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31 views

Does pseudospin break the crystal symmetry?

Pseudospin is a concept to describe a superposition of two quantum states. Sometimes, I see a pseudospin texture in the momentum space which breaks a crystal symmetry. A simple example is the ...
1
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0answers
20 views

For what choice of the basis atoms origin the Structure Factor is real?

I have a Cuprite Structure: Now, given that I described this structure as a Simple Cubic lattice with a 6 atoms basis, I have to choose the origin for these basis atoms such that the structure ...
1
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1answer
16 views

Metal-Insulator from number of atoms in the basis

I have an issue understanding what A&M means while saying this in chapter 12: It is a reassuring exercise to go through the periodic table looking up the crystal structure of all insulating ...
1
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0answers
15 views

Crystal Field Theory (CFT) in $s$-$p$ orbitals system

Crystal Field Theory (CFT) is a model that describes the breaking of degeneracies of electron orbital states, usually d or f orbitals (Source: wiki). My doubts: Is it possible to have CFT effect in ...
0
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1answer
25 views

Conventional unit cells and Bravais lattices

Conventional unit cell is defined in the following: A definition of a conventional unit cell of a lattice is one that contains the same point group symmetries as the overall lattice and is the ...
2
votes
1answer
64 views

Claim that DeBroglie relation doesn't work in crystal

In this Wikipedia article on Position and Momentum Space, https://en.wikipedia.org/wiki/Position_and_momentum_space there is a claim that "the de Broglie relation is not true in a crystal" in the ...
1
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1answer
60 views

Reciprocal lattice points and diffraction peaks

I am having a issue with the concept of what this question is asking. Question For a FCC crystal describe all the reciprocal lattice points corresponding to the two diffraction peaks. Here is my ...
0
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1answer
50 views

Two Dimensional Self-Reciprocal BravaisLattice

I've been reading Quantum States of Atoms Molecules and Solids by Morrison et al. for a condensed matter course. They make the claim that all 2D Bravais lattices are self-reciprocal, but I'm having ...
0
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0answers
37 views

What does it mean for a wavevector to terminate?

In the question we are told that the amplitude of a wave after being diffracted through a crystal is: $$ S\propto e^{i\mathbf{k\cdot r_D}}\sum_n e^{i(\mathbf{k-k_0)\cdot r_n}} $$ where $k_0$ is the ...
0
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1answer
67 views

Honeycomb lattice Brillouin zone structure and direct lattice periodic boundary conditions

One way to construct the Brillouin zone of the Honeycomb lattice is by obtaining the standard Wigner-Seitz cell by constructing the perpendicular bisectors of the reciprocal lattice vectors and ...
0
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1answer
45 views

Wigner Seitz cell

While searching the difference between primitive cell and unit cell I have seen that "Primitive unit cells contain only one lattice point, which is made up from the lattice points at each of the ...
0
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1answer
89 views

Momentum Space Representation of the Tight Binding Hamiltonian

I am trying to represent the tight-binding Hamiltonian \begin{equation} \hat{H}_{TB} = \sum_{\sigma} \sum_{\alpha,\beta} \sum_{\mathbf{R}_1,\mathbf{R}_2} t^{\alpha,\beta}_{\mathbf{R}_1,\mathbf{R}_2} \...
1
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0answers
28 views

How are the orientations and cuts of crystal substrates determined?

I have been looking at piezoelectric crystals, LiNbO3 primarily, so piezoelectric devices. But I have had trouble understanding the cuts and orientations that are referred to with the rotated cuts. ...
1
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2answers
75 views

What are the allowed wavenumbers in the finite size system?

Usually, we introduce wavenumber $\textbf{q}$ by Fourier transform, for example, an operator $A_{\textbf{q}}=1/\sqrt{N}*\sum_{i}e^{i \textbf{q}\cdot \textbf{r}_{i}}A_{i}$, where $N$ is number of sites,...
1
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1answer
56 views

Why does first photonic band go to zero at the centre of the Brillouin zone?

I have been plotting photonic band diagrams of various geometries recently and I identify if it is correct by looking if it goes to zero at the Brillouin zone centre, $\Gamma$. I realised early on ...
1
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1answer
111 views

What is the irreducible Brillouin zone for a rhombic unit cell?

So I realised that the rhombic unit cell is in fact not the same as a hexagonal unit cell. (I thought they both gave hexagonal lattices but the rhomic unit cell with two rods gives a honeycomb lattice ...
6
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1answer
214 views

Points of symmetry in $k$-space

Can you relate a point in the reciprocal space with a vector in real space? How do I find the family of planes that represent a point of symmetry in the Brillouin zone? For example, germanium has ...
0
votes
2answers
92 views

How is the a Fermi surface different from a Fermi sphere?

How is a Fermi surface (a surface in reciprocal space separating the occupied electron states from unoccupied states at $T=0$) different from a Fermi sphere? Is Fermi sphere a special case of Fermi ...
0
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1answer
48 views

Diffraction peaks and Miller indices

How do we find out if a diffraction peak is observable using miller-indices?
1
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1answer
86 views

BCC to FCC lattice conversion

In the book Condensed Matter Physics by Marder I have read that an FCC lattice can be obtained by expanding a bcc lattice along one axis by a factor of $\sqrt{2}$. How can I get that mathematically?
0
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0answers
60 views

Switch from sum to integral in non-Cartesian coordinates

when I do the BZ integral in honeycomb lattice numerically I need to calculate: $$\left| {\begin{array}{*{20}{c}} {\frac{{\partial {k_x}}}{{\partial {{k'}_1}}}}&{\frac{{\partial {k_y}}}{{\partial {...
1
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3answers
126 views

Why can we treat the Bloch hamiltonian as an effective hamiltonian?

Often when looking at topological insulators the hamiltonian is broken down into the Bloch hamiltonian and then analysed ignoring the creation/annhilation operators of the Bloch waves. Why is it okay ...
1
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0answers
49 views

How to evaluate the following Fourier calculation?

Let $\vec{k}$ and $\vec{r}$ represent coordinates in Fourier space and real space of a crystal. If there is no translational symmetry in the real space, is it possible to evaluate the following ...
0
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1answer
216 views

Efficient way to generate a crystal lattice in a box

I'm looking for a fast way to generate a crystal lattice. I know you can do a bunch of linear combinations of lattice vectors, but this can be costly. Here's my current code for doing so in Python, ...
0
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1answer
53 views

What is a dislocation defect in metals as opposed to a grain boundary?

Almost all metals found in nature are polycrystalline so that there must be grain boundaries. My understanding is that individual grains are tiny defectless crystals and different grains are rotated w....
0
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0answers
33 views

What is effect of choice of unit cell on 1st Brillouin zone?

Let's say we have a 1D lattice with $a$ as lattice constant: and hopping strength between two nearest neighbors (NN) is $t$. We can choose unit cell as one lattice site per unit cell. in k-space $k$...
1
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0answers
31 views

Growth of iron crystals

What determines how large a crystal of iron will grow when heat and pressure are applied? I was thinking about the Earth's core. Some have postulated that a large iron crystal is at the center of the ...
2
votes
2answers
159 views

Are phonons eigenstates of the momentum operator?

In the case of electrons in a periodic potential it can be demonstrated that the eigenstates of the Hamiltonian containing the periodic potential are the Bloch functions: $$\Psi_{n \mathbf{k}}(\mathbf{...
0
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1answer
99 views

Understanding Sums in the Reciprocal Lattice

So I'm trying to understand what appears to me to be a paradox in the Ashcroft & Mermin Solid State Physics book. In Eqn 5.8 it states $\exp(i\vec{k}\cdot\vec{R})=1$ where $\vec{R}$ is a direct ...
4
votes
3answers
224 views

Does heat transfer happen instantanoeusly in a perfect harmonic crystal?

A perfect harmonic crystal (PHC) has infinite thermal conductivity (see here for example, or also (1)). Does this mean that in a PHC there is instantaneous heat transfer? Fourier's law would seem to ...
1
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1answer
2k views

Primitive cell of graphene

I have spent hours on finding the primitive cells of honeycomb lattice of graphene. Based on the definition of graphene from most of solid state physics books, as I quoted from Wikipedia, and in which ...
3
votes
1answer
209 views

Tesellation: What does the trace of a rotation matrix means?

The crystallographic restriction theorem says that you cannot have a periodic lattice with $n$-fold rotation symmetry, with $n$ different from 1,2,3,4 and 6 (for 2D and 3D). There are many ways to ...
1
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2answers
115 views

Translations are normal subgroup of Space Group: Dresselhaus's proof not convincing

In Group Theory: Applications to the physics of condensed matter, eq. 9.15, Dresselhaus gives the following proof that the translation group is a normal subgroup of the space group: \begin{align*} \{...
0
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0answers
68 views

Gauss law in crystals, Madelung constant

I have a question regarding the calculation of the Madelung constant in ionic crystals. One of the well known methods for the calculation is the expanding-neutral shells method.I understand why the ...
1
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1answer
862 views

How to determine the degeneracy of an energy level for a periodic quantum system from its band structure over the Brillouin zone?

The following figure shows the 1st Brillouin zone of graphene (shaded area). At the $K$ and $K'$ points (called Dirac points), the upper (conduction) band touches the lower (valence) band, and ...
1
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0answers
128 views

Why use simple cubic reciprocal lattice vector to determine Miller indices?

Bragg diffraction: $$ d_{hkl}= \frac{\lambda}{2sin(\theta)}.$$ This equation would allow us to determine the lattice spacing ($d_{hkl}$) of parallel crystal planes given that we observe an intensity ...
0
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1answer
38 views

Behaviour of the crystal momentum under an externally applied electric field

Is there any equation which describes the behavior of the crystal momentum under the action of an externally applied electric field?
3
votes
1answer
234 views

What would an intuitive explanation for the $E$-$k$ diagram?

In all of my solid state books they seem to plot the $E-k$ diagram. But I don't understand why ? What is the physical significance of the $E$-$k$ diagram? Please don't give answers like it shows ...
3
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1answer
140 views

Fractal structure in colloidal systems

In describing the configuration of a colloidal system, one often deals with either, disordered fluid states, disordered jammed states or crystalline states (so an underlying lattice structure), but in ...
0
votes
1answer
410 views

Order parameter for liquid to crystalline solid transition under the broken continuous group of translation

The order parameter for liquid to crystalline solid transition is given by the Fourier transform of the density $$\tilde{\rho}(\textbf{k})=\int\rho(\textbf{r})e^{i\textbf{k}\cdot\textbf{r}}d^3\textbf{...
1
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0answers
213 views

Why is a one-Fe unit cell used in FeSe?

The true unit cell of FeSe has two Fe atoms and two Se atoms. However, a smaller unit cell with only one Fe atom is often used in the literature. This smaller--and seemingly incorrect--unit cell, ...
1
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0answers
138 views

Magnetic Bloch Functions and Diophantine Equation

I'm trying to understand the paper by Dana, Avron, and Zak [1] in which they prove the Diophantine equation for Hall Conductivity for an arbitrary periodic Hamiltonian using just magnetic translation ...
1
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1answer
674 views

How do I determine the $n$ integer in the Bragg's law?

I'm studying X-rays crystallography. The basic assumption is that if I let some X-rays scatter on a crystal the atoms act as sources of new waves. Considering two waves that hit two "neighbouring" ...
1
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0answers
31 views

Linear force vs. displacement relationship for crystalline solid

I did a series of density functional theory (DFT) calculations on crystalline silicon. These consisted of 50 total displacements of one atom from equilibrium in the x-direction until the displacement ...
1
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0answers
60 views

Fourier Transform of a HgTe 2D strip

I am currently trying to derive the Fourier transform of a 2D HgTe Hamiltonian, with $k_x $ PBC and vanishing boundary conditions in the y direction at 0 and L. Here is the Hamiltonian: $$ H = \sum_{...
1
vote
1answer
169 views

How to understand the recipirocal lattice vectors for BCC?

I know that if a set of vectors R constitutes a Bravis Lattice then the reciprocal vector Q is a wave vector such that Q.R = 2nPi where n is an integer. In this definition, Q represents a set of wave ...
1
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1answer
559 views

Bloch theorem and Bloch states

In deriving the Bloch theorem, they first give a Schrodinger equation, $$\hat H\psi_n(r)=E_n\psi_n(r)$$ $n=0,1,2…$ $\hat T_R$ is a translation operator commuting with $\hat H$. Its eigenvalue is $$\...
0
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2answers
181 views

RHEED (Reflection high-energy electron diffraction) spot size is small why does it can reveal the quality of large size sample

RHEED (Reflection high-energy electron diffraction) spot size is 60um to 1mm, so it reveals that the surface property of an area about 60um to 1mm. However, literature usually use RHEED pattern as an ...