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. P

    How to evaluate the strain-induced change in reciprocal space?

    In real space in crystal, strain-induced change can be written as follows: {\bf r'}=(1+\epsilon)\cdot {\bf r} But there is no way to evaluate the strain-induced change in reciprocal space. Can one calculate the strain-induced change in high-symmetry point in quasi-momentum space? I check almost...
  2. P

    Math Table or Method of Solving an infinite sum of reciprocal powers

    Is anybody aware of how to solve the following infinite sum: \sum\frac{1}{n^2} for all positive odd integers? Is this the sort of thing you just look up in a math table or solve? If math table, do I need a "sum of reciprocal powers" table or a "riemann zeta function" table? If...
  3. N

    Fourier of Basis Points (Basis in Reciprocal space) (Convolution Theorem)

    I came across this question where there is a FCC lattice. It states that the lattice is a convolution of the simple cubic (whose reciprocal lattice is itself) with a basis (that consists of 2 points). When finding the reciprocal of this BCC lattice, FourierTransform(BCC) =...
  4. U

    Circle Series Reciprocal: Taking the Reciprocal of an Infinite Series

    I understand how this works: \cos x = \frac{1}{0!} - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \frac{x^8}{8!} - \frac{x^{10}}{10!} + \ldots But what about this? \frac{1}{\cos x} = \frac{1}{0!} + \frac{x^2}{2!} + \frac{5x^4}{4!} + \frac{61x^6}{6!} + \frac{1385x^8}{8!} +...
  5. H

    How Can Optimal Bounds for Sum of Reciprocals Between Two Numbers Be Determined?

    given any two numbers a,b and an upper and lower bound for the sum of reciprocals of a certain class of integers between a and b, without any direct calculation how can optimal upper and lower bounds for the number of terms in the sum be found
  6. R

    Drawing Reciprocal Lattice: Methods & Steps

    What are the methods or steps of drawing reciprocal lattice?
  7. 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?
  8. M

    Understanding Reciprocal Lattices for Beginners

    here is a question on reciprocal lattices that I am stuck on for a simple cubic lattice, the unit cell is defined by a1=a(1,0,0) a2 = a(0,1,0) a3 = a(0,0,1), demonstrate that the reciprocal lattice of its reciprocal lattice is the original crystal lattice.From what I've found, i think the...
  9. P

    Prove Reciprocal Cubic Lattice of Cubic Lattice is Also Cubic

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  10. W

    What is the nature of the origin of reciprocal space, please?

    Hi, all I got a question about the origin of reciprocal space..what is the physical nature of the special point? Does that originate from diffraction? or Is that a diffraction points? We know that a 'normal' reciprocal point designates a group of parallel plane. How about the origin of...
  11. P

    The Reciprocal of a Derivative

    I have seen this sort of thing over the past few years and it is bothering me, something like this \frac{ds}{dx}=\frac{1}{\frac{dx}{ds}} But it seems to me that this sort of thing only works in certain situations. For example, take s(x) to be s(x)=x^{2} so that \frac{ds}{dx}=2x Now to get...
  12. N

    Calculating Residues of Reciprocal Polynomials

    I have need to calculate the residues of some functions of the form \frac{f(x)}{p(x)} where p(x) is a polynomial. To be more specific I have already calculated the 2 residues of \frac{1}{x^2+a^2}. That one was quite easy. Now I'm asked to calculate the residues of...
  13. I

    The reciprocal expectation value

    I am aware of the expectation value \left\langle\ r \right\rangle. But I was wondering what is physically meant by the expectation value: \left\langle\frac{1}{r}\right\rangle The reason I am asking is because calculating this (reciprocal) expectation value for the 1s state of hydrogen, one...
  14. U

    Showing A^-1 has eigenvalues reciprocal to A's eigenvalues

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  15. icystrike

    Sum of reciprocal of squares <Logic>

    Homework Statement https://www.physicsforums.com/attachment.php?attachmentid=21977&stc=1&d=1258886072 I think my proof is lousy and may be wrong. Please help me with it (= Thanks in advance Homework Equations My proof is of below. The Attempt at a Solution
  16. N

    Reciprocal Lorentz Contraction Mismatch

    Consider two groups of 3 spaceships each. The groups are numbered 1 and 2. The ships in group 1 are numbered 1A, 1B and 1C. The ships in group 2 are numbered 2A, 2B and 2C. In each group, ship A leads, followed directly in line, at fixed and equal proper intervals (as measured in each line's...
  17. U

    Symmetry elements and reciprocal lattices.

    Hi, There are some points I really want to clear up in this topic...I promise to finish my chain of doubts as quicly as possible! I'll put in my first questions... 1. Rotoinversion is a combination of inversion and rotation-- often it ends up as having the same effect on the crystal as...
  18. T

    Understanding Reciprocal Lattice Vectors and Orthogonality in Primitive Lattices

    f(\vec{r}) = f(\vec{r}+\vec{T}) \vec{T}= u_{1} \vec{a_{1}} + u_{2} \vec{a_{2}}+u_{3} \vec{a_{3}} u_{1},u_{2},u_{3} are integers. f(\vec{r}+\vec{T})= \sum n_{g} e^{(i\vec{G}.(\vec{r}+\vec{R}) )}= f(\vec{r}) e^{i\vec{G}.\vec{R} }= 1 \vec{G}.\vec{R} = 2\pi m we call...
  19. S

    Reciprocal of a cubic function

    Is it possible to have the reciprocal of a cubic function that does not have any vertical asymptotes?
  20. M

    Reciprocal of a quadratic function - math help

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  21. M

    Math problem involving reciprocal of linear functions

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  22. N

    Planes perpendicular to reciprocal vectors problems

    Homework Statement Given the shortest reciprocal lattice of fcc is (2PI/a)(+-x+-y+-z). How can i show that (111) is perpendicular to the shortest reciprocal lattice? And how to show that other planes such as (001),(010),(110) is not perpendicular to the shortest reciprocal lattice...
  23. M

    Why the name Reciprocal lattice ?

    Why the name Reciprocal lattice ?? Is it because the dimensions get bigger & we are able to understand the structure on a macro level & removing the infinities that occur in h, k, l plane system ??
  24. Beer w/Straw

    Reciprocal property inequality

    1. How to solve -1<1/(x+1)<2 2. Now if I switched both sign and took the reciprocal such as 1/-1>(x+1)/1>1/2 I get the right answer: x<-2 and x>-1/2. 3. Without using the reciprocal property would be lengthy and confusing, but I have very little and contradictory information found...
  25. R

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  26. N

    How Does the Reciprocal Lattice Relate to Real Space Lattice Vectors?

    Hey folks, Here's my problem: Knowing that for reciprocal lattice vectors K and real space lattice vectors R: and using the Kronecker delta: I need to prove b1, b1, b3 as shown http://www.doitpoms.ac.uk/tlplib/brillouin_zones/reciprocal.php" : I understand that for the...
  27. L

    Parallel resistanee is reciprocal of the sum?

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  28. S

    Vectors As Geometric Objects And Reciprocal Basis?

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  29. S

    Clueless about Reciprocal Lattice

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  30. B

    Estimating the sum of reciprocal powers using a given fourier series

    Homework Statement Let f(x) be defined by the following Fourier series for \left|x\right|: f(x) = \frac{\pi}{2} - \frac{4}{\pi}\sum_{1,3,...}\frac{cos\left(nx\right)}{n^{2}} Show that \sum_{1,3,...}\frac{1}{n^{2}} = \frac{\pi^{2}}{8} and \sum_{1,2,3,...}\frac{1}{n^{2}} =...
  31. W

    Understanding Reciprocal Space in Solid State Physics

    I am taking Solid State now and using Kittel as the textbook. Needless to say, I don't understand almost anything that's happening. I'm still stuck on Reciprocal Space here. If I have a lattice of atoms of spacing X, then in reciprocal space I get something like 2*pi/X spacing. My prof...
  32. B

    Understanding the Reciprocal Basis Problem

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  33. F

    Exploring Reciprocal Space in Crystallography

    Hello all, I don't understand some things concerning reciprocal space. I know how it appears from quantum mechanics (it comes from gas of free electrons model, then applying Fermi-statistics for it and solve Heisenberg equation and get some kx,y,z which have to be integer. After that we call the...
  34. F

    Covariant vectors vs reciprocal vectors

    If there is a contravariant vector v=aa+bb+cc with a reciprocal vector system where [abc]v=xb×c+ya×c+za×b would the vector expressed in the reciprocal vector system be a covariant vector? Is there any connection between the reciprocal vector system of a covariant vector and a...
  35. E

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  36. N

    Difference Between Inverse & Reciprocal

    Hello, can someone tell me the difference between reciprocal and the inverse. I have looked up the defenition and when I think I get a deep understanding I get confused. Thanks for all the help. Naicidrac
  37. C

    Finding the Antiderivative of a Reciprocal: Step-by-Step Guide

    1. Find the antiderivative of 1/(x+3)^2 Okayy i knwo this is an easy problem but i COMPLETELY forget how to do it. Ive tried using partial fractions but it doesn't seem to be working. I just need to know how to start the problem then i should be alright from there. Thank you!
  38. D

    How do I find the reciprocal primitive vector for a lattice?

    Homework Statement Here's a problem I'm having: The primitive vectors of the hexagonal lattice are: a1 = ck a2 = (a/2)i + ([a√3]/2)j a3 = (-a/2)i + ([a√3]/2)j Find the primitive vectors of the reciprocal lattice, i.e. b1, b2, and b3. Homework Equations I do know that the...
  39. I

    Band diagram in real space vs reciprocal space

    Hi, can anybody rigurously explain the relationship between the band diagram in k-space (I think I understand this one) and the diagram in real space (as is often used to explain the p-n-junction).
  40. T

    Reciprocal lattice and Fourier series

    First off, this is not a homework problem. I was reading Charles Kittel solid states book on Chapter 2, equation 3: electron number density, n(x), expanded in a Fourier series: n(x) = n_0 + \sum_{p} [C_p cos(\frac{2\pi p x}{a}) + S_p sin(\frac{2\pi p x}{a})] From this expansion, wouldn't...
  41. S

    Determining the Line of Symmetry of a Reciprocal Equation.

    Problem: How do you determine the line of symmetry of a reciprocal equation? Solution: For example, I'll graph the reciprocal function Y=1/(x+2) ^Just a quick sketch And the equation of the line of symmetry is simply -(x+2), which can be seen here: ^Also a quick sketch By adding a...
  42. J

    What is the method for finding the reciprocal function of a quadratic function?

    reciprocal functions ?? hey is this true with finding reciprocal functions : f(x) initial function , g(x) reciprocal function of f(x) if f(x) = a x^n then g(x) = ( xroot(n, x) ) / ( xroot(n, a) but now if f(x) = ax²+ bx + c how do i find its reciprocal function ?
  43. malawi_glenn

    What are the properties of reciprocal lattice vectors?

    Homework Statement "What does a reciprocal lattice vector represent in the real lattice?" The Attempt at a Solution The answer to that one is that the reciprocal lattice represent all possible k-values for the incoming radiation to be contained in the real lattice. Hence a...
  44. Z

    Visualization of a reciprocal lattice

    I have been learning kittel's solid physics, But find it hard to have a firm grab of what a reciprocal lattice is like and can't understand it's relationship with e original lattice.. Is there a picture that draws a lattice n its reciprocal lattice into e same picture? so that i can visualize...
  45. malawi_glenn

    What Is the Shortest Reciprocal Vector for a BCC Lattice?

    Homework Statement Find the shortest reciprocal vector G, given below, v_1,...,v_3 are integers. \vec{G} = \frac{2 \pi}{a}\left( (v_2 + v_3 )\vec{x} + (v_1 + v_3 )\vec{y} + (v_1 + v_2 )\vec{z} \right) Homework Equations x,y,z are ortonogal, length 1 l = l(v_1, v_2, v_3) =...
  46. D

    Reciprocal Derivative Identity

    Hi. I was just wondering, how can i prove the following identity: \frac{{dy}}{{dx}}\frac{{dx}}{{dy}} = 1 Its nothing that I am required to know, but i was just curious, so for all i know, it may be way out of anything that i can mathematically comprehend. The best I've been able to...
  47. T

    Reciprocal Salt Pairs: What Does It Mean?

    I understand that NaCl and NH4HCO3 are reciprocal salt pairs. What does this mean?
  48. J

    Understanding 3D Si Dispersion Relations & Reciprocal Lattice Vectors

    I am trying to understand 3D Si dispersion relations and reciprocal lattice vectors. My confusion is that when I look at dispersion relations the wave vector typically is normalized from 0 to 1 by a/2pi. I thought the edge of the first BZ was pi/a. Is this correct or is it 2pi/a for a diamond...
  49. P

    Principal of Superposition + Maxwell Reciprocal Theorem

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  50. D

    How can I sketch the reciprocal of a function with poles at x=-2 and x=2?

    Given the following graph: http://img245.imageshack.us/img245/2395/scan0001ou4.gif How can i sketch the reciprocal of that function? There are poles at x=-2 and x=2, so it means its reciprocal will have roots at -2 and 2 right? But that's not really enough information to compose a full...
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