Diagnostic Manual of Material Simulation

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## X-Ray scattering 

We will follow the treatment of [@ModernCondenseGirvin2019] and [@ClassicalElectJackso1999].

### X-Ray scattering experiment

Given a piece of material in your hand, how do you find out its atomic coordinates?
Before electron tunneling microscope, the standard approach is X-Ray
scattering, which allows you to determine the geometry of the material by
shining light on it.

{#fig-x-ray}
!!! figure

    ![X-Ray experiment for probing electron positions](assets/lattice/x_ray_setup.svg) 

The setup of a simplified  experiment is depicted in [](#fig-x-ray). Given a
atom at $\vec{r}_n$, which has an electron with a time dependent position
$\vec{r}_e(t)$ relative to the nuclei. We shine an electromagnetic planewave to
the atom with wave vector $\vec{k}$, which shakes the electron into a harmonic
motion. The oscillating electron then radiates an electromagnetic field 
that is received by a detector at $\vec{R}_D$, which measures the intensity of 
the radiated wave. By measuring the intensity of the radiation at different
$\vec{R}_D$, we aim to infer the location of the atom $\vec{r}_n$. This requires
a theoretical model of the measurement $\vec{E}(\vec{R}_D, t)$ as a function of $\vec{r}_n$, which is inverted 
to obtain $\vec{r}_n$



The experimental setup establishes a few assumptions that we will later exploit 
when modeling
1. the electron is considered a classical point charge.
2. the electron moves slowly, so magnetic field can be ignored.
3. the nuclei is too heavy to move or radiate.
4. the detector is far from the source, so we can make far field approximations.

Given these assumptions, we can build a model for the experiment.


### A point charge in an electric field

An X-Ray can be approximately modeled as a planewave, whose electrical component
takes the form

$$
\begin{equation}
\vec{\epsilon}(\vec{r}, t) = \vec{E}_{\mathrm{in}} \Re{e^{- i \omega t} e^{i \vec{k} \cdot \vec{r}}},
\end{equation}
$$

where $\vec{E}_{\mathrm{in}}$ is the magnitude and polarization of the field,
$\omega$ is the frequency, and $\vec{k}$ is the inverse wave length. We have
taken the real part of the complex wavefunction because the electric field is
real-valued, but it is customary in electromagnetism keep everything
complex and take the real part at the end, so we will do the same.

The X-Ray exerts a force on the electron, which moves according to Newton's 
law. Therefore, we can solve and ODE to obtain $\vec{r}_e(t)$

$$
\begin{equation}
\frac{\partial^2 \vec{r}_e(t)} {\partial t^2} = - \frac{e}{m_e} \vec{\epsilon}(\vec{r}_n + \vec{r}_e(t), t),  \label{eqn:x-ray-ode}
\end{equation}
$$

where $m_e$ is the mass of the electron. Let us also assume that the incident
field is transverse, which means that $\vec{E}_{\mathrm{in}} \cdot \vec{k} =
0$. Because the electron moves in the direction of the field, we also  have
$\vec{r}_e(t) \cdot \vec{k} = 0$. This allows us to write $\vec{\epsilon}(\vec{r}_n + \vec{r}_e(t), t) = \vec{\epsilon}(\vec{r}_n, t)$ and the solution to Eq. $\ref{eqn:x-ray-ode}$
is

$$
\begin{equation}
\vec{r}_e(t) = \frac{e}{ m_e \omega^2} \vec{E}_{\mathrm{in}} e^{i \vec{k} \cdot \vec{r}_n} e^{- i \omega t}.
\end{equation}
$$

We have implicitly and arbitrarily chosen a boundary condition because
different boundary conditions will lead to the same result.


### Radiation from an oscillating dipole 

An oscillating point charge radiates electromagnetic field. This can be seen by
solving the Maxwell equation with the source electron density 

$$
\begin{equation}
\rho(\vec{r}, t) = e \delta(\vec{r} - \vec{r}_e(t)).
\end{equation}
$$

The solution to this type of problems can be found in 9.1 and 9.2 in
[@ClassicalElectJackso1999], which approximates the vector potential given a
time harmonic source current

$$
\begin{equation}
\vec{A}(\vec{r}, t) = \frac{\mu_0}{ 4 \pi} \frac{e^{i \vec{k} \cdot \vec{r}}}{r} \int \vec{J}(\vec{r}', t) \mathrm{d} r', \label{eqn:vector-potential}
\end{equation}
$$

where $\mu_0$ is one of many annoying but inconsequential constants that you
should not care about. This approximation holds when 1) the source $\vec{J}$ is
localized in space and 2) $\vec{r}$ is far away from the source (far field),
and 3) the current is time harmonic, i.e., $\vec{J}(\vec{r}, t) =
\vec{J}(\vec{r}) e^{-i \omega t}$.



Since our source is given in terms of the charge density, we need to
relate it to the current density through the continuity relation.

{#cont-relation}
!!! theorem

    For the purpose of demonstrating the syntax for theorems, we write the 
    continuity relation as a theorem

    $$
    \begin{equation}
    \grad \cdot  \vec{J} (\vec{r}, t) = \frac{\partial \rho(\vec{r}, t)}{\partial t} = 
    (\nabla \delta) (\hat{r} - \hat{r}_e(t)) \cdot \frac{\partial \vec{r}_e(t)}{\partial t},
    \end{equation}
    $$

    which is indeed time harmonic because $\vec{r}_e(t)$ is time harmonic. 

We can combine Eq. [](#cont-relation) with integration by
part to obtain an electric dipole

$$
\begin{equation}
\vec{p}(t) = \int \vec{J} (\vec{r}', t) \mathrm{d} r' = - \int \vec{r}' \nabla
\cdot \vec{J}(\vec{r}', t) \mathrm{d} r' = - \int \vec{r}' \rho(\vec{r}', t)
\mathrm{d} r' = -e \vec{r}_e(t).
\end{equation}
$$

The integration by part works because the source current $\vec{J}$ is localized
in space, so the boundary term vanishes. Plugging this into the vector
potential in Eq. $\ref{eqn:vector-potential}$  yields 

$$
\begin{equation}
\vec{A}(\vec{r}, t) = - \frac{\mu_0 e}{ 4\pi} \frac{e^{i \vec{k} \cdot \vec{r}}}{r} \vec{r}_e(t).
\end{equation}
$$

The magnetic field can be obtained through the vector potential and
the electric field can be obtained from the fourth Maxwell equation.

$$
\begin{align}
\vec{H}(\vec{r}, t)  &=  \frac{1}{\mu_0} \nabla \times \vec{A}(\vec{r}, t),\\
\vec{E}(\vec{r}, t)  &=  \frac{i Z_0}{k} \nabla \times \vec{H}(\vec{r}, t),
\end{align}
$$

where $Z_0$ is an unimportant physical constant. This produces an electric
field at the detector position as  

$$
\begin{equation}
\vec{E}(\vec{R}_D, t) = \frac{e^2}{m_e c^2} \left(
\hat{n} \times (\hat{n} \times \vec{E}_{\mathrm{in}}) \right)
e^{i \vec{k} \cdot \vec{r}_n} e^{-i \omega t} 
\frac{e^{i |\vec{k}| |\vec{R}_D - \vec{r}_n|}}{|\vec{r}_D - \vec{r}_n|}.
\label{eqn:scattered-field}
\end{equation}
$$

### Phase Approximation 

In the form of $\ref{eqn:scattered-field}$, it is still inconvenient to solve
for $\vec{r}_n$ given the measurements $\vec{E}(\vec{R}_D, t)$, so we can
further approximate it with a Tylor expansion so that it looks like a
planewave.

$$
\begin{equation}
|\vec{k}| |\vec{R}_D - \vec{r}_n| 
= |\vec{k}| |\vec{R}_D| \sqrt{1 - 2 \frac{\vec{r}_n \cdot \vec{R}_D}{|\vec{R}_D|^2} + \frac{|\vec{r}|^2}{|\vec{R}_D|^2}}
\approx |\vec{k}| |\vec{R}_D| -  |\vec{k}| \frac{\vec{r}_n \cdot \vec{R}_D}{|\vec{R}_D|} 
= |\vec{k}| \hat{n} \cdot \vec{R}_D - |\vec{k}| \hat{n} \cdot \vec{r}_n
\end{equation}
$$

Defining $\vec{q} =  |\vec{k}| \hat{n} - \vec{k}$, which can be determined from the
position of the detector and the planewave direction, we find 

$$
\begin{equation}
\frac{e^{i |\vec{k}| |\vec{R}_D - \vec{r}_n|}}{|\vec{R}_D - \vec{r}_n|} \approx \frac{e^{i |\vec{k}|
|\vec{R}_D|}}{|\vec{R}_D|} e^{i (\vec{k} + \vec{q}) \cdot \vec{r}_n}.
\end{equation}
$$

With this approximation, we find that the dependence of the radiated field on
the position of the detector and the atom to be.

$$
\begin{equation}
\vec{E}(\vec{R}_D, t) \propto e^{- i \vec{q} \cdot \vec{r}_n}.
\end{equation}
$$

### Form factor

In the case where there are $N_e$ electrons, we have multiple sources of
radiation, and they superimpose due to the linearity of the Maxwell equations.
Thus, we can replace $e^{- i \vec{q} \cdot \vec{r}_n}$ by 

$$
\begin{equation}
f(\vec{q}) = \sum_{i = 1}^Z e^{- i \vec{q} \cdot \vec{r}_j},
\end{equation}
$$

which is called the form factor. More generally, when the electrons are not
point charges but charge densities $\rho$, the form factor is just the Fourier
transform of the density.

$$
\begin{equation}
f(\vec{q}) = \int \mathrm{d} \vec{r} e^{- i \vec{q} \cdot \vec{r}} \rho(\vec{r}).
\label{eqn:form-factor}
\end{equation}
$$

Since the detector measures $f(\vec{q})$, we can recover the electron density
by taking a inverse Fourier transform.

It is worth pointing out that the model we have developed for the X-Ray
scattering experiment breaks down when we have more than one electrons or when
the electron is not a point charge. Thus, we have not proven
$\ref{eqn:form-factor}$ as a relation between the measurement $f(\vec{q})$ and
the electron density $\rho$. 

    

## Bravais and reciprocal lattice 

Once we measure $f(\vec{q})$ experimentally, it turns out that it sharply
peaks on a lattice, which is shown in [](#recip).

{#recip style=\"width:400px\"}
!!! figure

    ![An example of a reciprocal lattice](assets/lattice/reciprocal.svg)

This lattice can be described as a set of integer linear combinations of three
basis vectors $G_1$, $G_2$, and $G_3$.

$$
\begin{equation}
\mathbb{L}^{*} = \left. \left\{ \sum_{i = 1}^{3} n_i \vec{G}_i  \right| n_i \in \mathbb{Z}, \forall i \in [\![3]\!]  \right\}.
\end{equation}
$$

This definition allows us to write down $f(\vec{q})$ approximately as

$$
\begin{equation}
f(\vec{q}) = \sum_{G \in \mathbb{L}^*} \overline{\rho}_{\vec{G}} \delta(\vec{q} - \vec{G}),
\end{equation}
$$

where the coefficients $\overline{\rho}$ are the intensity of the peaks. From
the frequency space data, we can reconstruct the electron density by inverting
$\ref{eqn:form-factor}$

$$
\begin{equation}
\rho(r) = \sum_{\vec{G} \in \mathbb{L}^*} \overline{\rho}_{\vec{G}} e^{i \vec{G} \cdot \vec{r}}, \label{eqn:series}
\end{equation}
$$

which is a Fourier series expansion whose expansion coefficients are $\overline{\rho}$.


If we visualize the density $\rho$ as in [](#bravais), we will see that it is also
repeated on a lattice, which is called the Bravais lattice.

{#bravais style=\"width:400px\"}
!!! figure

    ![A example of a Bravais lattice](assets/lattice/bravais.svg)


To describe the
Bravais lattice, we need to identify three primitive lattice translation
vectors $\vec{a}_1, \vec{a}_2, \vec{a}_3$. The lattice is defined as 
a set of vectors

$$
\begin{equation}
\mathbb{L} = \left. \left\{\sum_i^{3} n_i \vec{a}_i \right| n_i \in \mathbb{Z}, \forall i \right\}.
\end{equation}
$$

The periodicity of $\rho$ is now specified as 
$\rho(\vec{r}) = \rho(\vec{r} +
\vec{R}), \forall \vec{R} \in \mathbb{L}$. 
Plugging this into Eq. $\ref{eqn:series}$, we find that 
$
e^{i \vec{G} \cdot \vec{R}} = 1, \forall \vec{G} \in \mathbb{L}^*, \vec{R} \in \mathbb{L}
$.
For this relation to hold, one sufficient condition is 

$$
\begin{equation}
\vec{G}_i \cdot \vec{a}_j = 2\pi \delta_{i j},
\end{equation}
$$ 

which is used for relating the two lattices. In fact, these domains are important for 
defining Bloch waves and Wannier functions, which will be introduced in sec. [](#bloch-theorem) and [](#wannier-interpolation). 


## Unit cell and first Brillouin zone

When computing observables of a periodic systems such as the number electrons,
considering the full system always results in infinities. Instead, we consider
the number of electrons per period. This periodic is called a unit cell, which is conventionally defined as
semi-open set

$$
\begin{equation}
\Omega = \left. \left\{\sum_{i=1}^3 c_i \vec{a}_i \right| c_i \in [-1/2, 1/2) \right\},
\end{equation}
$$

which is illustrated as a shaded region in [](#recip). The definition of the
unit cell is not unique, but it should satisfy a requirement: tiling a unit
cell at each point in the Bravais should yield the full space. Mathematically,
this means 

$$
\begin{equation}
\mathbb{R}^3 = \mathbb{L} \times \Omega = \bigsqcup_{R \in \mathbb{L}} \left. \left\{\vec{r} + \vec{R} \right| \forall \vec{r} \in \Omega \right\}, \label{eqn:tiling}
\end{equation}
$$

where the notation $\mathbb{L} \times \Omega$ is formally a direct group
product. Note how the semi-open definition makes the union in
$\ref{eqn:tiling}$ disjoint.


A similar problem of divergence arises in the frequency space, so there is an
analogous domain called the first Brillouin zone. The definition is again not unique
as long as it tiles $\mathbb{R}^3$. A conventional definition is 

$$
\begin{equation}
\Omega^* = \left. \left\{\sum_{i=1}^3 c_i \vec{G}_i \right| c_i \in [-1/2, 1/2) \right\},
\end{equation}
$$

which is shown in [](#bravais).

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## Translation operator

### Notation

In this section, we introduce Bloch’s theorem, which is a direct consequence of
the lattice periodicity of the potential energy in crystalline solids. This
theorem provides the foundation for defining Bloch waves and, later, Wannier
functions.

Consider the single-particle Hamiltonian in $d$-dimensional Euclidean space:

$$
\begin{equation}
H = -\frac{\hbar^{2}}{2m}\nabla^{2} + V(\vec{r}),
\label{eqn:bloch-hamiltonian}
\end{equation}
$$

where the potential $V(\vec{r})$ is periodic with respect to a Bravais lattice
$\mathbb{L} \subset \mathbb{R}^d$.

Let $\{\vec{a}_1, \dots, \vec{a}_d\}$ denote a set of linearly independent
primitive lattice vectors, so that each lattice point can be written as

$$
\begin{equation}
\vec{R} = \sum_{i=1}^d n_i \vec{a}_i, \quad n_i \in \mathbb{Z}.
\end{equation}
$$

The lattice periodicity of the potential is expressed as

$$
\begin{equation}
V(\vec{r} + \vec{R}) = V(\vec{r}), \quad \forall \, \vec{R} \in \mathbb{L}.
\label{eqn:potential-periodicity}
\end{equation}
$$


### Translation operator

We define the **translation operator** $T_{\vec{R}}$, which acts on a wavefunction
$\psi(\vec{r})$ as

$$
\begin{equation}
(T_{\vec{R}} \psi)(\vec{r}) = \psi(\vec{r} + \vec{R}).
\label{eqn:translation-operator}
\end{equation}
$$

The set of all such operators $\{T_{\vec{R}} \mid \vec{R} \in \mathbb{L}\}$ forms an
Abelian group isomorphic to the lattice $\mathbb{L}$.

Our goal is to solve the stationary Schrödinger equation

$$
\begin{equation}
H \psi(\vec{r}) = E \psi(\vec{r}),
\label{eqn:schrodinger}
\end{equation}
$$

and determine how the lattice periodicity of $V$ constrains the form of
$\psi$.

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### Lemma: Translations commute with the Hamiltonian

{#bloch-lemma}
!!! theorem

    **Lemma.**  
    For every lattice translation $\vec{R} \in \mathbb{L}$, the operator $T_{\vec{R}}$
    commutes with the Hamiltonian $H$:

    $$
    \begin{equation}
    [H, T_{\vec{R}}] = 0.
    \label{eqn:commutation}
    \end{equation}
    $$

    **Proof.**  
    The kinetic energy operator $\hat{T} = -\frac{\hbar^2}{2m}\nabla^2$ is
    translation-invariant because derivatives are unaffected by constant shifts:

    $$
    \begin{equation}
    \nabla^2[\psi(\vec{r}+\vec{R})] = [\nabla^2 \psi](\vec{r}+\vec{R}),
    \end{equation}
    $$

    hence $[\hat{T}, T_{\vec{R}}] = 0$.  
    Using Eq. \eqref{eqn:potential-periodicity}, the potential also satisfies
    $V(\vec{r}+\vec{R}) = V(\vec{r})$, so that

    $$
    \begin{equation}
    H (T_{\vec{R}} \psi)(\vec{r}) = T_{\vec{R}} (H \psi)(\vec{r}),
    \end{equation}
    $$
    
    which proves Eq. \eqref{eqn:commutation}.
    

---

### Properties of the translation operator

The translation operator $T_{\vec{R}}$ has the following useful properties:

1. **Unitarity.**

   $$
   \begin{equation}
   \langle T_{\vec{R}}\psi \mid T_{\vec{R}}\phi \rangle = \langle \psi \mid \phi \rangle,
   \end{equation}
   $$

   which can be shown by the change of variables $\vec{r}' = \vec{r} + \vec{R}$.

2. **Adjoint property.**

   $$
   \begin{equation}
   T_{\vec{R}}^\dagger = T_{-\vec{R}}, \quad
   T_{\vec{R}}^\dagger T_{\vec{R}} = I.
   \end{equation}
   $$

---

### Simultaneous eigenstates of $H$ and $T_{\vec{R}}$

Because $H$ and $T_{\vec{R}}$ commute, they admit a common eigenbasis. Let
$\psi(\vec{r})$ be an eigenfunction of $T_{\vec{R}}$ with eigenvalue
$c(\vec{R})$:

$$
\begin{equation}
T_{\vec{R}} \psi(\vec{r}) = c(\vec{R}) \psi(\vec{r}),
\quad |c(\vec{R})| = 1.
\label{eqn:translation-eigen}
\end{equation}
$$

The group property
$T_{\vec{R}_1} T_{\vec{R}_2} = T_{\vec{R}_1 + \vec{R}_2}$ implies

$$
\begin{equation}
c(\vec{R}_1 + \vec{R}_2) = c(\vec{R}_1) c(\vec{R}_2),
\end{equation}
$$

and the continuous solutions of this functional equation are

$$
\begin{equation}
c(\vec{R}) = e^{-i \vec{k} \cdot \vec{R}},
\label{eqn:phase-eigenvalue}
\end{equation}
$$

where $\vec{k}$ is a real vector defined modulo any reciprocal lattice vector
$\vec{G} \in \mathbb{L}^*$.

---

## Bloch's Theorem

### Plane wave expansion

Consider the plane wave $e^{i \vec{l} \cdot \vec{r}}$.  
Acting with $T_{\vec{R}}$ yields

$$
\begin{equation}
T_{\vec{R}} e^{i \vec{l} \cdot \vec{r}}
= e^{i \vec{l} \cdot (\vec{r} - \vec{R})}
= e^{-i \vec{l} \cdot \vec{R}} e^{i \vec{l} \cdot \vec{r}},
\end{equation}
$$

so each plane wave is an eigenfunction of $T_{\vec{R}}$ with eigenvalue
$e^{-i \vec{l} \cdot \vec{R}}$.

Because the potential is periodic, momenta may be restricted to

$$
\begin{equation}
\vec{l} = \vec{k} + \vec{G}, \quad \vec{G} \in \mathbb{L}^*,
\end{equation}
$$

and since $e^{-i \vec{G} \cdot \vec{R}} = 1$ for all $\vec{R} \in \mathbb{L}$,
all plane waves of the form $e^{i (\vec{k} + \vec{G}) \cdot \vec{r}}$
share the same eigenvalue $e^{-i \vec{k} \cdot \vec{R}}$.

A general eigenstate of $T_{\vec{R}}$ can thus be expressed as

$$
\begin{equation}
\psi_{n\vec{k}}(\vec{r})
= \sum_{\vec{G}} \overline{U}_{n\vec{k}}(\vec{G}) \,
e^{i (\vec{k} + \vec{G}) \cdot \vec{r}}
= e^{i \vec{k} \cdot \vec{r}}
\underbrace{
\sum_{\vec{G}} \overline{U}_{n\vec{k}}(\vec{G}) e^{i \vec{G} \cdot \vec{r}}
}_{u_{n\vec{k}}(\vec{r})}.
\label{eqn:bloch-expansion}
\end{equation}
$$

The function $u_{n\vec{k}}(\vec{r})$ is **lattice-periodic** because
$e^{i \vec{G} \cdot (\vec{r} + \vec{R})} = e^{i \vec{G} \cdot \vec{r}}$, i.e.,

$$
\begin{equation}
u_{n\vec{k}}(\vec{r} + \vec{R}) = u_{n\vec{k}}(\vec{r}).
\end{equation}
$$

---

### Statement of Bloch’s theorem

{#bloch-theorem}
!!! theorem

    **Bloch’s theorem.**  
    For a Hamiltonian $H$ of the form in Eq. \eqref{eqn:bloch-hamiltonian},
    where the potential satisfies Eq. \eqref{eqn:potential-periodicity}, every
    eigenfunction $\psi_{n\vec{k}}$ of $H$ can be chosen as

    $$
    \begin{equation}
    \psi_{n\vec{k}}(\vec{r}) =
    e^{i \vec{k} \cdot \vec{r}} u_{n\vec{k}}(\vec{r}),
    \label{eqn:bloch-theorem}
    \end{equation}
    $$

    where $u_{n\vec{k}}(\vec{r})$ is periodic with the same lattice:

    $$
    u_{n\vec{k}}(\vec{r} + \vec{R}) = u_{n\vec{k}}(\vec{r}), \quad \forall \vec{R} \in \mathbb{L}.
    $$

    The wave vector $\vec{k}$ takes values in the first Brillouin zone
    $\Omega^*$ and is defined modulo reciprocal lattice vectors
    $\vec{G} \in \mathbb{L}^*$.


---

### Discussion

Equation \eqref{eqn:bloch-theorem} implies that the eigenstates of a
periodic Hamiltonian are **plane waves modulated by a periodic function**. The
periodic function $u_{n\vec{k}}(\vec{r})$ encodes the crystal’s internal
structure, while the phase factor $e^{i \vec{k} \cdot \vec{r}}$ carries the
crystal momentum. This result forms the basis for the concept of energy bands
in solid-state physics.

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## Monkhorst Pack grid

## Planewave Basis

## Exchange

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## Wannier interpolation

## Wannier localization

## Topological invariants

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## Two-band hybridization

## Band gap

## Graphene

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## Kohn Sham DFT

## Local density approximation

## Pseudopotential

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## Hubbard and magnetism

## Heisenberg model

## Bethe ansatz

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