Quantum gases. The ideal Fermi gas

[Petar Mali]

Relations for an ideal Fermi gas:

$$\frac{P}{k_BT}=\frac{1}{\lambda_D^3}f_{5/2}(\lambda)$$

$$\frac{1}{\upsilon}=\frac{1}{\lambda_D^3}f_{3/2}(\lambda)$$

But in some book books I find


$$\frac{P}{k_BT}=\frac{g}{\lambda_D^3}f_{5/2}(\lambda)$$

$$\frac{1}{\upsilon}=\frac{g}{\lambda_D^3}f_{3/2}(\lambda)$$

where $$g$$ is degeneration of spin I
guess.
$$g=2s+1$$

Can you tell me something about this

$$\lambda_D=\sqrt{\frac{2 \pi\hbar^2}{mk_BT}}$$

$$f_k(\lambda)=\sum^{\infty}_{n=1}(-1)^{n-1}\frac{\lambda^n}{n^k}$$

$$\lambda=e^{\frac{\mu}{\theta}}$$ - fugacity

 

[azntoon]

The relations given in your book are for a Fermi gas with spin degeneracy g, where g is the number of possible spin states. In this case, the pressure and volume equations can be written as:\frac{P}{k_BT}=\frac{g}{\lambda_D^3}f_{5/2}(\lambda)\frac{1}{\upsilon}=\frac{g}{\lambda_D^3}f_{3/2}(\lambda)where \lambda_D=\sqrt{\frac{2 \pi\hbar^2}{mk_BT}} is the de Broglie wavelength and f_k(\lambda)=\sum^{\infty}_{n=1}(-1)^{n-1}\frac{\lambda^n}{n^k} is the Fermi integral. The fugacity \lambda=e^{\frac{\mu}{\theta}} is the ratio of the chemical potential \mu to the thermal energy \theta.

 

[Entropia]

parameter

Quantum gases are systems of particles that behave according to the laws of quantum mechanics. One type of quantum gas is the ideal Fermi gas, which is a gas composed of fermions that obey Fermi-Dirac statistics. In this system, the particles have half-integer spin and follow the Pauli exclusion principle, meaning that no two particles can occupy the same quantum state.

The equations provided show the relations for an ideal Fermi gas, where P is the pressure, k_B is the Boltzmann constant, T is the temperature, \lambda_D is the thermal de Broglie wavelength, and f_{5/2} and f_{3/2} are statistical functions. These equations describe the behavior of the gas at a given temperature and density.

The degeneracy factor, g, in the second set of equations indicates the number of quantum states available to each particle, taking into account their spin. For a particle with spin s, the degeneracy is given by g=2s+1. This means that a particle with spin 1/2 has two quantum states available, while a particle with spin 1 has three quantum states available.

The fugacity parameter, \lambda, is a measure of the deviation of the gas from ideal behavior. In an ideal gas, \lambda=1, but in real gases, it can deviate from this value. This parameter is related to the chemical potential, \mu, and the temperature, \theta, through the equation \lambda=e^{\frac{\mu}{\theta}}.

Overall, these equations provide a comprehensive understanding of the behavior of an ideal Fermi gas and can be used to study various properties and phenomena in quantum gases.