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On the non-relativistic electromagnetic multipole-multipole interaction between two disjoint charge distributions
I
1.A
I J
Nuclear Physics A132 (1969) 1--4; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
ON THE NON-RELATIVISTIC ELECTROMAGNETIC MULTIPOLE-MULTIPOLE
INTERACTION
BETWEEN TWO DISJOINT CHARGE DISTRIBUTIONS KURT ALDER University of Basel, Basel, Switzerland and AAGE WINTHER The Niels Bohr Institute, University of Copenhagen, Demnark Received 16 April 1969 For reference purposes, an explicit and convenient formula for the non-relativistic electromagnetic interaction between two extended charge-current distributions is given in terms of the electric and magnetic multipole moments of the two systems.
Abstract:
A problem of rather general interest in atomic and nuclear physics is that of the electromagnetic interaction between two extended charge-current distributions 1). A convenient f o r m of this interaction can be obtained when the two charge-current distributions do not overlap, since then the distributions can be described by means o f the electric and magnetic multipole moments. z _
z
51 r
Fig. 1. Illustration of the charge-current distributions 1 and 2 and the coordinate systems in which their multipole moments are measured. Let us first consider the case where the two charge-current distributions are stationary. As is illustrated in fig. 1, we describe the charge-current distribution o f system 1 in a coordinate system $1 by p l ( r l ) and jl(r~), while the corresponding quantities o f system 2 are given by p2(r2) and j2(r2) in a coordinate system $2 with axes parallel to those o f S 1. The position vector o f the origin o f 1 with respect to the origin o f 2 is denoted by r, which we consider, for the m o m e n t , time independent. 1 July 1969
K. ALDER AND A. WINTHER
2
The non-relativistic interaction energy between system 1 and system 2 is given by V(1, 2)
ff
=
d3rldzr2
Pl(rl)P2(r2)--Jl(rl)j2(r2)/c2 Iv+r1--r2]
The expansion of i/Ir+r~-r21 into powers of either means of the well-known formula 1
- V
It+r,-,'21
4n
r~ '
(1)
q , r2 or l/r can be made by ,
Yal.l(rl) Y(~lm(r2 - r)
a~, 22~+1 I r - r = l ; ' + ' .).2
= Z 4n 12 ya~u~(r2)YT~(r+r, ) ;.2u2 222+1 It+vii 2~+1 "~- 2~ 2~4~1 ]rl--r2it~F2+l Y2.u(r2--P1)Y);(r)"
(2)
From these expansions, one may conclude that 1
[r+r,--r2[
;q 22 rl P2
-- Z a(2,~2]/1]/2]/)
a,2~
r a~+a=+l
lq#211
yalu,(rl)Yazuz(p2)yz,+&u(r),
(3)
provided that r > r~ -[-r 2 . The dependence of the coefficients a (2x, 22, ]/~, ]/2, ]/) on the indices ]/~, ]/2 and ]/ can be determined from the condition that the expression (3) is a scalar under rotation of the coordinate system. It must thus be of the form
a(2122]/1]/2]/)=(2, 22 ]/1
]/2
21+22) C(2t22) '
(4)
#
where the first factor is the 3-j symbol, which couples the three spherical harmonics in (3) to a scalar, while the second factor depends only on 2 x , 2 2 . The dependence of the coefficient C on 2, and 2 2 can be found by considering a special case, e.g., where the three vectors r, r 1 and r 2 are parallel. One finds V C(2,22) = (--1)a2(4n) ~:
(221 +222)[ (2).1+ 1 ) ! ( 2 2 2 + 1 ) ! "
(5)
The result is thus 1
]r-k-rl--r2]
_
~ C(2122 ) 2~ 22 ~"1J'2g/1'[/2# 1"/1 ]/2
21+22 t r 21 a r 22 2 ]/ / r2t +22+ 1 ×
(6)
The electric-electric multipole interaction originates from the first term in (1). In the stationary case, we obtain Vst.t(1, 2) = VE(1, 2)+ VM(1, 2),
(7)
MULTIPOLE-MULTI POLE INTERACTION
3
where VE(1,2)=
~
C(21"~2)( 2t
~'t a2111/12/2
I('/1
22
).,+2z)
~12
/'/
/
× ~ / l ( E j q 1/t),/ff2(E~.2//2) r)q +22+ 1 1 Yz,+~2u(r). (8) The electric multipole moments are defined, e.g., in ref. 2) (eq. II A.11). The magnetic-magnetic interaction which arises from the second term in (1) can be evaluated by inserting (6) in (1) and expanding the current vectorsj on the spherical unit vectors 1 (ex+_iey), e+t = + x./~
eo
= e=.
(9)
Rewriting the product of the spherical harmonics and the unit vectors in terms of vector spherical harmonics defined in ref. 3),
YJaM(r) = Z <2#lqldM> Y;.u(r)eq,
(10)
#q
and utilizing that divj = 0,
(ll)
one obtains
v.(1, 2) =
Z
22
P2
)q +)'2) #
/
1 x ./gt(M21 p,)./g2(M).2/~2) rZ~+z2+ ~
(12)
where the magnetic multipole moments are defined, e.g., in ref. 2) (eq. II A.39). It is noted that the expressions (8) and (12) are identical except for the fact that the summation in (8) includes the terms with 21 = 0 and 22 = 0. The formulae (8) and (12) can also be applied for time dependent charge-current distributions as long as the retardation effects can be neglected. In this approximation, the expressions for the interaction energy can be generalized to the case where the systems 1 and 2 have a relative velocity, i.e., where the vector r is a function of time and where ]rl << 1,
(13)
C
so that quadratic terms in this quantity can be neglected. For the electromagnetic interaction for the two moving systems, we may again use expression (1) with modified charge-current densities which take into account the translatory motion. If we evaluate the interaction in a rest system with respect to 2,
4
K. A L D E R A N D A. W I N T H E R
the charge-current density in system 1 is given by j'l(r,) = jl(r,)-}- I~pl(r,), p'l(r,) =
p,(r,)+ -fi1 ~ "j,(r,)
(14)
to lowest order in i'/c. Inserting (14) in (1), we obtain the following interaction energy:
r(1,2) = rs,a,(l,R)+ fi f f d3rtdar2 plj2-p2jl
[r+rl --rE I
(15)
Neglecting retardation effects, the first integral leads again to the results (7), (8) and (12). The second integral, however, gives rise to an interaction between the electric multipole moments of one system with the magnetic multipole moments of the other system. We may thus write V(1, 2) = VE(1, 2)+ VM(1, 2)+ VEM(1,2).
(16)
The electric-magnetic multipole expansion can be obtained by inserting (6) into (15) and expanding Jl and J2 on the unit vectors (9). By a similar technique to the one applied above for the magnetic-magnetic interaction, one obtains the following result: VEM(I, 2) =
+(2a ,.
Z
iC(2, 22){.~Yt'a(E21p,)~'(M2//~=)-dt'~(M2,
22 2 , + 2 2 - - 1 ) ] / /
22
.
2,(21+x )
,._
#,)dt'(Ek2/~2)}
Yz,+x,-,,zx+z,,,(')}.
(17)
It is noted that, for 21 = 0 or 22 = 0, the second term vanishes due to a vanishing 3-j symbol. In this case the interaction energy is proportional to 1 Yzz.(r) -- - -
+ 1)
LYz.(r),
(18)
where
L= --irxV. References
1) J. O. Hirschfelder, G. F. Curtiss and R. B. Bird, Molecular theory of gases and liquids (Wiley and Sons Ltd., New York, 1954) pp. 835 ft. 2) K. Alder, A. Bohr, T. Huus, B. Mottelson and A. Winther, Rev. Mod. Phys. 28 (1956) 432 3) A. R. Edmonds, Angular momentum in quantum mechanics (Princeton University Press, Princeton, 1957)
1.A
I J
Nuclear Physics A132 (1969) 1--4; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
ON THE NON-RELATIVISTIC ELECTROMAGNETIC MULTIPOLE-MULTIPOLE
INTERACTION
BETWEEN TWO DISJOINT CHARGE DISTRIBUTIONS KURT ALDER University of Basel, Basel, Switzerland and AAGE WINTHER The Niels Bohr Institute, University of Copenhagen, Demnark Received 16 April 1969 For reference purposes, an explicit and convenient formula for the non-relativistic electromagnetic interaction between two extended charge-current distributions is given in terms of the electric and magnetic multipole moments of the two systems.
Abstract:
A problem of rather general interest in atomic and nuclear physics is that of the electromagnetic interaction between two extended charge-current distributions 1). A convenient f o r m of this interaction can be obtained when the two charge-current distributions do not overlap, since then the distributions can be described by means o f the electric and magnetic multipole moments. z _
z
51 r
Fig. 1. Illustration of the charge-current distributions 1 and 2 and the coordinate systems in which their multipole moments are measured. Let us first consider the case where the two charge-current distributions are stationary. As is illustrated in fig. 1, we describe the charge-current distribution o f system 1 in a coordinate system $1 by p l ( r l ) and jl(r~), while the corresponding quantities o f system 2 are given by p2(r2) and j2(r2) in a coordinate system $2 with axes parallel to those o f S 1. The position vector o f the origin o f 1 with respect to the origin o f 2 is denoted by r, which we consider, for the m o m e n t , time independent. 1 July 1969
K. ALDER AND A. WINTHER
2
The non-relativistic interaction energy between system 1 and system 2 is given by V(1, 2)
ff
=
d3rldzr2
Pl(rl)P2(r2)--Jl(rl)j2(r2)/c2 Iv+r1--r2]
The expansion of i/Ir+r~-r21 into powers of either means of the well-known formula 1
- V
It+r,-,'21
4n
r~ '
(1)
q , r2 or l/r can be made by ,
Yal.l(rl) Y(~lm(r2 - r)
a~, 22~+1 I r - r = l ; ' + ' .).2
= Z 4n 12 ya~u~(r2)YT~(r+r, ) ;.2u2 222+1 It+vii 2~+1 "~- 2~ 2~4~1 ]rl--r2it~F2+l Y2.u(r2--P1)Y);(r)"
(2)
From these expansions, one may conclude that 1
[r+r,--r2[
;q 22 rl P2
-- Z a(2,~2]/1]/2]/)
a,2~
r a~+a=+l
lq#211
yalu,(rl)Yazuz(p2)yz,+&u(r),
(3)
provided that r > r~ -[-r 2 . The dependence of the coefficients a (2x, 22, ]/~, ]/2, ]/) on the indices ]/~, ]/2 and ]/ can be determined from the condition that the expression (3) is a scalar under rotation of the coordinate system. It must thus be of the form
a(2122]/1]/2]/)=(2, 22 ]/1
]/2
21+22) C(2t22) '
(4)
#
where the first factor is the 3-j symbol, which couples the three spherical harmonics in (3) to a scalar, while the second factor depends only on 2 x , 2 2 . The dependence of the coefficient C on 2, and 2 2 can be found by considering a special case, e.g., where the three vectors r, r 1 and r 2 are parallel. One finds V C(2,22) = (--1)a2(4n) ~:
(221 +222)[ (2).1+ 1 ) ! ( 2 2 2 + 1 ) ! "
(5)
The result is thus 1
]r-k-rl--r2]
_
~ C(2122 ) 2~ 22 ~"1J'2g/1'[/2# 1"/1 ]/2
21+22 t r 21 a r 22 2 ]/ / r2t +22+ 1 ×
(6)
The electric-electric multipole interaction originates from the first term in (1). In the stationary case, we obtain Vst.t(1, 2) = VE(1, 2)+ VM(1, 2),
(7)
MULTIPOLE-MULTI POLE INTERACTION
3
where VE(1,2)=
~
C(21"~2)( 2t
~'t a2111/12/2
I('/1
22
).,+2z)
~12
/'/
/
× ~ / l ( E j q 1/t),/ff2(E~.2//2) r)q +22+ 1 1 Yz,+~2u(r). (8) The electric multipole moments are defined, e.g., in ref. 2) (eq. II A.11). The magnetic-magnetic interaction which arises from the second term in (1) can be evaluated by inserting (6) in (1) and expanding the current vectorsj on the spherical unit vectors 1 (ex+_iey), e+t = + x./~
eo
= e=.
(9)
Rewriting the product of the spherical harmonics and the unit vectors in terms of vector spherical harmonics defined in ref. 3),
YJaM(r) = Z <2#lqldM> Y;.u(r)eq,
(10)
#q
and utilizing that divj = 0,
(ll)
one obtains
v.(1, 2) =
Z
22
P2
)q +)'2) #
/
1 x ./gt(M21 p,)./g2(M).2/~2) rZ~+z2+ ~
(12)
where the magnetic multipole moments are defined, e.g., in ref. 2) (eq. II A.39). It is noted that the expressions (8) and (12) are identical except for the fact that the summation in (8) includes the terms with 21 = 0 and 22 = 0. The formulae (8) and (12) can also be applied for time dependent charge-current distributions as long as the retardation effects can be neglected. In this approximation, the expressions for the interaction energy can be generalized to the case where the systems 1 and 2 have a relative velocity, i.e., where the vector r is a function of time and where ]rl << 1,
(13)
C
so that quadratic terms in this quantity can be neglected. For the electromagnetic interaction for the two moving systems, we may again use expression (1) with modified charge-current densities which take into account the translatory motion. If we evaluate the interaction in a rest system with respect to 2,
4
K. A L D E R A N D A. W I N T H E R
the charge-current density in system 1 is given by j'l(r,) = jl(r,)-}- I~pl(r,), p'l(r,) =
p,(r,)+ -fi1 ~ "j,(r,)
(14)
to lowest order in i'/c. Inserting (14) in (1), we obtain the following interaction energy:
r(1,2) = rs,a,(l,R)+ fi f f d3rtdar2 plj2-p2jl
[r+rl --rE I
(15)
Neglecting retardation effects, the first integral leads again to the results (7), (8) and (12). The second integral, however, gives rise to an interaction between the electric multipole moments of one system with the magnetic multipole moments of the other system. We may thus write V(1, 2) = VE(1, 2)+ VM(1, 2)+ VEM(1,2).
(16)
The electric-magnetic multipole expansion can be obtained by inserting (6) into (15) and expanding Jl and J2 on the unit vectors (9). By a similar technique to the one applied above for the magnetic-magnetic interaction, one obtains the following result: VEM(I, 2) =
+(2a ,.
Z
iC(2, 22){.~Yt'a(E21p,)~'(M2//~=)-dt'~(M2,
22 2 , + 2 2 - - 1 ) ] / /
22
.
2,(21+x )
,._
#,)dt'(Ek2/~2)}
Yz,+x,-,,zx+z,,,(')}.
(17)
It is noted that, for 21 = 0 or 22 = 0, the second term vanishes due to a vanishing 3-j symbol. In this case the interaction energy is proportional to 1 Yzz.(r) -- - -
+ 1)
LYz.(r),
(18)
where
L= --irxV. References
1) J. O. Hirschfelder, G. F. Curtiss and R. B. Bird, Molecular theory of gases and liquids (Wiley and Sons Ltd., New York, 1954) pp. 835 ft. 2) K. Alder, A. Bohr, T. Huus, B. Mottelson and A. Winther, Rev. Mod. Phys. 28 (1956) 432 3) A. R. Edmonds, Angular momentum in quantum mechanics (Princeton University Press, Princeton, 1957)