Topological classification of defects

[ayan849]

Please help somebody on this problem...

When we topologically classify the defects in ordered media, we consider the character of the fundamental group of the associated order parameter space. To construct those groups, we circumscribe the line defects by circles and the point defects by spheres.
My question is what is done for a surface (possibly infinite) defect, say domain walls. My query primary concerns crystal lattices. I want to characterize the essential defects in solid crystals--for dislocation and interstitial/vacancy, it is straightforward. But what to be done in case of grain/phase boundary?

 

[cgk]

When we topologically classify the defects in ordered media, we consider the character of the fundamental group of the associated order parameter space.

That is the most sciency statement I've ever heard in connection to defects. But seriously: What exactly are you trying to do? I always was of the impression that the classification goes like this:

0D defects: vacancies, interstitial. 0D since there is sizable spatial extend in zero directions.

1D: dislocations and so on. Basically given by a line--spatial extend only in one direction (from local topologic perspective, of course the actual dislocation lines can have complex 3D shapes)

2D: grain/phase boundaries and so on. Spatial extend in two directions (locally, idealized).

Of course, in practice grain and phase boundaries are not sharp. For example, a grain bounday will usually consist of at least 10-100 atomic layers; however, in relation to the spatial extension of the other two dimensions (which usually would me measured in µm to mm, and not nm), they can still be considered as essentially flat.

So are you looking for some hard mathematical definition of the defect dimension? I think that "circle around a dislocation" argument is just intended as a tool for the definition; if you just accept that a point is a 0-dimensional obeject, a line is a 1-dimensional object, and a sheet is a 2-dimensional object, you don't need such definitions.

 

[ayan849]

@ cgk
I think you are not aware of the topological classification of defects in ordered media. An ordered media is the generalization of crystalline materials. It can include liquid crystals to superfluids, ferromagnets and so on. There is a general classification scheme of defects occurring in those ordered structures, which revolves around algebraic topology of the order parameter space of that media. I want to know how 2D defects are classified in that scheme, because there are ample works done on 1D and 0D defects.

 

[edmondrobbins]

How to understand the "domain walls"

 

[element4]

@ayan849
The most popular reference for this topic is Mermin's review http://rmp.aps.org/abstract/RMP/v51/i3/p591_1" . Sorry I cannot provide you with a direct answer, Its some time ago I studied these topics.

 

[weejee]

I think you need to look for how many disconnected pieces there are in the order parameter space.