- Email: [email protected]
The electric dipole sum rule
I
2.I
[ Nuclear Physics 14 (1959160) 5 0 6 - - 5 1 1 ; ( ~ North-Holland Publishing Co., Amsterdam I
N o t to
be reproduced by photoprint or microfilm without written permission from the publisher
THE ELECTRIC DIPOLE SUM RULE GEOFFREY
I. O P A T
Physics Department, University o/ Melbourne, Australia R e c e i v e d 14 A u g u s t 1959 A b s t r a c t : A d e r i v a t i o n of t h e electric dipole s u m rul e is given, w h i c h t a k e s n u c l e a r recoil i n t o account and eliminates r e d u n d a n t coordinates.
1. I n t r o d u c t i o n
The electric dipole sum rule of Thomas, Reiche and Kuhn 1), aS developed b y Levinger and Bethe e), is one of the strongest rules used in the analysis of photo-nuclear experiments, because it gives an essentially model-independent expression for the integrated cross-section for electric dipole photon absorption. The presence of redundant coordinates together with the nuclear recoil effect leaves the status of this rule open to doubt. In this paper the rule, in the Levinger and Bethe form, is shown to hold by giving an exact treatment of recoil and redundant coordinates. Levinger and Bethe have discussed the effect ot exchange potentials on the dipole sum and so this point is not considered below. In using particle coordinates relative to the centre of mass of the system under consideration, an equation of constraint exists which is not present for laboratory coordinates. If ~i is the coordinate of the i th particle relative to centre of mass, and rni its mass, then m,~ = 0 (1) i
is the equation of constraint, the summation being over all particles. The effect of this equation is to limit the domain of integration in configuration space employed in the formation of matrix elements. In the past the distinction between centre-of-mass and laboratory coordinates has not always been made explicitly, with the result that a "hybrid" system has been used, expressing the operators in centre-of-mass coordinates but forming the matrix elements b y integration over laboratory coordinates. This is particularly true of the effective charge concept (Bethe 3)), which rightly belongs only to the centre-of-mass system, and its relation to the sum rule. One further point of objection that may be raised is that the solutions of SchrSdinger equation (2) below are not localised even for bound nuclear 506
THR ELECTRIC DIPOLE SUM RULE
507
systems, but extend into a region of macroscopic size. It is doubtful therefore if the matrix element of a position vector of a nucleon exists in the mathematical sense. Furthermore, the use of the electric dipole approximation implies that the system considered has a small extension. Thus the electric dipole approximation strictly applies to the interaction operator after centre-of-mass motion is removed. 2. D e r i v a t i o n of t h e S u m R u l e 2.1. T H E
COORDINATE
TRANSFORMATION
Let m~ be the mass of the i th particle with laboratory coordinates x~. The Schr6dinger equation for the system is given b y {
1 ~ 2 ~ ' ~-'72 I ~-'gP( .... dinate\-T(X) ~. --,i'~]/~i | - \diff. . . . . . . ))~ i
~-~ E ~ U ( x ) .
(2)
We carry out the transformation to internal or centre-of-mass coordinates g~ and the coordinate of the centre-of-mass X. It is to be noted that only (N--1) vectors ~ are employed below; the position operator of the N th particle, which is left unused, can always be found from eq. (1). Explicitly we have the transformation ~j = X , - -
E~4~iXi/i~/[,
(i =
1, 2 . . . . .
N;
f =
l, 2 .....
N--l),
(3)
i
(~)
x = E ~,x,/M, i
where
M = E ~,.
(5)
i
The wave-function T(x) expressed in the new coordinates is denoted q}(X, ~). Thus we find mi ( mi)
vx,-+~vx+~'
a,,-~
v~,,
(6)
where ~ ' indicates a sum over all particles other than the N tu and V,, is the gradient operator for coordinate x~ etc. The Schr6dinger equation becomes
_½~i2 ~ v x 2 ~ ,
,~,
.~/
~, ~ ~ ~>,z'-~ .v~, +~(~) ~(x, ~) = Eq,(x, ~).
(7)
The function q~(X, ~) can be shown to be separable into centre-of-mass and internal coordinates: ~(X, ~) = ~(X)~(~), (8) where ¢(X) satisfies
508
GEOFFREY
I.
OPAT
~2
----V~x6(X) 2M
= E°4(X),
(9)
and has the solution ¢ ( X ) =- V-½e 'K'x,
(10)
where K 2 = 2ME°/li 2 and V is the volume of a box on the surfaces of which ¢(X) satisfies periodic boundary conditions. The internal wave function satisfies
where E ' + E ° = E, E' being interpreted as the internal energy of the nucleus and E o as the kinetic energy of gross nuclear motion. It can readily be shown that the eigensolutions of equation (11) form a set of orthogonal normalisable functions in the coordinates ~ j ( / = 1. . . . N - - l ) . We now consider the matrix elements of g¢ defined by
From eq. (11), remembering the non-exchange character of ~ ( ~ ) we may show that 1 --1 1 . ~ (E',--E'o) = m-~.+ ~ ' < f l V ~ 1 0 > . (12) It follows that 1 m--; --
(E'f--E'o) 2
+ - -1 ~, m~ 1 • ~N
(13)
k
2.2. I N T E G R A T E D C R O S S - S E C T I O N
The expression for the integrated cross-section is given by Heitler 4) as
i -: f0
¢ d E = ~ - ~ [Hrol 2,
(14)
where
~e,~0.x, L~ 2kro
•
3
v,,/ %(x)&,. I
(15)
In this expression the photon field has been normalised to one photon per unit area per unit time and has polarisation in the direction of the unit vector u. The photon field is chosen to have periodic boundary conditions on the surface of the box referred to above. The volume of this box is made sufficiently large, so that the separation of the energy states is very much smaller than the width
THE ELECTRIC DIPOLE SUM RULE
509
of the excited nuclear states. That this last condition is required is implicit in the derivation of eq. (14). The wave number kr0 is the magnitude of the photon propagation vector, kto which has the direction of the photon beam; kto corresponds to the energy (E't--E'o). The charge of the i th particle q, has been expressed in rationalised MKS units and/% is the permeability of the vacuum in these units. The summation over f is over all final states of the system. The matrix element Hfo is expressed in terms of X and ~, by using eqs. (6) and (8) together with the inverse of eqs. (3) and (4), namely, Xj = X + ~ 3. (i = 1, 2 . . . . . N - - l ) ,
(16)
x N = X - - •' mk~le/m N.
(17)
k
We find
ih V/~°~ f¢f*(X)%v,*(~)[~' q¢ e*k,0'(x+'~,
[mJv'
"~' (~,.le-- ~ ) V , , )
+ q-~Ne;k'°''x-{'mk"/mN)U" ( - - ~ V x - - ~ ' - ~ V , , ) I $0(X)%v0(~)d*xdz,.
(18)
mN
Using the orthogonality of the yJ(~) and the $(X) together with the explicit form (10) we find on integrating over the dz x that "f0
i ' V ~-0~ ~ / ) f * ( ~ ) U
[i qii I¢t (Oikif Kf = Ko+kto,
I-- 0
if
(19j
Kf#Ko+kfo
where K o and K t are the initial and final vector wave-numbers of the centre of mass of the system. In eq. (19) the electric dipole approximation has also been employed, i.e. eik, o • E~ __+ 1
and exp[--ikr0- ~'mle~k/mN] --> 1. k
Note that in eq. (19) the summation i---- 1, 2 . . . . . N. We insert into eq. (14) one factor HIo given by eq. (19) and the other factor /Jr0 also given by eq. (19) but with the gradients removed'by using eq. (13). We now find I ---- --~#0hc ~ <01B*lf>, (20) where qs
(21)
510
and
GEOFFREY
• I~
(~)
(
I.
OPAT
'ink ~k)? =-u-I~' (qJ-- qN
; (2.2)
Q is the total charge of the system. Using the closure p r o p e r t y of the ~v(g) the s u m m a t i o n over f m a y be performed giving I = --~r/% ~c<01 B* A 10>. (23) Eq. (11) m a y be used to show t h a t f % * ( ~ ) u * • g~u • Vg~,~Oo(g)d~~ = _1c3 ~ jk,
(24)
whence
This reduces to the well-known result I --
~ o hce2 N Z 2Mp A
(26)
for a nuclear system, where Mp is the mass of a nucleon and Z, N are the p r o t o n and n e u t r o n numbers respectively, and A = N + Z . F o r an atomic system, neglecting the nuclear system because of its high mass, we obtain the well-known result
~#o ?~ce~
• z,
(27)
2m e
where m e is the electronic mass.
3. Possible Extension to Exchange Forces The presence of exchange forces has been a t t r i b u t e d to the existence of mesons in the nucleus. If it is possible to find an equation of the SchrSdinger t y p e which includes directly the meson and nucleon system together, and which contains no terms other t h a n the kinetic energy which fail to c o m m u t e with the position coordinates of the mesons and nucleons, an equation of the form (25) m a y be expected to hold. The mesons, having a lighter mass t h a n the nucleons, would increase the value of I above the value in eq. (26). In this case the increase in integrated cross section would be explained kinematically b y the presence of charged mesons r a t h e r t h a n the more usual w a y of Levinger and Bethe using exchange forces.
THE
ELECTRIC
DIPOLE
SUM
RULE
511
The author would like to thank Associate Professor C. B. O. Mohr and Dr. B. M. Spicer for their interest in the present work. He is also indebted to General Motors-Holden's Ltd. for a research fellowship. References I) W. Thomas, Naturwiss. 13 (1925) 627; \V. Kuhn, Zs. f. Phys. 33 (1925) 408; F. Reiche and W. Thomas, Zs. f. Phys. 34 (1925) 510 2) J. S. Levinger and H. A. Bethe, Phys. Rev. 78 (1950) 115 3) H. A. Bethe, Revs. Mod. Phys. 9 (1937) 221 4) W. Heitler, Quantum Theory of Radiation (3rd Edit. Oxford, 1954) Ch. IV
2.I
[ Nuclear Physics 14 (1959160) 5 0 6 - - 5 1 1 ; ( ~ North-Holland Publishing Co., Amsterdam I
N o t to
be reproduced by photoprint or microfilm without written permission from the publisher
THE ELECTRIC DIPOLE SUM RULE GEOFFREY
I. O P A T
Physics Department, University o/ Melbourne, Australia R e c e i v e d 14 A u g u s t 1959 A b s t r a c t : A d e r i v a t i o n of t h e electric dipole s u m rul e is given, w h i c h t a k e s n u c l e a r recoil i n t o account and eliminates r e d u n d a n t coordinates.
1. I n t r o d u c t i o n
The electric dipole sum rule of Thomas, Reiche and Kuhn 1), aS developed b y Levinger and Bethe e), is one of the strongest rules used in the analysis of photo-nuclear experiments, because it gives an essentially model-independent expression for the integrated cross-section for electric dipole photon absorption. The presence of redundant coordinates together with the nuclear recoil effect leaves the status of this rule open to doubt. In this paper the rule, in the Levinger and Bethe form, is shown to hold by giving an exact treatment of recoil and redundant coordinates. Levinger and Bethe have discussed the effect ot exchange potentials on the dipole sum and so this point is not considered below. In using particle coordinates relative to the centre of mass of the system under consideration, an equation of constraint exists which is not present for laboratory coordinates. If ~i is the coordinate of the i th particle relative to centre of mass, and rni its mass, then m,~ = 0 (1) i
is the equation of constraint, the summation being over all particles. The effect of this equation is to limit the domain of integration in configuration space employed in the formation of matrix elements. In the past the distinction between centre-of-mass and laboratory coordinates has not always been made explicitly, with the result that a "hybrid" system has been used, expressing the operators in centre-of-mass coordinates but forming the matrix elements b y integration over laboratory coordinates. This is particularly true of the effective charge concept (Bethe 3)), which rightly belongs only to the centre-of-mass system, and its relation to the sum rule. One further point of objection that may be raised is that the solutions of SchrSdinger equation (2) below are not localised even for bound nuclear 506
THR ELECTRIC DIPOLE SUM RULE
507
systems, but extend into a region of macroscopic size. It is doubtful therefore if the matrix element of a position vector of a nucleon exists in the mathematical sense. Furthermore, the use of the electric dipole approximation implies that the system considered has a small extension. Thus the electric dipole approximation strictly applies to the interaction operator after centre-of-mass motion is removed. 2. D e r i v a t i o n of t h e S u m R u l e 2.1. T H E
COORDINATE
TRANSFORMATION
Let m~ be the mass of the i th particle with laboratory coordinates x~. The Schr6dinger equation for the system is given b y {
1 ~ 2 ~ ' ~-'72 I ~-'gP( .... dinate\-T(X) ~. --,i'~]/~i | - \diff. . . . . . . ))~ i
~-~ E ~ U ( x ) .
(2)
We carry out the transformation to internal or centre-of-mass coordinates g~ and the coordinate of the centre-of-mass X. It is to be noted that only (N--1) vectors ~ are employed below; the position operator of the N th particle, which is left unused, can always be found from eq. (1). Explicitly we have the transformation ~j = X , - -
E~4~iXi/i~/[,
(i =
1, 2 . . . . .
N;
f =
l, 2 .....
N--l),
(3)
i
(~)
x = E ~,x,/M, i
where
M = E ~,.
(5)
i
The wave-function T(x) expressed in the new coordinates is denoted q}(X, ~). Thus we find mi ( mi)
vx,-+~vx+~'
a,,-~
v~,,
(6)
where ~ ' indicates a sum over all particles other than the N tu and V,, is the gradient operator for coordinate x~ etc. The Schr6dinger equation becomes
_½~i2 ~ v x 2 ~ ,
,~,
.~/
~, ~ ~ ~>,z'-~ .v~, +~(~) ~(x, ~) = Eq,(x, ~).
(7)
The function q~(X, ~) can be shown to be separable into centre-of-mass and internal coordinates: ~(X, ~) = ~(X)~(~), (8) where ¢(X) satisfies
508
GEOFFREY
I.
OPAT
~2
----V~x6(X) 2M
= E°4(X),
(9)
and has the solution ¢ ( X ) =- V-½e 'K'x,
(10)
where K 2 = 2ME°/li 2 and V is the volume of a box on the surfaces of which ¢(X) satisfies periodic boundary conditions. The internal wave function satisfies
where E ' + E ° = E, E' being interpreted as the internal energy of the nucleus and E o as the kinetic energy of gross nuclear motion. It can readily be shown that the eigensolutions of equation (11) form a set of orthogonal normalisable functions in the coordinates ~ j ( / = 1. . . . N - - l ) . We now consider the matrix elements of g¢ defined by
From eq. (11), remembering the non-exchange character of ~ ( ~ ) we may show that 1 --1 1 . ~ (E',--E'o)
(E'f--E'o) 2
(13)
k
2.2. I N T E G R A T E D C R O S S - S E C T I O N
The expression for the integrated cross-section is given by Heitler 4) as
i -: f0
¢ d E = ~ - ~ [Hrol 2,
(14)
where
~e,~0.x, L~ 2kro
•
3
v,,/ %(x)&,. I
(15)
In this expression the photon field has been normalised to one photon per unit area per unit time and has polarisation in the direction of the unit vector u. The photon field is chosen to have periodic boundary conditions on the surface of the box referred to above. The volume of this box is made sufficiently large, so that the separation of the energy states is very much smaller than the width
THE ELECTRIC DIPOLE SUM RULE
509
of the excited nuclear states. That this last condition is required is implicit in the derivation of eq. (14). The wave number kr0 is the magnitude of the photon propagation vector, kto which has the direction of the photon beam; kto corresponds to the energy (E't--E'o). The charge of the i th particle q, has been expressed in rationalised MKS units and/% is the permeability of the vacuum in these units. The summation over f is over all final states of the system. The matrix element Hfo is expressed in terms of X and ~, by using eqs. (6) and (8) together with the inverse of eqs. (3) and (4), namely, Xj = X + ~ 3. (i = 1, 2 . . . . . N - - l ) ,
(16)
x N = X - - •' mk~le/m N.
(17)
k
We find
ih V/~°~ f¢f*(X)%v,*(~)[~' q¢ e*k,0'(x+'~,
[mJv'
"~' (~,.le-- ~ ) V , , )
+ q-~Ne;k'°''x-{'mk"/mN)U" ( - - ~ V x - - ~ ' - ~ V , , ) I $0(X)%v0(~)d*xdz,.
(18)
mN
Using the orthogonality of the yJ(~) and the $(X) together with the explicit form (10) we find on integrating over the dz x that "f0
i ' V ~-0~ ~ / ) f * ( ~ ) U
[i qii I¢t (Oikif Kf = Ko+kto,
I-- 0
if
(19j
Kf#Ko+kfo
where K o and K t are the initial and final vector wave-numbers of the centre of mass of the system. In eq. (19) the electric dipole approximation has also been employed, i.e. eik, o • E~ __+ 1
and exp[--ikr0- ~'mle~k/mN] --> 1. k
Note that in eq. (19) the summation i---- 1, 2 . . . . . N. We insert into eq. (14) one factor HIo given by eq. (19) and the other factor /Jr0 also given by eq. (19) but with the gradients removed'by using eq. (13). We now find I ---- --~#0hc ~ <01B*lf>
(21)
510
and
GEOFFREY
• I~
(~)
(
I.
OPAT
'ink ~k)? =-u-I~' (qJ-- qN
; (2.2)
Q is the total charge of the system. Using the closure p r o p e r t y of the ~v(g) the s u m m a t i o n over f m a y be performed giving I = --~r/% ~c<01 B* A 10>. (23) Eq. (11) m a y be used to show t h a t f % * ( ~ ) u * • g~u • Vg~,~Oo(g)d~~ = _1c3 ~ jk,
(24)
whence
This reduces to the well-known result I --
~ o hce2 N Z 2Mp A
(26)
for a nuclear system, where Mp is the mass of a nucleon and Z, N are the p r o t o n and n e u t r o n numbers respectively, and A = N + Z . F o r an atomic system, neglecting the nuclear system because of its high mass, we obtain the well-known result
~#o ?~ce~
• z,
(27)
2m e
where m e is the electronic mass.
3. Possible Extension to Exchange Forces The presence of exchange forces has been a t t r i b u t e d to the existence of mesons in the nucleus. If it is possible to find an equation of the SchrSdinger t y p e which includes directly the meson and nucleon system together, and which contains no terms other t h a n the kinetic energy which fail to c o m m u t e with the position coordinates of the mesons and nucleons, an equation of the form (25) m a y be expected to hold. The mesons, having a lighter mass t h a n the nucleons, would increase the value of I above the value in eq. (26). In this case the increase in integrated cross section would be explained kinematically b y the presence of charged mesons r a t h e r t h a n the more usual w a y of Levinger and Bethe using exchange forces.
THE
ELECTRIC
DIPOLE
SUM
RULE
511
The author would like to thank Associate Professor C. B. O. Mohr and Dr. B. M. Spicer for their interest in the present work. He is also indebted to General Motors-Holden's Ltd. for a research fellowship. References I) W. Thomas, Naturwiss. 13 (1925) 627; \V. Kuhn, Zs. f. Phys. 33 (1925) 408; F. Reiche and W. Thomas, Zs. f. Phys. 34 (1925) 510 2) J. S. Levinger and H. A. Bethe, Phys. Rev. 78 (1950) 115 3) H. A. Bethe, Revs. Mod. Phys. 9 (1937) 221 4) W. Heitler, Quantum Theory of Radiation (3rd Edit. Oxford, 1954) Ch. IV