Verification of theoretical predictions of an oscillating behavior of the fine-structure constant \alpha ( \equiv e ^ { 2 } / \hbar c ) with cosmic time requires high precision \Delta \alpha / \alpha measurements at individual redshifts , while in earlier studies the mean \Delta \alpha / \alpha values averaged over wide redshift intervals were usually reported .
This requirement can be met via the single ion differential \alpha measurement ( SIDAM ) procedure proposed in Levshakov et al .
( 2005 ) .
We apply the SIDAM to the Fe ii lines associated with the damped Ly \alpha system observed at z _ { abs } = 1.15 in the spectrum of HE 0515–4414 .
The weighted mean \langle \Delta \alpha / \alpha \rangle calculated on base of carefully selected 34 Fe ii pairs \ { \lambda 1608 ,X\ } ( X = 2344 , 2374 , and 2586 Å ) is \langle \Delta \alpha / \alpha \rangle = ( -0.07 \pm 0.84 ) \times 10 ^ { -6 } ( 1 \sigma C.L . ) .
The precision of this estimate improves by a factor 2 the previous one reported for the same system by Quast et al .
( 2004 ) .
The obtained result represents an absolute improvement with respect to what has been done in the measurements of \Delta \alpha / \alpha .