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Questions tagged [crystals]

Crystals are solid material whose constituents, such as atoms, molecules or ions, are *arranged in a highly ordered microscopic pattern*, a crystal lattice that extends with regularity in all directions. Use for all crystallography and ordered structure topics.

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Symmetry Operation in Reciprocal Space

I have a set of k points spanning the entire Brillouin zone and I want to reduce it to the irreducible BZ. So for reduction, I use the point group symmetry operation. To verify, I use quantum espresso ...
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How is the point symmetry group of real space related to the reciprocal space?

I am taking a course of solid state physics. My teacher asked me that. I have been looking for the answer in the book Solid State Physics by Ashcroft & Mermin, and also in the book of Ibach & ...
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Effect of complex refractive index on reflectivity in a periodic structure?

I'm studying the effects of absorption in a 1D photonic structure with repeating alternate layers (a Distributed Bragg Reflector). To include absoprtion effects, the refractive index is defined as $...
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Solid state physics, meaning of momentum notated G in photon-phonon collision?

I have a problem in momentum conservation in the scattering of photon with phonon.In the process the momentum conservation is written as k'=k+G+q.where k'=wave vector of scattered photon,k=wave vector ...
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What is a ferroelectric metal?

I was reading online about ferroelectricity, and came upon the topic of ferroelectricity in metals. The main cited paper is the one below by Anderson and Blount in the 1960's Symmetry Considerations ...
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Representative Volume Element for crystals

I am trying to average the tensor properties of crystals with different types of unit cells. Due to symmetry elements for different crystals, the integration over a limited volume should be enough, ...
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Origin of XRD Peak Asymmetry

I have been doing an introductory XRD experiment in which we measure the crystal spacing of a simple lattice (LiF) and as usual obtained data for intensity against two theta. Ideally the peaks should ...
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Can we call a hexagonal system with two different atoms as inversion symmetric?

Graphene is two-dimensional honeycomb crystal lattice with the two Carbon atoms in its unit cell. Clearly, it is inversion symmetric. But, suppose we have two different atoms in the hexagonal system ...
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Inversion applicable to three-dimensional lattices only

I have just started my first course in solid state physics and while studying symmetries, Inversion is defined as "A point operation which is applicable to three-dimensional lattices only. This ...
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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 ...
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1answer
68 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} \...
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Density of states in Frequency Space vs K Space

Why do we use the density of states in frequency space when the density of states in k space is one state per unit k cubed (in 3 dimensions0?
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Single Crystal Growth- Czochralski process

Cz process operates with two opposite rotations (melt and seed's rotations). How these rotations prevent growing imposed by other facets' directions (Other than that perpendicular to the axis of ...
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Bond order correlation function

I am trying to compute the bond order correlation function, $g_6$. It is defined based on the bond order parameter: $$\psi_6(x_i) = \frac{1}{N_i}\sum_{i=1}^{N_i}{\exp(i6\theta_i^j)}$$ where $\theta_i^...
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Does the sequence of the $n$th Brillouin zones converges to any particular geometrical form?

I am currently studying solid state physics and we just saw the construction of the Brillouin zones. My question is the following: Does the sequence of the $n$th Brillouin zones converges to any ...
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Number of allowed (phonon)wave vectors in a poliatomic-basis lattice?

Lets say that my crystal is of size $N=N_1N_2N_3$ , this is the typical textbook example and after that they say that the number of allowed wavevectors is N. But tbh I dont really know if they are ...
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Reduced wave vector using Bloch's theorem

In every solid state physics book it says that the wave vectors appearing in Bloch's theorem can be confined to the first Brillouin zone and provide a hint on how to show this. Most of the times this ...
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Is there a mathematical way to find the lattice points in a crystal?

In my solid state physics class we need to be able to determine the number of lattice points per cell of various Bravais lattices. For example, the orthorhombic which has 4 lattice points per cell. I ...
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A computation problem on reciprocal lattice

I am reading David Tong’s lecture notes on Application of Quantum Mechanics. My confusion is about the following paragraph: Consider a function $f(\vec{x}) $, suppose $f(\vec{x})=f(\vec{x}+\vec{r})$...
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Drawing reciprocal lattice structures

I am currently trying to understand why and how reciprocal lattice relates to the diffraction plane, so I did some research on Wiki about reciprocal lattice, but I seem to be stuck as I would like to ...
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How does one construct the $n$th Brillouin Zone in a rectangular lattice geometrically?

What are the general rules of constructing the Brillouin Zone in a rectangular lattice geometrically? While the construction of the 1st and 2nd Brillouin zone is rather simple, starting to construct ...
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Terminology Trouble with respect to lattice structures

I am having an issue understanding terminology in a chemistry text book I am reading. I am reading about lattice structures and it keep interchanging with the word 'basis'. So for context, it ...
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35 views

Wave scattering by crystals

The structure factor is defined as (assuming orthogonal axes): $$S_{(hkl)} = \sum_{j \in \mathcal{U}} f_j \exp\{2\pi i (hx_j+ky_j+lz_j)\},$$ where $\mathcal{U}$ is a unit cell of the lattice and the ...
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Reciprocal lattice of fcc or bcc

If we want to find reciprocal vectors/points in reciprocal space of a bcc or fcc lattice we often use the non-primitive lattice vectors of the ordinary cubic (reciprocal) lattice. Looking at families ...
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35 views

Lattice structure of diamond

The previous question was about whether diamond can be described by an ordinary cube lattice instead of a FCC lattice. I now know that this is possible since we can still get the whole crystal in this ...
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3answers
28 views

A homogeneous isotropic liquid and a perfectly periodic crystal lattice

A homogeneous and isotropic liquid or a perfectly periodic crystal lattice which one has higher symmetry. I was thinking that both have same order of symmetry as if I am at a lattice point and ...
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Cesium chloride crystal structure

Cesium chloride has a cubic structure with eight $Cs$ atoms at each corner of each cube and one $Cl$ in the middle of the cube. It is stated in my solid state book that this crystal cannot be ...
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FCC and BCC lattice

Some crystals can be described by a FCC lattice or BCC lattice. For example diamond can be described as a FCC lattice with two basis vectors. Is it also possible to describe it using an ordinary cube ...
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Can we / should we use the Pauli principle to explain band structure?

Checking the wikipedia article on band structure, I got caught in major doubts... They try to give an intuitive explanation of the band structure relying heavily on Pauli Exclusion Principle: if a ...
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Can two atoms be a crystal?

In the physics literature, you can often find the term "two-ion crystal", when talking about two ions that are confined in a e.g. Paul trap. How is this possible? Shouldn't a crystal be a structure ...
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Viscosity table or diagram for melts of oxides

Is there a table or diagram to relatively determine which oxides have low 'melt-viscosity' (i.e relatively flow readily when melted)? Is it possible to determine such property based on phase diagrams ...
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1answer
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What is the relationship between directions in reciprocal and real space of a photonic crystal?

I am reading "Photonic crystals - molding the flow of light" by Joannopoulos et al. (available on-line). The figures below are reproduced from there. This is a diagram of a triangular lattice of air ...
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Group velocity in different dimensions

I am given the following equation for the group velocity in one-dimension: $$v_g = \frac{d\omega}{dk}.$$ In solid state physics one has $-\pi/a \leq k \leq \pi/a$ for a one-dimensional lattice with ...
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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. ...
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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,...
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1answer
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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 ...
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1answer
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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 ...
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Wave Superposition on a crystal

Does the principle of superposition apply for electromagnetic waves on a crystal? So I know that the principle applies for any wave but I don't understand why some books say that doesn't apply for ...
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Phase matching and the index ellipsoid

I've seen analytically all the process that brings to the condition for the phase matching (after I'll use PM) in an isotropic crystal, which is $n(\omega)=n(2\omega)$. My teacher hasn't shown it ...
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1answer
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What would happen if we had a crystal structure but only gravitational interactions?

The idea is simple. Let's say we arrange similar bodies (call them planets, ions, anything) in an infinite crystal structure, but the only possible interactions are gravitational interactions. A ...
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1answer
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Negative Miller indices and parallel planes

The integers are usually written in lowest terms, i.e. their greatest common divisor should be 1. https://en.m.wikipedia.org/wiki/Miller_index Does this mean, that parallel planes are generally ...
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TEM diffraction analysis- imaging artifact or distorted crystal structure

I've got some "nanobeam" diffraction data from TEM (transmission electron microscope). Analysis is a challenging because I forgot the golden rule... I didn't measure a "standard". Hence, I don't know ...
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Why don't marbles naturally arrange themselves like a crystal?

Most solids are crystalline in nature because the energy released during the formation of ordered structure is more than that released during the formation of disordered structure such that the ...
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Analytic derivation of honeycomb lattice Brillouin zone

My goal is to analytically derive the first Brillouin zone of the honeycomb lattice. Geometrically it's clear how to do this by just finding the space on the lattice nearest to a particular point of ...
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Why do quantum memories have to be so cold?

I understand that currently, atomic qubits, or just atomic ensembles are kept near absolute zero, in order to try to create an ideal gas structure. For higher temperatures the individual ions will ...
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1answer
178 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 ...
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How to estimate the mobility of atoms implanted in the interstitials and/or vacancies of a crystal?

I would like to compare the likelihood for implanted helium atoms to move from interstitials/vacancies of a crystal versus neon atoms. I read in the literature that He atoms located in interstitials/...
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2answers
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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 ...
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Spacing of atoms in a cubic meter [closed]

I want to figure out the spacing between neighbouring atoms in a cubic meter of a large amount of 10^17 atoms were placed in perfect cubic structures within the cubic meter. I’m not sure how to go ...
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Using Miller indices to determine interference from a particular plane in a crystal lattice

In terms of x-ray diffraction from crystals. I was wondering how you can use the miller indices to determine whether there will be constructive or destructive interference from a particular plane in ...