Basis (linear algebra)

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Basis_(linear_algebra) 

The same vector can be represented in two different bases (purple and red arrows).

In mathematics, a set B of vectors in a vector space V is called a basis if every element of V may be written in a unique way as a finite linear combination of elements of B. The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to B. The elements of a basis are called basis vectors.

Equivalently, a set B is a basis if its elements are linearly independent and every element of V is a linear combination of elements of B. In other words, a basis is a linearly independent spanning set.

A vector space can have several bases; however all the bases have the same number of elements, called the dimension of the vector space.

This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces.

Definition

A basis B of a vector space V over a field F (such as the real numbers R or the complex numbers C) is a linearly independent subset of V that spans V. This means that a subset B of V is a basis if it satisfies the two following conditions:

linear independence

for every finite subset ${\displaystyle \{\mathbf {v} _{1},\dotsc ,\mathbf {v} _{m}\}}$ of B, if ${\displaystyle c_{1}\mathbf {v} _{1}+\cdots +c_{m}\mathbf {v} _{m}=\mathbf {0} }$ for some ${\displaystyle c_{1},\dotsc ,c_{m}}$ in F, then ${\displaystyle c_{1}=\cdots =c_{m}=0}$;

spanning property

for every vector v in V, one can choose ${\displaystyle a_{1},\dotsc ,a_{n}}$ in F and ${\displaystyle \mathbf {v} _{1},\dotsc ,\mathbf {v} _{n}}$ in B such that ${\displaystyle \mathbf {v} =a_{1}\mathbf {v} _{1}+\cdots +a_{n}\mathbf {v} _{n}}$.

The scalars ${\displaystyle a_{i}}$ are called the coordinates of the vector v with respect to the basis B, and by the first property they are uniquely determined.

A vector space that has a finite basis is called finite-dimensional. In this case, the finite subset can be taken as B itself to check for linear independence in the above definition.

It is often convenient or even necessary to have an ordering on the basis vectors, for example, when discussing orientation, or when one considers the scalar coefficients of a vector with respect to a basis without referring explicitly to the basis elements. In this case, the ordering is necessary for associating each coefficient to the corresponding basis element. This ordering can be done by numbering the basis elements. In order to emphasize that an order has been chosen, one speaks of an ordered basis, which is therefore not simply an unstructured set, but a sequence, an indexed family, or similar; see § Ordered bases and coordinates below.

Examples

This picture illustrates the standard basis in R². The blue and orange vectors are the elements of the basis; the green vector can be given in terms of the basis vectors, and so is linearly dependent upon them.

The set R² of the ordered pairs of real numbers is a vector space under the operations of component-wise addition

$${\displaystyle (a,b)+(c,d)=(a+c,b+d)}$$

and scalar multiplication

$${\displaystyle \lambda (a,b)=(\lambda a,\lambda b),}$$

where ${\displaystyle \lambda }$ is any real number. A simple basis of this vector space consists of the two vectors e₁ = (1, 0) and e₂ = (0, 1). These vectors form a basis (called the standard basis) because any vector v = (a, b) of R² may be uniquely written as

$${\displaystyle \mathbf {v} =a\mathbf {e} _{1}+b\mathbf {e} _{2}.}$$

Any other pair of linearly independent vectors of R², such as (1, 1) and (−1, 2), forms also a basis of R².

More generally, if F is a field, the set ${\displaystyle F^{n}}$ of n-tuples of elements of F is a vector space for similarly defined addition and scalar multiplication. Let

$${\displaystyle \mathbf {e} _{i}=(0,\ldots ,0,1,0,\ldots ,0)}$$

be the n-tuple with all components equal to 0, except the ith, which is 1. Then ${\displaystyle \mathbf {e} _{1},\ldots ,\mathbf {e} _{n}}$ is a basis of ${\displaystyle F^{n},}$ which is called the standard basis of ${\displaystyle F^{n}.}$

A different flavor of example is given by polynomial rings. If F is a field, the collection F[X] of all polynomials in one indeterminate X with coefficients in F is an F-vector space. One basis for this space is the monomial basis B, consisting of all monomials:

$${\displaystyle B=\{1,X,X^{2},\ldots \}.}$$

Any set of polynomials such that there is exactly one polynomial of each degree (such as the Bernstein basis polynomials or Chebyshev polynomials) is also a basis. (Such a set of polynomials is called a polynomial sequence.) But there are also many bases for F[X] that are not of this form.

Properties

Many properties of finite bases result from the Steinitz exchange lemma, which states that, for any vector space V, given a finite spanning set S and a linearly independent set L of n elements of V, one may replace n well-chosen elements of S by the elements of L to get a spanning set containing L, having its other elements in S, and having the same number of elements as S.

Most properties resulting from the Steinitz exchange lemma remain true when there is no finite spanning set, but their proofs in the infinite case generally require the axiom of choice or a weaker form of it, such as the ultrafilter lemma.

If V is a vector space over a field F, then:

• If L is a linearly independent subset of a spanning set S ⊆ V, then there is a basis B such that
  $${\displaystyle L\subseteq B\subseteq S.}$$

  
• V has a basis (this is the preceding property with L being the empty set, and S = V).
• All bases of V have the same cardinality, which is called the dimension of V. This is the dimension theorem.
• A generating set S is a basis of V if and only if it is minimal, that is, no proper subset of S is also a generating set of V.
• A linearly independent set L is a basis if and only if it is maximal, that is, it is not a proper subset of any linearly independent set.

If V is a vector space of dimension n, then:

• A subset of V with n elements is a basis if and only if it is linearly independent.
• A subset of V with n elements is a basis if and only if it is a spanning set of V.

Coordinates

Let V be a vector space of finite dimension n over a field F, and

$${\displaystyle B=\{\mathbf {b} _{1},\ldots ,\mathbf {b} _{n}\}}$$

be a basis of V. By definition of a basis, every v in V may be written, in a unique way, as

$${\displaystyle \mathbf {v} =\lambda _{1}\mathbf {b} _{1}+\cdots +\lambda _{n}\mathbf {b} _{n},}$$

where the coefficients ${\displaystyle \lambda _{1},\ldots ,\lambda _{n}}$ are scalars (that is, elements of F), which are called the coordinates of v over B. However, if one talks of the set of the coefficients, one loses the correspondence between coefficients and basis elements, and several vectors may have the same set of coefficients. For example, ${\displaystyle 3\mathbf {b} _{1}+2\mathbf {b} _{2}}$ and ${\displaystyle 2\mathbf {b} _{1}+3\mathbf {b} _{2}}$ have the same set of coefficients {2, 3}, and are different. It is therefore often convenient to work with an ordered basis; this is typically done by indexing the basis elements by the first natural numbers. Then, the coordinates of a vector form a sequence similarly indexed, and a vector is completely characterized by the sequence of coordinates. An ordered basis is also called a frame, a word commonly used, in various contexts, for referring to a sequence of data allowing defining coordinates.

Let, as usual, ${\displaystyle F^{n}}$ be the set of the n-tuples of elements of F. This set is an F-vector space, with addition and scalar multiplication defined component-wise. The map

$${\displaystyle \varphi :(\lambda _{1},\ldots ,\lambda _{n})\mapsto \lambda _{1}\mathbf {b} _{1}+\cdots +\lambda _{n}\mathbf {b} _{n}}$$

is a linear isomorphism from the vector space ${\displaystyle F^{n}}$ onto V. In other words, ${\displaystyle F^{n}}$ is the coordinate space of V, and the n-tuple ${\displaystyle \varphi ^{-1}(\mathbf {v} )}$ is the coordinate vector of v.

The inverse image by ${\displaystyle \varphi }$ of ${\displaystyle \mathbf {b} _{i}}$ is the n-tuple ${\displaystyle \mathbf {e} _{i}}$ all of whose components are 0, except the ith that is 1. The ${\displaystyle \mathbf {e} _{i}}$ form an ordered basis of ${\displaystyle F^{n}}$, which is called its standard basis or canonical basis. The ordered basis B is the image by ${\displaystyle \varphi }$ of the canonical basis of ${\displaystyle F^{n}}$.

It follows from what precedes that every ordered basis is the image by a linear isomorphism of the canonical basis of ${\displaystyle F^{n}}$, and that every linear isomorphism from ${\displaystyle F^{n}}$ onto V may be defined as the isomorphism that maps the canonical basis of ${\displaystyle F^{n}}$ onto a given ordered basis of V. In other words it is equivalent to define an ordered basis of V, or a linear isomorphism from ${\displaystyle F^{n}}$ onto V.

Change of basis

Main article: Change of basis

Let V be a vector space of dimension n over a field F. Given two (ordered) bases ${\displaystyle B_{\text{old}}=(\mathbf {v} _{1},\ldots ,\mathbf {v} _{n})}$ and ${\displaystyle B_{\text{new}}=(\mathbf {w} _{1},\ldots ,\mathbf {w} _{n})}$ of V, it is often useful to express the coordinates of a vector x with respect to ${\displaystyle B_{\mathrm {old} }}$ in terms of the coordinates with respect to ${\displaystyle B_{\mathrm {new} }.}$ This can be done by the change-of-basis formula, that is described below. The subscripts "old" and "new" have been chosen because it is customary to refer to ${\displaystyle B_{\mathrm {old} }}$ and ${\displaystyle B_{\mathrm {new} }}$ as the old basis and the new basis, respectively. It is useful to describe the old coordinates in terms of the new ones, because, in general, one has expressions involving the old coordinates, and if one wants to obtain equivalent expressions in terms of the new coordinates; this is obtained by replacing the old coordinates by their expressions in terms of the new coordinates.

Typically, the new basis vectors are given by their coordinates over the old basis, that is,

$${\displaystyle \mathbf {w} _{j}=\sum _{i=1}^{n}a_{i,j}\mathbf {v} _{i}.}$$

If ${\displaystyle (x_{1},\ldots ,x_{n})}$ and ${\displaystyle (y_{1},\ldots ,y_{n})}$ are the coordinates of a vector x over the old and the new basis respectively, the change-of-basis formula is

$${\displaystyle x_{i}=\sum _{j=1}^{n}a_{i,j}y_{j},}$$

for i = 1, ..., n.

This formula may be concisely written in matrix notation. Let A be the matrix of the ${\displaystyle a_{i,j}}$, and

$${\displaystyle X={\begin{bmatrix}x_{1}\\\vdots \\x_{n}\end{bmatrix}}\quad {\text{and}}\quad Y={\begin{bmatrix}y_{1}\\\vdots \\y_{n}\end{bmatrix}}}$$

be the column vectors of the coordinates of v in the old and the new basis respectively, then the formula for changing coordinates is

$${\displaystyle X=AY.}$$

The formula can be proven by considering the decomposition of the vector x on the two bases: one has

$${\displaystyle \mathbf {x} =\sum _{i=1}^{n}x_{i}\mathbf {v} _{i},}$$

and

$${\displaystyle \mathbf {x} =\sum _{j=1}^{n}y_{j}\mathbf {w} _{j}=\sum _{j=1}^{n}y_{j}\sum _{i=1}^{n}a_{i,j}\mathbf {v} _{i}=\sum _{i=1}^{n}\left(\sum _{j=1}^{n}a_{i,j}y_{j}\right)\mathbf {v} _{i}.}$$

The change-of-basis formula results then from the uniqueness of the decomposition of a vector over a basis, here ${\displaystyle B_{\text{old}}}$; that is

$${\displaystyle x_{i}=\sum _{j=1}^{n}a_{i,j}y_{j},}$$

for i = 1, ..., n.

Related notions

Free module

Main articles: Free module and Free abelian group

If one replaces the field occurring in the definition of a vector space by a ring, one gets the definition of a module. For modules, linear independence and spanning sets are defined exactly as for vector spaces, although "generating set" is more commonly used than that of "spanning set".

Like for vector spaces, a basis of a module is a linearly independent subset that is also a generating set. A major difference with the theory of vector spaces is that not every module has a basis. A module that has a basis is called a free module. Free modules play a fundamental role in module theory, as they may be used for describing the structure of non-free modules through free resolutions.

A module over the integers is exactly the same thing as an abelian group. Thus a free module over the integers is also a free abelian group. Free abelian groups have specific properties that are not shared by modules over other rings. Specifically, every subgroup of a free abelian group is a free abelian group, and, if G is a subgroup of a finitely generated free abelian group H (that is an abelian group that has a finite basis), there is a basis ${\displaystyle \mathbf {e} _{1},\ldots ,\mathbf {e} _{n}}$ of H and an integer 0 ≤ k ≤ n such that ${\displaystyle a_{1}\mathbf {e} _{1},\ldots ,a_{k}\mathbf {e} _{k}}$ is a basis of G, for some nonzero integers ${\displaystyle a_{1},\ldots ,a_{k}}$. For details, see Free abelian group § Subgroups.

Analysis

In the context of infinite-dimensional vector spaces over the real or complex numbers, the term Hamel basis (named after Georg Hamel) or algebraic basis can be used to refer to a basis as defined in this article. This is to make a distinction with other notions of "basis" that exist when infinite-dimensional vector spaces are endowed with extra structure. The most important alternatives are orthogonal bases on Hilbert spaces, Schauder bases, and Markushevich bases on normed linear spaces. In the case of the real numbers R viewed as a vector space over the field Q of rational numbers, Hamel bases are uncountable, and have specifically the cardinality of the continuum, which is the cardinal number ${\displaystyle 2^{\aleph _{0}}}$, where ${\displaystyle \aleph _{0}}$ is the smallest infinite cardinal, the cardinal of the integers.

The common feature of the other notions is that they permit the taking of infinite linear combinations of the basis vectors in order to generate the space. This, of course, requires that infinite sums are meaningfully defined on these spaces, as is the case for topological vector spaces – a large class of vector spaces including e.g. Hilbert spaces, Banach spaces, or Fréchet spaces.

The preference of other types of bases for infinite-dimensional spaces is justified by the fact that the Hamel basis becomes "too big" in Banach spaces: If X is an infinite-dimensional normed vector space which is complete (i.e. X is a Banach space), then any Hamel basis of X is necessarily uncountable. This is a consequence of the Baire category theorem. The completeness as well as infinite dimension are crucial assumptions in the previous claim. Indeed, finite-dimensional spaces have by definition finite bases and there are infinite-dimensional (non-complete) normed spaces which have countable Hamel bases. Consider ${\displaystyle c_{00}}$, the space of the sequences ${\displaystyle x=(x_{n})}$ of real numbers which have only finitely many non-zero elements, with the norm ${\textstyle \|x\|=\sup _{n}|x_{n}|}$. Its standard basis, consisting of the sequences having only one non-zero element, which is equal to 1, is a countable Hamel basis.

Example

In the study of Fourier series, one learns that the functions {1} ∪ { sin(nx), cos(nx) : n = 1, 2, 3, ... } are an "orthogonal basis" of the (real or complex) vector space of all (real or complex valued) functions on the interval [0, 2π] that are square-integrable on this interval, i.e., functions f satisfying

$${\displaystyle \int _{0}^{2\pi }\left|f(x)\right|^{2}\,dx<\infty .}$$

The functions {1} ∪ { sin(nx), cos(nx) : n = 1, 2, 3, ... } are linearly independent, and every function f that is square-integrable on [0, 2π] is an "infinite linear combination" of them, in the sense that

$${\displaystyle \lim _{n\to \infty }\int _{0}^{2\pi }\left|a_{0}+\sum _{k=1}^{n}\left(a_{k}\cos \left(kx\right)+b_{k}\sin \left(kx\right)\right)-f(x)\right|^{2}dx=0}$$

for suitable (real or complex) coefficients a_k, b_k. But many square-integrable functions cannot be represented as finite linear combinations of these basis functions, which therefore do not comprise a Hamel basis. Every Hamel basis of this space is much bigger than this merely countably infinite set of functions. Hamel bases of spaces of this kind are typically not useful, whereas orthonormal bases of these spaces are essential in Fourier analysis.

Geometry

The geometric notions of an affine space, projective space, convex set, and cone have related notions of basis. An affine basis for an n-dimensional affine space is ${\displaystyle n+1}$ points in general linear position. A projective basis is ${\displaystyle n+2}$ points in general position, in a projective space of dimension n. A convex basis of a polytope is the set of the vertices of its convex hull. A cone basis consists of one point by edge of a polygonal cone. See also a Hilbert basis (linear programming).

Random basis

For a probability distribution in Rⁿ with a probability density function, such as the equidistribution in an n-dimensional ball with respect to Lebesgue measure, it can be shown that n randomly and independently chosen vectors will form a basis with probability one, which is due to the fact that n linearly dependent vectors x₁, ..., x_n in Rⁿ should satisfy the equation det[x₁ ⋯ x_n] = 0 (zero determinant of the matrix with columns x_i), and the set of zeros of a non-trivial polynomial has zero measure. This observation has led to techniques for approximating random bases.

Empirical distribution of lengths N of pairwise almost orthogonal chains of vectors that are independently randomly sampled from the n-dimensional cube [−1, 1]ⁿ as a function of dimension, n. Boxplots show the second and third quartiles of this data for each n, red bars correspond to the medians, and blue stars indicate means. Red curve shows theoretical bound given by Eq. (1) and green curve shows a refined estimate.

It is difficult to check numerically the linear dependence or exact orthogonality. Therefore, the notion of ε-orthogonality is used. For spaces with inner product, x is ε-orthogonal to y if ${\displaystyle \left|\left\langle x,y\right\rangle \right|/\left(\left\|x\right\|\left\|y\right\|\right)<\varepsilon }$ (that is, cosine of the angle between x and y is less than ε).

In high dimensions, two independent random vectors are with high probability almost orthogonal, and the number of independent random vectors, which all are with given high probability pairwise almost orthogonal, grows exponentially with dimension. More precisely, consider equidistribution in n-dimensional ball. Choose N independent random vectors from a ball (they are independent and identically distributed). Let θ be a small positive number. Then for

$${\displaystyle N\leq e^{\frac {\varepsilon ^{2}n}{4}}[-\ln(1-\theta )]^{\frac {1}{2}}}$$

(Eq. 1)

N random vectors are all pairwise ε-orthogonal with probability 1 − θ. This N growth exponentially with dimension n and ${\displaystyle N\gg n}$ for sufficiently big n. This property of random bases is a manifestation of the so-called measure concentration phenomenon.

The figure (right) illustrates distribution of lengths N of pairwise almost orthogonal chains of vectors that are independently randomly sampled from the n-dimensional cube [−1, 1]ⁿ as a function of dimension, n. A point is first randomly selected in the cube. The second point is randomly chosen in the same cube. If the angle between the vectors was within π/2 ± 0.037π/2 then the vector was retained. At the next step a new vector is generated in the same hypercube, and its angles with the previously generated vectors are evaluated. If these angles are within π/2 ± 0.037π/2 then the vector is retained. The process is repeated until the chain of almost orthogonality breaks, and the number of such pairwise almost orthogonal vectors (length of the chain) is recorded. For each n, 20 pairwise almost orthogonal chains were constructed numerically for each dimension. Distribution of the length of these chains is presented.

Proof that every vector space has a basis

Let V be any vector space over some field F. Let X be the set of all linearly independent subsets of V.

The set X is nonempty since the empty set is an independent subset of V, and it is partially ordered by inclusion, which is denoted, as usual, by ⊆.

Let Y be a subset of X that is totally ordered by ⊆, and let L_Y be the union of all the elements of Y (which are themselves certain subsets of V).

Since (Y, ⊆) is totally ordered, every finite subset of L_Y is a subset of an element of Y, which is a linearly independent subset of V, and hence L_Y is linearly independent. Thus L_Y is an element of X. Therefore, L_Y is an upper bound for Y in (X, ⊆): it is an element of X, that contains every element of Y.

As X is nonempty, and every totally ordered subset of (X, ⊆) has an upper bound in X, Zorn's lemma asserts that X has a maximal element. In other words, there exists some element L_max of X satisfying the condition that whenever L_max ⊆ L for some element L of X, then L = L_max.

It remains to prove that L_max is a basis of V. Since L_max belongs to X, we already know that L_max is a linearly independent subset of V.

If there were some vector w of V that is not in the span of L_max, then w would not be an element of L_max either. Let L_w = L_max ∪ {w}. This set is an element of X, that is, it is a linearly independent subset of V (because w is not in the span of L_max, and L_max is independent). As L_max ⊆ L_w, and L_max ≠ L_w (because L_w contains the vector w that is not contained in L_max), this contradicts the maximality of L_max. Thus this shows that L_max spans V.

Hence L_max is linearly independent and spans V. It is thus a basis of V, and this proves that every vector space has a basis.

This proof relies on Zorn's lemma, which is equivalent to the axiom of choice. Conversely, it has been proved that if every vector space has a basis, then the axiom of choice is true. Thus the two assertions are equivalent.

Spectroscopy

From Wikipedia, the free encyclopedia

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

 

An example of spectroscopy: a prism analyses white light by dispersing it into its component colors.

Spectroscopy is the general field of study that measures and interprets the electromagnetic spectra that result from the interaction between electromagnetic radiation and matter as a function of the wavelength or frequency of the radiation. Matter waves and acoustic waves can also be considered forms of radiative energy, and recently gravitational waves have been associated with a spectral signature in the context of the Laser Interferometer Gravitational-Wave Observatory (LIGO)

In simpler terms, spectroscopy is the precise study of color as generalized from visible light to all bands of the electromagnetic spectrum. Historically, spectroscopy originated as the study of the wavelength dependence of the absorption by gas phase matter of visible light dispersed by a prism.

Spectroscopy, primarily in the electromagnetic spectrum, is a fundamental exploratory tool in the fields of astronomy, chemistry, materials science, and physics, allowing the composition, physical structure and electronic structure of matter to be investigated at the atomic, molecular and macro scale, and over astronomical distances. Important applications include biomedical spectroscopy in the areas of tissue analysis and medical imaging.

Introduction

Spectroscopy is a branch of science concerned with the spectra of electromagnetic radiation as a function of its wavelength or frequency measured by spectrographic equipment, and other techniques, in order to obtain information concerning the structure and properties of matter. Spectral measurement devices are referred to as spectrometers, spectrophotometers, spectrographs or spectral analyzers. Most spectroscopic analysis in the laboratory starts with a sample to be analyzed, then a light source is chosen from any desired range of the light spectrum, then the light goes through the sample to a dispersion array (diffraction grating instrument) and is captured by a photodiode. For astronomical purposes, the telescope must be equipped with the light dispersion device. There are various versions of this basic setup that may be employed.

Spectroscopy as a science began with Isaac Newton splitting light with a prism and was called Optics. Therefore, it was originally the study of visible light which we call color that later under the studies of James Clerk Maxwell came to include the entire electromagnetic spectrum. Although color is involved in spectroscopy, it is not equated with the color of elements or objects which involve the absorption and reflection of certain electromagnetic waves to give objects a sense of color to our eyes. Rather spectroscopy involves the splitting of light by a prism, diffraction grating, or similar instrument, to give off a particular discrete line pattern called a “spectrum” unique to each different type of element. Most elements are first put into a gaseous phase to allow the spectra to be examined although today other methods can be used on different phases. Each element that is diffracted by a prism-like instrument displays either an absorption spectrum or an emission spectrum depending upon whether the element is being cooled or heated.

Until recently all spectroscopy involved the study of line spectra and most spectroscopy still does. Vibrational spectroscopy is the branch of spectroscopy that studies the spectra. However, the latest developments in spectroscopy can sometimes dispense with the dispersion technique. In biochemical spectroscopy, information can be gathered about biological tissue by absorption and light scattering techniques. Light scattering spectroscopy is a type of reflectance spectroscopy that determines tissue structures by examining elastic scattering. In such a case, it is the tissue that acts as a diffraction or dispersion mechanism.

Spectroscopic studies were central to the development of quantum mechanics, because the first useful atomic models described the spectra of Hydrogen which models include the Bohr model, the Schrödinger equation, and Matrix mechanics which all can produce the spectral lines of Hydrogen, therefore, providing the basis for discrete quantum jumps to match the discrete hydrogen spectrum. Also, Max Planck's explanation of blackbody radiation involved spectroscopy because he was comparing the wavelength of light using a photometer to the temperature of a Black Body. Spectroscopy is used in physical and analytical chemistry because atoms and molecules have unique spectra. As a result, these spectra can be used to detect, identify and quantify information about the atoms and molecules. Spectroscopy is also used in astronomy and remote sensing on Earth. Most research telescopes have spectrographs. The measured spectra are used to determine the chemical composition and physical properties of astronomical objects (such as their temperature, density of elements in a star, velocity, black holes and more). An important use for spectroscopy is in biochemistry. Molecular samples may be analyzed for species identification and energy content.

Theory

The central theory of spectroscopy is that light is made of different wavelengths and that each wavelength corresponds to a different frequency. The importance of spectroscopy is centered around the fact that every different element in the periodic table has a unique light spectrum described by the frequencies of light it emits or absorbs consistently appearing in the same part of the electromagnetic spectrum when that light is diffracted. This opened up an entire field of study with anything that contains atoms which is all matter. Spectroscopy is the key to understanding the atomic properties of all matter. As such spectroscopy opened up many new sub-fields of science yet undiscovered. The idea that each atomic element has its unique spectral signature enabled spectroscopy to be used in a broad number of fields each with a specific goal achieved by different spectroscopic procedures. These unique spectral lines for each element are so important in so many branches of science that the government carries a public Atomic Spectra Database that is continually updated with more precise measurements on its NIST website.

The broadening of the field of spectroscopy is due to the fact that any part of the electromagnetic spectrum may be used to analyze a sample from the infrared to the ultraviolet telling scientists different properties about the very same sample. For instance in chemical analysis, the most common types of spectroscopy include atomic spectroscopy, infrared spectroscopy, ultraviolet and visible spectroscopy, Raman spectroscopy and nuclear magnetic resonance. In nuclear magnetic resonance, the theory behind it is that frequency is analogous to resonance and its corresponding resonant frequency. Resonances by the frequency were first characterized in mechanical systems such as pendulums which have a frequency of motion noted famously by Galileo.

Classification of methods

A huge diffraction grating at the heart of the ultra-precise ESPRESSO spectrograph.

Spectroscopy is a sufficiently broad field that many sub-disciplines exist, each with numerous implementations of specific spectroscopic techniques. The various implementations and techniques can be classified in several ways.

Type of radiative energy

The types of spectroscopy are distinguished by the type of radiative energy involved in the interaction. In many applications, the spectrum is determined by measuring changes in the intensity or frequency of this energy. The types of radiative energy studied include:

• Electromagnetic radiation was the first source of energy used for spectroscopic studies. Techniques that employ electromagnetic radiation are typically classified by the wavelength region of the spectrum and include microwave, terahertz, infrared, near-infrared, ultraviolet-visible, x-ray, and gamma spectroscopy.
• Particles, because of their de Broglie waves, can also be a source of radiative energy. Both electron and neutron spectroscopy are commonly used. For a particle, its kinetic energy determines its wavelength.
• Acoustic spectroscopy involves radiated pressure waves.
• Dynamic mechanical analysis can be employed to impart radiating energy, similar to acoustic waves, to solid materials.

Nature of the interaction

The types of spectroscopy also can be distinguished by the nature of the interaction between the energy and the material. These interactions include:

• Absorption spectroscopy: Absorption occurs when energy from the radiative source is absorbed by the material. Absorption is often determined by measuring the fraction of energy transmitted through the material, with absorption decreasing the transmitted portion.
• Emission spectroscopy: Emission indicates that radiative energy is released by the material. A material's blackbody spectrum is a spontaneous emission spectrum determined by its temperature. This feature can be measured in the infrared by instruments such as the atmospheric emitted radiance interferometer. Emission can also be induced by other sources of energy such as flames, sparks, electric arcs or electromagnetic radiation in the case of fluorescence.
• Elastic scattering and reflection spectroscopy determine how incident radiation is reflected or scattered by a material. Crystallography employs the scattering of high energy radiation, such as x-rays and electrons, to examine the arrangement of atoms in proteins and solid crystals.
• Impedance spectroscopy: Impedance is the ability of a medium to impede or slow the transmittance of energy. For optical applications, this is characterized by the index of refraction.
• Inelastic scattering phenomena involve an exchange of energy between the radiation and the matter that shifts the wavelength of the scattered radiation. These include Raman and Compton scattering.
• Coherent or resonance spectroscopy are techniques where the radiative energy couples two quantum states of the material in a coherent interaction that is sustained by the radiating field. The coherence can be disrupted by other interactions, such as particle collisions and energy transfer, and so often require high intensity radiation to be sustained. Nuclear magnetic resonance (NMR) spectroscopy is a widely used resonance method, and ultrafast laser spectroscopy is also possible in the infrared and visible spectral regions.
• Nuclear spectroscopy are methods that use the properties of specific nuclei to probe the local structure in matter, mainly condensed matter, molecules in liquids or frozen liquids and bio-molecules.

Type of material

Spectroscopic studies are designed so that the radiant energy interacts with specific types of matter.

Atoms

Atomic spectroscopy was the first application of spectroscopy developed. Atomic absorption spectroscopy and atomic emission spectroscopy involve visible and ultraviolet light. These absorptions and emissions, often referred to as atomic spectral lines, are due to electronic transitions of outer shell electrons as they rise and fall from one electron orbit to another. Atoms also have distinct x-ray spectra that are attributable to the excitation of inner shell electrons to excited states.

Atoms of different elements have distinct spectra and therefore atomic spectroscopy allows for the identification and quantitation of a sample's elemental composition. After inventing the spectroscope, Robert Bunsen and Gustav Kirchhoff discovered new elements by observing their emission spectra. Atomic absorption lines are observed in the solar spectrum and referred to as Fraunhofer lines after their discoverer. A comprehensive explanation of the hydrogen spectrum was an early success of quantum mechanics and explained the Lamb shift observed in the hydrogen spectrum, which further led to the development of quantum electrodynamics.

Modern implementations of atomic spectroscopy for studying visible and ultraviolet transitions include flame emission spectroscopy, inductively coupled plasma atomic emission spectroscopy, glow discharge spectroscopy, microwave induced plasma spectroscopy, and spark or arc emission spectroscopy. Techniques for studying x-ray spectra include X-ray spectroscopy and X-ray fluorescence.

Molecules

The combination of atoms into molecules leads to the creation of unique types of energetic states and therefore unique spectra of the transitions between these states. Molecular spectra can be obtained due to electron spin states (electron paramagnetic resonance), molecular rotations, molecular vibration, and electronic states. Rotations are collective motions of the atomic nuclei and typically lead to spectra in the microwave and millimeter-wave spectral regions. Rotational spectroscopy and microwave spectroscopy are synonymous. Vibrations are relative motions of the atomic nuclei and are studied by both infrared and Raman spectroscopy. Electronic excitations are studied using visible and ultraviolet spectroscopy as well as fluorescence spectroscopy.

Studies in molecular spectroscopy led to the development of the first maser and contributed to the subsequent development of the laser.

Crystals and extended materials

The combination of atoms or molecules into crystals or other extended forms leads to the creation of additional energetic states. These states are numerous and therefore have a high density of states. This high density often makes the spectra weaker and less distinct, i.e., broader. For instance, blackbody radiation is due to the thermal motions of atoms and molecules within a material. Acoustic and mechanical responses are due to collective motions as well. Pure crystals, though, can have distinct spectral transitions, and the crystal arrangement also has an effect on the observed molecular spectra. The regular lattice structure of crystals also scatters x-rays, electrons or neutrons allowing for crystallographic studies.

Nuclei

Nuclei also have distinct energy states that are widely separated and lead to gamma ray spectra. Distinct nuclear spin states can have their energy separated by a magnetic field, and this allows for nuclear magnetic resonance spectroscopy.

Other types

Other types of spectroscopy are distinguished by specific applications or implementations:

• Acoustic resonance spectroscopy is based on sound waves primarily in the audible and ultrasonic regions.
• Auger electron spectroscopy is a method used to study surfaces of materials on a micro-scale. It is often used in connection with electron microscopy.
• Cavity ring-down spectroscopy
• Circular Dichroism spectroscopy
• Coherent anti-Stokes Raman spectroscopy is a recent technique that has high sensitivity and powerful applications for in vivo spectroscopy and imaging.
• Cold vapour atomic fluorescence spectroscopy
• Correlation spectroscopy encompasses several types of two-dimensional NMR spectroscopy.
• Deep-level transient spectroscopy measures concentration and analyzes parameters of electrically active defects in semiconducting materials.
• Dielectric spectroscopy
• Dual-polarization interferometry measures the real and imaginary components of the complex refractive index.
• Electron energy loss spectroscopy in transmission electron microscopy.
• Electron phenomenological spectroscopy measures the physicochemical properties and characteristics of the electronic structure of multicomponent and complex molecular systems.
• Electron paramagnetic resonance spectroscopy
• Force spectroscopy
• Fourier-transform spectroscopy is an efficient method for processing spectra data obtained using interferometers. Fourier-transform infrared spectroscopy is a common implementation of infrared spectroscopy. NMR also employs Fourier transforms.
• Gamma spectroscopy
• Hadron spectroscopy studies the energy/mass spectrum of hadrons according to spin, parity, and other particle properties. Baryon spectroscopy and meson spectroscopy are types of hadron spectroscopy.
• Hyperspectral imaging is a method to create a complete picture of the environment or various objects, each pixel containing a full visible, visible near infrared, near infrared, or infrared spectrum.
• Inelastic electron tunneling spectroscopy uses the changes in current due to inelastic electron-vibration interaction at specific energies that can also measure optically forbidden transitions.
• Inelastic neutron scattering is similar to Raman spectroscopy, but uses neutrons instead of photons.
• Laser-induced breakdown spectroscopy, also called laser-induced plasma spectrometry
• Laser spectroscopy uses tunable lasers and other types of coherent emission sources, such as optical parametric oscillators, for selective excitation of atomic or molecular species.
• Mass spectroscopy is a historical term used to refer to mass spectrometry. The current recommendation is to use the latter term. The term "mass spectroscopy" originated in the use of phosphor screens to detect ions.
• Mössbauer spectroscopy probes the properties of specific isotopic nuclei in different atomic environments by analyzing the resonant absorption of gamma rays. See also Mössbauer effect.
• Multivariate optical computing is an all optical compressed sensing technique, generally used in harsh environments, that directly calculates chemical information from a spectrum as analogue output.
• Neutron spin echo spectroscopy measures internal dynamics in proteins and other soft matter systems.
• Perturbed angular correlation (PAC) uses radioactive nuclei as probe to study electric and magnetic fields (hyperfine interactions) in crystals (condensed matter) and bio-molecules.
• Photoacoustic spectroscopy measures the sound waves produced upon the absorption of radiation.
• Photoemission spectroscopy
• Photothermal spectroscopy measures heat evolved upon absorption of radiation.
• Pump-probe spectroscopy can use ultrafast laser pulses to measure reaction intermediates in the femtosecond timescale.
• Raman optical activity spectroscopy exploits Raman scattering and optical activity effects to reveal detailed information on chiral centers in molecules.
• Raman spectroscopy
• Saturated spectroscopy
• Scanning tunneling spectroscopy
• Spectrophotometry
• Spin noise spectroscopy traces spontaneous fluctuations of electronic and nuclear spins.
• Time-resolved spectroscopy measures the decay rates of excited states using various spectroscopic methods.
• Time-stretch spectroscopy
• Thermal infrared spectroscopy measures thermal radiation emitted from materials and surfaces and is used to determine the type of bonds present in a sample as well as their lattice environment. The techniques are widely used by organic chemists, mineralogists, and planetary scientists.
• Transient grating spectroscopy measures quasiparticle propagation. It can track changes in metallic materials as they are irradiated.
• Ultraviolet photoelectron spectroscopy
• Ultraviolet–visible spectroscopy
• Vibrational circular dichroism spectroscopy
• Video spectroscopy
• X-ray photoelectron spectroscopy

Applications

UVES is a high-resolution spectrograph on the Very Large Telescope.

There are several applications of spectroscopy in the fields of medicine, physics, chemistry, and astronomy. Taking advantage of the properties of absorbance and with astronomy emission, spectroscopy can be used to identify certain states of nature. The uses of spectroscopy in so many different fields and for so many different applications has caused specialty scientific subfields. Such examples include:

• one of the first uses was for: Determining the atomic structure of a sample
• Next huge application was in astronomy: Studying spectral emission lines of the sun and distant galaxies
• Space exploration
• Cure monitoring of composites using optical fibers.
• Estimate weathered wood exposure times using near infrared spectroscopy.
• Measurement of different compounds in food samples by absorption spectroscopy both in visible and infrared spectrum.
• Measurement of toxic compounds in blood samples
• Non-destructive elemental analysis by X-ray fluorescence.
• Electronic structure research with various spectroscopes.
• Redshift to determine the speed and velocity of a distant object
• Determining the metabolic structure of a muscle
• Monitoring dissolved oxygen content in freshwater and marine ecosystems
• Altering the structure of drugs to improve effectiveness
• Characterization of proteins
• Respiratory gas analysis in hospitals
• Finding the physical properties of a distant star or nearby exoplanet using the Relativistic Doppler effect.
• In-ovo sexing: spectroscopy allows to determine the sex of the egg while it is hatching. Developed by French and German companies, both countries decided to ban chick culling, mostly done through a macerator, in 2022.

History

Main article: History of spectroscopy

The history of spectroscopy began with Isaac Newton's optics experiments (1666–1672). According to Andrew Fraknoi and David Morrison, "In 1672, in the first paper that he submitted to the Royal Society, Isaac Newton described an experiment in which he permitted sunlight to pass through a small hole and then through a prism. Newton found that sunlight, which looks white to us, is actually made up of a mixture of all the colors of the rainbow." Newton applied the word "spectrum" to describe the rainbow of colors that combine to form white light and that are revealed when the white light is passed through a prism.

Fraknoi and Morrison state that "In 1802, William Hyde Wollaston built an improved spectrometer that included a lens to focus the Sun's spectrum on a screen. Upon use, Wollaston realized that the colors were not spread uniformly, but instead had missing patches of colors, which appeared as dark bands in the spectrum." During the early 1800s, Joseph von Fraunhofer made experimental advances with dispersive spectrometers that enabled spectroscopy to become a more precise and quantitative scientific technique. Since then, spectroscopy has played and continues to play a significant role in chemistry, physics, and astronomy. Per Fraknoi and Morrison, "Later, in 1815, German physicist Joseph Fraunhofer also examined the solar spectrum, and found about 600 such dark lines (missing colors), are now known as Fraunhofer lines, or Absorption lines."

In quantum mechanical systems, the analogous resonance is a coupling of two quantum mechanical stationary states of one system, such as an atom, via an oscillatory source of energy such as a photon. The coupling of the two states is strongest when the energy of the source matches the energy difference between the two states. The energy E of a photon is related to its frequency ν by E = hν where h is Planck's constant, and so a spectrum of the system response vs. photon frequency will peak at the resonant frequency or energy. Particles such as electrons and neutrons have a comparable relationship, the de Broglie relations, between their kinetic energy and their wavelength and frequency and therefore can also excite resonant interactions.

Spectra of atoms and molecules often consist of a series of spectral lines, each one representing a resonance between two different quantum states. The explanation of these series, and the spectral patterns associated with them, were one of the experimental enigmas that drove the development and acceptance of quantum mechanics. The hydrogen spectral series in particular was first successfully explained by the Rutherford–Bohr quantum model of the hydrogen atom. In some cases spectral lines are well separated and distinguishable, but spectral lines can also overlap and appear to be a single transition if the density of energy states is high enough. Named series of lines include the principal, sharp, diffuse and fundamental series.