Momentum correlation functions at small relative momenta are calculated for light particles $left(n, p, d, tright)$ emitted from $^{197}$Au + $^{197}$Au collisions at different impact parameters and beam energies within the framework of the isospin-dependent quantum molecular dynamics model complemented by the $Lednickacute{y}$ and $Lyuboshitz$ analytical method. We first make sure our model is able to reproduce the FOPI data of proton-proton momentum correlation in a wide energy range from 0.4$A$ GeV to 1.5$A$ GeV. Then we explore more physics insights through the emission times and momentum correlations among different light particles. The specific emphasize is the effects of total pair momentum among different light particles, impact parameters and in-medium nucleon-nucleon cross section. Both two-deuteron and two-triton correlation functions are anti-correlation due to the final state interaction, and they are affected by in-medium nucleon-nucleon cross section for the higher total momentum of the particle pairs, but not for the lower ones. In addition, impact parameter and in-medium nucleon-nucleon cross section dependences of the emission source radii are extracted by fitting the momentum correlation functions. The results indicate that momentum correlation functions gating with total pair momentum is stronger for the smaller in-medium nucleon-nucleon cross section factor $left(etaright)$ or impact parameter $left(bright)$. Non-identical particle correlations ($np, pd, pt, $ and $dt$) are also investigated by the velocity-gated correlation functions which can give information of the particles emission sequence, and the result indicates that heavier ones $left(deuteron/tritonright)$ are, one the average, emitted earlier than protons, in the small relative momentum region.
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