Multiple scattering

L. Urbán

The G4MultipleScattering class simulates the multiple scattering of charged particles in material.It uses a new multiple scattering (MSC) model which does not use the Moliere formalism([Mol48]).This MSC model simulates the scattering of the particle after a given step , computes the mean path length correction and the mean lateral displacement as well.

Let us define a few notation first.

The true path length ('t' path length) is the total length travelled by the particle. All the physical processes restrict this 't' step.

The geometrical ( or 'z') path length is the straight distance between the starting and endpoint of the step , if there is no magnetic field. The geometry gives a constraint for this 'z' step. It should be noted, that the geometrical step length is meaningful in the case of magnetic field, too, but in this case it is a distance along a curved trajectory.

The mean properties of the multiple scattering process are determined by the transport mean free path , $\lambda$ , which is a function of the energy in a given material.Some of the mean properties - the mean lateral displacement and the second moment of $cos(\theta)$ - depend on the second transport mean free path, too. (The transport mean free path is called first transport mean free path as well.)

The 't' $\Rightarrow$ 'z' (true path length - geometrical path length) transformation is given by the simple equation

$\begin{displaymath} z = \lambda*(1.-exp(-t/\lambda)) \end{displaymath}$

(1)

which is an exact result for the mean values of z , if the differential cross section has an axial symmetry and the energy loss can be neglected . This formula and some other expressions for the first moments of the spatial distribution after a given 'true' path length t have been taken from the excellent paper of Fernandez-Varea et al. [Fer93], but the expressions have been calculated originally by Goudsmit and Saunderson [Goud40] and Lewis [Lew50]. Inverting eq. 1 the 'z' $\Rightarrow$ 't' transformation can be written as

$\begin{displaymath} t = -\lambda*ln(1.-z/\lambda) \end{displaymath}$

(2)

where $z < \lambda$ should be required (this condition is fulfilled if z has been computed from eq. 1).

The mean value of $cos(\theta)$ - $\theta$ is the scattering angle after a true step length t - is

$\begin{displaymath} <cos(\theta)> = exp(-t/\lambda) \end{displaymath}$

(3)

The transport mean free path values have been calculated by Liljequist et al. [Lil90],[Lil87] for electrons and positrons in the kinetic energy range $0.1 keV -- 20 MeV$ in 15 materials . The MSC model uses these values with an appropriate interpolation or extrapolation in the atomic number $Z$ and in the velocity of the particle $\beta$ , when it is necessary.

The quantity $cos(\theta)$ is sampled in the MSC model according to a model function $f(cos(\theta))$ . The shape of this function has been choosen in such a way, that $f(cos(\theta))$ reproduces the results of the direct simulation ot the particle transport rather well and eq. 3 is satisfied. The functional form of this model function is

$\begin{displaymath} f(x) = p \frac{(a + 1)^2 (a - 1)^2}{2 a} \frac{1}{(a-x)^3} + (1-p) \frac{1}{2} \end{displaymath}$

(4)

where $x= cos(\theta)$ , $0 \leq p \leq 1$ and $a > 1$ . The model parameters $p$ and $a$ depend on the path length t , the energy of the particle and the material.They are not independent parameters , they should satisfy the constraint

$\begin{displaymath} \frac{p}{a} = exp(-\frac{t}{\lambda}) \end{displaymath}$

(5)

which follows from eq. 3 .

The mean lateral displacement is given by a more complicated formula (see the paper [Fer93] ), but this quantity also can be calculated relatively easily and accurately.

It is worth to note that in this MSC model there is no step limitation originated from the multiple scattering process. Another important feature of this model that the total 'true' path length of the particle does not depend the length of the steps . Most of the algorithms used in simulations do not have these properties.

In the case of heavy charged particles ( $\mu,\pi,proton,etc.$ ) the mean transport free path is calculated from the $e+/e-$ $\lambda$ values with a 'scaling'.

In its present form the model computes and uses mean path length corrections and lateral displacements, the only random quantity is the scattering angle $\theta$ which is sampled according to the model function $f$ .

The G4MultipleScattering process has 'AlongStep' and 'PostStep' parts.

The AlongStepGetPhysicalInteractionLength function performs the
't' step $\Rightarrow$ 'z' step transformation . It should be called after the other physics GetPhysicalInteractionLength functions but before the GetPhysicalInteractionLength of the transportation process.The reason for this restriction is the following: The physics processes 'feel' the true path length travelled by the particle , the geometry (transport) uses the 'z' step length.If we want to compare the minimum step size coming from the physics with the constraint of the geometry, we have make the transformation.

The AlongStepDoIt function of the process performs the inverse, 'z' $\Rightarrow$ 't' transformation.This function should be called after the AlongStepDoIt of the transportation process , i.e. after the particle relocation determined by the geometrical step length, but before applying any other (physics) AlongStepDoIt.

The PostStepGetPhysicalInteractionLength part of the multiple scattering process is very simple , it sets the force flag to 'Forced' in order to ensure the call of the PostStepDoIt in every step and returns a big value as interaction length (that means that the multiple scattering process does not restrict the step size).

Bibliography

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