Dear Hao,

The competition between basal and prismatic slip in hcp metals is not controlled by the c/a ratio*, but by the energies of the basal and prismatic stacking faults which are responsible for dislocation dissociation in basal and prismatic planes. This has been shown, using tight binding modelling, by Legrand:

http://dx.doi.org/10.1080/13642818408227636

The original article is in French, but you can find summaries of this work in recent reviews, like the book of Ladislas Kubin (chap. 3.3, http://dx.doi.org/10.1093/acprof:oso/9780198525011.001.0001)

In transition metals of the column IV (Ti, Zr and Hf), the valence d-electrons induce a strong angular dependence of the atomic bonding, which is responsible for a high value of the basal stacking fault. As a consequence, prismatic dissociation of the dislocation, and hence prismatic glide, is favoured.

Going now to Ti-Al, you can have a look to this recent article published in Acta Materialia:

http://dx.doi.org/10.1016/j.actamat.2015.09.041

You will see that Al solute decreases the energy of the basal stacking fault, whereas it increases the energy of the prismatic fault, which should be the main reason why Al addition in Ti promotes basal slip.

* To convince yourself that you cannot directly link c/a ratio and basal/prismatic slip, think to Be. This element has the smallest c/a ratio among hcp metals. Hence the densest planes are the prismatic ones. But the easy glide system is nevertheless basal, because of the low energy basal stacking fault.

Slip should depend on geometrical conditions especially because they are related to bondings. However, in case of Ti(Al) you have two different atoms so that your anisotropic properties will change with the Ti-Al ratio and also the c/a ratio. I guess you ave to split this question into two. One related to single-element materials, and the other for two or more element materials. At least it wouldn't surprise me if there is some kind of thumb rule.  

Take a look at the Gamma Titanium Aluminide Alloys book by Appel et al. 

There you can find something related to your question specifically in chap. 5 and 6, for both single and two phases alloys, respectively.

Section 6.2 stacking fault energy etc see end of pg 112 http://research.iaun.ac.ir/pd/ebrahimzadeh/pdfs/UploadFile_6472.pdf Hull and Bacon Into to Dislocations

Many thanks for your kind help

Dear Hao, 

my answer is more fundamental than the previous answers and so I don't know if it is what you are looking for, but here it goes.

So, essentially, the AB packing sequence of the hcp lattice leads to an ideal c/a ratio of 1.633. Real hcp materials have c/a ratios that deviate from this ideal number, but there is always a 'configurational' driving force that tends to favor deformations that bring the real c/a ratio towards the ideal one. As I understand it, Ti-Al alloys have ratios around 1.60. This means that the system will yield more favorably under tension along the direction than along compression, because that increases the length of the vector and therefore increases the c/a ratio, taking it closer to 1.633.

Deformation is of course mediated by plastic slip. The problem is that in most hcp metals non-basal slip is very difficult to attain, which is one of the reasons why twinning is so prevalent when these metals are loaded along the axis. 

Conversely, you could have basal slip that tends to 'compress' the vectors, therefore increasing the c/a ratio. Hcp metals generally display very easy basal slip but because there are three distinct basal directions, these deformations are not trivial either.

Hope this helps.

Thank you for your kind help in understanding the fundemental mechanism of plastic deformation for hcp metals. Recently the influence of c/a ratio on the slip behavior of hcp metals was found in the classic textbook "Physical foundations of materials science". Thanks!

Dear Hao,

The competition between basal and prismatic slip in hcp metals is not controlled by the c/a ratio*, but by the energies of the basal and prismatic stacking faults which are responsible for dislocation dissociation in basal and prismatic planes. This has been shown, using tight binding modelling, by Legrand:

http://dx.doi.org/10.1080/13642818408227636

The original article is in French, but you can find summaries of this work in recent reviews, like the book of Ladislas Kubin (chap. 3.3, http://dx.doi.org/10.1093/acprof:oso/9780198525011.001.0001)

In transition metals of the column IV (Ti, Zr and Hf), the valence d-electrons induce a strong angular dependence of the atomic bonding, which is responsible for a high value of the basal stacking fault. As a consequence, prismatic dissociation of the dislocation, and hence prismatic glide, is favoured.

Going now to Ti-Al, you can have a look to this recent article published in Acta Materialia:

http://dx.doi.org/10.1016/j.actamat.2015.09.041

You will see that Al solute decreases the energy of the basal stacking fault, whereas it increases the energy of the prismatic fault, which should be the main reason why Al addition in Ti promotes basal slip.

* To convince yourself that you cannot directly link c/a ratio and basal/prismatic slip, think to Be. This element has the smallest c/a ratio among hcp metals. Hence the densest planes are the prismatic ones. But the easy glide system is nevertheless basal, because of the low energy basal stacking fault.

Dear Emmanuel,

Thanks for your help and making me aware of the role of stacking fault energy on slip behavior. Later I will carefully read the papers you suggested. Many thanks!