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Introduction
Introduction
Welcome to my library!
You'll find introductory information and core ideas needed to understand condensed matter and quantum many-body physics research. Before you start, you'll need to have a basic understanding of quantum mechanics. So, if you don't understand what this says:
The equation shown above is the time-dependent Schrödinger equation. For more information on the formalism and interpretation of fundamental quantum mechanics, you can reference David J. Griffith's "Introduction to Quantum Mechanics". For a more nuanced discussion, you can reference J. J. Sakurai's "Modern Quantum Mechanics".
The Big Picture
How does one study the internal properties of a solid? What are the fundamental building blocks of matter? Why does light reflect off of a mirror?
All of these questions are intricately related to what is generally termed "condensed matter" physics (CMP). You can think of CMP as a mix of thermodynamics, statistical mechanics, quantum mechanics, scattering theory, lattice theory, field theory, and most recently quantum information theory. Put simply, CMP is complex, studies systems with many variables (i.e. large degrees of freedom), and focuses on emergent properties. I like to believe that CMP fundamentally started with the advent of thermodynamics, which has origins dating back to the 17th century. There are many practical outcomes from the pioneers of thermodynamics, such as the refrigerator, the engine, and the thermometer. There are 3 fundamental laws and (at least) one crucial results that revolutionized our conceptual framework of the universe.
First Law of Thermodynamics
Second Law of Thermodynamics
Third Law of Thermodynamics
The Physical Picture of Thermodynamics
Let's discuss each thermodynamic law individually. But first, I should mention that I've hidden something. There is a fundamental assumption of thermodynamics, which is commonly referred to as the zeroth law. The zeroth law states: "When two objects are separately in thermodynamic equilibrium with a third object, then they are in equilibrium with each other." This sounds trivial. However, with this assumption and the three laws listed above, a plethora of physics can be characterized.
Let's discuss the first law. "The internal energy of a system is equivalent to the amount of heat inside it and is is lowered by the amount of work done by the system." The Work-Energy equivalence of classical mechanics makes the second aspect of this statement physically intuitive. However, it may be less clear that the amount of heat inside a system is equal to energy. Heat is the of the average kinetic energy for a distribution of particles. Thus, the internal energy of a system directly corresponds to the amount of heat inside. How can we observe and measure the heat? The second law of thermodynamics answers this question. It states, "The change in heat is equivalent to the temperature multiplied by the change in entropy." Intuitively, we know that temperature and heat are related. For instance, touching a hot stove is a bad idea because, for your hand to reach thermal equilibrium, a large transfer of energy must occur, which hurts! Although, the change in entropy is not as intuitive. There are many ways to discuss entropy, which will be covered in more detail later. For now, we are discussing thermodynamic entropy. You may think of thermodynamic entropy as the amount of heat transferred at a particular temperature, which is simply rearranging the second law. Then, entropy equates to heat transfer between two systems. Thus, the entropy of a truly isolated system can never go down because heat transfer would be zero in thermal equilibrium. This is the third law. At zero temperature, the entropy of an isolated system is equal to zero.
Why Should I Care?
Thermodynamic concepts are crucial to our understanding of quantum many-body physics in condensed matter systems. Thermalization, quantum entanglement, strongly-correlated particle dynamics, to name a few, all rely on these very intuitive ideas of internal energy, work, entropy, heat, and other thermodynamic variables. There is so much more beautiful physics packed inside these fundamental laws, but I will not go further than I have. If you'd like to learn more theory and applications of thermodynamics, I suggest reading "An Introduction to Thermal Physics" by Daniel V. Schroeder.
Scattering Theory
Scattering Theory
Introduction
I work in a research group that uses neutron scattering techniques to investigate the properties of quantum materials. With this in mind, take note that my discussion on scattering will primarily focus on neutrons. I'll do my best to mention electrons, photons, and other kinds of scattering particles too. However, let me explain why neutrons are useful. Neutrons are heavy fermions with zero charge, making them an ideal scattering probe for structural and magnetic properties of crystals. Neutron wavelengths can range roughly from 1 to 500 angstroms. 10 angstrom = 1 nanometer. In other words, neutrons can probe interactions at interatomic scales up-to the size large biological molecules such as lipids and proteins. Further, the neutron's lack of charge allows it to penetrate through the electron's orbital and probe deeply into the bulk, providing useful information about structural properties of a material. The neutron is a massive (non-relativistic) spin-1/2 Fermion, which allows us to study magnetic properties via spin interactions. These unique characteristics of the neutron make it highly sensitive to light elements in the presence of heavy element, as well as the ionization of atoms within material.
A General Overview of Elastic Scattering
The simplest form of scattering is elastic scattering, which conserves momentum and energy. Since neutrons are heavy enough to be non-relativistic, we can use Bragg's law to describe the elastic scattering of a neutron off of an atom in a crystal:
where d is the distance between two atoms, the sine function embeds the wave-like nature of the particle, n is an integer, and λ is the wave-length. Bragg's law relates to the world-renowned double-slit experiment, which validated the wave-like nature of the photon. It leads to bands of constructive and destructive interference patterns, depending on the angle of scattering, spacing between points, and wavelength of the beam. This simple equation is the best starting point for a deep intuition of microscopic scattering events. Unlike classical mechanics, the elastic collision is not fully-described with an expression of equivalent momentum values. In fact, the momentum dependence is fully embedded within the wavelength! This wave-like picture is the foundation of the scattering theory I will discuss throughout this section. Bragg's law is valid for when momentum and energy conservation hold true. In other words, the incoming particle and outgoing particle have equal energy and momentum. Below, I have provided a diagram of Bragg diffraction for scattering off of a square lattice of atoms, assuming that the incoming beam has traveled long enough to be considered a plane wave.
The Bragg plane spacing d is related to the interatomic lattice spacing a such that,
where (h,k,l) are called Miller indices of the momentum-space lattice points. Momentum-space is also called k-space or reciprocal-space. Reciprocal-space is commonly used because the units are inverse-angstrom when dropping Planck's reduced constant ℏ. Thus, to convert from reciprocal-space to momentum-space, all that is needed is a factor of Planck's reduced constant. An arbitrary point in reciprocal-space is defined by the wave-vector,
for a 3-dimensional crystal lattice. Note, the Miller indices are integer values. The wave-vector Q is simply the linear combination three unit-vectors in reciprocal-space.
where |k| is the magnitude of the incoming and outgoing wave-vector of the particle. This equation holds, given that the Laue condition is satisfied. The Laue condition states that the wave-vector Q is the difference between the incoming and outgoing wave-vectors:
An enlightening practice problem is to use Laue's condition to prove that the wave-vector equation of Bragg's law is equivalent to the relationship between the Bragg plane spacing and wave-length, first provided.
Elastic Nuclear Scattering
Since elastic scattering of a particle off atoms in a ``perfect" crystal obeys Bragg's law due to Laue's condition, the diffraction pattern may be modeled as the sum of coherent (constructive) scattering points in reciprocal-space. The coherent interference pattern for a system of N-atoms scattered as a function of wave-vector is defined:
where bn is the experimentally determined coherent-scattering length of the atom at index "n", Qm is the wave-vector in reciprocal-space and Rn is the real-space atomic position of the nth atom. The square-modulus is taken because that is what is actually measured in a scattering experiment. There is a loss of phase when measuring, which prevents us from mapping k-space directly to real-space due to degeneracy. This is called the phase problem.
The phase of the complex exponential is determined by the coherent scattering condition,
where z is an integer. The elastic scattering function discussed above is for nuclear scattering, which is modeled using the Fermi pseudo-potential:
which is a discrete sum of "atoms" that are considered to be Dirac-delta functions multiplied by the coherent scattering length of that specific atom.
Elastic Magnetic Scattering
As mentioned prior, particles can scatter via spin-spin interactions. This is termed "magnetic scattering" because it is extremely useful in characterizing magnetic properties of material at the quantum level. For instance, neutrons interact with spin systems by virtue of the spin dipole moment:
where gs is the magnetic g-factor, μB is the Bohr magneton, and S is the spin angular momentum. Aside from constant pre-factors, the interaction is mediated by the spin state S. For elastic magnetic scattering, the static structure factor is defined almost exactly the same as nuclear scattering, but the spin-spin interaction must be considered.
where the alpha and beta indices take values of x,y,z. It should be obvious that the only additional term is the spin-spin correlation at the very end. The spin-spin correlation represents the physical overlap of two spin states, indexed by n and m. The spin states are represented with Pauli matrices. Calculating the spin-spin correlation will be discussed in more detail within the framework of linear response theory, addressed later. For now, simply understand that the magnetic scattering can only occur if nuclear scattering is possible (i.e. the atom exists in that region of real-space) and if the spin-spin interaction is non-zero.
The Heisenberg Model
The Heisenberg Model
The Hubbard Model
The Hubbard Model
The t-J Model
The t-J Model
Space Groups / Point Groups / Topology
Space Groups / Point Groups / Topology
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