Question

Suppose a beam of 4.00 eV protons strikes a potential energy barrier of height 6.20 eV and thickness 0.560 nm, at a rate equivalent to a current of 1150 A. (a) How many years would you have to wait (on average) for one proton to be transmitted through the barrier? (b) How long would you have to wait if the beam consisted of electrons rather than protons?

Answer 1

Time taken by one proton  to be transmitted through the barrier,

t = 1 / nT

where n = rate of proton to reach the barrier

T = transmission coeffficient

n = l / q

n = 1150 / 1.6*10^(-19) = 7.18*10^21 / s

T = e^(-2bL)

here L = thickness

b = probability of tunneling

For proton,

b = sqrt [(8*pi^2*m*(U - E)) / h^2]

b = sqrt [(8*pi^2*1.67*10^-27*(6.2 - 5)*1.6*10^-19) / (6.62*10^-34)^2

b = 3.25*10^11 / m

so, T = e^(-2 * 3.25*10^11 * 0.56*10^(-9))

T = 6.66*10^(-159)

t = 1 / (7.18*10^21 *  6.66*10^(-159))

t = 2.09*10^136 s

(b)

For electron,

b = sqrt [(8*pi^2*m*(U - E)) / h^2]

b = sqrt [(8*pi^2*9.1*10^-31*(6.2 - 4)*1.6*10^-19) / (6.62*10^-34)^2

b = 7.59*10^9

T = e^(-2*7.59*10^9*0.56*10^(-9))

T = 0.0002017

t = 1 /  (7.18*10^21* 0.0002017)

t = 6.9*10^(-19) s