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Approximate transmission coefficient of a square barrier

  1. Feb 25, 2014 #1
    1. The problem statement, all variables and given/known data
    Two conductors are separated by an insulator. Model the insulator as a square barrier of height 0.01 keV and a width of 5nm. Determine the transmission coefficients for electrons of 7,000 meV.

    The only thing is I have to use the approximation formula for finding the transmission coefficient of a barrier with arbitrary shape.

    2. Relevant equations

    T(E)≈exp((-2/hbar)√(2m)∫√(U(x)-E)dx)


    3. The attempt at a solution

    T(E)≈exp((-2/1.973keVA/c)√(2(511keV/c))∫(U(x)-E)dx)

    I'm not sure what I'm supposed to do for the ∫(U(x)-E)dx part. I solved the problem using the formula for the transmission coefficient for a square barrier and I got T≈0.96x10^-38. I'm pretty certain this is correct because there was a very similar example in my book.

    That formula had a (U-E) where I used 0.01keV-7000meV=.003keV

    If I enter .003keV for U(x)-E then I would get ≈(.907)^x

    And this is where I get stuck because I should somehow get an answer that is close to T≈0.96x10^-38
    Any help would be appreciated!
     
  2. jcsd
  3. Feb 25, 2014 #2

    BvU

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    Isn't U-E a constant, so you can bring it outside the integral ?
     
  4. Feb 25, 2014 #3
    That's what I did. Still gave me the same answer
     
  5. Feb 25, 2014 #4

    BvU

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    Can't see it being done. Not telepathic and not clearvoyant. Show what you do, so we can discuss it....
     
  6. Feb 25, 2014 #5
    I showed you exactly what I did. The first calculation under "3. The Attempt at a Solution". I put 0.03keV for U-E and I entered it exactly like that in to my calculator. I told you the answer I got and the answer I'm looking for. Not sure how I can be any more specific.
     
  7. Feb 25, 2014 #6
    T(E)≈exp((-2/1.973keVA/c)√(2(511keV/c))∫(0.003keV)dx)=(0.907)^x

    It should look like 0.96x10^-38
     
  8. Feb 25, 2014 #7
    sorry I get exp(-.097x)
     
  9. Feb 25, 2014 #8

    TSny

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    Did you drop a square root somewhere? Compare what you wrote here with your original expression in the "Relevant Equations" section.

    Note, you wrote the units of the mass of the electron as kev/c. Is this correct?
     
  10. Feb 25, 2014 #9
    I made a mistake in that calculation. The answer I got from that equation is actually exp(-.097x) which is still not correct. Also I meant to write the unit of mass as keV/c^2
     
  11. Feb 25, 2014 #10

    TSny

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    I don't get .097. Again, did you drop a square root? I see a square root symbol occurring twice in

     
  12. Feb 25, 2014 #11
    Oh you're completely right. Now I got (.169)^x. Now if I change 5nm to 50A I get (.169)^50=.287x10^-38. This is a way better answer even though it's not the same it's at least the same order of magnitude.
     
  13. Feb 25, 2014 #12

    TSny

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    That's close to what I get, too. I get Exp[-1.77x] = Exp[-88.5] = .37x10-38
     
    Last edited: Feb 25, 2014
  14. Feb 25, 2014 #13
    cool. thanks for the help
     
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