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&SIGNAL: AVALANCHE


EXPONENTIAL

The multiplication factors are approximately exponentially distributed provided the mean multiplication is "average".

For large fields, deviations are observed which can be modeled better with a Polya distribution.

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


FIXED

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

[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.


GAUSSIAN

Not a physical choice for a single avalanche but sometimes used when many exponentially distributed avalanches overlap, resulting in a gamma distribution - which is similar in shape to a Gaussian.

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 normally distributed multiplication with mean 20000 and a sigma of 10000.

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


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 theta.

The Polya distribution is a gamma distribution that matches reasonably well the fluctuations in a cylindrically symmetric amplication region. The meaning of the parameter theta is described in the paper by G. D. Alkhazov, NIM 89 (1970) 155-165.

When theta is set to 0, an exponential distribution is obtained.

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


POLYA-TOWNSEND

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

The Polya distribution is a gamma distribution that matches reasonably well the fluctuations in a cylindrically symmetric amplication region. The meaning of the parameter theta is described in the paper by G. D. Alkhazov, NIM 89 (1970) 155-165.

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

The shapes can be examined with the following statements:

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
!opt log-y
Call plot_histogram(ref,`Multiplication`,`Polya distribution`)
Call plot_end

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

[The default value for theta 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 cluster.

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


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Formatted on 0099-12-08 at 15:52.