&SIGNAL: AVALANCHE

EXPONENTIAL

The multiplication factors are distributed exponentially with a mean (and also an RMS) equal to "mean".

For large mean multiplications, significant deviations are observed, especially at the low end high ends of the distribution. These can empirically be modeled with the Polya distribution which can be obtained by using POLYA-FIXED or POLYA-TOWNSEND

Although the exponential distribution has a non-zero probability at 0, the multiplication factor used in Garfield is always equal to at least 1: if a factor less than 1 is drawn, it is rejected and a new number is drawn.

The mean multiplication factor has to be strictly positive.

[Default: you have to specify the mean multiplication.]

FIXED

Results in a fixed multiplication by the factor that you specify.

The factor has to be larger than or equal to 1, the latter being equivalent to specifying NONE.

[Default: none, you have to specify the multiplication factor.]

FIXED-TOWNSEND

The Townsend coefficient is integrated over the drift-line, and the integral is exponentiated yielding a multiplication factor.

The multiplications obtained with this option are constant for a given drift path. They may vary when the Monte Carlo drift line integration routines are used since these introduce variations in the drift path.

This option is accessible only if Townsend data has been entered in the gas section.

GAUSSIAN

Not a physical choice for a single avalanche but sometimes used when many exponentially distributed avalanches overlap, resulting actually in a \Γ-distribution.

Please note that the 2nd argument is a relative, not an absolute, standard deviation. If you use for instance

avalanche gaussian 20000 0.5

then you'll end up with a normally distributed multiplication with a mean of 20000 and a sigma of 10000.

The Gaussian distribution has non-zero probability for factors less than 0. Since the multiplication factor as defined here has to be equal to 1 at least, Garfield will if needed repeatedly draw a random number until an acceptable number is found.

The mean is allowed to be negative, but the width must be strictly positive - a width of zero would be equivalent to using the FIXED option.

[Default: none, you have to specify both the mean and the relative standard deviation.]

NONE

States that there is no multiplication process.

POLYA-FIXED

The mean multiplication does not depend on the drift-line (you specify the number as first argument) but the fluctuations are according to a Polya distribution with parameter \θ.

The Polya distribution is a \Γ-distribution that matches reasonably well the fluctuations in a cylindrically symmetric amplification region. A physical interpretation of the parameter \θ is given in G.\ D.\ Alkhazov, NIM\ 89 (1970) 155-165.

When \θ is set to 0, an exponential distribution is obtained. If \θ\ <\ 0, the distribution is "concave", while for \θ\ >\ 0 the distribution assumes the more usual shape with a maximum.

The properties of this distribution can be examined using the RND_POLYA function, as shown in the following example:

Say "Please enter theta"
Parse Terminal theta
Call book_histogram(ref,100,0.0,5.0)
For i From 1 To 50000 Do
   Call fill_histogram(ref,rnd_polya(theta))
Enddo
!options log-y
Call plot_histogram(ref,`Multiplication`,`Polya distribution`)
Call plot_end

Although the Polya distribution has a non-zero probability at 0, the multiplication factor used in Garfield is always equal to at least 1. If a factor less than 1 is drawn, it is rejected and a new number is drawn.

The mean multiplication factor has to be strictly positive,

[The default value for the mean multiplication is 1, the default setting of the \θ parameter is 0.5.]

POLYA-TOWNSEND

The mean multiplication is obtained by integrating the Townsend coefficient over the current drift-line and the fluctuations are according to a Polya distribution with parameter \θ.

See POLYA-FIXED for comments on the Polya distribution.

Additional fluctuations are obtained when the Monte Carlo drift line integration routines are used since these introduce variations in the drift path.

This option is accessible only if Townsend data has been entered in the gas section.

[The default value for \θ is 0.5.]

TOWNSEND

The Townsend coefficient is integrated over the drift-line and the integral is exponentiated. The result is used as mean of an exponential distribution of multiplications.

This is similar to the EXPONENTIAL option, but here the mean of the distribution may be different for each ionisation cluster. Additional fluctuations are obtained when the Monte Carlo drift-line integration routines are used since these introduce variations in the drift path.

Although the exponential distribution has a non-zero probability at 0, the multiplication factor used in Garfield is always equal to at least 1. If a factor less than 1 is drawn, it is rejected and a new number is drawn.

This option is accessible only if Townsend data has been entered in the gas section.

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Formatted on 21/01/18 at 16:55.