Ideal Fermi Gas

We talk about basics for an ideal Fermi gas.

Fermi-Dirac distribution

The particles in an atom trap are isolated from the surroundings, thus the atom number $N$ and total energy content $E_{\rm {tot}}$ of the atomic cloud is fixed. However, it is convenient to consider the system to be in contact with a reservoir, with which it can exchange particles and energy (grand canonical ensemble). For non-interacting particles with single-particle energies $\epsilon _{i}$, the average occupation of state $i$ is

$f(\epsilon _{i},\mu ,T)={\frac {1}{e^{(\epsilon _{i}-\mu )/kT}+1}}\,.$

These is the Fermi-Dirac distribution. For a fixed number of particles $N$ one chooses the chemical potential $\mu$ such that $N=\sum _{i}f(\epsilon _{i},\mu ,T)$.

Fermi Energy

A very direct consequence of the Fermi - Dirac distribution is the existence of Fermi energy $E_{F}$, defined as the energy of the highest occupied state of the non-interacting Fermi gas at $T=0$. In this case, the FD distribution takes the simple form, we have the simplified Fermi-Dirac distribution

Failed to parse (unknown function "\begin{array}"): {\displaystyle f(\vec{r},\vec{p},T) = \frac{1}{e^{(\frac{\vec{p}^2}{2m} + V(\vec{r}) - \mu)/k_B T} + 1} \stackrel{T \rightarrow 0} \rightarrow \left\{ \begin{array}{ll} 1, & \hbox{$\frac{\vec{p}^2}{2m} + V(\vec{r}) < \mu$} \\ 0, & \hbox{$\frac{\vec{p}^2}{2m} + V(\vec{r}) > \mu$} \\ \end{array} \right. }

The (globally) largest momentum is $p_{F}\equiv \hbar k_{F}\equiv {\sqrt {2mE_{F}}}$, the Fermi momentum. The $E_{F}$ can be readily calculated from atom number conservation.

Locally, at position ${\vec {r}}$ in the trap, it is $p_{F}({\vec {r}})\equiv \hbar k_{F}({\vec {r}})\equiv {\sqrt {2m\epsilon _{F}({\vec {r}})}}\equiv \hbar (6\pi ^{2}n_{F}({\vec {r}}))^{1/3}$ with the local Fermi energy $\epsilon _{F}({\vec {r}})$ which equals $\mu ({\vec {r}},T=0)=E_{F}-V({\vec {r}})$. The value of $E_{F}$ is fixed by the number of fermions $N$, occupying the $N$ lowest energy states of the trap.

Density distributions

We assume that the thermal energy $kT\equiv 1/\beta$ is much larger than the quantum mechanical level spacings $\hbar \omega _{x,y,z}$ (Thomas-Fermi approximation). In this case, the occupation of a phase space cell $\left\{{\vec {r}},{\vec {p}}\right\}$ (which is the phase-space density times $h^3$) is given by

$f({\vec {r}},{\vec {p}})={\frac {1}{e^{({\frac {{\vec {p}}^{2}}{2m}}+V({\vec {r}})-\mu )/kT}+1}}\,.$

The density distribution of the thermal gas is

$n_{th}({\vec {r}})=-{\frac {1}{\lambda _{dB}^{3}}}\,{\rm {Li}}_{3/2}\left(-e^{\beta \left(\mu -V({\vec {r}})\right)}\right)\,.$

where ${\sqrt {\frac {2\pi \hbar ^{2}}{mk_{B}T}}}$ is the de Broglie wavelength. ${\rm {Li}}_{n}(z)$ is the $n^{th}$-order Polylogarithm, defined as

${\rm {Li}}_{n}(z)\;\equiv \;{\frac {1}{\pi ^{n}}}\int {\rm {d}}^{2n}r{\frac {1}{e^{{\vec {r}}^{2}}/z-1}}\;{\stackrel {n\neq 0}{=}}\;{\frac {1}{\Gamma (n)}}\int _{0}^{\infty }{\rm {d}}q{\frac {q^{n-1}}{e^{q}/z-1}}\,.$

where the first integral is over $2n$ dimensions, ${\vec {r}}$ is the radius vector in $2n$ dimensions, $n$ is any positive half-integer or zero and $\Gamma (n)$ is the Gamma-function. The Polylogarithm can be expressed as a sum ${\rm {Li}}_{n}(z)=\sum _{k=1}^{\infty }{\frac {z^{k}}{k^{n}}}$ which is often used as the definition of the Polylogarithm. This expression is valid for all complex numbers $n$ and $z$ where $|z|\leq 1$. The definition given in the text is valid for all $z\leq l$.

Special cases: ${\rm {Li}}_{0}(z)={\frac {1}{1/z-1}}$, ${\rm {Li_{1}}}(z)=-\ln(1-z)$. $f({\vec {r}},{\vec {p}})$ can be written as $-{\rm {Li}}_{0}(-\exp[\beta (\mu -{\frac {{\vec {p}}^{2}}{2m}}-V({\vec {r}}))])$. When integrating density distributions to obtain column densities, a useful formula is:

$\int _{-\infty }^{\infty }dx\;{\rm {Li}}_{n}(z\,e^{-x^{2}})={\sqrt {\pi }}\;{\rm {Li}}_{n+1/2}(z)\,.$

Limiting values: ${\rm {Li}}_{n}(z){\stackrel {z\ll 1}{\rightarrow }}z$ and $-{\rm {Li}}_{n}(-z){\stackrel {z\rightarrow \infty }{\rightarrow }}{\frac {1}{\Gamma (n+1)}}\;\ln ^{n}(z)$.}. Note that expression for $n$ is correct for any potential $V({\vec {r}})$. The constraint on the number of thermal particles is

$N_{th}=\int {{\text{d}}r}\;n_{th}({\vec {r}})\,.$

Thermodynamic properties

Thermodynamic properties can be calculated as the ensemble average given the FD distribution.

Trapped Fermi Gas

Local Density Approximation

A very important approximation for trapped fermi gas is the $Localdensityapproximation$.

Free space Fermi gas

As a simple demonstration of all the definition defined above, we firstly demonstrate the case for a $3D$ fermi gas with $V({\vec {r}})=0$.

At$T=0$, the distribution can be simplified as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(\vec{r},\vec{p},T) = \frac{1}{e^{(\frac{\vec{p}^2}{2m} - \mu)/k_B T} + 1} \stackrel{T \rightarrow 0} \rightarrow \left\{ \begin{array}{ll} 1, & \hbox{$\frac{\vec{p}^2}{2m} < E_F$} \\ 0, & \hbox{$\frac{\vec{p}^2}{2m} > E_F$} \\ \end{array} \right. }

We therefore readity obtain the important result:

${\begin{array}{ll}E_{F}&={\frac {\hbar ^{2}}{2m}}\left({\frac {3\pi ^{2}N}{V}}\right)^{2/3}\\E_{tot}&={\frac {3}{5}}NE_{F}\end{array}}$

The density in this case is homogeneous across the whole volume with

$n\sim k_{F}^{3}$

The fluctuation of density $\delta n\rightarrow 0$ when $T\rightarrow 0$. This specific characters for fermions suggest that zero temperature degenerate fermi gas can be treated as a crystallined structure with interatiomic distance on the order of $1/k_{F}$.

Harmonically Trapped Fermi gas

A more realistic example in the experiments is the harmonically trapped Fermi gas. Applying these distributions to particles confined in a harmonic trap, with trapping potential

$V({\vec {r}})={\frac {1}{2}}m(\omega _{x}^{2}x^{2}+\omega _{y}^{2}y^{2}+\omega _{z}^{2}z^{2})\,.$

For a harmonic potential, we obtain

$N_{th}=-\left({\frac {k_{B}T}{\hbar {\bar {\omega }}}}\right)^{3}{\rm {Li}}_{3}(-\,e^{\beta \mu })$

with ${\bar {\omega }}=(\omega _{x}\omega _{y}\omega _{z})^{1/3}$ the geometric mean of the trapping frequencies.

In the classical limit at high temperature, we recover the Maxwell-Boltzmann result of a gaussian distribution,

$n_{cl}({\vec {r}})={\frac {N}{\pi ^{3/2}\sigma _{x}\sigma _{y}\sigma _{z}}}e^{-\sum _{i}x_{i}^{2}/\sigma _{x_{i}}^{2}}\qquad {\rm {with}}\;\sigma _{x,y,z}^{2}={\frac {2k_{B}T}{m\omega _{x,y,z}^{2}}}\,.$

The regime of quantum degeneracy is reached when $\lambda _{dB}\approx n^{-1/3}$, or when the temperature $T\approx T_{\rm {deg}}$. The degeneracy temperature $T_{\rm {deg}}={\frac {\hbar ^{2}}{2mk_{B}}}n^{2/3}$ is around or below one $\mu {\rm {K}}$ for typical experimental conditions.

For {\bf bosons}, it is at this point that the ground state becomes macroscopically occupied and the condensate forms. For {\bf fermions}, the occupation of available phase space cells smoothly approaches unity without any sudden transition:

Accordingly, also the density profile changes smoothly from its gaussian form at high temperatures to its zero temperature shape:

Failed to parse (unknown function "\begin{array}"): {\displaystyle \begin{array} n_F(\vec{r}) &=& \Intp{p} \, f(\vec{r},\vec{p}) \stackrel{T\rightarrow 0}{\rightarrow} \int_{\left|\vec{p}\right|< \sqrt{2m(\mu-V(\vec{r}))}} \frac{{\rm d}^3\vec{p}}{(2\pi\hbar)^3}\nonumber\\ &=& \frac{1}{6\pi^2} \left(\frac{2m}{\hbar^2}\right)^{3/2} \left(\mu - V(\vec{r})\right)^{3/2}. \end{array}}

In terms of local Fermi energy, For a harmonic trap, we obtain

Failed to parse (unknown function "\begin{array}"): {\displaystyle \begin{array} N &=& \Int{r} \; n_F(\vec{r}) = \frac{1}{6} \left(\frac{E_F}{\hbar \bar{\omega}}\right)^3\nonumber\\ \Rightarrow E_F &=& \hbar \bar{\omega} (6 N)^{1/3} \end{array}}

and for the zero-temperature profile

Failed to parse (unknown function "\begin{array}"): {\displaystyle \begin{array} n_F(\vec{r}) &=& \frac{8}{\pi^2} \frac{N}{R_{Fx} R_{Fy} R_{Fz}} \; \left[\max \left(1 - \sum_i \frac{x_i^2}{R_{Fi}^2},0\right)\right]^{3/2} \end{array}}

with the Fermi radii $R_{F{x,y,z}}={\sqrt {\frac {2E_{F}}{m\omega _{x,y,z}^{2}}}}$. The profile of the degenerate Fermi gas has a rather flat top compared to the gaussian profile of a thermal cloud, as the occupancy of available phase space cells saturates at unity.

At finite $T\lesssim T_{F}$, we can understand the shape of the cloud by comparing $kT$ with the local Fermi energy $\epsilon _{F}({\vec {r}})$.

For the outer regions in the trap where $kT\gg \epsilon _{F}({\vec {r}})$, the gas shows a classical (Boltzmann) density distribution $n({\vec {r}})\propto e^{-\beta V({\vec {r}})}$. In the inner part of the cloud where $k_{B}T\ll \epsilon _{F}({\vec {r}})$, the density is of the zero-temperature form $n({\vec {r}})\propto (E_{F}-V({\vec {r}}))^{3/2}$.

The Polylogarithm smoothly interpolates between the two regimes. We notice here the difficulty of thermometry for very cold Fermi clouds: Temperature only affects the far wings of the density distribution where the signal to noise ratio is poor. While for thermal clouds above $T_{F}$, the size of the cloud is a direct measure of temperature, for cold Fermi clouds one needs to extract the temperature from the shape of the distribution's wings. Note that the validity of the above derivation required the Fermi energy $E_{F}$ to be much larger than the level spacing $\hbar \omega _{x,y,z}$. For example, in very elongated traps, and for low atom numbers, one can have a situation where this condition is violated in the tightly confining radial dimensions.