What is Brillouin zone: Definition and 32 Discussions

In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. In the same way the Bravais lattice is divided up into Wigner–Seitz cells in the real lattice, the reciprocal lattice is broken up into Brillouin zones. The boundaries of this cell are given by planes related to points on the reciprocal lattice. The importance of the Brillouin zone stems from the description of waves in a periodic medium given by Bloch's theorem, in which it is found that the solutions can be completely characterized by their behavior in a single Brillouin zone.
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 plane. Equivalently, this is the Voronoi cell around the origin of the reciprocal lattice.

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 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 of the symmetries in the point group of the lattice (point group of the crystal).
The concept of a Brillouin zone was developed by Léon Brillouin (1889–1969), a French physicist.

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  1. Rzbs

    I Electron scattering in the Brillouin zone boundary

    I what to know what is electron scattering in Brillouin zone boundary? What exactly happen for electron in Brillouin zone boundary; what happen for it in real space and what happen for it in reciprocal space? And is electron scattering from a Brillouin zone boundary could be a source for...
  2. phzrosmary

    A Raman spectroscopy and the phonon confined in the Brillouin zone

    Hello everyone, I need some confirmation on something: As far as I understood, the raman spectroscopy measures the inelastic scattering of a photon in a medium through the absorption or the emission of a phonon in the medium. The energy and the momentum is conserved...
  3. J

    How to think about a Brillouin Zone?

    Isn't a Brillouin zone a Wigner Seitz cell in reciprocal space? Is it just a collection of wave vectors? Will you have Brillouin zone boundaries in many different places in your real space crystal, and hence standing waves there? What causes the standing waves at the zone boundaries? Isn't a...
  4. V

    Is first Brillouin zone the same as Wigner-Seitz cell?

    Homework Statement Not a homework question, but I am attempting to understand what exactly the first Brillouin zone is. Homework EquationsThe Attempt at a Solution From my textbook, what I'm gathering is that one constructs the first Brillouin zone by constructing a "Wigner-Seitz" type cell in...
  5. T

    A Why is the Brillouin zone a torus?

    Recently, topological concepts are popular in solid state physics, and berry connection and berry curvature are introduced in band theory. The integration of berry curvature, i.e. chern number, is quantized because Brillouin zone is a torus. However, I cannot justify the argument that...
  6. M

    How big should a defected supercell be? DFT

    When studying defects using plane-wave basis density functional theory, it is necessary to ensure that the size of the supercell in which the defect is located is large enough to ensure that there is no interaction between the defect in question, and the periodically repeated defects that are a...
  7. S

    What is the first Brillouin zone and how do I construct it?

    HI, I read the other threads in this forum but I still didn't understand (and I have read Aschroft and Mermin as well as Kittel). So, could you pleae explain to me (preferably in a VERY detailed manner) what is the first Brillouin zone and how do I construct it? Thank you :)
  8. ZetaOfThree

    Volume of Brillouin zone is the same as Fourier primitive parallelepip

    In Kittel's solid state text, problem 2.3, he says that the volume of the Brillouin zone is the same as a primitive parallelepiped in Fourier space. Somehow I can't see why this is true. Can someone help me see why this is true? Also, is the same relationship true between Wigner-Seitz cells and...
  9. P

    What Is the Brillouin Zone in Solid State Physics?

    How can I define brillouin zone?
  10. S

    K vector in first brillouin zone

    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 talk about K vectors "living" in the first Brillouin Zone, when these vectors cannot be expressed on the vector form of reciprocal space ( r*=ha*kb*+lc* ...
  11. F

    Understanding Brillouin Zones - What do they represent physically?

    Hey all, Quick question about BZ's and it's probably a really dumb one. Solid State isn't my favorite class... What does it physically represent? Like is the area of the BZ divided by the area of a k-state (2π/L) equal to the number of charge carriers in the system? So if there was an...
  12. F

    Acoustic Wave Velocities in Brillouin Zone

    Acoustic Wave Velocities in Brillouin Zone - Phonon Spectrum of Ge Homework Statement Acoustic Wave Velocities in Brillouin Zone - Phonon Spectrum Diagram The exact problem I'm stuck on is Q3c on this exam paper. I have included an image of the problem below. I haven't had any trouble up...
  13. L

    Solid State - Phonons at Brillouin Zone Boundary

    Homework Statement Homework Equations {3.9b} A[2\mu -m\omega ^2 ]=2\mu Bcos(\frac{ka}{2}) B[2\mu -M\omega ^2 ]=2\mu Acos(\frac{ka}{2}) The Attempt at a Solution All I can think of is setting k =\frac{\pi}{a} so that B[2\mu -M\omega ^2 ]=A[2\mu -m\omega ^2 ] solve for omega...
  14. R

    Finding Distances to Brillouin Zone Borders from Origo

    Hi, i have problems understanding the Brilluoin zone to the bcc lattice First, i wonder what the distances are, from the origo to the brillouin zone plane borders. I firts thought it would be half the distance of the basis vectors (mine are 2pi/a, where a is my lattice constant, BUT its only...
  15. N

    Why are the K points in the Brillouin zone of graphene inequivalent?

    The Brillouin Zone (BZ) of both graphene and the (111) surface of metals like Ag(111) eihibit a hexagon, but I wonder why the BZ of graphene has two inequivalent ponits K and K', while the K point of BZ of the Ag(111) is equivalent. Thank you in advance!
  16. T

    Calculating Energy Levels for Lattice Using Analytic Continuation?

    When energy levels for a lattice are constructed the Bloch wave vector is evaluated along the edges of the irruducible zone. Like \Gamma - X - M path for a square lattice. I wonder why the calculation is NOT performed for values within the zone? And how the energy corresponding to an...
  17. I

    Group velocity at Brillouin zone boundary

    I am working on an assignment here; A linear chain with a two-atom primitive basis, both atoms of the same mass but different nearest neighbor separation and thus different force constants. I have made a rigorous calculation in order to find the dispersion relation ω(k), with extensive...
  18. D

    Ewald sphere vs Brillouin Zone

    Is it correct to say that the Ewald sphere and Brillouin Zone are both representations of k'=k+G? I'm comfortable with the construction of the Ewald sphere, but don't quite see how a BZ represents k'=k+G. Can anyone explain how the construction of a BZ represents k'=k+G and whether it...
  19. R

    Brillouin Zone Size: Debate Settled

    Hi guys, Me and a few of my coursemates are revising for a solid state exam but have hit a problem. Is the area of every 2D brillouin zone (independent of lattice type) (2Pi/a)^2? For a square lattice in 2D real space with lattice constant, a, the reciprocal lattice vectors can easily be found...
  20. K

    First Brillouin Zone: Unravelling Why [-pi/a,pi/a)

    Hey all, I have a question with first Brillouin zone. It's: It says that it's enough to consider values of K within the first brillouin zone, K= [-pi/a,pi/a). why is this easy to see, and why don't we choose the interval K= (0 , 2pi/a) instead? Why is the point K=pi/a not included (open...
  21. P

    Trying to understand the Brillouin zone

    I am having a really hard time understanding how the Brillouin zone relates to electron states and have a couple of questions that might help clear it up for me. For a band structure like this: https://wiki.fysik.dtu.dk/gpaw/_images/silicon_banddiagram.png I know that the different...
  22. J

    First brillouin zone construction

    Hi All, Is there any software or program to show the first brillouin zone and the miller index of zone boundary plane ? I use Mercury, but there is no such function.
  23. T

    Brillouin Zone & K Point Explained

    what is Brillouin zone and K point?
  24. R

    Brillouin zone: reciprocal lattice

    Can somebody explain me how can we visualise reciprocal lattice of a crystal lattice. Also what do we mean by wavevectors of a reciprocal lattice. What is its physical significance?
  25. P

    Solid State in tight binding approximation, Brillouin zone, Fermi

    Hello, In few days, I have an examination, and I still have some dark zone in my head! If somebody could help me by giving me some advices/answers/way of reflexion/books to consult, it could be very great! Here is my questions: How to determine energy levels and wavefunction of the...
  26. P

    Brillouin Zone of BCC lattice?

    I'm answering a question where it becomes necessary to know the closest face of the BZ in a bcc structure. The answer is given as +/- (2*pi) / (sqrt(2)*a) where a is the cubic lattice parameter. I would have thought the Answer would have been sqrt(3)*a / 4. Where does the pi come from...
  27. C

    Solid state - Energy of electron in Brillouin zone

    Homework Statement Using geometrical arguments or otherwise, derive how the energy of an electron in the second Brillouin zone may be less than the energy of an electron in the first zone. [3] Homework Equations The Attempt at a Solution I'm thinking this has something to do...
  28. A

    Wigner-Seitz = Brillouin Zone?

    I am confused about the relation between the Wigner-Seitz cell and the first Brillouin zone. My teacher explained that to find the Wigner-Seitz cell in real space, one draws lines between the lattice points and connects the perpendicular bisecting planes. This constructs the volume nearer to...
  29. C

    How Does the Brillouin Zone Influence Electron Wavefunctions in Solids?

    Hi, I'm following the book of Kaxiras on solid state physics and I'm a bit confused about Brillouin zone and solving Schroedinger equation in BZ. Please let me to write the logical statements I've understood (maybe correct ot not): 1. Crystals are made up of atoms located periodically in...
  30. C

    Copper oxide plane and brillouin zone

    I am having trouble understand the crystal structure or copper oxide and what the first brillouin zone looks like.
  31. N

    Help, first Brillouin zone and K points

    As it's said, the number of k point in a first Brillouin zone is determined by the number of lattice sites. For exmaple, a 2-d n by m square lattice, its 1st BZ contains m by n k values and I assume these k values are equally separated. My question is that how the layout of k point in the 1st...
  32. N

    Why only first brillouin zone?

    I realize this a fundamental flaw in my understanding but I can't seem to get my head around it. I understand that bragg reflection occurs at the zone boundary and so electron wavevectors are diffracted back into the 1st brillouin zone, but what is the justification for only considering the...