The amplitudes and the structure of the charge density wave in YBCO

We find unknown $s$- and $d$-wave amplitudes of the recently discovered charge density wave (CDW) in underdoped cuprates. To do so we perform a combined analysis of experimental data for ortho-II YBa${}_2$Cu${}_3$O${}_y$. The analysis includes data on nuclear magnetic resonance, resonant inelastic X-ray scattering, and hard X-ray diffraction. The amplitude of doping modulation found in our analysis is $3.5\cdot {10}^{-3}$ in a low magnetic field and $T=60$K, the amplitude is $6.5\cdot {10}^{-3}$ in a magnetic field of 30T and $T=1.3$K. The values are in units of elementary charge per unit cell of a CuO${}_2$ plane. We show that the data rule out a checkerboard pattern, and we also show that the data might rule out mechanisms of the CDW which do not include phonons.