Lattice Definition and 437 Threads

Let us have a 5 dimensional lattice in R^5 = R^3XR^2, where at each lattice point we have a point mass and all mass points are linked with springs in all 5 dimensions (edit, to nearest neighbors). Require that motion of the mass points is restricted to a two dimensional subspace, R^2, of R^5...

I need a proof that the set of natural numbers with the the relationship of divisibility form a distributive lattice with gcd as AND and lcm as OR. I know it can be shown that a AND (b OR c) >= (a AND b) OR (a AND c) for a general lattice, and that if we can show the opposite, that a AND (b OR...

I would like to know how to estimate the value (length) of the lattice constant in crystals in terms of fundamental parameters such as electron charge, electron mass, proton mass and Planck constant. (Alternatively, the formula may contain the Bohr radius, since I know how to calculate Bohr...

Hello, I am trying to work out a mean field theory for an antiferromagnetic Ising model on a square lattice. The Hamiltonian is: ## H = + J \sum_{<i,j>} s_{i} s_{j} - B \sum_{i} s_{i} ## ## J > 0 ## I'm running into issues trying to use ## <s_{i}> = m ## together with the self-consistency...

Say I have a large spacetime lattice set up on a supercomputer where I calculate the scattering cross section of two spinless electrons of equal and opposite momentum via lattice QED. To get the right results we must add the amplitudes for every possible "path" the field can evolve from initial...

Say we try and calculate the ground state energy of the bound state of a quark antiquark meson via lattice QCD. Say I look at one space time lattice point of one path. Do the fermi fields "live" on the lattice points? Do the boson fields "live" on the legs between the space time lattice points...

Greg Egan just proved something nice: the E10 lattice, famous in string theory and supergravity, can be described as the lattice of self-adjoint matrices with integral octonions as entries! I'm not sure this result is new, but I've been wanting it for quite a while and haven't seen it...

Background information: The wave function for an electron in a crystal lattice is modeled by a Bloch wave. A Bloch wave is a function with the periodicity of the lattice multiplied times a complex exponential function. This exponential function has a wave vector k, called the crystal momentum...

Hi there, I am trying to understand the dispersion curve(as shown below) of a 1D lattice with diatomic basic. Here are my questions 1) Can both optical and acoustic branch of phonon can simultaneously exist in crystal? 2)Why there is a band gap between optical and acoustic phonon...

So i don't really understand why the change in momentum of a crystal involves a reciprocal lattice vector. Surely it is just the change in momentum due to the change in the number and frequency of the phonons before and after whatever event/scattering/collision takes place. Can somebody please...

I have looked at the cation anion ratio of cubic, octahedral and tetrahedral arrangments on an internet site. By a mathematical derivation they find the minimum value for the cation anion ratios for cubic, octahedral and tetrahedral arrangments. My problem is that even though I get the...

I have seen in some books that the triclinic bravais lattice ( a≠b≠c , α≠β≠γ ) excludes explicitly the option that one angle equal 90°. For instance 90°≠α≠β≠γ=90°. If I got the definition of α, β and γ correctly, it would be a primitive cell with a pair of parallel faces as rectangles, and...

Suppose I have a cubic lattice of N^3 masses, M, each connected to six nearest neighbors with springs of constant k free to move but at rest. Now fire a single mass, m, with velocity v at surface of the lattice such that no rotation can be imparted to the cubic lattice. Let the fired mass bounce...

This is not a quiz but I am thinking how to write down a simple math formula to count the total number of squares present in a lattice of NxM points for my 12 year old nephew ? He'll sure be happy if I could turn this into, say, a common sense for pupils like him. :biggrin: For example, In a...

Hello, I am a first year graduate student in physics who is interested in getting involved in the field of lattice QCD. I purchased the text "Lattice methods for quantum chromodynamics" by DeGrand and Detar. I have never taken a course on quantum field theory, but I hoped that having...

Dummit and Foote give the Fourth or Lattice Isomorphism Theorem for Modules on page 349. I need some help with the proof of Fourth or Lattice Isomorphism Theorem for Modules ... hope someone will critique my attempted proof ... (I had considerable help from the proof of the theorem for groups...

Dummit and Foote give the Fourth or Lattice Isomorphism Theorem for Modules on page 349. The Theorem reads as follows:https://www.physicsforums.com/attachments/2981In the Theorem stated above we read: " ... ... There is a bijection between the submodules of $$M$$ which contain $$N$$ and the...

My problem for this thread is: Let $L$ be a Boolean lattice. Prove that $L$ is atomic if and only if the top element is the join of a set of atoms. For the forward implication, I am already done. I used Zorn's lemma to show that the set, $\mathcal{F}$, of the elements in $L$ which are the...

My problem is this: Let $L$ be a Boolean lattice. Prove that $L$ is atomic if and only if its order dual $L^{op}$ is atomic. My proof is going like this so far: Let $L$ be a Boolean lattice. Suppose first that $L$ is atomic. Then, by definition, every lowerset in $L$ contains an atom. Since...

The problem is this: Let $L$ be a Boolean lattice. For all $a<b\in L$, prove that the interval $[a,b] = \uparrow a \cap \downarrow b$ is a Boolean lattice under the partial ordering inherited from $L$. What I've managed to do so far: I used the fact that $[a,b]$ was a poset and showed that...

My A level Chemistry textbook defines Lattice Energy as "the enthalpy change when 1 mole of an ionic compound is formed from its gaseous ions under standard conditions"; a definition which I can't fully grasp because of the "standard conditions" part. How can gaseous ions exist under standard...

My problem is this: Let $L$ be a bounded, complemented, distributive lattice and let $x,y,z\in L$. Prove the following: 1. $x\wedge y = \bot \Leftrightarrow x\leq y^c$ 2. $x = (x^c)^c$ 3. $x\wedge y \leq z \Leftrightarrow y\leq x^c \vee z$ 4. $(x\vee y)^c = x^c \wedge y^c$ 5. $(x\wedge y)^c =...

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...

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.

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...

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'...

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...

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...

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?

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...

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

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...

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...

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...

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...

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...

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...

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...

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...

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)...

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...

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??

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?

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?

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...

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...

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

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...

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...

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)...