Minimum energy needed for a positrons to tunnel

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To determine the minimum kinetic energy required for positrons to tunnel through a 10 eV, 1 nm barrier, the relevant equations involve the tunneling probability p_T and the decay constant a. The decay constant a is calculated using the mass of the positron, the potential energy of the barrier, and the kinetic energy of the positrons. The discussion highlights that while theoretically, positrons can tunnel regardless of energy, the probability of tunneling becomes negligible at lower energies. The goal is to find the kinetic energy at which the average number of positrons successfully tunneling becomes significant.
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Homework Statement


10^12 positrons of the same kinetic energy are incident on 10eV, 1nm barrier. What is the minimum kinetic energy the positrons need before any of them can penetrate the barrier?

Homework Equations


p_T = e^(-2*a*L)
a = (sqrt(2m(V_0-E))/hbar

The Attempt at a Solution



For other problems I've plugged numbers into these equations to find non zero probabilities but this time I'm not sure how to solve it. I tried setting p_T to zero and rearranging for E but that just ended up as V_0=E.
 
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Theoretically, they can pass the barrier regardless of their energy. It is just that the probability will be vanishingly small.

What I would guess the problem wants you to do is to figure out when on average one positron will pass.

Can you clarify the question?
 

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