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. P

    How Can We Visualize the Primitive Vectors of a Basic Cubic Lattice?

    Do yoh have some nice picture to show why the primitive vectors of basic cubic lattice are \vec{a}_1=\frac{a}{2}(-\vec{e}_x+\vec{e}_y+\vec{e}_z) \vec{a}_2=\frac{a}{2}(\vec{e}_x-\vec{e}_y+\vec{e}_z) \vec{a}_3=\frac{a}{2}(\vec{e}_x+\vec{e}_y-\vec{e}_z) Thanks!
  2. M

    Lattice Diffusion : Techniques & Procedure

    Hello, I would like to know how the lattice diffusion is done & why is it temperature dependent? I have tried googling Lattice diffusion several times, but I couldn't find any complete online reading/video... Your contributions will be highly appreciated.
  3. M

    Why the name Reciprocal lattice ?

    Why the name Reciprocal lattice ?? Is it because the dimensions get bigger & we are able to understand the structure on a macro level & removing the infinities that occur in h, k, l plane system ??
  4. J

    First principle calculation of lattice structures

    Hi Well I have been thinking of this quite sometime, but i really dint know anyone who could answer my question. We all know that may elements exist in the form of crystals e.g. Si, Cu etc. And they have very specific crystal structure like Diamond for Si and face centered for Cu. Can we...
  5. Q

    Plane lattice proof of Leibniz series

    (nevermind, answered my own question after spending the time to type this up!) Hi, I was flipping through Hilbert's Geometry and the Imagination, and in it, he includes a proof of Leibniz' series ( pi/4 = 1 - 1/3 + 1/5 - 1/7 + ... ) which is carried out by estimating the area of a circle at...
  6. S

    Lattice diagrams and generator in Algebra

    Homework Statement I do not understand the following statement (Please, see the attachment): "C4 has trivial subgroups and only one cyclic subgroup of 2 elements, namely <b>. This is because both a and c can be verified to be generators of C4." The Attempt at a Solution The notation...
  7. M

    Introduction to Lattice Boltzmann for Non-Physicists: Q&A on Basics

    I want to use Lattice Boltzmann methods for a game-like simulation, and I need a good introduction. I am weak on the physics side, so any introduction that starts tossing around physics variables and equations without defining them is going to lose me. Any ideas for a good source? Some...
  8. V

    Atomic Radius Problem with 286 pm Lattice Parameter

    i have one problem on atomic radius. i read (http://en.wikipedia.org/wiki/Atomic_radius) the artical and try to understand it but i don't found any formula for it. The problem is: if the lattice parameter of alpha iron is 286 pm (Pico meter),what is its atomic radius? here we have only one...
  9. V

    Braggs law: lattice constant

    need help working out this problem. if you have a simple cubic lattice characterised using x-ray diffraction with a wavelegth of 1.6\dot{A}. The main peak in the scan is (222) and the angle is 32 degrees. By using braggs law to find the lattice constant do you just sub in the values and solve...
  10. T

    Solid State Physics-simple question on lattice vacancies

    Homework Statement If silver melts at 9620C and contains an intrinsic relative concentration of 3*10^(-7) lattice vacancies, estimate the energy required to remove an atom from the interior of the crystal lattice. Homework Equations The above is a past exam question and the equation...
  11. O

    Any Lattice Model for String Theory?

    Is there any lattice model for String Theory? If there is such models, can anyone any good literature? Many Thanks.
  12. M

    Can Bravais Lattices Be Expressed as a Combination of Primitive Vectors?

    I have three primitive vectors a1,a2,a3 for the body-centered cubic (bcc) Bravais can be chosen as a1=ax a2=ay a3=(a/2)(x+y+z) or, for instance, as b1=(a/2)(y+z-x) b2=(a/2)(z+x-y) b3=(a/2)(x+y-z) where x,y,z are unit vectors. Now I should show that any vector of the form...
  13. B

    Lattice QCD: topological charge

    I am reading a paper right now on lattice QCD that presents a "method that improves the cooling method and constructs an improved topological charge operator based on the product of link variables forming rectangular Wilson loops." Unfortunately I...
  14. L

    Difference between a boolean algebra and a complete lattice?

    What is the difference between a boolean algebra and a complete lattice? What are their similarities?
  15. F

    Calculate the lattice constant of caesium chloride

    Calculate the lattice constant of caesium chloride with ionic radii Cs+ 0.167nm and cl- is 0.181 nm I don't really get it, from all the structures i have seen the distance between one atom and the next is (a) or (a/2) but i don't think I am right.Any ideas. What is the difference between...
  16. P

    Brillouin Zone of BCC lattice?

    I'm answering a question where it becomes necessary to know the closest face of the BZ in a bcc structure. The answer is given as +/- (2*pi) / (sqrt(2)*a) where a is the cubic lattice parameter. I would have thought the Answer would have been sqrt(3)*a / 4. Where does the pi come from...
  17. Useful nucleus

    GULP-General Utility Lattice Program

    I wonder if somebody has ued it before. I have some probblems in compiling it to work in parallel and I hope that somebody can help in this!
  18. N

    How Does the Reciprocal Lattice Relate to Real Space Lattice Vectors?

    Hey folks, Here's my problem: Knowing that for reciprocal lattice vectors K and real space lattice vectors R: and using the Kronecker delta: I need to prove b1, b1, b3 as shown http://www.doitpoms.ac.uk/tlplib/brillouin_zones/reciprocal.php" : I understand that for the...
  19. W

    How can one define vortex on a lattice

    vortex in superfluidity is well known in a continuum system, a vortex is well defined along a closed (continuous) path, the phase change is well defined. However, on a lattice, i don't think the phase change from site to site can be well defined. but people are talking about vortex...
  20. F

    Quick Miller Index Lattice Problem

    Homework Statement Indicate the (111) and ([11]0) ([11] has an over line, so is negative, don't know how to do on forums!) planes in a cubic lattice in a diagram, then calculate the angle between them. The Attempt at a Solution I think I have drawn this right (see attatchment), but I am...
  21. S

    Ideal hcp lattice, ratio c/a = 1.633 proof

    Homework Statement q: show that for an ideal hcp structure the c/a ratio is equal to (8/3)^(1/2) = 1.633 this question has come up before in the forum but still it has not fully answered: Kouros Khamoushi Dec30-05, 12:27 AM This is the mathematical calculation ^ means to the power of...
  22. A

    One-dimensional lattice (electrostatics)

    Homework Statement Hey guys. So, I got this question in the pic. First of all, I drew what I think to be a one-dimensional lattice (in the green box) but I'm not sure, is it right? Second of all, I don't really understand the question, I mean I know that a potential energy of charge q is...
  23. S

    How do I calculate the volume density of atoms in a diamond lattice?

    1.Hey! I don't get the solution to the following question so I am hoping someone can explain! Ok so, the question is we have Silicon, which I have read has a DIAMOND LATTICE structure. Which basically seems to mean two face centered cubes come together. Now I need to calculate the VOLUME...
  24. H

    Face-centered cubic lattice.

    Homework Statement An element crystallises in a face-centered cubic lattice with a basis group of two atoms at [000] and [1/4 1/4 1/4]. The lattice constant is 3.55 x 10^-10 m (Q1) How many nearest and second-nearest neighbours does each atom have? (Q2) Calculate the average volume per...
  25. H

    Shear deformation of diamond cubic lattice

    I'm doing MD calculations and I am having trouble providing a satisfactory explanation for the following to my profs. I am trying to calculate c44 for a diamond cubic crystal (Si), and to do that I have to calculate energy vs shear strain. For each value of the shear strain, I have to do...
  26. M

    Galois extension, lattice of subfields

    K=\mathbb{Q}(\sqrt{2+\sqrt{2}}) is a Galois extension of \mathbb{Q} [I showed this]. Determine \text{Gal}(K/\mathbb{Q}) and describe the lattice of subfields \mathbb{Q} \subset F \subset K. I found that \text{Gal}(K/\mathbb{Q})=\mathbb{Z}_4. I do not know how to draw the lattice of subfields...
  27. S

    Solved Homework StatementMetal Fluoride Crystallizes in a Cubic Structure

    Homework Statement A metal fluoride crystallizes in a cubic structure such that the fluoride ions occupy lattice positions at the corners and on the faces while 4 metal atoms occupy positions within the body of the unit cells. The formula of the metal fluoride is? I don't know where...
  28. S

    Clueless about Reciprocal Lattice

    Hi, I'm a college student in Singapore... Currently doing some research on carbon nanotubes, and so happened one of the honors student in the same lab said something about reciprocal lattice after seeing my experimental results. I went to search more about it, but i am totally clueless...
  29. T

    Calculate lattice constant from x-rays

    Homework Statement When x-rays are diffracted off NaCl, constructive interference is observed at an angle of 16.3 degrees. Determine the lattice constant of NaCl. Frequency of the X-ray is 1.948 x 10^18 hz Theta = 16.3 degrees Homework Equations (As far as what we're told to use and what's in...
  30. M

    Distance between copper atoms in cubic crystal lattice

    Homework Statement The atomic mass number of copper is A = 64. Assume that atoms in solid copper form a cubic crystal lattice. To envision this, imagine that you place atoms at the centers of tiny sugar cubes, then stack the little sugar cubes to form a big cube. If you dissolve the sugar, the...
  31. P

    Deriving equation to describe lattice vibrations of a one dimensional crystal

    When setting up this derivation one assumes a chain of identical atoms. The interatomic interaction between atoms is short ranged and so only affects neighbouring atoms (see Hook and Hall, "Solid State Physics" chapter 2.3.1). The potential V(r) is expanded as a taylor series about r = a to...
  32. S

    Difference between bond length and lattice constant

    Can anyone explain to me the difference between the bond length and the lattice constant? I'm thinking specifically about cubic crystal systems. Thanks. -Alan
  33. C

    Packing fraction of body-centered cubic lattice - solid state physics

    packing fraction of body-centered cubic lattice -- solid state physics Homework Statement This is part of a series of short questions (i.e. prove everything in Kittel Ch. 1, Table 2): Prove that the packing fraction of a BCC (body-centered) cubic lattice is: 1/8 * pi * \sqrt{3}...
  34. T

    Lattice field theory in solid state physics

    I've gone through undergrad courses of QFT, Solid State Physics and Quantum Statistical Physics but the first one didn't cross path with second and third so I only got taste in QFT applications in Solid State Physics through reading Zee's "QFT in a nutshell". My first impressions was WOW! Solid...
  35. F

    Empty lattice energy bands (Kittel, problem 7.2)

    "empty lattice" energy bands (Kittel, problem 7.2) Homework Statement This is problem 2 of chapter 7 in Kittel's "Introduction to Solid State Physics". The student is required to sketch the free electron energy bands in the empty lattice approximation and in the reduced zone scheme, for...
  36. R

    Kondo Lattice vs. Single Impurity Kondo System

    In order for a dilute magnetic alloy to show the kondo effect (resistance minimum at low T), the magnetic impurites must be far apart and non-interacting. In the Kondo lattice, the magnetic impurities are much closer togeather (one impurity per unit cell), so why does this system show the...
  37. T

    How do you calculate the lattice parameters c and a of Hexagonal ZnO

    How do you calculate lattice parameters c and a of Hexagonal ZnO ? if you are given the wavelength= 1.5406 nm, diffraction angle= 48 and the miller indexes (102)
  38. E

    Lattice constant of martensite?

    Hi everyone- I am trying to run some MD simulations of martensitic iron. I am having a lot of trouble finding a lattice constant anywhere in the literature... I know the crystal structure is bct, so there will be some c/a ratio as well... any suggestions? Erin
  39. S

    Exploring the Carbon Dioxide Lattice: Facts and Information

    Can anyone help me with few info abt carbon di oxide lattice
  40. C

    Proof of Ira Gessel's Lattice Path Conjecture

    http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/gessel.html"
  41. L

    Why f0(600) is not seen on lattice computations?

    There is a lot of activity around the full identification of the light meson spectrum and the identification of glueballs. For these aims a lot of insight should come from lattice QCD. Presently, not all the resonances seen that appear on PDG review are obtained on lattice. What is the reason...
  42. B

    Cubic Lattice Atom Diameter Calculation

    Homework Statement Barium metal crystallizes in a body-centred cubic lattice. The density of the metal is 3.50gcm^-3. Calculate the radius(in pm) of a barium atom. M(Ba)=137.3g/mol , NA = 6.022x10^23/mol The Attempt at a Solution For 1 unit cell: m=2x137.3/6.022x10^23...
  43. I

    Can a Simple Cellular Lattice Model Unify All Observed Particles and Fields?

    Abstract: I propose a simple cellular lattice model with configuration space Aff(3)(Z^3). The model allows to obtain, in the continuous limit, all fermions of the SM, the gauge group of the SM, it's action on the fermionic sector, and an effective Lorentz metric. 1 a) Importance of a...
  44. E

    How close is LQG/SF to Wen's lattice spin models

    Wen's papers purport to show that given a 3D extended lattice spin model, the emergence of light, electrons, gravitons and others should work well. How similar is what he is doing with what LQG and SF have done? How hard would it be to construct a verion of LQG/SF that reproduces all of Wen's...
  45. humanino

    Calculating Static Potential with Lattice Calculations

    Hi, a few months ago, someone pointed me to an excellent primer on lattice calculations, available for free on arXiv. I was browsing and finding many results, so I was wondering if anybody had a good reference, including worked examples so one can readilly do such easy things as calculating...
  46. X

    One dimensional lattice dispersion relation

    I'm doing an experiment, in which I have a one dimensional lattice held up by strings. That is I have a series of n masses each of mass M each connected to each other by springs with spring constant C and unstretched length a. Each mass is suspended from the ceiling by a string of length L. I'm...
  47. jal

    Lattice and loop under one roof

    Two approaches under one roof? Would it be too much to say that the following paper has made it possible for the double tetra to be imbeded into the cube? http://arxiv.org/abs/0803.4483 Numerical techniques for solving the quantum constraint equation of generic lattice-refined models in...
  48. N

    Analysis of Electron Orbits in Magnetic Fields: Bravais Lattice

    Homework Statement IF we consider electrons in a crystal subject to a magnetic field. The electrons near the fermi energy wil obey open or closed orbits. Using semiclassical eqn of motion and band structure for a bravais lattice, discuss the behavour and derive all conserved quantities...
  49. D

    How do I find the reciprocal primitive vector for a lattice?

    Homework Statement Here's a problem I'm having: The primitive vectors of the hexagonal lattice are: a1 = ck a2 = (a/2)i + ([a√3]/2)j a3 = (-a/2)i + ([a√3]/2)j Find the primitive vectors of the reciprocal lattice, i.e. b1, b2, and b3. Homework Equations I do know that the...
  50. M

    Calculating Energy of Mono-energetic Electrons Diffracted by a Lattice

    Homework Statement "Mono-energetic electrons are diffracted by a lattice. Their wavelength is shown to be L=0.35nm. What are the energies of the electrons?" Homework Equations E=hf c=hL The Attempt at a Solution c/L gives 8.566exp17 h=6.626exp-34 I plug them in and...
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