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

Black-body radiation shift of atomic energy-levels:The $ (Z \alpha)^2\alpha T^2/m $ correction