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{{short description|Number of times a given event occurs per photon absorbed by a quantum system}}

In [[Physics:Particle physics|particle physics]], the '''quantum yield''' (denoted {{math|Φ}}) of a [[Physics:Radiation|radiation]]-induced process is the number of times a specific [[Physics:Event (particle physics)|event]] occurs per [[Physics:Photon|photon]] absorbed by the [[Physics:Quantum system|system]].<ref name=":2">{{Cite journal|last=Braslavsky|first=S. E.|date=2007-01-01|title=Glossary of terms used in photochemistry, 3rd edition (IUPAC Recommendations 2006)|journal=Pure and Applied Chemistry|volume=79|issue=3|pages=293–465|doi=10.1351/pac200779030293|s2cid=96601716 |issn=1365-3075|doi-access=free}}</ref>

<math display=block>\Phi(\lambda)=\frac{\text { number of events }}{\text { number of photons absorbed }}</math>

== Applications ==

=== Fluorescence spectroscopy ===
The [[Biology:Fluorescence|fluorescence]] quantum yield is defined as the [[Ratio|ratio]] of the number of photons emitted to the number of photons absorbed.<ref name="Lakowicz">Lakowicz, Joseph R. ''Principles of Fluorescence Spectroscopy'' (Kluwer Academic / Plenum Publishers 1999) p.10. {{ISBN|978-0-387-31278-1}}</ref>

<math display=block> \Phi = \frac{\rm \#\ photons\ emitted}{\rm \#\ photons\ absorbed}</math>

Fluorescence quantum yield is measured on a scale from 0 to 1.0, but is often represented as a [[Percentage|percentage]]. A quantum yield of 1.0 (100%) describes a process where each photon absorbed results in a photon emitted. Substances with the largest quantum yields, such as [[Physics:Rhodamine|rhodamine]]s, display the brightest emissions; however, compounds with quantum yields of 0.10 are still considered quite fluorescent.

Quantum yield is defined by the fraction of [[Physics:Excited state|excited state]] [[Physics:Fluorophore|fluorophore]]s that decay through fluorescence:

<math display=block>\Phi_f = \frac{k_f}{k_f + \sum k_\mathrm{nr}}</math>

where
*{{math|Φ{{sub|''f''}}}} is the fluorescence quantum yield,
*{{mvar|k{{sub|f}}}} is the rate constant for radiative relaxation (fluorescence),
*{{math|''k''{{sub|nr}}}} is the rate constant for all non-radiative relaxation processes.
Non-radiative processes are excited state decay mechanisms other than photon emission, which include: Förster resonance energy transfer, [[Chemistry:Internal conversion|internal conversion]], external conversion, and [[Physics:Intersystem crossing|intersystem crossing]]. Thus, the fluorescence quantum yield is affected if the rate of any non-radiative pathway changes. The quantum yield can be close to unity if the non-radiative decay rate is much smaller than the rate of radiative decay, that is {{math|''k{{sub|f}}'' > ''k''{{sub|nr}}}}.<ref name="Lakowicz" />

Fluorescence quantum yields are measured by comparison to a standard of known quantum yield.<ref name="Lakowicz" /> The [[Chemistry:Quinine|quinine]] salt ''quinine sulfate'' in a [[Chemistry:Sulfuric acid|sulfuric acid]] solution was regarded as the most common fluorescence standard,<ref>{{Cite journal|last=Brouwer|first=Albert M.|date=2011-08-31|title=Standards for photoluminescence quantum yield measurements in solution (IUPAC Technical Report)|journal=Pure and Applied Chemistry|language=|volume=83|issue=12|pages=2213–2228|doi=10.1351/PAC-REP-10-09-31|s2cid=98138291 |issn=1365-3075|doi-access=free}}</ref> however, a recent study revealed that the fluorescence quantum yield of this solution is strongly affected by the temperature, and should no longer be used as the standard solution. The quinine in 0.1M [[Chemistry:Perchloric acid|perchloric acid]] ({{math|1=Φ =}} 0.60) shows no temperature dependence up to 45 °C, therefore it can be considered as a reliable standard solution.<ref>{{Cite journal|last1=Nawara|first1=Krzysztof|last2=Waluk|first2=Jacek|date=2019-04-16|title=Goodbye to Quinine in Sulfuric Acid Solutions as a Fluorescence Quantum Yield Standard|url=https://doi.org/10.1021/acs.analchem.9b00583|journal=Analytical Chemistry|volume=91|issue=8|pages=5389–5394|doi=10.1021/acs.analchem.9b00583|pmid=30907575 |s2cid=85501014 |issn=0003-2700|url-access=subscription}}</ref>
{| class="wikitable"
|+Fluorescence quantum yield standards
!Compound
!Solvent
!<math>\lambda_{\mathrm{ex}}(\mathrm{nm})</math>
!<math>\Phi</math>
|-
|[[Chemistry:Quinine|Quinine]]
|0.1 M {{chem2|HClO4}}
|347.5
|0.60 ± 0.02
|-
|[[Chemistry:Fluorescein|Fluorescein]]
|0.1 M {{chem2|NaOH}}
|496
|0.95 ± 0.03
|-
|[[Chemistry:Tryptophan|Tryptophan]]
|Water
|280
|0.13 ± 0.01
|-
|[[Chemistry:Rhodamine 6G|Rhodamine 6G]]
|Ethanol
|488
|0.94
|}
Experimentally, relative fluorescence quantum yields can be determined by measuring fluorescence of a [[Physics:Fluorophore|fluorophore]] of known quantum yield with the same experimental parameters (excitation [[Wavelength|wavelength]], slit widths, [[Physics:Photomultiplier|photomultiplier]] voltage etc.) as the substance in question. The quantum yield is then calculated by:

<math display=block>\Phi = \Phi_\mathrm{R} \times \frac{\mathit{Int}}{\mathit{Int}_\mathrm{R}} \times \frac{1-10^{-A_\mathrm{R}}}{1-10^{-A}} \times \frac{{n}^2}{{n_\mathrm{R}}^2}</math>

where
*{{math|Φ}} is the quantum yield,
*{{mvar|Int}} is the area under the emission peak (on a wavelength scale),
*{{mvar|A}} is [[Physics:Absorbance|absorbance]] (also called "optical density") at the excitation wavelength,
*{{mvar|n}} is the [[Physics:Refractive index|refractive index]] of the [[Physics:Solvent|solvent]].
The subscript {{math|R}} denotes the respective values of the reference substance.<ref name=":0">Albert M. Brouwer, [http://www.iupac.org/publications/pac/83/12/2213/ Standards for photoluminescence quantum yield measurements in solution] (IUPAC Technical Report), [[Chemistry:Pure and Applied Chemistry|Pure Appl. Chem.]], Vol. 83, No. 12, pp. 2213–2228, 2011. doi:10.1351/PAC-REP-10-09-31.</ref><ref name=":1">{{Cite journal|last=Levitus|first=Marcia|date=2020-04-22|title=Tutorial: measurement of fluorescence spectra and determination of relative fluorescence quantum yields of transparent samples|journal=Methods and Applications in Fluorescence|volume=8|issue=3|pages=033001|doi=10.1088/2050-6120/ab7e10|issn=2050-6120|pmid=32150732|bibcode=2020MApFl...8c3001L |s2cid=212653274 }}</ref> The determination of fluorescence quantum yields in scattering media requires additional considerations and corrections.<ref>{{Cite journal|last=Lagorio|first=María Gabriela|date=2020-10-06|title=Determination of Fluorescence Quantum Yields in Scattering Media|url=https://doi.org/10.1088/2050-6120/aba69c|journal=Methods and Applications in Fluorescence|volume=8|issue=4|pages=043001|doi=10.1088/2050-6120/aba69c|pmid=32674086 |bibcode=2020MApFl...8d3001L |s2cid=220610164 |issn=2050-6120|url-access=subscription}}</ref>

==== FRET efficiency ====
Förster resonance energy transfer efficiency ({{mvar|E}}) is the quantum yield of the energy-transfer transition, i.e. the probability of the energy-transfer event occurring per donor excitation event:

<math display=block>E=\Phi_\mathrm{FRET}=\frac{k_\mathrm{ET}}{k_\mathrm{ET} + k_f + \sum k_\mathrm{nr}}</math>

where
*{{math|''k''{{sub|ET}}}} is the rate of energy transfer,
*{{mvar|k{{sub|f}}}} the radiative decay rate (fluorescence) of the donor,
*{{math|''k''{{sub|nr}}}} are non-radiative relaxation rates (e.g., internal conversion, intersystem crossing, external conversion etc.).<ref>{{Cite journal|last1=dos Remedios|first1=Cristobal G.|last2=Moens|first2=Pierre D.J.|date=September 1995|title=Fluorescence Resonance Energy Transfer Spectroscopy Is a Reliable "Ruler" for Measuring Structural Changes in Proteins|url=https://linkinghub.elsevier.com/retrieve/pii/S1047847785710428|journal=Journal of Structural Biology|language=en|volume=115|issue=2|pages=175–185|doi=10.1006/jsbi.1995.1042|pmid=7577238 |url-access=subscription}}</ref><ref>{{Cite web|date=2013-10-02|title=Fluorescence Resonance Energy Transfer|url=https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Fundamentals/Fluorescence_Resonance_Energy_Transfer|access-date=2020-11-30|website=Chemistry LibreTexts|language=en}}</ref>

==== Solvent and environmental effects ====
A fluorophore's environment can impact quantum yield, usually resulting from changes in the rates of non-radiative decay.<ref name="Lakowicz" /> Many fluorophores used to label macromolecules are sensitive to solvent polarity. The class of [[Chemistry:8-Anilinonaphthalene-1-sulfonic acid|8-anilinonaphthalene-1-sulfonic acid]] (ANS) probe molecules are essentially non-fluorescent when in aqueous solution, but become highly fluorescent in nonpolar solvents or when bound to proteins and membranes. The quantum yield of ANS is ~0.002 in [[Chemistry:Aqueous solution|aqueous]] buffer, but near 0.4 when bound to serum [[Biology:Albumin|albumin]].

=== Photochemical reactions ===
The quantum yield of a photochemical reaction describes the number of molecules undergoing a photochemical event per absorbed photon:<ref name=":2" /><ref>{{Cite journal |last=Swierk |first=John R. |date=2023-07-21 |title=The Cost of Quantum Yield |url=https://doi.org/10.1021/acs.oprd.3c00167 |journal=Organic Process Research & Development |volume=27 |issue=7 |pages=1411–1419 |doi=10.1021/acs.oprd.3c00167 |issn=1083-6160}}</ref>

<math display=block>\Phi=\frac{\rm \#\ molecules\ undergoing\ reaction\ of\ interest}{\rm \#\ photons\ absorbed\ by\ photoreactive\ substance}</math>

In a chemical [[Biology:Photodegradation|photodegradation]] process, when a molecule dissociates after absorbing a light [[Physics:Quantum|quantum]], the quantum yield is the number of destroyed molecules divided by the number of photons absorbed by the system. Since not all photons are absorbed productively, the typical quantum yield will be less than 1.

<math display=block> \Phi = \frac{\rm \#\ molecules \ decomposed} {\rm \#\ photons \ absorbed} </math>

Quantum yields greater than 1 are possible for photo-induced or radiation-induced [[Philosophy:Chain reaction#Chemical chain reactions|chain reactions]], in which a single photon may trigger a long [[Philosophy:Chain reaction|chain of transformations]]. One example is the reaction of [[Software:Hydrogen|hydrogen]] with [[Chemistry:Chlorine|chlorine]], in which as many as 10<sup>6</sup> molecules of [[Chemistry:Hydrogen chloride|hydrogen chloride]] can be formed per quantum of blue light absorbed.<ref>Laidler K.J., ''Chemical Kinetics'' (3rd ed., Harper & Row 1987) p.289 {{ISBN|0-06-043862-2}}</ref>

Quantum yields of photochemical reactions can be highly dependent on the structure, proximity and concentration of the reactive chromophores, the type of solvent environment as well as the wavelength of the incident light. Such effects can be studied with wavelength-tunable lasers and the resulting quantum yield data can help predict conversion and selectivity of photochemical reactions.<ref>{{cite journal |last1=Menzel |first1=Jan P. |last2=Noble |first2=Benjamin B. |last3=Blinco |first3=James P. |last4=Barner-Kowollik |first4=Christopher |title=Predicting wavelength-dependent photochemical reactivity and selectivity |journal=Nature Communications |date=2021 |volume=12 |issue=1 |pages=1691 |doi=10.1038/s41467-021-21797-x |pmid= 33727558|pmc=7966369 |bibcode=2021NatCo..12.1691M |doi-access=free }}</ref>

In optical spectroscopy, the quantum yield is the probability that a given [[Physics:Quantum state|quantum state]] is formed from the system initially prepared in some other quantum state. For example, a singlet to triplet transition quantum yield is the fraction of molecules that, after being photoexcited into a singlet state, cross over to the triplet state.

=== Photosynthesis ===
Quantum yield is used in modeling [[Earth:Photosynthesis|photosynthesis]]:<ref name="pmid18359752">{{cite journal|author=Skillman JB|year=2008|title=Quantum yield variation across the three pathways of photosynthesis: not yet out of the dark|journal=J. Exp. Bot.|volume=59|issue=7|pages=1647–61|doi=10.1093/jxb/ern029|pmid=18359752|doi-access=free}}</ref>

<math display=block> \Phi = \frac {\rm \mu mol\ CO_2 \ fixed} {\rm \mu mol\ photons \ absorbed} </math>

==See also==
*[[Physics:Quantum dot|Quantum dot]]

* [[Engineering:Quantum efficiency|Quantum efficiency]]
* Molecular [[Chemistry:Host–guest chemistry|Cage]]
* [[Engineering:Photoswitch|Photoswitches]]

== References ==
{{reflist}}

{{DEFAULTSORT:Quantum Yield}}
[[Category:Radiation]]
[[Category:Spectroscopy]]
[[Category:Photochemistry]]

{{Sourceattribution|Quantum yield}}

Physics:Quantum yield - Revision history

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