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subsection</span> </button> <ul id="toc-Parameters_of_interest-sublist" class="vector-toc-list"> <li id="toc-Target_type_dependence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Target_type_dependence"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Target type dependence</span> </div> </a> <ul id="toc-Target_type_dependence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Type_of_reaction_dependence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Type_of_reaction_dependence"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Type of reaction dependence</span> </div> </a> <ul id="toc-Type_of_reaction_dependence-sublist" class="vector-toc-list"> <li id="toc-Absorption_cross_section" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Absorption_cross_section"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2.1</span> <span>Absorption cross section</span> </div> </a> <ul id="toc-Absorption_cross_section-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Scattering_cross-section" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Scattering_cross-section"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2.2</span> <span>Scattering cross-section</span> </div> </a> <ul id="toc-Scattering_cross-section-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Incident_particle_energy_dependence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Incident_particle_energy_dependence"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Incident particle energy dependence</span> </div> </a> <ul id="toc-Incident_particle_energy_dependence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Target_temperature_dependence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Target_temperature_dependence"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Target temperature dependence</span> </div> </a> <ul id="toc-Target_temperature_dependence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Doppler_broadening" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Doppler_broadening"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>Doppler broadening</span> </div> </a> <ul id="toc-Doppler_broadening-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Link_to_reaction_rate_and_interpretation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Link_to_reaction_rate_and_interpretation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Link to reaction rate and interpretation</span> </div> </a> <ul id="toc-Link_to_reaction_rate_and_interpretation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Continuous_versus_average_cross_section" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Continuous_versus_average_cross_section"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Continuous versus average cross section</span> </div> </a> <ul id="toc-Continuous_versus_average_cross_section-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Microscopic_versus_macroscopic_cross_section" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Microscopic_versus_macroscopic_cross_section"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Microscopic versus macroscopic cross section</span> </div> </a> <ul id="toc-Microscopic_versus_macroscopic_cross_section-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mean_free_path" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mean_free_path"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Mean free path</span> </div> </a> <ul id="toc-Mean_free_path-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Within_stars" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Within_stars"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Within stars</span> </div> </a> <ul id="toc-Within_stars-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Typical_cross_sections" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Typical_cross_sections"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Typical cross sections</span> </div> </a> <ul id="toc-Typical_cross_sections-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">8</span> <span>External links</span> </div> </a> <ul id="toc-External_links-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">9</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" 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badge-Q70894304 mw-list-item" title=""><a href="https://de.wikipedia.org/wiki/Neutronenquerschnitt" title="Neutronenquerschnitt – German" lang="de" hreflang="de" data-title="Neutronenquerschnitt" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Secci%C3%B3n_transversal_de_neutrones" title="Sección transversal de neutrones – Spanish" lang="es" hreflang="es" data-title="Sección transversal de neutrones" 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/%D8%B3%D8%B7%D8%AD_%D9%85%D9%82%D8%B7%D8%B9_%D9%86%D9%88%D8%AA%D8%B1%D9%88%D9%86" title="سطح مقطع نوترون – Persian" lang="fa" hreflang="fa" data-title="سطح مقطع نوترون" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Neytron_kesimi" title="Neytron kesimi – Uzbek" lang="uz" hreflang="uz" data-title="Neytron kesimi" 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-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%86%DB%8C%D9%88%D9%B9%D8%B1%D9%88%D9%86_%DA%A9%D8%B1%D9%88%D8%B3_%D8%B3%DB%8C%DA%A9%D8%B4%D9%86" title="نیوٹرون کروس سیکشن – Western Punjabi" lang="pnb" hreflang="pnb" data-title="نیوٹرون کروس سیکشن" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Neutronski_presek" title="Neutronski presek – Serbian" lang="sr" hreflang="sr" data-title="Neutronski presek" 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-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/N%C3%B6tron_tesir_kesiti" title="Nötron tesir kesiti – Turkish" lang="tr" hreflang="tr" data-title="Nötron tesir kesiti" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%B8%AD%E5%AD%90%E6%88%AA%E9%9D%A2" title="中子截面 – Chinese" lang="zh" hreflang="zh" data-title="中子截面" 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 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.sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .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" style="background-color: #cee0f2">Science with <a href="/wiki/Neutron" title="Neutron">neutrons</a></th></tr><tr><td class="sidebar-image"><span typeof="mw:File"><a href="/wiki/File:Quark_structure_neutron.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/Quark_structure_neutron.svg/120px-Quark_structure_neutron.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/Quark_structure_neutron.svg/180px-Quark_structure_neutron.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/81/Quark_structure_neutron.svg/240px-Quark_structure_neutron.svg.png 2x" data-file-width="800" data-file-height="800" /></a></span></td></tr><tr><th class="sidebar-heading" style="background-color: #cee0f2"> Foundations</th></tr><tr><td class="sidebar-content" style="text-align: left"> <ul><li><a href="/wiki/Neutron_temperature" title="Neutron temperature">Neutron temperature</a></li> <li><a href="/wiki/Neutron_flux" title="Neutron flux">Flux</a>, <a href="/wiki/Neutron_radiation" title="Neutron radiation">Radiation</a>, <a href="/wiki/Neutron_transport" title="Neutron transport">Transport</a></li> <li><a class="mw-selflink selflink">Cross section</a>, <a href="/wiki/Neutron_absorption" class="mw-redirect" title="Neutron absorption">Absorption</a>, <a href="/wiki/Neutron_activation" title="Neutron activation">Activation</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background-color: #cee0f2"> <a href="/wiki/Neutron_scattering" title="Neutron scattering">Neutron scattering</a></th></tr><tr><td class="sidebar-content" style="text-align: left"> <ul><li><a href="/wiki/Neutron_diffraction" title="Neutron diffraction">Neutron diffraction</a> <ul><li><a href="/wiki/Small-angle_neutron_scattering" title="Small-angle neutron scattering">Small-angle neutron scattering</a></li> <li><a href="/wiki/Grazing-incidence_small-angle_neutron_scattering" class="mw-redirect" title="Grazing-incidence small-angle neutron scattering">GISANS</a></li> <li><a href="/wiki/Neutron_reflectometry" title="Neutron reflectometry">Reflectometry</a></li></ul></li> <li><a href="/wiki/Inelastic_neutron_scattering" class="mw-redirect" title="Inelastic neutron scattering">Inelastic neutron scattering</a> <ul><li><a href="/wiki/Neutron_triple-axis_scattering" class="mw-redirect" title="Neutron triple-axis scattering">Triple-axis spectrometer</a></li> <li><a href="/wiki/Neutron_time-of-flight_scattering" title="Neutron time-of-flight scattering">Time-of-flight spectrometer</a></li> <li><a href="/wiki/Neutron_backscattering" title="Neutron backscattering">Backscattering spectrometer</a></li> <li><a href="/wiki/Neutron_spin_echo" title="Neutron spin echo">Spin-echo spectrometer</a></li></ul></li></ul></td> </tr><tr><th class="sidebar-heading" style="background-color: #cee0f2"> Other applications</th></tr><tr><td class="sidebar-content" style="text-align: left"> <ul><li><a href="/wiki/Neutron_tomography" title="Neutron tomography">Neutron tomography</a></li> <li><a href="/wiki/Neutron_activation_analysis" title="Neutron activation analysis">Activation analysis</a>, <a href="/wiki/Prompt_gamma_neutron_activation_analysis" title="Prompt gamma neutron activation analysis">Prompt gamma activation analysis</a></li> <li>Fundamental research with neutrons: <a href="/wiki/Ultracold_neutrons" title="Ultracold neutrons">Ultracold neutrons</a>, <a href="/wiki/Neutron_interferometer" title="Neutron interferometer">Interferometry</a></li> <li><a href="/wiki/Fast_neutron_therapy" title="Fast neutron therapy">Fast neutron therapy</a></li> <li><a href="/wiki/Neutron_capture_therapy_of_cancer" title="Neutron capture therapy of cancer">Neutron capture therapy</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background-color: #cee0f2"> Infrastructure</th></tr><tr><td class="sidebar-content" style="text-align: left"> <ul><li><a href="/wiki/Neutron_source" title="Neutron source">Neutron sources</a>: <a href="/wiki/Research_reactor" title="Research reactor">Research reactor</a>, <a href="/wiki/Spallation" title="Spallation">Spallation</a>, <a href="/wiki/Neutron_moderator" title="Neutron moderator">Neutron moderator</a></li> <li>Neutron optics: <a href="/wiki/Neutron_reflector" title="Neutron reflector">Reflector</a>, <a href="/wiki/Supermirror" class="mw-redirect" title="Supermirror">Supermirror</a></li> <li><a href="/wiki/Neutron_detection" title="Neutron detection">Detection</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background-color: #cee0f2"> <a href="/wiki/Neutron_facilities" class="mw-redirect" title="Neutron facilities">Neutron facilities</a></th></tr><tr><td class="sidebar-content" style="text-align: left"> <ul><li>America: <a href="/wiki/High_Flux_Isotope_Reactor" title="High Flux Isotope Reactor">HFIR</a>, <a href="/wiki/Los_Alamos_Neutron_Science_Center" title="Los Alamos Neutron Science Center">LANSCE</a>, <a rel="nofollow" class="external text" href="http://www.nist.gov/ncnr/">NIST CNR</a> -<a href="/wiki/Spallation_Neutron_Source" title="Spallation Neutron Source">SNS</a></li> <li>Oceania: <a href="/wiki/Open-pool_Australian_lightwater_reactor" title="Open-pool Australian lightwater reactor">OPAL</a></li> <li>Asia: <a href="/wiki/J-PARC" title="J-PARC">J-PARC</a>, <a href="/wiki/High-Flux_Advanced_Neutron_Application_Reactor" title="High-Flux Advanced Neutron Application Reactor">HANARO</a></li> <li>Europe: <a href="/wiki/Helmholtz-Zentrum_Berlin" title="Helmholtz-Zentrum Berlin">BER II</a>, <a href="/wiki/FRM_II" title="FRM II">FRM II</a>, <a href="/wiki/Institut_Laue%E2%80%93Langevin" title="Institut Laue–Langevin">ILL</a>, <a href="/wiki/ISIS_Neutron_and_Muon_Source" title="ISIS Neutron and Muon Source">ISIS Neutron and Muon Source</a>, <a href="/wiki/Joint_Institute_for_Nuclear_Research" title="Joint Institute for Nuclear Research">JINR</a>, <a href="/wiki/Paul_Scherrer_Institute#Spallation_Neutron_Source_.28SINQ.29" title="Paul Scherrer Institute">SINQ</a></li> <li>Historic: <a href="/wiki/Intense_Pulsed_Neutron_Source" title="Intense Pulsed Neutron Source">IPNS</a>, <a rel="nofollow" class="external text" href="http://www.bnl.gov/bnlweb/history/HFBR_main.asp">HFBR</a></li> <li>Under construction: <a href="/wiki/European_Spallation_Source" title="European Spallation Source">ESS</a></li></ul></td> </tr><tr><td class="sidebar-navbar"><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 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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-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 article <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/Neutron_cross_section" title="Special:EditPage/Neutron cross section">improve this article</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 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=%22Neutron+cross+section%22">"Neutron cross section"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Neutron+cross+section%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Neutron+cross+section%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=%22Neutron+cross+section%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Neutron+cross+section%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Neutron+cross+section%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">September 2011</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>In <a href="/wiki/Nuclear_physics" title="Nuclear physics">nuclear physics</a>, the concept of a <b>neutron cross section</b> is used to express the likelihood of interaction between an incident <a href="/wiki/Neutron" title="Neutron">neutron</a> and a target nucleus. The neutron cross section σ can be defined as the area in cm<sup>2</sup> for which the number of neutron-nuclei reactions taking place is equal to the product of the number of incident neutrons that would pass through the area and the number of target nuclei.<sup id="cite_ref-McLane_1-0" class="reference"><a href="#cite_note-McLane-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (February 2022)">page needed</span></a></i>]</sup> In conjunction with the <a href="/wiki/Neutron_flux" title="Neutron flux">neutron flux</a>, it enables the calculation of the reaction rate, for example to derive the thermal <a href="/wiki/Power_(physics)" title="Power (physics)">power</a> of a <a href="/wiki/Nuclear_power_plant" title="Nuclear power plant">nuclear power plant</a>. The standard unit for measuring the cross section is the <a href="/wiki/Barn_(unit)" title="Barn (unit)">barn</a>, which is equal to 10<sup>−28</sup> m<sup>2</sup> or 10<sup>−24</sup> cm<sup>2</sup>. The larger the neutron cross section, the more likely a neutron will react with the nucleus. </p><p>An <a href="/wiki/Isotope" title="Isotope">isotope</a> (or <a href="/wiki/Nuclide" title="Nuclide">nuclide</a>) can be classified according to its neutron cross section and how it reacts to an incident neutron. Nuclides that tend to absorb a neutron and either <a href="/wiki/Radioactive_decay" title="Radioactive decay">decay</a> or keep the neutron in its nucleus are <a href="/wiki/Neutron_absorber" class="mw-redirect" title="Neutron absorber">neutron absorbers</a> and will have a <i>capture cross section</i> for that reaction. Isotopes that undergo <a href="/wiki/Nuclear_fission" title="Nuclear fission">fission</a> are <a href="/wiki/Fissile_material#Fissile_vs_fissionable" title="Fissile material">fissionable</a> fuels and have a corresponding <i>fission cross section</i>. The remaining isotopes will simply scatter the neutron, and have a <i>scatter cross section</i>. Some isotopes, like <a href="/wiki/Uranium-238" title="Uranium-238">uranium-238</a>, have nonzero cross sections of all three. </p><p>Isotopes which have a large scatter cross section and a low mass are good <a href="/wiki/Neutron_moderator" title="Neutron moderator">neutron moderators</a> (see chart below). Nuclides which have a large absorption cross section are <a href="/wiki/Neutron_poison" title="Neutron poison">neutron poisons</a> if they are neither fissile nor undergo decay. A poison that is purposely inserted into a nuclear reactor for controlling its <a href="/wiki/Effective_neutron_multiplication_factor" class="mw-redirect" title="Effective neutron multiplication factor">reactivity</a> in the long term and improve its <a href="/wiki/Shutdown_(nuclear_reactor)" title="Shutdown (nuclear reactor)">shutdown margin</a> is called a <i>burnable</i> poison. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Parameters_of_interest">Parameters of interest</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Neutron_cross_section&action=edit&section=1" title="Edit section: Parameters of interest"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The neutron cross section, and therefore the probability of an neutron-nucleus interaction, depends on: </p> <ul><li>the target type (<a href="/wiki/Hydrogen" title="Hydrogen">hydrogen</a>, <a href="/wiki/Uranium" title="Uranium">uranium</a>...),</li> <li>the type of <a href="/wiki/Nuclear_reaction" title="Nuclear reaction">nuclear reaction</a> (scattering, <a href="/wiki/Nuclear_fission" title="Nuclear fission">fission</a>...).</li> <li>the incident particle <a href="/wiki/Neutron_temperature" title="Neutron temperature">energy</a>, also called speed or temperature (thermal, fast...),</li></ul> <p>and, to a lesser extent, of: </p> <ul><li>its relative angle between the incident neutron and the target nuclide,</li> <li>the target nuclide temperature.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Target_type_dependence">Target type dependence</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Neutron_cross_section&action=edit&section=2" title="Edit section: Target type dependence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The neutron cross section is defined for a given type of target particle. For example, the capture cross section of deuterium <a href="/wiki/Deuterium" title="Deuterium"><sup>2</sup>H</a> is much smaller than that of common hydrogen <a href="/wiki/Hydrogen_atom" title="Hydrogen atom"><sup>1</sup>H</a>.<sup id="cite_ref-:0_2-0" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> This is the reason why some reactors use <a href="/wiki/Heavy_water_reactor" class="mw-redirect" title="Heavy water reactor">heavy water</a> (in which most of the hydrogen is deuterium) instead of ordinary <a href="/wiki/Light_water_reactor" class="mw-redirect" title="Light water reactor">light water</a> as <a href="/wiki/Neutron_moderator" title="Neutron moderator">moderator</a>: fewer neutrons are lost by capture inside the medium, hence enabling the use of <a href="/wiki/Natural_uranium" title="Natural uranium">natural uranium</a> instead of <a href="/wiki/Enriched_uranium" title="Enriched uranium">enriched uranium</a>. This is the principle of a <a href="/wiki/CANDU_reactor" title="CANDU reactor">CANDU reactor</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Type_of_reaction_dependence">Type of reaction dependence</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Neutron_cross_section&action=edit&section=3" title="Edit section: Type of reaction dependence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The likelihood of interaction between an incident neutron and a target nuclide, independent of the type of reaction, is expressed with the help of the total cross section <i>σ</i><sub>T</sub>. However, it may be useful to know if the incoming particle bounces off the target (and therefore continue travelling after the interaction) or disappears after the reaction. For that reason, the scattering and absorption cross sections <i>σ</i><sub>S</sub> and <i>σ</i><sub>A</sub> are defined and the total cross section is simply the sum of the two partial cross sections:<sup id="cite_ref-DOE_3-0" class="reference"><a href="#cite_note-DOE-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </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 \sigma _{\text{T}}=\sigma _{\text{S}}+\sigma _{\text{A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msub> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msub> <mo>+</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{\text{T}}=\sigma _{\text{S}}+\sigma _{\text{A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20f33bbb8596bd521933c0eb163821ba57f34b79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.951ex; height:2.343ex;" alt="{\displaystyle \sigma _{\text{T}}=\sigma _{\text{S}}+\sigma _{\text{A}}}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Absorption_cross_section">Absorption cross section</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Neutron_cross_section&action=edit&section=4" title="Edit section: Absorption cross section"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If the neutron is absorbed when approaching the nuclide, the atomic nucleus moves up on the table of isotopes by one position. For instance, <sup>235</sup>U becomes <sup>236*</sup>U with the * indicating the nucleus is highly energized. This energy has to be released and the release can take place through any of several mechanisms. </p> <ol><li>The simplest way for the release to occur is for the neutron to be ejected by the nucleus. If the neutron is emitted immediately, it acts the same as in other scattering events.</li> <li>The nucleus may emit gamma radiation.</li> <li>The nucleus may β<sup>−</sup> decay, where a neutron is converted into a proton, an electron and an electron-type antineutrino (the antiparticle of the neutrino)</li> <li>About 81% of the <sup>236*</sup>U nuclei are so energized that they undergo fission, releasing the energy as kinetic motion of the fission fragments, also emitting between one and five free neutrons.</li></ol> <ul><li>Nuclei that undergo fission as their predominant decay method after neutron capture include <sup>233</sup>U, <sup>235</sup>U, <sup>237</sup>U, <sup>239</sup>Pu, <sup>241</sup>Pu.</li> <li>Nuclei that predominantly absorb neutrons and then emit beta particle radiation lead to these isotopes, e.g., <sup>232</sup>Th absorbs a neutron and becomes <sup>233*</sup>Th, which beta decays to become <a href="/wiki/Isotopes_of_protactinium" title="Isotopes of protactinium"><sup>233</sup>Pa</a>, which in turn beta decays to become <sup>233</sup>U.</li> <li>Isotopes that undergo beta decay transmute from one element to another element. Those that undergo gamma or X-ray emission do not cause a change in element or isotope.</li></ul> <div class="mw-heading mw-heading4"><h4 id="Scattering_cross-section">Scattering cross-section</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Neutron_cross_section&action=edit&section=5" title="Edit section: Scattering cross-section"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The scattering cross-section can be further subdivided into coherent <a href="/wiki/Scattering" title="Scattering">scattering</a> and incoherent scattering, which is caused by the <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a> dependence of the scattering cross-section and, for a natural sample, presence of different <a href="/wiki/Isotope" title="Isotope">isotopes</a> of the same element in the sample. </p><p>Because <a href="/wiki/Neutron" title="Neutron">neutrons</a> interact with the <a href="/wiki/Nuclear_potential" class="mw-redirect" title="Nuclear potential">nuclear potential</a>, the scattering cross-section varies for different <a href="/wiki/Isotope" title="Isotope">isotopes</a> of the element in question. A very prominent example is <a href="/wiki/Hydrogen" title="Hydrogen">hydrogen</a> and its isotope <a href="/wiki/Deuterium" title="Deuterium">deuterium</a>. The total cross-section for hydrogen is over 10 times that of deuterium, mostly due to the large incoherent <a href="/wiki/Scattering_length" title="Scattering length">scattering length</a> of hydrogen. Some metals are rather transparent to neutrons, <a href="/wiki/Aluminum" class="mw-redirect" title="Aluminum">aluminum</a> and <a href="/wiki/Zirconium" title="Zirconium">zirconium</a> being the two best examples of this. </p> <div class="mw-heading mw-heading3"><h3 id="Incident_particle_energy_dependence">Incident particle energy dependence</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Neutron_cross_section&action=edit&section=6" title="Edit section: Incident particle energy dependence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Neutron_temperature" title="Neutron temperature">Neutron temperature</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:U235_Fission_cross_section.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/U235_Fission_cross_section.png/450px-U235_Fission_cross_section.png" decoding="async" width="450" height="344" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/U235_Fission_cross_section.png/675px-U235_Fission_cross_section.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e0/U235_Fission_cross_section.png/900px-U235_Fission_cross_section.png 2x" data-file-width="3195" data-file-height="2442" /></a><figcaption>U235 fission cross section</figcaption></figure> <p>For a given target and reaction, the cross section is strongly dependent on the neutron speed. In the extreme case, the cross section can be, at low energies, either zero (the energy for which the cross section becomes significant is called <a href="/wiki/Threshold_energy" title="Threshold energy">threshold energy</a>) or much larger than at high energies. </p><p>Therefore, a cross section should be defined either at a <a href="#Continuous_versus_average_cross_section">given energy or should be averaged</a> in an energy range (or group). </p><p>As an example, the plot on the right shows that the <a href="/wiki/Nuclear_fission" title="Nuclear fission">fission</a> cross section of <a href="/wiki/Uranium-235" title="Uranium-235">uranium-235</a> is low at high neutron energies but becomes higher at low energies. Such physical constraints explain why most operational <a href="/wiki/Nuclear_reactors" class="mw-redirect" title="Nuclear reactors">nuclear reactors</a> use a <a href="/wiki/Neutron_moderator" title="Neutron moderator">neutron moderator</a> to reduce the energy of the neutron and thus increase the probability of fission which is essential to produce energy and sustain the <a href="/wiki/Chain_reaction" title="Chain reaction">chain reaction</a>. </p><p>A simple estimation of energy dependence of any kind of cross section is provided by the Ramsauer model,<sup id="cite_ref-Ramsauer_4-0" class="reference"><a href="#cite_note-Ramsauer-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> which is based on the idea that the <i>effective</i> size of a neutron is proportional to the breadth of the <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a> of where the neutron is likely to be, which itself is proportional to the neutron's <a href="/wiki/Thermal_de_Broglie_wavelength" title="Thermal de Broglie wavelength">thermal de Broglie wavelength</a>. </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 \lambda (E)={\frac {h}{\sqrt {2mE}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <msqrt> <mn>2</mn> <mi>m</mi> <mi>E</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda (E)={\frac {h}{\sqrt {2mE}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4475a88d368d9394e22932fd3220f572df627de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:15.789ex; height:6.343ex;" alt="{\displaystyle \lambda (E)={\frac {h}{\sqrt {2mE}}}}"></span></dd></dl> <p>Taking <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 \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> as the effective radius of the neutron, we can estimate the area of the circle <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 \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"></span> in which neutrons hit the nuclei of effective radius <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> as </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 \sigma (E)\propto \pi (R+\lambda (E))^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo>∝</mo> <mi>π</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo>+</mo> <mi>λ</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma (E)\propto \pi (R+\lambda (E))^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a13e99a2cc3ef9b1e4ed89a75b3da983b6781146" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.753ex; height:3.176ex;" alt="{\displaystyle \sigma (E)\propto \pi (R+\lambda (E))^{2}}"></span></dd></dl> <p>While the assumptions of this model are naive, it explains at least qualitatively the typical measured energy dependence of the neutron absorption cross section. For neutrons of wavelength much larger than typical radius of atomic nuclei (1–10 fm, E = 10–1000 keV) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> can be neglected. For these low energy neutrons (such as thermal neutrons) the cross section <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 \sigma (E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma (E)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/625126d54f51953d556e435389f18908bdd5bbd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.915ex; height:2.843ex;" alt="{\displaystyle \sigma (E)}"></span> is inversely proportional to neutron velocity. </p><p>This explains the advantage of using a <a href="/wiki/Neutron_moderator" title="Neutron moderator">neutron moderator</a> in fission nuclear reactors. On the other hand, for very high energy neutrons (over 1 MeV), <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 \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> can be neglected, and the neutron cross section is approximately constant, determined just by the cross section of atomic nuclei. </p><p>However, this simple model does not take into account so called neutron resonances, which strongly modify the neutron cross section in the energy range of 1 eV–10 keV, nor the threshold energy of some nuclear reactions. </p> <div class="mw-heading mw-heading3"><h3 id="Target_temperature_dependence">Target temperature dependence</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Neutron_cross_section&action=edit&section=7" title="Edit section: Target temperature dependence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Cross sections are usually measured at 20 °C. To account for the dependence with temperature of the medium (viz. the target), the following formula is used:<sup id="cite_ref-DOE_3-1" class="reference"><a href="#cite_note-DOE-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </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 \sigma =\sigma _{0}\left({\frac {T_{0}}{T}}\right)^{\frac {1}{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>T</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma =\sigma _{0}\left({\frac {T_{0}}{T}}\right)^{\frac {1}{2}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19aaa3d96acaad848c50b0e6528e73999b3b2b32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.861ex; height:7.176ex;" alt="{\displaystyle \sigma =\sigma _{0}\left({\frac {T_{0}}{T}}\right)^{\frac {1}{2}},}"></span></dd></dl> <p>where <i>σ</i> is the cross section at temperature <i>T</i>, and <i>σ</i><sub>0</sub> the cross section at temperature <i>T</i><sub>0</sub> (<i>T</i> and <i>T</i><sub>0</sub> in <a href="/wiki/Kelvin" title="Kelvin">kelvins</a>). </p><p>The energy is defined at the most likely energy and velocity of the neutron. The neutron population consists of a Maxwellian distribution, and hence the mean energy and velocity will be higher. Consequently, also a Maxwellian correction-term <style data-mw-deduplicate="TemplateStyles:r1154941027">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>√π has to be included when calculating the cross-section <a rel="nofollow" class="external text" href="https://mragheb.com/NPRE%20402%20ME%20405%20Nuclear%20Power%20Engineering/Neutron%20Cross%20Sections.pdf">Equation 38</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Doppler_broadening">Doppler broadening</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Neutron_cross_section&action=edit&section=8" title="Edit section: Doppler broadening"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Doppler broadening of neutron resonances is a very important phenomenon and improves <a href="/wiki/Nuclear_reactor" title="Nuclear reactor">nuclear reactor</a> stability. The prompt temperature coefficient of most thermal reactors is negative, owing to the nuclear <a href="/wiki/Doppler_effect" title="Doppler effect">Doppler effect</a>. Nuclei are located in atoms which are themselves in continual motion owing to their thermal energy (temperature). As a result of these thermal motions, <a href="/wiki/Neutron" title="Neutron">neutrons</a> impinging on a target appears to the nuclei in the target to have a continuous spread in energy. This, in turn, has an effect on the observed shape of resonance. The <a href="/wiki/Resonance" title="Resonance">resonance</a> becomes shorter and wider than when the nuclei are at rest. </p><p>Although the shape of resonances changes with temperature, the total area under the resonance remains essentially constant. But this does not imply constant neutron absorption. Despite the constant area under resonance a resonance integral, which determines the absorption, increases with increasing target temperature. This, of course, decreases coefficient k (negative reactivity is inserted). </p> <div class="mw-heading mw-heading2"><h2 id="Link_to_reaction_rate_and_interpretation">Link to reaction rate and interpretation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Neutron_cross_section&action=edit&section=9" title="Edit section: Link to reaction rate and interpretation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Wiki_link_reaction_rate_XS.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wiki_link_reaction_rate_XS.svg/450px-Wiki_link_reaction_rate_XS.svg.png" decoding="async" width="450" height="338" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wiki_link_reaction_rate_XS.svg/675px-Wiki_link_reaction_rate_XS.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wiki_link_reaction_rate_XS.svg/900px-Wiki_link_reaction_rate_XS.svg.png 2x" data-file-width="512" data-file-height="384" /></a><figcaption>Interpretation of the reaction rate with the help of the cross section</figcaption></figure> <p>Imagine a spherical target (shown as the dashed grey and red circle in the figure) and a beam of particles (in blue) "flying" at speed <i>v</i> (vector in blue) in the direction of the target. We want to know how many particles impact it during time interval d<i>t</i>. To achieve it, the particles have to be in the green cylinder in the figure (volume <i>V</i>). The base of the cylinder is the geometrical cross section of the target perpendicular to the beam (surface <i>σ</i> in red) and its height the length travelled by the particles during d<i>t</i> (length <i>v</i> d<i>t</i>): </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 V=\sigma \,v\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mi>σ</mi> <mspace width="thinmathspace" /> <mi>v</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=\sigma \,v\,dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9310aed231b7fbd3a7dc6b777d08c30f040c4a7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.173ex; height:2.176ex;" alt="{\displaystyle V=\sigma \,v\,dt}"></span></dd></dl> <p>Noting <i>n</i> the <a href="/wiki/Number_density" title="Number density">number of particles per unit volume</a>, there are <i>n V</i> particles in the volume <i>V</i>, which will, per definition of <i>V</i>, undergo a reaction. Noting <i>r</i> the <a href="/wiki/Reaction_rate" title="Reaction rate">reaction rate</a> onto one target, it gives: </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 r\,dt=n\,V=n\,\sigma \,v\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mi>n</mi> <mspace width="thinmathspace" /> <mi>V</mi> <mo>=</mo> <mi>n</mi> <mspace width="thinmathspace" /> <mi>σ</mi> <mspace width="thinmathspace" /> <mi>v</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\,dt=n\,V=n\,\sigma \,v\,dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72f92b64d1a1468d564a5590a77b6c5954187152" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:20.326ex; height:2.176ex;" alt="{\displaystyle r\,dt=n\,V=n\,\sigma \,v\,dt}"></span></dd></dl> <p>It follows directly from the definition of the <a href="/wiki/Neutron_flux" title="Neutron flux">neutron flux</a><sup id="cite_ref-DOE_3-2" class="reference"><a href="#cite_note-DOE-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }"></span> = <i>n v</i></span>: </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 r=\sigma \,\Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>σ</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=\sigma \,\Phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/897608a6cba7ee0bd8219a806424c13972d59dcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.542ex; height:2.176ex;" alt="{\displaystyle r=\sigma \,\Phi }"></span></dd></dl> <p>Assuming that there is not one but <i>N</i> targets per unit volume, the reaction rate <i>R</i> per unit volume is: </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 R=N\,r=N\,\Phi \,\sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mi>N</mi> <mspace width="thinmathspace" /> <mi>r</mi> <mo>=</mo> <mi>N</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">Φ</mi> <mspace width="thinmathspace" /> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=N\,r=N\,\Phi \,\sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26fbe8271c418c924a5f22cf57ac6da8da602d41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.306ex; height:2.176ex;" alt="{\displaystyle R=N\,r=N\,\Phi \,\sigma }"></span></dd></dl> <p>Knowing that the typical nuclear radius <i>r</i> is of the order of 10<sup>−12</sup> cm, the expected nuclear cross section is of the order of <i>π r</i><sup>2</sup> or roughly 10<sup>−24</sup> cm<sup>2</sup> (thus justifying the definition of the <a href="/wiki/Barn_(unit)" title="Barn (unit)">barn</a>). However, if measured experimentally ( <i>σ</i> = <i>R</i> / (<i>Φ N</i>) ), the experimental cross sections vary enormously. As an example, for slow neutrons absorbed by the (n, γ) reaction the cross section in some cases (<a href="/wiki/Xenon-135" title="Xenon-135">xenon-135</a>) is as much as 2,650,000 barns, while the cross sections for transmutations by gamma-ray absorption are in the neighborhood of 0.001 barn (<a href="#Typical_cross_sections">§ Typical cross sections</a> has more examples). </p><p>The so-called <i>nuclear cross section</i> is consequently a purely conceptual quantity representing how big the nucleus should be to be consistent with this simple mechanical model. </p> <div class="mw-heading mw-heading2"><h2 id="Continuous_versus_average_cross_section">Continuous versus average cross section</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Neutron_cross_section&action=edit&section=10" title="Edit section: Continuous versus average cross section"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Cross sections depend strongly on the incoming particle speed. In the case of a beam with multiple particle speeds, the reaction rate <i>R</i> is integrated over the whole range of energy: </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 R=\int _{E}N\,\Phi (E)\,\sigma (E)\,dE}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mi>N</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=\int _{E}N\,\Phi (E)\,\sigma (E)\,dE}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad760a050cd595fb732d491589a0e5de5d472834" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.424ex; height:5.676ex;" alt="{\displaystyle R=\int _{E}N\,\Phi (E)\,\sigma (E)\,dE}"></span></dd></dl> <p>Where <i>σ</i>(<i>E</i>) is the continuous cross section, <i>Φ</i>(<i>E</i>) the differential flux and <i>N</i> the target atom density. </p><p>In order to obtain a formulation equivalent to the mono energetic case, an average cross section is defined: </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 \sigma ={\frac {\int _{E}\Phi (E)\,\sigma (E)\,dE}{\int _{E}\Phi (E)\,dE}}={\frac {\int _{E}\Phi (E)\,\sigma (E)\,dE}{\Phi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>E</mi> </mrow> <mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>E</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>E</mi> </mrow> <mi mathvariant="normal">Φ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ={\frac {\int _{E}\Phi (E)\,\sigma (E)\,dE}{\int _{E}\Phi (E)\,dE}}={\frac {\int _{E}\Phi (E)\,\sigma (E)\,dE}{\Phi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60fe2642b912ffb7eaa0b6dd150594baafdbc572" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:43.03ex; height:7.176ex;" alt="{\displaystyle \sigma ={\frac {\int _{E}\Phi (E)\,\sigma (E)\,dE}{\int _{E}\Phi (E)\,dE}}={\frac {\int _{E}\Phi (E)\,\sigma (E)\,dE}{\Phi }}}"></span></dd></dl> <p>Where <span class="nowrap"><i>Φ</i> = <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \int }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∫</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \int }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed07ec6b7c9532ddb859c43b95404125d5b34f25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.418ex; height:3.176ex;" alt="{\textstyle \int }"></span> <i>Φ</i>(<i>E</i>) d<i>E</i></span> is the integral flux. </p><p>Using the definition of the integral flux <i>Φ</i> and the average cross section <i>σ</i>, the same formulation as <a href="#Link_to_reaction_rate_and_interpretation">before</a> is found: </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 R=N\,\Phi \,\sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mi>N</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">Φ</mi> <mspace width="thinmathspace" /> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=N\,\Phi \,\sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f2e89f97f0f5f5b06a13e206bde86aec915d468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.708ex; height:2.176ex;" alt="{\displaystyle R=N\,\Phi \,\sigma }"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Microscopic_versus_macroscopic_cross_section">Microscopic versus macroscopic cross section</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Neutron_cross_section&action=edit&section=11" title="Edit section: Microscopic versus macroscopic cross section"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Up to now, the cross section referred to in this article corresponds to the microscopic cross section <i>σ</i>. However, it is possible to define the macroscopic cross section<sup id="cite_ref-DOE_3-3" class="reference"><a href="#cite_note-DOE-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <i>Σ</i> which corresponds to the total "equivalent area" of all target particles per unit volume: </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 \Sigma =N\,\sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Σ</mi> <mo>=</mo> <mi>N</mi> <mspace width="thinmathspace" /> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma =N\,\sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34330519fb3e4e926ab60d4e366bd59a1090d66c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.557ex; height:2.176ex;" alt="{\displaystyle \Sigma =N\,\sigma }"></span></dd></dl> <p>where <i>N</i> is the atomic density of the target. </p><p>Therefore, since the cross section can be expressed in cm<sup>2</sup> and the density in cm<sup>−3</sup>, the macroscopic cross section is usually expressed in cm<sup>−1</sup>. Using the equation derived <a href="#Link_to_reaction_rate_and_interpretation">above</a>, the reaction rate <i>R</i> can be derived using only the neutron flux <i>Φ</i> and the macroscopic cross section <i>Σ</i>: </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 R=\Sigma \,\Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mi mathvariant="normal">Σ</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=\Sigma \,\Phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64ada4dc17372ec2bfaff68c598aa30faf72ace5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.606ex; height:2.176ex;" alt="{\displaystyle R=\Sigma \,\Phi }"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Mean_free_path">Mean free path</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Neutron_cross_section&action=edit&section=12" title="Edit section: Mean free path"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Mean_free_path" title="Mean free path">mean free path</a> <i>λ</i> of a random particle is the average length between two interactions. The total length <i>L</i> that non perturbed particles travel during a time interval <i>dt</i> in a volume <i>dV</i> is simply the product of the length <i>l</i> covered by each particle during this time with the number of particles <i>N</i> in this volume: </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 L=l\,N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mi>l</mi> <mspace width="thinmathspace" /> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=l\,N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/892c5754e933bd833d4295fb23d7a3b37e0a090f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.825ex; height:2.176ex;" alt="{\displaystyle L=l\,N}"></span></dd></dl> <p>Noting <i>v</i> the speed of the particles and <i>n</i> is the number of particles per unit volume: </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 {\begin{aligned}l&=v\,dt\\N&=n\,dV\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>l</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>v</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mi>N</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>n</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}l&=v\,dt\\N&=n\,dV\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4ac7eec6663409bb123717bfa1bfcb620468fc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.698ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}l&=v\,dt\\N&=n\,dV\end{aligned}}}"></span></dd></dl> <p>It follows: </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 L=v\,dt\,n\,dV}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mi>v</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mspace width="thinmathspace" /> <mi>n</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=v\,dt\,n\,dV}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/124962e7e46ec2c9d64b1b7f094401d699952d2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.424ex; height:2.176ex;" alt="{\displaystyle L=v\,dt\,n\,dV}"></span></dd></dl> <p>Using the definition of the <a href="/wiki/Neutron_flux" title="Neutron flux">neutron flux</a><sup id="cite_ref-DOE_3-4" class="reference"><a href="#cite_note-DOE-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <i>Φ</i> </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 \Phi =n\,v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo>=</mo> <mi>n</mi> <mspace width="thinmathspace" /> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi =n\,v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc4435bf4731be7b8c270865317d052b20801251" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.686ex; height:2.176ex;" alt="{\displaystyle \Phi =n\,v}"></span></dd></dl> <p>It follows: </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 L=\Phi \,dt\,dV}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mi mathvariant="normal">Φ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=\Phi \,dt\,dV}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba254efaa91aa8ba1da3be409eda0cf473e66165" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.192ex; height:2.176ex;" alt="{\displaystyle L=\Phi \,dt\,dV}"></span></dd></dl> <p>This average length <i>L</i> is however valid only for unperturbed particles. To account for the interactions, <i>L</i> is divided by the total number of reactions <i>R</i> to obtain the average length between each collision <i>λ</i>: </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 \lambda ={\frac {L}{R}}={\frac {\Phi \,dt\,dV}{R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <mi>R</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Φ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> </mrow> <mi>R</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda ={\frac {L}{R}}={\frac {\Phi \,dt\,dV}{R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1061c0dd09d5e1a007a6b5215f891126e72584b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.499ex; height:5.509ex;" alt="{\displaystyle \lambda ={\frac {L}{R}}={\frac {\Phi \,dt\,dV}{R}}}"></span></dd></dl> <p>From <a href="#Microscopic_versus_macroscopic_cross_section">§ Microscopic versus macroscopic cross section</a>: </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 R=\Phi \,\Sigma \,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mi mathvariant="normal">Φ</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">Σ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=\Phi \,\Sigma \,dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8da2547afe7478d96d5356bfc8cfca4647bec13f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.048ex; height:2.176ex;" alt="{\displaystyle R=\Phi \,\Sigma \,dt}"></span></dd></dl> <p>It follows: </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 \lambda ={\frac {dV}{\Sigma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>V</mi> </mrow> <mi mathvariant="normal">Σ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda ={\frac {dV}{\Sigma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98115bfd78381c8dfa03893285712adf93663730" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.293ex; height:5.343ex;" alt="{\displaystyle \lambda ={\frac {dV}{\Sigma }}}"></span></dd></dl> <p>where <i>λ</i> is the mean free path and <i>Σ</i> is the macroscopic cross section. </p> <div class="mw-heading mw-heading2"><h2 id="Within_stars">Within stars</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Neutron_cross_section&action=edit&section=13" title="Edit section: Within stars"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Because <a href="/wiki/Isotopes_of_lithium" title="Isotopes of lithium"><sup>8</sup>Li</a> and <a href="/wiki/Isotopes_of_beryllium" title="Isotopes of beryllium"><sup>12</sup>Be</a> form natural stopping points on the table of isotopes for <a href="/wiki/Hydrogen" title="Hydrogen">hydrogen</a> <a href="/wiki/Nuclear_fusion" title="Nuclear fusion">fusion</a>, it is believed that all of the higher elements are formed in very hot stars where higher orders of fusion predominate. A star like the <a href="/wiki/Sun" title="Sun">Sun</a> produces <a href="/wiki/Energy" title="Energy">energy</a> by the fusion of simple <sup>1</sup>H into <a href="/wiki/Helium-4" title="Helium-4"><sup>4</sup>He</a> through a <a href="/wiki/Nuclear_fusion#Astrophysical_reaction_chains" title="Nuclear fusion">series of reactions</a>. It is believed that when the inner core exhausts its <sup>1</sup>H fuel, the Sun will contract, slightly increasing its core temperature until <sup>4</sup>He can fuse and become the main fuel supply. Pure <sup>4</sup>He fusion leads to <a href="/wiki/Beryllium-8" title="Beryllium-8"><sup>8</sup>Be</a>, which decays back to 2 <sup>4</sup>He; therefore the <sup>4</sup>He must fuse with isotopes either more or less massive than itself to result in an energy producing reaction. When <sup>4</sup>He fuses with <a href="/wiki/Deuterium" title="Deuterium"><sup>2</sup>H</a> or <a href="/wiki/Tritium" title="Tritium"><sup>3</sup>H</a>, it forms stable isotopes <sup>6</sup>Li and <sup>7</sup>Li respectively. The higher order isotopes between <sup>8</sup>Li and <a href="/wiki/Carbon-12" title="Carbon-12"><sup>12</sup>C</a> are synthesized by similar reactions between hydrogen, helium, and lithium isotopes. </p> <div class="mw-heading mw-heading2"><h2 id="Typical_cross_sections">Typical cross sections</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Neutron_cross_section&action=edit&section=14" title="Edit section: Typical cross sections"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Common_light_element_moderators,_reflectors_and_absorbers.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Common_light_element_moderators%2C_reflectors_and_absorbers.svg/300px-Common_light_element_moderators%2C_reflectors_and_absorbers.svg.png" decoding="async" width="300" height="303" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Common_light_element_moderators%2C_reflectors_and_absorbers.svg/450px-Common_light_element_moderators%2C_reflectors_and_absorbers.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Common_light_element_moderators%2C_reflectors_and_absorbers.svg/600px-Common_light_element_moderators%2C_reflectors_and_absorbers.svg.png 2x" data-file-width="648" data-file-height="655" /></a><figcaption>Scattering (full line) and absorption (dotted) crossections of light elements commonly used as neutron moderators, reflectors and absorbers, the data was obtained from database NEA N ENDF/B-VII.1 using JANIS software and plotted using matplotlib.</figcaption></figure> <p>Some cross sections that are of importance in a nuclear reactor are given in the following table. </p> <ul><li>The <i>thermal cross-section</i> is averaged using a Maxwellian spectrum.</li> <li>The <i>fast cross section</i> is averaged using the uranium-235 fission spectrum.</li></ul> <p>The cross sections were taken from the JEFF-3.1.1 library using JANIS software.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <table class="wikitable"> <tbody><tr> <th colspan="2" rowspan="2">Nucleon </th> <th colspan="3">Thermal cross section (barn) </th> <th colspan="3">Fast cross section (barn) </th></tr> <tr> <th>Scattering </th> <th>Capture </th> <th>Fission </th> <th>Scattering </th> <th>Capture </th> <th>Fission </th></tr> <tr> <td rowspan="3">Moderator </td> <td><a href="/wiki/Hydrogen_atom" title="Hydrogen atom"><sup>1</sup>H</a> </td> <td>20 </td> <td>0.2 </td> <td>- </td> <td>4 </td> <td>0.00004 </td> <td>- </td></tr> <tr> <td><a href="/wiki/Deuterium" title="Deuterium"><sup>2</sup>H</a> </td> <td>4 </td> <td>0.0003 </td> <td>- </td> <td>3 </td> <td>0.000007 </td> <td>- </td></tr> <tr> <td><a href="/wiki/Isotopes_of_carbon" title="Isotopes of carbon"><sup>12</sup>C</a> </td> <td>5 </td> <td>0.002 </td> <td>- </td> <td>2 </td> <td>0.00001 </td> <td>- </td></tr> <tr> <td rowspan="7">Structural <br />materials, <br />others </td> <td><a href="/wiki/Isotopes_of_gold" title="Isotopes of gold"><sup>197</sup>Au</a> </td> <td>8.2 </td> <td>98.7 </td> <td>- </td> <td>4 </td> <td>0.08 </td> <td>- </td></tr> <tr> <td><a href="/wiki/Isotopes_of_zirconium" title="Isotopes of zirconium"><sup>90</sup>Zr</a> </td> <td>5 </td> <td>0.006 </td> <td>- </td> <td>5 </td> <td>0.006 </td> <td>- </td></tr> <tr> <td><a href="/wiki/Iron-56" title="Iron-56"><sup>56</sup>Fe</a> </td> <td>10 </td> <td>2 </td> <td>- </td> <td>20 </td> <td>0.003 </td> <td>- </td></tr> <tr> <td><a href="/wiki/Isotopes_of_chromium" title="Isotopes of chromium"><sup>52</sup>Cr</a> </td> <td>3 </td> <td>0.5 </td> <td>- </td> <td>3 </td> <td>0.002 </td> <td>- </td></tr> <tr> <td><a href="/wiki/Isotopes_of_cobalt" title="Isotopes of cobalt"><sup>59</sup>Co</a> </td> <td>6 </td> <td>37.2 </td> <td>- </td> <td>4 </td> <td>0.006 </td> <td>- </td></tr> <tr> <td><a href="/wiki/Isotopes_of_nickel" title="Isotopes of nickel"><sup>58</sup>Ni</a> </td> <td>20 </td> <td>3 </td> <td>- </td> <td>3 </td> <td>0.008 </td> <td>- </td></tr> <tr> <td><a href="/wiki/Oxygen-16" title="Oxygen-16"><sup>16</sup>O</a> </td> <td>4 </td> <td>0.0001 </td> <td>- </td> <td>3 </td> <td>0.00000003 </td> <td>- </td></tr> <tr> <td rowspan="4">Absorber </td> <td><a href="/wiki/Isotopes_of_boron" title="Isotopes of boron"><sup>10</sup>B</a> </td> <td>2 </td> <td>200 </td> <td>- </td> <td>2 </td> <td>0.4 </td> <td>- </td></tr> <tr> <td><a href="/wiki/Isotopes_of_cadmium" title="Isotopes of cadmium"><sup>113</sup>Cd</a> </td> <td>100 </td> <td>30,000 </td> <td>- </td> <td>4 </td> <td>0.05 </td> <td>- </td></tr> <tr> <td><a href="/wiki/Xenon-135" title="Xenon-135"><sup>135</sup>Xe</a> </td> <td>400,000 </td> <td>2,000,000 </td> <td>- </td> <td>5 </td> <td>0.0008 </td> <td>- </td></tr> <tr> <td><a href="/wiki/Isotopes_of_indium" title="Isotopes of indium"><sup>115</sup>In</a> </td> <td>2 </td> <td>100 </td> <td>- </td> <td>4 </td> <td>0.02 </td> <td>- </td></tr> <tr> <td rowspan="3">Fuel </td> <td><a href="/wiki/Uranium-235" title="Uranium-235"><sup>235</sup>U</a> </td> <td>10 </td> <td>99 </td> <td>583<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> </td> <td>4 </td> <td>0.09 </td> <td>1 </td></tr> <tr> <td><a href="/wiki/Uranium-238" title="Uranium-238"><sup>238</sup>U</a> </td> <td>9 </td> <td>2 </td> <td>0.00002 </td> <td>5 </td> <td>0.07 </td> <td>0.3 </td></tr> <tr> <td><a href="/wiki/Plutonium-239" title="Plutonium-239"><sup>239</sup>Pu</a> </td> <td>8 </td> <td>269 </td> <td>748 </td> <td>5 </td> <td>0.05 </td> <td>2 </td></tr></tbody></table> <p>* <i>negligible, less than 0.1% of the total cross section and below the Bragg scattering cutoff</i> </p> <div class="mw-heading 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href="https://environmentalchemistry.com/yogi/periodic/crosssection.html">Periodic Table of Elements: Sorted by Cross Section (Thermal Neutron Capture)</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Neutron_cross_section&action=edit&section=16" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-McLane-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-McLane_1-0">^</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 .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFMcLaneDunfordRose2012" class="citation book cs1">McLane, Victoria; Dunford, Charles L.; Rose, Philip F. (2 December 2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rGxkmm5CioIC"><i>Neutron Cross Sections</i></a>. Elsevier. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-323-14222-9" title="Special:BookSources/978-0-323-14222-9"><bdi>978-0-323-14222-9</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1044711235">1044711235</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Neutron+Cross+Sections&rft.pub=Elsevier&rft.date=2012-12-02&rft_id=info%3Aoclcnum%2F1044711235&rft.isbn=978-0-323-14222-9&rft.aulast=McLane&rft.aufirst=Victoria&rft.au=Dunford%2C+Charles+L.&rft.au=Rose%2C+Philip+F.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DrGxkmm5CioIC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANeutron+cross+section" class="Z3988"></span></span> </li> <li id="cite_note-:0-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-:0_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120406064710/https://t2.lanl.gov/data/neutron7.html">"ENDF/B-VII Incident-Neutron Data"</a>. Los Alamos National Laboratory. 15 July 2007. Archived from <a rel="nofollow" class="external text" href="https://t2.lanl.gov/data/neutron7.html">the original</a> on 6 April 2012<span class="reference-accessdate">. Retrieved <span class="nowrap">2011-11-08</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=ENDF%2FB-VII+Incident-Neutron+Data&rft.pub=Los+Alamos+National+Laboratory&rft.date=2007-07-15&rft_id=https%3A%2F%2Ft2.lanl.gov%2Fdata%2Fneutron7.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANeutron+cross+section" class="Z3988"></span></span> </li> <li id="cite_note-DOE-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-DOE_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-DOE_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-DOE_3-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-DOE_3-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-DOE_3-4"><sup><i><b>e</b></i></sup></a></span> <span class="reference-text">DOE Fundamentals Handbook, Nuclear Physics and Reactor Theory, DOE-HDBK-1019/1-93 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20140319145623/https://energy.gov/sites/prod/files/2013/06/f2/h1019v1.pdf">"Archived copy"</a> <span class="cs1-format">(PDF)</span>. Archived from <a rel="nofollow" class="external text" href="https://www.standards.doe.gov/standards-documents/1000/1019-bhdbk-1993-v1">the original</a> on 2014-03-19<span class="reference-accessdate">. Retrieved <span class="nowrap">2023-03-13</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Archived+copy&rft_id=https%3A%2F%2Fwww.standards.doe.gov%2Fstandards-documents%2F1000%2F1019-bhdbk-1993-v1&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANeutron+cross+section" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_web" title="Template:Cite web">cite web</a>}}</code>: CS1 maint: archived copy as title (<a href="/wiki/Category:CS1_maint:_archived_copy_as_title" title="Category:CS1 maint: archived copy as title">link</a>)</span>.</span> </li> <li id="cite_note-Ramsauer-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Ramsauer_4-0">^</a></b></span> <span class="reference-text">R. W. Bauer, J. D. Anderson, S. M. Grimes, V. A. Madsen, Application of Simple Ramsauer Model to Neutron Total Cross Sections, <a rel="nofollow" class="external free" href="https://www.osti.gov/bridge/servlets/purl/641282-MK9s2L/webviewable/641282.pdf">https://www.osti.gov/bridge/servlets/purl/641282-MK9s2L/webviewable/641282.pdf</a></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">JANIS software, <a rel="nofollow" class="external free" href="https://www.oecd-nea.org/janis/">https://www.oecd-nea.org/janis/</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200910071709/https://www.oecd-nea.org/janis/">Archived</a> 2020-09-10 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.nndc.bnl.gov/atlas/atlasvalues.html">"Atlas of Neutron Resonances Thermal Cross Sections & Resonance Integrals"</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170220090327/https://www.nndc.bnl.gov/atlas/atlasvalues.html">Archived</a> from the original on 2017-02-20<span class="reference-accessdate">. Retrieved <span class="nowrap">2014-04-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Atlas+of+Neutron+Resonances+Thermal+Cross+Sections+%26+Resonance+Integrals&rft_id=https%3A%2F%2Fwww.nndc.bnl.gov%2Fatlas%2Fatlasvalues.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANeutron+cross+section" class="Z3988"></span></span> </li> </ol></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Neutron_cross_section&oldid=1258681474">https://en.wikipedia.org/w/index.php?title=Neutron_cross_section&oldid=1258681474</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Neutron" title="Category:Neutron">Neutron</a></li><li><a href="/wiki/Category:Nuclear_physics" title="Category:Nuclear physics">Nuclear physics</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:CS1_maint:_archived_copy_as_title" title="Category:CS1 maint: archived copy as title">CS1 maint: archived copy as title</a></li><li><a href="/wiki/Category:Webarchive_template_wayback_links" title="Category:Webarchive template wayback links">Webarchive template wayback links</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Articles_needing_additional_references_from_September_2011" title="Category:Articles needing additional references from September 2011">Articles needing additional references from September 2011</a></li><li><a href="/wiki/Category:All_articles_needing_additional_references" title="Category:All articles needing additional references">All articles needing additional references</a></li><li><a href="/wiki/Category:Wikipedia_articles_needing_page_number_citations_from_February_2022" title="Category:Wikipedia articles needing page number citations from February 2022">Wikipedia articles needing page number citations from February 2022</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 21 November 2024, at 00:35<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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