Suppose that a quantity of neutral hydrogen gas is heated to...

Suppose that a quantity of neutral hydrogen gas is heated to a temperature $T . T$ is sufficiently high that the hydrogen is completely ionized, but low enough that $k T / m_e c^2 \ll 1$ ( $m_e$ is the mass of the electron). In this gas, there will be a small density of positrons due to processes such as $\mathrm{e}^{-}+\mathrm{e}^{-} \leftrightarrow \mathrm{e}^{-}+\mathrm{e}^{-}+\mathrm{e}^{-}+\mathrm{e}^{+}$or $\mathrm{e}^{-}+\mathrm{p} \leftrightarrow \mathrm{e}^{-}+\mathrm{p}+\mathrm{e}^{-}+\mathrm{e}^{+}$in which electron-positron pairs are created and destroyed. For this problem, you need not understand these reactions in detail. Just assume that they are reactions that change the number of electrons and positrons, but in such a way that charge is always conserved. Suppose that the number density of protons is $10^{10} / \mathrm{cm}^3$. Find the chemical potentials for the electrons and positrons. Find the temperature at which the positron density is $1 / \mathrm{cm}^3$. Find the temperature at which it is $10^{10} / \mathrm{cm}^3$.

Suppose that a quantity of neutral hydrogen gas is heated to a temperature $T . T$ is sufficiently high that the hydrogen is completely ionized, but low enough that $k T / m_e c^2 \ll 1$ ( $m_e$ is the mass of the electron). In this gas, there will be a small density of positrons due to processes such as $\mathrm{e}^{-}+\mathrm{e}^{-} \leftrightarrow \mathrm{e}^{-}+\mathrm{e}^{-}+\mathrm{e}^{-}+\mathrm{e}^{+}$or $\mathrm{e}^{-}+\mathrm{p} \leftrightarrow \mathrm{e}^{-}+\mathrm{p}+\mathrm{e}^{-}+\mathrm{e}^{+}$in which electron-positron pairs are created and destroyed.

For this problem, you need not understand these reactions in detail. Just assume that they are reactions that change the number of electrons and positrons, but in such a way that charge is always conserved.
Suppose that the number density of protons is $10^{10} / \mathrm{cm}^3$. Find the chemical potentials for the electrons and positrons. Find the temperature at which the positron density is $1 / \mathrm{cm}^3$. Find the temperature at which it is $10^{10} / \mathrm{cm}^3$.

Key Concepts

Saha Equation and Equilibrium Conditions

The Saha equation is a tool used in astrophysics and plasma physics to describe the ionization state of a gas in thermal equilibrium. Although originally formulated for atomic ionization, similar equilibrium concepts apply to processes like electron-positron pair creation. It relates the number densities of particles to their chemical potentials, temperature, and other constants, thereby providing insight into the temperature thresholds at which specific particle densities become significant.

Fermionic Distribution and Boltzmann Approximation

For fermionic particles like electrons and positrons, the statistical distribution governs their number density at a given temperature. Under conditions where kT is much less than the particle rest mass (the non-relativistic regime), the Maxwell-Boltzmann approximation becomes applicable. This simplification allows one to relate the number densities to chemical potentials and temperature through exponential factors, which is essential for calculating low-abundance species.

Pair Production Equilibrium

Pair production processes, where particles such as electrons and positrons are created or annihilated, occur under thermal conditions and are subject to both energy and charge conservation. In equilibrium, the reaction conditions typically imply a relationship between the chemical potentials of the particles involved, often resulting in one being the negative of the other. This balance plays a key role in determining the densities of each species as temperature varies.

Chemical Potential in Equilibrium

This concept refers to the parameter that quantifies how the free energy of a system changes when the number of particles is varied, while keeping the temperature and volume constant. In systems undergoing chemical reactions, the chemical potentials of different species must be consistent with the reaction’s equilibrium conditions. It is central to determining the relative abundances of particles that can be created or destroyed in thermal equilibrium.

Charge Neutrality and Conservation Laws

In plasma and astrophysical contexts, overall charge neutrality is an important constraint. Even when processes like pair production occur, the requirement that the total electric charge remains conserved forces specific relationships between the chemical potentials of charged particles, such as electrons and positrons, ensuring that reactions conform to fundamental conservation laws.

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