{"id":18218,"date":"2018-07-10T19:17:20","date_gmt":"2018-07-10T19:17:20","guid":{"rendered":"http:\/\/sitepourvtc.com\/?page_id=18218"},"modified":"2023-01-27T07:37:13","modified_gmt":"2023-01-27T07:37:13","slug":"point-kinetics-equations","status":"publish","type":"page","link":"https:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/reactor-dynamics\/point-kinetics-equations\/","title":{"rendered":"Point Kinetics Equations"},"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\">To study the kinetic behavior of the system, engineers usually use <strong>point kinetics equations<\/strong>. The name <strong>point kinetics<\/strong> is used because, in this simplified formalism, the <strong>shape<\/strong> of the neutron flux and the neutron density <strong>distribution<\/strong> is <strong>ignored<\/strong>. The reactor is therefore <strong>reduced to a point<\/strong>. <\/div><\/div>\n<p>As we have seen in previous chapters, the number of neutrons is multiplied by a factor k<sub>eff<\/sub> from one neutron generation to the next. Therefore, the multiplication environment (<a href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/what-is-nuclear-reactor\/\">nuclear reactor<\/a>) behaves like an exponential system, which means the power increase is not linear but <strong>exponential<\/strong>.<\/p>\n<p><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/06\/Reactor-Period-Inhour-Equation-min.png\"><img loading=\"lazy\" class=\"alignright size-medium wp-image-18123\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/06\/Reactor-Period-Inhour-Equation-min-217x300.png\" alt=\"Reactor Period - Inhour Equation\" width=\"217\" height=\"300\" srcset=\"https:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/06\/Reactor-Period-Inhour-Equation-min-217x300.png 217w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/06\/Reactor-Period-Inhour-Equation-min.png 648w\" sizes=\"(max-width: 217px) 100vw, 217px\" \/><\/a>The<strong> effective multiplication factor<\/strong> in a multiplying system measures the change in the <a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-fission-chain-reaction\/neutron-generation-neutron-population\/\">fission neutron population<\/a> from one neutron generation to the subsequent generation.<\/p>\n<ul>\n<li><strong>k<\/strong><strong><sub>eff<\/sub><\/strong><strong> &lt; 1<\/strong>. Suppose the multiplication factor for a multiplying system is <strong>less than 1.0. In that case,<\/strong>\u00a0the <strong>number of neutrons decreases<\/strong> in time (with the <a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/fission\/delayed-neutrons\/mean-generation-time-with-delayed-neutrons\/\">mean generation time<\/a>), and the chain reaction will never be self-sustaining. This condition is known as <strong>the subcritical state<\/strong>.<\/li>\n<\/ul>\n<ul>\n<li><strong>k<\/strong><strong><sub>eff<\/sub><\/strong><strong> = 1<\/strong>. If the multiplication factor for a multiplying system is <strong>equal to 1.0<\/strong>, then there is <strong>no change in neutron population<\/strong> in time, and the chain reaction will be<strong> self-sustaining<\/strong>. This condition is known as <strong>the critical state<\/strong>.<\/li>\n<\/ul>\n<ul>\n<li><strong>k<\/strong><strong><sub>eff<\/sub><\/strong><strong> &gt; 1<\/strong>. If the multiplication factor for a multiplying system is <strong>greater than 1.0<\/strong>, then the multiplying system produces <strong>more neutrons<\/strong> than are needed to be self-sustaining. The number of neutrons is exponentially increasing in time (with the mean generation time). This condition is known as <strong>the supercritical state<\/strong>.<\/li>\n<\/ul>\n<p>But we have not yet discussed the <strong>duration of a neutron generation<\/strong>, which means<strong>\u00a0how many times in one second we have to multiply the neutron population by a factor k<sub>eff<\/sub><\/strong>. This time determines the <strong>speed of the exponential growth<\/strong>. But as was written, there are different types of neutrons: prompt neutrons and delayed neutrons, which completely change the kinetic behavior of the system. Therefore such a discussion will be not trivial.<\/p>\n<p>To study the kinetic behavior of the system, engineers usually use <strong>point kinetics equations<\/strong>. The name <strong>point kinetics<\/strong> is used because, in this simplified formalism, the <strong>shape<\/strong> of the neutron flux and the neutron density <strong>distribution<\/strong> is <strong>ignored<\/strong>. The reactor is therefore <strong>reduced to a point<\/strong>. The following section will introduce point kinetics and start with point kinetics in its<strong> simplest form<\/strong>.<\/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> Derivation of Simple Point Kinetics Equation<\/div><div   id="8n9s8rpp6h"    class=\"su-spoiler-content su-u-clearfix su-u-trim\">Let <strong><em>n(t)<\/em><\/strong> be the number of neutrons as a function of time <em>t<\/em> and <em>l <\/em>the <strong>prompt neutron lifetime, which<\/strong> is the <strong>average time from a prompt neutron emission<\/strong> to either <strong>its absorption<\/strong> (<a title=\"Nuclear Fission\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/fission\/\">fission<\/a> or <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<\/a>) or <strong>its escape<\/strong> from the system. The average number of neutrons that disappear during a unit time interval <em>dt <\/em>is<strong> <em>n.dt\/l. <\/em><\/strong>But each disappearance of a neutron contributes an average of <em>k<\/em> new neutrons.<\/p>\n<p>Finally, the change in the number of neutrons during a unit time interval <em>dt <\/em>is:<\/p>\n<p><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/06\/derivation-of-point-kinetics-equation-min.png\"><img loading=\"lazy\" class=\"aligncenter size-full wp-image-18120\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/06\/derivation-of-point-kinetics-equation-min.png\" alt=\"derivation of point kinetics equation-min\" width=\"353\" height=\"273\" srcset=\"https:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/06\/derivation-of-point-kinetics-equation-min.png 353w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/06\/derivation-of-point-kinetics-equation-min-300x232.png 300w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/06\/derivation-of-point-kinetics-equation-min-180x138.png 180w\" sizes=\"(max-width: 353px) 100vw, 353px\" \/><\/a><\/p>\n<p><strong>where:<\/strong><\/p>\n<p><strong>n(t) = transient reactor power<\/strong><\/p>\n<p><strong>n(0) = initial reactor power<\/strong><\/p>\n<p><strong>\u03c4<\/strong><strong><sub>e<\/sub><\/strong><strong> = reactor period<\/strong><\/p>\n<p><strong>The <a title=\"Reactor Period\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-fission-chain-reaction\/reactivity\/reactor-period-2\/\">reactor period<\/a>, <\/strong><strong>\u03c4<\/strong><strong><sub>e<\/sub><\/strong>, or <strong>e-folding time<\/strong>, is defined as the time required for the <a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/neutron-flux-neutron-intensity\/\">neutron density<\/a> to change by a factor e = 2.718. The reactor period is usually expressed in units of seconds or minutes. The <strong>smaller<\/strong> the value of <strong>\u03c4<\/strong><strong><sub>e<\/sub><\/strong>, the <strong>more rapid<\/strong> the change in reactor power. The reactor period may be positive or negative.<\/div><\/div><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>Simple Point Kinetics Equation without Delayed Neutrons<\/div><div   id="8n9s8rpp6h"    class=\"su-spoiler-content su-u-clearfix su-u-trim\">An equation governing the neutron kinetics of the system without source and with the absence of <a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/fission\/delayed-neutrons\/\">delayed neutrons<\/a> is <strong>the point kinetics equation<\/strong> (in a certain form). This equation states that the time change of the neutron population is equal to the <strong>excess of neutron production<\/strong> (by fission) <strong>minus neutron loss<\/strong> by absorption<strong> in one prompt neutron lifetime<\/strong>. The role of prompt neutron lifetime is evident. Shorter lifetimes give simply faster responses to multiplying systems.<\/p>\n<p>If there are neutrons in the system at t=0, that is, if n(0) &gt; 0, the solution of this equation gives the simplest form of point kinetics equation (without source and delayed neutrons).<\/p>\n<p><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/11\/point-kinetics-equation.png\"><img loading=\"lazy\" class=\"aligncenter size-full wp-image-12853\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/11\/point-kinetics-equation.png\" alt=\"point kinetics equation\" width=\"328\" height=\"185\" srcset=\"https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/11\/point-kinetics-equation.png 328w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/11\/point-kinetics-equation-300x169.png 300w\" sizes=\"(max-width: 328px) 100vw, 328px\" \/><\/a><\/p>\n<p>This simple point kinetics equation is often expressed in terms of reactivity and prompt generation time, <strong>\u039b<\/strong>, as:<\/p>\n<p><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/06\/point-kinetics-equation_1-min.png\"><img loading=\"lazy\" class=\"aligncenter size-full wp-image-18119\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/06\/point-kinetics-equation_1-min.png\" alt=\"point kinetics equation_1-min\" width=\"202\" height=\"122\" \/><\/a><\/p>\n<p>where<\/p>\n<ul>\n<li><strong>\u03c1<\/strong> = (k-1)\/k is the<a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-fission-chain-reaction\/reactivity\/\"> reactivity<\/a>, which describes the <strong>deviation of an effective multiplication factor from unity<\/strong>.<\/li>\n<li><strong>\u039b = l\/k<\/strong><strong><sub>eff<\/sub><\/strong><strong> = <a title=\"Prompt Generation Time \u2013 Mean Generation Time\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/fission\/prompt-neutrons\/prompt-generation-time-mean-generation-time\/\">prompt neutron generation time<\/a>,<\/strong> the average time from a prompt neutron emission to absorption that results only in fission.<\/li>\n<\/ul>\n<p>Both forms of the point kinetics equation are valid. The equation using <strong>\u039b, prompt neutron generation time, <\/strong>is usually better for calculations. This is because most reactivity transients are induced by changes in the absorption cross-section rather than in the fission cross-section. The prompt neutron lifetime is not constant during these transients, whereas the prompt generation time remains constant.<\/p>\n<p>Example:<\/p>\n<p>Let us consider that <strong>the prompt neutron lifetime is ~2 x 10<\/strong><strong><sup>-5<\/sup><\/strong>, and k (k<sub>\u221e<\/sub> \u2013 neutron multiplication factor) will be step increased <strong>by only 0.01%<\/strong> (<strong>i.e., 10pcm or ~1.5 cents<\/strong>), that is, k<sub>\u221e<\/sub>=1.0000 will increase to k<sub>\u221e<\/sub>=1.0001.<\/p>\n<p>It must be noted such reactivity insertion (10pcm)<strong> is very small<\/strong> in case of LWRs. The reactivity insertions <strong>of the order of one pcm<\/strong> are for LWRs <strong>practically unrealizable<\/strong>. In this case the reactor period will be:<\/p>\n<p><strong>T = l \/ (k<\/strong><strong><sub>\u221e <\/sub><\/strong><strong>&#8211; 1) = 2 x 10<\/strong><strong><sup>-5 <\/sup><\/strong><strong>\/ (1.0001 \u2013 1) = 0.2s<\/strong><\/p>\n<p><strong>This is a very short period. <\/strong>In one second, the neutron flux (and power) in <a href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/what-is-nuclear-reactor\/\">the reactor<\/a> would increase by a factor of e<sup>5<\/sup> = 2.718<sup>5<\/sup>. In 10 seconds, the reactor would pass through 50 periods, and the power would increase by e<sup>50<\/sup>.<\/p>\n<p>Furthermore, in the case of <a href=\"http:\/\/sitepourvtc.com\/fast-neutron-reactor\/\">fast reactors<\/a> in which prompt neutron lifetimes are <strong>of the order of 10<\/strong><strong><sup>-7<\/sup><\/strong><strong> seconds<\/strong>, the response of such a small reactivity insertion will be even more unimaginable. In the case of 10<sup>-7<\/sup>, the period will be:<\/p>\n<p><strong>T = l \/ (k<\/strong><strong><sub>\u221e <\/sub><\/strong><strong>&#8211; 1) = 10-7 \/ (1.0001 \u2013 1) = 0.001s<\/strong><\/p>\n<p><strong>Reactors with such kinetics would be very difficult to control.<\/strong> <strong>Fortunately, this behavior is not observed<\/strong> in any multiplying system. Actual reactor periods are observed to be considerably longer than computed above, and therefore the <a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-fission-chain-reaction\/\">nuclear chain reaction<\/a> can be <strong>controlled more easily<\/strong>. The longer periods are observed due to the presence of <a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/fission\/delayed-neutrons\/\"><strong>the delayed neutrons<\/strong><\/a><strong>.<\/strong><\/div><\/div><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>Simple Point Kinetics Equation with Delayed Neutrons<\/div><div   id="8n9s8rpp6h"    class=\"su-spoiler-content su-u-clearfix su-u-trim\">The simplest equation governing the neutron kinetics of the system with <a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/fission\/delayed-neutrons\/\">delayed neutrons<\/a> is the simple <strong>point kinetics equation with delayed neutrons<\/strong>. This equation states that the time change of the <a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-fission-chain-reaction\/neutron-generation-neutron-population\/\">neutron population<\/a> is equal to the <strong>excess of neutron production<\/strong> (by fission) <strong>minus neutron loss<\/strong> by absorption <strong>in one <\/strong><a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/fission\/delayed-neutrons\/mean-generation-time-with-delayed-neutrons\/\"><strong>mean generation time with delayed neutrons<\/strong><\/a><strong> (l<\/strong><strong><sub>d<\/sub><\/strong><strong>)<\/strong>.<\/p>\n<p><strong>l<\/strong><strong><sub>d<\/sub><\/strong><strong> = (1 \u2013 \u03b2).l<\/strong><strong><sub>p<\/sub><\/strong><strong> + \u2211l<\/strong><strong><sub>i<\/sub><\/strong><strong> . \u03b2<\/strong><strong><sub>i<\/sub><\/strong><strong> =&gt; l<\/strong><strong><sub>d<\/sub><\/strong><strong> = (1 \u2013 \u03b2).l<\/strong><strong><sub>p<\/sub><\/strong><strong> + \u2211\u03c4<\/strong><strong><sub>i<\/sub><\/strong><strong> . \u03b2<\/strong><strong><sub>i<\/sub><\/strong><\/p>\n<p>where<\/p>\n<ul>\n<li><strong>(1 \u2013 \u03b2)<\/strong> is the fraction of all <a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/atomic-nuclear-physics\/fundamental-particles\/neutron\/\">neutrons<\/a> emitted as <a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/fission\/prompt-neutrons\/\">prompt neutrons<\/a><\/li>\n<li><strong>l<\/strong><strong><sub>p<\/sub><\/strong> is the prompt neutron lifetime<\/li>\n<li><strong>\u03c4<\/strong><strong><sub>i <\/sub><\/strong>is the mean precursor lifetime, the inverse value of the decay constant<strong> \u03c4<\/strong><strong><sub>i<\/sub><\/strong><strong> = 1\/\u03bb<\/strong><strong><sub>i<\/sub><\/strong><\/li>\n<li>The weighted delayed generation time is given by<strong> \u03c4 = \u2211\u03c4<\/strong><strong><sub>i<\/sub><\/strong><strong> . \u03b2<\/strong><strong><sub>i<\/sub><\/strong><strong> \/ \u03b2 = 13.05 s<\/strong><\/li>\n<li>Therefore the weighted decay constant <strong>\u03bb = 1 \/ \u03c4 \u2248 0.08 s<\/strong><strong><sup>-1<\/sup><\/strong><\/li>\n<\/ul>\n<p>The number, <strong>0.08 s<\/strong><strong><sup>-1<\/sup><\/strong>, is relatively high and has <strong>a dominating effect on reactor time response<\/strong>, although delayed neutrons are a small fraction of all neutrons in <a href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/nuclear-reactor-core\/\">the core<\/a>. This is best illustrated by calculating a weighted mean generation time with delayed neutrons:<\/p>\n<p><strong>l<\/strong><strong><sub>d<\/sub><\/strong><strong> = (1 \u2013 \u03b2).l<\/strong><strong><sub>p<\/sub><\/strong><strong> + \u2211\u03c4<\/strong><strong><sub>i<\/sub><\/strong><strong> . \u03b2<\/strong><strong><sub>i<\/sub><\/strong><strong> = (1 \u2013 0.0065). 2 x 10<\/strong><strong><sup>-5<\/sup><\/strong><strong> + 0.085 = 0.00001987 + 0.085 \u2248 0.085<\/strong><\/p>\n<p>In short, <strong>the mean generation time with delayed neutrons<\/strong> is about <strong>~0.1 s<\/strong>, rather than ~<strong>10<\/strong><strong><sup>-5<\/sup><\/strong> as in section Prompt Neutron Lifetime, where the delayed neutrons were omitted.<\/p>\n<p>The role of <strong>l<\/strong><strong><sub>d<\/sub><\/strong> is evident, and longer lifetimes give simply slower responses to multiplying systems. The role of reactivity (k<sub>eff<\/sub> \u2013 1) is also evident, and higher reactivity gives the simply larger response of the multiplying system.<\/p>\n<p>If there are neutrons in the system at t=0, that is, if n(0) &gt; 0, the solution of this equation gives <strong>the simplest point kinetics equation with delayed neutrons (similarly to the <\/strong><a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/fission\/prompt-neutrons\/infinite-multiplying-system-without-source-and-delayed-neutrons\/\"><strong>case without delayed neutrons<\/strong><\/a><strong>):<\/strong><\/p>\n<p><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/11\/point-kinetics-equation-with-delayed-neutrons.png\"><img loading=\"lazy\" class=\"aligncenter size-full wp-image-12892\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/11\/point-kinetics-equation-with-delayed-neutrons.png\" alt=\"point kinetics equation with delayed neutrons\" width=\"282\" height=\"152\" \/><\/a><\/p>\n<p>Example:<\/p>\n<p>Let us consider that <strong>the mean generation time with delayed neutrons is ~0.085,<\/strong> and k (k<sub>\u221e<\/sub> \u2013 neutron multiplication factor) will be step increased <strong>by only 0.01%<\/strong> (<strong>i.e., 10pcm or ~1.5 cents<\/strong>). That is, k<sub>\u221e<\/sub>=1.0000 will increase to k<sub>\u221e<\/sub>=1.0001.<\/p>\n<p>It must be noted such reactivity insertion (10pcm)<strong> is very small<\/strong> in the case of LWRs (e.g., one step by control rods). The reactivity insertions <strong>of the order of one pcm<\/strong> are for LWRs <strong>practically unrealizable<\/strong>. In this case, the reactor period will be:<\/p>\n<p><strong>T = l<\/strong><strong><sub>d<\/sub><\/strong><strong> \/ (k<\/strong><strong><sub>\u221e<\/sub><\/strong><strong>-1) = 0.085 \/ (1.0001-1) = 850s<\/strong><\/p>\n<p>This is a very long period. In ~14 minutes, the <a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/neutron-flux-neutron-intensity\/\">neutron flux<\/a> (and power) in the reactor would increase by a factor of e = 2.718. This is a completely different dimension of the response on reactivity insertion in comparison with the case <a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/fission\/prompt-neutrons\/infinite-multiplying-system-without-source-and-delayed-neutrons\/\">without the presence of delayed neutrons<\/a>, where the reactor period was 1 second.<\/div><\/div><\/div>\n<p>Both simple point kinetics equations are only an approximation because they use many simplifications. The simple <strong>point kinetics equation with delayed neutrons <\/strong>completely fails for higher reactivity insertions, where is a significant difference between the production of prompt and delayed neutrons. Therefore a more accurate model is required. The <strong>exact point kinetics equations<\/strong>\u00a0that can be derived from the general neutron balance equations without making any approximations are:<\/p>\n<p><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/06\/point-kinetics-equations_1.png\"><img loading=\"lazy\" class=\"aligncenter size-full wp-image-18131\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/06\/point-kinetics-equations_1.png\" alt=\"point kinetics equations_1\" width=\"281\" height=\"147\" \/><\/a><\/p>\n<p>In the <strong>equation for neutrons<\/strong>, the first term on the right-hand side is the production of prompt neutrons in the present generation, <strong><em>k(1-\u03b2)n\/l<\/em><\/strong>, minus the total number of neutrons in the preceding generation, <strong><em>-n\/l<\/em><\/strong>. The second term is the production of delayed neutrons in the present generation. As can be seen, the rate of absorption of neutrons is the same as in the simple model (<strong><em>-n\/l<\/em><\/strong>). But a distinction is between the direct channel for prompt neutrons <strong><em>(1-\u03b2)<\/em><\/strong> production and the delayed channel resulting from radioactive decay of precursor nuclei (\u03bb<sub>i<\/sub>C<sub>i<\/sub>).<\/p>\n<p>In the <strong>equation for precursors<\/strong>, there is a balance between the production of the precursors of the i-th group and their decay after the decay constant \u03bb<sub>i<\/sub>. As can be seen, the decay rate of precursors is the radioactivity rate (\u03bb<sub>i<\/sub>C<sub>i<\/sub>). The production rate is proportional to the number of neutrons times <strong>\u03b2<\/strong><strong><sub>i<\/sub><\/strong>, which is defined as the fraction of the <a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/atomic-nuclear-physics\/fundamental-particles\/neutron\/\">neutrons<\/a> that appear as <strong>delayed neutrons in the i<em>th<\/em> group<\/strong>.<\/p>\n<p>As can be seen, the point kinetics equations include two differential equations, one for the neutron density <em>n(t)<\/em> and the other for precursors concentration <em>C(t)<\/em>.<\/p>\n<p>Again, the point kinetics equations are often expressed in terms of reactivity <strong>(\u03c1 = (k-1)\/k)<\/strong> and prompt generation time, <strong>\u039b<\/strong>, as:<\/p>\n<p><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/06\/point-kinetics-equations_2.png\"><img loading=\"lazy\" class=\"aligncenter size-full wp-image-18132\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/06\/point-kinetics-equations_2.png\" alt=\"point kinetics equations_2\" width=\"217\" height=\"134\" \/><\/a><\/p>\n<p>Both forms of the point kinetics equation are valid. The equation using <strong>\u039b, prompt neutron generation time, <\/strong>is usually better for calculations. This is because most reactivity transients are induced by changes in the absorption cross-section rather than in the fission cross-section. The prompt neutron lifetime is not constant during these transients, whereas the prompt generation time remains constant.<\/p>\n<figure id=\"attachment_18124\" aria-describedby=\"caption-attachment-18124\" style=\"width: 290px\" class=\"wp-caption alignright\"><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/06\/Reactor-Kinetics-change-in-reactivity-min.png\"><img loading=\"lazy\" class=\"size-medium wp-image-18124\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/06\/Reactor-Kinetics-change-in-reactivity-min-300x216.png\" alt=\"Reactor Kinetics - change in reactivity\" width=\"300\" height=\"216\" srcset=\"https:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/06\/Reactor-Kinetics-change-in-reactivity-min-300x216.png 300w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/06\/Reactor-Kinetics-change-in-reactivity-min.png 825w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/a><figcaption id=\"caption-attachment-18124\" class=\"wp-caption-text\">Solution of point kinetics equations for a cyclic change in reactivity<\/figcaption><\/figure>\n<p>The previous equation defines the reactivity of a reactor, which describes the <strong>deviation of an effective multiplication factor from unity<\/strong>. For critical conditions, the reactivity is equal to zero. The larger the absolute value of <strong>reactivity<\/strong> in the reactor core, the further the reactor is from <strong>criticality<\/strong>. The reactivity may be used to measure a <strong>reactor\u2019s relative departure from criticality<\/strong>. According to the reactivity, we can classify the different reactor states and the related consequences as follows:<\/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>prompt critical state<\/div><div   id="8n9s8rpp6h"    class=\"su-spoiler-content su-u-clearfix su-u-trim\"><strong>The <a title=\"Prompt Critical Reactor\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-fission-chain-reaction\/reactor-criticality\/prompt-critical-reactor\/\">prompt critical state<\/a> is defined as:<\/strong><\/p>\n<p><strong>k<\/strong><strong><sub>eff<\/sub><\/strong><strong> &gt; 1; \u03c1 \u2265 \u03b2<\/strong><strong><sub>eff<\/sub><\/strong>, where the reactivity of a reactor is higher than the <a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/fission\/delayed-neutrons\/effective-delayed-neutron-fraction-%ce%b2eff\/\">effective delayed neutron fraction<\/a>. In this case, the production of prompt neutrons alone is enough to balance neutron losses and increase the <a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-fission-chain-reaction\/neutron-generation-neutron-population\/\">neutron population<\/a>. The number of neutrons is <strong>exponentially increasing<\/strong> in time (as rapidly as the <a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/fission\/prompt-neutrons\/prompt-generation-time-mean-generation-time\/\">prompt neutron generation lifetime<\/a> ~<strong>10<\/strong><strong><sup>-5<\/sup><\/strong><strong>s<\/strong>).<\/div><\/div><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>delayed supercritical state<\/div><div   id="8n9s8rpp6h"    class=\"su-spoiler-content su-u-clearfix su-u-trim\"><strong>The <a title=\"Supercritical Reactor\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-fission-chain-reaction\/reactor-criticality\/supercritical-reactor\/\">prompt subcritical and delayed supercritical state<\/a> is defined as:<\/strong><\/p>\n<p><strong>k<sub>eff<\/sub>\u00a0&gt; 1; 0 &lt; \u03c1 &lt; \u03b2<sub>eff<\/sub><\/strong>, where the reactivity of a reactor is\u00a0<strong>higher than zero<\/strong>\u00a0and\u00a0<strong>lower than<\/strong>\u00a0the\u00a0<a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/fission\/delayed-neutrons\/effective-delayed-neutron-fraction-%ce%b2eff\/\">effective delayed neutron fraction<\/a>. In this case, the production of prompt neutrons alone is\u00a0<strong>insufficient<\/strong> to balance neutron losses, and the delayed neutrons are needed to sustain the chain reaction. The <a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-fission-chain-reaction\/neutron-generation-neutron-population\/\">neutron population<\/a>\u00a0increases, but\u00a0<strong>much more slowly<\/strong>\u00a0(as the\u00a0<a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/fission\/delayed-neutrons\/mean-generation-time-with-delayed-neutrons\/\">mean generation lifetime with delayed neutrons<\/a>\u00a0~0.1 s).<\/div><\/div><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>delayed critical<\/div><div   id="8n9s8rpp6h"    class=\"su-spoiler-content su-u-clearfix su-u-trim\"><strong>The <a title=\"Critical Reactor\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-fission-chain-reaction\/reactor-criticality\/critical-reactor\/\">prompt subcritical and delayed critical state<\/a> is defined as:<\/strong><\/p>\n<p><strong>k<sub>eff<\/sub>\u00a0= 1; \u03c1 = 0<\/strong>, where the reactivity of a reactor is\u00a0<strong>equal to zero<\/strong>. In this case, the production of prompt neutrons alone is insufficient to balance neutron losses, and the delayed neutrons are needed to sustain the chain reaction. There is no change in <a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-fission-chain-reaction\/neutron-generation-neutron-population\/\">neutron population<\/a> in time, and the chain reaction will be self-sustaining. This state is the same as the critical state from basic classification.<\/div><\/div><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>delayed subcritical state<\/div><div   id="8n9s8rpp6h"    class=\"su-spoiler-content su-u-clearfix su-u-trim\"><strong>The <a title=\"Subcritical Reactor\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-fission-chain-reaction\/reactor-criticality\/subcritical-reactor\/\">prompt subcritical and delayed subcritical state<\/a> is defined as:<\/strong><\/p>\n<p><strong>k<sub>eff<\/sub>\u00a0&lt; 1; \u03c1 &lt; 0<\/strong>, where the reactivity of a reactor is lower than zero. In this case, the production of all neutrons is insufficient to balance neutron losses, and the chain reaction is not self-sustaining. If the reactor core contains external or internal <a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/atomic-nuclear-physics\/fundamental-particles\/neutron\/neutron-sources\/\">neutron sources<\/a>, the reactor is in the state that is usually referred to as the\u00a0<a title=\"Subcritical Multiplication\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/reactor-dynamics\/subcritical-multiplication\/\"><strong>subcritical multiplication<\/strong><\/a>.<\/div><\/div><\/div>\n<h2>Approximate Solution of Point Kinetics Equations<\/h2>\n<p>Sometimes, it is convenient to predict qualitatively the behavior of a reactor. The exact solution can be obtained relatively easily using computers. Especially for illustration, the following approximations are discussed in the following sections:<\/p>\n<ul>\n<li>Prompt Jump Approximation<\/li>\n<li>Prompt Jump Approximation with One Group of Delayed Neutrons<\/li>\n<li>Constant Delayed Neutron Source Approximation<\/li>\n<\/ul>\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>Prompt Jump Approximation - Prompt Drop<\/div><div   id="8n9s8rpp6h"    class=\"su-spoiler-content su-u-clearfix su-u-trim\">As can be seen from the solution of exact point kinetics equation, any reactivity insertion (<strong>\u03c1<\/strong> &lt; <strong><em>\u03b2<\/em><\/strong>) causes at first a sharp change in prompt neutrons population, and then the neutron response is slowed as a result of the more slowly changing number of delayed neutrons. The rapid response results from the small value of prompt neutron generation time in the denominator of the point kinetics equation.<\/p>\n<p>Suppose we are interested in <strong>long-term behavior<\/strong> (asymptotic period) and not interested in the details of the prompt jump. In that case, we can simplify the point kinetics equations by assuming that the <strong>prompt jump takes place instantaneously<\/strong> in response to any reactivity change. This approximation is known as the <strong>Prompt Jump Approximation (PJA)<\/strong>. Due to prompt neutrons, the rapid power change is neglected, corresponding to taking <strong><em>dn\/dt |<sub>0<\/sub> = 0<\/em><\/strong> in the point kinetics equations. That means the point kinetics equations are as follows:<\/p>\n<p><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/07\/prompt-jump-approximation-equation.png\"><img loading=\"lazy\" class=\"aligncenter size-full wp-image-18188\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/07\/prompt-jump-approximation-equation.png\" alt=\"prompt jump approximation - equation\" width=\"233\" height=\"169\" srcset=\"https:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/07\/prompt-jump-approximation-equation.png 233w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/07\/prompt-jump-approximation-equation-220x161.png 220w\" sizes=\"(max-width: 233px) 100vw, 233px\" \/><\/a><\/p>\n<p>From the equation for neutron flux and the assumption that the delayed neutron precursor population does not respond instantaneously to a change in reactivity (i.e., C<sub>i,1<\/sub> = C<sub>i,2<\/sub>), it can be derived that the ratio of the neutron population just after and before the reactivity change is equal to:<\/p>\n<p><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/07\/prompt-jump-approximation-prompt-jump.png\"><img loading=\"lazy\" class=\"aligncenter size-full wp-image-18189\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/07\/prompt-jump-approximation-prompt-jump.png\" alt=\"prompt jump approximation - prompt jump\" width=\"240\" height=\"264\" \/><\/a><\/p>\n<p>The prompt-jump approximation is usually valid for smaller reactivity insertion, for example, for <strong>\u03c1 &lt; 0.5<em>\u03b2. <\/em><\/strong>It is usually used with another simplification and <strong>delayed precursor group approximation<\/strong>.<\/div><\/div><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>Prompt Jump Approximation with One Group of Delayed Neutrons<\/div><div   id="8n9s8rpp6h"    class=\"su-spoiler-content su-u-clearfix su-u-trim\">In the previous section, we have simplified the point kinetics equation using <strong>prompt jump approximation (PJA)<\/strong>. This eliminated the fast time scale due to prompt neutrons. This section considers that delayed neutrons are produced by <strong>one group of precursors<\/strong> with the same decay constant (averaged) and delayed neutron fraction. Point kinetics equation using PJA and one group of delayed neutrons becomes:<\/p>\n<p><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/07\/PJA-with-one-group.png\"><img loading=\"lazy\" class=\"aligncenter size-full wp-image-18190\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/07\/PJA-with-one-group.png\" alt=\"PJA with one group\" width=\"265\" height=\"143\" \/><\/a><\/p>\n<p>This simplification then leads to:<\/p>\n<p><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/07\/solution-PJA-one-group.png\"><img loading=\"lazy\" class=\"aligncenter size-full wp-image-18191\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/07\/solution-PJA-one-group.png\" alt=\"solution - PJA - one group\" width=\"282\" height=\"173\" \/><\/a><\/p>\n<p>Assuming that the reactivity is constant and n<sub>1<\/sub>\/n<sub>0<\/sub> can be determined from the prompt jump formula, this equation leads to a very simple formula:<\/p>\n<p><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/07\/solution-PJA-one-group2.png\"><img loading=\"lazy\" class=\"aligncenter size-full wp-image-18192\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/07\/solution-PJA-one-group2.png\" alt=\"solution - PJA - one group2\" width=\"298\" height=\"92\" \/><\/a><\/div><\/div><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>Constant Delayed Neutron Source Approximation<\/div><div   id="8n9s8rpp6h"    class=\"su-spoiler-content su-u-clearfix su-u-trim\">As can be seen from the solution of exact point kinetics equation, any reactivity insertion (<strong>\u03c1<\/strong> &lt; <strong><em>\u03b2<\/em><\/strong>) causes at first a sharp change in prompt neutrons population, and then the neutron response is slowed as a result of the more slowly changing number of delayed neutrons. The rapid response results from the small value of prompt neutron generation time in the denominator of the point kinetics equation.<\/p>\n<p>Suppose we are interested in short-term behavior and not interested in the details of the asymptotic behavior. In that case, we can simplify the point kinetics equations by assuming that the production of the delayed neutrons is constant and equal to the production at the beginning of the transient. This approximation is known as the <strong>Constant Delayed Neutron Source Approximation<\/strong> (CDS), in which changes in the number of delayed neutrons are neglected, corresponding to taking dC<sub>i<\/sub>(t)\/dt = 0 and C<sub>i<\/sub>(t) = C<sub>i,0<\/sub> in the point kinetics equations. That means the point kinetics equations are as follows:<\/p>\n<p><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/07\/constant-production-of-delayed-neutrons.png\"><img loading=\"lazy\" class=\"aligncenter size-full wp-image-18193\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/07\/constant-production-of-delayed-neutrons.png\" alt=\"constant production of delayed neutrons\" width=\"341\" height=\"351\" srcset=\"https:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/07\/constant-production-of-delayed-neutrons.png 341w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/07\/constant-production-of-delayed-neutrons-291x300.png 291w\" sizes=\"(max-width: 341px) 100vw, 341px\" \/><\/a><\/p>\n<p>The above equation can be solved analytically, and assuming that the reactivity is constant, the solution is given as:<\/p>\n<p><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/07\/constant-production-of-delayed-neutrons-2.png\"><img loading=\"lazy\" class=\"aligncenter size-full wp-image-18194\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/07\/constant-production-of-delayed-neutrons-2.png\" alt=\"constant production of delayed neutrons 2\" width=\"403\" height=\"89\" srcset=\"https:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/07\/constant-production-of-delayed-neutrons-2.png 403w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2017\/07\/constant-production-of-delayed-neutrons-2-300x66.png 300w\" sizes=\"(max-width: 403px) 100vw, 403px\" \/><\/a><\/div><\/div><\/div>\n<h2>Point Kinetics Equations &#8211; Notes<\/h2>\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>Prompt Neutron Lifetime<\/div><div   id="8n9s8rpp6h"    class=\"su-spoiler-content su-u-clearfix su-u-trim\"><strong>Prompt neutron lifetime, l<\/strong>, is the <strong>average time from a prompt neutron emission<\/strong> to either <strong>its absorption<\/strong> (fission or radiative capture) or\u00a0<strong>its escape<\/strong> from the system. This parameter is defined in multiplying or also in non-multiplying systems. In both systems, the prompt neutron lifetimes depend strongly on:<\/p>\n<ul>\n<li>the material composition of the system\n<ul>\n<li>multiplying \u2013 non-multiplying system<\/li>\n<li>the system with or without thermalization<\/li>\n<li>isotopic composition of the system<\/li>\n<\/ul>\n<\/li>\n<li>the geometric configuration of the system\n<ul>\n<li>homogeneous or heterogeneous system<\/li>\n<li>the shape of the entire system<\/li>\n<\/ul>\n<\/li>\n<li>size of the system<\/li>\n<\/ul>\n<p>In an infinite reactor (without escape), prompt neutron lifetime is the sum of <strong>the slowing downtime and the diffusion time<\/strong>.<\/p>\n<p><strong>l=t<sub>s<\/sub> + t<sub>d<\/sub><\/strong><\/p>\n<p>In an infinite thermal reactor <strong>t<sub>s<\/sub>\u00a0&lt;&lt; t<sub>d<\/sub><\/strong>\u00a0and therefore <strong>l \u2248 t<sub>d<\/sub><\/strong>. The typical prompt neutron lifetime <strong>in thermal reactors<\/strong> is on the order of<strong> 10<sup>\u22124<\/sup> seconds<\/strong>. Generally, the longer neutron lifetimes occur in systems in which the neutrons must be thermalized to be absorbed.<\/p>\n<p>Most neutrons are absorbed in higher energies, and the neutron thermalization is suppressed (e.g., in <a title=\"Fast Neutron Reactor\" href=\"http:\/\/sitepourvtc.com\/fast-neutron-reactor\/\">fast reactors<\/a>), having much shorter prompt neutron lifetimes. The typical prompt neutron lifetime <strong>in fast reactors<\/strong> is on the order of <strong>10<sup>\u22127<\/sup> seconds<\/strong>.<\/p>\n<figure id=\"attachment_12843\" aria-describedby=\"caption-attachment-12843\" style=\"width: 511px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/10\/diffusion-and-slowing-down-time.png\"><img loading=\"lazy\" class=\"size-full wp-image-12843\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/10\/diffusion-and-slowing-down-time.png\" alt=\"Slowing Down and Diffision Times - Prompt Neutron Lifetime\" width=\"521\" height=\"200\" srcset=\"https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/10\/diffusion-and-slowing-down-time.png 521w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/10\/diffusion-and-slowing-down-time-300x115.png 300w\" sizes=\"(max-width: 521px) 100vw, 521px\" \/><\/a><figcaption id=\"caption-attachment-12843\" class=\"wp-caption-text\">Slowing Down and Diffusion Times for Thermal Neutrons in an Infinite Medium<br \/>Source: Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.<\/figcaption><\/figure>\n<\/div><\/div><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>Prompt Generation Time - Mean Generation Time<\/div><div   id="8n9s8rpp6h"    class=\"su-spoiler-content su-u-clearfix su-u-trim\">\n<figure id=\"attachment_12813\" aria-describedby=\"caption-attachment-12813\" style=\"width: 467px\" class=\"wp-caption alignright\"><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/10\/Prompt-Neutron-Lifetimes-Reactor-Types.png\"><img loading=\"lazy\" class=\"size-full wp-image-12813\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/10\/Prompt-Neutron-Lifetimes-Reactor-Types.png\" alt=\"Prompt Neutron Lifetime - Reactor Types\" width=\"477\" height=\"417\" srcset=\"https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/10\/Prompt-Neutron-Lifetimes-Reactor-Types.png 477w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/10\/Prompt-Neutron-Lifetimes-Reactor-Types-300x262.png 300w\" sizes=\"(max-width: 477px) 100vw, 477px\" \/><\/a><figcaption id=\"caption-attachment-12813\" class=\"wp-caption-text\">Dependencies of asymptotic time on the reactivity required for different reactor types with different prompt neutron lifetimes.<\/figcaption><\/figure>\n<p>In multiplying systems, the absorption of a prompt fission neutron can initiate a fission reaction, l is equal to the average time between two generations of prompt neutrons (at k<sub>eff<\/sub>=1). This time is known as the <strong>prompt neutron generation time<\/strong>.<\/p>\n<p><strong>Prompt Neutron Generation Time<\/strong> (or <strong>Mean Generation Time<\/strong>), <strong>\u039b<\/strong>, is the average time <strong>from a prompt neutron emission<\/strong> <strong>to a capture that results only in fission<\/strong>. The prompt neutron generation time is designated as:<\/p>\n<p><strong>\u039b = l\/k<sub>eff<\/sub><\/strong><\/p>\n<p><strong>The prompt generation time<\/strong> changes with the fuel burnup in power reactors, and in LWRs increases with the fuel burnup. It is simple, and fresh uranium fuel contains much <a title=\"Fissile Material\" href=\"http:\/\/sitepourvtc.com\/glossary\/fissile-material\/\">fissile material<\/a> (in the case of uranium fuel, about 4% of <a title=\"Uranium 235\" href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/nuclear-fuel\/uranium\/uranium-235\/\"><sup>235<\/sup>U<\/a>). This causes significant excess of reactivity, and this <strong>excess must be compensated<\/strong> via <a title=\"Boric Acid \u2013 Chemical Shim\" href=\"http:\/\/sitepourvtc.com\/glossary\/boron-10\/boric-acid-chemical-shim\/\">chemical shim<\/a> (in case of <a title=\"PWR \u2013 Pressurized water reactor\" href=\"http:\/\/sitepourvtc.com\/pwr-pressurized-water-reactor\/\">PWRs<\/a>) or burnable absorbers.<\/p>\n<p>The prompt neutron lives much shorter, and the prompt neutron lifetime is low due to these factors (high probability of absorption in <a title=\"Nuclear Fuel\" href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/nuclear-fuel\/\">fuel<\/a> and high probability of absorption in <a title=\"Neutron moderator\" href=\"http:\/\/sitepourvtc.com\/neutron-moderator\/\">moderator<\/a>). With fuel burnup, the amount of fissile material as well as the absorption in moderator decreases, and therefore, the prompt neutron can \u201clive\u201d much longer.<\/div><\/div><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>Effect of Prompt Neutron Lifetime on Nuclear Safety<\/div><div   id="8n9s8rpp6h"    class=\"su-spoiler-content su-u-clearfix su-u-trim\">The prompt neutron lifetime belongs to <strong>key neutron-physical characteristics<\/strong> of <a title=\"Reactor Core\" href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/nuclear-reactor-core\/\">reactor core<\/a>. Its value depends especially on the type of <a title=\"Neutron moderator\" href=\"http:\/\/sitepourvtc.com\/neutron-moderator\/\">the moderator<\/a> and the <a title=\"Neutron Energy\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/atomic-nuclear-physics\/fundamental-particles\/neutron\/neutron-energy\/\">energy of the neutrons<\/a> causing fission. Its importance for nuclear reactor safety has been well known for a long time.<\/p>\n<p>The longer prompt neutron lifetimes can substantially improve the kinetic response of the reactor (<strong>the longer prompt neutron lifetime gives simply a slower power increase<\/strong>). For example, under RIA conditions (<strong>Reactivity-Initiated Accidents<\/strong>), reactors should withstand a jump-like insertion of relatively large (~1\u2009$ or even more) positive reactivity, and the PNL (prompt neutron lifetime) plays here the key role. Therefore the PNL should be verified in a reload safety evaluation process.<\/p>\n<p>In some cases (especially in some <a title=\"Fast Neutron Reactor\" href=\"http:\/\/sitepourvtc.com\/fast-neutron-reactor\/\">fast reactors<\/a>), reactor cores can be modified to increase the PNL and improve nuclear safety.<\/div><\/div><\/div>\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>Six Groups of Delayed Neutrons<\/div><div   id="8n9s8rpp6h"    class=\"su-spoiler-content su-u-clearfix su-u-trim\">\n<figure id=\"attachment_12829\" aria-describedby=\"caption-attachment-12829\" style=\"width: 173px\" class=\"wp-caption alignright\"><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/10\/Yield-of-Delayed-Neutrons.png\"><img loading=\"lazy\" class=\"size-medium wp-image-12829\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/10\/Yield-of-Delayed-Neutrons-183x300.png\" alt=\"Delayed neutron fraction - yield\" width=\"183\" height=\"300\" srcset=\"https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/10\/Yield-of-Delayed-Neutrons-183x300.png 183w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/10\/Yield-of-Delayed-Neutrons.png 505w\" sizes=\"(max-width: 183px) 100vw, 183px\" \/><\/a><figcaption id=\"caption-attachment-12829\" class=\"wp-caption-text\">Source: Keepin, G. R., Physics of Nuclear Kinetics. Addison-Wesley, 1965.<\/figcaption><\/figure>\n<p>Reactor-kinetic calculations considering many initial conditions would be correct, but they would also be very complicated. Therefore G. R. Keepin and his co-workers suggested <strong>to group together the precursors<\/strong> based on their half-lives. Therefore delayed neutrons are traditionally represented by <strong>six delayed neutron groups<\/strong>, whose yields and decay constants (\u03bb) are obtained from nonlinear least-squares fits experimental measurements. This model has the following disadvantages:<\/p>\n<ul>\n<li>All constants for each group of precursors are empirical fits to the data.<\/li>\n<\/ul>\n<ul>\n<li>They cannot be matched with decay constants of specific precursors.<\/li>\n<\/ul>\n<ul>\n<li>These constants are different for each <a title=\"Fissionable Material\" href=\"http:\/\/sitepourvtc.com\/glossary\/fissionable-material\/\">fissionable nuclide<\/a>.<\/li>\n<\/ul>\n<ul>\n<li>These constants also change with the <a title=\"Neutron Energy\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/atomic-nuclear-physics\/fundamental-particles\/neutron\/neutron-energy\/\">neutron energy<\/a> spectrum.<\/li>\n<\/ul>\n<div   id="8n9s8rpp6h"    class=\"imageframe-align-center\">\n<p>Although this six-group parameterization still satisfies the requirements of commercial organizations, higher accuracy of the delayed neutron yields and a better energy resolution in the delayed neutron spectra is desired.<\/p>\n<p>It was recognized that the half-lives in the six-group structure <strong>do not accurately reproduce<\/strong> the asymptotic die-away time constants associated with the three longest-lived dominant precursors: <strong><sup>87<\/sup>Br, <sup>137<\/sup>I, and <sup>88<\/sup>Br.<\/strong><\/p>\n<p><strong>This model may be insufficient,<\/strong> especially in the case of epithermal reactors, because virtually all delayed neutron activity measurements have been performed for fast or thermal-neutron-induced fission. In the case of <a title=\"Fast Neutron Reactor\" href=\"http:\/\/sitepourvtc.com\/fast-neutron-reactor\/\">fast reactors<\/a>, the <a title=\"Nuclear Fission\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/fission\/\">nuclear fission<\/a> of six fissionable isotopes of <a title=\"Uranium\" href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/nuclear-fuel\/uranium\/\">uranium<\/a> and <a title=\"Plutonium\" href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/nuclear-fuel\/plutonium\/\">plutonium<\/a> is important, and the accuracy and energy resolution may play an important role.<\/p>\n<\/div><\/div><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>Photoneutrons<\/div><div   id="8n9s8rpp6h"    class=\"su-spoiler-content su-u-clearfix su-u-trim\">In\u00a0<a href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/what-is-nuclear-reactor\/\">nuclear reactors<\/a>\u00a0the\u00a0<a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/atomic-nuclear-physics\/fundamental-particles\/photon\/gamma-ray\/\">gamma radiation<\/a>\u00a0plays a significant role also in\u00a0<strong>reactor kinetics<\/strong>\u00a0and in\u00a0<strong>a subcriticality control<\/strong>. Especially in nuclear reactors with D<sub>2<\/sub>O moderator (<a href=\"http:\/\/sitepourvtc.com\/candu-heavy-water-reactor\/\">CANDU reactors<\/a>) or Be reflectors (some experimental reactors). Neutrons can also be produced in <strong>(\u03b3, n) reactions,<\/strong>\u00a0and therefore they are usually referred to as\u00a0<strong>photoneutrons<\/strong>.<\/p>\n<p>A high-energy <a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/atomic-nuclear-physics\/fundamental-particles\/photon\/\">photon<\/a>\u00a0(<a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/atomic-nuclear-physics\/fundamental-particles\/photon\/gamma-ray\/\">gamma-ray<\/a>) can, under certain conditions, <strong>eject<\/strong>\u00a0a\u00a0<a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/atomic-nuclear-physics\/fundamental-particles\/neutron\/\">neutron<\/a> from a nucleus. It occurs when\u00a0<strong>its energy exceeds<\/strong>\u00a0<a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/atomic-nuclear-physics\/binding-energy\/\">the binding energy<\/a> of the neutron in the nucleus. Most nuclei have binding energies above <strong>6 MeV<\/strong>, above the energy of most gamma rays from <a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/fission\/\">fission<\/a>. On the other hand, <strong>few nuclei<\/strong> with sufficiently low binding energy are of\u00a0<strong>practical interest<\/strong>. These are: <strong><sup>2<\/sup>D,\u00a0<sup>9<\/sup>Be<\/strong>,\u00a0<sup>6<\/sup>Li,\u00a0<sup>7<\/sup>Li, and <sup>13<\/sup>C. As can be seen from the table, <strong>the lowest threshold<\/strong>\u00a0have\u00a0<strong><sup>9<\/sup>Be with 1.666 MeV<\/strong>\u00a0and\u00a0<strong><sup>2<\/sup>D with 2.226 MeV<\/strong>.<\/p>\n<figure id=\"attachment_12827\" aria-describedby=\"caption-attachment-12827\" style=\"width: 290px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/10\/Photoneutron-sources.png\"><img loading=\"lazy\" class=\"size-medium wp-image-12827\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/10\/Photoneutron-sources-300x146.png\" alt=\"Photoneutron sources\" width=\"300\" height=\"146\" srcset=\"https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/10\/Photoneutron-sources-300x146.png 300w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/10\/Photoneutron-sources-540x264.png 540w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/10\/Photoneutron-sources.png 543w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/a><figcaption id=\"caption-attachment-12827\" class=\"wp-caption-text\">Nuclides with low photodisintegration<br \/>threshold energies.<\/figcaption><\/figure>\n<p>In the case of deuterium, neutrons can be produced by the interaction of gamma rays (with a minimum energy of 2.22 MeV) with deuterium:<\/p>\n<p><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/11\/Photoneutron-deuterium.png\"><img loading=\"lazy\" class=\"aligncenter size-full wp-image-12882\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/11\/Photoneutron-deuterium.png\" alt=\"Photoneutron - deuterium\" width=\"232\" height=\"60\" \/><\/a><\/p>\n<p>Because gamma rays can be emitted by\u00a0<a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/fission\/fission-fragments\/\">fission products<\/a>\u00a0with certain delays, and\u00a0<strong>the process is very similar<\/strong>\u00a0to that through which a<strong> \u201ctrue\u201d delayed neutron<\/strong>\u00a0is emitted,\u00a0<strong>photoneutrons<\/strong> are usually treated no differently than regular delayed neutrons in the kinetic calculations. Photoneutron precursors can also be grouped by their decay constant, similarly to \u201creal\u201d precursors. The table below shows the relative importance of source neutrons in CANDU reactors by showing the makeup of the full power flux.<\/p>\n<p><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/11\/Photoneutron-balance.png\"><img loading=\"lazy\" class=\"aligncenter size-full wp-image-12883\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/11\/Photoneutron-balance.png\" alt=\"Photoneutron balance\" width=\"345\" height=\"162\" srcset=\"https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/11\/Photoneutron-balance.png 345w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/11\/Photoneutron-balance-300x141.png 300w\" sizes=\"(max-width: 345px) 100vw, 345px\" \/><\/a><\/p>\n<p>Although photoneutrons are important, especially in CANDU reactors, deuterium nuclei are always present <strong>(~0.0156%<\/strong>) <strong>in the light water of LWRs<\/strong>. Moreover, the capture of neutrons in the hydrogen nucleus of the water molecules in the moderator yields small amounts of D<sub>2<\/sub>O, enhancing the heavy water concentration. Therefore, in LWRs kinetic calculations, photoneutrons from D<sub>2<\/sub>O are treated as additional groups of delayed neutrons with characteristic decay constants \u03bb<sub>j<\/sub>\u00a0and effective group fractions.<\/p>\n<p>After a\u00a0<a href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/what-is-nuclear-reactor\/\">nuclear reactor<\/a> has been operating at full power for some time, there will be a considerable build-up of <a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/atomic-nuclear-physics\/fundamental-particles\/photon\/gamma-ray\/\">gamma rays<\/a>\u00a0from the\u00a0<a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/fission\/fission-fragments\/\">fission products<\/a>. This <strong>high gamma flux<\/strong>\u00a0from short-lived fission products will\u00a0<strong>decrease rapidly after shutdown<\/strong>. The photoneutron source decreases with the<strong>\u00a0decay of long-lived fission products<\/strong> that produce delayed high-energy gamma rays <strong>in the long term<\/strong>. The photoneutron source drops slowly, decreasing a little each day. The longest-lived fission product with gamma-ray energy above the threshold is <strong><sup>140<\/sup>Ba<\/strong>\u00a0with a half-life of\u00a0<strong>12.75 days.<\/strong><\/p>\n<p>The amount of fission products present in\u00a0<a href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/nuclear-fuel\/\">the fuel elements<\/a>\u00a0depends on\u00a0<strong>how long<\/strong> the reactor has been operated before shutdown and\u00a0<strong>at which power<\/strong> level has been the reactor operated before shutdown. <strong>Photoneutrons<\/strong> are usually a major source in a reactor and ensure\u00a0<strong>sufficient neutron flux<\/strong>\u00a0<strong>on source range detectors<\/strong> when the reactor is\u00a0<strong>subcritical<\/strong> in long-term shutdown.<\/p>\n<p>Compared with <strong>fission neutrons<\/strong>, which make a <strong><a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-fission-chain-reaction\/\">self-sustaining chain reaction<\/a>\u00a0possible<\/strong>,\u00a0<strong>delayed neutrons<\/strong>\u00a0make reactor\u00a0<strong>control possible,<\/strong>\u00a0and\u00a0<strong>photoneutrons<\/strong> are important\u00a0<strong>at low power operation<\/strong>.<\/div><\/div><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> Effective Delayed Neutron Fraction \u2013 \u03b2eff<\/div><div   id="8n9s8rpp6h"    class=\"su-spoiler-content su-u-clearfix su-u-trim\">See also: <a title=\"Delayed Neutrons Fraction\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/fission\/delayed-neutrons\/delayed-neutrons-fraction\/\">Delayed Neutron Fraction<\/a><\/p>\n<p>See also:\u00a0<a title=\"Effective Delayed Neutron Fraction \u2013 \u03b2eff\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/fission\/delayed-neutrons\/effective-delayed-neutron-fraction-%ce%b2eff\/\">Effective Delayed Neutron Fraction \u2013 \u03b2eff<\/a><\/p>\n<p>The delayed neutron fraction, <strong>\u03b2<\/strong>, is the fraction of delayed neutrons in the core <strong>at creation, that is, at high energies<\/strong>. But in the case of thermal reactors, the <a title=\"Nuclear Fission\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/fission\/\">fission<\/a> can be initiated <strong>mainly by a thermal neutron<\/strong>. Thermal neutrons are of practical interest in the study of thermal reactor behavior. <strong>The effective delayed neutron fraction<\/strong>\u00a0usually referred to as <strong>\u03b2<sub>eff<\/sub><\/strong>, is the same fraction at thermal energies.<\/p>\n<p>The effective delayed neutron fraction <strong>reflects the ability of the <a title=\"Nuclear Reactor\" href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/nuclear-reactor\/\">reactor<\/a><\/strong> to <strong>thermalize<\/strong> and <strong>utilize<\/strong> each <a title=\"Neutron\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/atomic-nuclear-physics\/fundamental-particles\/neutron\/\">neutron<\/a> produced. The <strong>\u03b2<\/strong> is not the same as the <strong>\u03b2<sub>eff<\/sub><\/strong>\u00a0due to the fact <strong>delayed neutrons do not have the same properties as <a title=\"Prompt Neutrons\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/fission\/prompt-neutrons\/\">prompt neutrons<\/a><\/strong> released directly from fission. In general, delayed neutrons have <strong>lower energies<\/strong> than prompt neutrons. <strong>Prompt neutrons<\/strong> have initial energy between <strong>1 MeV and 10 MeV<\/strong>, with an average energy of <strong>2 MeV<\/strong>. <strong>Delayed neutrons<\/strong> have initial energy between <strong>0.3 and 0.9 MeV<\/strong> with an average energy of <strong>0.4 MeV<\/strong>.<\/p>\n<p>Therefore, a delayed neutron <strong>traverses a smaller energy range<\/strong> in thermal reactors to become thermal. It is also <strong>less likely to be lost<\/strong> by leakage or parasitic absorption than the 2 MeV prompt neutron. On the other hand, <strong>delayed neutrons<\/strong> are also <strong>less likely to cause fast fission<\/strong>\u00a0because their average energy is less than the minimum required for fast fission to occur.<\/p>\n<p>These two effects (<strong>lower fast fission factor<\/strong> and <strong>higher fast non-leakage probability for delayed neutrons<\/strong>) tend to counteract each other and form a term called <strong>the importance factor (I)<\/strong>. The importance factor relates the average delayed neutron fraction to the effective delayed neutron fraction. As a result, the effective delayed neutron fraction is the product of the average delayed neutron fraction and the importance factor.<\/p>\n<p><strong>\u03b2<sub>eff<\/sub>\u00a0= \u03b2 . I<\/strong><\/p>\n<p>The delayed and prompt neutrons have a difference in their effectiveness in producing a subsequent fission event. Since the energy distribution of the delayed neutrons also differs from group to group, the different groups of delayed neutrons will also have different effectiveness. Moreover, a <a title=\"Nuclear Reactor\" href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/nuclear-reactor\/\">nuclear reactor <\/a>contains a mixture of <a title=\"Fissionable Material\" href=\"http:\/\/sitepourvtc.com\/glossary\/fissionable-material\/\">fissionable isotopes<\/a>. Therefore, the importance factor is insufficient in some cases, and an importance function must be defined.<\/p>\n<p><strong>For example:<\/strong><\/p>\n<p><strong>In a small thermal reactor with highly enriched fuel<\/strong>, the increase in fast non-leakage probability will dominate the decrease in the fast fission factor, and <strong>the importance factor will be greater than one<\/strong>.<\/p>\n<p><strong>In a large thermal reactor with low enriched fuel<\/strong>, the decrease in the fast fission factor will dominate the increase in the fast non-leakage probability, and <strong>the importance factor will be less\u00a0than one (about 0.97 for a commercial <a title=\"PWR \u2013 Pressurized water reactor\" href=\"http:\/\/sitepourvtc.com\/pwr-pressurized-water-reactor\/\">PWR<\/a>)<\/strong>.<\/p>\n<p><strong>In large <a title=\"Fast Neutron Reactor\" href=\"http:\/\/sitepourvtc.com\/fast-neutron-reactor\/\">fast reactors<\/a><\/strong>, the decrease in the fast fission factor will also dominate the increase in the fast non-leakage probability, and the <strong>\u03b2<sub>eff<\/sub><\/strong>\u00a0is less than \u03b2 by about 10%.<\/p>\n<figure id=\"attachment_12845\" aria-describedby=\"caption-attachment-12845\" style=\"width: 377px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/10\/kinetic-parameters-reactor-types.png\"><img loading=\"lazy\" class=\"size-full wp-image-12845\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/10\/kinetic-parameters-reactor-types.png\" alt=\"Table of main kinetic parameters.\" width=\"387\" height=\"163\" srcset=\"https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/10\/kinetic-parameters-reactor-types.png 387w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/10\/kinetic-parameters-reactor-types-300x126.png 300w\" sizes=\"(max-width: 387px) 100vw, 387px\" \/><\/a><figcaption id=\"caption-attachment-12845\" class=\"wp-caption-text\">Table of main kinetic parameters.<\/figcaption><\/figure>\n<\/div><\/div><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> Mean Generation Time with Delayed Neutrons<\/div><div   id="8n9s8rpp6h"    class=\"su-spoiler-content su-u-clearfix su-u-trim\"><strong>Mean Generation Time with Delayed Neutrons<\/strong>, <strong>l<sub>d<\/sub><\/strong>, is the weighted average of the prompt generation times and a delayed neutron generation time. The delayed neutron generation time, <strong>\u03c4<\/strong>, is the weighted average of mean precursor lifetimes of the six groups (or more groups) of delayed neutron precursors.<\/p>\n<p>It must be noted the true lifetime of delayed neutrons (the slowing downtime and the diffusion time) is very short compared with the mean lifetime of their precursors (t<sub>s<\/sub> + t<sub>d<\/sub>\u00a0&lt;&lt; \u03c4<sub>i<\/sub>). Therefore \u03c4<sub>i<\/sub> is also equal to the mean lifetime of a neutron from the ith group, that is, <strong>\u03c4<sub>i<\/sub> = l<sub>i,<\/sub><\/strong> and the equation for mean generation time with delayed neutrons is the following:<\/p>\n<p><strong>l<sub>d<\/sub> = (1 \u2013 \u03b2).l<sub>p<\/sub> + \u2211l<sub>i<\/sub> . \u03b2<sub>i<\/sub> =&gt; l<sub>d<\/sub> = (1 \u2013 \u03b2).l<sub>p<\/sub> + \u2211\u03c4<sub>i<\/sub> . \u03b2<sub>i<\/sub><\/strong><\/p>\n<p>where<\/p>\n<ul>\n<li><strong>(1 \u2013 \u03b2)<\/strong> is the fraction of all <a title=\"Neutron\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/atomic-nuclear-physics\/fundamental-particles\/neutron\/\">neutrons<\/a> emitted as <a title=\"Prompt Neutrons\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/fission\/prompt-neutrons\/\">prompt neutrons<\/a><\/li>\n<li><strong>l<sub>p<\/sub><\/strong>\u00a0is the prompt neutron lifetime<\/li>\n<li><strong>\u03c4<sub>i\u00a0<\/sub><\/strong>is the mean precursor lifetime, the inverse value of the decay constant<strong> \u03c4<sub>i<\/sub>\u00a0= 1\/\u03bb<sub>i<\/sub><\/strong><\/li>\n<li>The weighted delayed generation time is given by<strong> \u03c4 = \u2211\u03c4<sub>i<\/sub> . \u03b2<sub>i<\/sub>\u00a0\/ \u03b2 = 13.05 s<\/strong><\/li>\n<li>Therefore the weighted decay constant <strong>\u03bb = 1 \/ \u03c4 \u2248 0.08 s<sup>-1<\/sup><\/strong><\/li>\n<\/ul>\n<p>The number, <strong>0.08 s<sup>-1<\/sup><\/strong>, is relatively high and has <strong>a dominating effect on reactor time response<\/strong>, although delayed neutrons are a small fraction of all neutrons in <a title=\"Reactor Core\" href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/nuclear-reactor-core\/\">the core<\/a>. This is best illustrated by calculating a weighted mean generation time with delayed neutrons:<\/p>\n<p><strong>l<sub>d<\/sub> =\u00a0(1 \u2013 \u03b2).l<sub>p<\/sub> + \u2211\u03c4<sub>i<\/sub> . \u03b2<sub>i<\/sub> = (1 \u2013 0.0065). 2 x 10<sup>-5<\/sup> + 0.085 = 0.00001987 + 0.085 \u2248 0.085<\/strong><\/p>\n<p>In short, <strong>the mean generation time with delayed neutrons<\/strong> is about <strong>~0.1 s<\/strong>, rather than ~<strong>10-5<\/strong> as in section Prompt Neutron Lifetime, where the delayed neutrons were omitted.<\/div><\/div><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>Effect of Delayed Neutrons on Reactor Control<\/div><div   id="8n9s8rpp6h"    class=\"su-spoiler-content su-u-clearfix su-u-trim\">Despite the fact the <strong>number of delayed neutrons<\/strong> per <a title=\"Neutron\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/atomic-nuclear-physics\/fundamental-particles\/neutron\/\">fission neutron<\/a> <strong>is quite small (typically below 1%)<\/strong> and thus does not contribute significantly to the power generation, <strong>they play a crucial role in the reactor control<\/strong> and are essential from the point of view of reactor kinetics and <strong>reactor safety<\/strong>. Their presence completely <strong>changes the dynamic time response<\/strong> of a <a title=\"Nuclear Reactor\" href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/nuclear-reactor\/\">reactor<\/a> to some reactivity change, making it controllable by control systems such as <a title=\"Control Rods\" href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/control-rods\/\">the control rods<\/a>.<\/p>\n<p>Delayed neutrons allow to operate a reactor in <strong>a prompt subcritical<\/strong>, <strong>delayed critical condition<\/strong>. All power reactors are designed to operate in delayed critical conditions and are provided with safety systems to prevent them from ever achieving prompt criticality.<\/p>\n<p>For typical <a title=\"PWR \u2013 Pressurized water reactor\" href=\"http:\/\/sitepourvtc.com\/pwr-pressurized-water-reactor\/\">PWRs<\/a>, the prompt criticality occurs after positive reactivity insertion of <strong>\u03b2<sub>eff<\/sub><\/strong>(i.e.,\u00a0<strong>k<sub>eff<\/sub>\u00a0\u2248 1.006 or \u03c1 = +600 pcm)<\/strong>. In power reactors, such a reactivity insertion is <strong>practically impossible to insert <\/strong>(in case of normal and abnormal operation), especially when a reactor is in <strong>power operation mode<\/strong> and a reactivity insertion causes <strong>heating of a <a title=\"Reactor Core\" href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/nuclear-reactor-core\/\">reactor core<\/a><\/strong>. Due to the presence of<strong> reactivity feedbacks,<\/strong> the positive reactivity insertion is counterbalanced by the negative reactivity from <a title=\"Neutron moderator\" href=\"http:\/\/sitepourvtc.com\/neutron-moderator\/\">moderator<\/a> and fuel temperature coefficients. The presence of delayed neutrons is also of importance from this point of view because they also provide time for reactivity feedback to react on undesirable reactivity insertion.<\/div><\/div><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>Interactive chart \u2013 Infinite Multiplying System Without Source and Delayed Neutrons<\/div><div   id="8n9s8rpp6h"    class=\"su-spoiler-content su-u-clearfix su-u-trim\">Press the \u201c<strong>clear and run<\/strong>\u201d button and try to increase the power of the reactor.<\/p>\n<p>Compare the response of the reactor with the case of Infinite Multiplying System Without Source and without Delayed Neutrons (or set the \u03b2 = 0).<\/p>\n<form><input type=\"button\" value=\"clear and run\" onClick=\"clearturtle();run()\"><\/form>\r\n    \r\n<script src='http:\/\/jsxgraph.uni-bayreuth.de\/distrib\/jsxgraphcore.js' type='text\/javascript'><\/script><script src='http:\/\/jsxgraph.uni-bayreuth.de\/distrib\/GeonextReader.js' type='text\/javascript'><\/script><div id='box1' class='jxgbox' style='width:600px; height:500px;'><\/div><script type='text\/javascript'>\r\n\r\nvar brd = JXG.JSXGraph.initBoard('box1', {boundingbox: [0, 100, 15.75, -10], axis:true, showCopyright:<strong>bk8vietnamMọi quyền được bảo lưu cho trang web chính thức</strong>:false});\r\nvar t = brd.create('turtle',[4,3,70]);\r\nvar reac = brd.create('slider', [[4,+30], [9,+30],[-20,10,20]], {name:'pcm'});\r\nvar lifetime = brd.create('slider', [[4,+20], [9,+20],[1,20,200]], {name:'prompt neutron lifetime [10^{-6}.s]'});\r\nvar beta = brd.create('slider', [[4,+10], [9,+10],[0,0.6,0.7]], {name:'beta [%]'});\r\nvar e = brd.create('functiongraph', [function(x){return 10*Math.exp(reac.Value()*0.00001\/((1-0.01*beta.Value())*lifetime.Value() + 13.05*0.01*beta.Value()*1000000)\/0.000001*x);}],{strokeColor:'red'});\r\n\r\n\/\/\r\nvar p1 = brd.create('point', [2, 22], {strokeColor:'#A64D4D'});\r\nvar p2 = brd.create('point', [14, 90], {strokeColor:'#A64D4D'});\r\nvar Target = brd.create('line', [p1, p2], {name:'Target', strokecolor:'#A64D4D'});\r\nvar t1 = brd.create('text',[11,95,\"Target Power\"], {strokeColor:'#A64D4D', fontSize:20});\r\nvar tx = brd.create('text',[14,5,\"Time [s]\"], {strokeColor:'black', fontSize:18});\r\nvar ty = brd.create('text',[0,94,\"Power [%]\"], {strokeColor:'black', fontSize:18});\r\n\r\n\r\n\r\nt.hideTurtle();\r\n            \r\nvar A = 5;\r\nvar tau = 0.3;\r\n            \r\nfunction clearturtle() {\r\n  t.cs();\r\n  t.ht();\r\n}\r\n            \r\nfunction run() {\r\n  t.setPos(0,10);\r\n  t.setPenSize(4);\r\n  dx = 0.0134; \/\/ global\r\n  x = 0.0;  \/\/ global\r\n  loop();\r\n}\r\n             \r\nfunction loop() {\r\n  var dy = reac.Value()*0.00001\/((1-0.01*beta.Value())*lifetime.Value() + 13.05*0.01*beta.Value()*1000000)\/0.000001*t.Y()*dx;   \/\/ Exponential growth\r\n  t.moveTo([dx+t.X(),dy+t.Y()]);\r\n  x += dx;\r\n  if (x<20.0) {\r\n     setTimeout(loop,10);\r\n  }\r\n}\r\n<\/script><\/div><\/div><\/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. Clarendon Press; 1 edition, 1991, ISBN: 978-0198520467<\/li>\n<li>G.R.Keepin. Physics of Nuclear Kinetics. Addison-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. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.<\/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\"><\/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>Reactor Dynamics<a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/reactor-dynamics\/\" 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\"><\/div><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>As we have seen in previous chapters, the number of neutrons is multiplied by a factor keff from one neutron generation to the next. Therefore, the multiplication environment (nuclear reactor) behaves like an exponential system, which means the power increase is not linear but exponential. The effective multiplication factor in a multiplying system measures the &#8230; <a title=\"Point Kinetics Equations\" class=\"read-more\" href=\"https:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/reactor-dynamics\/point-kinetics-equations\/\" aria-label=\"More on Point Kinetics Equations\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":18114,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"generate_page_header":""},"_links":{"self":[{"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/pages\/18218"}],"collection":[{"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/types\/page"}],"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=18218"}],"version-history":[{"count":4,"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/pages\/18218\/revisions"}],"predecessor-version":[{"id":36962,"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/pages\/18218\/revisions\/36962"}],"up":[{"embeddable":true,"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/pages\/18114"}],"wp:attachment":[{"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/media?parent=18218"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}