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.

View More On Wikipedia.org
Recent contents Top users
  1. C

    How Does a Two Atoms Basis Lattice Act Independently?

    Homework Statement For a lattice with a two atoms basis, the two dispersion relations valid for Ka = ±∏ w2 = 2C/M2 and w2 = 2C/M1 Show that under these conditions the lattice acts as two independent lattices (one lattice per each atom) with one of the lattices moving while the other is...
  2. P

    How Does Periodic Potential Affect the Energy Spectrum of a Bose Gas?

    Homework Statement Suppose an ideal bose gas sees a periodic potential with a period a in both x and y directions. Its eigenstates are altered from the free-particle form. The lowest band has energies \epsilon_\vec{k}=2t(2-cos(k_xa)-cos(k_ya)) where t is an energy scale that depends on the...
  3. S

    Number of lattice points between y=ax+b and y=x^2?

    Is there a nice closed-form for this?
  4. J

    Heisenberg interaction Hamiltonian for square lattice

    Hi, I just started self studying solid state and I'm having trouble figuring out what the hamiltonian for a square lattice would be when considering the Heisenberg interaction. I reformulated the dot product into 1/2( Si+Si+δ+ +Si+δ+S-- ) + SizSi+δz and use Siz = S-ai+ai Si+ =...
  5. F

    Lattice Points on Circle: Determining the Number of Points on the Boundary

    Does any circle having irrational radius have no lattice points on its boundary ? Extended question: Is there any way to determine the number of lattice points lying on the boundary of a given circle ? *The centres of these circles are all (0,0) *
  6. Z

    Expanding atomic displacements in terms of all lattice wave modes?

    I find the expression: "In general, one can expand the atomic displacements in terms of all the lattice wave modes (resembles a Fourier series expansion)" at: https://courses.cit.cornell.edu/ece407/Lectures/handout17.pdf But I have not found the expression in any other literature. (In fact...
  7. J

    Question about conductors, wires and lattice of ions

    Hi guys, sorry as this is probably a silly question. This isn't homework, but it's something I'm confused about in my GCSE course. If a lattice of ions can't conduct electricity when solid, then why can a wire (composed of these lattices (I think?)) conduct electricity? The electrons in a...
  8. H

    Use of Vortex Lattice Method in Transonic Applications

    Hi, I'm looking for an aerodynamic code to model a transonic (say around M=0.8) wing for some basic structural sizing and performance estimation. I'm hoping to use the MATLAB based VLM code Tornado. It has an inbuilt Prandtl-Glauert correction option. I realize that this is technically...
  9. T

    Bloch Oscillation in 1D crystal Lattice

    Bear with me (Two part question), In the ideal case, an electron in a lattice under the influece of a static force will undergo bloch oscillations. A simple hamiltonian for this system would be: H=H° +Fx and V(x+d)=V(x) If I used the kronig-Penney Model would I be able to derive...
  10. P

    What is the correct method for finding the lattice constant of NaCl?

    Homework Statement NaCl is a simple cubic lattice, with the Na and Cl atoms alternating positions. Assuming that the atoms are hard spheres with nearest neighbors touching, what is the lattice constant of NaCl? (The effective radius of Na is 1.0 angstroms, and the effective radius of Cl is 1.8...
  11. A

    Plane waves and sites distance in a lattice

    Hi, i would ask you an opinion about a (maybe stupid) doubt. Let us think of a 1D lattice whose sites distant from each other "a"; a plane wave in the lattice is given by e^{ikja} where k is the momentum and j an integer label for each site. Now, we modify the lattice in this way: between...
  12. srfriggen

    Determine the subgroup lattice for Z8

    Homework Statement "Determine the subgroup lattice for Z8" Homework Equations <1>={1,2,3,4,5,6,7,0} <2>={2,4,6,0} <3>=<1>=<5>=<7> <4>={4,0} <6>={6,4,2,0}The Attempt at a Solution My book only mentions this topic in one sentence and shows a diagram for Z30, which looks like a cube. I don't...
  13. D

    Find the lattice type and base in 2D crystal

    Homework Statement Draw the position of the atoms in the two neighboring planes of the GaAs crystal perpendicular to vector [010]. Find the type of the lattice and the basis of 2D crystal which is made by singling out those two neighboring layers from 3D lattice. Homework Equations...
  14. L

    Calculating Magnetization for 2x2 Ising Model Lattice

    Homework Statement Calculate magnetisation for partition function ##Z=12+4\cosh (8\beta J)## for Ising model 2x2 lattice.Homework Equations F=-k_BTln Z M(H,T)=-\frac{\partial}{\partial H}(\frac{F}{k_BT}) The Attempt at a Solution For me it looks that magnetisation is zero. By just doing the...
  15. L

    The Role of Boundary Conditions in the 2D Ising Model: A Finite Lattice Study

    http://books.google.rs/books?id=vrcHC9XoHbsC&pg=PA252&lpg=PA252&dq=Nolting+Finite+Ising+lattice&source=bl&ots=5uRHp0iALf&sig=_YBUSvbCBbhNQJ5Zu1go9AsEkM8&hl=sr&sa=X&ei=y_E4UIEyyobiBPvFgMgM&ved=0CC0Q6AEwAA#v=onepage&q=Nolting%20Finite%20Ising%20lattice&f=false A finite lattice X with so...
  16. mcodesmart

    Integration of the reciprocal lattice

    I am studying Solid state physics from kittel and I am stuck at the following equation. I can see that the exponential term turns to the kroneckler delta, but I don't understand how the integral gives the volume of the specimen, Ω? What am I not seeing? ∫d3x f(x)eiK.x = \sum aG∫d3x ei(K+G).x =...
  17. jfy4

    Lattice field theories and the continuum limit

    I have heard generally that it is possible to put different physical theories on a lattice and after renormalization get the same continuum theory. I mean, different lattice theories that lead to the same continuum theory. Is this true for, say, qcd, or other particle theories? Are there...
  18. M

    Do lattice vibrations emit radiation?

    I know that accelerating electrons in a solid are responsible for the emission, absorption and reflection of light. However, what role do the lattice vibrations play? These oscillating atoms are charge distributions and should emit radiation too, right? As far as I can remember, the...
  19. fluidistic

    Solid state physics, exercise about reciprocal lattice

    Homework Statement Hi guys, I don't reach the correct answer to an exercise. I'm following Ashcroft's book. I must find that the reciprocal of the bcc Bravais lattice is a fcc one and the reciprocal of the fcc Bravais lattice is a bcc one. Homework Equations If a_1, a_2 and a_3 are...
  20. V

    Force and Electric field within a cubic lattice

    Homework Statement There are Cl+ ions at the corners of a cube of side a = .4nm and a Cs- ion at the center. a) What is the net force on the Cs- ion? b) What is the electric field at the center of the cube cause by all of the Cl ions? c) What is the electric potential at that point...
  21. P

    Theoretical Lattice Energy for MgF2

    Homework Statement Calculate the theoretical lattice energy for MgF2 (Born-Landé equation) Ionic radius Mg+2 (coordination number 6) = 86 pm Ionic radius F- (coordination number 3) = 116 pm Madelung constant = 2.408 n = 7 Homework Equations The Attempt at a Solution I'm...
  22. S

    Electron mass in a Kondo lattice

    I'm not a physicist, but I always thought that the rest mass of the electron was a physical constant and its apparent mass was only affected by relativistic effects. This abstract discusses heavy electrons in a Kondo lattice. In what sense are these electrons "heavy"? At low temperatures, they...
  23. P

    Need Advice on Spatial Statistics for a Lattice

    My professor gave me data points representing living cells on a rectangular plane and was told to analyze their spatial pattern, i.e., do the date points on the finite plane have a tendency to be clustered, random, or dispersed. I successfully accomplished this for points on a continuous plane...
  24. K

    Confusing on Bravais lattice and base vectors

    Hi there, I know that primitive cell is not unique and there are more than one way to define the primitive vectors but my question is when we said "primitive vectors" do we have to construct the Bravais lattice with choosing a proper basis first? My reasoning is suppose the crystal consist of...
  25. K

    About basis of the honeycomb lattice

    Hi there, I am reading the book "Condensed Matter Physics" second edition by Michael P. Marder. It stated in page 9 that one basis of the the honeycomb lattice is \vec{v}_1 = a [0 \ 1/(2\sqrt{3})], \qquad \vec{v}_2 = a [0 \ -1/(2\sqrt{3})] which is based on figure 1.5(B) in page...
  26. J

    Partition function to find expected occupancy of a lattice defect

    Homework Statement An impurity can be occupied by 0, 1 or 2 electrons. The impurity orbital in non-degenerate, except for the choice of electron spin. The energy of the impurity level is \epsilon, but to place the second electron on the site requires an additional energy \delta \epsilon...
  27. james.goetz

    Do theories of quantum gravity require that space-time is a lattice?

    Do theories of quantum gravity require that space-time is a lattice instead of a continuum? I guess this question has been addressed elsewhere, but I would appreciate hearing different points of view. Please dummy down the responses so a philosopher can understand it.
  28. M

    Deduce primitive lattice vectors from position vector.

    Homework Statement given the following position vector: R = (10n1 + 9n2 + 19n3)(a/10) x + 6(n2+n3)(a/5) y + 2(n3)a z where n1, n2 and n3 are integers Find the primitive lattice vectors. Homework Equations any position vector of a lattice point is of the type R= c1 a1 + c2 a2 + c3 a3; and...
  29. E

    Derivation of lattice parameter of zinc blende crystal structure

    I need to derive the lattice parameter in terms of the Zn-S separation distance, l. I looked up the value and I've found it to be a = \frac{4}{\sqrt{3}}l The way that I started my derivation was saying that each tetrahedron has a sulfide ion in the center, so then we can make a triangle from...
  30. C

    How to Compute Specific Heat for a 1-D Lattice?

    Homework Statement A 1-D lattice consists of a linear array of N particles (N>>1) interacting via spring-like nearest neighbor forces. The normal node frequencies are given by \omega_n=\omega_0\sqrt{\,\,2-2\cos\left(2\pi n/N\right)} where \omega_0 is a constant and n an integer ranging...
  31. S

    Is the wilson loop always positive in the lattice simulation?

    More precisely, my question is, Is the wilson loop used to calculate the heavy quark potential always positive in the lattice simulation? (i) as we are usually told, wilson loops of the following form are related with the heavy quark potentials ......... _________ <WC_______... |......|../\ ...
  32. S

    Is the wilson plaue always positive in the lattice simulation?

    More precisely, my question is, Is the wilson loop used to calculate the heavy quark potential always positive in the lattice simulation? (i) as we are usually told, wilson loops of the following form are related with the heavy quark potentials ......... _________ <WC_______... |......|../\ ...
  33. P

    Help Students Comprehend Reciprocal Lattice & Brillouin Zone

    How can I help my students to comprehend/understand the concept of reciprocal lattice (pictorially) and Brillouin zone?
  34. Fredrik

    Linear or conjugate operators and automorphisms on the lattice of subspaces

    I'm reading about symmetries in QM in "Geometry of quantum theory" by Varadarajan. In one of the proofs, he refers to theorem 2.1, which is stated without proof. He says that the theorem is proved in "Linear algebra and projective geometry" by Baer. That isn't very helpful, since he doesn't even...
  35. H

    The number of lattice point in FCC?

    The number of lattice point in FCC? Hi, I am having a trouble to figure the number of lattie point in FCC structure out. My textbook said simple cubic has 1 lattice point, BCC has 2 points, and FCC has 4 points. However there is no further explanation or figures. I am a kind of understand...
  36. H

    Winning Strategy for Player 1 in the 10x20 Lattice Game

    the 10 X 20 lattice game has the following rules: - two players alternate picking points (x,y) in the plane. the points must have integer coordinates(lattice points) and we must have 1 ≤ x ≤ 10 1 ≤ y ≤ 20 - first player must begin from (1,1). that is, player 1 can choose any point with one...
  37. Z

    What is an Optical Lattice and Quantum Dot?

    Who can tell me the concept of optical lattice and quantum dot? Or give me an advice of some thesis? Thank you very much.
  38. J

    Understanding Salt Crystal Lattice: Factors Influencing Crystal System Selection

    What decides that which salt will take up which crystal lattice? Some salts have cubic lattice , some have hexagonal , some have orthorhombic - but what factor decides which crystal system the salt will take over. Most of the examples i have seen consist of cubic lattice system. Is it becasue...
  39. B

    Analyzing Phonon Collisions in a Cubic Lattice

    Homework Statement In a simple cubic lattice of spacing 0.2nm a phonon traveling in the {1 0 0} direction with wavelength 0.42nm collides with another phonon of the same wavelength which is traveling in the {1 1 0} direction. Draw a reciprocal space diagram to show the magnitude and...
  40. D

    Exploring Wave Propagation in a 1D Uniform Lattice

    If we take a 1D uniform lattice I understand that we can derive a difference equation after using Hookes law and Newtons 2nd law as seen in the section labelled (Scalar wave equation in one space dimension, Derivation of the wave equation, From Hooke's law) in the link below...
  41. ElijahRockers

    Drawing the lattice curves of a transformation

    Homework Statement Draw the lattice lines for the following transformation. x=se^t y=se^-t The Attempt at a Solution Trying to draw this transformation. So first thing I did was let t be constant. I then used substitution to show that x=ye^{2c}. This is the set of all lines (as C is...
  42. C

    Sorting Lattice Energies: KF, MgS, RbI

    Homework Statement Place the following in order of decreasing magnitude of lattice energy. KF, MgS, RbI I'm not really sure how to do this without looking up the lattice energy values with each. And I don't think that is what our teacher wants us to do
  43. C

    Niobium atom lattice structure and its behavior

    Hi, I am trying to understand how do cluster of niobium atoms behave when I apply force to compress/stretch them. Niobium has a cubic lattice structure, 8 atoms on each corner and 1 in the center. So my question is let's say I have 2 of these cubes connected, and composed of 14 atoms. In...
  44. E

    LMTO - Creating Electronic Canonical bands of BCC lattice

    Hello, I need help with the Canonical band of BCC lattice (Linear Muffin Tin Orbitals method) I am trying to generate the Canonical (electronic) band for d-d orbital for BCC lattice; this can be done by diagonalizing the Structure constant matrix S(l,m,l',m'), (with l,m, l',m' being the...
  45. H

    Unit Cell & Reciprocal Lattice of a 2D Oblique Lattice

    Homework Statement a 2d oblique lattice is described by the following vectors: a = 3i b = 0.5i + j sketch the unit cell of this lattice Sketch accurately the reciprocal lattice of the cell and justify the direction and relative magnitudes of each reciprocal lattice vector The...
  46. F

    Cooling a lattice QCD gauge configuration

    Hello I am looking to learn about Lattice Gauge Theory. I am already understand the general theory, (that is, defining fermions and gauge fields on lattice sites and links, respectively), but am having trouble intuiting the results of such calculations. As an example, the following is a...
  47. M

    Reciprocal Lattice: X-Ray Diffraction Explained

    can anyone tell me why we use reciprocal lattice for understanding the diffraction of X rays from crystal? thanks in advance :)
  48. E

    Finding Components of Vectors in a Crystal Lattice

    Homework Statement Be the vectors a, b, c such as: | a | = | b | = | c | = 10.5 Angstron The angles between these vectors are: alpha = beta = gamma = 109.5 degree These vectors represent the lattice vectors of a crystal. Find out their components (a_1, a_2, a_3, b_1...
  49. Spinnor

    1D QED on a lattice, how much information?

    Suppose we were to simulate 1D QED on a 1D lattice. How much information do we need at each lattice site given the mass, charge, and spin of the particles (does spin make sense in 1-1D spacetime?)? The links between lattice sites represent the gauge field? How much information is needed at...
  50. E

    We have that for a crystal the lattice parameters

    Homework Statement We have that for a crystal the lattice parameters (in angstrom and degree) are: a b c (in Angstrom) 10.5 10.5 10.5 \alpha β γ (in degree) 109.5 109.5 109.5...
Back
Top