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. D

    Borne Haber Cycle/ Lattice Enthelpy for KBr

    Homework Statement Draw the Borne Haber cycle for KBr Homework Equations The Attempt at a Solution so the energy required to make the K into gas is +89kJ/mol and it takes +420kJ/mol to take 1 electron from K it takes +112kJ/mol to make Br a gas it takes -342kJ/mol to give...
  2. C

    Wilson Lattice Action and the Strong-Coupling Expansion

    Consider the Wilson lattice action for a Yang-Mills theory with two parameters - color N and coupling g. 1) The strong coupling expansion on the lattice is given in terms of \beta = N/g^2 . But what is the other parameter of the lattice theory? Is it N? In that case, does the \beta-expansion...
  3. R

    Eigenvalue eqn for a electron in a one-dimensional lattice

    Homework Statement An electron moves in a one-dimensional lattice with the separation between adjacent atoms being equal to a. a. Write down the momentum eigenvalue equation for the electron. b. Find the general form of the solutions of the eigenvalue equation. c. By requiring that the...
  4. H

    Calculating lattice enthelpy of FeF2

    Calculate the lattice enthalpy of FeF2 using the following data: Hlatt (FeCl2) = 2569 kJ/mol r(Cl-) = 181 pm r(F-) = 133 pm I started with Hlatt is proportional to [absolute value of the product of charge on cation and anion] divided by the distance between their nuclei. Since the charge...
  5. E

    Lattice parameter + valence + avg time between collisions

    valence Z=1, lattice param a = 4.05 conductivity = 1.76e7 . What's the avg time between collisions of a conduction electron in this metal (bcc structure).really have no idea lol
  6. D

    Answer: Calculating Lattice Binding Energy of MgO Crystal

    Homework Statement In a crystal lattice of MgO, the Mg is surrounded by 6 Oxygen atoms as shown. Assume that the ions all behave as point charges. If the system behaves as though there were separate interactions between Mg and each of the Oxygen ions what is the total binding energy of...
  7. C

    Commutators on a discrete QM lattice = ?

    Commutators on a discrete QM lattice = ? Please let me know if any of the following is unclear: I was thinking about how you could go about doing QM not in a continuous space but instead on a lattice, take 1D for simplicity. Let's use a finite (not countably infinite) number of positions say...
  8. C

    Periodic Boundary Conditions on non sq lattice

    Is it possible to impose boundary conditions on the other 2d lattices like a rhombic lattice? a hexagonal lattice? an oblique lattice? How does one typically index such lattices?
  9. A

    Is a Reciprocal Lattice Vector Perpendicular to Its Corresponding Crystal Plane?

    How do i prove that a reciprocal lattice vector G, where G= hb1+kb2+lb3 is perpendicular to a plane hkl in a crystal lattice? Is there Fourier involved?
  10. N

    Lattice length for Zincblende structures

    So we have the covalent Radii of Ga = 1.26 A and As = 1.19 A. And the structure is Zinc blende with arsenic occupying half of the tetrahedral sites. So its probably not close packed... The formula the works for me is, [4/sqrt(3)] x [ radius of Ga + radius of As ]. I can get the answer ( 5.66...
  11. S

    Primitive lattice vectors, reciprocal lattice, wigner seitz cell

    How can i develop a sketch of the lattice and reciprocal lattice from vector form a=i+4j b=3i i know how to draw the wigner site cell, but I am having problems developing a sketch from vectors. what is the method for working it out..please help
  12. S

    Bravais Lattice in Two Dimensions

    In the book Applied Physics by P.K.Mittal, on page#25 under the heading of "Bravais lattice in two dimensions", a paragraph says, "The number of point groups in two dimensions is 10." My 1st question is, Then how many in three dimensions? Paragraph further says, "10 groups in two...
  13. atyy

    Asymptotic safety on the lattice

    Interesting posts from Georg von Hippel at http://latticeqcd.blogspot.com/ Monday, July 11, 2011 Lattice 2011, Day One "Next was a talk by Jack Laiho on Asymptotic Safety and Quantum Gravity. The concept of asymptotic safety as introduced by Weinberg states that a perturbatively...
  14. N

    Murnaghan equation for lattice parameters?

    Murnaghan equation for lattice parameters? Hi everybody... I'm trying to determine lattice constant (a) for BN by calculating the total energy for different values of a using ABINIT, then chose the one which correspond the minimum value of energy, Is this convenient way to determine a ...
  15. A

    Leech lattice is a 'lie group?

    Leech lattice is a 'lie group?" My understanding of Lie groups is non-existent. But I'm trying to understand if the Leech lattice is a 'lie group?"
  16. E

    Choosing gauge for triangular lattice

    How to choose the "symmetrical gauge" for 2D triangular lattice under uniform magnetic field and staggered magnetic field?
  17. F

    (2,0,0) plane for simple cubic lattice

    Hi All, Does the plane of miller index (2,0,0) exist in simple cubic lattice? It is easy to understand that (1,0,0) plane exist. But for (2,0,0) plane, I feel confused. If (2,0,0) plane exists, the plane intersect with the x-axis at 1/2*a, if a is the lattice constant; and the plane is...
  18. N

    Why Use Reciprocal Lattice for Band Structure?

    Reciprocal lattice? Hi everybody... I'm read about the band structure and the dispersion curve,I've got informations about the reciprocal lattice and how we determine it, but what I want to know why we use the reciprocal lattice in band structure calculations? thanks
  19. A

    Silicon: Lattice spacing <111><100> Thermodynamics and Bloch theorem

    Hi, I had a question that maybe someone might know, and that although I have been researching it I am not finding enough information on the web that would solve the issue. (it's the end of the month, too... and being broke and in a hurry is a problem too...) The project is aimed at...
  20. T

    Diffraction on a body centered lattice, h+k+l even?

    This is in my book: exp[2\pi i(hx+ky+lz)][1+exp(\pi i(h+k+l))]=2exp[2\pi i(hx+ky+lz)]\cos^2[\frac{\pi}{2}(h+k+l)] And in order for the cosine not to be zero, h+k+l must be even when we want to see the reflection. But I think that the result should be exp[2\pi...
  21. S

    Continuum Conversion of Lattice Points via Taylor Series Expansion

    I consider an array of lattice points and construct a vector at each lattice points. How to convert this discrete system into a continuum one by using the Taylor series expansion by considering the lattice distance say \lambda? thanks in well advance?
  22. T

    MD-Orienting lattice in LAMMPS

    I am trying to reproduce results of a couette flow from a paper by Thompson & Troian. Which has a lattice orientation as follows: "x-direction of the fluid cell is aligned along the [11-2] orientation of the fcc lattice comprising the wall"& "each wall of the cell consists of atoms forming...
  23. N

    Optimizing Lattice Parameter (a) for Energy Efficiency and Temperature Control

    Hi... The variation of the total energy with increasing the lattice constant (a), the minimum value of the lattice constant correspond to what degree of temperature? Nawzad A.
  24. P

    Expectation value of spin in an Ising lattice

    Homework Statement I have to show that (the question says deduce from the fact that magnetization is monotonically increasing and a concave function for h>0) \left< \sigma^2_{j} \right> - \left \sigma_j \right>^2 \geq 0 and \left< \left( \sigma^_{j} \right> - \sigma_j \right)^2 \right> \geq 0...
  25. H

    Lattice Point Equidistant Proof

    Hello all, Could someone help me out with this problem? I tried using circle geometries, perpendicular bisectors, and some more pure algebra. Nothing has been "unifying." Here is the problem: Is it possible to have a point Q=(r,s), where r and s are rational, where the point Q is not...
  26. A

    Spacing between three dimentional lattice planes

    What, exactly,does spacing between three dimensional lattice planes mean?Does it refer to the formula d=2*pi/(translation vector)? *I am an engineering undergraduate and this word is in my syllabus in topic crystallography.I was wondering what it meant? Also, can someone tell me in...
  27. I

    Lattice Gauge Theory - Wilson Action

    I have one question, which can be very simple, but i can't answer it. I have Wilson action for ordinary SU(2) gauge field. In all of the books I had read, the proof, that calculations in lattice theory is true - is equality the Wilson action in continuum limit to the continuum action. We use...
  28. M

    Calculating the Lattice Misfit and Misfit Hardening of Ni and Ni3Al

    Homework Statement Ni lattice parameter a1=3.52*10^-10m, and for Ni3Al lattice pameter a2=3.525*10^-10m, 1-what is the lattice misfit between these phases? 2-if there is addition of Cr to 50a/o, the lattice parameter of lattice increased to a3= 3.525*10^-10m, what will be the change in...
  29. B

    MATLAB Lattice of harmonic oscillators - Matlab

    Can someone help me with my code? I am trying to generate a lattice of harmonic oscillators, and I have written the following algorithm: for n = 2:((N-1)/h) a(1,n) = -(x0-(x(2,n)-x(1,n))); x(1,n+1) = 2*x(1,n) -x(1,n-1) + (h^2)*a(1,n); t(1,n+1) = t(1,n) + (h); for i =...
  30. L

    Hexagonal lattice in reciprocal space

    Can anyone help me find the reciprical coordinates of the K and M points in a simple hexagonal brilloiun zone? I have the lattice vectors and the reciprocal vectors, but I can't get the right coords (in recip coords) for the K point (the corner of the hexagon) and the M point (half way between 2...
  31. E

    Clarification: relation spin network/spinfoam and Wen's Qbit lattice

    Wen has proposed string-net condensation as basis for SM. http://www.fqxi.org/data/documents/Wen%20Azores%20Talk.pdf he proposes qbit lattice as the basis of space, and his diagram looks similar to spin networks. What is the relation between his qbit spin-system lattice and...
  32. P

    Dynamics of electron in crystal lattice

    I'm not sure what is the correct equation for motion of electron in a crystal lattice under the influence of magnetic force. On may easily proof that for electric force the following equation holds (the proof might be found in http://ajp.aapt.org/resource/1/ajpias/v54/i2/p177_s1" ): \hbar...
  33. Y

    How the x-ray diffractometer caculate the lattice constant or miller indices?

    Bragg's law n.lambda=2d sin theta. d=1/{h^2/a^2+k^2/b^2+l^2/c^2} for orthorombic structure. thus d=n.lambda/2 sin theta=1/{h^2/a^2+k^2/b^2+l^2/c^2} lambda and theta are given. Look, there are two types/sets of unknowns in the d formula, they are miller indices h, k, l and lattice constant...
  34. B

    Coordination in a zincblende lattice

    Homework Statement Could someone please confirm my answers for the coordination numbers in Zincblende going from nearest to next nearest to next next nearest neighbours. I'm trying to estimate the value of the madelung constant for ZnS, but am getting too large a value. I'm fairly sure that...
  35. G

    Dynamical Matrix of a BCC Lattice

    Hi, So I have to find the dynamical matricx of a bcc lattice considering only nearest neighbours. Its like if the nearest neighbours were all attached to the atom in question by springs. So first we need the equations of motion for the lattice. There are 8 nearest neighbours, so the...
  36. M

    What is the Lattice Constant of Iron?

    Homework Statement The density of bcc iron is 7900 kg/m3, and its atomic wieght is 56 amu. Using this information, calculate the lattice constant of iron's cubic unit cell and the interatomic spacing (i.e. nearest neighbor distance). Homework Equations The Attempt at a Solution...
  37. K

    What measured values are used in the lattice QCD calculation of proton mass

    I've read that the mass of a proton has been calculated using QCD. But I assume some measured values are inputs to this calculation? If so can anyone provide a list of all of the values used (speed of light, Plank's constant, mass of other particles?)
  38. M

    Miller Indices of Simple Cubic Lattice

    I'm trying to get my head around indexing x-ray diffraction patterns and this thought experiment stumps me. Let's say I have a crystal with a simple cubic lattice structure: http://ecee.colorado.edu/~bart/book/sc.gif I align a narrow x-ray beam incident on the crystal so that the x-ray...
  39. P

    Can Spin Networks Overcome the Background Dependence in LQG Theory?

    Can one get the same results using a fine lattice that one gets with backgroundless (loops, strings, foam, spin network)?
  40. P

    Lattice Gauge Theory: Active Physics Research or Passe?

    Are Lattice Gauge Theories still considered an area of active physics research? (i.e., are people still producing PhDs in this subject?) Or has this research area become passe?
  41. M

    Deriving the Lattice Growth Formula: A Step-by-Step Guide

    Hi, I am trying to understand how one can derive the lattice growth formula of L (n) = n^2 ( n + 1)^2 / 4 I looked at 2x2 and 3x3 matrices and could count the squares and rectangles in the figure. L (2) = 9, L (3) =36 and so on. However, I am more interested in finding out...
  42. I

    How to calculate the lattice sums of BCC?

    How do I calculate the lattice sums A12 and A6 for a BCC structure? I have calculated the following so far: A12 = 8(1/1)^12 + 6(1/root2)^12 + 12(1/2)^12 + 16(1/root5.5)^12 + 8(1/root6)^12 = 8.097. Have i made any mistakes? Are my nearest neighbour values correct? Please help! o:)
  43. P

    Calculating Integral of Cubic Lattice Sum

    If I have cubic lattice and need to calculate sum \frac{1}{N}\sum_{\vec{k}}\frac{1}{\sqrt{1-\gamma^2(\vec{k})}}coth\frac{6SI\sqrt{1-\gamma^2(\vec{k})}}{2T} where \gamma(\vec{k})=\frac{1}{3}(cosk_xa+cosk_ya+cosk_za) I must go from sum to integral...
  44. M

    Quick calculation of lattice parameter and unit cell volume for germanium?

    Homework Statement Calculate dhkl and vc for the (111) direction of germanium. Homework Equations dhkl=a/sqrt(h2+k2+l2) vc=unit cell volume=a1dot(a2xa3) or vc=structure form factor/density The Attempt at a Solution dhkl=a/sqrt(3) I know how to solve this, I've just...
  45. G

    Estimating Energy to Remove Atom from Lattice Vacancies in Silver

    Hello everyone; I have little problem about a question. If silver melts at 962C and contains an intrinsic relative concentration of 3  107 of lattice vacancies, estimate the energy required to remove an atom from the interior of the crystal lattice. I don't even have a idea how to solve...
  46. A

    Lattice Field Theory - QCD. Trace?

    Hi guys, So I'm working on this project to simulate QCD using a computer using (hybrid) monte carlo. I follow the majority of what I've read thus far, though there are a few things I'm uncertain about. Firstly, the Wilson loop is often written as U_{P} and the invariant gauge action is...
  47. L

    Getting a matrix element in Lattice QCD

    I've been looking at how people calculate basic things on the lattice. How would you go about calculating a single number? By this I mean decay rate, cross section, amplitude or contribution to an amplitude. The reason I ask is that I've been looking at some lattice QCD code up close. The code...
  48. A

    Transfer matrix, section in Jan Smit's book on Lattice Fields theory

    Hi guys, Basically I'm working through Jan Smit's book, http://books.google.com/books?id=pFgUFfG7JygC&printsec=frontcover&dq=introduction+to+quantum+fields+on+a+lattice&hl=en&src=bmrr&ei=mabWTOjtMoK4hAfh0eGGBQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDIQ6AEwAA#v=onepage&q&f=false , and I'm...
  49. S

    Crystal-Bravis Lattice Definitions

    There is such thing as a orthorhombic body centered crystal lattice. I am wondering why this is the case see the image bellow, we can find a repeating pattern which has a smaller area. A unit cell - must be selected such that it has the highest symmetry and the smallest area, however i do...
  50. M

    What is a Lattice Point in Geometry?

    What exactly is a lattice point (in relation to geometry)? I seriously doubt my simple minded explanation suffices... A lattice point is the meeting of the y and x integers on the Cartesian plane. And if that's in essence correct, is the way to find the number of lattice points found by...
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