystal potential, in particular in the direction perpendicular to the two-dimensional plane. This effect is named in honour of Emmanuel Rashba, who discovered it with Valentin I. Sheka in 1959 for three-dimensional systems and afterward with Yurii A. Bychkov in 1984 for two-dimensional systems. Both the Rashba and Dresselhaus effects are concepts of the PhySH Physics Subject Headlines scheme.

Heat transfer physics describes the kinetics of energy storage, transport, and energy transformation by principal energy carriers: phonons, electrons, fluid particles, and photons. Heat is energy stored in temperature-dependent motion of particles including electrons, atomic nuclei, individual atoms, and molecules. Heat is transferred to and from matter by the principal energy carriers. The state of energy stored within matter, or transported by the carriers, is described by a combination of classical and quantum statistical mechanics. The energy is also transformed (converted) among various carriers. The heat transfer processes are governed by the rates at which various related physical phenomena occur, such as the rate of particle collisions in classical mechanics. These various states and kinetics determine the heat transfer, i.e., the net rate of energy storage or transport. Governing these process from the atomic level to macroscale are the laws of thermodynamics, including conservation of energy.

Electric dipole spin resonance (EDSR) is a method to control the magnetic moments inside a material using quantum mechanical effects like the spin–orbit interaction. Mainly, EDSR allows to flip the orientation of the magnetic moments through the use of electromagnetic radiation at resonant frequencies. EDSR was first proposed by Emmanuel Rashba.

References

1. L. Zhou; F. X. Lee; W. Wilcox; J. Christensen (2002). "Magnetic polarizability of hadrons from lattice QCD" (PDF). European Organization for Nuclear Research (CERN). Retrieved 25 May 2010.
2. Lide, David (1998). The CRC Handbook of Chemistry and Physics. The Chemical Rubber Publishing Company. pp. 12–17.
3. Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN   81-7758-293-3
4. Atkins, Peter; de Paula, Julio (2010). "17". Atkins' Physical Chemistry. Oxford University Press. pp. 622–629. ISBN   978-0-19-954337-3.
5. Electrodynamics of Continuous Media, L.D. Landau and E.M. Lifshitz, Pergamon Press, 1960, pp. 7 and 192.
6. C.E. Solivérez, Electrostatics and Magnetostatics of Polarized Ellipsoidal Bodies: The Depolarization Tensor Method, Free Scientific Information, 2016 (2nd edition), ISBN   978-987-28304-0-3, pp. 20, 23, 32, 30, 33, 114 and 133.
7. Anslyn, Eric; Dougherty, Dennis (2006). Modern Physical Organic Chemistry. University Science. ISBN   978-1-891389-31-3.
8. Schwerdtfeger, Peter (2006). "Computational Aspects of Electric Polarizability Calculations: Atoms, Molecules and Clusters". In G. Maroulis (ed.). Atomic Static Dipole Polarizabilities. IOS Press.
9. A. J. Silenko (18 Nov 2008). "Manifestation of tensor magnetic polarizability of the deuteron in storage ring experiments". Springer Berlin / Heidelberg. Bibcode:2008EPJST.162...59S. doi:10.1140/epjst/e2008-00776-9 . Retrieved 25 May 2010.