{{Short description|Concept relating to waves and signals}} {{other uses|Spectrum}} [[File:EM Spectrum Properties edit.svg|thumb|right|upright=1.2|Diagram illustrating the electromagnetic spectrum]] In the [[physical sciences]], '''spectrum''' describes any continuous range of either [[frequency]] or [[wavelength]] values. The term initially referred to the range of observed colors as white light is [[dispersion (optics)|disperse]]d through a [[prism (optics)|prism]] — introduced to [[optics]] by [[Isaac Newton]] in the 17th century.<ref>{{Open access}} OpenStax Astronomy, "Spectroscopy in Astronomy". OpenStax CNX. September 29, 2016 {{cite web |url=http://cnx.org/contents/1f92a120-370a-4547-b14e-a3df3ce6f083@3 |title=OpenStax CNX |access-date=2017-02-17 |url-status=live |archive-url=https://web.archive.org/web/20170217225151/http://cnx.org/contents/1f92a120-370a-4547-b14e-a3df3ce6f083@3 |archive-date=February 17, 2017 |df=mdy-all }}</ref><ref>{{cite journal|last1=Newton|first1=Isaac|title=A letter of Mr. Isaac Newton … containing his new theory about light and colours …|journal=Philosophical Transactions of the Royal Society of London|date=1671|volume=6|issue=80|pages=3075–3087|url=http://rstl.royalsocietypublishing.org/content/6/80/3075.full.pdf+html|bibcode=1671RSPT....6.3075N|doi=10.1098/rstl.1671.0072|doi-access=free|url-access=subscription}} The word "spectrum" to describe a band of colors that has been produced, by [[refraction]] or [[diffraction]], from a beam of light first appears on p. 3076.</ref>
The concept was later expanded to other [[wave]]s, such as [[sound wave]]s and [[sea wave]]s that also present a variety of frequencies and wavelengths (e.g., [[noise spectrum]], [[sea wave spectrum]]). Starting from [[Fourier analysis]], the concept of spectrum expanded to [[Signal processing|signal theory]], where the signal can be graphed as a function of frequency and [[Information theory|information]] can be placed in selected ranges of frequency. Presently, any quantity directly dependent on, and measurable along the [[Range of a function|range of]], a continuous [[independent variable]] can be graphed along its ''range'' or ''spectrum''. Examples are the range of ''electron energy'' in [[electron spectroscopy]] or the range of ''mass-to-charge ratio'' in [[mass spectrometry]].
== Etymology == {{excerpt|Spectrum#Etymology}}
== Electromagnetic spectrum == {{Main|Electromagnetic spectrum}}
[[File:Fluorescent lighting spectrum peaks labelled.svg|thumb|300px|Electromagnetic emission spectrum of a [[fluorescent lamp]]]]
Electromagnetic spectrum refers to the full range of all frequencies of [[electromagnetic radiation]]<ref>{{cite web |url=http://imagine.gsfc.nasa.gov/resources/dictionary.html#E |title=Electromagnetic spectrum |work=Imagine the Universe! Dictionary |publisher=NASA |access-date=June 3, 2015 |url-status=live |archive-url=https://web.archive.org/web/20150524072134/http://imagine.gsfc.nasa.gov/resources/dictionary.html#E |archive-date=May 24, 2015 |df=mdy-all }}</ref> and also to the characteristic distribution of electromagnetic radiation emitted or absorbed by that particular object. Devices used to measure an electromagnetic spectrum are called [[spectrograph]] or [[spectrometer]]. The [[visible spectrum]] is the part of the electromagnetic spectrum that can be seen by the [[human eye]]. The wavelength of visible light ranges from [[1 E-7 m|390 to 700 nm]].<ref>{{cite book | title = Biology: Concepts and Applications | first = Cecie |last=Starr | publisher = Thomson Brooks/Cole | year = 2005 | isbn = 0-534-46226-X | url = https://archive.org/details/biologyconceptsa06edstar| url-access = registration | page = [https://archive.org/details/biologyconceptsa06edstar/page/94 94] }}</ref> The [[absorption spectrum]] of a [[chemical element]] or [[chemical compound]] is the spectrum of frequencies or wavelengths of incident radiation that are absorbed by the compound due to electron transitions from a lower to a higher energy state. The [[emission spectrum]] refers to the spectrum of radiation emitted by the compound due to [[electron transitions]] from a higher to a lower energy state.
Light from many different sources contains various colors, each with its own brightness or intensity. A rainbow, or [[prism (optics)|prism]], sends these component colors in different directions, making them individually visible at different angles. A graph of the intensity plotted against the frequency (showing the brightness of each color) is the frequency spectrum of the light. When all the visible frequencies are present equally, the perceived color of the light is white, and the spectrum is a flat line. Therefore, flat-line spectra in general are often referred to as ''white'', whether they represent light or another type of wave phenomenon (sound, for example, or vibration in a structure).
In radio and telecommunications, the frequency spectrum can be shared among many different broadcasters. The [[radio spectrum]] is the part of the [[electromagnetic spectrum]] corresponding to frequencies lower below 300 GHz, which corresponds to wavelengths longer than about 1 mm. The [[microwave]] spectrum corresponds to frequencies between 300 MHz (0.3 [[hertz|GHz]]) and 300 GHz and wavelengths between one meter and one millimeter.<ref>Pozar, David M. (1993). ''Microwave Engineering'' Addison–Wesley Publishing Company. {{ISBN|0-201-50418-9}}.</ref><ref>Sorrentino, R. and Bianchi, Giovanni (2010) ''[https://books.google.com/books?id=6Hc30XnqdPwC Microwave and RF Engineering] {{webarchive|url=https://web.archive.org/web/20160805093112/https://books.google.com/books?id=6Hc30XnqdPwC |date=August 5, 2016 }}'', John Wiley & Sons, p. 4, {{ISBN|047066021X}}.</ref> Each broadcast radio and TV station transmits a wave on an assigned frequency range, called a ''channel''. When many broadcasters are present, the radio spectrum consists of the sum of all the individual channels, each carrying separate information, spread across a wide frequency spectrum. Any particular radio receiver will detect a single function of amplitude (voltage) vs. time. The radio then uses a [[tuned circuit]] or tuner to select a single channel or frequency band and [[modulation|demodulate]] or decode the information from that broadcaster. If we made a graph of the strength of each channel vs. the frequency of the tuner, it would be the frequency spectrum of the antenna signal.
In [[astronomical spectroscopy]], the strength, shape, and position of absorption and emission lines, as well as the overall [[spectral energy distribution]] of the continuum, reveal many properties of astronomical objects. [[Stellar classification]] is the categorisation of [[star]]s based on their characteristic electromagnetic spectra. The [[spectral flux density]] is used to represent the spectrum of a light-source, such as a star.
In [[radiometry]] and [[colorimetry]] (or [[color science]] more generally), the [[spectral power distribution]] (SPD) of a [[light source]] is a measure of the power contributed by each frequency or color in a light source. The light spectrum is usually measured at points (often 31) along the [[visible spectrum]], in wavelength space instead of frequency space, which makes it not strictly a spectral density. Some [[spectrophotometry|spectrophotometers]] can measure increments as fine as one to two [[nanometer]]s and even higher resolution devices with resolutions less than 0.5 nm have been reported.<ref>{{Cite journal |last1=Noui |first1=Louahab |last2=Hill |first2=Jonathan |last3=Keay |first3=Peter J |last4=Wang |first4=Robert Y |last5=Smith |first5=Trevor |last6=Yeung |first6=Ken |last7=Habib |first7=George |last8=Hoare |first8=Mike |date=2002-02-01 |title=Development of a high resolution UV spectrophotometer for at-line monitoring of bioprocesses |url=https://www.sciencedirect.com/science/article/pii/S0255270101001222 |journal=Chemical Engineering and Processing: Process Intensification |language=en |volume=41 |issue=2 |pages=107–114 |doi=10.1016/S0255-2701(01)00122-2 |bibcode=2002CEPPI..41..107N |issn=0255-2701|url-access=subscription }}</ref> the values are used to calculate other specifications and then plotted to show the spectral attributes of the source. This can be helpful in analyzing the color characteristics of a particular source.
== Mass spectrum == {{main|Mass spectrum}}
[[File:Titan atmosphere diagram.png|thumb|right|Mass spectrum of [[Titan (moon)|Titan]]'s [[ionosphere]]]] A plot of ion abundance as a function of [[mass-to-charge ratio]] is called a mass spectrum. It can be produced by a [[mass spectrometer]] instrument.<ref>{{GoldBookRef|title=mass spectrum|file= M03749}}</ref> The mass spectrum can be used to determine the quantity and [[Mass (mass spectrometry)|mass]] of atoms and molecules. [[Tandem mass spectrometry]] is used to determine molecular structure.
== Energy spectrum{{anchor|Energy}} == {{distinguish-redirect|Energy spectrum|Energy spectral density}} In physics, the energy spectrum of a particle is the number of particles or intensity of a particle beam as a function of particle energy. Examples of techniques that produce an energy spectrum are [[alpha-particle spectroscopy]], [[electron energy loss spectroscopy]], and [[mass-analyzed ion-kinetic-energy spectrometry]].
== Displacement == [[Oscillation|Oscillatory]] [[Displacement (geometry)|displacements]], including [[vibration]]s, can also be characterized spectrally. * For [[water wave]]s, see ''[[wave spectrum]]'' and ''[[tide spectrum]]''. * {{anchor|Sound|Acoustics}} [[Sound]] and non-audible [[acoustics|acoustic]] waves can also be characterized in terms of its spectral density, for example, [[timbre]] and [[musical acoustics]]. <gallery mode=packed> Munk ICCE 1950 Fig1.svg|Classification of the [[spectrum]] of ocean waves according to wave [[period (physics)|period]]<ref>{{cite journal |doi=10.9753/icce.v1.1 |title=Origin and Generation of Waves |date=2010 |last1=Munk |first1=Walter H. |journal=Coastal Engineering Proceedings |volume=1 |page=1 }}</ref> Tides Fourier Transform.png|Spectrum of tides measured at [[Fort Pulaski]] in 2012.<ref>{{Cite web |date=December 2013 |title=Datums - NOAA Tides & Currents |url=https://tidesandcurrents.noaa.gov/datums.html?id=8670870 |url-status=live |archive-url=https://web.archive.org/web/20221206114734/https://tidesandcurrents.noaa.gov/datums.html?id=8670870 |archive-date=2022-12-06 |access-date=2023-03-22 |website=tidesandcurrents.noaa.gov}}</ref> This [[Fourier transform]] was computed using [[SourceForge]]<ref>{{Cite web |title=A More Accurate Fourier Transform |url=https://sourceforge.net/projects/amoreaccuratefouriertransform/ |access-date=2023-03-22 |website=SourceForge |date=7 July 2015 |language=en}}</ref> </gallery>
=== Acoustical measurement === {{Main|Spectrogram|Spectrum analyzer}}
In [[acoustics]], a [[spectrogram]] is a visual representation of the frequency spectrum of sound as a function of time or another variable.
A source of sound can have many different frequencies mixed. A [[musical tone]]'s [[timbre]] is characterized by its [[harmonic spectrum]]. Sound in our environment that we refer to as ''noise'' includes many different frequencies. When a sound signal contains a mixture of all audible frequencies, distributed equally over the audio spectrum, it is called [[white noise]].<ref> {{cite web |url = http://www.yourdictionary.com/white-noise |title = white noise definition |work = yourdictionary.com |url-status = live |archive-url = https://web.archive.org/web/20150630015444/http://www.yourdictionary.com/white-noise |archive-date = June 30, 2015 |df = mdy-all }}</ref>
The [[spectrum analyzer]] is an instrument which can be used to convert the [[sound|sound wave]] of the musical note into a visual display of the constituent frequencies. This visual display is referred to as an acoustic [[spectrogram]]. Software based audio spectrum analyzers are available at low cost, providing easy access not only to industry professionals, but also to academics, students and the [[hobbyist]]. The acoustic spectrogram generated by the spectrum analyzer provides an [[acoustic signature]] of the musical note. In addition to revealing the fundamental frequency and its overtones, the spectrogram is also useful for analysis of the temporal [[ADSR envelope|attack]], [[ADSR envelope|decay]], [[ADSR envelope|sustain]], and [[ADSR envelope|release]] of the musical note.
<gallery mode=packed> Ultrasound range diagram.svg|Approximate frequency ranges corresponding to ultrasound, with rough guide of some applications Oh No Girl Spectrogram.jpg|[[Acoustic spectrogram]] of the words "Oh, no!" said by a young girl, showing how the discrete spectrum of the sound (bright orange lines) changes with time (the horizontal axis) Dolphin1.jpg|Spectrogram of dolphin vocalizations </gallery>
== Continuous versus discrete spectra {{anchor|Continuous|Discrete}}== [[Image:Simple spectroscope.jpg|thumb|300px|right|Continuous spectrum of an [[Incandescent light bulb|incandescent lamp]] (mid) and discrete spectrum lines of a [[Compact fluorescent lamp|fluorescent lamp]] (bottom)]]
In the [[physical sciences]], the spectrum of a [[physical quantity]] (such as [[energy]]) may be called '''''continuous''''' if it is non-zero over the whole spectrum domain (such as [[frequency]] or [[wavelength]]) or '''''discrete''''' if it attains non-zero values only in a [[discrete set]] over the [[independent variable]], with ''[[band gap]]s'' between pairs of ''[[spectral band]]s'' or ''[[spectral line]]s''.<ref>{{Cite journal |title=Continuous Spectrum - klinics.lib.kmutt.ac.th |url=https://klinics.lib.kmutt.ac.th/index.php/foundation/110-chemistry/2671-continuous-spectrum?format=pdf |format=PDF |journal=[[KMUTT: Thailands Science General]] |language=en-TH |volume=2 |issue=1 |pages=22 |archive-url=https://web.archive.org/web/20220820174604/https://klinics.lib.kmutt.ac.th/index.php/foundation/110-chemistry/2671-continuous-spectrum?format=pdf |archive-date=2022-08-20 |quote=In physics, a continuous spectrum usually means a set of achievable values for some physical quantity (such as energy or wavelength), best described as an interval of real numbers. It is the opposite of a discrete spectrum, a set of achievable values that are discrete in the mathematical sense where there is a positive gap between each value. |via=[[KMUTT]]}}</ref>
The classical example of a continuous spectrum, from which the name is derived, is the part of the [[emission spectrum|spectrum]] of the light emitted by [[excited state|excited]] [[atom]]s of [[hydrogen]] that is due to free [[electron]]s becoming bound to a hydrogen ion and emitting photons, which are smoothly spread over a wide range of wavelengths, in contrast to the [[hydrogen spectral series|discrete lines]] due to electrons falling from some bound [[quantum state]] to a state of lower energy. As in that classical example, the term is most often used when the range of values of a physical quantity may have both a continuous and a discrete part, whether at the same time or in different situations. In [[quantum system]]s, continuous spectra (as in [[bremsstrahlung]] and [[thermal radiation]]) are usually associated with free particles, such as atoms in a gas, electrons in an [[electron beam]], or [[conduction band]] electrons in a [[metal]]. In particular, the [[position operator|position]] and [[momentum]] of a free particle has a continuous spectrum, but when the particle is confined to a limited space its spectrum becomes discrete.
Often a continuous spectrum may be just a convenient model for a discrete spectrum whose values are too close to be distinguished, as in the [[phonon]]s in a [[crystal]].
The continuous and discrete spectra of physical systems can be modeled in [[functional analysis]] as different parts in the [[decomposition of spectrum (functional analysis)|decomposition of the spectrum]] of a [[linear operator]] acting on a [[function space]], such as the [[Hamiltonian (quantum mechanics)|Hamiltonian]] operator.
The classical example of a discrete spectrum (for which the term was first used) is the characteristic set of discrete [[spectral line]]s seen in the [[emission spectrum]] and [[absorption spectrum]] of isolated [[atom]]s of a [[chemical element]], which only absorb and emit light at particular [[wavelength]]s. The technique of [[spectroscopy]] is based on this phenomenon.
Discrete spectra are seen in many other phenomena, such as vibrating [[string (music)|string]]s, [[microwave]]s in a [[microwave cavity|metal cavity]], [[sound wave]]s in a [[pulsating star]], and [[resonance (particle physics)|resonances]] in high-energy [[particle physics]]. The general phenomenon of discrete spectra in physical systems can be mathematically modeled with tools of [[functional analysis]], specifically by the [[decomposition of spectrum (functional analysis)|decomposition of the spectrum]] of a [[linear operator]] acting on a [[functional space]].
=== In classical mechanics === In [[classical mechanics]], discrete spectra are often associated to [[wave]]s and [[oscillation]]s in a bounded object or domain. Mathematically they can be identified with the [[eigenvalue]]s of [[differential operator]]s that describe the evolution of some continuous variable (such as [[deformation (mechanics)|strain]] or [[pressure]]) as a function of time and/or space.
Discrete spectra are also produced by some [[non-linear oscillator]]s where the relevant quantity has a non-[[sinusoid]]al [[waveform]]. Notable examples are the sound produced by the [[vocal cords]] of mammals.<ref name=pulak> Hannu Pulakka (2005), [https://aaltodoc.aalto.fi/bitstream/handle/123456789/982/urn007925.pdf Analysis of human voice production using inverse filtering, high-speed imaging, and electroglottography]. Master's thesis, Helsinki University of Technology. </ref><ref name=lind>{{cite book | last1=Lindblom | first1=Björn | last2=Sundberg | first2=Johan | title=Springer Handbook of Acoustics | chapter=The Human Voice in Speech and Singing | publisher=Springer New York | publication-place=New York, NY | year=2007 | doi=10.1007/978-0-387-30425-0_16 | pages=669–712|isbn=978-0-387-30446-5}}</ref>{{rp|p.684}} and the [[stridulation]] organs of [[crickets]],<ref name=popov>{{cite journal | last1=Popov | first1=A. V. | last2=Shuvalov | first2=V. F. | last3=Markovich | first3=A. M. | title=The spectrum of the calling signals, phonotaxis, and the auditory system in the cricket Gryllus bimaculatus | journal=Neuroscience and Behavioral Physiology | publisher=Springer Science and Business Media LLC | volume=7 | issue=1 | year=1976 | issn=0097-0549 | doi=10.1007/bf01148749 | pages=56–62| pmid=1028002 | s2cid=25407842 }}</ref> whose spectrum shows a series of strong lines at frequencies that are integer multiples ([[harmonic]]s) of the [[fundamental frequency|oscillation frequency]].
A related phenomenon is the appearance of strong harmonics when a sinusoidal signal (which has the ultimate "discrete spectrum", consisting of a single spectral line) is modified by a non-linear [[filter (signal processing)|filter]]; for example, when a [[pure tone]] is played through an overloaded [[amplifier]],<ref name=klip>Paul V. Klipsch (1969), [http://www.elpee.info/Documenten/KlipschVervorming.pdf ''Modulation distortion in loudspeakers''] {{Webarchive|url=https://web.archive.org/web/20160304025155/http://www.elpee.info/Documenten/KlipschVervorming.pdf |date=2016-03-04 }} Journal of the Audio Engineering Society.</ref> or when an intense [[monochromatic]] [[laser]] beam goes through a [[non-linear optics|non-linear medium]].<ref>{{cite journal | last1=Armstrong | first1=J. A. | last2=Bloembergen | first2=N. | last3=Ducuing | first3=J. | last4=Pershan | first4=P. S. | title=Interactions between Light Waves in a Nonlinear Dielectric | journal=Physical Review | publisher=American Physical Society (APS) | volume=127 | issue=6 | date=1962-09-15 | issn=0031-899X | doi=10.1103/physrev.127.1918 | pages=1918–1939| bibcode=1962PhRv..127.1918A | doi-access=free }}</ref> In the latter case, if two arbitrary sinusoidal signals with frequencies ''f'' and ''g'' are processed together, the output signal will generally have spectral lines at frequencies {{abs|''mf'' + ''ng''}}, where ''m'' and ''n'' are any integers.
=== In quantum mechanics === {{main|Decomposition of spectrum (functional analysis) #Quantum physics}}
In [[quantum mechanics]], the discrete spectrum of an [[observable]] refers to the [[Spectrum_(functional_analysis)#Point_spectrum|pure point spectrum]] of [[eigenvalue]]s of the [[Linear operator|operator]] used to model that observable.<ref>{{cite web | last=Simon | first=B. | title=An Overview of Rigorous Scattering Theory | date=1978 |page=3| s2cid=16913591 }}</ref><ref>{{cite book | last=Teschl | first=G. | title=Mathematical Methods in Quantum Mechanics | publisher=American Mathematical Soc. |url=https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/schroe.pdf| publication-place=Providence, R.I | date=2009 | isbn=978-0-8218-4660-5 | chapter=5.2 The RAGE theorem}}</ref>
Discrete spectra are usually associated with systems that are [[bound state|bound]] in some sense (mathematically, confined to a [[compact space]]).{{Citation needed|date=November 2023|reason=Ambiguous use of [[compact space]]. Confined in what "sense"?}} The [[position operator|position]] and [[momentum operator]]s have continuous spectra in an infinite domain, but a discrete (quantized) spectrum in a compact domain and the same properties of spectra hold for [[angular momentum]], [[Hamiltonian (quantum mechanics)|Hamiltonians]] and other operators of quantum systems.
The [[quantum harmonic oscillator]] and the [[hydrogen atom]] are examples of physical systems in which the Hamiltonian has a discrete spectrum. In the case of the hydrogen atom the spectrum has both a continuous and a discrete part, the continuous part representing the [[ionization]].
<gallery mode=packed> Hydrogen spectrum.svg|The discrete part of the emission spectrum of hydrogen Solar Spectrum.png|Spectrum of sunlight above the atmosphere (yellow) and at sea level (red), revealing an absorption spectrum with a discrete part (such as the line due to {{chem|O|2}}) and a continuous part (such as the bands labeled {{chem|H|2|O}}) Deuterium lamp 1.png|Spectrum of light emitted by a [[deuterium]] lamp, showing a discrete part (tall sharp peaks) and a continuous part (smoothly varying between the peaks). The smaller peaks and valleys may be due to measurement errors rather than discrete spectral lines. </gallery>
== See also == * {{Section link|Spectrum (disambiguation)|Physics}}
== References == {{reflist}}
[[Category:Spectrum (physical sciences)| ]] [[Category:Structure]]