In a crude model of alkali metals, the charge of each...

In a crude model of alkali metals, the charge of each electron is treated as uniformly distributed over the volume of a sphere of radius $r_{\mathrm{s}}$ centred at each ion. Prove that the electrostatic energy per electron is then given by $$ \begin{aligned} U_{\text {coul }} &=-\frac{9 a_{0}}{5 r_{s}} \text { rydberg } \\ &=\frac{24.49}{r_{s} / a_{0}} \mathrm{eV} \end{aligned} $$ where $a_{0}$ is the Bohr radius.

In a crude model of alkali metals, the charge of each electron is treated as uniformly distributed over the volume of a sphere of radius $r_{\mathrm{s}}$ centred at each ion. Prove that the electrostatic energy per electron is then given by
$$
\begin{aligned}
U_{\text {coul }} &=-\frac{9 a_{0}}{5 r_{s}} \text { rydberg } \\
&=\frac{24.49}{r_{s} / a_{0}} \mathrm{eV}
\end{aligned}
$$
where $a_{0}$ is the Bohr radius.

Key Concepts

Use of Atomic Units and Energy Conversion

In many physical problems, particularly in atomic and condensed matter physics, natural units such as the Bohr radius and Rydberg energy are used to simplify equations and provide physical insight. The Bohr radius sets the scale for atomic dimensions, while the Rydberg energy characterizes the typical binding energy of electrons. Converting results between these natural units and SI or electron-volt units helps bridge theoretical models with experimental observations.

Coulomb's Law and Integration Techniques

Coulomb's law provides the fundamental interaction between two point charges as an inverse distance squared relation. In continuous distributions, integration techniques are employed to sum up these point interactions over spatial volumes, which is central to deriving quantities like the total electrostatic energy. The symmetry of the problem, such as in spherical coordinates, often leads to simplifications in these integrals.

Electrostatic Potential Energy

Electrostatic potential energy is the energy stored by the configuration of charged particles due to their mutual Coulomb interactions. When dealing with continuous charge distributions, the total energy is computed by integrating the interaction energy between infinitesimal charge elements, taking into account their spatial arrangement and the inverse-square nature of Coulomb's law.

Uniform Charge Distribution

This concept involves assuming that a fixed charge is spread out evenly over a given volume instead of being concentrated at a single point. In electrostatics, a uniform charge distribution simplifies the calculation of potentials and energies since the symmetry allows for analytical integration of Coulomb forces over the entire volume.

Self-Energy of a Uniformly Charged Sphere

The self-energy concept refers to the work required to assemble a charge distribution from infinitesimal charges brought in from infinity. For a uniformly charged sphere, this energy can be calculated by integrating the energy contributions over the volume of the sphere, resulting in a specific formula that relates the total energy to the sphere's size and total charge.

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