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. A

    MHB Proving Complete Lattice from Lower-Bounded & Maximal Conditions

    I need some help figuring out a problem dealing with lattices. The problem is this: Prove that any lower-bounded lattice satisfying the maximal condition is a complete lattice. I've been able to figure out some things so far. I know that a lattice is a meet- and join-semilattice, which...
  2. A

    Crystal Lattice of Spin-1 Particles: Chemical Potential?

    Homework Statement A crystal lattice consists of a spin 1 particle at each lattice point. Spin 1 particles can have z-components of magnetic moment that take on the values +μZ, 0, and -μZ. In an external magnetic field B, each spin can have an energy U = -μZB, so the possible energies are...
  3. X

    Is a Finite Lattice also a Complete Lattice?

    I'm not sure if I am using the right terms here, but: When X is a finite set and R is a relation... If (X,R) is a lattice, then (X,R) is also a complete lattice. Does this make sense? The question then is, why is is also automatically complete. I don't understand that.
  4. M

    Calculating the Lattice Mismatch of GaN/Sapphire

    Dear Physics Forum It has been reported that the lattice mismatch of GaN/Sapphire is ~13.9%. I have tried the following formula, but got the wrong answer: [(GaN-Al2O3)/GaN] x 100% where: GaN = 3.189 Angstroms Al2O3 = 4.765 Angstroms Obviously I am missing something huge...
  5. F

    Lattice systems and group symmetries

    Dear all, In Marder's Condensed matter physics, it uses matrix operations to explain how to justify two different lattice systems as listed in attachment. However, I cannot understand why the two groups are equivalent if there exists a single matrix S satisfying S-1RS-1+S-1a=R'+a'...
  6. Y

    Introduction level Solid State - Mean free time/path & lattice spacing

    Homework Statement Silver has a density of 10.5E3 kg/m3 and a resistivity of 1.6E-8 Ω*m at room temperature. On the basis of the classical free electron gas model, and assuming that each silver atom contributes one electron to the electron gas, calculate the average time, Tau, between...
  7. B

    Solid State: Diamond lattice and scattering

    I have the following homework question I am working on. I am given three scattering angles: 42.8, 73.2, 89. (in degrees) without the wavelength of the light used. I am to show that these are consistent with a diamond lattice. I started with Laue's Law: delta(k) = G and according to the...
  8. W

    Number of next-nearest and next-next-nearest neighbors in a SC lattice

    Unfortunately, my solids state physics textbook doesn't provide the numbers. However, I know the number of nearest neighbors in a SC structure is 6. If I'm not mistaken, the number of next-nearest-neighbors is 12 and the number of next-next-nearest neighbors is 8. Is that correct?
  9. A

    Intersection coordinates in lattice

    On the drawing below is a hexagonal lattice. For the basis vectors one can choose either the set of arrows in black or the set in yellow. The intersection coordinates of the plane in green seems to be the same regardless of choosing the black or the yellow basis. Why is that? My teacher said it...
  10. PsychonautQQ

    One dimensional diatomic lattice oscillations

    Suppose we allow two masses M1 and M2 in a one dimensional diatomic lattice to become equal. what happens to the frequency gap? what about in a monatomic lattice? Knowing that (M1)(A2) + (M2)(A1) = 0
  11. D

    Kinetic energy of a free electron in a lattice

    Homework Statement Show that for a simple square lattice (in 2-D) with the lattice spacing = a, the kinetic energy of a free electron at a corner (point A in the figure below) of the first Brillouin zone is higher than that of an electron at the midpoint of a side of the zone (point B in the...
  12. I

    Solid state physics, lattice constants, ionic radii, nacl

    Homework Statement NaCl (a0 = 5.64A° ), NaBr (a0 = 5.98A° ) and KCl (a0 = 6.30A° ) all have the same structure, which is the NaCl structure. (a) Assuming the spacings are determined by the ionic radii of the relevant ions, what would value would you expect for the lattice constant of...
  13. C

    Why does a silicon atom in a silicon lattice have 4 single bonds?

    Silicon has 14 electrons, this means if it fills up its first two shells it will have 4 electrons in the outermost shell (These are the valence electrons). This shell can have 18 electrons in it, so silicon can have 14 more electrons in its outermost shell. This means it could...
  14. aleksbooker

    Why do we *subtract* enthelpy of lattice formation?

    Hello all, I'm in gen chem 2 and we're going over how to calculate the enthalpy of lattice formation. The way given is to use the Born-Haber process and add the enthalpies of all the steps in between. e.g. Na_{(s)} --> Na^+_{(g)} + e^- (388kJ) There are three or four of these, and we combine...
  15. I

    Determining Type of Lattice from Powder Diffraction

    Homework Statement In a powder diffraction measurement, we obtain a measure of Bragg angles θ. (A powder sample contains small crystallines with all possible random orientations.) In a particular experiment with Al powder, the following data is obtained when X-ray radiation with wavelength λ =...
  16. PsychonautQQ

    Identifying BBravais Lattice with vectors Given

    Homework Statement Given that the primitive basis vectors of a lattice area (a/2)(I+J),(a/2)(j+k), (a/2)(k+i), where I j and k are the usual three unit vectors along Cartesian coordinates, what is the bravais lattice? Homework Equations The Attempt at a Solution So just drawing...
  17. B

    Green function for the particle hopping on a lattice, meaning?

    I am having trouble understanding something that I am sure is very basic. Let's say I have a particle that is hopping on a 1d lattice with a hard wall at x=0 in the presence of some potential - anything, say linear ##H_0=F*i## or Coulomb ##H_0=C/i## where i is the label of the site the particle...
  18. P

    The number of ways of placing M atoms on the interstices of a lattice

    Hi, N atoms are arranged to lie on a simple cubic crystal lattice. Then M of these atoms are moved from their lattice sites to lie at the interstices of the lattice, that is points which lie centrally between the lattice sites. Assume that the atoms are placed in the interstices in a way...
  19. Einj

    Van Hove singularity for a two dimensional lattice

    Hi everyone. Suppose we consider an electron in a two dimensional lattice, whose dispersion relation is given by: $$ \epsilon(k_x,k_y)=-J(\cos(k_x a)+\cos(k_y a)), $$ and where the wave vectors belong to the first Brillouin zone (k_i\in [-\pi/a,\pi/a]). In this case it turns out that the...
  20. J

    How Can I Calculate Reciprocal Lattice Vectors for a 2D Lattice?

    Homework Statement Si(001) has the following lattice vectors in a (2x1) reconstruction \vec{a'_1} = \vec{a_1} + \vec{a_2} \vec{a'_2} = -0.5 \vec{a_1} + 0.5 \vec{a_2} Calculate the reciprocal lattice vectors of the reconstructed unit cell, \vec{b'_1} and \vec{b'_2} in terms of...
  21. D

    Lattice of 1D anharmonic oscillators (Cannonical Ensemble)

    Homework Statement I have a system of N non-interacting anharmonic oscillators whose potential energy is given by, V(q) = cq^2 -gq^3 -fq^4 where c,f,g > 0 and f,g are small. Homework Equations The Hamiltonian is given by, H = \sum_{i=1}^N \big ( \frac{p^2_i}{2m} + V(q_i) \big )...
  22. D

    Worst load cases lattice structure

    Introduction Dear all, I'm working on an assignment to model a ship-to-shore crane for a FEM design course. Having modeled the crane, I now need to apply the load of the trolley (which is hoisting the container) on the boom (which in my case is a lattice structure made up of beam elements)...
  23. A

    Understanding Free Electron Kinetic Energy on a Square Lattice

    Homework Statement Show for a simple square lattice that the kinetic energy of a free electron is higher at the corner of the first zone than at the midpoint a side face by a factor of 2. Homework Equations Simple geometry. The Attempt at a Solution I think I know how to solve, but...
  24. M

    Understanding Lattice Points in a Primitive Cubic Cell

    Hello, Suppose I have a primitive cubic cell with 8 atoms, one on each corner of the cube. I don't understand how this consists of only one lattice point? Doesn't each corner have a lattice point, thus the cell would consist of 8 lattice points??
  25. S

    Schrodinger equation in the reciprocal lattice.

    Hi Everybody, I am learning solid state physics using a German book called "Festkorperphysik" written by Gross and Marx. Now, in page 336 the Schrodinger equation in momentum space is introduced: \left( \frac{\hbar^2 k^2}{2m} - E \right) C_\vec{k} + \sum_\vec{G} V_\vec{G}...
  26. atyy

    Lattice Simulations on a Sphere in Condensed Matter

    Most lattices I've come across in condensed matter, like the Kitaev model, are regular lattices and don't fit on a sphere. Are lattice simulations ever put on a sphere in condensed matter, and if so what sort of lattice is used?
  27. Hyo X

    Ordered lattice necessary for band structure?

    Is it possible for a disordered or amorphous structure to have band structure? I understand derivation of bands from Kronig-Penney model. E.g. does amorphous silicon have a band structure? While amorphous silicon oxide does not have a band structure?
  28. nomadreid

    Can one call a linear order a lattice? If not

    Can one call a linear order a lattice? If not... I have problems putting together the three ideas (1) the meets and joins of a lattice are unique, hence lattices must have discrete elements (2) the truth values of a logic are arranged in a lattice (3) there exist probability logics, whereby...
  29. nomadreid

    Lattice on the closed unit circle?

    Would either or both of these work as a lattice on the closed unit circle in the plane? (1) Using a linear order: Expressing points in polar coordinates (with angles 0≤θ<2π), define: (r,α) < (s,β) iff r<s or (r=s & α<β) (r,α) ≤ (s,β) iff (r,α) < (s,β) or (r=s & α=β) The meet and join...
  30. S

    Calculating 2θ for XRD Pattern of a FCC Lattice

    Homework Statement A wavelength of 0.7107 Angstroms is used to analyse a polycrystalline sample with a known FCC lattice structure. The interplanar spacing of the first peak is 0.3 A. Calculate 2θ for the first 3 peaks on the XRD pattern. First 3 peaks occur at (111), (200) and (220)...
  31. N

    Can Radio Waves Change the Electric or Magnetic Properties of Metals or Liquids?

    So mobile devices use radio waves so I had this thought that if there was evidence linking radio waves with changing organic tissue structure is the evidence that radio waves can be used to change the electric or magnetic properties of metals or liquids? Underwater walk talky jabbering for...
  32. R

    What's the difference between lattice vectors and basis vectors?

    Google has not been very useful, and Kittel has too little on crystallography. Actually, what's a good source on crystallography?
  33. M

    What is the Correct Crystal Lattice Structure?

    what is this answer choices: a. Primitive cubic with an octahedral hole b. Body centered cubic with an octahedral hole c. Face centered cubic with an octahedral hole d. None of the above e. Not enough information to determine We didn't talk about this in class, and this was a question...
  34. Y

    Harmonic vs Anharmonic Interactions in Lattice

    I am currently working my way through Kitel's Solid State Physics book. When discussing the consequences of the harmonic assumption (quadratic degree of freedom for interatomic lattice interactions), he states that 1) the lattice waves do not interact 2) a single wave does not change form...
  35. S

    Can we calculate three-point correlation in lattice qcd

    Is it feasible to calculate a three-point correlation on the lattice? Say, I have two quark fields separated at z_1+z_2 and 0, and a gluon field inserted at z_2. Also I need two gauge links to make this expression gauge invariant: \bar{\psi}(z_1+z_2) \Gamma(z_1+z_2; z_2) F^{\mu\nu}(z_2)...
  36. T

    What is the relation between Ultra-cold atom and optical lattice?

    Ultra cold atom is achieved by laser cooling. For optical lattice, it is achieved by the interference of counter-propagating laser beams. What is the relation between Ultra-cold atom and optical lattice? Why do people load Ultra-cold atom in optical lattice? Thank you for your answer.
  37. 1

    Scattering of Neutrons from 2d Crystal Lattice

    Homework Statement A two-dimensional rectangular crystal has a unit cell with sides a 6.28Å and b 3.14Å. A beam of monochromatic neutrons of wavelength 5.0 Å is used to examine the crystal. Using either the Laue condition for diffraction or Bragg's Law, determine whether it would be...
  38. anorlunda

    Can Lattice Size Changes Reveal the Quantization of Space-Time?

    In elementary particle theory, professor Susskind encourages us to think of space-time as divided into a lattice of cells. We use annihilation and creation operators in the Lagrangian to consume a particle in a cell and to create a new particle in an adjacent cell. Repeated application of...
  39. T

    Particle subject to a hopping potential between two atoms in a lattice

    Hello! Long time lurker, first time poster. This is the first of a couple of questions which has totally stumped me, although I have a feeling it's easier than it first seems. Homework Statement A particle is initially located at one of two atoms. The particle is subject to Hhop , a...
  40. M

    Determination of lattice energy of an ionic compound

    Hi, In my book it says that it is difficult to determine the lattice energy of NaCl so they use the Haber cycle which applies Hess' law. Lattice is the energy change when a solid ionic substance separates into ions in gas phase. We could simply increase tempetrature until NaCl breaks down...
  41. Y

    Partition function for hard spheres on a lattice

    Hi everyone, I'm reading some lecture notes on statistical physics and thermodynamics and I'm stuck at an expression for a partition function which I really don't understand. The chapter is on mean field theory and the discussion is about hard spheres on a lattice. The interaction of the hard...
  42. I

    MHB Use a subgroup lattice to compute a normalizer

    My question is at here: abstract algebra - Use a subgroup lattice to compute a normalizer - Mathematics Thank you!
  43. T

    Infinite Square Well for Bosons in an optical lattice

    I'm working on a research project and was wondering what you could use to experimentally create a periodic infinite square well (dirac comb?) in a direction orthogonal to a different potential, say a periodic potential. To help you understand what I'm trying to do picture a grid of atoms and...
  44. E

    Counting multiplicities of a particle lattice

    This is from Molecular Driving Forces, 2nd Ed. 5.3: Calculating the entropy of mixing. Consider a lattice with N sites and n green particles, and another lattice with M sites and m red particles. These lattices cannot exchange particles. This is state A. (a) What is the total number of...
  45. H

    Seek help for space groups in 2 dimensions Bravais lattice

    Dear experts, I'm not familiar with crystal structure theory. I'm seek expertise to figure out space groups in 2 dimensions Bravais lattice of the attached structures. In the figure, red and greens dots represent different atoms. I'll greatly appreciate your help. Struture 1...
  46. heycoa

    Solid State | Lattice constant | BCC -> HCP

    Homework Statement Sodium transforms from bcc to hcp at about T=23K. Assuming that the density remains fixed and the c/a ratio is ideal (1.633), calculate the hcp lattice constant a, given that the lattice constant a'=4.23 Angstrom in the cubic phase Homework Equations I can't find any...
  47. S

    How Does the Ratio of Lattice Points to Radius Behave as Radius Increases?

    Given a circle centered at the origin, how can one prove that the limit of the quotient of the number of lattice points on the circle over the radius goes to zero as radius goes to infinity?
  48. D

    Reciprocal Lattice: Visible Points in an X-Ray Experiment

    I got a bunch of questions about reciprocal lattice, I start with this one: In an x-ray experiment: For one specific orientation of your incident beam on your real lattice, only a portion of the points of your reciprocal lattice will become visible as your diffraction pattern right? See my...
  49. A

    Multiplicity of particles on a lattice

    Homework Statement A particle can exist in three microstates, with energies E0 < E1 < E2. Consider N >> 1 such particles, fixed on a lattice. There are now n0 particles with energy E0, n1 particles with energy E1 and n2 = N - n0 - n1 particles with energy E2. We have that n_j >> 1 for j = 0...
  50. F

    Calculating lattice energy on ionic compound

    Homework Statement Given the following thermodynamic data, calculate the lattice energy of CaBr2(s) caculate the lattice enegy: (A) Δ°Hf CaBr2(s) = -675 kJ/mol (B) Δ°Hf Ca(g) = 179 kJ/mol (C) Δ°Hf Br(g) = 112 kJ/mol (D) 1st ionization energy of Ca = 590 kJ/mol (E) 2nd ionization...
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