Synchrotron Mössbauer spectroscopy (SMS) was performed on an hcp-phase alloy of composition Fe 92 Ni 8 at a pressure of 21 GPa and a temperature of 11 K. Density functional theoretical calculations predict antiferromagnetism in both hcp Fe and hcp Fe-Ni. For hcp Fe, these calculations predict no hyperfine magnetic field, consistent with previous experiments. For hcp Fe-Ni, however, substantial hyperfine magnetic fields are predicted, but these were not observed in the SMS spectra. Two possible explanations are suggested. First, small but significant errors in the generalized gradient approximation density functional may lead to an erroneous prediction of magnetic order or of erroneous hyperfine magnetic fields in antiferromagnetic hcp Fe-Ni. Alternately, quantum fluctuations with periods much shorter than the lifetime of the nuclear excited state would prohibit the detection of moments by SMS. DOI: 10.1103/PhysRevLett.97.087202 PACS numbers: 75.50.Ee, 61.18.Fs, 71.15.Mb, 74.62.Fj Elemental iron, which has the body-centered cubic (bcc) crystal structure at ambient temperature and pressure, transforms to the hexagonal-close packed (hcp) phase at a pressure of approximately 13 GPa. The properties of hcp Fe are important for understanding the geophysics of the core of the Earth and for understanding the propagation of high-pressure shock waves through engineering materials. An antiferromagnetic (AFM) ground state has been predicted repeatedly for the hcp phase of iron [1][2][3][4][5][6][7][8], but, in the nearly 50 years since -Fe was first synthesized [9], Mössbauer effect experiments have never detected the presence of a hyperfine magnetic field (HMF) [10 -12], implying the absence of static magnetic moments or magnetic order. Recent density functional theory (DFT) calculations have suggested that the vanishing HMF in hcp iron can be explained by cancellation of a large core electron polarization by an equally large itinerant electron polarization in the afmII antiferromagnetic state [1,5,6]. This hypothesis neatly explains the null results of the Mössbauer measurements but sustains the possibility of antiferromagnetism in -iron. DFT calculations for the afmII structure have shown markedly better agreement with equation of state and elasticity measurements than nonmagnetic calculations [5] and have provided an explanation for the split Raman mode in -Fe [13] and a consistent calculation of this splitting [6]. These findings, taken together with the recent discovery of an unusual form of superconductivity in -Fe [14], lend new importance to determination of the correct magnetic ground state of -Fe, a topic with considerable history.In the present work, we tested the idea that -iron is antiferromagnetic yet exhibits no hyperfine field owing to the cancellation of up and down spin densities at iron nuclei. If indeed a delicate balance between core and conduction electron polarization exists, a magnetic perturbation should disrupt it, producing measurable hyperfine magnetic fields. Local hyperfine magnetic field p...