Peeter Joot's (OLD) Blog.

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Posts Tagged ‘occupation number’

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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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.

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PHY452H1S Basic Statistical Mechanics. Lecture 15: Grand Canonical/Fermion-Bosons. Taught by Prof. Arun Paramekanti

Posted by peeterjoot on March 14, 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.

Grand Canonical/Fermion-Bosons

Was mentioned that three dimensions confines us to looking at either Fermions or Bosons, and that two dimensions is a rich subject (interchange of two particles isn’t the same as one particle cycling around the other ending up in the same place — how is that different than a particle cycling around another in a two dimensional space?)

Definitions

  1. Fermion. Antisymmetric under exchange. n_k = 0, 1
  2. Boson. Symmetric under exchange. n_k = 0, 1, 2, \cdots

In either case our energies are

\begin{aligned}\epsilon_k = \frac{\hbar^2 k^2}{2m},\end{aligned} \hspace{\stretch{1}}(1.2.1)

For Fermions we’ll have occupation filling of the form fig. 1.1, where there can be only one particle at any given site (an energy level for that value of momentum). For Bosonic systems as in fig. 1.2, we don’t have a restriction of only one particle for each state, and can have any given number of particles for each value of momentum.

 

Fig 1.1: Fermionic energy level filling for free particle in a box

 

 

Fig 1.2: Bosonic free particle in a box energy level filling

 

Our Hamiltonian is

\begin{aligned}H = \sum_k \hat{n}_k \epsilon_k,\end{aligned} \hspace{\stretch{1}}(1.2.2)

where we have a number operator

\begin{aligned}N = \sum \hat{n}_k,\end{aligned} \hspace{\stretch{1}}(1.2.3)

such that

\begin{aligned}\left[{N},{H}\right] = 0.\end{aligned} \hspace{\stretch{1}}(1.2.4)

\begin{aligned}Z_{\mathrm{G}} = \sum_{N=0}^\infty e^{\beta \mu N}\sum_{n_k, \sum n_k = N} e^{-\beta \sum_k n_k \epsilon_k}.\end{aligned} \hspace{\stretch{1}}(1.2.5)

While the second sum is constrained, because we are summing over all n_k, this is essentially an unconstrained sum, so we can write

\begin{aligned}Z_{\mathrm{G}} &= \sum_{n_k}e^{\beta \mu \sum_k n_k}e^{-\beta \sum_k n_k \epsilon_k} \\ &= \sum_{n_k} \left( \prod_k e^{-\beta(\epsilon_k - \mu) n_k} \right) \\ &= \prod_{n} \left( \sum_{n_k} e^{-\beta(\epsilon_k - \mu) n_k} \right).\end{aligned} \hspace{\stretch{1}}(1.2.6)

Fermions

\begin{aligned}\sum_{n_k = 0}^1 e^{-\beta(\epsilon_k - \mu) n_k} = 1 + e^{-\beta(\epsilon_k - \mu)}\end{aligned} \hspace{\stretch{1}}(1.2.7)

Bosons

\begin{aligned}\sum_{n_k = 0}^\infty e^{-\beta(\epsilon_k - \mu) n_k} = \frac{1}{{1 - e^{-\beta(\epsilon_k - \mu)}}}\end{aligned} \hspace{\stretch{1}}(1.2.8)

(\epsilon_k - \mu \ge 0).

Our grand partition functions are then

\begin{aligned}Z_{\mathrm{G}}^f = \prod_k \left( 1 + e^{-\beta(\epsilon_k - \mu)} \right)\end{aligned} \hspace{\stretch{1}}(1.0.9a)

\begin{aligned}Z_{\mathrm{G}}^b = \prod_k \frac{1}{{ 1 - e^{-\beta(\epsilon_k - \mu)} }}\end{aligned} \hspace{\stretch{1}}(1.0.9b)

We can use these to compute the average number of particles

\begin{aligned}\left\langle{{n_k^f}}\right\rangle = \frac{1 \times 0 + e^{-\beta(\epsilon_k - \mu)} \times 1}{ 1 + e^{-\beta(\epsilon_k - \mu)} }=\frac{1}{{ 1 + e^{-\beta(\epsilon_k - \mu)} }}\end{aligned} \hspace{\stretch{1}}(1.0.10)

\begin{aligned}\left\langle{{n_k^b}}\right\rangle = \frac{1 \times 0 + e^{-\beta(\epsilon_k - \mu)} \times 1+e^{-2 \beta(\epsilon_k - \mu)} \times 2 + \cdots}{ 1+e^{-\beta(\epsilon_k - \mu)} +e^{-2 \beta(\epsilon_k - \mu)} }\end{aligned} \hspace{\stretch{1}}(1.0.11)

This chemical potential over temperature exponential

\begin{aligned}e^{\beta \mu} \equiv z,\end{aligned} \hspace{\stretch{1}}(1.0.12)

is called the fugacity. The denominator has the form

\begin{aligned}D = 1 + z e^{-\beta \epsilon_k}+ z^2 e^{-2 \beta \epsilon_k},\end{aligned} \hspace{\stretch{1}}(1.0.13)

so we see that

\begin{aligned}z \frac{\partial {D}}{\partial {z}} = z e^{-\beta \epsilon_k}+ 2 z^2 e^{-2 \beta \epsilon_k}+ 3 z^3 e^{-3 \beta \epsilon_k}+ \cdots\end{aligned} \hspace{\stretch{1}}(1.0.14)

Thus the numerator is

\begin{aligned}N = z \frac{\partial {D}}{\partial {z}},\end{aligned} \hspace{\stretch{1}}(1.0.15)

and

\begin{aligned}\left\langle{{n_k^b}}\right\rangle &= \frac{z \frac{\partial {D_k}}{\partial {z}} }{D_k} \\ &= z \frac{\partial {}}{\partial {z}} \ln D_k \\ &= \cdots \\ &= \frac{1}{{ e^{\beta(\epsilon_k - \mu)} - 1}}\end{aligned} \hspace{\stretch{1}}(1.0.16)

What is the density \rho?

For Fermions

\begin{aligned}\rho = \frac{N}{V} =\frac{1}{{V}} \sum_{\mathbf{k}}\frac{1}{{ e^{\beta(\epsilon_\mathbf{k} - \mu)} + 1}}\end{aligned} \hspace{\stretch{1}}(1.0.17)

Using a “particle in a box” quantization where k_\alpha = 2 \pi m_\alpha/L, in a d-dimensional space, we can approximate this as

\begin{aligned}\boxed{\rho = \int \frac{d^d k}{(2 \pi)^d}\frac{1}{{ e^{\beta(\epsilon_k - \mu)} - 1}}.}\end{aligned} \hspace{\stretch{1}}(1.0.18)

This integral is actually difficult to evaluate. For T \rightarrow 0 (\beta \rightarrow \infty, where

\begin{aligned}n_k = \Theta(\mu - \epsilon_k).\end{aligned} \hspace{\stretch{1}}(1.0.19)

This is illustrated in, where we also show the smearing that occurs as temperature increases fig. 1.3.

 

Fig 1.3: Occupation numbers for different energies

 

With

\begin{aligned}E_{\mathrm{F}} = \mu(T = 0),\end{aligned} \hspace{\stretch{1}}(1.0.20)

we want to ask what is the radius of the ball for which

\begin{aligned}\epsilon_k = E_{\mathrm{F}}\end{aligned} \hspace{\stretch{1}}(1.0.21)

or

\begin{aligned}E_{\mathrm{F}} = \frac{\hbar^2 k_{\mathrm{F}}^2}{2m},\end{aligned} \hspace{\stretch{1}}(1.0.22)

so that

\begin{aligned}k_{\mathrm{F}} = \sqrt{\frac{2 m E_{\mathrm{F}}}{\hbar^2}},\end{aligned} \hspace{\stretch{1}}(1.0.23)

so that our density where \epsilon_k = \mu is

\begin{aligned}\rho &= \int_{k \le k_{\mathrm{F}}} \frac{d^3 k}{(2 \pi)^3} \times 1 \\ &= \frac{1}{{(2\pi)^3}} 4 \pi \int^{k_{\mathrm{F}}} k^2 dk \\ &= \frac{4 \pi}{3} k_{\mathrm{F}}^3 \frac{1}{{(2 \pi)^3}},\end{aligned} \hspace{\stretch{1}}(1.0.24)

so that

\begin{aligned}k_{\mathrm{F}} = (6 \pi^2 \rho)^{1/3},\end{aligned} \hspace{\stretch{1}}(1.0.25)

Our chemical potential at zero temperature is then

\begin{aligned}\mu(T = 0) = \frac{\hbar^2}{2m} (6 \pi^2 \rho)^{2/3}.\end{aligned} \hspace{\stretch{1}}(1.0.26)

\begin{aligned}\rho^{-1/3} = \mbox{interparticle spacing}.\end{aligned} \hspace{\stretch{1}}(1.0.27)

We can convince ourself that the chemical potential must have the form fig. 1.4.

 

Fig 1.4: Large negative chemical potential at high temperatures

 

Given large negative chemical potential at high temperatures our number distribution will have the form

\begin{aligned}\left\langle{{n_k}}\right\rangle = e^{-\beta (\epsilon_k - \mu)} \propto e^{-\beta \epsilon_k}\end{aligned} \hspace{\stretch{1}}(1.0.28)

We see that the classical Boltzmann distribution is recovered for high temperatures.

We can also calculate the chemical potential at high temperatures. We’ll find that this has the form

\begin{aligned}e^{\beta \mu} = \frac{4}{3} \rho \lambda_T^3,\end{aligned} \hspace{\stretch{1}}(1.0.29)

where this quantity \lambda_T is called the Thermal de Broglie wavelength.

\begin{aligned}\lambda_T = \sqrt{\frac{ 2 \pi \hbar^2}{m k_{\mathrm{B}} T}}.\end{aligned} \hspace{\stretch{1}}(1.0.30)

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