Energy levels and density of states in one, two and three dimensions, electrical and thermal conductivities, Wiedemann–Franz law - One Line Questions

1. In 3D, if the number of electrons per unit volume is n, the Fermi energy E_F is related to n by n ∝ E_F^(3/2). If n doubles, E_F changes by a factor of: — 2^(2/3)
2. The proportionality constant in the Wiedemann–Franz law is known as the Lorenz number (L). What is its approximate value at room temperature for many metals? — 2.44 × 10⁻⁸ W Ω K⁻²
3. The Lorenz number L = κ / (σT). If the Wiedemann–Franz law holds, L should be: — A constant, independent of material and temperature.
4. An increase in impurity concentration in a metal typically leads to: — A decrease in electrical conductivity.
5. The Fermi energy (E_F) is the highest occupied energy level at absolute zero temperature. How does the density of states near E_F influence properties? — A higher DOS at E_F leads to higher electronic contributions to specific heat and conductivity.
6. Consider a 1D system where E = Ak. The density of states g(E) is proportional to: — Constant
7. In a 2D system with E ∝ k², the density of states g(E) is: — Constant
8. For a 2D square lattice with lattice constant 'a', and free electrons, the density of states per unit area is approximately: — Constant, independent of energy
9. If the temperature of a metal is increased, its electrical conductivity generally: — Decreases due to increased lattice scattering.
10. In a 2D electron gas, the density of states is constant. If the number of states per unit area up to energy E is N_s(E), then N_s(E) is proportional to: — E
11. For a free electron model in 1D, the dispersion relation is E(k) = ħ²k²/2m. The allowed values of k are quantized with spacing Δk = 2π/L, where L is the length of the system. The density of states per unit length is proportional to: — E^(-1/2)
12. Consider a 1D chain of N atoms, each contributing one free electron. If the length of the chain is L, the density of states per unit length in energy E is proportional to: — E^(-1/2)
13. In 3D, if the energy E is related to the wavevector k by E = ħ²k²/2m, and we consider a volume V, the density of states g(E) is proportional to: — E^(1/2)
14. The density of states in 3D for free electrons is g(E) = C E^(1/2). If the maximum energy is E_max, the total number of electrons in a volume V is proportional to: — E_max^(3/2)
15. The Wiedemann–Franz law establishes a relationship between which two transport properties of metals? — Electrical conductivity and thermal conductivity.
16. The Wiedemann–Franz law is generally a good approximation for metals at temperatures where which mechanism dominates heat transport? — Electron transport
17. At room temperature, the thermal conductivity of a metal is primarily due to: — Electrons
18. The Wiedemann–Franz law is a consequence of the fact that: — Electrons are the primary carriers of both heat and charge in metals.
19. What is the fundamental concept behind energy levels in solids? — Electrons occupy discrete, quantized energy states.
20. In metals, which charge carriers are primarily responsible for both electrical and thermal conductivity? — Free electrons
21. In a 1D system, if the energy E is related to the wavevector k by E = ħ²k²/2m, what is the form of g(E)? — g(E) = (L/π) * (m/2ħ²E)^(1/2)
22. In a three-dimensional system of free electrons, what is the relationship between the density of states g(E) and energy E? — g(E) ∝ E^(1/2)
23. For a two-dimensional free electron gas, how does the density of states vary with energy E? — g(E) is independent of E
24. In a one-dimensional system, how does the density of states typically vary with energy E, assuming a simple free electron model? — g(E) is proportional to E^(-1/2)
25. Thermal conductivity (κ) in metals is significantly contributed by electrons. A higher electron mobility implies: — Higher thermal conductivity.
26. The density of states function g(E) tells us: — How many quantum states are available within a given energy interval.
27. Why is the Wiedemann–Franz law more accurate for metals than for semiconductors? — In metals, electrons are the dominant carriers for both heat and charge; in semiconductors, phonons also play a significant role in heat transport.
28. Which of the following leads to a decrease in electrical conductivity? — Increased scattering of charge carriers by lattice vibrations.
29. If the mean free path of electrons in a metal increases, what happens to its electrical conductivity? — Increases
30. For a 2D free electron gas, the density of states is constant. This means that the number of available states between energy E and E+dE is: — Independent of E
31. What happens to the electrical conductivity of a pure metal as temperature increases? — It decreases.
32. The density of states in 3D is given by g(E) = C * E^(1/2). If we double the volume of the system, how does the density of states change? — It doubles.
33. What happens to the thermal conductivity of a pure metal as temperature increases? — It generally decreases slightly.
34. In 2D, if the energy E is proportional to k², how does the density of states g(E) behave? — It is constant.
35. In a 1D system, the density of states near the bottom of a band (low energy) is: — Low and increases with energy
36. Which of the following factors significantly influences the electrical conductivity of a metal? — Number of free charge carriers and their mobility.
37. The Wiedemann–Franz law is expected to break down at very low temperatures because: — Phonon contribution to thermal conductivity becomes significant.
38. At very high temperatures, the Wiedemann–Franz law may deviate because: — Phonon scattering of electrons increases significantly.
39. In insulators, thermal conductivity is primarily due to: — Phonons
40. The thermal conductivity of a semiconductor at room temperature is mainly due to: — Phonons
41. The Wiedemann–Franz law implies that materials that are good electrical conductors are also good thermal conductors. Which of the following is an exception? — Semiconductors at certain temperatures
42. Mathematically, the Wiedemann–Franz law states that the ratio of thermal conductivity (κ) to electrical conductivity (σ) is proportional to what? — Temperature (T)
43. What is electrical conductivity (σ) a measure of? — The ability of a material to conduct electric current.
44. What is thermal conductivity (κ) a measure of? — The ability of a material to conduct heat.
45. Electrical conductivity (σ) is inversely proportional to electrical resistivity (ρ). The formula for conductivity is often given by σ = ne²τ/m*, where 'n' is the carrier concentration, 'e' is the electron charge, 'τ' is the relaxation time, and 'm*' is the effective mass. What does 'τ' represent? — The average time between scattering events.
46. Which of the following is a consequence of the free electron model for density of states? — The density of states increases with energy in 3D.
47. Thermal conductivity in metals is significantly affected by: — The mean free path of electrons.
48. Which of the following best describes the density of states (DOS) function, g(E)? — The number of available energy states per unit volume per unit energy.
49. The electrical conductivity of a semiconductor is highly dependent on temperature because: — The number of charge carriers (electrons and holes) increases exponentially with temperature.
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