Vavilov theory

  Vavilov [] derived a more accurate straggling distribution by introducing the kinematic limit on the maximum transferable energy in a single collision, rather than using $E$_max= &inf; . Using the notations of [] we can write:

$f\; (\; \epsilon ,\; \delta s\; )$

$=$

$\{1\xi \}\phi$v( λv, κ, β2)

where

$\phi$v( λv, κ, β2)	$=$	$\{12\; \pi i\}\int$c-i&inf;c+i&inf;φ( s ) eλsds &sp;c ≥0
$\phi (\; s\; )$	$=$	$exp[\; \kappa (\; 1\; +\; \beta 2\gamma )\; ]\; exp[\; \psi (\; s\; )\; ],$
$\psi (\; s\; )$	$=$	$s\; ln\kappa +\; (\; s\; +\; \beta 2\kappa )[\; ln(s/\kappa )\; +\; E$1(s/κ) ]- κe-s/κ,

and

$E$1(z)	$=$	$\int$z&inf;t-1e-tdt &sp;(the exponential integral)
$\lambda$v	$=$	$\kappa [\; \{\epsilon -\epsilon \xi \}-\; \gamma \text{'}\; -\; \beta 2]$

The Vavilov parameters are simply related to the Landau parameter by $\lambda$_L= λ_v/κ- lnκ . It can be shown that as $\kappa \to 0$ , the distribution of the variable $\lambda$_L

approaches that of Landau. For $\kappa \le 0.01$ the two distributions are already practically identical. Contrary to what many textbooks report, the Vavilov distribution does not approximate the Landau distribution for small $\kappa$ , but rather the distribution of $\lambda$_L defined above tends to the distribution of the true $\lambda$ from the Landau density function. Thus the routine GVAVIV samples the variable $\lambda$_L rather than $\lambda$_v . For $\kappa \ge 10$ the Vavilov distribution tends to a Gaussian distribution (see next section).