Penelope Model

Total cross section

The total cross section of the Rayleigh scattering process is determined from an analytical parameterization. The atomic cross section for coherent scattering is given approximately by [Bor69]

(2)\[\sigma(E) = \pi r_{e}^{2} \int_{-1}^{1} \frac{1+\cos^{2}\theta}{2} [F(q,Z)]^{2} \ d \cos\theta,\]

where \(F(q,Z)\) is the atomic form factor, \(Z\) is the atomic number and \(q\) is the magnitude of the momentum transfer, i.e.

\[q \ = \ 2 \ \frac{E}{c} \ \sin \Big( \frac{\theta}{2} \Big).\]

In the numerical calculation the following analytical approximations are used for the form factor:

\[\begin{split}\begin{array}{rlll} F(q,Z) = f(x,Z) = & & & \\ & Z \ \frac{1+a_{1}x^{2}+a_{2}x^{3}+a_{3}x^{4}}{(1+a_{4}x^{2}+a_{5}x^{4})^{2}} & \mbox{or} & \\ & \max[f(x,Z),F_{K}(x,Z)] & \mbox{if} \ Z>10 \ \mbox{and} \ f(x,Z) < 2 & \\ % \begin{cases} % f(x,Z) = Z \ \frac{1+a_{1}x^{2}+a_{2}x^{3}+a_{3}x^{4}}{(1+a_{4}x^{2}+a_{5} % x^{4})^{2}} & \\ % \max[f(x,Z),F_{K}(x,Z)] & \textrm{if} \ Z>10 \ \textrm{and} \ % f(x,Z)<2\\ % \end{cases} \end{array}\end{split}\]

where

\[F_{K}(x,Z) = \frac{\sin(2b \arctan Q)}{bQ(1+Q^{2})^{b}},\]

with

\[x = 20.6074 \frac{q}{m_{e}c}, \quad Q = \frac{q}{2m_{e}ca}, \quad b = \sqrt{1-a^{2}}, \quad a = \alpha \Big( Z-\frac{5}{16} \Big ),\]

where \(\alpha\) is the fine-structure constant. The function \(F_{K}(x,Z)\) is the contribution to the atomic form factor due to the two K-shell electrons (see [eal94]). The parameters of expression \(f(x,Z)\) have been determined in Ref. [eal94] for \(Z=1\) to 92 by numerically fitting the atomic form factors tabulated in Ref. [eal75]. The integration of Eq.(2) is performed numerically using the 20-point Gaussian method. For this reason the initialization of the Penelope Rayleigh process is somewhat slower than the Low Energy Livermore process.

Form Factor for compounds and mixtures

In the case of compounds and mixtures, the form factor is calculated, by default, through a weighted sum of the atomic form factors of the elements composing the material and using their mass fractions as weights. However, this approach, which is called independent atom model (IAM) and is natively adoped by all the particle tracking codes, does not consider the inteference effect of the photons scattered by the bound electrons in molecules. As a consequence, the scattering pattern will not feature the peaks at small values of momentum transfer that characterize the considered material. To take into account the molecular interference effect (MI), form factors extracted from the measured scattering (diffraction) patterns can be used. A database of form factors including MI effect for a variety of materials, mainly biological tissue and plastics, is available. In order to use the files of the database, the user has to label the defined materials according to the following table:

material	label
fat	Fat_MI
water	Water_MI
collagen	BoneMatrix_MI
hydroxyapatite	Mineral_MI
PMMA	PMMA_MI
adipose	adipose_MI
glandular	glandular_MI
breast (50%fat + 50%water)	breast5050_MI
liver	liver_MI
kidney	kidney_MI
muscle	muscle_MI
heart	heart_MI
blood	blood_MI
bone	bone_MI
carcinoma	carcinoma_MI
white matter of brain	whiteMatter_MI
gray matter of brain	grayMatter_MI
fat (ext. to low q)	FatLowX_MI
collagen (ext. to low q)	BoneMatrixLowX_MI
dry bone (ext. to low q)	dryBoneLowX_MI
lexan	Lexan_MI
kapton	Kapton_MI
nylon	Nylon_MI
polyethylene	Polyethylene_MI
polystyrene	Polystyrene_MI
formaline	Formaline_MI
acetone	Acetone_MI
Hydrogen peroxide	Hperoxide_MI
CIRS30-70	CIRS30-70_MI
CIRS50-50	CIRS50-50_MI
CIRS70-30	CIRS70-30_MI
RMI454	RMI454_MI

Due to the tissue variability, the measured diffraction patterns of two samples of the same tissue type may differ significantly. To overcome this problem, a generic tissue can be decomposed in simpler basis materials with well-defined elemental composition. In particular, each tissue is considered as a composition of four components, namely fat, water, collagen or bone matrix, and hydroxyapatite (see [PaternoCC+18], [PaternoCGT20]). The form factor of a generic tissue can be then expressed through the mixture rule using tissues (molecules or supramolecules) instead of atoms as:

(3)\[\frac{F^2(q)}{W}=\sum_{i=1}^4\frac{a_iF_i^2(q)}{W_i},\]

where \(a_{i}\) is the mass fraction of i-th basis component.

In order to enable this functionality, the user has to create the mixture of basis materials (whose composition and density are defined in [PaternoCGT20] and label the material as MedMat_a1_a2_a3_a4, where ai are three digit numbers representing the mass fraction of the basis components. Then, the form factor of the material is automatically calculated according to Eq.(3).

In order to gain generality, the user has the faculty of providing the form factor of the materials he wants to consider. This is obtained by defining a material as G4ExtendedMaterial and registering G4MIData extension. The SetFilenameFF() method of this class allows to specify the path of the file with the form factor of the material. This functionality is particularly suited for modeling materials with partial crystalline beahviour, such as powder and polycrystals, which are characterize by a large number of sharp diffraction peaks.

Sampling of the final state

The angular deflection \(\cos\theta\) of the scattered photon is sampled from the probability distribution function

\[P(\cos\theta) = \frac{1+\cos^{2}\theta}{2} [F(q,Z)]^{2}.\]

For details on the sampling algorithm (which is quite heavy from the computational point of view) see Ref. [eal01]. The azimuthal scattering angle \(\phi\) of the photon is sampled uniformly in the interval \((0, 2\pi)\).

Bibliography

Bor69

M. Born. Atomic physics. Ed. Blackie and Sons, edition, 1969.

eal01

F. Salvat et al. Penelope - a code system for monte carlo simulation of electron and photon transport. Technical Report, Workshop Proceedings Issy-les-Moulineaux, France; AEN-NEA, 5-7 November 2001.

eal94(1,2)

J.Baró et al. Analytical cross sections for monte carlo simulation of photon transport. Radiat. Phys. Chem., 44():531, 1994.

eal75

J.H. Hubbel et al. Atomic form factors, incoherent scattering functions and photon scattering cross sections. Phys. Chem. Ref. Data, 4():471, 1975. Erratum: ibid. 6,615 (1977).

PaternoCC+18

G. Paternò, P. Cardarelli, A. Contillo, M. Gambaccini, and A. Taibi. Geant4 implementation of inter-atomic interference effect in small-angle coherent x-ray scattering for materials of medical interest. Physica Medica, 51:64–70, jul 2018. URL: https://doi.org/10.1016/j.ejmp.2018.04.395, doi:10.1016/j.ejmp.2018.04.395.

PaternoCGT20(1,2)

Gianfranco Paternò, Paolo Cardarelli, Mauro Gambaccini, and Angelo Taibi. Comprehensive data set to include interference effects in monte carlo models of x-ray coherent scattering inside biological tissues. Physics in Medicine & Biology, jul 2020. URL: https://doi.org/10.1088/1361-6560/aba7d2, doi:10.1088/1361-6560/aba7d2.