Foundations — Fermi-Dirac statistics — fermions, Fermi energy

This page builds every symbol, word, and picture the parent note leans on — starting from things a curious 12-year-old already knows (seats, temperature, probability) and ending exactly where the parent note begins. Nothing here is assumed; if the parent used it, we earn it here first.

0. The mental model: seats in a stadium

Imagine a stadium where every seat is a quantum state — a specific "slot" a particle is allowed to sit in, labelled by things like its momentum and spin. Seats are arranged by height: low seats = low energy, high seats = high energy.

Figure s01 — what to look at: every little box is one seat; its height up the page is its energy. The bottom three rows are shaded lavender with a coral dot inside each: those seats are filled. The pale grey rows above are empty. The dashed mint line is the water line — the boundary between full-below and empty-above. Notice that at the coldest temperature the filling is a perfectly sharp cut: full, full, full, then suddenly empty. That sharp cut is the whole story of the Fermi energy in one glance.

Two kinds of guests exist, and they behave completely differently:

• Fermions are antisocial — at most one per seat.
• Bosons are gregarious — any number may pile onto one seat.

We only care about fermions here. That "one per seat" rule is the seed of the entire topic.

1. Energy $\epsilon$ — the height of a seat

Why the topic needs it: fermions fill seats from the lowest energy upward, so we must be able to rank seats by height. The symbol $\epsilon$ is just "the height of the seat we're currently talking about."

2. Temperature $T$ and the Boltzmann constant $k_B$

Temperature is energy-per-jiggle only after we convert it. That conversion is the job of one constant:

At room temperature $T=300$ K, $k_{B}T\approx 0.025\text{eV}$ — much smaller than copper's $7\text{eV}$. Hold onto that: it is why metals are "cold" in the fermion sense even at 300 K.

3. Rate of change — the idea of a derivative

Before we meet the exponential we need one small piece of vocabulary, because the cleanest definition of the exponential uses it.

Why we need it here: some functions are best described not by a formula but by how fast they grow — and the exponential is the champion example.

4. The exponential $e^x$ — how nature weights energies

The formulas ahead are full of $e^{\text{something}}$. Before using it, we earn it.

Figure s02 — what to look at: the lavender curve is $e^x$. Three markers matter. The coral dot at $x=0$ sits at height $1$ — the break-even point. To its left (negative $x$) the curve hugs the floor, shrinking toward $0$. To its right (positive $x$) it rockets upward. The key visual habit: the sign of the exponent alone tells you whether the value is tiny or huge, before you compute anything.

We now know how to read the sign of an exponent: negative ⇒ $e^x\to 0$, positive ⇒ $e^x\to \infty$. Hold that habit — in the next two sections it becomes the on/off switch for whether a seat is filled.

5. Chemical potential $\mu$ — the (moving) water line

Why the topic needs it: with a fixed number of electrons, something must decide how high the filling goes. That "how high" is $\mu$. See Chemical Potential for the full story. We meet $\mu$ now, before writing any occupation formula, because $\mu$ is the reference point every seat's energy is measured against.

6. Building the occupation probability $f(\epsilon )$

Now we have every ingredient — a seat's energy $\epsilon$, the thermal energy $k_{B}T$, the exponential $e^x$, and the water line $\mu$ — so we can build the one formula the whole topic rests on.

Multiply top and bottom by $e^{+(\epsilon -\mu )/k_{B}T}$ to tidy it:

Now, with $f$ defined and derived, read its behaviour off the sign of the exponent $x=\frac{\epsilon -\mu }{k_{B}T}$ — "how far the seat sits above the water line, in thermal shakes":

So: negative exponent ⇒ high occupancy; positive exponent ⇒ suppressed occupancy. The exponent is a "distance above the water line" gauge, and its sign is the up/down switch for whether a seat is filled.

7. The floating water line — how $\mu$ drifts with $T$

Now that $f$ exists, we can say precisely how the water line moves.

8. Density of states $g(\epsilon )$ — how many seats at each height

Seats are not spread evenly by height. A stadium can have few seats near the floor and many at mid-height.

Before writing its formula we must name two plain physical quantities it contains:

Figure s03 — what to look at: the lavender curve rising like $\sqrt{\epsilon }$ is $g(\epsilon )$ — more seats at higher energy. The mint curve is $f(\epsilon )$, near $1$ on the left and dropping through the dashed $\mu$ line to $0$ on the right. The coral shaded region is their product $fg$ — the actual filled seats. Read the picture as a multiplication: wherever both "many seats" and "high chance filled" overlap, you get lots of electrons; that overlap is the coral hump.

9. The momentum labels: $\mathbf{p}$, $\mathbf{k}$, $k_F$

The parent note counts seats in momentum space. Here is the vocabulary.

Figure s04 — what to look at: each dot is one allowed momentum $\mathbf{k}$ (shown in 2-D for clarity; really 3-D). The coral dots inside the lavender circle are filled; the grey dots outside are empty. The circle's radius, marked by the slate arrow, is $k_F$. At $T=0$ the boundary is a crisp circle — the same sharp cut as the stadium water line, now drawn in momentum space. Each dot secretly holds two electrons (spin up + down), which is the $g_s=2$ from section 8.

10. Spin — the label that makes fermions fermions

Why the topic needs it: the exclusion rule is not an add-on; it is forced by electrons' half-integer spin making their combined wavefunction flip sign when you swap two of them. That sign flip makes two-in-one-seat mathematically impossible.

How the foundations feed the topic

Read the map as three stages, following the arrows top to bottom. Stage 1 (the rule): a quantum state is a seat, and half-integer spin forces Pauli exclusion, which caps each seat at $n=0$ or $1$. Stage 2 (the probability): feed energy, thermal energy $k_{B}T$, the exponential, the chemical potential and that occupation cap into one machine, and out comes the Fermi-Dirac probability $f$. Stage 3 (the payoff): cool $f$ to zero temperature to get the Fermi energy, and multiply $f$ by the density of states to convert probabilities into a real electron count.

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