Nuclear Physics 10 (1969) 16O---180;@North-Holland Publishing Co., .4msterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
A REMARK O N T H E I N T E R P R E T A T I O N OF T H E E L E C T R O N NUCLEUS SCATTERING EXPERIMENTS C. V I L L I
IStituto di Fisica dell'Universil~ di Trieste and Iaituto Nazionale di Fisica Nucleate, Sottoseaior~ di Trieste t R e c e i v e d 26 N o v e m b e r I958 T h e c o n s e q u e n c e s o n t h e i n t e r p r e t a t i o n of t h e e l e c t r o n - n u c l e u s ~ t t e r i n g a r i s i n g f r o m t h e f i n i t e size o f t h e p r o t o n a n d / o r of t h e electron, are discussed.
Abstract:
data,
It is known that the angular distributions of electrons elastically scattered by nuclei have been satisfactorily interpreted in terms of non-uniform densities of the nuclear charge z). In this kind of analysis the effects arising from the finite size of the protons bound in the nucleus are usually neglected or accounted for in an approximate way. The object of this note, suggested by an attempt carried out by Hofstadter 2) to allow for the finite size of the protons in Cz2, is to emphasize the importance that the experimental information on the structure of the elementary electron-proton interaction, brought to light by Hofstadter's experiments, might have for the correct interpretation of the electron-nucleus scattering data. The following discussion will be based on the first Born approximation not because this approximation is believed to be particularly reliable, but because it is ideally suited for a rapid outline and a simple formulation of the problem. As is well known, in this approximation the differential cross-section for the elastic scattering of electrons of energy E in the angle ~, by a nucleus of atomic number Z, reads a(0) = ~M(0)IF(q)I', 0) where
= (ze /2E)' cos' {o cosc' ½o
,(2)
is the Mott cross-section for a point nucleus and the quantity inside the vertical lines, usually called the form factor of the nucleus, is given by
F(q) = where q -----2E sin ~
fp(r) exp (¢q. r)dr,
(3)
is the momentum transfer tt in the collision and the
t T h i s r e s e a r c h h a s b e e n s p o n s o r e d b y t h e W r i g h t Air D e v e l o p m e n t C e n t e r of t h e A i r R e s e a r c h a n d D e v e l o p m e n t C o m m a n d , U n i t e d S t a t e s Air Force, t h r o u g h its E u r o p e a n Office, u n d e r C o n t r a c t NO. A F 61 (082)-164. t t A s y s t e m of u n i t s w h e r e ?~ = c ffi I will be u s e d t h r o u g h o u t . 166
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function p(r), normalized to unity [F(0) = 1], is generally identified with the charge distribution of the target nucleus. For our purposes it is important to stress that the primitive and more general definition of the form factor is not given b y eq. (3), b u t it follows naturally from the first Born approximation on which the cross-section (1) is based. In fact, elementary considerations show that the function F (q) can be defined simply as the ratio between the three-dimensional Fourier transform of the radial part of the potential energy q f (r) = -- ZeZ/(r) of an electron in the Coulomb field of an extended nucleus, and the same quantity evaluated for a point nucleus. For reasons which will become clear later, the form factor defined in this w a y will be indicated as "F(q)", i.e.
"F(q)"
~_. -~ qZ f/(r) exp (iq. r)dr.
(4)
We shall now prove that from eq. (4) emerge some conceptually critical questions, which are otherwise concealed b y the usual derivation of eq. (3) 9). Integrating b y parts twice the radial integral of eq. (4) and taking into account that, because of the finite size of the nucleus, we have lim
r](r) =
r-~o
l i m [ r d/(r)7 = 0, r-~0 k
dr 1
fV2l(r)
exp (iq. r)dr,
(5)
one finds 1 "F(q)"
=
-
(6)
from which one formally gets eq. (3) ["F(q)" =--F(q)] using the Poisson equation V2](r) ---- --4~p(r). The crucial point of the whole problem lies in the physical meaning to be attributed to the function p(r), because its identification with the charge distribution of the target nucleus is b y no means so obvious as m a y appear at first. Such an identification is physically legitimate only in so far as one implicitly assumes that the radial dependence of the electron-nucleus Coulomb potential energy arises from elementary interactions between point protons and a point electron. This fundamental circumstance, which will now be proved, reveals that it is in principle conflicting with Hofstadter's experimental evidence on the scattering of electrons b y protons, to consider as the nuclear charge density the function p(r), appearing in eq. (3). To be more specific, let us assume that one proton and one electron, whose individual densities will be indicated b y pp(r) and p,(r), be placed at the points r 1 and r 2 respectively, and construct the potential energy V(rx2)= --#v(rlz ) of the electron-proton system, where r13 = Irl--r~l and
v(r12) =
. I f ( I r ' - - r " l ) - X p p ( r l - - r ' ) p o ( r , - - r " ) d r ' d r ''.
(7)
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c. VILLI
Then, calling r z ~ r the distance of the electron from the centre of the nucleus, the radial dependence of the electron-nucleus potential energy is given b y
(8)
l (r) = fp (rl)v (rlz)dr I
where p (r) is the spherically symmetric distribution function of the nuclear charge. According to definition (6), for the evaluation of the form factor "F(q)" one must apply the operator V 2 = (az/ar~)+(e/r)(O/Or) to eq. (8), and then perform the three-dimensional Fourier transform of the function in tht~ w a y obtained. The Laplacian of [(r) leads of course to the Poisson equ~on v'/(r)
"¢(r)",
=
(9)
which, however, has not the well known classical significance, because "p(r)" turns out to be a complicated functional of the nuclear density p(r). We shall now prove that, as far as the fit of the electron-nucleus scattering data is concerned, the function "p(r)" plays -- so to say -- the role of an "apparent" density, which, besides the information on p(r), foreshadows also that concerning the structure of the elementary electron-proton interaction. To see how the latter comes into play, eq. (8) should be worked out. Defining the volume element as dr1 = (2ze]r)rldrxrx2dr~ and taking into account that, because of the absolute value sign on the limit of r12, i.e. Ira--r[ < ra~ < r14-r, the integration over rl must be split into two parts (a) rl ~ r and (b) rl _~ r, the final expression of eq. (8) reads
/(') = . LJorlP(rl)drxJ~.-.r,rl'v(r")drl'qL)~" 2~ r t"
t"+"~
l"°°
~"+"~ -[ r,t,(rx)drlJ,._rl~v(ra,)dra,j.
(,o)
According to definition (9), the radial part of the Laplacian, applied to eq. (10), leads to "p (r)" = Z (0)p (r) + p * if), (11) where
x(Ir-rll)
'If)
p*(r) = ~r
= {-[(r-rl)f)
(F--rl)-JI- (rl--r)'O(rl--lr)],
jp(rl)Kl(rlrl)drl+
f7
(12)
]
rxp(rl)K,(rlrl)dr I ,
(13)
the two functions Kx(rlrx) and K,(rlrx) being defined as
Kl(rlrx) -~ v(r--rl)--v(r+rl)-{- (r--r1)
Or(r--r1) Or
(r+rx)
Ov(r +rl) Or '
(14a)
K*(rlrl) = v ( r l - - r ) - - v ( r + r l ) - - (rx--r)
Ov(rl--r) Or
(r+rl)
Ov(r + rl) Or
(lib)
The normalization t o u n i t y
of "p(r)" and p(r) implies that it must be
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fp* (r)dr = 1--%(0).
169
(15)
In fact, inverting the integrations over r x and r, one has
j',*Cr)dr =
~ f or~p(r.)cp(r.)dr.
(16a)
where oo
r1
Using eqs. (14) and taking into account that for large interparticle separations v(rl, ) approaches to the r ~ law [%(00) = 1], eq. (16b) yields ~(ri)
r~[l-x(o)],
=
(17)
namely eq. (15) is verified. It is now easy to prove that the necessary and su]/icient condition/or the identification o/"p(r)" with the true nuclear charge density p(r), i.e. p*(r) = O, is that the law v(ra~) -----r ; 1 holds, namely that both the proton and the electron density be described in eq. (7) by a delta-/unction. Inspection of eqs. (13) and (14), in fact, shows that the condition p* (r) = 0 is fulfilled provided the electron-proton potential satisfies the differential equation dr(x) v(x) - --, (18) dx x -
whose solution v(x)----x -1 is the Coulomb law. The deceptive nature of the electron-nucleus scattering data will be clear by now. The presumptive structure of the interacting particles implies that the electron-nucleus differential cross-section is not given by eq. (1) with the form factor defined in eq. (3), but rather by the expression
a(O) = aM(O)l"F(q)"[',
(19)
" F ( q ) " = F(q)+F*(q)
(20)
where is the three-dimensional Fourier transform of eq. (17). Eq. (1) is evidently a particular case of eq. (10), valid for small momentum transfer or for structureless particles. It is, then, clear that the ratio between the measured and the Mott cross-sections allows the experimental determination of the behaviour, as a function of q, of the "apparent" form factor "F(q)", which disguises in a compficated manner the intrinsic features of the nuclear charge density p (r). In the light of this circumstance, a systematic re-examination of the electron-nucleus scattering data is an important task to accomplish in order to bring to light all physical information which probably they stiU conceal, and to dispeU the suspicion that in m a n y cases one has attributed to the nucleus also those effects which presumably belong to the structure of the elementary electron-proton interaction. The procedure to adopt for such an
170
C. VILLI
analysis is very simple, because the "apparent" density " p ( r ) " is nothing but the "experimental" density, so far improperly attributed to the nucleus and empirically determined from the experimental behaviour of [a (0)/a M(a) ] ½. Hence, the charge distribution of the target nucleus is obtained by solving the integral equation (17), which in compact form reads 1 /'~°°
"p (r)" = Z(0)P (r) + ~rJ. rip (rx)K(rlrx)drl'
(21)
the kernel K (fir1) being determined by the theoretical model used to construct the electron-proton interaction. Since, according to the above theorem, the validity of the Coulomb law implies K(r[rl) ~ O, eq. (21) discloses what the link is between the electron-nucleus and the electron-proton scattering data as far as the determination of p (r) is concerned. The solution of the integral equation (21) is somewhat complicated by the fact that, according to eqs. (14), the integration over r 1 must be split into two parts because K(rlrl) = Ka(r[rx) for r ~ r 1 and K(r[rl) -~ K~(r[rl) for r ~ r x. This mathematical problem, together with the extension of the preceding considerations up to the second Born approximation, will be examined in a forthcoming paper, where it will be shown that clear-cut information on the charge distributior, of the target nucleus can be obtained provided one takes into account the totality of Hofstadter's experimental evidence, namely that concerning the scattering of electrons b y both nuclei and protons. In this paper we shall restrict ourselves simply to support the above remarks with some corroborative evidence. With this aim in view, we shall examine, as an illustrative example, the limiting, and more unfavourable, case of a uniform distribution of Z protons confined into a nuclear sphere of radius R. Since in this case p(r) = Po -~ 3/(4~Rs) for r ~ R and p(r) ~ 0 for r > R, eq. (8) becomes (rs ~ r) (22)
](r, R ) = Po f v(rl~)drl.
To evaluate eq. (22) it is convenient to choose the volume element d r 1 in the same way as before, and then invert the integrations over rl and rx~, using the following rules: for r < R:
~d~'lff%~:d~'12----~f:d,12;c.._rld,l+f:rd,1,;1.._rd, 1 f : dr + r I ele-rd,,,j,rrll-r
J~, orl,[R
dr 1
~'1 S ~',
rl~r;
J~'l|--r
(23a)
(23b)
for r _~ R:
f~odr,f~,:dr,,~f~_Rdr,,f~ drl+fT"drl, f:._drl.
(24)
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Performing the operations indicated, eq. (22) ultimately reads
](r, R)
=P°[2r(f:+'+ f:-') ~,v(rx,)drx. ](r, R) = ~O---°f~+erl~(RS--rs--r2~+2rrls)v(rl~)dr 12 (r ~ R).
(25b)
Y J~--R
According to definition (9), the "apparent" density, following from eq. (25) is found to be: for r ~ R,
"'P(r)" = 2r P° [I'e~r12v(r")drl~--R(R+r)v(R+r)+R(R--r)v (R--v) ] , (26a) for r ~ R,
[/;,
]
_
. (26h)
It is readily verified t h a t eqs. (26) reduce to the mathematical expression of the step-function representing the rectangle-like shape of the assumed uniform distribution onlywhen both the proton and the electron are described as point objects and, consequently, their mutual interaction is governed by the Coulomb law v(rxs ) -----r~x. Of course, in this case eqs. (25) yield the well known result
I(r,R)=~-R
~,--
1
](r,R)~---
(r ~ R),
(27a)
(, ~ R).
(27b)
T
To be more Circumstantial, let us assume that the electron-proton interaction be described by the function 1 --J~e -krst
v(r18) -
rx8
ee
(28)
which contains, as particular cases, the modified Coulomb law examined (a) by Dpell 8) (~/= 1,~ = 0), (b) by Clementel and Villi ~,5) (~/> 1, ~ = 0) and (c) by Hofstadter 5) (7 ----- 1, ~ = {k). Since the function v(rxs), defined in eq. {q), is determined by the produ.ct of pp(r) and p,(r), no answer can be given to the question concerning the individual contribution of the proton and ~f the electron structure to the mutual interaction (28). For this reason, ono can interpret the "revised" Coulomb formula (28) in terms of a proton raC)del and, although it is conceivable that m a n y of the facts attributed to
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the proton size could pertain to a radius of the electron, eq. (28) m a y be thought of as obtained from eq. (7) by putting pc(r) = 0(r) and e-/Cr
pp(r) -----k[~lk+ (kr--2)¢] ~
-- (~7--1)0(r).
(29)
The case (a) (Yukawa-like density of the proton) has to be ruled out because confficting with the high energy electron-proton scattering and, as Hofstadter has discovered, only the eases (b) and (e) are capable of reproducing the data remarkably well. Since these two cases axe practically indistinguishable up to the energies so far available, we shall take into account only the latter one and evaluate the "apparent" density "p (r)", following from eq. (28) for = 1 and ~ = {k. In this ease eqs. (26) become: for r ~ R,
k:l'-J
"p(r)" = Po -Ll+½(kR+ 1) e-~R cosh kr--~(3+kR+kZR')e -~e sinhkr
(30a)
for r => R,
"p(r)" = Po [~kR(3+kR) cosh kR--½(3+kr+kS RS)sinh kR
I e-~" kr
(30b)
It is easily recognized that the last two terms in eq. (30a), and all terms in eq. (30b), represent the function indicated b y p* (r) in eq. (17). This function of course becomes zero if one suppresses the extended structure of the proton b y letting its size parameter k tend to infinity. Thus, in spite of the fact that we had intentionally confined ourselves to a uniform distribution of nuclear
2.0 ---''~'-~.
'='~POINT PROTON' "\'\.
..~ NOFSTADTER MO[:~L r,,.--
=,. Q4
0
rl2 in lO'~crn I 0.5
I 1.0
Fig. 1. Radial dependence of the electron-proton interaction for a point proton (continuous line) and for the exponential model of the proton (dotted line). The size parameter of t h e proton density has been computed from the relation ae = (12)tk -1, assuming a root-meansquare radius of the protbn equal to a e = 0.8× 10-Ls cm.
charge, the radial dependence of '.'p(r)" has all the characteristic/eatures o/a non-uni/orra distribution, being continuous at r = R and normalized to unity.
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The illusory nature of "p(r)" is nevertheless evident, and its non-uniform character has indeed nothing to do with the structure of the nucleus, but it arises entirely from the exponential density of the proton, responsible for the deviation from the r12 law, shown in fig. 1. The radial dependence of "p(r)" is surprising: it ezhibits some o/the ]eatures o/a smoothed uni]orm densitv with a di]]used boundary and a pronounced tail. This disconcerting situation, visualized in fig. 2, clearly suggests that neglecting the effects arising from tO
0.5
r
0
2
in lO't"lcm 4
6
Fig. 2. T h e c u r v e d outline of t h e s h a d o w e d area s h o w s t h e radial dependence "p(v)"]p 0 of t h e " a p p a r e n t " d e n s i t y for a u n i f o r m nuclear charge distribution, b r o u g h t a b o u t b y t h e finite size ~f the p r o t o n s . T h e p r o t o n h a s been described according to t h e H o f s t a d t e r exponential model. T h e size p a r a m e t e r k of t h e p r o t o n d e n s i t y h a s been c o m p u t e d f r o m the relation ap = (12)tk -1, a s s u m i n g a r o o t - m e a n - s q u a r e radius oi t h e p r o t o n equal to aF = 0.8 > 10 -is cm. Eqs. (30) h a v e b e e n evaluated for A ---- 100 a n d R ----- 1,2At × 10 -xa cm.
the structure of the electron-proton interaction leads to an overestimation of the diffuseness of the charge distributon of the target nucleus. This is of course true also in the more realistic case when p (r) is indeed non-uniform. For instance, inspection of eq. (21) with the kernels (14), specified according to eq. (28), namely
Kl(r[rl) = 2e-k'[(~k+k~r--~) sinh krl--k~r I cosh krl]
(r >~ ri),
(31a)
K=(rlri)=2e-~l[(~k+k~rx--~) sinh kr--k~r cosh kr] =Kx(rxlr) (r p(r) for r ~ R o, where R o is the root-mean-square radius of the "apparent" density. From the critical content of the above remarks a question arises spontaneously, namely whether instead of interpreting the data in terms of point electrons elastically scattered by a non-uniform distribution of point protons, they could also be explained from the opposite standpoint, that extended (or point) electrons be elastically scattered by a uniform distribu-
174
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tion of extended protons. The answer to this question can be given, without resorting to fitting subtleties, simply by taking advantage of the fact that eqs. (25) are valid whatever the radial dependence of v(rls ) may be. This remarkable circumstance suggests that we are here faced with a problem whose general mathematical nature is essentially determined by the assumed uniform character of the charge distribution. It is, in fact, readily established thane the potential energy R) = Vo/(r, R)
(32)
of a particle at a distance r from the centre of a system of particles uniformly distributed within a sphere of radius R, obeys the hyperbolic partial differential equation t
(0'20
0"40)
-~ r Or
aR'
R a-R ~ ( r , R ) = 0 ,
(33)
whose solutions, derived from particular specifications of the associated Cauchy problem, are in fact given by eqs. (25) for special choices of the function v(rxs). By virtue of eq. (33) it is possible to define the "apparent" density "p(r)" in a manner which is different from the one stated in eq. (9), namely
R) = --
"p(r)"
(34)
where the differential operator ~) is given by 02
4
a
= 0-~ + R a--R"
(35)
Eq. (34) provides us with the clue for answering the question concerning the conjectured possibility of explaining the electron-nucleus scattering data in terms of the finite size of the interacting particles only. In fact, applying the operator ~) to both sides of eq. (4) and taking into account eq. (34), the "apparent" form factor "F(q)" is found to obey the Bessel differential equation ( ~ + q ' ) "F" = O,
(36)
whose solutions are 3
"F(q)" = ~ [Cl(q)il(qR)+¢,(q)nx(qR)],
(37)
ix (qR) and nx (qR) being sphericalBessel and Neumann functions respectively, and Cx(q) and C s (q) two unknown functions independent of the nuclear radius R. These functaons of the momentum transfer contain the structure parameters characterizing the electron-proton interaction at small interparticle separation, and are determined by the proton model used to evaluate t This e q u a t i o n is of course valid also for particles interacting t h r o u g h a s h o r t range field.
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v(rxs ). For structureless particles one has Cx(q) = 1 and C2(q) = 0. In this case the electron-nucleus cross-section (19) identifies with eq. (1)and exhibits, in disagreement with experiment, points of zero diffraction corresponding to the zeros of the spherical Bessel function il(qR). This disappointing circumstance, which cannot be overcome even by taking into account that p* (r) is different from zero, compels one to rule out the rectangle-like charge distribution. In fact, the examination of the diffraction pattern determined by eq. (37) shows that in order to wash out the minima of the cross-section, Ca(q) and C2(q) should be oscillating functions of the momentum transfer. This behaviour is, however, unacceptable, because it would involve a sort of sharp boundary proton model, which is conflicting with the electron-proton scattering data. Any proton model with a diffused boundary, like the one suggested by Hofstadter or by Clementel and Villi, gives rise to an "apparent" density not sufficiently diffused to disentangle "F(q)" from the zeros of il(qR) and nx(qR ). The functions Cl(q) and C2(q), calculated either for the case (b) and (c), depend on the momentum transfer in such a way that the cross-section still possesses points of zero diffraction, although shifted towards large angles, where "F(q)" exhibits a more rapid decrease than when the proton density is described by a delta-function. This circumstance suggest~ that a nuclear charge p(r) less diffused than that implied by the assumption that p* (r) be zero, is involved in the fitting mechanism of the data. The influence on a(0) of the proton structure is obviously appreciable when the wave length of the impinging electron is smaller than the average interparticle separation within the nucleus. For this reason, structural effects due to the finite size of the protons are practically undetectable for scattering energies smaller than about 140 MeV. On the contrary, one should expect that at higher energies the interpretation of the data might lead, as a consequence of the neglecting of p*(r) in eq. (17), to a spurious energy dependence of the size parameters characterizing the spatial extension of "p(r)" ~ p(r). The situation shown in fig. 2 is not qualitatively altered when considering the charge distribution t>(r) extracted from the "experimental" density "p(r)" by solving the integral equation (21). One m a y argue that also in this case the modified Coulomb law (28) gives rise to fictitious diffuseness effects near the "surface" of the nucleus (r ~ R0). An exact evaluation of these effects is therefore highly desirable for a correct description of the skin thickness of the charge density of the nucleus. In connection with the above discussion, it is worth while calling attention to the fact that whenever in the theoretical picture of a system of interacting particles their individual structure is taken into account, effects arise which are to a certain extent indistinguishable from those due to a presumptive non-uniform character of the radial distribution of the assembly itself. To shed light in a very simple way on this peculiar circumstance, let us seek for
176
c. VILLI
a solution of eq. (34) in power series of r, valid for r ~ R. Since no odd power of the radial distance come into play, the power expansion reads nm~
(38)
3¢P(r, R) -~ ~ eo,(R)r ~", ~--0
where the unknown functions to. (R) axe determined b y solving the system of differential equations 2 (n-4-1) (2n-4-3)eo.+l(R) = ~)eo,,(R).
(39)
The solutions of eq. (34) having physical meaning must satisfy the evident condition ¢P(r, or) -~ 0. It follows that eqs. (39) cut off the expansion (38) at ainu = 1, all other coefficients being wn(R)-----0 for n > nrau = 1. Then, the most general solution of eq. (34), in the form (38), is found to be $/'(r, R) -~ - ~
3--
cx
,
(40)
where co and c1 are two arbitrary integration constants. Since the classical expression of the electron-nucleus potential is obtained from eq. (40) b y choosing co ~- --•Ze* and cx = 0 (eq. (27a)), one is inescapably led to the conclusion that the modified potential (40) with c1 :~ 0 has its only possible physical justification in the effects brought about b y the finite size of the particles t. On the strength of this conclusion it is interesting to evaluate the Coulomb energy of the nucleus
gc -- ½zfp(r)C"(r,
R)dr
(41)
and regard $/'(r, R) as the potential energy of one proton in the Coulomb field of the remaining ( Z - - I ) ones. Assuming in this case co = -b](Z--1)e*, it is readily established that eq. (40) leads to the classical expression E c ---- ~ Z ( Z - - 1 ) e * / R * , provided one introduces the effective nuclear radius R* = R + ] - ~5 _Cl, R
(42)
which accounts for the fact that the square root of cx plays the role of a new size parameter of the system, originating in the electromagnetic structure of the proton. If this theoretical background is ignored, it is clear that the modified expression (42) could also be interpreted assuming that the nuclear t T h e r e a d e r will find it i n t e r e s t i n g to e x a m i n e t h e case w h e n , w i t h o b v i o u s c h a n g e s in eq. (7), v (rll) is r e g a r d e d as t h e radial d e p e n d e n c e of t h e n u c l e o n - n u c l e o n potential, a n d eq. (40) as t h e p o t e n t i a l e n e r g y of o n e n u c l e o n in t h e s t r o n g , s h o r t r a n g e field o r i g i n a t e d b y t h e (.4 --1) n u c l e o n s b o u n d in t h e nucleus. I n t h i s case, eq. (40) is f o u n d to be c o n s i s t e n t w i t h t h e s a t u r a t i o n p r e s c r i p t i o n s of t h e n u c l e a r v o l u m e e n e r g y a n d d e n s i t y b y v i r t u e of t h e t e r m in c1 . A closer i n s p e c t i o n s h o w s t h a t t h i s t e r m , w h i c h p r e v e n t s t h e n u c l e a r collapse a t h i g h densities, implies in t h i s case a d e p e n d e n c e of t h e p o t e n t i a l e n e r g y of t h e n u c l e u s on t h e r a d i u s R w h i c h is similar to t h a t e x p e c t e d to be d e t e r m i n e d b y t w o - b o d y forces w i t h r e p u l s i v e core.
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charge, instead of being described like a sharp boundary drop of radius R =roAt, would be pictured rather like a uniform core with a tapering outward extension. In fact the former term in eq. (42) can be thought of as being related to a central core of uniform density and the latter one to a low density fringe. Thus, also in this case we are faced with effects formally attributable to a fictitious diffuseness of the nuclear charge, as shown b y the shadowed area in fig. 2. The value of the parameter c1 can be computed from eq. (42), using the energy difference between mirror nuclei in order to determine R* and b y regarding R as the radius of the equivalent uniform model consistent with the electron-nucleus scattering data. For instance, in the case of the two mirror nuclei Cn and B n one has R* = 3.19 × 10-18 cm. The value of R, extrapolated from Hofstadter's calculations 1), is equal to 2.95× 10-~s cm, and finally eq. (42) yields r* = ~/c t = 1.11×10 -~8 cm. The preceding considerations suggest that accurate evaluations of the Coulomb energy of mirror nuclei might provide an indirect clue for deciding whether or not an internal structure should be attributed to the electron. In fact, the possibility of explaining the Coulomb energy of the nucleus using both the charge distributions determined from the electron-nucleus scatterLug data and the Coulomb interaction between protons, constructed from proton models consistent with the electron-proton scattering experiments, depends upon the prejudicial assumption that the electron is a sort of point probe, having no influence at all on the electron-proton interaction at short distances. As has been already pointed out, this assumption in no w a y restricts the validity of the above considerations as far as they are confined to the study of the electron-proton or electron-nucleus interaction, but it m a y be found to be unacceptable when considering the Coulomb energy of the nucleus and, perhaps, the scattering at very small angles of high energy protons b y protons. If it is true that the electron-proton scattering data give a direct information on the structure of the proton, then, the deviation from the Coulomb law of the function v(r12), given in eq. (28), should be thought of as entirely determined b y the proton density (29). The obvious consequence of this preconceived assumption is that the proton-proton Coulomb interaction, which comes into play when calculating either the p - - p Mott cross-section or Ee, is indeed obtainable from eq. (7) simply by replacing pe(r) b y pp(r). To make clear this point, let us write eq. (7) as v(rlz ) = fpl(rl--r')G~([r'--r2l)dr', where
G2(lr'--r2l ) -----
fp~(r~-r")([r'--r"l)-Xdr'',
(43) (44)
the individual densities of the two interacting particles being defined as (i = L 2 )
178
c. VII2LI
p, (r) =
a,~, (r) +b, 0 (r).
(45)
Introducing the vectors x = r 2 - - r " and y----- ( r ' - - r l ) + x , the volume element d r " becomes equal to (2zt/z)xydxdy, where z = ]r'--rs[. Using the density (45) with i = 2, eq. (44) reads
c~(z) - ~a~ Z
If:
~,'¢~(~,)o+z
f.°
x~(x)d~
]
+ ~,
(46)
Z
and eq. (43) ultimately yields
,(rl,)
=
2r,----~a' [fo"' xSx(x)dxJ,,,__" r',,+"yGz(y)dy+f",,*¢'(*)d~J,-,,,VC,(y)dvJ " +bxG~(rx2 ).
(47)
From eq. (47) one obtains the potential (28) assuming ps(r) -~ pc(r)(a s = 0, b2 = 1) and identifying px(r) with the proton density pp(r), given in eq. (29). The proton-proton Coulomb potential is obtained from eq. (47) by putting px(r) ~ p ~ ( r ) = pp(r). In particular, assuming the Hofstadter (~7 = 1, = ~k) and the Clementel-ViUi (~/> 1, ~ = 0) model of the proton, the radial dependence of the p - - p Coulomb interaction is found to be 1
v(rx, ) = - - ~ ( k r , , ) rx2
(48)
where (krx, ---- z) for the former model, and
• cv(~) = 1 - ~ [ 4 - ~ ( 2 - ~ ) ] e - "
(495)
for the latter one. This way of constructing the Coulomb interaction between protons turns out to be conceptually wrong if the radial dependence of the electron-proton potential (28) is determined by the individual contributions of both the proton and the electron structure, because in this case eq. (29) must not be interpreted as the charge distribution o] the proton but rather as an "apparent" proton density "pp(r)". The very reason of this circumstance is readily established by noting that also for the electron-proton scattering process the primitive definition of the form factor is still given by eq. (4), with/(r) replaced by v(r12 ). From this definition one obtains eq. (6) irrespectiv~ of the conditions (5), provided the delta-function singularities are accounted for when calculating V2v(rts)= --4at "pp(rxm)". Assuming in eq. (47) pl(r) = pp(r) and p2(r) -~ p,(r), it is a simple matter to show that if the electron is not a point object, the form factor for the proton charge, determined from the experimental data, is the three-dimensional Fourier transform of the "apparent" density "pp.(r)". Of course, a similar situation arises also for the Mott scattering of protons by protons. In this case the fictitious
I N T E R P R E T A T I O N OF
THE
ELECTRON-NUCLEUS SCATTERING EXPERIMENTS
179
nature of the "apparent" density becomes more evident because of the identical nature of the two interacting particles. In fact, if one ignores the general definition (6), the paradoxical situation would arise that, while the target proton is believed to possess an internal structure, the incident one is implicitely assumed to be a point particle! From eqs. (9) and (48), the "apparent" density of the two-proton system is found to be k8 e-~r "pH(r) . . . . (3+
192z~
3kr+k2r2),
"Pcv(r)" ----- 8~------~(4+~/kr--4~)+(l+~f--2t/)~(r),
(50a) (50b)
and the corresponding form factors, brought about by the structure of the proton-proton Coulomb interaction, are given by ks
"FH(q)" -- (k~+q~)~" "Fcv(q)" =
(k2+q~)2--~q2(yq~-t- 2k2)
(51a) (51b)
The implications of eqs, (48) and (51) are expected to lead to experimental discrepancies if the electron is not a structureless particle, because one can readily prove that the true proton density pp(r) and the "apparent" density "pp(r)", determined from the electron-proton scattering data, are bound together by the integral equation (b1 ~ 0) "oP(r)" = Ze(0)0p(r)+ 2r
XPP(X)Ke(rlx)dx'
(52)
which is formally similar to eq. (21). In eq. (52) the parameter •e(0) and the kernel Ke(rlx ) are defined according to eqs. (12) and (14) respectively, where v(x) has to be .replaced b y the structure function of the electron Ge(x ) -~ G~(z), defined in eq. (44) [p2(r) ~_ 0e(r)]. Thus, the proton models suggested by Hofstadter and by Clementel and Villi refer only to the "apparent" density of the proton, and the very physical difference between them lies in the fact that, while the former implies an over-all attraction between the incident electron and the target proton, the latter implies a repulsi'¢e interaction at interpaxticle distances smaller than about 0.06× 10-xa cm. Whether "Üp(r)" ought to be really identified with pe(r), and consequently Ke(rlx ) is zero (point electron), is still an open problem. The most feasible and perhaps less sophisticated way to investigate whetherKe(r[x ) is, or not, equal to zero might be provided by a detailed examination of the Coulomb energies of nuclei. If the electron has a structure, one should expect it to be impossible to reproduce in agreement with the nuclear charge distributions,
180
co VXLLX
extracted from the electron-nucleus scattering data, the energy difference between mirror nuclei, computed in terms of the proton-proton Coulomb interaction (48), following from the assumption that " pp (r)" ---- p t , (r) [Ke (r]x) = 0]. Perhaps no discrepancy will ever be so welcome, at least b y those who secretly believe in the breakdown of electrodynamics. The author is indebted to Professor R. Hofstadter for m a n y stimulating discussions and to Professor L. Rosenfeld for the critical reading of the manuscript. References 1) 1t. Hofstadter, Revs. Mod. Phys. 28 (1956) 214; for more recent references see D. G. Ravenhall, Revs. Mod. Phys. 30 (1958) 430 2) 1t. Hofstadter, Annual Review of Nuclear Science, vol. V I I (1957) 3) S. D. Drell, Annals of Physics 4 (1958) 75 4) E. Clementel and C. Villi, Nuovo Cimento 4 (1956) 1207 5) R. Hofstadter, F. Bumiller and M. R. Yearian, Revs. Mod. Phys. 30 (1958) 482