Symmetry group

From Wikipedia, the free encyclopedia

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

A regular tetrahedron is invariant under twelve distinct rotations (if the identity transformation is included as a trivial rotation and reflections are excluded). These are illustrated here in the cycle graph format, along with the 180° edge (blue arrows) and 120° vertex (pink and orange arrows) rotations that permute the tetrahedron through the positions. The twelve rotations form the rotation (symmetry) group of the figure.

In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object. A frequent notation for the symmetry group of an object X is G = Sym(X).

For an object in a metric space, its symmetries form a subgroup of the isometry group of the ambient space. This article mainly considers symmetry groups in Euclidean geometry, but the concept may also be studied for more general types of geometric structure.

Introduction

We consider the "objects" possessing symmetry to be geometric figures, images, and patterns, such as a wallpaper pattern. For symmetry of physical objects, one may also take their physical composition as part of the pattern. (A pattern may be specified formally as a scalar field, a function of position with values in a set of colors or substances; as a vector field; or as a more general function on the object.) The group of isometries of space induces a group action on objects in it, and the symmetry group Sym(X) consists of those isometries which map X to itself (as well as mapping any further pattern to itself). We say X is invariant under such a mapping, and the mapping is a symmetry of X.

The above is sometimes called the full symmetry group of X to emphasize that it includes orientation-reversing isometries (reflections, glide reflections and improper rotations), as long as those isometries map this particular X to itself. The subgroup of orientation-preserving symmetries (translations, rotations, and compositions of these) is called its proper symmetry group. An object is chiral when it has no orientation-reversing symmetries, so that its proper symmetry group is equal to its full symmetry group.

Any symmetry group whose elements have a common fixed point, which is true if the group is finite or the figure is bounded, can be represented as a subgroup of the orthogonal group O(n) by choosing the origin to be a fixed point. The proper symmetry group is then a subgroup of the special orthogonal group SO(n), and is called the rotation group of the figure.

In a discrete symmetry group, the points symmetric to a given point do not accumulate toward a limit point. That is, every orbit of the group (the images of a given point under all group elements) forms a discrete set. All finite symmetry groups are discrete.

Discrete symmetry groups come in three types: (1) finite point groups, which include only rotations, reflections, inversions and rotoinversions – i.e., the finite subgroups of O(n); (2) infinite lattice groups, which include only translations; and (3) infinite space groups containing elements of both previous types, and perhaps also extra transformations like screw displacements and glide reflections. There are also continuous symmetry groups (Lie groups), which contain rotations of arbitrarily small angles or translations of arbitrarily small distances. An example is O(3), the symmetry group of a sphere. Symmetry groups of Euclidean objects may be completely classified as the subgroups of the Euclidean group E(n) (the isometry group of Rⁿ).

Two geometric figures have the same symmetry type when their symmetry groups are conjugate subgroups of the Euclidean group: that is, when the subgroups H₁, H₂ are related by H₁ = g⁻¹H₂g for some g in E(n). For example:

• two 3D figures have mirror symmetry, but with respect to different mirror planes.
• two 3D figures have 3-fold rotational symmetry, but with respect to different axes.
• two 2D patterns have translational symmetry, each in one direction; the two translation vectors have the same length but a different direction.

In the following sections, we only consider isometry groups whose orbits are topologically closed, including all discrete and continuous isometry groups. However, this excludes for example the 1D group of translations by a rational number; such a non-closed figure cannot be drawn with reasonable accuracy due to its arbitrarily fine detail.

One dimension

Main article: One-dimensional symmetry group

The isometry groups in one dimension are:

• the trivial cyclic group C₁
• the groups of two elements generated by a reflection; they are isomorphic with C₂
• the infinite discrete groups generated by a translation; they are isomorphic with Z, the additive group of the integers
• the infinite discrete groups generated by a translation and a reflection; they are isomorphic with the generalized dihedral group of Z, Dih(Z), also denoted by D_∞ (which is a semidirect product of Z and C₂).
• the group generated by all translations (isomorphic with the additive group of the real numbers R); this group cannot be the symmetry group of a Euclidean figure, even endowed with a pattern: such a pattern would be homogeneous, hence could also be reflected. However, a constant one-dimensional vector field has this symmetry group.
• the group generated by all translations and reflections in points; they are isomorphic with the generalized dihedral group Dih(R).

Two dimensions

Up to conjugacy the discrete point groups in two-dimensional space are the following classes:

• cyclic groups C₁, C₂, C₃, C₄, ... where C_n consists of all rotations about a fixed point by multiples of the angle 360°/n
• dihedral groups D₁, D₂, D₃, D₄, ..., where D_n (of order 2n) consists of the rotations in C_n together with reflections in n axes that pass through the fixed point.

C₁ is the trivial group containing only the identity operation, which occurs when the figure is asymmetric, for example the letter "F". C₂ is the symmetry group of the letter "Z", C₃ that of a triskelion, C₄ that of a chiral four-bladed pinwheel, and C₅, C₆, etc. are the symmetry groups of similar chiral pinwheels with five, six, etc. arms.

D₁ is the 2-element group containing the identity operation and a single reflection, which occurs when the figure has only a single axis of bilateral symmetry, for example the letter "A".

D₂, which is isomorphic to the Klein four-group, is the symmetry group of a non-equilateral rectangle. This figure has four symmetry operations: the identity operation, one twofold axis of rotation, and two nonequivalent mirror planes.

D₃, D₄ etc. are the symmetry groups of the regular polygons.

Within each of these symmetry types, there are two degrees of freedom for the center of rotation, and in the case of the dihedral groups, one more for the positions of the mirrors.

The remaining isometry groups in two dimensions with a fixed point are:

• the special orthogonal group SO(2) consisting of all rotations about a fixed point; it is also called the circle group S¹, the multiplicative group of complex numbers of absolute value 1. It is the proper symmetry group of a circle and the continuous equivalent of C_n. There is no geometric figure that has as full symmetry group the circle group, but for a vector field it may apply (see the three-dimensional case below).
• the orthogonal group O(2) consisting of all rotations about a fixed point and reflections in any axis through that fixed point. This is the symmetry group of a circle. It is also called Dih(S¹) as it is the generalized dihedral group of S¹.

Non-bounded figures may have isometry groups including translations; these are:

• the 7 frieze groups
• the 17 wallpaper groups
• for each of the symmetry groups in one dimension, the combination of all symmetries in that group in one direction, and the group of all translations in the perpendicular direction
• ditto with also reflections in a line in the first direction.

Three dimensions

See also: Point groups in three dimensions

Up to conjugacy the set of three-dimensional point groups consists of 7 infinite series, and 7 other individual groups. In crystallography, only those point groups are considered which preserve some crystal lattice (so their rotations may only have order 1, 2, 3, 4, or 6). This crystallographic restriction of the infinite families of general point groups results in 32 crystallographic point groups (27 individual groups from the 7 series, and 5 of the 7 other individuals).

The continuous symmetry groups with a fixed point include those of:

• cylindrical symmetry without a symmetry plane perpendicular to the axis. This applies, for example, to a bottle or cone.
• cylindrical symmetry with a symmetry plane perpendicular to the axis
• spherical symmetry

For objects with scalar field patterns, the cylindrical symmetry implies vertical reflection symmetry as well. However, this is not true for vector field patterns: for example, in cylindrical coordinates with respect to some axis, the vector field ${\displaystyle \mathbf{A}=A_{\rho }\hat{\mathit{\rho }}+A_{\varphi }\hat{\mathit{\varphi }}+A_z\hat{\mathit{z}}}$ has cylindrical symmetry with respect to the axis whenever ${\displaystyle A_{\rho },A_{\varphi },}$ and ${\displaystyle A_z}$ have this symmetry (no dependence on ${\displaystyle \varphi }$); and it has reflectional symmetry only when ${\displaystyle A_{\varphi }=0}$.

For spherical symmetry, there is no such distinction: any patterned object has planes of reflection symmetry.

The continuous symmetry groups without a fixed point include those with a screw axis, such as an infinite helix. See also subgroups of the Euclidean group.

Symmetry groups in general

See also: Automorphism and Automorphism group

In wider contexts, a symmetry group may be any kind of transformation group, or automorphism group. Each type of mathematical structure has invertible mappings which preserve the structure. Conversely, specifying the symmetry group can define the structure, or at least clarify the meaning of geometric congruence or invariance; this is one way of looking at the Erlangen programme.

For example, objects in a hyperbolic non-Euclidean geometry have Fuchsian symmetry groups, which are the discrete subgroups of the isometry group of the hyperbolic plane, preserving hyperbolic rather than Euclidean distance. (Some are depicted in drawings of Escher.) Similarly, automorphism groups of finite geometries preserve families of point-sets (discrete subspaces) rather than Euclidean subspaces, distances, or inner products. Just as for Euclidean figures, objects in any geometric space have symmetry groups which are subgroups of the symmetries of the ambient space.

Another example of a symmetry group is that of a combinatorial graph: a graph symmetry is a permutation of the vertices which takes edges to edges. Any finitely presented group is the symmetry group of its Cayley graph; the free group is the symmetry group of an infinite tree graph.

Group structure in terms of symmetries

Cayley's theorem states that any abstract group is a subgroup of the permutations of some set X, and so can be considered as the symmetry group of X with some extra structure. In addition, many abstract features of the group (defined purely in terms of the group operation) can be interpreted in terms of symmetries.

For example, let G = Sym(X) be the finite symmetry group of a figure X in a Euclidean space, and let H ⊂ G be a subgroup. Then H can be interpreted as the symmetry group of X⁺, a "decorated" version of X. Such a decoration may be constructed as follows. Add some patterns such as arrows or colors to X so as to break all symmetry, obtaining a figure X^# with Sym(X^#) = {1}, the trivial subgroup; that is, gX^# ≠ X^# for all non-trivial g ∈ G. Now we get:

${\displaystyle X^{+}=\bigcup _{h\in H}h{X}^{\mathrm{\#}}\text{satisfies}H=\mathrm{S}\mathrm{y}\mathrm{m}(X^{+}).}$

Normal subgroups may also be characterized in this framework. The symmetry group of the translation gX ⁺ is the conjugate subgroup gHg⁻¹. Thus H is normal whenever:

${\displaystyle \mathrm{S}\mathrm{y}\mathrm{m}(g{X}^{+})=\mathrm{S}\mathrm{y}\mathrm{m}(X^{+})\text{for all}g\in G;}$

that is, whenever the decoration of X⁺ may be drawn in any orientation, with respect to any side or feature of X, and still yield the same symmetry group gHg⁻¹ = H.

As an example, consider the dihedral group G = D₃ = Sym(X), where X is an equilateral triangle. We may decorate this with an arrow on one edge, obtaining an asymmetric figure X^#. Letting τ ∈ G be the reflection of the arrowed edge, the composite figure X⁺ = X^# ∪ τX^# has a bidirectional arrow on that edge, and its symmetry group is H = {1, τ}. This subgroup is not normal, since gX⁺ may have the bi-arrow on a different edge, giving a different reflection symmetry group.

However, letting H = {1, ρ, ρ²} ⊂ D₃ be the cyclic subgroup generated by a rotation, the decorated figure X⁺ consists of a 3-cycle of arrows with consistent orientation. Then H is normal, since drawing such a cycle with either orientation yields the same symmetry group H.

Micro black hole

From Wikipedia, the free encyclopedia

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

Micro black holes, also known as mini black holes and quantum mechanical black holes, are hypothetical tiny (<1 M_☉) black holes, for which quantum mechanical effects play an important role. The concept that black holes may exist that are smaller than stellar mass was introduced in 1971 by Stephen Hawking.

It is possible that such black holes were created in the high-density environment of the early universe (or Big Bang), or possibly through subsequent phase transitions (referred to as primordial black holes). They might be possible to observe through the particles they are expected to emit by Hawking radiation.

Some hypotheses involving additional space dimensions predict that micro black holes could be formed at energies as low as the teraelectronvolt (TeV) range, which are available in particle accelerators such as the Large Hadron Collider. Popular concerns have then been raised over end-of-the-world scenarios (see Safety of particle collisions at the Large Hadron Collider). However, such quantum black holes would instantly evaporate, either totally or leaving only a very weakly interacting residue. Beside the theoretical arguments, cosmic rays hitting the Earth do not produce any damage, although they reach energies in the range of hundreds of TeV.

Minimum mass of a black hole

In an early speculation, Stephen Hawking conjectured that a black hole would not form with a mass below about 10⁻⁸ kg (roughly the Planck mass). To make a black hole, one must concentrate mass or energy sufficiently that the escape velocity from the region in which it is concentrated exceeds the speed of light.

Some extensions of present physics posit the existence of extra dimensions of space. In higher-dimensional spacetime, the strength of gravity increases more rapidly with decreasing distance than in three dimensions. With certain special configurations of the extra dimensions, this effect can lower the Planck scale to the TeV range. Examples of such extensions include large extra dimensions, special cases of the Randall–Sundrum model, and string theory configurations like the GKP solutions. In such scenarios, black hole production could possibly be an important and observable effect at the Large Hadron Collider (LHC). It would also be a common natural phenomenon induced by cosmic rays.

All this assumes that the theory of general relativity remains valid at these small distances. If it does not, then other, currently unknown, effects might limit the minimum size of a black hole. Elementary particles are equipped with a quantum-mechanical, intrinsic angular momentum (spin). The correct conservation law for the total (orbital plus spin) angular momentum of matter in curved spacetime requires that spacetime is equipped with torsion. The simplest and most natural theory of gravity with torsion is the Einstein–Cartan theory. Torsion modifies the Dirac equation in the presence of the gravitational field and causes fermion particles to be spatially extended. In this case the spatial extension of fermions limits the minimum mass of a black hole to be on the order of 10¹⁶ kg, showing that micro black holes may not exist. The energy necessary to produce such a black hole is 39 orders of magnitude greater than the energies available at the Large Hadron Collider, indicating that the LHC cannot produce mini black holes. But if black holes are produced, then the theory of general relativity is proven wrong and does not exist at these small distances. The rules of general relativity would be broken, as is consistent with theories of how matter, space, and time break down around the event horizon of a black hole. This would prove the spatial extensions of the fermion limits to be incorrect as well. The fermion limits assume a minimum mass needed to sustain a black hole, as opposed to the opposite, the minimum mass needed to start a black hole, which in theory is achievable in the LHC under some conditions.

Stability

Hawking radiation

Main article: Hawking radiation

In 1975, Stephen Hawking argued that, due to quantum effects, black holes "evaporate" by a process now referred to as Hawking radiation in which elementary particles (such as photons, electrons, quarks and gluons) are emitted. His calculations showed that the smaller the size of the black hole, the faster the evaporation rate, resulting in a sudden burst of particles as the micro black hole suddenly explodes.

Any primordial black hole of sufficiently low mass will evaporate to near the Planck mass within the lifetime of the Universe. In this process, these small black holes radiate away matter. A rough picture of this is that pairs of virtual particles emerge from the vacuum near the event horizon, with one member of a pair being captured, and the other escaping the vicinity of the black hole. The net result is the black hole loses mass (due to conservation of energy). According to the formulae of black hole thermodynamics, the more the black hole loses mass, the hotter it becomes, and the faster it evaporates, until it approaches the Planck mass. At this stage, a black hole would have a Hawking temperature of ⁠T_P/8π⁠ (5.6×10³⁰ K), which means an emitted Hawking particle would have an energy comparable to the mass of the black hole. Thus, a thermodynamic description breaks down. Such a micro black hole would also have an entropy of only 4π nats, approximately the minimum possible value. At this point then, the object can no longer be described as a classical black hole, and Hawking's calculations also break down.

While Hawking radiation is sometimes questioned, Leonard Susskind summarizes an expert perspective in his book The Black Hole War: "Every so often, a physics paper will appear claiming that black holes don't evaporate. Such papers quickly disappear into the infinite junk heap of fringe ideas."

Conjectures for the final state

Conjectures for the final fate of the black hole include total evaporation and production of a Planck-mass-sized black hole remnant. Such Planck-mass black holes may in effect be stable objects if the quantized gaps between their allowed energy levels bar them from emitting Hawking particles or absorbing energy gravitationally like a classical black hole. In such case, they would be weakly interacting massive particles; this could explain dark matter.

Primordial black holes

Main article: Primordial black hole

Formation in the early Universe

Production of a black hole requires concentration of mass or energy within the corresponding Schwarzschild radius. It was hypothesized by Zel'dovich and Novikov first and independently by Hawking that, shortly after the Big Bang, the Universe was dense enough for any given region of space to fit within its own Schwarzschild radius. Even so, at that time, the Universe was not able to collapse into a singularity due to its uniform mass distribution and rapid growth. This, however, does not fully exclude the possibility that black holes of various sizes may have emerged locally. A black hole formed in this way is called a primordial black hole and is the most widely accepted hypothesis for the possible creation of micro black holes. Computer simulations suggest that the probability of formation of a primordial black hole is inversely proportional to its mass. Thus, the most likely outcome would be micro black holes.

Expected observable effects

A primordial black hole with an initial mass of around 10¹² kg would be completing its evaporation today; a less massive primordial black hole would have already evaporated. Under optimal conditions, the Fermi Gamma-ray Space Telescope satellite, launched in June 2008, might detect experimental evidence for evaporation of nearby black holes by observing gamma ray bursts. It is unlikely that a collision between a microscopic black hole and an object such as a star or a planet would be noticeable. The small radius and high density of the black hole would allow it to pass straight through any object consisting of normal atoms, interacting with only few of its atoms while doing so. It has, however, been suggested that a small black hole of sufficient mass passing through the Earth would produce a detectable acoustic or seismic signal. On the moon, it may leave a distinct type of crater, still visible after billions of years.

Human-made micro black holes

Feasibility of production

Further information: Kugelblitz (astrophysics)

In familiar three-dimensional gravity, the minimum energy of a microscopic black hole is 10¹⁶ TeV (equivalent to 1.6 GJ or 444 kWh), which would have to be condensed into a region on the order of the Planck length. This is far beyond the limits of any current technology. It is estimated that to collide two particles to within a distance of a Planck length with currently achievable magnetic field strengths would require a ring accelerator about 1,000 light years in diameter to keep the particles on track.

However, in some scenarios involving extra dimensions of space, the Planck mass can be as low as the TeV range. The Large Hadron Collider (LHC) has a design energy of 14 TeV for proton–proton collisions and 1,150 TeV for Pb–Pb collisions. It was argued in 2001 that, in these circumstances, black hole production could be an important and observable effect at the LHC or future higher-energy colliders. Such quantum black holes should decay emitting sprays of particles that could be seen by detectors at these facilities. A paper by Choptuik and Pretorius, published in 2010 in Physical Review Letters, presented a computer-generated proof that micro black holes must form from two colliding particles with sufficient energy, which might be allowable at the energies of the LHC if additional dimensions are present other than the customary four (three spatial, one temporal).

Safety

Main article: Safety of high-energy particle collision experiments

Hawking's calculation and more general quantum mechanical arguments predict that micro black holes evaporate almost instantaneously. Additional safety arguments beyond those based on Hawking radiation were given in the paper, which showed that in hypothetical scenarios with stable micro black holes massive enough to destroy Earth, such black holes would have been produced by cosmic rays and would have likely already destroyed astronomical objects such as planets, stars, or stellar remnants such as neutron stars and white dwarfs.

Black holes in quantum theories of gravity

It is possible, in some theories of quantum gravity, to calculate the quantum corrections to ordinary, classical black holes. Contrarily to conventional black holes, which are solutions of gravitational field equations of the general theory of relativity, quantum gravity black holes incorporate quantum gravity effects in the vicinity of the origin, where classically a curvature singularity occurs. According to the theory employed to model quantum gravity effects, there are different kinds of quantum gravity black holes, namely loop quantum black holes, non-commutative black holes, and asymptotically safe black holes. In these approaches, black holes are singularity-free.

Virtual micro black holes were proposed by Stephen Hawking in 1995 and by Fabio Scardigli in 1999 as part of a Grand Unified Theory as a quantum gravity candidate.

Near-infrared spectroscopy

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Near-infrared_spectroscopy 

Near-IR absorption spectrum of dichloromethane showing complicated overlapping overtones of mid IR absorption features.

Near-infrared spectroscopy (NIRS) is a spectroscopic method that uses the near-infrared region of the electromagnetic spectrum (from 780 nm to 2500 nm). Typical applications include medical and physiological diagnostics and research including blood sugar, pulse oximetry, functional neuroimaging, sports medicine, elite sports training, ergonomics, rehabilitation, neonatal research, brain computer interface, urology (bladder contraction), and neurology (neurovascular coupling). There are also applications in other areas as well such as pharmaceutical, food and agrochemical quality control, atmospheric chemistry, combustion propagation.

Theory

Near-infrared spectroscopy is based on molecular overtone and combination vibrations. Overtones and combinations exhibit lower intensity compared to the fundamental, as a result, the molar absorptivity in the near-IR region is typically quite small. (NIR absorption bands are typically 10–100 times weaker than the corresponding fundamental mid-IR absorption band.) The lower absorption allows NIR radiation to penetrate much further into a sample than mid infrared radiation. Near-infrared spectroscopy is, therefore, not a particularly sensitive technique, but it can be very useful in probing bulk material with little to no sample preparation.

The molecular overtone and combination bands seen in the near-IR are typically very broad, leading to complex spectra; it can be difficult to assign specific features to specific chemical components. Multivariate (multiple variables) calibration techniques (e.g., principal components analysis, partial least squares, or artificial neural networks) are often employed to extract the desired chemical information. Careful development of a set of calibration samples and application of multivariate calibration techniques is essential for near-infrared analytical methods.

History

Near-infrared spectrum of liquid ethanol.

The discovery of near-infrared energy is ascribed to William Herschel in the 19th century, but the first industrial application began in the 1950s. In the first applications, NIRS was used only as an add-on unit to other optical devices that used other wavelengths such as ultraviolet (UV), visible (Vis), or mid-infrared (MIR) spectrometers. In the 1980s, a single-unit, stand-alone NIRS system was made available.

In the 1980s, Karl Norris (while working at the USDA Instrumentation Research Laboratory, Beltsville, USA) pioneered the use NIR spectroscopy for quality assessments of agricultural products. Since then, use has expanded from food and agricultural to chemical, polymer, and petroleum industries; pharmaceutical industry; biomedical sciences; and environmental analysis.

With the introduction of light-fiber optics in the mid-1980s and the monochromator-detector developments in the early 1990s, NIRS became a more powerful tool for scientific research. The method has been used in a number of fields of science including physics, physiology, or medicine. It is only in the last few decades that NIRS began to be used as a medical tool for monitoring patients, with the first clinical application of so-called fNIRS in 1994.

Instrumentation

Instrumentation for near-IR (NIR) spectroscopy is similar to instruments for the UV-visible and mid-IR ranges. There is a source, a detector, and a dispersive element (such as a prism, or, more commonly, a diffraction grating) to allow the intensity at different wavelengths to be recorded. Fourier transform NIR instruments using an interferometer are also common, especially for wavelengths above ~1000 nm. Depending on the sample, the spectrum can be measured in either reflection or transmission.

Common incandescent or quartz halogen light bulbs are most often used as broadband sources of near-infrared radiation for analytical applications. Light-emitting diodes (LEDs) can also be used. For high precision spectroscopy, wavelength-scanned lasers and frequency combs have recently become powerful sources, albeit with sometimes longer acquisition timescales. When lasers are used, a single detector without any dispersive elements might be sufficient.

The type of detector used depends primarily on the range of wavelengths to be measured. Silicon-based CCDs are suitable for the shorter end of the NIR range, but are not sufficiently sensitive over most of the range (over 1000 nm). InGaAs and PbS devices are more suitable and have higher quantum efficiency for wavelengths above 1100 nm. It is possible to combine silicon-based and InGaAs detectors in the same instrument. Such instruments can record both UV-visible and NIR spectra 'simultaneously'.

Instruments intended for chemical imaging in the NIR may use a 2D array detector with an acousto-optic tunable filter. Multiple images may be recorded sequentially at different narrow wavelength bands.

Many commercial instruments for UV/vis spectroscopy are capable of recording spectra in the NIR range (to perhaps ~900 nm). In the same way, the range of some mid-IR instruments may extend into the NIR. In these instruments, the detector used for the NIR wavelengths is often the same detector used for the instrument's "main" range of interest.

NIRS as an analytical technique

The use of NIR as an analytical technique did not come from extending the use of mid-IR into the near-IR range, but developed independently. A striking way this was exhibited is that, while mid-IR spectroscopists use wavenumbers (cm⁻¹) when displaying spectra, NIR spectroscopists used wavelength (nm), as is used in ultraviolet–visible spectroscopy. Early practitioners of IR spectroscopy, who depended on assignment of absorption bands to specific bond types, were frustrated by the complexity of the bonding regions being measured. However, as a quantitative tool, the lower molar absorption levels in the bonding region tended to keep absorption maxima "on-scale", enabling quantitative work with little sample preparation. Techniques applied to extract the quantitative information from these complex spectra were unfamiliar to analytical chemists, and the technique was viewed with suspicion in academia.

Generally, quantitative NIR analysis is accomplished by selecting a group of calibration samples, for which the concentration of the analyte of interest has been determined by a reference method, and finding a correlation between various spectral features and those concentrations using a chemometric tool. The calibration is then validated by using it to predict the analyte values for samples in a validation set, whose values have been determined by the reference method but have not been included in the calibration. A validated calibration is then used to predict the values of samples. The complexity of the spectra are overcome by the use of multivariate calibration. The two tools most often used a multi-wavelength linear regression and partial least squares.

Applications

Typical applications of NIR spectroscopy include the analysis of food products, pharmaceuticals, combustion products, and a major branch of astronomical spectroscopy.

Astronomical spectroscopy

Near-infrared spectroscopy is used in astronomy for studying the atmospheres of cool stars where molecules can form. The vibrational and rotational signatures of molecules such as titanium oxide, cyanide, and carbon monoxide can be seen in this wavelength range and can give a clue towards the star's spectral type. It is also used for studying molecules in other astronomical contexts, such as in molecular clouds where new stars are formed. The astronomical phenomenon known as reddening means that near-infrared wavelengths are less affected by dust in the interstellar medium, such that regions inaccessible by optical spectroscopy can be studied in the near-infrared. Since dust and gas are strongly associated, these dusty regions are exactly those where infrared spectroscopy is most useful. The near-infrared spectra of very young stars provide important information about their ages and masses, which is important for understanding star formation in general. Astronomical spectrographs have also been developed for the detection of exoplanets using the Doppler shift of the parent star due to the radial velocity of the planet around the star.

Agriculture

Near-infrared spectroscopy is widely applied in agriculture for determining the quality of forages, grains, and grain products, oilseeds, coffee, tea, spices, fruits, vegetables, sugarcane, beverages, fats, and oils, dairy products, eggs, meat, and other agricultural products. It is widely used to quantify the composition of agricultural products because it meets the criteria of being accurate, reliable, rapid, non-destructive, and inexpensive. Abeni and Bergoglio 2001 apply NIRS to chicken breeding as the assay method for characteristics of fat composition.

Remote monitoring

Techniques have been developed for NIR spectroscopic imaging. Hyperspectral imaging has been applied for a wide range of uses, including the remote investigation of plants and soils. Data can be collected from instruments on airplanes, satellites or unmanned aerial systems to assess ground cover and soil chemistry.

Remote monitoring or remote sensing from the NIR spectroscopic region can also be used to study the atmosphere. For example, measurements of atmospheric gases are made from NIR spectra measured by the OCO-2, GOSAT, and the TCCON.

Materials science

Techniques have been developed for NIR spectroscopy of microscopic sample areas for film thickness measurements, research into the optical characteristics of nanoparticles and optical coatings for the telecommunications industry.

Medical uses

The application of NIRS in medicine centres on its ability to provide information about the oxygen saturation of haemoglobin within the microcirculation. Broadly speaking, it can be used to assess oxygenation and microvascular function in the brain (cerebral NIRS) or in the peripheral tissues (peripheral NIRS).

Cerebral NIRS

When a specific area of the brain is activated, the localized blood volume in that area changes quickly. Optical imaging can measure the location and activity of specific regions of the brain by continuously monitoring blood hemoglobin levels through the determination of optical absorption coefficients.

Infrascanner 1000, a NIRS scanner used to detect intracranial bleeding.

NIRS can be used as a quick screening tool for possible intracranial bleeding cases by placing the scanner on four locations on the head. In non-injured patients the brain absorbs the NIR light evenly. When there is an internal bleeding from an injury, the blood may be concentrated in one location causing the NIR light to be absorbed more than other locations, which the scanner detects.

So-called functional NIRS can be used for non-invasive assessment of brain function through the intact skull in human subjects by detecting changes in blood hemoglobin concentrations associated with neural activity, e.g., in branches of cognitive psychology as a partial replacement for fMRI techniques. NIRS can be used on infants, and NIRS is much more portable than fMRI machines, even wireless instrumentation is available, which enables investigations in freely moving subjects. However, NIRS cannot fully replace fMRI because it can only be used to scan cortical tissue, whereas fMRI can be used to measure activation throughout the brain. Special public domain statistical toolboxes for analysis of stand alone and combined NIRS/MRI measurement have been developed.

Example of data acquisition using fNIRS (Hitachi ETG-4000)

The application in functional mapping of the human cortex is called functional NIRS (fNIRS) or diffuse optical tomography (DOT). The term diffuse optical tomography is used for three-dimensional NIRS. The terms NIRS, NIRI, and DOT are often used interchangeably, but they have some distinctions. The most important difference between NIRS and DOT/NIRI is that DOT/NIRI is used mainly to detect changes in optical properties of tissue simultaneously from multiple measurement points and display the results in the form of a map or image over a specific area, whereas NIRS provides quantitative data in absolute terms on up to a few specific points. The latter is also used to investigate other tissues such as, e.g., muscle, breast and tumors. NIRS can be used to quantify blood flow, blood volume, oxygen consumption, reoxygenation rates and muscle recovery time in muscle.

By employing several wavelengths and time resolved (frequency or time domain) and/or spatially resolved methods blood flow, volume and absolute tissue saturation (${\displaystyle St{O}_2}$ or Tissue Saturation Index (TSI)) can be quantified. Applications of oximetry by NIRS methods include neuroscience, ergonomics, rehabilitation, brain-computer interface, urology, the detection of illnesses that affect the blood circulation (e.g., peripheral vascular disease), the detection and assessment of breast tumors, and the optimization of training in sports medicine.

The use of NIRS in conjunction with a bolus injection of indocyanine green (ICG) has been used to measure cerebral blood flow and cerebral metabolic rate of oxygen consumption (CMRO2). It has also been shown that CMRO2 can be calculated with combined NIRS/MRI measurements. Additionally metabolism can be interrogated by resolving an additional mitochondrial chromophore, cytochrome-c-oxidase, using broadband NIRS.

NIRS is starting to be used in pediatric critical care, to help manage patients following cardiac surgery. Indeed, NIRS is able to measure venous oxygen saturation (SVO2), which is determined by the cardiac output, as well as other parameters (FiO2, hemoglobin, oxygen uptake). Therefore, examining the NIRS provides critical care physicians with an estimate of the cardiac output. NIRS is favoured by patients, because it is non-invasive, painless, and does not require ionizing radiation.

Optical coherence tomography (OCT) is another NIR medical imaging technique capable of 3D imaging with high resolution on par with low-power microscopy. Using optical coherence to measure photon pathlength allows OCT to build images of live tissue and clear examinations of tissue morphology. Due to technique differences OCT is limited to imaging 1–2 mm below tissue surfaces, but despite this limitation OCT has become an established medical imaging technique especially for imaging of the retina and anterior segments of the eye, as well as coronaries.

A type of neurofeedback, hemoencephalography or HEG, uses NIR technology to measure brain activation, primarily of the frontal lobes, for the purpose of training cerebral activation of that region.

The instrumental development of NIRS/NIRI/DOT/OCT has proceeded tremendously during the last years and, in particular, in terms of quantification, imaging and miniaturization.

Peripheral NIRS

Peripheral microvascular function can be assessed using NIRS. The oxygen saturation of haemoglobin in the tissue (StO2) can provide information about tissue perfusion. A vascular occlusion test (VOT) can be employed to assess microvascular function. Common sites for peripheral NIRS monitoring include the thenar eminence, forearm and calf muscles.

Particle measurement

NIR is often used in particle sizing in a range of different fields, including studying pharmaceutical and agricultural powders.

Industrial uses

As opposed to NIRS used in optical topography, general NIRS used in chemical assays does not provide imaging by mapping. For example, a clinical carbon dioxide analyzer requires reference techniques and calibration routines to be able to get accurate CO₂ content change. In this case, calibration is performed by adjusting the zero control of the sample being tested after purposefully supplying 0% CO₂ or another known amount of CO₂ in the sample. Normal compressed gas from distributors contains about 95% O₂ and 5% CO₂, which can also be used to adjust %CO₂ meter reading to be exactly 5% at initial calibration.