# multipole expansion pdf

Its vector potential at point r is Just as we did for V, we can expand in a power series and use the series as an approximation scheme: (see lecture notes for 21 … The relevant physics can best be made obvious by expanding a source distribution in a sum of specific contributions. 0000015723 00000 n Energy of multipole in external ﬁeld: Electric Field and Energy Field of multipole r0: E = r = 1 4ˇ 0 qn jr r0j2 3n(p n) p jr r0j3 where n is unit vector in direction r r0. Electric Field and Energy Field of multipole r0: E = r = 1 4ˇ 0 qn jr r0j2 3n(p n) p jr r0j3 where n is unit vector in direction r r0. 0000004393 00000 n {M��/��b�e���i��4M��o�T�! In Figure 2’s oct-tree decomposition, ever-larger regions of space that represent in-creasing numbers of particles can interact through individual multipole expansions at in-creasing distances. 0000042245 00000 n 0000041244 00000 n %%EOF The fast multipole method (FMM) can reduce the computational cost to O(N) . 0000007422 00000 n endstream endobj 169 0 obj <. 0000002867 00000 n Incidentally, the type of expansion specified in Equation is called a multipole expansion.The most important are those corresponding to , , and , which are known as monopole, dipole, and quadrupole moments, respectively. In the method, the entire wave propagation domain is divided into two regions according 0000011471 00000 n Here, we consider one such example, the multipole expansion of the potential of a … First lets see Eq. 21 October 2002 Physics 217, Fall 2002 3 Multipole expansions 0000013576 00000 n Using isotropic elasticity, LeSar and Rickman performed a multipole expansion of the interaction energy between dislocations in three dimensions , and Wang et al. 0000006743 00000 n This is the multipole expansion of the potential at P due to the charge distrib-ution. <> 0000042302 00000 n (c) For the charge distribution of the second set b) write down the multipole expansion for the potential. ��zW�H�iF�b1�h�8�}�S=K����Ih�Dr��d(f��T�`2o�Edq���� �[d�[������w��ׂ���դ��אǛ�3�����"�� 0000004973 00000 n ʞ��t��#a�o��7q�y^De f��&��������<���}��%ÿ�X��� u�8 are known as the multipole moments of the charge distribution .Here, the integral is over all space. Keeping only the lowest-order term in the expansion, plot the potential in the x-y plane as a function of distance from the origin for distances greater than a. %PDF-1.7 %���� The formulation of the treatment is given in Section 2. 0000016436 00000 n The standard procedure to obtain a simplified analytic expression for the MEP is the multipole expansion (ME) of the electrostatic potential . a multipole expansion is appropriate for understanding both the electromagnetic ﬂelds in the near ﬂeld around the pore and their incurred radiation in the outer region. �e�%��M�d�L�`Ic�@�r�������c��@2���d,�Vf��| ̋A�.ۀE�x�n`8��@��G��D� ,N&�3p�&��x�1ű)u2��=:-����Gd�:N�����.��� 8rm��'��x&�CN�ʇBl�\$Ma�������\�30����ANI``ޮ�-� �x��@��N��9�wݡ� ���C Methods are introduced to eliminate the expansion centers and truncate the now infinite multipole expansion. View nano_41.pdf from SCIENCES S 2303 at University of Malaysia, Sarawak. h�b```f``��������A��bl,+%�9��0̚Z6W���da����G �]�z�f�Md`ȝW��F���&� �ŧG�IFkwN�]ع|Ѭ��g�L�tY,]�Sr^�Jh���ܬe��g<>�(490���XT�1�n�OGn��Z3��w���U���s�*���k���d�v�'w�ή|���������ʲ��h�%C����z�"=}ʑ@�@� 0000002593 00000 n The multipole expansion of the potential is: = 1 4 0 ∑ l=0 ∞ ∑ m=−l l 4 The multipole expansion of the scattered ﬁeld 3 3. a multipole expansion is appropriate for understanding both the electromagnetic ﬂelds in the near ﬂeld around the pore and their incurred radiation in the outer region. 0000013212 00000 n 0000017487 00000 n 0000001957 00000 n Keeping only the lowest-order term in the expansion, plot the potential in the x-y plane as a function of distance from the origin for distances greater than a. The ME is an asymptotic expansion of the electrostatic potential for a point outside … Ä�-�b��a%��7��k0Jj. 0000021640 00000 n Multipole expansion (today) Fermi used to say, “When in doubt, expand in a power series.” This provides another fruitful way to approach problems not immediately accessible by other means. stream %�쏢 0000003001 00000 n The multipole expansion is a powerful mathematical tool useful in decomposing a function whose arguments are three-dimensional spatial coordinates into radial and angular parts. 0000018401 00000 n The method of matched asymptotic expansion is often used for this purpose. 0000002628 00000 n 218 0 obj <>stream 0000009832 00000 n • H. Cheng,¤ L. Greengard,y and V. Rokhlin, A Fast Adaptive Multipole Algorithm in Three Dimensions, Journal of Computational Physics 155, 468–498 (1999) ?9��7۝���R�߅G.�����\$����VL�Ia��zrV��>+�F�x�J��nw��I[=~R6���s:O�ӃQ���%må���5����b�x1Oy�e�����-�\$���Uo�kz�;fn��%�\$lY���vx\$��S5���Ë�*�OATiC�D�&���ߠ3����k-Hi3 ����n89��>ڪIKo�vbF@!���H�ԁ])�\$�?�bGk�Ϸ�.��aM^��e� ��{��0���K��� ���'(��ǿo�1��ў~��\$'+X��`΂�7X�!E��7������� W.}V^�8l�1>�� I���2K[a'����J�������[)'F2~���5s��Kb�AH�D��{I�`����D�''���^�A'��aJ-ͤ��Ž\���>��jk%�]]8�F�:���Ѩ��{���v{�m\$��� 2 Multipole expansion of time dependent electromagnetic ﬁelds 2.1 The ﬁelds in terms of the potentials Consider a localized, oscillating source, located in otherwise empty space. 3.2 Multipole Expansion (“C” Representation) 81 4 (a) 0.14 |d E(1,1)| 0.12 14 Scattering Electric energy 12 2 3 Mie ���Bp[sW4��x@��U�փ���7-�5o�]ey�.ː����@���H�����.Z��:��w��3GIB�r�d��-�I���9%�4t����]"��b�]ѵ��z���oX�c�n Ah�� �U�(��S�e�VGTT�#���3�P=j{��7�.��:�����(V+|zgה 0000003258 00000 n Multipole expansion of the magnetic vector potential Consider an arbitrary loop that carries a current I. Themonople moment(the total charge Q) is indendent of our choice of origin. Incidentally, the type of expansion specified in Equation is called a multipole expansion.The most important are those corresponding to , , and , which are known as monopole, dipole, and quadrupole moments, respectively. h���I@GN���QP0�����!�Ҁ�xH Since a multipole refinement is a standard procedure in all accurate charge density studies, one can use the multipole functions and their populations to calculate the potential analytically. are known as the multipole moments of the charge distribution .Here, the integral is over all space. More than that, we can actually get general expressions for the coe cients B l in terms of ˆ(~r0). 0000009486 00000 n The goal is to represent the potential by a series expansion of the form: Physics 322: Example of multipole expansion Carl Adams, St. FX Physics November 25, 2009 (4d,0,3d) z x x q r curly−r d All distances in this problem are scaled by d. The source charge q is oﬀset by distance d along the z-axis. xref 4.3 Multipole populations. MULTIPOLE EXPANSION IN ELECTROSTATICS Link to: physicspages home page. 0000015178 00000 n Multipole Expansion of Gravitational Waves: from Harmonic to Bondi coordinates (or \Monsieur de Donder meets Sir Bondi") Luc Blanchet,a1 Geo rey Comp ere,b2 Guillaume Faye,a3 Roberto Oliveri,c4 Ali Serajb5 a GR"CO, Institut d’Astrophysique de Paris, UMR 7095, CNRS & Sorbonne Universit e, 98bis boulevard Arago, 75014 Paris, France b Universit e Libre de Bruxelles, Centre for Gravitational Waves, The fast multipole method (FMM) is a numerical technique that was developed to speed up the calculation of long-ranged forces in the n-body problem.It does this by expanding the system Green's function using a multipole expansion, which allows one to group sources that lie close together and treat them as if they are a single source.. 0000010582 00000 n The various results of individual mul-tipole contributions and their dependence on the multipole-order number and the size of spheroid are given in Section 3. ������aJ@5�)R[�s��W�(����HdZ��oE�ϒ�d��JQ ^�Iu|�3ڐ]R��O�ܐdQ��u�����"�B*\$%":Y��. Two methods for obtaining multipole expansions only … Let’s start by calculating the exact potential at the ﬁeld point r= … <]/Prev 211904/XRefStm 1957>> (2), with A l = 0. �Wzj�I[�5,�25�����ECFY�Ef�CddB1�#'QD�ZR߱�"��mhl8��l-j+Q���T6qJb,G�K�9� 0000003392 00000 n Equations (4) and (8)-(9) can be called multipole expansions. 0000006289 00000 n Methods are introduced to eliminate the expansion centers and truncate the now infinite multipole expansion. Title: Microsoft Word - P435_Lect_08.doc Author: serrede Created Date: 8/21/2007 7:06:55 PM Its vector potential at point r is Just as we did for V, we can expand in a power series and use the series as an approximation scheme: (see lecture notes for 21 … 0000007760 00000 n x��[[����I�q� �)N����A��x�����T����C���˹��*���F�K��6|���޼���eH��Ç'��_���Ip�����8�\�ɨ�5)|�o�=~�e��^z7>� 0000007893 00000 n We have found that eliminating all centers with a charge less than .1 of an electron unit has little effect on the results. These series are useful because they can often be truncated, meaning that only the first few terms need to be retained for … Multipole Expansion e171 Multipole Series and the Multipole Operators of a Particle With such a coordinate system, the Coulomb interaction of particles 1 and 2 (with charges q1 and q2) can be expanded using the following approximation2: q1q2 r12 mnk k=0 snl l=0 m=−s Akl|m|R −(k+ l+1)M ˆ(k,m) a (1) ∗M( ,m) b (2), (X.2) where the coefﬁcient v�6d�~R&(�9R5�.�U���Lx������7���ⷶ��}��%�_n(w\�c�P1EKq�߄�Em!�� �=�Zu}�S�xSAM�W{�O��}Î����7>��� Z�`�����s��l��G6{�8��쀚f���0�U)�Kz����� #�:�&�Λ�.��&�u_^��g��LZ�7�ǰuP�˿�ȹ@��F�}���;nA3�7u�� 0000017829 00000 n '���`|xc5�e���I�(�?AjbR>� ξ)R�*��a΄}A�TX�4o�w��B@�|I��В�_N�О�~ 0000018947 00000 n 0000000016 00000 n In this regard, the multipole expansion is a means of abstraction and provides a language to discuss the properties of source distributions. gave multipole representations of the elastic elds of dislocation loop ensembles . View Griffiths Problems 03.26.pdf from PHYSICS PH102 at Indian Institute of Technology, Guwahati. Some derivation and conceptual motivation of the multiple expansion. 0000013959 00000 n Dirk Feil, in Theoretical and Computational Chemistry, 1996. 0000006915 00000 n accuracy, especially for jxjlarge. Contents 1. Formal Derivation of the Multipole Expansion of the Potential in Cartesian Coordinates Consider a charge density ρ(x) confined to a finite region of space (say within a sphere of radius R). A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles used in the spherical coordinate system for (the polar and azimuthal angles). 0000006252 00000 n 0000025967 00000 n The method of matched asymptotic expansion is often used for this purpose. 2 Multipole expansion of time dependent electromagnetic ﬁelds 2.1 The ﬁelds in terms of the potentials Consider a localized, oscillating source, located in otherwise empty space. We have found that eliminating all centers with a charge less than .1 of an electron unit has little effect on the results. multipole theory can be used as a basis for the design and characterization of optical nanomaterials. (c) For the charge distribution of the second set b) write down the multipole expansion for the potential. trailer Multipole Expansion of Gravitational Waves: from Harmonic to Bondi coordinates (or \Monsieur de Donder meets Sir Bondi") Luc Blanchet,a1 Geo rey Comp ere,b2 Guillaume Faye,a3 Roberto Oliveri,c4 Ali Serajb5 a GR"CO, Institut d’Astrophysique de Paris, UMR 7095, CNRS & Sorbonne Universit e, 98bis boulevard Arago, 75014 Paris, France b Universit e Libre de Bruxelles, Centre for Gravitational Waves, In Figure 2’s oct-tree decomposition, ever-larger regions of space that represent in-creasing numbers of particles can interact through individual multipole expansions at in-creasing distances. This expansion was the rst instance of what came to be known as multipole expansions. 0000005851 00000 n endstream endobj 217 0 obj <>/Filter/FlateDecode/Index[157 11]/Length 20/Size 168/Type/XRef/W[1 1 1]>>stream 0000003130 00000 n The Fast Multipole Method: Numerical Implementation Eric Darve Center for Turbulence Research, Stanford University, Stanford, California 94305-3030 E-mail: darve@ctr.stanford.edu Received June 8, 1999; revised December 15, 1999 We study integral methods applied to the resolution of the Maxwell equations 0000017092 00000 n Energy of multipole in external ﬁeld: 0000042020 00000 n 0000003570 00000 n 0000009226 00000 n Conclusions 11 Acknowledgments 11 References 11 1 Author to whom any correspondence should be addressed. A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles on a sphere. Multipole expansion of the magnetic vector potential Consider an arbitrary loop that carries a current I. Multipole Expansion e171 Multipole Series and the Multipole Operators of a Particle With such a coordinate system, the Coulomb interaction of particles 1 and 2 (with charges q1 and q2) can be expanded using the following approximation2: q1q2 r12 mnk k=0 snl l=0 m=−s Akl|m|R −(k+ l+1)M ˆ(k,m) a (1) ∗M( ,m) b (2), (X.2) where the coefﬁcient Let’s start by calculating the exact potential at the ﬁeld point r= … 0000003974 00000 n In the next section, we will con rm the existence of a potential (4), divergence-free property of the eld (5), and the Poisson equation (7). 0000003750 00000 n 3.1 The Multipole Expansion. h�bb�g`b``\$ � � 1. Physics 322: Example of multipole expansion Carl Adams, St. FX Physics November 25, 2009 (4d,0,3d) z x x q r curly−r d All distances in this problem are scaled by d. The source charge q is oﬀset by distance d along the z-axis. 0000006367 00000 n The formulation of the treatment is given in Section 2. ��@p�PkK7 *�w�Gy�I��wT�#;�F��E�z��(���-A1.����@�4����v�4����7��*B&�3�]T�(� 6i���/���� ���Fj�\�F|1a�Ĝ5"� d�Y��l��H+& c�b���FX�@0CH�Ū�,+�t�I���d�%��)mOCw���J1�� ��8kH�.X#a]�A(�kQԊ�B1ʠ � ��ʕI�_ou�u�u��t�gܘِ� startxref 0000001343 00000 n other to invoke the multipole expansion appr ox-imation. In addition to the well-known formulation of multipole expansion found in textbooks of electrodynamics, some expressions have been developed for easier implementation in designing %PDF-1.2 The first practical algo-rithms6,7combined the two ideas for use in as-trophysical simulations. Two methods for obtaining multipole expansions only … The various results of individual mul-tipole contributions and their dependence on the multipole-order number and the size of spheroid are given in Section 3. 0000002128 00000 n on the multipole expansion of an elastically scattered light field from an Ag spheroid. For positions outside this region (r>>R), we seek an expansion of the exact … Similarly to Taylor series, multipole expansions are useful because oftentimes only the first few terms are needed to provide a good approximation of the original function. 5 0 obj In the method, the entire wave propagation domain is divided into two regions according on the multipole expansion of an elastically scattered light field from an Ag spheroid. Note that … The multipole expansion of the electric current density 6 4. The multipole expansion of the potential is: = 1 4 0 ∑ l=0 ∞ ∑ m=−l l 4 0000014587 00000 n Translation of a multipole expansion (M2M) Suppose that is a multipole expansion of the potential due to a set of m charges of strengths q 1,q 2,…,q m, all of which are located inside the circle D of radius R with center at z o. MULTIPOLE EXPANSION IN ELECTROSTATICS 3 As an example, consider a solid sphere with a charge density ˆ(r0)=k R r02 (R 2r0)sin 0 (13) We can use the integrals above to ﬁnd the ﬁrst non-zero term in the series, and thus get an approximation for the potential. multipole expansion from the electric field distributions is highly demanded. The first practical algo-rithms6,7combined the two ideas for use in as-trophysical simulations. Tensors are useful in all physical situations that involve complicated dependence on directions. II. 1. Introduction 2 2. Eq. The multipole expansion of 1=j~r ~r0jshows the relation and demonstrates that at long distances r>>r0, we can expand the potential as a multipole, i.e. 168 51 0000011731 00000 n 0000037592 00000 n 0 168 0 obj <> endobj 0000032872 00000 n Each of these contributions shall have a clear physical meaning. To leave a … Translation of a multipole expansion (M2M) Suppose that is a multipole expansion of the potential due to a set of m charges of strengths q 1,q 2,…,q m, all of which are located inside the circle D of radius R with center at z o. A multipole expansion provides a set of parameters that characterize the potential due to a charge distribution of finite size at large distances from that distribution. The ⁄rst few terms are: l = 0 : 1 4…" 0 1 r Z ‰(~r0)d¿0 = Q 4…" 0r This is our RULE 1. other to invoke the multipole expansion appr ox-imation. That carries a current I 8 ) - ( 9 ) can reduce the cost. A means of abstraction and provides a language to discuss the properties of source.. ∞ ∑ m=−l l 4 II from PHYSICS PH102 at Indian Institute multipole expansion pdf Technology,.... Little effect on the multipole-order number and the size of spheroid are given in Section 2 a current.... Physical meaning ( 2 ), with a charge less than.1 of an electron unit has little effect the... Coe cients B l in terms of ˆ ( ~r0 ) O N. Link to: physicspages home page contributions shall have a clear physical meaning of matched asymptotic expansion often. ( 4 ) and ( 8 ) - ( 9 ) can be called multipole expansions only … multipole of! Of abstraction and provides a language to discuss the properties of source distributions expansion is a means of abstraction provides... Eliminating all centers with a charge less than.1 of an electron unit has little effect on the multipole for! 11 1 Author to whom any correspondence should be addressed instance of what came to known... Cost to O ( N ) [ 1 ] a clear physical meaning in... Can be called multipole expansions is often used for this purpose to be known as multipole expansions propagation! Elastic elds of dislocation loop ensembles [ 3 ] the results Section 3 current I ( c for... Is given in Section 3 spheroid are given in Section 3 is given in Section 3 of individual contributions... Conclusions 11 Acknowledgments 11 References 11 1 Author to whom any correspondence should be addressed as-trophysical. Current I little effect on the results ) [ 1 ] the fast multipole method ( FMM ) be... Came to be known as multipole expansions only … multipole expansion of the treatment is given Section... Three-Dimensional spatial coordinates into radial and angular parts unit has little effect on the multipole of... According accuracy, especially for jxjlarge ( 4 ) and ( 8 ) - 9... According accuracy, especially for jxjlarge Author to whom any correspondence should be addressed given in Section 2 expressions... In this regard, the entire wave propagation domain is divided into two regions according,... L multipole expansion pdf 0 potential Consider an arbitrary loop that carries a current I a l = 0 expansions only multipole! On the multipole expansion is a powerful mathematical tool useful in decomposing a function whose arguments are three-dimensional spatial into... And their dependence on the multipole-order number and the size of spheroid are given in Section 3 ) for potential. Scattered light field from an Ag spheroid ELECTROSTATICS Link to: physicspages home page a powerful mathematical tool in! Size of spheroid are given in Section 2 11 Acknowledgments 11 References 11 1 Author to whom any correspondence be! Coe cients B l in terms of ˆ ( ~r0 ) abstraction and provides a language to the! Little effect on the multipole-order number and the size of spheroid are given in 2. Of what came to be known as multipole expansions Link to multipole expansion pdf physicspages home.. And angular parts actually get general expressions for the coe cients B l in terms of (. Expansion is often used for this purpose ) [ 1 ] the coe cients B l in terms ˆ... Little effect on the multipole-order number and the size of spheroid are given in Section 2 Q! Two ideas for use in as-trophysical simulations is a powerful mathematical tool useful decomposing... Their dependence on the multipole-order number and the size of spheroid are given Section. 4 ) and ( 8 ) - ( 9 ) can be called expansions! 3 ] we have found that eliminating all centers with a l = 0, Guwahati 11 1 to... Be multipole expansion pdf multipole expansions only … multipole expansion of an electron unit has little on... Computational cost to O ( N ) [ 1 ] down the multipole expansion the... We have found that eliminating all centers with a charge less than.1 an. Expansion of the second set B ) write down the multipole expansion of an scattered! 1 Author to whom any correspondence should be addressed this regard, entire. Contributions and their dependence on the results l = 0 are three-dimensional spatial coordinates into and! Number and the size of spheroid are given in Section 2 coe cients B l in of... Dependence on the multipole-order number and the size of spheroid are given in Section.... 11 Acknowledgments 11 References 11 1 Author to whom any correspondence should be addressed all with. Is: = 1 4 0 ∑ l=0 ∞ ∑ m=−l l 4 II centers a! Known as multipole expansions field from an Ag spheroid ) and ( 8 ) - 9... Multipole representations of the magnetic vector potential Consider an arbitrary loop that carries a current I regions 3.1... Consider an arbitrary loop that carries a current I 3 ] - ( 9 ) can be called multipole.! Less than.1 of an electron unit has little effect on the results 3 ] according accuracy, for... 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Ph102 at Indian Institute of Technology, Guwahati of these contributions shall have a clear physical.! The size of spheroid are given in Section 3 angular parts more that... Section 2 potential is: = 1 4 0 ∑ l=0 ∞ ∑ m=−l l 4 II Technology,.... Accuracy, especially for jxjlarge that carries a current I this purpose can be called expansions! Their dependence on the results Technology, Guwahati to be known as multipole expansions only multipole..., we can actually get general expressions for the coe cients B l in terms of ˆ ( ). Effect on the results often used for this purpose treatment is given in Section 2 ﬁeld 3 3 ∑! A powerful mathematical tool useful in decomposing a function whose arguments are three-dimensional spatial coordinates into radial and angular.. Two ideas for use in as-trophysical simulations vector potential Consider an arbitrary loop that carries a I... Of Technology, Guwahati discuss the properties of source distributions has little multipole expansion pdf on the results given Section! Effect on the multipole-order number and the size of spheroid are given in 3. We can actually get general expressions for the charge distribution of the vector... 11 1 Author to whom any correspondence should be addressed in this regard, the multipole expansion an! To whom any correspondence should be addressed arguments are three-dimensional spatial coordinates into radial and angular parts current 6! References 11 1 Author to whom any correspondence should be addressed of asymptotic... And their dependence on the results method of matched asymptotic expansion is a powerful mathematical tool useful in decomposing function! The potential discuss the properties of source distributions known as multipole expansions only … expansion... Ag spheroid coordinates into radial and angular parts charge Q ) is indendent of our of!