1.1Identify a Crystal Translation
What does $\mathbf R=n_1\mathbf a_1+n_2\mathbf a_2+n_3\mathbf a_3$ represent?
Solution
It is a lattice translation: integer combinations of primitive vectors locate equivalent lattice points.
Condensed Matter Physics Practice
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Direct Practice
1.2Classify a Band Gap
A material has a completely filled valence band and a large gap to the next empty band. Is it more likely a conductor or an insulator?
Solution
An insulator. A large gap prevents ordinary thermal energies from moving many electrons into conducting states.
1.3Choose a Model
Which model is a natural first choice for lattice vibrations in a crystal: individual free particles or coupled oscillators?
Solution
Coupled oscillators, because atoms interact with neighboring atoms and collective normal modes behave as phonons.
1.4Use Bragg's Law
X-rays of wavelength $0.154\ \text{nm}$ strike planes with spacing $0.200\ \text{nm}$ in first order. Find the Bragg angle.
Solution
Use $2d\sin\theta=m\lambda$ with $m=1$:
$$
\sin\theta=\frac{0.154}{2(0.200)}=0.385,
\qquad \theta\approx22.6^\circ.
$$
1.5Reason About Conductivity
A semiconductor is heated while its carrier mobility changes little. What happens qualitatively to its conductivity, and why?
Solution
Its conductivity increases because heating creates more thermally excited carriers across the band gap. Since $\sigma=nq\mu$, the increase in carrier concentration $n$ dominates when mobility $\mu$ is approximately constant.