  The next-to-leading order black-body radiation(BBR) shift to atomic
energy-levels, namely $ (Z\alpha)^2\alpha T^2/m $ correction, was studied by
using the nonrelativistic quantum electrodynamics(NRQED). We also estimate the
one-loop contribution of quadrupole and the two-loop contributions of BBR-shift
of the thermal(real) photon. These corrections have not been investigated
before. The order of magnitude BBR-shift indicates the one-loop contribution of
quadrupole is stronger than the previous result. And the two-loop contribution
of BBR-shift of the thermal(real) photon is tiny, but this next-to-leading
order BBR-shift may be as significant as the leading order in the
multi-electron atoms or cold ones.
