# Kronig-penney E-k diagram

1. Oct 22, 2009

### enaj

1. The problem statement, all variables and given/known data

reproduce the kronig-penney E-k diagram (nothing more specific than that)

2. Relevant equations
f($$\alpha*a$$)=P*SIN($$\alpha*a$$)/($$\alpha*a$$)+COS($$\alpha*a$$)
where P is the strength of the potential (I've assumed that it's 3$$\pi$$/2)
and -1<=f($$\alpha*a$$)<=1

3. The attempt at a solution
so far I've plotted f($$\alpha*a$$), and as far as I can tell it looks right. But when I try to plot the allowed energy values from that graph against the k values, I get a graph that's a little too regular (though it does have band gaps) and it has no allowed states below ~k=0.6
I may be calculating k wrong. As I understand it, it's equal to the square root of the the energy from the graph of f($$\alpha*a$$), since I'm using $$\alpha*a$$ as my variable, which is related to energy in the same way k is: $$\sqrt{ 2mE/\hbar^2}$$

I've attached the graphs I have so far (the spreadsheet I used to calculate them is much too big).

Can anyone see where I might have gone wrong? I've replotted this many times and I can't seem to find it.

#### Attached Files:

• ###### bandgapgraph.jpg
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Last edited: Oct 22, 2009
2. Oct 23, 2009

### jdwood983

My guess is that your software isn't capable of handling precision calculations. I'd recommend using another program, rather than OpenOffice/MS Excel.

As far as the graphs go, they look more or less right; though that non-existence of states below k=0.6 is a little strange--have you changed your value of P to see what effect that has on the graph?