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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} +

\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} +

\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 im 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 doesnt 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} +

\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} +

\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 im 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 doesnt make sense in a low temperature limit. Any help would be appreciated.

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