Question

# The potential energy difference between the tip of an electron scanning microscope and the surface is...

1. The potential energy difference between the tip of an electron scanning microscope and the surface is 2.09x10-18 J. The energy of an electron is 1.02x10-18 J. What length must the tip be from the surface for the probability of an electron tunneling from the tip to the surface is 0.75?

The probability of tunneling T is given by:

T = exp ( - 2GL)

which on rearranging gives:

L = - ln(T) / 2G

where ;

L = tunneling length

T = tunneling probability = 0.75

G = ( 2m (U - E) / h^2 )^0.5

where;

m = mass of electron = 9.1 X 10^-31 kg

U = potential energy barrier = 2.09 X 10^-18 J

E = energy of electron = 1.02 X 10^-18 J

h = reduced Planck's constant = 1.05 X 10^-34 J / s

Therefore;

G =  ( 2 X 9.1 X 10^-31 X (2.09 X 10^-18 - 1.02 X 10^-18) / (1.05 X 10^-34)^2 )^0.5

which gives;

G = 1.33 X 10^8

Substituting this in the equation;

L = - ln(T) / 2G

L = - ln(0.75) / ( 2 X 1.33 X 10^8)

L = 1.082 X 10^-9 m

L = 1.082 nm

which is the required length the tip must be from the surface.

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