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Hey folks,
studying for the qualifying exam and i wanted to run this by the forum. A model system has a density of states
g(E) = X for 0 < E < A
g(E) = 0 for A <= E <= B
g(E) = Y for B < E
the total number of particles in the system is given by N = X A.
Now this looks like a very simple model of a band gap. The first part of the problem asks us to consider particles that obey fermi-dirac statistics.
The first question asks for expressions for the total number of particles and the total energy. so we have
[tex] N = \int_0^{\infty} g(E) f(E) dE[/tex]
[tex] N = \int_0^{A} \frac{X dE}{e^{(E - \mu)/kT} + 1} + <br /> \int_B^{\infty} \frac{Y dE}{e^{(E - \mu)/kT} + 1}[/tex]
[tex] U = \int_0^{\infty} E g(E) f(E) dE[/tex]
[tex] U = \int_0^{A} \frac{X E dE}{e^{(E - \mu)/kT} + 1} + <br /> \int_B^{\infty} \frac{Y E dE}{e^{(E - \mu)/kT} + 1}[/tex]
Im fine with this much. Next part asks for the fermi energy as the temperature goes to 0. I know that the chemical potential at T=0 is equal to the fermi energy. and in this case it seems obvious that the fermi energy is A given that N = X A.
The third part is what I am having a bit of trouble with. We are asked to calculate the chemical potential at low but non-zero temperature. Now remember these problems are meant to be done without the use of calculators or integral tables. I see that I need to evaluate the integral for N and solve for the chemical potential. I can see a way to do it if i assume that T is low that
[tex] e^{(E - \mu) / kT} >> 1[/tex]
then the 1 in the denominator can be neglected but that is like switching to a classical distribution which doesn't make sense in a low temperature limit. Any help would be appreciated.
studying for the qualifying exam and i wanted to run this by the forum. A model system has a density of states
g(E) = X for 0 < E < A
g(E) = 0 for A <= E <= B
g(E) = Y for B < E
the total number of particles in the system is given by N = X A.
Now this looks like a very simple model of a band gap. The first part of the problem asks us to consider particles that obey fermi-dirac statistics.
The first question asks for expressions for the total number of particles and the total energy. so we have
[tex] N = \int_0^{\infty} g(E) f(E) dE[/tex]
[tex] N = \int_0^{A} \frac{X dE}{e^{(E - \mu)/kT} + 1} + <br /> \int_B^{\infty} \frac{Y dE}{e^{(E - \mu)/kT} + 1}[/tex]
[tex] U = \int_0^{\infty} E g(E) f(E) dE[/tex]
[tex] U = \int_0^{A} \frac{X E dE}{e^{(E - \mu)/kT} + 1} + <br /> \int_B^{\infty} \frac{Y E dE}{e^{(E - \mu)/kT} + 1}[/tex]
Im fine with this much. Next part asks for the fermi energy as the temperature goes to 0. I know that the chemical potential at T=0 is equal to the fermi energy. and in this case it seems obvious that the fermi energy is A given that N = X A.
The third part is what I am having a bit of trouble with. We are asked to calculate the chemical potential at low but non-zero temperature. Now remember these problems are meant to be done without the use of calculators or integral tables. I see that I need to evaluate the integral for N and solve for the chemical potential. I can see a way to do it if i assume that T is low that
[tex] e^{(E - \mu) / kT} >> 1[/tex]
then the 1 in the denominator can be neglected but that is like switching to a classical distribution which doesn't make sense in a low temperature limit. Any help would be appreciated.
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