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Solving for energy involving hyperbolic

  1. Dec 10, 2008 #1
    1. The problem statement, all variables and given/known data

    What energy (in eV) should a beam of electrons have so that 0.1% of them are able to tunnel through a barrier of height 7.0eV and 1.0 nm wide? Start with the equation for T(E) and set it up with 1/T(E) on one side and let E/U=x for the unknown. Solve the equation for x and then E.


    2. Relevant equations

    T(E) (1+.25(U2/(E(U-E)))sinh2([tex]\alpha[/tex]L))

    [tex]\alpha[/tex] = ((2m(U-E))1/2)/h


    3. The attempt at a solution

    I get
    0=x-x2-.25sinh2([tex]\alpha[/tex]L)

    this is after I make E/U=x. How can i solve for x and then solve for E?
     
  2. jcsd
  3. Dec 12, 2008 #2
    Have you tried to look for a numerical solution?

    EDIT: If you need to solve it analytically, you may need to rewrite the hyperbolic sine term another way:

    http://en.wikipedia.org/wiki/Hyperbolic_tangent#Standard_algebraic_expressions

    I actually derived this whole equation once. It required the use of an alternate expression for the hyperbolic sine term.

    The numerical solution seems easiest.
     
    Last edited: Dec 12, 2008
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