Exclusive measurements of high energy $gamma$-rays are performed in $rm ^{124}Ba$ and $rm ^{136}Ba$ at the same excitation energy ($sim$ 49 MeV), to study properties of the giant dipole resonance (GDR) over a wider $N/Z$ range. The high energy $gamma$-rays are measured in coincidence with the multiplicity of low energy $gamma$-rays to disentangle the effect of temperature ($T$) and angular momentum ($J$). The GDR parameters are extracted employing a simulated Monte Carlo statistical model analysis. The observed $gamma$-ray spectra of $rm ^{124}Ba$ can be explained with prolate deformation, whereas a single component Lorentzian function which corresponds to a spherical shape could explain the $gamma$-ray spectra from $rm ^{136}Ba$. The observed GDR width in $rm ^{136}Ba$ is narrower compared to that of $rm ^{124}Ba$. The statistical model best fit GDR cross sections are found to be in good agreement with the thermal shape fluctuation model (TSFM) calculations. Further, it is shown that the variation of GDR width with $T$ is well reproduced by the TSFM calculations over the temperature range of 1.1--1.7MeV.
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