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2014-08-10T15:22:27Z

JannAlderudt:

{{unreferenced|date=April 2012}}
[[File:AGOModra Leonids98.jpg|thumb|right|All-sky view of the 1998 [[Leonids]] shower. 156 meteors were captured in this 4 hour image.]]
In [[astronomy]], the '''Zenithal Hourly Rate''' ('''ZHR''') of a [[meteor shower]] is the number of meteors a single observer would see in one hour under a clear, dark sky (limiting [[apparent magnitude]] of 6.5) if the [[Radiant (meteor shower)|radiant]] of the shower were at the [[zenith]]. The rate that can effectively be seen is nearly always lower and decreases the closer the radiant is to the [[horizon]].

The formula to calculate the ZHR is:

<math> ZHR = \cfrac{\overline{HR} \cdot F \cdot r^{6.5-lm}}{\sin(hR)} </math>

where

<math>\overline{HR} = \cfrac{N}{T_{eff}} </math>

represents the hourly rate of the observer. N is the number of meteors observed, and Teff is the effective observation time of the observer.

Example: If the observer detected 12 meteors in 15 minutes, their hourly rate was 48. (12 divided by 0.25 hours).

<math> F = \cfrac{1}{1-k}</math>

This represents the field of view correction factor, where k is the percentage of the observer's field of view which is obstructed (by clouds, for example).

Example: If 20% of the observer's field of view were covered by clouds, k would be 0.2 and F would be 1.25. The observer should have seen 25% more meteors, therefore we multiply by F = 1.25.

<math> r^{6.5-lm} </math>

This represents the limiting magnitude correction factor. For every change of 1 magnitude in the limiting magnitude of the observer, the number of meteors observed changes by a factor of r. Therefore we must take this into account.

Example: If r is 2, and the observer's limiting magnitude is 5.5, we will have to multiply their hourly rate by 2 (2 to the power 6.5-5.5), to know how many meteors they would have seen if their limiting magnitude was 6.5.

<math> \sin(hR) </math>

This represents the correction factor for altitude of the radiant above the horizon (hR). The number of meteors seen by an observer changes as the sine of the radiant height in radians.

Example: If the radiant was at an average altitude of 30° during the observation period, we will have to divide the observer's hourly rate by 0.5 (sin 30°) to know how many meteors they would have seen if the radiant was at the zenith.

==See also==
*[[List of meteor showers]]

==External links==
*[http://www.namnmeteors.org/guidechap8.html North American Meteor Network] (NAMN)

[[Category:Meteoroids]]
[[Category:Observational astronomy]]

[[bg:Зенитно часово число]]
[[ca:Taxa horària zenital]]
[[de:Zenithal Hourly Rate]]
[[es:Tasa Horaria Zenital]]
[[fa:آهنگ ساعتی سرسویی]]
[[fr:Taux horaire zénithal]]
[[gl:Taxa Horaria Zenital]]
[[ko:정점 시율]]
[[it:Tasso orario zenitale]]
[[he:קצב זניתי לשעה]]
[[lb:Zenithal Hourly Rate]]
[[ms:Kadar Kemuncak Sejam]]
[[nl:Zenithal hourly rate]]
[[ja:天頂出現数]]
[[pl:Zenitalna liczba godzinna]]
[[pt:Taxa horária zenital]]
[[ru:Зенитное часовое число]]
[[sk:Zenitová hodinová frekvencia]]
[[sl:Zenitna urna frekvenca]]
[[sr:Зенитна часовна активност]]
[[zh:每小時天頂流星數]]

Main Page

2014-08-10T15:21:05Z

JannAlderudt:

{{Use dmy dates|date=March 2012}}
{{Use British English|date=March 2012}}
{{Refimprove|date=January 2008}}
{{Infobox scientist
|name = George Peacock
|image = George_Peacock.jpg
|image_size = 170px
|caption =
|birth_date = {{birth date|1791|4|9|df=y}}
|birth_place = Thornton Hall, Denton, [[County Durham]], [[England]]
|death_date = {{death date and age|1858|11|08|1791|4|9|df=y}}
|death_place = [[Pall Mall, London|Pall Mall]], [[London]], [[England]]
|residence = [[England]]
|citizenship =
|nationality = [[English people|English]]
|ethnicity =
|field = [[Mathematician]]
|work_institutions = [[University of Cambridge]]
|alma_mater = [[University of Cambridge]]
|doctoral_advisor =
|academic_advisors = [[John Hudson (mathematician)|John Hudson]][[Adam Sedgwick]]
|doctoral_students =
|notable_students = [[Augustus De Morgan]][[Arthur Cayley]][[George Biddell Airy]][[W. H. Thompson]]
|known_for = ''Treatise on Algebra''
|author_abbrev_bot =
|author_abbrev_zoo =
|influences =
|influenced =
|prizes =
|religion = [[Anglican]]
|footnotes = When he died his wife married his student [[W. H. Thompson]].
}}
'''George Peacock''' (9 April 1791 – 8 November 1858) was an [[English people|English]] [[mathematician]].

==Early life==
Peacock was born on 9 April 1791 at Thornton Hall, Denton, near [[Darlington]], County Durham.<ref>Harvey W. Becher, ‘Peacock, George (1791–1858)’, Oxford Dictionary of National Biography, Oxford University Press, 2004; online edn, May 2009 [http://www.oxforddnb.com/view/article/21673, accessed 2 May 2011]</ref> His father, the Rev. Thomas Peacock, was a clergyman of the [[Church of England]], incumbent and for 50 years curate of the parish of Denton, where he also kept a school. In early life Peacock did not show any precocity of genius, and was more remarkable for daring feats of climbing than for any special attachment to study. He received his elementary education from his father, and at 17 years of age, was sent to Richmond, to a school taught by a graduate of [[University of Cambridge|Cambridge University]] to receive instruction preparatory to entering that university. At this school he distinguished himself greatly both in classics and in the rather elementary mathematics then required for entrance at Cambridge. In 1809 he became a student of [[Trinity College, Cambridge]].<ref>{{Venn|id=PCK809G|name=Peacock, George}}</ref>

In 1812 Peacock took the rank of [[Second Wrangler]], and the second [[Smith's prize]], the senior wrangler being [[John Herschel]]. Two years later he became a candidate for a fellowship in his college and won it immediately, partly by means of his extensive and accurate knowledge of the classics. A fellowship then meant about pounds 200 a year, tenable for seven years provided the Fellow did not marry meanwhile, and capable of being extended after the seven years provided the Fellow took clerical orders, which Peacock did in 1819.

==Mathematical career==
The year after taking a Fellowship, Peacock was appointed a tutor and lecturer of his college, which position he continued to hold for many years. Peacock, in common with many other students of his own standing, was profoundly impressed with the need of reforming Cambridge's position ignoring the differential notation for calculus, and while still an undergraduate formed a league with [[Charles Babbage|Babbage]] and [[John Herschel|Herschel]] to adopt measures to bring it about. In 1815 they formed what they called the ''Analytical Society'', the object of which was stated to be to advocate the ''d'' 'ism of the Continent versus the ''dot''-age of the University.

The first movement on the part of the [[Analytical Society]] was to translate from the French the smaller work of [[Sylvestre François Lacroix|Lacroix]] on the differential and integral calculus; it was published in 1816. At that time the best manuals, as well as the greatest works on mathematics, existed in the French language. Peacock followed up the translation with a volume containing a copious ''Collection of Examples of the Application of the Differential and Integral Calculus'', which was published in 1820. The sale of both books was rapid, and contributed materially to further the object of the Society. In that time, high wranglers of one year became the examiners of the mathematical tripos three or four years afterwards. Peacock was appointed an examiner in 1817, and he did not fail to make use of the position as a powerful lever to advance the cause of reform. In his questions set for the examination the differential notation was for the first time officially employed in Cambridge. The innovation did not escape censure, but he wrote to a friend as follows: "I assure you that I shall never cease to exert myself to the utmost in the cause of reform, and that I will never decline any office which may increase my power to effect it. I am nearly certain of being nominated to the office of Moderator in the year 1818-1819, and as I am an examiner in virtue of my office, for the next year I shall pursue a course even more decided than hitherto, since I shall feel that men have been prepared for the change, and will then be enabled to have acquired a better system by the publication of improved elementary books. I have considerable influence as a lecturer, and I will not neglect it. It is by silent perseverance only, that we can hope to reduce the many-headed monster of prejudice and make the University answer her character as the loving mother of good learning and science." These few sentences give an insight into the character of Peacock: he was an ardent reformer and a few years brought success to the cause of the Analytical Society.

Another reform at which Peacock labored was the teaching of [[algebra]]. In 1830 he published a ''Treatise on Algebra'' which had for its object the placing of algebra on a true scientific basis, adequate for the development which it had received at the hands of the Continental mathematicians. To elevate astronomical science the Astronomical Society of London was founded, and the three reformers Peacock, Babbage and Herschel were again prime movers in the undertaking. Peacock was one of the most zealous promoters of an astronomical observatory at Cambridge, and one of the founders of the Philosophical Society of Cambridge.

In 1831 the British Association for the Advancement of Science (prototype of the American, French and Australasian Associations) held its first meeting in the ancient city of [[York]]. One of the first resolutions adopted was to procure reports on the state and progress of particular sciences, to be drawn up from time to time by competent persons for the information of the annual meetings, and the first to be placed on the list was a report on the progress of mathematical science. Dr. Whewell, the mathematician and philosopher, was a Vice-president of the meeting: he was instructed to select the reporter. He first asked Sir W. R. Hamilton, who declined; he then asked Peacock, who accepted. Peacock had his report ready for the third meeting of the Association, which was held in Cambridge in 1833; although limited to [[Algebra]], [[Trigonometry]], and the Arithmetic of Sines, it is one of the best of the long series of valuable reports which have been prepared for and printed by the Association.

In 1837 Peacock was appointed [[Lowndean Professor of Astronomy]] in the University of Cambridge, the chair afterwards occupied by [[John Couch Adams|Adams]], the co-discoverer of [[Neptune]], and later occupied by Sir [[Robert Stawell Ball|Robert Ball]], celebrated for his ''Theory of Screws''. An object of reform was the statutes of the University; he worked hard at it and was made a member of a commission appointed by the Government for the purpose.

He was elected a [[Fellow of the Royal Society]] in January 1818.<ref>{{cite web|url=http://royalsociety.org/DServe/dserve.exe?dsqIni=Dserve.ini&dsqApp=Archive&dsqCmd=Show.tcl&dsqDb=Persons&dsqPos=5&dsqSearch=%28%28text%29%3D%27peacock%27%29|title=Library Archive|publisher=The Royal Society|accessdate=28 August 2012}}</ref>

==Clerical Career==

He was ordained as a deacon in 1819, a priest in 1822 and appointed Vicar of Wymewold in 1826 (until 1835).<ref> {{cite web | url = http://www.theclergydatabase.org.uk/jsp/persons/DisplayCcePerson.jsp?PersonID=53533|title = Peacock, George (1819-1835)accessdate = 2012-29}} </ref>.

In 1839 he was appointed [[Dean of Ely]] cathedral, Cambridgeshire, a position he held for the rest of his life, some 20 years. Together with the architect [[Sir George Gilbert Scott]] he undertook a major restoration of the cathedral building. This included the installation of the boarded ceiling. <ref> {{cite web | url = http://www.elycathedral.org/history/the_story_cathedral.html| title = The Story of Ely Cathedral History & Heritage|accessdate = 2012-08-29}} </ref>

While holding this position he wrote a text book on algebra in two volumes, the one called ''Arithmetical Algebra'', and the other ''Symbolical Algebra''.

==Private life==
Politically he was a [[Whig (British political party)|Whig]].<ref>Radicals, Whigs and Conservatives: The Middle and Lower Classes in the Analytical Revolution at Cambridge in the Age of Aristocracy</ref>

His last public act was to attend a meeting of the university reform commission. He died in Ely on 8 November 1858 in the 68th year of his age and was buried in Ely cemetery. He had married Frances Elizabeth, the daughter of William Selwyn but had no children.

==Algebraic Theory==

Peacock's main contribution to mathematical analysis is his attempt to place algebra on a strictly logical basis. He founded what has been called the philological or symbolical school of mathematicians; to which [[Duncan Farquharson Gregory|Gregory]], [[Augustus De Morgan|De Morgan]] and [[George Boole|Boole]] belonged. His answer to Maseres and Frend was that the science of algebra consisted of two parts—arithmetical algebra and symbolical algebra—and that they erred in restricting the science to the arithmetical part. His view of arithmetical algebra is as follows: "In arithmetical algebra we consider symbols as representing numbers, and the operations to which they are submitted as included in the same definitions as in common arithmetic; the signs <math>+</math> and <math>-</math> denote the operations of addition and subtraction in their ordinary meaning only, and those operations are considered as impossible in all cases where the symbols subjected to them possess values which would render them so in case they were replaced by digital numbers; thus in expressions such as <math>a + b</math> we must suppose <math>a</math> and <math>b</math> to be quantities of the same kind; in others, like <math>a - b</math>, we must suppose <math>a</math> greater than <math>b</math> and therefore homogeneous with it; in products and quotients, like <math>ab</math> and <math>\frac{a}{b}</math> we must suppose the multiplier and divisor to be abstract numbers; all results whatsoever, including negative quantities, which are not strictly deducible as legitimate conclusions from the definitions of the several operations must be rejected as impossible, or as foreign to the science."

Peacock's principle may be stated thus: the elementary symbol of arithmetical algebra denotes a [[digital]], i.e., an integer number; and every combination of elementary symbols must reduce to a digital number, otherwise it is impossible or foreign to the science. If <math>a</math> and <math>b</math> are numbers, then <math>a + b</math> is always a number; but <math>a - b</math> is a number only when <math>b</math> is less than <math>a</math>. Again, under the same conditions, <math>ab</math> is always a number, but <math>\frac{a}{b}</math> is really a number only when <math>b</math> is an exact divisor of <math>a</math>. Hence the following dilemma: Either <math>\frac{a}{b}</math> must be held to be an impossible expression in general, or else the meaning of the fundamental symbol of algebra must be extended so as to include rational fractions. If the former horn of the dilemma is chosen, arithmetical algebra becomes a mere shadow; if the latter horn is chosen, the operations of algebra cannot be defined on the supposition that the elementary symbol is an integer number. Peacock attempts to get out of the difficulty by supposing that a symbol which is used as a multiplier is always an integer number, but that a symbol in the place of the multiplicand may be a fraction. For instance, in <math>ab</math>, <math>a</math> can denote only an integer number, but <math>b</math> may denote a rational fraction. Now there is no more fundamental principle in arithmetical algebra than that <math>ab = ba</math>; which would be illegitimate on Peacock's principle.

One of the earliest English writers on [[arithmetic]] is [[Robert Record]], who dedicated his work to King Edward the Sixth. The author gives his treatise the form of a dialogue between master and scholar. The scholar battles long over this difficulty, -- that multiplying a thing could make it less. The master attempts to explain the anomaly by reference to proportion; that the product due to a fraction bears the same proportion to the thing multiplied that the fraction bears to unity. But the scholar is not satisfied and the master goes on to say: "If I multiply by more than one, the thing is increased; if I take it but once, it is not changed, and if I take it less than once, it cannot be so much as it was before. Then seeing that a fraction is less than one, if I multiply by a fraction, it follows that I do take it less than once." Whereupon the scholar replies, "Sir, I do thank you much for this reason, -- and I trust that I do perceive the thing."

The fact is that even in arithmetic the two processes of [[multiplication]] and [[division (mathematics)|division]] are generalized into a common multiplication; and the difficulty consists in passing from the original idea of multiplication to the generalized idea of a ''[[tensor]]'', which idea includes compressing the [[magnitude (mathematics)|magnitude]] as well as stretching it. Let <math>m</math> denote an integer number; the next step is to gain the idea of the [[Multiplicative inverse|reciprocal]] of <math>m</math>, not as <math>\frac{1}{m}</math> but simply as <math>/m</math>. When <math>m</math> and <math>/n</math> are compounded we get the idea of a rational fraction; for in general <math>m/n</math> will not reduce to a number nor to the reciprocal of a number.

Suppose, however, that we pass over this objection; how does Peacock lay the foundation for general algebra? He calls it symbolical algebra, and he passes from arithmetical algebra to symbolical algebra in the following manner: "Symbolical algebra adopts the rules of arithmetical algebra but removes altogether their restrictions; thus symbolical subtraction differs from the same operation in arithmetical algebra in being possible for all relations of value of the symbols or expressions employed. All the results of arithmetical algebra which are deduced by the application of its rules, and which are general in form though particular in value, are results likewise of symbolical algebra where they are general in value as well as in form; thus the product of <math>a^{m}</math> and <math>a^{n}</math> which is <math>a^{m+n}</math> when <math>m</math> and <math>n</math> are whole numbers and therefore general in form though particular in value, will be their product likewise when <math>m</math> and <math>n</math> are general in value as well as in form; the series for <math>(a+b)^{n}</math> determined by the principles of arithmetical algebra when <math>n</math> is any whole number, ''if it be exhibited in a general form, without reference to a final term'', may be shown upon the same principle to the equivalent series for <math>(a+b)^n</math> when <math>n</math> is general both in form and value."

The principle here indicated by means of examples was named by Peacock the "principle of the permanence of equivalent forms," and at page 59 of the ''Symbolical Algebra'' it is thus enunciated: "Whatever algebraic forms are equivalent when the symbols are general in form, but specific in value, will be equivalent likewise when the symbols are general in value as well as in form."

For example, let <math>a</math>, <math>b</math>, <math>c</math>, <math>d</math> denote any integer numbers, but subject to the restrictions that <math>b</math> is less than <math>a</math>, and <math>d</math> less than <math>c</math>; it may then be shown arithmetically that <math>(a - b)(c - d)=ac + bd - ad - bc</math>. Peacock's principle says that the form on the left side is equivalent to the form on the right side, not only when the said restrictions of being less are removed, but when <math>a</math>, <math>b</math>, <math>c</math>, <math>d</math> denote the most general algebraic symbol. It means that <math>a</math>, <math>b</math>, <math>c</math>, <math>d</math> may be rational fractions, or surds, or imaginary quantities, or indeed [[Operator (mathematics)|operators]] such as <math>\frac{d}{dx}</math>. The [[Equivalence relation|equivalence]] is not established by means of the nature of the [[quantity]] denoted; the equivalence is assumed to be true, and then it is attempted to find the different interpretations which may be put on the symbol.

It is not difficult to see that the problem before us involves the fundamental problem of a rational logic or theory of knowledge; namely, how are we able to ascend from particular truths to more general truths. If <math>a</math>, <math>b</math>, <math>c</math>, <math>d</math> denote integer numbers, of which <math>b</math> is less than <math>a</math> and <math>d</math> less than <math>c</math>, then <math>(a - b)(c - d)=ac + bd - ad - bc</math>.

It is first seen that the above restrictions may be removed, and still the above equation holds. But the antecedent is still too narrow; the true scientific problem consists in specifying the meaning of the symbols, which, and only which, will admit of the forms being equal. It is not to find "some meanings", but the "most general meaning", which allows the equivalence to be true. Let us examine some other cases; we shall find that Peacock's principle is not a solution of the difficulty; the great logical process of generalization cannot be reduced to any such easy and arbitrary procedure. When <math>a</math>, <math>m</math>, <math>n</math> denote integer numbers, it can be shown that<math>a^{m}a^{n} = a^{m+n}</math>.

According to Peacock the form on the left is always to be equal to the form on the right, and the meanings of <math>a</math>, <math>m</math>, <math>n</math> are to be found by interpretation. Suppose that <math>a</math> takes the form of the incommensurate quantity <math>e</math>, the base of the natural system of [[logarithm]]s. A number is a degraded form of a complex quantity <math>p+q^{\sqrt{-1}}</math> and a complex quantity is a degraded form of a [[quaternion]]; consequently one meaning which may be assigned to <math>m</math> and <math>n</math> is that of quaternion. Peacock's principle would lead us to suppose that <math>e^{m}e^{n} = e^{m+n}</math>, <math>m</math> and <math>n</math> denoting quaternions; but that is just what Hamilton, the inventor of the quaternion generalization, denies. There are reasons for believing that he was mistaken, and that the forms remain equivalent even under that extreme generalization of <math>m</math> and <math>n</math>; but the point is this: it is not a question of conventional definition and formal truth; it is a question of objective definition and real truth. Let the symbols have the prescribed meaning, does or does not the equivalence still hold? And if it does not hold, what is the higher or more complex form which the equivalence assumes?

== References ==

{{More footnotes|date=January 2008}}
{{reflist}}
*{{Cite book | last1=Macfarlane | first1=Alexander | title=Lectures on Ten British Mathematicians of the Nineteenth Century | origyear=1916 | url=http://www.archive.org/details/lecturesontenbri00macf | publisher=Cornell University Library | series=Mathematical monographs | isbn=978-1-112-28306-2 | year=2009 | volume=17 | postscript={{inconsistent citations}}}} ([http://library.beau.org/gutenberg/etext06/tbmms10p.pdf complete text] at [[Project Gutenberg]])

==External links==
* {{MacTutor Biography|id=Peacock}}
* [http://etc.usf.edu/lit2go/contents/2900/2902/2902.pdf Biography of Peacock]

{{s-start}}
{{s-rel|en}}
{{succession box
| title = [[Dean of Ely]]
| years = 1839–1858
| before = [[James Wood (mathematician)|James Wood]]
| after = [[Harvey Goodwin]]
}}
{{s-end}}

{{Deans of Ely}}

{{Persondata 
| NAME = Peacock, George
| ALTERNATIVE NAMES =
| SHORT DESCRIPTION =
| DATE OF BIRTH = 9 April 1791
| PLACE OF BIRTH = Denton, [[County Durham]], [[England]]
| DATE OF DEATH = 8 November 1858
| PLACE OF DEATH = [[Pall Mall, London|Pall Mall]], [[London]], [[England]]
}}
{{DEFAULTSORT:Peacock, George}}
[[Category:1791 births]]
[[Category:1858 deaths]]
[[Category:People from County Durham]]
[[Category:19th-century mathematicians]]
[[Category:English mathematicians]]
[[Category:Mathematical analysts]]
[[Category:Fellows of the Royal Society]]
[[Category:Second Wranglers]]
[[Category:Lowndean Professors of Astronomy and Geometry]]
[[Category:Deans of Ely]]

[[cs:George Peacock]]
[[de:George Peacock]]
[[es:George Peacock]]
[[fr:George Peacock]]
[[ht:George Peacock]]
[[pms:George Peacock]]
[[ro:George Peacock]]

Main Page

2014-08-10T11:49:11Z

JannAlderudt:

In [[nuclear engineering]], a '''neutron moderator''' is a medium that reduces the speed of [[fast neutron]]s, thereby turning them into [[thermal neutron]]s capable of sustaining a [[nuclear chain reaction]] involving [[uranium-235]].

Commonly used moderators include [[water|regular (light) water]] (roughly 75% of the world's reactors), solid [[graphite]] (20% of reactors) and [[heavy water]] (5% of reactors).<ref>{{cite book
| last = Miller, Jr.
| first = George Tyler
| authorlink =
| title = Living in the Environment: Principles, Connections, and Solutions (12th Edition)
| publisher = [[The Thomson Corporation]]
| year = 2002
| location = Belmont
| pages = 345
| url =
| isbn = 0-534-37697-5}}</ref>
[[Beryllium]] has also been used in some experimental types, and [[hydrocarbon]]s have been suggested as another possibility.

{| class="wikitable" align="right"
|+Currently operating [[nuclear power]] reactors by moderator
|-
!Moderator!!Reactors!!Design!!Country
|-
|none ([[fast neutron reactor|fast]])||1||[[BN-600]]||Russia (1)
|-
|graphite||29||[[Advanced gas-cooled reactor|AGR]], [[Magnox]], [[RBMK]]|| United Kingdom (18), Russia (11)
|-
|heavy water||29||[[CANDU]]||Canada (17), South Korea (4), Romania (2), China (2), India (2), Argentina, Pakistan
|-
|light water||359||[[Pressurized water reactor|PWR]], [[Boiling water reactor|BWR]]||27 countries
|}

== Moderation ==
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 just under 15 minutes. 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.

Since the [[kinetic energy]], <math>E</math>, can be related to [[temperature]] via:

<math>E=\frac{1}{2}mv^2=\frac{3}{2}k_B T</math>

the characteristic [[neutron temperature]] of a several-MeV neutron is several tens of millions of degrees [[Celsius]].

Moderation is the process of the reduction of the initial high kinetic energy of the free neutron. Since energy is conserved, this reduction of the neutron kinetic energy takes place by transfer of energy to a material known as a moderator. It is also known as ''neutron slowing down'', since along with the reduction of energy comes a reduction in speed.

The probability of scattering of a neutron from a nucleus is given by the [[nuclear cross section|scattering cross section]]. The first couple of 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 [[Momentum#Special case: m1.3Dm2|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 1H 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
| authorlink =
| title = Nuclear reactor physics
| publisher = [[Wiley-VCH]]
| year = 2007
| location =
| pages = 29–31
| url = http://books.google.com/books?id=iolyNyJYEaYC
| isbn = 3-527-40679-4}}</ref>

This can be reasonably approximated to the very simple form <math>\xi\simeq \frac{2}{A+1}</math>.<ref name="
DB">{{cite book |last= Dobrzynski |first= L. |coauthors= 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</math>:<math> n=\frac{1}{\xi}(\ln E_0-\ln E)</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]].]]

===Choice of moderator materials===
Some nuclei have larger [[absorption cross section]]s 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 H2O and D2O, 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 kernels [[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 once moderated===
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.

==Reactor moderators==
In a [[thermal reactor|thermal nuclear reactor]], the nucleus of a heavy fuel element such as [[uranium]] absorbs a [[thermal neutron|slow-moving free neutron]], becomes unstable, and then splits ("[[Nuclear fission|fission]]s") into two smaller atoms ("[[fission product]]s"). The fission process for [[uranium-235|235U]] nuclei yields two fission products: two to three [[fast neutron|fast-moving free neutrons]], plus an amount of [[energy]] primarily manifested in the kinetic energy of the recoiling fission products. The free neutrons are emitted with a kinetic energy of ~2 MeV each. Because more [[free neutron]]s are released from a uranium fission event than thermal neutrons are required to initiate the event, the reaction can become self-sustaining — a [[chain reaction]] — under controlled conditions, thus liberating a tremendous amount of energy (see article [[nuclear fission]]).

[[Image:U235 Fission cross section.png|thumb|left|450px|[[Fission cross section]], measured in [[barn (unit)|barns]] (a unit equal to 10−28 m2), is a function of the energy (so-called [[excitation function]]) of the neutron colliding with a 235U nucleus. Fission probability decreases as neutron energy (and speed) increases. This explains why most reactors fueled with 235U need a moderator to sustain a chain reaction and why removing a moderator can shut down a reactor.]]

The probability of further fission events is determined by the [[nuclear cross section|fission cross section]], which is dependent upon the speed (energy) of the incident neutrons. For thermal reactors, high-energy neutrons in the MeV-range are much less likely to cause further fission. (Note: It is not ''impossible'' for fast neutrons to cause fission, just much less likely.) The newly released fast neutrons, moving at roughly 10% of the [[speed of light]], must be slowed down or "moderated," typically to speeds of a few kilometres per second, if they are to be likely to cause further fission in neighbouring [[uranium-235|235U]] nuclei and hence continue the chain reaction. This speed happens to be equivalent to temperatures in the few hundred celsius range.

In all moderated reactors, some neutrons of all energy levels will produce fission, including fast neutrons. Some reactors are more fully ''thermalised'' than others; for example, in a [[CANDU reactor]] nearly all fission reactions are produced by thermal neutrons, while in a [[pressurized water reactor]] (PWR) a considerable portion of the fissions are produced by higher-energy neutrons. In the proposed water-cooled [[supercritical water reactor]] (SCWR), the proportion of fast fissions may exceed 50%, making it technically a [[fast neutron reactor]].

A [[fast reactor]] uses no moderator, but relies on fission produced by unmoderated fast neutrons to sustain the chain reaction. In some fast reactor designs, up to 20% of fissions can come from direct fast neutron fission of [[uranium-238]], an isotope which is not [[fissile]] at all with thermal neutrons.

Moderators are also used in non-reactor neutron sources, such as [[plutonium]]-[[beryllium]] and [[spallation]] sources.

== Form and location ==

The form and location of the moderator can greatly influence the cost and safety of a reactor. Classically, moderators were precision-machined blocks of high purity graphite with embedded ducting to carry away heat. They were in the hottest part of the reactor, and therefore subject to [[corrosion]] and [[ablation]]. In some materials, including [[graphite]], the impact of the neutrons with the moderator can cause the moderator to accumulate dangerous amounts of [[Wigner energy]]. This problem led to the infamous [[Windscale fire]] at the Windscale Piles, a nuclear reactor complex in the United Kingdom, in 1957.

Some [[pebble-bed reactor]]s' moderators are not only simple, but also inexpensive{{Citation needed|date=September 2009}}: the nuclear fuel is embedded in spheres of reactor-grade [[pyrolytic carbon]], roughly of the size of [[tennis ball]]s. The spaces between the balls serve as ducting. The reactor is operated above the Wigner annealing temperature so that the graphite does not accumulate dangerous amounts of [[Wigner energy]].

In [[CANDU]] and [[pressurized water reactor|PWR]] reactors, the moderator is liquid water ([[heavy water]] for CANDU, [[Water|light water]] for PWR). In the event of a [[loss-of-coolant accident]] in a PWR, the moderator is also lost and the reaction will stop. This negative [[void coefficient]] is an important safety feature of these reactors. In CANDU the moderator is located in a separate heavy-water circuit, surrounding the pressurized heavy-water coolant channels. This design gives CANDU reactors a positive [[void coefficient]], although the slower neutron kinetics of heavy-water moderated systems compensates for this, leading to comparable safety with PWRs."<ref>[http://www.nuclearfaq.ca/Meneley_Muzumbdar_reactivity_review_CNS2009.pdf D.A. Meneley and A.P. Muzumdar, "Power Reactor Safety Comparison - a Limited Review", Proceedings of the CNS Annual Conference, June 2009]</ref>

==Moderator impurities==

Good moderators are also free of neutron-absorbing impurities such as [[boron]]. In commercial nuclear power plants the moderator typically contains dissolved boron. The boron concentration of the reactor coolant can be changed by the operators by adding boric acid or by diluting with water to manipulate reactor power. The German World War II nuclear program suffered a substantial setback when its inexpensive graphite moderators failed to work. At that time, most graphites were deposited on boron electrodes, and the German commercial graphite contained too much boron. Since the war-time German program never discovered this problem, they were forced to use far more expensive [[heavy water]] moderators. In the U.S., [[Leó Szilárd]], a former chemical engineer, discovered the problem.

== Non-graphite moderators ==

Some moderators are quite expensive, for example [[beryllium]], and reactor-grade heavy water. Reactor-grade heavy water must be 99.75% pure to enable reactions with unenriched uranium. This is difficult to prepare because heavy water and regular water form the same [[chemical bond]]s in almost the same ways, at only slightly different speeds.

The much cheaper light water moderator (essentially very pure regular water ) absorbs too many neutrons to be used with unenriched natural uranium, and therefore [[uranium enrichment]] or [[nuclear reprocessing]] becomes necessary to operate such reactors, increasing overall costs. Both enrichment and reprocessing are expensive and technologically challenging processes, and additionally both enrichment and several types of reprocessing can be used to create weapons-usable material, causing proliferation concerns. Reprocessing schemes that are more resistant to proliferation are currently under development.

The [[CANDU]] reactor's moderator doubles as a safety feature. A large tank of low-temperature, low-pressure heavy water moderates the neutrons and also acts as a heat sink in extreme [[loss of coolant|loss-of-coolant accident]] conditions. It is separated from the fuel rods that actually generate the heat. Heavy water is very effective at slowing down (moderating) neutrons, giving CANDU reactors their important and defining characteristic of high "neutron economy."

== Nuclear weapon design ==
{{Main|Uranium hydride bomb}}
Early speculation about [[nuclear weapon]]s assumed that an "atom bomb" would be a large amount of [[fissile]] material, moderated by a neutron moderator, similar in structure to a [[nuclear reactor]] or "pile".<ref>[http://nuclearweaponarchive.org/Nwfaq/Nfaq8.html#nfaq8.2.1 Nuclear Weapons Frequently Asked Questions - 8.2.1 Early Research on Fusion Weapons]</ref> Only the [[Manhattan project]] embraced the idea of a [[chain reaction]] of [[fast neutron]]s in pure metallic [[uranium]] or [[plutonium]]. Other moderated designs were also considered by the Americans; proposals included [[Uranium hydride bomb|using uranium hydride]] as the fissile material.<ref name="upshot">[http://www.nuclearweaponarchive.org/Usa/Tests/Upshotk.html Operation Upshot-Knothole]</ref><ref name="globalsecurity">[http://www.globalsecurity.org/wmd/systems/w48.htm W48] - globalsecurity.org</ref> In 1943 [[Robert Oppenheimer]] and [[Niels Bohr]] considered the possibility of using a "pile" as a weapon.<ref>[http://www.ask.ne.jp/~hankaku/english/np5y.html Atomic Bomb Chronology: 1942-1944]</ref> The motivation was that with a [[graphite]] moderator it would be possible to achieve the chain reaction without the use of any [[isotope separation]]. In August 1945, when information of the [[Atomic bombings of Hiroshima and Nagasaki|atomic bombing of Hiroshima]] was relayed to the scientists of the [[German nuclear program]], interned at Farm Hall in England, chief scientist [[Werner Heisenberg]] hypothesized that the device must have been "something like a nuclear reactor, with the neutrons slowed by many collisions with a moderator."<ref>[[Hans Bethe]] in ''[[Physics Today]]'' Vol 53 (2001) [http://www.nd.edu/~nsl/Lectures/phys205/pdf/Nuclear_warfare_3.pdf]</ref>

After the success of the Manhattan project, all major [[:Category:Nuclear weapons programs|nuclear weapons programs]] have relied on fast neutrons in their weapons designs. The notable exception is the ''[[Upshot-Knothole Ruth|Ruth]]'' and ''[[Upshot-Knothole Ray|Ray]]'' test explosions of [[Operation Upshot-Knothole]]. The aim of the [[University of California Radiation Laboratory]] design was to produce an explosion powerful enough to ignite a [[thermonuclear weapon]] with the minimal amount of fissile material. The [[Nuclear reactor core|core]] consisted of [[uranium hydride]], with [[hydrogen]], or in the case of ''Ray'', [[deuterium]] acting as the neutron moderator. The predicted [[Nuclear weapon yield|yield]] was 1.5 to 3 kt for ''Ruth'' and 0.5-1 kt for ''Ray''. The tests produced yields of 200 [[tons of TNT]] each; both tests were considered to be [[fizzle (nuclear test)|fizzles]].<ref name="upshot" /><ref name="globalsecurity" />

The main benefit of using a moderator in a nuclear explosive is that the amount of fissile material needed to reach [[Criticality (status)|criticality]] may be greatly reduced. Slowing of fast neutrons will increase the [[Nuclear cross section|cross section]] for [[neutron absorption]], reducing the [[critical mass]]. A side effect is however that as the chain reaction progresses, the moderator will be heated, thus losing its ability to cool the neutrons.

Another effect of moderation is that the time between subsequent neutron generations is increased, slowing down the reaction. This makes the containment of the explosion a problem; the [[inertia]] that is used to confine [[Nuclear weapon design#Implosion type weapon|implosion type]] bombs will not be able to confine the reaction. The end result may be a fizzle instead of a bang.

The explosive power of a fully moderated explosion is thus limited, at worst it may be equal to a chemical explosive of similar mass. Again quoting Heisenberg: ''"One can never make an explosive with slow neutrons, not even with the heavy water machine, as then the neutrons only go with thermal speed, with the result that the reaction is so slow that the thing explodes sooner, before the reaction is complete."''

While a nuclear bomb working on [[thermal neutron]]s may be impractical, modern weapons designs may still benefit from some level of moderation. A [[beryllium]] tamper used as a [[neutron reflector]] will also act as a moderator.<ref>[http://nuclearweaponarchive.org/Nwfaq/Nfaq4-1.html#Nfaq4.1.7.3 Nuclear Weapons Frequently Asked Questions - 4.1.7.3.2 Reflectors]</ref><ref name="killus">[http://unintentional-irony.blogspot.com/2007/07/n-moderation.html N Moderation]</ref>

==Materials used==
* [[Hydrogen]], as in ordinary "[[Water|light water]]." Because [[Hydrogen-1|protium]] also has a significant [[Neutron cross section|cross section]] for [[neutron capture]] only limited moderation is possible without losing too many neutrons. The less-moderated neutrons are relatively more likely to be captured by [[uranium-238]] and less likely to fission [[uranium-235]], so [[light water reactor]]s require [[enriched uranium]] to operate.
** There are also proposals to use the compound formed by the chemical reaction of metallic uranium and hydrogen ([[uranium hydride]]—UH3) as a combination fuel and moderator in [[Hydrogen Moderated Self-regulating Nuclear Power Module|a new type of reactor]].
** Hydrogen is also used in the form of cryogenic liquid [[methane]] and sometimes [[liquid hydrogen]] as a [[cold neutron]] source in some [[research reactor]]s: yielding a [[Maxwell–Boltzmann distribution]] for the neutrons whose maximum is shifted to much lower energies.
** Hydrogen combined with carbon as in [[paraffin wax]] was used in some early [[German nuclear energy project|German experiments]].
* [[Deuterium]], in the form of [[heavy water]], in [[heavy water reactor]]s, e.g. [[CANDU]]. Reactors moderated with heavy water can use unenriched [[natural uranium]].
* [[Carbon]], in the form of reactor-grade [[graphite]] or [[pyrolytic carbon]], used in e.g. [[RBMK]] and [[pebble-bed reactor]]s, or in compounds, e.g. [[carbon dioxide]] [http://www.bookrags.com/research/carbon-dioxide-chmc]. Lower-temperature reactors are susceptible to buildup of [[Wigner energy]] in the material. Like deuterium-moderated reactors, some of these reactors can use unenriched natural uranium.
** Graphite is also deliberately allowed to be heated to around 2000 K or higher in some [[research reactor]]s to produce a [[neutron temperature|hot neutron]] source: giving a [[Maxwell–Boltzmann distribution]] whose maximum is spread out to generate higher energy neutrons.
* [[Beryllium]], in the form of metal. Beryllium is expensive and toxic, so its use is limited.
* [[Lithium]]-7, in the form of a [[lithium fluoride]] salt, typically in conjunction with [[beryllium fluoride]] salt ([[FLiBe]]). This is the most common type of moderator in a [[Molten Salt Reactor]].

Other light-nuclei materials are unsuitable for various reasons. [[Helium]] is a gas and it requires special design to achieve sufficient density; [[lithium]]-6 and [[boron]]-10 absorb neutrons.

==References==
* {{cite book | title = DOE Fundamentals Handbook: Nuclear Physics and Reactor Theory. Vol. 2 (DOE-HDBK-1019/2-93) | date = January 1993 | publisher = [[U.S. Department of Energy]] | url = http://energy.gov/sites/prod/files/2013/06/f2/h1019v2.pdf | accessdate = November 29, 2013}}

=== Notes ===
{{reflist}}

==See also==
*[[Nuclear cross section]]
*[[Neutron reflector]]
{{Nuclear technology}}

{{DEFAULTSORT:Neutron Moderator}}
[[Category:Nuclear technology]]
[[Category:Neutron instrumentation|Moderator]]
[[Category:Neutron moderators| ]]