Booth, R. A.

Borchers, Julie A.

Hasz, K.

Ijiri, Yumi

Krycka, Kathryn L.

Majetich, S. A.

Rhyne, J. J.

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OberlinDigital Commons at OberlinFaculty & Staff Scholarship1-1-2015Reply to Comment on Origin of Surface CantingWithin Fe3O4 NanoparticlesKathryn L. KryckaJulie A. BorchersR. A. BoothYumi IjiriOberlin College, Yumi.Ijiri@oberlin.eduK. HaszSee next page for additional authorsFollow this and additional works at: https://digitalcommons.oberlin.edu/faculty_scholPart of the Physics CommonsThis Other is brought to you for free and open access by Digital Commons at Oberlin. It has been accepted for inclusion in Faculty & Staff Scholarshipby an authorized administrator of Digital Commons at Oberlin. For more information, please contact megan.mitchell@oberlin.edu.Repository CitationKrycka, K.L., J.A. Borchers, R.A.Booth, Y. Ijiri, K. Hasz, J.J. Rhyne, and S.A. Majetich. 2015. “Reply to Comment on Origin of SurfaceCanting Within Fe3O4 Nanoparticles.” Physical Review Letters 114: 149702.AuthorsKathryn L. Krycka, Julie A. Borchers, R. A. Booth, Yumi Ijiri, K. Hasz, J. J. Rhyne, and S. A. MajetichThis other is available at Digital Commons at Oberlin: https://digitalcommons.oberlin.edu/faculty_schol/3306Krycka et al. Reply: Measurable suppression of spin-flipscattering, explained by theM∥;Xð ~QÞM⊥;Yð ~QÞ cross term [1],is used to evaluate the extent of nanoparticle surface canting.We agree with [2] that jM∥ð ~QÞjjM⊥ð ~QÞjsinðθÞcos3ðθÞin Eq. (1) sums to zero over all θ quadrants. These terms,however, are multiplied by cosðδϕÞ, which changes signfor θ. In Eq. (1) [1] jcosðδϕÞj ¼ 1 for Q≧0.02 Å−1 sinceM⊥ð ~QÞ and M∥ð ~QÞ are centrosymmetric within eachparticle. (We incorrectly stated cosðδϕÞ ¼ 1 [1].) The signchange is a direct consequence of the symmetry present perself-assembled grain of nanoparticles. Specifically, long-range dipolar coupling can create a pattern of the canted shellmoments similar to Fig. 2(b) of Ref. [3] (“flower”)or Fig. 2(iv) of Ref. [4] over the submicron length scaleof a close-packed grain of nanoparticles, involving a netmoment along X̂∥ ~H and a gradual, average spatial variationamongM⊥;Yð ~QÞ shell moments with mirror symmetry aboutthe X axis. Although we did not discuss the characteristicsof the long-range, magnetic, nanoparticle structure inRef. [1] for brevity, particle clusters with similarly cantedshells were required to properly model the spin-flip scatter-ing [5]. The specific distribution, however, does not impactthe spin-flip scattering asymmetry ratio as evidenced by thefact that the spin-flip scattering shape was preserved whenviewed along any θ cut (Fig. 4 of Ref. [4]).Additional polarization dependent terms M⊥;Zð ~QÞM⊥;Yð ~QÞ andM⊥;Zð ~QÞM∥;Xð ~QÞ [Eq. (6) of Ref. [4]] appearonly as a difference between spin-flip cross sections. Sinceour spin-flip cross sections are equivalent within experi-mental resolution, we summed them together [1] as inRef. [5], rendering these extra terms irrelevant.Regarding our isotropic system with magnetic symmetryabout X̂∥ ~H, we clearly state that jM⊥;Yð ~QÞj ¼ jM⊥;Zð ~QÞj≡jM⊥ð ~QÞj in the Supplemental Material of Ref. [1] and inRef. [5]. Our system differs from that of the nanocomposite[3] in which magnetostatic dipolar interactions dominate.The M⊥;Yð ~QÞ and M⊥;Zð ~QÞ scattering from symmetricallyequivalent magnetization components along Y and Z arenot identical in this case [3] because the net moments alongY and Z sum to zero over the nanocomposite, resultingin zero spin-flip scattering along the coordinate axes. Incontrast, our nanoparticle system exhibits nonzero spin-flipscattering at all θ [4], which implies that the canted shellsmust have a net uniform moment throughout each particle[5], which is equivalent along Y and Z (on average), andthat jM⊥;Yð ~QÞj ¼ jM⊥;Zð ~QÞj. Moreover, any reduction inscattering due to long range coupling between nanopar-ticles that might suppress M⊥;Yð ~QÞ scattering would bysymmetry considerations suppress M⊥;Zð ~QÞ equally.The point of our energetic model was to evaluatethe relative contributions of Zeeman, magnetocrystallineanisotropy, exchange, and dipolar coupling to show thatTd tilting is energetically feasible and capable of explaining(a) scattering asymmetry and (b) temperature-dependent,canted shell formation at high magnetic field. Whensurveying a large collection of nanoparticles, we feel itmakes sense to consider the average parameters (i.e.,exchange coupling) per unit volume as a function of radialplacement. Specifically, 0.05 nm slices were chosen forcalculational convenience to be evenly divisible into thenanoparticle diameter.(i) As labeled, Eq. (2) [1] stems from estimating theminimum size of the anisotropy energy necessary tooutcompete the Zeeman energy, irrespective of exchangeand the specific anisotropy symmetry. Later, when specifictotal energy contributions are computed, the simplestappropriate model of anisotropy is included as in Eq. (5)in Ref. [1]. The point remains that in all of these cases,the necessary anisotropy constant would be many times(50–100 times greater) than that of the bulk.(ii) While in general, dipolar contributions within ananoparticle and for multiple next nearest neighbors couldbe important, we found that in this system the individualnanoparticle contribution was negligible in comparisonto other energy terms, and considerations of 52 nearestnanoparticle neighbors yielded nearly indistinguishableresults from those for 18 nearest neighbors, justifyingthe truncation.(iii) As clearly stated in Ref. [1], we perform ourcalculation using an average shell canting angle as thiscorresponds to experimental measurement and it representsthe global average for a collection of many nanoparticles.We do not dispute that Fe3O4 has cubic anisotropy,consistent with Eq. (5) [1]. We consider the tilt away fromonly the most energetically favorable magnetocrystallineaxis (most ∥ ~H), rather than from all remaining axes, whichwould be a second order effect.(iv) Oh shell spins were set ∥ ~H because calculationsrevealed that only the Td spins showed appreciable cantingaway from ~H (Table I of Ref. [1]).In summary, our small-angle neutron scatteringasymmetry analysis combined with this heuristic energymodel provides a reasonable explanation for the observednanoparticle shell canting.This work utilized facilities supported in part by NationalScience Foundation Grants No. DMR-0944772 (neutroninstrumentation used by all authors) and No. DMR-1104489 (Y. I.) and Department of Energy Grant No.DE-FG02-08ER46481 (S. M.).K. L. Krycka,1,* J. A. Borchers,1 R. A. Booth,2 Y. Ijiri,3K. Hasz, 3 J. J. Rhyne4, and S. A. Majetich21NIST Center for Neutron ResearchGaithersburg, Maryland 20899, USA2Carnegie Mellon UniversityPittsburgh, Pennsylvania 15213, USAPRL 114, 149702 (2015) P HY S I CA L R EV I EW LE T T ER Sweek ending10 APRIL 20150031-9007=15=114(14)=149702(2) 149702-1 © 2015 American Physical Society3Oberlin CollegeOberlin, Ohio 44074, USA4U.S. Department of EnergyWashington, DC 20585, USAReceived 11 November 2014; published 7 April 2015DOI: 10.1103/PhysRevLett.114.149702PACS numbers: 75.25.-j, 75.60.Ej, 75.75.Lf*kathryn.krycka@nist.gov[1] K. L. Krycka, J. A. Borchers, R. A. Booth, Y. Ijiri, K. Hasz,J. J.Rhyne, andS. A.Majetich,OriginofSurfaceCantingwithinFe3O4 Nanoparticles, Phys. Rev. Lett. 113, 147203 (2014).[2] A. Michels, D. Honecker, S. Erokhin, and D.Berkov, Comment on “Origin of Surface Canting withinFe3O4 Nanoparticles”, Phys. Rev. Lett. 114, 149701(2015).[3] A. Michels, S. Erokhin, D. Berkov, and N. Gorn, Micro-magnetic simulation of magnetic small-angle neutron scatter-ing from two-phase nanocomposites, J. Magn. Magn. Mater.350, 55 (2014).[4] K. Krycka, J. Borchers, Y. Ijiri, R. Booth, andS. Majetich, Polarization-analyzed small-angle neutron scat-tering. II. Mathematical angular analysis, J. Appl. Crystal-logr. 45, 554 (2012).[5] K. L. Krycka, R. A. Booth, C. R. Hogg, Y. Ijiri,J. A. Borchers, W. C. Chen, S. M. Watson, M. Laver, T. R.Gentile, L. R. Dedon, S. Harris, J. J. Rhyne, and S. A.Majetich, Core-Shell Magnetic Morphology of StructurallyUniform Magnetite Nanoparticles, Phys. Rev. Lett. 104,207203 (2010).PRL 114, 149702 (2015) P HY S I CA L R EV I EW LE T T ER Sweek ending10 APRIL 2015149702-2

Reply to Comment on Origin of Surface Canting Within Fe3O4 Nanoparticles

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Reply to Comment on Origin of Surface Canting Within Fe3O4 Nanoparticles

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