The C₆D₆ detection system coupling with the pulse height weighting technique is widely used for experimentally measuring the neutron capture cross section. The thickness of sample used in the experiment directly affects the neutron beam time and the reliability of the experimental data. In the present work, we compare the lutetium (Lu) neutron capture reaction cross sections among the samles with different thickness, obtained by the C₆D₆ detection system of the back-streaming white neutron beam line at China spallation Neutron Source (CSNS back-n). The light response of the detection system is simulated with the consideration of the sample thickness by GEANT4 Monte Carlo simulation code. The 4^th order polynomial pulse weight functions for different samples are determined by using the above light response function. In the experiment, the high precision capture yield distributions in the resonance energy region are obtained by measuring the longer flight distance and background. The experimental resonance parameters are deduced by analyzing the capture yield distribution with the R-matrix theory. The comparisons of the results of capture yield and the resonance parameters between the two groups show that the resonance curve of 1.06mm ^natLu sample changes due to its thickness effect, and there is a large difference between the experimental resonance parameters and ENDF/B-VIII.0 database. However, the experimental results of 0.207mm ^natLu sample can well accord with the ENDF/B-VIII. 0 data.