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:
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Answer: 288.8 m
Explanation:
We have the following data:
is the time it takes to the child to reach the bottom of the slope
is the initial velocity (the child started from rest)
is the angle of the slope
is the length of the slope
Now, the Force exerted on the sled along the ramp is:
(1)
Where is the mass of the sled and
its acceleration
In addition, if we draw a free body diagram of this sled, the force along the ramp will be:
(2)
Where is the acceleration due gravity
Then:
(3)
Finding :
(4)
(5)
(6)
Now, we will use the following kinematic equations to find :
(7)
(8)
Where is the final velocity
Finding from (7):
(9)
(10)
Substituting (10) in (8):
(11)
Finding :