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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How to prove Bloch function is periodic in reciprocal lattice?

How to prove Bloch function is periodic in reciprocal lattice? I saw in some textbooks this formula: $$ \Psi_{\mathbf{k}} (\mathbf{r}) = \sum_{\mathbf{G}} ...
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Neutron diffraction on Salty Ice VII

Can someone explain in layman's term how this figure shows the presence of Cl and Li ions in Ice VII ? What can we deduce about the crystal from this data ? The whole figure and caption are ...
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How many are the points which are $n$th nearest to a certain point in a hexagonal lattice

Suppose there are infinite points arranged as hexagonal lattice. The question is the one as the title. First we choose a point called $A$. Then when we count the $n$th nearest points to $A$, what ...
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Bragg diffraction and lattice planes

Crystalline substances show, for certain sharply defined wavelength and incident directions, very sharp peaks of scattered X-ray radiation. From the illustration below we see that we get constructive ...
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164 views

Point group symmetries and unit cell

I was wondering if the unit cell (of a given lattice) had to have every point group symmetries of the lattice it defines ? I guess there is no unique way to define a unit cell and that it may not have ...
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Notations for high symmetry points in the 1st Brillouin zone

I am trying to understand how I should interpret the letters like Г,K,M,T etc., that are usually there in the electronic band structure diagrams. So, let's assume we have graphene with its hexagonal ...
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What is the lattice sum and how can I calculate it for a general reciprocal lattice vector?

In lecture we were given the following definition: Lattice sum $$\sum_{n=1}^N e^{-i \mathbf{Q}\mathbf{R}_n}$$ where $\mathbf Q=\mathbf k-\mathbf k'$ is the scattering vector. First question: What ...
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Lattice sums (solid state physics)

The potential in a crystal doesn't only depend on the distance between 2 atoms, but also with the interaction to all $j^{th}$ neighbour-atoms. The Lennard-Jones-Potential then becomes ...
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Crystal momentum and the vector potential

I noticed that the Aharonov–Bohm effect describes a phase factor given by $e^{\frac{i}{\hbar}\int_{\partial\gamma}q A_\mu dx^\mu}$. I also recognize that electrons in a periodic potential gain a phase ...
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161 views

Structure factor of crystals (X-ray crystallography)

How can one prove that the degree of each node in a distance graph must be at least four in order to obtain a unique solution to an exact distance geometry problem with sparse distance data? The ...
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158 views

Coordinate system for crystallographic groups

In the International Tables for Crystallography for each crystallographic group an asymmetric unit is supplied (mathematicians call this a fundamental domain of the group). This region is a bounded ...
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40 views

What exactly is the reciprocal lattice and how is it connected to the Ewald sphere?

I want to understand what the reciprocal lattice is and how it is connected to the Ewald sphere. I know a very similar question has been asked on this site already: Reciprocal lattices. The top ...
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Momentum operator in effective mass approximation

When we calculate the band structure of some solid then we often find that in the bottom of the conduction band the dispersion looks approximately quadratic with some new effective mass: $$E(k) = ...
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Atomic nearest neighbor notation

I recently got a correction to a paper that I am writing. The correction references a section in which I talk about nearest neighbors. The comment says: Do you mean NN, NNN, etc., or NN, 2NN, 3NN? ...
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How does crystal lattice explain electrical conductance?

From http://education.jlab.org In a metal, the atoms are arranged in a crystal-like configuration. ... Now, in a metal, the valence band is relatively close to the conduction band - ...
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Honeycomb Bravais Lattice with Basis

I just had my second solid state physics lecture and we were talking about bravais lattices. As far as I understand a Bravais lattice is an infinite network of points that looks the same from each ...
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Occupancies of atoms in a crystalline structure from wyckoff positions of space groups

I am trying to fit an XRD pattern using the ILL program: FullProf and in one of the codes I have to enter the positions and occupancies of the atoms in the unit cell. I am working on the YSZ ...
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218 views

Lattice geometry and dispersion relation

Is there a general theorem which gives some information about which influence have the lattice geometry (for example sub-lattice structure, square lattice, honeycomb lattice, lattice symmetries, ...) ...
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FCC-to-BCC phase transition in NaCl, Buckingham or Lennard-Jones potentials?

Background The transformation from B1 (face centered cubic (FCC) type) to B2 (body centered cubic (body centered cubic (BCC) type) structures is one of the best documented high pressure phase ...
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Construction of a lattice structure and the Wyckoff positions

I would like to build a unit cell of a Cmcm (no. 63) lattice structure. It is a orthorhombic crystal and my lattice vectors are $\vec a_1 = (1,0,0)\,,\; \vec a_2 = (0,3,0)\,, \;\vec a_3 = (0,0,2)\,.$ ...
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Space groups and crystals. Simple example for the diamond structure

I try to learn the basis notions in crystallography and space group. But I already fail at the beginning. Let me give you an example: Consider the diamond structure (space group $Fd\overline 3 m$, ...
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Lattice parameters and basis vectors of crystal lattice structures

Does someone know where I can find lattice parameters and basis vectors of crystal lattice structures (Strukturbericht Designation) for different materials? In particular I am searching the lattice ...
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34 views

How does the surface of a material always break inversion symmetry?

I am trying to visualize this for an HCP structure. Take the profile view as such: just working in 2d. So my understanding is if we can take a point (x,y) -> (-x,-y) and get the same crystal than ...
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Explanation of crystal formation in a Fitzroy's storm glass with the change in weather conditions

I was recently reading about Admiral Robert FitzRoy's storm glass and how it was used to predict the weather conditions based on the formation of camphor crystals in a sealed glass tube containing ...
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214 views

Do indirect optical transitions “cool” the material a little?

So I'm reading in Ashcroft and Mermin about indirect optical transitions: So, a photon comes in, and it only excites the electron across the indirect band gap if a phonon with the appropriate wave ...
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42 views

Hartree-Fock orbitals of a periodic crystal -> Bloch waves?

I am wondering how I can see that the Hartree-Fock orbitals of a periodic crystal obey the Bloch theorem? My problem is that the Hartree-Fock Hamiltonian does not have the form $-\frac 1 2 ...
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What is the growth mode of thermal-evaporated aluminum on organic subtrates?

Does aluminum grow in an island mode or layer-by-layer mode on organic substrates which is amorphous? For example, on PMMA, fullerene or surfactant thin films. Can I predict the mode by comparing the ...
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Would High Explosive helped free Shackleton's ship, trapped in pack ice?

This question is based on physical concepts regarding presssure waves and crystallography, but I thought it might be helpful to include a brief historical context first. From Wikipedia: Loss of Polar ...
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Commutation of photonic crystal master equations

In studies of photonic crystal's, there are two "master equations" for the electric and magnetic fields: ...
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Dopants vs impurities

The question is related to terminology. What is the difference between dopants and impurities in condensed matter (semiconductor) physics?
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Nonzero stress on crystal at equilibrium volume?

Using a first principles computational method such as DFT, you can calculate the energy of a unit cell at different volumes to obtain a parabolic energy vs. volume curve. The minimum of this curve ...
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Analytical derivation of photonic bandstructure in photonic crystal

It seems the usual way to find the photonic bands of a photonic crystal is to setup the "master equation" $$\nabla\times\left(\frac{1}{\epsilon(\mathbf{r})}\nabla\times\mathbf{H}(\mathbf{r}) ...
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Physical meaning of crystal momentum

I am familiar with some of the definitions of crystal momentum and I am familiar with how it is related to Bloch's theorem. I also am familiar that crystal momentum is not the momentum of each ...
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460 views

Reciprocal Lattice of a non-bravais lattice

Is a reciprocal lattice defined for non-Bravais lattices? I'm trying to work out one for HCP structure and not figuring it out.
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Structure Factor for a Simple BCC Lattice

This is an example of a general misunderstanding I am having. The structure factor is given by $$S=\sum_{j}f_je^{i\mathbf{G}.\mathbf{x_j}}$$ where the index $j$ denotes a sum over the atoms within a ...
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Number of atoms in HCP unit cell

Whenever the atomic packing factor for the hexagonal close-packed (HCP) crystal structure is discussed, such as in this wikipedia article, it is stated that the (effective) number $N$ of atoms in a ...
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Is it possible for a crystal to have different structures at different temperatures?

For instance, suppose it is a 50-50 alloy of two metals that is BCC at room temperature $T_0$. If I raise (or lower) the temperature, is it possible for the bonds in the crystal to rearrange and form ...
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Is it possible to see a diffraction pattern from a BCC crystal made of 2 randomly-distributed species?

Suppose I have a 50-50 mix of two different kinds of atoms in a crystal. The crystal is BCC and the atoms have different scattering factors $f$. I know that if the crystal is organized in the regular ...
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Process of x-ray crystallography

This is a pretty basic question but I would just like some confirmation of what I suspect is true. As I understand it, the basic idea behind x-ray crystallography is that we take our crystal and hit ...
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41 views

Understanding FCC and BCC Bravais Lattices

A book I am reading states that one possible definition of a Bravais Lattice is that the surroundings will look the same from whichever lattice point you observe from. Consider for example the simple ...
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Using Huygen's principle to draw diagram of wave propagation within uniaxial crystal?

"A beam of unpolarised plane wave is incident normal to the surface of a slab of the crystal, with optics axis 45 degrees to the surface. Based on Huygens’ principle, draw the beams propagating in the ...
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Which crystal defects might single, very bright Reflexes in Laue Images indicate?

In a Laue X-ray crystallography , if you observe a single Laue reflex very bright, which is not one of the main reflexes, can that indicate a certain crystal defect or is it a peculiarity in the ...
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If pencil tip is heated why doesn't it write?

Why doesn't a pencil write if its tip is heated in a candle flame?
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Bloch Functions as an implication of the Crystallographic Restriction Theorem?

I'm studying Bloch Functions and it seems to me safe to assume that they are the most general Eigenfunction of a Hamiltionian with the crystal periodicity. Now the only considerations made in deriving ...
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Why are reciprocal lattice vector periodic, and time-frequency not?

k-space vectors are related to each other by $k=k'+G$, where $G$ is the reciprocal lattice vector $G=2\pi/a$. This means that the frequency of oscillation in real space of a plane wave $e^{ikx}$ is ...
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How to determine the basis of any crystal

In the following example, I have a rutile TiO2. Here the basis of O with respect to Ti is (0.3*x, 0.3*y, 0*z), where a,b,c are the lattice vectors. So in order to build a crystal, I need the basis and ...
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What is temperature dependence (Schottky defects)?

The problem requires us to use the following property to determine the temperature dependence: For a system in thermal equilibrium in a heat bath at temperature $T$ the Helmholtz free energy is a ...
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Can a piezoelectric material only vibrate at one frequency?

Given a material X with piezoelectric properties which is used in a ultrasonic transducer, what determines the working frequency of said transducer? If hypothetically such a transducer is specified ...
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Calculating potential energy per ion of an one dimensional ionic crystal

The problem states to calculate "the potential energy, per ion, for an infinite 1D ionic crystal with separation $a$"; the crystal is a 1D lattice of alternating charges, likes so: ...
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Whether raman spectroscopy is dependent on crystallographic orientation

If I have three bars of Si, and each has the (100) (110) and (111) surface pointing up at the laser respectively, will I get a different raman spectra for each of them? In my opinion, I shouldn't ...