Variation of exchange interaction in the GeNi2-xCoxO4 systems and in the CoRh2O4 nanoparticle
Identifieur interne : 000928 ( Istex/Corpus ); précédent : 000927; suivant : 000929Variation of exchange interaction in the GeNi2-xCoxO4 systems and in the CoRh2O4 nanoparticle
Auteurs : R. Masrour ; M. Hamedoun ; A. Hourmatallah ; K. Bouslykhane ; N. BenzakourSource :
- Physica Scripta [ 0031-8949 ] ; 2008.
Abstract
The magnetic properties of the spinels GeNi2-xCoxO4 systems in the range 0 x 2 have been studied by mean field theory (MFT) and high-temperature series expansions. The nearest neighbouring and the next-neighbouring super-exchange interactions J1 (x) and J2 (x) are evaluated for the spinel GeNi2-xCoxO4 systems in the range 0 x 2. The exchanges interactions are calculated for different sizes of CoRh2O4 nanoparticle by using the MFT. The intra-planar and the inter-planar interactions and the exchange energy are deduced for GeNi2-xCoxO4 and CoRh2O4. The second theory has been applied to the spinel GeNi2-xCoxO4 systems, combined with the Pad approximant method, we have obtained the magnetic phase diagrams (TN versus dilution x) in the range 0 x 2. The obtained theoretical results are in agreement with experimental ones obtained by magnetic measurements. The critical exponents associated with the magnetic susceptibility () and the correlation lengths () are deduced. The obtained theoretical results are in agreement with the experimental data obtained by magnetic measurements. The critical exponent associated to the correlation lengths () is deduced for different sizes of CoRh2O4 nanoparticle.
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DOI: 10.1088/0031-8949/78/02/025702
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<record><TEI wicri:istexFullTextTei="biblStruct"><teiHeader><fileDesc><titleStmt><title xml:lang="en">Variation of exchange interaction in the GeNi2-xCoxO4 systems and in the CoRh2O4 nanoparticle</title><author><name sortKey="Masrour, R" sort="Masrour, R" uniqKey="Masrour R" first="R" last="Masrour">R. Masrour</name><affiliation><mods:affiliation>Laboratoire de Physique du Solide, Universit Sidi Mohammed Ben Abdellah, Facult des sciences, BP 1796, Fs, Morocco</mods:affiliation></affiliation><affiliation><mods:affiliation>E-mail: rachidmasrour@hotmail.com</mods:affiliation></affiliation></author><author><name sortKey="Hamedoun, M" sort="Hamedoun, M" uniqKey="Hamedoun M" first="M" last="Hamedoun">M. Hamedoun</name><affiliation><mods:affiliation>Laboratoire de Physique du Solide, Universit Sidi Mohammed Ben Abdellah, Facult des sciences, BP 1796, Fs, Morocco</mods:affiliation></affiliation><affiliation><mods:affiliation>Expert auprs de l'Acadmie Hassan II des Sciences et Techniques, Rabat, Morocco</mods:affiliation></affiliation></author><author><name sortKey="Hourmatallah, A" sort="Hourmatallah, A" uniqKey="Hourmatallah A" first="A" last="Hourmatallah">A. Hourmatallah</name><affiliation><mods:affiliation>Laboratoire de Physique du Solide, Universit Sidi Mohammed Ben Abdellah, Facult des sciences, BP 1796, Fs, Morocco</mods:affiliation></affiliation><affiliation><mods:affiliation>Equipe de Physique du Solide, Ecole Normale Suprieure, BP 5206, Bensouda, Fes, Morocco</mods:affiliation></affiliation></author><author><name sortKey="Bouslykhane, K" sort="Bouslykhane, K" uniqKey="Bouslykhane K" first="K" last="Bouslykhane">K. Bouslykhane</name><affiliation><mods:affiliation>Laboratoire de Physique du Solide, Universit Sidi Mohammed Ben Abdellah, Facult des sciences, BP 1796, Fs, Morocco</mods:affiliation></affiliation></author><author><name sortKey="Benzakour, N" sort="Benzakour, N" uniqKey="Benzakour N" first="N" last="Benzakour">N. Benzakour</name><affiliation><mods:affiliation>Laboratoire de Physique du Solide, Universit Sidi Mohammed Ben Abdellah, Facult des sciences, BP 1796, Fs, Morocco</mods:affiliation></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">ISTEX</idno><idno type="RBID">ISTEX:2C01C83D03782FBE4D582F705AE2B3DA133B2022</idno><date when="2008" year="2008">2008</date><idno type="doi">10.1088/0031-8949/78/02/025702</idno><idno type="url">https://api.istex.fr/document/2C01C83D03782FBE4D582F705AE2B3DA133B2022/fulltext/pdf</idno><idno type="wicri:Area/Istex/Corpus">000928</idno><idno type="wicri:explorRef" wicri:stream="Istex" wicri:step="Corpus" wicri:corpus="ISTEX">000928</idno></publicationStmt><sourceDesc><biblStruct><analytic><title level="a" type="main" xml:lang="en">Variation of exchange interaction in the GeNi2-xCoxO4 systems and in the CoRh2O4 nanoparticle</title><author><name sortKey="Masrour, R" sort="Masrour, R" uniqKey="Masrour R" first="R" last="Masrour">R. Masrour</name><affiliation><mods:affiliation>Laboratoire de Physique du Solide, Universit Sidi Mohammed Ben Abdellah, Facult des sciences, BP 1796, Fs, Morocco</mods:affiliation></affiliation><affiliation><mods:affiliation>E-mail: rachidmasrour@hotmail.com</mods:affiliation></affiliation></author><author><name sortKey="Hamedoun, M" sort="Hamedoun, M" uniqKey="Hamedoun M" first="M" last="Hamedoun">M. Hamedoun</name><affiliation><mods:affiliation>Laboratoire de Physique du Solide, Universit Sidi Mohammed Ben Abdellah, Facult des sciences, BP 1796, Fs, Morocco</mods:affiliation></affiliation><affiliation><mods:affiliation>Expert auprs de l'Acadmie Hassan II des Sciences et Techniques, Rabat, Morocco</mods:affiliation></affiliation></author><author><name sortKey="Hourmatallah, A" sort="Hourmatallah, A" uniqKey="Hourmatallah A" first="A" last="Hourmatallah">A. Hourmatallah</name><affiliation><mods:affiliation>Laboratoire de Physique du Solide, Universit Sidi Mohammed Ben Abdellah, Facult des sciences, BP 1796, Fs, Morocco</mods:affiliation></affiliation><affiliation><mods:affiliation>Equipe de Physique du Solide, Ecole Normale Suprieure, BP 5206, Bensouda, Fes, Morocco</mods:affiliation></affiliation></author><author><name sortKey="Bouslykhane, K" sort="Bouslykhane, K" uniqKey="Bouslykhane K" first="K" last="Bouslykhane">K. Bouslykhane</name><affiliation><mods:affiliation>Laboratoire de Physique du Solide, Universit Sidi Mohammed Ben Abdellah, Facult des sciences, BP 1796, Fs, Morocco</mods:affiliation></affiliation></author><author><name sortKey="Benzakour, N" sort="Benzakour, N" uniqKey="Benzakour N" first="N" last="Benzakour">N. Benzakour</name><affiliation><mods:affiliation>Laboratoire de Physique du Solide, Universit Sidi Mohammed Ben Abdellah, Facult des sciences, BP 1796, Fs, Morocco</mods:affiliation></affiliation></author></analytic><monogr></monogr><series><title level="j">Physica Scripta</title><title level="j" type="abbrev">Phys. Scr.</title><idno type="ISSN">0031-8949</idno><imprint><publisher>IOP Publishing</publisher><date type="published" when="2008">2008</date><biblScope unit="volume">78</biblScope><biblScope unit="issue">2</biblScope><biblScope unit="page" from="1">1</biblScope><biblScope unit="page" to="4">4</biblScope><biblScope unit="production">Printed in the UK</biblScope></imprint><idno type="ISSN">0031-8949</idno></series><idno type="istex">2C01C83D03782FBE4D582F705AE2B3DA133B2022</idno><idno type="DOI">10.1088/0031-8949/78/02/025702</idno><idno type="articleID">257512</idno><idno type="articleNumber">025702</idno></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">0031-8949</idno></seriesStmt></fileDesc><profileDesc><textClass></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract">The magnetic properties of the spinels GeNi2-xCoxO4 systems in the range 0 x 2 have been studied by mean field theory (MFT) and high-temperature series expansions. The nearest neighbouring and the next-neighbouring super-exchange interactions J1 (x) and J2 (x) are evaluated for the spinel GeNi2-xCoxO4 systems in the range 0 x 2. The exchanges interactions are calculated for different sizes of CoRh2O4 nanoparticle by using the MFT. The intra-planar and the inter-planar interactions and the exchange energy are deduced for GeNi2-xCoxO4 and CoRh2O4. The second theory has been applied to the spinel GeNi2-xCoxO4 systems, combined with the Pad approximant method, we have obtained the magnetic phase diagrams (TN versus dilution x) in the range 0 x 2. The obtained theoretical results are in agreement with experimental ones obtained by magnetic measurements. The critical exponents associated with the magnetic susceptibility () and the correlation lengths () are deduced. The obtained theoretical results are in agreement with the experimental data obtained by magnetic measurements. The critical exponent associated to the correlation lengths () is deduced for different sizes of CoRh2O4 nanoparticle.</div></front></TEI><istex><corpusName>iop</corpusName><author><json:item><name>R Masrour</name><affiliations><json:string>Laboratoire de Physique du Solide, Universit Sidi Mohammed Ben Abdellah, Facult des sciences, BP 1796, Fs, Morocco</json:string><json:string>E-mail: rachidmasrour@hotmail.com</json:string></affiliations></json:item><json:item><name>M Hamedoun</name><affiliations><json:string>Laboratoire de Physique du Solide, Universit Sidi Mohammed Ben Abdellah, Facult des sciences, BP 1796, Fs, Morocco</json:string><json:string>Expert auprs de l'Acadmie Hassan II des Sciences et Techniques, Rabat, Morocco</json:string></affiliations></json:item><json:item><name>A Hourmatallah</name><affiliations><json:string>Laboratoire de Physique du Solide, Universit Sidi Mohammed Ben Abdellah, Facult des sciences, BP 1796, Fs, Morocco</json:string><json:string>Equipe de Physique du Solide, Ecole Normale Suprieure, BP 5206, Bensouda, Fes, Morocco</json:string></affiliations></json:item><json:item><name>K Bouslykhane</name><affiliations><json:string>Laboratoire de Physique du Solide, Universit Sidi Mohammed Ben Abdellah, Facult des sciences, BP 1796, Fs, Morocco</json:string></affiliations></json:item><json:item><name>N Benzakour</name><affiliations><json:string>Laboratoire de Physique du Solide, Universit Sidi Mohammed Ben Abdellah, Facult des sciences, BP 1796, Fs, Morocco</json:string></affiliations></json:item></author><language><json:string>eng</json:string></language><originalGenre><json:string>paper</json:string></originalGenre><abstract>The magnetic properties of the spinels GeNi2-xCoxO4 systems in the range 0 x 2 have been studied by mean field theory (MFT) and high-temperature series expansions. The nearest neighbouring and the next-neighbouring super-exchange interactions J1 (x) and J2 (x) are evaluated for the spinel GeNi2-xCoxO4 systems in the range 0 x 2. The exchanges interactions are calculated for different sizes of CoRh2O4 nanoparticle by using the MFT. The intra-planar and the inter-planar interactions and the exchange energy are deduced for GeNi2-xCoxO4 and CoRh2O4. The second theory has been applied to the spinel GeNi2-xCoxO4 systems, combined with the Pad approximant method, we have obtained the magnetic phase diagrams (TN versus dilution x) in the range 0 x 2. The obtained theoretical results are in agreement with experimental ones obtained by magnetic measurements. The critical exponents associated with the magnetic susceptibility () and the correlation lengths () are deduced. The obtained theoretical results are in agreement with the experimental data obtained by magnetic measurements. The critical exponent associated to the correlation lengths () is deduced for different sizes of CoRh2O4 nanoparticle.</abstract><qualityIndicators><score>5.294</score><pdfVersion>1.4</pdfVersion><pdfPageSize>595.276 x 841.89 pts (A4)</pdfPageSize><refBibsNative>true</refBibsNative><keywordCount>0</keywordCount><abstractCharCount>1205</abstractCharCount><pdfWordCount>2646</pdfWordCount><pdfCharCount>14715</pdfCharCount><pdfPageCount>4</pdfPageCount><abstractWordCount>179</abstractWordCount></qualityIndicators><title>Variation of exchange interaction in the GeNi2-xCoxO4 systems and in the CoRh2O4 nanoparticle</title><genre><json:string>research-article</json:string></genre><host><volume>78</volume><publisherId><json:string>ps</json:string></publisherId><pages><total>4</total><last>4</last><first>1</first></pages><issn><json:string>0031-8949</json:string></issn><issue>2</issue><genre><json:string>journal</json:string></genre><language><json:string>unknown</json:string></language><title>Physica Scripta</title></host><publicationDate>2008</publicationDate><copyrightDate>2008</copyrightDate><doi><json:string>10.1088/0031-8949/78/02/025702</json:string></doi><id>2C01C83D03782FBE4D582F705AE2B3DA133B2022</id><score>0.07844219</score><fulltext><json:item><original>true</original><mimetype>application/pdf</mimetype><extension>pdf</extension><uri>https://api.istex.fr/document/2C01C83D03782FBE4D582F705AE2B3DA133B2022/fulltext/pdf</uri></json:item><json:item><original>false</original><mimetype>application/zip</mimetype><extension>zip</extension><uri>https://api.istex.fr/document/2C01C83D03782FBE4D582F705AE2B3DA133B2022/fulltext/zip</uri></json:item><istex:fulltextTEI uri="https://api.istex.fr/document/2C01C83D03782FBE4D582F705AE2B3DA133B2022/fulltext/tei"><teiHeader><fileDesc><titleStmt><title level="a" type="main" xml:lang="en">Variation of exchange interaction in the GeNi2-xCoxO4 systems and in the CoRh2O4 nanoparticle</title></titleStmt><publicationStmt><authority>ISTEX</authority><publisher>IOP Publishing</publisher><availability><p>2008 The Royal Swedish Academy of Sciences</p></availability><date>2008</date></publicationStmt><sourceDesc><biblStruct type="inbook"><analytic><title level="a" type="main" xml:lang="en">Variation of exchange interaction in the GeNi2-xCoxO4 systems and in the CoRh2O4 nanoparticle</title><author xml:id="author-1"><persName><forename type="first">R</forename><surname>Masrour</surname></persName><email>rachidmasrour@hotmail.com</email><affiliation>Laboratoire de Physique du Solide, Universit Sidi Mohammed Ben Abdellah, Facult des sciences, BP 1796, Fs, Morocco</affiliation></author><author xml:id="author-2"><persName><forename type="first">M</forename><surname>Hamedoun</surname></persName><affiliation>Laboratoire de Physique du Solide, Universit Sidi Mohammed Ben Abdellah, Facult des sciences, BP 1796, Fs, Morocco</affiliation><affiliation>Expert auprs de l'Acadmie Hassan II des Sciences et Techniques, Rabat, Morocco</affiliation></author><author xml:id="author-3"><persName><forename type="first">A</forename><surname>Hourmatallah</surname></persName><affiliation>Laboratoire de Physique du Solide, Universit Sidi Mohammed Ben Abdellah, Facult des sciences, BP 1796, Fs, Morocco</affiliation><affiliation>Equipe de Physique du Solide, Ecole Normale Suprieure, BP 5206, Bensouda, Fes, Morocco</affiliation></author><author xml:id="author-4"><persName><forename type="first">K</forename><surname>Bouslykhane</surname></persName><affiliation>Laboratoire de Physique du Solide, Universit Sidi Mohammed Ben Abdellah, Facult des sciences, BP 1796, Fs, Morocco</affiliation></author><author xml:id="author-5"><persName><forename type="first">N</forename><surname>Benzakour</surname></persName><affiliation>Laboratoire de Physique du Solide, Universit Sidi Mohammed Ben Abdellah, Facult des sciences, BP 1796, Fs, Morocco</affiliation></author></analytic><monogr><title level="j">Physica Scripta</title><title level="j" type="abbrev">Phys. Scr.</title><idno type="pISSN">0031-8949</idno><imprint><publisher>IOP Publishing</publisher><date type="published" when="2008"></date><biblScope unit="volume">78</biblScope><biblScope unit="issue">2</biblScope><biblScope unit="page" from="1">1</biblScope><biblScope unit="page" to="4">4</biblScope><biblScope unit="production">Printed in the UK</biblScope></imprint></monogr><idno type="istex">2C01C83D03782FBE4D582F705AE2B3DA133B2022</idno><idno type="DOI">10.1088/0031-8949/78/02/025702</idno><idno type="articleID">257512</idno><idno type="articleNumber">025702</idno></biblStruct></sourceDesc></fileDesc><profileDesc><creation><date>2008</date></creation><langUsage><language ident="en">en</language></langUsage><abstract><p>The magnetic properties of the spinels GeNi2-xCoxO4 systems in the range 0 x 2 have been studied by mean field theory (MFT) and high-temperature series expansions. The nearest neighbouring and the next-neighbouring super-exchange interactions J1 (x) and J2 (x) are evaluated for the spinel GeNi2-xCoxO4 systems in the range 0 x 2. The exchanges interactions are calculated for different sizes of CoRh2O4 nanoparticle by using the MFT. The intra-planar and the inter-planar interactions and the exchange energy are deduced for GeNi2-xCoxO4 and CoRh2O4. The second theory has been applied to the spinel GeNi2-xCoxO4 systems, combined with the Pad approximant method, we have obtained the magnetic phase diagrams (TN versus dilution x) in the range 0 x 2. The obtained theoretical results are in agreement with experimental ones obtained by magnetic measurements. The critical exponents associated with the magnetic susceptibility () and the correlation lengths () are deduced. The obtained theoretical results are in agreement with the experimental data obtained by magnetic measurements. The critical exponent associated to the correlation lengths () is deduced for different sizes of CoRh2O4 nanoparticle.</p></abstract><textClass><classCode scheme="">75.10.Hk</classCode><classCode scheme="">75.10.Nr</classCode><classCode scheme="">75.25.z</classCode><classCode scheme="">75.30.Et</classCode></textClass></profileDesc><revisionDesc><change when="2008">Published</change></revisionDesc></teiHeader></istex:fulltextTEI><json:item><original>false</original><mimetype>text/plain</mimetype><extension>txt</extension><uri>https://api.istex.fr/document/2C01C83D03782FBE4D582F705AE2B3DA133B2022/fulltext/txt</uri></json:item></fulltext><metadata><istex:metadataXml wicri:clean="corpus iop not found" wicri:toSee="no header"><istex:xmlDeclaration>version="1.0" encoding="UTF-8"</istex:xmlDeclaration><istex:docType SYSTEM="http://ej.iop.org/dtd/iopv1_5_2.dtd" name="istex:docType"></istex:docType><istex:document><article artid="pscr257512"><article-metadata><jnl-data jnlid="ps"><jnl-fullname>Physica Scripta</jnl-fullname><jnl-abbreviation>Phys. Scr.</jnl-abbreviation><jnl-shortname>PhysScr</jnl-shortname><jnl-issn>0031-8949</jnl-issn><jnl-coden>PHSTBO</jnl-coden><jnl-imprint>IOP Publishing</jnl-imprint><jnl-web-address>stacks.iop.org/PhysScr</jnl-web-address></jnl-data><volume-data><year-publication>2008</year-publication><volume-number>78</volume-number></volume-data><issue-data><issue-number>2</issue-number><coverdate>August 2008</coverdate></issue-data><article-data><article-type type="paper" sort="regular"></article-type><type-number type="paper" numbering="article" artnum="025702">025702</type-number><article-number>257512</article-number><first-page>1</first-page><last-page>4</last-page><length>4</length><pii></pii><doi>10.1088/0031-8949/78/02/025702</doi><copyright>2008 The Royal Swedish Academy of Sciences</copyright><ccc>0031-8949/08/025702+04$30.00</ccc><printed>Printed in the UK</printed></article-data></article-metadata><header><title-group><title>Variation of exchange interaction in the <inline-eqn><math-text><upright>GeNi</upright><sub>2-<italic>x</italic></sub><upright>Co</upright><sub><italic>x</italic></sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> systems and in the <inline-eqn><math-text><upright>CoRh</upright><sub>2</sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> nanoparticle</title><short-title>Variation of exchange interaction in the <inline-eqn><math-text><upright>GeNi</upright><sub>2-<italic>x</italic></sub><upright>Co</upright><sub><italic>x</italic></sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> systems and in the <inline-eqn><math-text><upright>CoRh</upright><sub>2</sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> nanoparticle</short-title><ej-title>Variation of exchange interaction in the GeNi2-xCoxO4 systems and in the CoRh2O4 nanoparticle</ej-title></title-group><author-group><author address="pscr257512ad1" email="pscr257512ea1"><first-names>R</first-names><second-name>Masrour</second-name></author><author address="pscr257512ad1" second-address="pscr257512ad2"><first-names>M</first-names><second-name>Hamedoun</second-name></author><author address="pscr257512ad1" second-address="pscr257512ad3"><first-names>A</first-names><second-name>Hourmatallah</second-name></author><author address="pscr257512ad1"><first-names>K</first-names><second-name>Bouslykhane</second-name></author><author address="pscr257512ad1"><first-names>N</first-names><second-name>Benzakour</second-name></author><short-author-list>R Masrour <italic>et al</italic></short-author-list></author-group><address-group><address id="pscr257512ad1" showid="yes">Laboratoire de Physique du Solide, <orgname>Université Sidi Mohammed Ben Abdellah</orgname>, Faculté des sciences, BP 1796, Fès, <country>Morocco</country></address><address id="pscr257512ad2" showid="yes"><orgname>Expert auprès de l'Académie Hassan II des Sciences et Techniques</orgname>, Rabat, <country>Morocco</country></address><address id="pscr257512ad3" showid="yes">Equipe de Physique du Solide, <orgname>Ecole Normale Supérieure</orgname>, BP 5206, Bensouda, Fes, <country>Morocco</country></address><e-address id="pscr257512ea1"><email mailto="rachidmasrour@hotmail.com">rachidmasrour@hotmail.com</email></e-address></address-group><history received="7 August 2007" accepted="5 June 2008" online="25 July 2008"></history><abstract-group><abstract><heading>Abstract</heading><p indent="no">The magnetic properties of the spinels <inline-eqn><math-text><upright>GeNi</upright><sub>2-<italic>x</italic></sub><upright>Co</upright><sub><italic>x</italic></sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> systems in the range 0 ≦ <italic>x</italic> ≦ 2 have been studied by mean field theory (MFT) and high-temperature series expansions. The nearest neighbouring and the next-neighbouring super-exchange interactions <italic>J</italic><sub>1</sub> (<italic>x</italic>) and <italic>J</italic><sub>2</sub> (<italic>x</italic>) are evaluated for the spinel <inline-eqn><math-text><upright>GeNi</upright><sub>2-<italic>x</italic></sub><upright>Co</upright><sub><italic>x</italic></sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> systems in the range 0 ≦ <italic>x</italic> ≦ 2. The exchanges interactions are calculated for different sizes of <inline-eqn><math-text><upright>CoRh</upright><sub>2</sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> nanoparticle by using the MFT. The intra-planar and the inter-planar interactions and the exchange energy are deduced for <inline-eqn><math-text><upright>GeNi</upright><sub>2-<italic>x</italic></sub><upright>Co</upright><sub><italic>x</italic></sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> and <inline-eqn><math-text><upright>CoRh</upright><sub>2</sub><upright>O</upright><sub>4</sub></math-text></inline-eqn>. The second theory has been applied to the spinel <inline-eqn><math-text><upright>GeNi</upright><sub>2-<italic>x</italic></sub><upright>Co</upright><sub><italic>x</italic></sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> systems, combined with the Padé approximant method, we have obtained the magnetic phase diagrams (<inline-eqn><math-text><italic>T</italic><sub><upright>N</upright></sub></math-text></inline-eqn> versus dilution <inline-eqn><math-text><italic>x</italic></math-text></inline-eqn>) in the range 0 ≦ <italic>x</italic> ≦ 2. The obtained theoretical results are in agreement with experimental ones obtained by magnetic measurements. The critical exponents associated with the magnetic susceptibility (γ) and the correlation lengths (ν) are deduced. The obtained theoretical results are in agreement with the experimental data obtained by magnetic measurements. The critical exponent associated to the correlation lengths (ν) is deduced for different sizes of <inline-eqn><math-text><upright>CoRh</upright><sub>2</sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> nanoparticle.</p></abstract></abstract-group><classifications><class-codes scheme="pacs"><code>75.10.Hk</code><code>75.10.Nr</code><code>75.25.+z</code><code>75.30.Et</code></class-codes></classifications></header><body refstyle="numeric"><sec-level1 id="pscr257512s1" label="1"><heading>Introduction</heading><p indent="no">The materials with spinel structures, with the formula <inline-eqn><math-text><upright>AB</upright><sub>2</sub><upright>X</upright><sub>4</sub></math-text></inline-eqn> are of continued interest because of their wide variety of physical properties and potential applications in nanoscience and technology as high-density magnetic recording media, magnetic carriers in ferro fluids, magnetically guided drug carriers, etc [<cite linkend="pscr257512bib1">1</cite>]. This is essentially related to (i) the existence of two types of crystallographic sublattices, tetrahedral (A) and octahedral (B), available for the metal ions; (ii) the great flexibility of the structure in hosting various metal ions, differently distributed between the two sublattices, with a large possibility of reciprocal substitution between them. In the nickel <inline-eqn><math-text><upright>GeNi</upright><sub>2</sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> and cobalt <inline-eqn><math-text><upright>GeCo</upright><sub>2</sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> spinels, <inline-eqn><math-text><upright>Ni</upright><sup>2+</sup></math-text></inline-eqn> and <inline-eqn><math-text><upright>Co</upright><sup>2+</sup></math-text></inline-eqn> ions form a network of tetrahedral sharing corners like a pyrochlore lattice. These systems are a geometrically frustrated magnet if the <inline-eqn><math-text><upright>Ni</upright><sup>2+</sup></math-text></inline-eqn> and <inline-eqn><math-text><upright>Co</upright><sup>2+</sup></math-text></inline-eqn> spins are coupled antiferromagnetically. Former studies have revealed a two-step magnetic transition at <inline-eqn><math-text><italic>T</italic><sub><upright>N</upright>1</sub> = 12.13 <upright>K</upright></math-text></inline-eqn> and <inline-eqn><math-text><italic>T</italic><sub><upright>N</upright>2</sub> = 11.46 <upright>K</upright></math-text></inline-eqn>, and did not reveal a structural transition at <inline-eqn><math-text><italic>T</italic><sub><upright>N</upright>1</sub></math-text></inline-eqn> and <inline-eqn><math-text><italic>T</italic><sub><upright>N</upright>2</sub></math-text></inline-eqn>, which is significantly different from the spinels with half-integer spins [<cite linkend="pscr257512bib2">2</cite>]. Recently, a muon-spin relaxation <inline-eqn><math-text>(μ <sup>+</sup><upright>SR</upright>)</math-text></inline-eqn> study of <inline-eqn><math-text><upright>GeNi</upright><sub>2</sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> showed evidence for phase separation into short-range- and long-range-ordered regions at low temperature [<cite linkend="pscr257512bib3">3</cite>]. Moreover, a cusp appears in the specific heat at 11 K, which implies that a third magnetic phase transition may be present in <inline-eqn><math-text><upright>GeNi</upright><sub>2</sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> [<cite linkend="pscr257512bib4">4</cite>]. On the other hand, <inline-eqn><math-text><upright>GeCo</upright><sub>2</sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> with a half-integer spin <inline-eqn><math-text>(<italic>S</italic>= 3/2)</math-text></inline-eqn> magnetic sublattice undergoes a lattice distortion at a single magnetic transition <inline-eqn><math-text><italic>T</italic><sub><upright>N</upright></sub> = 20 <upright>K</upright></math-text></inline-eqn> [<cite linkend="pscr257512bib5">5</cite>]. <inline-eqn><math-text><upright>CoRh</upright><sub>2</sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> (structure: <inline-eqn></inline-eqn>) with <inline-eqn><math-text><italic>T</italic><sub><upright>N</upright></sub> ≈ 27 <upright>K</upright></math-text></inline-eqn> is an anlogous of <inline-eqn><math-text><upright>Co</upright><sub>3</sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> (structure: <inline-eqn></inline-eqn>), where <inline-eqn><math-text><upright>Co</upright><sup>3 +</sup></math-text></inline-eqn> is replaced by non-magnetic <inline-eqn><math-text><upright>Rh</upright><sup>3 +</sup></math-text></inline-eqn> <inline-eqn><math-text>(4<upright>d</upright><sup>6</sup></math-text></inline-eqn>) ions [<cite linkend="pscr257512bib6">6</cite>].</p><p>In the first case, we have calculated the first and the second nearest neighbours exchange interactions <italic>J</italic><sub>1</sub>(<italic>x</italic>) and <italic>J</italic><sub>2</sub>(<italic>x</italic>), respectively on the basis of magnetic results in <inline-eqn><math-text><upright>GeNi</upright><sub>2-<italic>x</italic></sub><upright>Co</upright><sub><italic>x</italic></sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> for 0 ≦ <italic>x</italic> ≦ 2 [<cite linkend="pscr257512bib7">7</cite>]. The second case, we have applied the same theory to the <inline-eqn><math-text><upright>CoRh</upright><sub>2</sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> nanoparticle for calculating the exchange interactions with different sizes [<cite linkend="pscr257512bib8">8</cite>]. The values of the intra-plane and inter-plane interactions <inline-eqn><math-text><italic>J</italic><sub><upright>aa</upright></sub></math-text></inline-eqn>, <inline-eqn><math-text><italic>J</italic><sub><upright>ab</upright></sub></math-text></inline-eqn> and <inline-eqn><math-text><italic>J</italic><sub><upright>ac</upright></sub></math-text></inline-eqn>, respectively, and the interaction energy of the magnetic structure are deduced from the values of <italic>J</italic><sub>1</sub> (<italic>x</italic>) and <italic>J</italic><sub>2</sub> (<italic>x</italic>) for the systems <inline-eqn><math-text><upright>GeNi</upright><sub>2-<italic>x</italic></sub><upright>Co</upright><sub><italic>x</italic></sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> in the range 0 ≦ <italic>x</italic> ≦ 2 (see table <tabref linkend="pscr257512tab1">1</tabref>) and for different sizes for the <inline-eqn><math-text><upright>CoRh</upright><sub>2</sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> nanoparticle (see table <tabref linkend="pscr257512tab2">2</tabref>).</p><table id="pscr257512tab1" frame="topbot" position="float" width="page" place="top"><caption type="table" id="pscr257512tc1" label="Table 1"><p>The Curie–Weiss temperature <inline-eqn></inline-eqn>, the Néel temperature <inline-eqn></inline-eqn>, the values of the first, second, intra-plane, inter-plane exchange integrals and the energy of <inline-eqn><math-text><upright>GeNi</upright><sub>2-<italic>x</italic></sub><upright>Co</upright><sub><italic>x</italic></sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> as a function of dilution <inline-eqn><math-text><italic>x</italic></math-text></inline-eqn>.</p></caption><tgroup cols="9"><colspec colnum="1" colname="col1" align="left"></colspec><colspec colnum="2" colname="col2" align="char" char="."></colspec><colspec colnum="3" colname="col3" align="char" char="."></colspec><colspec colnum="4" colname="col4" align="char" char="."></colspec><colspec colnum="5" colname="col5" align="char" char="."></colspec><colspec colnum="6" colname="col6" align="char" char="."></colspec><colspec colnum="7" colname="col7" align="char" char="."></colspec><colspec colnum="8" colname="col8" align="char" char="."></colspec><colspec colnum="9" colname="col9" align="char" char="."></colspec><thead><row><entry><inline-eqn><math-text><italic>x</italic></math-text></inline-eqn></entry><entry><inline-eqn></inline-eqn> [<cite linkend="pscr257512bib7">7</cite>]</entry><entry><inline-eqn></inline-eqn> [<cite linkend="pscr257512bib7">7</cite>]</entry><entry><inline-eqn></inline-eqn></entry><entry><inline-eqn></inline-eqn></entry><entry><inline-eqn></inline-eqn></entry><entry><inline-eqn></inline-eqn></entry><entry><inline-eqn></inline-eqn></entry><entry><inline-eqn></inline-eqn></entry></row></thead><tbody><row><entry>2.00</entry><entry>23</entry><entry>40</entry><entry><inline-eqn><math-text>−2.30</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−1.150</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−4.60</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−18.40</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−4.60</math-text></inline-eqn></entry><entry>27.60</entry></row><row><entry>1.80</entry><entry>22.60</entry><entry>37.5</entry><entry><inline-eqn><math-text>−2.26</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−1.130</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−4.52</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−18.08</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−4.52</math-text></inline-eqn></entry><entry>27.12</entry></row><row><entry>1.45</entry><entry>21.6</entry><entry>29</entry><entry><inline-eqn><math-text>−2.16</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−1.080</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−4.32</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−17.28</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−4.32</math-text></inline-eqn></entry><entry>25.92</entry></row><row><entry>1.20</entry><entry>21.2</entry><entry>26</entry><entry><inline-eqn><math-text>−2.12</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−1.060</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−4.24</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−16.96</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−4.24</math-text></inline-eqn></entry><entry>25.44</entry></row><row><entry>1.00</entry><entry>20.1</entry><entry>23</entry><entry><inline-eqn><math-text>−2.01</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−1.005</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−4.02</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−16.08</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−4.02</math-text></inline-eqn></entry><entry>24.12</entry></row><row><entry>0.85</entry><entry>20</entry><entry>19</entry><entry><inline-eqn><math-text>−2.00</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−1.000</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−4.00</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−16.00</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−4.00</math-text></inline-eqn></entry><entry>24.00</entry></row><row><entry>0.65</entry><entry>19</entry><entry>16</entry><entry><inline-eqn><math-text>−1.90</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.950</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−3.80</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−15.20</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−3.80</math-text></inline-eqn></entry><entry>22.80</entry></row><row><entry>0.50</entry><entry>18</entry><entry>10</entry><entry><inline-eqn><math-text>−1.80</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.900</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−3.60</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−14.40</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−3.60</math-text></inline-eqn></entry><entry>21.60</entry></row><row><entry>0.30</entry><entry>16.2</entry><entry>3</entry><entry><inline-eqn><math-text>−1.62</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.810</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−3.24</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−12.96</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−3.24</math-text></inline-eqn></entry><entry>19.44</entry></row><row><entry>0.20</entry><entry>16</entry><entry>0.3</entry><entry><inline-eqn><math-text>−1.60</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.800</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−3.20</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−12.80</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−3.20</math-text></inline-eqn></entry><entry>19.20</entry></row><row><entry>0.00</entry><entry>14</entry><entry><inline-eqn><math-text>−3</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−1.40</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.700</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−2.80</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−11.20</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−2.80</math-text></inline-eqn></entry><entry>16.80</entry></row></tbody></tgroup></table><table id="pscr257512tab2" frame="topbot" position="float" width="page" place="top"><caption type="table" id="pscr257512tc2" label="Table 2"><p>The Curie–Weiss temperature <inline-eqn></inline-eqn>, the Néel temperature <inline-eqn></inline-eqn>, the values of the first, second, intra-plane, inter-plane exchange integrals and the energy of <inline-eqn><math-text><upright>CoRh</upright><sub>2</sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> nanoparticle.</p></caption><tgroup cols="9"><colspec colnum="1" colname="col1" align="left"></colspec><colspec colnum="2" colname="col2" align="char" char="."></colspec><colspec colnum="3" colname="col3" align="char" char="."></colspec><colspec colnum="4" colname="col4" align="char" char="."></colspec><colspec colnum="5" colname="col5" align="char" char="."></colspec><colspec colnum="6" colname="col6" align="char" char="."></colspec><colspec colnum="7" colname="col7" align="char" char="."></colspec><colspec colnum="8" colname="col8" align="char" char="."></colspec><colspec colnum="9" colname="col9" align="char" char="."></colspec><thead><row><entry><inline-eqn></inline-eqn></entry><entry><inline-eqn></inline-eqn> [<cite linkend="pscr257512bib8">8</cite>]</entry><entry><inline-eqn></inline-eqn> [<cite linkend="pscr257512bib8">8</cite>]</entry><entry><inline-eqn></inline-eqn></entry><entry><inline-eqn></inline-eqn></entry><entry><inline-eqn></inline-eqn></entry><entry><inline-eqn></inline-eqn></entry><entry><inline-eqn></inline-eqn></entry><entry><inline-eqn></inline-eqn></entry></row></thead><tbody><row><entry>Bulk</entry><entry><inline-eqn><math-text>−44.23</math-text></inline-eqn></entry><entry><inline-eqn><math-text>27±0.5</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−1.474</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.737</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−2.948</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−11.792</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−2.948</math-text></inline-eqn></entry><entry>17.688</entry></row><row><entry>70</entry><entry><inline-eqn><math-text>−42.80</math-text></inline-eqn></entry><entry>27±0.5</entry><entry><inline-eqn><math-text>−1.426</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.713</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−2.852</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−11.408</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−2.852</math-text></inline-eqn></entry><entry>17.112</entry></row><row><entry>50</entry><entry><inline-eqn><math-text>−42.05</math-text></inline-eqn></entry><entry>27±0.5</entry><entry><inline-eqn><math-text>−1.401</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.700</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−2.802</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−11.204</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−2.802</math-text></inline-eqn></entry><entry>16.806</entry></row><row><entry>32</entry><entry><inline-eqn><math-text>−41.84</math-text></inline-eqn></entry><entry>27±0.5</entry><entry><inline-eqn><math-text>−1.394</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.697</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−2.788</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−11.152</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−2.788</math-text></inline-eqn></entry><entry>16.728</entry></row></tbody></tgroup></table><p>In recent works [<cite linkend="pscr257512bib9" range="pscr257512bib9,pscr257512bib10,pscr257512bib11">9–11</cite>], we have used the high-temperature series expansions (HTSE) to study the thermal and disorder variation of the short-range order (SRO) in the particular B-spinels <inline-eqn><math-text><upright>ZnCr</upright><sub>2x</sub><upright>Al</upright><sub>2 - 2x</sub><upright>S</upright><sub>4</sub></math-text></inline-eqn> and <inline-eqn><math-text><upright>Zn</upright><sub><italic>x</italic></sub><upright>Cd</upright><sub>1 - <italic>x</italic></sub><upright>Cr</upright><sub>2</sub><upright>S</upright><sub>4</sub></math-text></inline-eqn> compounds. The Padé approximant (PA) [<cite linkend="pscr257512bib12">12</cite>] analysis of the HTSE of the correlation length has been shown to be a useful method for the study of the critical region [<cite linkend="pscr257512bib13">13</cite>,<cite linkend="pscr257512bib14">14</cite>]. The model that we used in this present work is valid in the case of a spinel structure. We have used the HTSE technique for determining the Néel temperature <inline-eqn><math-text><italic>T</italic><sub><upright>N</upright></sub></math-text></inline-eqn> and the critical exponents <inline-eqn><math-text>γ</math-text></inline-eqn> and <inline-eqn><math-text>ν</math-text></inline-eqn> associated with the magnetic susceptibility <inline-eqn><math-text>χ</math-text></inline-eqn> and the correlation length <inline-eqn><math-text>ξ</math-text></inline-eqn>, respectively, for <inline-eqn><math-text><upright>GeNi</upright><sub>2-<italic>x</italic></sub><upright>Co</upright><sub><italic>x</italic></sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> in the range <inline-eqn></inline-eqn>.</p><p>In the <inline-eqn><math-text><upright>CoRh</upright><sub>2</sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> nanoparticle, the temperature shift has been observed in numerous experimental studies and it has also been investigated in theory [<cite linkend="pscr257512bib15" range="pscr257512bib15,pscr257512bib16,pscr257512bib17,pscr257512bib18">15–18</cite>]. Specifically, the Néel temperature <inline-eqn><math-text><italic>T</italic><sub><upright>N</upright></sub></math-text></inline-eqn> must be regarded as a diameter-dependent parameter, <inline-eqn></inline-eqn> which approaches the bulk critical temperature <inline-eqn></inline-eqn> as the scale factor <italic>L</italic> (the diameter of nanoparticle) approaches <inline-eqn><math-text>∞</math-text></inline-eqn>. It has been shown that the approach of <inline-eqn></inline-eqn> to <inline-eqn></inline-eqn> can also be described by a simple power law [<cite linkend="pscr257512bib19">19</cite>] characterized by a shift exponent λ defined by:
<display-eqn id="pscr257512eqn1" textype="equation" number="yes" notation="LaTeX" eqnnum="1"></display-eqn>the shift exponent <inline-eqn><math-text>λ</math-text></inline-eqn> is given by <inline-eqn></inline-eqn>, where <inline-eqn><math-text>ν <sub><upright>b</upright></sub></math-text></inline-eqn> is the correlation length critical exponent. In figure <figref linkend="pscr257512fig1">1</figref>, we exhibit the dependence of the shift <inline-eqn></inline-eqn> with <inline-eqn></inline-eqn> in a log–log scale to determine the exponent <inline-eqn><math-text>λ</math-text></inline-eqn> by using equation (<eqnref linkend="pscr257512eqn1">1</eqnref>).</p><figure id="pscr257512fig1" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic position="indented"><graphic-file version="print" format="EPS" width="19.6pc" printcolour="no" filename="physscr_78_2_025702eps/pscr257512fig1.eps"></graphic-file><graphic-file version="ej" format="JPEG" printcolour="no" filename="physscr_78_2_025702img/pscr257512fig1.jpg"></graphic-file></graphic><caption type="figure" id="pscr257512fc1" label="Figure 1"><p indent="no">Plot showing log–log shift of reduced critical temperature <inline-eqn></inline-eqn> versus diameter <inline-eqn></inline-eqn> of <inline-eqn><math-text><upright>CoRh</upright><sub>2</sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> nanoparticle.</p></caption></figure></sec-level1><sec-level1 id="pscr257512s2" label="2"><heading>Theoretical methods</heading><sec-level2 id="pscr257512s2.1" label="2.1"><heading>Calculation of the values of the exchange integrals from mean field approximation</heading><p indent="no">Starting from the well-known Heisenberg model, the Hamiltonian of the system is given by:
<display-eqn id="pscr257512eqn2" textype="equation" number="yes" notation="LaTeX" eqnnum="2"></display-eqn>where <inline-eqn><math-text><italic>J</italic><sub><italic>ij</italic></sub></math-text></inline-eqn> is the exchange integral between the spins situated at sites <italic>i</italic> and <italic>j</italic>. <inline-eqn></inline-eqn> is the spin operator of the spin localized at the site <italic>i</italic>. In this work, we consider the nearest neighbour <inline-eqn></inline-eqn> and the next nearest-neighbour <inline-eqn></inline-eqn> interactions <inline-eqn><math-text><italic>J</italic><sub>1</sub></math-text></inline-eqn> and <inline-eqn><math-text><italic>J</italic><sub>2</sub></math-text></inline-eqn>, respectively. In the case of spinels containing the magnetic moment only in the octahedral sublattice, the mean field approximation leads to a simple relation between the <inline-eqn><math-text>ϕ</math-text></inline-eqn> angle of helimagnetic ordering and the Néel temperature <inline-eqn><math-text><italic>T</italic><sub><upright>N</upright></sub></math-text></inline-eqn>, respectively, and the considered two exchange integrals <inline-eqn><math-text><italic>J</italic><sub>1</sub></math-text></inline-eqn> and <inline-eqn><math-text><italic>J</italic><sub>2</sub></math-text></inline-eqn>.</p><p>The Néel temperature <inline-eqn><math-text><italic>T</italic><sub><upright>N</upright></sub></math-text></inline-eqn> and the <inline-eqn><math-text>ϕ</math-text></inline-eqn> angle of helimagnetic ordering describing the <inline-eqn><math-text><upright>GeNi</upright><sub>2-<italic>x</italic></sub><upright>Co</upright><sub><italic>x</italic></sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> systems and <inline-eqn><math-text><upright>CoRh</upright><sub>2</sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> nanoparticle are given by [<cite linkend="pscr257512bib20">20</cite>]:
<display-eqn id="pscr257512eqn3" textype="equation" number="yes" notation="LaTeX" eqnnum="3"></display-eqn>where <inline-eqn></inline-eqn> is the eigenvalue of the matrix formed by the Fourier transform of the exchange integral and <inline-eqn><math-text>(<italic>S</italic>= 3/2)</math-text></inline-eqn>.
<display-eqn id="pscr257512eqn4" textype="equation" number="yes" notation="LaTeX" eqnnum="4"></display-eqn>where <inline-eqn><math-text><italic>K</italic><sub><upright>B</upright></sub></math-text></inline-eqn> is the Boltzmann's constant.</p><p>By using the experimental values of <inline-eqn><math-text><italic>T</italic><sub><upright>N</upright></sub></math-text></inline-eqn> obtained by magnetic measurement in the spinel <inline-eqn><math-text><upright>GeNi</upright><sub>2-<italic>x</italic></sub><upright>Co</upright><sub><italic>x</italic></sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> [<cite linkend="pscr257512bib7">7</cite>] and in the <inline-eqn><math-text><upright>CoRh</upright><sub>2</sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> nanoparticle [<cite linkend="pscr257512bib8">8</cite>]. We have deduced the values of exchange integrals <italic>J</italic><sub>1</sub> (<italic>x</italic>) and <italic>J</italic><sub>2</sub> (<italic>x</italic>) in the range 0 ≦ <italic>x</italic> ≦ 2. From these values, we have derived the variation of the intra-plane coupling and the coupling between nearest and next-nearest plane with the concentration <inline-eqn><math-text><italic>x</italic></math-text></inline-eqn> in the spinel <inline-eqn><math-text><upright>GeNi</upright><sub>2-<italic>x</italic></sub><upright>Co</upright><sub><italic>x</italic></sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> system with 0 ≦ <italic>x</italic> ≦ 2. The obtained values are given in table (1). The values of corresponding classical exchange energy for the magnetic structure [<cite linkend="pscr257512bib20">20</cite>] and the values of the intra-plane and inter-plane interactions <inline-eqn><math-text><italic>J</italic><sub><upright>aa</upright></sub> = 2<italic>J</italic><sub>1</sub></math-text></inline-eqn>, <inline-eqn><math-text><italic>J</italic><sub><upright>ab</upright></sub> = 4<italic>J</italic><sub>1</sub> + 8<italic>J</italic><sub>2</sub></math-text></inline-eqn> and <inline-eqn><math-text><italic>J</italic><sub><upright>ac</upright></sub> = 4<italic>J</italic><sub>2</sub></math-text></inline-eqn>, respectively, are also given in the same table for the <inline-eqn><math-text><upright>GeNi</upright><sub>2-<italic>x</italic></sub><upright>Co</upright><sub><italic>x</italic></sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> system with 0 ≦ <italic>x</italic> ≦ 2. The MFT is applied to the <inline-eqn><math-text><upright>CoRh</upright><sub>2</sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> nanoparticle for calculating the parameters <italic>J</italic><sub>1</sub> (<italic>x</italic>), <italic>J</italic><sub>2</sub> (<italic>x</italic>), the intra-plane, inter-plane interactions and the exchange energy of the magnetic structure of <inline-eqn><math-text><upright>CoRh</upright><sub>2</sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> nanoparticle for different sizes (see table <tabref linkend="pscr257512tab2">2</tabref>).</p></sec-level2><sec-level2 id="pscr257512s2.2" label="2.2"><heading>HTSE</heading><p indent="no">In this section, we shall derive HTSE for both the zero field magnetic susceptibility <inline-eqn><math-text>χ</math-text></inline-eqn> and the order sixth in <inline-eqn><math-text>β</math-text></inline-eqn>. The relationship between the magnetic susceptibility per spin and the correlation functions may be expressed as follows:
<display-eqn id="pscr257512eqn5" textype="equation" number="yes" notation="LaTeX" eqnnum="5"></display-eqn>where <inline-eqn></inline-eqn> and <inline-eqn><math-text><italic>N</italic></math-text></inline-eqn> is the number of magnetic ions; <inline-eqn></inline-eqn> is the correlation function between spins at sites <italic>i</italic> and <italic>j</italic>.</p><p>In the previous work [<cite linkend="pscr257512bib16">16</cite>], a relation between the susceptibility and the first three correlation functions is given in the case of the B-spinel lattice with a particular ordering vector <inline-eqn></inline-eqn>. In the ferromagnetic case, we get <inline-eqn><math-text><italic>k</italic>= 0</math-text></inline-eqn>. HTSE of χ(<italic>T</italic>) gives the function:
<display-eqn id="pscr257512eqn6" textype="equation" number="yes" notation="LaTeX" eqnnum="6"></display-eqn>HTSE of <inline-eqn><math-text>ξ <sup>2</sup></math-text></inline-eqn> gives the function:
<display-eqn id="pscr257512eqn7" textype="equation" number="yes" notation="LaTeX" eqnnum="7"></display-eqn>where <inline-eqn></inline-eqn> and <inline-eqn></inline-eqn>. The series coefficients <inline-eqn></inline-eqn> and <inline-eqn><math-text><italic>b</italic>(<italic>m</italic>,<italic>n</italic>)</math-text></inline-eqn> are given in [<cite linkend="pscr257512bib21">21</cite>].</p><p>Figure <figref linkend="pscr257512fig2">2</figref> shows a magnetic phase diagram of spinel <inline-eqn><math-text><upright>GeNi</upright><sub>2-<italic>x</italic></sub><upright>Co</upright><sub><italic>x</italic></sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> systems. We can see good agreement between the magnetic phase diagram obtained by the HTSE technique and the experimental ones, in particular, in the case of the last systems of which the phase diagrams have been established well by different methods [<cite linkend="pscr257512bib22" range="pscr257512bib22,pscr257512bib23,pscr257512bib24,pscr257512bib25">22–25</cite>].</p><figure id="pscr257512fig2" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic position="indented"><graphic-file version="print" format="EPS" width="19.9pc" printcolour="no" filename="physscr_78_2_025702eps/pscr257512fig2.eps"></graphic-file><graphic-file version="ej" format="JPEG" printcolour="no" filename="physscr_78_2_025702img/pscr257512fig2.jpg"></graphic-file></graphic><caption type="figure" id="pscr257512fc2" label="Figure 2"><p indent="no">Magnetic phase diagram of <inline-eqn><math-text><upright>GeNi</upright><sub>2-<italic>x</italic></sub><upright>Co</upright><sub><italic>x</italic></sub><upright>O</upright><sub>4</sub></math-text></inline-eqn>. The various phases are the PM phase and AFM phase <inline-eqn></inline-eqn>. The solid circles represent the present experimental results. The solid squares represent the experimental points deduced by magnetic measurements [<cite linkend="pscr257512bib7">7</cite>].</p></caption></figure><p>The simplest assumption that one can make concerning the nature of the singularity of the magnetic susceptibility χ(<italic>T</italic>) and the correlation length are that at the neighbourhood of the critical point, the above two functions exhibit an asymptotic behaviour:
<display-eqn id="pscr257512eqn8" textype="equation" number="yes" notation="LaTeX" eqnnum="8"></display-eqn><display-eqn id="pscr257512eqn9" textype="equation" number="yes" notation="LaTeX" eqnnum="9"></display-eqn></p><p>Estimates of <inline-eqn><math-text><italic>T</italic><sub><upright>N</upright></sub></math-text></inline-eqn>, <inline-eqn><math-text>γ</math-text></inline-eqn> and <inline-eqn><math-text>ν</math-text></inline-eqn> for <inline-eqn><math-text><upright>GeNi</upright><sub>2-<italic>x</italic></sub><upright>Co</upright><sub><italic>x</italic></sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> have been obtained by using the HTSE method and the PA method [<cite linkend="pscr257512bib19">19</cite>]. The simple pole corresponds to <inline-eqn><math-text><italic>T</italic><sub><upright>N</upright></sub></math-text></inline-eqn> and the residues to the critical exponents <inline-eqn><math-text>γ</math-text></inline-eqn> and <inline-eqn><math-text>ν</math-text></inline-eqn>. The obtained central values are <inline-eqn><math-text>γ = 1.37 ± 0.01</math-text></inline-eqn> and <inline-eqn><math-text>ν = 0.69 ± 0.01</math-text></inline-eqn>. These values of <inline-eqn><math-text>γ</math-text></inline-eqn> and <inline-eqn><math-text>ν</math-text></inline-eqn> are nearest to the one of the Heisenberg model and insensitive to the dilution.</p></sec-level2></sec-level1><sec-level1 id="pscr257512s3" label="3"><heading>Discussions and conclusions</heading><p indent="no"><inline-eqn><math-text><italic>J</italic><sub>1</sub> (<italic>x</italic>)</math-text></inline-eqn> and <inline-eqn><math-text><italic>J</italic><sub>2</sub> (<italic>x</italic>)</math-text></inline-eqn> have been determined from the MFT using the experimental data of <inline-eqn><math-text><italic>T</italic><sub><upright>N</upright></sub></math-text></inline-eqn> given in [<cite linkend="pscr257512bib7">7</cite>] for the <inline-eqn><math-text><upright>GeNi</upright><sub>2-<italic>x</italic></sub><upright>Co</upright><sub><italic>x</italic></sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> systems. We used the same theory to determine the values of exchange interactions with different sizes of <inline-eqn><math-text><upright>CoRh</upright><sub>2</sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> nanoparticle. From the values obtained, we have deduced the intra-plane and inter-plane interactions <inline-eqn><math-text><italic>J</italic><sub><upright>aa</upright></sub></math-text></inline-eqn>, <inline-eqn><math-text><italic>J</italic><sub><upright>bb</upright></sub></math-text></inline-eqn> and <inline-eqn><math-text><italic>J</italic><sub><upright>ac</upright></sub></math-text></inline-eqn>, respectively, the energy of the magnetic structure (see table <tabref linkend="pscr257512tab1">1</tabref> for the spinels <inline-eqn><math-text><upright>GeNi</upright><sub>2-<italic>x</italic></sub><upright>Co</upright><sub><italic>x</italic></sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> systems with 0 ≦ <italic>x</italic> ≦ 2 and table <tabref linkend="pscr257512tab2">2</tabref> with different sizes <inline-eqn><math-text><italic>L</italic> (<upright>nm</upright>)</math-text></inline-eqn> or <inline-eqn><math-text><upright>CoRh</upright><sub>2</sub><upright>O</upright><sub>4</sub></math-text></inline-eqn>). From the results given in table <tabref linkend="pscr257512tab1">1</tabref>, one can see that <inline-eqn><math-text><italic>J</italic><sub>1</sub> (<italic>x</italic>)</math-text></inline-eqn> and <inline-eqn><math-text><italic>J</italic><sub>2</sub> (<italic>x</italic>)</math-text></inline-eqn> increase with the absolute value when <inline-eqn><math-text><italic>x</italic></math-text></inline-eqn> decreases. The signs of <inline-eqn><math-text><italic>J</italic><sub>1</sub> (<italic>x</italic>)</math-text></inline-eqn> and <inline-eqn><math-text><italic>J</italic><sub>2</sub> (<italic>x</italic>)</math-text></inline-eqn> are negative in the whole range of concentration. For <inline-eqn><math-text><italic>x</italic>= 2</math-text></inline-eqn>, a long-range antiferromagnetic (AFM) order is observed in the two systems. In <inline-eqn><math-text><upright>GeNi</upright><sub>2-<italic>x</italic></sub><upright>Co</upright><sub><italic>x</italic></sub><upright>O</upright><sub>4</sub></math-text></inline-eqn>, the introduction of magnetic impurities in the nonmagnetic does not perturb this AFM order for the whole range of dilution <inline-eqn><math-text><italic>x</italic></math-text></inline-eqn>. The exchange integrals and the exchange energy of the magnetic structure decrease in absolute value when the dilution and the size of the nanoparticle decrease. We see that <inline-eqn></inline-eqn>.</p><p>HTSE extrapolated with the PA method is shown to be a convenient method for providing valid estimations of the critical temperatures for real system. By applying this method to the magnetic susceptibility χ (<italic>T</italic>), we have estimated the critical temperature <inline-eqn><math-text><italic>T</italic><sub><upright>N</upright></sub></math-text></inline-eqn> for each dilution <inline-eqn><math-text><italic>x</italic></math-text></inline-eqn>. The obtained magnetic phase diagram of spinel <inline-eqn><math-text><upright>GeNi</upright><sub>2-<italic>x</italic></sub><upright>Co</upright><sub><italic>x</italic></sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> systems is presented in figure <figref linkend="pscr257512fig2">2</figref>. Several thermodynamic phases may appear including the paramagnetic (PM) and the AFM in the range 0 ≦ <italic>x</italic> ≦ 2 for the <inline-eqn><math-text><upright>GeNi</upright><sub>2-<italic>x</italic></sub><upright>Co</upright><sub><italic>x</italic></sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> systems. In this figure, we have included, for comparison, the experimental results obtained by magnetic measurement. From this figure, one can see good agreement between the theoretical phase diagram and the experimental results.</p><p>In this figure, the initial <inline-eqn><math-text><italic>T</italic><sub><upright>N</upright></sub></math-text></inline-eqn> increases with an increase in the number of <inline-eqn><math-text><italic>AB</italic></math-text></inline-eqn> interactions suggesting these interactions stabilize a long-range order. This order is not simple and would take the form of an AFM interaction between small clusters. The higher <inline-eqn><math-text><italic>T</italic><sub><upright>N</upright></sub></math-text></inline-eqn> value for <inline-eqn><math-text><upright>GeCo</upright><sub>2</sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> than for the other samples could be due to a new kind of AFM ordering, resulting from new balancing between different exchange interactions. We saw for all dopings less than <inline-eqn><math-text><italic>x</italic>= 0.6</math-text></inline-eqn>, a falls in temperature is due to a separation into short-ranged- and long-ranged-ordered states.</p><p>On the other hand, the value of critical exponents <inline-eqn><math-text>γ</math-text></inline-eqn> and <inline-eqn><math-text>ν</math-text></inline-eqn> associated to the magnetic susceptibility χ(<italic>T</italic>) and the correlation length <inline-eqn></inline-eqn> have been estimated in the range of the composition 0 ≦ <italic>x</italic> ≦ 2. The sequence of [M, N] PA to series of χ (<italic>T</italic>) and ξ (<italic>T</italic>) has been evaluated. By examining the behaviour of these PA, the convergence was found to be quite rapid. Estimates of the critical exponents associated with magnetic susceptibility and the correlation length for <inline-eqn><math-text><upright>GeNi</upright><sub>2-<italic>x</italic></sub><upright>Co</upright><sub><italic>x</italic></sub><upright>O</upright><sub>4</sub></math-text></inline-eqn> systems are found to be <inline-eqn><math-text>γ = 1.37 ± 0.01</math-text></inline-eqn> and <inline-eqn><math-text>ν = 0.69 ± 0.01</math-text></inline-eqn>, respectively. In the magnetic order, the values of <inline-eqn><math-text>γ</math-text></inline-eqn> and <inline-eqn><math-text>ν</math-text></inline-eqn> are nearest to the one of the Heisenberg model and insensitive to the dilution. 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The nearest neighbouring and the next-neighbouring super-exchange interactions J1 (x) and J2 (x) are evaluated for the spinel GeNi2-xCoxO4 systems in the range 0 x 2. The exchanges interactions are calculated for different sizes of CoRh2O4 nanoparticle by using the MFT. The intra-planar and the inter-planar interactions and the exchange energy are deduced for GeNi2-xCoxO4 and CoRh2O4. The second theory has been applied to the spinel GeNi2-xCoxO4 systems, combined with the Pad approximant method, we have obtained the magnetic phase diagrams (TN versus dilution x) in the range 0 x 2. The obtained theoretical results are in agreement with experimental ones obtained by magnetic measurements. The critical exponents associated with the magnetic susceptibility () and the correlation lengths () are deduced. The obtained theoretical results are in agreement with the experimental data obtained by magnetic measurements. The critical exponent associated to the correlation lengths () is deduced for different sizes of CoRh2O4 nanoparticle.</abstract><classification authority="pacs">75.10.Hk</classification><classification authority="pacs">75.10.Nr</classification><classification authority="pacs">75.25.z</classification><classification authority="pacs">75.30.Et</classification><relatedItem type="host"><titleInfo><title>Physica Scripta</title></titleInfo><titleInfo type="abbreviated"><title>Phys. Scr.</title></titleInfo><genre type="journal">journal</genre><identifier type="ISSN">0031-8949</identifier><identifier type="PublisherID">ps</identifier><identifier type="CODEN">PHSTBO</identifier><identifier type="URL">stacks.iop.org/PhysScr</identifier><part><date>2008</date><detail type="volume"><caption>vol.</caption><number>78</number></detail><detail type="issue"><caption>no.</caption><number>2</number></detail><extent unit="pages"><start>1</start><end>4</end><total>4</total></extent></part></relatedItem><identifier type="istex">2C01C83D03782FBE4D582F705AE2B3DA133B2022</identifier><identifier type="DOI">10.1088/0031-8949/78/02/025702</identifier><identifier type="articleID">257512</identifier><identifier type="articleNumber">025702</identifier><accessCondition type="use and reproduction" contentType="copyright">2008 The Royal Swedish Academy of Sciences</accessCondition><recordInfo><recordContentSource>IOP</recordContentSource><recordOrigin>2008 The Royal Swedish Academy of Sciences</recordOrigin></recordInfo></mods></metadata><serie></serie></istex></record>Pour manipuler ce document sous Unix (Dilib)
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