Magnetic ground states in the three <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant="normal">s</mml:mi></mml:mrow><mml:mrow><mml:mn>6</mml:mn><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>(</mml:mo><mml:mrow><mml:mn>5</mml:mn><mml:msup><mml:mi>d</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:math> double perovskites <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:msub><mml:mi mathvariant="normal">a</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mi>M</mml:mi><mml:mi>Os</mml:mi><mml:msub><mml:mi mathvariant="normal">O</mml:mi><mml:mn>6</mml:mn></mml:msub></mml:mrow></mml:math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>(</mml:mo><mml:mrow><mml:mi>M</mml:mi><mml:mo>=</mml:mo><mml:mi>Mg</mml:mi><mml:mo>,</mml:mo><mml:mtext>Zn</mml:mtext><mml:mo>,</mml:mo><mml:mtext>and Cd</mml:mtext><mml:mo>)</mml:mo></mml:mrow></mml:math> from Néel order to its suppression

Three closely related double perovskites (DP) based on the $5{d}^{2}$ ion, $\mathrm{O}{\mathrm{s}}^{6+},\mathrm{B}{\mathrm{a}}_{2}M\mathrm{Os}{\mathrm{O}}_{6}$, with $M=\mathrm{Mg}$, Zn, and Cd have been prepared and characterized using x-ray (XRD) and neutron diffraction (ND), dc magnetization, heat capacity, and muon spin relaxation $(\ensuremath{\mu}\mathrm{SR})$ techniques. All three are cubic, Fm-$3m$, at ambient temperature from XRD with $\mathrm{\ensuremath{\Delta}}d/d\ensuremath{\sim}5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$ resolution. For both $M=\mathrm{Mg}$ and Zn, ND data at 3.5 K and lower, $\mathrm{\ensuremath{\Delta}}d/d=2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$, resolution show no signs of a distortion. The results are compared with the known DP material, $\mathrm{B}{\mathrm{a}}_{2}\mathrm{CaOs}{\mathrm{O}}_{6}$, which shows antiferromagnetic (AF) order below ${T}_{\mathrm{N}}=49\phantom{\rule{0.16em}{0ex}}\mathrm{K}$ and a moderate frustration index, $f\ensuremath{\sim}4$, $(f=|{\ensuremath{\theta}}_{\mathrm{CW}}|/{T}_{\mathrm{N}})$, where ${\ensuremath{\theta}}_{\mathrm{CW}}$ is the Curie-Weiss temperature. $\mathrm{B}{\mathrm{a}}_{2}\mathrm{MgOs}{\mathrm{O}}_{6}$ with a unit cell constant ${a}_{0}=8.0757(1)\phantom{\rule{0.16em}{0ex}}\AA{}$, 3% smaller than for $\mathrm{B}{\mathrm{a}}_{2}\mathrm{CaOs}{\mathrm{O}}_{6}$, also shows N\'eel order below ${T}_{\mathrm{N}}=51\phantom{\rule{0.16em}{0ex}}\mathrm{K}$ with $f\ensuremath{\sim}2$. However, $\mathrm{B}{\mathrm{a}}_{2}\mathrm{ZnOs}{\mathrm{O}}_{6},{a}_{0}=8.0975(1)\phantom{\rule{0.16em}{0ex}}\AA{}$, 0.27% larger than $\mathrm{B}{\mathrm{a}}_{2}\mathrm{MgOs}{\mathrm{O}}_{6}$, does not show N\'eel order from either heat capacity or $\ensuremath{\mu}\mathrm{SR}$ data. A zero field cooled/field cooled (ZFC/FC) irreversibility occurs near 30 K and a broad heat capacity anomaly is detected at a similar temperature. The $\ensuremath{\mu}\mathrm{SR}$ data are consistent with a weak spin ordering with an onset below 28 K but with a coexisting dynamic component. $\mathrm{B}{\mathrm{a}}_{2}\mathrm{CdOs}{\mathrm{O}}_{6}$ with ${a}_{0}=8.3190(1)\phantom{\rule{0.16em}{0ex}}\AA{}$, 0.5% smaller than $\mathrm{B}{\mathrm{a}}_{2}\mathrm{CaOs}{\mathrm{O}}_{6}$, shows no evidence for any type of order/spin freezing to 0.47 K from any of the measurement techniques applied. The results for $M=\mathrm{Zn}$ and Cd appear to lie outside of the mean field theory of Chen and Balents [Phys. Rev. B 84, 094420 (2011)] for cubic ${d}^{2}$ DP with strong spin orbit coupling, but $\mathrm{B}{\mathrm{a}}_{2}\mathrm{MgOs}{\mathrm{O}}_{6}$, along with $\mathrm{B}{\mathrm{a}}_{2}\mathrm{CaOs}{\mathrm{O}}_{6}$, is likely one of the three predicted AF phases. The remarkable contrast between the doppelg\"anger pairs $M=\mathrm{Mg}$/Zn and $M=\mathrm{Ca}$/Cd may be traceable to differences in electronic structure of the diamagnetic $M$ ions. All of the super-super exchange pathways in these DP materials involve Os--O--$M$--O--Os linkages.

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