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Here are some quick questions: |
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The answers are sometimes (and possibly only indirectly) contained
in the links. |
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Write down the free enthalpy of a crystal with N atoms for
- 1 vacancy
- n vacancies
in terms of the relevant quantities "G", "H",
"S" for a single vacancy. |
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The binomial coefficient as defined below gives the number of possibilities that one has in selecting n
different elements from a given set of N elements with the condition that
different arrangements of the same elements do not
count ("Indistinguishable arrangements").
Example: Set = {1,2,3,4,5,6,7,8,9}; i.e. N = 9, Selected elements = {1,5,8},
i.e. n
= 3 = equivalent to {5,1, 8}; {1,8,5}, ...
How do you have to phrase
the question for the vacancy arrangement so that the answer is immediately obvious? |
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æ
è | N
k | ö
ø |
= | Binomial
coefficient |
= |
N!
(N – k)! · k! |
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Write the entropy of mixing with the binomial coefficient
from above and then express it with the help of the Stirling equation. |
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What is the chemical potential µ, and what must be valid for n vacancies
in equilibrium? |
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In the famous Boltzmann entropy equation S = kB · ln P the number
P could be a probability for a state; i.e. a number between 0 and 1, or the number of arrangements,
i.e. a huge number. Explain why this doesn't matter. |
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How is the formation entropy defined
formally? Why and how is it connected to a single vacancy? What does it describe or measure in practical terms? How large
is it (order of magnitude)? |
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Give some numbers for formation enthalpies of vacancies in common crystals.
Give some numbers for formation
enthalpies of self-interstitials in common crystals.
Draw some conclusion. |
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What kind of difference DH = Hi – HV
of the formation enthalpy of vacancy and self interstitial produces a concentration ratio of nV/ni
> 10? Do a quick and dirty estimation! |
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Write down the basic equation for the concentration of single vacancies. Produce a graph of this equation
with some numbers on the axes:
- in a direct representation.
- in an Arrhenius
representation. |
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Describe how you can tell from the nV curves for two
crystals in one Arrhenius plot which one has the larger HV
and SV. |
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What is the relation between the Boltzmann distribution (or Boltzmann factor) and the vacancy concentration?
How does on have to pose the question for the vacancy concentration to obtain the result directly from this? |
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Write down the free enthalpy
n2v and the resulting concentration c2v for divacancies
in a crystal with N atoms. |
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Generalize for nxv and cxv, i.e. for multiple
vacancy clusters. Use two different approaches for this. |
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Discuss the relative concentration of cxv/c1v
in equilibrium. |
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Describe the use of the mass action law
for obtaining vacancy concentrations. |
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© H. Föll (Defects - Script)
With frame
2.1.1 Simple Vacancies and Interstitials