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. 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...
  2. 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...
  3. J

    Lattice Energy Comparison: Al2O3 vs. AlF3 - Exploring the Differences

    will the lattice energy of Al2O3 be greater than the lattice energy of AlF3. If so, why? If not, why?
  4. X

    Schools What schools have lattice gauge theory?

    I'm applying to grad school this year and I'm thinking I might be interested in exploring lattice gauge theory. Honestly, I don't know very much about the subject but it sounds very interesting to me. Does anybody know which schools have lattice gauge theory and which schools are really good...
  5. Demystifier

    Understanding Particles in Lattice QCD: Challenges and Potential Solutions"

    Can someone explain to me how the concept of particle is defined in lattice QCD? Here are the reasons why it seems problematic to me: 1) Lattice QCD is based on functional-integral formulation of QFT, which does not contain any operators in the Hilbert space. In particular, it does not contain...
  6. N

    Coordination number in triangular lattice

    Hi. I have a question on configuration numbers of graphs in lattices, often used in high dimension expansion of Ising model. A 2 point graph in a hypercubic d-dimensional lattice of N sites has a configuration number Nd, a square in the same lattice has Nd(d-1)/2. My question is: which is the...
  7. R

    Modifying Tornado Vortex Lattice Code for Velocity Output

    I was wondering if anyone has any experience with the vortex lattice code Tornado. I would like to modify the source code of Tornado so that it outputs velocity vectors instead of the pressure distribution. I am assuming this is possible because for the vortex lattice method to work the...
  8. C

    Designing a Core Lattice: Thermal-Hydraulics & Neutronics Considerations

    When designing a core lattice, which one of the thermal-hydraulics and neutronics consideration are important?
  9. A

    Finding Orthonormal Basis of Hilbert Space wrt Lattice of Subspaces

    I have a Hilbert space H; given a closed subspace U of H let PU denote the orthogonal projection onto U. I also have a lattice L of closed subspaces of H, such that for all U and U' in L, PU and PU' commute. The problem is to find an orthonormal basis B of H, such that for every element b of B...
  10. D

    Certifying shortest vector in a lattice

    Given an n dimensional integer lattice, is it possible to certify a vector as being a shortest one in time polynomial in n? If we then fix the dimension n, is it possible to certify the shortest-ness in time strictly less than O(k^n) where k is the length of the largest basics vector?
  11. M

    Lattice bolztmann - what exactly are regularized boundaries

    I'm picking thru some D2Q9 code, and saw some boundaries call Regularized boundaries that handled walls for north, east, west, south walls. For each wall the following is computed (east for example): Rho, velocity, and pi(have no clue what this is). Not much to go and haven't been successful...
  12. M

    Question about integral used in Lattice Boltzmann text

    I would have asked in math, but I was hoping the context of lattice Boltzmann may make my question clearer. Given f is the number density of particles, v velocity, and u equilibrium velocity. In a book(http://www.ndsu.edu/fileadmin/physics.ndsu.edu/Wagner/LBbook.pdf equation 3.14), he...
  13. R

    Drawing Reciprocal Lattice: Methods & Steps

    What are the methods or steps of drawing reciprocal lattice?
  14. R

    Brillouin zone: reciprocal lattice

    Can somebody explain me how can we visualise reciprocal lattice of a crystal lattice. Also what do we mean by wavevectors of a reciprocal lattice. What is its physical significance?
  15. E

    Matter as excitations of spacetime lattice?

    It is a common theme among background independent quantum gravity theories that there should be some sort of discretization, or fuzziness, of the spacetime manifold occurring on Planckian scales. It has occurred to me that if we take this discretization to consist of a lattice of sorts, might...
  16. M

    Understanding Reciprocal Lattices for Beginners

    here is a question on reciprocal lattices that I am stuck on for a simple cubic lattice, the unit cell is defined by a1=a(1,0,0) a2 = a(0,1,0) a3 = a(0,0,1), demonstrate that the reciprocal lattice of its reciprocal lattice is the original crystal lattice.From what I've found, i think the...
  17. N

    Lattice discrete Fourier transforms

    Homework Statement Hi Say I have a 5x5 lattice, where each entry (or we can call it site) contains the number 1. Now, on the lattice we have a function g(R), which is equal to the number on the site. In this case g(R)=1 for all sites (here R is a vector from the point (3,3), which denotes the...
  18. F

    Diamond Lattice: Tetahedral Bond Angles & Direction Cosines

    hello how can I find the angles between the tetahedral bonds of a diamond lattice & what are the direction cosines of the nearest-neighbor bonds with the atom at the origin along the x, y, z axes ?
  19. P

    Prove Reciprocal Cubic Lattice of Cubic Lattice is Also Cubic

    Homework Statement Show that the reciprocal cubic of cubic lattice is also cubic. Homework Equations cos\alpha*=\frac{cos\beta cos\gamma-cos\alpha}{sin\beta sin\gamma} cos\beta*=\frac{cos\alpha cos\gamma-cos\beta}{sin\alpha sin\gamma} cos\gamma*=\frac{cos\alpha...
  20. P

    Matrices of simple face and cubic centered cubic lattice

    S= \begin{bmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \end{bmatrix} for simple cubic I= \begin{bmatrix} -\frac{a}{2} & \frac{a}{2} & \frac{a}{2} \\ \frac{a}{2} & -\frac{a}{2} & \frac{a}{2} \\ \frac{a}{2} & \frac{a}{2} & -\frac{a}{2} \end{bmatrix}...
  21. C

    BRS: Subgroup lattice of a Permutation Group via GAP

    I am somewhat distracted so this post will not be what it should, given that GAP is one of my interests. For those who don't already know: GAP is a powerful open source software package for computational algebra, especially computational group theory and allied subjects. This long running...
  22. G

    Need guidance for learning lattice Boltzmann

    Hello This term i must do a project for my advanced fluid mechanics course,in this project i must solve navier stokes equations in a cavity using lattice Boltzmann method,but the problem is i have no background in lattice Boltzmann,and i need some sources like some useful pdfs or textbooks...
  23. J

    Calculating the interplanar distance d111 for an FCC lattice

    Homework Statement As a part of a lab report, I need to calculate the distance of the (111) planes of an FCC lattice made out of spheres with diameter D. Homework Equations The Attempt at a Solution The course assistant has given me the value of \frac{\sqrt{6}}{3}D. I can...
  24. R

    What is the basic idea behind lattice theory?

    What is the basic idea behind lattice theory or computer-based QFT calculations? For example, take a scalar field, and the functional path integral: W[J(x)]=\int [d\phi(x)]e^{i\int \mathcal L \mbox{ }d^4x+i\int J(x)\phi(x)d^4x} W[J] is the starting point for all types of...
  25. S

    What is the lattice enthalpy of MgCl2?

    can someone please tell me what actually is lattice enthalpy. Is it energy released when one mole of bonds are broken or the energy released when one of an ionic compound is broken into the constituents? So do we write U or 2U in the born habber cycle of MgCl2?
  26. H

    What is the Lattice Parameter c of an HCP Crystal Structure?

    Homework Statement I'm trying to figure out the lattice parameter, c, of the HCP crystal structure. Here are a couple links showing the structure...
  27. N

    Hamiltonian of a metal lattice

    Hi guys I have the Hamiltonian, which describes my lattice of NxN metal atoms, and their mutual coupling. What I need is the density of states of this lattice, and I am quite sure that there is a way to find it from my Hamiltonian; I just need to find out how. What I thought was that I can...
  28. Spinnor

    Use 3D lattice and vector field(s) to represent curved manifold?

    I would like to try and map a small piece of a 3 dimensional curved manifold using a flat 3 dimensional space, and a vector field. Will the following work? Take a 3 dimensional cube of size a*a*a that lies in a 6 dimensional space, R^6, with coordinates x1,x2,x3,x4,x5,x6. Let this cube be a...
  29. J

    What's the recoil energy in optical lattice?

    Will anyone give me an explanation helping me understanding it?
  30. C

    Difference between 1D lattice and 2D lattice on BEC

    I studied in AMO physics. nowaday, I study about BEC. I'm wonder, Difference between 1D lattice and 2D lattice on BEC. In the web, they just explain what they do using that. Maybe just short word, or sentence, give me a huge knowledge. Thanks you, and Have a nice day!
  31. edpell

    Higgs and Lattice calculations

    My understanding is that the folks who do particle mass calculations using the formalism of a lattice get results that fit measured values well without using a Higgs field. Is this correct? If so, do we have any use for and/or reason to believe in the Higgs field/particle?
  32. N

    How to calculate lattice parameters?

    Dear Friends, I read some articles about alloys. Most articles have a table listing lattice parameters (a,b,c-axis) and unit cell volumes. But they never explain how those data come from. How do we get those data? By comparing XRD images with PDF databases? Or by calculation? Example...
  33. N

    Metals with what type of crystal lattice less prone to brittle fracture?

    Metals with what type of crystal lattice less prone to brittle fracture? Why?
  34. U

    Nearest Neighbor and Second Nearest Neighbor Distances in FCC Lattice?

    Homework Statement An element crystalises in a face-centred cubic lattice with a basis group of two atoms at 000 and 1/4 1/4 1/4. The lattice constant is 3.55Angstroms. i) what is the separation of nearest neighbor atoms ii) how many nearest and second nearest neighbors does each atom...
  35. S

    Lattice Energy - Hard question - I keep getting wrong answer

    Homework Statement What is the lattice energy of KI? Given the information below. Heat of formation for KI = -328 kJ/mol Heat of sublimation for K = 89.20 kJ/mol Ionization energy for K = 419 kJ/mol Bond dissociation energy for I2 = 149 kJ/mol Electron affinity for I = - 295...
  36. S

    What is the lattice energy of LiI? Given the information below.

    Homework Statement What is the lattice energy of LiI? Given the information below. Heat of formation for LiI = -270 kJ/mol Heat of sublimation for Li = 159.40 kJ/mol Ionization energy for Li = 520 kJ/mol Bond dissociation energy for I2 = 149 kJ/mol Electron affinity for I = -...
  37. I

    What Are Some Resources for Learning About Lattice Theory Coloring?

    I don't know if this is the proper place but I'll put it here. I just started doing my OURE on lattice coloring. I really don't know much about either (lattices or coloring) but I am interested in both. I do know a little about graphs and graph coloring(but not very much knowledge about it)...
  38. Y

    Counting Lattice Points in a Circle: A Math Contest Question

    In a math contest, the question goes somehow like this: A lattice point is a point wherein the value of (x,y) is an integer. Determine the total number of lattice points in a circle which has a radius of 6 and the its center is at the origin. Any one knows the solution or shortcut for this?
  39. C

    De Haas-van Alphen effect in a 2d lattice

    when calculating the frequencies of 1\H's oscillation using De Haas-van Alphen, we need to find the extremal cross-section of the fermi surface perpendicular to the magnetic field direction. in 3d i can understand this. but when talking about a 2d lattice where the magnetic field is...
  40. T

    Showing reciprocol lattice is perpenducular to a plane

    Homework Statement If a_1 a_2 a_3 are the unit vectors of a real space lattice, then the so-called “reciprocal lattice” is defined by the unit vectors b_1 b_2 b_3 where: b_1 = \frac{2\pi a_2 \times a_3}{a_1 \cdot a_2 \times a_3} b_2 = \frac{2\pi a_3 \times a_1}{a_1 \cdot a_2 \times...
  41. S

    Does the Powder XRD Data Indicate a Face-Centred Cubic Structure?

    1. Homework Statement Using the powder XRD data below, show that the substance has a face centred cubic structure. (xray lamda = 0.154056 nm) Peak No.------2(theta) 1 -------------38.06 2 -------------44.24 3 -------------64.34 4 -------------68.77 5 -------------73.07 2...
  42. S

    Determine crystal lattice structure from powder XRD

    Homework Statement Using the powder XRD data below, show that the substance has a face centred cubic structure. (xray lamda = 0.154056 nm) Peak No.------2(theta) 1 -------------38.06 2 -------------44.24 3 -------------64.34 4 -------------68.77 5 -------------73.07 Homework...
  43. P

    Disoersion relation and lattice constants

    Dispersion relation and lattice constants I need to be able to calculate the period (which I believe is the lattice constant) for a 1D crystal given the energy wavevector relation. Is this possible? I also have to find the Bravais lattice of a 2D crystal give a similar relation. What is it...
  44. I

    Solid State: Mean Square Lattice Strain

    Homework Statement Okay, this is from Kittel's Introduction to Solid State Physics (8th ed.) and it's driving me crazy. The problem is: In the Debye approximation, consider \langle (\tfrac{\partial R}{\partial x})^2 \rangle =\tfrac{1}{2}\Sigma K^2u^2_0 as the mean square strain, and show...
  45. M

    Lattice Vibrations and em waves

    Can EM waves falling on a solid contribute towards lattice vibrations? If yes, then i)when is the energy used in lattice vibrations, ii)when is it used in excitation of electron into higher energy level, and finally, iii)when is the energy utilized for slight vibration of the electron...
  46. T

    Understanding Reciprocal Lattice Vectors and Orthogonality in Primitive Lattices

    f(\vec{r}) = f(\vec{r}+\vec{T}) \vec{T}= u_{1} \vec{a_{1}} + u_{2} \vec{a_{2}}+u_{3} \vec{a_{3}} u_{1},u_{2},u_{3} are integers. f(\vec{r}+\vec{T})= \sum n_{g} e^{(i\vec{G}.(\vec{r}+\vec{R}) )}= f(\vec{r}) e^{i\vec{G}.\vec{R} }= 1 \vec{G}.\vec{R} = 2\pi m we call...
  47. Spinnor

    Standard Model Lagrange Density, 2D vectors, Lattice Theory.

    In the article "The Lattice Theory of Quark Confinement", by Claudio Rebbi (Scientific American) there is a graphic representing the chromoelectric field. The caption reads: "Chromoelectric field is a gauge field similar in principle to the electromagnetic field but more complicated...
  48. J

    Entangled protons in a crystal lattice

    Can entangled protons be maintained in a crystal lattice for an indefinite period of time? The following papers seem to support this contention. I need the help of the Forum’s readers to critique this research and the assertions the authors are making. Dr. Francois Fillauxa and Dr Alain...
  49. E

    Wen bosonic lattice model, chiral interactions, noncommutative space

    Hello, I know that Wen might be able to get W Z gauge bosons and chiral neutrinos using his lattice bosonic models he is unable to get chiral interactions using standard bosonic lattice on minkowski space. If instead he used Connes' noncommutative space would he be able to get chiral...
  50. N

    Understanding Phonons on a Lattice: Seeking Insight from Niles

    Hi all My book says: "The reason that phonons on a lattice do not carry momentum is that a phonon coordinate (except for wavevector K=0) involves relative coordinates of the atoms". I can't quite figure this statement out. I understand the words, but I cannot see why it is an...
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