Editing Neutron moderator (section)

== Moderation ==
[[Neutron diffraction|Neutrons]] are normally bound into an [[atomic nucleus]] and do not exist free for long in nature. The unbound neutron has a [[half-life]] of [[Free neutron decay|10 minutes and 11 seconds]]. The release of neutrons from the nucleus requires exceeding the [[binding energy]] of the neutron, which is typically 7-9 [[MeV]] for most [[isotopes]]. [[Neutron source]]s generate free neutrons by a variety of nuclear reactions, including [[nuclear fission]] and [[nuclear fusion]]. Whatever the source of neutrons, they are released with energies of several MeV.

According to the [[equipartition theorem]], the average [[kinetic energy]], <math>\bar{E}</math>, can be related to [[temperature]], <math>T</math>, via:
:<math>\bar{E}=\frac{1}{2}m_n \langle v^2 \rangle=\frac{3}{2}k_B T</math>, 
where <math>m_n</math> is the neutron mass, <math>\langle v^2 \rangle</math> is the average squared neutron speed, and <math>k_B</math> is the [[Boltzmann constant]].<ref>{{cite book|last1=Kratz|first1=Jens-Volker|last2=Lieser|first2=Karl Heinrich|title=Nuclear and Radiochemistry: Fundamentals and Applications|date=2013|publisher=John Wiley & Sons|isbn=9783527653355|edition=3|url=https://books.google.com/books?id=gXNwAAAAQBAJ&q=neutron+moderator+kinetic+energy+%22boltzmann+constant%22&pg=PT346|access-date=27 April 2018}}</ref><ref>{{cite book|last1=De Graef|first1=Marc|last2=McHenry|first2=Michael E.|title=Structure of Materials: An Introduction to Crystallography, Diffraction and Symmetry|date=2012|publisher=Cambridge University Press|isbn=9781139560474|page=324|url=https://books.google.com/books?id=NMUgAwAAQBAJ&q=neutron+moderator+kinetic+energy+%22boltzmann+constant%22&pg=PA324|access-date=27 April 2018}}</ref> The characteristic [[neutron temperature]] of several-MeV neutrons is several tens of billions [[kelvin]].

Moderation is the process of the reduction of the initial high speed (high kinetic energy) of the free neutron. Since energy is conserved, this reduction of the neutron speed takes place by transfer of energy to a material called a ''moderator''.

The probability of scattering of a neutron from a nucleus is given by the [[nuclear cross section|scattering cross section]]. The first few collisions with the moderator may be of sufficiently high energy to excite the nucleus of the moderator. Such a collision is [[inelastic collision|inelastic]], since some of the kinetic energy is transformed to [[potential energy]] by exciting some of the internal [[Degrees of freedom (physics and chemistry)|degrees of freedom]] of the nucleus to form an [[Nuclear isomer|excited state]]. As the energy of the neutron is lowered, the collisions become predominantly [[elastic collision|elastic]], i.e., the total kinetic energy and momentum of the system (that of the neutron and the nucleus) is conserved.

Given the [[elastic collision|mathematics of elastic collisions]], as neutrons are very light compared to most nuclei, the most efficient way of removing kinetic energy from the neutron is by choosing a moderating nucleus that has near identical mass.

[[Image:Elastischer stoß.gif|frame|center|Elastic collision of equal masses]]

A collision of a neutron which has mass of 1 with a <sup>1</sup>H nucleus (a [[proton]]) could result in the neutron losing virtually all of its energy in a single head-on collision. More generally, it is necessary to take into account both glancing and head-on collisions. The ''mean logarithmic reduction of neutron energy per collision'', <math>\xi</math>, depends only on the atomic mass, <math>A</math>, of the nucleus and is given by:

<math>\xi= \ln\frac{E_0}{E}=1-\frac{(A-1)^2}{2A}\ln\left(\frac{A+1}{A-1}\right)</math>.<ref name="
Weston">{{cite book
  | last = Stacey.
  | first = Weston M
  | title = Nuclear reactor physics
  | publisher = [[Wiley-VCH]]
  | year = 2007
  | pages = 29–31
  | url = https://books.google.com/books?id=9gLXk-LRvZwC
  | isbn = 978-3-527-40679-1}}</ref>

This can be reasonably approximated to the very simple form <math>\xi\simeq \frac{2}{A+2/3}</math>.<ref name="
DB">{{cite book |last= Dobrzynski |first= L. |author2=K. Blinowski  |title= Neutrons and Solid State Physics|publisher= Ellis Horwood Limited |year= 1994 |isbn= 0-13-617192-3}}</ref> From this one can deduce <math>n</math>, the expected number of collisions of the neutron with nuclei of a given type that is required to reduce the kinetic energy of a neutron from <math>E_0</math> to <math>E_1</math>
:<math> n=\frac{1}{\xi}(\ln E_0-\ln E_1)</math>.<ref name="DB" />

[[Image:Translational motion.gif|frame|right|In a system at thermal equilibrium, neutrons (red) are elastically scattered by a hypothetical moderator of free hydrogen nuclei (blue), undergoing thermally activated motion. Kinetic energy is transferred between particles. As the neutrons have essentially the same mass as [[protons]] and there is no absorption, the velocity distributions of both particles types would be well-described by a single [[Maxwell–Boltzmann distribution]].]]

===Materials===
Some nuclei have larger [[Neutron cross section#Absorption cross section|absorption cross sections]] than others, which removes free neutrons from the [[flux]]. Therefore, a further criterion for an efficient moderator is one for which this parameter is small. The ''moderating efficiency'' gives the ratio of the [[Nuclear cross section#Macroscopic cross section|macroscopic cross sections]] of scattering, <math>\Sigma_s</math>, weighted by <math>\xi</math> divided by that of absorption, <math>\Sigma_a</math>: i.e., <math>\frac{\xi\Sigma_s}{\Sigma_a}</math>.<ref name="Weston" /> For a compound moderator composed of more than one element, such as light or heavy water, it is necessary to take into account the moderating and absorbing effect of both the hydrogen isotope and oxygen atom to calculate <math>\xi</math>. To bring a neutron from the fission energy of <math>E_0</math> 2 MeV to an <math>E</math> of 1 eV takes an expected <math>n</math> of 16 and 29 collisions for H<sub>2</sub>O and D<sub>2</sub>O, respectively. Therefore, neutrons are more rapidly moderated by light water, as H has a far higher <math>\Sigma_s</math>. However, it also has a far higher <math>\Sigma_a</math>, so that the moderating efficiency is nearly 80 times higher for heavy water than for light water.<ref name="Weston" />

''The ideal moderator is of low mass, high scattering cross section, and low absorption cross section''.
{| class="table" 
 !
 ![[Hydrogen]]
 ![[Deuterium]]
 ![[Beryllium]]
 ![[Carbon]]
 ![[Oxygen]]
 ![[Uranium]]
 |-
 |Mass of nuclei [[atomic mass unit|u]]
 |1
 |2
 |9
 |12
 |16
 |238
 |-
 |Energy decrement <math>\xi</math>
 |1
 |0.7261
 |0.2078
 |0.1589
 |0.1209
 |0.0084
 |-
 |Number of Collisions
 |18
 |25
 |86
 |114
 |150
 |2172
 |}

===Distribution of neutron velocities===
After sufficient impacts, the speed of the neutron will be comparable to the speed of the nuclei given by thermal motion; this neutron is then called a [[thermal neutron]], and the process may also be termed ''thermalization''. Once at equilibrium at a given temperature the distribution of speeds (energies) expected of rigid spheres scattering elastically is given by the [[Maxwell–Boltzmann distribution]]. This is only slightly modified in a real moderator due to the speed (energy) dependence of the absorption cross-section of most materials, so that low-speed neutrons are preferentially absorbed,<ref name="DB" /><ref>[http://www.ncnr.nist.gov/resources/n-lengths/ Neutron scattering lengths and cross sections] V.F. Sears, ''Neutron News'' 3, No. 3, 26-37 (1992)</ref> so that the true neutron velocity distribution in the core would be slightly hotter than predicted.