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| {{Science with neutrons}}
| | During the Neolithic Revolution, humans started consuming large amounts of dairy products, beans, cereals, alcohol and salt. Mediterranean diets are typically high in olive oil consumption as well as cereals,dairy products(primarily yoghurt and cheese)legumes,vegetables and fish. Following a healthy diet signifies changing your diet plan, lowering the amounts but still getting all the proper vitamins and minerals. Prepare to leave your unhealthy habits behind in exchange of a better lifestyle, best trim healthy whole body, and days active. Heat a medium size pan with 1 tablespoons of tallow over medium heat Now put ready cut small pieces of carrot in to the pan fry it for 3 minutes and remove them from pan. <br><br>Since farming and the cultivation of plants like wheat have only been known to man for about 10,000 years, our bodies have not adapted to properly digest these food sources. And also what's more: regular people from many areas of life are actually advocating this product, from common buyers to authorised physicians. Ready-made paleo foods items are starting to find their way into conventional grocery stores. Provide help to immune system after the disease attacked. Copyright (c) 2014 Keith Frame - Article Source: out more about diet and weight loss pill at or my review website at. <br><br>The reason for this is because the cows are free to roam around and eat all the grass. Meals such as grilled or roasted pork, beef and chicken along with broccoli, carrots and sweet potatoes are part of the Paleo diet. Both fruit and veggies contain a number of vitamins. You might know, I arrived so close to giving up now this healthy diet. Diaz is qualified to handle any oral health need, including Invisalign, cosmetic dentistry, laser dentistry and dental implants in Pembroke Pines. <br><br>His photos quickly went viral, with admirers of his sleek physique warring with fans who expressed concern that the NBA's Most Valuable Player had shredded too much fat. Chives: Used in salads, soups, stews and appetizers. When we do feel hungry, we usually also feel panic or anxiety. Advocates who have adhered to the paleo diet can attest to the range of paleo foods there is to choose from -- many are already on your list of routinely consumed foods. I want to start off by saying, the Paleo diet was first put together in the mid 1970. <br><br>Walleye has omega 3 fatty acids, vitamins, potassium and other minerals, hence certainly a worthwhile proposition. * No need to rack your mind, working out caloric intake. The time period when our elders used to feed on such diet was from many years ago. You want to find recipes that will fit into your easy paleo diet plan. Summary Our bodies are designed to react to what we put in them. <br><br>Those cave men and women led very different lives than we do today. Desserts are put together from fruits like bananas and berries. This further frustrates the addict who attempts to cooperate in treatment and yet receives no benefit from that treatment. It is not an easy thing to have new ideas for meal and having lack of options but there are still some good choices from the paleo diet which might be used. The sick emotion lingered in my mind for times and I have not experienced a cheeseburger given that.<br><br>If you loved this information and you would certainly like to obtain even more information relating to [http://gritsandgroceries.info/ paleo bread recipes] kindly see the web site. |
| {{portal|Physics}}
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| {{refimprove|date=September 2011}}
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| In [[nuclear physics|nuclear]] and [[particle physics]], the concept of a '''neutron cross section''' is used to express the likelihood of interaction between an incident [[neutron]] and a target nucleus. In conjunction with the [[neutron flux]], it enables the calculation of the reaction rate, for example to derive the thermal [[power (physics)|power]] of a [[nuclear power plant]]. The standard unit for measuring the cross section is the [[barn (unit)|barn]], which is equal to 10<sup>−28</sup> m<sup>2</sup> or 10<sup>−24</sup> cm<sup>2</sup>.
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| == Parameters of interest ==
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| The neutron cross section, and therefore the probability of an interaction, depends on:
| |
| * the target type ([[hydrogen]], [[uranium]]…),
| |
| * the type of [[nuclear reaction]] (scattering, [[Nuclear fission|fission]]…).
| |
| * the incident particle [[Neutron temperature|energy]], also called speed or temperature (thermal, fast…),
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| and, to a lesser extent, of: | |
| * its relative angle between the incident neutron and the target nuclide,
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| * the target nuclide temperature.
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| | |
| === Target type dependence ===
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| | |
| The neutron cross section is defined for a given type of target particle. For example, the capture cross section of [[hydrogen]]-2 (referred to as [[deuterium]]) is much smaller than that of common hydrogen-1.<ref>{{cite web|url=http://t2.lanl.gov/data/neutron7.html |title=ENDF/B-VII Incident-Neutron Data |publisher=T2.lanl.gov |date=2007-07-15 |accessdate=2011-11-08}}</ref> This is the reason why some reactors use [[heavy water reactor|heavy water]] (in which most of the hydrogen is deuterium) instead of ordinary [[light water reactor|light water]] as [[neutron moderator|moderator]]: fewer neutrons are lost by capture inside the medium, hence enabling the use of [[natural uranium]] instead of [[enriched uranium]]. This is the principle of a [[CANDU reactor]].
| |
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| === Type of reaction dependence ===
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| | |
| The likelihood of interaction between an incident neutron and a target nuclide, independent of the type of reaction, is expressed with the help of the total cross section ''σ''<sub>''T''</sub>. However, it may be useful to know if the incoming particle bounces off the target (and therefore continue travelling after the interaction) or disappears after the reaction. For that reason, the scattering and absorption cross sections ''σ''<sub>''S''</sub> and ''σ''<sub>''A''</sub> are defined and the total cross section is simply the sum of the two partial cross sections:<ref name=DOE/>
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| :<math> \sigma_T = \sigma_S + \sigma_A </math>
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| ==== Absorption cross section ====
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| | |
| If the neutron is absorbed when approaching the nuclide, the atomic nucleus moves up on the table of isotopes by one position. For instance, U-235 becomes U-236* with the * indicating the nucleus is highly energized. This energy has to be released and the release can take place through any of several mechanisms.
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| | |
| # The simplest way for the release to occur is for the neutron to be ejected by the nucleus. If the neutron is emitted immediately, it acts the same as in other scattering events.
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| # The nucleus may emit gamma radiation.
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| # The nucleus may β<sup>-</sup> decay, where a neutron is converted into a proton, an electron and an electron-type antineutrino (the antiparticle of the neutrino)
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| # About 81% of the U-236* nuclei are so energized that they undergo fission, releasing the energy as kinetic motion of the fission fragments, also emitting between one and five free neutrons.
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| | |
| * Nuclei that undergo fission as their predominant decay method after neutron capture include U-233, U-235, U-237, Pu-239, Pu-241.
| |
| * Nuclei that predominantly absorb neutrons and then emit Beta particle radiation lead to these isotopes, e.g., Th-232 absorbs a neutron and becomes Th-233*, which emits a Beta particle and becomes [[Protactinium|Pa]]-233, which emits another Beta particle to become U-233.
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| * Isotopes that undergo Beta emission transmute from one element to another element, those that undergo gamma or X-ray emission don't change in element or isotope.
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| | |
| ==== Scattering cross-section ====
| |
| The scattering cross-section can be further subdivided into coherent [[scattering]] and incoherent scattering, which is caused by the [[spin (physics)|spin]] dependence of the scattering cross-section and for a natural sample, presence of different [[isotope]]s of the same element in the sample.
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| | |
| Since [[neutrons]] interact with the [[nuclear potential]], the scattering cross-section varies for different [[isotope]]s of the element in question. A very prominent example is [[hydrogen]] and its isotope [[deuterium]]. The total cross-section for hydrogen is over 10 times that of deuterium, mostly due to the large incoherent [[scattering length]] of hydrogen. Metals tend to be rather transparent to neutrons, [[aluminum]] and [[zirconium]] being the two best examples of this.
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| | |
| === Incident particle energy dependence ===
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| {{Main|Neutron temperature}}
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| [[Image:U235 Fission cross section.png|thumb|right|450px|U235 fission cross section]]
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| | |
| For a given target and reaction, the cross section is strongly dependent on the neutron speed. In the extreme case, the cross section can be, at low energies, either zero (the energy for which the cross section becomes significant is called [[threshold energy]]) or much larger than at high energies.
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| | |
| Therefore, a cross section should be defined either at a given energy or should be averaged in an energy range (or group). See [[#Continuous versus average cross section|here]] for more details.
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| | |
| As an example, the plot on the right shows that the [[Nuclear fission|fission]] cross section of the [[uranium 235]] is low at high neutron energies but becomes higher at low energies. Such physical constraint explains why most operational [[nuclear reactors]] use a [[neutron moderator]] to reduce the energy of the neutron and thus increase the probability of fission, essential to produce energy and sustain the [[chain reaction]].
| |
| | |
| A simple estimation of energy dependence of any kind of cross section is provided by the Ramsauer Model,
| |
| <ref name=Ramsauer> | |
| R. W. Bauer, J. D. Anderson, S. M. Grimes, V. A. Madsen, Application of Simple Ramsauer Model to Neutron Total Cross Sections, http://www.osti.gov/bridge/servlets/purl/641282-MK9s2L/webviewable/641282.pdf</ref> which is based on idea that effective size of neutron is given by its [[Thermal de Broglie wavelength|de Broglie wavelength]].
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| :<math> \lambda(E) = \frac {\hbar} {\sqrt{2mE}} </math>
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| | |
| Taking <math> \lambda </math> as effective radius of neutron, we can estimate area of circle <math> \sigma </math> in which neutron hit nuclei of effective radius <math>R </math> as
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| | |
| <math> \sigma(E) \propto \pi (R+\lambda(E))^2 </math>
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| | |
| While assumption of this model are naive, it explains at least qualitatively typical measured energy dependence of neutron absorption cross section. For neutron of wavelength much larger than typical radius of atomic nuclei (1–10 fm, E = 10–1000 keV) the <math>R</math> can be neglected. For these low energy neutrons (such as thermal neutrons) cross section <math>\sigma(E)</math> is inversely proportional to neutron energy.
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| This explains the advantage of using [[neutron moderator]] in fission nuclear reactor. On the other hand, for very high energy neutrons (over 1 MeV), <math>\lambda</math> can be neglected, and neutron cross section is approximately constant, determined just by cross section of atomic nuclei.
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| However, this simple model does not take into account so called neutron resonances, which strongly modify neutron cross section in energy range of 1 eV–10 keV, nor threshold energy of some nuclear reactions.
| |
| | |
| === Target temperature dependence ===
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| | |
| Cross sections are usually measured at 20°C. To account for the dependence with temperature of the medium (viz. the target), the following formula is used:<ref name=DOE>DOE Fundamentals Handbook, Nuclear Physics and Reactor Theory, DOE-HDBK-1019/1-93 http://www.hss.doe.gov/nuclearsafety/techstds/docs/handbook/h1019v1.pdf</ref>
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| :<math> \sigma = \sigma_0 \, \left(\frac{T_0}{T} \right)^\frac{1}{2} </math>
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| | |
| Where ''σ'' is the cross section at temperature ''T'' and ''σ''<sub>''0''</sub> the cross section at temperature ''T''<sub>''0''</sub> (''T'' and ''T''<sub>''0''</sub> in [[Kelvin]])
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| | |
| == Link to reaction rate and interpretation ==
| |
| [[Image:Wiki link reaction rate XS.svg|thumb|right|450px|Interpretation of the reaction rate with the help of the cross section]]
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| | |
| Let us imagine a spherical target (outlined in grey in the figure) and a beam of particles (in blue) “flying” at speed ''v'' (vector in black) in the direction of the target. We want to know how many particles impact it during time interval ''dt''. To achieve it, the particles have to be in the black cylinder in the figure (volume ''V''). The base of the cylinder is the geometrical cross section of the target perpendicular to the beam (surface ''σ'' in red) and its height the length travelled by the particles during ''dt'' (length ''v dt''):
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| :<math> V = \sigma \, v \, dt </math>
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| | |
| Noting ''n'' the number of particles per unit volume, there are ''n V'' particles in the volume ''V'', which will, per definition of ''V'', undergo a reaction. Noting ''r'' the [[reaction rate]] onto one target, it gives:
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| :<math> r \, dt = n \, V = n \, \sigma \, v \, dt </math>
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| It follows directly from the definition of the [[neutron flux]]<ref name=DOE/> ''Φ'' = ''n v'':
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| :<math> r = \sigma \, \Phi </math>
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|
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| Assuming that there is not one but ''N'' targets per unit volume, the reaction rate ''R'' per unit volume is:
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| :<math> R = N \, r = N \, \Phi \, \sigma </math>
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| Knowing that the typical nuclear radius ''r'' is of the order of 10<sup>−12</sup> cm, the expected nuclear cross section is of the order of ''π r''<sup>2</sup> or roughly 10<sup>−24</sup> cm<sup>2</sup> (thus justifying the definition of the [[Barn (unit)|barn]]). However, if measured experimentally ( ''σ'' = ''R'' / (''Φ N'') ), the experimental cross sections vary enormously. As an example, for slow neutrons absorbed by the (n, γ) reaction the cross section in some cases is as much as 1,000 barns, while the cross sections for transmutations by gamma-ray absorption are in the neighborhood of 0.001 barn (See [[#Typical cross sections|here]] for more example of cross sections).
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| The “nuclear cross section” is consequently a purely conceptual quantity representing how big the nucleus should be to be consistent with this simple mechanical model.
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| == Continuous versus average cross section ==
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| | |
| Cross sections depend strongly on the incoming particle speed. In the case of a beam with multiple particle speeds, the reaction rate ''R'' is integrated over the whole range of energy:
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| :<math> R = \int_E N \, \Phi (E) \, \sigma (E) \, dE </math>
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| | |
| Where ''σ(E)'' is the continuous cross section, ''Φ(E)'' the differential flux and ''N'' the target atom density.
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| | |
| In order to obtain a formulation equivalent to the mono energetic case, an average cross section is defined:
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| :<math> \sigma =\frac{\int_E \Phi (E) \, \sigma (E) \, dE }{\int_E \Phi (E) \, dE}=\frac{\int_E \Phi (E) \, \sigma (E) \, dE}{\Phi} </math>
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| | |
| Where ''Φ''= <math> \int </math> ''Φ(E) dE'' is the integral flux.
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| | |
| Using the definition of the integral flux ''Φ'' and the average cross section ''σ'', the same formulation as [[#Link to reaction rate and interpretation|before]] is found:
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| :<math> R = N \, \Phi \, \sigma </math>
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| | |
| == Microscopic versus macroscopic cross section ==
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| | |
| Up to now, the cross section referred to in this article corresponds to the microscopic cross section ''σ''. However, it is possible to define the macroscopic cross section<ref name=DOE/> ''Σ'' which corresponds to the total “equivalent area” of all target particles per unit volume:
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| :<math> \Sigma = N \, \sigma </math>
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| where ''N'' is the atomic density of target.
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| Therefore, since the cross section can be expressed in cm<sup>2</sup> and the density in cm<sup>−3</sup>, the macroscopic cross section is usually expressed in cm<sup>−1</sup>. Using equation derived in [[#Link to reaction rate and interpretation]], the reaction rate per unit volume ''R'' can be derived using only the neutron flux ''Φ'' and the macroscopic cross section ''Σ'':
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| :<math> R = \Sigma \, \Phi </math>
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| == Mean free path ==
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| | |
| The “mean free path” ''λ'' of a random particle is the average length between two interactions. The total length ''L'' that non perturbed particles travel during a time interval ''dt'' in a volume ''dV'' is simply the product of the length ''l'' covered by each particle during this time with the number of particles ''N'' in this volume:
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| :<math> L = l \, N </math>
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| | |
| Noting ''v'' the speed of the particles and ''n'' is the number of particles per unit volume:
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| :<math> l = v \, dt </math>
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| :<math> N = n \, dV </math>
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| It follows:
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| :<math> L = v \, dt \, n \, dV </math>
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| | |
| Using the definition of the [[neutron flux]]<ref name=DOE /> ''Φ''
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| :<math> \Phi = n \, v </math>
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| It follows:
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| :<math> L = \Phi \, dt \, dV </math>
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| | |
| This average length ''L'' is however valid only for unperturbed particles. To account for the interactions, ''L'' is divided by the total number of reactions ''R'' to obtain the average length between each collision ''λ'':
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| :<math> \lambda = \frac{L}{R}= \frac{\Phi \, dt \, dV}{R} </math>
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| | |
| From [[#Microscopic versus macroscopic cross section]]:
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| :<math> R = \Phi \, \Sigma \, dt \, dV </math>
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| It follows:
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| :<math> \lambda = \frac{1}{\Sigma} </math>
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| | |
| where ''λ'' is the mean free path and ''Σ'' is the macroscopic cross section.
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| ==Within stars==
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| Because [[lithium]]-8 and [[beryllium]]-12 form natural stopping points on the table of isotopes for [[hydrogen]] [[Nuclear fusion|fusion]], it is believed that all of the higher elements are formed in very hot stars where higher orders of fusion predominate. A star like the [[Sun]] produces [[energy]] by the fusion of simple H-1 into [[helium]]-4 through a [[Nuclear fusion#Astrophysical reaction chains|series of reactions]]. It is believed that when the inner core exhausts its H-1 fuel, the Sun will contract, slightly increasing its core temperature until He-4 can fuse and become the main fuel supply. Pure He-4 fusion leads to Be-8, which decays back to 2 He-4; therefore the He-4 must fuse with isotopes either more or less massive than itself to result in an energy producing reaction. When He-4 fuses with [[Deuterium|H-2]] or [[Tritium|H-3]], it forms stable isotopes Li-6 and Li-7 respectively. The higher order isotopes between Li-8 and C-12 are synthesized by similar reactions between hydrogen, helium, and lithium isotopes.
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| == Typical cross sections ==
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| In the following, some cross sections which are of importance in a nuclear reactor are given. The thermal cross-section is averaged using a Maxwellian spectrum and the fast cross section is averaged using the uranium-235 fission spectrum. The cross sections are taken from the Jeff-3.1.1 library using Janis software.<ref>Janis 3.3, http://www.oecd-nea.org/janis/</ref>
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| [[File:Common light element moderators, reflectors and absorbers.svg|thumb|Width300px|Scattering (full line) and absorption (dotted) crossections of light element commonly used as neutron moderators, reflectors and absorbers, the data was obtained from database NEA N ENDF/B-VII.1 using janis software and ploted using mathplotlib]]
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| {| border="1" cellpadding="5" cellspacing="0" style="margin:auto; text-align:center;"
| |
| |-
| |
| | colspan="2" rowspan="2" |
| |
| | colspan="3" | Thermal cross section (barn)
| |
| | colspan="3" | Fast cross section (barn)
| |
| |-
| |
| | Scattering
| |
| | Capture
| |
| | Fission
| |
| | Scattering
| |
| | Capture
| |
| | Fission
| |
| |-
| |
| | rowspan="3" | Moderator
| |
| | H-1
| |
| | 2E+1
| |
| | 2E-1
| |
| | -
| |
| | 4E+0
| |
| | 4E-5
| |
| | -
| |
| |-
| |
| | H-2
| |
| | 4E+0
| |
| | 3E-4
| |
| | -
| |
| | 3E+0
| |
| | 7E-6
| |
| | -
| |
| |-
| |
| | C (nat)
| |
| | 5E+0
| |
| | 2E-3
| |
| | -
| |
| | 2E+0
| |
| | 1E-5
| |
| | -
| |
| |-
| |
| | rowspan="5" | Structural materials, others
| |
| | Zr-90
| |
| | 5E+0
| |
| | 6E-3
| |
| | -
| |
| | 5E+0
| |
| | 6E-3
| |
| | -
| |
| |-
| |
| | Fe-56
| |
| | 1E+1
| |
| | 2E+0
| |
| | -
| |
| | 2E+1
| |
| | 3E-3
| |
| | -
| |
| |-
| |
| | Cr-52
| |
| | 3E+0
| |
| | 5E-1
| |
| | -
| |
| | 3E+0
| |
| | 2E-3
| |
| | -
| |
| |-
| |
| | Ni-58
| |
| | 2E+1
| |
| | 3E+0
| |
| | -
| |
| | 3E+0
| |
| | 8E-3
| |
| | -
| |
| |-
| |
| | O-16
| |
| | 4E+0
| |
| | 1E-4
| |
| | -
| |
| | 3E+0
| |
| | 3E-8
| |
| | -
| |
| |-
| |
| | rowspan="4" | Absorber
| |
| | B-10
| |
| | 2E+0
| |
| | 2E+3
| |
| | -
| |
| | 2E+0
| |
| | 4E-1
| |
| | -
| |
| |-
| |
| | Cd-113
| |
| | 1E+2
| |
| | 3E+4
| |
| | -
| |
| | 4E+0
| |
| | 5E-2
| |
| | -
| |
| |-
| |
| | Xe-135
| |
| | 4E+5
| |
| | 2E+6
| |
| | -
| |
| | 5E+0
| |
| | 8E-4
| |
| | -
| |
| |-
| |
| | In-115
| |
| | 2E+0
| |
| | 1E+2
| |
| | -
| |
| | 4E+0
| |
| | 2E-1
| |
| | -
| |
| |-
| |
| | rowspan="3" | Fuel
| |
| | U-235
| |
| | 1E+1
| |
| | 6E+1
| |
| | 3E+2
| |
| | 4E+0
| |
| | 9E-2
| |
| | 1E+0
| |
| |-
| |
| | U-238
| |
| | 9E+0
| |
| | 2E+0
| |
| | 2E-5
| |
| | 5E+0
| |
| | 7E-2
| |
| | 3E-1
| |
| |-
| |
| | Pu-239
| |
| | 8E+0
| |
| | 4E-2
| |
| | 7E+2
| |
| | 5E+0
| |
| | 5E-2
| |
| | 2E+0
| |
| |}
| |
| | |
| ==External links==
| |
| * [http://www.ncnr.nist.gov/resources/n-lengths/ Neutron scattering lengths and cross-sections]
| |
| * [http://environmentalchemistry.com/yogi/periodic/crosssection.html Periodic Table of Elements: Sorted by Cross Section (Thermal Neutron Capture)]
| |
| <!-- What does this mean? "Coherent scattering: In this nuclear process reemission occurs at the same frequency as the incident radiation." -->
| |
| | |
| ==References==
| |
| | |
| {{reflist}}
| |
| | |
| [[Category:Nuclear physics]]
| |
| [[Category:Neutron]]
| |
During the Neolithic Revolution, humans started consuming large amounts of dairy products, beans, cereals, alcohol and salt. Mediterranean diets are typically high in olive oil consumption as well as cereals,dairy products(primarily yoghurt and cheese)legumes,vegetables and fish. Following a healthy diet signifies changing your diet plan, lowering the amounts but still getting all the proper vitamins and minerals. Prepare to leave your unhealthy habits behind in exchange of a better lifestyle, best trim healthy whole body, and days active. Heat a medium size pan with 1 tablespoons of tallow over medium heat Now put ready cut small pieces of carrot in to the pan fry it for 3 minutes and remove them from pan.
Since farming and the cultivation of plants like wheat have only been known to man for about 10,000 years, our bodies have not adapted to properly digest these food sources. And also what's more: regular people from many areas of life are actually advocating this product, from common buyers to authorised physicians. Ready-made paleo foods items are starting to find their way into conventional grocery stores. Provide help to immune system after the disease attacked. Copyright (c) 2014 Keith Frame - Article Source: out more about diet and weight loss pill at or my review website at.
The reason for this is because the cows are free to roam around and eat all the grass. Meals such as grilled or roasted pork, beef and chicken along with broccoli, carrots and sweet potatoes are part of the Paleo diet. Both fruit and veggies contain a number of vitamins. You might know, I arrived so close to giving up now this healthy diet. Diaz is qualified to handle any oral health need, including Invisalign, cosmetic dentistry, laser dentistry and dental implants in Pembroke Pines.
His photos quickly went viral, with admirers of his sleek physique warring with fans who expressed concern that the NBA's Most Valuable Player had shredded too much fat. Chives: Used in salads, soups, stews and appetizers. When we do feel hungry, we usually also feel panic or anxiety. Advocates who have adhered to the paleo diet can attest to the range of paleo foods there is to choose from -- many are already on your list of routinely consumed foods. I want to start off by saying, the Paleo diet was first put together in the mid 1970.
Walleye has omega 3 fatty acids, vitamins, potassium and other minerals, hence certainly a worthwhile proposition. * No need to rack your mind, working out caloric intake. The time period when our elders used to feed on such diet was from many years ago. You want to find recipes that will fit into your easy paleo diet plan. Summary Our bodies are designed to react to what we put in them.
Those cave men and women led very different lives than we do today. Desserts are put together from fruits like bananas and berries. This further frustrates the addict who attempts to cooperate in treatment and yet receives no benefit from that treatment. It is not an easy thing to have new ideas for meal and having lack of options but there are still some good choices from the paleo diet which might be used. The sick emotion lingered in my mind for times and I have not experienced a cheeseburger given that.
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