A mathematica notepad is used to handle calculation of the integral η = ∫((d 
                    N (ξ))/d)ξ 
            M (ξ)  dξ more easily. An algorithm used to calculate the electromagnetic shower number  is shown below only valid for energies greater than 100 MeV in GeV ( 0.1 GeV), using a radiation lenght (t) of 4; Radiation lenghts of  t=  6, 8, 10, and 15 are approached at the final section. Thus we have to define some values that will be later reproduce functions of interest that depend on the incident kinetic energy (ξ) of charged particles.

M =  (Subscript[m, o ]) (t^a) (E^((- b t)));

(Subscript[m, o ]) =  
        5.51 ξ (Sqrt[Z]) 
          (b^( a  +  1))/ Gamma[a + 1];

b = 0.634 -  0.0021 Z;

a = 1.77 - 0.52  Log[ξ]; For Z = 13

=2.0 - 
        (Z/340) + ( .664 - (Z/340)) Log[ξ]; 
    for Z ≥  26

Z = 13;

t = 4;

the gamma function Γ(a+1) exhibits a discontinuous character for our argument as shown below, rising to infinity at about ξ=205.8 where the argument of the gamma function is zero. This then, gives us the upper limit on our fucntion M(ξ).

Plot [
      Gamma[2.77 - 0.52  Log[ξ]], {ξ,  .1, 
        200}]

Both functions contributing to electromagnetic shower production are shown below by themselves as a function of the energy ξ.

Plot[(Subscript[m, o ]), {ξ,  .1, 120}]

Plot[(t^a) (E^((- b t))), {ξ,  .1, 
        10}]

As discussed  before, the number of particles created does turns around high energy following a (1/ξ)dependence

Plot[M, {ξ,  .1, 120}]

Plot[M, {ξ,  .1,  .5}]

We want to find the place where pair production factor is equal to one to separate events not discernable by our counter.

M /. ξ ->  .33043

A plot of the final electromagnetic component created is shown below as a fucntion of energy ξ.

Plot[ .2955  (ξ^(-2.477)) M , {ξ,  .1,  .5}]

A numerical integration has to be done as shown below for discernable events.

NIntegrate[ .2955  (ξ^(-2.477)) M, {ξ,  .1,  .33043}]

The total number of events are those that can be discernable and those that cannot, as denoted by both integrals below respectively.

\(∫\_ .1\% .33043 .2955 \( ξ\^\(-2.477\)\)\ [\(m\_o\) \((ξ)\)\ \(t\^\(\(\ \)\(a \((ξ)\)\)\)\) E\^\((\(-\ b\)\ t)\)] \[DifferentialD]ξ = 2.257\)

\(∫\_ .33043\%∞ .2955 \( ξ\^\(-2.477\)\) \[DifferentialD]ξ = 1.02683\)

So for t= 4 number of events expected= 2.257+1.02683= .151 which is10.9 % of original 30 particles / ((m^2)s sr)

For t= 6

\(∫\_ .1\% .44824 .2955 \( ξ\^\(-2.477\)\)\ [\(m\_o\) \((ξ)\)\ \(t\^\(\(\ \)\(a \((ξ)\)\)\)\) E\^\((\(-\ b\)\ t)\)] \[DifferentialD]ξ = 2.31\)

\(∫\_ .44824\%∞ .2955 \( ξ\^\(-2.477\)\) \[DifferentialD]ξ = 0.654479\)

Plot[M, {ξ,  .1,  .5}]

M /. ξ ->  .44824

So for t= 6, the number of events expected= 2.31+.6544 =2.9644 which is 9.88% of original 30 particles / ((m^2)s sr)

For t= 8

Plot[M, {ξ,  .1, 2}]

M /. ξ -> 1.1045

\(∫\_ .1\%1.1045 .2955 \( ξ\^\(-2.477\)\)\ [\(m\_o\) \((ξ)\)\ \(t\^\(\(\ \)\(a \((ξ)\)\)\)\) E\^\((\(-\ b\)\ t)\)] \[DifferentialD]ξ = 1.815\)

(∫_1.1045^∞) .2955  (ξ^(-2.477)) \[DifferentialD]ξ = 0.172751

So for t=8, the number of events expected = 1.815 + 0.17275 = 1.9878 which is 6.62 % of original 30 particles / ((m^2)s sr)

For t=10

Plot[M, {ξ,  .1, 30}]

This one never reaches one, which implies that the energy range valid narrows as radiation lenghts increase. So only that range where the formula is properly described should be taken into account (i.e. ξ ≤ 16)

M /. ξ -> 16

\(∫\_ .1\%16 .2955 \( ξ\^\(-2.477\)\)\ [\(m\_o\) \((ξ)\)\ \(t\^\(\(\ \)\(a \((ξ)\)\)\)\) E\^\((\(-\ b\)\ t)\)] \[DifferentialD]ξ = 1.046\)

(∫_16^∞) .2955  (ξ^(-2.477)) \[DifferentialD]ξ =  .0033319

So for t=10, the total number of events expected = 1.046 + .00333 = 1.04933 which is 3.5 % of original 30 particles / ((m^2)s sr)

For t=15

Plot[M, {ξ,  .1, 20}]

same situation as previously, again take only that valid range.

M /. ξ -> 6

\(∫\_ .1\%6 .2955 \( ξ\^\(-2.477\)\)\ [\(m\_o\) \((ξ)\)\ \(t\^\(\(\ \)\(a \((ξ)\)\)\)\) E\^\((\(-\ b\)\ t)\)] \[DifferentialD]ξ = 0.136698\)

(∫_6^∞) .2955  (ξ^(-2.477)) \[DifferentialD]ξ =  .0141856

So for t=15, the total number of events expected = .137 + .014 = .151 which is 0.5 % of original 30 particles / ((m^2)s)