Results obtained several years ago using the high-resolution 93.3 keV Mössbauer resonance in 67 ZnO and β ′ -brass have been reanalyzed with the notion that the clock hypothesis of Special Relativity Theory is not sufficient, but that a maximal acceleration am exists and that an acceleration a contributes to the temperature dependence of the center shift by a term ±(1/2)(a/am) 2 . The significance of the sign of this term is discussed in detail. For both substances a lower limit of am > 1.5 · 10 21 m/s 2 is inferred which is more than two orders of magnitude larger than the value am = 1 · 10 19 m/s 2 suggested by 57 Fe rotor experiments.