Critical attenuation measurements are reported for two anomalous acoustic modes near the 250°K transition of BaMnF 4 . A mechanism for the transition involving rotations of the MnF 6 octahedra about the orthorhombic b axis is proposed, with a pair of phonons at the Brillouin zone S points 7r(0,1/6 ,± 1/c) being the soft modes. This mechanism, which suggests strong two-dimensional correlations, appears consistent with previous work on BaMnF 4 as well as with the ultrasonic data.I report here measurements of ultrasonic attenuation in the vicinity of the 250°K antiferrodistortive transition of BaMnF 4 . These yield the first determination of critical attenuation exponents for an antiferrodistortive transition in a material not of the perovskite structure and thus provide a new test of recent general theories of ultrasonic anomalies associated with this important class of structural phase transitionSo I also present a proposed mechanism for the transition which appears consistent with the experimental work done so far on BaMnF 4 . In particular, this mechanism suggests critical fluctuations which are strongly correlated in two dimensions, and this provides an explanation for the observed longitudinal-mode attenuation exponent being significantly larger than in the perovskites.The 250°K transition was first observed in ultrasonic attenuation studies by Spencer, Guggenheim, and Kominiak. 1 Ryan and Scott 2 investigated the transition by Raman scattering and showed that it is antiferrodistortive and involves a doubling of the unit cell. Recently the author 3 reported ultrasonic velocity measurements for the pure modes propagating along the principal axes and showed that there are sharp velocity dips for the longitudinal mode propagating along the c axis (v cc ) and for the shear mode propagating along the c (orb) axis and polarized along the b (or c) axis {v ch = v bc ). There has not been a direct experimental determination of the structure of the low-temperature phase, so that the actual mechanism of the transition is not established.In the present work the attenuation of the two modes with sharp velocity anomalies was measured by the pulse-echo technique for temperatures between 190 and 300°K. Within about 0.5°K of the transition the data exhibit rounding effects, presumably due to sample imperfections. 4 Data were obtained at frequencies between 14 and 90 MHz. For the shear mode (a cb ) data were obtained and analyzed both above and below the transition temperature T 0 . For the longitudinal mode (a cc ) the data could only be analyzed above T 0 because of domain effects below T Q0 Typical critical attenuation data are shown in Fig. 1, which is a log-log plot of the longitudinalmode critical attenuation versus reduced temperature e for T > T 0 . The critical attenuation exponent rj defined by a =A (co)e T) is r) = ^7 CC > = 2.2 ± 0.3. The frequency dependence A(w) of the attenuation (at fixed temperature) does not appear to follow 50