From e0c6872cf40896c7be36b11dcc744620f10adf1d Mon Sep 17 00:00:00 2001 From: Norbert Preining Date: Mon, 2 Sep 2019 13:46:59 +0900 Subject: Initial commit --- info/examples/lwc/apa/latexexa-raw.xml | 1054 ++++++++++++++++++++++++++++++++ 1 file changed, 1054 insertions(+) create mode 100644 info/examples/lwc/apa/latexexa-raw.xml (limited to 'info/examples/lwc/apa/latexexa-raw.xml') diff --git a/info/examples/lwc/apa/latexexa-raw.xml b/info/examples/lwc/apa/latexexa-raw.xml new file mode 100644 index 0000000000..095b37c922 --- /dev/null +++ b/info/examples/lwc/apa/latexexa-raw.xml @@ -0,0 +1,1054 @@ + + + + + + +]> + + +Simulation of Energy Loss Straggling +Maria Physicist + +January 17, 1999 + + +
+ +Introduction + + + + + (a+b)2 + +Due to the statistical nature of ionisation energy loss, large fluctuations can occur in +the amount of energy deposited by a particle traversing an absorber element. +Continuous processes such as multiple scattering and energy loss play a relevant role +in the longitudinal and lateral development of electromagnetic and hadronic +showers, and in the case of sampling calorimeters the measured resolution +can be significantly affected by such fluctuations in their active layers. The +description of ionisation fluctuations is characterised by the significance parameter +κ, +which is proportional to the ratio of mean energy loss to the maximum +allowed energy transfer in a single collision with an atomic electron + + κ= ξEmax + +Emax is the +maximum transferable energy in a single collision with an atomic electron. + + Emax= 2meβ2γ21+2γme/mx+me/mx 2, + where +γ=E/mx, +E is energy and +mx the mass of the +incident particle, β2=1-1/γ2 +and me is the +electron mass. ξ +comes from the Rutherford scattering cross section and is defined as: + + ξ=2πz2e4NAvZρδxmeβ2c2A =153.4 z2β2 ZAρδxkeV , + +where +zcharge of the incident particle +NAvAvogadro's number +Zatomic number of the material +Aatomic weight of the material +ρdensity +δxthickness of the material + + +κ +measures the contribution of the collisions with energy transfer close to +Emax. For a given absorber, +κ tends towards large +values if δx is large +and/or if β is small. +Likewise, κ tends +towards zero if δx is +small and/or if β +approaches 1. +The value of κ +distinguishes two regimes which occur in the description of ionisation fluctuations +: + + +A +large +number +of +collisions +involving +the +loss +of +all +or +most +of +the +incident +particle +energy +during +the +traversal +of +an +absorber. +As +the +total +energy +transfer +is +composed +of +a +multitude +of +small +energy +losses, +we +can +apply +the +central +limit +theorem +and +describe +the +fluctuations +by +a +Gaussian +distribution. +This +case +is +applicable +to +non-relativistic +particles +and +is +described +by +the +inequality +κ>10 +(i.e. +when +the +mean +energy +loss +in +the +absorber +is +greater +than +the +maximum +energy +transfer +in +a +single +collision). + + +Particles +traversing +thin +counters +and +incident +electrons +under +any +conditions. +The +relevant +inequalities +and +distributions +are +0.01<κ<10, +Vavilov +distribution, +and +κ<0.01, +Landau +distribution. +An additional regime is defined by the contribution of the collisions +with low energy transfer which can be estimated with the relation +ξ/I0, +where I0 +is the mean ionisation potential of the atom. Landau theory assumes that +the number of these collisions is high, and consequently, it has a restriction +ξ/I01. In GEANT (see +URL http://wwwinfo.cern.ch/asdoc/geant/geantall.html), the limit of Landau theory has +been set at ξ/I0=50. +Below this limit special models taking into account the atomic structure of the material are +used. This is important in thin layers and gaseous materials. Figure shows the behaviour +of ξ/I0 as +a function of the layer thickness for an electron of 100 keV and 1 GeV of kinetic +energy in Argon, Silicon and Uranium. + +
+ +The variable ξ/I0 +can be used to measure the validity range of the Landau +theory. It depends on the type and energy of the particle, +Z, +A +and the ionisation potential of the material and the layer thickness. +
+In the following sections, the different theories and models for the energy loss +fluctuation are described. First, the Landau theory and its limitations are discussed, +and then, the Vavilov and Gaussian straggling functions and the methods in the thin +layers and gaseous materials are presented. + +
+
+ +Landau theory +For a particle of mass mx traversing +a thickness of material δx, +the Landau probability distribution may be written in terms of the universal Landau +function φ(λ) +as: + + f(ε,δx) = 1ξφ(λ) + +where + + φ(λ) = 12πi + c+ic-iexpulnu+λuduc0 + + λ = ε-εξ -γ-β2-ln ξEmax + + γ = 0.422784...=1-γ + + γ = 0.577215...(Euler's constant) + + ε = average energy loss + + ε = actual energy loss + + + + +Restrictions +The Landau formalism makes two restrictive assumptions : + + +The +typical +energy +loss +is +small +compared +to +the +maximum +energy +loss +in +a +single +collision. +This +restriction +is +removed +in +the +Vavilov +theory +(see +section +). + + +The +typical +energy +loss +in +the +absorber +should +be +large +compared +to +the +binding +energy +of +the +most +tightly +bound +electron. +For +gaseous +detectors, +typical +energy +losses +are +a +few +keV +which +is +comparable +to +the +binding +energies +of +the +inner +electrons. +In +such +cases +a +more +sophisticated +approach +which +accounts +for +atomic +energy +levels +is +necessary +to +accurately +simulate +data +distributions. +In +GEANT, +a +parameterised +model +by +L. +Urbán +is +used +(see +section +). +In addition, the average value of the Landau distribution is infinite. +Summing the Landau fluctuation obtained to the average energy from the +dE/dx +tables, we obtain a value which is larger than the one coming from the table. The +probability to sample a large value is small, so it takes a large number of steps +(extractions) for the average fluctuation to be significantly larger than zero. This +introduces a dependence of the energy loss on the step size which can affect +calculations. +A solution to this has been to introduce a limit on the value of the +variable sampled by the Landau distribution in order to keep the average +fluctuation to 0. The value obtained from the GLANDO routine is: + + δdE/dx=ε-ε=ξ(λ-γ+β2+ln ξEmax ) + +In order for this to have average 0, we must impose that: + + λ=-γ-β2-ln ξEmax + +This is realised introducing a λmax(λ) +such that if only values of λλmax +are accepted, the average value of the distribution is +λ. +A parametric fit to the universal Landau distribution has been performed, with following result: + + λmax=0.60715+1.1934λ+(0.67794+0.052382λ)exp(0.94753+0.74442λ) + only values +smaller than λmax +are accepted, otherwise the distribution is resampled. + + +
+
+ +Vavilov theory +Vavilov derived a more accurate straggling distribution by introducing the kinematic +limit on the maximum transferable energy in a single collision, rather than using +Emax=. Now +we can write: + + f ε,δs = 1ξφv λv,κ,β2 + +where + + φv λv,κ,β2 = 12πi + c+ic-iφseλsdsc0 + + φs = expκ(1+β2γ)expψ s, + + ψ s = slnκ+(s+β2κ)ln(s/κ)+E +1(s/κ)-κe-s/κ, + +and + + E1(z) = + zt-1e-tdt(the exponential integral) + + λv = κε-εξ -γ-β2 + +The Vavilov parameters are simply related to the Landau parameter by +λL=λv/κ-lnκ. It can be shown that +as κ0, the distribution of +the variable λL approaches +that of Landau. For κ0.01 +the two distributions are already practically identical. Contrary to what many textbooks +report, the Vavilov distribution does not approximate the Landau distribution for small +κ, but rather the +distribution of λL +defined above tends to the distribution of the true +λ from +the Landau density function. Thus the routine GVAVIV samples the variable +λL rather +than λv. +For κ10 +the Vavilov distribution tends to a Gaussian distribution (see next section). + +
+
+ +Gaussian Theory +Various conflicting forms have been proposed for Gaussian straggling functions, but most +of these appear to have little theoretical or experimental basis. However, it has been shown +that for κ10 +the Vavilov distribution can be replaced by a Gaussian of the form: + + f(ε,δs) 1ξ2πκ 1-β2/2exp(ε-ε)22 κξ2(1-β2/2) + +thus implying + + mean = ε + + σ2 = ξ2κ (1-β2/2)=ξE +max(1-β2/2) + + +
+
+ +Urbán model +The method for computing restricted energy losses with +δ-ray +production above given threshold energy in GEANT is a Monte Carlo method that +can be used for thin layers. It is fast and it can be used for any thickness of a +medium. Approaching the limit of the validity of Landau's theory, the loss +distribution approaches smoothly the Landau form as shown in Figure . +
+ +Energy loss distribution for a 3 GeV electron in Argon as given by +standard GEANT. The width of the layers is given in centimeters. +
+It is assumed that the atoms have only two energy levels with binding energy +E1 and +E2. +The particle--atom interaction will then be an excitation with energy loss +E1 or +E2, or +an ionisation with an energy loss distributed according to a function +g(E)1/E2: + + g(E)=(Emax+I)IEmax 1E2 (1) + +The macroscopic cross-section for excitations +(i=1,2) is + + Σi=C fiEi ln(2mβ2γ2/Ei)-β2ln(2mβ2γ2/I)-β2 (1-r) (2) +and +the macroscopic cross-section for ionisation is + + Σ3=C EmaxI(Emax+I)ln(Emax+II )r (3) +Emax +is the GEANT cut for δ-production, +or the maximum energy transfer minus mean ionisation energy, if it is smaller than +this cut-off value. The following notation is used: +r,Cparameters of the model +Eiatomic energy levels +Imean ionisation energy +fioscillator strengths + +The model has the parameters fi, +Ei, +C and +r(0r1). The oscillator +strengths fi and the +atomic level energies Ei +should satisfy the constraints + + f1+f2 = 1 (4) + + f1lnE1+f2lnE2 = lnI (5) + +The parameter C +can be defined with the help of the mean energy loss +dE/dx in the following way: The +numbers of collisions (ni, +i = 1,2 for the excitation and 3 for the ionisation) follow the Poisson distribution with a mean +number <ni>;. In a step +Δx the mean number +of collisions is + <ni>;=ΣiΔx (6) +The +mean energy loss dE/dx +in a step is the sum of the excitation and ionisation contributions + + dEdx Δx=Σ1E1+Σ2E2+Σ3 + IEmax+IEg(E)dEΔx (7) +From +this, using the equations (), (), () and (), one can define the parameter +C + + C=dEdx (8) + +The following values have been chosen in GEANT for the other parameters: + + f2=0 ifZ2 +2/ZifZ>2 + f1=1-f2 + E2=10Z2eV E1= IE2f2 1f1 + r=0.4 + + With these values +the atomic level E2 +corresponds approximately the K-shell energy of the atoms and +Zf2 the number of +K-shell electrons. r +is the only variable which can be tuned freely. It determines the relative contribution +of ionisation and excitation to the energy loss. +The energy loss is computed with the assumption that the step length (or the relative +energy loss) is small, and---in consequence---the cross-section can be considered +constant along the path length. The energy loss due to the excitation is + + ΔEe=n1E1+n2E2 (9) +where +n1 and +n2 +are sampled from Poisson distribution as discussed above. The +loss due to the ionisation can be generated from the distribution +g(E) by +the inverse transformation method: + + u=F(E) = + IEg(x)dx + + E=F-1(u) = I1-u EmaxEmax+I (10) + + (11) + +where u is a uniform random +number between F(I)=0 and +F(Emax+I)=1. The contribution from the +ionisations will be + ΔEi= + j=1n3 I1-uj EmaxEmax+I (12) +where +n3 is the +number of ionisation (sampled from Poisson distribution). The energy loss in a step will +then be ΔE=ΔEe+ΔEi. + + + +Fast simulation for n316 +If the number of ionisation n3 +is bigger than 16, a faster sampling method can be used. The possible energy loss +interval is divided in two parts: one in which the number of collisions is large and the +sampling can be done from a Gaussian distribution and the other in which +the energy loss is sampled for each collision. Let us call the former interval +[I,αI] the interval A, +and the latter [αI,Emax] the +interval B. α lies +between 1 and Emax/I. +A collision with a loss in the interval A happens with the probability + + P(α)= + IαIg(E)dE=(Emax+I)(α-1)Emaxα (13) +The +mean energy loss and the standard deviation for this type of collision are + + <ΔE(α)>;= 1P(α) + IαIEg(E)dE=Iαlnαα-1 (14) +and + + σ2(α)= 1P(α) + IαIE2g(E)dE=I2α1- αln2α(α-1)2 (15) +If the +collision number is high, we assume that the number of the type A collisions can be +calculated from a Gaussian distribution with the following mean value and standard +deviation: + + <nA>; = n3P(α) (16) + + σA2 = n3P(α)(1-P(α)) (17) + +It is further assumed that the energy loss in these collisions has a Gaussian +distribution with + + <ΔEA>; = nA<ΔE(α)>; (18) + + σE,A2 = nAσ2(α) (19) + +The energy loss of these collision can then be sampled from the Gaussian +distribution. +The collisions where the energy loss is in the interval B are sampled directly from + + ΔEB= + i=1n3-nA αI1-ui Emax+I-αIEmax+I (20) +The +total energy loss is the sum of these two types of collisions: + + ΔE=ΔEA+ΔEB (21) + +The approximation of equations (), (), () and () can be used under the following +conditions: + + <nA>;-cσA 0 (22) + + <nA>;+cσA n3 (23) + + <ΔEA>;-cσE,A 0 (24) + +where c4. From +the equations (), () and () and from the conditions () and () the following limits can be +derived: + αmin=(n3+c2)(Emax+I)n3(Emax+I)+c2I ααmax=(n3+c2)(Emax+I)c2(Emax+I)+n3I (25) +This +conditions gives a lower limit to number of the ionisations +n3 for which the fast +sampling can be done: + n3c2 (26) +As +in the conditions (), () and () the value of +c is as minimum +4, one gets n316. +In order to speed the simulation, the maximum value is used for +α. +The number of collisions with energy loss in the interval B (the number of interactions +which has to be simulated directly) increases slowly with the total number of collisions +n3. +The maximum number of these collisions can be estimated as + + nB,max=n3-nA,minn3(<nA>;-σA) (27) +From the previous +expressions for <nA>; and +σA one can derive the +condition + nBnB,max= 2n3c2n3+c2 (28) +The following +values are obtained with c=4: +n3nB,maxn3nB,max +16 16 200 29.63 +20 17.78 500 31.01 +50 24.24 1000 31.50 +100 27.59 32.00 + + + + + +Special sampling for lower part of the spectrum +If the step length is very small (5 +mm in gases, +2-3 μm in solids) +the model gives 0 energy loss for some events. To avoid this, the probability of 0 energy loss is +computed + P(ΔE=0)=e-(<n1>;+<n2>;+<n3>;) (29) +If the +probability is bigger than 0.01 a special sampling is done, taking into account the fact that in +these cases the projectile interacts only with the outer electrons of the atom. An energy level +E0=10 eV is chosen +to correspond to the outer electrons. The mean number of collisions can be calculated from + + <n>;= 1E0 dEdx Δx (30) +The number +of collisions n +is sampled from Poisson distribution. In the case of the thin layers, all the +collisions are considered as ionisations and the energy loss is computed as + + ΔE= + i=1n E01- EmaxEmax+E0 ui (31) + + + +
+
+ +References + + +L.Landau. +On +the +Energy +Loss +of +Fast +Particles +by +Ionisation. +Originally +published +in +J. +Phys., +8:201, +1944. +Reprinted +in +D.ter +Haar, +Editor, +L.D.Landau, +Collected +papers, +page +417. +Pergamon +Press, +Oxford, +1965. + + +B.Schorr. +Programs +for +the +Landau +and +the +Vavilov +distributions +and +the +corresponding +random +numbers. +Comp. +Phys. +Comm., +7:216, +1974. + + +S.M.Seltzer +and +M.J.Berger. +Energy +loss +straggling +of +protons +and +mesons. +In +Studies +in +Penetration +of +Charged +Particles +in +Matter, +Nuclear +Science +Series 39, +Nat. +Academy +of +Sciences, +Washington +DC, +1964. + + +R.Talman. +On +the +statistics +of +particle +identification +using +ionization. +Nucl. +Inst. +Meth., +159:189, +1979. + + +P.V.Vavilov. +Ionisation +losses +of +high +energy +heavy +particles. +Soviet +Physics +JETP, +5:749, +1957. +
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