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class="vector-toc-numb">2.2</span> <span>Elastic and inelastic scattering</span> </div> </a> <ul id="toc-Elastic_and_inelastic_scattering-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mathematical_framework" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mathematical_framework"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Mathematical framework</span> </div> </a> <ul id="toc-Mathematical_framework-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Theoretical_physics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Theoretical_physics"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Theoretical physics</span> </div> </a> <ul id="toc-Theoretical_physics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Electromagnetics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Electromagnetics"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Electromagnetics</span> </div> </a> <ul id="toc-Electromagnetics-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">5</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">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>External links</span> </div> </a> <ul id="toc-External_links-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 cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Scattering</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 45 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-45" 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">45 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Verstrooiing" title="Verstrooiing – Afrikaans" lang="af" hreflang="af" data-title="Verstrooiing" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%A8%D8%B9%D8%AB%D8%B1" title="تبعثر – Arabic" lang="ar" hreflang="ar" data-title="تبعثر" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/S%C9%99p%C9%99l%C9%99n%C9%99n_radiasiya" title="Səpələnən radiasiya – Azerbaijani" lang="az" hreflang="az" data-title="Səpələnən radiasiya" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A6%BF%E0%A6%95%E0%A7%8D%E0%A6%B7%E0%A7%87%E0%A6%AA%E0%A6%A3" title="বিক্ষেপণ – Bangla" lang="bn" hreflang="bn" data-title="বিক্ষেপণ" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A0%D0%B0%D1%81%D1%81%D0%B5%D1%8F%D0%BD%D0%BD%D0%B5_%D0%BC%D1%96%D0%BA%D1%80%D0%B0%D1%87%D0%B0%D1%81%D1%86%D1%96%D1%86" title="Рассеянне мікрачасціц – Belarusian" lang="be" hreflang="be" data-title="Рассеянне мікрачасціц" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B7%D1%81%D0%B5%D0%B9%D0%B2%D0%B0%D0%BD%D0%B5" title="Разсейване – Bulgarian" lang="bg" hreflang="bg" data-title="Разсейване" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Dispersi%C3%B3_(f%C3%ADsica)" title="Dispersió (física) – Catalan" lang="ca" hreflang="ca" data-title="Dispersió (física)" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A1%D0%B0%D0%BF%D0%B0%D0%BB%D0%B0%D0%BD%D0%B0%D0%B2" title="Сапаланав – Chuvash" lang="cv" hreflang="cv" data-title="Сапаланав" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Spredning_(fysik)" title="Spredning (fysik) – Danish" lang="da" hreflang="da" data-title="Spredning (fysik)" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Streuung_(Physik)" title="Streuung (Physik) – German" lang="de" hreflang="de" data-title="Streuung (Physik)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Hajumine" title="Hajumine – Estonian" lang="et" hreflang="et" data-title="Hajumine" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A3%CE%BA%CE%AD%CE%B4%CE%B1%CF%83%CE%B7" title="Σκέδαση – Greek" lang="el" hreflang="el" data-title="Σκέδαση" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Dispersi%C3%B3n_(f%C3%ADsica)" title="Dispersión (física) – Spanish" lang="es" hreflang="es" data-title="Dispersión (física)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%BE%D8%B1%D8%A7%DA%A9%D9%86%D8%AF%DA%AF%DB%8C" title="پراکندگی – Persian" lang="fa" hreflang="fa" data-title="پراکندگی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Diffusion_des_ondes" title="Diffusion des ondes – French" lang="fr" hreflang="fr" data-title="Diffusion des ondes" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Scaipeadh_(fisic)" title="Scaipeadh (fisic) – Irish" lang="ga" hreflang="ga" data-title="Scaipeadh (fisic)" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%82%B0%EB%9E%80" title="산란 – Korean" lang="ko" hreflang="ko" data-title="산란" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%95%E0%A5%80%E0%A4%B0%E0%A5%8D%E0%A4%A3%E0%A4%A8" title="प्रकीर्णन – Hindi" lang="hi" hreflang="hi" data-title="प्रकीर्णन" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Raspr%C5%A1enje" title="Raspršenje – Croatian" lang="hr" hreflang="hr" data-title="Raspršenje" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Penyebaran" title="Penyebaran – Indonesian" lang="id" hreflang="id" data-title="Penyebaran" 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://it.wikipedia.org/wiki/Scattering" title="Scattering – Italian" lang="it" hreflang="it" data-title="Scattering" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%99%D7%96%D7%95%D7%A8" title="פיזור – Hebrew" lang="he" hreflang="he" data-title="פיזור" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Sz%C3%B3r%C3%A1s_(fizika)" title="Szórás (fizika) – Hungarian" lang="hu" hreflang="hu" data-title="Szórás (fizika)" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B5%E0%B4%BF%E0%B4%B8%E0%B4%B0%E0%B4%A3%E0%B4%82" title="വിസരണം – Malayalam" lang="ml" hreflang="ml" data-title="വിസരണം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Serakan" title="Serakan – Malay" lang="ms" hreflang="ms" data-title="Serakan" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Verstrooiing_(natuurkunde)" title="Verstrooiing (natuurkunde) – Dutch" lang="nl" hreflang="nl" data-title="Verstrooiing (natuurkunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%95%A3%E4%B9%B1" title="散乱 – Japanese" lang="ja" hreflang="ja" data-title="散乱" 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-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Spredning" title="Spredning – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Spredning" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Spreiing" title="Spreiing – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Spreiing" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Yorug%27likning_sochilishi" title="Yorug'likning sochilishi – Uzbek" lang="uz" hreflang="uz" data-title="Yorug'likning sochilishi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Dispers%C3%A3o_(f%C3%ADsica)" title="Dispersão (física) – Portuguese" lang="pt" hreflang="pt" data-title="Dispersão (física)" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A0%D0%B0%D1%81%D1%81%D0%B5%D1%8F%D0%BD%D0%B8%D0%B5" title="Рассеяние – Russian" lang="ru" hreflang="ru" data-title="Рассеяние" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Scattering" title="Scattering – Simple English" lang="en-simple" hreflang="en-simple" data-title="Scattering" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Rozptyl_(%C5%BEiarenie)" title="Rozptyl (žiarenie) – Slovak" lang="sk" hreflang="sk" data-title="Rozptyl (žiarenie)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Sipanje" title="Sipanje – Slovenian" lang="sl" hreflang="sl" data-title="Sipanje" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A0%D0%B0%D1%81%D0%B5%D1%98%D0%B0%D1%9A%D0%B5" title="Расејање – Serbian" lang="sr" hreflang="sr" data-title="Расејање" 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://sh.wikipedia.org/wiki/Raspr%C5%A1enje" title="Raspršenje – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Raspršenje" 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-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Sironta" title="Sironta – Finnish" lang="fi" hreflang="fi" data-title="Sironta" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Spridning_(fysik)" title="Spridning (fysik) – Swedish" lang="sv" hreflang="sv" data-title="Spridning (fysik)" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%9A%E0%AE%BF%E0%AE%A4%E0%AE%B1%E0%AE%BF%E0%AE%AF_%E0%AE%95%E0%AE%A4%E0%AE%BF%E0%AE%B0%E0%AF%8D%E0%AE%B5%E0%AF%80%E0%AE%9A%E0%AF%8D%E0%AE%9A%E0%AF%81" title="சிதறிய கதிர்வீச்சு – Tamil" lang="ta" hreflang="ta" data-title="சிதறிய கதிர்வீச்சு" 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://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%81%E0%B8%A3%E0%B8%B0%E0%B9%80%E0%B8%88%E0%B8%B4%E0%B8%87" title="การกระเจิง – Thai" lang="th" hreflang="th" data-title="การกระเจิง" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Sa%C3%A7%C4%B1lma" title="Saçılma – Turkish" lang="tr" hreflang="tr" data-title="Saçılma" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D0%BE%D0%B7%D1%81%D1%96%D1%8E%D0%B2%D0%B0%D0%BD%D0%BD%D1%8F_%D1%87%D0%B0%D1%81%D1%82%D0%B8%D0%BD%D0%BE%D0%BA_%D1%96_%D1%85%D0%B2%D0%B8%D0%BB%D1%8C" title="Розсіювання частинок і хвиль – Ukrainian" lang="uk" hreflang="uk" data-title="Розсіювання частинок і хвиль" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/T%C3%A1n_x%E1%BA%A1" title="Tán xạ – Vietnamese" lang="vi" hreflang="vi" data-title="Tán xạ" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%95%A3%E5%B0%84" title="散射 – Chinese" lang="zh" hreflang="zh" data-title="散射" 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://www.wikidata.org/wiki/Special:EntityPage/Q210028#sitelinks-wikipedia" title="Edit interlanguage links" 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.sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar nomobile nowraplinks"><tbody><tr><th class="sidebar-title"><a class="mw-selflink selflink">Scattering</a></th></tr><tr><td class="sidebar-image"><span typeof="mw:File"><a href="/wiki/File:Electron-scattering.svg" class="mw-file-description" title="Electron scattering"><img alt="Electron scattering" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Electron-scattering.svg/150px-Electron-scattering.svg.png" decoding="async" width="150" height="129" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Electron-scattering.svg/225px-Electron-scattering.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Electron-scattering.svg/300px-Electron-scattering.svg.png 2x" data-file-width="976" data-file-height="841" /></a></span><div class="sidebar-caption" style="padding-bottom:0.4em;border-bottom:1px solid #aaa;"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Feynman_diagram" title="Feynman diagram">Feynman diagram</a> of scattering between two electrons by emission of a virtual <a href="/wiki/Photon" title="Photon">photon</a></div></div></td></tr><tr><td class="sidebar-content hlist" style="padding-bottom:0.5em;"> <ul><li><a href="/wiki/Bragg%27s_law" title="Bragg's law">Bragg diffraction</a></li> <li><a href="/wiki/Brillouin_scattering" title="Brillouin scattering">Brillouin</a></li> <li><a href="/wiki/Compton_scattering" title="Compton scattering">Compton</a></li> <li><a href="/wiki/Dynamic_light_scattering" title="Dynamic light scattering">Dynamic light</a></li> <li><a href="/wiki/Kikuchi_line_(solid_state_physics)" class="mw-redirect" title="Kikuchi line (solid state physics)">Kikuchi lines</a></li> <li><a href="/wiki/Light_scattering_by_particles" title="Light scattering by particles">Light scattering by particles</a></li> <li><a href="/wiki/Mie_scattering" title="Mie scattering">Mie</a></li> <li><a href="/wiki/Neutron_scattering" title="Neutron scattering">Neutron</a></li> <li><a href="/wiki/Powder_diffraction" title="Powder diffraction">Powder diffraction</a></li> <li><a href="/wiki/Raman_scattering" title="Raman scattering">Raman</a></li> <li><a href="/wiki/Rayleigh_scattering" title="Rayleigh scattering">Rayleigh</a></li> <li><a href="/wiki/Rutherford_scattering" class="mw-redirect" title="Rutherford scattering">Rutherford</a></li> <li><a href="/wiki/Small-angle_scattering" title="Small-angle scattering">Small-angle</a></li> <li><a href="/wiki/Tyndall_effect" title="Tyndall effect">Tyndall</a></li> <li><a href="/wiki/Thomson_scattering" title="Thomson scattering">Thomson</a></li> <li><a href="/wiki/Wolf_effect" title="Wolf effect">Wolf effect</a></li> <li><a href="/wiki/X-ray_crystallography" title="X-ray crystallography">X-ray crystallography</a></li></ul></td> </tr><tr><td class="sidebar-navbar" style="border-top:1px solid #aaa;"><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 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href="/wiki/Template:Scattering" title="Template:Scattering"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Scattering" title="Template talk:Scattering"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Scattering" title="Special:EditPage/Template:Scattering"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Wine_glass_in_LCD_projectors_beam.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Wine_glass_in_LCD_projectors_beam.jpg/220px-Wine_glass_in_LCD_projectors_beam.jpg" decoding="async" width="220" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Wine_glass_in_LCD_projectors_beam.jpg/330px-Wine_glass_in_LCD_projectors_beam.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Wine_glass_in_LCD_projectors_beam.jpg/440px-Wine_glass_in_LCD_projectors_beam.jpg 2x" data-file-width="5009" data-file-height="3339" /></a><figcaption><a href="/wiki/Wine_glass" title="Wine glass">Wine glass</a> in <a href="/wiki/LCD_projector" title="LCD projector">LCD projectors</a> light beam makes the beam scatter.</figcaption></figure> <p>In physics, <b>scattering</b> is a wide range of physical processes where moving particles or radiation of some form, such as <a href="/wiki/Light" title="Light">light</a> or <a href="/wiki/Sound" title="Sound">sound</a>, are forced to deviate from a straight <a href="/wiki/Trajectory" title="Trajectory">trajectory</a> by localized non-uniformities (including particles and radiation) in the medium through which they pass. In conventional use, this also includes deviation of reflected radiation from the angle predicted by the <a href="/wiki/Law_of_reflection" class="mw-redirect" title="Law of reflection">law of reflection</a>. Reflections of radiation that undergo scattering are often called <i><a href="/wiki/Diffuse_reflection" title="Diffuse reflection">diffuse reflections</a></i> and unscattered reflections are called <i><a href="/wiki/Specular" class="mw-redirect" title="Specular">specular</a></i> (mirror-like) reflections. Originally, the term was confined to light scattering (going back at least as far as <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> in the 17th century<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>). As more "ray"-like phenomena were discovered, the idea of scattering was extended to them, so that <a href="/wiki/William_Herschel" title="William Herschel">William Herschel</a> could refer to the scattering of "heat rays" (not then recognized as electromagnetic in nature) in 1800.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> <a href="/wiki/John_Tyndall" title="John Tyndall">John Tyndall</a>, a pioneer in light scattering research, noted the connection between light scattering and acoustic scattering in the 1870s.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Near the end of the 19th century, the scattering of <a href="/wiki/Cathode_ray" title="Cathode ray">cathode rays</a> (electron beams)<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> and X-rays<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> was observed and discussed. With the discovery of subatomic particles (e.g. <a href="/wiki/Ernest_Rutherford" title="Ernest Rutherford">Ernest Rutherford</a> in 1911<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>) and the development of quantum theory in the 20th century, the sense of the term became broader as it was recognized that the same mathematical frameworks used in light scattering could be applied to many other phenomena. </p><p>Scattering can refer to the consequences of <a href="/wiki/Particle_collision" class="mw-redirect" title="Particle collision">particle-particle collisions</a> between molecules, atoms, <a href="/wiki/Electron" title="Electron">electrons</a>, photons and other particles. Examples include: <a href="/wiki/Cosmic_ray" title="Cosmic ray">cosmic ray</a> scattering in the Earth's upper atmosphere; particle collisions inside <a href="/wiki/Particle_accelerator" title="Particle accelerator">particle accelerators</a>; electron scattering by gas atoms in fluorescent lamps; and <a href="/wiki/Neutron_scattering" title="Neutron scattering">neutron scattering</a> inside <a href="/wiki/Nuclear_reactor" title="Nuclear reactor">nuclear reactors</a>.<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><p>The types of non-uniformities which can cause scattering, sometimes known as <i>scatterers</i> or <i>scattering centers</i>, are too numerous to list, but a small sample includes <a href="/wiki/Particle" title="Particle">particles</a>, <a href="/wiki/Liquid_bubble" class="mw-redirect" title="Liquid bubble">bubbles</a>, <a href="/wiki/Droplet" class="mw-redirect" title="Droplet">droplets</a>, <a href="/wiki/Density" title="Density">density</a> fluctuations in <a href="/wiki/Fluid" title="Fluid">fluids</a>, <a href="/wiki/Crystallite" title="Crystallite">crystallites</a> in <a href="/wiki/Polycrystal" class="mw-redirect" title="Polycrystal">polycrystalline</a> solids, defects in <a href="/wiki/Monocrystal" class="mw-redirect" title="Monocrystal">monocrystalline</a> solids, <a href="/wiki/Surface_roughness" title="Surface roughness">surface roughness</a>, <a href="/wiki/Cell_(biology)" title="Cell (biology)">cells</a> in organisms, and textile <a href="/wiki/Fiber" title="Fiber">fibers</a> in clothing. The effects of such features on the path of almost any type of propagating wave or moving particle can be described in the framework of <a href="/wiki/Scattering_theory" class="mw-redirect" title="Scattering theory">scattering theory</a>. </p><p>Some areas where scattering and scattering theory are significant include radar sensing, <a href="/wiki/Medical_ultrasound" title="Medical ultrasound">medical ultrasound</a>, <a href="/wiki/Semiconductor_wafer" class="mw-redirect" title="Semiconductor wafer">semiconductor wafer</a> inspection, <a href="/wiki/Polymerization" title="Polymerization">polymerization</a> process monitoring, acoustic tiling, free-space communications and <a href="/wiki/Computer-generated_imagery" title="Computer-generated imagery">computer-generated imagery</a>.<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> Particle-particle scattering theory is important in areas such as <a href="/wiki/Particle_physics" title="Particle physics">particle physics</a>, <a href="/wiki/Atomic,_molecular,_and_optical_physics" title="Atomic, molecular, and optical physics">atomic, molecular, and optical physics</a>, <a href="/wiki/Nuclear_physics" title="Nuclear physics">nuclear physics</a> and <a href="/wiki/Astrophysics" title="Astrophysics">astrophysics</a>. In particle physics the quantum interaction and scattering of fundamental particles is described by the Scattering Matrix or <a href="/wiki/S-Matrix" class="mw-redirect" title="S-Matrix">S-Matrix</a>, introduced and developed by <a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">John Archibald Wheeler</a> and <a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Werner Heisenberg</a>.<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>Scattering is quantified using many different concepts, including <a href="/wiki/Cross_section_(physics)" title="Cross section (physics)">scattering cross section</a> (σ), <a href="/wiki/Attenuation_coefficient" title="Attenuation coefficient">attenuation coefficients</a>, the <a href="/wiki/Bidirectional_scattering_distribution_function" title="Bidirectional scattering distribution function">bidirectional scattering distribution function</a> (BSDF), <a href="/wiki/S-matrix" title="S-matrix">S-matrices</a>, and <a href="/wiki/Mean_free_path" title="Mean free path">mean free path</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Single_and_multiple_scattering">Single and multiple scattering</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Scattering&action=edit&section=1" title="Edit section: Single and multiple scattering"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Zodiacal_Glow_Lightens_Paranal_Sky.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Zodiacal_Glow_Lightens_Paranal_Sky.jpg/220px-Zodiacal_Glow_Lightens_Paranal_Sky.jpg" decoding="async" width="220" height="145" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Zodiacal_Glow_Lightens_Paranal_Sky.jpg/330px-Zodiacal_Glow_Lightens_Paranal_Sky.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Zodiacal_Glow_Lightens_Paranal_Sky.jpg/440px-Zodiacal_Glow_Lightens_Paranal_Sky.jpg 2x" data-file-width="5616" data-file-height="3696" /></a><figcaption><a href="/wiki/Zodiacal_light" title="Zodiacal light">Zodiacal light</a> is a faint, diffuse glow visible in the <a href="/wiki/Night_sky" title="Night sky">night sky</a>. The phenomenon stems from the <a href="/wiki/Light_scattering_by_particles" title="Light scattering by particles">scattering</a> of <a href="/wiki/Sunlight" title="Sunlight">sunlight</a> by <a href="/wiki/Interplanetary_dust_cloud" title="Interplanetary dust cloud">interplanetary dust</a> spread throughout <a href="/wiki/Invariable_plane" title="Invariable plane">the plane</a> of the <a href="/wiki/Solar_System" title="Solar System">Solar System</a>.<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></figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Scattering" title="Special:EditPage/Scattering">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">January 2017</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>When radiation is only scattered by one localized scattering center, this is called <i>single scattering</i>. It is more common that scattering centers are grouped together; in such cases, radiation may scatter many times, in what is known as <i>multiple scattering</i>.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> The main difference between the effects of single and multiple scattering is that single scattering can usually be treated as a random phenomenon, whereas multiple scattering, somewhat counterintuitively, can be modeled as a more deterministic process because the combined results of a large number of scattering events tend to average out. Multiple scattering can thus often be modeled well with <a href="/wiki/Diffusion_theory" class="mw-redirect" title="Diffusion theory">diffusion theory</a>.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>Because the location of a single scattering center is not usually well known relative to the path of the radiation, the outcome, which tends to depend strongly on the exact incoming trajectory, appears random to an observer. This type of scattering would be exemplified by an electron being fired at an atomic nucleus. In this case, the atom's exact position relative to the path of the electron is unknown and would be unmeasurable, so the exact trajectory of the electron after the collision cannot be predicted. Single scattering is therefore often described by probability distributions. </p><p>With multiple scattering, the randomness of the interaction tends to be averaged out by a large number of scattering events, so that the final path of the radiation appears to be a deterministic distribution of intensity. This is exemplified by a <a href="/wiki/Light_beam" title="Light beam">light beam</a> passing through thick <a href="/wiki/Fog" title="Fog">fog</a>. Multiple scattering is highly analogous to <a href="/wiki/Diffusion" title="Diffusion">diffusion</a>, and the terms <i>multiple scattering</i> and <i>diffusion</i> are interchangeable in many contexts. Optical elements designed to produce multiple scattering are thus known as <i>diffusers</i>.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Coherent_backscattering" title="Coherent backscattering">Coherent backscattering</a>, an enhancement of <a href="/wiki/Backscattering" class="mw-redirect" title="Backscattering">backscattering</a> that occurs when coherent radiation is multiply scattered by a random medium, is usually attributed to <a href="/wiki/Weak_localization" title="Weak localization">weak localization</a>. </p><p>Not all single scattering is random, however. A well-controlled laser beam can be exactly positioned to scatter off a microscopic particle with a deterministic outcome, for instance. Such situations are encountered in <a href="/wiki/Radar" title="Radar">radar</a> scattering as well, where the targets tend to be macroscopic objects such as people or aircraft. </p><p>Similarly, multiple scattering can sometimes have somewhat random outcomes, particularly with coherent radiation. The random fluctuations in the multiply scattered intensity of coherent radiation are called <a href="/wiki/Speckle_pattern" class="mw-redirect" title="Speckle pattern">speckles</a>. Speckle also occurs if multiple parts of a coherent wave scatter from different centers. In certain rare circumstances, multiple scattering may only involve a small number of interactions such that the randomness is not completely averaged out. These systems are considered to be some of the most difficult to model accurately. </p><p>The description of scattering and the distinction between single and multiple scattering are tightly related to <a href="/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality">wave–particle duality</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Theory">Theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Scattering&action=edit&section=2" title="Edit section: Theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Scattering" title="Special:EditPage/Scattering">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a> in this section. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Scattering%22">"Scattering"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Scattering%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Scattering%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Scattering%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Scattering%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Scattering%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">August 2023</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>Scattering theory is a framework for studying and understanding the scattering of <a href="/wiki/Wave" title="Wave">waves</a> and <a href="/wiki/Elementary_particle" title="Elementary particle">particles</a>. Wave scattering corresponds to the collision and scattering of a wave with some material object, for instance (sunlight) scattered by <a href="/wiki/Rain_drop" class="mw-redirect" title="Rain drop">rain drops</a> to form a <a href="/wiki/Rainbow" title="Rainbow">rainbow</a>. Scattering also includes the interaction of <a href="/wiki/Billiard_balls" class="mw-redirect" title="Billiard balls">billiard balls</a> on a table, the <a href="/wiki/Rutherford_scattering" class="mw-redirect" title="Rutherford scattering">Rutherford scattering</a> (or angle change) of <a href="/wiki/Alpha_particle" title="Alpha particle">alpha particles</a> by <a href="/wiki/Gold" title="Gold">gold</a> <a href="/wiki/Atomic_nucleus" title="Atomic nucleus">nuclei</a>, the Bragg scattering (or diffraction) of electrons and X-rays by a cluster of atoms, and the <a href="/wiki/Inelastic_scattering" title="Inelastic scattering">inelastic scattering</a> of a fission fragment as it traverses a thin foil. More precisely, scattering consists of the study of how solutions of <a href="/wiki/Partial_differential_equations" class="mw-redirect" title="Partial differential equations">partial differential equations</a>, propagating freely "in the distant past", come together and interact with one another or with a <a href="/wiki/Boundary_condition" class="mw-redirect" title="Boundary condition">boundary condition</a>, and then propagate away "to the distant future". </p><p>The <b>direct scattering problem</b> is the problem of determining the distribution of scattered radiation/particle flux basing on the characteristics of the scatterer. The <a href="/wiki/Inverse_scattering_problem" title="Inverse scattering problem">inverse scattering problem</a> is the problem of determining the characteristics of an object (e.g., its shape, internal constitution) from measurement data of radiation or particles scattered from the object. </p> <div class="mw-heading mw-heading3"><h3 id="Attenuation_due_to_scattering">Attenuation due to scattering</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Scattering&action=edit&section=3" title="Edit section: Attenuation due to scattering"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Xsection2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Xsection2.png/288px-Xsection2.png" decoding="async" width="288" height="270" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/2/2e/Xsection2.png 1.5x" data-file-width="354" data-file-height="332" /></a><figcaption>Equivalent quantities used in the theory of scattering from composite specimens, but with a variety of units</figcaption></figure> <p>When the target is a set of many scattering centers whose relative position varies unpredictably, it is customary to think of a range equation whose arguments take different forms in different application areas. In the simplest case consider an interaction that removes particles from the "unscattered beam" at a uniform rate that is proportional to the incident number of particles per unit area per unit time (<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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span>), i.e. that </p> <dl><dd><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 {\frac {dI}{dx}}=-QI\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>I</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi>Q</mi> <mi>I</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dI}{dx}}=-QI\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02f635379cb0102cb002e4fd8d01ef2286977896" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-right: -0.387ex; width:11.685ex; height:5.509ex;" alt="{\displaystyle {\frac {dI}{dx}}=-QI\,\!}"></span></dd></dl> <p>where <i>Q</i> is an interaction coefficient and <i>x</i> is the distance traveled in the target. </p><p>The above ordinary first-order <a href="/wiki/Differential_equation" title="Differential equation">differential equation</a> has solutions of the form: </p> <dl><dd><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=I_{o}e^{-Q\Delta x}=I_{o}e^{-{\frac {\Delta x}{\lambda }}}=I_{o}e^{-\sigma (\eta \Delta x)}=I_{o}e^{-{\frac {\rho \Delta x}{\tau }}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>Q</mi> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mrow> </msup> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mrow> <mi>λ</mi> </mfrac> </mrow> </mrow> </msup> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>η</mi> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mrow> <mi>τ</mi> </mfrac> </mrow> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=I_{o}e^{-Q\Delta x}=I_{o}e^{-{\frac {\Delta x}{\lambda }}}=I_{o}e^{-\sigma (\eta \Delta x)}=I_{o}e^{-{\frac {\rho \Delta x}{\tau }}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24a67318bcb33c434092304b2e08b4c214ad2f33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:47.877ex; height:4.009ex;" alt="{\displaystyle I=I_{o}e^{-Q\Delta x}=I_{o}e^{-{\frac {\Delta x}{\lambda }}}=I_{o}e^{-\sigma (\eta \Delta x)}=I_{o}e^{-{\frac {\rho \Delta x}{\tau }}},}"></span></dd></dl> <p>where <i>I</i><sub>o</sub> is the initial flux, path length Δx ≡ <i>x</i> − <i>x</i><sub>o</sub>, the second equality defines an interaction <a href="/wiki/Mean_free_path" title="Mean free path">mean free path</a> λ, the third uses the number of targets per unit volume η to define an area <a href="/wiki/Cross_section_(physics)" title="Cross section (physics)">cross-section</a> σ, and the last uses the target mass density ρ to define a density mean free path τ. Hence one converts between these quantities via <i>Q</i> = 1/<i>λ</i> = <i>ησ</i> = <i>ρ/τ</i>, as shown in the figure at left. </p><p>In electromagnetic absorption spectroscopy, for example, interaction coefficient (e.g. Q in cm<sup>−1</sup>) is variously called <a href="/wiki/Opacity_(optics)" class="mw-redirect" title="Opacity (optics)">opacity</a>, <a href="/wiki/Absorption_coefficient" class="mw-redirect" title="Absorption coefficient">absorption coefficient</a>, and <a href="/wiki/Attenuation_coefficient" title="Attenuation coefficient">attenuation coefficient</a>. In nuclear physics, area cross-sections (e.g. σ in <a href="/wiki/Barn_(unit)" title="Barn (unit)">barns</a> or units of 10<sup>−24</sup> cm<sup>2</sup>), density mean free path (e.g. τ in grams/cm<sup>2</sup>), and its reciprocal the <a href="/wiki/Mass_attenuation_coefficient" title="Mass attenuation coefficient">mass attenuation coefficient</a> (e.g. in cm<sup>2</sup>/gram) or <i>area per nucleon</i> are all popular, while in electron microscopy the <a href="/wiki/Inelastic_mean_free_path" title="Inelastic mean free path">inelastic mean free path</a><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> (e.g. λ in nanometers) is often discussed<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> instead. </p> <div class="mw-heading mw-heading3"><h3 id="Elastic_and_inelastic_scattering">Elastic and inelastic scattering</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Scattering&action=edit&section=4" title="Edit section: Elastic and inelastic scattering"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The term "elastic scattering" implies that the internal states of the scattering particles do not change, and hence they emerge unchanged from the scattering process. In inelastic scattering, by contrast, the particles' internal state is changed, which may amount to exciting some of the electrons of a scattering atom, or the complete annihilation of a scattering particle and the creation of entirely new particles. </p><p>The example of scattering in <a href="/wiki/Quantum_chemistry" title="Quantum chemistry">quantum chemistry</a> is particularly instructive, as the theory is reasonably complex while still having a good foundation on which to build an intuitive understanding. When two atoms are scattered off one another, one can understand them as being the <a href="/wiki/Bound_state" title="Bound state">bound state</a> solutions of some differential equation. Thus, for example, the <a href="/wiki/Hydrogen_atom" title="Hydrogen atom">hydrogen atom</a> corresponds to a solution to the <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a> with a negative inverse-power (i.e., attractive Coulombic) <a href="/wiki/Central_potential" class="mw-redirect" title="Central potential">central potential</a>. The scattering of two hydrogen atoms will disturb the state of each atom, resulting in one or both becoming excited, or even <a href="/wiki/Ionization" title="Ionization">ionized</a>, representing an inelastic scattering process. </p><p>The term "<a href="/wiki/Deep_inelastic_scattering" title="Deep inelastic scattering">deep inelastic scattering</a>" refers to a special kind of scattering experiment in particle physics. </p> <div class="mw-heading mw-heading3"><h3 id="Mathematical_framework">Mathematical framework</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Scattering&action=edit&section=5" title="Edit section: Mathematical framework"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, scattering theory deals with a more abstract formulation of the same set of concepts. For example, if a <a href="/wiki/Differential_equation" title="Differential equation">differential equation</a> is known to have some simple, localized solutions, and the solutions are a function of a single parameter, that parameter can take the conceptual role of <a href="/wiki/Time" title="Time">time</a>. One then asks what might happen if two such solutions are set up far away from each other, in the "distant past", and are made to move towards each other, interact (under the constraint of the differential equation) and then move apart in the "future". The scattering matrix then pairs solutions in the "distant past" to those in the "distant future". </p><p>Solutions to differential equations are often posed on <a href="/wiki/Manifold" title="Manifold">manifolds</a>. Frequently, the means to the solution requires the study of the <a href="/wiki/Spectrum_(functional_analysis)" title="Spectrum (functional analysis)">spectrum</a> of an <a href="/wiki/Operator_theory" title="Operator theory">operator</a> on the manifold. As a result, the solutions often have a spectrum that can be identified with a <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a>, and scattering is described by a certain map, the <a href="/wiki/S_matrix" class="mw-redirect" title="S matrix">S matrix</a>, on Hilbert spaces. Solutions with a <a href="/wiki/Discrete_spectrum_(physics)" class="mw-redirect" title="Discrete spectrum (physics)">discrete spectrum</a> correspond to <a href="/wiki/Bound_state" title="Bound state">bound states</a> in quantum mechanics, while a <a href="/wiki/Continuous_spectrum" class="mw-redirect" title="Continuous spectrum">continuous spectrum</a> is associated with scattering states. The study of inelastic scattering then asks how discrete and continuous spectra are mixed together. </p><p>An important, notable development is the <a href="/wiki/Inverse_scattering_transform" title="Inverse scattering transform">inverse scattering transform</a>, central to the solution of many <a href="/wiki/Exactly_solvable_model" class="mw-redirect" title="Exactly solvable model">exactly solvable models</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Theoretical_physics">Theoretical physics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Scattering&action=edit&section=6" title="Edit section: Theoretical physics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Scattering_theory_illust.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Scattering_theory_illust.png/220px-Scattering_theory_illust.png" decoding="async" width="220" height="911" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Scattering_theory_illust.png/330px-Scattering_theory_illust.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Scattering_theory_illust.png/440px-Scattering_theory_illust.png 2x" data-file-width="480" data-file-height="1988" /></a><figcaption>Top: the <a href="/wiki/Real_part" class="mw-redirect" title="Real part">real part</a> of a <a href="/wiki/Plane_wave" title="Plane wave">plane wave</a> travelling upwards. Bottom: The real part of the field after inserting in the path of the plane wave a small transparent disk of <a href="/wiki/Index_of_refraction" class="mw-redirect" title="Index of refraction">index of refraction</a> higher than the index of the surrounding medium. This object scatters part of the wave field, although at any individual point, the wave's frequency and wavelength remain intact.</figcaption></figure> <p>In <a href="/wiki/Mathematical_physics" title="Mathematical physics">mathematical physics</a>, <b>scattering theory</b> is a framework for studying and understanding the interaction or scattering of solutions to <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equations</a>. In <a href="/wiki/Acoustics" title="Acoustics">acoustics</a>, the differential equation is the <a href="/wiki/Wave_equation" title="Wave equation">wave equation</a>, and scattering studies how its solutions, the <a href="/wiki/Sound_wave" class="mw-redirect" title="Sound wave">sound waves</a>, scatter from solid objects or propagate through non-uniform media (such as sound waves, in <a href="/wiki/Sea_water" class="mw-redirect" title="Sea water">sea water</a>, coming from a <a href="/wiki/Submarine" title="Submarine">submarine</a>). In the case of classical <a href="/wiki/Electrodynamics" class="mw-redirect" title="Electrodynamics">electrodynamics</a>, the differential equation is again the wave equation, and the scattering of <a href="/wiki/Light" title="Light">light</a> or <a href="/wiki/Radio_wave" title="Radio wave">radio waves</a> is studied. In <a href="/wiki/Particle_physics" title="Particle physics">particle physics</a>, the equations are those of <a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">Quantum electrodynamics</a>, <a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">Quantum chromodynamics</a> and the <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a>, the solutions of which correspond to <a href="/wiki/Fundamental_particle" class="mw-redirect" title="Fundamental particle">fundamental particles</a>. </p><p>In regular <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, which includes <a href="/wiki/Quantum_chemistry" title="Quantum chemistry">quantum chemistry</a>, the relevant equation is the <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a>, although equivalent formulations, such as the <a href="/wiki/Lippmann-Schwinger_equation" class="mw-redirect" title="Lippmann-Schwinger equation">Lippmann-Schwinger equation</a> and the <a href="/wiki/Faddeev_equation" class="mw-redirect" title="Faddeev equation">Faddeev equations</a>, are also largely used. The solutions of interest describe the long-term motion of free atoms, molecules, photons, electrons, and protons. The scenario is that several particles come together from an infinite distance away. These reagents then collide, optionally reacting, getting destroyed or creating new particles. The products and unused reagents then fly away to infinity again. (The atoms and molecules are effectively particles for our purposes. Also, under everyday circumstances, only photons are being created and destroyed.) The solutions reveal which directions the products are most likely to fly off to and how quickly. They also reveal the probability of various reactions, creations, and decays occurring. There are two predominant techniques of finding solutions to scattering problems: <a href="/wiki/Partial_wave_analysis" class="mw-redirect" title="Partial wave analysis">partial wave analysis</a>, and the <a href="/wiki/Born_approximation" title="Born approximation">Born approximation</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Electromagnetics">Electromagnetics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Scattering&action=edit&section=7" title="Edit section: Electromagnetics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-More_citations_needed_section plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Scattering" title="Special:EditPage/Scattering">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a> in this section. Unsourced material may be challenged and removed.</span> <span class="date-container"><i>(<span class="date">January 2020</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Electron-scattering.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Electron-scattering.png/220px-Electron-scattering.png" decoding="async" width="220" height="191" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/e/e8/Electron-scattering.png 1.5x" data-file-width="285" data-file-height="248" /></a><figcaption>A <a href="/wiki/Feynman_diagram" title="Feynman diagram">Feynman diagram</a> of scattering between two electrons by emission of a virtual <a href="/wiki/Photon" title="Photon">photon</a></figcaption></figure> <p><a href="/wiki/Electromagnetic_radiation" title="Electromagnetic radiation">Electromagnetic waves</a> are one of the best known and most commonly encountered forms of radiation that undergo scattering.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> Scattering of light and radio waves (especially in radar) is particularly important. Several different aspects of electromagnetic scattering are distinct enough to have conventional names. Major forms of elastic light scattering (involving negligible energy transfer) are <a href="/wiki/Rayleigh_scattering" title="Rayleigh scattering">Rayleigh scattering</a> and <a href="/wiki/Mie_theory" class="mw-redirect" title="Mie theory">Mie scattering</a>. Inelastic scattering includes <a href="/wiki/Brillouin_scattering" title="Brillouin scattering">Brillouin scattering</a>, <a href="/wiki/Raman_scattering" title="Raman scattering">Raman scattering</a>, inelastic <a href="/wiki/X-ray" title="X-ray">X-ray</a> scattering and <a href="/wiki/Compton_scattering" title="Compton scattering">Compton scattering</a>. </p><p>Light scattering is one of the two major physical processes that contribute to the visible appearance of most objects, the other being absorption. Surfaces described as <i>white</i> owe their appearance to multiple scattering of light by internal or surface inhomogeneities in the object, for example by the boundaries of transparent microscopic crystals that make up a stone or by the microscopic fibers in a sheet of paper. More generally, the <a href="/wiki/Gloss_(material_appearance)" class="mw-redirect" title="Gloss (material appearance)">gloss</a> (or <a href="/wiki/Lustre_(mineralogy)" title="Lustre (mineralogy)">lustre</a> or <a href="/wiki/Gloss_(paint)" class="mw-redirect" title="Gloss (paint)">sheen</a>) of the surface is determined by scattering. Highly scattering surfaces are described as being dull or having a matte finish, while the absence of surface scattering leads to a glossy appearance, as with polished metal or stone. </p><p>Spectral absorption, the selective absorption of certain colors, determines the color of most objects with some modification by <a href="/wiki/Elastic_scattering" title="Elastic scattering">elastic scattering</a>. The apparent blue color of <a href="/wiki/Veins" class="mw-redirect" title="Veins">veins</a> in skin is a common example where both spectral absorption and scattering play important and complex roles in the coloration. Light scattering can also create color without absorption, often shades of blue, as with the sky (Rayleigh scattering), the human blue <a href="/wiki/Iris_(anatomy)" title="Iris (anatomy)">iris</a>, and the feathers of some birds (Prum et al. 1998). However, resonant light scattering in <a href="/wiki/Nanoparticles" class="mw-redirect" title="Nanoparticles">nanoparticles</a> can produce many different highly saturated and vibrant hues, especially when <a href="/wiki/Surface_plasmon_resonance" title="Surface plasmon resonance">surface plasmon resonance</a> is involved (Roqué et al. 2006).<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>Models of light scattering can be divided into three domains based on a dimensionless size parameter, <i>α</i> which is defined as: <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 \alpha =\pi D_{\text{p}}/\lambda ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mi>π</mi> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>p</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>λ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =\pi D_{\text{p}}/\lambda ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b71643594b110b96f5186ddce3f89bad5bed8e4f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.153ex; height:3.009ex;" alt="{\displaystyle \alpha =\pi D_{\text{p}}/\lambda ,}"></span> where <i>πD</i><sub>p</sub> is the circumference of a particle and <i>λ</i> is the wavelength of incident radiation in the medium. Based on the value of <i>α</i>, these domains are: </p> <ul><li><i>α</i> ≪ 1: <a href="/wiki/Rayleigh_scattering" title="Rayleigh scattering">Rayleigh scattering</a> (small particle compared to wavelength of light);</li> <li><i>α</i> ≈ 1: <a href="/wiki/Mie_scattering" title="Mie scattering">Mie scattering</a> (particle about the same size as wavelength of light, valid only for spheres);</li> <li><i>α</i> ≫ 1: geometric scattering (particle much larger than wavelength of light).</li></ul> <p>Rayleigh scattering is a process in which electromagnetic radiation (including light) is scattered by a small spherical volume of variant refractive indexes, such as a particle, bubble, droplet, or even a density fluctuation. This effect was first modeled successfully by <a href="/wiki/Lord_Rayleigh" class="mw-redirect" title="Lord Rayleigh">Lord Rayleigh</a>, from whom it gets its name. In order for Rayleigh's model to apply, the sphere must be much smaller in diameter than the <a href="/wiki/Wavelength" title="Wavelength">wavelength</a> (<i>λ</i>) of the scattered wave; typically the upper limit is taken to be about 1/10 the wavelength. In this size regime, the exact shape of the scattering center is usually not very significant and can often be treated as a sphere of equivalent volume. The inherent scattering that radiation undergoes passing through a pure gas is due to microscopic density fluctuations as the gas molecules move around, which are normally small enough in scale for Rayleigh's model to apply. This scattering mechanism is the primary cause of the blue color of the Earth's sky on a clear day, as the shorter blue wavelengths of sunlight passing overhead are more strongly scattered than the longer red wavelengths according to Rayleigh's famous 1/<i>λ</i><sup>4</sup> relation. Along with absorption, such scattering is a major cause of the attenuation of radiation by the <a href="/wiki/Earth%27s_atmosphere" class="mw-redirect" title="Earth's atmosphere">atmosphere</a>.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> The degree of scattering varies as a function of the ratio of the particle diameter to the wavelength of the radiation, along with many other factors including <a href="/wiki/Polarization_(waves)" title="Polarization (waves)">polarization</a>, angle, and <a href="/wiki/Coherence_(physics)" title="Coherence (physics)">coherence</a>.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>For larger diameters, the problem of electromagnetic scattering by spheres was first solved by <a href="/wiki/Gustav_Mie" title="Gustav Mie">Gustav Mie</a>, and scattering by spheres larger than the Rayleigh range is therefore usually known as Mie scattering. In the Mie regime, the shape of the scattering center becomes much more significant and the theory only applies well to spheres and, with some modification, <a href="/wiki/Spheroids" class="mw-redirect" title="Spheroids">spheroids</a> and <a href="/wiki/Ellipsoids" class="mw-redirect" title="Ellipsoids">ellipsoids</a>. Closed-form solutions for scattering by certain other simple shapes exist, but no general closed-form solution is known for arbitrary shapes. </p><p>Both Mie and Rayleigh scattering are considered elastic scattering processes, in which the energy (and thus wavelength and frequency) of the light is not substantially changed. However, electromagnetic radiation scattered by moving scattering centers does undergo a <a href="/wiki/Doppler_shift" class="mw-redirect" title="Doppler shift">Doppler shift</a>, which can be detected and used to measure the velocity of the scattering center/s in forms of techniques such as <a href="/wiki/Lidar" title="Lidar">lidar</a> and <a href="/wiki/Radar" title="Radar">radar</a>. This shift involves a slight change in energy. </p><p>At values of the ratio of particle diameter to wavelength more than about 10, the laws of <a href="/wiki/Geometric_optics" class="mw-redirect" title="Geometric optics">geometric optics</a> are mostly sufficient to describe the interaction of light with the particle. Mie theory can still be used for these larger spheres, but the solution often becomes numerically unwieldy. </p><p>For modeling of scattering in cases where the Rayleigh and Mie models do not apply such as larger, irregularly shaped particles, there are many numerical methods that can be used. The most common are <a href="/wiki/Finite_element_method" title="Finite element method">finite-element methods</a> which solve <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a> to find the distribution of the scattered electromagnetic field. Sophisticated software packages exist which allow the user to specify the refractive index or indices of the scattering feature in space, creating a 2- or sometimes 3-dimensional model of the structure. For relatively large and complex structures, these models usually require substantial execution times on a computer. </p><p><a href="/wiki/Electrophoresis" title="Electrophoresis">Electrophoresis</a> involves the migration of <a href="/wiki/Macromolecule" title="Macromolecule">macromolecules</a> under the influence of an electric field.<sup id="cite_ref-ElectrophoresisWyatt_21-0" class="reference"><a href="#cite_note-ElectrophoresisWyatt-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Electrophoretic_light_scattering" title="Electrophoretic light scattering">Electrophoretic light scattering</a> involves passing an electric field through a liquid which makes particles move. The bigger the charge is on the particles, the faster they are able to move.<sup id="cite_ref-ElectrophoreticMalvPan_22-0" class="reference"><a href="#cite_note-ElectrophoreticMalvPan-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </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="/w/index.php?title=Scattering&action=edit&section=8" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 20em;"> <ul><li><a href="/wiki/Attenuation#Light_scattering" title="Attenuation">Attenuation#Light scattering</a></li> <li><a href="/wiki/Backscattering" class="mw-redirect" title="Backscattering">Backscattering</a></li> <li><a href="/wiki/Bragg_diffraction" class="mw-redirect" title="Bragg diffraction">Bragg diffraction</a></li> <li><a href="/wiki/Brillouin_scattering" title="Brillouin scattering">Brillouin scattering</a></li> <li><a href="/wiki/Characteristic_mode_analysis" title="Characteristic mode analysis">Characteristic mode analysis</a></li> <li><a href="/wiki/Compton_scattering" title="Compton scattering">Compton scattering</a></li> <li><a href="/wiki/Coulomb_scattering" class="mw-redirect" title="Coulomb scattering">Coulomb scattering</a></li> <li><a href="/wiki/Deep_scattering_layer" title="Deep scattering layer">Deep scattering layer</a></li> <li><a href="/wiki/Diffuse_sky_radiation" title="Diffuse sky radiation">Diffuse sky radiation</a></li> <li><a href="/wiki/Doppler_effect" title="Doppler effect">Doppler effect</a></li> <li><a href="/wiki/Dynamic_Light_Scattering" class="mw-redirect" title="Dynamic Light Scattering">Dynamic Light Scattering</a></li> <li><a href="/wiki/Electron_diffraction" title="Electron diffraction">Electron diffraction</a></li> <li><a href="/wiki/Electron_scattering" title="Electron scattering">Electron scattering</a></li> <li><a href="/wiki/Electrophoretic_light_scattering" title="Electrophoretic light scattering">Electrophoretic light scattering</a></li> <li><a href="/wiki/Extinction_(astronomy)" title="Extinction (astronomy)">Extinction</a></li> <li><a href="/wiki/Haag%E2%80%93Ruelle_scattering_theory" class="mw-redirect" title="Haag–Ruelle scattering theory">Haag–Ruelle scattering theory</a></li> <li><a href="/wiki/Kikuchi_line_(solid_state_physics)" class="mw-redirect" title="Kikuchi line (solid state physics)">Kikuchi line</a></li> <li><a href="/wiki/Light_scattering_by_particles" title="Light scattering by particles">Light scattering by particles</a></li> <li><a href="/wiki/Linewidth" class="mw-redirect" title="Linewidth">Linewidth</a></li> <li><a href="/wiki/Mie_scattering" title="Mie scattering">Mie scattering</a></li> <li><a href="/wiki/Mie_theory" class="mw-redirect" title="Mie theory">Mie theory</a></li> <li><a href="/wiki/Molecular_scattering" class="mw-redirect" title="Molecular scattering">Molecular scattering</a></li> <li><a href="/wiki/Mott_scattering" title="Mott scattering">Mott scattering</a></li> <li><a href="/wiki/Neutron_scattering" title="Neutron scattering">Neutron scattering</a></li> <li><a href="/wiki/Phase_space_measurement_with_forward_modeling" title="Phase space measurement with forward modeling">Phase space measurement with forward modeling</a></li> <li><a href="/wiki/Photon_diffusion" title="Photon diffusion">Photon diffusion</a></li> <li><a href="/wiki/Powder_diffraction" title="Powder diffraction">Powder diffraction</a></li> <li><a href="/wiki/Raman_scattering" title="Raman scattering">Raman scattering</a></li> <li><a href="/wiki/Rayleigh_scattering" title="Rayleigh scattering">Rayleigh scattering</a></li> <li><a href="/wiki/Resonances_in_scattering_from_potentials" title="Resonances in scattering from potentials">Resonances in scattering from potentials</a></li> <li><a href="/wiki/Rutherford_scattering" class="mw-redirect" title="Rutherford scattering">Rutherford scattering</a></li> <li><a href="/wiki/Small-angle_scattering" title="Small-angle scattering">Small-angle scattering</a></li> <li><a href="/wiki/Scattering_amplitude" title="Scattering amplitude">Scattering amplitude</a></li> <li><a href="/wiki/Scattering_from_rough_surfaces" title="Scattering from rough surfaces">Scattering from rough surfaces</a></li> <li><a href="/wiki/Scintillation_(physics)" title="Scintillation (physics)">Scintillation (physics)</a></li> <li><a href="/wiki/S-Matrix" class="mw-redirect" title="S-Matrix">S-Matrix</a></li> <li><a href="/wiki/Tyndall_effect" title="Tyndall effect">Tyndall effect</a></li> <li><a href="/wiki/Thomson_scattering" title="Thomson scattering">Thomson scattering</a></li> <li><a href="/wiki/Wolf_effect" title="Wolf effect">Wolf effect</a></li> <li><a href="/wiki/X-ray_crystallography" title="X-ray crystallography">X-ray crystallography</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Scattering&action=edit&section=9" 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 mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output 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Springer-Verlag. pp. 80–93. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-50496-6" title="Special:BookSources/3-540-50496-6"><bdi>3-540-50496-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Particle+Physics%3A+Concepts+and+Phenomena&rft.pages=80-93&rft.pub=Springer-Verlag&rft.date=1990&rft.isbn=3-540-50496-6&rft.aulast=Nachtmann&rft.aufirst=Otto&rfr_id=info%3Asid%2Fen.wikipedia.org%3AScattering" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation news cs1"><a rel="nofollow" class="external text" href="http://www.eso.org/public/images/potw1348a/">"Zodiacal Glow Lightens Paranal Sky"</a>. <i>ESO Picture of the Week</i>. <a href="/wiki/European_Southern_Observatory" title="European Southern Observatory">European Southern Observatory</a><span class="reference-accessdate">. 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Butler (1999). <i>Multiple Scattering in Solids</i>. <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-98853-5" title="Special:BookSources/978-0-387-98853-5"><bdi>978-0-387-98853-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Multiple+Scattering+in+Solids&rft.pub=Springer&rft.date=1999&rft.isbn=978-0-387-98853-5&rft.aulast=Gonis&rft.aufirst=Antonios&rft.au=William+H.+Butler&rfr_id=info%3Asid%2Fen.wikipedia.org%3AScattering" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGonisWilliam_H._Butler1999" class="citation book cs1">Gonis, Antonios; William H. Butler (1999). <i>Multiple Scattering in Solids</i>. <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-98853-5" title="Special:BookSources/978-0-387-98853-5"><bdi>978-0-387-98853-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Multiple+Scattering+in+Solids&rft.pub=Springer&rft.date=1999&rft.isbn=978-0-387-98853-5&rft.aulast=Gonis&rft.aufirst=Antonios&rft.au=William+H.+Butler&rfr_id=info%3Asid%2Fen.wikipedia.org%3AScattering" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStover1995" class="citation book cs1">Stover, John C. (1995). <i>Optical Scattering: Measurement and Analysis</i>. SPIE Optical Engineering Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8194-1934-7" title="Special:BookSources/978-0-8194-1934-7"><bdi>978-0-8194-1934-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Optical+Scattering%3A+Measurement+and+Analysis&rft.pub=SPIE+Optical+Engineering+Press&rft.date=1995&rft.isbn=978-0-8194-1934-7&rft.aulast=Stover&rft.aufirst=John+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AScattering" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">R. F. 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"Copper and silver nanocrystals in lustre lead glazes: development and optical properties". <i><a href="/wiki/Journal_of_the_European_Ceramic_Society" title="Journal of the European Ceramic Society">Journal of the European Ceramic Society</a></i>. <b>26</b> (16): 3813–3824. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jeurceramsoc.2005.12.024">10.1016/j.jeurceramsoc.2005.12.024</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+European+Ceramic+Society&rft.atitle=Copper+and+silver+nanocrystals+in+lustre+lead+glazes%3A+development+and+optical+properties&rft.volume=26&rft.issue=16&rft.pages=3813-3824&rft.date=2006&rft_id=info%3Adoi%2F10.1016%2Fj.jeurceramsoc.2005.12.024&rft.aulast=Roqu%C3%A9&rft.aufirst=Josep&rft.au=J.+Molera&rft.au=P.+Sciau&rft.au=E.+Pantos&rft.au=M.+Vendrell-Saz&rfr_id=info%3Asid%2Fen.wikipedia.org%3AScattering" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><a href="/wiki/John_H._Seinfeld" title="John H. Seinfeld">Seinfeld</a>, John H.; Pandis, Spyros N. (2006). Atmospheric Chemistry and Physics - From Air Pollution to Climate Change (2nd Ed.). John Wiley and Sons, Inc. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-82857-2" title="Special:BookSources/0-471-82857-2">0-471-82857-2</a></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPrumRodolfo_H._TorresScott_WilliamsonJan_Dyck1998" class="citation journal cs1">Prum, Richard O.; <a href="/wiki/Rodolfo_H._Torres" title="Rodolfo H. Torres">Rodolfo H. Torres</a>; Scott Williamson; Jan Dyck (1998). "Coherent light scattering by blue feather barbs". <i><a href="/wiki/Nature_(journal)" title="Nature (journal)">Nature</a></i>. <b>396</b> (6706): 28–29. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1998Natur.396...28P">1998Natur.396...28P</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2F23838">10.1038/23838</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:4393904">4393904</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=Coherent+light+scattering+by+blue+feather+barbs&rft.volume=396&rft.issue=6706&rft.pages=28-29&rft.date=1998&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A4393904%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1038%2F23838&rft_id=info%3Abibcode%2F1998Natur.396...28P&rft.aulast=Prum&rft.aufirst=Richard+O.&rft.au=Rodolfo+H.+Torres&rft.au=Scott+Williamson&rft.au=Jan+Dyck&rfr_id=info%3Asid%2Fen.wikipedia.org%3AScattering" class="Z3988"></span></span> </li> <li id="cite_note-ElectrophoresisWyatt-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-ElectrophoresisWyatt_21-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation news cs1"><a rel="nofollow" class="external text" href="https://www.wyatt.com/library/theory/electrophoretic-light-scattering-theory.html">"Understanding Electrophoretic Light Scattering"</a>. <i>Wyatt Technology</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Wyatt+Technology&rft.atitle=Understanding+Electrophoretic+Light+Scattering&rft_id=https%3A%2F%2Fwww.wyatt.com%2Flibrary%2Ftheory%2Felectrophoretic-light-scattering-theory.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AScattering" class="Z3988"></span></span> </li> <li id="cite_note-ElectrophoreticMalvPan-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-ElectrophoreticMalvPan_22-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation news cs1"><a rel="nofollow" class="external text" href="https://www.malvernpanalytical.com/en/products/technology/light-scattering">"Light Scattering"</a>. <i>Malvern Panalytical</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Malvern+Panalytical&rft.atitle=Light+Scattering&rft_id=https%3A%2F%2Fwww.malvernpanalytical.com%2Fen%2Fproducts%2Ftechnology%2Flight-scattering&rfr_id=info%3Asid%2Fen.wikipedia.org%3AScattering" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Scattering&action=edit&section=10" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output 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/></span></span></div> <div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/scattering" class="extiw" title="wiktionary:scattering">scattering</a></b></i> in Wiktionary, the free dictionary.</div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, 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href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Quantum_mechanics_topics" title="Template:Quantum mechanics topics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Quantum_mechanics_topics" title="Template talk:Quantum mechanics topics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Quantum_mechanics_topics" title="Special:EditPage/Template:Quantum mechanics topics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Quantum_mechanics" style="font-size:114%;margin:0 4em"><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Background</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics">Introduction</a></li> <li><a href="/wiki/History_of_quantum_mechanics" title="History of quantum mechanics">History</a> <ul><li><a href="/wiki/Timeline_of_quantum_mechanics" title="Timeline of quantum mechanics">Timeline</a></li></ul></li> <li><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a></li> <li><a href="/wiki/Old_quantum_theory" title="Old quantum theory">Old quantum theory</a></li> <li><a href="/wiki/Glossary_of_elementary_quantum_mechanics" title="Glossary of elementary quantum mechanics">Glossary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fundamentals</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Born_rule" title="Born rule">Born rule</a></li> <li><a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">Bra–ket notation</a></li> <li><a href="/wiki/Complementarity_(physics)" title="Complementarity (physics)"> Complementarity</a></li> <li><a href="/wiki/Density_matrix" title="Density matrix">Density matrix</a></li> <li><a href="/wiki/Energy_level" title="Energy level">Energy level</a> <ul><li><a href="/wiki/Ground_state" title="Ground state">Ground state</a></li> <li><a href="/wiki/Excited_state" title="Excited state">Excited state</a></li> <li><a href="/wiki/Degenerate_energy_levels" title="Degenerate energy levels">Degenerate levels</a></li> <li><a href="/wiki/Zero-point_energy" title="Zero-point energy">Zero-point energy</a></li></ul></li> <li><a href="/wiki/Quantum_entanglement" title="Quantum entanglement">Entanglement</a></li> <li><a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a></li> <li><a href="/wiki/Wave_interference" title="Wave interference">Interference</a></li> <li><a href="/wiki/Quantum_decoherence" title="Quantum decoherence">Decoherence</a></li> <li><a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">Measurement</a></li> <li><a href="/wiki/Quantum_nonlocality" title="Quantum nonlocality">Nonlocality</a></li> <li><a href="/wiki/Quantum_state" title="Quantum state">Quantum state</a></li> <li><a href="/wiki/Quantum_superposition" title="Quantum superposition">Superposition</a></li> <li><a href="/wiki/Quantum_tunnelling" title="Quantum tunnelling">Tunnelling</a></li> <li><a href="/wiki/Scattering_theory" class="mw-redirect" title="Scattering theory">Scattering theory</a></li> <li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry in quantum mechanics</a></li> <li><a href="/wiki/Uncertainty_principle" title="Uncertainty principle">Uncertainty</a></li> <li><a href="/wiki/Wave_function" title="Wave function">Wave function</a> <ul><li><a href="/wiki/Wave_function_collapse" title="Wave function collapse">Collapse</a></li> <li><a href="/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality">Wave–particle duality</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Formulations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Formulations</a></li> <li><a href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg</a></li> <li><a href="/wiki/Interaction_picture" title="Interaction picture">Interaction</a></li> <li><a href="/wiki/Matrix_mechanics" title="Matrix mechanics">Matrix mechanics</a></li> <li><a href="/wiki/Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger</a></li> <li><a href="/wiki/Path_integral_formulation" title="Path integral formulation">Path integral formulation</a></li> <li><a href="/wiki/Phase-space_formulation" title="Phase-space formulation">Phase space</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Equations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon</a></li> <li><a href="/wiki/Dirac_equation" title="Dirac equation">Dirac</a></li> <li><a href="/wiki/Weyl_equation" title="Weyl equation">Weyl</a></li> <li><a href="/wiki/Majorana_equation" title="Majorana equation">Majorana</a></li> <li><a href="/wiki/Rarita%E2%80%93Schwinger_equation" title="Rarita–Schwinger equation">Rarita–Schwinger</a></li> <li><a href="/wiki/Pauli_equation" title="Pauli equation">Pauli</a></li> <li><a href="/wiki/Rydberg_formula" title="Rydberg formula">Rydberg</a></li> <li><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">Interpretations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_Bayesianism" title="Quantum Bayesianism">Bayesian</a></li> <li><a href="/wiki/Consistent_histories" title="Consistent histories">Consistent histories</a></li> <li><a href="/wiki/Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen</a></li> <li><a href="/wiki/De_Broglie%E2%80%93Bohm_theory" title="De Broglie–Bohm theory">de Broglie–Bohm</a></li> <li><a href="/wiki/Ensemble_interpretation" title="Ensemble interpretation">Ensemble</a></li> <li><a href="/wiki/Hidden-variable_theory" title="Hidden-variable theory">Hidden-variable</a> <ul><li><a href="/wiki/Local_hidden-variable_theory" title="Local hidden-variable theory">Local</a> <ul><li><a href="/wiki/Superdeterminism" title="Superdeterminism">Superdeterminism</a></li></ul></li></ul></li> <li><a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds</a></li> <li><a href="/wiki/Objective-collapse_theory" title="Objective-collapse theory">Objective collapse</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Relational_quantum_mechanics" title="Relational quantum mechanics">Relational</a></li> <li><a href="/wiki/Transactional_interpretation" title="Transactional interpretation">Transactional</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Wigner_interpretation" title="Von Neumann–Wigner interpretation">Von Neumann–Wigner</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Experiments</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bell_test" title="Bell test">Bell test</a></li> <li><a href="/wiki/Davisson%E2%80%93Germer_experiment" title="Davisson–Germer experiment">Davisson–Germer</a></li> <li><a href="/wiki/Delayed-choice_quantum_eraser" title="Delayed-choice quantum eraser">Delayed-choice quantum eraser</a></li> <li><a href="/wiki/Double-slit_experiment" title="Double-slit experiment">Double-slit</a></li> <li><a href="/wiki/Franck%E2%80%93Hertz_experiment" title="Franck–Hertz experiment">Franck–Hertz</a></li> <li><a href="/wiki/Mach%E2%80%93Zehnder_interferometer" title="Mach–Zehnder interferometer">Mach–Zehnder interferometer</a></li> <li><a href="/wiki/Elitzur%E2%80%93Vaidman_bomb_tester" title="Elitzur–Vaidman bomb tester">Elitzur–Vaidman</a></li> <li><a href="/wiki/Popper%27s_experiment" title="Popper's experiment">Popper</a></li> <li><a href="/wiki/Quantum_eraser_experiment" title="Quantum eraser experiment">Quantum eraser</a></li> <li><a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach</a></li> <li><a href="/wiki/Wheeler%27s_delayed-choice_experiment" title="Wheeler's delayed-choice experiment">Wheeler's delayed choice</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_nanoscience" class="mw-redirect" title="Quantum nanoscience">Science</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_biology" title="Quantum biology">Quantum biology</a></li> <li><a href="/wiki/Quantum_chemistry" title="Quantum chemistry">Quantum chemistry</a></li> <li><a href="/wiki/Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li> <li><a href="/wiki/Quantum_cosmology" title="Quantum cosmology">Quantum cosmology</a></li> <li><a href="/wiki/Quantum_differential_calculus" title="Quantum differential calculus">Quantum differential calculus</a></li> <li><a href="/wiki/Quantum_dynamics" title="Quantum dynamics">Quantum dynamics</a></li> <li><a href="/wiki/Quantum_geometry" title="Quantum geometry">Quantum geometry</a></li> <li><a href="/wiki/Measurement_problem" title="Measurement problem">Quantum measurement problem</a></li> <li><a href="/wiki/Quantum_mind" title="Quantum mind">Quantum mind</a></li> <li><a href="/wiki/Quantum_stochastic_calculus" title="Quantum stochastic calculus">Quantum stochastic calculus</a></li> <li><a href="/wiki/Quantum_spacetime" title="Quantum spacetime">Quantum spacetime</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_technology" class="mw-redirect" title="Quantum technology">Technology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_algorithm" title="Quantum algorithm">Quantum algorithms</a></li> <li><a href="/wiki/Quantum_amplifier" title="Quantum amplifier">Quantum amplifier</a></li> <li><a href="/wiki/Quantum_bus" title="Quantum bus">Quantum bus</a></li> <li><a href="/wiki/Quantum_cellular_automaton" title="Quantum cellular automaton">Quantum cellular automata</a> <ul><li><a href="/wiki/Quantum_finite_automaton" title="Quantum finite automaton">Quantum finite automata</a></li></ul></li> <li><a href="/wiki/Quantum_channel" title="Quantum channel">Quantum channel</a></li> <li><a href="/wiki/Quantum_circuit" title="Quantum circuit">Quantum circuit</a></li> <li><a href="/wiki/Quantum_complexity_theory" title="Quantum complexity theory">Quantum complexity theory</a></li> <li><a href="/wiki/Quantum_computing" title="Quantum computing">Quantum computing</a> <ul><li><a href="/wiki/Timeline_of_quantum_computing_and_communication" title="Timeline of quantum computing and communication">Timeline</a></li></ul></li> <li><a href="/wiki/Quantum_cryptography" title="Quantum cryptography">Quantum cryptography</a></li> <li><a href="/wiki/Quantum_electronics" class="mw-redirect" title="Quantum electronics">Quantum electronics</a></li> <li><a href="/wiki/Quantum_error_correction" title="Quantum error correction">Quantum error correction</a></li> <li><a href="/wiki/Quantum_imaging" title="Quantum imaging">Quantum imaging</a></li> <li><a href="/wiki/Quantum_image_processing" title="Quantum image processing">Quantum image processing</a></li> <li><a href="/wiki/Quantum_information" title="Quantum information">Quantum information</a></li> <li><a href="/wiki/Quantum_key_distribution" title="Quantum key distribution">Quantum key distribution</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Quantum_logic_gate" title="Quantum logic gate">Quantum logic gates</a></li> <li><a href="/wiki/Quantum_machine" title="Quantum machine">Quantum machine</a></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li> <li><a href="/wiki/Quantum_metamaterial" title="Quantum metamaterial">Quantum metamaterial</a></li> <li><a href="/wiki/Quantum_metrology" title="Quantum metrology">Quantum metrology</a></li> <li><a href="/wiki/Quantum_network" title="Quantum network">Quantum network</a></li> <li><a href="/wiki/Quantum_neural_network" title="Quantum neural network">Quantum neural network</a></li> <li><a href="/wiki/Quantum_optics" title="Quantum optics">Quantum optics</a></li> <li><a href="/wiki/Quantum_programming" title="Quantum programming">Quantum programming</a></li> <li><a href="/wiki/Quantum_sensor" title="Quantum sensor">Quantum sensing</a></li> <li><a href="/wiki/Quantum_simulator" title="Quantum simulator">Quantum simulator</a></li> <li><a href="/wiki/Quantum_teleportation" title="Quantum teleportation">Quantum teleportation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Extensions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_fluctuation" title="Quantum fluctuation">Quantum fluctuation</a></li> <li><a href="/wiki/Casimir_effect" title="Casimir effect">Casimir effect</a></li> <li><a href="/wiki/Quantum_statistical_mechanics" title="Quantum statistical mechanics">Quantum statistical mechanics</a></li> <li><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a> <ul><li><a href="/wiki/History_of_quantum_field_theory" title="History of quantum field theory">History</a></li></ul></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li> <li><a href="/wiki/Relativistic_quantum_mechanics" title="Relativistic quantum mechanics">Relativistic quantum mechanics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Schr%C3%B6dinger%27s_cat" title="Schrödinger's cat">Schrödinger's cat</a> <ul><li><a href="/wiki/Schr%C3%B6dinger%27s_cat_in_popular_culture" title="Schrödinger's cat in popular culture">in popular culture</a></li></ul></li> <li><a href="/wiki/Wigner%27s_friend" title="Wigner's friend">Wigner's friend</a></li> <li><a href="/wiki/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox" title="Einstein–Podolsky–Rosen paradox">EPR paradox</a></li> <li><a href="/wiki/Quantum_mysticism" title="Quantum mysticism">Quantum mysticism</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" 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