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{{Use American English|date=January 2019}}{{Short description|Higher-order interactions of magnetic moments of chemicals
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Magnetic materials with strong [[spin-orbit interaction]], such as: LaFeAsO,<ref name="LaFeAsO">{{cite journal | last1=Cricchio | first1=Francesco | last2=Grånäs | first2=Oscar | last3=Nordström | first3=Lars | title=Low spin moment due to hidden multipole order from spin-orbital ordering in LaFeAsO | journal=Physical Review B | publisher=American Physical Society (APS) | volume=81 | issue=14 | date=13 April 2010 | issn=1098-0121 | doi=10.1103/physrevb.81.140403 | page=140403(R)| bibcode=2010PhRvB..81n0403C }}</ref><ref>{{cite journal | last1=Gonnelli | first1=R. S. | last2=Daghero | first2=D. | last3=Tortello | first3=M. | last4=Ummarino | first4=G. A. | last5=Stepanov | first5=V. A. | last6=Kim | first6=J. S. | last7=Kremer | first7=R. K. | title=Coexistence of two order parameters and a pseudogaplike feature in the iron-based superconductor LaFeAsO<sub>1−x</sub>F<sub>x</sub> | journal=Physical Review B | volume=79 | issue=18 | date=29 May 2009 | issn=1098-0121 | doi=10.1103/physrevb.79.184526 | page=184526| arxiv=0807.3149 | s2cid=118546381 }}</ref> PrFe<sub>4</sub>P<sub>12</sub>,<ref name="PrFe4P12">{{cite journal | last1=Kiss | first1=Annamária | last2=Kuramoto | first2=Yoshio | title=On the Origin of Multiple Ordered Phases in PrFe<sub>4</sub>P<sub>12</sub> | journal=Journal of the Physical Society of Japan | publisher=Physical Society of Japan | volume=74 | issue=9 | date=15 September 2005 | issn=0031-9015 | doi=10.1143/jpsj.74.2530 | pages=2530–2537| arxiv=cond-mat/0504014 | bibcode=2005JPSJ...74.2530K | s2cid=119350615 }}</ref><ref>{{cite journal | last1=Sato | first1=Hidekazu | last2=Sakakibara | first2=Toshiro | last3=Tayama | first3=Takashi | last4=Onimaru | first4=Takahiro | last5=Sugawara | first5=Hitoshi | last6=Sato | first6=Hideyuki | title=Angle-Resolved Magnetization Study of the Multipole Ordering in PrFe<sub>4</sub>P<sub>12</sub> | journal=Journal of the Physical Society of Japan | publisher=Physical Society of Japan | volume=76 | issue=6 | date=15 June 2007 | issn=0031-9015 | doi=10.1143/jpsj.76.064701 | page=064701| bibcode=2007JPSJ...76f4701S }}</ref> YbRu<sub>2</sub>Ge<sub>2</sub>,<ref name="YbRu2Ge2">{{cite journal | last1=Takimoto | first1=Tetsuya | last2=Thalmeier | first2=Peter | title=Theory of induced quadrupolar order in tetragonal YbRu<sub>2</sub>Ge<sub>2</sub>| journal=Physical Review B | volume=77 | issue=4 | date=8 January 2008 | issn=1098-0121 | doi=10.1103/physrevb.77.045105 | page=045105| arxiv=0708.2872 | bibcode=2008PhRvB..77d5105T | s2cid=119203279 }}</ref> UO<sub>2</sub>,<ref name="UO2">{{cite journal | last1=Pi | first1=Shu-Ting | last2=Nanguneri | first2=Ravindra | last3=Savrasov | first3=Sergey | title=Calculation of Multipolar Exchange Interactions in Spin-Orbital Coupled Systems | journal=Physical Review Letters | volume=112 | issue=7 | date=20 February 2014 | issn=0031-9007 | doi=10.1103/physrevlett.112.077203 | page=077203| pmid=24579631 | arxiv=1308.1488 | bibcode=2014PhRvL.112g7203P | s2cid=42262386 }}</ref><ref>{{cite journal | last1=Giannozzi | first1=Paolo | last2=Erdös | first2=Paul | title=Theoretical analysis of the 3-k magnetic structure and distortion of uranium dioxide | journal=Journal of Magnetism and Magnetic Materials | publisher=Elsevier BV | volume=67 | issue=1 | year=1987 | issn=0304-8853 | doi=10.1016/0304-8853(87)90722-0 | pages=75–87| bibcode=1987JMMM...67...75G }}</ref><ref>{{cite book | last1=Mironov | first1=V.S | last2=Chibotaru | first2=L.F | last3=Ceulemans | first3=A | title=Advances in Quantum Chemistry | chapter=First-order Phase Transition in UO<sub>2</sub>: The Interplay of the 5f<sup>2</sup>–5f<sup>2</sup> Superexchange Interaction and Jahn–Teller Effect | publisher=Elsevier | year=2003 | isbn=978-0-12-034844-2 | issn=0065-3276 | doi=10.1016/s0065-3276(03)44040-9 | pages=599–616|volume=44}}</ref><ref>{{cite journal | last1=Carretta | first1=S. | last2=Santini | first2=P. | last3=Caciuffo | first3=R. | last4=Amoretti | first4=G. | title=Quadrupolar Waves in Uranium Dioxide | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=105 | issue=16 | date=11 October 2010 | issn=0031-9007 | doi=10.1103/physrevlett.105.167201 | page=167201| pmid=21231002 | bibcode=2010PhRvL.105p7201C }}</ref><ref>{{cite journal | last1=Caciuffo | first1=R. | last2=Santini | first2=P. | last3=Carretta | first3=S. | last4=Amoretti | first4=G. | last5=Hiess | first5=A. | last6=Magnani | first6=N. | last7=Regnault | first7=L.-P. | last8=Lander | first8=G. H. | title=Multipolar, magnetic, and vibrational lattice dynamics in the low-temperature phase of uranium dioxide | journal=Physical Review B | volume=84 | issue=10 | date=6 September 2011 | issn=1098-0121 | doi=10.1103/physrevb.84.104409 | page=104409| arxiv=1312.5113 | bibcode=2011PhRvB..84j4409C | s2cid=118624728 }}</ref> NpO<sub>2</sub>,<ref name="NpO2">{{cite journal | last1=Santini | first1=P. | last2=Amoretti | first2=G. | title=Magnetic-Octupole Order in Neptunium Dioxide? | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=85 | issue=10 | date=4 September 2000 | issn=0031-9007 | doi=10.1103/physrevlett.85.2188 | pages=2188–2191| pmid=10970494 | bibcode=2000PhRvL..85.2188S }}</ref><ref>{{cite journal | last1=Santini | first1=P. | last2=Carretta | first2=S. | last3=Magnani | first3=N. | last4=Amoretti | first4=G. | last5=Caciuffo | first5=R. | title=Hidden Order and Low-Energy Excitations in NpO<sub>2</sub> | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=97 | issue=20 | date=14 November 2006 | issn=0031-9007 | doi=10.1103/physrevlett.97.207203 | page=207203| pmid=17155710 | bibcode=2006PhRvL..97t7203S }}</ref><ref>{{cite journal | last1=Kubo | first1=Katsunori | last2=Hotta | first2=Takashi | title=Microscopic theory of multipole ordering in NpO<sub>2</sub> | journal=Physical Review B | publisher=American Physical Society (APS) | volume=71 | issue=14 | date=29 April 2005 | issn=1098-0121 | doi=10.1103/physrevb.71.140404 | page=140404(R)| arxiv=cond-mat/0409116 | bibcode=2005PhRvB..71n0404K | s2cid=119391692 }}</ref> Ce<sub>1−x</sub>La<sub>x</sub>B<sub>6</sub>,<ref name="Ce1−xLaxB6">{{cite journal | last1=Mannix | first1=D. | last2=Tanaka | first2=Y. | last3=Carbone | first3=D. | last4=Bernhoeft | first4=N. | last5=Kunii | first5=S. | title=Order Parameter Segregation in Ce<sub>0.7</sub>La<sub>0.3</sub>B<sub>6</sub>:4f Octopole and 5d Dipole Magnetic Order | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=95 | issue=11 | date=8 September 2005 | issn=0031-9007 | doi=10.1103/physrevlett.95.117206 | page=117206| pmid=16197044 | bibcode=2005PhRvL..95k7206M }}</ref> URu<sub>2</sub>Si<sub>2</sub><ref name="URu2Si2">{{cite journal | last1=Chandra | first1=P. | last2=Coleman | first2=P. | last3=Mydosh | first3=J. A. | last4=Tripathi | first4=V. | title=Hidden orbital order in the heavy fermion metal URu<sub>2</sub>Si<sub>2</sub> | journal=Nature | publisher=Springer Nature | volume=417 | issue=6891 | year=2002 | issn=0028-0836 | doi=10.1038/nature00795 | pages=831–834| pmid=12075346 | arxiv=cond-mat/0205003 | bibcode=2002Natur.417..831C | s2cid=11902278 }}</ref><ref>{{cite journal | last1=Cricchio | first1=Francesco | last2=Bultmark | first2=Fredrik | last3=Grånäs | first3=Oscar | last4=Nordström | first4=Lars | title=Itinerant Magnetic Multipole Moments of Rank Five as the Hidden Order in URu<sub>2</sub>Si<sub>2</sub> | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=103 | issue=10 | date=1 August 2009 | issn=0031-9007 | doi=10.1103/physrevlett.103.107202 | page=107202| pmid=19792338 | arxiv=0904.3883 | bibcode=2009PhRvL.103j7202C | s2cid=20622071 }}</ref><ref>{{cite journal | last1=Ikeda | first1=Hiroaki | last2=Suzuki | first2=Michi-To | last3=Arita | first3=Ryotaro | last4=Takimoto | first4=Tetsuya | last5=Shibauchi | first5=Takasada | last6=Matsuda | first6=Yuji | title=Emergent rank-5 nematic order in URu<sub>2</sub>Si<sub>2</sub> | journal=Nature Physics | volume=8 | issue=7 | date=3 June 2012 | issn=1745-2473 | doi=10.1038/nphys2330 | pages=528–533| arxiv=1204.4016 | bibcode=2012NatPh...8..528I | s2cid=119108102 }}</ref><ref>{{cite journal | last1=Kiss | first1=Annamária | last2=Fazekas | first2=Patrik | title=Group theory and octupolar order in URu<sub>2</sub>Si<sub>2</sub> | journal=Physical Review B | publisher=American Physical Society (APS) | volume=71 | issue=5 | date=23 February 2005 | issn=1098-0121 | doi=10.1103/physrevb.71.054415 | page=054415| arxiv=cond-mat/0411029 | bibcode=2005PhRvB..71e4415K | s2cid=118892596 }}</ref><ref>{{cite journal | last1=Rau | first1=Jeffrey G. | last2=Kee | first2=Hae-Young | title=Hidden and antiferromagnetic order as a rank-5 superspin in URu<sub>2</sub>Si<sub>2</sub> | journal=Physical Review B | volume=85 | issue=24 | date=13 June 2012 | issn=1098-0121 | doi=10.1103/physrevb.85.245112 | page=245112| arxiv=1203.1047 | bibcode=2012PhRvB..85x5112R | s2cid=118313829 }}</ref> and many other compounds, are found to have magnetic ordering constituted by high rank multipoles, e.g. quadruple,
== Tensor operator expansion ==
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