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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Solid State (FCC Lattice Structure) [closed]
How many atoms per mm2 surface area are there in (110) plane for lead which has FCC structure. The radius of atom is 0.74nm.
I'm not able to proceed
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Non-symmorphic space group representation
What is the representation of non-symmorphic group operation?
I know that this results in phase factor multiplication in the original point group representation but do i need to multiply this phase ...
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In semiconductors, can distant particles in the same $k$-space recombine? Is this a wave phenomenon, ignoring physical separation?
I'm trying to understand recombination in semiconductors from a wave perspective. I'm considering the wave vector $k$ not just as momentum, but as a descriptor of the wave function's phase. In direct ...
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Electron momentum in a one-dimensional lattice and conservation issue
A one-dimensional lattice is a periodic array of atoms or ions where any two adjacent ions are separated by a fixed distance, the lattice spacing $a$. The Hamiltonian of an electron moving in this ...
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How to enumerate all Miller indices in the unit cell?
Taking the unit cell of $\rm NaCl$ as an example, I want to compute the XRD pattern of its crystal structure. One step is to enumerate all miller indices within a limited sphere as follows:
$$\frac{1}{...
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Possibility of calculating phonon dispersion from crystal structure
Is it in principle possible to calculate the dispersion relation of phonons in a crystal from the crystal structure?
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How might one produce a glory -- a circular rainbow -- in crystal?
A glory is the technical name for a full circle rainbow, such as is sometimes (but rarely) seen from below above a waterfall, or below from above a cloud. I am asking this question here. because it ...
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What is convention of predefined coordination system for defining elastic constant of a specific material
I try to find the elastic constants of LiTaO3 from online databases. I did find a page that lists them https://www.roditi.com/SingleCrystal/Lithium-Tantalate/LiTaO3-Properties.html
As I know, the ...
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Difference between Bravais lattice, point lattice and space lattice
I am good at crystallographic terminologies. Can somebody explain to me what is the difference between Bravais lattice, point lattice, and space lattice, if any?
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Why a crystal lattice with short axes would have a diffraction pattern in which spots would appear far apart using Bragg's law?
With the aid of Bragg's law, explain why a crystal lattice with short axes would have a diffraction pattern in which spots would appear far apart?
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Dispersion Relation for one dimensional monoatomic lattice in Kronig-Penny model and in Tight-Binding Approximation model
In the tight binding approximation model, we have the dispersion relation for a one-dimensional atomic lattice given as:
$$E(k) = E_0 - \alpha - 2\beta \cos(ka)$$
Here, $\alpha$ and $\beta$ are ...
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Nearly Free Electron Model (Perturbation Theory)
I am having difficulty understanding the degenerate perturbation theory treatment of the nearly free electron model. So for a free electron, the energy dispersion is relation is $E^{0}=\frac{h^{2}k^{2}...
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Structural notation for multi-element FCC crystal structures
I was wondering if different compound FCC structures share any kind of indicator or structural notation which I could use to find and categorise them.
To clarify my problem: the FCC L12 structure, ...
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At temperature $T>0K$, are all normal vibrational modes present simultaneously in a one-dimensional solid?
I am studying Debye theory of Specific heat.
hyperphysics has this picture and there it says
"Considering a solid to be a periodic array of mass points, there are constraints on both the minimum ...
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Impact of labeling in a bloch-crystal with orbital basis
If I'm diagonalizing a hamiltonian of electrons in a crystal that is written in the orbital basis, does it matter whether I calculate the matrix element between one atom and another atom (or the image ...
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Question about the elasticity matrix in metals
The most general anisotropic linear elastic material has 21 elastic constants. I am working with an HCP material and I found that it has 5 independent elastic constants. I am programming a subroutine ...
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First brillouin in twisted bilayer graphene
When it comes to the reciprocal space of twist bilayer graphene, there is a very typical picture:
In this picutre it shows the hoppings bewteen the nearest Dirac point between two layers. And there ...
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Is heat treatment the only means known to man for increasing grain size in steel?
It is well known that steel grains tend to grow larger under heat treatment. Is it possible to enlarge grain size through any other means?
I cannot seem to find anything via web search. If steel is ...
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Greens Theorem for periodic functions
Ashcroft and Mermin supply the following proof of their equations (I.1/2), which get used often in computing integrals over the first Brillouin zone (in computing current densities etc.).
I find ...
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Why haven't we found thermal superconductors?
First of all I want to apologize if this is a stupid question. I'm a layman who's merely very interested in physics, without a degree to my name.
I was trying to research electric superconductors ...
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Complex values for the dispersion relation obtained through an $s$-band only tight binding model for diamond cubic crystal
Any given atom in a diamond cubic lattice (Like Si or Ge) has four nearest neighbours at at a distance $\sqrt{3}a/4$, being $a$ the lattice constant. The translation vectors to these neighbours can be ...
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Question on the elementary cell of a rectangular/square lattice in magnetic firld?
In all references on the rectangular/square lattice in the presence of a magnetic field, they mention that you get a periodic structure of the model if and only if the flux per plaquette is a rational ...
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How to calculate a momentum space of a semi-finite lattice?
If we have a 2D square lattice of lattice constant a whose $x$ axis has only $N_x$ cells each with one atom and no with spin degeneracy, and periodic boundary conditions on $y$ with $N_y$ cells along ...
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How to interpret huge number of phonons?
In my condensed matter class we have seen how to treat the problem of vibration in a lattice in quantum mechanics. After heavy calculation we derive that the number of phonons with a given crystal ...
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Fourier transform on 1D bipartite lattice
I have a 1-D lattice Hamiltonian like
$$
H=-t\sum_{\langle ij \rangle} (c^\dagger_i c_j + h.c.),
$$
I have to do its Fourier transform for 1-D bipartite lattice, I can think of two possible ways of ...
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Exciton and Trion Physics with Maxima and Minima at $\Gamma$ point in Brillouin zone
Much of the literature on TMDs mentions valence band maxima and conduction band minima occurring at $K$ and $K'$ points in the Brillouin zone, say for $MoS_2$. From this here you can carry out the ...
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First brillouin zone propagation wave vector components in special surface cuts in cubic systems
I'm trying to figure out a context in an article about the allowed in plane propagation wave vector , in specific surface cuts or orientations such as (001) (110) (111) where the authors gave the ...
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How to construct the Wigner-Seitz cell if you don't know it is a two-atom basis?
Assuming we don't know that e.g., Graphene has a two-atom basis. How does the Wigner-Seitz Construction lead me to a two-atomic base?
So, assuming we do not distinguish jet between the green and red ...
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Apparent Contradiction in Bloch's Theorem
As far as I learned, there are two ways to state the Bloch's theorem. One is that the translation of a Bloch function is proportional to a complex phase,
$$ \psi(\mathbf{r} + \mathbf{R}) = e^{i\mathbf{...
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Integral of diverging function
I'm reading the chapter 10 on excitons of "Quantum Theory of Optical and Electronic Properties of Semiconductors" by Hartmut Haug and and Stephan W. Koch. In the second section of this ...
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What is Bloch's theorem really saying (isomorphism between Bloch eigenbasis and $k$-space)?
(Throughout this development, I neglect spin.)
Bloch's theorem, strictly speaking (according to Ashcroft and Mermin), says that given a (one-electron) Hamiltonian
$$\hat{H} = \frac{\hat{\mathbf{P}}^2}{...
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Understanding a sum in the Bravais lattice
In our solid-state physics course one frequently uses the identity
$$ \sum\limits_{\vec{R}}e^{i\vec{R}(\vec{k} - \vec{k}')} = N\delta_{\vec{k},\vec{k}'} $$
where the summation goes over the Bravais ...
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Anharmonic effects in crystals, help with intuition
I've been reading a bit about how it is necessary to consider anharmonic effects in crystals if one wants to properly understand things like thermal expansion etc. So for example here:
So the cubic ...
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Is the sum in $k$, in a Fourier expansion, on the reciprocal lattice set?
I was wondering about the expansion in a Fourier series of a function,
$$ f(\textbf{r}) = \sum_{\textbf{k}} f_{\textbf{k}} e^{i \textbf{k} \cdot \textbf{r}}, $$
in the context of condensed matter ...
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What does the radius of Fermi-Surface or the wavevector says about the occupied electronic states?
I have the following task:
Consider a simple cubic structure with a lattice constant of 0.3 nm and one atom per unit cell; the
macroscopic crystal has a size of 1 cm³. Assume that each atom ...
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How phonon emerges from the quantum mechanics of the lattice?
In all textbooks and lecture notes I've seen so far, a phonon is introduced by imposing the (second) quantization condition on the classical Hamiltonian of the bodies connected with springs.
However, ...
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How to derive symmetry invariant terms in Hamiltonian from a space group?
If a space group of some crystal is know, how can we derive the Hamilton of it in spin form to do more theoretical calculations?
For example, for a space group No.5 (C2), whose generators are
$x,y,z$
...
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Restoring force in a crystal
Consider a simple, cubic crystal, such as NaCl. Imagine displacing slightly one Na+ ion in any direction, then releasing it. The ion will quickly return to its equilibrium position. What is the ...
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Question about Debye's and Einstein's Heat Capacity Theory
I am reading the phonon part of Omar's textbook, but
I am a bit confused over the way through which the distribution of the frequencies of oscillators are determined. In Einstein's model all ...
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Kronig-Penney model in the presence of small external potential perturbation
Kronig-Penney model in 1D is a nice and simple way to understand the physics of
a crystal lattice. Usually, one relies on using the Bloch theorem to obtain the dispersion relation and associated wave ...
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Central Equation and the Formal Proof of Bloch's Theorem by Kittel
I am a bit confused over Kittel's derivation of Bloch's theorem:
My question lies in eq. 29. The k+G is only a subset of possible k value that satisfy the periodic boundary condition. Why is it ...
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Relation between reciprocal vector and gradient of a plane $(hkl)$
Suppose we have a unit cell with lattice vector $\vec a_1,\;\vec a_2,\;and\;\vec a_3$ (crystallographic axes)
Consider $(hkl)$ plane. This plane makes an intercept of $\frac{1}{h}$ unit along $\vec ...
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The expression of elastic energy in this paper
Elastic properties of $\rm{Ni_{2}MnGa}$ from first-principles calculations
Hello,
I am reading a paper investigating the linear elasticity of a crystal. However, I am a little bit confused over the ...
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How to find a proper basis defined on (111) plane such that only one component of piezoelectric field is nonzero?
I am considering a material which has cubic crystal structure (in particular, zincblende). The space group is $T_d^2 F \bar{4}3m$. In particular, I am concerned with finding the piezoelectric field in ...
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Is the expression of elastic energy in this paper correct?
Elastic properties of Ni2MnGa from first-principles calculations
I am reading a paper investigating the linear elasticity of a crystal. However, I am a little bit confused over the expression of ...
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Basis of body centered cubic latice
For the past few days I've been studying solid state physics and more specifically I have been busy with the crystal structures. I was just making some exercices and on of them asked me to give the ...
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How do you cut a single crystal into one of a different orientation?
I am reading about laser-cutting of single crystal wafers to produce wafers of a different orientation.
Is this as simple as cutting along the plane of the desired orientation? Meaning: with a 100 ...
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The relation between the energy in semiconductor crystal and wave vector
the solution of kroning-penny model inside the semiconductor crystall was something like this
where this solution can be approximated in the conduction band and valence band by a parabola where
\...
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The difference between $E-k$ diagram and simplified Energy band digram
I know form the the Kroning-Penny Model the $E-k$ diagram solution which something like this
or like this
but While dealing with semiconductors we always draw this energy band diagram
My question ...
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Why tetragonal Tanabe-Sugano diagram is not just a splitting of octahedral one?
It is known that electron energy levels will be split when the symmetry reduces. For example, $T_1$ symmetry will split into $E$ and $A_2$ when the octahedral crystal field has a tetragonal distortion....
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