# What is Lattice: Definition and 508 Discussions

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.

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1. ### I What is the size of the quotient group L/pZ^m?

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...
2. ### Efficient MCNP Lattice Source Help: Defining Universes and Tallies in Cell File

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...
3. ### I Proof regarding congruence relation

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...
4. ### I Question on proof ##\Lambda^{\perp}(AU) = U^{-1} \Lambda^{\perp}(A)##

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...
5. ### I Can all possible periodic arrangements of lattices be arranged as a Bravais lattice just by taking a different motif?

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...
6. ### I The missing factor of 2π in reciprocal lattice calculations

##\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...
7. ### Thermal lattice Boltzmann model ignoring source term -- python code help please

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...
8. ### I Finite many Lattice Points in Sphere?

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)...
9. ### I Prove that a triangle with lattice points cannot be equilateral

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 =...
10. ### MCNP lattice of the fuel assembly input file?

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?
11. ### I Question regarding fundamental region of a lattice

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...
12. ### I Time evolution of the electromagnetic wavefunction on a lattice

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...
13. ### A Hamiltonian V and T of a lattice?

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...
14. ### A Definition of the Wilson action on a lattice plaquette?

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##...
15. ### I Why is the dual of Z^n again Z^n ?

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...
16. ### A Gauge theory on a lattice: intertwiners, gauge potentials...

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...
17. ### A Proof of the inequality of a reduced basis

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...
18. ### I Diophantine approximation via Lattice Reduction

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...
19. ### I Exploring the Relationship Between Lattice Successive Minima and Basis Vectors

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...
20. ### A Question regarding proof of convex body theorem

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...
21. ### B ETMC lattice results are closer to the measured muon g-2

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.
22. ### Bragg angle with an inclined plane through a square lattice

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...
23. ### MCNP4 help: f4 tally in lattice

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...
24. ### I A question about potential in the periodic lattice of a solid

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.
25. ### I Simulating physics: the current status of lattice field theories

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...
26. ### I Problems involving combinatorics of lattice with certain symmetries

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...
27. ### Renormalization Group:NiemeijerVan Leeuwen Method-Ising Square Lattice

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...
28. ### What would be the Wigner-Seitz cell of this lattice?

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...
29. ### Big Robot- Lattice boom crane - calculating maximum stress

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...
30. ### Chemistry Determine Lattice Energy of PbCl2

Here's my attempt at solving it using Hess's Law. I get -1770kJ when I add up all the numbers. The correct answer is supposed to be -2484 kJ.
31. ### I How do shockwaves in a 1D linear lattice work?

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...
32. ### Is the potential energy of an ion in a lattice ##U_0## or ##U_0/2##?

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...
33. ### MCNP: How to use value zero in a fully specified fill

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...
34. ### MHB Proving Complemented Distributive Lattice Property

Show that in a complemented distributive lattice a ≤ b ⇔ a ∗ bʹ = 0 ⇔ aʹ ⊕ b = 1 ⇔ aʹ ≤ bʹ. can somebody help me prove this.
35. ### A Ising lattice seen by an inertial observer

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...
36. ### Silicon FCC -- Why are so many atoms shown in the lattice?

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...
37. ### Engineering Lattice Diagram for Transmission Line

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...
38. ### Fortran Density of state of a linear lattice

PROGRAM DensityOfState IMPLICIT NONE INTEGER :: omega , t0 , a0 ,kx omega=1 ; t0 =1 ; a0=1 REAL :: kx , eta , w , ek eta=0.0015 REAL,PARAMETER :: pi=3.14 COMPLEX , PARAMETER ::i=(0,1) COMPLEX :: Energy COMPLEX :: Density COMPLEX :: GreenFunc...
39. ### A Where Can I Find the Proof of Asymptotically Helical Trajectory in a Lattice?

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...
40. ### A Green's function calculation of an infinite lattice with periodicity in 1D

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...

43. ### A Is the "op" lattice ##\mathscr{L_H}^\perp## also atomistic....?

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...
44. ### Energy gaps for quasi-free electrons in a 2D lattice

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...
45. ### I What is the Relationship Between Lattice Points and Basis Balls?

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...
46. ### A Exploring Basis Vector Relationships in Incompatible Propositions

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...
47. ### Number of k-cells for a 1D lattice in reciprocal space

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 ?
48. ### Code for BCC Lattice Algorithm: Start Here

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?
49. ### Some questions about reciprocal lattice vectors

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...
50. ### Ising Model for a Lattice Gas

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 ##...