We make a detailed pulse-wise study of gamma-ray bursts ( GRBs ) with known redshift detected by Fermi /Gamma Ray Burst Monitor ( GBM ) .
The sample contains 19 GRBs with 43 pulses .
We find that the average peak energy is correlated to the radiated energy ( the Amati relation ) for individual pulses with a correlation coefficient of 0.86 , which is slightly better than the correlation for the full GRBs .
As the present correlation holds within GRBs , it is a strong evidence supporting the reliability of such a correlation .
We investigate several aspects of this correlation .
( i ) We divide our sample into redshift bins and study the evolution of the correlation .
Though there is a marginal indication of evolution of the correlation , we can conclude that the present data is consistent with no evolution .
( ii ) We compare the correlation in the first or single pulses of these GRBs to that of the rest of the pulses , and confirm that the correlation is unaffected by the fact that first/single pulses are generally harder than the rest .
Finally , we conclude that the pulse-wise Amati correlation is more robust and it has the potential of refining the correlation so that GRB study could be used as a cosmological tool .