Awareness Ideas Blog

What is Energy Loss?

by admin on November 6, 2007

Energy loss is the action of transferring otherwise usable energy into a form that is no longer useable, either by losing control over the method the energy is transferred and converted, losing energy to waste as a function of that conversion or transfer or by the waste in conduction between mediums.

Want a technical look at the nature of energy loss? See this webpage…
http://wwwasd.web.cern.ch/wwwasd/geant/geant4_public/G4UsersDocuments/UsersGuides/PhysicsReferenceManual/html/node6.html

(text copied below for your convenience)
The energy loss processes are very similar for e+/e- , and charged hadrons , so it is quite natural to have a common description for them and we have chosen this way.

Energy loss of electrons/positrons

The G4eEnergyLoss class computes the continuous energy loss
of electrons/positrons. The continuous energy loss is calculated as a sum of the contribution of the different processes. At present there are two processes contributing to the continuous energy loss , they are:

the ionisation process (class G4eIonisation ) and the bremsstrahlung process (class G4eBremsstrahlung).

The class G4eIonisation calculates the contribution due to ionisation and it simulates the ‘discrete’ part of the ionisation , the Moller and Bhabha scattering or delta ray production by e+/e-.

The class G4eBremsstrahlung computes the energy loss contribution due to soft bremsstrahlung and it simulates the ‘discrete’ or hard bremsstrahlung. The formulae used to compute the continuous energy loss are the same than those in GEANT3, but the two programs are not the same. The main difference between GEANT3 and GEANT4 is that while in GEANT3 the continuous energy loss, delta ray production and bremsstrahlung are three independent processes , in GEANT4 these processes are closely connected, they form a unified scheme.

The G4eEnergyLoss class constructs dE/dx and range tables for every material. First the dE/dx tables are costructed and filled , the energy loss class simply summes up of the contributions computed by the ionisation and bremsstrahlung processes. After this step G4eEnergyLoss creates range tables of e+/e- for every material and computes the inverse of these range tables as well. All the tables are constructed at the beginning of the GEANT4 run , at initialisation time. Later, during the simulation the energy loss process performs two tasks : it imposes a limit on the stepsize of the particle and computes the energy loss after a step travelled by the particle.

Step limit due to the continuous energy loss

The continuous energy loss imposes a limit on the stepsize. The reason of this is the energy dependence of the cross sections. It is generally assumed in MC programs that the cross sections of the particles are approximately constant during a step , i.e. the step size should be so small that the relative difference of the cross sections at the beginning of the step and at the end should be small enough.In principle one has to use very small steps in order to have an accurate simulation , but the computing time increases if the stepsize decreases. As a good compromise the stepsize is limited here by the requirement that the stopping range of the particle can decrease by not more than 20 % during the step. This condition works fine for a particle of kinetic energy > 0.5 MeV – 1. MeV , but for low energy it gives very short step sizes. To cure this problem a lower limitation on the stepsize is also introduced. The choice of this lower limit is quite natural , the stepsize can not be smaller than the cut in range parameter of the program. The stepsize limit varies smoothly with decreasing energy from the value given by the condition to the lowest possible value cut in range . It is worthwile to mention , that the lower limit used here is much more natural and physical , than the one is used in GEANT3 at the automatic calculation of the tracking parameters.

Computation of the energy loss

The computation of the mean energy loss after a given step is done by using the range and inverse range tables. The mean loss can be written as

Where T0 is the kinetic energy, r0 the range at the beginning of the step step, the function fT(r) is the inverse of the range table (i.e. it gives the kinetic energy of the particle for a range value of r) . After the mean energy loss has been calculated the process computes the actual energy loss , i.e. the loss with fluctuation. The fluctuation is computed from a fluctuation model , from the model GLANDZ of the GEANT3 code [GEANT3] , [La95] .

Energy loss of muons

The energy loss of muons is computed by the class G4MuEnergyLoss. The scheme of the computation is the same as in the case of e+/e-, the only difference is that now there are three processes contributing to the total continuous energy loss , these are

the ionisation process (class G4MuIonisation), the bremsstrahlung process (class G4MuBremsstrahlung) and the direct pair production of muons (class G4MuPairProduction). The G4MuIonisation class computes the contribution to the continuous energy loss due to ionisation and simulates the corresponding ‘discrete’ process, the knock-on electron or -ray production by muons.

The G4MuBremsstrahlung class calculates the continuous loss due to the soft bremsstrahlung and simulates the ‘discrete’, hard bremsstrahlung.

The G4MuPairProduction class gives the contribution to the continuous energy loss due to sub-cutoff e+/e- pairs and performs the simulation of the pair production.

Energy loss of charged hadrons

The continuous energy loss of the charged hadrons are calculated by the class G4hEnergyLoss . The algorithm is the same than in the case of e+/e- energy loss. Here there is only one process , the ionisation (class G4hIonisation) which contributes to the continuous energy loss .

The G4hIonisation class computes the continuous energy loss and simulates the -ray producion by hadrons.

In the case of energy loss of the hadrons two dE/dx , range and inverse range tables are constructed only , these are the tables for proton and antiproton . The energy loss for othe charged hadrons are computed from these tables at the scaled kinetic energy Tscaled

Where T is the kinetic energy of the particle, Mproton and Mparticle are the masses of the proton and particle, respectively.

There is an important constraint for this process: the cut in range must be the same for all the charged hadrons . This condition is meaningful physically and it makes possible to use the proton/antiproton tables only when computing the energy loss.

Or check out this page for another look at energy loss:
http://www.irs.inms.nrc.ca/inms/irs/papers/MSKB/node10.html

To take the energy loss of electrons during the step into consideration, we have to replace in the exponent of Eq. (3) by

Here, $N$ is the number of atoms per unit volume and $t$ the path-length. The dependence of the single-scattering cross section $\sigma(\cos \chi,t^\prime)$ on the energy is expressed through the dependence on the path-length $t^\prime$. Although in principle it is possible to use $G_l$ instead of $\lambda Q_l$ to calculate the $q^{(2+)}$-surface, this approach would not be efficient on a present-day computer due to the large increase of memory required to score the pre-calculated data. In fact, when we take energy loss into account, the dependence of the multiple-scattering distribution on the screening parameter $\eta$ and the material in which the transport takes place. Therefore, one additional dimension will be needed to store the pre-calculated data. We will therefore approximate $G_l$ in such a way that the application of the method presented in the last section becomes possible. alone is replaced by a dependence on the energy

We rewrite Eq. (39)


$\displaystyle G_l$ $\textstyle =$ $\displaystyle 2 \pi N \int_{E_{\rm f}}^{E_{\rm i}} {{\rm d}E \over S(E)} \sigma... ... {\rm d}(\cos\chi)\,\tilde{\sigma}(\cos \chi,E) \Big[ 1 - P_l (\cos \chi) \Big]$
$\textstyle =$ $\displaystyle b_c \int_{E_{\rm f}}^{E_{\rm i}} {{\rm d}E \over S(E) \beta^2} Q_l(E)~.$ (40)


Here, $E_{\rm i}$ and $E_{\rm f}$ are the initial and final kinetic energies of the electron, $S(E)$ the restricted collision stopping power
4, $\sigma_{\rm tot}(E)$ the total elastic cross section and $\tilde{\sigma}$ is again normalized to unity. To arrive at the second equation, we have made use of the fact that $\sigma_{\rm tot}$ is proportional to $1/\beta^2$ where $\beta$ is the electron’s velocity in units of the velocity of light and introduced the short hand notation $b_c$ for the product of all constants in the total elastic cross section times $2 \pi N$. If we now neglect the very weak (logarithmic) energy dependence of $S \beta^2$, Eq. (40) becomes


\begin{displaymath} G_l \approx {b_c \Delta E \over \tilde{\beta}^2 S(\tilde{E})... ...over \Delta E} \int_{E_{\rm f}}^{E_{\rm i}} {\rm d}E\,Q_l(E)~. \end{displaymath} (41)


Here, $\Delta E = E_{\rm i} - E_{\rm f}$ is the energy loss during the step under consideration, $\tilde{E} = (E_{\rm i}+E_{\rm f})/2$ the average energy of the electron and $\tilde{\beta}$ the velocity calculated from $\tilde{E}$. To carry out the $E$-integration we can perform a power series expansion in $\Delta E$,


\begin{displaymath} \int_{E_{\rm f}}^{E_{\rm i}} {\rm d}E\,Q_l(E) \approx \Delta... ...\prime}(\tilde{E}) \over Q_l(\tilde{E})} \pm \cdots \right]~, \end{displaymath} (42)


where $Q_l^{\prime \prime}$ is the second derivative of $Q_l$ with respect to $E$. Using Eq. (18) for $Q_l$ and neglecting terms of the order of $\eta$ and terms small compared to $\ln(1/\eta)$, we arrive at the result


$\displaystyle G_l$ $\textstyle \approx$ $\displaystyle \lambda_{{\rm eff}} Q_l(\tilde{E})~,$
$\displaystyle \lambda_{{\rm eff}}$ $\textstyle =$ $\displaystyle {b_c \Delta E \over \tilde{\beta}^2 S(\tilde{E}) }~ \left[1 + {4 ... ...u}^2 \over 3 (2 + \tilde{\tau})^2} ~{\epsilon^2 \over (2 - \epsilon)^2} \right]$ (43)


where $\tilde{\tau}$ is the ratio of the average electron kinetic energy to it’s rest mass energy and $\epsilon=\Delta E/E_{\rm i}$ the energy loss fraction. That means, when energy loss is taken into account, the multiple-scattering distribution is to a good approximation equivalent to the multiple-scattering distribution without energy loss resulting from $\lambda_{{\rm eff}}$ elastic collisions of electrons with the energy $\tilde{E}$. With this observation we can easily apply the theory developed in the previous sections to realistic calculations with electron energy loss taken into account.

Figure 4: Ratio of the exact-MS distribution to the MS-distribution with energy loss taken into account according to Eq. (43) for 100 keV and 10 MeV electrons in water and energy loss fractions $\epsilon $ of 10, 25 and 33%. For other energies and materials, similar dependence on the energy-loss fraction $\epsilon $ is observed.



To test the accuracy of the approximations leading to Eq. (43), we have calculated the multiple-scattering distribution resulting from the exact GS-moments $G_l$ and compared it to the distribution obtained with the approximated $G_l$’s given in Eq. (43) for various energies, materials and energy-loss fractions. The energy integration in (40) was done by a 32-point Gauss-Legendre quadrature. The disagreement between the exact and approximated distributions increases with increasing $\epsilon $. The maximum deviation found for $\epsilon=25$% was of the order of 1%. For $\epsilon \le 10$% the agreement was almost perfect. The ratio of the approximated to the exact distribution for $\epsilon=10$, 25 and 33% is shown for typical cases in Fig. 4.

Energy loss

To take the energy loss of electrons during the step into consideration, we have to replace $\lambda Q_l$ in the exponent of Eq. (3) by


\begin{displaymath} G_l = 2 \pi N \int_0^t {\rm d}t^\prime\,\int_{-1}^1 {\rm d}... ...,\sigma(\cos \chi,t^\prime) \Big[ 1 - P_l (\cos \chi) \Big]~. \end{displaymath} (39)


Here, $N$ is the number of atoms per unit volume and $t$ the path-length. The dependence of the single-scattering cross section $\sigma(\cos \chi,t^\prime)$ on the energy is expressed through the dependence on the path-length $t^\prime$. Although in principle it is possible to use $G_l$ instead of $\lambda Q_l$ to calculate the $q^{(2+)}$-surface, this approach would not be efficient on a present-day computer due to the large increase of memory required to score the pre-calculated data. In fact, when we take energy loss into account, the dependence of the multiple-scattering distribution on the screening parameter $\eta$ and the material in which the transport takes place. Therefore, one additional dimension will be needed to store the pre-calculated data. We will therefore approximate $G_l$ in such a way that the application of the method presented in the last section becomes possible. alone is replaced by a dependence on the energy

We rewrite Eq. (39)


$\displaystyle G_l$ $\textstyle =$ $\displaystyle 2 \pi N \int_{E_{\rm f}}^{E_{\rm i}} {{\rm d}E \over S(E)} \sigma... ... {\rm d}(\cos\chi)\,\tilde{\sigma}(\cos \chi,E) \Big[ 1 - P_l (\cos \chi) \Big]$
$\textstyle =$ $\displaystyle b_c \int_{E_{\rm f}}^{E_{\rm i}} {{\rm d}E \over S(E) \beta^2} Q_l(E)~.$ (40)


Here, $E_{\rm i}$ and $E_{\rm f}$ are the initial and final kinetic energies of the electron, $S(E)$ the restricted collision stopping power
4, $\sigma_{\rm tot}(E)$ the total elastic cross section and $\tilde{\sigma}$ is again normalized to unity. To arrive at the second equation, we have made use of the fact that $\sigma_{\rm tot}$ is proportional to $1/\beta^2$ where $\beta$ is the electron’s velocity in units of the velocity of light and introduced the short hand notation $b_c$ for the product of all constants in the total elastic cross section times $2 \pi N$. If we now neglect the very weak (logarithmic) energy dependence of $S \beta^2$, Eq. (40) becomes


\begin{displaymath} G_l \approx {b_c \Delta E \over \tilde{\beta}^2 S(\tilde{E})... ...over \Delta E} \int_{E_{\rm f}}^{E_{\rm i}} {\rm d}E\,Q_l(E)~. \end{displaymath} (41)


Here, $\Delta E = E_{\rm i} - E_{\rm f}$ is the energy loss during the step under consideration, $\tilde{E} = (E_{\rm i}+E_{\rm f})/2$ the average energy of the electron and $\tilde{\beta}$ the velocity calculated from $\tilde{E}$. To carry out the $E$-integration we can perform a power series expansion in $\Delta E$,


\begin{displaymath} \int_{E_{\rm f}}^{E_{\rm i}} {\rm d}E\,Q_l(E) \approx \Delta... ...\prime}(\tilde{E}) \over Q_l(\tilde{E})} \pm \cdots \right]~, \end{displaymath} (42)


where $Q_l^{\prime \prime}$ is the second derivative of $Q_l$ with respect to $E$. Using Eq. (18) for $Q_l$ and neglecting terms of the order of $\eta$ and terms small compared to $\ln(1/\eta)$, we arrive at the result


$\displaystyle G_l$ $\textstyle \approx$ $\displaystyle \lambda_{{\rm eff}} Q_l(\tilde{E})~,$
$\displaystyle \lambda_{{\rm eff}}$ $\textstyle =$ $\displaystyle {b_c \Delta E \over \tilde{\beta}^2 S(\tilde{E}) }~ \left[1 + {4 ... ...u}^2 \over 3 (2 + \tilde{\tau})^2} ~{\epsilon^2 \over (2 - \epsilon)^2} \right]$ (43)


where $\tilde{\tau}$ is the ratio of the average electron kinetic energy to it’s rest mass energy and $\epsilon=\Delta E/E_{\rm i}$ the energy loss fraction. That means, when energy loss is taken into account, the multiple-scattering distribution is to a good approximation equivalent to the multiple-scattering distribution without energy loss resulting from $\lambda_{{\rm eff}}$ elastic collisions of electrons with the energy $\tilde{E}$. With this observation we can easily apply the theory developed in the previous sections to realistic calculations with electron energy loss taken into account.

Figure 4: Ratio of the exact-MS distribution to the MS-distribution with energy loss taken into account according to Eq. (43) for 100 keV and 10 MeV electrons in water and energy loss fractions $\epsilon $ of 10, 25 and 33%. For other energies and materials, similar dependence on the energy-loss fraction $\epsilon $ is observed.
\begin{figure} \begin{center} \epsfig {figure=figures/fig4/fig4.eps,height=14cm}\end{center}\end{figure}

To test the accuracy of the approximations leading to Eq. (43), we have calculated the multiple-scattering distribution resulting from the exact GS-moments $G_l$ and compared it to the distribution obtained with the approximated $G_l$’s given in Eq. (43) for various energies, materials and energy-loss fractions. The energy integration in (40) was done by a 32-point Gauss-Legendre quadrature. The disagreement between the exact and approximated distributions increases with increasing $\epsilon $. The maximum deviation found for $\epsilon=25$% was of the order of 1%. For $\epsilon \le 10$% the agreement was almost perfect. The ratio of the approximated to the exact distribution for $\epsilon=10$, 25 and 33% is shown for typical cases in Fig. 4.

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