The definition "twin" does not specify a mirror operation as the only one. You also can use a 180° rotation.

A Σ3 boundary is for cubic crytsals from my point of view fixed by 60° around . It describes a misorientation between two crystals of the same phase. The term twin is often indicated as flat plane but also this has not to be if you consider quartz which shows nice penetration twins. The misorientation is well described, only the interface is irregular. 

I don't like to use Σ3 boundaries as description for twin boundaries because of the Brandon criterion which opens the door for quite differently oriented grains (more than 8° are technically allowed) whereas twins are quite exact and usually better than 1°.

First you should make yourself clear what CSL actually means: Coincident Site Lattice, i.e. we are talking about a translation lattice as mathematical abstraction of a crystal and not about a structure or atomic positions. For very simple metals this difference might be not mean much but one should keep this in mind for slightly more complex phases.

CSL is an exclusively geometric construct. It only describes which combinations in misorientations exist so that an interface has a certain correlation related to the formed crystals. In so far, the CSL model is an attempt to understand the formation of specific boundaries. Does CSL doesn't work as nice as we would like has its reason in the fact that finally everything happens on the atomic level, with atoms, or caused by their bondings. Which CSL is finally preferred by twinning does not depend on the lattice, i.e. practically each CSL could be a twin plane if the atoms are "strangely" distributed. For fcc the energetically best arrangement is Sigma 3, but you are right, other phases can use also Sigma1, and some rare phases even have different twins since their interface energy is obviously quite similar.

Sometimes coherency is defined based on Sigma3 boundary. If deviation from sigma3 happens it may call incoherent. In FCC alloys occasionally sigma3 and sigma 11 forms twin boundaries. One twin can show both coherent and incoherent in different parts. Sometime deformation-induced dislocation and its interaction with twin boundary results in formation or deviation from coherent sigma3 boundary.

Hi,

(1) CSL is defined in three dimensional space at a given orientation relationship rather than defining CSL structure in the grain boundary, so the boundary is not called  Σ1 boundary, and for your case it has a full name like  Σ3[-110](111) twin boundary .  However, the detailed CSL structure in the boundary is much important! 

(2) CSL structure usually happens at a special orientation relationship (OR). At this special OR, grain boundary tend to pass through density CSL points to lower the interracial energy. For Σ3 OR, the twin plane contain most density CSL points, and all of the lattice points at the plane are coincident, thus it is called coherent twin boundary.  At this OR, an arbitrary plane  may not pass through density CSL points, so these planes  would be called incoherent twin boundary.

Dear Xinfu Gu, the image does show a CSL boundary. But shouldn't a twin boundary need to have a mirror orientation for the twin and parent? Also, If I were to say all Σ3 boundaries in FCC materials are twin boundaries, would I be correct or does it work only the other way around?

Dear Mathew,

(1) Yes, twin boundary only be possible at twin orientation relationship. 

(2) Some people use Σ3 boundary as twin boundary. However, I do not suggest so. At least you should say Σ3 twin boundary or Σ3 (111) twin boundary, because at the same OR, you could have many other boundaries.

Thank you so much Xinfu Gu. 

The definition "twin" does not specify a mirror operation as the only one. You also can use a 180° rotation.

A Σ3 boundary is for cubic crytsals from my point of view fixed by 60° around . It describes a misorientation between two crystals of the same phase. The term twin is often indicated as flat plane but also this has not to be if you consider quartz which shows nice penetration twins. The misorientation is well described, only the interface is irregular. 

I don't like to use Σ3 boundaries as description for twin boundaries because of the Brandon criterion which opens the door for quite differently oriented grains (more than 8° are technically allowed) whereas twins are quite exact and usually better than 1°.

hi

I have had the same question as M Rohit Mathew. Thank you for these explanations. now it  is a bit more clear for me this problem, even if i still have a lot of questions about. in my work on the deformation mechanism in fatigue of a martensitic steel, I'll probably confronted to the problem how to relate the slip resistance to the CSL boundaries? Does anyone knows something on this? 

Dear Gulzar,

     The slip resistance of the CSL boundaries for room temperature deformation can be considered considered very different than that of random high angle boundaries. Some of the special boundaries such as the Σ3 twin boundary in FCC polycrystals have been shown to show very good dislocation accommodation and pileup during deformation. Numerous experiments and simulation studies on these boundaries show that these can significantly enhance strain hardening behaviour and show crack deflection and thus improve toughness as well. The low energy CSL boundaries become more important for high temperature deformations such a creep and associated grain boundary sliding phenomenon where these boundaries show very low mobility. Since these boundaries have low excess energy, there is increased resistance to corrosion attack as well. I hope this can help with some of your your questions.

thank you very much for your answer, it really helps.

Dear Rohit,

You expressed that "Also if any boundary is introduced into material, some excess energy is inherent, hence how can it ever be called coherent since it introduces mismatch?"

Creating a boundary certainly increases the total free energy of the system. However, in case of a twin boundary where a mirror image of atoms can be found on either side of the boundary this increase in the free energy is not due to mismatch but due to the changes in the bond angles.

Regards

Dear Ali Arslan,

Thank you so much for that answer! That was the perfect answer to the question I had asked.

I am glad that it helped.

Dear Rohit,

It's exactly the topic that I'm in, i.e. twinning process and twin boundary.

(1) The difination of CSL Σ3 boundary is that, two side of the boundary are of misorientations such as 60° - ( equivalently 70.53°-, ..., 180°- and 180°-).

(2) Furthermore, according to the crystallographic theory of twins, the {111} twin in FCC structure can be defined by 180°-{111} and 180°-. Hence, now we can say that CSL Σ3 = {111} twin bounday in FCC.

(3) About the mismatch of the twin boundary, from my opinion, it is maybe arisen from the relaxion mechanism.

Best luck!

Xianbing Zhang

According to the book " Concise dictionary of materials science, written by Vladimir Novikov" :

- A coherent twin boundary is the interface between the twin parts that coincides with the plane of the perfect joining of their lattices. Coherent twin boundary,

being a special high-angle boundary with Σ = 3 (see CSL-boundary), is

characterized by low energy and a mobility significantly lower than those

of general grain boundaries. Under an optical microscope, coherent twin

boundary looks like a thin, straight line.

- An incoherent twin boundary is a twin boundary whose plane does not coincide with the twinning plane (see twin). A boundary of this kind is always joined to either a coherent twin boundary or the boundaries of the twinned grain. The energy and mobility of an incoherent twin boundary are rather close

to those of general high-angle grain boundaries, in contrast to a coherent

twin boundary.

Good luck.

hi

i agree with Gulzar Seidametova answer.

arun

See Figure 2 in Article Structure and motion of junctions between coherent and incoh...

For coherent sigma 3 in copper, the boundary plane is parallel to {111} plane.