Approximate transmission coefficient of a square barrier

[bobby.pdx]

Homework Statement

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.

Homework Equations

T(E)≈exp((-2/hbar)√(2m)∫√(U(x)-E)dx)

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!

 

[BvU]

Isn't U-E a constant, so you can bring it outside the integral ?

 

[bobby.pdx]

That's what I did. Still gave me the same answer

 

[BvU]

Can't see it being done. Not telepathic and not clearvoyant. Show what you do, so we can discuss it...

 

[bobby.pdx]

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 into 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.

 

[bobby.pdx]

T(E)≈exp((-2/1.973keVA/c)√(2(511keV/c))∫(0.003keV)dx)=(0.907)^x

It should look like 0.96x10^-38

 

[bobby.pdx]

sorry I get exp(-.097x)

 

[TSny]

T(E)≈exp((-2/1.973keVA/c)√(2(511keV/c))∫(0.003keV)dx)=(0.907)^x

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?

 

[bobby.pdx]

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

 

[TSny]

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

T(E)≈exp((-2/hbar)√(2m)∫√(U(x)-E)dx)

 

[bobby.pdx]

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.

 

[TSny]

That's close to what I get, too. I get Exp[-1.77x] = Exp[-88.5] = .37x10^-38

 

[bobby.pdx]

cool. thanks for the help