# Condensed Matter Physics These notes were written for the class PHY 801 at Michigan State University in Spring 2026. #### Primary texts for the course - Simon, "The Oxford Solid State Basicsˮ - Ashcroft and Mermin, "Solid State Physicsˮ #### Other good books that may serve as useful points of reference - Kittel, "Introduction to Solid State Physicsˮ - Girvin and Yang, “Modern Condensed Matter Physicsˮ #### Useful online resources - Delft open-source courses - Akhmerov and van der Sar, “Open Solid State Notesˮ - Akhmerov, Sau and van Heck, “Topology in Condensed Matter: Tying Quantum Knotsˮ - Hyperphyiscs - The Physics of Solid State - MIT OpenCourseWare - Physics for Solid-State Applications - Dan Arovas - Lecture Notes on Condensed Matter Physics (a Work in Progress) - Steve Simon - Resources for those using the Oxford Solid State Basics - [Statistical Mechanics - My personal notes](https://kaedon.net/l/rpch) - [Condensed Matter Physics (2022) - My personal notes from an undergraduate level class](https://dracentis.github.io/pdfs/CondensedMatter.pdf) #### Problem Sets: [Problem_set_1.pdf](https://kaedon.net/l/3p4e) These notes are very closely related to [Statistical Mechanics](https://kaedon.net/l/^rpch). However, there are some notable notation differences between the two texts. Some of these differences include: total energy $U$ instead of $E$, multiplicity function $W$ instead of $\Omega$ and many others. The texts are self consistent, but cross comparison should be done carefully. **File 0 **  Problem_set_1.pdf ## 1 Canonical models of solids ### 1.1 Fundamental Statistical Mechanics **Definition 1.1.1 **  **Energy** denoted $U$ with SI unit Joules (J) is a conserved quantity that is transferred between systems by [work](https://kaedon.net/l/^d2z2) and [heat](https://kaedon.net/l/^h7n0). **Definition 1.1.2 **  **Work** is energy transferred to a system by macroscopic forces. **Definition 1.1.3 **  **Heat** is energy transferred to a system by microscopic forces. **Definition 1.1.4 **  The **multiplicity function** $W$ of a system is the number of possible microstates for a given macrostate. **Definition 1.1.5 **  The **Boltzmann constant** denoted $k_B$ is the the proportionality factor fixed at exactly $k_B=1.380649\times 10^{-23}\text{J}/\text{K}$ that defines temperature as it is related to the statistical probability of energy states in a system. **Definition 1.1.6 **  The **entropy** $S$ of a system is the Boltzmann constant times the natural log of the multiplicity function. \[S = k_B\log W\] **Definition 1.1.7 **  The **temperature** $T$ in units of kelvin (K) and **thermodynamic temperature** $\beta$ in units of joules (J) of a system are defined in terms of the derivative of energy $U$ with respect to entropy $S$. \[T = \frac{\partial U}{\partial S} = \frac{1}{k_B\beta},\quad\beta = \frac{1}{k_B}\frac{\partial S}{\partial U} = \frac{1}{k_B T}\] **Definition 1.1.8 **  The **microcanonical ensemble** is the ensemble of statistical mechanics where the macrostates are described by the total energy $U$, volumes $\mathbf{V}$ and particle numbers $\mathbf{N}$. The probability of each possible microstate $\mathscr{p}_i$ is assumed to be the same, so it is simply one over the multiplicity function $W$, which can be written exactly as the number of states that match the macrostate. The [canonical ensemble](https://kaedon.net/l/^kjk0#r449) and the [grand canonical ensemble](https://kaedon.net/l/^kjk0#829f) can be derived by considering a system inside a large reservoir in the microcanonical ensemble. \[\mathscr{p}_i = \frac{1}{W(U,\mathbf{V},\mathbf{N})}\] **Definition 1.1.9 **  The **canonical ensemble** is the ensemble of statistical mechanics where the macrostates are described by the temperature $T$, volumes $\mathbf{V}$ and particles numbers $\mathbf{N}$. The probability of a particular microstate $i$ is written in terms of the energy of the microstate $E_i$, the thermodynamic temperature $\beta$ and the **partition function** $z$. \[\mathscr{p}_i = \frac{1}{W(T,\mathbf{V},\mathbf{N})}=\frac{e^{-\beta E_i}}{\sum_{j}{e^{-\beta E_j}}} = \frac{e^{-\beta E_i}}{z}\] \[z = \sum_{j}{e^{-\beta E_j}} = \sum_{j}{e^{-E_j/(k_BT)}}\] **Definition 1.1.10 **  The **grand-canonical ensemble** is the ensemble of statistical mechanics where the macrostates are described by the temperature $T$, volumes $\mathbf{V}$, and chemical potentials $\mathbf{\mu}$. The probability of a particular microstate $i$ is written in terms of the energy of the microstate $E_i$, the particle numbers of the microstate $\mathbf{N}_i$, the thermodynamic temperature $\beta$, the chemical potentials $\mathbf{\mu}$ and the **grand partition function** $\mathscr{z}$. \[\mathscr{p}_i = \frac{1}{W(\mathbf{T},\mathbf{V},\mathbf{\mu})}=\frac{e^{-\beta(E_i-\mathbf{\mu}\cdot\mathbf{N}_i)}}{\sum_\mathbf{N}{\sum_j{e^{-\beta(E_j-\mathbf{\mu}\cdot\mathbf{N})}}}}=\frac{e^{-\beta(E_i-\mathbf{\mu}\cdot\mathbf{N}_i)}}{\mathscr{z}}\] \[\mathscr{z} = \sum_\mathbf{N}{\sum_j{e^{-\beta(E_j-\mathbf{\mu}\cdot\mathbf{N})}}}\] ### 1.2 Boltzmann Solids **Definition 1.2.1 **  **Energy** denoted $U$ with SI unit Joules (J) is a conserved quantity that is transferred between systems by [work](https://kaedon.net/l/^d2z2) and [heat](https://kaedon.net/l/^h7n0). **Definition 1.2.2 **  The **heat capacity** denoted $C$ is the derivative of total energy $U$ in terms of temperature $T$ of a system. \[C = \frac{\partial U}{\partial T}\] **Definition 1.2.3 **  The **heat capacity at constant pressure** denoted $C_P$ is the derivative of total energy $U$ in terms of temperature $T$ of a system while pressure $P$ is held constant. \[C_P = \left(\frac{\partial U}{\partial T}\right)_P\] **Definition 1.2.4 **  The **heat capacity at constant volume** denoted $C_V$ is the derivative of total energy $U$ in terms of temperature $T$ of a system while pressure $V$ is held constant. \[C_V = \left(\frac{\partial U}{\partial T}\right)_V\] **Definition 1.2.5 **  The **coefficient of thermal expansion** denoted $\alpha$ is the derivative of volume $V$ in terms of temperature $T$ of a system while pressure $P$ is held constant divided by the volume of the system. \[\alpha = \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P>0\] **Definition 1.2.6 **  The **isothermal compressibility** denoted $\kappa$ is the negative derivative of volume $V$ in terms of pressure $P$ of a system while temperature $T$ is held constant divided by the volume of the system. \[\kappa = -\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T>0\] **Result 1.2.7 **  **Mayer's relation** states that the [heat capacity at constant pressure](https://kaedon.net/l/5tja) $C_P$ and [heat capacity at constant volume](https://kaedon.net/l/79an) $C_V$ must differ by a positive nonzero value determined by the temperature $T$, volume $V$, [coefficient of thermal expansion](https://kaedon.net/l/^kjk0#mpdm) $\alpha$ and [isothermal compressibility](https://kaedon.net/l/5kpm) $\kappa$ of the system. \[C_P-C_V = \frac{TV\alpha^2}{\kappa}= -T\left(\frac{\partial V}{\partial T}\right)^2_P\left(\frac{\partial P}{\partial V}\right)_T>0\] **Definition 1.2.8 **  An **intensive quantity** is a variable of a system that *does not* scale with the size of the system., **Definition 1.2.9 **  An **extensive quantity** is a variable of a system that *does* scale with the size of the system. **Definition 1.2.10 **  The **specific heat** denoted $c$ is the derivative of total energy $U$ in terms of temperature $T$ scaled per unit mass $M$ of the system such that it is an [intensive quantity](https://kaedon.net/l/9wj5). \[c = \frac{C}{M} = \frac{1}{M}\frac{\partial U}{\partial T}\] **Definition 1.2.11 **  The **molar specific heat** denoted $c_{mol}$ is the derivative of total energy $U$ in terms of temperature $T$ scaled to the number of moles $N_{mol}$ in the system such that it is an [intensive quantity](https://kaedon.net/l/9wj5). It can also be defined in terms of the [specific heat](https://kaedon.net/l/cna6) $c$ and molar mass $m_{mol}$. \[c_{mol} = \frac{C}{N_{mol}} = \frac{c}{m_{mol}} = \frac{1}{N_{mol}}\frac{\partial U}{\partial T}\] **Definition 1.2.12 **  **Result 1.2.13 **  **Law 1.2.14 **  The **Dulong Petit law** states that the [molar specific heat](https://kaedon.net/l/^kjk0#fm30) for most bulk materials is a constant at high temperatures. \[C = N_{atoms}\frac{\partial u}{\partial T} = 3k_BN_{atoms}\] \[c_{mol}=3k_BN_{mol} = 3R\] ### 1.3 Einstein Solids **Definition 1.3.1 **  wao ### 1.4 Debye Solids **Definition 1.4.1 **  ### 1.5 Electronic Transport and Response (Drude and Sommerfeld) **Definition 1.5.1 **  **Definition 1.5.2 **  **Definition 1.5.3 **  **1.5.4 **  ## 2 Geometry and Constitution of Solids **Section 2.1 **  **Section 2.2 **  **Section 2.3 **  **Section 2.4 **  ## 3 Electrons in Solids **Section 3.1 **  **Section 3.2 **  **Section 3.3 **  **Section 3.4 **  ## 4 Disorder and Defects **Section 4.1 **  **Section 4.2 **  **Section 4.3 **  ## 5 Topology **Section 5.1 **  **Section 5.2 **  **Section 5.3 **