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Quadrupole Excitation in Tunnel Splitting Oscillation in Nanoparticle M n 12

Quadrupole Excitation in Tunnel Splitting Oscillation in Nanoparticle M n 12 Quadrupole Excitation in Tunnel Splitting Oscillation in Nanoparticle 𝐌𝐧12 <meta name="citation_title" content="Quadrupole Excitation in Tunnel Splitting Oscillation in Nanoparticle M n 12 " /> Hindawi Publishing Corporation Home Journals About Us About this Journal Submit a Manuscript Table of Contents Journal Menu Abstracting and Indexing Aims and Scope Article Processing Charges Articles in Press Author Guidelines Bibliographic Information Contact Information Editorial Board Editorial Workflow Free eTOC Alerts Reviewers Acknowledgment Subscription Information Open Special Issues Published Special Issues Special Issue Guidelines Abstract Full-Text PDF Full-Text HTML Full-Text ePUB Linked References How to Cite this Article Complete Special Issue Advances in Condensed Matter Physics Volume 2012 (2012), Article ID 530765, 4 pages doi:10.1155/2012/530765 Research Article Quadrupole Excitation in Tunnel Splitting Oscillation in Nanoparticle 𝐌 𝐧 1 2 Yousef Yousefi and Khikmat Kh. Muminov Physical-Technical Institute Named after S.U.Umarov, Academy of Sciences of Republic of Tajikistan, Aini Ave 299/1, Dushanbe, Tajikistan Received 25 January 2012; Revised 14 March 2012; Accepted 14 March 2012 Academic Editor: Rosa Lukaszew Copyright © 2012 Yousef Yousefi and Khikmat Kh. Muminov. This is an open access article distributed under the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We analyze the interference between tunneling paths that occur for a spin system with special Hamiltonian both for dipole and quadrupole excitations. Using an instanton approach, we find that as the strength of the second-order transverse anisotropy is increased, the tunnel splitting for both excitations is modulated, with zeros occurring periodically and the number of quenching points for quadrupole excitation decreasing. This effect results from the interference of four tunneling paths connecting easy-axis spin orientations and occurs in the absence of any magnetic field. 1. Introduction Berry phase effects play an important role in spin dynamics. Tunneling of a spin (or magnetic particle) between degenerate orientations can be calculated by such Berry phase effects via the interference between tunneling paths, with quenching of the tunnel splitting occurring when tunneling paths destructively interfere [ 1 , 2 ]. The first conclusive experimental evidence of tunneling in molecular magnet Mn 12 O 12 (CH 3 COO) 16 (H 2 O) 4 called Mn 12 was provided by Friedman [ 3 ]. He applies a magnetic field along the easyaxis of the crystal at a variety of temperatures and found a curve punctuated by sharp steps at regular intervals in the parameter πœ† . Also this observation that is confirmed by Hernandez et al. [ 4 ] and Thomas et al. [ 5 ] shows a series of steps in the hysteresis loops in Mn 12 in low temperature. The steps observed in the hysteresis loops at nearly equal intervals of magnetic field are due to enhanced relaxation of the magnetization at the resonant fields when levels on opposite side of the anisotropy barrier coincide in energy (see Figure 1 ). Figure 1: The first curve was obtained from numerical calculation of the tunnel splitting for Mn 12 and the second curve is the magnetization plot of this magnetic molecule in low temperature. The common feature of molecular magnets responsible for their interesting behavior is the strong spin-spin coupling between metallic ions in the core of the molecule. In this magnet molecule, there are 4 Mn 4+ atoms (spin 3/2) surrounded by 8 Mn 3+ atoms (spin 2), and they retain a relative orientation such as to give the whole molecule a spin of 10 (= 8 ( 2 ) −4(3/2)) [ 3 ]. The Mn 12 single-molecule magnet has a fourfold transverse magnetic anisotropy and displays resonant tunneling between two easy-axis orientations [ 3 ]. Here we consider the interference that occurs between tunneling paths in such a system and how that interference can be modified by the presence of a second-order transverse anisotropy perturbation and multipole excitations. Such a perturbation could potentially be induced by the application of uniaxial pressure to a sample of Mn 12 . We find that for both excitations the tunnel splitting is periodically quenched as a function of the strength of the perturbation. This interference effect takes place in the absence of any magnetic field. In the present study, the fourth-order anisotropy term is the primary transverse anisotropy in the problem, producing four interfering paths, but the interference is modulated by the strength of the second-order anisotropy, leading to periodic quenches as that term is varied. In particular, we consider a spin governed by the Hamiltonian   𝐽 𝐻 = 2 −  𝐽 2 𝑧 ξ‚€  𝐽 + π‘˜ 4 + +  𝐽 4 −  ξ‚€  𝐽 + πœ† 2 π‘₯ −  𝐽 2 𝑦  , ( 1 ) where the  𝐽 𝑠 are standard spin operators. We restrict the values of π‘˜ and πœ† such that the z -axis is the spin easy axis. In the above Hamiltonian, the leading term −  𝐽 2 𝑧 established the double well potential and the easy axis and the term fourth-order transverse anisotropy  𝐽 π‘˜ ( 4 + +  𝐽 4 − ) are responsible for 4-fold rotational symmetries. The third term is a second-order transverse anisotropy that is present in many low-symmetry SMMs. In this paper, firstly we calculated periodically depended of spin tunneling for Mn 12 in SU ( 2 ) group, in other word, we considered only dipole excitation in Hamiltonian. Due to the symmetry of spin operators in Hamiltonian, for being more accurate, we have to consider other multipole excitation. Then, we consider both dipole and quadrupole excitations. 2. Theory and Calculations The instanton method is an efficient way of calculating tunnel splitting, both for particles [ 6 ] and for spin [ 7 ]. It is based on evaluating the path integral for a certain propagator in the steepest-descent approximation and is designed to be asymptotically correct in the semiclassical limit ( 𝐽 → ∞ or ℏ → 0 ). Instantons are classical paths that run between degenerate classical minima of the energy. By classical we mean that the path obeys the principle of least action and satisfies energy conservation. However, a path along which energy is conserved can not run between two minima and still have real coordinates and momenta. Hence, one must enlarge the notion of a classical path and allow the coordinates and/or momenta to become complex. In the continuous approach the tunnel splitting can be computed as the functional integral [ 8 ] ξ€œ e x p ( − 𝑆 ( Μ‡ π‘₯ , π‘₯ , 𝑑 ) ) 𝑑 πœ‡ ( π‘₯ ) , ( 2 ) where ∏ 𝑑 πœ‡ ( π‘₯ ) = 𝑁 𝑖 = 1 𝑑 π‘₯ 𝑖 . In Su ( 2 ) group 𝑁 = 2 and in Su ( 3 ) group 𝑁 = 4 , and 𝑁 is the number of degree of freedoms, and the action is given by 𝑆 ( Μ‡ π‘₯ , π‘₯ , 𝑑 ) = 𝑆 𝐾 + 𝑆 𝐷 + 𝑆 𝐡 . ( 3 ) The first term, 𝑆 𝐾 , is kinetic term which has the properties of a Berry phase, which can give rise to interference between different trajectories. The second term, 𝑆 𝐷 , is dynamical term and the final term is boundary term this term depends explicitly on the boundary values of the path. When the functional action is specified, the instanton recipe for calculating the tunnel splitting is as follows. Let there be a number of instantons, that is, least action paths, labeled by k, and let the actions for these various paths be 𝑆 π‘˜ c l . The tunneling amplitude is given by  Δ = k 𝐷 π‘˜ ξ€· e x p − 𝑆 c l k ξ€Έ . ( 4 ) The prefactor 𝐷 π‘˜ results from integrating the Gaussian fluctuations around the k th instanton. For nanoparticle Mn 12 , the tunneling amplitude calculated and obtained the following relation [ 3 ]: Δ ≈ 4 𝑒 − 𝑆 𝑅 ξ€· c o s ( 𝐽 πœ‹ ) c o s 𝐽 𝑆 𝐼 ξ€Έ , ( 5 ) where 𝑆 𝐼 ξ‚΅ ξ€œ = R e 3 πœ‹ / 2 πœ‹ / 2 ξ‚Ά , 𝑃 ( πœ™ ) 𝑑 πœ™ ( 6 ) and 𝑃 ( πœ™ ) is solution of classical enrgy equation. 𝑆 𝑅 is the real part of the Euclidean action and the same for all instantons. The factor c o s ( 𝐽 πœ‹ ) ensures that the tunnel splitting is zero for half-integer spin. The classical energy is (the x -axis is chosen as the azimuth because the-path will not encounter it and πœ™ is measured from the z axis) 𝐸 c l ( πœƒ , πœ™ ) = 𝐽 2 ξ€· 1 − s i n 2 πœƒ s i n 2 πœ™ ξ€Έ + πœ† 𝐽 2 ξ€· c o s 2 πœƒ − s i n 2 πœƒ c o s 2 πœ™ ξ€Έ + 2 π‘˜ 𝐽 4 ξ€· c o s 4 πœƒ + s i n 4 πœƒ c o s 4 πœ™ − 6 c o s 2 πœƒ s i n 2 πœƒ c o s 2 πœ™ ξ€Έ . ( 7 ) The minimum of energy obtained at πœ‹ πœƒ = 2 πœ‹ πœ™ = 2 , 3 πœ‹ 2 . ( 8 ) The minima of enregy are 𝐸 c l ξ‚€ πœ‹ 2 , πœ‹ 2  𝐸 = 0 c l ξ‚€ πœ‹ 2 , 3 πœ‹ 2  = 0 . ( 9 ) Making the substitution 𝑃 = c o s πœƒ , the instanton must satisfy the constraint 𝐽 2 ξ€· ξ€· 1 − 1 − 𝑃 2 ξ€Έ s i n 2 πœ™ ξ€Έ + πœ† 𝐽 2 ξ€· 𝑃 2 − ξ€· 1 − 𝑃 2 ξ€Έ c o s 2 πœ™ ξ€Έ + 2 π‘˜ 𝐽 4 ξ‚€ 𝑃 4 + ξ€· 1 − 𝑃 2 ξ€Έ 2 c o s 4 πœ™ − 6 𝑃 2 ξ€· 1 − 𝑃 2 ξ€Έ c o s 2 πœ™  = 0 . ( 1 0 ) The four solutions for the above equation are: 𝑃 ξƒŽ ( πœ™ ) = ± √ 𝐴 ( πœ™ ) ± 𝑖 𝐡 ( πœ™ ) , 𝐢 ( πœ™ ) ( 1 1 ) where 𝐴 ( πœ™ ) = 2 π‘Ž c o s 2 ξ€· πœ™ ( 7 + c o s 2 πœ™ ) − πœ† 1 + c o s 2 ξ€Έ − s i n 2 πœ™ 𝐡 ( πœ™ ) = π‘Ž c o s 2 − ξ€· πœ† ξ€· πœ™ ( 1 + π‘˜ − πœ† + π‘˜ c o s 2 πœ™ ) × ( 3 5 + 2 8 c o s 2 πœ™ + c o s 4 πœ™ ) 1 + c o s 2 πœ™ ξ€Έ − 2 π‘Ž c o s 2 πœ™ ( 7 + c o s 2 πœ™ ) + s i n 2 πœ™ ξ€Έ 2 ξ€· 𝐢 ( πœ™ ) = 4 π‘Ž c o s 4 πœ™ + 6 c o s 2 ξ€Έ , πœ™ + 1 ( 1 2 ) and π‘Ž = 𝐽 2 π‘˜ . Solutions ( 11 ), despite conserving energy, fail to meet the boundary conditions 𝑃 ( πœ‹ / 2 ) = 𝑃 ( 3 πœ‹ / 2 ) = 0 . These solutions gives rise to what Garg called boundary jump instantons [ 9 ]. As a final note, we consider only the following value of 𝑃 ( πœ™ ) : 𝑃 ξƒŽ ( πœ™ ) = √ 𝐴 ( πœ™ ) + 𝑖 𝐡 ( πœ™ ) . 𝐢 ( πœ™ ) ( 1 3 ) Substituting above relation in ( 6 ) and using from ( 5 ), splitting of level energy calculated. Up this section, we have just considered the dipole excitation of spin system. Because we will analyze the periodical dependence of spin tunneling, by relation ( 5 ), plotted function C o s ( 𝐽 𝑆 𝐼 ) versus πœ† . If quadrupole excitation is considered, the classical energy is changed in the following form: 𝐸 c l ( πœƒ , πœ™ ) = 𝐽 2 ξ€· 1 − s i n 2 πœƒ c o s 2 2 𝑔 s i n 2 πœ™ ξ€Έ + πœ† 𝐽 2 c o s 2 ξ€· 2 𝑔 c o s 2 πœƒ − s i n 2 πœƒ c o s 2 πœ™ ξ€Έ + 2 π‘˜ 𝐽 4 c o s 4 ξ€· 2 𝑔 c o s 4 πœƒ + s i n 4 πœƒ c o s 2 πœ™ − 6 c o s 2 πœƒ s i n 2 πœƒ c o s 2 πœ™ ξ€Έ . ( 1 4 ) The minimum of energy obtained at πœ‹ πœƒ = 2 , πœ‹ 𝑔 = 4 , πœ‹ πœ™ = 2 , 3 πœ‹ 2 . ( 1 5 ) Also minima of enrgy are 𝐸 c l ξ‚€ πœ‹ 2 , πœ‹ 2 , πœ‹ 4  𝐸 = 0 c l ξ‚€ πœ‹ 2 , 3 πœ‹ 2 , πœ‹ 4  = 0 . ( 1 6 ) Making the substitution 𝑃 = C o s πœƒ , the instanton must satisfy the constraint: 𝐽 2 ξ€· ξ€· 1 − 1 − 𝑃 2 ξ€Έ c o s 2 𝑔 s i n 2 πœ™ ξ€Έ + πœ† 𝐽 2 c o s 2 ξ€· 𝑃 2 𝑔 2 − ξ€· 1 − 𝑃 2 ξ€Έ c o s 2 πœ™ ξ€Έ + 2 π‘˜ 𝐽 4 c o s 4 ξ‚€ 𝑃 2 𝑔 4 + ξ€· 1 − 𝑃 2 ξ€Έ 2 c o s 4 πœ™ − 6 𝑃 2 ξ€· 1 − 𝑃 2 ξ€Έ c o s 2 πœ™  = 0 . ( 1 7 ) Similar to previous section, solution that has the following form is considered: ξƒŽ 𝑃 ( πœ™ , 𝑔 ) = √ 𝐴 ( πœ™ , 𝑔 ) + 𝑖 𝐡 ( πœ™ , 𝑔 ) , 𝐢 ( πœ™ , 𝑔 ) ( 1 8 ) where 𝐴 ( πœ™ , 𝑔 ) = 2 π‘Ž c o s 2 πœ™ s e c 2 − 1 2 𝑔 ( 7 + c o s 2 πœ™ ) 2 ( 1 + 3 πœ† + ( πœ† − 1 ) c o s 2 πœ™ ) s e c 4 𝐡 ξ€· 2 𝑔 ( πœ™ , 𝑔 ) = 8 π‘Ž 1 + 6 c o s 2 πœ™ + c o s 4 πœ™ ξ€Έ × ξ€· 1 + 2 π‘˜ c o s 4 2 𝑔 c o s 4 πœ™ − c o s 2 ξ€· 2 𝑔 πœ† c o s 2 πœ™ + s i n 2 πœ™ − ξ€· ξ€Έ ξ€Έ πœ† + s i n 2 πœ™ + c o s 2 πœ™ ξ€· πœ† − 2 π‘Ž c o s 2 2 𝑔 ( 7 + c o s 2 πœ™ ) ξ€Έ ξ€Έ 2 𝐢 ξ€· ( πœ™ , 𝑔 ) = 4 π‘Ž 1 + 6 c o s 2 πœ™ + c o s 4 πœ™ ξ€Έ c o s 2 2 𝑔 . ( 1 9 ) Substituting relation ( 19 ) in ( 6 ) and using ( 5 ), splitting of energy level was calculated. In this section, we consider both dipole and quadrupole excitations of spin system. Like the previous section, plotted function c o s ( 𝐽 𝑆 𝐼 ) versus πœ† . In this plot, π‘Ž = 1 0 − 4 and π‘˜ = 1 0 − 6 are considered. As seen from Figures 2 and 3 , when we added the quadrapole excitation to Hamilton, the number of quenching point reduced from 5 to 4, also by using Figure 1 we see that the number of steps is 4; that is equal to the number of quenching points and then these steps in magnetization graphic are quenching point is in spin tunneling phenomena. Figure 2: Red curve is R e ( 𝑃 ) / πœ‹ and blue curve is c o s ( 𝐽 𝑆 𝐼 ) versus πœ† obtained by using ( 11 ). In this graph, only dipole excitation was considered. Figure 3: Red curve is R e ( 𝑃 ) / πœ‹ and blue curve is c o s ( 𝐽 𝑆 𝐼 ) versus πœ† obtained using ( 16 ). In this graph both dipole and quadrupole excitations are considered. As mentioned in the introduction, there are step changes in the experimental hysteresis loop of this crystal that have been explained considering only dipolar interaction between molecules. For a more accurate description of the experimental results, we found that we have to consider both dipolar and quadrupole excitations. 3. Discussion In this paper, we consider the spin tunneling phenomena to parameter πœ† in both groups SU ( 2 ) and SU ( 3 ) . Graph of function R e ( 𝑃 ) versus πœ† in SU ( 2 ) group is a straight line started from 5 and with increasing πœ† up to 0.04, this straight line decreases and in point πœ† = 0 . 0 4 the value is zero and remains in this value. But above function graph in SU ( 3 ) group, started less than 4, is not straight line and in point πœ† = 0 . 0 4 the value is zero and remains in this value. The number of quenching points in SU ( 2 ) group (dipole excitation) is 5, these points decreases to 4, when quadrupole excitation is considered. References M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proceedings of the Royal Society of London A , vol. 392, pp. 45–57, 1984. M. S. Foss-Feig and J. R. Friedman, “Geometric-phase-effect tunnel-splitting oscillations in single-molecule magnets with fourth-order anisotropy induced by orthorhombic distortion,” EPL , vol. 86, no. 2, Article ID 27002, 2009. View at Publisher · View at Google Scholar · View at Scopus J. R. Friedman, M. P. Sarachik, J. Tejada, and R. Ziolo, “Macroscopic measurement of resonant magnetization tunneling in high-spin molecules,” Physical Review Letters , vol. 76, no. 20, pp. 3830–3833, 1996. View at Scopus J. M. Hernández, X. X. Zhang, F. Luis, J. Bartolomé, J. Tejada, and R. Ziolo, “Field tuning of thermally activated magnetic quantum tunnelling in Mn 12 —Ac molecules,” Europhysics Letters , vol. 35, no. 4, p. 301, 1996. L. Thomas, F. Lionti, R. Ballou, D. Gatteschi, R. Sessoli, and B. Barbara, “Macroscopic quantum tunnelling of magnetization in a single crystal of nanomagnets,” Nature , vol. 383, no. 6596, pp. 145–147, 1996. View at Publisher · View at Google Scholar · View at Scopus B. Felsager, Geometry Particle, and Fields , Springer, New York, NY, USA, 1998. E. M. Chudnovsky and L. Gunther, “Quantum tunneling of magnetization in small ferromagnetic particles,” Physical Review Letters , vol. 60, no. 8, pp. 661–664, 1988. View at Publisher · View at Google Scholar · View at Scopus E. M. Chudovsky and X. Martinez, “Non-Kramers freezing and unfreezing of tunneling in the biaxial spin model,” Europhysics Letters , vol. 50, no. 3, pp. 395–401, 2000. A. Garg, “Spin tunneling in magnetic molecules: quasisingular perturbations and discontinuous SU(2) instantons,” Physical Review B , vol. 67, no. 5, Article ID 054406, 13 pages, 2003. var _gaq = _gaq || []; _gaq.push(['_setAccount', 'UA-8578054-2']); _gaq.push(['_setDomainName', 'hindawi.com']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 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