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High-energy physics detectors, like Low Gain Avalanche Detectors (LGADs) that will be used as fast timing detectors in the High Luminosity LHC experiments, have to exhibit a significant radiation tolerance. Thereby the impact of radiation on the highly boron-doped gain layer that enables the internal charge multiplication, is of special interest, since due to the so-called Acceptor Removal Effect (ARE) a radiation-induced deactivation of active boron dopants takes place. In this paper we present defect-spectroscopy measurements (Deep-Level Transient Spectroscopy and Thermally Stimulated Current technique) on neutron irradiated p-type silicon pad diodes of different resistivity as well as LGADs irradiated at fluences up to 1 x 10^15 neq/cm2. Thereby we show that while for the silicon pad diodes irradiated with electrons, neutrons or protons the determination of defect electronic properties and defect introduction rates is straightforward, DLTS and TSC measurements on LGADs are strongly influenced by the impact of the gain layer. It is shown that the measurability of the capacitance of the gain layer shows a strong frequency and temperature dependence leading to a capacitance drop in DLTS and non-reliable measurement results. With TSC defects formed in the LGADs can be very nicely observed and compared to the defects formed in the silicon pad diodes. However the exact assignment of defects to the gain layer or bulk region remains challenging and the charge amplification effect of the LGADs impacts the exact determination of defect concentrations. Additionally, we will demonstrate that depending on the TSC measurement conditions defect induced residual internal electric fields are built up in the irradiated LGADs that are influencing the current signal of carriers emitted from the defect states.