Unveiling complex magnetic structures of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>4</mml:mn> <mml:mi>f</mml:mi> <mml:mtext>−</mml:mtext> <mml:mn>4</mml:mn> <mml:mi>d</mml:mi> </mml:mrow> </mml:math> exchange co

The magnetic structures of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mi mathvariant="normal">B</a:mi> <a:msub> <a:mi mathvariant="normal">a</a:mi> <a:mn>3</a:mn> </a:msub> <a:mi>R</a:mi> <a:mi mathvariant="normal">R</a:mi> <a:msub> <a:mi mathvariant="normal">u</a:mi> <a:mn>2</a:mn> </a:msub> <a:msub> <a:mi mathvariant="normal">O</a:mi> <a:mn>9</a:mn> </a:msub> </a:mrow> </a:math> <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mrow> <g:mo>(</g:mo> <g:mi>R</g:mi> <g:mo>=</g:mo> <g:mi>Tb</g:mi> <g:mo>,</g:mo> <g:mi>Er</g:mi> <g:mo>)</g:mo> </g:mrow> </g:math> have been investigated using high-intensity neutron powder diffraction (NPD) supported by dc and ac susceptibility studies. The NPD data reveal antiferromagnetic (AFM) ground states below <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:msub> <h:mi>T</h:mi> <h:mi mathvariant="normal">N</h:mi> </h:msub> <h:mo> </h:mo> <h:mrow> <h:mo>∼</h:mo> <h:mn>9.8</h:mn> </h:mrow> </h:math> and 6 K in <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"> <j:mrow> <j:mi mathvariant="normal">B</j:mi> <j:msub> <j:mi mathvariant="normal">a</j:mi> <j:mn>3</j:mn> </j:msub> <j:mi>TbR</j:mi> <j:msub> <j:mi mathvariant="normal">u</j:mi> <j:mn>2</j:mn> </j:msub> <j:msub> <j:mi mathvariant="normal">O</j:mi> <j:mn>9</j:mn> </j:msub> </j:mrow> </j:math> and <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"> <o:mrow> <o:mi mathvariant="normal">B</o:mi> <o:msub> <o:mi mathvariant="normal">a</o:mi> <o:mn>3</o:mn> </o:msub> <o:mi>ErR</o:mi> <o:msub> <o:mi mathvariant="normal">u</o:mi> <o:mn>2</o:mn> </o:msub> <o:msub> <o:mi mathvariant="normal">O</o:mi> <o:mn>9</o:mn> </o:msub> </o:mrow> </o:math> , respectively, consistent with the magnetic transitions observed in dc and ac susceptibility measurements. In <t:math xmlns:t="http://www.w3.org/1998/Math/MathML"> <t:mrow> <t:mi mathvariant="normal">B</t:mi> <t:msub> <t:mi mathvariant="normal">a</t:mi> <t:mn>3</t:mn> </t:msub> <t:mi>TbR</t:mi> <t:msub> <t:mi mathvariant="normal">u</t:mi> <t:mn>2</t:mn> </t:msub> <t:msub> <t:mi mathvariant="normal">O</t:mi> <t:mn>9</t:mn> </t:msub> </t:mrow> </t:math> , the magnetic structure is characterized by a propagation vector, <y:math xmlns:y="http://www.w3.org/1998/Math/MathML"> <y:mrow> <y:mi>k</y:mi> <y:mo>=</y:mo> <y:mo>(</y:mo> <y:mn>0</y:mn> <y:mo>,</y:mo> <y:mspace width="0.16em"/> <y:mn>0</y:mn> <y:mo>,</y:mo> <y:mspace width="0.16em"/> <y:mn>0</y:mn> <y:mo>)</y:mo> </y:mrow> </y:math> , where Tb and Ru order ferromagnetically in the <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML"> <bb:mrow> <bb:mi mathvariant="italic">ab</bb:mi> </bb:mrow> </bb:math> plane. Neighboring planes of Tb are coupled antiferromagnetically along the <db:math xmlns:db="http://www.w3.org/1998/Math/MathML"> <db:mi>c</db:mi> </db:math> direction as are neighboring planes of Ru, which leads to an AFM order between Tb and Ru as well. In contrast, the magnetic structure of <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML"> <eb:mrow> <eb:mi mathvariant="normal">B</eb:mi> <eb:msub> <eb:mi mathvariant="normal">a</eb:mi> <eb:mn>3</eb:mn> </eb:msub> <eb:mi>ErR</eb:mi> <eb:msub> <eb:mi mathvariant="normal">u</eb:mi> <eb:mn>2</eb:mn> </eb:msub> <eb:msub> <eb:mi mathvariant="normal">O</eb:mi> <eb:mn>9</eb:mn> </eb:msub> </eb:mrow> </eb:math> is characterized by <jb:math xmlns:jb="http://www.w3.org/1998/Math/MathML"> <jb:mrow> <jb:mi>k</jb:mi> <jb:mo>=</jb:mo> <jb:mo>(</jb:mo> <jb:mn>0.5</jb:mn> <jb:mo>,</jb:mo> <jb:mspace width="0.16em"/> <jb:mn>0</jb:mn> <jb:mo>,</jb:mo> <jb:mspace width="0.16em"/> <jb:mn>0</jb:mn> <jb:mo>)</jb:mo> </jb:mrow> </jb:math> , with Er and Ru moments exhibiting a canted AFM ordering. Interestingly, the collinear (noncollinear) AFM order of Tb (Er) accompanied by the collinear (noncollinear) magnetic order of Ru in <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML"> <mb:mrow> <mb:mi mathvariant="normal">B</mb:mi> <mb:msub> <mb:mi mathvariant="normal">a</mb:mi> <mb:mn>3</mb:mn> </mb:msub> <mb:mi>TbR</mb:mi> <mb:msub> <mb:mi mathvariant="normal">u</mb:mi> <mb:mn>2</mb:mn> </mb:msub> <mb:msub> <mb:mi mathvariant="normal">O</mb:mi> <mb:mn>9</mb:mn> </mb:msub> </mb:mrow> </mb:math> ( <rb:math xmlns:rb="http://www.w3.org/1998/Math/MathML"> <rb:mrow> <rb:mi mathvariant="normal">B</rb:mi> <rb:msub> <rb:mi mathvariant="normal">a</rb:mi> <rb:mn>3</rb:mn> </rb:msub> <rb:mi>ErR</rb:mi> <rb:msub> <rb:mi mathvariant="normal">u</rb:mi> <rb:mn>2</rb:mn> </rb:msub> <rb:msub> <rb:mi mathvariant="normal">O</rb:mi> <rb:mn>9</rb:mn> </rb:msub> </rb:mrow> </rb:math> ) along with the concurrent ordering of the rare earth and Ru in both compounds suggests a strong coupling between <wb:math xmlns:wb="http://www.w3.org/1998/Math/MathML"> <wb:mrow> <wb:mn>4</wb:mn> <wb:mi>f</wb:mi> </wb:mrow> </wb:math> and <xb:math xmlns:xb="http://www.w3.org/1998/Math/MathML"> <xb:mrow> <xb:mn>4</xb:mn> <xb:mi>d</xb:mi> </xb:mrow> </xb:math> electrons. Both compounds show weak magnetic ordering above