Research Article Semiclassical Modeling of Isotropic Non...

Hindawi Publishing CorporationPhysics Research InternationalVolume 2013 Article ID 634073 4 pageshttpdxdoiorg1011552013634073

Research ArticleSemiclassical Modeling of Isotropic Non-Heisenberg Magnets forSpin 119878 = 1 and Linear Quadrupole Excitation Dynamics

Yousef Yousefi and Khikmat Kh Muminov

SU Umarov Physical-Technical Institute Academy of Sciences of Republic of Tajikistan Aini Avenue 2991Dushanbe Tajikistan

Correspondence should be addressed to Yousef Yousefi yousof54yahoocom

Received 25 January 2013 Accepted 10 March 2013

Academic Editor Ali Hussain Reshak

Copyright copy 2013 Y Yousefi and K Kh Muminov This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Equations describing one-dimensional non-Heisenberg model are studied by use of generalized coherent states in real parameter-ization and then dissipative spin wave equation for dipole and quadrupole branches is obtained if there is a small linear excitationfrom the ground state Finally it is shown that for such exchange-isotropy Hamiltonians optical branch of spin wave is nondis-sipative

1 Introduction

Many condensed matter systems are fully described by useof effective continuum field models Topologically nontrivialfield configurations have an important role in modeling sys-tems with reduced spatial dimensionality [1] Magnetic sys-tems are usually modeled with the help of the Heisenbergexchange interaction [2ndash4]

However for spin 119878 gt 12 the general isotropic exchangegoes beyond the purely Heisenberg interaction bilinear inspin operators

119894and includes higher order terms of the

type (119894119895)119899 with 119899 up to 2119878 [5] Due to the spin states the

2119878 + 1 complex parameters are necessary to describe eachof them and this corresponds with the 4119878 + 2 degrees offreedom Two degrees of freedom are omitted one becauseof normalization condition and the other for arbitrary phasedecrease hence 4S parameters are required to completelymodeled the remainder 4119878 degrees of freedom of spin states[6]

Particularly in case 119878 = 1 with the isotropic nearestneighbor exchange on a lattice is derived by use of the Hamil-tonian

= minussum

119894

119869 (119878119894

119878119894+1) + 119870(

119878119894

119878119894+1)

2

(1)

Here 119909119894 119910

119894 119911

119894are the spin operators acting at a site

119894 119869 and 119870 are respectively the bilinear (Heisenberg) andbiquadratic exchange integrals The model (1) has beendiscussed recently in connection with 119878 = 1 bosonic gasesin optical lattices [7] and in the context of the deconfinedquantum criticality [8 9] Hamiltonian (1) is a special formpresented in [10] and because of importance of quadrupoleexcitation in ferromagnetic Materials it is considered hereThis paper does not consider the antiferromagnetic andnematic states

Considering the effects of both dipole and quadrupolebranches gives a nonlinear approximation If higher ordermultipole effects are considered the approximation is moreaccurate but at the same time deriving the equations is toocomplicated In this paper only the effect of quadrupolebranch forHamiltonians described by (1) is considered Studyof isotropic and anisotropic spin Hamiltonian with non-Heisenberg terms is complicated due to quadrupole excita-tion dynamics [5 11 12] Antiferromagnetic property of thisexcitation in states near the ground proves the existence of itandDzyaloshinskii calculated the effect of this excitation [13]Also numerical calculations more accurately justify labora-tory results if the effect of quadrupole excitation in nanopar-ticles Fe

8and Mn

12is considered [14 15] In addition this

method may be promising for description of the multispin

2 Physics Research International

configuration of the Fe in the different ligand coordinations[16]

To calculate effect of quadrupole excitation at first it isnecessary to obtain classical equivalent of Hamiltonian (1)and then to find out the solution of spin wave it is necessaryto analyze resulted equations in case there is small linearexcitation from the ground states Therefore the stages ofprocess are the following

(1) Obtaining coherent states for spin 119904 = 1 which arecoherent states of SU(3) group

(2) Calculation of average values of spin operator(3) Classical spinHamiltonian equation is obtained using

previously calculated values(4) Calculating Lagrangian equation by use of Feynman

path integral over coherent states and then computingclassical equations of motion

(5) It is necessary to substitute resulted Hamiltonian inclassical equations of motion to obtain nonlinearequations of magnets Solutions of these nonlinearequations result in soliton description of magnet thatis not interested here

(6) Now ground states of magnets calculated and thennonlinear equations are linearized above the groundstates for small linear excitation

(7) Finally spin wave equations and dispersion equationsmust be calculated

In this paper we write coherent states in real parametersbecause each parameter in this representation related to onedegree of freedom In complex parameterization each para-meter related to two or more degrees of freedom Then inphysical problems the first representation is very helpful

In next sections we developmathematical descriptions ofthe pervious stages and analyze these descriptions

2 Theory and Calculations

In quantum mechanics coherent states are special kinds ofquantum states that their dynamics are very close to their cor-responding classical system Type of coherent state is used inproblem and depends on the operators symmetry in Hamil-tonian Due to the operators symmetry in Hamiltonian (1)for more detailed description and considering all multipolarexcitation of coherent states we used coherent states in SU(3)group Vacuum state of this group is (1 0 0)119879 and coherentstate for a single site in this group is introduced as [11]

1003816100381610038161003816120595⟩ = 1198631(120579 120601) 119890

minus119894120574119911

1198902119894119892119909119910

|0⟩ (2)

where 1198631(120579 120601) is wigner function and 119876119909119910 is quadrupole

moment Two angles 120579 and 120601 the Euler angles determinethe direction of classical spin vector in spherical coordinatesystem The angle 120574 determines the direction of quadrupolemoment around the spin vector and parameter 119892 showschange of magnitude of spin vector and average value ofquadrupole moment Two angles 120601 and 120574 change between 0to 120587 and angle 120579 changes between minus120587 to 120587

In order to derive Lagrangian frompath integral we drivethe path integral from the following transition amplitude

119875 (1205951 1199051 120595 119905) = ⟨120595

1

1003816100381610038161003816 exp(minus119894

ℎ

( (1199051minus 119905)))

1003816100381610038161003816120595⟩ (3)

Using completeness relation and doing some mathemati-cal work the following equation for transition amplitudeobtained

119875 (1205951 1199051 120595 119905)

= lim119899rarrinfin

sum

119895

int119863120583119895(120595) exp(minus119894

ℎ

int

1199051

119905

119871119895(120579 120601 119892 120574) 119889120591)

(4)

In the pervious equation 119871 is Lagrangian and has the fol-lowing form [11]

119871 = ℎ cos 2119892 (cos 120579120601119905+ 120574119905) minus 119867 (120579 120601 119892 120574) (5)

119909119905= (120597119909120597119905) (119909 = 120579 120601) and 119867(120579 120601 119892 120574) is classical

Hamiltonian By use of (4) and (5) and the action stationaryprinciple the classical equations of motion are obtained

When Lagrangian is obtained from path integral anothertwo terms appear one is kinetic term and has ldquoBerry phaserdquoproperties that are important in spin tunneling phenomenaand the other depends on boundary condition values In thispaper we do not consider these two terms

Now classical equivalent of spin vector and their productsmust be computed so that the classical equivalent of Hamil-tonian (1) is obtained So consider the following

= ⟨12059510038161003816100381610038161198781003816100381610038161003816120595⟩

(6)

as classical spin vector and also consider the following

119894119895

=1

2(119894119895+ 119895119894minus4

3120575119894119895119868) (7)

components of quadrupole moment Because we can writeany coherent state as product of single site coherent statesnamely

1003816100381610038161003816120595⟩ = prod

119894

1003816100381610038161003816120595⟩119894 (8)

Then spin operators in ground state of nonsingle ionsHamiltonian can be commuted in different lattices [11] so

⟨1205951003816100381610038161003816 119894

119899119895

119899+1

1003816100381610038161003816120595⟩ = ⟨1205951003816100381610038161003816 119894

119899

1003816100381610038161003816120595⟩ ⟨1205951003816100381610038161003816 119894

119899+1

1003816100381610038161003816120595⟩ (9)

where |120595⟩ = |120595⟩119899|120595⟩119899+1

The average spin values in SU(3) group for coherent states

(2) are defined as [17]

119878+= 119890119894120601 cos (2119892) sin 120579

119878minus= 119890minus119894120601 cos (2119892) sin 120579

119878119911= cos (2119892) cos 120579

(10)

Physics Research International 3

And also

1198782= cos2 (2119892)

1199022= sin2 (2119892)

1198782+ 1199022= 1

(11)

In the pervious relation 1198782 is related to dipole momentand 1199022 is related to quadrupole moment Classical Hamilto-nian can be obtained from the average calculation of Hamil-tonian (1) over coherent states The classical continuous limitof Hamiltonian in SU(3) group is

119867 = minus int119889119909

1198860

(119869cos2 (2119892) + 119870cos4 (2119892)

minus1198862

0

2(41198922

119909sin2 (2119892) (119869 + 2119870cos2 (2119892))

+ cos2 (2119892) (119869 + 2119870cos2 (2119892))

times (1205792

119909+ 1206012

119909sin2120579)))

(12)

where 1198860is length of crystal sites The pervious classi-

cal Hamiltonian is substituted in equation of motion thatobtained from the Lagrangian and the result is classical equa-tions of motion

1

1205960

120579119905= minus1198862

0cos (2119892) (119869 + 119870 + 119870 cos (4119892)) 120601

119909119909sin 120579

1

1205960

120601119905=1198862

0cos (2119892) (119869+119870+119870 cos (4119892)) (1206012

119909cos 120579+120579

119909119909csc 120579)

1

1205960

119892119905= 0

1

1205960

120574119905= 4 cos (2119892) (119869 + 119870 + 119870 cos (4119892))

+(119870cos3 (2119892) (161198922119909minus81205792

119909minus51206012

119909

+31206012

119909cos (2120579)minus2120579119909119909)+cos (2119892)

times (81198922

119909(119869 minus 119870) minus 2119869120579

2

119909+ 81198922

119909119870 cos (4119892)

+1

21198691206012

119909(minus3 + cos (2120579)) minus 119869120579119909119909cot 120579)

+ 4119892119909119909119869 sin (2119892)

+2119892119909119909119870(sin (2119892) + sin (6119892))) 1198862

0

(13)

In these equations 1205960is ℎ1198860 These equations describe

nonlinear dynamics of non-Heisenberg ferromagnetic chaincompletely If we omitted quadrupole excitation (119892 = 0) inthe pervious equations these equations reduced to Landau-Lifshitz equationsThen in comparison with Landau-Lifshitz

equations these equations are more complete and containmore degrees of freedom Note that solutions of these equa-tions are different forms of magnetic solitons

In this paper only the linearized form of (13) for smallexcitation above the ground states is considered To thisendat first classical ground states must be calculated so inabove Hamiltonian only nonderivative part is considered

1198670= minusint

119889119909

1198860

(119869 cos22119892 + 119870 cos42119892) (14)

To find the smallest value of the 1198670we vary it respect to

all the parameters with the ground state is obtained at

119892 = 0 119892 =120587

2 (15)

In this paper only dispersion of spin wave in neighbor-hood of the ground states is studied For this purpose smalllinear excitations from the ground states as shown in (15) aredefined

2119892 997888rarr 120587 + 119892 (16)

In this situation the linearized classical equationsmotionare

1

1205960

120579119905= minus1198862

0(119869 + 2119870) 120601119909119909

1

1205960

120601119905= 1198862

0(119869 + 2119870) 120579119909119909

1

1205960

119892119905= 0

1

1205960

120574119905= minus4 (119869 + 2119870)

(17)

Consider functions 120579 and 120601 as plane waves to obtain dis-persion equation

120601 = 1206010119890119894(120596119905minus119896119909)

+ 1206010119890minus119894(120596119905minus119896119909)

120579 = 1205790119890119894(120596119905minus119896119909)

+ 1205790119890minus119894(120596119905minus119896119909)

(18)

Substitution of these equations in (17) results in disper-sion equation of spin wave near the ground states

1205962

1= (119869 + 2119870)

21198862

011989641205962

0

1205962= minus4 (119869 + 2119870)1205960

(19)

It is evident from (19) that in addition to the dispersionacoustic branch there exist nondispersion optical brancheswhich is related to the dipole and quadrupole excitations

3 Conclusion

In this paper describing equations of one-dimensional iso-tropic non-HeisenbergHamiltonians are obtained using real-parameter coherent states It is shown that both dipole and


quadrupole excitations have different dispersion if there issmall linear excitation from the ground state

In addition it is shown that for isotropic ferromagnetsthe magnitude of average quadrupole moment is constant(119892119905= 0) and its dynamic is rotational dynamics around the

classical spin vector (120574119905= 0)

References

[1] NManton and P SutcliffeTopological Solitons Cambridge uni-versity press New York NY USA 2004

[2] E L Nagaev ldquoAnomalous magnetic structures and phase tran-sitions in non-Heisenberg magnetic materialsrdquo Soviet Physicsvol 25 no 1 pp 31ndash75 1982

[3] E L Nagaev Magnets with Nonsimple Exchange InteractionsNauka Moscow Russia 1988

[4] VM Loktev and V S Ostrovski ldquoPeculiarities of the statics anddynamics of magnetic insulators with single-ion anisotropyrdquoLow Temperature Physics vol 20 no 1 article 775 26 pages1994

[5] B A Ivanov A Yu Galkin R S Khymyn and A YuMerkulovldquoNonlinear dynamics and two-dimensional solitons for spin-1 ferromagnets with biquadratic exchangerdquo Physical Review Bvol 77 no 6 Article ID 064402 11 pages 2008

[6] V S Ostrovskii ldquoNonlinear dynamics of highly anisotropicspin-1 magnetic materialsrdquo Journal of Experimental and Theo-retical Physics vol 64 no 5 p 999 1986

[7] A Imambekov M Lukin and E Demler ldquoSpin-exchange inter-actions of spin-one bosons in optical lattices Singlet nematicand dimerized phasesrdquo Physical Review A vol 68 no 6 ArticleID 063602 24 pages 2003

[8] K Harada N Kawashima andM Troyer ldquoDimer-quadrupolarquantumphase transition in the quasi-one-dimensional heisen-bergmodel with biquadratic interactionrdquo Journal of the PhysicalSociety of Japan vol 76 Article ID 013703 4 pages 2007

[9] T Grover and T Senthil ldquoQuantum spin nematics dimeriza-tion and deconfined criticality in quasi-1D spin-one magnetsrdquoPhysical Review Letters vol 98 Article ID 247202 4 pages 2007

[10] N Papanicolaou ldquoUnusual phases in quantum spin-1 systemsrdquoNuclear Physics B vol 305 no 3 pp 367ndash395 1988

[11] O K Abdulloev and K KMuminov ldquoSemiclassical descriptionof anisotropic magnets acted upon by constant external mag-netic fieldsrdquo Physics of the Solid State vol 36 no 1 pp 93ndash971994

[12] A Y Fridman O A Kosmachev and B A Ivanov ldquoSpin nema-tic state for a spin S = 32 isotropic non-Heisenberg magnetrdquoPhysical Review Letters vol 106 Article ID 097202 2011

[13] I E Dzyaloshinskii ldquoExternal magnetic fields of antiferromag-netsrdquo Solid State Communications vol 82 no 7 pp 579ndash5801992

[14] A Garg ldquoSpin tunneling in magnetic molecules quasisingularperturbations and discontinuous SU(2) instantonsrdquo PhysicalReview B vol 67 Article ID 054406 13 pages 2003

[15] M S Foss-Feig and J R Friedman ldquoGeometric-phase-effecttunnel-splitting oscillations in single-molecule magnets withfourth-order anisotropy induced by orthorhombic distortionrdquoEurophysics Letters vol 86 no 1 article 27002 2009

[16] M Matusiewicz M Czerwinski J Kasperczyk and I V KitykldquoDescription of spin interactions in model [Fe

6S6]4+ superclus-

terrdquo Journal of Chemical Physics vol 111 no 14 pp 6446ndash64551999

[17] V G Makhankov M A Granados and A V MakhankovldquoGeneralized coherent states and spin 119878 ge 1 systemsrdquo Journalof Physics A vol 29 no 12 2005

Submit your manuscripts athttpwwwhindawicom

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Advances in Condensed Matter Physics

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AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


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Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


configuration of the Fe in the different ligand coordinations[16]

To calculate effect of quadrupole excitation at first it isnecessary to obtain classical equivalent of Hamiltonian (1)and then to find out the solution of spin wave it is necessaryto analyze resulted equations in case there is small linearexcitation from the ground states Therefore the stages ofprocess are the following

(1) Obtaining coherent states for spin 119904 = 1 which arecoherent states of SU(3) group

(2) Calculation of average values of spin operator(3) Classical spinHamiltonian equation is obtained using

previously calculated values(4) Calculating Lagrangian equation by use of Feynman

path integral over coherent states and then computingclassical equations of motion

(5) It is necessary to substitute resulted Hamiltonian inclassical equations of motion to obtain nonlinearequations of magnets Solutions of these nonlinearequations result in soliton description of magnet thatis not interested here

(6) Now ground states of magnets calculated and thennonlinear equations are linearized above the groundstates for small linear excitation

(7) Finally spin wave equations and dispersion equationsmust be calculated

In this paper we write coherent states in real parametersbecause each parameter in this representation related to onedegree of freedom In complex parameterization each para-meter related to two or more degrees of freedom Then inphysical problems the first representation is very helpful

In next sections we developmathematical descriptions ofthe pervious stages and analyze these descriptions

2 Theory and Calculations

In quantum mechanics coherent states are special kinds ofquantum states that their dynamics are very close to their cor-responding classical system Type of coherent state is used inproblem and depends on the operators symmetry in Hamil-tonian Due to the operators symmetry in Hamiltonian (1)for more detailed description and considering all multipolarexcitation of coherent states we used coherent states in SU(3)group Vacuum state of this group is (1 0 0)119879 and coherentstate for a single site in this group is introduced as [11]

1003816100381610038161003816120595⟩ = 1198631(120579 120601) 119890

minus119894120574119911

1198902119894119892119909119910

|0⟩ (2)

where 1198631(120579 120601) is wigner function and 119876119909119910 is quadrupole

moment Two angles 120579 and 120601 the Euler angles determinethe direction of classical spin vector in spherical coordinatesystem The angle 120574 determines the direction of quadrupolemoment around the spin vector and parameter 119892 showschange of magnitude of spin vector and average value ofquadrupole moment Two angles 120601 and 120574 change between 0to 120587 and angle 120579 changes between minus120587 to 120587

In order to derive Lagrangian frompath integral we drivethe path integral from the following transition amplitude

119875 (1205951 1199051 120595 119905) = ⟨120595

1

1003816100381610038161003816 exp(minus119894

ℎ

( (1199051minus 119905)))

1003816100381610038161003816120595⟩ (3)

Using completeness relation and doing some mathemati-cal work the following equation for transition amplitudeobtained

119875 (1205951 1199051 120595 119905)

= lim119899rarrinfin

sum

119895

int119863120583119895(120595) exp(minus119894

ℎ

int

1199051

119905

119871119895(120579 120601 119892 120574) 119889120591)

(4)

In the pervious equation 119871 is Lagrangian and has the fol-lowing form [11]

119871 = ℎ cos 2119892 (cos 120579120601119905+ 120574119905) minus 119867 (120579 120601 119892 120574) (5)

119909119905= (120597119909120597119905) (119909 = 120579 120601) and 119867(120579 120601 119892 120574) is classical

Hamiltonian By use of (4) and (5) and the action stationaryprinciple the classical equations of motion are obtained

When Lagrangian is obtained from path integral anothertwo terms appear one is kinetic term and has ldquoBerry phaserdquoproperties that are important in spin tunneling phenomenaand the other depends on boundary condition values In thispaper we do not consider these two terms

Now classical equivalent of spin vector and their productsmust be computed so that the classical equivalent of Hamil-tonian (1) is obtained So consider the following

= ⟨12059510038161003816100381610038161198781003816100381610038161003816120595⟩

(6)

as classical spin vector and also consider the following

119894119895

=1

2(119894119895+ 119895119894minus4

3120575119894119895119868) (7)

components of quadrupole moment Because we can writeany coherent state as product of single site coherent statesnamely

1003816100381610038161003816120595⟩ = prod

119894

1003816100381610038161003816120595⟩119894 (8)

Then spin operators in ground state of nonsingle ionsHamiltonian can be commuted in different lattices [11] so

⟨1205951003816100381610038161003816 119894

119899119895

119899+1

1003816100381610038161003816120595⟩ = ⟨1205951003816100381610038161003816 119894

119899

1003816100381610038161003816120595⟩ ⟨1205951003816100381610038161003816 119894

119899+1

1003816100381610038161003816120595⟩ (9)

where |120595⟩ = |120595⟩119899|120595⟩119899+1

The average spin values in SU(3) group for coherent states

(2) are defined as [17]

119878+= 119890119894120601 cos (2119892) sin 120579

119878minus= 119890minus119894120601 cos (2119892) sin 120579

119878119911= cos (2119892) cos 120579

(10)


And also

1198782= cos2 (2119892)

1199022= sin2 (2119892)

1198782+ 1199022= 1

(11)


119867 = minus int119889119909

1198860

(119869cos2 (2119892) + 119870cos4 (2119892)

minus1198862

0

2(41198922

119909sin2 (2119892) (119869 + 2119870cos2 (2119892))

+ cos2 (2119892) (119869 + 2119870cos2 (2119892))

times (1205792

119909+ 1206012

119909sin2120579)))

(12)



1

1205960

120579119905= minus1198862

0cos (2119892) (119869 + 119870 + 119870 cos (4119892)) 120601

119909119909sin 120579

1

1205960

120601119905=1198862

0cos (2119892) (119869+119870+119870 cos (4119892)) (1206012

119909cos 120579+120579

119909119909csc 120579)

1

1205960

119892119905= 0

1

1205960

120574119905= 4 cos (2119892) (119869 + 119870 + 119870 cos (4119892))

+(119870cos3 (2119892) (161198922119909minus81205792

119909minus51206012

119909

+31206012

119909cos (2120579)minus2120579119909119909)+cos (2119892)

times (81198922

119909(119869 minus 119870) minus 2119869120579

2

119909+ 81198922

119909119870 cos (4119892)

+1

21198691206012


+ 4119892119909119909119869 sin (2119892)

+2119892119909119909119870(sin (2119892) + sin (6119892))) 1198862

0

(13)





1198670= minusint

119889119909

1198860

(119869 cos22119892 + 119870 cos42119892) (14)



119892 = 0 119892 =120587

2 (15)


2119892 997888rarr 120587 + 119892 (16)


1

1205960

120579119905= minus1198862

0(119869 + 2119870) 120601119909119909

1

1205960

120601119905= 1198862

0(119869 + 2119870) 120579119909119909

1

1205960

119892119905= 0

1

1205960

120574119905= minus4 (119869 + 2119870)

(17)


120601 = 1206010119890119894(120596119905minus119896119909)

+ 1206010119890minus119894(120596119905minus119896119909)

120579 = 1205790119890119894(120596119905minus119896119909)

+ 1205790119890minus119894(120596119905minus119896119909)

(18)


1205962

1= (119869 + 2119870)

21198862

011989641205962

0

1205962= minus4 (119869 + 2119870)1205960

(19)


3 Conclusion






References

















6S6]4+ superclus-








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




And also

1198782= cos2 (2119892)

1199022= sin2 (2119892)

1198782+ 1199022= 1

(11)


119867 = minus int119889119909

1198860

(119869cos2 (2119892) + 119870cos4 (2119892)

minus1198862

0

2(41198922

119909sin2 (2119892) (119869 + 2119870cos2 (2119892))

+ cos2 (2119892) (119869 + 2119870cos2 (2119892))

times (1205792

119909+ 1206012

119909sin2120579)))

(12)



1

1205960

120579119905= minus1198862

0cos (2119892) (119869 + 119870 + 119870 cos (4119892)) 120601

119909119909sin 120579

1

1205960

120601119905=1198862

0cos (2119892) (119869+119870+119870 cos (4119892)) (1206012

119909cos 120579+120579

119909119909csc 120579)

1

1205960

119892119905= 0

1

1205960

120574119905= 4 cos (2119892) (119869 + 119870 + 119870 cos (4119892))

+(119870cos3 (2119892) (161198922119909minus81205792

119909minus51206012

119909

+31206012

119909cos (2120579)minus2120579119909119909)+cos (2119892)

times (81198922

119909(119869 minus 119870) minus 2119869120579

2

119909+ 81198922

119909119870 cos (4119892)

+1

21198691206012


+ 4119892119909119909119869 sin (2119892)

+2119892119909119909119870(sin (2119892) + sin (6119892))) 1198862

0

(13)





1198670= minusint

119889119909

1198860

(119869 cos22119892 + 119870 cos42119892) (14)



119892 = 0 119892 =120587

2 (15)


2119892 997888rarr 120587 + 119892 (16)


1

1205960

120579119905= minus1198862

0(119869 + 2119870) 120601119909119909

1

1205960

120601119905= 1198862

0(119869 + 2119870) 120579119909119909

1

1205960

119892119905= 0

1

1205960

120574119905= minus4 (119869 + 2119870)

(17)


120601 = 1206010119890119894(120596119905minus119896119909)

+ 1206010119890minus119894(120596119905minus119896119909)

120579 = 1205790119890119894(120596119905minus119896119909)

+ 1205790119890minus119894(120596119905minus119896119909)

(18)


1205962

1= (119869 + 2119870)

21198862

011989641205962

0

1205962= minus4 (119869 + 2119870)1205960

(19)


3 Conclusion






References

















6S6]4+ superclus-








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







References

















6S6]4+ superclus-








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



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