Initialization

Needs["CleanSlate`"]

CleanSlate["Global`",  Verbose  ->  False]

Off[General::spell]

Off[Needs::nocont]

Needs["Graphics`"]

Needs["Miscellaneous`"]

WorkSpace

Data for α radiation

Calculations for polyethylene

Units:  unless specified otherwise,
	energy -> eV, length -> m, density -> (g/(cm^3)), mass -> g

e0  =  4.86 (10^6);

r0  =  3.3 (10^(-2));

elements =  {Carbon,  Aluminum,  Copper,  Silver,  Lead};

a  =  AtomicWeight[elements];

ρ  =  Density[elements] ((Meter^3)/((10^3) Kilogram));

z  =  AtomicNumber[elements];

r  =  
      r0 1.512 (10^(-4)) ((a (Sqrt[10  +  z]))/(ρ z));

rate  =  (e0/r);

e[t_]  :=  e0 (10^(-6))  -  t rate (10^(-12))

Note:  e is in MeV

Plot[Evaluate[Table[e[t][[n]],  {n,  1,  5}]], 
       {t,  0,  18},  
      PlotLabel  ->  "Energy loss for α radiation",  
      Frame  ->  True,  
      FrameLabel  -> 
         {"Thickness (in μm)",  "Energy of α (in MeV)"}, 
       PlotStyle  -> 
         {{Dashing[{ .1, 0}]},  {Dashing[{ .08,  .08}]}, 
           {Dashing[{ .04,   .04}]},  {Dashing[{ .02,   .02}]}, 
           {Dashing[{ .01,   .01}]}},  PlotRange  ->  {0,  Automatic}, 
       PlotLegend  -> 
         {"C",  "Al",  "Cu",  "Ag",  "Pb"},  
      LegendPosition  ->  {1,  - .5},  
      TextStyle  ->  {FontSize  ->  24}];

m  =  Convert[4.0026 AMU,  Gram] (1/Gram);

q  =  2 ElectronCharge (1/Coulomb);

radius[t_,  b_]  :=  
      ((Sqrt[2 e[t] m])/(q b)) 1.2658 (10^(-6))

Note:  numerical factors are present for dimensional consistency

Table[
      Plot3D[radius[t, b][[n]],  {t,  0,  7},  {b,   .5,  1},  
        PlotLabel  ->  elements[[n]],  
        AxesLabel  -> 
           {"Thickness (in μm)",  "B field (in T)",  
            "Radius (in cm)"}],  {n,  1,  5}];

Data for β radiation

Calculations for C, Al, Cu, Ag, and Pb

e0  =  1.7 (10^6);

s  =  {1.7,  1.6,  1.53,  1.467,  1.33} (10^6);

rate  =  s ρ (10^2);

v[t_]  :=  
      c (Sqrt[ 1  -  (((m^2) (c^4))/e)[t]^2 2.567 (10^(-20))])

Note:  numerical factors are present for dimensional consistency

c  =  SpeedOfLight (Second/Meter);

m  =  ElectronMass ((10^3)/Kilogram);

q  =  ElectronCharge (1/Coulomb);

radius[t_,  b_]  :=  ((m v[t])/(q b))

Plot[Evaluate[Table[e[t][[n]],  {n,  1,  5}]], 
       {t,  0,  4000},  
      PlotLabel  ->  "Energy loss for β radiation",  
      Frame  ->  True,  
      FrameLabel  -> 
         {"Thickness (in μm)",  "Energy of β (in MeV)"}, 
       PlotStyle  -> 
         {{Dashing[{ .1, 0}]},  {Dashing[{ .08,  .08}]}, 
           {Dashing[{ .04,   .04}]},  {Dashing[{ .02,   .02}]}, 
           {Dashing[{ .01,   .01}]}},  PlotRange  ->  {0,  Automatic}, 
       PlotLegend  -> 
         {"C",  "Al",  "Cu",  "Ag",  "Pb"},  
      LegendPosition  ->  {1,  - .5},  
      TextStyle  ->  {FontSize  ->  24}];

Table[
      Plot3D[radius[t, b][[n]],  {t,  0,  1000}, 
         {b,   .5,  1},  PlotLabel  ->  elements[[n]],  
        AxesLabel  -> 
           {"Thickness (in μm)",  "B field (in T)",  
            "Radius (in cm)"}],  {n,  1,  5}];

Calculations for polyethylene

s  =  2.076 (10^6);

ρ  =   .9;

rate  =  s ρ (10^2);

Plot3D[radius[t,  b],  {t,  0,  2500},  {b,   .5,  1},  
      PlotLabel  ->  "Energy loss for polyethylene",  
      AxesLabel  -> 
         {"Thickness (in μm)",  "B field (in Tesla)",  
          "Radius (in cm)"}];

Plot[radius[0,  b],  {b,   .5,  1},  
      PlotLabel  ->  "Radius of β particles",  
      Frame  ->  True,  
      FrameLabel  ->  {"B field (in T)",  "Radius (in cm)"}];