Approximations in use

A charged particle traversing a medium is deflected by many scatters, mostly at small angle. These scatters are due to the interaction with the Coulomb field of the nuclei, and to a lesser degree, to the electron field, hence the name of Coulomb scattering. For hadronic projectiles, however, the strong interaction contributes to multiple scattering. Multiple scattering is well described by Molière theory []. Molière multiple scattering theory is used by default in GEANT (see [PHYS325]). We define $\theta$₀= θ_plane^rms= θ_space^rms/2 as the r.m.s. of the angle between the directions projected on a plane of a particle before and after traversing a thickness t of absorber. In this case a simple form for the multiple Coulomb scattering of singly charged particles which is widely used is:

$\theta$₀≈{15 MeVE β²}{tX₀}

where $X$₀ is the radiation length of the absorber. This form was proposed by [], []. It can introduce large errors because it ignores significant dependences from pathlength and Z. To compensate for this, a similar formula was proposed [], []:

$\theta$₀≈{14.1 MeVE β²}{tX₀}[ 1 + 0.038 log( {tX₀}) ]

A form taking into account the $\beta$ and z dependence of the particle has been proposed by []:

$\theta$₀= {S₂E β²}{tX₀}[ 1 + εlog₁₀( {t Z_inc²X₀β²}) ]

The problem with the formulae () and () is that if the distance t in the absorber is travelled in two steps $t\; =\; t$₁+ t₂ , the r.m.s. angle $\theta$₀(t) = θ₀(t₁+t₂) ≠θ₀²(t₁) + θ₀²(t₂) , limiting their use in a MonteCarlo like GEANT.

Much of the difficulty in approximating multiple Coulomb scattering in terms of the radiation length is that the number of radiation lengths is a poor measure of the scattering. A better expression, which is used in GEANT has been proposed by []:

$\theta$₀²= {χ_c²1+F²}[{1+νν}log( 1+ν) -1 ]

where

$\nu$	$=$	$\{\Omega$02 ( 1-F)}
$F$	$=$	$fraction\; of\; the\; tracks\; in\; the\; sample$
$\Omega$0	$=$	$\{\chi$c21.167 χα2}&sp;is the mean number of scatters

$\chi$_c² and $\chi$_α² are parameters of the Molière theory, for which the reader is invited to see [PHYS325], and $1.167\; \approx exp(2\gamma -1)$ where $\gamma$ is the Euler's constant $=\; 0.57721\; ...$ . This form, which is the result of empirical fits, is derived from the screened Rutherford cross section, which has the form $2\chi$_α²/(χ_α²+θ₀²) . For F anywhere from $0.9$ to $0.995$ this expression represents Molière scattering to better than $2$ for ₀< 10⁸ , which covers singly charged relativistic particles ($\beta \approx 1$ ) in the range ₀< 100 .

In GEANT we have adopted the values of $F=0.98$ and $\Omega$₀= 40,000

scatters, obtaining the following formula:

$\theta$₀= 2.557 χ_cc{tE β²}

where $\chi$_cc is another parameter of the Molière theory for which the reader is referred to [PHYS325]. This algorithm is implemented by the routine GMGAUS.

F.Carminati

PHYS325