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. S

    I Understanding the Derivation of Reciprocal Lattice Basis from Equations 5 and 6

    may someone explain to me or show me the steps of how equations 7a, 7b, 7c were determined from equations 5 and 6
  2. F

    A Bloch theory on hexagonal lattice

    Hello, i am new in condensed matter physics and studying the article 'Engineering artificial graphene in a two-dimensional electron gas' (Gibertini et al., 2009). I read a lot about bloch theorem lately and i got the main idea but still i don't understand the component of the equation. In the...
  3. S

    Find the sampling matrix and sampling structure for R, G and B components

    Hello, everyone. :) All I could gather is that, if I'm correct, lattices are spans of the column vectors of the matrix within the "LAT()" notation and the X and Y occurrences are unit placeholders (such as the pixel unit (since this is in the context of image processing)). And, as an attempt...
  4. P

    A Half filling? Free electron gas? 2D tight binding lattice?

    It is well known that the 2D free electron gas fermi momentum can be expressed as follows, k_F=\left(2\pi n\right)^{1/2} where n is the electron surface density. Assuming this 2D electron system can be considered as 2-D tight-binding square lattice whose eigenergy can be written as...
  5. dRic2

    I Lattice specific heat, help understanding a passage in Ziman's book

    Hi, I don't understand why the author in calculating the expression for the specific heat, divides by ##1/V## (the total volume). Also, in calculating the fraction of modes with frequency in the range ##v##, ##v+dv## he divides by ##V## (eq. 2.65, 2.67) Thanks, Ric
  6. P

    MCNP error zero lattice element hit

    I have been encountering "zero lattice element hit" when I try to run it in MCNP5. The input file is provided below. The square lattice runs fine, but the hexagonal does not. Hope someone can help me here. Thank you.
  7. dRic2

    I Bloch functions and momentum of electrons in a lattice

    Hi, I'm a bit confused about Bloch functions. This is what, I think, I understood: can someone please tell me what's wrong? From Bloch's theorem we know that the wave-function of an electron inside a periodical lattice can be written as ##ψ_k(r)=u_k(r)e^{ik⋅r}##. We hope that far from a lattice...
  8. J

    Help with lattice hinge calculations

    Can anyone confirm the claim on this site that n is greater or equal to 23 when Theta is 90 degrees and t is 3? My calculations indicate that the argument of arccosine in the numerator has to be very close to 1, but the numerator of the argument (k_laset+t) is 3.1 whereas the denominator...
  9. T

    Speed usage in Lattice Boltzmann Method

    One thing that confuses me is the physical speed and sound speed. The lattice sound speed cs=1/sqrt{3} corresponds to the physical sound speed for isothermal flow (sqt{RT}). Why isn't the physical speed (e.g. inlet speed up of lid cavity) converted and use accoringly? $$c_p=\sqrt{RT}≈330m/s...
  10. C

    I Is a photon simply a vibration of the spacetime lattice?

    Is a photon simply a propagating vibration of the spacetime lattice similar to gravitational waves but at a different wavelength and amplitude, and the electron that creates it plucks a single lattice string rather than a bunch? Therefore it has no mass and travels differently through spacetime...
  11. T

    Reciprocal lattice for FCC and diffraction peaks

    My issue is more with the steps to approach rather than the calculations. I was just wondering if someone could confirm my approach to be correct. As it asking for the reciprocal lattice of an FCC I assume this would mean I need to use the points on the BCC to calculate the geometrical...
  12. L

    A Doubts in a lattice translation example

    I have two question about a exemple given in the Sakurai's quantum mechanics book, section 4.3. Let's consider an electron in a periodic potential ##V(x + a) = V(x)##, that has the form of a wave. We will take the potential to go to infinity between two latteces sites, such that its form change...
  13. L

    Can we study an odd number sized lattice model at half filling?

    So one can numerically study (I am interested in exact diagonalization) any 1D lattice model with ##L## sites and ##N## number of particles. At half filling, ##L/N = 2##. My question to a professor was that can we study a system of size ##L = 31## at half filling? He replied yes, there is a way...
  14. Kyueong-Hwang

    What are the lattice points for this?

    In picture, there is crystal structure with two atoms. Question is, what is the lattice points? Find a set of lattice points. I think green and pink points are both lattice points. Am i right? This is our full hw.
  15. nomadreid

    I Lattice of truth values for a paraconsistent logic?

    Usually the truth values of propositions of a logic are structured into a lattice, with 0 (False) on (say) the bottom and 1(True) on (say) the top, and the connecting lines being implication. In paraconsistent logics, there is at least one node which is not implied by 0. Can one safely say that...
  16. Another

    One-dimensional with lattice constant (Problem)

    In question (a) Calculate the average position ? https://en.wikipedia.org/wiki/Expected_value#Finite_case ##<x> =\sum x_ip_i## I saw the answer and wondered. ## \bar{x} = <x> = \sum_{n=0}^{N} \frac{N!}{n!(N-n)!} x_n p^n q^{N-n}## so ##x_n = (2n-N)a## a is lattice constant why x_n = (2n-N)a ...
  17. F

    I Does Lattice QCD use virtual gluons?

    I'm given to understand that perturbation methods don't work in QCD because the coupling constant is too large. So they use supercomputers to calculate equations at various points on a lattice. Does this lattice method still take into account the virtual gluons that we might see in perturbation...
  18. J

    A Lattice parameter difference: PBE vs PBE-D3 functional

    Hi guys! Does anyone know why the lattice parameters of a crystal calculated with PBE-D3 functional are lower than those calculated with PBE? Thanks in advance! :)
  19. W

    I Empty Lattice approximation/Nearly-free electron model

    Hi all, I am having trouble understanding the some ideas presented in some notes I've been reading, help is greatly appreciated! I've uploaded screenshots of the material I'm referring to below, the last two images are what I'm mainly referencing, and the first few are to provide context...
  20. L

    A How to calculate the on-site interaction in optical lattice

    We know the on-site interaction in optical lattice can be expressed as U=4pi*ħ2 a/m ∫d3x |w(x)|4 It's simply understood in a 3d system. But how to calculate this term in 1d and 2d system?
  21. O

    Building a Honeycomb Structure in MCNP: Error Solving

    Hi, I am trying to build a honeycomb structure in MCNP. The code is given below. Hexagonal lattice 1 0 -1 -19 29 fill=1 imp:n=1 2 1 -2.26 -10 11 -12 13 -14 15 u=1 lat=2 imp:n=1 3 0 1:19:-29 imp:n=0 1 c/z 50 50 200 19 pz 31 29 pz -31 10 px 1 11 px -1 12 p 1...
  22. atyy

    I Lattice standard model (Wang & Wen)

    There is a proposal for a generalized lattice standard model by Juven Wang and Xiao-Gang Wen. Could this be correct? I've put this under BTSM because it also mentions grand unified models. https://arxiv.org/abs/1809.11171 A Non-Perturbative Definition of the Standard Models Juven Wang...
  23. L

    A Relative magnetization and a Face Centered Cubic lattice

    In case of simple cubic lattice relative magnetization is given by \sigma=1-\frac{1}{S}\frac{v}{(2\pi)^3}\int^{\frac{\pi}{a}}_{-\frac{\pi}{a}}\int^{\frac{\pi}{a}}_{-\frac{\pi}{a}}\int^{\frac{\pi}{a}}_{-\frac{\pi}{a}}\mbox{d} k_x\mbox{d} k_y\mbox{d}k_z(\mbox{e}^{\frac{E(\vec{k})}{kT}}-1)^{-1}...
  24. Mr Davis 97

    Showing that subgroups of G form a lattice

    Homework Statement Prove that the set of all subgroups of a group ##G## is a lattice with respect to the partial order relation given by containment. Note: You need not prove that containment is a partial order relation but you do need to prove that if ##H\leq G## and ##K\leq G## then there...
  25. Vigardo

    Wind loads for circular cross-section lattice structures?

    Dear experts, How should I compute the wind loads for a Shukhow tower, i.e. a circular cross-section steel lattice structure? The structural members should be rectangular or circular. I´ve been checking EUROCODEs EN 1991-1-4 (2005) and EN 1993-3-1 (2006) without success. Is there any other...
  26. diegzumillo

    A Graph or lattice topology discretization

    Mathematicians, I summon thee to help me identify which field deals with this stuff. I come here not as a physicist but as a sunday programmer trying to solve some numerical problems. I set out to model a lattice version of a smooth space. A discretization procedure not uncommon in physics, but...
  27. W

    I Debye model and reciprocal space

    Hi everyone, I need a little help understanding how periodic reciprocal space applies to the Debye model for solids. Many thanks in advance! If we start with the general derivation of a dispersion relation for a 1D system, with atoms coupled by springs, one gets the following relation $$\omega...
  28. diegzumillo

    A Confusion with finite temperature and lattice formulation

    I think the title sums up pretty well my doubts. I learned QFT from Peskin and Schroeder and other common sources, all implicitly defined QFT at zero temperature. Then I started learning about lattice QCD, how to define the action, how to find continuum limits, the importance of the dependence...
  29. SchroedingersLion

    Setting up an FCC lattice in a simulation

    Greetings, for a Monte Carlo simulation of N hard spheres, I want to set up the initial configuration as a fcc lattice. I am thinking about how to do it most efficiently. Here is the lattice structure: The 3 lattice vectors, each one pointing to a different type of the 12 nearest neighbors...
  30. A

    Lattice Diagram of a transmission line

    In lattice diagram, if the voltage wave is generated at Junction that there is an overhead transmission line before it, will the wave be in two directions with the same magnitude, or one in neglected ? Picture may illustrate .
  31. J

    I Computing Polyakov Loops in Lattice QCD (Basic Question)

    Hi PhysicsForums, I have a pretty basic question about extracting physical parameters from lattice QCD simulations. As described in "Quantum Chromodynamics on the Lattice" by Gattringer and Lang, it seems we should be able to extract the static quark/anti-quark potential by considering the...
  32. Clara Chung

    How Many Atoms to Draw in an FCC Lattice Problem?

    Homework Statement Homework EquationsThe Attempt at a Solution Attempted in the picture
  33. TheQuestionGuy14

    B Is Spacetime a Discrete Lattice? - Prof. Sila Beane's Experiment

    Recently I found an experiment done by Prof. Sila Beane. In this experiment he simulates a tiny porportion of the universe using a QCD Lattice where spacetime is a discrete lattice. He inputs the GZK cutoff point of a cosmic ray into the lattice and what happens is, the result is that the...
  34. Ramil

    QFT on the lattice and Green's functions

    Homework Statement [/B] I'm trying to write a program for caclulating Green's function using Monte Carlo method (Metropolis algorithm) in scalar field theory with a potential λφ4 in 4D. I'm writing it in python. N_t, N_x, N_y, N_z - total number of lattice sites in each directions. Field...
  35. G

    Maximum Wavelength and Scattering Angle for Triangular Lattice

    Homework Statement A triangular lattice of lattice spacing ##a=2 ## angstroms is irradiated with x-rays at time zero of wavelength 20 angstroms at an incident angle of ##\alpha =135##. 1) What is the maximum wavelength of the incident x-rays? 2) What is the scattering angle ##\Omega## for...
  36. fluidistic

    I Mean speed of electrons in a periodic potential / lattice

    Hello people, I have 3 questions related to the mean speed of electrons in a period potential /lattice. I've read Ashcroft and Mermin's page 139 as well as the Apendix E. From what I understood, if one applies the momentum operator on the wavefunction of a Bloch electron, one doesn't get a...
  37. charlesmartin14

    A What Other Phenomena Involve Nuclear Excitations Coupling with Lattice Phonons?

    Mossbauer spectroscopy is an example of a nuclear (de)excitation coupling to a lattice phonons Are there any other examples of this phenomena ?
  38. diegzumillo

    A Meaning of phase transitions in lattice gauge theories

    Hi all Decided to post in 'beyond standard model' because lattice gauge theories are usually used to explore these models. Hope that's all right. So in lattice simulations, phase transitions in beta mass plane seem to have an important meaning. I understand phase transitions from a...
  39. J

    I Ordered lattice of topologies on the 3 point set

    So I made this diagram of the order imposed on the set of topologies on a 3 point set. Each small circle is a topology with the circle elements being the open sets. The larger circles join isomorphic topologies together. The ordering was messy when I tried to draw inclusion from each topology to...
  40. S

    B Electron gyromagnetic ratio & lattice qft

    The electron gyromagnetic ratio of 1.0011596522 is computed using perturbation method and Feynman diagrams that is said to produce a value to better than one part in 10^10, or about three parts in 100 billion. Does the nonperturbative lattice QFT also able to compute it? What is the counterpart...
  41. MathematicalPhysicist

    Effect of Lattice Vibrations on Vacancy Formation.

    Homework Statement I am not sure I understand how did they solve question (b). Here's the solution to question (b): Homework EquationsThe Attempt at a Solution For the first approximation I plugged everything to the identity in (a) and indeed got the approximation as it's written in the...
  42. S

    A Lattice non-perturbative definition

    Dear All I would like to understand a paper for Xiao-Gang Wen " A lattice non-perturbative definition of an SO(10) chiral gauge theory and its induced standard model" . Can anyone help me by suggesting another reference that is a little bit easier. Thank you
  43. T

    A Asking about lattice translation as a discrete symmetry

    I've read Modern Quantum Mechanics J.J. Sakurai 2nd Edition. In topic about lattice translation as a discrete sysmmetry(page 287) , they discuss only in case of ground state. If I want to study more in case of including excited states, where can I find this information? thanks.
  44. A

    I How to simulate a lattice with boundary conditions?

    Hello, My question is very simple but I do not have a lot of experience with simulation. I want to write some code to simulate a lattice with boundary conditions and then I will perform calculations with the Hubbard model to find different kinds of properties of interest. I would like to know...
  45. L

    2D lattice, primitive cell choice

    Homework Statement When calculating the Fourier coefficients of the potential of the following lattice (the potential is a sum of deltas at the atom sites): I get the wrong coefficients if I choose the following primitve cell, with primitve vectors a1,a2: And the right coefficients if I...
  46. Leonardo Machado

    A Convergence of lattice Ising model

    Hello everyone. I'm working on a program to solve 2D Ising model of magnetic materials, using a system with 10x10 spins for simplicity at a temperature of 1E-8 K. I'm using this parameters to get a faster result of m=1 and guarantee it is correct. but... For now i already pass 300 Monte Carlo's...
  47. S

    I How can I obtain the reciprocal lattice of graphene?

    I have a question about reciprocal lattice of graphene. When we see LEED pattern, we can know that reciprocal lattice of graphene is honeycomb. But how can we know theorically that it is honeycomb? Hexagonal lattice or other bravais lattice has just lattice vectors which don`t contain baises. So...
  48. M

    Solve MCNP Error: Low Sampling Efficiency with Lattice Source

    Hi, I have a sphere that it contains many sub-spheres. I want to define these small spheres as volumetric source. But when I run MCNP, it doesn't work. MCNP error: the sampeling effeiciency is too low Maybe someone can help me. 100 0 10 200 1 -1 -10 fill=2 300 1 -1 -2 3 -4 5 -6 7...
  49. V

    Fourier transform of periodic potential in crystal lattice

    Homework Statement Homework Equations I'm not sure. The Attempt at a Solution I started on (i) -- this is where I've gotten so far. I am asked to compute the Fourier transform of a periodic potential, ##V(x)=\beta \cos(\frac{2\pi x}{a})## such that...
  50. binbagsss

    Elliptic functions, periodic lattice, equivalence classes

    Homework Statement ##\Omega = {nw_1+mw_2| m,n \in Z} ## ##z_1 ~ z_2 ## is defined by if ##z_1-z_2 \in \Omega ## My notes say ##z + \Omega## are the cosets/ equivalence classes , denoted by ##[z] = {z+mw_1+nw_2} ## Homework Equations above The Attempt at a Solution So equivalance classe...
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