Multipole

Multipoles (from lat. Multum - many and Greek. Πόλος - pole) - certain configurations of point sources ( charges ). The simplest examples of a multipole are a point charge — a zero-order multipole; two opposite in sign signs of charge, equal in absolute value - a dipole , or a multipole of the first order; 4 identical in absolute value charges placed at the vertices of the parallelogram, so that each side of it connects charges of the opposite sign (or two identical but oppositely directed dipoles) - a quadrupole , or a second-order multipole. The name multipole includes the designation of the number of charges (in Latin) that form the multipole, for example, an octupole (octu - 8) means that the multipole includes 8 charges.

The separation of such configurations is associated with the expansion of the field ^[1] from complex field-limited systems of field sources (including the case of a continuous distribution of sources) into multipoles - with the so-called 'multipole expansion'.

By field can be meant an electrostatic or magnetostatic field, as well as fields similar to them (for example, Newtonian gravitational field) ^[2] .

Such a decomposition can often be used for an approximate description of the field from a complex system of sources at a large (much larger than the size of this system itself) distance from it; In this case, it is important that the multipole field of each next order decreases with a distance much faster than the previous ones, so it is often possible to limit ourselves to several (depending on the distance and the required accuracy) terms (lower orders) of the multipole expansion. In the other case, for various reasons, the multipole expansion is convenient even when summing all orders (then it is an infinite series); in this case, it gives an exact expression of the field not only at large, but in principle at any distances from the source system (with the exception of its internal areas).

In addition to static (or approximately static) fields, often in connection with multipole moments we speak of multipole radiation — radiation considered as due to a change in time of multipole moments of the emitter system. This case is distinguished by the fact that in it fields of different orders decrease with distance equally quickly, differing depending on the angle.

Content

Multiple Scalar Field Decomposition

The system of point resting charges

The electrostatic potential of a charge system at a point $\mathbf {R}$ ${\ displaystyle \ mathbf {R}}$

\varphi (\mathbf {R} )=\sum \limits _{i}{\frac {q_{i}}{|\mathbf {R} -\mathbf {r} _{i}|}},

{\ displaystyle \ varphi (\ mathbf {R}) = \ sum \ limits _ {i} {\ frac {q_ {i}} {| \ mathbf {R} - \ mathbf {r} _ {i} |}} ,}

Where $q_{i}$ ${\ displaystyle q_ {i}}$ - charges $\mathbf {r} _{i}$ ${\ displaystyle \ mathbf {r} _ {i}}$ - their coordinates. Expanding this potential in a Taylor series , we obtain

\varphi (\mathbf {R} )=\sum \limits _{l=0}^{\infty }\varphi ^{(l)}(\mathbf {R} ),

{\ displaystyle \ varphi (\ mathbf {R}) = \ sum \ limits _ {l = 0} ^ {\ infty} \ varphi ^ {(l)} (\ mathbf {R}),}

called multipole decomposition , where the notation

\varphi ^{(l)}(\mathbf {R} )={\frac {(-1)^{l}}{l!}}\sum \limits _{i}q_{i}r_{i}^{\alpha _{1}}\dots r_{i}^{\alpha _{n}}{\frac {\partial ^{l}}{\partial R_{i}^{\alpha _{1}}\dots \partial R_{i}^{\alpha _{n}}}}\left({\frac {1}{R}}\right)

{\ displaystyle \ varphi ^ {(l)} (\ mathbf {R}) = {\ frac {(-1) ^ {l}} {l!}} \ sum \ limits _ {i} q_ {i} r_ {i} ^ {\ alpha _ {1}} \ dots r_ {i} ^ {\ alpha _ {n}} {\ frac {\ partial ^ {l}} {\ partial R_ {i} ^ {\ alpha _ {1}} \ dots \ partial R_ {i} ^ {\ alpha _ {n}}}} \ left ({\ frac {1} {R}} \ right)}

- $2^{l}$ ${\ displaystyle 2 ^ {l}}$ floor potentials $l$ ${\ displaystyle l}$ called the order of the term of the multipole expansion. The 0th order term has the form

\varphi ^{(0)}(\mathbf {R} )={\frac {\sum \limits _{i}q_{i}}{R}},

{\ displaystyle \ varphi ^ {(0)} (\ mathbf {R}) = {\ frac {\ sum \ limits _ {i} q_ {i}} {R}},}

which coincides with the potential of a point charge $Q=\sum \limits _{i}q_{i}$ ${\ displaystyle Q = \ sum \ limits _ {i} q_ {i}}$ (monopole potential). 1st order term equals

\varphi ^{(1)}(\mathbf {R} )={\frac {\left(\sum \limits _{i}q_{i}\mathbf {r} _{i}\right)\mathbf {n} }{R^{2}}},

{\ displaystyle \ varphi ^ {(1)} (\ mathbf {R}) = {\ frac {\ left (\ sum \ limits _ {i} q_ {i} \ mathbf {r} _ {i} \ right) \ mathbf {n}} {R ^ {2}}},}

Where $\mathbf {n} =\mathbf {R} /R$ ${\ displaystyle \ mathbf {n} = \ mathbf {R} / R}$ Is a unit vector directed along $\mathbf {R}$ ${\ displaystyle \ mathbf {R}}$ . If we introduce the dipole moment of the charge system as $\mathbf {d} =\sum \limits _{i}q_{i}\mathbf {r} _{i}$ ${\ displaystyle \ mathbf {d} = \ sum \ limits _ {i} q_ {i} \ mathbf {r} _ {i}}$ then the system $\varphi ^{(1)}(\mathbf {R} )$ ${\ displaystyle \ varphi ^ {(1)} (\ mathbf {R})}$ coincides with the potential of a point dipole . Thus, the potential in the 1st order of expansion in multipoles has the form

\varphi (\mathbf {R} )={\frac {q}{R}}+{\frac {\mathbf {d} \mathbf {n} }{R^{2}}}+O\left({\frac {1}{R^{3}}}\right).

{\ displaystyle \ varphi (\ mathbf {R}) = {\ frac {q} {R}} + {\ frac {\ mathbf {d} \ mathbf {n}} {R ^ {2}}} + O \ left ({\ frac {1} {R ^ {3}}} \ right).}

If a $q=0$ ${\ displaystyle q = 0}$ , then the dipole moment does not depend on the choice of the origin. If a $q\neq 0$ ${\ displaystyle q \ neq 0}$ , then you can choose a coordinate system centered at a point $\mathbf {R} _{0}=\mathbf {d} /q$ ${\ displaystyle \ mathbf {R} _ {0} = \ mathbf {d} / q}$ then the dipole moment becomes equal to zero. Such a system is called a center of charge system. The next expansion term has the form

\varphi ^{(2)}(\mathbf {R} )={\frac {D^{\alpha _{1}\alpha _{2}}n_{\alpha _{1}}n_{\alpha _{2}}}{2R^{3}}},

{\ displaystyle \ varphi ^ {(2)} (\ mathbf {R}) = {\ frac {D ^ {\ alpha _ {1} \ alpha _ {2}} n _ {\ alpha _ {1}} n_ { \ alpha _ {2}}} {2R ^ {3}}},}

Where $D^{\alpha _{1}\alpha _{2}}=\sum \limits _{i}q_{i}(3r_{i}^{\alpha _{1}}r_{i}^{\alpha _{2}}-\delta ^{\alpha _{1}\alpha _{2}}\mathbf {r} _{i}^{2})$ ${\ displaystyle D ^ {\ alpha _ {1} \ alpha _ {2}} = \ sum \ limits _ {i} q_ {i} (3r_ {i} ^ {\ alpha _ {1}} r_ {i} ^ {\ alpha _ {2}} - \ delta ^ {\ alpha _ {1} \ alpha _ {2}} \ mathbf {r} _ {i} ^ {2})}$ Is the quadrupole moment of the charge system. We introduce the matrix $D$ ${\ displaystyle D}$ quadrupole moment. Then the potential in the 2nd order of expansion in multipoles will take the form

\varphi (\mathbf {R} )={\frac {q}{R}}+{\frac {\mathbf {d} \mathbf {n} }{R^{2}}}+{\frac {\mathbf {n} D\mathbf {n} }{2R^{3}}}+O\left({\frac {1}{R^{4}}}\right).

{\ displaystyle \ varphi (\ mathbf {R}) = {\ frac {q} {R}} + {\ frac {\ mathbf {d} \ mathbf {n}} {R ^ {2}}} + {\ frac {\ mathbf {n} D \ mathbf {n}} {2R ^ {3}}} + O \ left ({\ frac {1} {R ^ {4}}} \ right).}

Matrix $D$ ${\ displaystyle D}$ is traceless , that is $\mathrm {tr} D=0$ ${\ displaystyle \ mathrm {tr} D = 0}$ . In addition, it is symmetrical , i.e. $D^{T}=D$ ${\ displaystyle D ^ {T} = D}$ . Therefore, it can be reduced to a diagonal view by turning the axes of the Cartesian coordinates.

Generally contribution $l$ ${\ displaystyle l}$ th order in potential can be represented as:

\varphi ^{(l)}(\mathbf {R} )={\frac {d^{\alpha _{1}\dots \alpha _{l}}n_{\alpha _{1}}\dots n_{\alpha _{l}}}{l!R^{l+1}}},

{\ displaystyle \ varphi ^ {(l)} (\ mathbf {R}) = {\ frac {d ^ {\ alpha _ {1} \ dots \ alpha _ {l}} n _ {\ alpha _ {1}} \ dots n _ {\ alpha _ {l}}} {l! R ^ {l + 1}}},}

Where $d^{\alpha _{1}\dots \alpha _{l}}$ ${\ displaystyle d ^ {\ alpha _ {1} \ dots \ alpha _ {l}}}$ - $2^{l}$ ${\ displaystyle 2 ^ {l}}$ is the total moment of the charge system, which is an irreducible tensor $l$ ${\ displaystyle l}$ th order. This tensor is symmetric in any pair of indices and vanishes when minimized in any pair of indices.

Distributed Charge System

If the charge is distributed with some density $\rho (\mathbf {r} )$ ${\ displaystyle \ rho (\ mathbf {r})}$ , then passing to the continuous limit (or directly deriving from the original formulas) in the formulas for the discrete distribution, we can obtain a multipole expansion and in this case:

\varphi (\mathbf {R} )=\int \limits _{(V)}{\frac {\rho (\mathbf {r} )}{|\mathbf {R} -\mathbf {r} |}}d^{3}\mathbf {r} ,

{\ displaystyle \ varphi (\ mathbf {R}) = \ int \ limits _ {(V)} {\ frac {\ rho (\ mathbf {r})} {| \ mathbf {R} - \ mathbf {r} |}} d ^ {3} \ mathbf {r},}

Where $V$ ${\ displaystyle V}$ - the volume in which the distributed charge is located. Then the multipole moments have the form:

q=\int \limits _{(V)}\rho (\mathbf {r} )d^{3}\mathbf {r} ,

{\ displaystyle q = \ int \ limits _ {(V)} \ rho (\ mathbf {r}) d ^ {3} \ mathbf {r},}

\mathbf {d} =\int \limits _{(V)}\rho (\mathbf {r} )\mathbf {r} d^{3}\mathbf {r} ,

{\ displaystyle \ mathbf {d} = \ int \ limits _ {(V)} \ rho (\ mathbf {r}) \ mathbf {r} d ^ {3} \ mathbf {r},}

D^{\alpha _{1}\alpha _{2}}=\int \limits _{(V)}\rho (\mathbf {r} )(3r^{\alpha _{1}}r^{\alpha _{2}}-\delta ^{\alpha _{1}\alpha _{2}}\mathbf {r} ^{2})d^{3}\mathbf {r}

{\ displaystyle D ^ {\ alpha _ {1} \ alpha _ {2}} = \ int \ limits _ {(V)} \ rho (\ mathbf {r}) (3r ^ {\ alpha _ {1}} r ^ {\ alpha _ {2}} - \ delta ^ {\ alpha _ {1} \ alpha _ {2}} \ mathbf {r} ^ {2}) d ^ {3} \ mathbf {r}}

The formulas for the multipole potentials remain unchanged. The case of a discrete system of charges can be obtained by substituting their distribution density, which can be expressed in terms of δ-functions :

\rho (\mathbf {r} )=\sum _{i}q_{i}\delta (\mathbf {r} -\mathbf {r} _{i}).

{\ displaystyle \ rho (\ mathbf {r}) = \ sum _ {i} q_ {i} \ delta (\ mathbf {r} - \ mathbf {r} _ {i}).}

When calculating the potential, a useful formula ${\frac {1}{|R-r|}}={\frac {1}{R}}\sum _{n=0}^{\infty }P_{n}(x)\left({\frac {r}{R}}\right)^{n}$ ${\ displaystyle {\ frac {1} {| Rr |}} = {\ frac {1} {R}} \ sum _ {n = 0} ^ {\ infty} P_ {n} (x) \ left ({ \ frac {r} {R}} \ right) ^ {n}}$ where $P_{n}(x)$ ${\ displaystyle P_ {n} (x)}$ - Legendre polynomials , $x=\cos \theta$ ${\ displaystyle x = \ cos \ theta}$ . ^[3]

Multipole Decomposition of Electrostatic Field

The electrostatic field strength of the system of charges is equal to the gradient of the electrostatic potential, taken with the opposite sign

\mathbf {E} (\mathbf {R} )=-\nabla \varphi (\mathbf {R} ).

{\ displaystyle \ mathbf {E} (\ mathbf {R}) = - \ nabla \ varphi (\ mathbf {R}).}

Substituting the multipole expansion of the potential into this formula, we obtain the multipole expansion of the electrostatic field strength

\mathbf {E} (\mathbf {R} )=\sum \limits _{l=0}^{\infty }\mathbf {E} ^{(l)}(\mathbf {R} ),

{\ displaystyle \ mathbf {E} (\ mathbf {R}) = \ sum \ limits _ {l = 0} ^ {\ infty} \ mathbf {E} ^ {(l)} (\ mathbf {R}), }

Where

(\mathbf {E} ^{(l)}(\mathbf {R} ))_{\alpha _{0}}={\frac {(-1)^{l+1}}{l!}}\sum \limits _{i}q_{i}r_{i}^{\alpha _{1}}\dots r_{i}^{\alpha _{n}}{\frac {\partial ^{l+1}}{\partial R_{i}^{\alpha _{0}}\partial R_{i}^{\alpha _{1}}\dots \partial R_{i}^{\alpha _{n}}}}\left({\frac {1}{R}}\right)

{\ displaystyle (\ mathbf {E} ^ {(l)} (\ mathbf {R})) _ {\ alpha _ {0}} = {\ frac {(-1) ^ {l + 1}} {l !}} \ sum \ limits _ {i} q_ {i} r_ {i} ^ {\ alpha _ {1}} \ dots r_ {i} ^ {\ alpha _ {n}} {\ frac {\ partial ^ {l + 1}} {\ partial R_ {i} ^ {\ alpha _ {0}} \ partial R_ {i} ^ {\ alpha _ {1}} \ dots \ partial R_ {i} ^ {\ alpha _ {n}}}} \ left ({\ frac {1} {R}} \ right)}

- electric field $2^{l}$ ${\ displaystyle 2 ^ {l}}$ -fields.

In particular, the field of a point charge (monopole) has the form:

\mathbf {E} ^{(0)}(\mathbf {R} )={\frac {Q}{R^{2}}}\mathbf {n} ,

{\ displaystyle \ mathbf {E} ^ {(0)} (\ mathbf {R}) = {\ frac {Q} {R ^ {2}}} \ mathbf {n},}

which corresponds to the law of Coulomb .

Point dipole field:

\mathbf {E} ^{(1)}(\mathbf {R} )={\frac {3(\mathbf {n} \mathbf {d} )\mathbf {n} -\mathbf {d} }{R^{3}}}.

{\ displaystyle \ mathbf {E} ^ {(1)} (\ mathbf {R}) = {\ frac {3 (\ mathbf {n} \ mathbf {d}) \ mathbf {n} - \ mathbf {d} } {R ^ {3}}}.}

Point quadrupole field:

\mathbf {E} ^{(2)}(\mathbf {R} )={\frac {5\mathbf {n} (\mathbf {n} D\mathbf {n} )-2D\mathbf {n} }{2R^{4}}}.

{\ displaystyle \ mathbf {E} ^ {(2)} (\ mathbf {R}) = {\ frac {5 \ mathbf {n} (\ mathbf {n} D \ mathbf {n}) -2D \ mathbf { n}} {2R ^ {4}}}.}

Thus, the electric field of the system of resting charges in the 2nd order of multipole expansion has the form:

\mathbf {E} (\mathbf {R} )={\frac {Q}{R^{2}}}\mathbf {n} +{\frac {3(\mathbf {n} \mathbf {d} )\mathbf {n} -\mathbf {d} }{R^{3}}}+{\frac {5\mathbf {n} (\mathbf {n} D\mathbf {n} )-2D\mathbf {n} }{2R^{4}}}+O\left({\frac {1}{R^{5}}}\right).

{\ displaystyle \ mathbf {E} (\ mathbf {R}) = {\ frac {Q} {R ^ {2}}} \ mathbf {n} + {\ frac {3 (\ mathbf {n} \ mathbf { d}) \ mathbf {n} - \ mathbf {d}} {R ^ {3}}} + {\ frac {5 \ mathbf {n} (\ mathbf {n} D \ mathbf {n}) -2D \ mathbf {n}} {2R ^ {4}}} + O \ left ({\ frac {1} {R ^ {5}}} \ right).}

From this formula, it is easy to obtain the normal (radial) component of the electric field

E_{n}(\mathbf {R} )=\mathbf {n} \mathbf {E} (\mathbf {R} )={\frac {Q}{R^{2}}}+{\frac {2\mathbf {n} \mathbf {d} }{R^{3}}}+{\frac {3\mathbf {n} D\mathbf {n} }{2R^{4}}}+O\left({\frac {1}{R^{5}}}\right).

{\ displaystyle E_ {n} (\ mathbf {R}) = \ mathbf {n} \ mathbf {E} (\ mathbf {R}) = {\ frac {Q} {R ^ {2}}} + {\ frac {2 \ mathbf {n} \ mathbf {d}} {R ^ {3}}} + {\ frac {3 \ mathbf {n} D \ mathbf {n}} {2R ^ {4}}} + O \ left ({\ frac {1} {R ^ {5}}} \ right).}

The tangential component can be found by subtracting the normal

E_{\tau }(\mathbf {R} )=E(\mathbf {R} )-\mathbf {n} \mathbf {E} (\mathbf {R} )={\frac {(\mathbf {n} \mathbf {d} )\mathbf {n} -\mathbf {d} }{R^{3}}}+{\frac {\mathbf {n} (\mathbf {n} D\mathbf {n} )-D\mathbf {n} }{R^{4}}}+O\left({\frac {1}{R^{5}}}\right).

{\ displaystyle E _ {\ tau} (\ mathbf {R}) = E (\ mathbf {R}) - \ mathbf {n} \ mathbf {E} (\ mathbf {R}) = {\ frac {(\ mathbf {n} \ mathbf {d}) \ mathbf {n} - \ mathbf {d}} {R ^ {3}}} + {\ frac {\ mathbf {n} (\ mathbf {n} D \ mathbf {n }) -D \ mathbf {n}} {R ^ {4}}} + O \ left ({\ frac {1} {R ^ {5}}} right).}

If the normal (radial) component reflects a spherically symmetric distribution of charges, then the tangential component represents a non-spherical contribution to the electrostatic field . Thus, the quadrupole moment is interesting for studying not only when the total charge and dipole moment of the system are equal to zero, but also in the case when the Coulomb contribution is nonzero. Then, in accordance with the formula for the tangential component, the quadrupole moment characterizes the degree of non-sphericity of the electric field in the center of charge system. This is how the electric quadrupole moments of atomic nuclei were measured and it was concluded that they lack spherical symmetry.

Multipole Decomposition of a Static Magnetic Field

The vector potential of charges moving at a constant speed has the form:

\mathbf {A} (\mathbf {R} )={\frac {1}{c}}\sum \limits _{i}{\frac {q_{i}\mathbf {v} _{i}}{|\mathbf {R} -\mathbf {r} _{i}|}}

{\ displaystyle \ mathbf {A} (\ mathbf {R}) = {\ frac {1} {c}} \ sum \ limits _ {i} {\ frac {q_ {i} \ mathbf {v} _ {i }} {| \ mathbf {R} - \ mathbf {r} _ {i} |}}}

It likewise decomposes into a multipole expansion:

\mathbf {A} (\mathbf {R} )=\sum \limits _{l=1}^{\infty }\mathbf {A} ^{(l)}(\mathbf {R} ).

{\ displaystyle \ mathbf {A} (\ mathbf {R}) = \ sum \ limits _ {l = 1} ^ {\ infty} \ mathbf {A} ^ {(l)} (\ mathbf {R}). }

The row begins with $l=1$ ${\ displaystyle l = 1}$ , since there are no magnetic charges (magnetic charges in the physics of fundamental interactions were not detected, although they can be used as a model for describing phenomena in solid state physics). This term corresponds to a magnetic dipole (point circular circuit with current):

\mathbf {A} ^{(1)}(\mathbf {R} )={\frac {[\mathbf {m} ,\mathbf {R} ]}{R^{3}}},

{\ displaystyle \ mathbf {A} ^ {(1)} (\ mathbf {R}) = {\ frac {[\ mathbf {m}, \ mathbf {R}]} {R ^ {3}}},}

Where $\mathbf {m}$ ${\ displaystyle \ mathbf {m}}$ - the magnetic moment of the current system (moving charges):

\mathbf {m} ={\frac {1}{2c}}\sum \limits _{i}q_{i}[\mathbf {r} _{i},\mathbf {v} _{i}]

{\ displaystyle \ mathbf {m} = {\ frac {1} {2c}} \ sum \ limits _ {i} q_ {i} [\ mathbf {r} _ {i}, \ mathbf {v} _ {i }]}

Literature

Landau L.D. , Lifshits E.M. Field Theory. - 7th edition, revised. - M .: Nauka , 1988 .-- 512 p. - (“ Theoretical Physics ”, Volume II). - ISBN 5-02-014420-7 .
Prokhorov A.M. (ed.). Physical Encyclopedia . - M .: Soviet Encyclopedia , 1992. - T. 3. - 672 p. - ISBN 5-85270-034-7 .
Denisov V.I. Chapter II. Stationary electromagnetic fields // Lectures on electrodynamics. Tutorial. - 2nd ed .. - M .: Publishing House UC DO, 2007. - 272 p. - ISBN 978-5-88800-330-5 .

Notes

↑ Of course, the presented field can be both potential and tension.
↑ For fields like gravitational, without negative charges, multipole expansion contains only even orders. In this case, negative charges in multipoles of even orders (for example, in a quadrupole) are considered in this case purely formally.
↑ Li Tsung-tao. Mathematical methods in physics. - M.: Mir, 1965. - p. 146