The fundamental property of a crystal is its triple periodicity and a crystal may be generated by repeating a certain unit of pattern through the translations of a certain lattice called the direct lattice. The macroscopic geometric properties of a crystal are a direct consequence of the existence of this lattice on a microscopic scale. Let us for instance consider the natural faces of a crystal. These faces are parallel to sets of lattice planes. The lateral extension of these faces depends on the local physico-chemical conditions during growth but not on the geometric properties of the lattice.
To describe the morphology of a crystal, the simplest way is to associate, with each set of lattice planes parallel to a natural face, a vector drawn from a given origin and normal to the corresponding lattice planes. To complete the description it suffices to give to each vector a length directly related to the spacing of the lattice planes.
As we shall see in the next section this polar diagram is the geometric basis for the reciprocal lattice. On the other hand, the basic tool to study a crystal is the diffraction of a wave with a wavelength of the same order of magnitude as that of the lattice spacings.
The nature of the diffraction pattern is governed by the triple periodicity and the positions of the diffraction spots depend directly on the properties of the lattice. This operation transforms the direct space into an associated space, the reciprocal spaceand we shall see that the diffraction spots of a crystal are associated with the nodes of its reciprocal lattice.
The reciprocal lattice is therefore an essential concept for the study of crystal lattices and their diffraction properties. This concept and the relation of the direct and reciprocal lattices through the Fourier transform was first introduced in crystallography by P.
Ewald Let abc be the basic vectors defining the unit cell of the direct lattice. The basic vectors of the reciprocal lattice are defined by: 2. Referring to Fig. From the definition of the reciprocal lattice vectors, we may therefore already draw the following conclusions:.
The basic vectors of the reciprocal lattice possess therefore the properties that we were looking for in the introduction. We shall see in the next section that with each family of lattice planes of the direct lattice a reciprocal lattice vector may be thus associated.
For practical purposes the definition equations 2. From relations 2. The two sets of equations 2. These relations are symmetrical and show that the reciprocal lattice of the reciprocal lattice is the direct lattice.
Let M be a reciprocal lattice point whose coordinates hkl have no common divider M is the first node on the reciprocal lattice row OMand P a point in direct space. We may write: 2. Let us look for the locus of all points P of direct space such that the scalar product should be constant.
Using 2. Since all numbers in the left hand side are integers, we find that C is also an integer. With each value of C we may associate a lattice plane and thus generate a set of direct lattice planes which are all normal to the reciprocal vector OM Fig. The distance of one of these planes to the origin is given by:. The lattice planes have, as expected, an equal spacing: 2. Equation 2. This is the fundamental relation of the reciprocal lattice which shows that with any node M of the reciprocal lattice whose numerical coordinates have no common divider we may associate a set of direct lattice planes normal to OM.
Their spacing is inversely proportional to the parameter along the reciprocal row OM. In order that the correspondence between direct and reciprocal lattice should be fully established, the converse of the preceding theorem should also be demonstrated.
This will be done in paragraph 2.
It is interesting at this point to give an interpretation to the reciprocal lattice points whose numerical coordinates have a common divider.
It only takes a minute to sign up. I don't understand some of the terminology in this question. I googled reciprocal vectors and got an article on reciprocal lattices, but I'm not sure if that is what they are talking about in this question. Does anyone have a link for a clear explanation? From Reciprocal Lattice. This wiki article would be a good start. Sign up to join this community. The best answers are voted up and rise to the top.
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I'm a little stumped by this one, even now that I understand the terminology. Which proof and solution do you mean? TZakrevskiy TZakrevskiy Of course, you can't legitimately carry out this cancellation but it does help to explain the name. Sign up or log in Sign up using Google.
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A concrete example for this is the structure determination by means of diffraction. As will become apparent later it is useful to introduce the concept of the reciprocal lattice. The initial Bravais lattice of a reciprocal lattice is usually referred to as the direct lattice.
The definition is fine so far but we are of course interested in a more concrete representation of the actual reciprocal lattice. Now we can write eq. This is a nice result. Is there such a basis at all? Therefore we multiply eq.
It remains invariant under cyclic permutations of the indices. Now we will exemplarily construct the reciprocal-lattice of the fcc structure.
The reciprocal lattice
One may be tempted to use the vectors which point along the edges of the conventional cubic unit cell but they are not primitive translation vectors. Those reach only the lattice points at the vertices of the cubic structure but not the ones at the faces.
Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. I am working in C and I am trying to figure out how to take the reciprocal of the vector velocity. But I cannot do this because I cannot take an int and divide it by a vector. I have tried to look for an answer to this, but I have not fared well Mathematically speaking, the reciprocal of a vector is not well-defined.
You can take the reciprocal of the magnitude of a vector, and you can create a new vector whose components are the reciprocals of the components of the original vector, but the notion of the reciprocal of a vector itself isn't meaningful. Learn more. How do you take the reciprocal of a vector? Ask Question. Asked 9 years, 6 months ago. Active 9 years, 6 months ago. Viewed 4k times.
Is this a custom vector-type under your control? The one from XNA? Something else?
The Reciprocal Lattice
What does the reciprocal of a vector mean? Gabe: I imagine the OP wants a new vector with the individual components being reciprocals of the original. Ani: I was thinking normal, but your guess is as good too. Active Oldest Votes. Vector2 Velocity Vector2 Reciprocal Reciprocal. Sqrt Math. Pow 1. Martheen Martheen 3, 4 4 gold badges 20 20 silver badges 44 44 bronze badges. Depending on which operation you want to do, the code will be different. I just want the magnitude, because the direction does not matter to me for this problem.
Ryan Then you don't want the reciprocal of a vectorwhich, as templatetypedef wrote, is not defined. In any application I can think of, the vector would stay pointing the same direction and you just divide by the magnitude. I think this even appears in Stewart. Lao Tsu- I have not seen this notation before. I see why it might be intuitive, but since a Google search for "reciprocal vector" turns up pages primarily pertaining to lattice cryptography, I'm not sure that it's standardized.
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It only takes a minute to sign up. So, I learned that I can create a reciprocal lattice from direct lattice, using the following formulas:. You don't need to. If, on the other hand, you do want to stick to the three-dimensional formalism which then lets you use the same formulas for both cases then you need to supplement your basis of the plane with a third vector to make the span three-dimensional.
So to understand why you get what you get, it's important to understand what the reciprocal lattice is solving. Or, similarly, one has the same problem with the moment-of-inertia tensor: it has principal axes about which there is a simple moment of inertia, but those axes might not be orthogonal. Well there is an analogue to the above formula, but it requires being more precise with the mathematics.
A covector sometimes also called a one-form is a linear mapping from vectors to scalars. We assume that these exist in a one-to-one correspondence, so that for every covector there is a vector, too. So we need a notation that makes all of this easy to see and understand.
The letters are not variables, they do not stand in for numbers to be substituted in later; they are just symbols to help us tell apart different things. And then we write a tensor with the symbols which identify which space it belongs to. And when we want to contract a tensor, we repeat the index top and bottom. In other words, what do these "dual components" of the vector really mean?
In fact, it's this reciprocal lattice basis. So we take our covectors and we cast them back into vectors. The orientation tensor, from its antisymmetry, gives us a vector orthogonal to the lattice vectors, fulfilling step 1 above. This now just needs to be normalized by its own dot product. What happens when we go from 3D to 2D, or 4D to 3D? Well, we need to remove one of those dimensions of the orientation tensor. But there's a really easy way to do that: apply it to whatever dimension we're removing.
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Reciprocal lattice in 2D Ask Question. Asked 3 years, 3 months ago.In physicsthe 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-spacedue to the relationship between the Pontryagin duals momentum and position. The reciprocal of a reciprocal lattice is the original direct lattice, since the two are Fourier transforms of each other.
Mathematically, direct and reciprocal lattice vectors represent covariant and contravariant vectorsrespectively. 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 diffractiondue to the Laue conditionsthe 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. Assuming a two-dimensional Bravais lattice. As such, any function which exhibits the same periodicity of the lattice can be expressed as a Fourier series with angular frequencies taken from the reciprocal lattice.
This reciprocal lattice is itself a Bravais lattice, and the reciprocal of the reciprocal lattice is the original lattice, which reveals the Pontryagin duality of their respective vector spaces. Using column vector representation of reciprocal primitive vectors, the formulae above can be rewritten using matrix inversion :. This method appeals to the definition, and allows generalization to arbitrary dimensions. The cross product formula dominates introductory materials on crystallography.
This can simplify certain mathematical manipulations, and expresses reciprocal lattice dimensions in units of spatial frequency. It is a matter of taste which definition of the lattice is used, as long as the two are not mixed. The direction of the reciprocal lattice vector corresponds to the normal to the real space planes.
The magnitude of the reciprocal lattice vector is given in reciprocal length and is equal to the reciprocal of the interplanar spacing of the real space planes. Reciprocal lattices for the cubic crystal system are as follows. The cubic lattice is therefore said to be self-dual, having the same symmetry in reciprocal space as in real space. Consider an FCC compound unit cell. Locate a primitive unit cell of the FCC; i. Now take one of the vertices of the primitive unit cell as the origin.
Give the basis vectors of the real lattice. Then from the known formulae, you can calculate the basis vectors of the reciprocal lattice.
Note that the basis vectors of a real BCC lattice and the reciprocal lattice of an FCC resemble each other in direction but not in magnitude. The simple hexagonal lattice is therefore said to be self-dual, having the same symmetry in reciprocal space as in real space. One path to the reciprocal lattice of an arbitrary collection of atoms comes from the idea of scattered waves in the Fraunhofer long-distance or lens back-focal-plane limit as a Huygens-style sum of amplitudes from all points of scattering in this case from each individual atom.StatSoft Italia srl StatSoft Japan Inc.
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What is the problem the reciprocal lattice solves?
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