The role of the Boltzmann transport equation in radiation damage calculations

Williams, M.M.R.

doi:10.1016/0149-1970(79)90004-0

Cited by 67 publications

(5 citation statements)

References 57 publications

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“…Besides the evident motivation of this choice based on our model, this is a standard way to avoid singularities in calculating total cross sections (see e.g. Williams (1979), section 5.1). The upper limit of integration is due to the fact that the primary electron has larger energy than the secondary electron and that the binding energy B was introduced into the scattering processes (usually the upper limit is e /2).…”

Section: Discussionmentioning

confidence: 99%

Deterministic model for dose calculation in photon radiotherapy

2006

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We present a model for dose calculation in photon radiotherapy based on deterministic transport equations. The model consists of two coupled equations, one for photon and one for electron transport and an equation for the absorbed dose. No assumptions are made with respect to the geometry or the homogeneity of the irradiated medium, so irradiation of any heterogenous medium can be simulated. To get a mathematically simpler model, approximations to the exact equations are presented which keep the essential physical contents of the exact equations, but which can be solved with less numerical effort. The results of dose calculations for simple examples are presented and discussed.

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Section: Discussionmentioning

confidence: 99%

Deterministic model for dose calculation in photon radiotherapy

2006

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“…It is obvious from the derivation of the forward and the backward equations from a common general equation, Equation (2), that their solutions are identical. Some conditions that are necessary to be fulfilled for this equivalence are discussed in [7].…”

Section: Backward Master Equationmentioning

confidence: 99%

“…The corresponding equations are different, even if their solutions, at the level of the Green's function, are identical. The relationship between the forward and backward equations has been discussed substantially in the literature [6][7][8][9].…”

Section: Introductionmentioning

confidence: 99%

Symmetries and Asymmetries in Branching Processes

Pázsit

2023

Symmetry

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As is known in stochastic particle theory, the same random process can be described by two different master equations for the evolution of the probability density, namely, by a forward or a backward master equation. These are the generalised analogues of the direct and adjoint equations of traditional transport theory. At the level of the first moment, these two equations show considerable resemblance to each other, but they become increasingly different with increasing moment order. The purpose of this paper is to demonstrate this increasing asymmetry and to discuss the underlying reasons. It is argued that since the reason of the different forms of the forward and the backward equations lies in the lack of invariance of the process to time reversal, the reason for the increasing asymmetry between the two forms for higher-order moments or processes with several variables (particle types) can be related to the increasing level of the violation of the invariance to time reversal, as is illustrated with some examples.

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“…23 The Method of Moments has been used in the nuclear engineering community to predict gamma-ray and neutron penetration in shields 24 and radiation damage in solids. 25 In this paper, we present the Method of Moments as an approximate electron dose calculation algorithm that overcomes the approximations of the Fermi-Eyges theory listed above, and that is less computationally intensive than algorithms based on the direct solution of the transport equation.…”

Section: Introductionmentioning

confidence: 99%

“…This procedure is algebraically equivalent to earlier work on the Method of Moments. [18][19][20][21][22][23][24][25] ͒ ͑2͒ The spatial moments of the fluence are used to represent accurately the scalar fluence and the dose. ͑See Sec.…”

Section: Introductionmentioning

confidence: 99%

Electron dose calculations using the Method of Moments

et al. 1997

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The Method of Moments is generalized to predict the dose deposited by a prescribed source of electrons in a homogeneous medium. The essence of this method is (i) to determine, directly from the linear Boltzmann equation, the exact mean fluence, mean spatial displacements, and mean-squared spatial displacements, as functions of energy; and (ii) to represent the fluence and dose distributions accurately using this information. Unlike the Fermi-Eyges theory, the Method of Moments is not limited to small-angle scattering and small angle of flight, nor does it require that all electrons at any specified depth z have one specified energy E(z). The sole approximation in the present application is that for each electron energy E, the scalar fluence is represented as a spatial Gaussian, whose moments agree with those of the linear Boltzmann solution. Numerical comparisons with Monte Carlo calculations show that the Method of Moments yields expressions for the depth-dose curve, radial dose profiles, and fluence that are significantly more accurate than those provided by the Fermi-Eyges theory.

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The role of the Boltzmann transport equation in radiation damage calculations

Cited by 67 publications

References 57 publications

Deterministic model for dose calculation in photon radiotherapy

Deterministic model for dose calculation in photon radiotherapy

Symmetries and Asymmetries in Branching Processes

Electron dose calculations using the Method of Moments

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