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| [[Image:CrystalBallFunction.svg|right|thumb|Examples of the Crystal Ball function.]]
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| The '''Crystal Ball function''', named after the [[Crystal Ball (detector)|Crystal Ball]] Collaboration (hence the capitalized initial letters), is a [[probability density function]] commonly used to model various [[lossy process]]es in [[high-energy physics]]. It consists of a [[Gaussian function|Gaussian]] core portion and a [[power-law]] low-end tail, below a certain threshold. The function itself and its first [[derivative]] are both [[Continuous function|continuous]].
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| The Crystal Ball function is given by:
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| :<math>f(x;\alpha,n,\bar x,\sigma) = N \cdot \begin{cases} \exp(- \frac{(x - \bar x)^2}{2 \sigma^2}), & \mbox{for }\frac{x - \bar x}{\sigma} > -\alpha \\
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| A \cdot (B - \frac{x - \bar x}{\sigma})^{-n}, & \mbox{for }\frac{x - \bar x}{\sigma} \leqslant -\alpha \end{cases}</math>
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| where
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| :<math>A = \left(\frac{n}{\left| \alpha \right|}\right)^n \cdot \exp\left(- \frac {\left| \alpha \right|^2}{2}\right)</math>,
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| :<math>B = \frac{n}{\left| \alpha \right|} - \left| \alpha \right|</math>,
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| :<math>N = \frac{1}{\sigma (C + D)}</math>
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| :<math>C = \frac{n}{\left| \alpha \right|} \cdot \frac{1}{n-1} \cdot \exp\left(- \frac {\left| \alpha \right|^2}{2}\right)</math>
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| :<math>D = \sqrt{\frac{\pi}{2}} \left(1 + \operatorname{erf}\left(\frac{\left| \alpha \right|}{\sqrt 2}\right)\right)</math> | |
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| <math>N</math> (Skwarnicki 1986) is a normalization factor and <math>\alpha</math>, <math>n</math>, <math>\bar x</math> and <math>\sigma</math> are parameters which are fitted with the data. erf is the [[error function]].
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| ==External links==
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| * J. E. Gaiser, [http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-r-255.pdf Appendix-F Charmonium Spectroscopy from Radiative Decays of the J/Psi and Psi-Prime, Ph.D. Thesis], SLAC-R-255 (1982). (This is a 205 page document in .pdf form – the function is defined on p. 178.)
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| * M. J. Oreglia, [http://www.slac.stanford.edu/pubs/slacreports/slac-r-236.html A Study of the Reactions psi prime --> gamma gamma psi, Ph.D. Thesis], SLAC-R-236 (1980), Appendix D.
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| * T. Skwarnicki, [http://inspirehep.net/record/230779/files/f31-86-02.pdf A study of the radiative CASCADE transitions between the Upsilon-Prime and Upsilon resonances, Ph.D Thesis], DESY F31-86-02(1986), Appendix E.
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| {{ProbDistributions|continuous-infinite}}
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| {{DEFAULTSORT:Crystal Ball Function}}
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| [[Category:Probability distributions]]
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| [[Category:Functions and mappings]]
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| [[Category:Continuous distributions]]
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