Electroweak interaction

From Wikipedia, the free encyclopedia

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

In particle physics, the electroweak interaction or electroweak force is the unified description of two of the fundamental interactions of nature: electromagnetism (electromagnetic interaction) and the weak interaction. Although these two forces appear very different at everyday low energies, the theory models them as two different aspects of the same force. Above the unification energy, on the order of 246 GeV,^[a] they would merge into a single force. Thus, if the temperature is high enough – approximately 10¹⁵ K – then the electromagnetic force and weak force merge into a combined electroweak force.

During the quark epoch (shortly after the Big Bang), the electroweak force split into the electromagnetic and weak force. It is thought that the required temperature of 10¹⁵ K has not been seen widely throughout the universe since before the quark epoch, and currently the highest human-made temperature in thermal equilibrium is around 5.5×10¹² K (from the Large Hadron Collider).

Sheldon Glashow, Abdus Salam, and Steven Weinberg were awarded the 1979 Nobel Prize in Physics for their contributions to the unification of the weak and electromagnetic interaction between elementary particles, known as the Weinberg–Salam theory. The existence of the electroweak interactions was experimentally established in two stages, the first being the discovery of neutral currents in neutrino scattering by the Gargamelle collaboration in 1973, and the second in 1983 by the UA1 and the UA2 collaborations that involved the discovery of the W and Z gauge bosons in proton–antiproton collisions at the converted Super Proton Synchrotron. In 1999, Gerardus 't Hooft and Martinus Veltman were awarded the Nobel prize for showing that the electroweak theory is renormalizable.

History

After the Wu experiment in 1956 discovered parity violation in the weak interaction, a search began for a way to relate the weak and electromagnetic interactions. Extending his doctoral advisor Julian Schwinger's work, Sheldon Glashow first experimented with introducing two different symmetries, one chiral and one achiral, and combined them such that their overall symmetry was unbroken. This did not yield a renormalizable theory, and its gauge symmetry had to be broken by hand as no spontaneous mechanism was known, but it predicted a new particle, the Z boson. This received little notice, as it matched no experimental finding.

In 1964, Salam and John Clive Ward had the same idea, but predicted a massless photon and three massive gauge bosons with a manually broken symmetry. Later around 1967, while investigating spontaneous symmetry breaking, Weinberg found a set of symmetries predicting a massless, neutral gauge boson. Initially rejecting such a particle as useless, he later realized his symmetries produced the electroweak force, and he proceeded to predict rough masses for the W and Z bosons. Significantly, he suggested this new theory was renormalizable. In 1971, Gerard 't Hooft proved that spontaneously broken gauge symmetries are renormalizable even with massive gauge bosons.

Formulation

Main article: Mathematical formulation of the Standard Model

Weinberg's weak mixing angle θ_W, and relation between coupling constants g, g′, and e. Adapted from Lee (1981).

The pattern of weak isospin, T₃, and weak hypercharge, Y_w, of the known elementary particles, showing the electric charge, Q, along the weak mixing angle. The neutral Higgs field (circled) breaks the electroweak symmetry and interacts with other particles to give them mass. Three components of the Higgs field become part of the massive W and Z bosons.

Mathematically, electromagnetism is unified with the weak interactions as a Yang–Mills field with an SU(2) × U(1) gauge group, which describes the formal operations that can be applied to the electroweak gauge fields without changing the dynamics of the system. These fields are the weak isospin fields W₁, W₂, and W₃, and the weak hypercharge field B. This invariance is known as electroweak symmetry.

The generators of SU(2) and U(1) are given the name weak isospin (labeled T) and weak hypercharge (labeled Y) respectively. These then give rise to the gauge bosons that mediate the electroweak interactions – the three W bosons of weak isospin (W₁, W₂, and W₃), and the B boson of weak hypercharge, respectively, all of which are "initially" massless. These are not physical fields yet, before spontaneous symmetry breaking and the associated Higgs mechanism.

In the Standard Model, the observed physical particles, the W^±
and Z⁰
bosons, and the photon, are produced through the spontaneous symmetry breaking of the electroweak symmetry SU(2) × U(1)_y to U(1)_em, effected by the Higgs mechanism (see also Higgs boson), an elaborate quantum-field-theoretic phenomenon that "spontaneously" alters the realization of the symmetry and rearranges degrees of freedom.

The electric charge arises as the particular linear combination (nontrivial) of Y_w (weak hypercharge) and the T₃ component of weak isospin (${\displaystyle Q=T_3+\frac{1}{2}{Y}_{\mathrm{W}}}$) that does not couple to the Higgs boson. That is to say: the Higgs and the electromagnetic field have no effect on each other, at the level of the fundamental forces ("tree level"), while any other combination of the hypercharge and the weak isospin must interact with the Higgs. This causes an apparent separation between the weak force, which interacts with the Higgs, and electromagnetism, which does not. Mathematically, the electric charge is a specific combination of the hypercharge and T₃ outlined in the figure.

U(1)_em (the symmetry group of electromagnetism only) is defined to be the group generated by this special linear combination, and the symmetry described by the U(1)_em group is unbroken, since it does not directly interact with the Higgs.

The above spontaneous symmetry breaking makes the W₃ and B bosons coalesce into two different physical bosons with different masses – the Z⁰
boson, and the photon (γ),

${\displaystyle \left(\begin{array}{c}\gamma \\ Z^0\end{array}\right)=\left(\begin{array}{cc}\cos {\theta }_{\text{W}}& \sin {\theta }_{\text{W}}\\ -\sin {\theta }_{\text{W}}& \cos {\theta }_{\text{W}}\end{array}\right)\left(\begin{array}{c}B\\ W_3\end{array}\right),}$

where θ_w is the weak mixing angle. The axes representing the particles have essentially just been rotated, in the (W₃, B) plane, by the angle θ_w. This also introduces a mismatch between the mass of the Z⁰
and the mass of the W^±
particles (denoted as m_z and m_w, respectively),

${\displaystyle m_{\text{Z}}=\frac{m_{\text{W}}}{\cos {\theta }_{\text{W}}}.}$

The W₁ and W₂ bosons, in turn, combine to produce the charged massive bosons W^±
:

${\displaystyle W^{\pm }=\frac{1}{\sqrt{2}}(W_1\mp i{W}_2).}$

Lagrangian

Before electroweak symmetry breaking

The Lagrangian for the electroweak interactions is divided into four parts before electroweak symmetry breaking manifests,

${\displaystyle {\mathcal{L}}_{\mathrm{E}\mathrm{W}}={\mathcal{L}}_g+{\mathcal{L}}_f+{\mathcal{L}}_h+{\mathcal{L}}_y.}$

The ${\displaystyle {\mathcal{L}}_g}$ term describes the interaction between the three W vector bosons and the B vector boson,

${\displaystyle {\mathcal{L}}_g=-\frac{1}{4}{W}_a^{\mu \nu }{W}_{\mu \nu }^a-\frac{1}{4}{B}^{\mu \nu }{B}_{\mu \nu },}$

where ${\displaystyle W_a^{\mu \nu }}$ (${\displaystyle a=1,2,3}$) and ${\displaystyle B^{\mu \nu }}$ are the field strength tensors for the weak isospin and weak hypercharge gauge fields.

${\displaystyle {\mathcal{L}}_f}$ is the kinetic term for the Standard Model fermions. The interaction of the gauge bosons and the fermions are through the gauge covariant derivative,

${\displaystyle {\mathcal{L}}_f={\overline{Q}}_{j}iD/Q_j+{\overline{u}}_{j}iD/u_j+{\overline{d}}_{j}iD/d_j+{\overline{L}}_{j}iD/L_j+{\overline{e}}_{j}iD/e_j,}$

where the subscript j sums over the three generations of fermions; Q, u, and d are the left-handed doublet, right-handed singlet up, and right handed singlet down quark fields; and L and e are the left-handed doublet and right-handed singlet electron fields. The Feynman slash ${\displaystyle D/}$ means the contraction of the 4-gradient with the Dirac matrices, defined as

${\displaystyle D/\equiv {\gamma }^{\mu }{D}_{\mu },}$

and the covariant derivative (excluding the gluon gauge field for the strong interaction) is defined as

${\displaystyle D_{\mu }\equiv {\mathrm{\partial }}_{\mu }-i\frac{g^{\prime }}{2}Y{B}_{\mu }-i\frac{g}{2}{T}_j{W}_{\mu }^j.}$

Here ${\displaystyle Y}$ is the weak hypercharge and the ${\displaystyle T_j}$ are the components of the weak isospin.

The ${\displaystyle {\mathcal{L}}_h}$ term describes the Higgs field ${\displaystyle h}$ and its interactions with itself and the gauge bosons,

${\displaystyle {\mathcal{L}}_h=|D_{\mu }h{|}^2-\lambda {\left(|h{|}^2-\frac{v^2}{2}\right)}^2,}$

where ${\displaystyle v}$ is the vacuum expectation value.

The ${\displaystyle {\mathcal{L}}_y}$ term describes the Yukawa interaction with the fermions,

${\displaystyle {\mathcal{L}}_y=-y_u^{ij}{ϵ}^{ab}{h}_b^{†}{\overline{Q}}_{ia}{u}_j^c-y_d^{ij}h{\overline{Q}}_i{d}_j^c-y_e^{ij}h{\overline{L}}_i{e}_j^c+\mathrm{h}.\mathrm{c}.,}$

and generates their masses, manifest when the Higgs field acquires a nonzero vacuum expectation value, discussed next. The ${\displaystyle y_k^{ij},}$ for ${\displaystyle k\in \{\mathrm{u},\mathrm{d},\mathrm{e}\},}$ are matrices of Yukawa couplings.

After electroweak symmetry breaking

The Lagrangian reorganizes itself as the Higgs field acquires a non-vanishing vacuum expectation value dictated by the potential of the previous section. As a result of this rewriting, the symmetry breaking becomes manifest. In the history of the universe, this is believed to have happened shortly after the hot big bang, when the universe was at a temperature 159.5±1.5 GeV (assuming the Standard Model of particle physics).

Due to its complexity, this Lagrangian is best described by breaking it up into several parts as follows.

${\displaystyle {\mathcal{L}}_{\mathrm{E}\mathrm{W}}={\mathcal{L}}_{\mathrm{K}}+{\mathcal{L}}_{\mathrm{N}}+{\mathcal{L}}_{\mathrm{C}}+{\mathcal{L}}_{\mathrm{H}}+{\mathcal{L}}_{\mathrm{H}\mathrm{V}}+{\mathcal{L}}_{\mathrm{W}\mathrm{W}\mathrm{V}}+{\mathcal{L}}_{\mathrm{W}\mathrm{W}\mathrm{V}\mathrm{V}}+{\mathcal{L}}_{\mathrm{Y}}.}$

The kinetic term ${\displaystyle {\mathcal{L}}_K}$ contains all the quadratic terms of the Lagrangian, which include the dynamic terms (the partial derivatives) and the mass terms (conspicuously absent from the Lagrangian before symmetry breaking)

${\displaystyle \begin{array}{r}{\mathcal{L}}_{\mathrm{K}}=\sum _f\overline{f}(i\mathrm{\partial }/-m_f)f-\frac{1}{4}{A}_{\mu \nu }{A}^{\mu \nu }-\frac{1}{2}{W}_{\mu \nu }^{+}{W}^{-\mu \nu }+m_W^2{W}_{\mu }^{+}{W}^{-\mu }\\ -\frac{1}{4}{Z}_{\mu \nu }{Z}^{\mu \nu }+\frac{1}{2}{m}_Z^2{Z}_{\mu }{Z}^{\mu }+\frac{1}{2}({\mathrm{\partial }}^{\mu }H)({\mathrm{\partial }}_{\mu }H)-\frac{1}{2}{m}_H^2{H}^2,\end{array}}$

where the sum runs over all the fermions of the theory (quarks and leptons), and the fields ${\displaystyle A_{\mu \nu },}$ ${\displaystyle Z_{\mu \nu },}$ ${\displaystyle W_{\mu \nu }^{-},}$ and ${\displaystyle W_{\mu \nu }^{+}\equiv (W_{\mu \nu }^{-}{)}^{†}}$ are given as

${\displaystyle X_{\mu \nu }^a={\mathrm{\partial }}_{\mu }{X}_{\nu }^a-{\mathrm{\partial }}_{\nu }{X}_{\mu }^a+g{f}^{abc}{X}_{\mu }^b{X}_{\nu }^c,}$

with ${\displaystyle X}$ to be replaced by the relevant field (${\displaystyle A,}$ ${\displaystyle Z,}$ ${\displaystyle W^{\pm }}$) and f ^abc by the structure constants of the appropriate gauge group.

The neutral current ${\displaystyle {\mathcal{L}}_{\mathrm{N}}}$ and charged current ${\displaystyle {\mathcal{L}}_{\mathrm{C}}}$ components of the Lagrangian contain the interactions between the fermions and gauge bosons,

${\displaystyle {\mathcal{L}}_{\mathrm{N}}=e{J}_{\mu }^{\mathrm{e}\mathrm{m}}{A}^{\mu }+\frac{g}{\cos {\theta }_W}(J_{\mu }^3-{\sin }^2{\theta }_W{J}_{\mu }^{\mathrm{e}\mathrm{m}})Z^{\mu },}$

where ${\displaystyle e=g\sin {\theta }_{\mathrm{W}}=g^{\prime }\cos {\theta }_{\mathrm{W}}.}$ The electromagnetic current ${\displaystyle J_{\mu }^{\mathrm{e}\mathrm{m}}}$ is

${\displaystyle J_{\mu }^{\mathrm{e}\mathrm{m}}=\sum _f{q}_f\overline{f}{\gamma }_{\mu }f,}$

where ${\displaystyle q_f}$ is the fermions' electric charges. The neutral weak current ${\displaystyle J_{\mu }^3}$ is

${\displaystyle J_{\mu }^3=\sum _f{T}_f^3\overline{f}{\gamma }_{\mu }\frac{1-{\gamma }^5}{2}f,}$

where ${\displaystyle T_f^3}$ is the fermions' weak isospin.

The charged current part of the Lagrangian is given by

${\displaystyle {\mathcal{L}}_{\mathrm{C}}=-\frac{g}{\sqrt{2}}\left[{\overline{u}}_i{\gamma }^{\mu }\frac{1-{\gamma }^5}{2}{M}_{ij}^{\mathrm{C}\mathrm{K}\mathrm{M}}{d}_j+{\overline{\nu }}_i{\gamma }^{\mu }\frac{1-{\gamma }^5}{2}{e}_i\right]W_{\mu }^{+}+\mathrm{h}.\mathrm{c}.,}$

where ${\displaystyle \nu }$ is the right-handed singlet neutrino field, and the CKM matrix ${\displaystyle M_{ij}^{\mathrm{C}\mathrm{K}\mathrm{M}}}$ determines the mixing between mass and weak eigenstates of the quarks.

${\displaystyle {\mathcal{L}}_{\mathrm{H}}}$ contains the Higgs three-point and four-point self interaction terms,

${\displaystyle {\mathcal{L}}_{\mathrm{H}}=-\frac{g{m}_{\mathrm{H}}^2}{4m_{\mathrm{W}}}{H}^3-\frac{g^2{m}_{\mathrm{H}}^2}{32m_{\mathrm{W}}^2}{H}^4.}$

${\displaystyle {\mathcal{L}}_{\mathrm{H}\mathrm{V}}}$ contains the Higgs interactions with gauge vector bosons,

${\displaystyle {\mathcal{L}}_{\mathrm{H}\mathrm{V}}=\left(g{m}_{\mathrm{H}\mathrm{V}}+\frac{g^2}{4}{H}^2\right)\left(W_{\mu }^{+}{W}^{-\mu }+\frac{1}{2{\cos }^2{\theta }_{\mathrm{W}}}{Z}_{\mu }{Z}^{\mu }\right).}$

${\displaystyle {\mathcal{L}}_{\mathrm{W}\mathrm{W}\mathrm{V}}}$ contains the gauge three-point self interactions,

${\displaystyle {\mathcal{L}}_{\mathrm{W}\mathrm{W}\mathrm{V}}=-ig\left[\left(W_{\mu \nu }^{+}{W}^{-\mu }-W^{+\mu }{W}_{\mu \nu }^{-}\right)\left(A^{\nu }\sin {\theta }_{\mathrm{W}}-Z^{\nu }\cos {\theta }_{\mathrm{W}}\right)+W_{\nu }^{-}{W}_{\mu }^{+}\left(A^{\mu \nu }\sin {\theta }_{\mathrm{W}}-Z^{\mu \nu }\cos {\theta }_{\mathrm{W}}\right)\right].}$

${\displaystyle {\mathcal{L}}_{\mathrm{W}\mathrm{W}\mathrm{V}\mathrm{V}}}$ contains the gauge four-point self interactions,

${\displaystyle \begin{array}{rl}{\mathcal{L}}_{\mathrm{W}\mathrm{W}\mathrm{V}\mathrm{V}}=-\frac{g^2}{4}\{& [2W_{\mu }^{+}{W}^{-\mu }+(A_{\mu }\sin {\theta }_{\mathrm{W}}-Z_{\mu }\cos {\theta }_{\mathrm{W}}{)}^2{]}^2\\ & -[W_{\mu }^{+}{W}_{\nu }^{-}+W_{\nu }^{+}{W}_{\mu }^{-}+\left(A_{\mu }\sin {\theta }_{\mathrm{W}}-Z_{\mu }\cos {\theta }_{\mathrm{W}}\right)\left(A_{\nu }\sin {\theta }_{\mathrm{W}}-Z_{\nu }\cos {\theta }_{\mathrm{W}}\right){]}^2\}.\end{array}}$

${\displaystyle {\mathcal{L}}_{\mathrm{Y}}}$ contains the Yukawa interactions between the fermions and the Higgs field,

${\displaystyle {\mathcal{L}}_{\mathrm{Y}}=-\sum _f\frac{g{m}_f}{2m_{\mathrm{W}}}\overline{f}fH.}$

Lattice gauge theory

From Wikipedia, the free encyclopedia

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

In physics, lattice gauge theory is the study of gauge theories on a spacetime that has been discretized into a lattice.

Gauge theories are important in particle physics, and include the prevailing theories of elementary particles: quantum electrodynamics, quantum chromodynamics (QCD) and particle physics' Standard Model. Non-perturbative gauge theory calculations in continuous spacetime formally involve evaluating an infinite-dimensional path integral, which is computationally intractable. By working on a discrete spacetime, the path integral becomes finite-dimensional, and can be evaluated by stochastic simulation techniques such as the Monte Carlo method. When the size of the lattice is taken infinitely large and its sites infinitesimally close to each other, the continuum gauge theory is recovered.

Basics

In lattice gauge theory, the spacetime is Wick rotated into Euclidean space and discretized into a lattice with sites separated by distance ${\displaystyle a}$ and connected by links. In the most commonly considered cases, such as lattice QCD, fermion fields are defined at lattice sites (which leads to fermion doubling), while the gauge fields are defined on the links. That is, an element U of the compact Lie group G (not algebra) is assigned to each link. Hence, to simulate QCD with Lie group SU(3), a 3×3 unitary matrix is defined on each link. The link is assigned an orientation, with the inverse element corresponding to the same link with the opposite orientation. And each node is given a value in ${\displaystyle {\mathbb{C}}^3}$ (a color 3-vector, the space on which the fundamental representation of SU(3) acts), a Dirac spinor, an n_f vector, and a Grassmann variable.

Thus, the composition of links' SU(3) elements along a path (i.e. the ordered multiplication of their matrices) approximates a path-ordered exponential (geometric integral), from which Wilson loop values can be calculated for closed paths.

Yang–Mills action

The Yang–Mills action is written on the lattice using Wilson loops (named after Kenneth G. Wilson), so that the limit ${\displaystyle a\to 0}$ formally reproduces the original continuum action. Given a faithful irreducible representation ρ of G, the lattice Yang–Mills action, known as the Wilson action, is the sum over all lattice sites of the (real component of the) trace over the n links e₁, ..., e_n in the Wilson loop,

${\displaystyle S=\sum _F-\mathrm{\Re }\{{\chi }^{(\rho )}(U(e_1)\cdots U(e_n))\}.}$

Here, χ is the character. If ρ is a real (or pseudoreal) representation, taking the real component is redundant, because even if the orientation of a Wilson loop is flipped, its contribution to the action remains unchanged.

There are many possible Wilson actions, depending on which Wilson loops are used in the action. The simplest Wilson action uses only the 1×1 Wilson loop, and differs from the continuum action by "lattice artifacts" proportional to the small lattice spacing ${\displaystyle a}$. By using more complicated Wilson loops to construct "improved actions", lattice artifacts can be reduced to be proportional to ${\displaystyle a^2}$, making computations more accurate.

Measurements and calculations

This result of a Lattice QCD computation shows a meson, composed out of a quark and an antiquark. (After M. Cardoso et al.)

Quantities such as particle masses are stochastically calculated using techniques such as the Monte Carlo method. Gauge field configurations are generated with probabilities proportional to ${\displaystyle e^{-\beta S}}$, where ${\displaystyle S}$ is the lattice action and ${\displaystyle \beta }$ is related to the lattice spacing ${\displaystyle a}$. The quantity of interest is calculated for each configuration, and averaged. Calculations are often repeated at different lattice spacings ${\displaystyle a}$ so that the result can be extrapolated to the continuum, ${\displaystyle a\to 0}$.

Such calculations are often extremely computationally intensive, and can require the use of the largest available supercomputers. To reduce the computational burden, the so-called quenched approximation can be used, in which the fermionic fields are treated as non-dynamic "frozen" variables. While this was common in early lattice QCD calculations, "dynamical" fermions are now standard. Simulating lattice gauge theories is one hypothetical application of quantum computers.

The results of lattice QCD computations show that in a meson not only the particles (quarks and antiquarks), but also the "fluxtubes" of the gluon fields are important.

Quantum triviality

Lattice gauge theory is also important for the study of quantum triviality by the real-space renormalization group. The most important information in the RG flow are the fixed points. The possible macroscopic states of the system, at a large scale, are given by this set of fixed points. If these fixed points correspond to a free field theory, the theory is said to be trivial or noninteracting.

The concept of triviality has been applied to the physics of the Higgs boson. A comparatively simple toy model of the "Higgs sector" of the Standard Model is provided by φ⁴ theory, i.e., a scalar field theory whose Lagrangian includes an interaction term that is quartic in the field φ. It is conjectured, based on evidence including lattice gauge calculations, that this theory is "trivial": if the regularization cutoff is taken to zero distance or infinite energy, the theory will describe only free particles. Using this as a model for the Higgs boson, the conjectured triviality implies that the cutoff cannot be removed. Because there is an upper bound on how high the cutoff energy can be, there is an upper bound on the interaction strength, which ultimately implies an upper bound on the mass of the Higgs boson.

Other applications

Originally, solvable two-dimensional lattice gauge theories had already been introduced in 1971 as models with interesting statistical properties by the theorist Franz Wegner, who worked in the field of phase transitions.

When only 1×1 Wilson loops appear in the action, lattice gauge theory can be shown to be exactly dual to spin foam models.

Analytical chemistry

From Wikipedia, the free encyclopedia

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

Gas chromatography laboratory

Analytical chemistry (or chemical analysis) is the branch of chemistry concerned with the development and application of methods to identify the chemical composition of materials and quantify the amounts of components in mixtures. It focuses on methods to identify unknown compounds, possibly in a mixture or solution, and quantify a compound's presence in terms of amount of substance (in any phase), concentration (in aqueous or solution phase), percentage by mass or number of moles in a mixture of compounds (or partial pressure in the case of gas phase).

It encompasses both classical techniques (e.g. titration, gravimetric analysis) and modern instrumental approaches (e.g. spectroscopy, chromatography, mass spectrometry, electrochemical methods). Modern analytical chemistry is deeply intertwined with data analysis and chemometrics, and is increasingly shaped by trends such as automation, miniaturization, and real-time sensing, with applications across fields as diverse as biochemistry, medicinal chemistry, forensic science, archaeology, nutritional science, agricultural chemistry, chemical synthesis, metallurgy, chemical engineering and materials science.

In the age of "big data", analytical chemistry, along with chemometrics and bioinformatics, is becoming central to interpreting complex results from high-throughput techniques like gas chromatography-mass spectrometry (GCMS), high-performance liquid chromatography, inductively coupled plasma mass spectrometry, and high-resolution mass spectrometry. There is also a strong trend towards miniaturization, automation, and the development of real-time, point-of-care diagnostic sensors.

History

Gustav Kirchhoff (left) and Robert Bunsen (right)

Analytical chemistry has been important since the early days of chemistry, providing methods for determining which elements and chemicals are present in the object in question. During this period, significant contributions to analytical chemistry included the development of systematic elemental analysis by Justus von Liebig and systematized organic analysis based on the specific reactions of functional groups.

The first instrumental analysis was flame emissive spectrometry, developed by Robert Bunsen and Gustav Kirchhoff, who discovered rubidium (Rb) and caesium (Cs) in 1860.

Most of the major developments in analytical chemistry took place after 1900. During this period, instrumental analysis became progressively dominant in the field. In particular, many of the basic spectroscopic and spectrometric techniques were discovered in the early 20th century and refined in the late 20th century.

The separation sciences follow a similar timeline of development and have also became increasingly transformed into high-performance instruments.^[6] In the 1970s many of these techniques began to be used together as hybrid techniques to achieve a complete characterization of samples.

Starting in the 1970s, analytical chemistry became progressively more inclusive of biological questions (bioanalytical chemistry), whereas it had previously been largely focused on inorganic or small organic molecules. Lasers have been increasingly used as probes and even to initiate and influence a wide variety of reactions. The late 20th century also saw an expansion of the application of analytical chemistry from somewhat academic chemical questions to forensic, environmental, industrial and medical questions, such as in histology.

Modern analytical chemistry is dominated by instrumental analysis. Many analytical chemists focus on a single type of instrument. Academics tend to either focus on new applications and discoveries or on new methods of analysis. The discovery of a chemical present in blood that increases the risk of cancer would be a discovery that an analytical chemist might be involved in. An effort to develop a new method might involve the use of a tunable laser to increase the specificity and sensitivity of a spectrometric method. Many methods, once developed, are kept purposely static so that data can be compared over long periods of time. This is particularly true in industrial quality assurance (QA), forensic, and environmental applications. Analytical chemistry plays an increasingly important role in the pharmaceutical industry where, aside from QA, it is used in the discovery of new drug candidates and in clinical applications where understanding the interactions between the drug and the patient are critical.

The 21st century has been defined by the digitalization of analytical chemistry. The handling of large datasets ("big data") from instruments like Orbitrap mass spectrometers has made advanced data analysis, including machine learning, an essential skill. This era also focuses strongly on sustainability, leading to the green chemistry subfield of Green Analytical Chemistry, which aims to minimize the environmental impact of chemical analyses.

Classical methods

The presence of copper in this qualitative analysis is indicated by the bluish-green color of the flame.

Although modern analytical chemistry is dominated by sophisticated instrumentation, the roots of analytical chemistry and some of the principles used in modern instruments are from traditional techniques, many of which are still used today. These techniques also tend to form the backbone of most undergraduate analytical chemistry educational labs.

Qualitative analysis

Chemical tests

Further information: Chemical test

There are numerous qualitative chemical tests; examples include the acid test for gold and the Kastle-Meyer test for the presence of blood.

Flame test

Further information: Flame test

Inorganic qualitative analysis generally refers to a systematic scheme to confirm the presence of certain aqueous ions or elements by performing a series of reactions that eliminate a range of possibilities and then confirm suspected ions with a confirming test. Sometimes small carbon-containing ions are included in such schemes. With modern instrumentation, these tests are rarely used but can be useful for educational purposes and in fieldwork or other situations where access to state-of-the-art instruments is not available or expedient.

Quantitative analysis

Further information: Quantitative analysis (chemistry)

Quantitative analysis is the measurement of the quantities of particular chemical constituents present in a substance. Quantities can be measured by mass (gravimetric analysis) or volume (volumetric analysis).

Gravimetric analysis

Further information: Gravimetric analysis

Gravimetric analysis involves determining the amount of material present by weighing the sample before and/or after some transformation. A common example used in undergraduate education is the determination of the amount of water in a hydrate by heating the sample to remove the water such that the difference in weight is due to the loss of water.

Volumetric analysis

Further information: Titration

Titration involves the gradual addition of a measurable reactant to an exact volume of a solution being analyzed until some equivalence point is reached. Titration is a family of techniques used to determine the concentration of an analyte. Titrating accurately to either the half-equivalence point or the endpoint of a titration allows the chemist to determine the amount of moles used, which can then be used to determine a concentration or composition of the titrant. Most familiar to those who have taken chemistry during secondary education is the acid-base titration involving a color-changing pH indicator, such as phenolphthalein. There are many other types of titrations, including potentiometric titrations and precipitation titrations. Chemists might also create titration curves by systematically testing the pH after every added drop in order to understand different properties of the titrant.

Instrumental methods

Main article: Instrumental analysis

Block diagram of an analytical instrument showing the stimulus and measurement of response

Spectroscopy

Further information: Spectroscopy

Spectroscopy measures the interaction of the molecules with electromagnetic radiation. Spectroscopy consists of many different applications such as time-resolved raman spectroscopy, atomic absorption spectroscopy, atomic emission spectroscopy, ultraviolet-visible spectroscopy, X-ray spectroscopy, fluorescence spectroscopy, infrared spectroscopy, Raman spectroscopy, dual polarization interferometry, nuclear magnetic resonance spectroscopy, photoemission spectroscopy, Mössbauer spectroscopy and so on.

Mass spectrometry

Further information: Mass spectrometry

An accelerator mass spectrometer used for radiocarbon dating and other analysis

Mass spectrometry measures mass-to-charge ratio of molecules using electric and magnetic fields. In a mass spectrometer, a small amount of sample is ionized and converted to gaseous ions, where they are separated and analyzed according to their mass-to-charge ratios.

There are several ionization methods: electron ionization, chemical ionization, electrospray ionization, fast atom bombardment, matrix-assisted laser desorption/ionization, and others. Also, mass spectrometry is categorized by approaches of mass analyzers: magnetic-sector, quadrupole mass analyzer, quadrupole ion trap, time-of-flight, Fourier transform ion cyclotron resonance, and so on.

Electrochemical analysis

Further information: Electroanalytical method

Electroanalytical methods measure the potential (volts) and/or current (amps) in an electrochemical cell containing the analyte. These methods can be categorized according to which aspects of the cell are controlled and which are measured. The four main categories are potentiometry (the difference in electrode potentials is measured), coulometry (the transferred charge is measured over time), amperometry (the cell's current is measured over time), and voltammetry (the cell's current is measured while actively altering the cell's potential).

Thermal analysis

Further information: Calorimetry and Thermal analysis

Calorimetry and thermogravimetric analysis measure the interaction of a material and heat.

Separation

Separation of black ink on a thin-layer chromatography plate

Further information: Separation process, Chromatography, and Electrophoresis

Separation processes are used to decrease the complexity of material mixtures. Chromatography, electrophoresis and field flow fractionation are representative of this field.

Chromatographic assays

Further information: Chromatography

Chromatography can be used to determine the presence of substances in a sample, as different components in a mixture have different tendencies to adsorb onto the stationary phase or dissolve in the mobile phase. Thus, different components of the mixture move at different speeds. Different components of a mixture can therefore be identified by their respective R_ƒ values, which is the ratio between the migration distance of the substance and the migration distance of the solvent front during chromatography.

In combination with the instrumental methods, chromatography can be used in the quantitative determination of substances. There are different types of chromatography that differ from the media they use to separate the analyte and the sample. In thin-layer chromatography, the analyte mixture moves up and separates along the coated sheet under the volatile mobile phase. In gas chromatography, the gas phase separates the volatile analytes. A common method of chromatography using liquid as a mobile phase is high-performance liquid chromatography.

Hybrid techniques

Combinations of the above techniques produce a "hybrid" or "hyphenated" technique. Several examples are in popular use today and new hybrid techniques are under development. For example, gas chromatography-mass spectrometry, gas chromatography-infrared spectroscopy, liquid chromatography-mass spectrometry, liquid chromatography-NMR spectroscopy, liquid chromatography-infrared spectroscopy, and capillary electrophoresis-mass spectrometry.

Hyphenated separation techniques refer to a combination of two (or more) techniques to detect and separate chemicals from solutions. Most often the other technique is some form of chromatography. Hyphenated techniques are widely used in chemistry and biochemistry. A slash is sometimes used instead of hyphen, especially if the name of one of the methods contains a hyphen itself.

Microscopy

Further information: Microscopy

Fluorescence microscope image of two mouse cell nuclei in prophase (scale bar is 5 μm)

The visualization of single molecules, single cells, biological tissues, and nanomaterials is an important and attractive approach in analytical science. Also, hybridization with other traditional analytical tools is revolutionizing analytical science. Microscopy can be categorized into three different fields: optical microscopy, electron microscopy, and scanning probe microscopy. Recently, this field is rapidly progressing because of the rapid development of the computer and camera industries.

Lab-on-a-chip

Further information: Microfluidics and Lab-on-a-chip

Devices that integrate (multiple) laboratory functions on a single chip of only millimeters to a few square centimeters in size and that are capable of handling extremely small fluid volumes down to less than picoliters.

Data analysis and chemometrics

The vast amount of data produced by modern analytical instruments has made computational data analysis an integral part of the field. The field of chemometrics uses statistical and mathematical methods to design optimal experimental procedures and to extract meaningful information from chemical data.

Key areas include:

• Multivariate calibration: Used to develop models that correlate instrument responses (e.g., spectra) to analyte concentrations, essential in techniques like near-infrared spectroscopy.
• Pattern recognition: Employed to classify samples based on their analytical profile, with applications in food authenticity and medical diagnostics.
• Machine learning and artificial intelligence: These techniques are increasingly used for predictive modeling, optimizing analytical methods, and automating data interpretation.

Errors

Main article: Approximation error

Error can be defined as numerical difference between observed value and true value. The experimental error can be divided into two types, systematic error and random error. Systematic error results from a flaw in equipment or the design of an experiment while random error results from uncontrolled or uncontrollable variables in the experiment.

In error the true value and observed value in chemical analysis can be related to each other by the equation

${\displaystyle {\epsilon }_{\mathrm{a}}=|x-\overline{x}|}$

where

• ${\displaystyle {\epsilon }_{\mathrm{a}}}$ is the absolute error.
• ${\displaystyle x}$ is the true value.
• ${\displaystyle \overline{x}}$ is the observed value.

An error of a measurement is an inverse measure of accurate measurement (i.e., smaller the error greater the accuracy of the measurement).

Errors can be expressed relatively. Given the relative error (${\displaystyle {\epsilon }_{\mathrm{r}}}$):

${\displaystyle {\epsilon }_{\mathrm{r}}=\frac{{\epsilon }_{\mathrm{a}}}{|x|}=\left|\frac{x-\overline{x}}{x}\right|}$

The percent error can also be calculated:

${\displaystyle {\epsilon }_{\mathrm{r}}\times 100\mathrm{\%}}$

To use these values in a function, it may be useful to calculate the error of the function. If ${\displaystyle f}$ is a function with ${\displaystyle N}$ variables, the propagation of uncertainty must be calculated in order to know the error in ${\displaystyle f}$:

${\displaystyle {\epsilon }_{\mathrm{a}}(f)\approx \sum _{i=1}^N\left|\frac{\mathrm{\partial }f}{\mathrm{\partial }{x}_i}\right|{\epsilon }_{\mathrm{a}}(x_i)=\left|\frac{\mathrm{\partial }f}{\mathrm{\partial }{x}_1}\right|{\epsilon }_{\mathrm{a}}(x_1)+\left|\frac{\mathrm{\partial }f}{\mathrm{\partial }{x}_2}\right|{\epsilon }_{\mathrm{a}}(x_2)+\dots +\left|\frac{\mathrm{\partial }f}{\mathrm{\partial }{x}_N}\right|{\epsilon }_{\mathrm{a}}(x_N)}$

Standards

See also: Analytical quality control

Standard curve

A calibration curve plot showing limit of detection (LOD), limit of quantification (LOQ), dynamic range, and limit of linearity (LOL)

A general method for analysis of concentration involves the creation of a calibration curve. This allows for the determination of the amount of a chemical in a material by comparing the results of an unknown sample to those of a series of known standards. If the concentration of an element or compound in a sample exceeds the detection range of the technique, it can simply be diluted in a pure solvent. If the amount in the sample is below an instrument's range of measurement, the method of addition can be used. In this method, a known quantity of the element or compound under study is added, and the difference between the concentration added and the concentration observed is the amount actually in the sample.

Internal standards

Sometimes an internal standard is added at a known concentration directly to an analytical sample to aid in quantitation. The amount of analyte present is then determined relative to the internal standard as a calibrant. An ideal internal standard is an isotopically enriched analyte which gives rise to the method of isotope dilution.

Standard addition

The method of standard addition is used in instrumental analysis to determine the concentration of a substance (analyte) in an unknown sample by comparison to a set of samples of known concentration, similar to using a calibration curve. Standard addition can be applied to most analytical techniques and is used instead of a calibration curve to solve the matrix effect problem.

Signals and noise

One of the most important components of analytical chemistry is maximizing the desired signal while minimizing the associated noise. The analytical figure of merit is known as the signal-to-noise ratio (S/N or SNR).

Noise can arise from environmental factors as well as from fundamental physical processes.

Thermal noise

Main article: Johnson–Nyquist noise

Thermal noise results from the motion of charge carriers (usually electrons) in an electrical circuit generated by their thermal motion. Thermal noise is white noise, meaning that the power spectral density is constant throughout the frequency spectrum.

The root mean square value of the thermal noise in a resistor is given by

${\displaystyle v_{\mathrm{R}\mathrm{M}\mathrm{S}}=\sqrt{4k_{\mathrm{B}}TR\mathrm{\Delta }f},}$

where k_B is the Boltzmann constant, T is the temperature, R is the resistance, and ${\displaystyle \mathrm{\Delta }f}$ is the bandwidth of the frequency ${\displaystyle f}$.

Shot noise

Main article: Shot noise

Shot noise is a type of electronic noise that occurs when the finite number of particles (such as electrons in an electronic circuit or photons in an optical device) is small enough to give rise to statistical fluctuations in a signal.

Shot noise is a Poisson process, and the charge carriers that make up the current follow a Poisson distribution. The root mean square current fluctuation is given by

${\displaystyle i_{\mathrm{R}\mathrm{M}\mathrm{S}}=\sqrt{2eI\mathrm{\Delta }f}}$

where e is the elementary charge and I is the average current. Shot noise is white noise.

Flicker noise

Main article: Flicker noise

Flicker noise is electronic noise with a 1/ƒ frequency spectrum; as f increases, the noise decreases. Flicker noise arises from a variety of sources, such as impurities in a conductive channel, generation, and recombination noise in a transistor due to base current, and so on. This noise can be avoided by modulation of the signal at a higher frequency, for example, through the use of a lock-in amplifier.

Environmental noise

Noise in a thermogravimetric analysis; lower noise in the middle of the plot results from less human activity (and environmental noise) at night

Environmental noise arises from the surroundings of the analytical instrument. Sources of electromagnetic noise are power lines, radio and television stations, wireless devices, compact fluorescent lamps and electric motors. Many of these noise sources are narrow bandwidth and, therefore, can be avoided. Temperature and vibration isolation may be required for some instruments.

Noise reduction

Noise reduction can be accomplished either in computer hardware or software. Examples of hardware noise reduction are the use of shielded cable, analog filtering, and signal modulation. Examples of software noise reduction are digital filtering, ensemble average, boxcar average, and correlation methods.

Applications

A U.S. Food and Drug Administration scientist uses a portable near-infrared spectroscopy device to inspect lactose for adulteration with melamine

Analytical chemistry has applications across science and industry. It is fundamental to forensic science (e.g., DNA fingerprinting and toxicology), bioanalysis (e.g., measuring drug concentrations in pharmacokinetic studies), clinical analysis (e.g., blood glucose monitoring and COVID-19 PCR testing), environmental monitoring (e.g., testing for pollutants in water and air), and materials science (e.g., quality control of semiconductors and nanomaterials).

Great effort is being put into shrinking the analysis techniques to chip size. Although few examples of such systems compete with traditional analysis techniques, potential advantages include size/portability, speed, and cost. Micro total analysis system (μTAS) or lab-on-a-chip. Microscale chemistry reduces the amount of chemicals used.

Many developments improve the analysis of biological systems. Examples of rapidly expanding fields in this area are genomics, DNA sequencing and related research in genetic fingerprinting and DNA microarray; proteomics, the analysis of protein concentrations and modifications, especially in response to various stressors, at various developmental stages, or in various parts of the body; metabolomics, which deals with metabolites; transcriptomics, including mRNA and associated fields; lipidomics, dealing with lipids and its related fields; peptidomics, dealing with peptides and its related fields; and metallomics, dealing with metal concentrations and especially with their binding to proteins and other molecules.

Analytical chemistry has played a critical role in the understanding of basic science to a variety of practical applications, such as biomedical applications, environmental monitoring, quality control of industrial manufacturing, and forensic science.

The recent developments in computer automation and information technologies have extended analytical chemistry into several new biological fields. For example, automated DNA sequencing machines were the basis for completing human genome projects, leading to the birth of genomics. Protein identification and peptide sequencing by mass spectrometry opened a new field of proteomics. In addition to automating specific processes, there is effort to automate larger sections of lab testing, such as in companies like Emerald Cloud Lab and Transcriptic.

Analytical chemistry has been an indispensable area in the development of nanotechnology. Surface characterization instruments, electron microscopes and scanning probe microscopes enable scientists to visualize atomic structures with chemical characterizations.