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Posts Tagged ‘density’

A final pre-exam update of my notes compilation for ‘PHY452H1S Basic Statistical Mechanics’, Taught by Prof. Arun Paramekanti

Posted by peeterjoot on April 22, 2013

Here’s my third update of my notes compilation for this course, including all of the following:

April 21, 2013 Fermi function expansion for thermodynamic quantities

April 20, 2013 Relativistic Fermi Gas

April 10, 2013 Non integral binomial coefficient

April 10, 2013 energy distribution around mean energy

April 09, 2013 Velocity volume element to momentum volume element

April 04, 2013 Phonon modes

April 03, 2013 BEC and phonons

April 03, 2013 Max entropy, fugacity, and Fermi gas

April 02, 2013 Bosons

April 02, 2013 Relativisitic density of states

March 28, 2013 Bosons

plus everything detailed in the description of my previous update and before.

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Relativisitic Fermi gas

Posted by peeterjoot on April 20, 2013

[Click here for a PDF of this post with nicer formatting (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]

Question: Relativisitic Fermi gas ([1], pr 9.3)

Consider a relativisitic gas of N particles of spin 1/2 obeying Fermi statistics, enclosed in volume V, at absolute zero. The energy-momentum relation is

\begin{aligned}\epsilon = \sqrt{(p c)^2 + \epsilon_0^2},\end{aligned} \hspace{\stretch{1}}(1.1)

where \epsilon_0 = m c^2, and m is the rest mass.

Find the Fermi energy at density n.

With the pressure P defined as the average force per unit area exerted on a perfectly-reflecting wall of the container.

Set up expressions for this in the form of an integral.

Define the internal energy U as the average \epsilon - \epsilon_0.
Set up expressions for this in the form of an integral.

Show that P V = 2 U/3 at low densities, and P V = U/3 at high densities. State the criteria for low and high densities.

There may exist a gas of neutrinos (and/or antineutrinos) in the cosmos. (Neutrinos are massless Fermions of spin 1/2.) Calculate the Fermi energy (in eV) of such a gas, assuming a density of one particle per \text{cm}^3.

Attempt exact evaluation of the various integrals.

Answer

We’ve found [3] that the density of states associated with a 3D relativisitic system is

\begin{aligned}\mathcal{D}(\epsilon) = \frac{4 \pi V}{(c h)^3} \epsilon \sqrt{\epsilon^2 -\epsilon_0^2},\end{aligned} \hspace{\stretch{1}}(1.0.2)

For a given density n, we can find the Fermi energy in the same way as we did for the non-relativisitic energies, with the exception that we have to integrate from a lowest energy of \epsilon_0 instead of 0 (the energy at \mathbf{p} = 0). That is

\begin{aligned}n &= \frac{N}{V} \\ &= \left( 2 \frac{1}{{2}} + 1 \right)\frac{4 \pi}{(c h)^3} \int_{\epsilon_0}^{\epsilon_{\mathrm{F}}}d\epsilon \epsilon \sqrt{ \epsilon^2 -\epsilon_0^2} \\ &= \frac{8 \pi}{(c h)^3}\frac{1}{{3}} {\left.{{\left( x^2 - \epsilon_0^2 \right)^{3/2}}}\right\vert}_{{\epsilon_0}}^{{\epsilon_{\mathrm{F}}}} \\ &= \frac{8 \pi}{3 (c h)^3}\left( \epsilon_{\mathrm{F}}^2 - \epsilon_0^2 \right)^{3/2}.\end{aligned} \hspace{\stretch{1}}(1.0.2)

Solving for \epsilon_{\mathrm{F}}/\epsilon_0 we have

\begin{aligned}\frac{\epsilon_{\mathrm{F}}}{\epsilon_0} =\sqrt{\left( \frac{3 (c h)^3 n}{8 \pi \epsilon_0^3} \right)^{2/3}+ 1}.\end{aligned} \hspace{\stretch{1}}(1.0.2)

We’ll see the constant factor above a number of times below and designate it

\begin{aligned}n_0 = \frac{8 \pi}{3} \left( \frac{\epsilon_0}{c h} \right)^3,\end{aligned} \hspace{\stretch{1}}(1.0.2)

so that the Fermi energy is

\begin{aligned}\frac{\epsilon_{\mathrm{F}}}{\epsilon_0} =\sqrt{\left( \frac{n}{n_0} \right)^{2/3}+ 1}.\end{aligned} \hspace{\stretch{1}}(1.0.2)

For the pressure calculation, let’s suppose that we have a configuration with a plane in the x,y orientation as in fig. 1.1.

Fig 1.1: Pressure against x,y oriented plane

 

It’s argued in [4] section 6.4 that the pressure for such a configuration is

\begin{aligned}P = n \int p_z u_z f(\mathbf{u}) d^3 \mathbf{u},\end{aligned} \hspace{\stretch{1}}(1.7)

where n is the number density and f(\mathbf{u}) is a normalized distribution function for the velocities. The velocity and momentum components are related by the Hamiltonian equations. From the Hamiltonian eq. 1.1 we find \footnote{ Observe that by squaring and summing one can show that this is equivalent to the standard relativisitic momentum p_x = \frac{m v_x}{\sqrt{ 1 - \mathbf{u}^2/c^2}}.} (for the x-component which is representative)

\begin{aligned}u_x \\ &= \frac{\partial {\epsilon}}{\partial {p_x}} \\ &= \frac{\partial {}}{\partial {p_x}}\sqrt{(p c)^2 +\epsilon_0^2} \\ &= \frac{ p_x c^2 }{\sqrt{(p c)^2 +\epsilon_0^2}}.\end{aligned} \hspace{\stretch{1}}(1.8)

For \alpha \in \{1, 2, 3\} we can summarize these velocity-momentum relationships as

\begin{aligned}\frac{u_\alpha}{c} = \frac{ c p_\alpha }{ \epsilon }.\end{aligned} \hspace{\stretch{1}}(1.9)

Should we attempt to calculate the pressure with this parameterization of the velocity space we end up with convergence problems, and can’t express the results in terms of f^+_\nu(z). Let’s try instead with a distribution over momentum space

\begin{aligned}P=n \int \frac{(c p_z)^2}{\epsilon} f(c \mathbf{p}) d^3 (c \mathbf{p}).\end{aligned} \hspace{\stretch{1}}(1.10)

Here the momenta have been scaled to have units of energy since we want to express this integral in terms of energy in the end. Our normalized distribution function is

\begin{aligned}f(c \mathbf{p})\propto \frac{\frac{1}{{ z^{-1} e^{\beta \epsilon} + 1 }}}{\int \frac{1}{{ z^{-1} e^{\beta \epsilon} + 1 }} d^3 (c \mathbf{p})},\end{aligned} \hspace{\stretch{1}}(1.11)

but before evaluating anything, we first want to change our integration variable from momentum to energy. In spherical coordinates our volume element takes the form

\begin{aligned}d^3 (c \mathbf{p}) &= 2 \pi (c p)^2 d (c p) \sin\theta d\theta \\ &= 2 \pi (c p)^2 \frac{d (c p)}{d \epsilon} d \epsilon \sin\theta d\theta.\end{aligned} \hspace{\stretch{1}}(1.12)

Implicit derivatives of

\begin{aligned}c^2 p^2 = \epsilon^2 - \epsilon_0^2,\end{aligned} \hspace{\stretch{1}}(1.13)

gives us

\begin{aligned}\frac{d (c p)}{d\epsilon}= \frac{\epsilon}{c p}=\frac{\epsilon}{\sqrt{\epsilon^2 -\epsilon_0^2}}.\end{aligned} \hspace{\stretch{1}}(1.0.14)

Our momentum volume element becomes

\begin{aligned}d^3 (c \mathbf{p}) \\ &= 2 \pi (c p)^2 \frac{\epsilon}{\sqrt{\epsilon^2 - \epsilon_0^2 }}d \epsilon \sin\theta d\theta \\ &= 2 \pi \left( \epsilon^2 - \epsilon_0^2 \right)\frac{\epsilon}{\sqrt{\epsilon^2 - \epsilon_0^2 }}d \epsilon \sin\theta d\theta \\ &= 2 \pi \epsilon \sqrt{ \epsilon^2 - \epsilon_0^2} d \epsilon \sin\theta d\theta.\end{aligned} \hspace{\stretch{1}}(1.0.14)

For our distribution function, we can now write

\begin{aligned}f(c \mathbf{p}) d^3 (c \mathbf{p})= C \frac{\epsilon \sqrt{ \epsilon^2 - \epsilon_0^2} d \epsilon }{ z^{-1} e^{\beta \epsilon} + 1 }\frac{ 2 \pi \sin\theta d\theta }{ 4 \pi \epsilon_0^3 },\end{aligned} \hspace{\stretch{1}}(1.0.14)

where C is determined by the requirement \int f(c \mathbf{p}) d^3 (c \mathbf{p}) = 1

\begin{aligned}C^{-1} = \int_{0}^\infty \frac{(y + 1)\sqrt{ (y + 1)^2 - 1} dy }{ z^{-1} e^{\beta \epsilon_0 (y + 1)} + 1 }.\end{aligned} \hspace{\stretch{1}}(1.0.14)

The z component of our momentum can be written in spherical coordinates as

\begin{aligned}(c p_z)^2= (c p)^2 \cos^2\theta= \left( \epsilon^2 - \epsilon_0^2 \right)\cos^2\theta,\end{aligned} \hspace{\stretch{1}}(1.0.18)

Noting that

\begin{aligned}\int_0^\pi \cos^2\theta \sin\theta d\theta =-\int_0^\pi \cos^2\theta d(\cos\theta)= \frac{2}{3},\end{aligned} \hspace{\stretch{1}}(1.0.19)

all the bits come together as

\begin{aligned}P &= \frac{C n}{3 \epsilon_0^3 } \int_{\epsilon_0}^\infty\left( \epsilon^2 - \epsilon_0^2 \right)^{3/2} \frac{1}{{ z^{-1} e^{\beta \epsilon} + 1 }} d \epsilon \\ &= \frac{n \epsilon_0}{3} \int_{1}^\infty\left( x^2 - 1 \right)^{3/2} \frac{1}{{ z^{-1} e^{\beta \epsilon_0 x} + 1 }} dx.\end{aligned} \hspace{\stretch{1}}(1.0.19)

Letting y = x - 1, this is

\begin{aligned}P= \frac{C n \epsilon_0}{3} \int_{0}^\infty \frac{ \left( (y + 1)^2 - 1 \right)^{3/2} } { z^{-1} e^{\beta \epsilon_0 (y + 1)} + 1 } dy.\end{aligned} \hspace{\stretch{1}}(1.0.19)

We could conceivable expand the numerators of each of these integrals in power series, which could then be evaluated as a sum of f^+_\nu(z e^{-\beta \epsilon_0}) terms.

Note that above the Fermi energy n also has an integral representation

\begin{aligned}n &= \left(2\left( \frac{1}{{2}} \right) + 1\right)\int_{\epsilon_0}^\infty d\epsilon \mathcal{D}(\epsilon) \frac{1}{{ z^{-1} e^{\beta \epsilon} + 1}} \\ &= \frac{8 \pi}{(c h)^3} \int_{\epsilon_0}^\infty d\epsilon\frac{\epsilon \sqrt{\epsilon^2 - \epsilon_0^2} }{ z^{-1} e^{\beta \epsilon} + 1} \\ &= \frac{8 \pi \epsilon_0^3}{(c h)^3} \int_{0}^\infty dy\frac{(y + 1)\sqrt{(y + 1)^2 - 1} }{ z^{-1} e^{\beta \epsilon_0 (y + 1)} + 1},\end{aligned} \hspace{\stretch{1}}(1.0.19)

or

\begin{aligned}\boxed{n = \frac{3 n_0}{C}.}\end{aligned} \hspace{\stretch{1}}(1.0.23)

Observe that we can use this result to remove the dependence of pressure on this constant C

\begin{aligned}\boxed{\frac{P}{n_0 \epsilon_0}= \int_{0}^\infty dy \frac{ \left( (y + 1)^2 - 1 \right)^{3/2} } { z^{-1} e^{\beta \epsilon_0 (y + 1)} + 1 }.}\end{aligned} \hspace{\stretch{1}}(1.0.24)

Now for the average energy difference from the rest energy \epsilon_0

\begin{aligned}U &= \left\langle{{\epsilon - \epsilon_0}}\right\rangle \\ &= \int_{\epsilon_0}^\infty d\epsilon \mathcal{D}(\epsilon) f(\epsilon) (\epsilon - \epsilon_0) \\ &= \frac{8 \pi V}{(c h)^3}\int_{\epsilon_0}^\infty d\epsilon \frac{ \epsilon(\epsilon - \epsilon_0) \sqrt{ \epsilon^2 - \epsilon_0 } }{ z^{-1} e^{\beta \epsilon} + 1} \\ &= \frac{8 \pi V \epsilon_0^4}{(c h)^3}\int_{0}^\infty dy\frac{ y ( y - 1 ) \sqrt{ (y + 1)^2 - 1 }}{ z^{-1} e^{\beta \epsilon} + 1}.\end{aligned} \hspace{\stretch{1}}(1.0.24)

So the average energy density difference from the rest energy, relative to the rest energy, is

\begin{aligned}\boxed{\frac{\left\langle{{\epsilon - \epsilon_0}}\right\rangle}{V \epsilon_0} =3 n_0 \int_{0}^\infty dy \frac { y (y + 1)\sqrt{(y + 1)^2 - 1} } { z^{-1} e^{\beta \epsilon_0 (y + 1)} + 1 }.}\end{aligned} \hspace{\stretch{1}}(1.0.26)

From eq. 1.0.24 and eq. 1.0.26 we have

\begin{aligned}\frac{1}{{n_0}} &= 3 \frac{V \epsilon_0} {\left\langle{{\epsilon - \epsilon_0}}\right\rangle} \int_{0}^\infty \frac { y (y + 1)\sqrt{(y + 1)^2 - 1} dy } { z^{-1} e^{\beta \epsilon_0 (y + 1)} + 1 } \\ &= \frac{\epsilon_0}{P} \int_{0}^\infty \frac{ \left( (y + 1)^2 - 1 \right)^{3/2} } { z^{-1} e^{\beta \epsilon_0 (y + 1)} + 1 } dy,\end{aligned} \hspace{\stretch{1}}(1.0.26)

or

\begin{aligned}P V =\frac{U}{3}\frac{ \int_{0}^\infty \frac{ \left( (y + 1)^2 - 1 \right)^{3/2} } { z^{-1} e^{\beta \epsilon_0 (y + 1)} + 1 } dy}{ \int_{0}^\infty \frac { y (y + 1)\sqrt{(y + 1)^2 - 1} dy } { z^{-1} e^{\beta \epsilon_0 (y + 1)} + 1 }}.\end{aligned} \hspace{\stretch{1}}(1.0.26)

This ratio of integrals is supposed to resolve to 1 and 2 in the low and high density limits. To consider this let’s perform one final non-dimensionalization, writing

\begin{aligned}\begin{aligned} \\ x &= \beta \epsilon_0 y \\ \theta &= \frac{1}{{\beta \epsilon_0}} = \frac{k_{\mathrm{B}} T}{\epsilon_0} \\ \bar{\mu} &= \mu - \epsilon_0 \\ \bar{z} &= e^{\beta \bar{\mu}}.\end{aligned}\end{aligned} \hspace{\stretch{1}}(1.0.29)

The density, pressure, and energy take the form

\begin{aligned}\frac{n}{n_0}= 3 \theta\int_{0}^\infty dx\frac{(\theta x + 1)\sqrt{(\theta x + 1)^2 - 1} }{ \bar{z}^{-1} e^{x} + 1}\end{aligned} \hspace{\stretch{1}}(1.0.30a)

\begin{aligned}\frac{P}{n_0 \epsilon_0}= \theta \int_{0}^\infty dx \frac{ \left( (\theta x + 1)^2 - 1 \right)^{3/2} } { \bar{z}^{-1} e^{x} + 1 }\end{aligned} \hspace{\stretch{1}}(1.0.30b)

\begin{aligned}\frac{\left\langle{{\epsilon - \epsilon_0}}\right\rangle}{V \epsilon_0 n_0} =3 \theta^2 \int_{0}^\infty dx \frac { x (\theta x + 1)\sqrt{(\theta x + 1)^2 - 1} } { \bar{z}^{-1} e^{x} + 1 }.\end{aligned} \hspace{\stretch{1}}(1.0.30c)

We can rewrite the square roots in the number density and energy density expressions by expanding out the completion of the square

\begin{aligned}(1 + \theta x) \sqrt{ (1 + \theta x)^2 - 1}=(1 + \theta x) \sqrt{ 1 + \theta x + 1 }\sqrt{ 1 + \theta x - 1 }= \sqrt{2 \theta} x^{1/2} (1 + \theta x) \sqrt{ 1 + \frac{\theta x}{2}},\end{aligned} \hspace{\stretch{1}}(1.0.30c)

Expanding the distribution about \bar{z} e^{-x} = 0, we have

\begin{aligned}\frac{1}{ \bar{z}^{-1} e^{x} + 1}=\frac{\bar{z} e^{-x}}{ 1 + \bar{z} e^{-x}}=z e^{-x} \sum_{s = 0}^\infty (-1)^s \left( \bar{z} e^{-x} \right)^s,\end{aligned} \hspace{\stretch{1}}(1.0.32)

allowing us to write, in the low density limit with respect to \bar{z}

\begin{aligned}\frac{n}{n_0}= 3 \sqrt{2}\theta^{3/2} \sum_{s=0}^\infty(-1)^s\bar{z}^{s + 1}\int_{0}^\infty dx x^{1/2}(1 + \theta x) \sqrt{ 1 + \frac{\theta x}{2}} e^{-x(1 + s)} \end{aligned} \hspace{\stretch{1}}(1.0.33a)

\begin{aligned}\frac{P}{n_0 \epsilon_0}= \theta\sum_{s=0}^\infty(-1)^s\bar{z}^{s + 1} \int_{0}^\infty dx\left( (\theta x + 1)^2 - 1 \right)^{3/2} e^{-x(1 + s)} \end{aligned} \hspace{\stretch{1}}(1.0.33b)

\begin{aligned}\frac{\left\langle{{\epsilon - \epsilon_0}}\right\rangle}{V \epsilon_0 n_0} =3 \sqrt{2} \theta^{5/2} \sum_{s=0}^\infty(-1)^s\bar{z}^{s + 1} \int_{0}^\infty dx x^{3/2} (1 + \theta x) \sqrt{ 1 + \frac{\theta x}{2}} e^{-x(1 + s)} .\end{aligned} \hspace{\stretch{1}}(1.0.33c)

Low density result

An exact integration of the various integrals above is possible in terms of special functions. However, that attempt (included below) introduced an erroneous extra factor of \theta. Given that this end result was obtained by tossing all but the lowest order terms in \theta and \bar{z}, let’s try that right from the get go.

For the pressure we have an integrand containing a factor

\begin{aligned}\left( (\theta x + 1)^2 -1 \right)^{3/2}&= \left( \theta x + 1 - 1 \right)^{3/2}\left( \theta x + 1 + 1 \right)^{3/2} \\ &= \theta^{3/2} x^{3/2} 2^{3/2} \left( 1 + \frac{\theta x}{2} \right)^{3/2} \\ &= 2 \sqrt{2} \theta^{3/2} x^{3/2} \left( 1 + \frac{\theta x}{2} \right)^{3/2}\approx2 \sqrt{2} \theta^{3/2} x^{3/2} \end{aligned} \hspace{\stretch{1}}(1.0.33c)

Our pressure, to lowest order in \theta and \bar{z} is then

\begin{aligned}\frac{P}{\epsilon_0 n_0} = 2 \sqrt{2} \theta^{5/2} \bar{z} \int_0^\infty x^{3/2} e^{-x} dx= 2 \sqrt{2} \theta^{5/2} \bar{z} \Gamma(5/2).\end{aligned} \hspace{\stretch{1}}(1.0.33c)

Our energy density to lowest order in \theta and \bar{z} from eq. 1.0.33c is

\begin{aligned}\frac{U}{V \epsilon_0 n_0} &= 3 \sqrt{2} \theta^{5/2} \bar{z} \int_{0}^\infty dx x^{3/2} e^{-x} \\ &= 3 \sqrt{2} \theta^{5/2} \bar{z} \Gamma(5/2).\end{aligned} \hspace{\stretch{1}}(1.0.33c)

Comparing these, we have

\begin{aligned}\frac{1}{{\epsilon_0 n_0\sqrt{2} \theta^{5/2} \bar{z} \Gamma(5/2)}} &= 3 \frac{V}{U} \\ &= \frac{2}{P},\end{aligned} \hspace{\stretch{1}}(1.0.37)

or in this low density limit

\begin{aligned}\boxed{P V = \frac{2}{3} U.}\end{aligned} \hspace{\stretch{1}}(1.0.38)

High density limit

For the high density limit write \bar{z} = e^y, so that the distribution takes the form

\begin{aligned}f(\bar{z}) &= \frac{1}{ \bar{z}^{-1} e^{x} + 1} \\ &= \frac{1}{ e^{x - y} + 1}.\end{aligned} \hspace{\stretch{1}}(1.0.39)

This can be approximated by a step function, so that

\begin{aligned}\frac{P}{n_0 \epsilon_0}\approx \int_{0}^y \theta dx\left( (\theta x + 1)^2 - 1 \right)^{3/2} \end{aligned} \hspace{\stretch{1}}(1.0.40a)

\begin{aligned}\frac{U}{V \epsilon_0 n_0} \approx3 \int_{0}^\infty \theta dx \theta x (\theta x + 1)\sqrt{(\theta x + 1)^2 - 1} \end{aligned} \hspace{\stretch{1}}(1.0.40b)

With a change of variables u = \theta x + 1, we have

\begin{aligned}\begin{aligned}\frac{P}{n_0 \epsilon_0} &\approx \int_{1}^{\theta y + 1x} du\left( u^2 - 1 \right)^{3/2} \\ &=\frac{1}{8} \left((2 \theta y (\theta y+2)-3) \sqrt{\theta y (\theta y+2)} (\theta y+1)+3 \ln \left(\theta y+\sqrt{\theta y (\theta y+2)}+1\right)\right) \\ &\approx\frac{1}{4} \left( \theta \ln \bar{z} \right)^4\end{aligned}\end{aligned} \hspace{\stretch{1}}(1.0.41a)

\begin{aligned}\begin{aligned}\frac{U}{V \epsilon_0 n_0} &\approx3 \int_{1}^{\theta y + 1x} (u^2 - u)\sqrt{u^2 - 1} \\ &=\frac{3}{24} \left(\sqrt{\theta y (\theta y+2)} (\theta y (2 \theta y (3 \theta y+5)-1)+3)-3 \left(\ln \left(\theta y+\sqrt{\theta y (\theta y+2)}+1\right)\right)\right) \\ &\approx\frac{3}{4} \left( \theta \ln \bar{z} \right)^4\end{aligned}\end{aligned} \hspace{\stretch{1}}(1.0.41b)

Comparing both we have

\begin{aligned}\frac{4}{\epsilon_0 n_0 \left( \theta \ln \bar{z} \right) } = \frac{1}{{P}} = \frac{3 V}{U},\end{aligned} \hspace{\stretch{1}}(1.0.42)

or

\begin{aligned}\boxed{P V = \frac{1}{{3}} U.}\end{aligned} \hspace{\stretch{1}}(1.0.43)

\begin{aligned}{\left.{{\epsilon_{\mathrm{F}}}}\right\vert}_{{n = 1/(0.01)^3}} = 6.12402 \times 10^{-35} \text{J} \times 6.24150934 \times 10^{18} \frac{\text{eV}}{\text{J}} = 3.82231 \times 10^{-16} \text{eV}\end{aligned} \hspace{\stretch{1}}(1.0.43)

Wow. That’s pretty low!

Pressure integral

Of these the pressure integral is yields directly to Mathematica

\begin{aligned}\begin{aligned} \int_{0}^\infty & dx\left( (\theta x + 1)^2 - 1 \right)^{3/2} e^{-x(1 + s)} \\ &=\frac{3 \theta e^{(s+1)/\theta}}{(s + 1)^2} K_2\left( \frac{s+1}{\theta } \right) \\ &=\frac{3 \sqrt{\frac{\pi }{2}} \theta ^{3/2}}{(s+1)^{5/2}}+\frac{45 \sqrt{\frac{\pi }{2}} \theta ^{5/2}}{8 (s+1)^{7/2}}+\frac{315 \sqrt{\frac{\pi }{2}} \theta ^{7/2}}{128 (s+1)^{9/2}}-\frac{945 \sqrt{\frac{\pi }{2}} \theta ^{9/2}}{1024 (s+1)^{11/2}}+\frac{31185 \sqrt{\frac{\pi }{2}} \theta ^{11/2}}{32768 (s+1)^{13/2}} + \cdots\end{aligned}\end{aligned} \hspace{\stretch{1}}(1.0.45)

where K_2(z) is a modified Bessel function [5] of the second kind as plotted in fig. 1.2.

Fig 1.2: Modified Bessel function of the second kind

 

Plugging this into the series for the pressure, we have

\begin{aligned}\frac{P}{n_0 \epsilon_0}= 3 \left( \frac{k_{\mathrm{B}} T}{\epsilon_0} \right)^2\sum_{s=0}^\infty(-1)^s\frac{\left( \bar{z} e^{\epsilon_0/k_{\mathrm{B}} T} \right)^{s + 1}}{(s + 1)^2}K_2\left( (s+1) \epsilon_0/k_{\mathrm{B}} T \right).\end{aligned} \hspace{\stretch{1}}(1.0.46)

Plotting the summands 3 (-1)^s \frac{\theta^2}{(s + 1)^2} \left( \bar{z} e^{ 1/\theta} \right)^{s + 1} K_2\left((s+1)/\theta\right) for \bar{z} = 1 in fig. 1.4 shows that this mix of exponential Bessel and quadratic terms decreases with s.

Plotting this sum in fig. 1.3 numerically to 10 terms, shows that we have a function that appears roughly polynomial in \bar{z} and \theta.

Fig 1.3: Pressure to ten terms in z and theta

 

Fig 1.4: Pressure summands

 

For small \bar{z} it can be seen graphically that there is very little contribution from anything but the s = 0 term of this sum. An expansion in series for a few terms in \bar{z} and \theta gives us

\begin{aligned}\begin{aligned}\frac{P}{\epsilon_0 n_0}&=\sqrt{\pi} \theta^{5/2} \left(\frac{3 \bar{z}}{\sqrt{2}}-\frac{3 \bar{z}^2}{8}+\frac{\bar{z}^3}{3 \sqrt{6}}-\frac{3 \bar{z}^4}{32 \sqrt{2}}+\frac{3 \bar{z}^5}{25 \sqrt{10}}\right) \\ &+\sqrt{\pi} \theta^{7/2} \left(\frac{45 \bar{z}}{8 \sqrt{2}}\right) -\frac{45 \bar{z}^2}{128}+\frac{5 \bar{z}^3}{24 \sqrt{6}}-\frac{45 \bar{z}^4}{1024 \sqrt{2}}+\frac{9 \bar{z}^5}{200 \sqrt{10}}\\ &+\sqrt{\pi} \theta^{9/2} \left(\frac{315 \bar{z}}{128 \sqrt{2}}-\frac{315 \bar{z}^2}{4096}+\frac{35 \bar{z}^3}{1152 \sqrt{6}}-\frac{315 \bar{z}^4}{65536 \sqrt{2}}+\frac{63 \bar{z}^5}{16000 \sqrt{10}}\right).\end{aligned}\end{aligned} \hspace{\stretch{1}}(1.0.47)

This allows a k_{\mathrm{B}} T \ll m c^2 and \bar{z} \ll 1 approximation of the pressure

\begin{aligned}\frac{P}{\epsilon_0 n_0} = \frac{3}{2} \sqrt{2 \pi} \bar{z} \theta^{5/2}.\end{aligned} \hspace{\stretch{1}}(1.0.48)

Number density integral

For the number density, it appears that we can evaluate the integral using integration from parts applied to eq. 1.0.30.30

\begin{aligned}\frac{n}{n_0}= \theta\int_{0}^\infty dx\frac{3 (\theta x + 1)\sqrt{(\theta x + 1)^2 - 1} }{ \bar{z}^{-1} e^{x} + 1}=\theta\int_{0}^\infty dx\left( \frac{d}{dx} \left( (\theta x + 1)^2 - 1 \right) ^{3/2} \right)\frac{1}{ \bar{z}^{-1} e^{x} + 1}={\left.{{\theta\left( (\theta x + 1)^2 - 1 \right)^{3/2}\frac{1}{ \bar{z}^{-1} e^{x} + 1}}}\right\vert}_{{0}}^{{\infty}}-\theta\int_{0}^\infty dx\left( (\theta x + 1)^2 - 1 \right)^{3/2}\frac{ -\bar{z}^{-1} e^{x} }{ \left( \bar{z}^{-1} e^{x} + 1 \right)^2}=\theta\int_{0}^\infty dx\left( (\theta x + 1)^2 - 1 \right)^{3/2}\frac{ \bar{z} e^{-x} }{ \left( 1 + \bar{z} e^{-x} \right)^2}.\end{aligned} \hspace{\stretch{1}}(1.0.48)

Expanding in series, gives us

\begin{aligned}\frac{n}{n_0}=\theta\sum_{s = 0}^\infty\binom{-2}{s}\bar{z}^{s + 1} \int_{0}^\infty dx\left( (\theta x + 1)^2 - 1 \right)^{3/2} e^{-x(s + 1)}=3 \theta^2\sum_{s = 0}^\infty\binom{-2}{s}\frac{\left( \bar{z} e^{1/\theta} \right)^{s + 1}}{(s + 1)^2}K_2\left( \frac{s+1}{\theta } \right).\end{aligned} \hspace{\stretch{1}}(1.0.48)

Here the binomial coefficient has the meaning given in the definitions of \statmechchapcite{nonIntegralBinomialSeries}, where for negative integral values of b we have

\begin{aligned}\binom{b}{s}\equiv(-1)^s \frac{-b}{-b + s} \binom{-b+s}{-b}.\end{aligned} \hspace{\stretch{1}}(1.0.51)

Expanding in series to a couple of orders in \theta and \bar{z} we have

\begin{aligned}\frac{n}{n_0} = \frac{\sqrt{2 \pi}}{36} \theta^{1/2} \left(\left(2 \sqrt{3} \bar{z} - 9/\sqrt{2} \right) \bar{z} +18 \right) \bar{z}+\frac{5 \sqrt{ 2 \pi}}{576} \theta^{3/2} \left(\left(4 \sqrt{3} \bar{z} - 27/\sqrt{2}\right) \bar{z} +108 \right) \bar{z}+ \cdots\end{aligned} \hspace{\stretch{1}}(1.0.52)

To first order in \theta and \bar{z} this is

\begin{aligned}\frac{n}{n_0} = \frac{1}{{2}} \sqrt{ 2 \pi } \bar{z} \theta^{1/2},\end{aligned} \hspace{\stretch{1}}(1.0.53)

which allows a relation to pressure

\begin{aligned}P V = 3 N (k_{\mathrm{B}} T)^2 /\epsilon_0.\end{aligned} \hspace{\stretch{1}}(1.0.54)

It’s kind of odd seeming that this is quadratic in temperature. Is there an error?

Energy integral

Starting from eq. 1.0.30c and integrating by parts we have

\begin{aligned}\frac{\left\langle{{\epsilon - \epsilon_0}}\right\rangle}{V \epsilon_0 n_0} &= 3 \theta^2 \int_{0}^\infty dx \frac { x (\theta x + 1)\sqrt{(\theta x + 1)^2 - 1} } { \bar{z}^{-1} e^{x} + 1 } \\ &= -\theta^2 \int_{0}^\infty dx\left( (\theta x + 1)^2 - 1 \right)^{3/2}\frac{d}{dx} \left( \frac{x} { \bar{z}^{-1} e^{x} + 1 } \right) \\ &= -\theta^2 \int_{0}^\infty dx\left( (\theta x + 1)^2 - 1 \right)^{3/2}\left( \frac{1} { \bar{z}^{-1} e^{x} + 1 } - \frac{x \bar{z}^{-1} e^{x} } { \left( \bar{z}^{-1} e^{x} + 1 \right)^2 } \right) \\ &= \theta^2 \int_{0}^\infty dx \left( (\theta x + 1)^2 - 1 \right)^{3/2} \frac{ (x - 1)\bar{z}^{-1} e^{x} - 1} { \left( \bar{z}^{-1} e^{x} + 1 \right)^2 } \\ &= \theta^2 \int_{0}^\infty dx \left( (\theta x + 1)^2 - 1 \right)^{3/2} \frac{ (x - 1)\bar{z} e^{-x} - \bar{z}^2 e^{-2 x}} { \left( 1 + \bar{z} e^{-x} \right)^2 } \\ &= \theta^2\sum_{s=0}^\infty \binom{-2}{s} \int_{0}^\infty dx \left( (\theta x + 1)^2 - 1 \right)^{3/2} \left( (x - 1)\bar{z} e^{-x} - \bar{z}^2 e^{-2 x} \right) (\bar{z} e^{-x})^s \\ &= \theta^2\sum_{s=0}^\infty \binom{-2}{s} \bar{z}^{s + 1} \int_{0}^\infty dx \left( (\theta x + 1)^2 - 1 \right)^{3/2} \left( (x - 1) e^{-x(s + 1)} - \bar{z} e^{-x(s + 2)} \right).\end{aligned} \hspace{\stretch{1}}(1.0.54)

The integral with the factor of x doesn’t have a nice closed form as before (if you consider the K_2 a nice closed form), but instead evaluates to a confluent hypergeometric function [6]. That integral is

\begin{aligned}\int_0^{\infty } x \left((\theta x+1)^2-1\right)^{3/2} e^{-x (1+s)} dx = \frac{15 \sqrt{\pi } \theta^3 U\left(-\frac{3}{2},-4,\frac{2 (s+1)}{\theta }\right)}{8 (s+1)^5},\end{aligned} \hspace{\stretch{1}}(1.0.54)

and looks like fig. 1.5. Series expansion shows that this hypergeometricU function has a \theta^{3/2} singularity at the origin

Fig 1.5: Plot of HypergeometricU, and with theta^5 scaling

 

\begin{aligned}U\left(-\frac{3}{2},-4,\frac{2 (s+1)}{\theta }\right)=\frac{2 \sqrt{2} \sqrt{s+1} s+2 \sqrt{2} \sqrt{s+1}}{\theta^{3/2}}+\frac{21 \sqrt{s+1}}{2 \sqrt{2} \sqrt{\theta }}+ \cdots\end{aligned} \hspace{\stretch{1}}(1.57)

so our multiplication by \theta^5 brings us to zero as seen in the plot. Evaluating the complete integral yields the unholy mess

\begin{aligned}\frac{\left\langle{{\epsilon - \epsilon_0}}\right\rangle}{V \epsilon_0 n_0} &= \sum_{s=0}^\infty \theta^2 (-1)^s (s+1) \bar{z}^{s+1} \Bigl( \\ &\frac{105 \sqrt{\pi } \theta^3 U\left(-\frac{1}{2},-4,\frac{2 (s+1)}{\theta }\right)}{16 (s+1)^5} \\ &- \frac{3 \sqrt{\pi } \theta^2 U\left(-\frac{1}{2},-2,\frac{2 (s+1)}{\theta }\right)}{2 (s+1)^3} \\ &- \frac{3 \sqrt{\pi } \theta^2 \bar{z} U\left(-\frac{1}{2},-2,\frac{2 (s+2)}{\theta }\right)}{2 (s+2)^3} \\ &+\frac{(\theta -2) (-3 \theta +2 s+2) e^{\frac{s+1}{\theta }} K_2\left(\frac{s+1}{\theta }\right)}{\theta (s+1)^2} \\ &-\frac{2 (\theta -2) e^{\frac{s+1}{\theta }} K_1\left(\frac{s+1}{\theta }\right)}{\theta (s+1)} \\ &+\frac{\bar{z} (-3 \theta +2 s+4) e^{\frac{s+2}{\theta }} K_2\left(\frac{s+2}{\theta }\right)}{(s+2)^2} \\ &-\frac{2 \bar{z} e^{\frac{s+2}{\theta }} K_1\left(\frac{s+2}{\theta }\right)}{s+2} \Bigr),\end{aligned} \hspace{\stretch{1}}(1.58)

to first order in \bar{z} and \theta this is

\begin{aligned}\frac{\left\langle{{\epsilon - \epsilon_0}}\right\rangle}{V \epsilon_0 n_0} =\frac{9}{4} \sqrt{2 \pi} \bar{z} \theta^{7/2}.\end{aligned} \hspace{\stretch{1}}(1.59)

Comparing pressure and energy we have for low densities (where \bar{z} \approx 0)

\begin{aligned}\frac{1}{{\epsilon_0 n_0 \sqrt{2 \pi} \bar{z} \theta^{5/2}}} = \frac{3}{2} \frac{1}{{P}} = \frac{9}{4} \theta \frac{V}{U},\end{aligned} \hspace{\stretch{1}}(1.0.60)

or

\begin{aligned}\theta P V = \frac{2}{3} U.\end{aligned} \hspace{\stretch{1}}(1.0.61)

It appears that I’ve picked up an extra factor of \theta somewhere, but at least I’ve got the 2/3 low density expression. Given that I’ve Taylor expanded everything anyways around \bar{z} and \theta this could likely have been done right from the get go, instead of dragging along the messy geometric integrals. Reworking this part of this problem like that was done above.

References

[1] Kerson Huang. Introduction to statistical physics. CRC Press, 2001.

[2] Peeter Joot. Basic statistical mechanics., chapter {Non integral binomial coefficient}. \natexlab{a}. URL http://sites.google.com/site/peeterjoot2/math2013/phy452.pdf.

[3] Peeter Joot. Basic statistical mechanics., chapter {Relativisitic density of states}. \natexlab{b}. URL http://sites.google.com/site/peeterjoot2/math2013/phy452.pdf.

[4] RK Pathria. Statistical mechanics. Butterworth Heinemann, Oxford, UK, 1996.

[5] Wolfram. BesselK, \natexlab{a}. URL http://reference.wolfram.com/mathematica/ref/BesselK.html. [Online; accessed 11-April-2013].

[6] Wolfram. HyperGeometricU, \natexlab{b}. URL http://reference.wolfram.com/mathematica/ref/HypergeometricU.html. [Online; accessed 17-April-2013].

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PHY452H1S Basic Statistical Mechanics. Lecture 19: Bosons. Taught by Prof. Arun Paramekanti

Posted by peeterjoot on March 28, 2013

[Click here for a PDF of this post with nicer formatting (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]

Disclaimer

Peeter’s lecture notes from class. May not be entirely coherent.

Fermions summary

We’ve considered a momentum sphere as in fig. 1.1, and performed various appromations of the occupation sums fig. 1.2.

Fig 1.1: Summation over momentum sphere

Fig 1.2: Fermion occupation

\begin{aligned}\epsilon \sim T^2\end{aligned} \hspace{\stretch{1}}(1.0.1.1)

\begin{aligned}C \sim T\end{aligned} \hspace{\stretch{1}}(1.0.1.1)

\begin{aligned}P \sim \text{constant}\end{aligned} \hspace{\stretch{1}}(1.0.1.1)

The physics of Fermi gases has an extremely wide range of applicability. Illustrating some of this range, here are some examples of Fermi temperatures (from E_{\mathrm{F}} = k_{\mathrm{B}} T_{\mathrm{F}})

  1. Electrons in copper: T_{\mathrm{F}} \sim 10^4 \mbox{K}
  2. Neutrons in neutron star: T_{\mathrm{F}} \sim 10^7 - 10^8 \mbox{K}
  3. Ultracold atomic gases: T_{\mathrm{F}} \sim (10 - 100) \mbox{n K}

Bosons

We’d like to work with a fixed number of particles, but the calculations are hard, so we move to the grand canonical ensemble

\begin{aligned}n_{\mathrm{B}}(\mathbf{k}) = \frac{1}{{ e^{\beta(\epsilon_\mathbf{k} - \mu)} - 1 }}\end{aligned} \hspace{\stretch{1}}(1.2)

Again, we’ll consider free particles with energy as in fig. 1.3, or

\begin{aligned}\epsilon_\mathbf{k} = \frac{\hbar^2 k^2}{2 m}.\end{aligned} \hspace{\stretch{1}}(1.3)

Fig 1.3: Free particle energy momentum distribution

 

Again introducing fugacity z = e^{\beta \mu}, we have

\begin{aligned}n_{\mathrm{B}}(\mathbf{k}) = \frac{1}{{ z^{-1} e^{\beta \epsilon_\mathbf{k}} - 1 }}\end{aligned} \hspace{\stretch{1}}(1.4)

We’ll consider systems for which

\begin{aligned}N = \sum_\mathbf{k} n_{\mathrm{B}}(\mathbf{k}) = \text{fixed}\end{aligned} \hspace{\stretch{1}}(1.5)

Observe that at large energies we have

\begin{aligned}n_{\mathrm{B}}(\text{large} \, \mathbf{k}) \sim z e^{-\beta \epsilon_\mathbf{k}}\end{aligned} \hspace{\stretch{1}}(1.6)

For small energies

\begin{aligned}n_{\mathrm{B}}(\mathbf{k} \rightarrow 0) \sim \frac{1}{{z^{-1} - 1}} = \frac{z}{1 - z}\end{aligned} \hspace{\stretch{1}}(1.7)

Observe that we require z < 1 (or \mu < 0) so that the number distribution is strictly positive for all energies. This tells us that the fugacity is a function of temperature, but there will be a point at which it must saturate. This is illustrated in fig. 1.4.

Fig 1.4: Density times cubed thermal de Broglie wavelength

 

Let’s calculate this density (assumed fixed for all temperatures)

\begin{aligned}\rho &= \frac{N}{V} \\ &= \int \frac{d^3 \mathbf{k}}{(2 \pi)^3} \frac{1}{{z^{-1} e^{\beta \epsilon_\mathbf{k}} -1 }} \\ &= \frac{2}{(2 \pi)^2} \int_0^\infty k^2 dk \frac{1}{{z^{-1} e^{\beta \hbar^2 k^2/2m} -1 }} \\ &= \frac{2}{(2 \pi)^2} \left( \frac {2 m} {\beta \hbar^2} \right)^{3/2}\int_0^\infty \left( \frac {\beta \hbar^2} {2 m} \right)^{3/2}k^2 dk \frac{1}{{z^{-1} e^{\beta \hbar^2 k^2/2m} -1 }}\end{aligned} \hspace{\stretch{1}}(1.8)

With the substitution

\begin{aligned}x^2 = \beta \frac{\hbar^2 k^2}{2m},\end{aligned} \hspace{\stretch{1}}(1.9)

we find

\begin{aligned}\rho \lambda^3 &= \frac{2}{(2 \pi)^2} \left( \frac {2 \not{{m}}} {\not{{\beta \hbar^2}}} \right)^{3/2}\left( \frac{ 2 \pi \not{{\hbar^2 \beta}}}{\not{{m}}} \right)^{3/2}\int_0^\infty x^2 dx \frac{1}{{z^{-1} e^{x^2} -1 }} \\ &= \frac{4}{\sqrt{\pi}} \int_0^\infty dx \frac{x^2}{z^{-1} e^{x^2} - 1 } \\ &\equiv g_{3/2}(z).\end{aligned} \hspace{\stretch{1}}(1.10)

This implicitly defines a relationship for the fugacity as a function of temperature z = z(T).

It can be shown that

\begin{aligned}g_{3/2}(z) = z + \frac{z^2}{2^{3/2}}+ \frac{z^3}{3^{3/2}}+ \cdots\end{aligned} \hspace{\stretch{1}}(1.11)

As z \rightarrow 1 we end up with a zeta function, for which we can look up the value

\begin{aligned}g_{3/2}(z \rightarrow 1) = \sum_{n = 1}^\infty \frac{1}{{n^{3/2}}} = \zeta(3/2) \approx 2.612\end{aligned} \hspace{\stretch{1}}(1.12)

where the Riemann zeta function is defined as

\begin{aligned}\zeta(s) = \sum_{ n = 1 } \frac{1}{{n^s}}.\end{aligned} \hspace{\stretch{1}}(1.13)

\begin{aligned}g_{3/2}(z) = \rho \lambda^3\end{aligned} \hspace{\stretch{1}}(1.14)

At high temperatures we have

\begin{aligned}\rho \lambda^3 \rightarrow 0\end{aligned} \hspace{\stretch{1}}(1.15)

(as T does down, \rho \lambda^3 goes up)

Looking at g_{3/2}(z = 1) = \rho \lambda^3(T_{\mathrm{c}}) leads to

\begin{aligned}\boxed{k_{\mathrm{B}} T_{\mathrm{c}} = \left( \frac{\rho}{\zeta(3/2)} \right)^{2/3} \frac{ 2 \pi \hbar^2}{m}.}\end{aligned} \hspace{\stretch{1}}(1.16)

How do I satisfy number conservation?

We have a problem here since as T \rightarrow 0 the 1/\lambda^3 \sim T^{3/2} term in \rho above drops to zero, yet g_{3/2}(z) cannot keep increasing without bounds to compensate and keep the density fixed. The way to deal with this was worked out by

  1. Bose (1924) for photons (examining statistics for symmetric wave functions).
  2. Einstein (1925) for conserved particles.

To deal with this issue, we (somewhat arbitrarily, because we need to) introduce a non-zero density for \mathbf{k} = 0. This is an adjustment of the approximation so that we have

\begin{aligned}\sum_{\mathbf{k}} \rightarrow \int \frac{d^3 \mathbf{k}}{(2 \pi)^3} \qquad \mbox{Except around k = 0},\end{aligned} \hspace{\stretch{1}}(1.17)

as in fig. 1.5, so that

Fig 1.5: Momentum sphere with origin omitted

 

\begin{aligned}\sum_\mathbf{k} = \left( \mbox{Contribution at k = 0} \right)+ V \int \frac{d^3 \mathbf{k}}{(2 \pi)^3}.\end{aligned} \hspace{\stretch{1}}(1.18)

Given this, we have

\begin{aligned}N= N_{\mathbf{k} = 0}+ V \int \frac{d^3 \mathbf{k}}{(2 \pi)^3} n_{\mathrm{B}}(\mathbf{k})\end{aligned} \hspace{\stretch{1}}(1.19)

We can illustrate this as in fig. 1.6.

Fig 1.6: Boson occupation vs momentum

 

\begin{aligned}\rho= \rho_{\mathbf{k} = 0}+ \frac{1}{{\lambda^3}} g_{3/2}(z)= \rho_{\mathbf{k} = 0}+ \frac{ \lambda(T_{\mathrm{c}}) }{ \lambda(T)}\frac{1}{{ \lambda^3(T_{\mathrm{c}})}}g_{3/2}(z)\end{aligned} \hspace{\stretch{1}}(1.20)

At T > T_{\mathrm{c}} we have \rho_{\mathbf{k} = 0}, whereas at T < T_{\mathrm{c}} we must introduce a non-zero density if we want to be able to keep a constant density constraint.

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An updated compilation of notes, for ‘PHY452H1S Basic Statistical Mechanics’, Taught by Prof. Arun Paramekanti

Posted by peeterjoot on March 27, 2013

Here’s my second update of my notes compilation for this course, including all of the following:

March 27, 2013 Fermi gas

March 26, 2013 Fermi gas thermodynamics

March 26, 2013 Fermi gas thermodynamics

March 23, 2013 Relativisitic generalization of statistical mechanics

March 21, 2013 Kittel Zipper problem

March 18, 2013 Pathria chapter 4 diatomic molecule problem

March 17, 2013 Gibbs sum for a two level system

March 16, 2013 open system variance of N

March 16, 2013 probability forms of entropy

March 14, 2013 Grand Canonical/Fermion-Bosons

March 13, 2013 Quantum anharmonic oscillator

March 12, 2013 Grand canonical ensemble

March 11, 2013 Heat capacity of perturbed harmonic oscillator

March 10, 2013 Langevin small approximation

March 10, 2013 Addition of two one half spins

March 10, 2013 Midterm II reflection

March 07, 2013 Thermodynamic identities

March 06, 2013 Temperature

March 05, 2013 Interacting spin

plus everything detailed in the description of my first update and before.

Posted in Math and Physics Learning. | Tagged: , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | 1 Comment »

PHY452H1S Basic Statistical Mechanics. Lecture 16: Fermi gas. Taught by Prof. Arun Paramekanti

Posted by peeterjoot on March 27, 2013

[Click here for a PDF of this post with nicer formatting (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]

Disclaimer

Peeter’s lecture notes from class. May not be entirely coherent.

Fermi gas

Review

Continuing a discussion of [1] section 8.1 content.

We found

\begin{aligned}n_{\mathbf{k}} = \frac{1}{{e^{\beta(\epsilon_k - \mu)} + 1}}\end{aligned} \hspace{\stretch{1}}(1.2.1)

With no spin

\begin{aligned}\int n_\mathbf{k} \times \frac{d^3 k}{(2\pi)^3} = \rho\end{aligned} \hspace{\stretch{1}}(1.2.2)

Fig 1.1: Occupancy at low temperature limit

 

Fig 1.2: Volume integral over momentum up to Fermi energy limit

 

\begin{aligned}\epsilon_{\mathrm{F}} = \frac{\hbar^2 k_{\mathrm{F}}^2}{2m}\end{aligned} \hspace{\stretch{1}}(1.2.3)

gives

\begin{aligned}k_{\mathrm{F}} = (6 \pi^2 \rho)^{1/3}\end{aligned} \hspace{\stretch{1}}(1.2.4)

\begin{aligned}\sum_\mathbf{k} n_\mathbf{k} = N\end{aligned} \hspace{\stretch{1}}(1.2.5)

\begin{aligned}\mathbf{k} = \frac{2\pi}{L}(n_x, n_y, n_z)\end{aligned} \hspace{\stretch{1}}(1.2.6)

This is for periodic boundary conditions \footnote{I filled in details in the last lecture using a particle in a box, whereas this periodic condition was intended. We see that both achieve the same result}, where

\begin{aligned}\Psi(x + L) = \Psi(x)\end{aligned} \hspace{\stretch{1}}(1.2.7)

Moving on

\begin{aligned}\sum_{k_x} n(\mathbf{k}) = \sum_{p_x} \Delta p_x n(\mathbf{k})\end{aligned} \hspace{\stretch{1}}(1.2.8)

with

\begin{aligned}\Delta k_x = \frac{2 \pi}{L} \Delta p_x\end{aligned} \hspace{\stretch{1}}(1.2.9)

this gives

\begin{aligned}\sum_{k_x} n(\mathbf{k}) = \sum_{n_x} \frac{L}{2\pi} \Delta k_x \rightarrow \frac{L}{2\pi} \int d k_x\end{aligned} \hspace{\stretch{1}}(1.2.10)

Over all dimensions

\begin{aligned}\sum_{\mathbf{k}} n_\mathbf{k} = \left( \frac{L}{2\pi} \right)^3 \left( \int d^3 \mathbf{k} \right)n(\mathbf{k})=N\end{aligned} \hspace{\stretch{1}}(1.2.11)

so that

\begin{aligned}\rho = \int \frac{d^3 \mathbf{k}}{(2 \pi)^3}\end{aligned} \hspace{\stretch{1}}(1.2.12)

Again

\begin{aligned}k_{\mathrm{F}} = (6 \pi^2 \rho)^{1/3}\end{aligned} \hspace{\stretch{1}}(1.2.13)

Example: Spin considerations

{example:basicStatMechLecture16:1}{

\begin{aligned}\sum_{\mathbf{k}, m_s} = N\end{aligned} \hspace{\stretch{1}}(1.2.14)

\begin{aligned}\sum_{\mathbf{k}, m_s} \frac{1}{{e^{\beta(\epsilon_k - \mu)} + 1}} = (2 S + 1)\left( \int \frac{d^3 \mathbf{k}}{(2 \pi)^3} n(\mathbf{k}) \right)L^3\end{aligned} \hspace{\stretch{1}}(1.2.15)

This gives us

\begin{aligned}k_{\mathrm{F}} = \left( \frac{ 6 \pi^2 \rho }{2 S + 1} \right)^{1/3}\end{aligned} \hspace{\stretch{1}}(1.2.16)

and again

\begin{aligned}\epsilon_{\mathrm{F}} = \frac{\hbar^2 k_{\mathrm{F}}^2}{2m}\end{aligned} \hspace{\stretch{1}}(1.2.17)

}

High Temperatures

Now we want to look at the at higher temperature range, where the occupancy may look like fig. 1.3

Fig 1.3: Occupancy at higher temperatures

 

\begin{aligned}\mu(T = 0) = \epsilon_{\mathrm{F}}\end{aligned} \hspace{\stretch{1}}(1.2.18)

\begin{aligned}\mu(T \rightarrow \infty) \rightarrow - \infty\end{aligned} \hspace{\stretch{1}}(1.2.19)

so that for large T we have

\begin{aligned}\frac{1}{{e^{\beta(\epsilon_k - \mu)} + 1}} \rightarrow e^{-\beta(\epsilon_k - \mu)}\end{aligned} \hspace{\stretch{1}}(1.2.20)

\begin{aligned}\rho &= \int \frac{d^3 \mathbf{k}}{(2 \pi)^3} e^{\beta \mu} e^{-\beta \epsilon_k} \\ &= e^{\beta \mu} \int \frac{d^3 \mathbf{k}}{(2 \pi)^3} e^{-\beta \epsilon_k} \\ &= e^{\beta \mu} \int dk \frac{4 \pi k^2}{(2 \pi)^3} e^{-\beta \hbar^2 k^2/2m}.\end{aligned} \hspace{\stretch{1}}(1.2.21)

Mathematica (or integration by parts) tells us that

\begin{aligned}\frac{1}{{(2 \pi)^3}} \int 4 \pi^2 k^2 dk e^{-a k^2} = \frac{1}{{(4 \pi a )^{3/2}}},\end{aligned} \hspace{\stretch{1}}(1.2.22)

so we have

\begin{aligned}\rho &= e^{\beta \mu} \left( \frac{2m}{ 4 \pi \beta \hbar^2} \right)^{3/2} \\ &= e^{\beta \mu} \left( \frac{2 m k_{\mathrm{B}} T 4 \pi^2 }{ 4 \pi h^2} \right)^{3/2} \\ &= e^{\beta \mu} \left( \frac{2 m k_{\mathrm{B}} T \pi }{ h^2} \right)^{3/2}\end{aligned} \hspace{\stretch{1}}(1.2.23)

Introducing \lambda for the thermal de Broglie wavelength, \lambda^3 \sim T^{-3/2}

\begin{aligned}\lambda \equiv \frac{h}{\sqrt{2 \pi m k_{\mathrm{B}} T}},\end{aligned} \hspace{\stretch{1}}(1.2.24)

we have

\begin{aligned}\rho = e^{\beta \mu} \frac{1}{{\lambda^3}}.\end{aligned} \hspace{\stretch{1}}(1.2.25)

Does it make any sense to have density as a function of temperature? An inappropriately extended to low temperatures plot of the density is found in fig. 1.4 for a few arbitrarily chosen numerical values of the chemical potential \mu, where we see that it drops to zero with temperature. I suppose that makes sense if we are not holding volume constant.

Fig 1.4: Density as a function of temperature

 

We can write

\begin{aligned}\boxed{e^{\beta \mu} = \left( \rho \lambda^3 \right)}\end{aligned} \hspace{\stretch{1}}(1.2.26)

\begin{aligned}\frac{\mu}{k_{\mathrm{B}} T} = \ln \left( \rho \lambda^3 \right)\sim -\frac{3}{2} \ln T\end{aligned} \hspace{\stretch{1}}(1.2.27)

or (taking \rho (and/or volume?) as a constant) we have for large temperatures

\begin{aligned}\mu \propto -T \ln T\end{aligned} \hspace{\stretch{1}}(1.2.28)

The chemical potential is plotted in fig. 1.5, whereas this - k_{\mathrm{B}} T \ln k_{\mathrm{B}} T function is plotted in fig. 1.6. The contributions to \mu from the k_{\mathrm{B}} T \ln (\rho h^3 (2 \pi m)^{-3/2}) term are dropped for the high temperature approximation.

Fig 1.5: Chemical potential over degenerate to classical range

Fig 1.6: High temp approximation of chemical potential, extended back to T = 0

Pressure

\begin{aligned}P = - \frac{\partial {E}}{\partial {V}}\end{aligned} \hspace{\stretch{1}}(1.2.29)

For a classical ideal gas as in fig. 1.7 we have

Fig 1.7: Ideal gas pressure vs volume

 

\begin{aligned}P = \rho k_{\mathrm{B}} T\end{aligned} \hspace{\stretch{1}}(1.2.30)

For a Fermi gas at T = 0 we have

\begin{aligned}E &= \sum_\mathbf{k} \epsilon_k n_k \\ &= \sum_\mathbf{k} \epsilon_k \Theta(\mu_0 - \epsilon_k) \\ &= \frac{V}{(2\pi)^3} \int_{\epsilon_k < \mu_0} \frac{\hbar^2 \mathbf{k}^2}{2 m} d^3 \mathbf{k} \\ &= \frac{V}{(2\pi)^3} \int_0^{k_{\mathrm{F}}} \frac{\hbar^2 \mathbf{k}^2}{2 m} d^3 \mathbf{k} \\ &= \frac{V}{(2\pi)^3} \frac{\hbar^2}{2 m} \int_0^{k_{\mathrm{F}}} k^2 4 \pi k^2 d k\propto k_{\mathrm{F}}^5\end{aligned} \hspace{\stretch{1}}(1.2.31)

Specifically,

\begin{aligned}E(T = 0) = V \times \frac{3}{5} \underbrace{\epsilon_{\mathrm{F}}}_{\sim k_{\mathrm{F}}^2}\underbrace{\rho}_{\sim k_{\mathrm{F}}^3}\end{aligned} \hspace{\stretch{1}}(1.2.32)

or

\begin{aligned}\frac{E}{N} = \frac{3}{5} \epsilon_{\mathrm{F}}\end{aligned} \hspace{\stretch{1}}(1.2.33)

\begin{aligned}E = \frac{3}{5} N \frac{\hbar^2}{2 m} \left( 6 \pi^2 \frac{N}{V} \right)^{2/3} = a V^{-2/3},\end{aligned} \hspace{\stretch{1}}(1.2.34)

so that

\begin{aligned}\frac{\partial {E}}{\partial {V}} = -\frac{2}{3} a V^{-5/3}.\end{aligned} \hspace{\stretch{1}}(1.2.35)

\begin{aligned}P &= -\frac{\partial {E}}{\partial {V}}  \\ &= \frac{2}{3} \left( a V^{-2/3} \right)V^{-1} \\ &= \frac{2}{3} \frac{E}{V} \\ &= \frac{2}{3} \left( \frac{3}{5} \epsilon_{\mathrm{F}} \rho \right) \\ &= \frac{2}{5} \epsilon_{\mathrm{F}} \rho.\end{aligned} \hspace{\stretch{1}}(1.2.36)

We see that the pressure ends up deviating from the classical result at low temperatures, as sketched in fig. 1.8. This low temperature limit for the pressure 2 \epsilon_{\mathrm{F}} \rho/5 is called the degeneracy pressure.

Fig 1.8: Fermi degeneracy pressure

 

References

[1] RK Pathria. Statistical mechanics. Butterworth Heinemann, Oxford, UK, 1996.

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PHY452H1S Basic Statistical Mechanics. Lecture 17: Fermi gas thermodynamics. Taught by Prof. Arun Paramekanti

Posted by peeterjoot on March 26, 2013

[Click here for a PDF of this post with nicer formatting (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]

Disclaimer

Peeter’s lecture notes from class. May not be entirely coherent.

Fermi gas thermodynamics

  • Energy was found to be

    \begin{aligned}\frac{E}{N} = \frac{3}{5} \epsilon_{\mathrm{F}}\qquad \text{where} \quad T = 0.\end{aligned} \hspace{\stretch{1}}(1.2.1)

  • Pressure was found to have the form fig. 1.1

    Fig 1.1: Pressure in Fermi gas

  • The chemical potential was found to have the form fig. 1.2.

    \begin{aligned}e^{\beta \mu} = \rho \lambda_{\mathrm{T}}^3\end{aligned} \hspace{\stretch{1}}(1.0.2a)

    \begin{aligned}\lambda_{\mathrm{T}} = \frac{h}{\sqrt{ 2 \pi m k_{\mathrm{B}} T}},\end{aligned} \hspace{\stretch{1}}(1.0.2b)

    so that the zero crossing is approximately when

    \begin{aligned}e^{\beta \times 0} = 1 = \rho \lambda_{\mathrm{T}}^3.\end{aligned} \hspace{\stretch{1}}(1.0.3)

    That last identification provides the relation T \sim  T_{\mathrm{F}}. FIXME: that bit wasn’t clear to me.

    Fig 1.2: Chemical potential in Fermi gas

How about at other temperatures?

  • \mu(T) = ?
  • E(T) = ?
  • C_{\mathrm{V}}(T) = ?

We had

\begin{aligned}N = \sum_k \frac{1}{{e^{\beta (\epsilon_k - \mu)} + 1}} = \sum_{\mathbf{k}} n_{\mathrm{F}}(\epsilon_\mathbf{k})\end{aligned} \hspace{\stretch{1}}(1.0.4)

\begin{aligned}E(T) =\sum_k \epsilon_\mathbf{k} n_{\mathrm{F}}(\epsilon_\mathbf{k}).\end{aligned} \hspace{\stretch{1}}(1.0.5)

FIXME: references to earlier sections where these were derived.

We can define a density of states

\begin{aligned}\sum_\mathbf{k} &= \sum_\mathbf{k} \int_{-\infty}^\infty d\epsilon  \delta(\epsilon  - \epsilon_\mathbf{k}) \\ &= \int_{-\infty}^\infty d\epsilon \sum_\mathbf{k}\delta(\epsilon  - \epsilon_\mathbf{k}),\end{aligned} \hspace{\stretch{1}}(1.0.6)

where the liberty to informally switch the order of differentiation and integration has been used. This construction allows us to write a more general sum

\begin{aligned}\sum_\mathbf{k} f(\epsilon_\mathbf{k}) &= \sum_\mathbf{k} \int_{-\infty}^\infty d\epsilon  \delta(\epsilon  - \epsilon_\mathbf{k}) f(\epsilon_\mathbf{k}) \\ &= \sum_\mathbf{k}\int_{-\infty}^\infty d\epsilon \delta(\epsilon  - \epsilon_\mathbf{k})f(\epsilon_\mathbf{k}) \\ &=\int_{-\infty}^\infty d\epsilon  f(\epsilon_\mathbf{k})\left( \sum_\mathbf{k} \delta(\epsilon  - \epsilon_\mathbf{k}) \right).\end{aligned} \hspace{\stretch{1}}(1.0.7)

This sum, evaluated using a continuum approximation, is

\begin{aligned}N(\epsilon ) &\equiv \sum_\mathbf{k}\delta(\epsilon  - \epsilon_\mathbf{k}) \\ &= \frac{V}{(2 \pi)^3} \int d^3 \mathbf{k} \delta\left( \epsilon  - \frac{\hbar^2 k^2}{2 m} \right) \\ &= \frac{V}{(2 \pi)^3} 4 \pi \int_0^\infty k^2 dk \delta\left( \epsilon  - \frac{\hbar^2 k^2}{2 m} \right)\end{aligned} \hspace{\stretch{1}}(1.0.8)

Using

\begin{aligned}\delta(g(x)) = \sum_{x_0} \frac{\delta(x - x_0)}{\left\lvert {g'(x_0)} \right\rvert},\end{aligned} \hspace{\stretch{1}}(1.0.9)

where the roots of g(x) are x_0, we have

\begin{aligned}N(\epsilon ) &= \frac{V}{(2 \pi)^3} 4 \pi \int_0^\infty k^2 dk \delta\left( k - \frac{\sqrt{2 m \epsilon }}{\hbar} \right)\frac{m \hbar }{ \hbar^2 \sqrt{2 m \epsilon }} \\ &= \frac{V}{(2 \pi)^3} 2 \pi \frac{2 m \epsilon }{\hbar^2}\frac{2 m \hbar }{ \hbar^2 \sqrt{2 m \epsilon }} \\ &= V \left( \frac{2 m}{\hbar^2} \right)^{3/2} \frac{1}{{4 \pi^2}} \sqrt{\epsilon }.\end{aligned} \hspace{\stretch{1}}(1.0.10)

In 2D this would be

\begin{aligned}N(\epsilon ) \sim  V \int dk k \delta \left( \epsilon  - \frac{\hbar^2 k^2}{2m} \right) = V \frac{\sqrt{2 m \epsilon }}{\hbar} \frac{m \hbar}{\hbar^2 \sqrt{ 2 m \epsilon }} \sim  V\end{aligned} \hspace{\stretch{1}}(1.0.11)

and in 1D

\begin{aligned}N(\epsilon ) &\sim  V \int dk \delta \left( \epsilon  - \frac{\hbar^2 k^2}{2m} \right) \\ &= V \frac{m \hbar}{\hbar^2 \sqrt{ 2 m \epsilon }} \\ &\sim  \frac{1}{{\sqrt{\epsilon }}}.\end{aligned} \hspace{\stretch{1}}(1.0.12)

What happens when we have linear energy momentum relationships?

Suppose that we have a linear energy momentum relationship like

\begin{aligned}\epsilon_\mathbf{k} = v \left\lvert {\mathbf{k}} \right\rvert.\end{aligned} \hspace{\stretch{1}}(1.0.13)

An example of such a relationship is the high velocity relation between the energy and momentum of a particle

\begin{aligned}\epsilon_\mathbf{k} = \sqrt{ m_0^2 c^4 + p^2 c^2 } \sim  \left\lvert {\mathbf{p}} \right\rvert c.\end{aligned} \hspace{\stretch{1}}(1.0.14)

Another example is graphene, a carbon structure of the form fig. 1.3. The energy and momentum for such a structure is related in roughly as shown in fig. 1.4, where

Fig 1.3: Graphene bond structure

 

Fig 1.4: Graphene energy momentum dependence

 

\begin{aligned}\epsilon_\mathbf{k} = \pm v_{\mathrm{F}} \left\lvert {\mathbf{k}} \right\rvert.\end{aligned} \hspace{\stretch{1}}(1.0.15)

Continuing with the 3D case we have

FIXME: Is this (or how is this) related to the linear energy momentum relationships for Graphene like substances?

\begin{aligned}N = V \int_0^\infty\underbrace{n_{\mathrm{F}}(\epsilon )}_{1/(e^{\beta (\epsilon  - \mu)} + 1)}\underbrace{N(\epsilon )}_{\epsilon ^{1/2}}\end{aligned} \hspace{\stretch{1}}(1.0.16)

\begin{aligned}\rho &= \frac{N}{V} \\ &= \left( \frac{2m}{\hbar^2 } \right)^{3/2} \frac{1}{{ 4 \pi^2}}\int_0^\infty d\epsilon  \frac{\epsilon ^{1/2}}{z^{-1} e^{\beta \epsilon } + 1} \\ &= \left( \frac{2m}{\hbar^2 } \right)^{3/2} \frac{1}{{ 4 \pi^2}}\left( k_{\mathrm{B}} T \right)^{3/2}\int_0^\infty dx \frac{x^{1/2}}{z^{-1} e^{x} + 1}\end{aligned} \hspace{\stretch{1}}(1.0.17)

where z = e^{\beta \mu} as usual, and we write x = \beta \epsilon . For the low temperature asymptotic behavior see [1] appendix section E. For z large it can be shown that this is

\begin{aligned}\int_0^\infty dx \frac{x^{1/2}}{z^{-1} e^{x} + 1}\approx \frac{2}{3}\left( \ln z \right)^{3/2}\left( 1 + \frac{\pi^2}{8} \frac{1}{{(\ln z)^2}} \right),\end{aligned} \hspace{\stretch{1}}(1.0.18)

so that

\begin{aligned}\rho &\approx  \left( \frac{2m}{\hbar^2 } \right)^{3/2} \frac{1}{{ 4 \pi^2}}\left( k_{\mathrm{B}} T \right)^{3/2}\frac{2}{3}\left( \ln z \right)^{3/2}\left( 1 + \frac{\pi^2}{8} \frac{1}{{(\ln z)^2}} \right) \\ &= \left( \frac{2m}{\hbar^2 } \right)^{3/2} \frac{1}{{ 4 \pi^2}}\frac{2}{3}\mu^{3/2}\left( 1 + \frac{\pi^2}{8} \frac{1}{{(\beta \mu)^2}} \right) \\ &= \left( \frac{2m}{\hbar^2 } \right)^{3/2} \frac{1}{{ 4 \pi^2}}\frac{2}{3}\mu^{3/2}\left( 1 + \frac{\pi^2}{8} \left( \frac{k_{\mathrm{B}} T}{\mu} \right)^2 \right) \\ &= \rho_{T = 0}\left( \frac{\mu}{ \epsilon_{\mathrm{F}} } \right)^{3/2}\left( 1 + \frac{\pi^2}{8} \left( \frac{k_{\mathrm{B}} T}{\mu} \right)^2 \right)\end{aligned} \hspace{\stretch{1}}(1.0.19)

Assuming a quadratic form for the chemical potential at low temperature as in fig. 1.5, we have

Fig 1.5: Assumed quadratic form for low temperature chemical potential

 

\begin{aligned}1 &= \left( \frac{\mu}{ \epsilon_{\mathrm{F}} } \right)^{3/2}\left( 1 + \frac{\pi^2}{8} \left( \frac{k_{\mathrm{B}} T}{\mu} \right)^2 \right) \\ &= \left( \frac{\epsilon_{\mathrm{F}} - a T^2}{ \epsilon_{\mathrm{F}} } \right)^{3/2}\left( 1 + \frac{\pi^2}{8} \left( \frac{k_{\mathrm{B}} T}{\epsilon_{\mathrm{F}} - a T^2} \right)^2 \right) \\ &\approx  \left( 1 - \frac{3}{2} a \frac{T^2}{\epsilon_{\mathrm{F}}} \right)\left( 1 + \frac{\pi^2}{8} \frac{(k_{\mathrm{B}} T)^2}{\epsilon_{\mathrm{F}}^2} \right) \\ &\approx  1 - \frac{3}{2} a \frac{T^2}{\epsilon_{\mathrm{F}}} + \frac{\pi^2}{8} \frac{(k_{\mathrm{B}} T)^2}{\epsilon_{\mathrm{F}}^2},\end{aligned} \hspace{\stretch{1}}(1.0.20)

or

\begin{aligned}a = \frac{\pi^2}{12} \frac{k_{\mathrm{B}}^2}{\epsilon_{\mathrm{F}}},\end{aligned} \hspace{\stretch{1}}(1.0.21)

We have used a Taylor expansion (1 + x)^n \approx  1 + n x for small x, for an end result of

\begin{aligned}\mu = \epsilon_{\mathrm{F}} - \frac{\pi^2}{12} \frac{(k_{\mathrm{B}} T)^2}{\epsilon_{\mathrm{F}}}.\end{aligned} \hspace{\stretch{1}}(1.0.22)

References

[1] RK Pathria. Statistical mechanics. Butterworth Heinemann, Oxford, UK, 1996.

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