All Questions
84 questions
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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 ...
- 173
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0
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42
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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 ...
- 1
1
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0
answers
56
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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, ...
- 51
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0
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15
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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 ...
-1
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1
answer
77
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Translational invariance $\neq $ Galilean invariance?
I have the impression that some literature say that Galilean invariance is broken by a uniform lattice. That is, although a uniform lattice like a tight binding model is translationally invariant, it ...
- 2,165
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1
answer
123
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Why are there triclinic and monoclinic lattices, but biclinic is never mentioned?
When classifying the Bravais lattices we have the
triclinic (point group ${\rm C_i}$) and the monoclinic $({\rm C_{2h}})$ cases, but we do not see the "biclinic" case listed. Why not?
It ...
- 5,781
1
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1
answer
221
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Can the lattice of an element with face-centered cubic (FCC) crystal structure be regarded as simple cubic (SC)?
According to Introduction to Solid State Physics by C. Kittel,
An ideal crystal is constructed by the infinite repetition of identical groups of atoms. A group is called the basis. The set of ...
- 483
1
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1
answer
223
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Crystal field Hamiltonian using Stevens operators
I am trying to explicitly find (or understand how to find at first) Hamilton operators for crystal fields in different symmetries, e.g. $T_d$, using Stevens operators. The Hamiltonian is then of the ...
- 11
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79
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Why can't hexagonal lattice be body-centered or face-centered?
There are 14 bravais lattices, extending 7 primitive lattice by optionally adding points on base centers, face centers or body centers for 7 crystal systems. I know some combination are not allowed, e....
- 11
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0
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51
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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 ...
- 143
3
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1
answer
1k
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Space group of the honeycomb lattice
The question is simple: what is the space group of the 2-dimensional honeycomb (graphene) lattice?
I tried googling it. One source (PDF) says it is $P6/mmm$ (No. 191) (on the 11th page); another says ...
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1
answer
35
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Enumerating all the crystallographic directions over a half sphere in a crystal
Im taking alook at this paper [1] where there is the following statement:
CO2 was treated as a linear rigid molecule and energies were computed for 61 orientations at each center-of-mass position. ...
- 113
5
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2
answers
218
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How does one show that the ${\bf k}$-vector labeling a Bloch state is an arbitrary real vector?
I'm frustrated that I can't understand something that must be simple and fundamental. I'd appreciate any answer to the question, but also any clarifications of how my presentation of the theorem/proof,...
- 1,330
1
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0
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88
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Understanding of Band Inversion in $\rm Bi_2Se_3$ Topological Insulator [closed]
I'm currently trying to understand Fig. 2 in this Paper (http://dx.doi.org/10.1103/PhysRevB.82.045122) which aims to explain the origin of band inversion in Bi2Se3. I really want to get an ...
- 39
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1
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154
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Point group symmetry: difference between $C_6$ and $C_{6v}$
I am having trouble understanding the difference between $C_6$ and $C_{6v}$ symmetry: why don't all structures with $C_6$ symmetry also have $C_{6v}$ symmetry? Does $C_6$ only apply to 2D structures? ...
- 119
2
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1
answer
124
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Is the formation of crystal due to internal symmetry or spacetime symmetry?
In the contexts of field theory, we have internal symmetries and spacetime symmetries. Referring to crystal, people would say it is due to space translation symmetry. However, I don't think the ...
- 31
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1
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234
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How can I compute the coordinates of The $k$-points of Silicon?
I would like to know how to find the coordinates of the high symmetry points or $k$-points of Silicon?
1
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1
answer
69
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Why aren't there any $S_8$ or $S_{12}$ point groups in crystallography?
When considering improper rotations (roto-reflections), we can derive that if $n$ is odd, then $C_n$ and $\sigma_h$ (reflection plane normal to $C_n$ axis) must exist. Similarly, we can also derive ...
4
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3
answers
2k
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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?
- 53
3
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1
answer
325
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How to unfold electronic bands? How do the reduced/extended schemes work? Confusion regarding the quasi-momentum
One of the simplest models for electronic band structures is a tight binding model on a one dimensional chain with spacing $a$, one atom per cell and interactions only for nearest neighbours. This ...
- 423
5
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1
answer
401
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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 ...
- 257
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0
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64
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Symmetries of lattice model of Weyl semimetal
Lattice model of (Time Reversal symmetry breaking) Weyl semimetal is given by Hamiltonian:
$$t_x\big((\cos(k_x a)-\cos(k_0 a)) + t_y(\cos k_y b -1) + t_z(\cos {k_z c} -1) \big)\sigma_x + t_y(\sin(k_y ...
6
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1
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833
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What’s the intuition behind snowflake symmetry? [duplicate]
What’s a more rigorous description of why snowflakes are so symmetric?
The general explanations of why they’re symmetrical are:
Theyre not. The branches actually vary.
Snowflakes are somewhat ...
- 3,395
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41
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Given a crystal with mirror symmetry along a lattice plane, how to find the correspond plane in first Brillouin zone
Given a crystal with mirror symmetry along a lattice plane, how to find the correspond invariant plane in first Brillouin zone?
- 263
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424
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Symmetries of the high symmetry points in the Brillouin zone
The monolayer $MoS_2$ belongs to the $D_{3h}$ point group and can be described with a hexagonal lattice. The $\Gamma$-point in the Brillouin zone has the full symmetry of the point group, $D_{3h}$, ...
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3
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1k
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Why is the translational symmetry broken?
In the book Condensed Matter Field Theory by Altland, on page 5, it is given that
$$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}\...
- 2,313
2
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1
answer
595
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Most symmetric unit cell in a two-dimensional arrangement
Consider the two dimensional structure below, where two atoms A (white) and B (black) are displayed like a brick wall.
I need to find the most symmetric primitive unit cell of this structure and the ...
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1
answer
2k
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Why phonons are Goldstone modes?
I read this in the lecture notes by David Tong:
"Gapless excitations often dominate the low-temperature behaviour of a system, where they are the only excitations that are not Boltzmann ...
- 83
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2
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2k
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What is a simple example illustrating the concept of "commensurate" and "incommensurate" order in condensed matter physics?
In a wide range of contexts in condensed matter physics, e.g this paper, the concepts of commensurate and incommensurate orders are invoked to describe particular ordered phases. I think I have some ...
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1
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600
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Bravais lattice point groups
I'm trying to describe Bravais lattice point groups as permutations of lattice points. In doing so, I encounter a problem: I can only find descriptions of Bravais lattice point groups in terms of ...
- 1,510
34
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1
answer
900
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Why are snowflakes flat?
There have been many questions and excellent answers in this community about the symmetry of snowflakes, e.g., here and here. There is however one aspect of snowflakes that does not seem clearly ...
- 65k
5
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1
answer
491
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Bloch's theorem for a lattice with sublattices
Bloch's theorem states the following: suppose we have a Hamiltonian
$$ H = \frac{p^2}{2m} + V(x) $$
where $V(x + a) = V(x)$, then the wavefunctions take the form $\psi_k(x) = e^{ikx} u_k(x)$ and $u_k(...
- 4,064
2
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2
answers
1k
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Why do the symmetries of a simple cubic lattice not include a 4-fold rotational axis through the lattice points?
When I read about classifying lattices based on symmetry, for the simple cubic lattice (or, as Wikipedia calls it, a primitive cubic lattice), there are only three 4-fold axes of rotational symmetry, ...
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1
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89
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What are good introductory texts to symmetry groups of molecules and crystal structures?
When reading about spectroscopy and non-linear optics I've stumbled a few times over short mentionings of symmetry groups for molecules or crystals.
E.g. to argue that a molecule like CCl$_4$ has a ...
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0
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59
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Symmetry of spherical ice crystals
Page no. 291 of Hidden Unity in Nature's Laws by John C. Taylor says the following -
"Take a spherical water drop.
No special direction is picked out by such a drop. If we rotate it
nothing has ...
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1
answer
76
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What does it mean to assign group operations to distinct sets for space groups?
I am trying to understand space groups in crystallography. In Internation tables for crystallography, for a nonsymmorphic space group, they list some symmetry operations. 8 of them are listed under ...
- 1,802
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0
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115
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Find the density of states in X points of Silicon
The problem statement is given verbatim
In Si, the dispersion relation at the [001] X points is: $$E=\frac{\hbar^2}{2}\left(\frac{k_x^2+k_y^2}{m_t}+\frac{(k_z-G)^2}{m_l}\right)$$ where G is the ...
- 305
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1
answer
248
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What happens to the space group of a crystal when introducing a non-trivial basis?
I am trying to understand crystallography and the space groups of crystals, but I have one major question bugging me. The book I am using adresses different lattice symmetries and applications of ...
- 1,802
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1
answer
571
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How is a non-primitive unit cell/lattice helpful?
I am starting with the basics of X-ray crystallography, and I have encountered something I'm not able to rationalize.
As I understand it, the unit cell is the smallest parallelepiped enclosing the (a?)...
- 153
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1
answer
693
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Symmetries of the square lattice
According to the literature, the square lattice has $C_{4v}$ symmetry. This point group does not contain inversion. However, the square lattice is obviously inversion-symmetric.
Is this because ...
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2
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295
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How can crystals be isotropic?
In cubic crystals where $a=b=c$, there are rotational invariances that leave the system unchanged. If some of the electrons are responsible for many properties of solids and that they are free to move ...
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35
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Why do some lines connecting points of high symmetry not have a name?
This page lists a few Brillouin-Zones. You can click on them and see a 3D model you can rotate. I noticed that some lines connecting points of high symmetry have special names, others don't. Are they ...
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0
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40
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Why are the six prongs of a snowflake the same?
My understanding is that the variety in snowflakes is determined by pressure, temperature, and humidity. This is why the six prongs grow same way, because they are close enough that they are in the ...
- 1,276
3
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1
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498
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What is the meaning of vertical bars in paths of high symmetry points?
I am a new to the study of high symmetry paths. After looking at the Silicon path that is $Γ—X—U|K—Γ—L—W—X$, I am not able to understand the meaning of $U|K$ in this path?
4
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2
answers
2k
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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 ...
- 4,064
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0
answers
291
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Bloch Electron velocity at Brillouin Zone edge
Ashcroft and Mermin in their discussion on the energy levels near a single Bragg plane (Chapter 9: Electrons in a weak periodic potential) mentions the following while calculating the velocity near ...
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0
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32
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Crystals, symmetries in mixture
What would happen to the crystal structure of ice, if i freeze salt water?
What changes in symmetry can i expect? I do not know the terminology for this stuff...but in what 'subclass' or 'subfamily', ...
- 484
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0
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193
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Notation of basis functions for irreducible representations
In character tables for symmetry groups, there are typically basis functions for each irreducible representation given. There are basis functions given like $xy$, $S_x$ or $R$. Could someone explain ...
- 171
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2
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1k
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Does the space group P63/m (No. 176) have C6 rotation symmetry?
Recently I'm working on a compound with space group P63/m. The top view of its structure is shown below (where only atoms of z=1/4 are shown).
From the list of space groups (Wiki: List of Space ...
- 308
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1
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29
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What space group describes a 1-dimensional crystal with reflection symmetry along axis?
I'm trying to understand the symmetry of an effectively 1-dimensional system, but I'm confused about how the 1-dimensional ``line groups'' are classified. If you have a system along the $z$-axis which ...
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