IntRoduction to solid state physics

Week 8: Midterm Exam

Here are 10 sample problems related to Week 8: Midterm Exam I, covering the topics from Weeks 1-7 on solid-state physics, including crystal structures, x-ray diffraction, crystal binding, free electron theory, energy bands, and thermal properties of solids. Each problem includes a detailed description and the key concepts required to solve it.

Problem 1: Calculate Atomic Packing Factor (APF) for Face-Centered Cubic (FCC)

Concepts: Crystal structures, atomic packing factor

Given an FCC crystal with a lattice constant a=0.4 nm, calculate the atomic packing factor (APF). The atomic packing factor is the fraction of volume in a crystal structure occupied by atoms. For FCC, the number of atoms per unit cell is 4. The APF is calculated as the ratio of atoms' volume to the unit cell's volume.

Problem 2: Use Bragg's Law to Calculate Diffraction Angle

Concepts: X-ray diffraction, Bragg's law

An X-ray with a wavelength of 1.54 A˚ is incident on a crystal. The interplanar spacing ddd is 2.0 A˚. Calculate the first-order diffraction angle using Bragg's Law. Bragg’s Law is given by nλ=2dsin⁡(θ), where n is the order of diffraction, λ is the wavelength, d is the interplanar spacing, and θ is the diffraction angle. This problem helps in understanding the conditions for constructive interference in X-ray diffraction.

Problem 3: Cohesive Energy of an Ionic Crystal

Concepts: Crystal binding, cohesive energy

Calculate the cohesive energy per ion pair in an ionic crystal, assuming the ions have charges +q and −q, separated by a distance r=0.3 nm. Use Coulomb’s law, assuming the Madelung constant A=1.747 for NaCl. Cohesive energy is the energy required to break a crystal into its individual ions. It is calculated using Coulomb’s law, accounting for the Madelung constant, which represents the contribution of the entire lattice.

Problem 4: Calculate the Fermi Energy for a Metal

Concepts: Free electron theory, Fermi energy

Given an electron density of 8.5×10^28 electrons/m^3, calculate the Fermi energy for a metal using the free electron model. The Fermi energy is the highest energy state occupied by electrons at absolute zero. It is calculated using EF={(ℏ^2)/2m}x(3π^2n)^(2/3), where n is the electron density, m is the electron mass, and ℏ is the reduced Planck’s constant.

Problem 5: Calculate the Energy Band Gap of a Semiconductor

Concepts: Energy bands, band gap

In a semiconductor, the conduction band minimum is at 1.2 eV, and the valence band maximum is at 0.0 eV. Calculate the energy band gap. The band gap is the energy difference between the conduction band and the valence band. This is a key factor in determining the electrical conductivity of semiconductors. The band gap plays a crucial role in electron excitation for intrinsic semiconductors.

Problem 6: Calculate Heat Capacity using the Debye Model

Concepts: Thermal properties, Debye model, heat capacity

Given a Debye temperature TD=200 K and a solid at T=50 K, calculate the heat capacity using the Debye model. The Debye model accounts for phonons in a solid, approximating the heat capacity at low temperatures. The heat capacity can be found using the Debye integral and the ratio T/TD​, which affects the number of active phonon modes.

Problem 7: Calculate the Drift Velocity of Electrons in a Metal

Concepts: Free electron theory, drift velocity

A metal has a current density J=1×106 A/m^2, electron density n=8.5×10^28 electrons/m^3, and the electron charge is e=1.6×10−19 C. Calculate the drift velocity of the electrons. Drift velocity is the average velocity of electrons in a material due to an applied electric field. It can be calculated using vd=Jne, where J is the current density, n is the electron density, and e is the charge of an electron.

Problem 8: Calculate Phonon Density of States for a 1D Lattice

Concepts: Phonons, density of states

In a 1D lattice, the maximum phonon frequency is 10 THz. Calculate the phonon density of states (DOS) at a frequency ω=5 THz. The density of states for phonons in a 1D lattice can be calculated as:

​, where ωmax​ is the maximum phonon frequency and ω is the phonon frequency of interest.

Problem 9: Calculate the Debye Temperature for a Solid

Concepts: Debye temperature, phonons

Given the sound velocity vs=5000 m/s, number of atoms N=1×10^28, and volume V=1×10^(−6) m^3, calculate the Debye temperature for the solid. The Debye temperature provides an estimate for the highest energy phonons in a solid. It is calculated using 

​where vs​ is the sound velocity, N is the number of atoms, and V is the volume.

Problem 10: Determine the Specific Heat of a Solid Using the Dulong-Petit Law

Concepts: Thermal properties, Dulong-Petit law

Calculate the molar specific heat of a solid at high temperatures using the Dulong-Petit law. The Dulong-Petit law gives the specific heat of many solids at high temperatures, predicting Cv=3R, where R is the universal gas constant. This law applies well to solids at temperatures well above the Debye temperature.

Summary:

These problems are designed to test a range of concepts covered in Weeks 1-7, from crystal structures and X-ray diffraction to energy bands, free electron theory, and thermal properties of solids. They integrate key formulas and principles from solid-state physics, challenging students to apply their understanding to solve real-world problems related to the structure and behavior of materials.