Normal-state pseudogap in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_8$ characterized by impurity scattering

A major difficulty in resolving the discrepancies between angle-resolved photoemission spectroscopy (ARPES) studies on the normal-state pseudogap and high-precision electronic heat capacity and NMR measurements has been that the ARPES results are exclusively for ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_8$ while the heat capacity and NMR data are primarily for ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ and ${\mathrm{Y}}_{1-x}{\mathrm{Ca}}_x{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7\mathrm{-}\mathrm{\delta }}.$ Here we determine the density of states near the Fermi energy for ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_8$ from the impurity depression of $T_c$ and using the scaling behavior of the spin susceptibility and entropy above and below $T_c$ we are able to generate the full temperature dependence of these quantities in the normal and superconducting states. The detailed behavior is almost identical to that for ${\mathrm{Y}}_{1-x}{\mathrm{Ca}}_x{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7\mathrm{-}\mathrm{\delta }}$ showing that the direct comparison of these two cuprates is valid.