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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Recommendation of book/article on nonlinear optical crystals
I'm reading Boyd's book but it doesn't have details on the different types of nonlinear optical crystals: only a few pages.
Could someone recommend books or articles on it?
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Nielsen-Ninomiya theorem
The Nielsen-Ninomiya theorem states that for a local, Hermitian, translationally invariant lattice fermion theory in even-dimensional spacetime, the number of left-handed Weyl fermions is equal to the ...
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Nonvanishing matrix elements of electric dipole transition in $O_h$ symmetry from $E_g$-symmetric initial state to $T_{1u}$-symmetric final state
I am trying to work out Q6.3(b) of Dresselhaus's Group Theory: Application to the Physics of Condensed Matter. The question is to find the non-vanishing matrix elements for an electric dipole ...
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How do I calculate the electric susceptibility of a semiconductor crystal if I have the absorbance data? [closed]
I need help with this.
I tried to calculate the electric susceptibility from absorbance data across a wavelength range, as you can see on the plot. I converted the wavelength to photon energy. The ...
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Non-linear crystals' interaction with light
This is really just a general question because we've been seeing non-linear crystals in a crystallography class, very briefly.
I was wondering how can we possibly understand the unique way non-linear ...
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Why phonons outside the First Brillouin zone can be represented by phonons inside the First Brillouin zone?
I cannot understand clearly that why phonons outside the First Brillouin zone (in terms of wave numbers) can be represented by phonons within the First Brillouin zone.
For the 1-D monoatomic case, ...
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How to get a periodic structure for a Hexagonal lattice in a perpendicular magnetic field?
I am interested in the problem of a hexagonal lattice placed into a homogeneous magnetic field perpendicular to the lattice plane. My question is about the choices of the magnetic field and vector ...
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Any interesting lattice properties with alloys where $m_2=2 m_1$?
I was creating a homework assignment for some easy heat capacity problems and I found that the rounded molar masses of some metallic elements are nice multiples.
For instance, looking at some rounded ...
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The definition of the primitive vectors?
I was studying Solid State Physics by Kittel, when he defined the primitive transilation vectors he mentioned that: "The lattice is said to be primitive if any two points from which the atomic
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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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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 $(V=0)$, $$[\hat ...
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Can longitudinal/transverse phonons propagate along any direction in the crystal? Interpreting 2D centered rectangular crystal dispersion curves
So I'm looking at the above centered rectangular lattice with atoms of mass $M$ and nearest neighbor interaction strength $A = -\kappa$. The differential equation governing the system is
\begin{align}
...
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Why are rectangular and centered rectangular lattices considered two different Bravais lattices?
They are both lattices, i.e., all the vectors are in the form of $m \vec{e}_1 + n\vec{e}_2 $ with $m$, $n$ integral.
The point groups are the same, i.e., D2
In which sense are they different? A ...
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Derive magnetic space group from lattice space group
Consider we have a simple cubic lattice, space group P23. At each corner of the cubic lattice there is one atom.
Now, if we assign a spin to each atom, and let the spins align in a ferromagnetic ...
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Binding energy in metallic crystals
I have a question regarding metallic bonds.
Let's say that I have an iron crystal (Body-centered cubic bavais lattice), the bonds between iron atoms are (obviously) metallic bonds but i dont know how ...
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In solid physics, why can the wavefunction in real space be real-valued when only the $\Gamma(k=0)$ point is considered? [duplicate]
Packages like the Quantum Espresso use this technique, but I cannot figure out the reason. The Bloch's theorum states that for periodic systems, we have $$\psi(\vec{r}) = e^{i\vec{k}\cdot\vec{r}} u(\...
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Convert between the spatial and Miller-Bravais representations
Story: I am postprocessing my atomistic simulation results done in Lammps by Ovito. One of the interesting features of Ovito is DXA (Dislocation Analysis), which gives information about the ...
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How can we prove that $\langle p_x\rangle=\hbar k_x$ where $\hbar$ is the reduced Planck's Constant for an essentially free electron? [closed]
In my solid state electronics course, I came across this question from the Ben G Streetman and Bannerjee book on 'Solid State Electronic devices'.
Assuming the potential energy $U$ is constant for ...
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Non-zero resitivity in perfect crystal
It is very commonly stated that a perfect crystal has zero resistivity at zero temperature due to its translation invariance. However, in Critical drag as a mechanism for resistivity by Else and ...
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Polarization Under Periodic Boundary Condition
I have a state given as,
$$|{\psi_{0k}}\rangle = \exp(-ik\cdot \hat{r}) |\psi_0\rangle$$
where $k = \frac{2\pi}{L} $ and $ |\psi_{0k}\rangle$ is the PBC ground eigenstate of Hamiltonian $$\hat{H_k}= \...
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What is the expected value of orbital angular momentum $𝐿$ in a octahedral crystal field, where 2 electrons occupy the $𝑡_{2𝑔}$ orbitals?
What is the expected value of the orbital angular momentum 𝐿 for a regular octahedral crystal field, where only 2 electrons occupy the 𝑡_2𝑔 orbitals? Since these orbitals are degenerate and the 𝑡...
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Converting conventional to primitive by hand
I am trying to obtain primitive vectors from conventional ones in my crystal with tetragonal symmetry. I know the conventional $3$-vectors that describe the crystal structure and I can obtain ...
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What is the equation that determine the crystal structure?
Is there a governing equation(s) that the crystal structure can be derived from?
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What is a crystal system, compared to a Bravais lattice?
In my condensed matter class I have been told there are 7 crystal systems: Triclinic, Monoclinic, Orthorhombic, Tetragonal, Trigonal, Hexagonal, Cubic.
I am also told there are 14 Bravais lattices. I ...
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Inverting Phonon Creation / Annihilation Operators for General 3D Lattice
My lecture notes, which cite textbooks by Ashcroft/Mermin and Cohen/Louie as sources, give the following definitions for the creation and annihilation operators for phonons in a 3D lattice, with ...
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What is special of the transition metals?
The title of a book I am now reading is 'multiplets of transition-metal ions in crystals'.
I am curious about one point. Why transition-metal? What is the point of emphasizing it? It seems to me that ...
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Confused about selection rules in optical transitions
This question is motivated by this paper in particular (let me know if it is not open access). They measure transitions between electronic levels inside an ion, placed inside a crystal. In Fig. 2, ...
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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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What causes these 2nd order knot shapes and why?
I twisted a (broken) rubber band around multiple revolutions. Initially it created what I call first order twists, where the "wave-length" of the twist got shorter and shorter across the ...
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Madelung constant convergence and order of summation for 3D infinite-series
I am trying to calculate the Madelung constants for the ordering of an experimental NaAsS2 structure in a fully undistorted face-centered cubic lattice. I originally wrote python code that will ...
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Structure factor of FCC
I know that the structure factor of FCC in conventional basis is
\begin{equation}
S=1+e^{iπ(h+k)}+e^{iπ(l+k)}+e^{iπ(h+l)}
\end{equation}
Now if I change the basis to the primitive basis, the new ...
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Analytical Equivalent Crystal Theory
I need some help in the calculation of the surface energy of a high index plane crystal using the Analytical Equivalent Crystal Theory (AECT) Method.
I am new to the find so, it would be appreciated ...
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Can a Wigner crystal form spontaneously, or it will always need an external confining force field?
As far as I know, a Wigner crystal could be formed spontaneously from a free electron "cloud" with the right conditions (low temperature and electron density). Would it be stable and not ...
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Does a cube under high pressure transform into a ball?
Will a material in the shape of a cube, under high pressure, crumble into the shape of a ball?
One would expect that there will develop strains and stresses, after which the corners crumble and ...
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Non-trivial "energy coverings" of reciprocal space
My question is about sheets of the Fermi surface and their mathematical properties.
As far as I understand, in the one-electron approximation with a weak periodic potential (Bloch approximation) you ...
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How to get Miller indexes from a diffraction pattern?
Working with a single crystal, what is the first thing the diffractometer has access to? What is the output I see and what are the steps from this two levels of information?
If I'm correct, the output ...
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How do I unwrap Wyckoff positions of a crystal into the XYZ coordinates?
I am trying to understand how to use the Wyckoff positions from the Bilbao Crystallographic Server and I am wondering if someone could help with some confusion.
The example material I am looking at is ...
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How can electrons hop large distances if they are connected to the atom which is stationary in an lattice?
How electrons in valency and flow as they are connected to the atom and the atom is stationary but the electron travels way more distance than the size of atom that causes conduction?
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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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Is there an easier way to generate Brillouin zone $\mathbf{k}$-points of FCC lattice?
I am new to working with 3D lattices and am wondering if there are any well-developed methods for generating all $\mathbf{k}$-points inside the first Brillouin zone (BZ) of an FCC lattice.
I use the ...
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Is it possible to have a crystal structure where you are getting peaks of different intensities at different $2\theta$ values with same $hkl$ planes?
Is it possible to have a crystal structure. where you are getting peaks of different intensities at different 2thetha values with same h k l planes? What will you be calling that system to be a single ...
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Boundary conditions in quantum mechanics
I'm having some trouble understanding the role of boundary conditions in (non-relativistic) quantum mechanics.
EDIT:
The following text talks a bit about Bloch's theorem, but this was just supposed to ...
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Confusion between trigonal and hexagonal systems
I'm studying space groups.
It's quite clear (I think) why trigonal and hexagonal systems collapse in the same primitive Bravais lattice, while are different when we introduce non-primitive unit cells, ...
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Confusion about Derivation for Bragg's Law/Condition
I am currently learning about Bragg's Law and its derivation. However, the assumptions/logic behind the derivation seems flawed to me. It is probably likely that I am incorrectly understanding some of ...
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Why the Moire Brillouin zone (BZ) of twist bilayer graphene is hexangular?
Usually, we have the definition of the basis vectors of Moire BZ, which is
\begin{align}
\boldsymbol{b}_1^m=\boldsymbol{b}_1-\boldsymbol{b}_1^\theta,\quad\boldsymbol{b}_2^m=\boldsymbol{b}_2-\...
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Most generic form of refractive index tensors
The refractive index of a material is in general a $3x3$ tensor (as in the case of birefringent crystals). From literature, it seems that in the case of transparent crystals, this tensor is in general ...
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Electron microscopy and Interatomic distances of miller planes
I am studying Transmission Electron Microscopy (TEM), and have been seeing in articles TEM images of different materials typically come accompanied by these diffraction patterns, caused by the ...
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Symmertry of $R$-tensor of Stark Effect in diamond structure
Currently, I am studying the effects of electric fields on color centers in diamonds. However, I have encountered a problem: when addressing the Stark effect caused by the electric field, I use the R ...
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Why must solids in equilibrium become crystalline?
In Landau's Statistical Physics's analysis of solids, he begins with the remark that solids are caracterized by their atoms' small oscillations about equilibrium positions. However, he states that ...
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Brillouin zone(s) the Fermi surface lies within
I'm trying to understand how you can calculate which Brillouin zone(s) the fermi surface lies within. The way that seems reasonable for me is to:
1: Calculate the fermi radius.
2:Look in the "...
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