Fano factor
Encyclopedia
In statistics
, the Fano factor , like the coefficient of variation
, is a measure of the dispersion
of a probability distribution
of a Fano noise
. It is named after Ugo Fano
.
The Fano factor is defined as
where
is the variance and 
is the mean of a random process in some time window 
. The Fano factor can be viewed as a kind of noise-to-signal ratio; it is a measure of the reliability with which the random variable
could be estimated from a time window that on average contains several random events.
For a Poisson process
, the variance in the count equals the mean count, so
(normalization).
If the time window is chosen to be infinity, the Fano factor is similar to the variance-to-mean ratio (VMR) which in statistics
is also known as the Index of dispersion.
In particle detectors
, the Fano Factor results from the energy loss in a collision not being purely statistical. The process giving rise to each individual charge carrier is not independent as the number of ways an atom may be ionized is limited by the discrete electron shells. The net result is a better energy resolution than predicted by purely statistical considerations. For example, if
is the average energy for a particle to produce a charge carrier in a detector, then the FWHM
resolution for measuring the particle energy
is
.
The Fano factor is material specific. Some theoretical values are:
Measuring the Fano factor is difficult because many factors contribute to the resolution, but some experimental values are:
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, the Fano factor , like the coefficient of variation
Coefficient of variation
In probability theory and statistics, the coefficient of variation is a normalized measure of dispersion of a probability distribution. It is also known as unitized risk or the variation coefficient. The absolute value of the CV is sometimes known as relative standard deviation , which is...
, is a measure of the dispersion
Statistical dispersion
In statistics, statistical dispersion is variability or spread in a variable or a probability distribution...
of a probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
of a Fano noise
Fano noise
Fano noise is a fluctuation of an electric charge obtained in a detector , arising from processes in the detector....
. It is named after Ugo Fano
Ugo Fano
Ugo Fano was an Italian American physicist, a leader in theoretical physics in the 20th century.- Biography :Ugo Fano was born into a wealthy Jewish family in Turin, Italy...
.
The Fano factor is defined as
where
is the variance and is the mean of a random process in some time window . The Fano factor can be viewed as a kind of noise-to-signal ratio; it is a measure of the reliability with which the random variableRandom variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
could be estimated from a time window that on average contains several random events.
For a Poisson process
Poisson process
A Poisson process, named after the French mathematician Siméon-Denis Poisson , is a stochastic process in which events occur continuously and independently of one another...
, the variance in the count equals the mean count, so
(normalization).If the time window is chosen to be infinity, the Fano factor is similar to the variance-to-mean ratio (VMR) which in statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
is also known as the Index of dispersion.
In particle detectors
Particle detector
In experimental and applied particle physics, nuclear physics, and nuclear engineering, a particle detector, also known as a radiation detector, is a device used to detect, track, and/or identify high-energy particles, such as those produced by nuclear decay, cosmic radiation, or reactions in a...
, the Fano Factor results from the energy loss in a collision not being purely statistical. The process giving rise to each individual charge carrier is not independent as the number of ways an atom may be ionized is limited by the discrete electron shells. The net result is a better energy resolution than predicted by purely statistical considerations. For example, if
is the average energy for a particle to produce a charge carrier in a detector, then the FWHMFull width at half maximum
Full width at half maximum is an expression of the extent of a function, given by the difference between the two extreme values of the independent variable at which the dependent variable is equal to half of its maximum value....
resolution for measuring the particle energy
is.The Fano factor is material specific. Some theoretical values are:
| Si: | 0.115 |
| Ge: | 0.13 |
| GaAs: | 0.10 |
| Diamond: | 0.08 |
Measuring the Fano factor is difficult because many factors contribute to the resolution, but some experimental values are:
| Ar (gas): | 0.20 +/- 0.01/0.02 |
| Xe (gas): | 0.13 to 0.29 |
| CZT Cadmium zinc telluride Cadmium zinc telluride, or CZT, is a compound of cadmium, zinc and tellurium or, more strictly speaking, an alloy of cadmium telluride and zinc telluride. A direct bandgap semiconductor, it is used in a variety of applications, including radiation detectors, photorefractive gratings,... : |
0.089±0.005 |
See also
- Fano noiseFano noiseFano noise is a fluctuation of an electric charge obtained in a detector , arising from processes in the detector....
- Index of dispersion - equivalent to the Fano factor for an infinite time window
- Ugo FanoUgo FanoUgo Fano was an Italian American physicist, a leader in theoretical physics in the 20th century.- Biography :Ugo Fano was born into a wealthy Jewish family in Turin, Italy...