A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every two elements have a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet). An example is given by the natural numbers, partially ordered by divisibility, for which the unique supremum is the least common multiple and the unique infimum is the greatest common divisor.
Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities. Since the two definitions are equivalent, lattice theory draws on both order theory and universal algebra. Semilattices include lattices, which in turn include Heyting and Boolean algebras. These "lattice-like" structures all admit order-theoretic as well as algebraic descriptions.
Hello,
I have a question that I would like to ask here.
Let ##L = \left\{ x \in \mathbb{Z}^m : Ax = 0 \text{ mod } p \right\}##, where ##A \in \mathbb{Z}_p^{n \times m}##, ##rank(A) = n##, ## m \geq n## and ##Ax = 0## has ##p^{m-n}## solutions, why is then ##|L/p\mathbb{Z}^m| = p^{m-n}##?
I...
This is what I hate about MCNP, not a lot of documentation. How do I define all of a universe as a source and a tally? I have a lattice like the below code.
How do I get this code to work with tallies for positions 1,2, and 3 in the lattice; and a source for the 2's. I get the error "sampling...
Let ##\Lambda## be a lattice and ##a, b \in \mathbb{R}^n##, then
$$a \equiv b \text{ mod } \Lambda \Leftrightarrow a- b \in \Lambda$$
I want to prove the statement.
For the left to right direction I would say, ##a \equiv b \text{ mod } \Lambda \Leftarrow a = b +k\Lambda##, where ##k \in...
Say we have as special lattice ## \Lambda^{\perp}(A) = \left\{z \in \mathbf{Z^m} : Az = 0 \in \mathbf{Z_q^n}\right\}##. We define ##U \in \mathbf{Z^{m \times m}}## as an invertible matrix then I want to proof the following fact:
$$ \Lambda^{\perp}(AU) = U^{-1} \Lambda^{\perp}(A) $$
My idea:
Let...
or Are all naturally occurring crystals with periodic arrangement of lattices Bravais lattices?
From two days, I have been trying to understand Bravais lattices and what it's importance is and after a lot of research, I came to know that they are a periodic arrangement of lattice points with...
##\require{physics}## Recently, I wet my feet in X-ray diffraction a bit more than what is usually covered in standard solid state physics textbooks at the undergrad level, like Kittel. Two good books that I chanced upon included Christopher Hammond, The Basics of Crystallography and Diffraction...
LBM model for phase change- relevant equations found here. Also here.
#Thermal LBM
#solves 1D 1 phase phase-change
#D2Q5 Lattice
nx=100 # the number of nodes in x direction lattice direction
ny=5 # the number of nodes in y...
Hello,
I am wondering if in an n-ball the number of lattice points is finite.
First, we have a ball which is bounded by the radius. The distance between two lattice points is given by the successive minimum. Theoretically, one could now draw a ball* around each lattice point in the (big)...
I assumed three points for a triangle P1 = (a, c), P2 = (c, d), P3 = (b, e)
and of course:
a, b, c, d, e∈Z
Using the distance formula between each of the points and setting them equal:
\sqrt { (b - a)^2 + (e - d)^2 } = \sqrt { (c - a)^2 + (d - d)^2 } = \sqrt { (b - c)^2 + (e - d)^2 }(e+d)2 =...
There is an input file for a simple 16 x 16 lattice fuel assembly. I have a message blocking the run of the code;
"bad trouble in subroutine newcel of mcrun source particle no 1 random number 6647299061401 zero lattice element hit."
What is wrong?
Hello,
I want to prove that for any lattice ##\Lambda = \Lambda(B)## (the ##B## is a basis) the orthogonalized parallelepiped ##P(B^*) = B^*\left[-0.5,0.5\right)^n## is a fundamental region of a lattice.
If I wanted to show this, I would try to establish a volume argument here. After all, the...
The Maxwell wavefunction of a photon is given in [here] as follows:
Because the curl operation mixes 3 different components, this wavefunction only works for a minimum of 3 space dimensions, with each grid point having 6 component numbers ##{E^1, E^2, E^3, B^1, B^2, B^3}##, and with the...
A toy model of a QFT lattice (in 1 dimension) is given in [here] (at 5:55):
We assume that ##\Psi## is a vector set of four complex numbers having some values at every point on the grid, for instance:
$$\Psi_{100} =
\begin{bmatrix}
1+2i \\
3+4i \\
5+6i \\
7+8i
\end{bmatrix}$$
and...
The definition of the Wilson action relating to discrete Yang-Mills model is:
$$ S_{plaq} (\sigma) := \frac{1}{2}\sum_{plaq}\|I_N - \sigma_p\|^2 $$
(from [here] at 5:55)
It is mentioned that ##\sigma_p## is some kind of a matrix. Could anyone give an explicit example of what a ##\sigma_p##...
Hello,
how can one proof that the dual of ##\mathbb{Z}^n## is ##\mathbb{Z}^n##?
My idea:
The definition of a dual lattice says, that it is as set of all lattice vectors ##x \in span(\Lambda)## such that ##\langle x , y \rangle## is an integer. When we now consider ##\mathbb{Z}^n## we see that...
Hi Pfs
i am interested in spin networks (a pecular lattices) and i found two ways to define them. they both take G = SU(2) as the Lie group.
in the both ways the L oriented edges are colored with G representations (elements of G^L
the difference is about the N nodes.
1) in the first way the...
I would like to show that a LLL-reduced basis satisfies the following property (Reference):
My Idea:
I also have a first approach for the part ##dist(H,b_i) \leq || b_i ||## of the inequality, which I want to present here based on a picture, which is used to explain my thought:
So based...
With some help from another thread, I learned how to solve a simultaneous diophantine approximation involving log(2), log(3), log(5) etc. This method is based on Mathematica's LatticeReduce function. At first, I was quite happy to use it as a black box to work on some hobby math exploration, but...
Hello,
I've been thinking a bit about the definition of the ##i##-th successive minima of a lattice (denoted with ##\lambda_i(\Lambda)##), and I would argue that the ##i##-th successive minimum is at most as large as the largest lattice basis vector ##b_i##.
More formally...
Hello,
I am currently working on the proof of Minkowski's convex body theorem. The statement of the corollary here is the following:
Now in the proof the following is done:
My questions are as follows: First, why does the equality ##vol(S/2) = 2^{-m} vol(S)## hold here and second what...
Jester points out a new lattice result https://arxiv.org/abs/2206.15084 from the Extended Twisted Mass Collaboration (ETMC) that is closer to the measured muon g-2.
Davide Castelvecchi has a news item in Nature summarizing various results about the possibility of a muon g-2 anomaly.
Suppose a square lattice. The planes are such as the image below:
I light wave incides perpendicular to the square lattice.
The first maximum occurs for bragg angle (angle with the plane (griding angle) as ##\theta_B = 30°## (blue/green), green/blue in the figure).
The angle that the...
Hi all, I'm new to the forum. Maybe you guys can give me hand with this.
I am using MCNP4 to model a 17x17 fuel element. I want to know the average neutron flux in a specific pincell but so far everything I try results in error. This is my input (text file is attached too):
c CELL CARDS
1 1...
In 3D period lattice, can we separate variable and write potential as V=V(x)+V(y)+V(z)?Then we can reduce the 3D problems into 1D problems. I ask this question because in Solid State Physics books they often consider the 1D problems.
I recently watched this video by David Tong on computer simulation of quantum fields on lattices, fermionic fields in particular. He said it was impossible to simulate a fermionic field on a lattice so that the action be local, Hermitian and translation-invariant unless extra fermions get...
I was reading about numerical methods in statistical physics, and some examples got me thinking about what seems to be combinatorics, an area of math I hardly understand at all beyond the very basics. In particular, I was thinking about how one would go about directly summing the partition...
Hello, I have to solve this problem. I will apply the Niemeijer Van Leeuwen method once I have the probability distribution
proper to the renormalization group ,P(s,s'). For example, in the case of a triangular lattice, this distribution is:
where I is the block index. However, it is very...
I know WS cell only contains one lattice point, so we would have to trace bisectors, and obtain some kind of irregular shape.
Anyways, I wanted to check if what I did is okay. It is considering a fictitious point as the center of the (non-primitive) unit cell, which would be one of those...
Hello brilliant engineers.
How to calculate the bending stress of a lattice boom crane design?
It’s clear how to calculate a single chord in a lattice box, at least as a cantilevered tube. However, when placed in a box lattice what is the proper approach?
I’m constructing this crane in...
I am struggling to understand shocks in a one dimensional lattice with a linear spring connecting the masses. Say I have a one dimensional lattice with a linear spring constant, k and lattice spacing a. If the particles in the lattice has mass, m then my speed of sound c is a*sqrt(k/m). That is...
Potential energy in a two-dimensional crystal
Consider the potential energy of a given ion due to the full infinite plane. Call it##U_{0}##. If we sum over all ions (or a very large number##N##) to find the total##U##of these ions, we obtain##N U_{0}##. However, we have counted each pair twice...
The MCNP6.2 manual (page 3-37) says: "There are two nj values that can be used in the lattice array that have special meanings. A zero in the level-zero (real world) lattice means that the lattice element does not exist, making it possible, in effect, to specify a non-rectangular array."
How...
We have an Ising lattice on the x-axis . on every site there is an atom which can be up or down. i suppose that there are N atoms (repetedly with the same values). Each sequence of spins has an energy H with a probability exp(-H/k T)
i suppose that there is a device attached to each atom...
https://www.researchgate.net/figure/1-Silicon-crystallographic-structure-It-has-the-diamond-structure-which-is-two-fcc_fig4_34172659 the fcc silicon lattice is shown.
My question is:
Since the silicon atom has 4 valence electrons and requires 4 more to be completed, why are so many atoms shown...
The propagation speed of the wave is C/sqrt(9) = up if the length of the transmission line is 2m then every 10ns it will pass through the middle of the transmission line. But the switch in the circuit is opened after 5ns so after the current wave bounces off of the Load for the first time it...
In the article 'Cellular vacuum' (Int. J. Theor. Phys. 21: 537-551, 1982), Minsky writes: "One can prove that any bounded packet which moves within a regular lattice must have an asymptotically helical trajectory...". But he gives no references whatsoever.
I had no success in a search on the...
I am currently trying to compute the Green's function matrix of an infinite lattice with a periodicity in 1 dimension in the tight binding model. I have matrix ##V## that describes the hopping of electrons within each unit cell, and a matrix ##W## that describes the hopping between unit cells...
Hi,
The enthalpy of crystallization of $KCl$ (Potassium Chloride) is + 715 kJ/mol. The enthalpies of hydration for Potassium and Chloride are -322 and -363 kJ/mol respectively. So, enthalpy of solution of $KCl \Delta H_{sol}=\Delta H_{lattice} - \Delta H_{hyd}$
$\Delta H_{sol}=715 kJ/mol -685...
For an ionic lattice, the contribution to the electric potential energy from a single ion will be ##U_i = \sum_{j\neq i} U_{ij}##, which can be expressed as$$\begin{align*}U_i &= -6 \left( \frac{z_+ z_- e^2}{4\pi \varepsilon_0 r_0} \right) + 12 \left( \frac{z_+ z_- e^2}{4\pi \varepsilon_0...
Let ##\mathscr{L_H}## be the usual lattice of subspaces of Hilbert space ##\mathscr{H}##, where for ##p,q\in\mathscr{H}## we write ##p\leq q## iff ##p## is a subspace of ##q##. Then, as discussed by, e.g., Beltrametti&Cassinelli https://books.google.com/books?id=yWoq_MRKAgcC&pg=PA98, this...
Hi!
Situation: quasi-free electron in a 2D lattice, considering atomic potential V(r) = exp{-|r|/b} (r is the distance from the atom)
I'm trying to compute the first five energy gaps at point (10),
firstly I don't understand the meaning of calculated 5 energy gaps at one point and usually we...
Hi, take a look at the picture from my textbook, specifically the bottom part:
there are five lattice points, shouldn't that mean that there are also 5 "small basis balls"? Or can they be "shared"? If so, they are not all oriented in the same way - is that not important since there's no...
If propositions ##p,q\in{\mathscr L}_{\mathcal H}## (i.e., the lattice of subspaces of ##\mathcal H##) are incompatible, then ##\hat p\hat q\neq\hat q\hat p##. But since it's a lattice, there exists a unique glb ##p\wedge q=q\wedge p##. How are they mathematically related?
In particular, I...
hi guys
our solid state professor sent us a work sheet that included this example :
i solved it not sure its correct tho :
is it that simple , or this is not the right approach for it ?
I need to write a code for my simulation for which I have to create bcc lattice. Can anybody suggest how I can start since I am jumbled up this 3d lattice coordinate writing?
hi guys
our solid state physics professor introduced to us this new concept of reciprocal lattice , and its vectors in k space ( i am still an undergrad)
i find these concepts some how hard to visualize , i mean i don't really understand the k vector of the wave it elf and what it represents...
Hi everyone,
even before addressing the following points I have a serious issue in understandig the text of the Exercise.My idea was to model this system with a lattice gas. Given that each site can host 2 atoms I have 3 possibilities for each site: I'll call'em ##S_{11} S_{00}## and ##...