Talk:Reciprocal lattice

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Untitled

There a request for reciprocal lattice primitive cell from Wikipedia:ACF Regionals 2000 answers

If this is another name for a page that already exists could someone more knowledgeable than me make a redirect. Or if not could some make an article.--BirgitteSB 16:29, July 12, 2005 (UTC)

Why are the definitions repeated? I'm not sure if I'm qualified, but I think that the introduction could be made clearer. Hithisishal (talk) 03:22, 16 February 2008 (UTC)[reply]

What is the point of the redirect-sentence at the top? It suggests that there is some other article about general dual lattices, but it redirects to itself! --AasmundE 06:52, March 8, 2009 (UTC) —Preceding unsigned comment added by 129.241.133.230 (talk)

Formula incorrect?

Aren't the reciprocal lattice vector equations on this page incorrect? I thought (and I have checked some other sources such as http://www.chemsoc.org/ExemplarChem/entries/2003/bristol_cook/reciprocallattice.htm which verify this) that the bottom line didn't change, unlike what is presented on the current page. 212.2.167.48 10:11, 12 November 2007 (UTC)[reply]

The formulae are correct. The three expressions in the denominators are equivalent (a scalar triple product). You can check this by writing out the result component-wise. The result equals the (signed) volume of the ~~reciprocal~~ lattice primitive cell. WilliamDParker 13:15, 14 November 2007 (UTC)[reply]

Lattice means: discrete set of points

Is the extension to arbitrary arrangements of atoms justified? I have not seen this in the literature so far. In my view, a lattice is a discrete set of points, see Lattice (group). This would mean that a nonperiodic arrangement has no reciprocal lattice; also the animated C60 example does not fit here. --Anastasius zwerg (talk) 20:31, 18 August 2008 (UTC)[reply]

I agree that this should be limited to lattices whaich spatially periodic (in normal Euclidean space). Perhaps a new article could be started that linked to some broader view that some wish to take. The c60 image, while beautiful, should go. Hess88 (talk) —Preceding undated comment added 14:56, 14 September 2009 (UTC).[reply]

Somewhat tangentially related to the idea of a lattice being a discrete, periodic structure, I had a question about the formal definition of the reciprocal lattice. Given that the mathematical definition, e^(ik.r)=1, will always be true when the dot product of k and r is zero (ie. they are orthogonal), doesn't any given r vector have an entire, continuous plane of orthogonal k vectors (given that the magnitude doesn't affect the dot product in this case)? The mapping, then would be from a discrete space to a continuous one. It seems to me that, for the formal definition to be rigorous, a further restriction is needed on the definition of the k vectors, similar to what is said later in the article about "discrete mathematics" (that the product of magnitudes of k and r be an integer or something like that). I am relatively new to the topic, so forgive me if I am not explaining myself properly or am completely off the mark. --Chris —Preceding unsigned comment added by 68.195.78.160 (talk) 08:47, 22 November 2009 (UTC)[reply]

That is correct, and there is a restriction. The restriction is the periodicity of the lattice basis electron clouds which is used to derive the definition. IMO that's a good question. See Kittel, Intro to solid state physics, 8th ed, beginning of chap 2. --129.6.130.22 (talk) 12:35, 21 June 2010 (UTC)[reply]

Another point regarding the line "curiously the scalar triple product definition dominates most intro texts" line in the introductory paragraph. It probably follows from the usual derivation procedure used to introduce the interaction between the incident wave and electron clouds, which usually begins by stressing the independence of the primitive lattice vectors and 3D nature of the lattice. So a combined form is mathematically/geometrically astute, but not very intuitive upon a comparison with the original independent primitives presentation. Just an opinion. --129.6.130.22 (talk) 12:35, 21 June 2010 (UTC)[reply]

Diagrams

It would be beneficial to people new to the concept to have some diagrams of lattices in normal space alongside the reciprocal versions. e.g., in 2d square and hexagonal lattices and some simple 3d, e.g., bcc fcc lattices.Pondermotive (talk) 02:56, 10 March 2011 (UTC)[reply]

I agree, it would be very helpful to see an example! I'm particularly interested in an example for a finite discrete lattice. 10:57, 26 January 2012 (EST) — Preceding unsigned comment added by 130.207.197.185 (talk)

Article is misconceived and poorly written

The subject of lattices (the kind that are discrete subgroups of Euclidean space Rⁿ) is a part of mathematics. That should be the way this article is introduced and basically slanted.

This subject has ample application to chemistry and physics, and naturally these ought to be given significant discussion in the article.

This article gets off to an extremely poor start. In the sentence that begins: "Consider a set of points R constituting a Bravais lattice, and a plane wave defined by:", there appear the letters K and r whose meaning is not stated. This certainly is not a "mathematical" description of anything, since a mathematical description doesn't leave letters undefined!!! This is simply an issue of clarity.

In the section "Generalization of a dual lattice" -- a very strange title for an article whose name is "Reciprocal lattice" -- at least two definitions of "dual" lattice are given, with no mention of how they are connected to one another.

Further, the identification of a (finite dimensional real) vector space V with its dual V^* is *not* a question of a choice of "Haar measure" but rather the choice of an inner product on V.

If it is felt necessary for there to be a separate article on the applications of reciprocal lattices to physicis -- fine. But reciprocal or dual lattices is a mathematical subject and the underlying narrative, and above all the definitions, should be stated in a careful mathematical way -- completely unlike this article.Daqu (talk) 13:15, 26 August 2011 (UTC)[reply]

Then fix it. — Preceding unsigned comment added by 134.226.252.160 (talk) 15:58, 27 August 2011 (UTC)[reply]

Got a mathematically focused dual lattices page started here: https://wiki.riteme.site/wiki/Dual_lattice ... I'm sure there problems with it, so let's make it better. :-) Lorenzo (talk) September 2020. —Preceding undated comment added 06:23, 13 September 2020 (UTC)[reply]

can anyone understand from this abrakadabra what simple reciprocal lattice is?

stop putting rubbishes in science — Preceding unsigned comment added by 134.7.190.150 (talk) 09:00, 4 June 2015 (UTC)[reply]

Assessment comment

The comment(s) below were originally left at Talk:Reciprocal lattice/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

This article is unsatisfactory. An article called "reciprocal lattice" should start with a discussion of the general concept, not with a definition of the reciprocal lattice of a particular kind of lattice. It provides little or no background and is almost completely unintelligible to the introductory reader. I graded it as "start" for these reasons, although the criteria are described more in terms of quantity of content rather than understandability and usefulness.Mark Foskey 01:38, 3 November 2007 (UTC)

Last edited at 01:39, 3 November 2007 (UTC). Substituted at 04:01, 30 April 2016 (UTC)

Inconsistency in Mathematical Description section?

First line would seem to imply $\mathbf {n} \in \mathbb {Z} ^{2}$ whereas seventh line states $\mathbf {n} ,\mathbf {k} \in \mathbb {Z} ^{3}$ . — Preceding unsigned comment added by 2607:F140:6000:E:25CE:DD76:E54F:579 (talk) 22:44, 3 January 2017 (UTC)[reply]

We also appear to have changed the definition of R after the first few lines of mathematics Bubsir (talk) 18:05, 24 November 2017 (UTC)[reply]

I agree. I've tried to lessen the worst offences; but this section probably needs revamping properly by someone who favours this more algebraic approach. NeilOnWiki (talk) 19:07, 22 January 2021 (UTC)[reply]

Presentation of examples

The wording of the examples in the section "Reciprocal lattices of various crystals" is a bit problematic- it's phrased midway between homework-set and bad-solved-example, and not much is said beyond "the way to calculate is to calculate in the way to calculate"
For example (the BCC-FCC subsection):

Consider an FCC compound unit cell. Locate a primitive unit cell of the FCC, i.e., a unit cell with one lattice point. 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.

I would try to get back to this and rewrite, but I'm currently studying this topic, so it will have to wait until I can understand an actual source better.
NoePol (talk) 16:39, 19 March 2017 (UTC)[reply]

Unquestionably one of the WORST introductions of any technical article in Wikipedia

The first paragraph of the Introduction is this:

"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 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 vectors, respectively."

Is this deliberately intended to be as confusing as possible? In any case, it certainly is.

I hope that someone who knows how to express technical ideas clearly and is knowledgeable about dual lattices will please replace this gobbledegook with clear writing.

It is essential to recognize that proper grammar and correct spelling are nowhere near enough to ensure clear writing.50.205.142.50 (talk) 16:42, 10 June 2020 (UTC)[reply]

Deleted Proof that the reciprocal lattice of the reciprocal lattice is the direct lattice

I took the liberty and deleted this section, as I went through it and really do not see any understanding that it contributes. That the reciprocal of the reciprocal is the original lattice follows already from the reciprocal lattice being the set of all points $\mathbf {G} _{m}$ that fulfill

\mathbf {G} _{m}\cdot \mathbf {R} _{n}=2\pi N

for

N\in \mathbb {Z}

where $\mathbf {R} _{n}$ are all direct lattice points. Clearly, this expression is symmetric in $\mathbf {R} _{n}$ and $\mathbf {G} _{m}$ . Seattle Jörg (talk) 14:47, 16 June 2020 (UTC)[reply]

Comment: it's zero unless m=n. Add a kronecker delta, this is important. — Preceding unsigned comment added by 132.239.74.44 (talk) 23:05, 28 October 2020 (UTC)[reply]

I think (like me initially) you may have been thrown by the notation. It should be more obvious now (see latest version) why the delta isn't needed. NeilOnWiki (talk) 19:12, 22 January 2021 (UTC)[reply]

Moved 'Reciprocal space' to become first section

I've moved the Reciprocal space section to the start of the article body, since it's the natural habitat for the reciprocal lattice; hence a modified version can (I think) provide a gentler, more contextualised explanation than the gorier mathematical detail subsequently. My hope is to reword it and extend it, so it's more widely accessible to a scientifically minded reader and not too offensive to a mathematical one. This will take me a few days and meanwhile there are interim drafts in the "Reciprocal lattice" section of my sandbox. NeilOnWiki (talk) 17:48, 15 January 2021 (UTC)[reply]

And now I've completed the bulk of my intended rewording, which seems to have grown way beyond what I originally envisaged. The main section title is now Wave-based description; and I've tried to express things more clearly, rather than add anything new. I'm painfully aware of (at least) three short-comings: lack of reliable sources (which I don't have decent access to); such a large section throwing the article out of balance (which I think is best resolved with someone better informed on the topic, who can view and edit the page as a whole); and a diagram would help explain the outlined construction (though the existing diagrams may be enough). I hope this more geometrical wave-based approach will help improve the article; and (as intended) at least be more accessible, while being sound mathematically. NeilOnWiki (talk) 17:33, 22 January 2021 (UTC)[reply]