The doped topological insulator ${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$ has attracted considerable attention as a new platform for studying novel properties of spin-triplet and topological superconductivity. In this work, we performed synchrotron x-ray diffraction measurements on ${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$ ($0.24\le x\le 0.46$) to investigate the coupling between the superconducting order parameter and crystal lattice. In the crystals in which the vector order parameter ($\mathit{d}$ vector) is tilted from the crystal high-symmetry directions as evidenced by nematic diamagnetic susceptibility, we find a sizable lattice distortion ($\sim 100\mathrm{pp}\mathrm{m}$) associated with the onset of superconductivity. In contrast, in crystals with the $\mathit{d}$ vector aligned along the high-symmetry directions, we find no appreciable change in lattice constant. Together with a pronounced vestigial behavior of the distortion, the results are clear evidence for an odd-parity $E_u$ order parameter that couples with trigonal lattice. Furthermore, in the crystal with $x=0.46$ where diamagnetic susceptibility is isotropic in the plane, no lattice distortion accompanying the superconducting transition is found, which is in line with a chiral superconducting state in the highly doped region. Our work shows that lattice distortion can be a powerful diagnosing quantity for nematic superconductivity with two-component order parameter.

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• Superconducting order parameter
• Unconventional superconductors
• X-ray scattering