Lattice Definition and 510 Threads

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. SquidgyGuff

    Potential Energy of a 2D Crystal Lattice

    Homework Statement Use a computer to calculate numerically the potential energy, per ion, for an ifinite 2D square, ionic crystal with separation a; that is, a plane of equally spaced charges of magnitude e and alternating sign. Homework EquationsThe Attempt at a Solution The closest I can...
  2. L

    Bravais lattices in 2 dimensions (and 3 dimensions)

    I'm reading M. Omar Ali's Elementary Solid State Physics and in it, in Subsection 1.4 The Fourteen Bravais Lattices and the Seven Crystal Systems he says that "..., but one cannot place many such pentagons side by side so that they fit tightly and cover the whole area. In fact, it can be...
  3. Helios

    Why an Infinite Lattice Cannot Be Stable: An Analysis of Peter Donis' Answers

    Questions I have arrive from other forum by request. I asked: Why can't an infinite motionless lattice be stable or even possible? We set the stage with an infinite lattice and an exactly zero cosmo-constant ( which I gather results in infinite Minkowski space ) universe, what happens? Can't a...
  4. G

    Reciprocal of the reciprocal lattice is the original lattice

    In my course we are currently studyinh Bravis lattices. We were told that the reciprocal of the reciprocal lattice is the original lattice. This is very easy to prove when given an example of a SC/BCC/FCC lattice, however, is there a formal proof for this?
  5. C

    A Dark Matter effect in Lattice Space

    There are not found the WIMPs until now. Ma be the effect of Dark Matter is because of a defect in topology otherwise than time dilation in General Relativity. The lattice gauge theory formulated on a grid or lattice of points in space and time is successfully used in the quantum...
  6. G

    How to find reciprocal lattice vectors

    So I know that the basis vectors of an FCC in a symmetric form are: a = \frac{a}{2}(\hat{x} + \hat{y}) b = \frac{a}{2}(\hat{y} + \hat{z}) c = \frac{a}{2}(\hat{x} + \hat{z}) And that the reciprocal lattice vectors are the basis vectors of the BCC cells. I'm having a hard time doing the...
  7. atyy

    Status of lattice standard model

    What is the "consensus" status of the existence of a lattice standard model? These two sets of notes don't seem to be in agreement. Wiese's 2009 notes http://www.itp.uni-hannover.de/saalburg/Lectures/wiese.pdf say "Thanks to a recent breakthrough in lattice gauge theory, the standard model is...
  8. Hepth

    A modern intro to lattice QCD?

    Does anyone have a good reference to a more modern introduction to lqcd? By modern I mean including unquenched, staggered quark, etc. Most intros I find are quite dated and I'm interested in learning about newer technology and methods/improved. Figured there may be some lattice theorists here...
  9. R

    Determine the lattice parameter of Tetragonal system

    As a part of our exercise we need to determine the lattice parameter of the given xrd peaks (http://bama.ua.edu/~mweaver/courses/MTE481/Lab2_PP.pdf . the last page of the link), here's the peak data: 2theta 23.73 39.23 46.36 56.69 62.31 71.1 76.15 84.35 89.2 97.29 102.21 110.83 115.9 luckily...
  10. S

    Atoms as spheres in packing fraction of crystal lattice

    Why are atoms taken to be spheres, and not of some other shape, in the calculation of the packing fraction of different crystal lattices? In other words, what experimental evidence and theoretical reasoning motivates this form of the atomic shape?
  11. S

    Figuring out Bravais lattice from primitive basis vectors

    Homework Statement Given that the primitive basis vectors of a lattice are ##\mathbf{a} = \frac{a}{2}(\mathbf{i}+\mathbf{j})##, ##\mathbf{b} = \frac{a}{2}(\mathbf{j}+\mathbf{k})##, ##\mathbf{c} = \frac{a}{2}(\mathbf{k}+\mathbf{i})##, where ##\mathbf{i}##, ##\mathbf{j}##, and ##\mathbf{k}## are...
  12. K

    Lattice points and lattice basis

    Hi! I'm struggling in identifying the lattice points and atom basis. As I understand in a cube, there are 8 lattice points, on on each corner of a cube. But in 2d it is any square between 4 points which are the lattice points. Is this correct? So if the points on the corners are the lattice...
  13. O

    Building a Non-Ideal HCP Lattice in MD Simulation

    Dear Friends, I'm trying to bulid a non-ideal hcp lattice in order to use in MD simualtion. I have12 lattice basis. but 2 of them are negative values. Can anyone tell me how I can convert the negative values to positive? 0.66667 0.33333 -0.0416 0.66667 0.33333 -0.1668
  14. R

    A couple of quick questions on screening / lattice deformation

    If a material is doped (introducing positive dopants to the lattice) there will be a build up of negative cahrge around them, screening. I have a couple of questions that seem obvious but I'm still a little unsure: 1) how would more screening effect resistivity? I would assume that a build up...
  15. gracy

    Is the Calculation of Lattice Energy Correct at Time 2:40 in This Video?

    In the following video,is he right at time at time 2:40 I don't think so .Because lattice energy of fe2o3 should be 36 times greater than Nacl (without considering radius factor)
  16. Matt atkinson

    Statistical Physics: Cubic lattice of two molecules

    Homework Statement A mixture of two substances exists on a cubic lattice of N sites, each of which is occupied by either an A molecule or a B molecule. The number of A molecules is NA and the number of B molecules is NB, such that NA + NB = N. The energy of interaction is k_BT\chi_{AA} between...
  17. Arsenic&Lace

    Are microcanonical algorithms still used in lattice QCD?

    I stumbled across a paper which stated that the relation between statistical mechanics and field theory is exploited to recast lattice QCD in terms of a "microcanonical ensemble" of sorts. I was curious to know if this was still a commonly used technique. The paper in question...
  18. AL-Hassan Naser

    Free electron or empty lattice schrodinger equation solution

    in the solution for free electron we start with X(x) = A exp (ikx) + B exp (-ikx) then using boundary conditions we eliminate B if the wave is traveling in the positive direction and vice versa my questions are: 1. what is the boundary condition used? 2. is it X(-inf) = 0? because this would...
  19. K

    2D-Diamond and 2D-Center Rectangular Lattice Comparison

    Hi All, Using (Mirror + Translation in 2D). I see that Diamond and Center Rectangular Lattice in 2D has the same Symmetry i.e: 1)Horizontal Mirror Plane 2)Vertical Mirror Plane 3)2-Fold Rotation Axis. and Diamond has Smaller Area. Then why we say Center Rectangular has higher Symmetry and we...
  20. Fyj

    Finding the force in a simple harmonic lattice

    Homework Statement If there are a large number of ions oscillating in a straight line, we can pick the nth one oscillating about its equilibrium a*n. The potential of the entire lattice is then U = 0.5*K[u(an)-u([n+1]a)]^2 - summed over all n. How do I use Force = -dU/du(an) to derive that...
  21. N

    Why Is the Length of the Reciprocal Lattice Vector Equal to 2π/d_hkl?

    Hi. This is a very simple and stupid question: Why is the length of the reciprocal lattice vector ##G_{hkl}## equal to ##2 \pi / d_{hkl}##, where ##d_{hkl}## is the distance between the ##(hkl)## planes. Just like the length of the wave vector ##k## equals ##2 \pi / \lambda## I remember that...
  22. H

    Bulk Modulus and its derivative in a fcc lattice

    The bulk modulus B = - V (∂P/∂V). At constant temperature the pressure is given by P= -∂U/∂V, where U is the total energy. We can write B in terms of the energy per particle u = U/N and volume per particle v = V/N : B = v...
  23. N

    Lattice wave dispersion relation

    Hi. A very quick question. Why is it impossible for a wave to travel on a linear one-atomic chain if its wavelength equals the lattice constant? I.e. the lattice points vibrate with a wavelength equal to the distance between them? Here's what I mean...
  24. A

    Transition Metals Lattice Constants - Reference Guide

    Hello every one, Can some one tell me about a reference (paper or book) that possible to find lattice constants of the transition metals. many thanks, Ave
  25. K

    Vortex Lattice Method: Find Airfoil Equation for Designing

    Hello! I have some queries regarding Vortex Lattice Method. I want to design an airfoil using Vortex Lattice Method. For that, I need to find out the equation of airfoil first. So how exactly am I going to do that? Please refer the following link...
  26. K

    Laser Lattice: Understanding Frequency & Wave Vector Effects

    Hi all, Recently I am reading an introduction on using laser to create the so-called optical lattices and a periodic potential to trap the atoms or as a grating, some applications like to act on the cold atoms. I don't have much background on laser but there are few concepts I don't...
  27. Spinnor

    Masses and springs in R^3XR^2 and K.G. Eq. on a lattice?

    Let us have a 5 dimensional lattice in R^5 = R^3XR^2, where at each lattice point we have a point mass and all mass points are linked with springs in all 5 dimensions (edit, to nearest neighbors). Require that motion of the mass points is restricted to a two dimensional subspace, R^2, of R^5...
  28. X

    Natural numbers distributive lattice

    I need a proof that the set of natural numbers with the the relationship of divisibility form a distributive lattice with gcd as AND and lcm as OR. I know it can be shown that a AND (b OR c) >= (a AND b) OR (a AND c) for a general lattice, and that if we can show the opposite, that a AND (b OR...
  29. E

    MCNP: How to specify a small source in a large lattice

    I am working on an input file in MCNPX/6 that uses a CT scan lattice geometry. I want to specify a small source in a large universe (lung). Right now I have a source uniformly distributed through the universe. The existing documentation is vague on this topic. Is it possible to contain the...
  30. Demystifier

    Lattice constant from first principles

    I would like to know how to estimate the value (length) of the lattice constant in crystals in terms of fundamental parameters such as electron charge, electron mass, proton mass and Planck constant. (Alternatively, the formula may contain the Bohr radius, since I know how to calculate Bohr...
  31. C

    Solving Antiferromagnetic Ising Model on Square Lattice

    Hello, I am trying to work out a mean field theory for an antiferromagnetic Ising model on a square lattice. The Hamiltonian is: ## H = + J \sum_{<i,j>} s_{i} s_{j} - B \sum_{i} s_{i} ## ## J > 0 ## I'm running into issues trying to use ## <s_{i}> = m ## together with the self-consistency...
  32. Spinnor

    Lattice QED, most likely path of fields --> no interactions?

    Say I have a large spacetime lattice set up on a supercomputer where I calculate the scattering cross section of two spinless electrons of equal and opposite momentum via lattice QED. To get the right results we must add the amplitudes for every possible "path" the field can evolve from initial...
  33. K

    Vortex Lattice Method: Best Resources to Learn

    can anybody tell me which book or video lecture to study for understanding this method?? any help would be much appreciated.thank you.
  34. Spinnor

    Lattice QCD, path integral, single "path", what goes on at a point?

    Say we try and calculate the ground state energy of the bound state of a quark antiquark meson via lattice QCD. Say I look at one space time lattice point of one path. Do the fermi fields "live" on the lattice points? Do the boson fields "live" on the legs between the space time lattice points...
  35. john baez

    Building the E10 lattice with integer octonions

    Greg Egan just proved something nice: the E10 lattice, famous in string theory and supergravity, can be described as the lattice of self-adjoint matrices with integral octonions as entries! I'm not sure this result is new, but I've been wanting it for quite a while and haven't seen it...
  36. E

    Crystal momentum in a lattice.

    Background information: The wave function for an electron in a crystal lattice is modeled by a Bloch wave. A Bloch wave is a function with the periodicity of the lattice multiplied times a complex exponential function. This exponential function has a wave vector k, called the crystal momentum...
  37. G

    Help needed to understand dispersion curve of a 1D lattice with diatomic basic

    Hi there, I am trying to understand the dispersion curve(as shown below) of a 1D lattice with diatomic basic. Here are my questions 1) Can both optical and acoustic branch of phonon can simultaneously exist in crystal? 2)Why there is a band gap between optical and acoustic phonon...
  38. C

    Why is the change in momentum of a crystal include reciprocal lattice vectors?

    So i don't really understand why the change in momentum of a crystal involves a reciprocal lattice vector. Surely it is just the change in momentum due to the change in the number and frequency of the phonons before and after whatever event/scattering/collision takes place. Can somebody please...
  39. fisher garry

    Problem about lattice structure proof

    I have looked at the cation anion ratio of cubic, octahedral and tetrahedral arrangments on an internet site. By a mathematical derivation they find the minimum value for the cation anion ratios for cubic, octahedral and tetrahedral arrangments. My problem is that even though I get the...
  40. D

    (Bravais?) lattice with one angle equal 90

    I have seen in some books that the triclinic bravais lattice ( a≠b≠c , α≠β≠γ ) excludes explicitly the option that one angle equal 90°. For instance 90°≠α≠β≠γ=90°. If I got the definition of α, β and γ correctly, it would be a primitive cell with a pair of parallel faces as rectangles, and...
  41. Spinnor

    Cubic lattice, masses and springs, fire little mass at it.

    Suppose I have a cubic lattice of N^3 masses, M, each connected to six nearest neighbors with springs of constant k free to move but at rest. Now fire a single mass, m, with velocity v at surface of the lattice such that no rotation can be imparted to the cubic lattice. Let the fired mass bounce...
  42. Medicol

    Counting squares of NxM lattice

    This is not a quiz but I am thinking how to write down a simple math formula to count the total number of squares present in a lattice of NxM points for my 12 year old nephew ? He'll sure be happy if I could turn this into, say, a common sense for pupils like him. :biggrin: For example, In a...
  43. V

    Getting introduced to Lattice QCD

    Hello, I am a first year graduate student in physics who is interested in getting involved in the field of lattice QCD. I purchased the text "Lattice methods for quantum chromodynamics" by DeGrand and Detar. I have never taken a course on quantum field theory, but I hoped that having...
  44. Math Amateur

    MHB Proof of Fourth or Lattice Isomorphism Theorem for Modules

    Dummit and Foote give the Fourth or Lattice Isomorphism Theorem for Modules on page 349. I need some help with the proof of Fourth or Lattice Isomorphism Theorem for Modules ... hope someone will critique my attempted proof ... (I had considerable help from the proof of the theorem for groups...
  45. Math Amateur

    MHB Fourth or Lattice Isomorphism Theorem for Modules - clarification

    Dummit and Foote give the Fourth or Lattice Isomorphism Theorem for Modules on page 349. The Theorem reads as follows:https://www.physicsforums.com/attachments/2981In the Theorem stated above we read: " ... ... There is a bijection between the submodules of M which contain N and the submodules...
  46. A

    MHB Is a Boolean Lattice Atomic if the Top Element is the Join of Atoms?

    My problem for this thread is: Let $L$ be a Boolean lattice. Prove that $L$ is atomic if and only if the top element is the join of a set of atoms. For the forward implication, I am already done. I used Zorn's lemma to show that the set, $\mathcal{F}$, of the elements in $L$ which are the...
  47. A

    MHB Another Boolean Lattice Problem

    My problem is this: Let $L$ be a Boolean lattice. Prove that $L$ is atomic if and only if its order dual $L^{op}$ is atomic. My proof is going like this so far: Let $L$ be a Boolean lattice. Suppose first that $L$ is atomic. Then, by definition, every lowerset in $L$ contains an atom. Since...
  48. A

    MHB Proving Boolean Lattice Complementarity in [a,b]

    The problem is this: Let $L$ be a Boolean lattice. For all $a<b\in L$, prove that the interval $[a,b] = \uparrow a \cap \downarrow b$ is a Boolean lattice under the partial ordering inherited from $L$. What I've managed to do so far: I used the fact that $[a,b]$ was a poset and showed that...
  49. P

    Exploring the Mystery Of Lattice Energy Under Standard Conditions

    My A level Chemistry textbook defines Lattice Energy as "the enthalpy change when 1 mole of an ionic compound is formed from its gaseous ions under standard conditions"; a definition which I can't fully grasp because of the "standard conditions" part. How can gaseous ions exist under standard...
  50. A

    MHB Proving Properties of Lattices: How to Use DeMorgan's Laws

    My problem is this: Let $L$ be a bounded, complemented, distributive lattice and let $x,y,z\in L$. Prove the following: 1. $x\wedge y = \bot \Leftrightarrow x\leq y^c$ 2. $x = (x^c)^c$ 3. $x\wedge y \leq z \Leftrightarrow y\leq x^c \vee z$ 4. $(x\vee y)^c = x^c \wedge y^c$ 5. $(x\wedge y)^c =...
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