6.9 Ground State of a System of Electrons

So far, only the physics of bosons has been discussed. However, by far the most important particles in physics are electrons, and electrons are fermions. The electro­nic structure of matter determines almost all engineering physics: the strength of materials, all chemistry, electrical conduction and much of heat conduction, power systems, electro­nics, etcetera. It might seem that nuclear engineering is an exception because it primarily deals with nuclei. However, nuclei consist of protons and neutrons, and these are spin $\leavevmode\kern.03em \raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em /\kern-.2em\lower.4ex\hbox{\the\scriptfont0 2}\kern.05em$ fermions just like electrons. The analysis below applies to them too.

Non­interacting electrons in a box form what is called a “free-electron gas.” The valence electrons in a block of metal are often modeled as such a free-electron gas. These electrons can move relatively freely through the block. As long as they do not try to get off the block, that is. Sure, a valence electron experi­ences repulsions from the surrounding electrons, and attractions from the nuclei. However, in the inter­ior of the block these forces come from all directions and so they tend to average away.

Of course, the electrons of a “free” electron gas are confined. Since the term “non­inter­acting-el­ectron gas” would be correct and under­standable, there were few possible names left. So “free-electron gas” it was.

At absolute zero temperature, a system of fermions will be in the ground state, just like a system of bosons. However, the ground state of a macroscopic system of electrons, or any other type of fermions, is dramati­cally different from that of a system of bosons. For a system of bosons, in the ground state all bosons crowd together in the single-particle state of lowest energy. That was illustrated in figure 6.2. Not so for electrons. The Pauli exclusion principle allows only two electrons to go into the lowest energy state; one with spin up and the other with spin down. A system of $I$ electrons needs at least $I$$\raisebox{.5pt}{$/$}$2 spatial states to occupy. Since for a macroscopic system $I$ is a some gigantic number like 10$\POW9,{20}$, that means that a gigantic number of states needs to be occupied.

Figure 6.11: Ground state of a system of non­interacting electrons, or other fermions, in a box.

In the system ground state, the electrons crowd into the $I$$\raisebox{.5pt}{$/$}$2 spatial states of lowest energy. Now the energy of the spatial states increases with the distance from the origin in wave number space. Therefore, the electrons occupy the $I$$\raisebox{.5pt}{$/$}$2 states closest to the origin in this space. That is shown to the left in figure 6.11. Every red spatial state is occupied by 2 electrons, while the black states are unoccupied. The occupied states form an octant of a sphere. Of course, in a real macroscopic system, there would be many more states than a figure could show.

The spectrum to the right in figure 6.11 shows the occupied energy levels in red. The width of the spectrum indicates the density of states, the number of single-particle states per unit energy range.

Key Points

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Non­interacting electrons in a box are called a free-electron gas.

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In the ground state, the $I$$\raisebox{.5pt}{$/$}$2 spatial states of lowest energy are occupied by two electrons each. The remaining states are empty.

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The ground state applies at absolute zero temperature.