Difference between revisions of "Ideal Bose Gas"
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where <math> \lambda_T = (2\pi\hbar^2/mkT)^{1/2}</math> is the thermal de Broglie wavelength. | where <math> \lambda_T = (2\pi\hbar^2/mkT)^{1/2}</math> is the thermal de Broglie wavelength. | ||
| − | Some typical | + | Some typical parameters for |
* Classical gas | * Classical gas | ||
** Atom density <math> n \sim 10^{25} \text{m}^{-3}</math> | ** Atom density <math> n \sim 10^{25} \text{m}^{-3}</math> | ||
| Line 28: | Line 28: | ||
\,. | \,. | ||
</math> | </math> | ||
| + | At high temperature, the chemical potential lies below <math> \epsilon_{min} </math>. | ||
==== Density of States ==== | ==== Density of States ==== | ||
==== Thermaldynamic Properties ==== | ==== Thermaldynamic Properties ==== | ||
The thermaldynamic properties can be readily calculated from the Bose distributions and sum over all the states. | The thermaldynamic properties can be readily calculated from the Bose distributions and sum over all the states. | ||
| − | + | The total energy | |
:<math> | :<math> | ||
E = \underset{\nu}{\sum}\langle n_{\nu} \rangle E_{\nu} | E = \underset{\nu}{\sum}\langle n_{\nu} \rangle E_{\nu} | ||
Revision as of 21:21, 3 May 2017
A Bose–Einstein condensate (BEC) is a state of matter of a dilute gas of bosons cooled to temperatures very close to 0K (usually ~100nK in experiments). Under such conditions, a large fraction of bosons occupies the lowest quantum state, at which point macroscopic quantum phenomena become apparent.
Contents
Overview
In this section, we summarize some basic and useful thermodynamic results for Bose-Einstein condensation in a uniform, non-interacting gas of bosons. Most of the discussion here will be limited to 3D case. Physics for lower dimensions will be mentioned in the end.
Thermodynamics of a Bose Gas
Phase Space Density
The fundamental difference between a BEC and a classical gas is the occupancy of a single-particle state. In a classical gas, the mean occupation number for a single quantum state satisfies the Boltzmann distribution 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 \langle n_{\nu} \rangle = e^{-(\epsilon_{\nu}-\mu)/kT} } which is much less than unity. This feature is qualitatively captured by the 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 \textit{Phase-space Density} } defined as (3D, homogeneous gas)
- 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 \rho_D = n\lambda_T^3 \,, }
where 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 \lambda_T = (2\pi\hbar^2/mkT)^{1/2}} is the thermal de Broglie wavelength.
Some typical parameters for
- Classical gas
- Atom density
- Interatomic distance 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 (1/n)^{1/3} \sim 3 \text{nm} }
- Thermal de Broglie wavelength 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 \lambda_T \sim 10^{-2} \text{nm}（T = 300K) }
- 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 \rho_D \sim 10^{-8} }
- BEC in dilute gas
- Atom density 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 n \sim 10^{20} \text{m}^{-3}}
- Interatomic distance 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 (1/n)^{1/3} \sim 10^{2} \text{nm} }
- Thermal de Broglie wavelength 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 \lambda_T \sim 10^{3} \text{nm}（T = 100nK) }
- 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 \rho_D \sim 10^{2} }
The Bose-Einstein Distribution
For non-interacting bosons in thermodynamic equilibrium, the mean occupation number of the single-particle state 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 \nu } is
- 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 \langle n_{\nu} \rangle = \frac{1}{e^{(\epsilon_{\nu}-\mu)/kT}-1} \,. }
At high temperature, the chemical potential lies below 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 \epsilon_{min} } .
Density of States
Thermaldynamic Properties
The thermaldynamic properties can be readily calculated from the Bose distributions and sum over all the states. The total energy
- 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 E = \underset{\nu}{\sum}\langle n_{\nu} \rangle E_{\nu} \,. }
