The Crystal Ball function, named after the

Crystal Ball A crystal ball is a crystal or glass ball commonly used in fortune-telling. It is generally associated with the performance of clairvoyance and scrying through crystal gazing. Used since Antiquity, crystal balls have had a broad reputation with ...

Collaboration (hence the capitalized initial letters), is a

probability density function In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...

(PDF) commonly used to model various lossy processes in

high-energy physics Particle physics or high-energy physics is the study of fundamental particles and forces that constitute matter and radiation. The field also studies combinations of elementary particles up to the scale of protons and neutrons, while the stu ...

such as

Bremsstrahlung In particle physics, bremsstrahlung (; ; ) is electromagnetic radiation produced by the deceleration of a charged particle when deflected by another charged particle, typically an electron by an atomic nucleus. The moving particle loses kinetic ...

by electrons. It consists of a

Gaussian Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...

core portion and a

power-law In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a relative change in the other quantity proportional to the change raised to a constant exponent: one quantity var ...

low-end tail, below a certain threshold. The function itself and its first

derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...

are both

continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...

. The Crystal Ball function is given by: :

f(x;\alpha,n,\bar x,\sigma) = N \cdot \begin \exp(- \frac), & \mbox\frac > -\alpha \\
 A \cdot (B - \frac)^, & \mbox\frac \leqslant -\alpha \end ,

where :

A = \left(\frac\right)^n \cdot \exp\left(- \frac \right)

, :

B = \frac  - \left,  \alpha \

, :

N = \frac

, :

C = \frac \cdot \frac \cdot \exp\left(- \frac \right)

, :

D = \sqrt \left(1 + \operatorname\left(\frac\right)\right)

, with the

error function In mathematics, the error function (also called the Gauss error function), often denoted by , is a function \mathrm: \mathbb \to \mathbb defined as: \operatorname z = \frac\int_0^z e^\,\mathrm dt. The integral here is a complex Contour integrat ...

erf. The parameters of the function (that are usually determined by a fit) are: *

N

is a normalization factor (Skwarnicki 1986) *

\alpha>0

defines the point where the PDF changes from a power-law to a Gaussian distribution *

n>1

is the power of the power-law tail *

\bar x

and

\sigma

are the

mean A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...

and the

standard deviation In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...

of the Gaussian

External links

* J. E. Gaiser
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.) * M. J. Oreglia
A Study of the Reactions psi prime --> gamma gamma psi, Ph.D. Thesis
SLAC-R-236 (1980), Appendix D. * T. Skwarnicki
A study of the radiative CASCADE transitions between the Upsilon-Prime and Upsilon resonances, Ph.D Thesis
DESY F31-86-02(1986), Appendix E. {{DEFAULTSORT:Crystal Ball Function Functions and mappings Continuous distributions Experimental particle physics