Nature + Arrhenius

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There are so many processes   that are thermally activated. There are so many  processes that have Arrhenius-like behavior.   That are Arrhenius-like. And if you go to  Dartmouth then they'll give you goodie bags   with live crickets. And actually I really  hope not. But this is one of the labs that   they have where they take crickets and they  measure the number of times a cricket chirps.   And they're like, well okay. Let's measure the  cricket chirp over 13 seconds. We're gonna cool   them down, hopefully not too cold, and then we're  going to heat them up, hopefully not too hot.   Because crickets are nice, right? And so  then-- and they ca-- but look at that. And   they count it. And then what do they do? Well they  didn't know about Arrhenius yet until somebody   from MIT went and visited. So the first thing  they did is they plotted the data. Look at that.   Chirps per 13 seconds plotted. And they're all  sitting there trying to fit a straight line to   it. And then someone from this class is up  there visiting. They're like, you know what   i think, this looks like a thermally activated  process. So i think it's probably exponential.   And then they fit this nice exponential and  it fits the the cricket tripping beautifully.   And you can go even further because you see  if you got this far. Well now you see this   is a line. This is a line and we're going to do  that a lot when we go into reaction kinetics.   If you have a exponential and you take a log,  that's a line versus 1 over T. Right? That's a   line versus 1 over T. And so that's another way  you could look at data. They didn't do it there.   But, you know, you could plot for example--  you could plot 1 over T versus the log of   the number of vacancies. But the lattice-- the  number of vacancies is what we want. That ratio   is the concentration. That concentration  is in equilibrium at some temperature.   Okay. The lattice-- the number of lattice sites  is simply how many lattice sites you have,   in whatever volume you have, for whatever crystal  structure you have, for whatever element you have.   We'll see that in a few examples. So that's just  a concert-- it's the number of sites you have in   the chunk of material. And then instead of-- The  question this equation tells you the answer to,   is how many of those have a vacancy?  Because it's a thermally activated process.   And if you plot that log in Nv versus  temperature you get this really nice   linear line. And the slope of that  line is equal to minus E vacancy divided by R or it could be kB. R.  Let's write this again per mole.

Or it could be kB if it's  per atom. You will see both.   You will see both. And this  intercept-- intercept-- is equal to the-- let's see-- the intercept  is equal-- what do i have here? The   log of n. Did i write it right? Log of n.

Okay. Alright.

Now, okay. Oh yeah. What else can you  do? Well before we go on to the defects,   this explains the doping. I kept calling  the doping in semiconductors a thermally   activated process. But look at what happens.  This is the carrier concentration in that   conduction band. The thing you've  been you've been learning about,   right? And thinking about. But look at it  now versus temperature. It's a straight line.   It's a straight line. This is-- this  is experimentally what you observe.   And the reason is because it's a thermally  activated process. And in fact, in this case,   what is the activation energy? Right? The  activation energy for getting an electron into   the conduction band is the gap. Right? And so now  you say germanium has a smaller gap than silicon,   which has a smaller gap than gallium arsenide.  The slopes are different. The slopes are different   because the energy that it takes in that activated  process is the gap. That's why the slopes are   different. Right? Okay. Alright. 

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