Bethe-Bloche Equation

Create a Bethe-Bloch function

#Planck's constant:
h <- 6.62607004e-34

#Speed of light m/s
c <- 299792458

#Fine structure constant
alpha <- 1/137

#Mass of an electron Mass/GeV

m.e <- 0.005

#Density n, atomic number Z, the fraction of the speed of light the particle is moving at, beta, and the particle's velocity v are specified as parameters to the equation


dE.dx <- function(n,Z,v,beta){
  -4 * pi * h^2 * c^2 * alpha^2 * ((n * Z)/(m.e * v^2)) * log(((2 * beta^2 * gamma^2 * c^2 * m.e)/(I.e)) - beta^2,base=exp(1))
}

#For an electron traversing a silicon detector:

v <- seq(0.1*c,c,100000)

beta <- v/c

#Lorentz factor

gamma <- 1/(sqrt(1-(v^2/c^2)))

n <- 1



Z <- 14

#Effective ionization potential of the material

I.e <- 10 * Z

electron.y = dE.dx(n=n,Z=Z,v=v,beta=beta)

require(latex2exp)

## Loading required package: latex2exp

m.e <- 273.13*m.e

pion.y = dE.dx(n=n,Z=Z,v=v,beta=beta)



plot(x=beta*gamma, y=-pion.y,type="l",main="Bethe-Bloch Curve of a Pion moving through Silicon", xlab = TeX("$\\beta\\cdot\\gamma$"),ylab=TeX("$-dE/dx$"),col="blue",cex.main=0.8)

plot(x=beta*gamma, y=-electron.y,type="l",main="Bethe-Bloch Curve of an Electron moving through Silicon", xlab = TeX("$\\beta\\cdot\\gamma$"),ylab=TeX("$-dE/dx$"),col="red",cex.main=0.8)

v <- seq(0.8*c,c,100000)

beta <- v/c

#Lorentz factor

gamma <- 1/(sqrt(1-(v^2/c^2)))

n <- 1

m.e <- 0.005

electron.y = dE.dx(n=n,Z=Z,v=v,beta=beta)

m.e <- 273.13*m.e

pion.y = dE.dx(n=n,Z=Z,v=v,beta=beta)

plot(x=beta*gamma, y=-pion.y,type="l",main="Bethe-Bloch Curve of a Pion moving through Silicon \nat Speeds Upwards of 80% of the Speed of Light", xlab = TeX("$\\beta\\cdot\\gamma$"),ylab=TeX("$-dE/dx$"),col="blue",cex.main=0.8)

plot(x=beta*gamma, y=-electron.y,type="l",main="Bethe-Bloch Curve of an Electron moving through Silicon \nat Speeds Upwards of 80% of the Speed of Light", xlab = TeX("$\\beta\\cdot\\gamma$"),ylab=TeX("$-dE/dx$"),col="red",cex.main=0.8)

v <- seq(0.9*c,c,100000)

beta <- v/c

#Lorentz factor

gamma <- 1/(sqrt(1-(v^2/c^2)))

n <- 1

m.e <- 0.005

electron.y = dE.dx(n=n,Z=Z,v=v,beta=beta)

m.e <- 273.13*m.e

pion.y = dE.dx(n=n,Z=Z,v=v,beta=beta)

plot(x=beta*gamma, y=-pion.y,type="l",main="Bethe-Bloch Curve of a Pion moving through Silicon \nat Speeds Upwards of 90% of the Speed of Light", xlab = TeX("$\\beta\\cdot\\gamma$"),ylab=TeX("$-dE/dx$"),col="blue",cex.main=0.8)

plot(x=beta*gamma, y=-electron.y,type="l",main="Bethe-Bloch Curve of an Electron moving through Silicon \nat Speeds Upwards of 90% of the Speed of Light", xlab = TeX("$\\beta\\cdot\\gamma$"),ylab=TeX("$-dE/dx$"),col="red",cex.main=0.8)