An important consideration in materials physics is the number of defects that are unavoidable due to statistical considerations.
Defects influence structural, electronic and optical properties. The equilibrium statistical physics of defects in a material can be modeled as a collection of N two-state systems. Every atomic site can be either, (a) occupied with an atom, in which case its energy is zero, or (b) occupied by a defect, in which case its energy is e. The formation energy (€) is the energy needed to remove an atom from the regular lattice position to an interstitial position, from which it diffuses out of the crystal
(a) Determine the partition function for this system, in terms of N, E, and T.
(b) What is the Helmholtz free energy (F) and the internal energy (U) for a sample of N atoms?
(c) What is the average number of defects, in terms of N, E, and T?
(d) Assume the formation energy is 1 eV. If you have a collection of spherical copper nanocrystals with a radius of 50 nm at room temperature, what is the minimum number of nanocrystals that you would need in a sample before you could expect (on average) that only one of them would have a single defect and the others would be defect free?
(a) Determine the partition function for this system, in terms of N, E, and T.
(b) What is the Helmholtz free energy (F) and the internal energy (U) for a sample of N atoms?
(c) What is the average number of defects, in terms of N, E, and T?
(d) Assume the formation energy is 1 eV. If you have a collection of spherical copper nanocrystals with a radius of 50 nm at room temperature, what is the minimum number of nanocrystals that you would need in a sample before you could expect (on average) that only one of them would have a single defect and the others would be defect free?
1 answer:
You might be interested in
Are you familiar with any basic calculus? If so, we can just look at this derivative and see what's happening with our units here..
Here it shows that acceleration is the derivative of velocity with respect to time. In other words, we can say that:
We can read that equation as: "acceleration is the change of velocity divided by the change in time (aka the time interval)."
If you're not familiar with calculus, we can use a simple equation of motion:
where:
vf = final velocity
vi = initial velocity
a = acceleration
t = observed time interval
We can rearrange this equation to find:
This is the same exact thing we wrote before!