(a) For a maser, show that a population inversion corresponds to a negative excitation temperature. (b) The negative excitation temperature gives a negative optical depth. Show from the radiative transfer equation that this implies amplification.

Population inversion is a crucial concept in understanding masers. It happens when more atoms or molecules occupy a higher energy state than a lower one, denoted as \(N_2\) for excited state and \(N_1\) for ground state.
According to the Boltzmann distribution, the ratio of populations of two energy states, expressed through excitation temperature \(T_e\), is calculated using: \[ e^{-\frac{E_2 - E_1}{kT_e}} = \frac{N_2}{N_1} \] For a population inversion, \(N_2/N_1 > 1\), making:
\[ e^{-\frac{E_2 - E_1}{kT_e}} > 1 \] This implies:
\[ -\frac{E_2 - E_1}{kT_e} > 0 \]
Therefore, the excitation temperature \(T_e\) must be negative.
In simpler terms, a higher population in the excited state translates to a negative excitation temperature, a characteristic feature of population inversion in masers.

Optical depth, \(\tau\), measures how much light is absorbed as it travels through a medium.
It's determined by the population densities and absorption coefficient. For masers, with a negative excitation temperature \(T_e\), the absorption coefficient turns negative. This leads to:
\[ \tau = -\frac{u_0}{c} \frac{N_2 - N_1}{\frac{E_2 - E_1}{kT_e}} < 0 \] Here, \(u_0\) is the central frequency, and \(c\) is the speed of light.
This negative optical depth indicates more particles in the excited state, implying less absorption and potentially more amplification of light, rather than attenuation, an essential aspect of the maser behavior.

The radiative transfer equation helps in understanding the change in intensity of radiation as it travels through a medium. It is expressed as:
\[ \frac{dI}{ds} = j - \frac{u_0}{c} (N_2 - N_1) I \] In this equation, \(I\) represents the radiation intensity, \(j\) is the emission coefficient, and the second term represents the absorption or stimulation emission.
For a negative optical depth, the term \[ -\frac{u_0}{c}(N_2 - N_1) I\] becomes positive because \(N_2 - N_1\) is positive under population inversion.
Thus, \[ \frac{dI}{ds} > 0 \] This implies that the radiation intensity \(I\) increases as it travels through the medium, indicating amplification.
Simply put, instead of weakening, the radiation grows stronger, a defining characteristic of masers.

Amplification is the key function of masers, which stands for 'Microwave Amplification by Stimulated Emission of Radiation'.
This phenomenon is driven by population inversion where more particles are in excited states than in lower ones.
Due to the negative excitation temperature, the maser medium exhibits negative optical depth.
This results in less absorption and more stimulated emission, leading to amplification.
Mathematically, as explained by the radiative transfer equation, the increase in radiation intensity (\( dI/ds > 0 \)) indicates that the radiation is being amplified.
In practical terms, masers use this principle to produce highly coherent and powerful microwave signals. They are used in various applications, including astronomy, atomic clocks, and deep space communication.