{"id":359,"date":"2015-11-21T18:09:34","date_gmt":"2015-11-21T18:09:34","guid":{"rendered":"http:\/\/sitepourvtc.com\/?p=359"},"modified":"2022-03-06T11:12:24","modified_gmt":"2022-03-06T11:12:24","slug":"neutron-cross-section","status":"publish","type":"post","link":"https:\/\/sitepourvtc.com\/neutron-cross-section\/","title":{"rendered":"Neutron Cross-section"},"content":{"rendered":"<div   id="8n9s8rpp6h"    class=\"su-quote su-quote-style-default\"><div   id="8n9s8rpp6h"    class=\"su-quote-inner su-u-clearfix su-u-trim\">In general, <strong>the neutron cross-section<\/strong> is an effective area that quantifies <strong>the likelihood<\/strong> of certain interaction between an incident neutron and a target object. It must be noted this likelihood does not depend on real target dimensions, because we are not describing geometrical cross-section. In the vicinity of target nucleus, neutron is subjected to <a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/atomic-nuclear-physics\/fundamental-interactions-fundamental-forces\/strong-interaction-strong-force\/\">strong nuclear forces<\/a> of target nucleons. The interaction strongly depends on many variables, such as type of target nucleus and the neutron energy. For example, the likelihood that a thermal neutron will be absorbed by <a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/reactor-operation\/xenon-135\/\">xenon-135<\/a> is about a million times higher than it will be scattered.<\/div><\/div>\n<p>At\u00a0 this point we have to distinguish between two basic types of nuclear or neutron cross-sections.<\/p>\n<ul>\n<li><strong>Microscopic Cross-section<\/strong>. The effective target area in m<sup>2<\/sup> presented by a single nucleus to an incident neutron beam is denoted the microscopic cross section, <strong>\u03c3<\/strong>. The microscopic cross-sections characterize interactions with single isotopes and are a part of data libraries, such as ENDF\/B-VII.1.<\/li>\n<li><strong>Macroscopic Cross-section<\/strong>. The <strong>macroscopic cross-section<\/strong> represents the <strong>effective target area of all of the nuclei<\/strong> contained in the volume of the material (such as fuel pellet). The units are given in <strong>cm<sup>-1<\/sup><\/strong>. It is the probability of neutron-nucleus interaction per centimeter of neutron travel. These data are commonly used by codes for <a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/neutron-diffusion-theory\/numerical-solution-diffusion-equation\/\">reactor core analyses and design<\/a>. These codes are based on <strong>pre-computed assembly homogenized macroscopic cross-sections.<\/strong><\/li>\n<\/ul>\n<figure id=\"attachment_13580\" aria-describedby=\"caption-attachment-13580\" style=\"width: 290px\" class=\"wp-caption alignright\"><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2016\/01\/nuclear-cross-section-min.png\"><img loading=\"lazy\" class=\"wp-image-13580 size-medium\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2016\/01\/nuclear-cross-section-min-300x300.png\" alt=\"nuclear cross-sections, microscopic cross-sections\" width=\"300\" height=\"300\" srcset=\"https:\/\/sitepourvtc.com\/wp-content\/uploads\/2016\/01\/nuclear-cross-section-min-300x300.png 300w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2016\/01\/nuclear-cross-section-min-150x150.png 150w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2016\/01\/nuclear-cross-section-min-66x66.png 66w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2016\/01\/nuclear-cross-section-min.png 900w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/a><figcaption id=\"caption-attachment-13580\" class=\"wp-caption-text\">Various microscopic cross-section for uranium-235 and incident thermal neutron.<\/figcaption><\/figure>\n<h2>Barn &#8211; Unit of Cross-section<\/h2>\n<p>The cross-section is typically denoted <strong>\u03c3<\/strong> and measured in units of area [m<sup>2<\/sup>]. But a square meter (or centimeter) is tremendously large in comparison to the effective area of a nucleus, and it has been suggested that a physicist once referred to the measure of a square meter as being &#8220;as big as a barn&#8221; when applied to nuclear processes. The name has persisted and microscopic cross sections are expressed in terms of barns. The standard unit for measuring a nuclear cross section is the <strong>barn<\/strong>, which is equal to <strong>10<sup>\u221228<\/sup> m\u00b2 or 10<sup>\u221224<\/sup> cm\u00b2<\/strong>. It can be seen the concept of a nuclear cross section can be quantified physically in terms of <strong>&#8220;characteristic target area&#8221;<\/strong> where a larger area means a larger probability of interaction.<\/p>\n<h3>Typical Values of Microscopic Cross-sections<\/h3>\n<ul>\n<li><strong>Uranium 235<\/strong> is a\u00a0<a title=\"Fissile Material\" href=\"http:\/\/sitepourvtc.com\/glossary\/fissile-material\/\">fissile isotope<\/a>\u00a0and its\u00a0<a title=\"Cross-sections of Uranium\" href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/nuclear-fuel\/uranium\/cross-sections-of-uranium\/\">fission cross-section<\/a>\u00a0for thermal neutrons is about\u00a0<strong>585 barns<\/strong>\u00a0(for 0.0253 eV neutron). For fast neutrons its fission cross-section is\u00a0<strong>on the order of barns<\/strong>.<\/li>\n<li><strong>Xenon-135<\/strong>\u00a0is a product of<a href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/nuclear-fuel\/uranium\/uranium-235\/uranium-235-fission\/\">\u00a0U-235 fission<\/a>\u00a0and has a\u00a0<strong>very large\u00a0<\/strong><a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/neutron-capture-radiative-capture\/neutron-capture-cross-section\/\"><strong>neutron capture cross-section<\/strong><\/a>\u00a0(about<strong>\u00a02.6 x 10<\/strong><strong><sup>6<\/sup><\/strong><strong>\u00a0barns<\/strong>).<\/li>\n<li><strong>Boron<\/strong> is commonly used as a <strong>neutron absorber<\/strong>\u00a0due to the high\u00a0<a title=\"Neutron cross-section\" href=\"http:\/\/sitepourvtc.com\/neutron-cross-section\/\">neutron cross-section<\/a>\u00a0of isotope \u00a0<strong><sup>10<\/sup>B<\/strong>. Its\u00a0<strong>(n,alpha) reaction<\/strong>\u00a0cross-section for\u00a0<a title=\"Neutron Energy\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/atomic-nuclear-physics\/fundamental-particles\/neutron\/neutron-energy\/\">thermal neutrons<\/a>\u00a0is about\u00a0<strong>3840 barns<\/strong>\u00a0(for 0.025 eV neutron).<\/li>\n<li><strong>Gadolinium<\/strong> is commonly used as a<strong>\u00a0neutron absorber<\/strong>\u00a0due to very high neutron absorbtion\u00a0<a title=\"Neutron cross-section\" href=\"http:\/\/sitepourvtc.com\/neutron-cross-section\/\">cross-section<\/a>\u00a0of two isotopes\u00a0<strong><sup>155<\/sup>Gd<\/strong>\u00a0and\u00a0<strong><sup>157<\/sup>Gd<\/strong>. \u00a0<sup>155<\/sup>Gd\u00a0has 61 000 barns for\u00a0<a title=\"Neutron Energy\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/atomic-nuclear-physics\/fundamental-particles\/neutron\/neutron-energy\/\">thermal neutrons\u00a0<\/a>(for 0.025 eV neutron) and\u00a0<sup>157<\/sup>Gd\u00a0has even 254 000 barns.<\/li>\n<\/ul>\n<p>See also: <a href=\"https:\/\/www.oecd-nea.org\/jcms\/pl_39910\/janis\">JANIS (Java-based Nuclear Data Information Software)<\/a><\/p>\n<div   id="8n9s8rpp6h"    class=\"collapsible-container\">\n<h2>Microscopic Cross-section<\/h2>\n<p>The extent to which <a title=\"Neutron\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/atomic-nuclear-physics\/fundamental-particles\/neutron\/\">neutrons<\/a> interact with nuclei is described in terms of quantities known as <strong>cross-sections<\/strong>. <strong>Cross-sections<\/strong> are used to express the <strong>likelihood of particular interaction<\/strong> between an incident neutron and a target nucleus. It must be noted this likelihood does not depend on real target dimensions. In conjunction with the neutron flux, it enables the calculation of the reaction rate, for example to derive the <strong>thermal power of a nuclear power plant<\/strong>. The standard unit for measuring the <strong>microscopic cross-section<\/strong> (\u03c3-sigma) is the <strong>barn<\/strong>, which is equal to <strong>10<sup>-28<\/sup> m<sup>2<\/sup><\/strong>. This unit is very small, therefore barns (abbreviated as \u201cb\u201d) are commonly used.<\/p>\n<p><strong>The cross-section \u03c3<\/strong> can be interpreted as the <strong>effective \u2018target area\u2019<\/strong> that a nucleus interacts with an incident neutron. The larger the effective area, the greater the probability for reaction. This cross-section is usually known as <strong>the microscopic cross-section<\/strong>.<\/p>\n<p>The concept of the microscopic cross-section is therefore introduced to represent the probability of a <a title=\"Neutron Nuclear Reactions\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/\">neutron-nucleus reaction<\/a>. Suppose that a thin \u2018film\u2019 of atoms (one atomic layer thick) with N<sub>a<\/sub> atoms\/cm<sup>2<\/sup> is placed in a monodirectional beam of intensity I<sub>0<\/sub>. Then the number of interactions C per cm<sup>2<\/sup> per second will be proportional to the intensity I<sub>0<\/sub> and the atom density N<sub>a<\/sub>. We define the proportionality factor as the microscopic cross-section \u03c3:<\/p>\n<p style=\"text-align: center;\"><strong>\u03c3<sub>t<\/sub> = C\/N<sub>a<\/sub>.I<sub>0<\/sub><\/strong><\/p>\n<p>In order to be able to determine the microscopic cross section, <strong>transmission measurements<\/strong> are performed on plates of materials. Assume that if a neutron collides with a nucleus it will either be <a title=\"Neutron Elastic Scattering\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/neutron-elastic-scattering\/\">scattered<\/a> into a different direction or be <a title=\"Neutron Absorption\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/neutron-absorption\/\">absorbed<\/a> (without fission absorption). Assume that there are N (nuclei\/cm<sup>3<\/sup>) of the material and there will then be N.dx per cm<sup>2<\/sup> in the layer dx.<br \/>\nOnly the neutrons that have not interacted will remain traveling in the x direction. This causes the intensity of the uncollided beam will be attenuated as it penetrates deeper into the material.<\/p>\n<p><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/12\/CR.png\"><img loading=\"lazy\" class=\" size-medium wp-image-13408 aligncenter\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/12\/CR-300x249.png\" alt=\"CR\" width=\"300\" height=\"249\" srcset=\"https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/12\/CR-300x249.png 300w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/12\/CR.png 391w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/a><\/p>\n<p>Then, according to the definition of the microscopic cross section, the reaction rate per unit area is N\u03c3 \u0399(x)dx. This is equal to the decrease of the beam intensity, so that:<\/p>\n<p style=\"text-align: center;\"><strong>-dI = N.\u03c3.\u0399(x).dx<\/strong><\/p>\n<p>and<\/p>\n<p style=\"text-align: center;\"><strong>\u0399(x) = \u0399<sub>0<\/sub>e<sup>-N.\u03c3.x<\/sup><\/strong><\/p>\n<p>It can be seen that whether a neutron will interact with a certain volume of material depends not only on <strong>the microscopic cross-section<\/strong> of the individual nuclei but also on <strong>the density of nuclei<\/strong> within that volume. It depends on the <strong>N.\u03c3 factor<\/strong>. This factor is therefore widely defined and it is known <strong>as the macroscopic cross section<\/strong>.<\/p>\n<p>The difference between the microscopic and macroscopic cross sections is extremely important.\u00a0The <strong>microscopic cross section<\/strong> represents the effective target area of a\u00a0<strong>single nucleus<\/strong>, while the <strong>macroscopic cross section<\/strong> represents the effective target area of\u00a0<strong>all of <\/strong><strong>the nuclei<\/strong> contained in certain volume.<\/p>\n<div   id="8n9s8rpp6h"    class=\"su-accordion su-u-trim\"><div   id="8n9s8rpp6h"    class=\"su-spoiler su-spoiler-style-default su-spoiler-icon-plus su-spoiler-closed\" data-scroll-offset=\"0\"><div   id="8n9s8rpp6h"    class=\"su-spoiler-title\" tabindex=\"0\" role=\"button\"><span class=\"su-spoiler-icon\"><\/span>Nuclear Radius<\/div><div   id="8n9s8rpp6h"    class=\"su-spoiler-content su-u-clearfix su-u-trim\"><strong>Typical nuclear radii<\/strong> are of the order <strong>10<sup>\u221214<\/sup> m<\/strong>. Assuming spherical shape, nuclear radii can be calculated according to following formula:<\/p>\n<p>r = r<sub>0<\/sub> . A<sup>1\/3<\/sup><\/p>\n<p>where r<sub>0<\/sub> = 1.2 x 10<sup>-15\u00a0<\/sup>m = 1.2 fm<\/p>\n<p>If we use this approximation, we therefore expect the <strong>geometrical cross-sections<\/strong> of nuclei to be of the order of \u03c0r<sup>2<\/sup> or <strong>4.5 x 10<sup>\u221230\u00a0<\/sup>m\u00b2 for hydrogen<\/strong> nuclei or <strong>1.74 x 10<sup>\u221228<\/sup> m\u00b2 for <a title=\"Uranium 238\" href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/nuclear-fuel\/uranium\/uranium-238\/\"><sup>238<\/sup>U<\/a><\/strong> nuclei.<\/p>\n<p>Since there are many nuclear reaction from the incident particle point of view, but, in nuclear reactor physics, neutron-nuclear reactions are of particular interest. In this case the neutron cross-section must be defined.<\/div><\/div>\n<div   id="8n9s8rpp6h"    class=\"su-spoiler su-spoiler-style-default su-spoiler-icon-plus su-spoiler-closed\" data-scroll-offset=\"0\"><div   id="8n9s8rpp6h"    class=\"su-spoiler-title\" tabindex=\"0\" role=\"button\"><span class=\"su-spoiler-icon\"><\/span>Total Cross-section<\/div><div   id="8n9s8rpp6h"    class=\"su-spoiler-content su-u-clearfix su-u-trim\">In general, <strong>nuclear cross-sections<\/strong> can be measured for all possible interaction processes together, in this case they are called <strong>total cross-sections (\u03c3<sub>t<\/sub>)<\/strong>. The total cross section is the sum of all the partial cross sections such as:<\/p>\n<ul>\n<li><a title=\"Elastic Scattering Cross-section\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/neutron-elastic-scattering\/elastic-scattering-cross-section\/\">elastic scattering cross-section<\/a> (\u03c3<sub>s<\/sub>)<\/li>\n<li><a title=\"Inelastic Scattering Cross-section\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/neutron-inelastic-scattering\/inelastic-scattering-cross-section\/\">inelastic scattering cross-section<\/a> (\u03c3<sub>i<\/sub>)<\/li>\n<li><a title=\"Neutron Absorption Cross-section\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/neutron-absorption\/neutron-absorption-cross-section\/\">absorption cross-section<\/a> (\u03c3<sub>a<\/sub>)<\/li>\n<li><a title=\"Neutron Capture Cross-section\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/neutron-capture-radiative-capture\/neutron-capture-cross-section\/\">radiative capture cross-section<\/a> (\u03c3<sub>\u03b3<\/sub>)<\/li>\n<li><a title=\"Nuclear Fission\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/fission\/\">fission cross-section<\/a>\u00a0(\u03c3<sub>f<\/sub>)<\/li>\n<\/ul>\n<p style=\"text-align: center;\"><strong>\u03c3<sub>t<\/sub> = \u03c3<sub>s<\/sub> + \u03c3<sub>i<\/sub> + \u03c3<sub>\u03b3<\/sub> + \u03c3<sub>f<\/sub> + \u2026&#8230;<\/strong><\/p>\n<p>The total cross-section measures the probability that an interaction of any type will occur when neutron interacts with a target.<\/div><\/div>\n<\/div>\n<p><strong>Microscopic cross-sections<\/strong> constitute a key parameters of <a title=\"Nuclear Fuel\" href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/nuclear-fuel\/\">nuclear fuel<\/a>. In general, neutron cross-sections are essential for the reactor core calculations and are a part of data libraries, such as ENDF\/B-VII.1.<\/p>\n<p>The neutron cross-section is variable and depends on:<\/p>\n<ul>\n<li><strong><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2014\/11\/cross-sections_nuclides.png\"><img loading=\"lazy\" class=\"size-full wp-image-381 alignright\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2014\/11\/cross-sections_nuclides.png\" alt=\"Table of cross-sections\" width=\"732\" height=\"441\" srcset=\"https:\/\/sitepourvtc.com\/wp-content\/uploads\/2014\/11\/cross-sections_nuclides.png 732w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2014\/11\/cross-sections_nuclides-300x180.png 300w\" sizes=\"(max-width: 732px) 100vw, 732px\" \/><\/a>Target nucleus<\/strong> (hydrogen, <a title=\"Boron 10\" href=\"http:\/\/sitepourvtc.com\/glossary\/boron-10\/\">boron<\/a>, <a title=\"Uranium\" href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/nuclear-fuel\/uranium\/\">uranium<\/a>, etc.). Each isotop has its own set of cross-sections.<\/li>\n<li><strong>Type of the reaction<\/strong> (<a title=\"Neutron Capture \u2013 Radiative Capture\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/neutron-capture-radiative-capture\/\">capture<\/a>, <a title=\"Nuclear Fission\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/fission\/\">fission<\/a>, etc.). Cross-sections are different for each nuclear reaction.<\/li>\n<li><strong><a title=\"Neutron Energy\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/atomic-nuclear-physics\/fundamental-particles\/neutron\/neutron-energy\/\">Neutron energy<\/a> <\/strong>(<a title=\"Thermal Neutron\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/atomic-nuclear-physics\/fundamental-particles\/neutron\/thermal-neutron\/\">thermal neutron<\/a>, resonance neutron, fast neutron). For a given target and reaction type, the cross-section is strongly dependent on the neutron energy. In the common case, the cross section is usually much larger at low energies than at high energies. This is why most nuclear reactors use a <a title=\"Neutron Moderator\" href=\"http:\/\/sitepourvtc.com\/neutron-moderator\/\">neutron moderator <\/a>to reduce the energy of the neutron and thus increase the probability of fission, essential to produce energy and sustain the chain reaction.<\/li>\n<li><strong>Target energy<\/strong> (temperature of target material \u2013 <a title=\"Doppler Broadening\" href=\"http:\/\/sitepourvtc.com\/glossary\/doppler-broadening\/\">Doppler broadening<\/a>). This dependency is not so significant, but the target energy strongly influences inherent safety of <a title=\"Nuclear Reactor\" href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/nuclear-reactor\/\">nuclear reactors <\/a>due to a Doppler broadening of resonances.<\/li>\n<\/ul>\n<p><strong>Microscopic cross-section varies with incident neutron energy<\/strong>. Some nuclear reactions exhibit <strong>very specific dependency<\/strong> on incident neutron energy. This dependency will be described on the example of the <a title=\"Neutron Capture \u2013 Radiative Capture\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/neutron-capture-radiative-capture\/\">radiative capture reaction<\/a>. The likelihood of a neutron radiative capture is represented by the radiative capture cross section as <strong>\u03c3<sub>\u03b3<\/sub><\/strong>. The following dependency is typical for radiative capture, it definitely does not mean, that it is typical for other types of reactions (see <a title=\"Elastic Scattering Cross-section\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/neutron-elastic-scattering\/elastic-scattering-cross-section\/\">elastic scattering cross-section<\/a> or <a title=\"Charged Particle Ejection\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/charged-particle-ejection\/\">(n,alpha) reaction cross-section<\/a>).<\/p>\n<p>The capture cross-section can be divided into three regions according to the incident neutron energy. These regions will be discussed separately.<\/p>\n<ul>\n<li><strong>1\/v Region<\/strong><\/li>\n<li><strong>Resonance Region<\/strong><\/li>\n<li><strong>Fast Neutrons Region<\/strong><\/li>\n<\/ul>\n<div   id="8n9s8rpp6h"    class=\"su-accordion su-u-trim\">\n<div   id="8n9s8rpp6h"    class=\"su-spoiler su-spoiler-style-default su-spoiler-icon-plus su-spoiler-closed\" data-scroll-offset=\"0\"><div   id="8n9s8rpp6h"    class=\"su-spoiler-title\" tabindex=\"0\" role=\"button\"><span class=\"su-spoiler-icon\"><\/span>Charts of Cross-sections<\/div><div   id="8n9s8rpp6h"    class=\"su-spoiler-content su-u-clearfix su-u-trim\">\n<a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/11\/neutron-absorption-capture-scattering-fission-min.png\"><img loading=\"lazy\" class=\"aligncenter size-medium wp-image-13131\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/11\/neutron-absorption-capture-scattering-fission-min-300x144.png\" alt=\"Uranium 238. Neutron absorption and scattering. Comparison of cross-sections.\" width=\"300\" height=\"144\" srcset=\"https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/11\/neutron-absorption-capture-scattering-fission-min-300x144.png 300w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/11\/neutron-absorption-capture-scattering-fission-min-1024x491.png 1024w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/11\/neutron-absorption-capture-scattering-fission-min.png 1248w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/a><div   id="8n9s8rpp6h"    class=\"su-spacer\" style=\"height:10px\"><\/div><div   id="8n9s8rpp6h"    class=\"su-spacer\" style=\"height:10px\"><\/div><a title=\"Uranium 238\" href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/nuclear-fuel\/uranium\/uranium-238\/\">Uranium 238<\/a>. Comparison of cross-sections.<br \/>\nSource: JANIS (Java-based Nuclear Data Information Software); The JEFF-3.1.1 Nuclear Data Library<div   id="8n9s8rpp6h"    class=\"su-spacer\" style=\"height:15px\"><\/div><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/08\/gadolinium-cross-sections.png\"><img loading=\"lazy\" class=\"aligncenter size-medium wp-image-12520\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/08\/gadolinium-cross-sections-300x206.png\" alt=\"Gadolinium 155 and 157. Comparison of radiative capture cross-sections.\" width=\"300\" height=\"206\" srcset=\"https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/08\/gadolinium-cross-sections-300x206.png 300w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/08\/gadolinium-cross-sections.png 889w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/a><div   id="8n9s8rpp6h"    class=\"su-spacer\" style=\"height:10px\"><\/div><div   id="8n9s8rpp6h"    class=\"su-spacer\" style=\"height:10px\"><\/div><a title=\"Gadolinium\" href=\"http:\/\/sitepourvtc.com\/glossary\/gadolinium\/\">Gadolinium <\/a>155 and 157. Comparison of radiative capture cross-sections.<br \/>\nSource: JANIS (Java-based Nuclear Data Information Software); The JEFF-3.1.1 Nuclear Data Library<div   id="8n9s8rpp6h"    class=\"su-spacer\" style=\"height:15px\"><\/div>\n<figure id=\"attachment_12069\" aria-describedby=\"caption-attachment-12069\" style=\"width: 290px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/05\/1v-Law.png\"><img loading=\"lazy\" class=\"size-medium wp-image-12069\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/05\/1v-Law-300x221.png\" alt=\"1\/v Law\" width=\"300\" height=\"221\" srcset=\"https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/05\/1v-Law-300x221.png 300w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/05\/1v-Law-220x161.png 220w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/05\/1v-Law.png 737w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/a><figcaption id=\"caption-attachment-12069\" class=\"wp-caption-text\">For thermal neutrons (in 1\/v region), absorption cross sections increases as the velocity (kinetic energy) of the neutron decreases.<br \/>Source: JANIS 4.0<\/figcaption><\/figure>\n<\/div><\/div>\n<div   id="8n9s8rpp6h"    class=\"su-spoiler su-spoiler-style-default su-spoiler-icon-plus su-spoiler-closed\" data-scroll-offset=\"0\"><div   id="8n9s8rpp6h"    class=\"su-spoiler-title\" tabindex=\"0\" role=\"button\"><span class=\"su-spoiler-icon\"><\/span>1\/v Region<\/div><div   id="8n9s8rpp6h"    class=\"su-spoiler-content su-u-clearfix su-u-trim\">In the common case, the cross section is usually <strong>much larger at low energies<\/strong> than at high energies. For <a title=\"Thermal Neutron\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/atomic-nuclear-physics\/fundamental-particles\/neutron\/thermal-neutron\/\">thermal neutrons<\/a> (in 1\/v region), also radiative capture cross-sections increase as the velocity (kinetic energy) of the neutron decreases. Therefore the <a title=\"Law 1\/v\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/atomic-nuclear-physics\/fundamental-particles\/neutron\/interactions-neutrons-matter\/law-1v\/\">1\/v Law<\/a> can be used to determine shift in capture cross-section, if the neutron is in equilibrium with a surrounding medium. This phenomenon is due to the fact the nuclear force between the target nucleus and the neutron has a longer time to interact.<\/p>\n<p><img class=\"mathtex-equation-editor aligncenter\" src=\"http:\/\/chart.apis.google.com\/chart?cht=tx&amp;chl=%5Csigma_a%20%5Csim%20%5Cfrac%7B1%7D%7Bv%7D%7D%7D%20%5Csim%20%5Cfrac%7B1%7D%7B%5Csqrt%7BE%7D%7D%7D%7D%7D%20%5Csim%20%5Cfrac%7B1%7D%7B%5Csqrt%7BT%7D%7D%7D%7D%7D\" alt=\"\\sigma_a \\sim \\frac{1}{v}}} \\sim \\frac{1}{\\sqrt{E}}}}} \\sim \\frac{1}{\\sqrt{T}}}}}\" align=\"absmiddle\" \/><\/p>\n<p>This law is aplicable only for absorbtion cross-section and only in the 1\/v region.<\/p>\n<p><strong>Example of cross- sections in 1\/v region:<\/strong><\/p>\n<p>The absorbtion cross-section for 238U at 20\u00b0C = 293K (~0.0253 eV) is:<\/p>\n<p><img class=\"mathtex-equation-editor\" src=\"http:\/\/chart.apis.google.com\/chart?cht=tx&amp;chl=%5Csigma_a(293K)%20%3D%202.68b%20\" alt=\"\\sigma_a(293K) = 2.68b \" align=\"absmiddle\" \/>.<\/p>\n<p>The absorbtion cross-section for 238U at 1000\u00b0C = 1273K is equal to:<\/p>\n<p><img class=\"mathtex-equation-editor alignleft\" src=\"http:\/\/chart.apis.google.com\/chart?cht=tx&amp;chl=%5Csigma_a(1273K)%20%3D%20%5Csigma_a(293K)%20%5Ccdot%20%5Cfrac%20%7BT_0%7D%7BT_1%7D%20%3D%202.68%20%5Ccdot%20%5Cfrac%7B293%7D%7B1273%7D%20%3D%20%200.617b\" alt=\"\\sigma_a(1273K) = \\sigma_a(293K) \\cdot \\frac {T_0}{T_1} = 2.68 \\cdot \\frac{293}{1273} = 0.617b\" align=\"absmiddle\" \/><\/p>\n<p>This cross-section reduction is caused only due to the shift of temperature of surrounding medium.<\/p>\n<\/div><\/div>\n<div   id="8n9s8rpp6h"    class=\"su-spoiler su-spoiler-style-default su-spoiler-icon-plus su-spoiler-closed\" data-scroll-offset=\"0\"><div   id="8n9s8rpp6h"    class=\"su-spoiler-title\" tabindex=\"0\" role=\"button\"><span class=\"su-spoiler-icon\"><\/span>Resonance Region<\/div><div   id="8n9s8rpp6h"    class=\"su-spoiler-content su-u-clearfix su-u-trim\">\n<figure id=\"attachment_12633\" aria-describedby=\"caption-attachment-12633\" style=\"width: 290px\" class=\"wp-caption alignright\"><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/09\/Compound_nucleus.png\"><img loading=\"lazy\" class=\"size-medium wp-image-12633\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/09\/Compound_nucleus-300x199.png\" alt=\"Compound state - resonance\" width=\"300\" height=\"199\" srcset=\"https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/09\/Compound_nucleus-300x199.png 300w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/09\/Compound_nucleus.png 719w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/a><figcaption id=\"caption-attachment-12633\" class=\"wp-caption-text\">Energy levels of compound state. For neutron absorption reaction on 238U the first resonance E1 corresponds to the excitation energy of 6.67eV. E0 is a base state of 239U.<\/figcaption><\/figure>\n<p><strong>The largest cross-sections<\/strong> are usually at neutron energies, that lead to long-lived states of the <a title=\"What is Nuclear Resonance \u2013 Compound Nucleus\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/compound-nucleus-reactions\/what-is-nuclear-resonance-compound-nucleus\/\">compound nucleus<\/a>. The compound nuclei of these certain energies are referred to as <strong>nuclear resonances<\/strong> and its formation is typical <strong>in the resonance region<\/strong>. The widths of the resonances increase in general with increasing energies. At higher energies the widths may reach the order of the distances between resonances and then no resonances can be observed. The narrowest resonances are usually compound states of heavy nuclei (such as <a title=\"Fissionable Material\" href=\"http:\/\/sitepourvtc.com\/glossary\/fissionable-material\/\">fissionable nuclei<\/a>).<\/p>\n<p>Since the <strong>mode of decay<\/strong> of the compound nucleus <strong>does not depend on the way the compound nucleus was formed<\/strong>, the nucleus sometimes emits a <a title=\"Gamma Rays \/ Gamma Radiation\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/atomic-nuclear-physics\/fundamental-particles\/photon\/gamma-ray\/\">gamma ray<\/a> (radiative capture) or sometimes emits a <a title=\"Neutron\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/atomic-nuclear-physics\/fundamental-particles\/neutron\/\">neutron<\/a> (<a title=\"Elastic and Inelastic Scattering\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/neutron-elastic-scattering\/elastic-and-inelastic-scattering\/\">scattering<\/a>). In order to understand the way, how a nucleus will stabilize itself, we have to understand the behaviour of compound nucleus.<\/p>\n<figure id=\"attachment_12819\" aria-describedby=\"caption-attachment-12819\" style=\"width: 290px\" class=\"wp-caption alignright\"><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/10\/ground-state-compound-nucleus.png\"><img loading=\"lazy\" class=\"size-medium wp-image-12819\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/10\/ground-state-compound-nucleus-300x163.png\" alt=\"ground state compound nucleus - excitation\" width=\"300\" height=\"163\" srcset=\"https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/10\/ground-state-compound-nucleus-300x163.png 300w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/10\/ground-state-compound-nucleus.png 736w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/a><figcaption id=\"caption-attachment-12819\" class=\"wp-caption-text\">The position of the energy levels during the formation of a compound nucleus. Ground state and energy states.<\/figcaption><\/figure>\n<p>The compound nucleus emits a neutron only after one neutron obtains an energy in collision with other nucleon greater than its <a title=\"Nuclear Binding Energy\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/atomic-nuclear-physics\/binding-energy\/nuclear-binding-energy\/\">binding energy<\/a> in the nucleus. It have some delay, because the excitation energy of the compound nucleus <strong>is divided<\/strong> among several nucleons. It is obvious the average time that elapses before a neutron can be emitted is much longer for nuclei <strong>with large number of nucleons<\/strong> than when only a few nucleons are involved. It is a consequence of sharing the excitation energy among a large number of nucleons.<\/p>\n<p>This\u00a0is the reason the <strong>radiative capture<\/strong> is comparatively <strong>unimportant in light nuclei<\/strong> but becomes increasingly <strong>important in the heavier nuclei<\/strong>.<\/p>\n<p>It is obvious the compound states (resonances) are observed at low excitation energies. This is due to the fact, the energy gap between the states is large. At high excitation energy, the gap between two compound states is very small and\u00a0the widths of resonances may reach the order of the distances between resonances. Therefore at high energies\u00a0no resonances can be observed and the cross section in this energy region is continuous and smooth.<\/p>\n<p>The lifetime of a compound nucleus is inversely proportional to its total width. <strong>Narrow resonances<\/strong> therefore correspond to capture while the wider resonances are due to scattering.<\/p>\n<p>See also: <a title=\"What is Nuclear Resonance \u2013 Compound Nucleus\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/compound-nucleus-reactions\/what-is-nuclear-resonance-compound-nucleus\/\">Nuclear Resonance<\/a><\/p>\n<\/div><\/div>\n<div   id="8n9s8rpp6h"    class=\"su-spoiler su-spoiler-style-default su-spoiler-icon-plus su-spoiler-closed\" data-scroll-offset=\"0\"><div   id="8n9s8rpp6h"    class=\"su-spoiler-title\" tabindex=\"0\" role=\"button\"><span class=\"su-spoiler-icon\"><\/span>Fast Neutron Region<\/div><div   id="8n9s8rpp6h"    class=\"su-spoiler-content su-u-clearfix su-u-trim\"><strong>The radiative capture cross-section<\/strong> at energies above the resonance region <strong>drops rapidly<\/strong> to very small values. This rapid drop is caused by the compound nucleus, which is formed in more highly-excited states. In these <strong>highly-excited states<\/strong> it is more likely that one neutron obtains an <strong>energy<\/strong> in collision with other nucleon <strong>greater than its <a title=\"Nuclear Binding Energy\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/atomic-nuclear-physics\/binding-energy\/nuclear-binding-energy\/\">binding energy<\/a><\/strong> in the nucleus. <strong>The neutron emission becomes dominant<\/strong> and gamma decay becomes less important. Moreover, at high energies, <a title=\"Neutron Inelastic Scattering\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/neutron-inelastic-scattering\/\">the inelastic scattering<\/a> and<strong> (n,2n) reaction<\/strong> are highly probable at the expense of both <a title=\"Neutron Elastic Scattering\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/neutron-elastic-scattering\/\">elastic scattering<\/a> and radiative capture.<\/p>\n<\/div><\/div>\n<\/div>\n<h2>Macroscopic Cross-section<\/h2>\n<p>The difference between the <strong>microscopic cross-section<\/strong> and <strong>macroscopic cross-section<\/strong>\u00a0is very important and is restated for clarity. The <strong>microscopic cross section<\/strong> represents the <strong>effective target area of a single target nucleus<\/strong> for an incident particle. The units are given in <strong>barns or cm<sup>2<\/sup><\/strong>.<\/p>\n<p>While the <strong>macroscopic cross-section<\/strong> represents the <strong>effective target area of all of the nuclei<\/strong> contained in the volume of the material. The units are given in <strong>cm<sup>-1<\/sup><\/strong>.<\/p>\n<p>A macroscopic cross-section is derived from <strong>microscopic cross-section<\/strong> and the <a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/atomic-number-density\/\"><strong>atomic number density<\/strong><\/a>:<\/p>\n<p style=\"text-align: center;\"><strong>\u03a3=\u03c3.N<\/strong><\/p>\n<p>Here <strong>\u03c3<\/strong>, which has units of m<sup>2<\/sup>, is the microscopic cross-section. Since the units of N (nuclei density) are nuclei\/m<sup>3<\/sup>, the macroscopic cross-section \u03a3 have units of m<sup>-1<\/sup>, thus in fact is an incorrect name, because it is not a correct unit of cross-sections. In terms of \u03a3<sub>t<\/sub> (the total cross-section), the equation for the intensity of a neutron beam can be written as<\/p>\n<p style=\"text-align: center;\"><strong>-dI = N.\u03c3.\u03a3<sub>t<\/sub>.dx<\/strong><\/p>\n<p>Dividing this expression by I(x) gives<\/p>\n<p style=\"text-align: center;\"><strong>-d\u0399(x)\/I(x) = \u03a3<sub>t<\/sub>.dx<\/strong><\/p>\n<p>Since dI(x) is the number of neutrons that collide in dx, the quantity &#8211;<strong>d\u0399(x)\/I(x)<\/strong> represents the probability that a neutron that has survived without colliding until x, will collide in the next layer dx.\u00a0It follows that the probability P(x) that a neutron will travel a distance x without any interaction in the material, which is characterized by \u03a3<sub>t<\/sub>, is:<\/p>\n<p style=\"text-align: center;\"><strong>P(x) = e<sup>-\u03a3<sub>t<\/sub>.x<\/sup><\/strong><\/p>\n<p>From this equation, we can derive the probability that a neutron will make its <strong>first collision in dx<\/strong>. It will be the quantity<strong> P(x)dx<\/strong>. If the probability of the first collision in dx is independent of its past history, the required result will be equal to the probability that a neutron survives up to layer x without any interaction (~\u03a3<sub>t<\/sub>dx) times the probability that the neutron will interact in the additional layer dx (i.e., ~e<sup>-\u03a3<sub>t<\/sub>.x<\/sup>).<\/p>\n<p style=\"text-align: center;\"><strong>P(x)dx = \u03a3<sub>t<\/sub>dx . e<sup>-\u03a3<sub>t<\/sub>.x<\/sup> = \u03a3<sub>t<\/sub> e<sup>-\u03a3<sub>t<\/sub>.x<\/sup> dx<\/strong><\/p>\n<h2>Mean Free Path<\/h2>\n<p>From the equation for the probability of the<strong> first collision in dx<\/strong> we can calculate <strong>the mean free path<\/strong> that is traveled by a neutron between two collisions. This quantity is usually designated by the symbol <strong>\u03bb<\/strong> and it is equal to the average value of x, the distance traveled by a neutron without any interaction, over the interaction probability distribution.<\/p>\n<p style=\"text-align: center;\"><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/12\/mean-free-path-equation.png\"><img loading=\"lazy\" class=\"alignnone size-full wp-image-13422\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/12\/mean-free-path-equation.png\" alt=\"mean free path - equation\" width=\"490\" height=\"82\" srcset=\"https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/12\/mean-free-path-equation.png 490w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/12\/mean-free-path-equation-300x50.png 300w\" sizes=\"(max-width: 490px) 100vw, 490px\" \/><\/a><\/p>\n<p>whereby one can distinguish<strong> \u03bb<sub>s<\/sub>, \u03bb<sub>a<\/sub>, \u03bb<sub>f<\/sub><\/strong>, etc. This quantity is also known as the <strong>relaxation length<\/strong>, because it is the distance in which the intensity of the neutrons that have not caused a reaction has decreased with a factor e.<\/p>\n<p>For materials with high <a title=\"Neutron Absorption Cross-section\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/neutron-absorption\/neutron-absorption-cross-section\/\">absorption cross-section<\/a>, the mean free path is <strong>very short<\/strong> and neutron absorption occurs mostly<strong> on the surface<\/strong> of the material. This surface absorption is called <strong>self-shielding<\/strong> because the outer layers of atoms shield the inner layers.<\/p>\n<h2>Macroscopic Cross-section of Mixtures and Molecules<\/h2>\n<p>Most materials are composed of several chemical elements and compounds. Most of chemical elements\u00a0contains several<strong> isotopes<\/strong> of these elements (e.g., <a title=\"Gadolinium\" href=\"http:\/\/sitepourvtc.com\/glossary\/gadolinium\/\">gadolinium<\/a> with its six stable isotopes). For this reason most materials involve many cross-sections. Therefore, to include all the isotopes within a given material, it is necessary to determine the\u00a0macroscopic cross section for each isotope and then sum all the individual macroscopic cross-sections.<\/p>\n<p>In this section both factors (different<strong> atomic densities<\/strong> and different <strong>cross-sections<\/strong>) will be considered in the calculation of the <strong>macroscopic cross-section of mixtures<\/strong>.<\/p>\n<p>First, consider the Avogadro\u2019s number N<sub>0<\/sub> = <strong>6.022 x 10<sup>23<\/sup><\/strong>, is the number of\u00a0particles (molecules, atoms)\u00a0that is contained in the amount of substance given by one mole.\u00a0Thus if M\u00a0is\u00a0the <strong>molecular weight<\/strong>, the ratio <strong>N<sub>0<\/sub>\/M<\/strong>\u00a0equals to the number of molecules in 1g of the\u00a0mixture. The number of molecules per cm<sup>3<\/sup> in the material of density \u03c1 and the macroscopic cross-section for mixtures are given by following equations:<\/p>\n<p style=\"text-align: center;\"><strong>N<sub>i<\/sub> = \u03c1<sub>i<\/sub>.N<sub>0<\/sub> \/ M<sub>i<\/sub><\/strong><\/p>\n<p><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/12\/Macroscopic-Cross-section-for-mixtures.png\"><img loading=\"lazy\" class=\"aligncenter size-full wp-image-13424\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/12\/Macroscopic-Cross-section-for-mixtures.png\" alt=\"Macroscopic Cross-section for Mixtures and Molecules\" width=\"239\" height=\"69\" \/><\/a><\/p>\n<figure id=\"attachment_13044\" aria-describedby=\"caption-attachment-13044\" style=\"width: 290px\" class=\"wp-caption alignright\"><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/11\/Elastic-Scaterring-on-Molecule-min.png\"><img loading=\"lazy\" class=\"size-medium wp-image-13044\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/11\/Elastic-Scaterring-on-Molecule-min-300x155.png\" alt=\"Scattering of slow neutrons by molecules is greater than by free nuclei.\" width=\"300\" height=\"155\" srcset=\"https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/11\/Elastic-Scaterring-on-Molecule-min-300x155.png 300w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/11\/Elastic-Scaterring-on-Molecule-min-1024x531.png 1024w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/11\/Elastic-Scaterring-on-Molecule-min.png 1067w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/a><figcaption id=\"caption-attachment-13044\" class=\"wp-caption-text\">Scattering of slow neutrons by molecules is greater than by free nuclei.<\/figcaption><\/figure>\n<p>Note that, in some cases, the cross-section of the molecule<strong> is not equal<\/strong> to the sum of cross-sections of its<strong> individual nuclei<\/strong>. For example the cross-section of<a title=\"Neutron Elastic Scattering\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/neutron-elastic-scattering\/\"> neutron elastic scattering<\/a> of water exhibits anomalies for <a title=\"Thermal Neutron\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/atomic-nuclear-physics\/fundamental-particles\/neutron\/thermal-neutron\/\">thermal neutrons<\/a>. It occurs, because the kinetic energy of an incident neutron is of the order or less than <strong>the chemical binding energy<\/strong> and therefore the scattering of slow neutrons by water (H<sub>2<\/sub>O) is greater than by free nuclei (2H + O).<\/p>\n<div   id="8n9s8rpp6h"    class=\"su-accordion su-u-trim\"><div   id="8n9s8rpp6h"    class=\"su-spoiler su-spoiler-style-default su-spoiler-icon-plus su-spoiler-closed\" data-scroll-offset=\"0\"><div   id="8n9s8rpp6h"    class=\"su-spoiler-title\" tabindex=\"0\" role=\"button\"><span class=\"su-spoiler-icon\"><\/span>Example - Macroscopic cross-section for boron carbide in control rods<\/div><div   id="8n9s8rpp6h"    class=\"su-spoiler-content su-u-clearfix su-u-trim\">A <a title=\"Control Rods\" href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/control-rods\/\"><strong>control rod<\/strong><\/a> usually contains solid <strong>boron carbide<\/strong> with natural boron. <a title=\"Boron 10\" href=\"http:\/\/sitepourvtc.com\/glossary\/boron-10\/\">Natural boron<\/a> consists primarily of two stable isotopes,<strong><sup>11<\/sup>B<\/strong> (80.1%) and<a title=\"Boron 10\" href=\"http:\/\/sitepourvtc.com\/glossary\/boron-10\/\"><strong> <sup>10<\/sup>B<\/strong> <\/a>(19.9%). Boron carbide has a density of <strong>2.52 g\/cm<sup>3<\/sup><\/strong>.<\/p>\n<p>Determine the <strong>total macroscopic cross-section<\/strong> and the <strong>mean free path<\/strong>.<\/p>\n<p>Density:<br \/>\nM<sub>B<\/sub> = 10.8<br \/>\nM<sub>C<\/sub> = 12<br \/>\nM<sub>Mixture<\/sub> = 4 x 10.8 + 1&#215;12 g\/mol<br \/>\nN<sub>B4C<\/sub> = \u03c1 . N<sub>a<\/sub> \/ M<sub>Mixture<\/sub><br \/>\n= (2.52 g\/cm<sup>3<\/sup>)x(6.02&#215;10<sup>23<\/sup> nuclei\/mol)\/ (4 x 10.8 + 1&#215;12 g\/mol)<br \/>\n= <strong>2.75&#215;10<sup>22<\/sup> molecules of B4C\/cm<sup>3<\/sup><\/strong><\/p>\n<p>N<sub>B<\/sub> = 4 x 2.75&#215;10<sup>22<\/sup> atoms of boron\/cm<sup>3<\/sup><br \/>\nN<sub>C<\/sub> = 1 x 2.75&#215;10<sup>22<\/sup> atoms of carbon\/cm<sup>3<\/sup><\/p>\n<p>N<sub>B10<\/sub> = 0.199 x 4 x 2.75&#215;10<sup>22<\/sup> = 2.18&#215;10<sup>22<\/sup> atoms of 10B\/cm<sup>3<\/sup><br \/>\nN<sub>B11<\/sub> = 0.801 x 4 x 2.75&#215;10<sup>22<\/sup> = 8.80&#215;10<sup>22<\/sup> atoms of 11B\/cm<sup>3<\/sup><br \/>\nN<sub>C<\/sub> = 2.75&#215;10<sup>22<\/sup> atoms of 12C\/cm<sup>3<\/sup><\/p>\n<p><strong>the microscopic cross-sections<\/strong><\/p>\n<p>\u03c3<sub>t<\/sub><sup>10B<\/sup> = 3843 b of which \u03c3<sub>(n,alpha)<\/sub><sup>10B<\/sup> = 3840 b<br \/>\n\u03c3<sub>t<\/sub><sup>11B<\/sup> = 5.07 b<br \/>\n\u03c3<sub>t<\/sub><sup>12C<\/sup> = 5.01 b<\/p>\n<p><strong>the macroscopic cross-section<\/strong><\/p>\n<p><strong>\u03a3<sub>t<\/sub><sup>B4C<\/sup> <\/strong>= 3843&#215;10<sup>-24<\/sup> x 2.18&#215;10<sup>22<\/sup> + 5.07&#215;10<sup>-24<\/sup> x 8.80&#215;10<sup>22<\/sup> + 5.01&#215;10<sup>-24<\/sup> x 2.75&#215;10<sup>22<\/sup><br \/>\n= 83.7 + 0.45 + 0.14 = <strong>84.3 cm<sup>-1<\/sup><\/strong><\/p>\n<p><strong>the mean free path<\/strong><\/p>\n<p><strong>\u03bb<sub>t<\/sub> <\/strong>= 1\/\u03a3<sub>t<\/sub><sup>B4C<\/sup> = 0.012 cm = <strong>0.12 mm<\/strong> (compare with B4C pellets diameter in control rods which may be around 7mm)<br \/>\n<strong>\u03bb<sub>a<\/sub> \u2248 0.12 mm<\/strong><\/p>\n<\/div><\/div>\n<div   id="8n9s8rpp6h"    class=\"su-spoiler su-spoiler-style-default su-spoiler-icon-plus su-spoiler-closed\" data-scroll-offset=\"0\"><div   id="8n9s8rpp6h"    class=\"su-spoiler-title\" tabindex=\"0\" role=\"button\"><span class=\"su-spoiler-icon\"><\/span>Example - Atomic number density of 235U in uranium fuel<\/div><div   id="8n9s8rpp6h"    class=\"su-spoiler-content su-u-clearfix su-u-trim\">It was written the <strong>macroscopic cross-section<\/strong> is derived from <strong>microscopic cross-section<\/strong> and the <strong>atomic number density (N)<\/strong>:<\/p>\n<p style=\"text-align: center;\"><strong>\u03a3=\u03c3.N<\/strong><\/p>\n<p>In this equation, the <strong>atomic number density<\/strong> plays the crucial role as the microscopic cross-section, because in the <a title=\"Reactor Core\" href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/nuclear-reactor-core\/\">reactor core<\/a> the atomic number density of certain materials (e.g., water as the <a title=\"Neutron Moderator\" href=\"http:\/\/sitepourvtc.com\/neutron-moderator\/\">moderator<\/a>) can be simply changed leading into certain <strong>reactivity changes<\/strong>. In order to understand the nature of these <strong>reactivity changes<\/strong>, we must understand the term the atomic number density.<\/p>\n<p>See theory: <a title=\"Atomic Number Density\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/atomic-number-density\/\">Atomic Number Density<\/a><\/p>\n<p>Most of <a title=\"PWR \u2013 Pressurized water reactor\" href=\"http:\/\/sitepourvtc.com\/pwr-pressurized-water-reactor\/\">PWRs<\/a> use the <b>uranium fuel<\/b>, which is in the form of <b>uranium dioxide <\/b>(UO<sub>2<\/sub>). Typically, the fuel have enrichment of \u03c9<sub>235<\/sub> = 4% [grams of <a title=\"Uranium 235\" href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/nuclear-fuel\/uranium\/uranium-235\/\"><sup>235<\/sup>U<\/a>\u00a0per gram of <a title=\"Uranium\" href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/nuclear-fuel\/uranium\/\">uranium<\/a>] of isotope <a title=\"Uranium 235\" href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/nuclear-fuel\/uranium\/uranium-235\/\"><sup>235<\/sup>U<\/a>.<\/p>\n<p><strong>Calculate the atomic number density of <a title=\"Uranium 235\" href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/nuclear-fuel\/uranium\/uranium-235\/\"><sup>235<\/sup>U<\/a><\/strong><strong>\u00a0(N235U), when:<\/strong><\/p>\n<ul>\n<li>the molecular weight of the enriched uranium M<sub>UO2<\/sub> = 237.9 + 32 = <strong>269.9 g\/mol<\/strong><\/li>\n<li>the uranium density <span style=\"font-size: large;\">\u2374<\/span><sub>UO2<\/sub> = <strong>10.5 g\/cm<sup>3<\/sup><\/strong><\/li>\n<\/ul>\n<p style=\"text-align: left;\"><strong>N<sub>UO2<\/sub> = <span style=\"font-size: large;\">\u2374<\/span><sub>UO2<\/sub>\u00a0. N<sub>A<\/sub>\u00a0\/ M<sub>UO2<\/sub><\/strong><\/p>\n<p><strong>N<sub>UO2\u00a0<\/sub><\/strong>= (10.5 g\/cm<sup>3<\/sup>) x (6.02&#215;10<sup>23<\/sup>\u00a0nuclei\/mol)\/ 269.9<br \/>\n<strong>N<sub>UO2\u00a0<\/sub><\/strong>= 2.34 x\u00a010<sup>22<\/sup> molecules of UO2\/cm<sup>3<\/sup><\/p>\n<p>N<sub>U<\/sub> = 1 x 2.34&#215;10<sup>22<\/sup> atoms of uranium\/cm<sup>3<\/sup><br \/>\nN<sub>O<\/sub> = 2 x 2.34&#215;10<sup>22<\/sup> atoms of oxide\/cm<sup>3<\/sup><\/p>\n<p><strong>N<sub>235U<\/sub><\/strong> = \u03c9<sub>235<\/sub>.N<sub>A<\/sub>.<span style=\"font-size: large;\">\u2374<\/span><sub>UO2<\/sub>\/M<sub>235U<\/sub> x (M<sub>U<\/sub>\/M<sub>UO2<\/sub>)<\/p>\n<p><strong>N<sub>235U<\/sub><\/strong> = 0.04 x 6.02&#215;10<sup>23<\/sup>\u00a0x 10.5 \/ 235 x 237.9 \/ 269.9 =<strong>9.48\u00a0x\u00a010<sup>20<\/sup>\u00a0atoms of 235U\/cm<sup>3<\/sup><\/strong><\/p>\n<\/div><\/div><\/div>\n<h2>Doppler Broadening of Resonances<\/h2>\n<p>In general, <a title=\"Doppler Broadening\" href=\"http:\/\/sitepourvtc.com\/glossary\/doppler-broadening\/\">Doppler broadening<\/a> is the broadening of spectral lines due to the <strong>Doppler effect<\/strong> caused by a distribution of kinetic energies of molecules or atoms. In reactor physics a particular case of this phenomenon is the <strong>thermal Doppler broadening of the resonance capture cross sections<\/strong> of the <a title=\"Fertile Material\" href=\"http:\/\/sitepourvtc.com\/glossary\/fertile-material\/\">fertile material<\/a> (e.g.,<a title=\"Uranium 238\" href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/nuclear-fuel\/uranium\/uranium-238\/\"> <sup>238<\/sup>U<\/a> or<a title=\"Plutonium 240\" href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/nuclear-fuel\/plutonium\/plutonium-240\/\"> <sup>240<\/sup>Pu<\/a>) caused by <strong>thermal motion of target nuclei<\/strong> in the <a title=\"Nuclear Fuel\" href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/nuclear-fuel\/\">nuclear fuel<\/a>.<\/p>\n<figure id=\"attachment_378\" aria-describedby=\"caption-attachment-378\" style=\"width: 290px\" class=\"wp-caption alignright\"><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2014\/11\/Doppler_broadening.jpg\"><img loading=\"lazy\" class=\"size-medium wp-image-378\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2014\/11\/Doppler_broadening-300x161.jpg\" alt=\"Doppler effect\" width=\"300\" height=\"161\" srcset=\"https:\/\/sitepourvtc.com\/wp-content\/uploads\/2014\/11\/Doppler_broadening-300x161.jpg 300w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2014\/11\/Doppler_broadening.jpg 624w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/a><figcaption id=\"caption-attachment-378\" class=\"wp-caption-text\">Doppler effect improves reactor stability. Broadened resonance (heating of a fuel) results in a higher probability of absorbtion, thus causes negative reactivity insertion (reduction of reactor power).<\/figcaption><\/figure>\n<p>The Doppler broadening of resonances is <strong>very important phenomenon<\/strong>, which <strong>improves reactor stability<\/strong>, because it accounts for the dominant part of the<strong> fuel temperature coefficient<\/strong> (the change in reactivity per degree change in fuel temperature) in thermal reactors and makes a substantial contribution in <a title=\"Fast Neutron Reactor\" href=\"http:\/\/sitepourvtc.com\/fast-neutron-reactor\/\">fast reactors<\/a> as well. This coefficient is also called the <strong>prompt temperature coefficient<\/strong> because it causes an <strong>immediate response<\/strong> on changes in fuel temperature. The prompt temperature coefficient of most thermal reactors<strong> is negative<\/strong>.<\/p>\n<p>See also: <a title=\"Doppler Broadening\" href=\"http:\/\/sitepourvtc.com\/glossary\/doppler-broadening\/\">Doppler Broadening<\/a><\/p>\n<h2>Self-Shielding<\/h2>\n<p>It was written, in some cases the amount of absorption reactions is dramatically reduced despite the <strong>unchanged microscopic cross-section<\/strong> of the material. This phenomena is commonly known as <strong>the <a title=\"Self-shielding\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/self-shielding\/\">resonance self-shielding<\/a><\/strong> and also <strong>contributes to to the reactor stability<\/strong>. There are two types of self-shielding.<\/p>\n<ul>\n<li><strong>Energy Self-shielding.<\/strong><\/li>\n<li><strong>Spatial Self-shielding.<\/strong><\/li>\n<\/ul>\n<p>See also: <a title=\"Self-shielding\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/self-shielding\/\">Resonance Self-shielding<\/a><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/12\/self-shielding-min.png\"><img loading=\"lazy\" class=\"aligncenter size-large wp-image-13381\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/12\/self-shielding-min-1024x512.png\" alt=\"Self-shielding - neutron cross-section\" width=\"1024\" height=\"512\" srcset=\"https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/12\/self-shielding-min-1024x512.png 1024w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/12\/self-shielding-min-300x150.png 300w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/12\/self-shielding-min.png 1200w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/a>An increase in temperature from T<sub>1<\/sub> to T<sub>2<\/sub> causes the broadening of spectral lines of resonances. Although the area under the resonance remains the same, the broadening of spectral lines causes an<strong> increase in <a title=\"Neutron Flux \u2013 Neutron Intensity\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/neutron-flux-neutron-intensity\/\">neutron flux<\/a><\/strong> in the fuel \u03c6<sub>f<\/sub>(E), which in turn increases the absorption as the temperature increases.<\/p>\n<\/div>\n<div   id="8n9s8rpp6h"    class=\"su-divider su-divider-style-dotted\" style=\"margin:20px 0;border-width:2px;border-color:#999999\"><\/div>\n<div   id="8n9s8rpp6h"     class=\"lgc-column lgc-grid-parent lgc-grid-100 lgc-tablet-grid-100 lgc-mobile-grid-100 lgc-equal-heights \"><div   id="8n9s8rpp6h"     class=\"inside-grid-column\">\u00a0 <div   id="8n9s8rpp6h"    class=\"su-accordion su-u-trim\"><div   id="8n9s8rpp6h"    class=\"su-spoiler su-spoiler-style-default su-spoiler-icon-plus su-spoiler-closed\" data-scroll-offset=\"0\"><div   id="8n9s8rpp6h"    class=\"su-spoiler-title\" tabindex=\"0\" role=\"button\"><span class=\"su-spoiler-icon\"><\/span>References:<\/div><div   id="8n9s8rpp6h"    class=\"su-spoiler-content su-u-clearfix su-u-trim\"><strong>Nuclear and Reactor Physics:<\/strong>\n<ol>\n<li>J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading,\u00a0MA (1983).<\/li>\n<li>J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.<\/li>\n<li>W. M. Stacey, Nuclear Reactor Physics, John Wiley &amp; Sons, 2001, ISBN: 0- 471-39127-1.<\/li>\n<li>Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering,\u00a0Springer; 4th edition, 1994, ISBN:\u00a0978-0412985317<\/li>\n<li>W.S.C. Williams. Nuclear and Particle Physics.\u00a0Clarendon Press; 1 edition, 1991, ISBN:\u00a0978-0198520467<\/li>\n<li>G.R.Keepin. Physics of Nuclear Kinetics.\u00a0Addison-Wesley Pub. Co; 1st edition, 1965<\/li>\n<li>Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.<\/li>\n<li>U.S. Department of Energy, Nuclear Physics and Reactor Theory.\u00a0DOE Fundamentals Handbook,\u00a0Volume 1 and 2.\u00a0January\u00a01993.<\/li>\n<\/ol>\n<p><strong><\/strong><strong>Advanced Reactor Physics:<\/strong><\/p>\n<ol>\n<li>K. O. Ott, W. A. Bezella, Introductory Nuclear Reactor Statics, American Nuclear Society, Revised edition (1989), 1989, ISBN: 0-894-48033-2.<\/li>\n<li>K. O. Ott, R. J. Neuhold, Introductory Nuclear Reactor Dynamics, American Nuclear Society, 1985, ISBN: 0-894-48029-4.<\/li>\n<li><span style=\"font-weight: 400;\">D. L. Hetrick, Dynamics of Nuclear Reactors, American Nuclear Society, 1993, ISBN: 0-894-48453-2.\u00a0<\/span><\/li>\n<li>E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport, American Nuclear Society, 1993, ISBN: 0-894-48452-4.<\/li>\n<\/ol>\n<\/div><\/div><\/div><div   id="8n9s8rpp6h"    class=\"su-divider su-divider-style-dotted\" style=\"margin:15px 0;border-width:2px;border-color:#999999\"><\/div><\/div><\/div><div   id="8n9s8rpp6h"    class=\"su-divider su-divider-style-default\" style=\"margin:15px 0;border-width:2px;border-color:#999999\"><\/div><div   id="8n9s8rpp6h"     class=\"lgc-column lgc-grid-parent lgc-grid-33 lgc-tablet-grid-33 lgc-mobile-grid-100 lgc-equal-heights \"><div   id="8n9s8rpp6h"     class=\"inside-grid-column\">\n<h2>See previous:<\/h2>\n<a href=\"\" class=\"su-button su-button-style-flat\" style=\"color:#606060;background-color:#ffffff;border-color:#cccccc;border-radius:10px;-moz-border-radius:10px;-webkit-border-radius:10px\" target=\"_self\"><span style=\"color:#606060;padding:7px 20px;font-size:16px;line-height:24px;border-color:#ffffff;border-radius:10px;-moz-border-radius:10px;-webkit-border-radius:10px;text-shadow:0px 0px 0px #000000;-moz-text-shadow:0px 0px 0px #000000;-webkit-text-shadow:0px 0px 0px #000000\"><i class=\"sui sui-link\" style=\"font-size:16px;color:#5d5d5d\"><\/i> <\/span><\/a><\/div><\/div><div   id="8n9s8rpp6h"     class=\"lgc-column lgc-grid-parent lgc-grid-33 lgc-tablet-grid-33 lgc-mobile-grid-100 lgc-equal-heights \"><div   id="8n9s8rpp6h"     class=\"inside-grid-column\">\n<h2>See above:<\/h2>\n<p>Neutron Nuclear Reactions<a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/\" class=\"su-button su-button-style-flat\" style=\"color:#606060;background-color:#ffffff;border-color:#cccccc;border-radius:10px;-moz-border-radius:10px;-webkit-border-radius:10px\" target=\"_self\"><span style=\"color:#606060;padding:7px 20px;font-size:16px;line-height:24px;border-color:#ffffff;border-radius:10px;-moz-border-radius:10px;-webkit-border-radius:10px;text-shadow:0px 0px 0px #000000;-moz-text-shadow:0px 0px 0px #000000;-webkit-text-shadow:0px 0px 0px #000000\"><i class=\"sui sui-link\" style=\"font-size:16px;color:#5d5d5d\"><\/i> <\/span><\/a><\/p><\/div><\/div><div   id="8n9s8rpp6h"     class=\"lgc-column lgc-grid-parent lgc-grid-33 lgc-tablet-grid-33 lgc-mobile-grid-100 lgc-equal-heights \"><div   id="8n9s8rpp6h"     class=\"inside-grid-column\">\n<h2>See next:<\/h2>\n<a href=\"\" class=\"su-button su-button-style-flat\" style=\"color:#606060;background-color:#ffffff;border-color:#cccccc;border-radius:10px;-moz-border-radius:10px;-webkit-border-radius:10px\" target=\"_self\"><span style=\"color:#606060;padding:7px 20px;font-size:16px;line-height:24px;border-color:#ffffff;border-radius:10px;-moz-border-radius:10px;-webkit-border-radius:10px;text-shadow:0px 0px 0px #000000;-moz-text-shadow:0px 0px 0px #000000;-webkit-text-shadow:0px 0px 0px #000000\"><i class=\"sui sui-link\" style=\"font-size:16px;color:#5d5d5d\"><\/i> <\/span><\/a><\/div><\/div>\n","protected":false},"excerpt":{"rendered":"<p>At\u00a0 this point we have to distinguish between two basic types of nuclear or neutron cross-sections. Microscopic Cross-section. The effective target area in m2 presented by a single nucleus to an incident neutron beam is denoted the microscopic cross section, \u03c3. The microscopic cross-sections characterize interactions with single isotopes and are a part of data &#8230; <a title=\"Neutron Cross-section\" class=\"read-more\" href=\"https:\/\/sitepourvtc.com\/neutron-cross-section\/\" aria-label=\"More on Neutron Cross-section\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"generate_page_header":""},"categories":[8],"tags":[17],"_links":{"self":[{"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/posts\/359"}],"collection":[{"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/comments?post=359"}],"version-history":[{"count":7,"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/posts\/359\/revisions"}],"predecessor-version":[{"id":31567,"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/posts\/359\/revisions\/31567"}],"wp:attachment":[{"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/media?parent=359"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/categories?post=359"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/tags?post=359"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}