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Review of the Monte Carlo Method (MCM)
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Although the Monte Carlo method (MCM) is seen as the gold standard in the dose calculation, its application is limited by the computational effort. This is linked to the large number of particles that must be simulated in order to reduce the numerical uncertainty inherent in the complexity of the system. In this review the method is described and the problem of numerical uncertainty is reviewed.

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Random Walk
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To explore the properties of Monte Carlo suppose that we want to simulate the behavior of a drunked.

He moves unidimensionally and can take steps to the right and to the left.

The distances traveled in each direction depend on the objects along the way. These are distributed randomly.

Each time he reaches an object, he reverses the direction in which he moves.

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Random walk with variable pitch
Equation

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The simplest case is that of a particular movimg along an axis that can impact some object, after which it will reverse its direction of advance.

If the probability of reaching a distance between x and x+dx is P(x), the probability of impact will be equal to

P(x)-P(x+dx)\sim-\displaystyle\frac{dP}{dx}dx

If this probability is proproposal to the probability itself P, one has

\displaystyle\frac{dP}{dx}=-\displaystyle\frac{dx}{\lambda}P(x)

and get the probability function

$p(x)dx = \displaystyle\frac{1}{\lambda}e^{-x/\lambda}dx$

We will call \lambda the free path.

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Simulador random walk variable pitch
Php

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In order to obtain the distribution of the particulars according to the position, it is possible to perform an iteration in which

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0. A starting position and direction is defined

1. It is displaced by a distance generated randomly as a function of the distance probability in a direction

2. the direction is reversed

3. continued in 1

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If we assume that we expect a definite time and that the particle moves at constant speed, we can determine the position it has after a given time or after a definite total path.

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Summary
Description

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Playing with the simulator we noticed that

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1. It only makes sense to consider distributions of possible positions

2. The distribution is based on determining positions in discrete ranges

3. Ranges of smaller size require a greater number of iterations

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Total effective section and free path
Equation

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The total effective section \sigma is related to the effective section that the particle offers to the incident beam and thus directly affects the free path. If it is multiplied by the concentration of the particles c it can be shown that the free path is

$\lambda=\displaystyle\frac{1}{c\,\sigma}$

with which it is possible to estimate the probability of impact with the total effective section:

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Compton Scattering
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Compton scattering occurs when a photon interacts with a charged particle, in particular with an electron. In the process the photon loses energy and deviates by putting the electron in motion:

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Compton Scattering
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Compton scattering occurs when a photon interacts with an electron by transferring the first energy to the second (inelastic interaction). The wavelength of the photon after the scattering can be calculated by

$\lambda_2=\lambda+\lambda_c(1-\cos\theta)$



where

$\lambda_c=\displaystyle\frac{h}{m_ec}$

Compton wave length and \theta the angle of deviation of the photon is.

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Compton scattering differential effective section
Equation

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In the case of Compton scattering, the differential effective section is according to Klein-Nishina

$\displaystyle\frac{d\sigma_{KN}}{d\Omega}=\displaystyle\frac{3}{16\pi}\displaystyle\frac{\sigma_T}{(1+\epsilon(1-\cos\theta))^2}\left(\epsilon(1-cos\theta)+\displaystyle\frac{1}{1+\epsilon(1-\cos\theta)}-\cos^2\theta\right)$



where

$\sigma_T=\displaystyle\frac{8\pi}{3}r_0^2$



is the Thomson total effective section and the

$\epsilon=\displaystyle\frac{E}{m_ec^2}$

is the normalized energy.

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Total effective section for Compton scattering
Equation

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If the differential effective section is taken according to Klein-Nishina

$\displaystyle\frac{d\sigma_{KN}}{d\Omega}=\displaystyle\frac{3}{16\pi}\displaystyle\frac{\sigma_T}{(1+\epsilon(1-\cos\theta))^2}\left(\epsilon(1-cos\theta)+\displaystyle\frac{1}{1+\epsilon(1-\cos\theta)}-\cos^2\theta\right)$



and integrates in the solid angle

$d\Omega=2\pi \sin\theta d\theta$



the total effective section is obtained

$\sigma_{KN}=\displaystyle\frac{3}{4}\sigma_T\left(\displaystyle\frac{(1+\epsilon)}{\epsilon^3}\left(\displaystyle\frac{2\epsilon(1+\epsilon)}{1+2\epsilon}-\log(1+2\epsilon)\right)+\displaystyle\frac{\log(1+2\epsilon)}{2\epsilon}-\displaystyle\frac{(1+3\epsilon)}{(1+2\epsilon)^2}\right)$



where

$\sigma_T=\displaystyle\frac{8\pi}{3}r_0^2$



is the effective section of Thomson and the

$\epsilon=\displaystyle\frac{E}{m_ec^2}$

is the normalized energy.

At the limit of small \epsilon\ll1 we have that the total section is

\sigma_{KN}\sim\sigma_T\left(1-2\epsilon+\displaystyle\frac{26}{5}\epsilon^2\ldots\right)

and in the limit \epsilon\gg 1 the total effective section is

\sigma_{KN}\sim\displaystyle\frac{3}{8}\displaystyle\frac{\sigma_T}{\epsilon}\left(\log(2\epsilon)+\displaystyle\frac{1}{2}\right)

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Simulador random walk with Compton scattering
Php

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The Klein-Nishina model can be studied in numerical form. This is shown

- the total effective section as a function of photon energy

- the differential section as a function of the angle for the minimum, medium and maximum energies defined

- what would be the total effective section in a one-dimensional system that gives according to the energy transmission or reflection

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