Maria Physicist January 14, 1999

Introduction Due to the statistical nature of ionisation energy loss, large fluctuations can occur in the amount of energy deposited by a particle traversing an absorber element. Continuous processes such as multiple scattering and energy loss play a relevant role in the longitudinal and lateral development of electromagnetic and hadronic showers, and in the case of sampling calorimeters the measured resolution can be significantly affected by such fluctuations in their active layers. The description of ionisation fluctuations is characterised by the significance parameter $\kappa$, which is proportional to the ratio of mean energy loss to the maximum allowed energy transfer in a single collision with an atomic electron $\kappa =\frac{\xi }{E_{\max }}$ $E_{\max }$ is the maximum transferable energy in a single collision with an atomic electron. ....

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 $E_{\max }=\infty$. Now we can write: $f\left(\epsilon ,,,\delta ,s\right)=\frac{1}{\xi }{\phi }_v\left({\lambda }_v,,,\kappa ,,,{\beta }^2\right)$ where ${\phi }_v\left({\lambda }_v,,,\kappa ,,,{\beta }^2\right)=\frac{1}{2\pi i}{\int }_{c+i\infty }^{c-i\infty }\phi \left(s\right)e^{\lambda s}dsc\ge 0$$$$\phi \left(s\right)=exp\left[\kappa ,(1+{\beta }^2\gamma )\right]exp\left[\psi ,\left(s\right)\right],$$$$\psi \left(s\right)=sln\kappa +(s+{\beta }^2\kappa )\left[ln,(s/\kappa ),+,E_1,(s/\kappa )\right]-\kappa e^{-s/\kappa },$ and $E_1\left(z\right)={\int }_{\infty }^z{t}^{-1}{e}^{-t}dt\text{(the exponential integral)}$$$${\lambda }_v=\kappa \left[\frac{\epsilon -\underset{}{\overset{⌅}{\epsilon }}}{\xi },-,\gamma ,\prime ,-,{\beta }^2\right]$ The Vavilov parameters are simply related to the Landau parameter by ${\lambda }_L={\lambda }_v/\kappa -ln\kappa$. 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).

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References L.Landau. On the Energy Loss of Fast Particles by Ionisation. Originally published in J. Phys., 8:201, 1944. Reprinted in D.ter Haar, Editor, L.D.Landau, Collected papers, page 417. Pergamon Press, Oxford, 1965. B.Schorr. Programs for the Landau and the Vavilov distributions and the corresponding random numbers. Comp. Phys. Comm., 7:216, 1974. S.M.Seltzer and M.J.Berger. Energy loss straggling of protons and mesons. In Studies in Penetration of Charged Particles in Matter, Nuclear Science Series 39, Nat. Academy of Sciences, Washington DC, 1964. R.Talman. On the statistics of particle identification using ionization. Nucl. Inst. Meth., 159:189, 1979. P.V.Vavilov. Ionisation losses of high energy heavy particles. Soviet Physics JETP, 5:749, 1957.