To determine which reaction in fcc/bcc crystals with lattice parameter 'a' is energetically favorable, we must consider the crystallographic structure and the nature of the lattice dislocations involved.

In crystallography, the energetics of dislocation reactions in lattice structures such as fcc (face-centered cubic) and bcc (body-centered cubic) depend on the balance of Burgers vectors involved in the reactions. The Burgers vector is crucial in determining the energy associated with dislocations.

To analyze the reactions given in the options:

• \(\frac{a}{2}[\bar{1}10] + \frac{a}{2}[0\bar{1}1]\): This vector sum results in a Burgers vector that is typically representative of a \([111]\) direction. This direction tends to be energetically favorable in fcc crystals because it aligns with close-packed planes.
• \(\frac{a}{2}[\bar{1}10] + \frac{a}{2}[110]\): The resultant of this vector addition is along the \([100]\) direction. This direction often corresponds to a higher energy state due to lower atomic packing in fcc and bcc structures.
• \(\frac{a}{2}[111] + \frac{a}{2}[11\bar{1}]\): The result of this combination is along a more energetically unfavorable direction due to less efficient atomic packing.
• \(\frac{a}{2}[111] + \frac{a}{2}[111]\): This adds up to \([222]\), which is not a simple closed-packed direction in these lattice structures, indicating a higher energy.

The reaction \(\frac{a}{2}[\bar{1}10] + \frac{a}{2}[0\bar{1}1]\) produces a resultant vector that aligns with the \([111]\) direction, which is one of the closest packed directions in fcc and bcc lattices, making it energetically favorable.

Therefore, the correct and energetically favorable reaction is \(\frac{a}{2}[\bar{1}10] + \frac{a}{2}[0\bar{1}1]\).