What is Reciprocal: Definition and 165 Discussions

In physics, the reciprocal lattice represents the Fourier transform of another lattice (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is usually a periodic spatial function in real-space and is also known as the direct lattice. While the direct lattice exists in real-space and is what one would commonly understand as a physical lattice, the reciprocal lattice exists in reciprocal space (also known as momentum space or less commonly as K-space, due to the relationship between the Pontryagin duals momentum and position). The reciprocal lattice of a reciprocal lattice is equivalent to the original direct lattice, because the defining equations are symmetrical with respect to the vectors in real and reciprocal space. Mathematically, direct and reciprocal lattice vectors represent covariant and contravariant vectors, respectively.
The reciprocal lattice plays a very fundamental role in most analytic studies of periodic structures, particularly in the theory of diffraction. In neutron and X-ray diffraction, due to the Laue conditions, the momentum difference between incoming and diffracted X-rays of a crystal is a reciprocal lattice vector. The diffraction pattern of a crystal can be used to determine the reciprocal vectors of the lattice. Using this process, one can infer the atomic arrangement of a crystal.
The Brillouin zone is a Wigner-Seitz cell of the reciprocal lattice.

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  1. Wrichik Basu

    I The missing factor of 2π in reciprocal lattice calculations

    ##\require{physics}## Recently, I wet my feet in X-ray diffraction a bit more than what is usually covered in standard solid state physics textbooks at the undergrad level, like Kittel. Two good books that I chanced upon included Christopher Hammond, The Basics of Crystallography and Diffraction...
  2. brotherbobby

    Inequality involving a reciprocal - where's the mistake?

    Problem Statement : Solve the inequality : ##\left( \dfrac{1}{3} \right)^x<9##. Attempts: I copy and paste my attempt below using Autodesk Sketchbook##^{\circledR}##. The two attempts are shown in colours black and blue. Issue : On checking, the first attempt in black turns out to be...
  3. U

    I Prove series identity (Alternating reciprocal factorial sum)

    This alternating series indentity with ascending and descending reciprocal factorials has me stumped. \frac{1}{k! \, n!} + \frac{-1}{(k+1)! \, (n-1)!} + \frac{1}{(k+2)! \, (n-2)!} \cdots \frac{(-1)^n}{(k+n)! \, (0)!} = \frac{1}{(k-1)! \, n! \, (k+n)} Or more compactly, \sum_{r=0}^{n} (...
  4. karush

    MHB Find Half a Number's Reciprocal Increased by Half its Reciprocal

    $\tiny{3.1.2}$ The reciprocal of half a number increased by half the recipical of the number is $\dfrac{1}{2}$ $\begin{array}{rl} n= & \textit{the number} \\ \\ \dfrac{n}{2}= &\textit{half the number}\\ \\ \dfrac{2}{n} = &\textit{the reciprocal...
  5. Mark44

    Fast reciprocal square root algorithm

    I ran into an interesting video on Youtube yesterday, about a fast way to compute the reciprocal of the square root of a number. I.e., to compute this function: ##f(x)= \frac 1 {\sqrt x}## The presenter is John Carmack. If you search for "Fast Inverse Square Root — A Quake III Algorithm" you'll...
  6. AN630078

    Solving Reciprocal Trigonometric Equation cot^2θ+5cosecθ=4

    cot^2θ+5cosecθ=4 cot^2θ+5cosecθ-4=0 cosec^2θ+5cosec-4-1=0 cosec^2θ+5cosec-5=0 Let u=cosecθ u^2+5u-5=0 Solve using the quadratic formula; u=(-5± 3√5)/2 u=(-5+ 3√5)/2=0.8541... Substitute cosecθ=u Therefore, cosecθ=0.8541 1/sinθ=0.8541 sinθ=1/0.8541=1.170... which is not true since sin x cannot be...
  7. J

    I Expected value of a variable and its reciprocal

    If E(X1)>E(X2), does it mean that E(1/X1)<E(1/X2)? Thanks!
  8. patric44

    Number of k-cells for a 1D lattice in reciprocal space

    hi guys our solid state professor sent us a work sheet that included this example : i solved it not sure its correct tho : is it that simple , or this is not the right approach for it ?
  9. patric44

    Some questions about reciprocal lattice vectors

    hi guys our solid state physics professor introduced to us this new concept of reciprocal lattice , and its vectors in k space ( i am still an undergrad) i find these concepts some how hard to visualize , i mean i don't really understand the k vector of the wave it elf and what it represents...
  10. S

    I Understanding the Derivation of Reciprocal Lattice Basis from Equations 5 and 6

    may someone explain to me or show me the steps of how equations 7a, 7b, 7c were determined from equations 5 and 6
  11. dRic2

    I Are electron bands symmetric in the reciprocal space?

    Hi, in the lecture notes my professor gave us, it is stated that, due to Kramers theorem, the energy in a band must satisfy this condition: $$E(-k) = E(k)$$ But, judging from actual pictures of band structures I don't find this condition to be true. Here's a (random) picture I guess it looks...
  12. H

    A Diffusion equation and a system of equations with reciprocal unknowns?

    So the normal diffusion equation looks like \frac{\partial c}{\partial t} = k\frac{\partial}{\partial x}\left(\frac{\partial c}{\partial x}\right) I know how to get explicit and implicit solutions to this equation using finite differences. However, I am trying to do the same for an equation of...
  13. P

    A Connection between Reciprocal Space and Cotangent Space

    [Moderator's note: Link to reference protected by copyright removed.]
  14. T

    Reciprocal lattice for FCC and diffraction peaks

    My issue is more with the steps to approach rather than the calculations. I was just wondering if someone could confirm my approach to be correct. As it asking for the reciprocal lattice of an FCC I assume this would mean I need to use the points on the BCC to calculate the geometrical...
  15. W

    I Debye model and reciprocal space

    Hi everyone, I need a little help understanding how periodic reciprocal space applies to the Debye model for solids. Many thanks in advance! If we start with the general derivation of a dispersion relation for a 1D system, with atoms coupled by springs, one gets the following relation $$\omega...
  16. karush

    MHB Finding the Solution to a Differential Equation with Reciprocal Relationship

    $\textsf{Find the solution of:}$ $\displaystyle\frac{dy}{dx}=\frac{1}{e^y-x}$ ok i kinda don't know what the first step is was going to multiply both sides by the denominator but
  17. G

    Maximum Wavelength and Scattering Angle for Triangular Lattice

    Homework Statement A triangular lattice of lattice spacing ##a=2 ## angstroms is irradiated with x-rays at time zero of wavelength 20 angstroms at an incident angle of ##\alpha =135##. 1) What is the maximum wavelength of the incident x-rays? 2) What is the scattering angle ##\Omega## for...
  18. G

    Electron Diffraction Intensity and Reciprocal Lattic

    Homework Statement Low-energy electron diffraction (LEED) experiments are carried on to study a deposition of argon (Ar) and xenon (Xe) on the surface of a graphite single crystal. In the regime of vapor pressure considered, 75% of Ar and 25% of Xe are adsorbed on the (hexagonal) crystalline...
  19. B

    Reciprocal Function is Unbounded

    Homework Statement Let ##f : (0,\infty) \to \Bbb{R} - \{0\}##. If ##\frac{1}{f}## is unbounded on every interval containing ##x=3##, will ##\lim_{x \to 3} f(x) = 0##? Homework EquationsThe Attempt at a Solution Consider the function $$f(x) = \begin{cases} 1, & x \in \Bbb{Q} \cap (0,\infty)...
  20. RJLiberator

    Reciprocal Theorem proof in 3 variables

    Homework Statement w is a function of three variables x, y, and z. Prove that \frac{\partial w}{\partial x}_{y,z} = \frac{1}{\frac{\partial x}{\partial w}}_{y,z} Homework EquationsThe Attempt at a Solution w=w(x,y,z) dx = \frac{\partial w}{\partial x}_{y,z}dx +\frac{\partial w}{\partial...
  21. S

    I How can I obtain the reciprocal lattice of graphene?

    I have a question about reciprocal lattice of graphene. When we see LEED pattern, we can know that reciprocal lattice of graphene is honeycomb. But how can we know theorically that it is honeycomb? Hexagonal lattice or other bravais lattice has just lattice vectors which don`t contain baises. So...
  22. N

    I Real world process that results in reciprocal wavefunction

    Two Questions from a newbie. A) Is there a easily implemented process or reaction that results in a particle with reciprocal wave function of input particle? B) Is there a easily implemented process or reaction that results in a particle A transferring it's wavenumber and angular frequency to...
  23. DaTario

    I How to obtain the inverse (reciprocal) of a line segment?

    Hi All, Which are the ways one can geometrically obtain, given a line segment AB with length x and an unitary segment OC, a line segment with length 1/x ? Straight edge and compass are allowed (also some auxilliary curve). Best wishes, DaTario
  24. Schaus

    Proving Reciprocal Identities: (secx+1)/(sin2x) = (tanx)/2cosx-2cos2x

    Homework Statement (secx+1)/(sin2x) = (tanx)/2cosx-2cos2x) Homework EquationsThe Attempt at a Solution Left Side ((1+cosx)/cosx)/2sinxcosx ((1+cosx)/cosx) x (1/2sinxcosx) cancel the a cosx from both to get (1/2sinxcosx) This is all I could manage with left side so I tried right side Right...
  25. 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...
  26. F

    The reciprocal relationship between frequency and period

    I was asked by a friend to explain why the frequency, ##f## and period, ##T## of a wave. The initial explanation I gave to them was as follows: Heuristically, the period of a wave is defined as ##T=\frac{\text{number of units time}}{\text{cycle}}##, and its frequency as ##f=\frac{\text{number...
  27. M

    What algebraic property can I use here?

    Homework Statement [/B] example problem: 5 = [(x)(4+x)] / (4-x) answer: 5 Homework Equations Unsure what to use. 3. The Attempt at a Solution Not sure what my professor did, but I thought that if i multiply by the reciprocal of something, I have to balance by multiplying the other side as...
  28. A

    I Why does (100) and (010) have the shortest reciprocal lattic

    Can anyone explain why the two planes (100) and (010) in the HCP structure have the shortest reciprocal lattice vectors? I mean it says {001}, but why is (001) not included?
  29. I

    MHB Prove Positive Fraction + Inverse is ≥2

    Problem: Prove that any positive fraction plus its inverse is greater than or equal to two. Proof: \frac{a}{b}+\frac{b}{a}\ge2 \frac{a^2+b^2}{ab}\ge2 {a^2+b^2}\ge2ab a^2+b^2 - 2ab\ge0 a^2 - 2ab + b^2\ge0 (a-b)^2\ge0 This is true for all a and b: Case 1: a>b\therefore a-b>0; (a-b)^2>0...
  30. 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...
  31. amind

    Sum involving reciprocal of binomial coeffients

    Homework Statement If $$a_n = \sum_{r=0}^{n} \frac{1}{\binom{n}{r}}$$ Find $$\sum_{r=0}^{n} \frac{r}{\binom{n}{r}}$$ in terms of an and n 2. The attempt at a solution Let $$f(x) =\sum_{r=0}^{n} \frac{x^r}{\binom{n}{r}}$$ Then, an = f(1). Observe that f'(1) is the required sum. I was thinking...
  32. sinkersub

    A: Reciprocal series, B: Laurent Series and Cauchy's Formula

    Problem A now solved! Problem B: I am working with two equations: The first gives me the coefficients for the Laurent Series expansion of a complex function, which is: f(z) = \sum_{n=-\infty}^\infty a_n(z-z_0)^n This first equation for the coefficients is: a_n = \frac{1}{2πi} \oint...
  33. ryanuser

    Cos(-pi/4) & Sec(pi/4): What's the Difference?

    Why 1 divided by cos(pi/4)=cos(-pi/4)? Is it wrong to say 1/cos(pi/4)=sec(pi/4)? Thanks
  34. B

    Classical Demonstration of Onsager reciprocal relations

    Hello everyone, I am working on the Onsager reciprocal relations, more precisely on the demonstration of those relations. I try to understand the Onsager original paper (1931) but it's really not easy (although he says that the examples are "extremely simple"). I was wondering if any of you...
  35. 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...
  36. B

    Non-uniqueness of the k-vector in Bloch state

    How to understand that Bloch wave solutions can be completely characterized by their behaviour in a single Brillouin zone? Given Bloch wave: \begin{equation*} \psi_{\mathbf{k}}(\mathbf{r}) = u_{\mathbf{k}}(\mathbf{r}) \exp (i\mathbf{k}\mathbf{r}) \end{equation*} I can write wavefunction for...
  37. N

    X''(t) required for reciprocal of X'(t) to be constant

    Hey all, was watching my downloads today and saw that my download speed was dropping at such a rate that the eta of the download was almost constant, so I was wondering what negative acceleration would be required for the eta to be constant, if time left is size/speed, speed is first derivative...
  38. G

    Reciprocal of the reciprocal lattice is the original lattice

    In my course we are currently studyinh Bravis lattices. We were told that the reciprocal of the reciprocal lattice is the original lattice. This is very easy to prove when given an example of a SC/BCC/FCC lattice, however, is there a formal proof for this?
  39. G

    How to find reciprocal lattice vectors

    So I know that the basis vectors of an FCC in a symmetric form are: a = \frac{a}{2}(\hat{x} + \hat{y}) b = \frac{a}{2}(\hat{y} + \hat{z}) c = \frac{a}{2}(\hat{x} + \hat{z}) And that the reciprocal lattice vectors are the basis vectors of the BCC cells. I'm having a hard time doing the...
  40. D

    Proving the reciprocal relation between partial derivatives

    If three variables x,y and z are related via some condition that can be expressed as $$F(x,y,z)=constant$$ then the partial derivatives of the functions are reciprocal, e.g. $$\frac{\partial x}{\partial y}=\frac{1}{\frac{\partial y}{\partial x}}$$ Is the correct way to prove this the following...
  41. F

    General Formula for the Reciprocal of a Sum of Reciprocals

    I know that the reduced mass, μ, of an object is: \mu = \frac{1}{\frac{1}{m_1} + \frac{1}{m_2}} \mu = \frac{m_1 m_2}{ m_1 + m_2 } But is there a general formula (or a simplified expression) for finding the value of: \frac{1}{ \frac{1}{x_1} + \frac{1}{x_2} + ... + \frac{1}{x_n} } ? Thank you.
  42. Chillguy

    Using Reciprocal to Determine Miller Indices

    If we know the reciprocal space basis of a BCC lattice b_1=\frac{2\pi}{a}(\vec{x}+\vec{y}),b_2=\frac{2\pi}{a}(\vec{z}+\vec{y}),b_3=\frac{2\pi}{a}(\vec{x}+\vec{z}) how do we go about finding the shortest reciprocal lattice vector and its corresponding miller index? To me all the constants in...
  43. N

    Why Is the Length of the Reciprocal Lattice Vector Equal to 2π/d_hkl?

    Hi. This is a very simple and stupid question: Why is the length of the reciprocal lattice vector ##G_{hkl}## equal to ##2 \pi / d_{hkl}##, where ##d_{hkl}## is the distance between the ##(hkl)## planes. Just like the length of the wave vector ##k## equals ##2 \pi / \lambda## I remember that...
  44. naima

    Maxwell's equation in reciprocal space

    I found this http://people.sissa.it/~benassi/capitolo1/node2.html in the vacuum the equations would be q B = q E = 0 ##q \times E = (\omega / c) B## ##q \times B = -(\omega / c) E## Is there a typo? there is no t derivative. If E = 0 B would have to be null. Has B to be allways orthogonal to E...
  45. SpiderET

    Is relativity reciprocal or directional?

    I was wondering whether is there some real evidence for directional relativity in this article or is this rather a pure speculation. Actually it sounds as common sense for me, that the relativity effects are taking place on the moving object and not on both objects reciprocally.
  46. P

    Reciprocal Lattices and Ewald Sphere: Solving for a* and c* in a Tetragonal Cell

    Homework Statement I have to draw a reciprocal lattice of a tetragonal cell with parameters a=3A and c=5A, a body-centred lattice. How do I find a* and c*? I also have to draw an Ewald sphere, and lamda=1.5. However, if I use my solutions (I think they're wrong, see below) I get something that...
  47. E

    MHB State the range of the reciprocal function?

    State the range of the reciprocal function of f(x) = - (x+3)^2 - 1. I'm not sure if I did this right. I wrote that y is above/equal to -1 and below/equal to 0. Is this correct? Also, how would you graph the reciprocal function of f(x) if there is no VA and only a HA?
  48. A

    Why is Reciprocal Space Used in Crystallography?

    Hello ! I have read several threads on this topic but I don't seem to fully understand the reason for using the reciprocal space in crystallography . Can anyone please provide more information on this subject ?