Examples of the Crystal Ball function. The Crystal Ball function , named after the Crystal Ball Collaboration (hence the capitalized initial letters), is a probability density function (PDF) commonly used to model various lossy processes in high-energy physics such as Bremsstrahlung by electrons. It consists of a Gaussian core portion and a power-law low-end tail, below a certain threshold. The function itself and its first derivative are both continuous .
The Crystal Ball function is given by:
f ( x ; α , n , x ¯ , σ ) = N ⋅ { exp ( − ( x − x ¯ ) 2 2 σ 2 ) , for x − x ¯ σ > − α A ⋅ ( B − x − x ¯ σ ) − n , for x − x ¯ σ ⩽ − α , {\displaystyle 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 \\A\cdot (B-{\frac {x-{\bar {x}}}{\sigma }})^{-n},&{\mbox{for }}{\frac {x-{\bar {x}}}{\sigma }}\leqslant -\alpha \end{cases}},} where
A = ( n | α | ) n ⋅ exp ( − | α | 2 2 ) {\displaystyle A=\left({\frac {n}{\left|\alpha \right|}}\right)^{n}\cdot \exp \left(-{\frac {\left|\alpha \right|^{2}}{2}}\right)} ,B = n | α | − | α | {\displaystyle B={\frac {n}{\left|\alpha \right|}}-\left|\alpha \right|} ,N = 1 σ ( C + D ) {\displaystyle N={\frac {1}{\sigma (C+D)}}} ,C = n | α | ⋅ 1 n − 1 ⋅ exp ( − | α | 2 2 ) {\displaystyle C={\frac {n}{\left|\alpha \right|}}\cdot {\frac {1}{n-1}}\cdot \exp \left(-{\frac {\left|\alpha \right|^{2}}{2}}\right)} ,D = π 2 ( 1 + erf ( | α | 2 ) ) {\displaystyle D={\sqrt {\frac {\pi }{2}}}\left(1+\operatorname {erf} \left({\frac {\left|\alpha \right|}{\sqrt {2}}}\right)\right)} ,with the error function erf.
The parameters of the function (that are usually determined by a fit) are:
N {\displaystyle N} is a normalization factor (Skwarnicki 1986)α > 0 {\displaystyle \alpha >0} defines the point where the PDF changes from a power-law to a Gaussian distributionn > 1 {\displaystyle n>1} is the power of the power-law tailx ¯ {\displaystyle {\bar {x}}} and σ {\displaystyle \sigma } are the mean and the standard deviation of the GaussianThis page is based on this
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