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. binbagsss

    Elliptic functions proof -- convergence series on lattice

    Homework Statement Hi I am looking at the proof attached for the theorem attached that: If ##s \in R##, then ##\sum'_{w\in\Omega} |w|^-s ## converges iff ##s > 2## where ##\Omega \in C## is a lattice with basis ##{w_1,w_2}##. For any integer ##r \geq 0 ## : ##\Omega_r := {mw_1+nw_2|m,n \in...
  2. M

    I Reciprocal lattice in connection with band theory

    I need details on this topic ,this is my assignment but my solid state physics is not so good,and don't know much about it but i have to do this assignment ,i have material on reciprocal lattice but for only including in assignment ,not for my understanding,so i directly need any material on...
  3. F

    I Proof that lattice points can't form an equilateral triangle

    From Courant's Differential and Integral Calculus p.13, In an ordinary system of rectangular co-ordinates, the points for which both co-ordinates are integers are called lattice points. Prove that a triangle whose vertices are lattice points cannot be equilateral. Proof: Let ##A=(0,0)...
  4. hominnimoh

    I Fe Lattice: Band Structure & Fermi Surface Location

    This is schematic band structure for the dxz and dyz orbitals using a 2D square Fe lattice. What is the location in the Brillouin zone of the hole-like Fermi surface and the electron-like Fermi surface?
  5. S

    A Lattice Boltzmann simulation with arbitrary equilibrium func

    Hi everyone, I plan to do a simulation of a Boltzmann equation with experimentally known scattering between two particles. Initially I intend to incorporate the scattering into the collision integral and use Lattice Boltzmann Equation (LBE) afterwards. But I only see LGBK (DnQb) which requires...
  6. T

    I Dispersion relation in tight binding model

    Hamiltonian of tight binding model in second quantization is given as H = -t \sum_{<i,j>} a_i^{\dagger} a_j After changing basis it is H = \sum_{\vec{k}} E_{\vec{k}} a_{\vec{k}}^{\dagger} a_{\vec{k}} where E_{\vec{k}} = -t \sum_{\vec{b}} e^{i \vec{k} \cdot \vec{b}} where \vec{b} is a nearest...
  7. J

    Solving for A in a Simple Cubic Lattice

    Homework Statement 1. For a simple cubic lattice with a lattice constant of a, the energy band can be expressed as:E = Acos(kxa)cos(kya)cos(kza) + B. (a) Suppose the effective mass for the electron at conduction band is m* = -ħ2/2a2, find A. Homework EquationsThe Attempt at a Solution I...
  8. E

    I Is it Possible to Index a Cubic Lattice with h^2+k^2+l^2=7?

    Hi everyone, I've been given a problem where I have to index reflections from a cubic lattice, the procedure is simple enough but I'm getting a case where I get : h^2+k^2+l^2=7 I've taken to many books, but most either don't mention the topic or say they are simply 'forbidden' reflections. I...
  9. M

    Lattice Models for Fluids - Regular Solution Model

    1. Problem Statement: For the regular solution model, develop the equations for the compositions of the coexisting phases in a binary system and plot the phase boundary as a function of χ/RT.2. This question stems from Sandler's Introduction to Applied Statistical Thermodynamics. The Attempt...
  10. kaio marques

    I Who can identify a Bravais Lattice?

    I have a real hard time trying to imagine why a face centered cubic cell originates a Bravais lattice. Could you try to explain it? I have also been trying to figure out if a side-centered cell is a Bravais lattice as well. This cell is a simple cubic cell with additional point at the midpoints...
  11. hkazmi25

    In bcc lattice why XRD 100 peak is not observed ?

    Can someone tell me that in bcc lattice why xrd 100 peak does not appear?
  12. Einj

    A Can the Vortices in a Superfluid Form a Triangular Lattice?

    Hello everyone, I am not an expert in the topic so I apologize in advance if the question has an easy answer. I think it is well known that if we have a set of N vortices in a superfluid their mutual interaction is logarithmic. Meaning that if \vec x_n is the position of the n-th vortex, then...
  13. G

    Using F4 Tally on Hexagonal Lattice Geometry

    How can i use F4 tally in hexagonal lattice geometry which composed of circular fuel assembly? For example one hexagonal lattice includes circle which is cell number 5 and another hexagonal lattice includes circle which is cell number 7 etc. I want to use all these cells in one f4 tally. Please...
  14. A

    Building a Lattice in MCNP: Seeking Help

    HI, i´m trying to build a lattice in mcnp. Actually it works but there are still lots of red lines in the plot. Maybe someone can help me. The Lattice code looks like that : lattice 1 0 2 $ outer space 2 3 -0.00126 -2 #4 #39 #41 #42 #43 #44 #45 #46 #47 $ universe 4 2 -8.4 -1 $ ball to enable...
  15. K

    A Question about LQCD and parallelization

    Hi there, I'm currently working on a relatively simple code to do some lattice simulations. I have access to a computing cluster at school and have been learning how to use OpenMP to parallelize my code (each node has 16 cores). I'm currently not planning to use MPI. My main question is...
  16. magnetic flux

    Physics Outlook after M.Sc. or PhD in Lattice QCD

    I will soon start a Master thesis in Lattice QCD. There I will spends lots of time developing C++ code and running it on different supercomputers. After that I consider doing a PhD if the Master thesis runs well. For the experimentalists I can see a lot of almost-engineering jobs where they...
  17. D

    A Basic questions on crystal lattice: k-vectors and Brillouin

    Hi all, I’m brushing up my skills on solid state physics and I have a few questions about crystal lattices: 1. What’s the spacing between allowed kx, ky, and kz states for a lattice of dimensions La x Lb x Lc? My attempt: The spacing is kx, ky, and kz: k_x = \frac{2\pi}{L_a}, \qquad k_y =...
  18. A

    I How Is the Ion Lattice Approximation Justified in Electron Dynamics?

    To solve the full many-body electron problem one often uses the approximation that the dynamics of the electron system, is that of N interacting electrons living in a periodic lattice of positive ion cores. What justifies this approximation and does it have a particular name?
  19. E

    I Fermi sphere and density of states

    Hello! When computing the density of states of electrons in a lattice, a material with dimensions L_x, L_y, L_z can be considered. The allowed \mathbf{k} vectors will have components k_x = \displaystyle \frac{\pi}{L_x}p k_y = \displaystyle \frac{\pi}{L_y}q k_z = \displaystyle \frac{\pi}{L_z}r...
  20. F

    Perturbation matrix: free electron model on a square lattice

    Homework Statement Nearly free electron model in a 2D lattice. Consider a divalent 2D metal with a square lattice and one atom per primitive lattice cell. The periodic potential has two Fourier components V10 and V11, corresponding to G = (1,0) and (1,1). Both are negative and mod(V10) >...
  21. F

    Distinct modes of oscillation in a 3D crystal lattice

    Homework Statement Silicon crystalises in a cubic structure whose lattice is face-centred, with a basis [000],[1/4,1/4,1/4]. How many optic/acoustic modes are to be found in the phonon dispersion diagram for silicon. How many distinct branches would you expect along the [100] direction...
  22. C

    Decimation of a triangular lattice

    Homework Statement Consider the network of spins shown below. The Hamiltonian is given by $$H = - \sum_{\langle i j k \rangle} [J \sigma_i \sigma_j \sigma_k + J_0]$$ with ##J,J_o \geq 0## and ##\langle i j k \rangle## denoting spin in the same triangle (the triangles under consideration are...
  23. C

    Triangular lattice and criticality

    Homework Statement Consider a two dimensional triangular lattice, each point of which can either contain a particle of a gas or be empty. The Hamiltonian characterising the system is defined in terms of the particle occupation numbers ##\left\{n_i \right\}_{i=1,...,N}## which can either be 0 or...
  24. Z

    A Can't crystal of simple lattice be antiferromagnetic?

    Is it correct that a crystall of simple lattice (lattice with primitive cells each having only one atom) cannot be antiferromagnetic? In other words, the antiparallelism must occur between atoms within each primitive cell. Thanks.
  25. A. Neumaier

    A Is Poincare symmetry the real thing?

    This is a continuation of a side issue from another thread.
  26. A

    Other Comparing Lattice QCD vs QED Research Experience

    I have the opportunity to pursue a research project in either lattice QCD or QED. In the case of QED, I'd be writing the code myself, will be able to understand the whole thing, et cetera. There shouldn't be any mystery in this. It almost seems like a long exercise, and there isn't anything I'd...
  27. I

    A Can a Tree Visit Every Cell in a Cubic Lattice Blindly?

    I need to generate a tree in a cubic lattice that, from any cell, visits every other cell in the lattice just once. This visit must be blind, that is, it is not allowed to mark the cell as visited. Thanks in advance for any solution or reference.
  28. thegirl

    I Drawing a reciprocal lattice, also basis

    Hey could anyone please explain how you go about drawing a reciprocal lattice? For example a 2d rectangular lattice to it's reciprocal form? Also... I don't know if this is correct but if you have a 2d rectangular lattice with lattice vectors L=n1a1 + n2a2 would the reciprocal lattice vectors...
  29. F

    Superconductor structure and contents

    Hi. What are superconductor ions? Are they naturally occurring ions inside the lattice structure of a superconductor when it's manufactured or are they ions introduced artificially to a superconductor's lattice ? And do the ions have spin?
  30. F

    B Quark Seeding: Info & Possibilities

    Hi. I'm wondering if anyone has any info on "quark seeding" like: Is it possible to dope the crystal lattice of a solid material by replacing electrons with quarks ?
  31. M

    I Partitions of Euclidean space, cubic lattice, convex sets

    If the Euclidean plane is partitioned into convex sets each of area A in such a way that each contains exactly one vertex of a unit square lattice and this vertex is in its interior, is it true that A must be at least 1/2? If not what is the greatest lower bound for A? The analogous greatest...
  32. Adoniram

    Nearest Neighbors in solid state, but with basis

    Homework Statement What is the area of the primitive cell for the lattice shown below? The nearest neighbor separation is "a." Homework Equations Here's the lattice we were given on our handout, and I have added the lines to indicate the square lattice (in red), the basis (in purple), and...
  33. R

    MCNP- repeated structures and lattice

    i have a simulation problem about a fuel assembly, after running this warring pop up:- " non-lattice cell in lattice universe "; and visual editor crash with one warning message " warning. 2 surfaces were deleted for being the same as others." So, what may be the problem with the input?
  34. P

    Specific heat of 1D lattice

    Homework Statement Analyze the specific heat of a one dimensional lattice of identical atoms: Show within Debye approximation that the specific heat at low temperatures ( ≪ Θ) is proportional to T/ΘD . Here ΘD=ℏD/ kB = ℏvs/KBa is the Debye temperature valid for 1D, kB the Boltzmann...
  35. H

    Lattice energy and energy required to vaporize MgO ?

    I understand that the lattice energy is the energy released when an solid ionic compound forms or it is the energy required to separate completely a mole of a solid ionic compound into its gaseous ions. So is second definition the same thing as vaporizing an ionic compound and so we can...
  36. N

    Number of ways in a 3D lattice

    Hi! If I have points A and B in a lattice in the plane, and the closest path between them is n + m steps (for example 4 steps upwards and 5 steps to the right), there are C(9,(5-4)) = 9 combinations of paths between them. I have to choose the 4 ways upwards (or the 5 ways to the right) of the 9...
  37. U

    Can someone explain peierls distortion/transition to me?

    Peierls distortion states that for a 1 dimensional polymer (like polyacyteline) with lattice spacing a they should have a half-filled conduction band, why? And how does changing the lattice space to 2a cause it to form an energy gap? In my mind it should be the same as the first case...
  38. Gvido_Anselmi

    Can Lattice Formulations Accommodate Modern Quantum Field Theories?

    Hello everybody, I have three quite mathematical questions in modern QFT. 1) Why it's supposed that N=2 SUSY Yang-Mills probably cannot be put on a lattice? 2) What is the recent status of lattice approach to conformal quantum field theories? This question is motivated by the following...
  39. H

    Determine the lattice parameter

    Homework Statement Given this image http://postimg.org/image/m9dyiqwy3/ find the lattice parametre of the reciprocal lattice and the coordinates of the point x. Homework Equations a*[/B]=2π(b×c)/(a⋅b×c) b*=2π(c×a)/(b⋅c×a) c*=2π(a×b)/(c⋅a×b) Which are the vectors of the reciprocal lattice a is...
  40. C

    Spins on a 3D cubic lattice

    Homework Statement Consider a system of rotors, each of them placed at one node of a 3D cubic lattice. The system is known as a Heisenberg spin model. Each rotor is represented by a vector of unit length ##\mathbf S(\mathbf r)##, where ##\mathbf r## is its position on the lattice. The...
  41. F

    Is a Bloch wave periodic in reciprocal space?

    A Bloch wave has the following form.. ## \Psi_{nk}(r)=e^{ik\cdot r}u_{nk}(r)## The ##u_{nk}## part is said to be periodic in real space. But what about reciprocal space? I've had a hard time finding a direct answer to this question, but I seem to remember reading somewhere that the entire...
  42. gonadas91

    Field Theory vs Lattice: Exploring Differences in Calculations and Results"

    Hello guys! I just just wondering a general thing about calculations done in the field theory and those made in the lattice. In the field theory we have some results that in principle should match with the lattice ones in the thermodynamic limit. However, when we tried to solve the same problem...
  43. O

    Substitutional impurity in a lattice?

    Homework Statement Boron atoms at a concentration of 3.60E+16 atoms/cm3 are added to pure intrinsic silicon as a substitutional impurity. Assume that the boron atoms are distributed homogeneously through the silicon in a cubic array. Enter the fraction, by weight, accounted for by impurity...
  44. O

    Lattice constant, space between planes?

    Homework Statement the volume density of a body centered cubic lattice is 5.3*10^22 Calculate the lattice constant, effective radius of atom, surface density on 110 plane, and distance between two nearest parallel 110 planes. Homework Equations 4/a^3= volume density 2/sqrt2*a^2 The Attempt at...
  45. O

    What is the lattice constant of a zincblende structure with two types of atoms?

    Homework Statement A material contains two types of atoms (A and B) in a zincblende structure. Atom A is group V with a hard sphere radius of 2.5 Å. Atom B is group III with a hard sphere radius of 1.6 Å. (a) What is the lattice constant of this material assuming that nearest neighbour atoms...
  46. O

    Lattice constant and volume density

    Homework Statement a) A material is composed of two atoms, A with effecitve radius 2.00 angstroms and B with effective radius 3.10 angstroms. The lattice is a body-centred lattice. b)Enter the volume density of either the A or B atoms in atoms/cm3 Homework Equations .5*sqrt3=(r1+r2) surface...
  47. J

    Software for drawing group lattice diagrams?

    Hi, I was wondering if there is code already available to draw group lattice diagrams if I already know what the subgroup structure of the group and its subgroups are. For example, it's easy to determine the subgroup lattice for cyclic groups simply using divisors via Lagrange's Theorem...
  48. D

    Isn't Born-Landé equation double the electrostatic energy?

    I'm confused as to how the Born-Landé equation can be extrapolated to find the electrostatic potential for an ionic lattice without halving it, as each interaction is otherwise counted twice. As I understand it, and according to Wikipedia, the electric potential energy in a charge configuration...
  49. SquidgyGuff

    Potential Energy of a 2D Crystal Lattice

    Homework Statement Use a computer to calculate numerically the potential energy, per ion, for an ifinite 2D square, ionic crystal with separation a; that is, a plane of equally spaced charges of magnitude e and alternating sign. Homework EquationsThe Attempt at a Solution The closest I can...
  50. L

    Bravais lattices in 2 dimensions (and 3 dimensions)

    I'm reading M. Omar Ali's Elementary Solid State Physics and in it, in Subsection 1.4 The Fourteen Bravais Lattices and the Seven Crystal Systems he says that "..., but one cannot place many such pentagons side by side so that they fit tightly and cover the whole area. In fact, it can be...
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