Piezoelectricity in GaAs I am studying the piezoelectricity in Gallium arsenide (GaAs), $[110]$ and $[\bar{1}10]$ oriented. Piezoelectricity is usually described microscopically by a 3 index tensor, $d_{ijk}$ where
i,j and k equal to x,y and z.
$[110]$ and $[\bar{1}10]$ have different piezoelectric behaviour but they are linked by a parity operation on the x-axis: $$x\rightarrow -x$$
Can I assume the following identity:$$d^{[110]}_{ijk}=(-)^{\delta_{xi}+\delta_{xj}+\delta_{xk}}d^{[\bar110]}_{ijk}$$
Derived by applying the same transformation.
 A: I think the problem is a general tensor analysis one rather semiconductor physics (GaAs), since no particular values of any d-symbol is of any importance? Anyway, here is my effort to produce an answer.
The piezoelectric coefficient $d_{ijk}=d^{[110]}_{ijk}$ is a third rank Cartesian  tensor which transforms from the reference frame $[x_1, x_2, x_3]$ to $[\bar {x}_1,\bar {x}_2,\bar {x}_3]$ giving $\bar {d}_{pqr}= d^{[\bar110]}_{pqr}$ according to the general rule:
${d}^{110}_{ijk} =\Sigma \bar {d}^{\bar {1}10}_{pqr}a_{pi}a_{qj}a_{prk} $
where $a_{\lambda\mu}$ are the cosines between coordinates in the two frames, and the summation is over repeated indices. 
In your particular coordinate transformation $[- x,y,z]$ to $[x,y,z]$ which transforms $\bar {d}_{pqr}$  to $d_{ijk}$ you need to bear in mind that $a_{pi}=\delta_{pi}$ etc so you get the general relationship with the $\delta$ symbols multiplying each other in each term of the summation . They are not exponents (powers) to base (-1).   
The expression you have written doesn't seem to give the correct relationship between $d^{[110]}_{ijk}$ and $d_{pqr}=d^{[\bar {1}10]}_{pqr}$.
I hope this helps.
