Character table

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Character_table

In group theory, a branch of abstract algebra, a character table is a two-dimensional table whose rows correspond to irreducible representations, and whose columns correspond to conjugacy classes of group elements. The entries consist of characters, the traces of the matrices representing group elements of the column's class in the given row's group representation. In chemistry, crystallography, and spectroscopy, character tables of point groups are used to classify e.g. molecular vibrations according to their symmetry, and to predict whether a transition between two states is forbidden for symmetry reasons. Many university level textbooks on physical chemistry, quantum chemistry, spectroscopy and inorganic chemistry devote a chapter to the use of symmetry group character tables.

Definition and example

The irreducible complex characters of a finite group form a character table which encodes much useful information about the group G in a compact form. Each row is labelled by an irreducible character and the entries in the row are the values of that character on any representative of the respective conjugacy class of G (because characters are class functions). The columns are labelled by (representatives of) the conjugacy classes of G. It is customary to label the first row by the character of the trivial representation, which is the trivial action of G on a 1-dimensional vector space by ${\displaystyle \rho (g)=1}$ for all ${\displaystyle g\in G}$. Each entry in the first row is therefore 1. Similarly, it is customary to label the first column by the identity. The entries of the first column are the values of the irreducible characters at the identity, the degrees of the irreducible characters. Characters of degree 1 are known as linear characters.

Here is the character table of C₃ = <u>, the cyclic group with three elements and generator u:

	(1)	(u)	(u2)
1	1	1	1
χ1	1	ω	ω2
χ2	1	ω2	ω

where ω is a primitive third root of unity. The character table for general cyclic groups is (a scalar multiple of) the DFT matrix.

Another example is the character table of ${\displaystyle S_{3}}$:

	(1)	(12)	(123)
χtriv	1	1	1
χsgn	1	−1	1
χstand	2	0	−1

where (12) represents conjugacy class consisting of (12),(13),(23), and (123) represents conjugacy class consisting of (123),(132).

The first row of the character table always consists of 1s, and corresponds to the trivial representation (the 1-dimensional representation consisting of 1×1 matrices containing the entry 1). Further, the character table is always square because (1) irreducible characters are pairwise orthogonal, and (2) no other non-trivial class function is orthogonal to every character. (A class function is one that is constant on conjugacy classes.) This is tied to the important fact that the irreducible representations of a finite group G are in bijection with its conjugacy classes. This bijection also follows by showing that the class sums form a basis for the center of the group algebra of G, which has dimension equal to the number of irreducible representations of G.

Orthogonality relations

Main article: Schur orthogonality relations

The space of complex-valued class functions of a finite group G has a natural inner-product:

${\displaystyle \left\langle \alpha ,\beta \right\rangle :={\frac {1}{\left|G\right|}}\sum _{g\in G}\alpha (g){\overline {\beta (g)}}}$

where ${\displaystyle {\overline {\beta (g)}}}$ means the complex conjugate of the value of ${\displaystyle \beta }$ on ${\displaystyle g}$. With respect to this inner product, the irreducible characters form an orthonormal basis for the space of class-functions, and this yields the orthogonality relation for the rows of the character table:

${\displaystyle \left\langle \chi _{i},\chi _{j}\right\rangle ={\begin{cases}0&{\mbox{ if }}i\neq j,\\1&{\mbox{ if }}i=j.\end{cases}}}$

For ${\displaystyle g,h\in G}$ the orthogonality relation for columns is as follows:

${\displaystyle \sum _{\chi _{i}}\chi _{i}(g){\overline {\chi _{i}(h)}}={\begin{cases}\left|C_{G}(g)\right|,&{\mbox{ if }}g,h{\mbox{ are conjugate }}\\0&{\mbox{ otherwise.}}\end{cases}}}$

where the sum is over all of the irreducible characters ${\displaystyle \chi _{i}}$ of G and the symbol ${\displaystyle \left|C_{G}(g)\right|}$ denotes the order of the centralizer of ${\displaystyle g}$.

For an arbitrary character ${\displaystyle \chi _{i}}$, it is irreducible if and only if ${\displaystyle \left\langle \chi _{i},\chi _{i}\right\rangle =1}$.

The orthogonality relations can aid many computations including:

• Decomposing an unknown character as a linear combination of irreducible characters, i.e. # of copies of irreducible representation V_i in V = ${\displaystyle \left\langle \chi ,\chi _{i}\right\rangle }$.
• Constructing the complete character table when only some of the irreducible characters are known.
• Finding the orders of the centralizers of representatives of the conjugacy classes of a group.
• Finding the order of the group, ${\displaystyle \left|G\right|=\left|Cl(g)\right|*\sum _{\chi _{i}}\chi _{i}(g){\overline {\chi _{i}(g)}}}$, for any g in G.

If the irreducible representation V is non-trivial, then ${\displaystyle \sum _{g}\chi (g)=0}$.

More specifically, consider the regular representation which is the permutation obtained from a finite group G acting on itself. The characters of this representation are ${\displaystyle \chi (e)=\left|G\right|}$ and ${\displaystyle \chi (g)=0}$ for ${\displaystyle g}$ not the identity. Then given an irreducible representation ${\displaystyle V_{i}}$,

${\displaystyle \left\langle \chi _{\text{reg}},\chi _{i}\right\rangle ={\frac {1}{\left|G\right|}}\sum _{g\in G}\chi _{i}(g){\overline {\chi _{\text{reg}}(g)}}={\frac {1}{\left|G\right|}}\chi _{i}(1){\overline {\chi _{\text{reg}}(1)}}=\operatorname {dim} V_{i}}$.

Then decomposing the regular representations as a sum of irreducible representations of G, we get ${\displaystyle V_{\text{reg}}=\oplus V_{i}^{\operatorname {dim} V_{i}}}$. From which we conclude

${\displaystyle \left|G\right|=\operatorname {dim} V_{\text{reg}}=\sum (\operatorname {dim} V_{i})^{2}}$

over all irreducible representations ${\displaystyle V_{i}}$. This sum can help narrow down the dimensions of the irreducible representations in a character table. For example, if the group has order 10 and 4 conjugacy classes (for instance, the dihedral group of order 10) then the only way to express the order of the group as a sum of four squares is ${\displaystyle 10=1^{2}+1^{2}+2^{2}+2^{2}}$, so we know the dimensions of all the irreducible representations.

Properties

Complex conjugation acts on the character table: since the complex conjugate of a representation is again a representation, the same is true for characters, and thus a character that takes on non-trivial complex values has a conjugate character.

Certain properties of the group G can be deduced from its character table:

• The order of G is given by the sum of the squares of the entries of the first column (the degrees of the irreducible characters). (See Representation theory of finite groups#Applying Schur's lemma.) More generally, the sum of the squares of the absolute values of the entries in any column gives the order of the centralizer of an element of the corresponding conjugacy class.
• All normal subgroups of G (and thus whether or not G is simple) can be recognised from its character table. The kernel of a character χ is the set of elements g in G for which χ(g) = χ(1); this is a normal subgroup of G. Each normal subgroup of G is the intersection of the kernels of some of the irreducible characters of G.
• The number of irreducible representations of G equals the number of conjugacy classes that G has.
• The commutator subgroup of G is the intersection of the kernels of the linear characters of G.
• If G is finite, then since the character table is square and has as many rows as conjugacy classes, it follows that G is abelian iff each conjugacy class is a singleton iff the character table of G is ${\displaystyle |G|\times |G|}$ iff each irreducible character is linear.
• It follows, using some results of Richard Brauer from modular representation theory, that the prime divisors of the orders of the elements of each conjugacy class of a finite group can be deduced from its character table (an observation of Graham Higman).

The character table does not in general determine the group up to isomorphism: for example, the quaternion group Q and the dihedral group of 8 elements (D₄) have the same character table. Brauer asked whether the character table, together with the knowledge of how the powers of elements of its conjugacy classes are distributed, determines a finite group up to isomorphism. In 1964, this was answered in the negative by E. C. Dade.

The linear representations of G are themselves a group under the tensor product, since the tensor product of 1-dimensional vector spaces is again 1-dimensional. That is, if ${\displaystyle \rho _{1}:G\to V_{1}}$ and ${\displaystyle \rho _{2}:G\to V_{2}}$ are linear representations, then ${\displaystyle \rho _{1}\otimes \rho _{2}(g)=(\rho _{1}(g)\otimes \rho _{2}(g))}$ defines a new linear representation. This gives rise to a group of linear characters, called the character group under the operation ${\displaystyle [\chi _{1}*\chi _{2}](g)=\chi _{1}(g)\chi _{2}(g)}$. This group is connected to Dirichlet characters and Fourier analysis.

Outer automorphisms

The outer automorphism group acts on the character table by permuting columns (conjugacy classes) and accordingly rows, which gives another symmetry to the table. For example, abelian groups have the outer automorphism ${\displaystyle g\mapsto g^{-1}}$, which is non-trivial except for elementary abelian 2-groups, and outer because abelian groups are precisely those for which conjugation (inner automorphisms) acts trivially. In the example of ${\displaystyle C_{3}}$ above, this map sends ${\displaystyle u\mapsto u^{2},u^{2}\mapsto u,}$ and accordingly switches ${\displaystyle \chi _{1}}$ and ${\displaystyle \chi _{2}}$ (switching their values of ${\displaystyle \omega }$ and ${\displaystyle \omega ^{2}}$). Note that this particular automorphism (negative in abelian groups) agrees with complex conjugation.

Formally, if ${\displaystyle \phi \colon G\to G}$ is an automorphism of G and ${\displaystyle \rho \colon G\to \operatorname {GL} }$ is a representation, then ${\displaystyle \rho ^{\phi }:=g\mapsto \rho (\phi (g))}$ is a representation. If ${\displaystyle \phi =\phi _{a}}$ is an inner automorphism (conjugation by some element a), then it acts trivially on representations, because representations are class functions (conjugation does not change their value). Thus a given class of outer automorphisms, it acts on the characters – because inner automorphisms act trivially, the action of the automorphism group Aut descends to the quotient Out.

This relation can be used both ways: given an outer automorphism, one can produce new representations (if the representation is not equal on conjugacy classes that are interchanged by the outer automorphism), and conversely, one can restrict possible outer automorphisms based on the character table.

Finding the vibrational modes of water molecule using character table

To find the total number of vibrational modes of water molecule, the irreducible representation ${\displaystyle \Gamma _{irreducible}}$ needs to calculate from the character table of water molecule first.

Finding Γ_reducible from the Character Table of H₂O molecule

Water (${\displaystyle {\ce {H2O}}}$) molecule falls under the point group ${\displaystyle C_{2v}}$. Below is the character table of ${\displaystyle C_{2v}}$ point group, which is also the character table for water molecule.

Character table for ${\displaystyle C_{2v}}$ point group

	${\displaystyle E}$	${\displaystyle C_{2}}$	${\displaystyle \sigma _{v}}$	${\displaystyle \sigma '_{v}}$
${\displaystyle A_{1}}$	1	1	1	1	${\displaystyle z}$	${\displaystyle x^{2},y^{2},z^{2}}$
${\displaystyle A_{2}}$	1	1	-1	-1	${\displaystyle R_{z}}$	${\displaystyle xy}$
${\displaystyle B_{1}}$	1	-1	1	-1	${\displaystyle R_{y},x}$	${\displaystyle xz}$
${\displaystyle B_{2}}$	1	-1	-1	1	${\displaystyle R_{x},y}$	${\displaystyle yz}$

In here, the first row describes the possible symmetry operations of this point group and the first column represents the Mulliken symbols. The fifth and sixth columns are called as functions.

Functions:

• ${\displaystyle x}$, ${\displaystyle y}$ and ${\displaystyle z}$ are related to translational movement and IR active bands.
• ${\displaystyle R_{x}}$, ${\displaystyle R_{y}}$ and ${\displaystyle R_{z}}$ are related to rotation about respective axis.
• Quadratic functions (such as ${\displaystyle x^{2}+y^{2}}$, ${\displaystyle x^{2}-y^{2}}$, ${\displaystyle x^{2}}$, ${\displaystyle y^{2}}$,${\displaystyle z^{2}}$, ${\displaystyle xy}$, ${\displaystyle yz}$,${\displaystyle zx}$) are related to Raman active bands.

When determining the characters for a representation, assign ${\displaystyle 1}$ if it remains unchanged, ${\displaystyle 0}$ if it moved, and ${\displaystyle (-1)}$ if it reversed its direction. A simple way to determine the characters for the reducible representation ${\displaystyle \Gamma _{reducible}}$, is to multiply the number of unshifted atom(s) with 'contribution per atom' along each of three axis (${\displaystyle x,y,z}$) when a symmetry operation is carried out.

Unless otherwise stated, for the identity operation ${\displaystyle E}$, 'contribution per unshifted atom' for each atom is always ${\displaystyle 3}$, as none of the atom(s) change their position during this operation. For any reflective symmetry operation ${\displaystyle \sigma }$, 'contribution per atom' is always ${\displaystyle 1}$, as for any reflection, an atom remains unchanged along with two axis and reverse its direction along with the other axis. For the inverse symmetry operation ${\displaystyle i}$, 'contribution per unshifted atom' is always ${\displaystyle -3}$, as each of three axis of an atom reverse its direction during this operation. An easiest way to calculate 'contribution per unshifted atom' for ${\displaystyle C_{n}}$ and ${\displaystyle S_{n}}$ symmetry operation is to use below formulas.

${\displaystyle C_{n}=2\cos \theta +1}$

${\displaystyle S_{n}=2\cos \theta -1}$

Where, ${\displaystyle \theta ={\frac {360}{n}}}$

A simplified version of above statements is summarized in the table below

Operation	Contribution per unshifted atom
${\displaystyle E}$	3
${\displaystyle C_{2}}$	-1
${\displaystyle C_{3}}$	0
${\displaystyle C_{4}}$	1
${\displaystyle C_{6}}$	2
${\displaystyle \sigma _{xy/yz/zx}}$	1
${\displaystyle i}$	-3
${\displaystyle S_{3}}$	-2
${\displaystyle S_{4}}$	-1
${\displaystyle S_{6}}$	0

Character of ${\displaystyle \Gamma _{reducible}}$ for any symmetry operation ${\displaystyle =}$ Number of unshifted atom(s) during this operation ${\displaystyle \times }$ Contribution per unshifted atom along each of three axis

Finding the characters for ${\displaystyle \Gamma _{red}}$

${\displaystyle C_{2v}}$	${\displaystyle E}$	${\displaystyle C_{2}}$	${\displaystyle \sigma _{v(xz)}}$	${\displaystyle \sigma '_{v(yz)}}$
Number of unshifted atom(s)	3	1	3	1
Contribution per unshifted atom	3	-1	1	1
${\displaystyle \Gamma _{red}}$	9	-1	3	1

Calculating the irreducible representation Γ_irreducible from the reducible representation Γ_reducible along with the character table

From the above discussion, a new character table for water molecule (${\displaystyle C_{2v}}$ point group) can be written as

New character table for ${\displaystyle {\ce {H2O}}}$ molecule including ${\displaystyle \Gamma _{red}}$

	${\displaystyle E}$	${\displaystyle C_{2}}$	${\displaystyle \sigma _{v(xz)}}$	${\displaystyle \sigma '_{v(yz)}}$
${\displaystyle A_{1}}$	1	1	1	1
${\displaystyle A_{2}}$	1	1	-1	-1
${\displaystyle B_{1}}$	1	-1	1	-1
${\displaystyle B_{2}}$	1	-1	-1	1
${\displaystyle \Gamma _{red}}$	9	-1	3	1

Using the new character table including ${\displaystyle \Gamma _{red}}$, the reducible representation for all motion of the ${\displaystyle {\ce {H2O}}}$ molecule can be reduced using below formula

${\displaystyle N={\frac {1}{h}}\sum _{x}(X_{i}^{x}\times X_{r}^{x}\times n^{x})}$

where,

${\displaystyle h=}$ order of the group,

${\displaystyle X_{i}^{x}=}$ character of the ${\displaystyle \Gamma _{reducible}}$ for a particular class,

${\displaystyle X_{r}^{x}=}$ character from the reducible representation for a particular class,

${\displaystyle n^{x}=}$ the number of operations in the class

So,

${\displaystyle N_{A_{1}}={\frac {1}{4}}[\{9\times 1\times 1\}+\{(-1)\times 1\times 1\}+\{3\times 1\times 1\}+\{1\times 1\times 1\}]=3}$

${\displaystyle N_{A_{2}}={\frac {1}{4}}[\{9\times 1\times 1\}+\{(-1)\times 1\times 1\}+\{3\times (-1)\times 1\}+\{1\times (-1)\times 1\}]=1}$

${\displaystyle N_{B_{1}}={\frac {1}{4}}[\{9\times 1\times 1\}+\{(-1)\times (-1)\times 1\}+\{3\times 1\times 1\}+\{1\times (-1)\times 1\}]=3}$

${\displaystyle N_{B_{2}}={\frac {1}{4}}[\{9\times 1\times 1\}+\{(-1)\times (-1)\times 1\}+\{3\times (-1)\times 1\}+\{1\times 1\times 1\}]=2}$

So, the reduced representation for all motions of water molecule will be

${\displaystyle \Gamma _{irreducible}=3A_{1}+A_{2}+3B_{1}+2B_{2}}$

Translational motion for water molecule

Translational motion will corresponds with the reducible representations in the character table, which have ${\displaystyle x}$, ${\displaystyle y}$ and ${\displaystyle z}$ function

For ${\displaystyle {\ce {H2O}}}$molecule

${\displaystyle A_{1}}$	${\displaystyle z}$
${\displaystyle A_{2}}$
${\displaystyle B_{1}}$	${\displaystyle x}$
${\displaystyle B_{2}}$	${\displaystyle y}$

As only the reducible representations ${\displaystyle B_{1}}$, ${\displaystyle B_{2}}$ and ${\displaystyle A_{1}}$ correspond to the ${\displaystyle x}$, ${\displaystyle y}$ and ${\displaystyle z}$ function,

${\displaystyle \Gamma _{translational}=A_{1}+B_{1}+B_{2}}$

Rotational motion for water molecule

Rotational motion will corresponds with the reducible representations in the character table, which have ${\displaystyle R_{x}}$, ${\displaystyle R_{y}}$ and ${\displaystyle R_{z}}$ function

For ${\displaystyle {\ce {H2O}}}$ molecule

${\displaystyle A_{1}}$
${\displaystyle A_{2}}$	${\displaystyle R_{z}}$
${\displaystyle B_{1}}$	${\displaystyle R_{y}}$
${\displaystyle B_{2}}$	${\displaystyle R_{x}}$

As only the reducible representations ${\displaystyle B_{2}}$, ${\displaystyle B_{1}}$ and ${\displaystyle A_{2}}$ correspond to the ${\displaystyle x}$, ${\displaystyle y}$ and ${\displaystyle z}$ function,

${\displaystyle \Gamma _{rotational}=A_{2}+B_{1}+B_{2}}$

Total vibrational modes for water molecule

Total vibrational mode, ${\displaystyle \Gamma _{vibrational}=\Gamma _{irreducible}-\Gamma _{translational}-\Gamma _{rotational}}$

${\displaystyle =(3A_{1}+A_{2}+3B_{1}+2B_{2})-(A_{1}+B_{1}+B_{2})-(A_{2}+B_{1}+B_{2})}$

${\displaystyle =2A_{1}+B_{1}}$

So, total ${\displaystyle (2+1)=3}$ vibrational modes are possible for water molecules and two of them are symmetric vibrational modes (as ${\displaystyle 2A_{1}}$) and the other vibrational mode is antisymmetric (as ${\displaystyle 1B_{1}}$)

Checking whether the water molecule is IR active or Raman active

There is some rules to be IR active or Raman active for a particular mode.

• If there is a ${\displaystyle x}$, ${\displaystyle y}$ or ${\displaystyle z}$ for any irreducible representation, then the mode is IR active
• If there is a quadratic functions such as ${\displaystyle x^{2}+y^{2}}$, ${\displaystyle x^{2}-y^{2}}$, ${\displaystyle x^{2}}$, ${\displaystyle y^{2}}$,${\displaystyle z^{2}}$, ${\displaystyle xy}$, ${\displaystyle yz}$ or ${\displaystyle xz}$ for any irreducible representation, then the mode is Raman active
• If there is no ${\displaystyle x}$, ${\displaystyle y}$, ${\displaystyle z}$ nor quadratic functions for any irreducible representation, then the mode is neither IR active nor Raman active

As the vibrational modes for water molecule ${\displaystyle \Gamma _{vibrational}}$ contains both ${\displaystyle x}$, ${\displaystyle y}$ or ${\displaystyle z}$ and quadratic functions, it has both the IR active vibrational modes and Raman active vibrational modes.

Similar rules will apply for rest of the irreducible representations ${\displaystyle \Gamma _{irreducible},\Gamma _{translational},\Gamma _{rotational}}$

Chirality (chemistry)

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Chirality_(chemistry) 

Two enantiomers of a generic amino acid that are chiral

 

(S)-Alanine (left) and (R)-alanine (right) in zwitterionic form at neutral pH

In chemistry, a molecule or ion is called chiral (/ˈkaɪrəl/) if it cannot be superposed on its mirror image by any combination of rotations, translations, and some conformational changes. This geometric property is called chirality (/kaɪˈrælɪti/). The terms are derived from Ancient Greek χείρ (cheir) 'hand'; which is the canonical example of an object with this property.

A chiral molecule or ion exists in two stereoisomers that are mirror images of each other, called enantiomers; they are often distinguished as either "right-handed" or "left-handed" by their absolute configuration or some other criterion. The two enantiomers have the same chemical properties, except when reacting with other chiral compounds. They also have the same physical properties, except that they often have opposite optical activities. A homogeneous mixture of the two enantiomers in equal parts is said to be racemic, and it usually differs chemically and physically from the pure enantiomers.

Chiral molecules will usually have a stereogenic element from which chirality arises. The most common type of stereogenic element is a stereogenic center, or stereocenter. In the case of organic compounds, stereocenters most frequently take the form of a carbon atom with four distinct groups attached to it in a tetrahedral geometry. A given stereocenter has two possible configurations, which give rise to stereoisomers (diastereomers and enantiomers) in molecules with one or more stereocenter. For a chiral molecule with one or more stereocenter, the enantiomer corresponds to the stereoisomer in which every stereocenter has the opposite configuration. An organic compound with only one stereogenic carbon is always chiral. On the other hand, an organic compound with multiple stereogenic carbons is typically, but not always, chiral. In particular, if the stereocenters are configured in such a way that the molecule has an internal plane of symmetry, then the molecule is achiral and is known as a meso compound. Less commonly, other atoms like N, P, S, and Si can also serve as stereocenters, provided they have four distinct substituents (including lone pair electrons) attached to them.

Molecules with chirality arising from one or more stereocenters are classified as possessing central chirality. There are two other types of stereogenic elements that can give rise to chirality, a stereogenic axis (axial chirality) and a stereogenic plane (planar chirality). Finally, the inherent curvature of a molecule can also give rise to chirality (inherent chirality). These types of chirality are far less common than central chirality. BINOL is a typical example of an axially chiral molecule, while trans-cyclooctene is a commonly cited example of a planar chiral molecule. Finally, helicene possesses helical chirality, which is one type of inherent chirality.

Chirality is an important concept for stereochemistry and biochemistry. Most substances relevant to biology are chiral, such as carbohydrates (sugars, starch, and cellulose), the amino acids that are the building blocks of proteins, and the nucleic acids. In living organisms, one typically finds only one of the two enantiomers of a chiral compound. For that reason, organisms that consume a chiral compound usually can metabolize only one of its enantiomers. For the same reason, the two enantiomers of a chiral pharmaceutical usually have vastly different potencies or effects.

Definition

The chirality of a molecule is based on the molecular symmetry of its conformations. A conformation of a molecule is chiral if and only if it belongs to the C_n, D_n, T, O, I point groups (the chiral point groups). However, whether the molecule itself is considered to be chiral depends on whether its chiral conformations are persistent isomers that could be isolated as separated enantiomers, at least in principle, or the enantiomeric conformers rapidly interconvert at a given temperature and timescale through low-energy conformational changes (rendering the molecule achiral). For example, despite having chiral gauche conformers that belong to the C₂ point group, butane is considered achiral at room temperature because rotation about the central C–C bond rapidly interconverts the enantiomers (3.4 kcal/mol barrier). Similarly, cis-1,2-dichlorocyclohexane consists of chair conformers that are nonidentical mirror images, but the two can interconvert via the cyclohexane chair flip (~10 kcal/mol barrier). As another example, amines with three distinct substituents (R¹R²R³N:) are also regarded as achiral molecules because their enantiomeric pyramidal conformers rapidly invert and interconvert through a planar transition state (~6 kcal/mol barrier).

However, if the temperature in question is low enough, the process that interconverts the enantiomeric chiral conformations becomes slow compared to a given timescale. The molecule would then be considered to be chiral at that temperature. The relevant timescale is, to some degree, arbitrarily defined: 1000 seconds is sometimes employed, as this is regarded as the lower limit for the amount of time required for chemical or chromatographic separation of enantiomers in a practical sense. Molecules that are chiral at room temperature due to restricted rotation about a single bond (barrier to rotation ≥ ca. 23 kcal/mol) are said to exhibit atropisomerism.

A chiral compound can contain no improper axis of rotation (S_n), which includes planes of symmetry and inversion center. Chiral molecules are always dissymmetric (lacking S_n) but not always asymmetric (lacking all symmetry elements except the trivial identity). Asymmetric molecules are always chiral.^[5]

The following table shows some examples of chiral and achiral molecules, with the Schoenflies notation of the point group of the molecule. In the achiral molecules, X and Y (with no subscript) represent achiral groups, whereas X_R and X_S or Y_R and Y_S represent enantiomers. Note that there is no meaning to the orientation of an S₂ axis, which is just an inversion. Any orientation will do, so long as it passes through the center of inversion. Also note that higher symmetries of chiral and achiral molecules also exist, and symmetries that do not include those in the table, such as the chiral C₃ or the achiral S₄.

Molecular symmetry and chirality

Rotational axis (Cn)	Improper rotational elements (Sn)
	Chiral no Sn	Achiral mirror plane S1 = σ	Achiral inversion center S2 = i
C1	C1	Cs	Ci
C2	C2 (Note: This molecule has only one C2 axis: perpendicular to line of three C, but not in the plane of the figure.)	C2v	C2h Note: This also has a mirror plane.

Stereogenic centers

Main article: Stereogenic center

Many chiral molecules have point chirality, namely a single chiral stereogenic center that coincides with an atom. This stereogenic center usually has four or more bonds to different groups, and may be carbon (as in many biological molecules), phosphorus (as in many organophosphates), silicon, or a metal (as in many chiral coordination compounds). However, a stereogenic center can also be a trivalent atom whose bonds are not in the same plane, such as phosphorus in P-chiral phosphines (PRR′R″) and sulfur in S-chiral sulfoxides (OSRR′), because a lone-pair of electrons is present instead of a fourth bond.

Chirality can also arise from isotopic differences between atoms, such as in the deuterated benzyl alcohol PhCHDOH; which is chiral and optically active ([α]_D = 0.715°), even though the non-deuterated compound PhCH₂OH is not.

If two enantiomers easily interconvert, the pure enantiomers may be practically impossible to separate, and only the racemic mixture is observable. This is the case, for example, of most amines with three different substituents (NRR′R″), because of the low energy barrier for nitrogen inversion.

1,1′-Bi-2-naphthol is an example of a molecule lacking point chirality.

While the presence of a stereogenic center describes the great majority of chiral molecules, many variations and exceptions exist. For instance it is not necessary for the chiral substance to have a stereogenic center. Examples include 1-bromo-3-chloro-5-fluoroadamantane, methylethylphenyltetrahedrane, certain calixarenes and fullerenes, which have inherent chirality. The C₂-symmetric species 1,1′-bi-2-naphthol (BINOL), 1,3-dichloroallene have axial chirality. (E)-cyclooctene and many ferrocenes have planar chirality.

When the optical rotation for an enantiomer is too low for practical measurement, the species is said to exhibit cryptochirality.

Chirality is an intrinsic part of the identity of a molecule, so the systematic name includes details of the absolute configuration (R/S, D/L, or other designations).

Manifestations of chirality

• Flavor: the artificial sweetener aspartame has two enantiomers. L-aspartame tastes sweet whereas D-aspartame is tasteless.
• Odor: R-(–)-carvone smells like spearmint whereas S-(+)-carvone smells like caraway.
• Drug effectiveness: the antidepressant drug Citalopram is sold as a racemic mixture. However, studies have shown that only the (S)-(+) enantiomer is responsible for the drug's beneficial effects.
• Drug safety: D‑penicillamine is used in chelation therapy and for the treatment of rheumatoid arthritis whereas L‑penicillamine is toxic as it inhibits the action of pyridoxine, an essential B vitamin.

In biochemistry

Many biologically active molecules are chiral, including the naturally occurring amino acids (the building blocks of proteins) and sugars.

The origin of this homochirality in biology is the subject of much debate. Most scientists believe that Earth life's "choice" of chirality was purely random, and that if carbon-based life forms exist elsewhere in the universe, their chemistry could theoretically have opposite chirality. However, there is some suggestion that early amino acids could have formed in comet dust. In this case, circularly polarised radiation (which makes up 17% of stellar radiation) could have caused the selective destruction of one chirality of amino acids, leading to a selection bias which ultimately resulted in all life on Earth being homochiral.

Enzymes, which are chiral, often distinguish between the two enantiomers of a chiral substrate. One could imagine an enzyme as having a glove-like cavity that binds a substrate. If this glove is right-handed, then one enantiomer will fit inside and be bound, whereas the other enantiomer will have a poor fit and is unlikely to bind.

L-forms of amino acids tend to be tasteless, whereas D-forms tend to taste sweet. Spearmint leaves contain the L-enantiomer of the chemical carvone or R-(−)-carvone and caraway seeds contain the D-enantiomer or S-(+)-carvone. The two smell different to most people because our olfactory receptors are chiral.

Chirality is important in context of ordered phases as well, for example the addition of a small amount of an optically active molecule to a nematic phase (a phase that has long range orientational order of molecules) transforms that phase to a chiral nematic phase (or cholesteric phase). Chirality in context of such phases in polymeric fluids has also been studied in this context.

In inorganic chemistry

Delta-ruthenium-tris(bipyridine) cation

 

Main article: Complex (chemistry): Isomerism

Chirality is a symmetry property, not a property of any part of the periodic table. Thus many inorganic materials, molecules, and ions are chiral. Quartz is an example from the mineral kingdom. Such noncentric materials are of interest for applications in nonlinear optics.

In the areas of coordination chemistry and organometallic chemistry, chirality is pervasive and of practical importance. A famous example is tris(bipyridine)ruthenium(II) complex in which the three bipyridine ligands adopt a chiral propeller-like arrangement. The two enantiomers of complexes such as [Ru(2,2′-bipyridine)₃]²⁺ may be designated as Λ (capital lambda, the Greek version of "L") for a left-handed twist of the propeller described by the ligands, and Δ (capital delta, Greek "D") for a right-handed twist (pictured). Also cf. dextro- and levo- (laevo-).

Chiral ligands confer chirality to a metal complex, as illustrated by metal-amino acid complexes. If the metal exhibits catalytic properties, its combination with a chiral ligand is the basis of asymmetric catalysis.

Methods and practices

The term optical activity is derived from the interaction of chiral materials with polarized light. In a solution, the (−)-form, or levorotatory form, of an optical isomer rotates the plane of a beam of linearly polarized light counterclockwise. The (+)-form, or dextrorotatory form, of an optical isomer does the opposite. The rotation of light is measured using a polarimeter and is expressed as the optical rotation.

Enantiomers can be separated by chiral resolution. This often involves forming crystals of a salt composed of one of the enantiomers and an acid or base from the so-called chiral pool of naturally occurring chiral compounds, such as malic acid or the amine brucine. Some racemic mixtures spontaneously crystallize into right-handed and left-handed crystals that can be separated by hand. Louis Pasteur used this method to separate left-handed and right-handed sodium ammonium tartrate crystals in 1849. Sometimes it is possible to seed a racemic solution with a right-handed and a left-handed crystal so that each will grow into a large crystal.

Miscellaneous nomenclature

• Any non-racemic chiral substance is called scalemic. Scalemic materials can be enantiopure or enantioenriched.
• A chiral substance is enantiopure when only one of two possible enantiomers is present so that all molecules within a sample have the same chirality sense. Use of homochiral as a synonym is strongly discouraged.
• A chiral substance is enantioenriched or heterochiral when its enantiomeric ratio is greater than 50:50 but less than 100:0.
• Enantiomeric excess or e.e. is the difference between how much of one enantiomer is present compared to the other. For example, a sample with 40% e.e. of R contains 70% R and 30% S (70% − 30% = 40%).

History

The rotation of plane polarized light by chiral substances was first observed by Jean-Baptiste Biot in 1812, and gained considerable importance in the sugar industry, analytical chemistry, and pharmaceuticals. Louis Pasteur deduced in 1848 that this phenomenon has a molecular basis. The term chirality itself was coined by Lord Kelvin in 1894. Different enantiomers or diastereomers of a compound were formerly called optical isomers due to their different optical properties. At one time, chirality was thought to be restricted to organic chemistry, but this misconception was overthrown by the resolution of a purely inorganic compound, a cobalt complex called hexol, by Alfred Werner in 1911.

In the early 1970s, various groups established that the human olfactory organ is capable of distinguishing chiral compounds.

Transparency and translucency

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Transparency_and_translucency 

Dichroic filters are created using optically transparent materials.

In the field of optics, transparency (also called pellucidity or diaphaneity) is the physical property of allowing light to pass through the material without appreciable scattering of light. On a macroscopic scale (one in which the dimensions are much larger than the wavelengths of the photons in question), the photons can be said to follow Snell's Law. Translucency (also called translucence or translucidity) allows light to pass through, but does not necessarily (again, on the macroscopic scale) follow Snell's law; the photons can be scattered at either of the two interfaces, or internally, where there is a change in index of refraction. In other words, a translucent material is made up of components with different indices of refraction. A transparent material is made up of components with a uniform index of refraction. Transparent materials appear clear, with the overall appearance of one color, or any combination leading up to a brilliant spectrum of every color. The opposite property of translucency is opacity.

When light encounters a material, it can interact with it in several different ways. These interactions depend on the wavelength of the light and the nature of the material. Photons interact with an object by some combination of reflection, absorption and transmission. Some materials, such as plate glass and clean water, transmit much of the light that falls on them and reflect little of it; such materials are called optically transparent. Many liquids and aqueous solutions are highly transparent. Absence of structural defects (voids, cracks, etc.) and molecular structure of most liquids are mostly responsible for excellent optical transmission.

Materials which do not transmit light are called opaque. Many such substances have a chemical composition which includes what are referred to as absorption centers. Many substances are selective in their absorption of white light frequencies. They absorb certain portions of the visible spectrum while reflecting others. The frequencies of the spectrum which are not absorbed are either reflected or transmitted for our physical observation. This is what gives rise to color. The attenuation of light of all frequencies and wavelengths is due to the combined mechanisms of absorption and scattering.

Transparency can provide almost perfect camouflage for animals able to achieve it. This is easier in dimly-lit or turbid seawater than in good illumination. Many marine animals such as jellyfish are highly transparent.

Comparisons of 1. opacity, 2. translucency, and 3. transparency; behind each panel is a star.

Etymology

• late Middle English: from Old French, from medieval Latin transparent- ‘shining through’, from Latin transparere, from trans- ‘through’ + parere ‘be visible’.
• late 16th century (in the Latin sense): from Latin translucent- ‘shining through’, from the verb translucere, from trans- ‘through’ + lucere ‘to shine’.
• late Middle English opake, from Latin opacus ‘darkened’. The current spelling (rare before the 19th century) has been influenced by the French form.

Introduction

With regard to the absorption of light, primary material considerations include:

• At the electronic level, absorption in the ultraviolet and visible (UV-Vis) portions of the spectrum depends on whether the electron orbitals are spaced (or "quantized") such that they can absorb a quantum of light (or photon) of a specific frequency, and does not violate selection rules. For example, in most glasses, electrons have no available energy levels above them in range of that associated with visible light, or if they do, they violate selection rules, meaning there is no appreciable absorption in pure (undoped) glasses, making them ideal transparent materials for windows in buildings.
• At the atomic or molecular level, physical absorption in the infrared portion of the spectrum depends on the frequencies of atomic or molecular vibrations or chemical bonds, and on selection rules. Nitrogen and oxygen are not greenhouse gases because there is no molecular dipole moment.

With regard to the scattering of light, the most critical factor is the length scale of any or all of these structural features relative to the wavelength of the light being scattered. Primary material considerations include:

• Crystalline structure: whether the atoms or molecules exhibit the 'long-range order' evidenced in crystalline solids.
• Glassy structure: scattering centers include fluctuations in density or composition.
• Microstructure: scattering centers include internal surfaces such as grain boundaries, crystallographic defects and microscopic pores.
• Organic materials: scattering centers include fiber and cell structures and boundaries.

Main article: Light scattering

 

General mechanism of diffuse reflection

Diffuse reflection - Generally, when light strikes the surface of a (non-metallic and non-glassy) solid material, it bounces off in all directions due to multiple reflections by the microscopic irregularities inside the material (e.g., the grain boundaries of a polycrystalline material, or the cell or fiber boundaries of an organic material), and by its surface, if it is rough. Diffuse reflection is typically characterized by omni-directional reflection angles. Most of the objects visible to the naked eye are identified via diffuse reflection. Another term commonly used for this type of reflection is "light scattering". Light scattering from the surfaces of objects is our primary mechanism of physical observation.

Light scattering in liquids and solids depends on the wavelength of the light being scattered. Limits to spatial scales of visibility (using white light) therefore arise, depending on the frequency of the light wave and the physical dimension (or spatial scale) of the scattering center. Visible light has a wavelength scale on the order of a half a micrometer. Scattering centers (or particles) as small as one micrometer have been observed directly in the light microscope (e.g., Brownian motion).

Transparent ceramics

Optical transparency in polycrystalline materials is limited by the amount of light which is scattered by their microstructural features. Light scattering depends on the wavelength of the light. Limits to spatial scales of visibility (using white light) therefore arise, depending on the frequency of the light wave and the physical dimension of the scattering center. For example, since visible light has a wavelength scale on the order of a micrometer, scattering centers will have dimensions on a similar spatial scale. Primary scattering centers in polycrystalline materials include microstructural defects such as pores and grain boundaries. In addition to pores, most of the interfaces in a typical metal or ceramic object are in the form of grain boundaries which separate tiny regions of crystalline order. When the size of the scattering center (or grain boundary) is reduced below the size of the wavelength of the light being scattered, the scattering no longer occurs to any significant extent.

In the formation of polycrystalline materials (metals and ceramics) the size of the crystalline grains is determined largely by the size of the crystalline particles present in the raw material during formation (or pressing) of the object. Moreover, the size of the grain boundaries scales directly with particle size. Thus a reduction of the original particle size well below the wavelength of visible light (about 1/15 of the light wavelength or roughly 600/15 = 40 nanometers) eliminates much of light scattering, resulting in a translucent or even transparent material.

Computer modeling of light transmission through translucent ceramic alumina has shown that microscopic pores trapped near grain boundaries act as primary scattering centers. The volume fraction of porosity had to be reduced below 1% for high-quality optical transmission (99.99 percent of theoretical density). This goal has been readily accomplished and amply demonstrated in laboratories and research facilities worldwide using the emerging chemical processing methods encompassed by the methods of sol-gel chemistry and nanotechnology.

Translucency of a material being used to highlight the structure of a photographic subject

Transparent ceramics have created interest in their applications for high energy lasers, transparent armor windows, nose cones for heat seeking missiles, radiation detectors for non-destructive testing, high energy physics, space exploration, security and medical imaging applications. Large laser elements made from transparent ceramics can be produced at a relatively low cost. These components are free of internal stress or intrinsic birefringence, and allow relatively large doping levels or optimized custom-designed doping profiles. This makes ceramic laser elements particularly important for high-energy lasers.

The development of transparent panel products will have other potential advanced applications including high strength, impact-resistant materials that can be used for domestic windows and skylights. Perhaps more important is that walls and other applications will have improved overall strength, especially for high-shear conditions found in high seismic and wind exposures. If the expected improvements in mechanical properties bear out, the traditional limits seen on glazing areas in today's building codes could quickly become outdated if the window area actually contributes to the shear resistance of the wall.

Currently available infrared transparent materials typically exhibit a trade-off between optical performance, mechanical strength and price. For example, sapphire (crystalline alumina) is very strong, but it is expensive and lacks full transparency throughout the 3–5 micrometer mid-infrared range. Yttria is fully transparent from 3–5 micrometers, but lacks sufficient strength, hardness, and thermal shock resistance for high-performance aerospace applications. Not surprisingly, a combination of these two materials in the form of the yttrium aluminium garnet (YAG) is one of the top performers in the field.

Absorption of light in solids

When light strikes an object, it usually has not just a single frequency (or wavelength) but many. Objects have a tendency to selectively absorb, reflect or transmit light of certain frequencies. That is, one object might reflect green light while absorbing all other frequencies of visible light. Another object might selectively transmit blue light while absorbing all other frequencies of visible light. The manner in which visible light interacts with an object is dependent upon the frequency of the light, the nature of the atoms in the object, and often the nature of the electrons in the atoms of the object.

Some materials allow much of the light that falls on them to be transmitted through the material without being reflected. Materials that allow the transmission of light waves through them are called optically transparent. Chemically pure (undoped) window glass and clean river or spring water are prime examples of this.

Materials which do not allow the transmission of any light wave frequencies are called opaque. Such substances may have a chemical composition which includes what are referred to as absorption centers. Most materials are composed of materials which are selective in their absorption of light frequencies. Thus they absorb only certain portions of the visible spectrum. The frequencies of the spectrum which are not absorbed are either reflected back or transmitted for our physical observation. In the visible portion of the spectrum, this is what gives rise to color.

Absorption centers are largely responsible for the appearance of specific wavelengths of visible light all around us. Moving from longer (0.7 micrometer) to shorter (0.4 micrometer) wavelengths: red, orange, yellow, green and blue (ROYGB) can all be identified by our senses in the appearance of color by the selective absorption of specific light wave frequencies (or wavelengths). Mechanisms of selective light wave absorption include:

• Electronic: Transitions in electron energy levels within the atom (e.g., pigments). These transitions are typically in the ultraviolet (UV) and/or visible portions of the spectrum.
• Vibrational: Resonance in atomic/molecular vibrational modes. These transitions are typically in the infrared portion of the spectrum.

UV-Vis: Electronic transitions

In electronic absorption, the frequency of the incoming light wave is at or near the energy levels of the electrons within the atoms which compose the substance. In this case, the electrons will absorb the energy of the light wave and increase their energy state, often moving outward from the nucleus of the atom into an outer shell or orbital.

The atoms that bind together to make the molecules of any particular substance contain a number of electrons (given by the atomic number Z in the periodic chart). Recall that all light waves are electromagnetic in origin. Thus they are affected strongly when coming into contact with negatively charged electrons in matter. When photons (individual packets of light energy) come in contact with the valence electrons of atom, one of several things can and will occur:

• A molecule absorbs the photon, some of the energy may be lost via luminescence, fluorescence and phosphorescence.
• A molecule absorbs the photon which results in reflection or scattering.
• A molecule cannot absorb the energy of the photon and the photon continues on its path. This results in transmission (provided no other absorption mechanisms are active).

Most of the time, it is a combination of the above that happens to the light that hits an object. The states in different materials vary in the range of energy that they can absorb. Most glasses, for example, block ultraviolet (UV) light. What happens is the electrons in the glass absorb the energy of the photons in the UV range while ignoring the weaker energy of photons in the visible light spectrum. But there are also existing special glass types, like special types of borosilicate glass or quartz that are UV-permeable and thus allow a high transmission of ultra violet light.

Thus, when a material is illuminated, individual photons of light can make the valence electrons of an atom transition to a higher electronic energy level. The photon is destroyed in the process and the absorbed radiant energy is transformed to electric potential energy. Several things can happen then to the absorbed energy: it may be re-emitted by the electron as radiant energy (in this case the overall effect is in fact a scattering of light), dissipated to the rest of the material (i.e. transformed into heat), or the electron can be freed from the atom (as in the photoelectric and Compton effects).

Infrared: Bond stretching

Normal modes of vibration in a crystalline solid

The primary physical mechanism for storing mechanical energy of motion in condensed matter is through heat, or thermal energy. Thermal energy manifests itself as energy of motion. Thus, heat is motion at the atomic and molecular levels. The primary mode of motion in crystalline substances is vibration. Any given atom will vibrate around some mean or average position within a crystalline structure, surrounded by its nearest neighbors. This vibration in two dimensions is equivalent to the oscillation of a clock’s pendulum. It swings back and forth symmetrically about some mean or average (vertical) position. Atomic and molecular vibrational frequencies may average on the order of 10¹² cycles per second (Terahertz radiation).

When a light wave of a given frequency strikes a material with particles having the same or (resonant) vibrational frequencies, then those particles will absorb the energy of the light wave and transform it into thermal energy of vibrational motion. Since different atoms and molecules have different natural frequencies of vibration, they will selectively absorb different frequencies (or portions of the spectrum) of infrared light. Reflection and transmission of light waves occur because the frequencies of the light waves do not match the natural resonant frequencies of vibration of the objects. When infrared light of these frequencies strikes an object, the energy is reflected or transmitted.

If the object is transparent, then the light waves are passed on to neighboring atoms through the bulk of the material and re-emitted on the opposite side of the object. Such frequencies of light waves are said to be transmitted.

Transparency in insulators

An object may be not transparent either because it reflects the incoming light or because it absorbs the incoming light. Almost all solids reflect a part and absorb a part of the incoming light.

When light falls onto a block of metal, it encounters atoms that are tightly packed in a regular lattice and a "sea of electrons" moving randomly between the atoms. In metals, most of these are non-bonding electrons (or free electrons) as opposed to the bonding electrons typically found in covalently bonded or ionically bonded non-metallic (insulating) solids. In a metallic bond, any potential bonding electrons can easily be lost by the atoms in a crystalline structure. The effect of this delocalization is simply to exaggerate the effect of the "sea of electrons". As a result of these electrons, most of the incoming light in metals is reflected back, which is why we see a shiny metal surface.

Most insulators (or dielectric materials) are held together by ionic bonds. Thus, these materials do not have free conduction electrons, and the bonding electrons reflect only a small fraction of the incident wave. The remaining frequencies (or wavelengths) are free to propagate (or be transmitted). This class of materials includes all ceramics and glasses.

If a dielectric material does not include light-absorbent additive molecules (pigments, dyes, colorants), it is usually transparent to the spectrum of visible light. Color centers (or dye molecules, or "dopants") in a dielectric absorb a portion of the incoming light. The remaining frequencies (or wavelengths) are free to be reflected or transmitted. This is how colored glass is produced.

Most liquids and aqueous solutions are highly transparent. For example, water, cooking oil, rubbing alcohol, air, and natural gas are all clear. Absence of structural defects (voids, cracks, etc.) and molecular structure of most liquids are chiefly responsible for their excellent optical transmission. The ability of liquids to "heal" internal defects via viscous flow is one of the reasons why some fibrous materials (e.g., paper or fabric) increase their apparent transparency when wetted. The liquid fills up numerous voids making the material more structurally homogeneous.

Light scattering in an ideal defect-free crystalline (non-metallic) solid which provides no scattering centers for incoming light will be due primarily to any effects of anharmonicity within the ordered lattice. Light transmission will be highly directional due to the typical anisotropy of crystalline substances, which includes their symmetry group and Bravais lattice. For example, the seven different crystalline forms of quartz silica (silicon dioxide, SiO₂) are all clear, transparent materials.

Optical waveguides

Propagation of light through a multi-mode optical fiber

 

A laser beam bouncing down an acrylic rod, illustrating the total internal reflection of light in a multimode optical fiber

Optically transparent materials focus on the response of a material to incoming light waves of a range of wavelengths. Guided light wave transmission via frequency selective waveguides involves the emerging field of fiber optics and the ability of certain glassy compositions to act as a transmission medium for a range of frequencies simultaneously (multi-mode optical fiber) with little or no interference between competing wavelengths or frequencies. This resonant mode of energy and data transmission via electromagnetic (light) wave propagation is relatively lossless.

An optical fiber is a cylindrical dielectric waveguide that transmits light along its axis by the process of total internal reflection. The fiber consists of a core surrounded by a cladding layer. To confine the optical signal in the core, the refractive index of the core must be greater than that of the cladding. The refractive index is the parameter reflecting the speed of light in a material. (Refractive index is the ratio of the speed of light in vacuum to the speed of light in a given medium. The refractive index of vacuum is therefore 1.) The larger the refractive index, the more slowly light travels in that medium. Typical values for core and cladding of an optical fiber are 1.48 and 1.46, respectively.

When light traveling in a dense medium hits a boundary at a steep angle, the light will be completely reflected. This effect, called total internal reflection, is used in optical fibers to confine light in the core. Light travels along the fiber bouncing back and forth off of the boundary. Because the light must strike the boundary with an angle greater than the critical angle, only light that enters the fiber within a certain range of angles will be propagated. This range of angles is called the acceptance cone of the fiber. The size of this acceptance cone is a function of the refractive index difference between the fiber's core and cladding. Optical waveguides are used as components in integrated optical circuits (e.g. combined with lasers or light-emitting diodes, LEDs) or as the transmission medium in local and long haul optical communication systems.

Mechanisms of attenuation

See also: Light scattering

 

Light attenuation by ZBLAN and silica fibers

Attenuation in fiber optics, also known as transmission loss, is the reduction in intensity of the light beam (or signal) with respect to distance traveled through a transmission medium. Attenuation coefficients in fiber optics usually use units of dB/km through the medium due to the very high quality of transparency of modern optical transmission media. The medium is usually a fiber of silica glass that confines the incident light beam to the inside. Attenuation is an important factor limiting the transmission of a signal across large distances. In optical fibers the main attenuation source is scattering from molecular level irregularities (Rayleigh scattering) due to structural disorder and compositional fluctuations of the glass structure. This same phenomenon is seen as one of the limiting factors in the transparency of infrared missile domes. Further attenuation is caused by light absorbed by residual materials, such as metals or water ions, within the fiber core and inner cladding. Light leakage due to bending, splices, connectors, or other outside forces are other factors resulting in attenuation.

As camouflage

Many animals of the open sea, like this Aurelia labiata jellyfish, are largely transparent.

 

Further information: List of camouflage methods

Many marine animals that float near the surface are highly transparent, giving them almost perfect camouflage. However, transparency is difficult for bodies made of materials that have different refractive indices from seawater. Some marine animals such as jellyfish have gelatinous bodies, composed mainly of water; their thick mesogloea is acellular and highly transparent. This conveniently makes them buoyant, but it also makes them large for their muscle mass, so they cannot swim fast, making this form of camouflage a costly trade-off with mobility. Gelatinous planktonic animals are between 50 and 90 percent transparent. A transparency of 50 percent is enough to make an animal invisible to a predator such as cod at a depth of 650 metres (2,130 ft); better transparency is required for invisibility in shallower water, where the light is brighter and predators can see better. For example, a cod can see prey that are 98 percent transparent in optimal lighting in shallow water. Therefore, sufficient transparency for camouflage is more easily achieved in deeper waters. For the same reason, transparency in air is even harder to achieve, but a partial example is found in the glass frogs of the South American rain forest, which have translucent skin and pale greenish limbs. Several Central American species of clearwing (ithomiine) butterflies and many dragonflies and allied insects also have wings which are mostly transparent, a form of crypsis that provides some protection from predators.