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On the electric charge current density of a nuclear system
P h y s i c a I X , no 3
M a a r t 1942
ON T H E ELECTRIC CHARGE CURRENT DENSITY OF A NUCLEAR SYSTEM*) hy A. PAIS Physisch Laboratorium der Rijks-Universiteit te Utrecht Summary T h e p r o j e c t i v e e n e r g y m o m e n t u m t e n s o r is c o m p u t e d for a n u c l e a r s y s t e m in w h i c h t h e s h o r t r a n g e forces are d e s c r i b e d b y t h e m i x e d m e s o n field t h e o r y . F r o m this t e n s o r t h e H a m i l t o n i ~ n a n d t h e electric .charge c u r r e n t d e n s i t v of t h e s y s t e m are d e r i v e d . T h e H a m i l t o n i a n is t r a n s f o r m e d b y s e p a r a t i n g off t h e l o n g i t u d i n a l e l e c t r o m a g n e t i c field a n d t h e s t a t i c m e s o n field s u c c e s s i v e l y a n d t h e t r a n s f o r m a t i o n of t h e c u r r e n t a n d t h e d e n s i t y of electric c h a r g e is discussed in detail. E x p r e s s i o n s are g i v e n for t h e e l e c t r i c dipole a n d q u a d r u p o l e m o m e n t a n d for t h e m a g n e t i c d i p o l e m o m e n t of t h e s y s t e m .
§ 1: Introduction. As has been shown by M 0 11 e r 2), one may adequately describe the ,,mixed" meson theory of nuclear forces by taking the meson field wave function to be a S-vector, provided that the source densities producing this field are appropriately defined. This ,,mixed" theory which is based on the assumption that the nuclear forces are established through the intermediary of a particular mixture of ~ vector and a pseudoscalar meson field 2) seems to be well suited to account for the experimental facts concerning nuclear forces and ~-disintegration 4) as well as the properties of mesons in cosmic rays 5). In a previous paper e) we have considered the projective interpretation of the 5-dimensional meson field theories. As was seen there, an essential feature of this projective treatment consists in its allowing the field equations, including the electromagnetic interaction terms, to be put in a more compact form. This formalism, therefore, is especially useful for the consideration of problems concerning the interaction with the electromagnetic field; the more so as one can define for any system a ,,projective energy momenturr/ *) "Fhis paper constitutes a revised part of the author's thesis t), (chap. 2, §§ 5--7). I
4 0 7
- -
408
A. PAlS
tensor" which comprises both its energy m o m e n t u m 4-tensor and its electric charge current density. This projective tensor can be obtained by means of the general prescription given in another paper 7), further quoted .as I. In the present paper it will be calculated for a nuclear system, assuming the short range forces between nucleons to be described by the mixed theory, (section 2). In section 3 the Hamiltonian obtained in this way is transformed by successively separating off the longitudinal electromagnetic field and the ,,static" meson field 8) ; in performing the latter transformation, the compact form in which the vector and pseudoscalar meson variables and the equations they have to satisfy m a y be comprised will prove to be useful. The field equations of the mixed theory in projective form are *) F~v = Uvl~ - - U~lv - - xXr~ A U~1 + S ~ ,
.x -
g
c
~c v ~ x '
(1)
g2 ~Ft xA F~'vll, = - - ~2U~' + M y - - X F ~ " A X , ,
(2)
M r = gl LF* x A ct~"q~. Here X~ = (0, 0, X~,); ui is the nucleon 5-undor, A the hermitizing matrix and • the isotopic spin vector, the eigenvalue + 1 (-- l) of x3 denoting neutron (proton) states. Considering (1) as defining F~,, (2) can be derived from the following Lagrange function (throughout this paper we always write down tensors instead of the corresponding densities) L = Ltl) + L(,,), iz2 L(/) = 1 F~F~, ~ + 2 - U~U~ - - U~M~,
~x = m c / h ,
(3) L(~) = Re --=-hc(~FtAct~,~ll~, + XFtAkXF) ' k =
~c/1
~
+'ra
l--'r3 mN c2 + ----V--mp c 2)
l--xa 2
ie c "hc" v ' ~ ' ~ °
• ) See loc. oil s) equ. (22), (23), (26) and (27). The following notations are used: AlL/z, A II/z, A j/z are the covariant gauge derivative, covariant derivative and derivative of A respectively; ~., v, . . . . . 1 , . . . . . , 5; i, k, . . . . . 1, . . . . 4. × is the gravitational constant. F u r t h e r A is a symbolic vector in isotopic spin space having components AI, A2, A3; A B = t~ AIB i. A and A denote vector multiplication in ordinary and isotopic spin space respectively.
ELECTRIC CHARGE C U R R E N T D E N S I T Y OF A N U C L E A R SYSTEM
409
m, m2v, mp are the meson, neutron and proton mass respectively;
=0 = =~X~.-~ is a n u m b e r such that A~ is antihermitian (see I p. | 157). By performing a phase transformation *), (1)--(3) become f.v = uvlL~,--u~,l.~
f~lL~ --
+ S~,~,
(4)
~z2 ut' + M~',
(5)
~t2 Lit) = I f~v f ~ + -2- u~u~ - - u ~ M ~ .
(6)
(4) and (5) are decomposed into ,,4-space" relations by putting f ~ __ .c~• ;,;k f~k +
~
Xt~ ~]. i _g~,
u~. = "52 u , + ~ _u X . ,
(7) (8)
where [ is the universal 4-pseudoscalar 4- I det gik [~, for which in the case of special relativity ~=
~
(9)
holds: __g~and u are the meson 4-pseudovector and 4-pseudoscalar respectively, while fi~ and u~ refer to the vector mesons. F r o m (4), (5), (7) and (8) one infers that f~k = U.~l:i - - Ui,_k + S~k + [~ u X ~ ] , ftklk k =
--
~L2 U i ' +
M i,
= -- u,_, + s!0~,
g klLk=--~2u+M'°'--[l~xikf'~],
Sik = g2 +, x A 0etaj +, (1 O)
M' = gl +*'cA =~+,
(11)
S ! °) = g--&+* x A ~_oO~,+, (12)
Vt
M'°'----g,+txA_~o~b.
('13)
~_0is a pseudoscalar matrix; ~bis the nucleon 4-undor. The terms in square brackets are negligibly small, as Xik = c -I ~/2× ~ik, where ~ik is the electromagnetic field tensor. § 2. The projective energy m o m e n t u m tensor. With the help of the prescription given in I we now derive an expression for the energy m o m e n t u m 5-tensor Tvv, using L m in its form (3). For the terms of R ~ (cf. I (s2) and (75)--(77)) corresponding with L m we find *) Cf. loc. ctt. o) equ. (6).
A. PAIS
4'1'0
The terms in L(tI containing X~ explicitly are k2
X 2 Xt~ A U ~ (Uv~ - - U~,~v+ S~,~) + ~- Xt~,A U~ . XU, A U ~ • Therefore, (~'/8' X ~ denotes differentiation with respect to those X ~ that occur explicitly), X~ ~'L(tI
~ LIt)
~,X~ = X ~ ~ X~ = - - x F ~ .
( x ~ A U~),
with ~e X v - T.~v can now immediately be obtained b y using I field equations (1) and (2) and those of the nucleons:
(80)
and the
T.~ = F~'P F~p + ~2 U ~ Uv - - F~P Svp - - M ~ Uv - - L¢I) ~ +
+ ~, { F ~ ( X ~ A U J + F~(X~AU~)} +
To.I%,
where T(,).%corresponds with L(,). The application of the phase transformation yields T.% = f~P f~p + ~2 U/z Uv _ _ f~p Svp - - M ~ u~ - - L(/I 8{ +
+ ~ {fw~(X~Au~) + f~A(X~AuA)} + TIll%.
(14)
The energy momentum 4-tensor Tik and the electric charge current density sk can be found at once from T ~ . In fact one has
~{,_ v'2× ~,~{X ~ T.~,, c
where T'~ = ~ " ~ k ~
1 .
~p~
is the energy momentum tensor of the electromagnetic field. Using (7) and (8) we get
~' f~p ~
f~p = (~,,f" -- ~.g_'x~). (~" f,~ -- ~ g_~ x J = f" fk~ + E2 gi g , .
E L E C T R I C C H A R G E C U R R E N T D E N S I T Y OF A N U C L E A R S Y S T E M
411
The cross t e r m s disappear on account of, ¥~' XP -- -fP.iXp = 0. Similarly ,f~ f~P y~., S~p = fiz S,~ + ~2 gl _S~01. The t e r m s in ( 14) containing Xv or X~ explicitly disappear if we cont r a c t with ¥~i y~.,, so t h a t the result is T!k ---- fi~ fkt - - fg S,~ - - M i uk + ~2 u s u , +
+ ~2 [g, g, _ g, _s~0~]- - L~ ~', + T(,0'., + T(,):,.
(15)
T(.0!, corresponds with L(.); according to (6)--(8). L(n expressed in 4-dimensional quantities is equal to
41 fik Ilk + 2- HkU k - U~ M k + ~2
~ u 2 - - u M(0~ . gk gk + -~
F r o m ( 15) we now derive an expression for the H a m i l t o n i a n of the s y s t e m in the case of special relativity. In this case use m a y be made of (9) and of the following representation for the matrices s). ~ i = O2~i,(i---- 1 , 2 , 3 ) ,
ea=iPa,
A =ipa,
_~0=--Pa-
As A is antihermitian, ~ has to be real; we take ~ = 1. In order to avoid the occurrence of singular t e r m s of the ~-function t y p e in the static nuclear interaction potential the scalar - - ] Svv S ~'~ must be a d d e d to the L a g r a n g e function *). I n t r o d u c i n g the v e c t o r n o t a t i o n corresponding with t h a t of M o l l e r and R o s e n f e 1 d a) for the meson field variables and for the nuclear source densities: -. & f,~ ---- F, G, uk = U . - - ~ r ; S,~ =- "r, g. M, = iVI, - - N.
E=W;
gk=F,--O,
S_!° ) = : P , - Q ,
M (°~=R,
we then get for the H a m i l t o n i a n : H = f (H(/) + H(,)) dv + H('. I ,
1 -'2
~2
~r2)}__(~,~
~1~)
(16)
l ( q 2 - - § 2)+
+ 2 ( ~^2 + o ~ + ~ 2 ~ 2 ) _ ( R ~ + 0 o ) +
1 (Q2__ ~2),
H(~I ---2 *) Cf. M o i l e r
2), p. 26.
]
412
A. PAIS
H('~) "has been represented in configuration-space of the nucleons, the index (i) referring to the i-th nucleon; coIO is a small term which will not be written down explicitly. (~, ~) is the electromagnetic field, ~ is-the electromagnetic vector potential. The electric charge-current density is to be calculated b y the same methods and it is the term Xf# (X~ A u;0 in (14) that here plays an essential part. We obtain 1 --
us
]k = e ~b*A ~ 0 c
-
e
~ +-+ - ~ {(f'kAu,) s +
~2
(_g~A_u)s} +
~k.
E.~ is a small term which, apart from a contribution of the ,,D ir a c-type" due to the nucleons, whose divergence vanishes, (compare I (102)) originates from the first four terms of (14) and from the terms of T(,)~.~that symmetrize the third and fourth term of (14) ; ~ktik = 0 as a consequence of the general theorem proved in I and it can be seen that this relation still holds if all terms proportional to ×½ are ignored. Therefore, from now on omitting all small terms, we may write in the case of special relativity, using the vector ^ _~ notation s k = s, p" S =
Snucl +
Sines = e z,
-
pl0
+ --
'
p = p..~ + p~os=
- ~ 1--0 - -* - ~
e iX ~ B ( x - - x ( ' l ) +
+
e ~-Zc (UAF--WA
(17)
~,)3. (18)
For the sake of the developments in the next section we introduce ^ ^~ the 5-vector s ~ that may be connected with Sm~ ^
^k
S~mes = Y.P'k Sines = ~ § 3. T r a n s f o r m a t i o n
o/
the
e
(fvP'AILlv)3.
Hamiltonian.
a)
of the longitudinal electromagnetic In effecting this separation we put
=
~[ll -{-
.L,
(19) S e p a r a t io n
field.
E L E C T R I C C H A R G E C U R R E N T D E N S I T Y OF A N U C L E A R S Y S T E M
'4'13
where ~tl is the longitudinal and ~ x is the transversal part of the electric field; similarly for ~ . ~i. is the part of the electrostatic potential which is created by the system of nucleons and meson fields, while ~ex is the contribution of other sources ] eventually present. ~1, ~11 and ~Bi, now must be eliminated from H. This problem has been extensively dealt with by several authors; we will here merely state the results. For the first term of H we get
fH(n dv = fHu) dv - - f Smes~.L dv
(20)
where H(t ) is obtained from H(t ) by putting = rot U + S,
(21)
~2 V ---=- - div F + N,
(22)
r' = - - grad it' + P,
(23)
in the place of G, ~2V and r respectively, that is to say by replacing the gauge derivatives, o'ccurring in the ,,identities" that define the ,,derived" variables 8) G, V and I" in terms of the canonical meson variables, by ordinary derivatives. Performing the same substitution in the second term of (20) and aMling Sm~ the quantity so obtained from Sinesone gets
f Sm~ ~ . dv = f sm~ ~ . d r + W ( ~ ) , where the second term generally may be neglected as it is propor: tional to the square of the perturbation parameter (e2/l~c)½. The second term of (16) gives n¢',0 = H¢,0 - - f s.~a ~_L dv with
and the last term 1
fH(,)dv = fH(,)±dv + E + f p~e. dv, HI,). = ~-(~.2L + g21. E is the Coulomb energy of the system. Infinite electrostatic self energy terms have been suppressed. Consequently the total
414
A. PAIS
result is (in the following we write if, ~, ?3, HI, I instead of ~±, ~±, ?3ex, HI,)± respectively)
H =fHct) dv + H~,o + fHc, I dv + ~ - - f s ~i dv + f p ?3 dr. (24) The quantization of the present scheme may be performed in the usual way; there is one point within this formalism, though which in this connection requires closerexamination, especially with regard to the calculations of section 3b, namely the commutability of the pro~ective"meson variables. The commutation rules "for the canonical meson variables are hc [U,,~ (~, O, F~,. (-7', t)] -- -7- ~ ( ; - ~') ~'~ ~m, (25) [0= (x,-t), ~F.(x', t)]
= h_~c~ ( x - x') 8 . .
(26)
"t'
from which the commutation rules holding for the derived variables Gi,k, VI and I'i,k immediately follow, taking into account (21 )--(23). Thus the following commutators (and only "these ones) do not vanish identically (U stands for U~,k etc.)" [U, F], [U, V], [F, G], [G, V], [~F, q>] and [I', ~]. Therefore none of the commutators that can be formed by means of the projective variables *): [/;,~,l, up,~], []~,,,~,/0*,k] and [u.~,~, U,,k] is identically equal to zero. (The prime in ];,~ indicates that the 4-space quantities that can be derived from ] ~ refer to the space-time point (x', t) ; similarly % refers to the argument (x, t)). The question thus arises whether it might be practically useful to "establish explicit expressions for the commutators of the projective variables **) ; though this of course does not present any difficulty at all, such a proceeding would not lead to any simplification of the further calculations. We will therefore not consider these commutation relations here, but follow another method of calculation which *) T h e q u a n t i t i e s @v,^Up used here are not the s a m e as the/Izv, up occurring ill (7) and (8) as the latter refer to G*,, V, ~ e t c . while the former refer to I-~, X', ~ e t c . The s a m e holds
2. ^2.
for the q u a n t i t i e s / i k , gi, lti t h a t will occur in the following; as G, V, F henceforward no more a p p e a r this will not give rise to confusion. **) A n a t t e m p t in this direction has been m a d e ~), viz. by i n t r o d u c i n g the explicit expression for []~*4,k, up, l], where I/L4 = Yr.4/i.Lv" (In this connection we r e m a r k t h a t in loc. cir. t) p. 45, footnote, only those t e r m s h a v e been written down t h a t give a n o n v a n i s h i n g eorttribution to s~)).
E L E C T R I C CHARGE C U R R E N T D E N S I T Y OF A N U C L E A R SYSTEM
415
only makes explicit use of the canonical commlttation rules (25), (26) and which seems quite practical. b) S e p a r a t i o n of the static meson field. The equations determining the static fields are z) : F=o - - g rra,d_~~
} fik
C--= rot U + S,
_____~,,+ _~!o), J fdv:U"t~'--ut*'+SJ""
&
r---- - - grad * + P, ] ~i
~,-- 0,
~-
div F = - - ~2 ~r + N, &
-g
}~% =--
F2
+
rot G = - - ix2 U, -g
div F = - - ~2 ~,. The equations in the first column define the static fields; those in the second one are the same but now written ill tensor form, while in the third column the 5-tensor form is indicated. Any,,tensor" labeled with ° is the same function of the static variables as the corresponding tensor in the former equations is of the ordinary variables. Of course this covariant form of the equations has no further meaning as the process of splitting off the static fields is not invariant; it will be seen, however, that it is useful to work with these equations in their 5-tensor form. As has been shown b y M ~ l l e r and R o s e n f e l d 3 ) the static part of the fields may be separated from all variables b y means of a canonical transformation:
(27)
7~ = S - ' AS,
where the unitary operator S which transforms the function .4 of the .%
,,old" variables (from now on indicated by U, V . . . . .
) to the same function of the , , n e w " variables (0, V . . . . . ) is g i v e n b y i S = exp. -hc K , o ..¢.
-+
~-
-+
K = f dv [F U - - U F + ¢I' O] = f dv [i~4 u~ - - ~t, f~,], (28)
where f ~ -- T.~4f~,. The transformation of the terms of H which do
416
A. PAlS
'"
not depend on the electromagnetic field has been treated b y the mentioned authors. F u r t h e r f H ( ~ I dv is of course not affected b y this transformation, so we only have to consider more closely the Coulomb energy and the last two terms of (24). Thus all depends, on the -%
transformation of ~ and s. In this paper we will compute that part of -%
and s which does not depend on the non-static meson fields, the ,,field-independent" (f.i.) part, this being the only part which is of interest for problems in which free mesons are not involved. ~
-%
First we will transform ~mes and sines and this will be done b y ma~king use of the 5-vector s ~ from which both can be derived *). As a consequence of (27) **)
~
e
S_ I
~1
li
Il
0
= s ~ , + s~, + ~ , + z ~
~ K, ~.~,l "
Terms for which l > 3 need not be taken into account, as they are of higher order than the second in gl and g2. We now introduce a set of variables marked with 1 that refer to free mesons. Of course : 1
1
1 1 ~ = 0, ~ = ~ , ,~=~', ® =#,
while I
I
V=--~-2divF,
I
5=rotU,
F=--grad~,
SO
1
5=5+
1
1
S, V = V +
~-2N, r = f +
P,
1 :mes =;rues
e [,~
q- -~C
--
~_2~A N + ~A~] 3
The second term in the development (29) is given by
~
i = ~ [K, ~kos].
^ ^k *) s ~ e s i s related to Skmesin the s a m e way as S~e s to -~mes, see (19). **) {A, B } l ~ [ A , [ A , . . . . [ A , B]] . . . . J; l is the n u m b e r of brackets.
(30)
ELECTRIC
CHARGE
CURRENT
DENSITY
OF A NUCLEAR
SYSTEM
417
Keeping in mind the commutativity propel,ties of the-projective variables this gives s~)-
e i . f d v ' {(f;,Au.)3 °' • [u'O, lv.] + ({l'OAfV~')3. [/•," uv] hc • •c
(f~Afv~)3. [u'O, %] - - ( u*"' P A u v ) 3
--
-
. [G,/v"]}.
-
(31)
The commutators occurring in the integrals are composed of quantities with the same isotopic index which has been omitted. The contribution of each term to the 4-vector sh) is then found by inner multiplication with y~' and with the help of (7) and (8) and of the relations y.~ X~, = 0, ),~iX~ = 0 and "(#i'(~k = ~ . Keeping in mind (25) and (26) the Contribution of the first term is found to be (apart from the factor e/hc)
top(t)'--(FA 3; tos(,)--(FAV)3 = - - ( F A V ) 3 - - ~ - 2 ( F A N ) 3 and the secon.d to p(,): -- ( F A U h - ('FA 'I'h; 1
I
tos m" (G~6)s + (FACF)3 + (g~6)3 + (PA~)3; The third term gives after multiplication with y/L~ and keeping in mind that ~4 = ~, = 0
i fdv' (f~Afk'), " [u'*,uk], •
~C
so it does not contribute to Pit) while it gives for ~a) i 3 3 -- Z f dv' (#; A F)3 . EU',, VII -- ----i~ -2 y~ f dv, (~; A ~)3 EU;, div ~]" 1~c,=1 hc ,=t Now [U;, div/7]
/~c ~
hc Z
~)
~X 'l
therefore the third term gives for s(u after a partial integration
(~A~r)3 + ~-2 (FAN)3. Finally the last term of (31) becomes after multiplyingwith y# _
fi}i f d v " {({l'kAut)s. [l~,/"] + ~ ' Au)s Eg~,gl]} Physica I X
27
418
A. PAIS
f r o m which we infer t h a t it does not c o n t r i b u t e to p(~). The c o n t r i b u tion to ~(t) can be f o u n d in a similar w a y to the t r e a t m e n t of the preceding term. The result is
Consequently the complete result of (31) becomes e
-*
p(,) ---- px = ~c ( " A ~ + ~AI~' - - ~ A 'l')s,
(32)
where
Now we must find the t h i r d t e r m of (29) i
=
[K,
In order to compute this commutator, P(1) and ~(t) as given by (32) and (33) are again comprised in 5-vector form: e
o
1
1
~} =-fi-£ {f;~Au,~ + f ~ ' A u ~ - - f;~'Au;~ + f~'~'Au;~}s. s~l m a y now be calculated in the same w a y as the previous c o m m u t a t o r was found. I t gives rise to f.i. t e r m s as well as to t e r m s q u a d r a tic in the meson field c o m p o n e n t s or of t h i r d order in g and .we obtain for the e
2~
--t
f.i. p a r t of Pc21 -------Pexch = ~cc (U A F)3. f.i. part of sl2 ) = Sezch- - - ~
(35).
__ U - - ~-2 F A N + P A ~),, (36),
with -~
-%
-t
Thus from (32) and (35) it is seen that, apart from terms quadratic
ELECTRIC
CHARGE
CURRENT
DENSITY
OF A NUCLEAR
SYSTEM
419
in the meson field components and from t e r l m of higher order t h a n the second in gl a n d g2 Pm~ = pm~ + px + P~,,~h
(38)
and likewise from (30), (33) and (36) t h a t 1
Sm,s =
sines +
+
xch.
(39)
(38) and (39)are evident if one remembers t h a t a n y field variable .~ occurring in ames is approximately the sum of a static part A and 1
a new free meson variable A. Inserting this one gets: a) a part depending only on the free meson variables: Pines;b) a part depending b o t h on .~ and A" ax ; c) f.i. part of Pine.: Oexc,- Similarly for sine,. The same transformation must now be applied to =k S.ua. This calculation is quite straightforward and it can be seen t h a t the f.i. part of s~.~ is simply Snuak. Thus, summarizing the result of the transformation we have, only considering/.i, terms o/.no higher order than
the second in gl and g2 : P:
Pexch +
Onuel,
S :
~Sexch + s~ uel.
F r o m these relations we m a y obtain ill the same approximation the electric and magnetic multipole m o m e n t s of the nuclear system. Of these we will here compute the most interesting ones, viz. the electric dipole m o m e n t : ~ = f p xdv, the electric quadrupole m o m e n t : £~h
:
½f-P x~ x~ dr,
the magnetic dipole m o m e n t : 9~ = ½ f x A "~dr. To this purpose the static meson variables must be expressed in terms of the nucleon variables; we have *) =
z
Ft i
A
¢(",
9=g~ Z.(o¢(% i
•i, = g--~z ¢('~. (~(0 9('1) ¢(% • ) Cf. loc. cir. 3) e q u . (14) a n d (30) a n d 1o¢. t i t *) e q u . (40).
420
A. PAIS
where ->
~{;} = grad{O, ,~{0 =
1
4= I~ o
o
-
exp. - - [z [ x - - x{;}[.
;{'}1
-
.2~.
G, F a n d r m a y be o b t a i n e d b y m e a n s of these e x p r e s s i o n s a n d of t h e s t a t i c field e q u a t i o n s . W i t h t h e help of _ _
& -~r(ik)
f dv e{0 {p{~} - - e-~:{~} "
'
8~l~
f dv-~ ~{o e{~} __ _ _ '
'
_+
x+{,~},
16=F
we get *)
e 8~
gl g2. X {,¢o A'¢"I)]. (~l~}A .~2{~,,I). e-~J'l "
t~c
V.
~,~
r (~)
4~
e
32nhc"
=
e
g~ g2 ~3 (~{0A~{% x+c~kl ~z
i,k
"
~"
(~iOA~_{%,} " e-~'c~k~
•
r (~)
'
(~(,}
+ 8=h---T'g~ ('¢'A'¢% ("{"A;~*')• ~
2~,] +
l •
.
.
+ 5~-£c ~• (~{'}A~{k})3(;+{,k} A~-{,~}) gf + ¢~ d'} d h}
r(~k}
+
• r{ik}
I s h o u l d like t o t h a n k prof• L. R o s e n f e l d for his i n t e r e s t in this w o r k . Received F e b r u a r y 10th, 1942.
*) In loc. cir. ~) the terms in the last line of 9~ have been forgotten. {These terms vanish in the centre of gravity s y s t e m of the deuteron; they do therefore not alter t h e considcrations of loc. cir. chap. 3).
ELECTRIC CHARGE C U R R E N T D E N S I T Y OF A N U C L E A R SYSTEM
421
REFERENCES 1) A. P a i s, Thesis Utrecht, 1941. 2) C. M o i l e r , Proc. Copenhagen 18, no. 6, 1941. 3) C. M o i l e r andL. Rosenfeld, Proc. Copenhagen 17, no. 8, 1940, a n d a s e q u e l to this paper which is in course of publication. 4) S. R o z e n t a l , Proe. Copenhagen 18, no. 7, 1941. 5) S. R o z e n t a l , to appear in Phys. Rev. 6) A. P a i s , Physicag, 267, 1942. 7) A. P a i s, Physica 8, 1137, 1941. 8) F . J . B e l i n f a n t e , PhysicaT, 765,1940.
M a a r t 1942
ON T H E ELECTRIC CHARGE CURRENT DENSITY OF A NUCLEAR SYSTEM*) hy A. PAIS Physisch Laboratorium der Rijks-Universiteit te Utrecht Summary T h e p r o j e c t i v e e n e r g y m o m e n t u m t e n s o r is c o m p u t e d for a n u c l e a r s y s t e m in w h i c h t h e s h o r t r a n g e forces are d e s c r i b e d b y t h e m i x e d m e s o n field t h e o r y . F r o m this t e n s o r t h e H a m i l t o n i ~ n a n d t h e electric .charge c u r r e n t d e n s i t v of t h e s y s t e m are d e r i v e d . T h e H a m i l t o n i a n is t r a n s f o r m e d b y s e p a r a t i n g off t h e l o n g i t u d i n a l e l e c t r o m a g n e t i c field a n d t h e s t a t i c m e s o n field s u c c e s s i v e l y a n d t h e t r a n s f o r m a t i o n of t h e c u r r e n t a n d t h e d e n s i t y of electric c h a r g e is discussed in detail. E x p r e s s i o n s are g i v e n for t h e e l e c t r i c dipole a n d q u a d r u p o l e m o m e n t a n d for t h e m a g n e t i c d i p o l e m o m e n t of t h e s y s t e m .
§ 1: Introduction. As has been shown by M 0 11 e r 2), one may adequately describe the ,,mixed" meson theory of nuclear forces by taking the meson field wave function to be a S-vector, provided that the source densities producing this field are appropriately defined. This ,,mixed" theory which is based on the assumption that the nuclear forces are established through the intermediary of a particular mixture of ~ vector and a pseudoscalar meson field 2) seems to be well suited to account for the experimental facts concerning nuclear forces and ~-disintegration 4) as well as the properties of mesons in cosmic rays 5). In a previous paper e) we have considered the projective interpretation of the 5-dimensional meson field theories. As was seen there, an essential feature of this projective treatment consists in its allowing the field equations, including the electromagnetic interaction terms, to be put in a more compact form. This formalism, therefore, is especially useful for the consideration of problems concerning the interaction with the electromagnetic field; the more so as one can define for any system a ,,projective energy momenturr/ *) "Fhis paper constitutes a revised part of the author's thesis t), (chap. 2, §§ 5--7). I
4 0 7
- -
408
A. PAlS
tensor" which comprises both its energy m o m e n t u m 4-tensor and its electric charge current density. This projective tensor can be obtained by means of the general prescription given in another paper 7), further quoted .as I. In the present paper it will be calculated for a nuclear system, assuming the short range forces between nucleons to be described by the mixed theory, (section 2). In section 3 the Hamiltonian obtained in this way is transformed by successively separating off the longitudinal electromagnetic field and the ,,static" meson field 8) ; in performing the latter transformation, the compact form in which the vector and pseudoscalar meson variables and the equations they have to satisfy m a y be comprised will prove to be useful. The field equations of the mixed theory in projective form are *) F~v = Uvl~ - - U~lv - - xXr~ A U~1 + S ~ ,
.x -
g
c
~c v ~ x '
(1)
g2 ~Ft xA F~'vll, = - - ~2U~' + M y - - X F ~ " A X , ,
(2)
M r = gl LF* x A ct~"q~. Here X~ = (0, 0, X~,); ui is the nucleon 5-undor, A the hermitizing matrix and • the isotopic spin vector, the eigenvalue + 1 (-- l) of x3 denoting neutron (proton) states. Considering (1) as defining F~,, (2) can be derived from the following Lagrange function (throughout this paper we always write down tensors instead of the corresponding densities) L = Ltl) + L(,,), iz2 L(/) = 1 F~F~, ~ + 2 - U~U~ - - U~M~,
~x = m c / h ,
(3) L(~) = Re --=-hc(~FtAct~,~ll~, + XFtAkXF) ' k =
~c/1
~
+'ra
l--'r3 mN c2 + ----V--mp c 2)
l--xa 2
ie c "hc" v ' ~ ' ~ °
• ) See loc. oil s) equ. (22), (23), (26) and (27). The following notations are used: AlL/z, A II/z, A j/z are the covariant gauge derivative, covariant derivative and derivative of A respectively; ~., v, . . . . . 1 , . . . . . , 5; i, k, . . . . . 1, . . . . 4. × is the gravitational constant. F u r t h e r A is a symbolic vector in isotopic spin space having components AI, A2, A3; A B = t~ AIB i. A and A denote vector multiplication in ordinary and isotopic spin space respectively.
ELECTRIC CHARGE C U R R E N T D E N S I T Y OF A N U C L E A R SYSTEM
409
m, m2v, mp are the meson, neutron and proton mass respectively;
=0 = =~X~.-~ is a n u m b e r such that A~ is antihermitian (see I p. | 157). By performing a phase transformation *), (1)--(3) become f.v = uvlL~,--u~,l.~
f~lL~ --
+ S~,~,
(4)
~z2 ut' + M~',
(5)
~t2 Lit) = I f~v f ~ + -2- u~u~ - - u ~ M ~ .
(6)
(4) and (5) are decomposed into ,,4-space" relations by putting f ~ __ .c~• ;,;k f~k +
~
Xt~ ~]. i _g~,
u~. = "52 u , + ~ _u X . ,
(7) (8)
where [ is the universal 4-pseudoscalar 4- I det gik [~, for which in the case of special relativity ~=
~
(9)
holds: __g~and u are the meson 4-pseudovector and 4-pseudoscalar respectively, while fi~ and u~ refer to the vector mesons. F r o m (4), (5), (7) and (8) one infers that f~k = U.~l:i - - Ui,_k + S~k + [~ u X ~ ] , ftklk k =
--
~L2 U i ' +
M i,
= -- u,_, + s!0~,
g klLk=--~2u+M'°'--[l~xikf'~],
Sik = g2 +, x A 0etaj +, (1 O)
M' = gl +*'cA =~+,
(11)
S ! °) = g--&+* x A ~_oO~,+, (12)
Vt
M'°'----g,+txA_~o~b.
('13)
~_0is a pseudoscalar matrix; ~bis the nucleon 4-undor. The terms in square brackets are negligibly small, as Xik = c -I ~/2× ~ik, where ~ik is the electromagnetic field tensor. § 2. The projective energy m o m e n t u m tensor. With the help of the prescription given in I we now derive an expression for the energy m o m e n t u m 5-tensor Tvv, using L m in its form (3). For the terms of R ~ (cf. I (s2) and (75)--(77)) corresponding with L m we find *) Cf. loc. ctt. o) equ. (6).
A. PAIS
4'1'0
The terms in L(tI containing X~ explicitly are k2
X 2 Xt~ A U ~ (Uv~ - - U~,~v+ S~,~) + ~- Xt~,A U~ . XU, A U ~ • Therefore, (~'/8' X ~ denotes differentiation with respect to those X ~ that occur explicitly), X~ ~'L(tI
~ LIt)
~,X~ = X ~ ~ X~ = - - x F ~ .
( x ~ A U~),
with ~e X v - T.~v can now immediately be obtained b y using I field equations (1) and (2) and those of the nucleons:
(80)
and the
T.~ = F~'P F~p + ~2 U ~ Uv - - F~P Svp - - M ~ Uv - - L¢I) ~ +
+ ~, { F ~ ( X ~ A U J + F~(X~AU~)} +
To.I%,
where T(,).%corresponds with L(,). The application of the phase transformation yields T.% = f~P f~p + ~2 U/z Uv _ _ f~p Svp - - M ~ u~ - - L(/I 8{ +
+ ~ {fw~(X~Au~) + f~A(X~AuA)} + TIll%.
(14)
The energy momentum 4-tensor Tik and the electric charge current density sk can be found at once from T ~ . In fact one has
~{,_ v'2× ~,~{X ~ T.~,, c
where T'~ = ~ " ~ k ~
1 .
~p~
is the energy momentum tensor of the electromagnetic field. Using (7) and (8) we get
~' f~p ~
f~p = (~,,f" -- ~.g_'x~). (~" f,~ -- ~ g_~ x J = f" fk~ + E2 gi g , .
E L E C T R I C C H A R G E C U R R E N T D E N S I T Y OF A N U C L E A R S Y S T E M
411
The cross t e r m s disappear on account of, ¥~' XP -- -fP.iXp = 0. Similarly ,f~ f~P y~., S~p = fiz S,~ + ~2 gl _S~01. The t e r m s in ( 14) containing Xv or X~ explicitly disappear if we cont r a c t with ¥~i y~.,, so t h a t the result is T!k ---- fi~ fkt - - fg S,~ - - M i uk + ~2 u s u , +
+ ~2 [g, g, _ g, _s~0~]- - L~ ~', + T(,0'., + T(,):,.
(15)
T(.0!, corresponds with L(.); according to (6)--(8). L(n expressed in 4-dimensional quantities is equal to
41 fik Ilk + 2- HkU k - U~ M k + ~2
~ u 2 - - u M(0~ . gk gk + -~
F r o m ( 15) we now derive an expression for the H a m i l t o n i a n of the s y s t e m in the case of special relativity. In this case use m a y be made of (9) and of the following representation for the matrices s). ~ i = O2~i,(i---- 1 , 2 , 3 ) ,
ea=iPa,
A =ipa,
_~0=--Pa-
As A is antihermitian, ~ has to be real; we take ~ = 1. In order to avoid the occurrence of singular t e r m s of the ~-function t y p e in the static nuclear interaction potential the scalar - - ] Svv S ~'~ must be a d d e d to the L a g r a n g e function *). I n t r o d u c i n g the v e c t o r n o t a t i o n corresponding with t h a t of M o l l e r and R o s e n f e 1 d a) for the meson field variables and for the nuclear source densities: -. & f,~ ---- F, G, uk = U . - - ~ r ; S,~ =- "r, g. M, = iVI, - - N.
E=W;
gk=F,--O,
S_!° ) = : P , - Q ,
M (°~=R,
we then get for the H a m i l t o n i a n : H = f (H(/) + H(,)) dv + H('. I ,
1 -'2
~2
~r2)}__(~,~
~1~)
(16)
l ( q 2 - - § 2)+
+ 2 ( ~^2 + o ~ + ~ 2 ~ 2 ) _ ( R ~ + 0 o ) +
1 (Q2__ ~2),
H(~I ---2 *) Cf. M o i l e r
2), p. 26.
]
412
A. PAIS
H('~) "has been represented in configuration-space of the nucleons, the index (i) referring to the i-th nucleon; coIO is a small term which will not be written down explicitly. (~, ~) is the electromagnetic field, ~ is-the electromagnetic vector potential. The electric charge-current density is to be calculated b y the same methods and it is the term Xf# (X~ A u;0 in (14) that here plays an essential part. We obtain 1 --
us
]k = e ~b*A ~ 0 c
-
e
~ +-+ - ~ {(f'kAu,) s +
~2
(_g~A_u)s} +
~k.
E.~ is a small term which, apart from a contribution of the ,,D ir a c-type" due to the nucleons, whose divergence vanishes, (compare I (102)) originates from the first four terms of (14) and from the terms of T(,)~.~that symmetrize the third and fourth term of (14) ; ~ktik = 0 as a consequence of the general theorem proved in I and it can be seen that this relation still holds if all terms proportional to ×½ are ignored. Therefore, from now on omitting all small terms, we may write in the case of special relativity, using the vector ^ _~ notation s k = s, p" S =
Snucl +
Sines = e z,
-
pl0
+ --
'
p = p..~ + p~os=
- ~ 1--0 - -* - ~
e iX ~ B ( x - - x ( ' l ) +
+
e ~-Zc (UAF--WA
(17)
~,)3. (18)
For the sake of the developments in the next section we introduce ^ ^~ the 5-vector s ~ that may be connected with Sm~ ^
^k
S~mes = Y.P'k Sines = ~ § 3. T r a n s f o r m a t i o n
o/
the
e
(fvP'AILlv)3.
Hamiltonian.
a)
of the longitudinal electromagnetic In effecting this separation we put
=
~[ll -{-
.L,
(19) S e p a r a t io n
field.
E L E C T R I C C H A R G E C U R R E N T D E N S I T Y OF A N U C L E A R S Y S T E M
'4'13
where ~tl is the longitudinal and ~ x is the transversal part of the electric field; similarly for ~ . ~i. is the part of the electrostatic potential which is created by the system of nucleons and meson fields, while ~ex is the contribution of other sources ] eventually present. ~1, ~11 and ~Bi, now must be eliminated from H. This problem has been extensively dealt with by several authors; we will here merely state the results. For the first term of H we get
fH(n dv = fHu) dv - - f Smes~.L dv
(20)
where H(t ) is obtained from H(t ) by putting = rot U + S,
(21)
~2 V ---=- - div F + N,
(22)
r' = - - grad it' + P,
(23)
in the place of G, ~2V and r respectively, that is to say by replacing the gauge derivatives, o'ccurring in the ,,identities" that define the ,,derived" variables 8) G, V and I" in terms of the canonical meson variables, by ordinary derivatives. Performing the same substitution in the second term of (20) and aMling Sm~ the quantity so obtained from Sinesone gets
f Sm~ ~ . dv = f sm~ ~ . d r + W ( ~ ) , where the second term generally may be neglected as it is propor: tional to the square of the perturbation parameter (e2/l~c)½. The second term of (16) gives n¢',0 = H¢,0 - - f s.~a ~_L dv with
and the last term 1
fH(,)dv = fH(,)±dv + E + f p~e. dv, HI,). = ~-(~.2L + g21. E is the Coulomb energy of the system. Infinite electrostatic self energy terms have been suppressed. Consequently the total
414
A. PAIS
result is (in the following we write if, ~, ?3, HI, I instead of ~±, ~±, ?3ex, HI,)± respectively)
H =fHct) dv + H~,o + fHc, I dv + ~ - - f s ~i dv + f p ?3 dr. (24) The quantization of the present scheme may be performed in the usual way; there is one point within this formalism, though which in this connection requires closerexamination, especially with regard to the calculations of section 3b, namely the commutability of the pro~ective"meson variables. The commutation rules "for the canonical meson variables are hc [U,,~ (~, O, F~,. (-7', t)] -- -7- ~ ( ; - ~') ~'~ ~m, (25) [0= (x,-t), ~F.(x', t)]
= h_~c~ ( x - x') 8 . .
(26)
"t'
from which the commutation rules holding for the derived variables Gi,k, VI and I'i,k immediately follow, taking into account (21 )--(23). Thus the following commutators (and only "these ones) do not vanish identically (U stands for U~,k etc.)" [U, F], [U, V], [F, G], [G, V], [~F, q>] and [I', ~]. Therefore none of the commutators that can be formed by means of the projective variables *): [/;,~,l, up,~], []~,,,~,/0*,k] and [u.~,~, U,,k] is identically equal to zero. (The prime in ];,~ indicates that the 4-space quantities that can be derived from ] ~ refer to the space-time point (x', t) ; similarly % refers to the argument (x, t)). The question thus arises whether it might be practically useful to "establish explicit expressions for the commutators of the projective variables **) ; though this of course does not present any difficulty at all, such a proceeding would not lead to any simplification of the further calculations. We will therefore not consider these commutation relations here, but follow another method of calculation which *) T h e q u a n t i t i e s @v,^Up used here are not the s a m e as the/Izv, up occurring ill (7) and (8) as the latter refer to G*,, V, ~ e t c . while the former refer to I-~, X', ~ e t c . The s a m e holds
2. ^2.
for the q u a n t i t i e s / i k , gi, lti t h a t will occur in the following; as G, V, F henceforward no more a p p e a r this will not give rise to confusion. **) A n a t t e m p t in this direction has been m a d e ~), viz. by i n t r o d u c i n g the explicit expression for []~*4,k, up, l], where I/L4 = Yr.4/i.Lv" (In this connection we r e m a r k t h a t in loc. cir. t) p. 45, footnote, only those t e r m s h a v e been written down t h a t give a n o n v a n i s h i n g eorttribution to s~)).
E L E C T R I C CHARGE C U R R E N T D E N S I T Y OF A N U C L E A R SYSTEM
415
only makes explicit use of the canonical commlttation rules (25), (26) and which seems quite practical. b) S e p a r a t i o n of the static meson field. The equations determining the static fields are z) : F=o - - g rra,d_~~
} fik
C--= rot U + S,
_____~,,+ _~!o), J fdv:U"t~'--ut*'+SJ""
&
r---- - - grad * + P, ] ~i
~,-- 0,
~-
div F = - - ~2 ~r + N, &
-g
}~% =--
F2
+
rot G = - - ix2 U, -g
div F = - - ~2 ~,. The equations in the first column define the static fields; those in the second one are the same but now written ill tensor form, while in the third column the 5-tensor form is indicated. Any,,tensor" labeled with ° is the same function of the static variables as the corresponding tensor in the former equations is of the ordinary variables. Of course this covariant form of the equations has no further meaning as the process of splitting off the static fields is not invariant; it will be seen, however, that it is useful to work with these equations in their 5-tensor form. As has been shown b y M ~ l l e r and R o s e n f e l d 3 ) the static part of the fields may be separated from all variables b y means of a canonical transformation:
(27)
7~ = S - ' AS,
where the unitary operator S which transforms the function .4 of the .%
,,old" variables (from now on indicated by U, V . . . . .
) to the same function of the , , n e w " variables (0, V . . . . . ) is g i v e n b y i S = exp. -hc K , o ..¢.
-+
~-
-+
K = f dv [F U - - U F + ¢I' O] = f dv [i~4 u~ - - ~t, f~,], (28)
where f ~ -- T.~4f~,. The transformation of the terms of H which do
416
A. PAlS
'"
not depend on the electromagnetic field has been treated b y the mentioned authors. F u r t h e r f H ( ~ I dv is of course not affected b y this transformation, so we only have to consider more closely the Coulomb energy and the last two terms of (24). Thus all depends, on the -%
transformation of ~ and s. In this paper we will compute that part of -%
and s which does not depend on the non-static meson fields, the ,,field-independent" (f.i.) part, this being the only part which is of interest for problems in which free mesons are not involved. ~
-%
First we will transform ~mes and sines and this will be done b y ma~king use of the 5-vector s ~ from which both can be derived *). As a consequence of (27) **)
~
e
S_ I
~1
li
Il
0
= s ~ , + s~, + ~ , + z ~
~ K, ~.~,l "
Terms for which l > 3 need not be taken into account, as they are of higher order than the second in gl and g2. We now introduce a set of variables marked with 1 that refer to free mesons. Of course : 1
1
1 1 ~ = 0, ~ = ~ , ,~=~', ® =#,
while I
I
V=--~-2divF,
I
5=rotU,
F=--grad~,
SO
1
5=5+
1
1
S, V = V +
~-2N, r = f +
P,
1 :mes =;rues
e [,~
q- -~C
--
~_2~A N + ~A~] 3
The second term in the development (29) is given by
~
i = ~ [K, ~kos].
^ ^k *) s ~ e s i s related to Skmesin the s a m e way as S~e s to -~mes, see (19). **) {A, B } l ~ [ A , [ A , . . . . [ A , B]] . . . . J; l is the n u m b e r of brackets.
(30)
ELECTRIC
CHARGE
CURRENT
DENSITY
OF A NUCLEAR
SYSTEM
417
Keeping in mind the commutativity propel,ties of the-projective variables this gives s~)-
e i . f d v ' {(f;,Au.)3 °' • [u'O, lv.] + ({l'OAfV~')3. [/•," uv] hc • •c
(f~Afv~)3. [u'O, %] - - ( u*"' P A u v ) 3
--
-
. [G,/v"]}.
-
(31)
The commutators occurring in the integrals are composed of quantities with the same isotopic index which has been omitted. The contribution of each term to the 4-vector sh) is then found by inner multiplication with y~' and with the help of (7) and (8) and of the relations y.~ X~, = 0, ),~iX~ = 0 and "(#i'(~k = ~ . Keeping in mind (25) and (26) the Contribution of the first term is found to be (apart from the factor e/hc)
top(t)'--(FA 3; tos(,)--(FAV)3 = - - ( F A V ) 3 - - ~ - 2 ( F A N ) 3 and the secon.d to p(,): -- ( F A U h - ('FA 'I'h; 1
I
tos m" (G~6)s + (FACF)3 + (g~6)3 + (PA~)3; The third term gives after multiplication with y/L~ and keeping in mind that ~4 = ~, = 0
i fdv' (f~Afk'), " [u'*,uk], •
~C
so it does not contribute to Pit) while it gives for ~a) i 3 3 -- Z f dv' (#; A F)3 . EU',, VII -- ----i~ -2 y~ f dv, (~; A ~)3 EU;, div ~]" 1~c,=1 hc ,=t Now [U;, div/7]
/~c ~
hc Z
~)
~X 'l
therefore the third term gives for s(u after a partial integration
(~A~r)3 + ~-2 (FAN)3. Finally the last term of (31) becomes after multiplyingwith y# _
fi}i f d v " {({l'kAut)s. [l~,/"] + ~ ' Au)s Eg~,gl]} Physica I X
27
418
A. PAIS
f r o m which we infer t h a t it does not c o n t r i b u t e to p(~). The c o n t r i b u tion to ~(t) can be f o u n d in a similar w a y to the t r e a t m e n t of the preceding term. The result is
Consequently the complete result of (31) becomes e
-*
p(,) ---- px = ~c ( " A ~ + ~AI~' - - ~ A 'l')s,
(32)
where
Now we must find the t h i r d t e r m of (29) i
=
[K,
In order to compute this commutator, P(1) and ~(t) as given by (32) and (33) are again comprised in 5-vector form: e
o
1
1
~} =-fi-£ {f;~Au,~ + f ~ ' A u ~ - - f;~'Au;~ + f~'~'Au;~}s. s~l m a y now be calculated in the same w a y as the previous c o m m u t a t o r was found. I t gives rise to f.i. t e r m s as well as to t e r m s q u a d r a tic in the meson field c o m p o n e n t s or of t h i r d order in g and .we obtain for the e
2~
--t
f.i. p a r t of Pc21 -------Pexch = ~cc (U A F)3. f.i. part of sl2 ) = Sezch- - - ~
(35).
__ U - - ~-2 F A N + P A ~),, (36),
with -~
-%
-t
Thus from (32) and (35) it is seen that, apart from terms quadratic
ELECTRIC
CHARGE
CURRENT
DENSITY
OF A NUCLEAR
SYSTEM
419
in the meson field components and from t e r l m of higher order t h a n the second in gl a n d g2 Pm~ = pm~ + px + P~,,~h
(38)
and likewise from (30), (33) and (36) t h a t 1
Sm,s =
sines +
+
xch.
(39)
(38) and (39)are evident if one remembers t h a t a n y field variable .~ occurring in ames is approximately the sum of a static part A and 1
a new free meson variable A. Inserting this one gets: a) a part depending only on the free meson variables: Pines;b) a part depending b o t h on .~ and A" ax ; c) f.i. part of Pine.: Oexc,- Similarly for sine,. The same transformation must now be applied to =k S.ua. This calculation is quite straightforward and it can be seen t h a t the f.i. part of s~.~ is simply Snuak. Thus, summarizing the result of the transformation we have, only considering/.i, terms o/.no higher order than
the second in gl and g2 : P:
Pexch +
Onuel,
S :
~Sexch + s~ uel.
F r o m these relations we m a y obtain ill the same approximation the electric and magnetic multipole m o m e n t s of the nuclear system. Of these we will here compute the most interesting ones, viz. the electric dipole m o m e n t : ~ = f p xdv, the electric quadrupole m o m e n t : £~h
:
½f-P x~ x~ dr,
the magnetic dipole m o m e n t : 9~ = ½ f x A "~dr. To this purpose the static meson variables must be expressed in terms of the nucleon variables; we have *) =
z
Ft i
A
¢(",
9=g~ Z.(o¢(% i
•i, = g--~z ¢('~. (~(0 9('1) ¢(% • ) Cf. loc. cir. 3) e q u . (14) a n d (30) a n d 1o¢. t i t *) e q u . (40).
420
A. PAIS
where ->
~{;} = grad{O, ,~{0 =
1
4= I~ o
o
-
exp. - - [z [ x - - x{;}[.
;{'}1
-
.2~.
G, F a n d r m a y be o b t a i n e d b y m e a n s of these e x p r e s s i o n s a n d of t h e s t a t i c field e q u a t i o n s . W i t h t h e help of _ _
& -~r(ik)
f dv e{0 {p{~} - - e-~:{~} "
'
8~l~
f dv-~ ~{o e{~} __ _ _ '
'
_+
x+{,~},
16=F
we get *)
e 8~
gl g2. X {,¢o A'¢"I)]. (~l~}A .~2{~,,I). e-~J'l "
t~c
V.
~,~
r (~)
4~
e
32nhc"
=
e
g~ g2 ~3 (~{0A~{% x+c~kl ~z
i,k
"
~"
(~iOA~_{%,} " e-~'c~k~
•
r (~)
'
(~(,}
+ 8=h---T'g~ ('¢'A'¢% ("{"A;~*')• ~
2~,] +
l •
.
.
+ 5~-£c ~• (~{'}A~{k})3(;+{,k} A~-{,~}) gf + ¢~ d'} d h}
r(~k}
+
• r{ik}
I s h o u l d like t o t h a n k prof• L. R o s e n f e l d for his i n t e r e s t in this w o r k . Received F e b r u a r y 10th, 1942.
*) In loc. cir. ~) the terms in the last line of 9~ have been forgotten. {These terms vanish in the centre of gravity s y s t e m of the deuteron; they do therefore not alter t h e considcrations of loc. cir. chap. 3).
ELECTRIC CHARGE C U R R E N T D E N S I T Y OF A N U C L E A R SYSTEM
421
REFERENCES 1) A. P a i s, Thesis Utrecht, 1941. 2) C. M o i l e r , Proc. Copenhagen 18, no. 6, 1941. 3) C. M o i l e r andL. Rosenfeld, Proc. Copenhagen 17, no. 8, 1940, a n d a s e q u e l to this paper which is in course of publication. 4) S. R o z e n t a l , Proe. Copenhagen 18, no. 7, 1941. 5) S. R o z e n t a l , to appear in Phys. Rev. 6) A. P a i s , Physicag, 267, 1942. 7) A. P a i s, Physica 8, 1137, 1941. 8) F . J . B e l i n f a n t e , PhysicaT, 765,1940.