Up to now,at least 806 carbon allotropes have been proposed theoretically.Three interesting carbon allotropes (named Pbam-32,P6/mmm,and I4-3d) were recently uncovered based on a random sampling strategy combined with space group and graph theory.The calculation results show that they are superhard and remarkably stable compared with previously proposed metastable phases.This indicates that they are likely to be synthesized in experiment.We use the factor group analysis method to analyze their Γ-point vibrational modes.Owing to their large number of atoms in primitive unit cells (32 atoms in Pbam-32,36 atoms in P6/mmm,and 94 atoms in I4-3d),they have many Raman-and infraredactive modes.There are 48 Raman-active modes and 37 infrared-active modes in Pbam-32,24 Raman-active modes and 14 infrared-active modes in P6/mmm,and 34 Raman-active modes and 35 Raman-and infrared-active modes in I4-3d.Their calculated Raman spectra can be divided into middle frequency range from 600 cm-1 to 1150 cm-1 and high frequency range above 1150 cm-1.Their largest infrared intensities are 0.82,0.77,and 0.70 (D/(A))2/amu for Pbam,P6/mmm,and I4-3d,respectively.Our calculated results provide an insight into the lattice vibrational spectra of these sp3 carbon allotropes and suggest that the middle frequency Raman shift and infrared spectrum may play a key role in identifying newly proposed carbon allotropes.