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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How to justify unit cell normalization of Bloch functions?
As is widely known, non-square-integrable wavefunctions don't belong to the Hilbert space, and therefore cannot represent physical states.
This is the case for e.g. oscillating wavefunctions and ...
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How to determine the $xyz$ coordinate system of a crystal group?
I am reading the book Non-Linear Optics by Robert Boyd. I have a question about how the crystal coordinates relate to the Cartesian coordinates used in the optics.
As an example, if I consider BaTiO3. ...
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Why do some minerals prefer diamond structures while others prefer wurtzite structures?
Carbon can exist as both diamond and wurtzite structures (lonsdaleite), but diamonds are much more common than lonsdaleites in nature. Some materials like GaN preferentially form wurtzite structures. ...
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Quasi-periodic potential and Bloch's theorem
Let's look at a physical system of a particle in a one dimensional periodic potential $V(x)$. When the potential satisfies the periodicity condition of the form
$$ V(x + n b) = V(x),$$
this leads to ...
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Unitary matrix that transforms Bloch wavefunctions of k-points connected by crystal symmetries
In the following link https://docs.abinit.org/theory/wavefunctions/#symmetry-properties, there's Eqn. (8) that describes for degenerate states how wavefunctions of k-points connected by symmetry $\...
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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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Does a thermally expanding torus experience internal stress?
I'm trying to learn continuum mechanics and thermo-mechanics.
As we know, heating an object increases the mean atomic distance $a_0$ of the atoms in a rigid body. Let's assume it is a linear elastic ...
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Why crystal wave number $k$ is the constant of motion?
I thought if there is an operator $\hat A$ which commutes with Hamiltonian $\hat H$, the eigenvalue of the corresponding observable $A$ should be the constant of motion.
In free space $(V=0)$, $$[\hat ...
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Different ways to defining Fourier transform of a lattice model
I have a Kagome 2D lattice as shown in the figure below:
There are three atoms in one unit-cell A, B, and C (shown in circle). An interaction Hamiltonian can be
$$H = H_1+H_2+H_3\equiv -t\sum_{\...
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Simulation of a dispersive crystal mirror
I am trying to simulate a simple setup where I have a point source of broadband light whose light is incident upon a spherical crystal at a central angle $\theta_i$. Assuming Bragg diffraction some of ...
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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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What groups of symmetry are most suited for filling uniformely a spherical 3D space, whilst possessing the lowest possible surface-to-volume ratio?
I am looking for the closest known approximate solution to Kelvin foams problem that would obey a spherical symmetry.
One alternative way of formulating it: I am looking for an equivalent of Weaire–...
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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.
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Now, in a metal, the valence band is relatively close to the
conduction band - ...
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Why does a lattice have to have an inversion center?
Indeed all lattices have inversion symmetry, but my teacher said a lattice has to have an inversion center: why? If a lattice doesn't have inversion symmetry, what would happen?
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Determining Brillouin Zone for a crystal with multiple atoms
The Brillouin Zone (BZ) refers to a region of reciprocal space corresponding to the primitive cell. That is, a Brillouin Zone is a subset of the reciprocal space which contains all the information ...
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Does a crystalline ferromagnetic solid break the rotational symmetry twice?
Both Heisenberg ferromagnets and crystalline solids break the rotational symmetry in space. Now consider a crystalline ferromagnetic solid. By virtue of being in a crystalline phase, it already broke ...
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Conservation of crystal momentum
I am trying to convince myself that crystal momentum is conserved in a periodic lattice modulo a reciprocal lattice vector.
Consider a Hamiltonian $H$ which is periodic under translations of a ...
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Why does Sugar form a crystal structure?
I always wondered about the question why Sugar dissolves like salt, without being ionic.
Now my question is: How does sugar forms a crystal if it is a covalent compound? Are there very strong IMF's?...
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Connection between reciprocal space and momentum
Given a lattice in real space with lattice vectors $\vec{a}$, $\vec{b}$ and $\vec{c}$, we can do the following.
We take any function $f(\vec{r})$ and take the Fourier transform to obtain $\tilde{f}(\...
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How to interpret band structure of $\rm Si$?
I found it very instructive to see how the $E$-$k$ relationship of a free particle can be roughly identified from the extended band structure of a solid: The following is the outcome of the one ...
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Space group setting of a crystal structure
I was reading a PDF of a crystal phase in order to draw its structure, when I noticed that it was, apparently, ambiguously described.
The PDF lists two descriptions of the monoclinic structure:
1) ...
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What limits the doping concentration in a semiconductor?
Si and Ge can be blended in any ratio, $\mathrm{Si}_x\mathrm{Ge}_{1-x},\ 0\le x\le 1$. So do
InxGa1-x.
So what exactly causes doping impurities inside Si/Ge/etc. to saturate at $\sim 10^{-19}\ \...
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What physical phenomenon occured in this experiment?
I noticed a strange behavior of my experiment. I have a glass container with a fully saturated solution of CuSO4, which I use to grow CuSO4 crystals, by waiting for the water to evaporate. Recently I ...
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Bloch theorem, Energy, Free electron
I'm trying to learn on my own a bit of solid physics to tackle semiconductors afterwards. I'm struggling with the Energy versus $k$ diagrams for a free electron which shows that for a single value of $...
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The 2011 Nobel Prize in Chemistry
As far as I understand a new pattern of crystal growth has been found experimentally. How does it relate to the known 2D and 3D nucleation and growth of crystals? The dominating theory of crystal ...
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Why is the translational symmetry broken?
In the book Condensed Matter Field Theory by Altland, on page 5, it is given that
$$H[\pi, \phi]=\int d x\left(\frac{\pi^{2}}{2 m}+\frac{k_{\mathrm{s}} a^{2}}{2}\left(\partial_{x} \phi\right)^{2}\...
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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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Why are HCP materials brittle while FCC materials are ductile?
Why are hexagonal close packed materials brittle, While face centered cubic is ductile. Is it related to crystal planes?
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Nature of tetragonal distortion in Jahn-Teller effect
I am wondering: If I have a regular octahedron as my starting point, oriented along the x-y-z axis, and now Jahn-Teller suggest I elongate or compress along the $z$-axis, what happens along the other ...
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Does sugar have a quasi-liquid film on its surface at room temperature?
Background
According to an article from Physics Today, ice is slippery because there is a “liquid or liquid-like layer” on its surface.
There are 3 mechanisms that can cause this layer to exist, each ...
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How does the reciprocal lattice takes into account the basis of a crystal structure?
I am reading about solid state physics, and I think I got right the concept of crystal lattice. We first define a Bravais lattice as the set of vectors spanned by $\{\vec{a}_1,\vec{a}_2,\vec{a}_3 \}$ ...
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Differences in crystals used for SHG and OPO
SHG: second harmonic generation
OPO: optical parametric oscillator
As we know these are non linear cavities (optical cavities with non linear crystals in them along a part of light beam's path) used ...
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Vacancy formation energy and chemical potential
TLDR: What exactly is the difference between vacancy formation energy and chemical potential? Does the vacancy formation energy include, e.g., the energetic cost of bond-breaking, or is this ...
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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 ...
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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 ...
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Overview and doubts about Bloch's theorem and the concept of partial density of states
So I have a large confusion with QM as applied to solid state. The following is a summary of what I know, what I think I know, and what I know I don't know. I hope to stir a discussion that will help ...
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Topological Insulators: is HgX a special case?
I got confused by reading this article:
Francois Virot, Roland Hayn, Manuel Richter, and Jeroen van den Brink. “Engineering topological surface-states: HgS, HgSe and HgTe.” arXiv preprint arXiv:...
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What is the difference between lattice vectors and basis vectors?
Google has not been very useful in this regard. It seems no one has clearly defined terms and Kittel has too little on this.
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Group theory: The energy splitting of the five-fold $d$ orbitals in a $D_{3h}$ crystal field
I am trying to figure out how the degerenate 5-fold $d$ orbitals is split in a $D_{3h}$ crystal field. A practical case would be a subtitutional $\rm Fe$ impurity in the hexagonal graphene lattice. ...
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Current operator for Bloch electrons [closed]
I'm trying to understand why in some lectures or review (e.g. in QHE lectures by Tong, or in the review Topological Field Theory of Time-Reversal Invariant Insulators by Qi), they say (without a proof)...
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Why are crystalline structures in solids so common?
Is there any sort of theorem or paper that shows that periodic arrays gives ground states? Or any other theoretical reason why crystals are so common?
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Why aren't the eigenvectors of a tight-binding Hamiltonian periodic?
I try to calculate the Berry connection for a simple graphene model and stumbled across the following question. Suppose I have a tight binding Hamiltonian (further details here or here):
$$H = \begin{...
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What are these geometric patterns on ice?
In a winter day, I noticed the water frozen inside a canal in our building. As you see there are very nice geometric patterns formed by the ice, with specific angles. What is the physical ...
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Symmetry of the dielectric tensor
In the book Principles of Optics by Max Born, in chapter XIV, the rate of change in the electric energy density $w_{e}$ is generalised to
\begin{equation}
\frac{dw_{e}}{dt} = \frac{1}{4\pi}\sum_{kl}\,...
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Algorithm for identifying planes in a Bravais Lattice
I have a lattice with Lattice Vectors $(\vec{t}_1,\vec{t}_2,\vec{t}_3)$ which are NOT orthogonal in general.
How can I identify the atoms/unit cells that belong to a plane - that is normal to a given ...
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What is crystal field anisotropy or effect ? It forces the magnetic moment to point in particular local direction..
Can you give a basic explanation of what is crystal field anisotropy ?
What is the reason to arise ?
In spin ice it forces the dipoles to point in the local 111 direction.
For partially filled rare ...
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Are the molecules of lattice in quantum entanglement when they vibrate as a phonon?
As all the molecules in the lattice are vibrating together to form a wave and the phonon is a quantum phenomenon, it makes sense to me that all the molecules inside the lattice are in the entangled ...
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Choice of lattice for a crystal
Is the choice of a lattice for describing a crystal arbritary ( within the constraint of system symmetries)?
It seems to me that in order to correctly depict the crystal momentum of modes, it is ...
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Miller indices for hexagonal crystal systems
To find the draw the direction for a given Miller index say, [1234] we first convert this miller index consisting of 4 indices into one containing 3 indices.
To do so, we have a set of formulae ...
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How the number of atoms in the basis affects the density of states?
When dealing with phonons and specific heat of solids, it seems the really important quantity to obtain is the density of states $N(\omega)$. When we have it, we can find the internal energy as
$$U(T)...
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