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. C

    What Frequency Does an Atom in a Cubic Lattice Absorb Radiation?

    [SOLVED] SHM: atom in a lattice Homework Statement From Advanced Physics by Adams and Allday. Spread 3.34, Q 5. An atom of mass 5 \times 10^{-26} kg is in a cubic lattice with all bonds between adjacent pairs of atoms having a spring constant about 100 N/m. Estimate the natural frequency...
  2. Q

    Does any lattice or lattice shape has a periodic boundary condition?

    If not, then what are the conditions for us to construct a periodic boundary condition(PBC)? If so, then please help me construct a PBC for the lattice shape in the attachments. I want to ask that what lattice site m's left neighbor is and what lattice site i's down neighbor is.From the...
  3. P

    Arranging Compounds by Increasing Lattice Energy

    Homework Statement Order the following compounds in order of increasing lattice energy: AlN, BaTe, KCl, Rb2Se Homework Equations The Attempt at a Solution Ok I know that as the distance between the two nuclei decreases and as the charge of the cation and anion increase the lattice energy...
  4. B

    Book for studying lattice models

    Could anyone suggest a book for studying lattice models. Especially xy and Heisenberg models.
  5. D

    Calculate the lattice constant of a body-centered cubic iron

    Calculate the lattice constant of a body-centered cubic iron crystal using the molar mass of iron, the density of iron and the Avogadro number.
  6. T

    Reciprocal lattice and Fourier series

    First off, this is not a homework problem. I was reading Charles Kittel solid states book on Chapter 2, equation 3: electron number density, n(x), expanded in a Fourier series: n(x) = n_0 + \sum_{p} [C_p cos(\frac{2\pi p x}{a}) + S_p sin(\frac{2\pi p x}{a})] From this expansion, wouldn't...
  7. F

    Largest Lattice Energy: NaCl, CaO, Li2O, or CaCl2?

    Homework Statement Which one of the following ionic compounds has the largest lattice energy? NaCl CaO Li2O CaCl2 Homework Equations Ionic/Atomic size The smaller the ionic size, the larger the lattice energy. Size increases from right to left, increases top to bottom of...
  8. Jim Kata

    Why not put gravity on a lattice and be done with it

    It seems to me that the UV divergences in a quantum theory of gravity could be canceled by just requiring the distance between any two points be finite. I know loop quantum gravity basically addresses this through the quantization of area and volume, but you don't even need loop quantum gravity...
  9. N

    Square lattice electron gas

    Hi all, I'm struggling to understand the relationship between electrical conductivity and Bragg reflection in a 2D square lattice free electron gas with lattice spacing a. Is it the case that Bragg reflection in the electron gas results in electrical resistivity? My understanding of...
  10. J

    Find the Lattice Constant of Atoms - Get Help Now!

    Will someone please give me a link that explains how to find the "lattice constant" of certain atoms. Or could someone please walk me through the process of how you find the lattice constants. Thanks in advance:smile:
  11. malawi_glenn

    What are the properties of reciprocal lattice vectors?

    Homework Statement "What does a reciprocal lattice vector represent in the real lattice?" The Attempt at a Solution The answer to that one is that the reciprocal lattice represent all possible k-values for the incoming radiation to be contained in the real lattice. Hence a...
  12. Z

    Visualization of a reciprocal lattice

    I have been learning kittel's solid physics, But find it hard to have a firm grab of what a reciprocal lattice is like and can't understand it's relationship with e original lattice.. Is there a picture that draws a lattice n its reciprocal lattice into e same picture? so that i can visualize...
  13. L

    Lattice points : Convex region symm. about the origin

    Let R be a convex region symmetrical about the origin with area greater than 4. Show that R must contain a lattice point different from the origin. This is the 2-D case of Minkowski's theorem, right ? How about the n-dimensional version ? The n-dimensional version is : Given a convex...
  14. J

    Understanding 3D Si Dispersion Relations & Reciprocal Lattice Vectors

    I am trying to understand 3D Si dispersion relations and reciprocal lattice vectors. My confusion is that when I look at dispersion relations the wave vector typically is normalized from 0 to 1 by a/2pi. I thought the edge of the first BZ was pi/a. Is this correct or is it 2pi/a for a diamond...
  15. O

    Calculating Lattice Energy: LiF vs. CsF

    I'm having some problems with the calculations associated with this problem: Which compound in each of the following pairs of ionic substances has the most exothermic lattice energy? (a) LiF CsF I know I need to use Coulomb's law, but I don't know how to calculate r...
  16. M

    Exploring CoIr Alloys: Crystal Structures & Lattice Parameters

    I am looking for informations about CoIr alloys. Who can tell me about crystal structure, lattice parameters of CoIr alloy? Please share me some references about this material. Thanks! :!)
  17. P

    Lattice Energy and the Born Haber Cycle

    Ok most questions (that I have come across) when dealing with the Born Haber Cycle give the Sublimation energy, the Dissociation energy, the Electron Affinity, the Ionization Energy, and the Formation Energy. I know that to get the Formation Energy it is: Hf= Hs+Hi.e.+1/2Hd+He.a.+U (Lattice...
  18. P

    Lattice Energy and the Born Haber Cycle

    Ok most questions (that I have come across) when dealing with the Born Haber Cycle give the Sublimation energy, the Dissociation energy, the Electron Affinity, the Ionization Energy, and the Formation Energy. I know that to get the Formation Energy it is: Hf= Hs+Hi.e.+1/2Hd+He.a.+U (Lattice...
  19. D

    Lattice Waves: Atom Chain Fixed at Ends or Free is Space?

    My textbook treats lattice waves as if they are waves propagating through a mass-spring chain. Kind of like this: M -/\/\/- M -/\/\/- M -/\/\/- M The book talks about the motion of the nth atom. It says that the motion of the nth atom has been simplified by only considering the motion...
  20. E

    How similar is Wen's spin lattice model to LQG's 4D spin foam theories?

    Is there a way to get a local bosonic theory out of spin foam formalism? Xiao-Gang Wen's suggestion, first published 6 Jul 2004 http://arxiv.org/abs/cond-mat/0407140v2 Page 8 "Remarkably, it appears that the theory of loop quantum gravity can be reformulated in terms of a particular kind of...
  21. jal

    Lattice refining loop quantum cosmology, anisotropic models and stability

    CONGRATULATIONS TO Martin Bojowald! http://arxiv.org/PS_cache/arxiv/pdf/0704/0704.1137v1.pdf Lattice refining loop quantum cosmology, anisotropic models and stability Martin Bojowald∗ 09 April 2007 Standard Theory SU(3)xSU(2)xU(1).…. String …. Have not done it. He is the first to...
  22. C

    Understanding Phonons & Quantum Sound Waves in a Lattice

    I'm looking for some physical intuition as to why sound waves are quantized. I know two mathematical procedures for deriving phonons in a lattice: 1) impose the canonical commutation relations on the system ad-hoc, and 2) apply the Schrodinger equation to the lattice. But neither of these...
  23. R

    How Do You Calculate Work Done in Lattice Compression?

    I'm given the energy for a crystal lattice is U(Ro) = (-2Nq^2 ln(2) (1-1/n))/Ro for equilibrium separation (Ro). I need to show that the work to compress the crystal from Ro --> Ro(1-x) is 1/2Cx^2 where C = (n-1)q^2 ln(2)/ Ro. Any hints about where to start? I thought it would just...
  24. T

    Atoms & Coatoms of Lattice of Subgroups of \mathbb{Z}

    Homework Statement How do the atoms and coatoms of the lattice of all subgroups of the group \mathbb{Z}(+,-,0) look like? Homework Equations Let (L,\le) be a lattice and e, f \in L is the minimum (maximum) elements of L. Then we say that a, b \in L is the atom (coatom) of L if a covers...
  25. P

    Trigonal lattice with each angle equal to 120degree

    1. What happens when the angles between the three sides of a trigonal reach 120degrees? I know that at 90degrees, it becomes a simple cubic. At 109degrees, it becomes a body centered cubic. At 60 degrees, it becomes a face centered cubic. What about 120degrees? I think that perhaps it should...
  26. B

    Reciprocal lattice diffraction

    Question 1 — Construct the crystal lattice from the diffraction pattern drawn on page 5 of this exam paper making sure you include the (110) and (220) planes. Explain the procedure used in reconstructing the crystal lattice. What Bravais lattice is represented by the diffraction pattern...
  27. B

    Bragg equation to determine the lattice constants

    Does anyone have a link or know a website where there is a table of values of lattice constants for NaCl, LiF etc? I'm asking because I was supposed to be provided with a table of values for writing my lab report, but unfortunately the lecturer seemed to have got 'array'. Any help and...
  28. V

    Stoichiometric and non stoichiometric defects in the crystal lattice

    friends i studied abt the crystalline structures of compounds but i am just failing to understand the defects of the crystal lattice so i hope some one will help me in sorting out this problem of mineo:)
  29. A

    What's the difference between a superlattice, lattice and sublattice?

    What's the difference between a superlattice, lattice and sublattice?
  30. F

    Proving Bravais Lattice Volume?

    Proving Bravais Lattice Volume?!? Hi guys, So with a Bravais lattice, you have 3 basis vectors: a1, a2, and a3. I know that you would get the volume of the lattice as a scalar product of the three: V = a1 dot [a2 x a3]. How would you start going about PROVING this? A little direction...
  31. C

    Help Me Solve a Lab Question Involving Atoms in a Lattice

    this is one of lab question but i don't even understand please help me the question is 1. consider spherical atoms of radius R arranged in a cube closest-packed lattice. if a small cube is constructed so that its corners are at the centers of the corner atoms, then one-eighth of each corner...
  32. B

    How Do You Convert a 2D Crystal Lattice to a 2D Reciprocal Lattice?

    I know this might be a really stupid question, but to convert a crystal lattice 2D representation to a 2D reciprocal lattice do you justdo you just invert the scaling. I know this is a pretty poor explanation so I will try and illustrate what I mean. Let's say that you have a reciprocal lattice...
  33. W

    Electrical Analogue Simulating a crystal Lattice

    hey guys...im stuck with this experiment of an Electrical Analogue Simulating a crystal Lattice...could anyone explain me the whole concept of relating a set of LC filters to a crystal lattice?
  34. S

    Uncovering the Mystery of A in Lattice Energy Equation

    I've just come across a problem and the equation within this problem isn't entirely explained in the text. I've searched on the net for this same 'lattice energy' equation, but there have been no results. Here is the equation: E = Aexp^ \frac{-r} { \alpha} - \frac{Me^2} {4 \pi \epsilon_0 r}...
  35. T

    Theretical aspects of the non-planar DOUBLET LATTICE METHOD

    Hello, I have been programming a doublet lattice code for nonplanar surfaces with zero thickness. I have some questions about LANDAHL's formulation. 1) How the dihedral angle is measured? This question may appear silly, but please explain. Suppose to have a wing with positive dihedral...
  36. N

    Bravais Lattice of FeAl Crystal: What is It?

    could some one please help me with this question, What is the bravais lattice of FeAl crystal I know Iron is bcc, and FeAl is bcc with the Al in the (1/2,1/2,1/2) position, but what is the bravais lattice for this crystal?? I would appreciate any pointers newo
  37. C

    Prove that the formula is valid for rectangles with sides parallel to the coordinate axes

    Let P be a polygon whose vertices are lattice points. The area of P is Z + \frac{1}{2}B - 1 . Z is the number of lattice points inside the polygon, and B is the number on the boundary. (a) Prove that the forumula is valid for rectangles with sides parallel to the coordinate axes. (b)...
  38. T

    Why Does FCC's Reciprocal Lattice Cube Side Equal 4π/a?

    If we are studying FCC in the direct lattice, Why does the length of the cube side in the reciprocal lattice equal to 4*Pi/a Where a is the lattice constant, a*=|G|=2*Pi/a Sqrt(4) = 4*Pi/a Where a* is the length of the cube site in reciprocal lattice Note: this thing is repeated in 2...
  39. A

    What distinguishes a crystal lattice from a unit cell?

    what's the difference between those two?
  40. S

    The Power of Idle Home PCs: Lattice QCD Project

    Has there ever been/is there a project for lattice QCD which uses idle home PCs to assist in QCD calculations, similar to SETI@home? Just wondering as it seems like the kind of project that is perfectly suited to the @home structure.
  41. J

    Enthelpy change & lattice energy - check please

    Write the reaction where the enthalpy change is the lattice energy for silver floride. Make sure you include the physical states. Ag F (ag) yields Ag Positive(g) + F negative(g) Could you please check this Thanks Could you help with this one: write the equation for the reaction...
  42. P

    Crystal Planes, Miller Indices: Cubic Lattice

    Just when I thought I understood the concept of planes and miller indices, I got stuck on a 'test your understanding' Q in my book. I can't understand that there can be two or more (110) planes in a crystal lattice? I thought there can be only one such plane. The question asks me to find the...
  43. R

    Electron-positron lattice model

    Does anyone have any comments (pro-con) on the "electron-positron lattice model of vacuum space" by nuclear physicist Dr. Menahem Simhony ? An overview of the model is found at this web page: http://www.epola.co.uk/ From what I can gather, the model predicts that the vacuum space is a...
  44. T

    Doublet Lattice FORTRAN code

    Hello everybody, I need a FORTRAN code (the source code) with the capability to calculate the generalized aerodynamic matrices given the modes of the structure. In particular, I need a code that is able to handle 3D structures made by assembling trapezoidal plate elements. Does anyone...
  45. S

    Geometry - Lattice in Circle Problem

    Hi. Can someone please help me with the following problem: What is the radius of the largest circle whose interior contains exactly n lattice points. Find the solution for n=0 through 9, and a general solution.
  46. T

    What Free Libraries Can I Use to Draw Crystal Lattice Structures in 3D Using C?

    Hello, my name is Kenichi, from japan. Inorder to draw Crystal Lattice 3D Structures, what libraries for free out there to be used with C programming language ? I am still a C novice. I wonder, in chemical simulation research do you also program using C language plus some libraries to...
  47. W

    A question about an explanation of the electrical resistance of perfect lattice

    Perfect periodic ion lattice has no electrical resistance.As we know,this can be easily shown by solving the Schrodinger equation using Bloch therom.Yet,another explanation is that "in a periodic array of scatterers a wave can propagate without attenuation because of the coherent constructive...
  48. C

    Lattice Points and Equilateral Triangles: A Proof

    Hello all In a ordinary syatem of rectangular coordinates, the points for which both coordinates are integers are called lattice points . Prove that a triangle whose vertices are lattice points cannot be equilateral. Ok so I know that in a equilateral triangle the angle measures are...
  49. E

    Exploring the Densest Space-Filling Polyhedra Lattice

    What is the densiest space-filling polyhedra lattice?My first guess would be the cubic one, but i am not sure.
  50. S

    Life on the Lattice: Matthew Nobes' QCD Technical Blog

    There is a new site called "Life on the Lattice" by Matthew Nobes which is what you might call a technical blog about Lattice QCD. He gets into some detail, but he is a clear guide, and it's not beyond the knowledge level of some I've seen post here. Give it a try, it's at...
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