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Homework 2

This slot is reserved so later problem sets can drop in without a redesign.

Q1 · Conventional cell

3D Unit Cell & Atom Counter

Toggle the Bravais type. Fractional cuts slice atoms on the cell boundary so you can see ⅛, ½, and 1.

Structure

Show Fractional Cuts

Cuts clip every atom against the six faces of this cell. Wireframes are the pieces owned by neighbors.

Live count

corners · 1/8
faces · 1/2
body · 1
diamond internals · 1

Q2 · Intercepts & arrows

Miller Indices 3D Sketcher

x forward, y right, z up. Planes from reciprocal intercepts; directions scale and shift so they stay inside the box.

Plane (hkl)

h k l

Draw plane

Direction [uvw]

u v w

Draw direction

Paste any triple

Q3 & Q4 · Density then transitions

Silicon Calculator & E–k Simulator

Part A is the conventional-cell count you already sketched. Part B is why Si needs a phonon and GaAs does not.

Part A · Si atomic density

Lattice constant a 5.431 Å

Conservation reminder

A photon is nearly vertical in this plot: large ΔE, tiny Δk. A phonon is nearly horizontal: small ΔE, large Δk. Silicon’s conduction-band minimum is near X, so both are required.

These E(k) curves are teaching parabolas, not DFT. Heights match the usual textbook gaps (Si 1.12 eV, GaAs 1.42 eV) and the Γ–X offset.

Electron sits at the valence-band maximum (k = 0). Press Excite to conserve energy — and, in Si, crystal momentum.

Q5 · Occupancy

Dynamic Fermi–Dirac Distribution

f(E) on the horizontal axis, energy vertical — the same orientation as a band diagram. The amber pivot is always f = ½.

Temperature 300 K

Band-edge indicator

E_c

E_v

E_F

Shades are qualitative tails, not calibrated integrals. For a neutrality solve, use the Fermi Level Visualizer on the catalog.