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class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-History_and_uses_of_the_term_absorbance" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History_and_uses_of_the_term_absorbance"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>History and uses of the term absorbance</span> </div> </a> <button aria-controls="toc-History_and_uses_of_the_term_absorbance-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle History and uses of the term absorbance subsection</span> </button> <ul id="toc-History_and_uses_of_the_term_absorbance-sublist" class="vector-toc-list"> <li id="toc-Beer-Lambert_law" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Beer-Lambert_law"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Beer-Lambert law</span> </div> </a> <ul id="toc-Beer-Lambert_law-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Absorbance_for_non-scattering_samples" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Absorbance_for_non-scattering_samples"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Absorbance for non-scattering samples</span> </div> </a> <ul id="toc-Absorbance_for_non-scattering_samples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Absorbance_for_scattering_samples" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Absorbance_for_scattering_samples"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Absorbance for scattering samples</span> </div> </a> <ul id="toc-Absorbance_for_scattering_samples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Optics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Optics"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Optics</span> </div> </a> <ul id="toc-Optics-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Mathematical_definitions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mathematical_definitions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Mathematical definitions</span> </div> </a> <button aria-controls="toc-Mathematical_definitions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Mathematical definitions subsection</span> </button> <ul id="toc-Mathematical_definitions-sublist" class="vector-toc-list"> <li id="toc-Absorbance_of_a_material" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Absorbance_of_a_material"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Absorbance of a material</span> </div> </a> <ul id="toc-Absorbance_of_a_material-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spectral_absorbance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spectral_absorbance"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Spectral absorbance</span> </div> </a> <ul id="toc-Spectral_absorbance-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Relationship_with_attenuation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relationship_with_attenuation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Relationship with attenuation</span> </div> </a> <button aria-controls="toc-Relationship_with_attenuation-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Relationship with attenuation subsection</span> </button> <ul id="toc-Relationship_with_attenuation-sublist" class="vector-toc-list"> <li id="toc-Attenuance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Attenuance"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Attenuance</span> </div> </a> <ul id="toc-Attenuance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Attenuation_coefficient" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Attenuation_coefficient"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Attenuation coefficient</span> </div> </a> <ul id="toc-Attenuation_coefficient-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Measurements" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Measurements"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Measurements</span> </div> </a> <button aria-controls="toc-Measurements-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Measurements subsection</span> </button> <ul id="toc-Measurements-sublist" class="vector-toc-list"> <li id="toc-Logarithmic_vs._directly_proportional_measurements" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Logarithmic_vs._directly_proportional_measurements"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Logarithmic vs. directly proportional measurements</span> </div> </a> <ul id="toc-Logarithmic_vs._directly_proportional_measurements-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Instrument_measurement_range" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Instrument_measurement_range"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Instrument measurement range</span> </div> </a> <ul id="toc-Instrument_measurement_range-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Method_of_measurement" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Method_of_measurement"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Method of measurement</span> </div> </a> <ul id="toc-Method_of_measurement-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Shade_number" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Shade_number"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Shade number</span> </div> </a> <ul id="toc-Shade_number-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left"> <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents"> <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet 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Available in 20 languages"> <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-20" aria-hidden="true"><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">20 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://web.archive.org/web/20240918212750/https://ar.wikipedia.org/wiki/%D8%A7%D9%85%D8%AA%D8%B5%D8%A7%D8%B5%D9%8A%D8%A9" title="امتصاصية – Arabic" lang="ar" hreflang="ar" data-title="امتصاصية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://web.archive.org/web/20240918212750/https://ca.wikipedia.org/wiki/Absorb%C3%A0ncia" title="Absorbància – Catalan" lang="ca" hreflang="ca" data-title="Absorbància" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://web.archive.org/web/20240918212750/https://cs.wikipedia.org/wiki/Absorbance" title="Absorbance – Czech" lang="cs" hreflang="cs" data-title="Absorbance" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://web.archive.org/web/20240918212750/https://de.wikipedia.org/wiki/Extinktion_(Optik)" title="Extinktion (Optik) – German" lang="de" hreflang="de" data-title="Extinktion (Optik)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://web.archive.org/web/20240918212750/https://es.wikipedia.org/wiki/Absorbancia" title="Absorbancia – Spanish" lang="es" hreflang="es" data-title="Absorbancia" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://web.archive.org/web/20240918212750/https://eo.wikipedia.org/wiki/Optika_denso" title="Optika denso – Esperanto" lang="eo" hreflang="eo" data-title="Optika denso" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://web.archive.org/web/20240918212750/https://fr.wikipedia.org/wiki/Absorbance" title="Absorbance – French" lang="fr" hreflang="fr" data-title="Absorbance" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://web.archive.org/web/20240918212750/https://ko.wikipedia.org/wiki/%ED%9D%A1%EA%B4%91%EB%8F%84" title="흡광도 – Korean" lang="ko" hreflang="ko" data-title="흡광도" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://web.archive.org/web/20240918212750/https://id.wikipedia.org/wiki/Absorbansi" title="Absorbansi – Indonesian" lang="id" hreflang="id" data-title="Absorbansi" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://web.archive.org/web/20240918212750/https://it.wikipedia.org/wiki/Assorbanza" title="Assorbanza – Italian" lang="it" hreflang="it" data-title="Assorbanza" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://web.archive.org/web/20240918212750/https://ja.wikipedia.org/wiki/%E5%90%B8%E5%85%89%E5%BA%A6" title="吸光度 – Japanese" lang="ja" hreflang="ja" data-title="吸光度" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://web.archive.org/web/20240918212750/https://pt.wikipedia.org/wiki/Absorb%C3%A2ncia" title="Absorbância – Portuguese" lang="pt" hreflang="pt" data-title="Absorbância" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://web.archive.org/web/20240918212750/https://ro.wikipedia.org/wiki/Absorban%C8%9B%C4%83" title="Absorbanță – Romanian" lang="ro" hreflang="ro" data-title="Absorbanță" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://web.archive.org/web/20240918212750/https://sk.wikipedia.org/wiki/Absorbancia" title="Absorbancia – Slovak" lang="sk" hreflang="sk" data-title="Absorbancia" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://web.archive.org/web/20240918212750/https://so.wikipedia.org/wiki/Nuugid" title="Nuugid – Somali" lang="so" hreflang="so" data-title="Nuugid" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://web.archive.org/web/20240918212750/https://sr.wikipedia.org/wiki/%D0%9E%D0%BF%D1%82%D0%B8%D1%87%D0%BA%D0%B0_%D0%B3%D1%83%D1%81%D1%82%D0%B8%D0%BD%D0%B0" title="Оптичка густина – Serbian" lang="sr" hreflang="sr" data-title="Оптичка густина" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://web.archive.org/web/20240918212750/https://sh.wikipedia.org/wiki/Opti%C4%8Dka_gustina" title="Optička gustina – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Optička gustina" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://web.archive.org/web/20240918212750/https://ta.wikipedia.org/wiki/%E0%AE%92%E0%AE%B3%E0%AE%BF%E0%AE%AF%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D_%E0%AE%85%E0%AE%9F%E0%AE%B0%E0%AF%8D%E0%AE%A4%E0%AF%8D%E0%AE%A4%E0%AE%BF" title="ஒளியியல் அடர்த்தி – Tamil" lang="ta" hreflang="ta" data-title="ஒளியியல் அடர்த்தி" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://web.archive.org/web/20240918212750/https://th.wikipedia.org/wiki/%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B8%94%E0%B8%B9%E0%B8%94%E0%B8%81%E0%B8%A5%E0%B8%B7%E0%B8%99" title="ความดูดกลืน – Thai" lang="th" hreflang="th" data-title="ความดูดกลืน" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://web.archive.org/web/20240918212750/https://zh.wikipedia.org/wiki/%E5%90%B8%E5%85%89%E5%BA%A6" title="吸光度 – Chinese" lang="zh" hreflang="zh" data-title="吸光度" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://web.archive.org/web/20240918212750/https://www.wikidata.org/wiki/Special:EntityPage/Q907315#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages"> 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<div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/web/20240918212750/https://en.wikipedia.org/w/index.php?title=Optical_density&redirect=no" class="mw-redirect" title="Optical density">Optical density</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Logarithm of ratio of incident to transmitted radiant power through a sample</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about a quantitative expression. For the process itself, see <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Absorption_(electromagnetic_radiation)" title="Absorption (electromagnetic radiation)">Absorption (electromagnetic radiation)</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Optical density" redirects here. For other uses, see <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Refractive_index" title="Refractive index">Refractive index</a>, <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Nucleic_acid_quantitation" title="Nucleic acid quantitation">Nucleic acid quantitation</a>, and <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Neutral-density_filter" title="Neutral-density filter">Neutral-density filter</a>.</div> <p><b>Absorbance</b> is defined as "the logarithm of the ratio of incident to transmitted radiant power through a sample (excluding the effects on cell walls)".<sup id="cite_ref-GoldBook_1-0" class="reference"><a href="#cite_note-GoldBook-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Alternatively, for samples which scatter light, absorbance may be defined as "the negative logarithm of one minus absorptance, as measured on a uniform sample".<sup id="cite_ref-:0_2-0" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The term is used in many technical areas to quantify the results of an experimental measurement. While the term has its origin in quantifying the absorption of light, it is often entangled with quantification of light which is “lost” to a detector system through other mechanisms. What these uses of the term tend to have in common is that they refer to a logarithm of the ratio of a quantity of light incident on a sample or material to that which is detected after the light has interacted with the sample. </p><p>The term <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Absorption_(electromagnetic_radiation)" title="Absorption (electromagnetic radiation)">absorption</a> refers to the physical process of absorbing light, while absorbance does not always measure only absorption; it may measure <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Attenuation" title="Attenuation">attenuation</a> (of transmitted radiant power) caused by absorption, as well as reflection, scattering, and other physical processes. Sometimes the term "attenuance" or "experimental absorbance" is used to emphasize that radiation is lost from the beam by processes other than absorption, with the term "internal absorbance" used to emphasize that the necessary corrections have been made to eliminate the effects of phenomena other than absorption.<sup id="cite_ref-:1_3-0" class="reference"><a href="#cite_note-:1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc"/> <div class="mw-heading mw-heading2"><h2 id="History_and_uses_of_the_term_absorbance">History and uses of the term absorbance</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/web/20240918212750/https://en.wikipedia.org/w/index.php?title=Absorbance&action=edit&section=1" title="Edit section: History and uses of the term absorbance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Beer-Lambert_law">Beer-Lambert law</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/web/20240918212750/https://en.wikipedia.org/w/index.php?title=Absorbance&action=edit&section=2" title="Edit section: Beer-Lambert law"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The roots of the term absorbance are in the <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Beer-Lambert_law#Beer–Lambert_law" class="mw-redirect" title="Beer-Lambert law">Beer–Lambert law</a>. As light moves through a medium, it will become dimmer as it is being "extinguished". Bouguer recognized that this extinction (now often called attenuation) was not linear with distance traveled through the medium, but related by what we now refer to as an exponential function. </p><p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{0}}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/893d08e90ea73781dc133414d661529d0651ca80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.077ex; height:2.509ex;" alt="{\displaystyle I_{0}}"></span> is the intensity of the light at the beginning of the travel and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{d}}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/f2cb44b76d459d859d2e9957dafbb9b8e2313b8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.115ex; height:2.509ex;" alt="{\displaystyle I_{d}}"></span> is the intensity of the light detected after travel of a distance <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span>,</span> the fraction transmitted, <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>,</span> is given by </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T={\frac {I_{d}}{I_{0}}}=\exp(-\mu d)\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>μ</mi> <mi>d</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"/> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T={\frac {I_{d}}{I_{0}}}=\exp(-\mu d)\,,}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/3a52f8cc93055eaeac8e68ae8eb6ae9cb53209d9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.606ex; height:5.676ex;" alt="{\displaystyle T={\frac {I_{d}}{I_{0}}}=\exp(-\mu d)\,,}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> is called an <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Propagation_constant#Attenuation_constant" title="Propagation constant">attenuation constant</a> (a term used in various fields where a signal is transmitted though a medium) or coefficient. The amount of light transmitted is falling off exponentially with distance. Taking the natural logarithm in the above equation, we get </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\ln(T)=\ln {\frac {I_{0}}{I_{d}}}=\mu d\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mi>μ</mi> <mi>d</mi> <mspace width="thinmathspace"/> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\ln(T)=\ln {\frac {I_{0}}{I_{d}}}=\mu d\,.}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/d1d5d14ed1b1ba1a2d97f6887b356dea26edac50" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.706ex; height:5.843ex;" alt="{\displaystyle -\ln(T)=\ln {\frac {I_{0}}{I_{d}}}=\mu d\,.}"></span> </p><p>For scattering media, the constant is often divided into two parts,<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =\mu _{s}+\mu _{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =\mu _{s}+\mu _{a}}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/07cc776fca79322e5c56aa4229c0725eb53ab704" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.249ex; height:2.509ex;" alt="{\displaystyle \mu =\mu _{s}+\mu _{a}}"></span>,</span> separating it into a scattering coefficient <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{s}}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/8369676bc9600b382e8a6af5a9369f2e619cd5c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.405ex; height:2.176ex;" alt="{\displaystyle \mu _{s}}"></span> and an absorption coefficient <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{a}}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/ea3dc3301612e553675341c662acee2c98f8607f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.504ex; height:2.176ex;" alt="{\displaystyle \mu _{a}}"></span>,</span> obtaining </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\ln(T)=\ln {\frac {I_{0}}{I_{s}}}=(\mu _{s}+\mu _{a})d\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>d</mi> <mspace width="thinmathspace"/> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\ln(T)=\ln {\frac {I_{0}}{I_{s}}}=(\mu _{s}+\mu _{a})d\,.}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/f4cca018d7767c20560e92e7db82f7ff8225e8d4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.825ex; height:5.843ex;" alt="{\displaystyle -\ln(T)=\ln {\frac {I_{0}}{I_{s}}}=(\mu _{s}+\mu _{a})d\,.}"></span> </p><p>If a size of a detector is very small compared to the distance traveled by the light, any light that is scattered by a particle, either in the forward or backward direction, will not strike the detector. (Bouguer was studying astronomical phenomena, so this condition was met.) In such case, a plot of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\ln(T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\ln(T)}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/99de6cb85f316b2869ce590bebdb189fa12b78cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.58ex; height:2.843ex;" alt="{\displaystyle -\ln(T)}"></span> as a function of wavelength will yield a superposition of the effects of absorption and scatter. Because the absorption portion is more distinct and tends to ride on a background of the scatter portion, it is often used to identify and quantify the absorbing species. Consequently, this is often referred to as <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Absorption_spectroscopy" title="Absorption spectroscopy">absorption spectroscopy</a>, and the plotted quantity is called "absorbance", symbolized as <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {A} }</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/ff6366939c4ebbd4e8494d0dedc54c4b8dd7135a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \mathrm {A} }"></span>.</span> Some disciplines by convention use decadic (base 10) absorbance rather than Napierian (natural) absorbance, resulting in: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {A} _{10}=\mu _{10}d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {A} _{10}=\mu _{10}d}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/141ec1e7713a260a2748b1ebe5ce5dd6576898a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.212ex; height:2.676ex;" alt="{\displaystyle \mathrm {A} _{10}=\mu _{10}d}"></span> (with the subscript 10 usually not shown). </p> <div class="mw-heading mw-heading3"><h3 id="Absorbance_for_non-scattering_samples">Absorbance for non-scattering samples</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/web/20240918212750/https://en.wikipedia.org/w/index.php?title=Absorbance&action=edit&section=3" title="Edit section: Absorbance for non-scattering samples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Within a homogeneous medium such as a solution, there is no scattering. For this case, researched extensively by <a href="/web/20240918212750/https://en.wikipedia.org/wiki/August_Beer" title="August Beer">August Beer</a>, the concentration of the absorbing species follows the same linear contribution to absorbance as the path-length. Additionally, the contributions of individual absorbing species are additive. This is a very favorable situation, and made absorbance an absorption metric far preferable to absorption fraction (absorptance). This is the case for which the term "absorbance" was first used. </p><p>A common expression of the <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Beer%27s_law" class="mw-redirect" title="Beer's law">Beer's law</a> relates the attenuation of light in a material as: <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {A} =\varepsilon \ell c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>=</mo> <mi>ε</mi> <mi>ℓ</mi> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {A} =\varepsilon \ell c}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/3692c0ee4e27e839b04ce84cface5b9a7083046f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.902ex; height:2.176ex;" alt="{\displaystyle \mathrm {A} =\varepsilon \ell c}"></span>,</span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {A} }</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/ff6366939c4ebbd4e8494d0dedc54c4b8dd7135a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \mathrm {A} }"></span> is the <b>absorbance;</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"></span> is the <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Molar_attenuation_coefficient" class="mw-redirect" title="Molar attenuation coefficient">molar attenuation coefficient</a> or <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Molar_absorptivity" class="mw-redirect" title="Molar absorptivity">absorptivity</a> of the attenuating species; <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }"></span> is the optical path length; and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> is the concentration of the attenuating species. </p> <div class="mw-heading mw-heading3"><h3 id="Absorbance_for_scattering_samples">Absorbance for scattering samples</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/web/20240918212750/https://en.wikipedia.org/w/index.php?title=Absorbance&action=edit&section=4" title="Edit section: Absorbance for scattering samples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For samples which scatter light, absorbance is defined as "the negative logarithm of one minus absorptance (absorption fraction: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>) as measured on a uniform sample".<sup id="cite_ref-:0_2-1" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> For decadic absorbance,<sup id="cite_ref-:1_3-1" class="reference"><a href="#cite_note-:1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> this may be symbolized as <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {A} _{10}=-\log _{10}(1-\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {A} _{10}=-\log _{10}(1-\alpha )}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/943e7e18b487da1b21fe39cfe6cfb8ec395afb1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.061ex; height:2.843ex;" alt="{\displaystyle \mathrm {A} _{10}=-\log _{10}(1-\alpha )}"></span>.</span> If a sample both transmits and <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Diffuse_reflectance_spectroscopy#Remission" title="Diffuse reflectance spectroscopy">remits light</a>, and is not luminescent, the fraction of light absorbed <span class="nowrap">(<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>),</span> remitted <span class="nowrap">(<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>),</span> and transmitted <span class="nowrap">(<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>)</span> add to 1: <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha +R+T=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>+</mo> <mi>R</mi> <mo>+</mo> <mi>T</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha +R+T=1}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/5eeb05d30c873931a399f28ecd19f43d28257470" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.83ex; height:2.343ex;" alt="{\displaystyle \alpha +R+T=1}"></span>.</span> Note that <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-\alpha =R+T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mo>=</mo> <mi>R</mi> <mo>+</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-\alpha =R+T}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/4522c255b9cb2a15e0aad6a491405b93ab9447af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.83ex; height:2.343ex;" alt="{\displaystyle 1-\alpha =R+T}"></span>,</span> and the formula may be written as <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {A} _{10}=-\log _{10}(R+T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo>+</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {A} _{10}=-\log _{10}(R+T)}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/22b31ecf8cd420f11f2d3d45769e669b149c5d2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.811ex; height:2.843ex;" alt="{\displaystyle \mathrm {A} _{10}=-\log _{10}(R+T)}"></span>.</span> For a sample which does not scatter, <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=0}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/b7e38203eea12f5613721e2fc5e185b8eef9e800" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.025ex; height:2.176ex;" alt="{\displaystyle R=0}"></span>,</span> and <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-\alpha =T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mo>=</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-\alpha =T}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/4428ea0c8309d08b4bdd2d41b2ae06df3aef58f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.225ex; height:2.343ex;" alt="{\displaystyle 1-\alpha =T}"></span>,</span> yielding the formula for absorbance of a material discussed below. </p><p>Even though this absorbance function is very useful with scattering samples, the function does not have the same desirable characteristics as it does for non-scattering samples. There is, however, a property called <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Representative_layer_theory#Absorbing_Power:_The_Scatter_Corrected_Absorbance_of_a_sample" title="Representative layer theory">absorbing power</a> which may be estimated for these samples. The <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Representative_layer_theory#Absorbing_Power:_The_Scatter_Corrected_Absorbance_of_a_sample" title="Representative layer theory">absorbing power</a> of a single unit thickness of material making up a scattering sample is the same as the absorbance of the same thickness of the material in the absence of scatter.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Optics">Optics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/web/20240918212750/https://en.wikipedia.org/w/index.php?title=Absorbance&action=edit&section=5" title="Edit section: Optics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Optics" title="Optics">optics</a>, <b>absorbance</b> or <b>decadic absorbance</b> is the <i><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Common_logarithm" title="Common logarithm">common logarithm</a></i> of the ratio of incident to <em>transmitted</em> <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Radiant_power" class="mw-redirect" title="Radiant power">radiant power</a> through a material, and <b>spectral absorbance</b> or <b>spectral decadic absorbance</b> is the common logarithm of the ratio of incident to <em>transmitted</em> <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Radiant_power" class="mw-redirect" title="Radiant power">spectral radiant power</a> through a material. Absorbance is <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Dimensionless_quantity" title="Dimensionless quantity">dimensionless</a>, and in particular is not a length, though it is a monotonically increasing function of path length, and approaches zero as the path length approaches zero. </p> <div class="mw-heading mw-heading2"><h2 id="Mathematical_definitions">Mathematical definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/web/20240918212750/https://en.wikipedia.org/w/index.php?title=Absorbance&action=edit&section=6" title="Edit section: Mathematical definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Absorbance_of_a_material">Absorbance of a material</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/web/20240918212750/https://en.wikipedia.org/w/index.php?title=Absorbance&action=edit&section=7" title="Edit section: Absorbance of a material"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>absorbance</b> of a material, denoted <span class="texhtml mvar" style="font-style:italic;">A</span>, is given by<sup id="cite_ref-GoldBook_1-1" class="reference"><a href="#cite_note-GoldBook-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\log _{10}{\frac {\Phi _{\text{e}}^{\text{i}}}{\Phi _{\text{e}}^{\text{t}}}}=-\log _{10}T,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msubsup> <msubsup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>t</mtext> </mrow> </msubsup> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mi>T</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\log _{10}{\frac {\Phi _{\text{e}}^{\text{i}}}{\Phi _{\text{e}}^{\text{t}}}}=-\log _{10}T,}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/d178c951e237eab4770ad959a7e62162fda39bd7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.365ex; height:6.176ex;" alt="{\displaystyle A=\log _{10}{\frac {\Phi _{\text{e}}^{\text{i}}}{\Phi _{\text{e}}^{\text{t}}}}=-\log _{10}T,}"></span> </p><p>where </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Phi _{\text{e}}^{\text{t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>t</mtext> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \Phi _{\text{e}}^{\text{t}}}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/f18cea8a744ff3dcaf6ac33e1f9398299589cc5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.64ex; height:2.676ex;" alt="{\textstyle \Phi _{\text{e}}^{\text{t}}}"></span> is the <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Radiant_flux" title="Radiant flux">radiant flux</a> <em>transmitted</em> by that material,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Phi _{\text{e}}^{\text{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \Phi _{\text{e}}^{\text{i}}}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/bfcb84df15b69d761f76585a113a8750b1aa98cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.64ex; height:2.843ex;" alt="{\textstyle \Phi _{\text{e}}^{\text{i}}}"></span> is the <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Radiant_flux" title="Radiant flux">radiant flux</a> <em>received</em> by that material, and</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle T=\Phi _{\text{e}}^{\text{t}}/\Phi _{\text{e}}^{\text{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <msubsup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>t</mtext> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msubsup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle T=\Phi _{\text{e}}^{\text{t}}/\Phi _{\text{e}}^{\text{i}}}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/26f83946ddcdb4d02c23978e464139bc2b0fa596" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.178ex; height:3.009ex;" alt="{\textstyle T=\Phi _{\text{e}}^{\text{t}}/\Phi _{\text{e}}^{\text{i}}}"></span> is the <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Transmittance" title="Transmittance">transmittance</a> of that material.</li></ul> <p>Absorbance is a <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Dimensionless" class="mw-redirect" title="Dimensionless">dimensionless</a> quantity. Nevertheless, the <b>absorbance unit</b> or <b>AU</b> is commonly used in <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Ultraviolet%E2%80%93visible_spectroscopy" title="Ultraviolet–visible spectroscopy">ultraviolet–visible spectroscopy</a> and its <a href="/web/20240918212750/https://en.wikipedia.org/wiki/High-performance_liquid_chromatography" title="High-performance liquid chromatography">high-performance liquid chromatography</a> applications, often in derived units such as the milli-absorbance unit (mAU) or milli-absorbance unit-minutes (mAU×min), a unit of absorbance integrated over time.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>Absorbance is related to <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Optical_depth" title="Optical depth">optical depth</a> by </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\frac {\tau }{\ln 10}}=\tau \log _{10}e\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>τ</mi> <mrow> <mi>ln</mi> <mo>⁡</mo> <mn>10</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>τ</mi> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mi>e</mi> <mspace width="thinmathspace"/> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {\tau }{\ln 10}}=\tau \log _{10}e\,,}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/be3be8f1344dcf8cd9714422054bffac895a5f9c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.369ex; height:4.843ex;" alt="{\displaystyle A={\frac {\tau }{\ln 10}}=\tau \log _{10}e\,,}"></span> </p><p>where <span class="texhtml mvar" style="font-style:italic;">τ</span> is the optical depth. </p> <div class="mw-heading mw-heading3"><h3 id="Spectral_absorbance">Spectral absorbance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/web/20240918212750/https://en.wikipedia.org/w/index.php?title=Absorbance&action=edit&section=8" title="Edit section: Spectral absorbance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Spectral absorbance in frequency</b> and <b>spectral absorbance in wavelength</b> of a material, denoted <span class="texhtml"><i>A<sub>ν</sub></i></span> and <span class="texhtml"><i>A<sub>λ</sub></i></span> respectively, are given by<sup id="cite_ref-GoldBook_1-2" class="reference"><a href="#cite_note-GoldBook-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}A_{\nu }&=\log _{10}{\frac {\Phi _{{\text{e}},\nu }^{\text{i}}}{\Phi _{{\text{e}},\nu }^{\text{t}}}}=-\log _{10}T_{\nu }\,,\\A_{\lambda }&=\log _{10}{\frac {\Phi _{{\text{e}},\lambda }^{\text{i}}}{\Phi _{{\text{e}},\lambda }^{\text{t}}}}=-\log _{10}T_{\lambda }\,,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> <mo>,</mo> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msubsup> <msubsup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> <mo>,</mo> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>t</mtext> </mrow> </msubsup> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mspace width="thinmathspace"/> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> <mo>,</mo> <mi>λ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msubsup> <msubsup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> <mo>,</mo> <mi>λ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>t</mtext> </mrow> </msubsup> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> <mspace width="thinmathspace"/> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}A_{\nu }&=\log _{10}{\frac {\Phi _{{\text{e}},\nu }^{\text{i}}}{\Phi _{{\text{e}},\nu }^{\text{t}}}}=-\log _{10}T_{\nu }\,,\\A_{\lambda }&=\log _{10}{\frac {\Phi _{{\text{e}},\lambda }^{\text{i}}}{\Phi _{{\text{e}},\lambda }^{\text{t}}}}=-\log _{10}T_{\lambda }\,,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/92bb391ad8ebc9837a465bfc700511769e35eca9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:31.022ex; height:14.843ex;" alt="{\displaystyle {\begin{aligned}A_{\nu }&=\log _{10}{\frac {\Phi _{{\text{e}},\nu }^{\text{i}}}{\Phi _{{\text{e}},\nu }^{\text{t}}}}=-\log _{10}T_{\nu }\,,\\A_{\lambda }&=\log _{10}{\frac {\Phi _{{\text{e}},\lambda }^{\text{i}}}{\Phi _{{\text{e}},\lambda }^{\text{t}}}}=-\log _{10}T_{\lambda }\,,\end{aligned}}}"></span> </p><p>where </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Phi _{\mathrm {e} ,\nu }^{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mo>,</mo> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \Phi _{\mathrm {e} ,\nu }^{t}}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/22bd69b7f0a5679d3c152ba11dfdf9fe2a8d171a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.969ex; height:3.176ex;" alt="{\textstyle \Phi _{\mathrm {e} ,\nu }^{t}}"></span> is the <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Radiant_flux" title="Radiant flux">spectral radiant flux in frequency</a> <em>transmitted</em> by that material;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Phi _{\mathrm {e} ,\nu }^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mo>,</mo> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \Phi _{\mathrm {e} ,\nu }^{i}}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/70f898f5f4acd83bf56d806628b1a709b2a6dbd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.969ex; height:3.176ex;" alt="{\textstyle \Phi _{\mathrm {e} ,\nu }^{i}}"></span> is the spectral radiant flux in frequency <em>received</em> by that material;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle T_{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle T_{\nu }}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/a8fe9c93df9233befbd628219f5b92336343f22f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.461ex; height:2.509ex;" alt="{\textstyle T_{\nu }}"></span> is the <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Transmittance" title="Transmittance">spectral transmittance in frequency</a> of that material;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Phi _{\mathrm {e} ,\lambda }^{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mo>,</mo> <mi>λ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \Phi _{\mathrm {e} ,\lambda }^{t}}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/67d3be747c1f0fde16f2044ae9b2ee3f20d812cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:4.056ex; height:3.509ex;" alt="{\textstyle \Phi _{\mathrm {e} ,\lambda }^{t}}"></span> is the <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Radiant_flux" title="Radiant flux">spectral radiant flux in wavelength</a> <em>transmitted</em> by that material;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Phi _{\mathrm {e} ,\lambda }^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mo>,</mo> <mi>λ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \Phi _{\mathrm {e} ,\lambda }^{i}}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/d34c4f57d99ef4f0a618840bc5e54a1b45c1a994" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:4.056ex; height:3.509ex;" alt="{\textstyle \Phi _{\mathrm {e} ,\lambda }^{i}}"></span> is the spectral radiant flux in wavelength <em>received</em> by that material; and</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle T_{\lambda }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle T_{\lambda }}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/c8d1300bbd4c573f30e2792d4ef18116318a701c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.548ex; height:2.509ex;" alt="{\textstyle T_{\lambda }}"></span> is the <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Transmittance" title="Transmittance">spectral transmittance in wavelength</a> of that material.</li></ul> <p>Spectral absorbance is related to spectral optical depth by </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}A_{\nu }&={\frac {\tau _{\nu }}{\ln 10}}=\tau _{\nu }\log _{10}e\,,\\A_{\lambda }&={\frac {\tau _{\lambda }}{\ln 10}}=\tau _{\lambda }\log _{10}e\,,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mrow> <mi>ln</mi> <mo>⁡</mo> <mn>10</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mi>e</mi> <mspace width="thinmathspace"/> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> <mrow> <mi>ln</mi> <mo>⁡</mo> <mn>10</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mi>e</mi> <mspace width="thinmathspace"/> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}A_{\nu }&={\frac {\tau _{\nu }}{\ln 10}}=\tau _{\nu }\log _{10}e\,,\\A_{\lambda }&={\frac {\tau _{\lambda }}{\ln 10}}=\tau _{\lambda }\log _{10}e\,,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/0e15ca52ac952ae048d9efb5e0c944ca1be8c76d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.06ex; margin-bottom: -0.278ex; width:25.316ex; height:9.843ex;" alt="{\displaystyle {\begin{aligned}A_{\nu }&={\frac {\tau _{\nu }}{\ln 10}}=\tau _{\nu }\log _{10}e\,,\\A_{\lambda }&={\frac {\tau _{\lambda }}{\ln 10}}=\tau _{\lambda }\log _{10}e\,,\end{aligned}}}"></span> </p><p>where </p> <ul><li><span class="texhtml mvar" style="font-style:italic;">τ<sub>ν</sub></span> is the spectral optical depth in frequency, and</li> <li><span class="texhtml mvar" style="font-style:italic;">τ<sub>λ</sub></span> is the spectral optical depth in wavelength.</li></ul> <p>Although absorbance is properly unitless, it is sometimes reported in "absorbance units", or AU. Many people, including scientific researchers, wrongly state the results from absorbance measurement experiments in terms of these made-up units.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Relationship_with_attenuation">Relationship with attenuation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/web/20240918212750/https://en.wikipedia.org/w/index.php?title=Absorbance&action=edit&section=9" title="Edit section: Relationship with attenuation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Attenuance">Attenuance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/web/20240918212750/https://en.wikipedia.org/w/index.php?title=Absorbance&action=edit&section=10" title="Edit section: Attenuance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Absorbance is a number that measures the <i>attenuation</i> of the transmitted radiant power in a material. Attenuation can be caused by the physical process of "absorption", but also reflection, scattering, and other physical processes. Absorbance of a material is approximately equal to its attenuance<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="This term desperately needs a definition, otherwise the whole point of the fine distinctions in this section will be lost. Is attenuance a numerical measure of the physical process of attenuation, or are they also, like absorbance and absorption, unexpectedly distinct? (April 2015)">clarification needed</span></a></i>]</sup> when both the absorbance is much less than 1 and the emittance of that material (not to be confused with <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Radiant_exitance" title="Radiant exitance">radiant exitance</a> or <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Emissivity" title="Emissivity">emissivity</a>) is much less than the absorbance. Indeed, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{\mathrm {e} }^{\mathrm {t} }+\Phi _{\mathrm {e} }^{\mathrm {att} }=\Phi _{\mathrm {e} }^{\mathrm {i} }+\Phi _{\mathrm {e} }^{\mathrm {e} }\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msubsup> <mspace width="thinmathspace"/> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{\mathrm {e} }^{\mathrm {t} }+\Phi _{\mathrm {e} }^{\mathrm {att} }=\Phi _{\mathrm {e} }^{\mathrm {i} }+\Phi _{\mathrm {e} }^{\mathrm {e} }\,,}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/821addb692e94e9f0bc94cfa9531e59683f999ef" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.746ex; height:2.843ex;" alt="{\displaystyle \Phi _{\mathrm {e} }^{\mathrm {t} }+\Phi _{\mathrm {e} }^{\mathrm {att} }=\Phi _{\mathrm {e} }^{\mathrm {i} }+\Phi _{\mathrm {e} }^{\mathrm {e} }\,,}"></span> </p><p>where </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Phi _{\mathrm {e} }^{\mathrm {t} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \Phi _{\mathrm {e} }^{\mathrm {t} }}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/005eaf78d78ea73caa04b731782f4f376145fa3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.64ex; height:2.676ex;" alt="{\textstyle \Phi _{\mathrm {e} }^{\mathrm {t} }}"></span> is the radiant power transmitted by that material,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Phi _{\mathrm {e} }^{\mathrm {att} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \Phi _{\mathrm {e} }^{\mathrm {att} }}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/fae39e95a7e8002cdb381b362a87b22b7808fea9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.012ex; height:2.676ex;" alt="{\textstyle \Phi _{\mathrm {e} }^{\mathrm {att} }}"></span> is the radiant power attenuated by that material,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Phi _{\mathrm {e} }^{\mathrm {i} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \Phi _{\mathrm {e} }^{\mathrm {i} }}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/3c8310892faa35157e8c516d21cef3e6786ebd0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.64ex; height:2.843ex;" alt="{\textstyle \Phi _{\mathrm {e} }^{\mathrm {i} }}"></span> is the radiant power received by that material, and</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Phi _{\mathrm {e} }^{\mathrm {e} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \Phi _{\mathrm {e} }^{\mathrm {e} }}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/1945e04231a74006e77d0245ea31933a4aee1a46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.64ex; height:2.509ex;" alt="{\textstyle \Phi _{\mathrm {e} }^{\mathrm {e} }}"></span> is the radiant power emitted by that material.</li></ul> <p>This is equivalent to </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T+\mathrm {ATT} =1+E\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">T</mi> <mi mathvariant="normal">T</mi> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>E</mi> <mspace width="thinmathspace"/> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T+\mathrm {ATT} =1+E\,,}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/3e279c0881035446960897153ab8c9e578140d37" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.487ex; height:2.509ex;" alt="{\displaystyle T+\mathrm {ATT} =1+E\,,}"></span> </p><p>where </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle T=\Phi _{\mathrm {e} }^{\mathrm {t} }/\Phi _{\mathrm {e} }^{\mathrm {i} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <msubsup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msubsup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle T=\Phi _{\mathrm {e} }^{\mathrm {t} }/\Phi _{\mathrm {e} }^{\mathrm {i} }}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/b69b12a92854a20a4cd68a7ad06be3ac105efa0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.178ex; height:3.009ex;" alt="{\textstyle T=\Phi _{\mathrm {e} }^{\mathrm {t} }/\Phi _{\mathrm {e} }^{\mathrm {i} }}"></span> is the transmittance of that material,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathrm {ATT} =\Phi _{\mathrm {e} }^{\mathrm {att} }/\Phi _{\mathrm {e} }^{\mathrm {i} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">T</mi> <mi mathvariant="normal">T</mi> </mrow> <mo>=</mo> <msubsup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msubsup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \mathrm {ATT} =\Phi _{\mathrm {e} }^{\mathrm {att} }/\Phi _{\mathrm {e} }^{\mathrm {i} }}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/c327167c3c560d1f1181844b231ce0c5b96a15ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.012ex; height:3.009ex;" alt="{\textstyle \mathrm {ATT} =\Phi _{\mathrm {e} }^{\mathrm {att} }/\Phi _{\mathrm {e} }^{\mathrm {i} }}"></span> is the <em>attenuance</em> of that material,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle E=\Phi _{\mathrm {e} }^{\mathrm {e} }/\Phi _{\mathrm {e} }^{\mathrm {i} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <msubsup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msubsup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle E=\Phi _{\mathrm {e} }^{\mathrm {e} }/\Phi _{\mathrm {e} }^{\mathrm {i} }}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/6deab11d3f8061c5cd87927013a832831281574a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.317ex; height:3.009ex;" alt="{\textstyle E=\Phi _{\mathrm {e} }^{\mathrm {e} }/\Phi _{\mathrm {e} }^{\mathrm {i} }}"></span> is the emittance of that material.</li></ul> <p>According to the <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Beer%E2%80%93Lambert_law" title="Beer–Lambert law">Beer–Lambert law</a>, <span class="texhtml"><i>T</i> = 10<sup>-<i>A</i></sup></span>, so </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {ATT} =1-10^{-A}+E\approx A\ln 10+E,\quad {\text{if}}\ A\ll 1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">T</mi> <mi mathvariant="normal">T</mi> </mrow> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>A</mi> </mrow> </msup> <mo>+</mo> <mi>E</mi> <mo>≈</mo> <mi>A</mi> <mi>ln</mi> <mo>⁡</mo> <mn>10</mn> <mo>+</mo> <mi>E</mi> <mo>,</mo> <mspace width="1em"/> <mrow class="MJX-TeXAtom-ORD"> <mtext>if</mtext> </mrow> <mtext> </mtext> <mi>A</mi> <mo>≪</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {ATT} =1-10^{-A}+E\approx A\ln 10+E,\quad {\text{if}}\ A\ll 1,}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/5216ddd5a920b6f9c604c4a176b1b013cae00c2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:48.843ex; height:3.009ex;" alt="{\displaystyle \mathrm {ATT} =1-10^{-A}+E\approx A\ln 10+E,\quad {\text{if}}\ A\ll 1,}"></span></li></ul> <p>and finally </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {ATT} \approx A\ln 10,\quad {\text{if}}\ E\ll A.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">T</mi> <mi mathvariant="normal">T</mi> </mrow> <mo>≈</mo> <mi>A</mi> <mi>ln</mi> <mo>⁡</mo> <mn>10</mn> <mo>,</mo> <mspace width="1em"/> <mrow class="MJX-TeXAtom-ORD"> <mtext>if</mtext> </mrow> <mtext> </mtext> <mi>E</mi> <mo>≪</mo> <mi>A</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {ATT} \approx A\ln 10,\quad {\text{if}}\ E\ll A.}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/9fc2d4e1b9cc77ed0f50f91448398af91e12c6c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.055ex; height:2.509ex;" alt="{\displaystyle \mathrm {ATT} \approx A\ln 10,\quad {\text{if}}\ E\ll A.}"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Attenuation_coefficient">Attenuation coefficient</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/web/20240918212750/https://en.wikipedia.org/w/index.php?title=Absorbance&action=edit&section=11" title="Edit section: Attenuation coefficient"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Absorbance of a material is also related to its <i><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Attenuation_coefficient" title="Attenuation coefficient">decadic attenuation coefficient</a></i> by </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\int _{0}^{l}a(z)\,\mathrm {d} z\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msubsup> <mi>a</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"/> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>z</mi> <mspace width="thinmathspace"/> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\int _{0}^{l}a(z)\,\mathrm {d} z\,,}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/0f9052538d6e764ab1aa3f2f7a47d877a6e14498" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.344ex; height:6.343ex;" alt="{\displaystyle A=\int _{0}^{l}a(z)\,\mathrm {d} z\,,}"></span> </p><p>where </p> <ul><li><span class="texhtml mvar" style="font-style:italic;">l</span> is the thickness of that material through which the light travels, and</li> <li><span class="texhtml"><i>a</i>(<i>z</i>)</span> is the <i>decadic attenuation coefficient</i> of that material at <span class="texhtml mvar" style="font-style:italic;">z</span>.</li></ul> <p>If <i>a</i>(<i>z</i>) is uniform along the path, the attenuation is said to be a <i>linear attenuation</i>, and the relation becomes <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=al.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>a</mi> <mi>l</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=al.}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/0ca0efb6ebb61943e6cf3e1e7c86cd9604715ce3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.412ex; height:2.176ex;" alt="{\displaystyle A=al.}"></span> </p><p>Sometimes the relation is given using the <i><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Molar_attenuation_coefficient" class="mw-redirect" title="Molar attenuation coefficient">molar attenuation coefficient</a></i> of the material, that is its attenuation coefficient divided by its <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Molar_concentration" title="Molar concentration">molar concentration</a>: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\int _{0}^{l}\varepsilon c(z)\,\mathrm {d} z\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msubsup> <mi>ε</mi> <mi>c</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"/> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>z</mi> <mspace width="thinmathspace"/> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\int _{0}^{l}\varepsilon c(z)\,\mathrm {d} z\,,}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/644c3875b744a5acfe963d9188bb34787150e1a6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.205ex; height:6.343ex;" alt="{\displaystyle A=\int _{0}^{l}\varepsilon c(z)\,\mathrm {d} z\,,}"></span> </p><p>where </p> <ul><li><span class="texhtml mvar" style="font-style:italic;">ε</span> is the <i>molar attenuation coefficient</i> of that material, and</li> <li><span class="texhtml"><i>c</i>(<i>z</i>)</span> is the molar concentration of that material at <span class="texhtml mvar" style="font-style:italic;">z</span>.</li></ul> <p>If <span class="texhtml"><i>c</i>(<i>z</i>)</span> is uniform along the path, the relation becomes </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\varepsilon cl\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>ε</mi> <mi>c</mi> <mi>l</mi> <mspace width="thinmathspace"/> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\varepsilon cl\,.}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/fa2bec134b469726b3cb58d101ce53f552103b87" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.659ex; height:2.176ex;" alt="{\displaystyle A=\varepsilon cl\,.}"></span> </p><p>The use of the term "molar absorptivity" for molar attenuation coefficient is discouraged.<sup id="cite_ref-GoldBook_1-3" class="reference"><a href="#cite_note-GoldBook-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Measurements">Measurements</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/web/20240918212750/https://en.wikipedia.org/w/index.php?title=Absorbance&action=edit&section=12" title="Edit section: Measurements"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Logarithmic_vs._directly_proportional_measurements">Logarithmic vs. directly proportional measurements</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/web/20240918212750/https://en.wikipedia.org/w/index.php?title=Absorbance&action=edit&section=13" title="Edit section: Logarithmic vs. directly proportional measurements"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The amount of light transmitted through a material diminishes <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Exponential_function" title="Exponential function">exponentially</a> as it travels through the material, according to the Beer–Lambert law (<span class="texhtml"><i>A</i> = (<i>ε</i>)(<i>l</i>)</span>). Since the absorbance of a sample is measured as a logarithm, it is directly proportional to the thickness of the sample and to the concentration of the absorbing material in the sample. Some other measures related to absorption, such as transmittance, are measured as a simple ratio so they vary exponentially with the thickness and concentration of the material. </p> <table class="wikitable sortable" style="text-align: center;"> <caption>Absorbances and equivalent transmittances </caption> <tbody><tr> <th scope="col">Absorbance: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle -\log _{10}\left(\Phi _{\mathrm {e} }^{\mathrm {t} }/\Phi _{\mathrm {e} }^{\mathrm {i} }\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>−</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msubsup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle -\log _{10}\left(\Phi _{\mathrm {e} }^{\mathrm {t} }/\Phi _{\mathrm {e} }^{\mathrm {i} }\right)}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/90ad418c301c09a39e66a714ff1270103d02d4e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.616ex; height:3.176ex;" alt="{\textstyle -\log _{10}\left(\Phi _{\mathrm {e} }^{\mathrm {t} }/\Phi _{\mathrm {e} }^{\mathrm {i} }\right)}"></span> </th> <th scope="col">Transmittance: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Phi _{\mathrm {e} }^{\mathrm {t} }/\Phi _{\mathrm {e} }^{\mathrm {i} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msubsup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \Phi _{\mathrm {e} }^{\mathrm {t} }/\Phi _{\mathrm {e} }^{\mathrm {i} }}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/9edf55f9bf2db888f60c63f805f6a2167ba0cc16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.443ex; height:3.009ex;" alt="{\textstyle \Phi _{\mathrm {e} }^{\mathrm {t} }/\Phi _{\mathrm {e} }^{\mathrm {i} }}"></span> </th></tr> <tr> <td>0 </td> <td>1 </td></tr> <tr> <td>0.1 </td> <td>0.79 </td></tr> <tr> <td>0.25 </td> <td>0.56 </td></tr> <tr> <td>0.5 </td> <td>0.32 </td></tr> <tr> <td>0.75 </td> <td>0.18 </td></tr> <tr> <td>0.9 </td> <td>0.13 </td></tr> <tr> <td>1 </td> <td>0.1 </td></tr> <tr> <td>2 </td> <td>0.01 </td></tr> <tr> <td>3 </td> <td>0.001 </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Instrument_measurement_range">Instrument measurement range</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/web/20240918212750/https://en.wikipedia.org/w/index.php?title=Absorbance&action=edit&section=14" title="Edit section: Instrument measurement range"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Any real measuring instrument has a limited range over which it can accurately measure absorbance. An instrument must be calibrated and checked against known standards if the readings are to be trusted. Many instruments will become non-linear (fail to follow the Beer–Lambert law) starting at approximately 2 AU (~1% transmission). It is also difficult to accurately measure very small absorbance values (below <span class="nowrap"><span data-sort-value="6996100000000000000♠"></span>10<sup>−4</sup></span>) with commercially available instruments for chemical analysis. In such cases, <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Laser_absorption_spectrometry" title="Laser absorption spectrometry">laser-based absorption techniques</a> can be used, since they have demonstrated detection limits that supersede those obtained by conventional non-laser-based instruments by many orders of magnitude (detection has been demonstrated all the way down to <span class="nowrap"><span data-sort-value="6987500000000000000♠"></span>5<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−13</sup></span>). The theoretical best accuracy for most commercially available non-laser-based instruments is attained in the range near 1 AU. The path length or concentration should then, when possible, be adjusted to achieve readings near this range. </p> <div class="mw-heading mw-heading3"><h3 id="Method_of_measurement">Method of measurement</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/web/20240918212750/https://en.wikipedia.org/w/index.php?title=Absorbance&action=edit&section=15" title="Edit section: Method of measurement"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Typically, absorbance of a dissolved substance is measured using <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Absorption_spectroscopy" title="Absorption spectroscopy">absorption spectroscopy</a>. This involves shining a light through a solution and recording how much light and what wavelengths were transmitted onto a detector. Using this information, the wavelengths that were absorbed can be determined.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> First, measurements on a "blank" are taken using just the solvent for reference purposes. This is so that the absorbance of the solvent is known, and then any change in absorbance when measuring the whole solution is made by just the solute of interest. Then measurements of the solution are taken. The transmitted spectral radiant flux that makes it through the solution sample is measured and compared to the incident spectral radiant flux. As stated above, the spectral absorbance at a given wavelength is </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{\lambda }=\log _{10}\!\left({\frac {\Phi _{\mathrm {e} ,\lambda }^{\mathrm {i} }}{\Phi _{\mathrm {e} ,\lambda }^{\mathrm {t} }}}\right)\!.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mspace width="negativethinmathspace"/> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mo>,</mo> <mi>λ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msubsup> <msubsup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mo>,</mo> <mi>λ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msubsup> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="negativethinmathspace"/> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{\lambda }=\log _{10}\!\left({\frac {\Phi _{\mathrm {e} ,\lambda }^{\mathrm {i} }}{\Phi _{\mathrm {e} ,\lambda }^{\mathrm {t} }}}\right)\!.}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/5ece9af2b44101ff801dcfda735ac5c7d87cec47" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:20.1ex; height:7.676ex;" alt="{\displaystyle A_{\lambda }=\log _{10}\!\left({\frac {\Phi _{\mathrm {e} ,\lambda }^{\mathrm {i} }}{\Phi _{\mathrm {e} ,\lambda }^{\mathrm {t} }}}\right)\!.}"></span> </p><p>The absorbance spectrum is plotted on a graph of absorbance vs. wavelength.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>An <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Ultraviolet-visible_spectroscopy#Ultraviolet–visible_spectrophotometer" class="mw-redirect" title="Ultraviolet-visible spectroscopy">Ultraviolet-visible spectroscopy#Ultraviolet–visible spectrophotometer</a> will do all this automatically. To use this machine, solutions are placed in a small <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Cuvette" title="Cuvette">cuvette</a> and inserted into the holder. The machine is controlled through a computer and, once it has been "blanked", automatically displays the absorbance plotted against wavelength. Getting the absorbance spectrum of a solution is useful for determining the concentration of that solution using the Beer–Lambert law and is used in <a href="/web/20240918212750/https://en.wikipedia.org/wiki/HPLC" class="mw-redirect" title="HPLC">HPLC</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Shade_number">Shade number</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/web/20240918212750/https://en.wikipedia.org/w/index.php?title=Absorbance&action=edit&section=16" title="Edit section: Shade number"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some filters, notably <a href="/web/20240918212750/https://en.wikipedia.org/wiki/Welding" title="Welding">welding</a> glass, are rated by shade number (SN), which is 7/3 times the absorbance plus one:<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathrm {SN} &={\frac {7}{3}}A+1\\&={\frac {7}{3}}(-\log _{10}T)+1\,.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">N</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>3</mn> </mfrac> </mrow> <mi>A</mi> <mo>+</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd/> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>3</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> <mspace width="thinmathspace"/> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathrm {SN} &={\frac {7}{3}}A+1\\&={\frac {7}{3}}(-\log _{10}T)+1\,.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://web.archive.org/web/20240918212750im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/bccf743eac3789c9c4a70b7dbfbb1fcbbe14d008" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.558ex; margin-bottom: -0.28ex; width:24.797ex; height:10.843ex;" alt="{\displaystyle {\begin{aligned}\mathrm {SN} &={\frac {7}{3}}A+1\\&={\frac {7}{3}}(-\log _{10}T)+1\,.\end{aligned}}}"></span> </p><p>For example, if the filter has 0.1% transmittance (0.001 transmittance, which is 3 absorbance units), its shade number would be 8. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/web/20240918212750/https://en.wikipedia.org/w/index.php?title=Absorbance&action=edit&section=17" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Absorptance" title="Absorptance">Absorptance</a></li> <li><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Tunable_Diode_Laser_Absorption_Spectroscopy" class="mw-redirect" title="Tunable Diode Laser Absorption Spectroscopy">Tunable Diode Laser Absorption Spectroscopy</a> (TDLAS)</li> <li><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Densitometry" title="Densitometry">Densitometry</a></li> <li><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Neutral_density_filter" class="mw-redirect" title="Neutral density filter">Neutral density filter</a></li> <li><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Mathematical_descriptions_of_opacity" title="Mathematical descriptions of opacity">Mathematical descriptions of opacity</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/web/20240918212750/https://en.wikipedia.org/w/index.php?title=Absorbance&action=edit&section=18" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-GoldBook-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-GoldBook_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-GoldBook_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-GoldBook_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-GoldBook_1-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><a href="/web/20240918212750/https://en.wikipedia.org/wiki/International_Union_of_Pure_and_Applied_Chemistry" title="International Union of Pure and Applied Chemistry">IUPAC</a>, <i><a href="/web/20240918212750/https://en.wikipedia.org/wiki/IUPAC_books#Gold_Book" class="mw-redirect" title="IUPAC books">Compendium of Chemical Terminology</a></i>, 2nd ed. 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A Dictionary of Units of Measurement"</a>. Unc.edu. Archived from <a rel="nofollow" class="external text" href="https://web.archive.org/web/20240918212750/http://www.unc.edu/~rowlett/units/dictS.html">the original</a> on 1998-12-03<span class="reference-accessdate">. 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.navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Protein_structural_analysis" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2" style="background:#eee8aa;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Template:Protein_structural_analysis" title="Template:Protein structural analysis"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Template_talk:Protein_structural_analysis" title="Template talk:Protein structural analysis"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Special:EditPage/Template:Protein_structural_analysis" title="Special:EditPage/Template:Protein structural analysis"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Protein_structural_analysis" style="font-size:114%;margin:0 4em"><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Protein_structure" title="Protein structure">Protein structural analysis</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#eee8aa;">High resolution</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Cryo-electron_microscopy" class="mw-redirect" title="Cryo-electron microscopy">Cryo-electron microscopy</a></li> <li><a href="/web/20240918212750/https://en.wikipedia.org/wiki/X-ray_crystallography" title="X-ray crystallography">X-ray crystallography</a></li> <li><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Protein_nuclear_magnetic_resonance_spectroscopy" class="mw-redirect" title="Protein nuclear magnetic resonance spectroscopy">NMR</a></li> <li><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Electron_crystallography" title="Electron crystallography">Electron crystallography</a></li> <li><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Electron_paramagnetic_resonance" title="Electron paramagnetic resonance">EPR</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#eee8aa;">Medium resolution</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Fiber_diffraction" title="Fiber diffraction">Fiber diffraction</a></li> <li><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Mass_spectrometry" title="Mass spectrometry">Mass spectrometry</a></li> <li><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Biological_Small-Angle_X-ray_Scattering" class="mw-redirect" title="Biological Small-Angle X-ray Scattering">SAXS</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#eee8aa;">Spectroscopic</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Protein_nuclear_magnetic_resonance_spectroscopy" class="mw-redirect" title="Protein nuclear magnetic resonance spectroscopy">NMR</a></li> <li><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Circular_dichroism" title="Circular dichroism">Circular dichroism</a></li> <li><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Dual-polarization_interferometry" title="Dual-polarization interferometry">Dual-polarization interferometry</a></li> <li><a class="mw-selflink selflink">Absorbance</a></li> <li><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Fluorescence" title="Fluorescence">Fluorescence</a></li> <li><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Fluorescence_anisotropy" title="Fluorescence anisotropy">Fluorescence anisotropy</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#eee8aa;">Translational Diffusion</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Ultracentrifuge#Analytical_ultracentrifuge" title="Ultracentrifuge">Analytical ultracentrifugation</a></li> <li><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Size_exclusion_chromatography" class="mw-redirect" title="Size exclusion chromatography">Size exclusion chromatography</a></li> <li><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Dynamic_light_scattering" title="Dynamic light scattering">Light scattering</a></li> <li><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Protein_nuclear_magnetic_resonance_spectroscopy" class="mw-redirect" title="Protein nuclear magnetic resonance spectroscopy">NMR</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#eee8aa;">Rotational Diffusion</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Fluorescence_anisotropy" title="Fluorescence anisotropy">Fluorescence anisotropy</a></li> <li><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Flow_birefringence" title="Flow birefringence">Flow birefringence</a></li> <li><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Dielectric_relaxation" class="mw-redirect" title="Dielectric relaxation">Dielectric relaxation</a></li> <li><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Protein_nuclear_magnetic_resonance_spectroscopy" class="mw-redirect" title="Protein nuclear magnetic resonance spectroscopy">NMR</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#eee8aa;">Chemical</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Hydrogen%E2%80%93deuterium_exchange" title="Hydrogen–deuterium exchange">Hydrogen–deuterium exchange</a></li> <li><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Site-directed_mutagenesis" title="Site-directed mutagenesis">Site-directed mutagenesis</a></li> <li><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Chemical_modification" title="Chemical modification">Chemical modification</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#eee8aa;">Thermodynamic</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Equilibrium_unfolding" title="Equilibrium unfolding">Equilibrium unfolding</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#eee8aa;">Computational</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Protein_structure_prediction" title="Protein structure prediction">Protein structure prediction</a></li> <li><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Docking_(molecular)" title="Docking (molecular)">Molecular docking</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2" style="background:#eee8aa;"><div><div style="float:left;"><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Protein_tertiary_structure" title="Protein tertiary structure">←Tertiary structure</a></div><div style="float:right;"><a href="/web/20240918212750/https://en.wikipedia.org/wiki/Protein_quaternary_structure" title="Protein quaternary structure">Quaternary structure→</a></div></div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"><style 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