Briloner Heute bestellen, versandkostenfrei The first Brillouin zone is the locus of points in reciprocal space that are closer to the origin of the reciprocal lattice than they are to any other reciprocal lattice points (see the derivation of the Wigner-Seitz cell) The first Brillouin zone is the smallest volume entirely enclosed by planes that are the perpendicular bisectors of the reciprocal lattice vectors drawn from the origin. The concept of Brillouin zone is particularly important in the consideration of the electronic structure of solids

6.15: The first Brillouin zone Last updated; Save as PDF Page ID 52348; No headers. Since there are only N distinct values of the coefficients (corresponding to one period of the Fourier transform), we typically restrict k to the N values in the range \(-N/2 < n \leq N/2\), i.e. \[ -\frac{\pi}{a_{0}} < k \leq \frac{\pi}{a_{0}} .\ The first Brillouin zone is defined as the set of points reached from the origin without crossing any Bragg plane (except that the points lying on the Bragg planes are common to two or more zones). The second Brillouin zone is the set of points that can be reached from the first zone by crossing only one Bragg plane * The first Brillouin zone of an fcc lattice has the same shape (a truncated octahedron) as the Wigner-Seitz cell of a bcc lattice*. Some crystals with an fcc Bravais lattice are Al, Cu, C (diamond), Si, Ge, Ni, Ag, Pt, Au, Pb, NaCl. Cut-out pattern to make a paper model of the fcc Brillouin zone. Punkte hoher Symmetrie des fcc-Gitter The first Brillouin zone of a simple cubic lattice The first Brillouin zone is just the hexagon obtained by following the geometrical prescription given above. The two-dimensional square lattice is even simpler. The two lattice vectors aand bare equal in length and separated by a 90˚ angle

The first Brillouin zone is defined as (these are all correct, but not all are useful in all contexts): The Wigner-Seitz cell of the reciprocal lattice. In the construction described in the wiki article, the cell around the (arbitrarily chosen) origin is the first Brilliouin zone, and the immediately adjacent cells are the second This first part contains the introduc... This is part 1 of a two-part video on how to make sense of Brillouin zones, a central concept in the physics of solids 1 Brillouin zone Quantum ESPRESSO (QE) support for the de nition of high symmetry lines inside the Brillouin zone (BZ) is still rather limited. However QE can calculate the coordinates of the vertexes of the BZ and of particular points inside the BZ. These notes show the shape and orientation of the BZ used by QE

The set of all such wavevectors defines the first Brillouin zone. Additional Brillouin zones may be defined as copies of the first zone, shifted by some reciprocal lattice vector. Thermodynamics. The thermodynamic properties of a solid are directly related to its phonon structure 4. First Brillouin zone of fcc lattice The shortest G's are the eight vectors: ( 1, 1, 1) 2 a . The boundaries of the central cell in the reciprocal lattice are determined for the most part by the eight planes normal to these vectors at their midpoints. But the corners of th So the first Brillouin zone can be defined as a set of points in reciprocal space that can be reached from a specific point of origin without crossing any Bragg planes. So what are Bragg planes? A Bragg plane, or in this case, a Bragg line, is a Bragg line which perpendicularly bisects a reciprocal lattice vector-- a vector which connects two lattice points This region is known as the Brillouin zone (sometimes called the first or the central Brillouin zone). It is usually possible to restrict attention to k values inside the zone. Discontinuities occur only on the boundaries. If the zone is repeated indefinitely, all k-space will be filled

Brillouin zone, rectangle lattice (Kittel ISSP 9-2) A 2D metal has one atom of valence one in a simple rectangular primitive cell a 2 Å; b = 4 Å. (a) Draw the first Brillouin zone. Give its dimensions, in cm-1. (b) Calculate the radius of the free electron Fermi sphere, in cm-1. (c) Draw this sphere to scale on a drawing of the first. The locus of points in reciprocal space that have no Bragg Planes between them and the origin defines the first Brillouin Zone. It is equivalent to the Wigner-Seitz unit cell of the reciprocal lattice. In the picture below the first Zone is shaded red. Now draw on the Bragg Planes corresponding to the next nearest neighbours This is the set of points one reaches with a straight line from the origin and passing through n-1 Bragg planes. In this terminology, the Brillouin zone defined above is the first Brillouin zone. The n-th Brillouin zone is a shell around lower Brillouin zones and its shape becomes for higher values of nrapidly rather complicated (see figure) How can I find the shape of first Brillouin zone of any crystal structure that is not a common crystal At first, I curious why in the log window are marked positions of only 9 atoms,.

Brillouin zonesBand structures holds: -E(k) = E(-k) - One value of E for eachkwithin one band -E(k) is a periodical function ofk, it is sufficient to be displayed within the interval (-/a; /a) -the first Brillouin zone the first Brillouin zone-Wigner-Seitz cell in the reciprocal lattic First Brillouin zone is all points in space which could be reached by not crossing any plane calculated in 3. My program generates first Brillouin zones, using steps 2-4 of this algorithm. Link to program explanations can be found here. This program also calculates Fermi surfaces. Explanations on Fermi surfaces can be found here The first Brillouin zone of an hexagonal lattice is hexagonal again. Some crystals with an (simple) hexagonal Bravais lattice are Mg, Nd, Sc, Ti, Zn, Be, Cd, Ce, Y. Cut-out pattern to make a paper model of the hexagonal Brillouin zone The **first** **Brillouin** **Zone** (BZ) is the Wigner-Seitz cell of the reciprocal lattice, which can be constructed by applying Voronoi decomposition to a lattice. The reciprocal lattices (dots) and.. This Demonstration shows the construction of the first Brillouin zone (BZ) of a single-walled carbon nanotube (SWNT) superimposed on the 2D hexagonal first BZ of graphene. The first BZ of a SWNT is given by an irreducible set of equidistant lines (also called cutting-lines or 1D BZs) whose spacing and length are related to the chiral indices of.

The first Brillouin zone has the shape of a truncated octahedron. It can be visualized as a set of eight hexagonal planes halfway between the centre of the cell and the lattice points at the corner, and six square planes halfway to the lattice points in the center of the next cell in this lecture we will try to understand brillouin zone, bragg plane and also will try to understand brillouin zones for simple cubic lattice.reciprocal lat.. The first Brillouin zone is defined to be the Wigner-Seitz primitive cell of the reciprocal lattice, or it could be defined as the set of points in k space that can be reached from the origin without crossing any Bragg plane. The second Brillouin zone is the set of points that can be reached from the first zone by crossing only one Bragg plane

* 6*.1 Geometric Properties of the First Brillouin Zone. The crystal structure of silicon is known as diamond structure which is adopted by solids with four symmetrically placed covalent bonds. The diamond structure can be described by a face-centered cubic (FCC) lattice with a basis of two atoms where one is placed at and the other at ¼ ¼ ¼. Brillouin Zones, in 3D: 1) FCC first Brillouin zone example. Note standard labeling of high symmetry points , L, X, etc. Center (k = 0) is always called , other labels by historical convention for specific Bravais lattices 2) BCC first Brillouin zone, two images below. (image from Wikipedia The real space and reciprocal space primitive translation vectors are: $\large \vec{a}_1 = a\hat{x}$ $\large \vec{a}_2 = a\hat{y}$ $\large \vec{a}_3 = c\hat{z}$

The first Brillouin zone (BZ) of a real space Bravais lattice is the Wigner-Seitz primitive cell of the reciprocal lattice. The volume of the primitive cell of the reciprocal lattice is [math](2\pi)^3/V[/math] where [math]V=a_1\cdot (a_2\times a_3.. Brillouin Zones (a) Consider a cubic lattice with lattice constant a. Describe the first Brillouin zone. Given an arbitrary wavevector k, write an expression for an equivalent wavevector within the first Brillouirn zone (there are several possible expressions you can write) First Brillouin Zone as I understand it, extends until half the length of the first neighboring lattice point of the reciprocal space. Answers and Replies Jun 30, 2014 #2 DrDu. Science Advisor. 6,148 818. If I interprete your question correctly, you seem to be confusing the reciprocal space and reciprocal lattice Download Wolfram Player. Three steps are needed to construct successive Brillouin zones for 2D lattices. [more] 1. Select one reciprocal lattice point as the origin. 2. Draw the bisectors perpendicular to the line segments connecting the origin with the other reciprocal lattice points (the Bragg lines). 3. The Brillouin zone consists of those. In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell of the reciprocal lattice in the frequency domain. It is found by the same method as for the Wigner-Seitz cell in the Bravais lattice.The importance of the Brillouin zone stems from the Bloch wave description of waves in a periodic medium, in which it is found that the solutions can be.

- A Brillouin zone is defined as a Wigner~Secitz primitive cell in the reciprocal lattice [1,2]. The construction of the first Brillouin zone for two different 2D lattices are shown below: The construction procedure for 3D lattices is basically the same as for 2D lattices
- zone will be m+1. • A Brillouin zone is formed by polygons (polyhedra) having the same number (ﬁg. 3 A). • As anticipated, the ﬁrst Brillouin zone is also the ﬁrst W-S cell (no line is crossed). • The different portions of a Brillouin zone are reduced to the ﬁrst Brillouin zone in the normal way, i.e., using the repeated W-S.
- The second Brillouin Zone is the region of reciprocal space in which a point has one Bragg Plane between it and the origin. This area is shaded yellow in the picture below. Note that the areas of the first and second Brillouin Zones are the same
- First-Brillouin-zone integration areas for anisotropic superconducting states J S Millan 1, I R Ortiz , L A Perez2, C Wang3 . 1. Facultad de Ingeniería, Universidad Autónoma del Carmen, Cd. del Carmen, 24180
- The first two planes are part of the Brillouin zone boundary, whereas the last three ones are inside of the Brillouin zone.Figure 6.3 gives a sketch of the irreducible wedge placed within the first octant of the first Brillouin zone
- first brillouin zone free download. BZFlag - Multiplayer 3D Tank Game OpenSource OpenGL Multiplayer Multiplatform Battle Zone capture the Flag. 3D first person tank game
- The reduced zone scheme arises when we represent all of the bands and their gaps within the first brillouin zone by translating them in with the appropriate reciprocal lattice vector. Posted by strail at 12:15. Email This BlogThis! Share to Twitter Share to Facebook Share to Pinterest

the topological invariant can be identified by studying the first Brillouin Zone. As an example, a simple tight-binding model with similar structure as the homework problem of planar 2 is analyzed following Ref. 14 The Bravais unit consists of three sites, with nearest neighbor hopping amplitude t * Zone de Brillouin (Fr)*.Brillouin-Zone (Ge).Zona di Brillouin (It).ブリュアンゾーン (Ja).Zona de Brillouin (Sp).. Definition. A Brillouin zone is a particular choice of the unit cell of the reciprocal lattice. It is defined as the Wigner-Seitz cell (also called Dirichlet or Voronoi domain of influence) of the reciprocal lattice.It is constructed as the set of points enclosed by the.

- The first Brillouin zone would then be the area which is enclosed by the perpendicular bisectors (and is the grey area shaded on the image above). The higher zones are constructed in the same way; a slightly variant explanation is given on this site
- On Brillouin Zones 3 M: two geodesics °1 and °2 focus at some point y 2 M if °1(T) = y = °2(T).This gives rise to a decomposition of the tangent space of M at x into regions where the same number of geodesics focus. In order to study focusing of geodesics on a manifold (M;g)with metric g via Brillouin zones, we do the following.Choose a base-pointp0 in M and construct the universal cover X.
- First Brillouin zone: lt;p|>In |mathematics| and |solid state physics|, the first |Brillouin zone| is a uniquely define... World Heritage Encyclopedia, the aggregation of the largest online encyclopedias available, and the most definitive collection ever assembled
- Those who know some solid state physics should know what the first brillouin zone is. How do I plot the dispersion relation in the 1st brillouin zone so that the curves can 'fold back'? For instan..
- 2-D Hexagonal lattice to 6th Brillouin zone (pdf file, 0.60 MB) Zone folding. An important property of the Brillouin Zones is that, because the reciprocal lattice is periodic, there exists for any point outside the first zone a unique reciprocal lattice vector that will translate that point back inside the first zone

Figure: 2D Brillouin zone of a surface with cubic symmetry with a Monkhorst-Pack grid. The thin square indicates the conventional first Brillouin zone, the thick square marks the Brillouin zone as realized in the fhi98md code. The location of one special k-point (out of 64) within its tile is marked by the cross The first segments-π/a k π/a is called the first Brillouin zone, the two segments -π/a k -2π/a and π/a k 2π/a correspond to the second Brillouin zone and so on. Therefore the total number of states in the first Brillouin zone of one-dimensional crystal of length L will be L/a The first Brillouin zone is the first volume enclosed by the planes that perpendicularlly bisect all reciprocal lattice vectors, with the origin taken as a point in the reciprocal lattice. It is easy to show that the first Brillouin zone is simply the Wigner-Seitz cell of a crystal's reciprocal lattice

Definition of brillouin zone in the Definitions.net dictionary. Meaning of brillouin zone. What does brillouin zone mean? Information and translations of brillouin zone in the most comprehensive dictionary definitions resource on the web ** First brillouin zone for hexagonal lattice**. GitHub Gist: instantly share code, notes, and snippets

Sampling the Brillouin-zone: Martijn MARSMAN Institut fu¨r Materialphysik and Center for Computational Materials Science Universitat Wien, Sensengasse 8, A-1090 Wien, Austria¨ ienna imulation ackage b-initio M. MARSMAN, SAMPLING THE BRILLOUIN-ZONE Page 1 Overview • introduction • k-point meshes • Smearing methods • What to do in practic We will first establish the geometrical construction of the (first) Brillouin zone and then use it to establish the diffraction condition. Brillouin zones will feature again in later parts of this course such as those on the electronic structure of solids and on phonons A diagram of the first Brillouin zone of a face-centred cubic (FCC) lattice, with pointsof high symmetry marked. Päiväys: 5. toukokuuta 2008: Lähde: Oma teos: Tekijä: Inductiveload: Käyttöoikeus (Tämän tiedoston käyttäminen The region in k space is known as first brillouin zone. When the electrons have k value between pi/a and 2pi/a along X axis direction, then the region is known as second brillouin zone. In this zone, the reflection and energy gap occurs at 2pi/a in both positive and negative direction. In this way we can form third, fourth etc Brillouin zone construct the first Brillouin zone; determine its high symmetry points; find a irreducible polyhedron and verify that its conforms to the pointgroup symmetry of the spacegroup. Constructing and irreducible Brillouin zone polyhedron for a face centered cubic lattice can be accomplished with, e.g.

First, choose a lattice you wish to visualize, by clicking the right mouse button, then clicking on Lattices and then clicking on your favorite lattice. Note that these lattices are already reciprocal lattices. At this point both Brillouin zone and Fermi surface are displayed pirmoji Brijueno zona statusas T sritis fizika atitikmenys: angl. first Brillouin zone vok. erste Brillouinsche Zone, f rus. первая зона Бриллюэна, f pranc. première zone de Brillouin, Bragg reflection occurs at the first Brillouin zone boundaries గ At గ a from ECE 614 at Oregon State Universit * First Brillouin Zone Adding or subtracting a reciprocal lattice vector G from k leaves the wavefunction unchanged - in other words our system is periodic in reciprocal-space too*. We only need to study the behaviour in the reciprocal-space unit cell, to know how it behaves everywhere

* However, away from the center of the first Brillouin zone, the lack of periodicity in the space makes the allocation of these eigenvalues with respect to their corresponding Brillouin zone non-trivial*. The energy-band dispersion for our Penrose-tiled potential is shown in figure 3 Nano-scale wave dispersion beyond the **First** **Brillouin** **Zone** is often observed as descending branches and inflection points when plotting frequency or phase velocity against the wave number. Modeling..

Nano-scale wave dispersion beyond the First Brillouin Zone is often observed as descending branches and inflection points when plotting frequency or phase velocity against the wave number Brillouin zone. In the propagation of any type of wave motion through a crystal lattice, the frequency is a periodic function of wave vector k.This function may be complicated by being multivalued; that is, it may have more than one branch

The first Brillouin zone is a restricted set of values of k with the property that no two of them are equivalent, yet every possible k is equivalent to one (and only one) vector in the first Brillouin zone. Bloch wave-Wikipedia 2) Calculate the coordinates of the corners of the Wigner-Seitz cell and the first Brillouin zone. 3) Determine the coordinates of the corner and the equations of the lines from the origin to the corner, from the orgiin to the center of the edge, and from the center of the edge to the corner of the first quadrant (high symmetry directions) for both the Wigner-Seitz cell and the first Brillouin. Definisi dalam bahasa Inggris: Brillouin Zone. Arti Lain dari BZ Selain Zona Brillouin, BZ memiliki arti lain. Mereka tercantum di sebelah kiri bawah. Silakan gulir ke bawah dan klik untuk melihat masing-masing. Untuk semua arti dari BZ, silahkan klik More

Brillouin Zone courses provides a comprehensive and comprehensive pathway for students to see progress after the end of each module. With a team of extremely dedicated and quality lecturers, Brillouin Zone courses will not only be a place to share knowledge but also to help students get inspired to explore and discover many creative ideas from themselves ** It was established that within the first Brillouin zone of graphene there is a sufficiently large region**, $ \vert k_{x}\vert \le \pi/3a\sqrt{3}$ ; $ \vert k_{y}\vert \le \pi/3a$ , in which it is possible to construct an approximate dispersion law

** Hey guys, I just realized that there is a gap somewhere in my understanding of K vectors and reciprocal space**. My question is how can we. Hemispherical imaging of the first Brillouin zone (FBZ) of a single-network diamond photonic crystal. ( a) Image of the hemispherical reflectance of a single-scale domain. The shadow of the glass pipette holding the scale is seen at 09.00. The white-dashed circles indicate scattering angles of 5°, 30°, 60° and 90° Fermi Surfaces and Brillouin Zones. Consider a two-dimensional lattice which has two electrons per unit cell. The area of the Fermi circle for the electrons then has an area equal to the area of the first Brillouin zone, 1 (lattice unit) 2 . The radius of the Fermi circle is thus. 1/ (π) 1/2

First Brillouin zone of FCC lattice, a truncated octahedron, showing symmetry labels for high symmetry lines and points. Other lattices have different types of high-symmetry points. They can be found in the illustrations below. Triclinic Primitive triclinic(TRI) Triclinic Lattice type 1a (TRI1a) Triclini Exploring the Brillouin zone. When we first discussed periodic systems, we talked about the unit cell, which is repeated forever in space. We did not at the time talk about what that meant for reciprocal space. Let's examine the dispersion relation for the 1D chain of identical oscillators The Brillouin Zone Slide 14 The Brillouin zone is the Wigner‐Seitz unit cell constructed from the reciprocal lattice. It is essentially a map of the periodicity of the lattice as a function of direction. Shown here is the Brillouin zone for a face‐ centered cubic lattice. It is a truncate From the k -vector table there is a link to the corresponding Brillouin-zone figure. Each table consists of two main parts. The first two columns under the heading ' k -vector description' refer to the description of k -vectors found in Tables 3.9 and 3.11 of CDML [2]. It consists of labels of k-vectors (Column 1) and their parameter. Because of translational symmetry, electromagnetic fields confined within 2D periodic optical structures can be represented within the first Brillouin zone (BZ). In contrast, the wavevectors of scattered electromagnetic fields outside the lattice are constrained by the 3D light cone, the free-photon dispersion in the surrounding medium. Here, we report that light-cone surface lattice.

In a TB approximation including only first nearest neighbor s-orbital, the band structure, i.e. the electron energy dispersion in the Brillouin zone of the crystal, is given by (1) where ss is the overlap integral between s-orbitals, R is the translation vector of the lattice, and k is the crystal momentum— the quantum number for periodic systems 8. There are several pages where you can find scripts/simulations to generate the first Brillouin zone for square and hexagonal 2D lattices. I wonder if there is a tool to generate the Brillouin for other 2D lattices like the tiles presented here and here. PS: I am aware that not all the tiles can be used to represent a 2D lattice as they don't. As a result, the first Brillouin zone is often called simply the Brillouin zone. (In general, the n-th Brillouin zone consists of the set of points that can be reached from the origin by crossing exactly n − 1 distinct Bragg planes.) A related concept is that of the irreducible Brillouin zone, which is the first Brillouin zone reduced by all. st st 5. Sketch the first Brillouin Zone (1 BZ) for a 2D square of real lattice with a = b= 28. Draw the incident wave vector, Ko, scattered wave vector, K and vector G at boundary of the 1 BZ to show the (0, -1) is the valid lattice point for constructive interference Construction of brillouin zones • The brillouin zones are constructed from the planes which are the perpendicular or bisectors of all reciprocal lattice vectors • The first zones is the smallest volume about the origin enclosed by these planes • The second zone is the volume between the first zone and next set. 9

Constructing **Brillouin** **Zones**. I read one paper about **Brillouin** **Zones** that said they are a significant feature of crystal structures and their constructing for a two dimensional lattice is easier than in a three dimensional lattices. with two dimensional square lattice, **Brillouin** **Zones** constructed from the perpendicular bisectors of the vectors. There are also second, third, etc., Brillouin zones, corresponding to a sequence of disjoint regions (all with the same volume) at increasing distances from the origin, but these are used less frequently.As a result, the first Brillouin zone is often called simply the Brillouin zone.In general, the n-th Brillouin zone consists of the set of points that can be reached from the origin by. Reduced Brillouin zone Since the crystal momentum is only well defined modulo G, we can limit the momentum to the first Brillouin zone (e.g. for a 1D system,-π/a≤k <π/a) without loss of information. We can plot the eigen-energy ϵn,k as a function of k in the first Brillouin zone. This way of plotting ϵn,k is known as the reduced Brillouin. Volume of Brillouin zone show that the volume of the first. Need more help! Volume of Brillouin zone show that the volume of the first Brillouin zone is (2π) 3 /V c, where V c is the volume of a crystal primitive cell. Recall the vector identity (e x a) x (a x b) = (e ∙ a x b)a Brillouin zone: | | ||| | The reciprocal lattices (dots) and corresponding first... World Heritage Encyclopedia, the aggregation of the largest online encyclopedias available, and the most definitive collection ever assembled

Brillouin zone and Léon Brillouin · See more » Locus (mathematics) In geometry, a locus (plural: loci) (Latin word for place, location) is a set of all points (commonly, a line, a line segment, a curve or a surface), whose location satisfies or is determined by one or more specified conditions first Brillouin zone of this 1D lattice Lecture 5 6 5.1.1 Brilloiun Zone in 1D: extended, reduced and repeated Extended Reduced Repeated BZ boundaries. 3 Lecture 5 7 Reduction to the first Brillouin zone This general demand of periodicity implies that the possible electron states ar This Demonstration considers the construction of the Brillouin zone (BZ) -bands electronic dispersion relations for a 2D honeycomb crystal lattice of graphene under the tight binding (TB) approximation. Plots are shown for the electron energy dispersion for and *-bands in the first and extended Brillouin zones as contour plots at equidistant. Numerical integrations over the Brillouin zone (BZ) occur in several contexts, such as the sum of single particle energies. Care must be taken when performing this integration. In practice the first BZ is divided into a uniform mesh of k-points parallel to the three primitive reciprocal lattice vectors,. Note: Questaal works with dimensionless reciprocal lattice vectors

Lecture 4 Jan 16 2013 Primitive cell Primitive cellWigner-Seitz cell (WS) First Brillouin zone The Wigner-Seitz primitive cell of the reciprocal lattice is known as the first Brillouin zone. 29 pages, published by , 2015-05-04 06:04:02 . Tags: cel In order to interpret optical transitions observed in CER, PR, and PzR spectra, the electronic band structure for the four crystals has been calculated from the first principles within the density functional theory for various points of Brillouin zone including K and H points The first Brillouin zone is the locus of points in reciprocal space that are closer to the origin of the reciprocal lattice than they are to any other reciprocal lattice points (see the derivation of the Wigner-Seitz cell ). Another definition is as the set of points in k -space that can be reached from the origin without crossing any Bragg.

Abstract First Brillouin zone-centre phonon frequencies, elastic stiffness and mechanical properties of the Ln2Hf2O7 [Ln: La, Nd, Sm and Eu] pyrochlore structure were predicted by using an eight parameter bond-bending force constant model. One of the preliminary results of our study is that all the examined compounds are mechanically stable, and the elastic stiffness constants, and bulk and. Show that the volume of the first Brillouin zone is equal to (2π)3/Vc, where Vc is the volume of the crystal primitive cell. Note that the volume of the Brillouin zoneis equal to the volume of the primitive parallelepiped in Fourier space and recall thevector identity (c×a)×(a×b) = (c·a×b)a. 6 Brillouin Zone Sampling The choice of k-points used in a DFPT calculation deserves careful consideration. In a single-point calculation, the space group symmetry is used to determine an irreducible set from a Monkhorst-Pack grid of k-points [ 60 ]