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Electron scattering and effective charge
Nuclear Physics 40 (1963) 282--292; ( ~ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
ELECTRON SCATTERING AND EFFECTIVE CHARGE J. D. WALECKA and R. S. WILLEY
Institute of Theoretical Physics, Department of Physics, Stanford University, Stanford, California ? Received 9 July 1962
Abstract: The possibility of studying single-particle nuclear transition charge densities with inelastic electron scattering is discussed and it is pointed out that such experiments lead to valuable information about the nature of the "effective charge." A few simple calculations of cross sections are carried out for various models of the effective charge and it is shown that the momentum transfer dependence of the cross section distinguishes between them.
1. Introduction
The development of inelastic electron scattering as a quantitative tool for investigating nuclear structure 1-3) opens up many interesting possibilities. One of these is the elucidation of the nature of the effective charge. Now it is well-known that many oddneutron and odd-proton nuclei show electric moments and transition probabilities larger than the single particle values. It has been suggested by many people that one can describe this phenomenon, while maintaining the mathematically tractable shell model of the nucleus, by assigning to the nucleon an effective charge 4). There have been several mechanisms proposed for the generation of this effective charge. The simplest is that of Wilets and Blieden 5). They suggested that because of the attractive nature of the nuclear force, the odd nucleon will tend to surround itself with other nucleons and hence will acquire an effective charge by polarizing nuclear matter in much the same manner as an electron has its charge changed by an electromagnetic polarization of the vacuum. A second method has been put forward by Elliot, Lane and others 6). They point out that if the odd particle is weakly coupled to the collective oscillations of the positively charged core, then electric moments have matrix elements between what are predominantly single particle states, not only directly, as in the case of the proton, but also by connecting to the different core states which are mixed into the wave function. Such a process is equivalent to the single particle acquiring an effective charge (we shall carry out a similar calculation in the next section). It is a simple extension of this model to show that almost identical results are obtained if the extra nucleon is weakly coupled to a permanently, ellipsoidally deformed core; the one difference being that in the latter case the nucleon only acquires an additional charge for quadrupole moments and transitions. t Work supported in part by the U. S. Air Force through the Air Force Office of Scientific Research. 282
ELECTRON SCATTERING AND EFFECTIVE CHARGE
,'~8~
A more fundamental explanation of this phenomenon has been put forth by deShalit and others 7, s). De-Shalit has shown that under quite general conditions, the inclusion of a residual neutron-proton interaction of the Wigner type can mix in excited proton states and allow the electric multipole operator actually to connect the admixed states while having the appearance of giving the odd neutron an effective charge. Since the wave length of the emitted 7-ray is much larger than nuclear dimensions in the transitions of interest (and infinite, of course, for static moments) all the above theories reduce to the statement that the multipole operator is that of the odd nucleon with a constant total electric charge of e*. The different models will predict different values of this constant (it is in general a function of the transition) but since it is usually determined empirically, the results are essentially the same. The purpose of this note is to point out the fact that just as elastic electron scattering through momentum transfers large compared to the reciprocal of the nuclear dimension can give us detailed information about the spatial nature of the ground state charge distribution in the nucleus 9,1 0), SO tOO can inelastic scattering between essentially single particle states and of comparable momentum transfer distinguish between different spatial distributions of the transition charge density and hence between the different models. We shall actually carry out a few very simple calculations to show how the models can differ but first it is necessary to say a few words about inelastic electron scattering. The cross section for inelastic electron scattering has been discussed by many authors a~-~6). If we assume that the electrons are relativistic, that the energy loss to the nucleus is negligible , and that we can use Born approximation neglecting nuclear recoil, we find t da (Sf ,- J 3 = 8~a2 / 2K z COS 2 d--~ -~ t
1
½o
[(Jfl]dg s(q)llJi)l 2
s:o" 2J i +----~
q- K2( I q- sin2 ½0j~ l )Z 2 j ilq_------1([(jfHT~,(q)Hji)[2.Fl(jf[[TMag.(q)Hji)[2)} ,
(1)
where ~ is the fine structure constant,
E 1 "~ E 2
= hlklc,
q2 = ( k 2 _ _ k l ) 2 = 4k 2 sin 2 ½0.
The first sum is Coulomb scattering and dt'j~t(q) is just the Jth multipole moment of the nuclear charge density operator
•-'[[JM(q) = ( d x pN(x)js(qx) Ys~t(f2x). d
(2)
The second sum represents the exchange of transverse photons and expressions for the usual transverse multipole operators can be found in refs. 11' 16). It has been shown t All the angular momentum notation follows ref. 17).
284
s.D.
WALECKA
AND
R . S. Vc'ILLEY
that for collective transitions, the Coulomb term will dominate as long as al, 16) hco << hqc << 2Me 2, where ha) is the energy of the transition and M is the nucleon mass, and provided one is not at backward angles, where 2 cos 2 ½0 << 1 + sin 2 ½0. For single particle transitions, as was originally pointed out by Schiff tl), the situation is no longer so simple. An extensive analysis of the contributions of the various terms in eq. (1) to single particle transitions has been carried out by one of us is) and a specific example is discussed in sect. 4. The general conclusions can be summarized as follows. Coulomb scattering dominates the single particle transitions if i) The lowest multipole which contributes to the transition is electric, that is, the transition is parity-favoured. ii) One stays close ("close" depending on the transition involved, see sect. 4) to the forward direction. We shall henceforth keep only the Coulomb term with the above restrictions in mind and calculate tile reduced transition probability for some of the models discussed previously to show how the results will differ. 2. Discussion of the Models
2.1. THE POLARIZATION MODEL We discuss first the polarization model. We imagine that the transition is that of a single particle going between two levels, but the particle now has a complex structure as it is a nucleon which has surrounded itself with some positive charge through the action of the nuclear force. We therefore rewrite eq. (2) as •J}'~/(q) =
4-~/sf
dO~ Y*M(K2,)e' ' ' p
fe~'t'-'.)p(lx-xpl)d(x-xO,
(4)
where xp is the position of the odd nucleon and p(lx-xpl) is the single particle charge density referred to that point. If we now define
f(q2) _ f e,~.j,p(y) dy,
(5)
. M ~ (q) = f(q2)js(qXp) YsM(f2p).
(6)
we have
This is just the singe particle transition operator back again, only now it is multiplied by the form factor of the effective charge. We can write in general f(q2) = e * [ l _ { < r 2 > q 2 + . . . ] . e
(7)
ELECTRON SCATTERING AND EFFECTIVE CHARGE
285
Evidently f(0) = e*/e and if the effective charge were a point charge, f(q2) would be independent of q~-. If the odd particle is a proton, we must include in p(y) the spatial distribution of the intrinsic proton charge ~s). 2.2. WEAK COUPLING TO SURFACE OSCILLATIONS We next ask for the transition multipole operator in the case where the odd nucleon is weakly coupled to surface oscillations of the core. If the nuclear potential of the core is assumed to have the same shape as the core, the interaction between the odd particle and the core is
H' = - aVo
0 Z Q," r,, l
(8)
where a is the equilibrium radius of the core, Vo is the depth of the potential and Q~ • Y~is the tensor scalar product of Y,,,(t2,) referring to the particle and Q,. = (-1)mq,_. = ( - 1)r" ( ~ f
r [ a , _ ' + ( - 1)a , .+] "
referring to the core 16). We now write our states as [jLJM), where j represents the particle quantum numbers, L those of the surface vibration quanta, and J is the total angular momentum. If H' is weak enough to be treated as a perturbation, we can write for our initial and final states
~],m~ = [jiOji mi) + ~ [j"Ljl mi)(jaLji milH'lJiOjt mi), [%, + hCOL] ~[Jjfmf ~
]jfOjfmf)+ ~ ]jnLjfmf)(jnLjfmflH'ljfOjfmf) jaL
(9)
8 j r - - [ S J a "/ff hOJL]
We have used the fact that H ' is a scalar under rotations and that it can only create one surface oscillation quantum from the vacuum. The multipole moments of the charge distribution are 3Z . "~JM(q) = G Jj(qa)qzM-b ½(1 -[-'r3)jz(qXp) Yj~(~p),
(10)
where the first term is from the core 16) and the second term is from the odd particle. Now taking matrix elements of this operator between the states given above (to first order in tt') and noting the selection rules on the matrix dements involved we find
(Ttdflld[s(q)]lT j,) = (jfOjf[[,..,~'jl[jiOji) + (jfOjfii'gs[ljfJji)(jfJjimilH']jiOjimi) 8ji -- ejf-- h~s q- (jr 0jr mdH'lj i Jjf mf)(j i Jjfll,/gjl[ji 0ji ) % - % - hoj s
(11)
~86
$. D. WALECKA AND R. S. WILLEY
Making use o f the rules for the matrix elements of tensor operators in coupled schemes ~7) we find that (~j~ IIdt's(q)ll wj~) = (Jr] Id[)'P'(q) + ~')'P'(q)ll Ji),
(12)
where ~'~'~t" + ~'~'~/ = ½(1 + za)js(qxp)YsM(12p)
(3z)(,,~) + Vo -~
2
2h~o~
.
,~ (ncoj),2----7"- ( e j f - e i l ,2 ) Js(qa)ab(xv- a) YjM(~2p)
(13)
and the matrix element is taken between the singleparticle states t. Thus mixing in the collective states and allowing for the collective transitions effectively modifies the single particle operator. Since the entire angular dependence of the operator is contained in YsM(Op), we have all the selection rules of single particle transitions (/-forbiddenhess for example) even when the transition is really taking place by another mechanism. If we concentrate on odd neutrons, the form factor is characteristicallyjs(qa) and the effective charge is proportional to [(hcos) 2 - (ejf-ejf)2] - 1 so that it will depend on the process, being in general somewhat less for static moments than for transitions. (We have tacitly assumed hoga > lejf-e~,l in making a perturbation expansion in H'.) The odd nucleon likes to pull the surface along with it and hence acquires a charge by the polarization of the core. The form factor for transitions is then characteristically the form factor for core transitions. 2.3. COUPLING TO CORE OF ELLIPSOIDAL DEFORMATION A trivial extension o f the above calculation is the case when the odd particle is coupled to a core which has a small permanent ellipsoidal deformation. The nuclear potential is t h e n
V~p,(rOc~) = - VoO[a(1 + 2 E ~2'O(--~--fl--~)Y2M'(O, ~b))--r]
(14)
M'
O(x) = {1, O,
x > 0, x
The core-particle perturbation is therefore
U' = -- 21Io aft(r-- a ) ~ .
Y2,
(15)
where 2 is the deformation parameter and ~ o2 • II2 is the scalar tensor product o f (-1)M~o(-a-fl-7) referring to the core and Y2u'(0q0 referring to the particle. The unperturbed states are again labelled by [jLJM) where L refers to the rotational t The observation that the result of a configuration mixing calculation such as this can be cast in the form of a modified single particle operator is due to de-Shalit 7).
ELECTRON SCATTERING AND EFFECTIVE CHARGE
287
state of the deformed core. In this case the calculation yields eq. (12) with ½(1 +
+6j2
,3)jj(qxp) rj (Op)
Vo \ 4 r c / \ 5 ] [3h2~2_(ei_sj,)2J2(qa)a6(Xp-a)Y2M(g2P)
"
(16)
The only difference from the previous case is that there is only an extra term for J = 2 if we have ellipsoidal deformations. The energy of the core is given by [h2L(L + 1)]/2I where I is the moment of inertia and we have again made the assumption that the deformation is small enough so that 3h2/I > [eJf-zji[. 2.4. THE DE-SHALIT C A L C U L A T I O N F O R T H E MULTIPOLE O P E R A T O R
Finally, we discuss the de-Shalit calculation for the multipole operator 7) (we shall actually oversimplify things a little for convenience). We restrict the discussion to single odd neutrons outside of closed proton shells (noloJo). We assume a residual neutron-proton interaction of the Wigner type Vnp = -- Z K
fK(rnrp)CK(n)" CK(P),
(17)
[Ckq = ~/4zc/(2k+ 1 ) Y j . Using the odd-group modelwe write the states as ]jndpJM) where j~ refers to the neutron and Jp to the protons. We now start with the states [L0jimj), [jfOjfmO and mix in all the states reached by treating g = Z Vnp np
(18)
as a perturbation. The transition multipole operator is that of the protons
~ s u = Z Js(qxi)YsM(Oi) • protons
The calculation proceeds just as in subsects. 2.2 and 2.3. The result can again be written in the form of eq. (12) with -s.. _
I
~ s M - 2J +-~1
Z
2~p
nlj>nolojo 8p x (no loJolljj(qx)Yj(Ox)llnlj)(no loJollfj(r,y)Cj(f2y)llnlj),
(19)
where 8p is the proton excitation energy to the higher shell (nlj). (Again note that because &the Cj~(On), all the single particle selection rules hold.) We cannot immediately read off the q dependence because the admixture of states depends on thefs (or on the force). We can, however, say that the q dependence is that characteristic of a
proton transition to a higher shell.
288
$. D. WALECKA AND R. S. WILLEY
3. Discussion of the Calculations Knowing the operator dt'~/~/(q) we immediately obtain the transition probability for real ? emission, if the transitions are electric, (_Dfi ~--- 8~0~C
(-2 J-+ 1)!!
~
[(Jf[[Qs[[Ji)]
(20) QSM = Lim ( 2 J + 1)[!
~o
qS
dts~(q).
We see, therefore, that according to the four models discussed the odd particle acquires an effective charge for electric emission, the value of the charge depending on the model. Each model, however, predicts a different q dependence of the inelastic electron scattering cross section. In the weak coupling to the collective motion of the core, the q dependence is given exactly byjs(qa ) where a is the equilibrium radius of the core. In the fourth model, where proton states are admixed by a neutron-proton interaction, we see that the q dependence is that characteristic of a proton transition between different proton shells. In the polarization model, the q dependence is that of the single particle operator js(qx)Ysu(f2x) taken between the single particle states and then multiplied by the form factor of the effective charge. In these last two models one must, of course, know the radial wave functions well before the q dependence can be calculated t. 4. Numerical Example In order to get an estimate of the magnitude of the effects discussed above we have carried out a numerical calculation for 017. The shell model is expected to have validity here 4) and the configuration of the ground state should be 016 + ld~ neutron. The first excited state is thought to be 016 +2s½ neutron, as indicated in fig. 1. The mean life of the first excited state is zo) ¥ = 2.55 x 10 - l ° see. Assuming harmonic oscillator radial wave functions with an oscillator parameter ,t 1
/~
h
-
- 3.2 fm 2
Mo~o
and using a single particle operator
Q2M = e*x2y2M(Ox),
(21)
t H. Kendall and L Talmi la) have tried to fit the data on scattering from Vzl. They work with the reduced 7 ray transition probabilities (eq. (20)) obtained by extrapolating the cross section to zero momentum transfer. The different models discussed above give different values for these transition probabilities (and actually slightly different procedures for extracting them from the data). Our proposal is really complementary to this, as we advocate looking at the shape of the cross sections themselves to distinguish between the models. *t This result is obtained by interpolating the results lo, z2) of elastic electron scattering from Oin to 0 is.
ELECTRON
SCATTERING
AND
289
EFFECTIVE CHARGE
A ~
LEVEL SCHEME FOR MeV 3.06
I/2-
u
O. 87 "{'=2"55x10
0
-io
secI/2+
II
6 512+
Fig. 1. Level scheme for 80~ 1T. T h e data are taken f r o m refs. ~0,~1).
.io-~0
L
I
1
I
~
I
*co= 197 MeV
Lu--
~
Io--0.51el
--
\ I0"" (0 °}
I g (fff~')l~
P 0.2 0.4 (11-5"} (2:5.1°)
0.6 0.8 [55.0"} (47.2")
1
l
1.0
1.2 (T3.9")
[60 °)
1.4 (89 °}
Fig. 2. Inelastic electron scattering cross section to the first excited state of e l l T h e incident electron energy is 197 MeV. The C o u l o m b cross section (d
--1.91(eh/2Mc).
I
I
I
I
Eo= 197 MeV
0"87T2$ o-_=
T io-_==
~m
~-,, (0'=l
I
~
r q(fr~)7
,
0.2 0.4 0.6 0.8 (11.5") (23.1") (35.0")(47.2")
f
I
1.0 1.2 1.4 {60 =) (73.9")(89*)
Fig. 3. Inelastic electron scattering cross section for a single p r o t o n o f charge e m a k i n g a ld i -+ 2s½ transition. T h e incident electron energy is 197 MoV. T h e c o n t r i b u t i o n o f t h e transverse multipoles (daE2, daM 3) was c o m p u t e d for a charge e a n d m o m e n t / ~ p = +2.79(eti/2Mc).
I
1
I
I
I
I
I
Eo - 197 MeV
G87
2SII=
0 " ~ - - " Ids/= Coulomb Cross Section
io-_=
I \,x
~X 2p, Ip protont m n ~ t ~ n ~ "vocuum p o l o r i z a t i o l l ~ \ \ \
io-_~
io"* {0°}
I
0.2 04 0.6 0.8 (11.5") (23,1") (35.0"} H72")
1.0
1.2
1.4
(600)
(73.9 °)
(89")
Fig. 4. T h e C o u l o m b cross section for inelastic electron scattering to t h e first excited state of O z7 c o m p u t e d for various m o d e l s o f t h e effective charge. T h e m o d e l s a n d p a r a m e t e r s u s e d are discussed in sect. 4 o f the text.
ELECTRON SCATTERING AND EFFECTIVE CHARGE
291
one finds
The q u a ~ u p o l e moment of the ground state is Q / e = - 2 . 7 1 fm 2 and a similar calculation gives = 0.42.
--
e
Q
(Note that this is consistent with our assertion that e* should be less for static moments.) We now turn to the cross section for inelastic electron scattering. In fig. 2, we show the cross section for scattering to the s~ level assuming that the extra neutron has a point e* = 0.51e and including the contributions of the allowed transverse multipoles E2 and M3. (We have taken an electron energy of 197 MeV [K = lfm -1 ] and have used e = 0 for the transverse multipoles.) For purposes of comparison, we have plotted in fig. 3 the cross sections calculated for a single proton of charge e making the ld t --* 2s½ transition. We see that if one stays at angles < 75 °, the cross section is dominated by the Coulomb scattering. We have calculated the Coulomb scattering cross section using harmonic oscillator wave functions and the value of fl given above for the various models we have discussed and the results are given in fig. 4. The curves have all been normalized to fit the experimental lifetime (or equivalently, to give the same e*). For the polarization model, curve I, we have usedf(q 2) = e -*~2~" with b = 1.5 fin, a reasonable correlation distance. In curve II, the result of coupling to the collective modes of the core, we have taken a = 3.55 fm. For the results o f t b e de-Shalit calculation (eq. (19)) we have plotted two simple extremes. In the first we assume the force is such that the protons only make a l p ~ 2p transition, and in the second, only a l p ~ i f transition. The curves are again normalized to give the same e*. The result is that there are significant differences in the shapes of the various curves. These differences are greater than the (calculable) background contribution of the transverse photons as can be seen from the figures. Since the cross sections are large enough to be seen, it would be interesting to have some experimental data bearing on this question.
References
1) 2) 3) 4) 5) 6) 7) 8)
J. H. Fregeau and R. Hofstadter, Phys. Rev. 99 (1955) 1503 H. Crannel, R. Helm, H. Kendall, J. Oeser and M. Yearian, Phys. Rev. 123 (1960) 923 W. C. Barber, F. Berthold, G. Fricke and F. E. Gudden, Phys. Key. 120 (1960) 2081 I. Talmi and I. Unna, Ann. Rev. Nucl. Sci. 10 (1960) 353 R. Blieden and L. Wilets, Bull. Am. Phys. Soc. 5 (1958) 244 J. P. Elliot and A. M. Lane, Handbuch der Physik (Springer-Verlag, Berlin, 1957) p. 364 A. de-Shalit, Phys. Rev. 113 (1958) 547 G. Barton, D. M. Brink and L M. Delves, Nuclear Physics 14 (1959) 256
9.9~ 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22)
J. D. WALECKAAND R. S. WILLEY R. Hofstadter, Rev. Mod. Phys. 28 (1956) 214 L. R. B. Elton, Nuclear sizes (Oxford University Press, London, 1961) L. I. Schiff, Phys. Rev. 96 (1954) 765 G. Morpttrgu, Nuovo Cim. 3 (1956) 430 K. Aider, A. Bohr, T. Huns, B. Mottelson and A. Winther, Rev. Mod. Phys. 28 (1956) 475 L. J. Tassie, Nuovo Cim. 5 (1957) 1 4 9 7 K. W. McVoy and L. Van Hove, Phys. Roy. 125 (1962) 1034 J. D. Walecka, Phys. Rev. 126 (1962) 653 A. R. Edmonds, Angular momentum in quantum mechanics (Princeton University Press, Princeton, New Jersey, 1957) R. S. Willey, to be published H. Kendall and I. Talmi, to be published F. Ajzenberg-Selove and T. Lauritsen, Nuclear Physics 11 (1959) 1 Nuclear spectroscopy, Part B., ed. by F. Ajzenberg-Selove (Academic Press, New York, 1960) F. Lacoste and G. R. Bishop, Nuclear Physics 26 (1961) 511
ELECTRON SCATTERING AND EFFECTIVE CHARGE J. D. WALECKA and R. S. WILLEY
Institute of Theoretical Physics, Department of Physics, Stanford University, Stanford, California ? Received 9 July 1962
Abstract: The possibility of studying single-particle nuclear transition charge densities with inelastic electron scattering is discussed and it is pointed out that such experiments lead to valuable information about the nature of the "effective charge." A few simple calculations of cross sections are carried out for various models of the effective charge and it is shown that the momentum transfer dependence of the cross section distinguishes between them.
1. Introduction
The development of inelastic electron scattering as a quantitative tool for investigating nuclear structure 1-3) opens up many interesting possibilities. One of these is the elucidation of the nature of the effective charge. Now it is well-known that many oddneutron and odd-proton nuclei show electric moments and transition probabilities larger than the single particle values. It has been suggested by many people that one can describe this phenomenon, while maintaining the mathematically tractable shell model of the nucleus, by assigning to the nucleon an effective charge 4). There have been several mechanisms proposed for the generation of this effective charge. The simplest is that of Wilets and Blieden 5). They suggested that because of the attractive nature of the nuclear force, the odd nucleon will tend to surround itself with other nucleons and hence will acquire an effective charge by polarizing nuclear matter in much the same manner as an electron has its charge changed by an electromagnetic polarization of the vacuum. A second method has been put forward by Elliot, Lane and others 6). They point out that if the odd particle is weakly coupled to the collective oscillations of the positively charged core, then electric moments have matrix elements between what are predominantly single particle states, not only directly, as in the case of the proton, but also by connecting to the different core states which are mixed into the wave function. Such a process is equivalent to the single particle acquiring an effective charge (we shall carry out a similar calculation in the next section). It is a simple extension of this model to show that almost identical results are obtained if the extra nucleon is weakly coupled to a permanently, ellipsoidally deformed core; the one difference being that in the latter case the nucleon only acquires an additional charge for quadrupole moments and transitions. t Work supported in part by the U. S. Air Force through the Air Force Office of Scientific Research. 282
ELECTRON SCATTERING AND EFFECTIVE CHARGE
,'~8~
A more fundamental explanation of this phenomenon has been put forth by deShalit and others 7, s). De-Shalit has shown that under quite general conditions, the inclusion of a residual neutron-proton interaction of the Wigner type can mix in excited proton states and allow the electric multipole operator actually to connect the admixed states while having the appearance of giving the odd neutron an effective charge. Since the wave length of the emitted 7-ray is much larger than nuclear dimensions in the transitions of interest (and infinite, of course, for static moments) all the above theories reduce to the statement that the multipole operator is that of the odd nucleon with a constant total electric charge of e*. The different models will predict different values of this constant (it is in general a function of the transition) but since it is usually determined empirically, the results are essentially the same. The purpose of this note is to point out the fact that just as elastic electron scattering through momentum transfers large compared to the reciprocal of the nuclear dimension can give us detailed information about the spatial nature of the ground state charge distribution in the nucleus 9,1 0), SO tOO can inelastic scattering between essentially single particle states and of comparable momentum transfer distinguish between different spatial distributions of the transition charge density and hence between the different models. We shall actually carry out a few very simple calculations to show how the models can differ but first it is necessary to say a few words about inelastic electron scattering. The cross section for inelastic electron scattering has been discussed by many authors a~-~6). If we assume that the electrons are relativistic, that the energy loss to the nucleus is negligible , and that we can use Born approximation neglecting nuclear recoil, we find t da (Sf ,- J 3 = 8~a2 / 2K z COS 2 d--~ -~ t
1
½o
[(Jfl]dg s(q)llJi)l 2
s:o" 2J i +----~
q- K2( I q- sin2 ½0j~ l )Z 2 j ilq_------1([(jfHT~,(q)Hji)[2.Fl(jf[[TMag.(q)Hji)[2)} ,
(1)
where ~ is the fine structure constant,
E 1 "~ E 2
= hlklc,
q2 = ( k 2 _ _ k l ) 2 = 4k 2 sin 2 ½0.
The first sum is Coulomb scattering and dt'j~t(q) is just the Jth multipole moment of the nuclear charge density operator
•-'[[JM(q) = ( d x pN(x)js(qx) Ys~t(f2x). d
(2)
The second sum represents the exchange of transverse photons and expressions for the usual transverse multipole operators can be found in refs. 11' 16). It has been shown t All the angular momentum notation follows ref. 17).
284
s.D.
WALECKA
AND
R . S. Vc'ILLEY
that for collective transitions, the Coulomb term will dominate as long as al, 16) hco << hqc << 2Me 2, where ha) is the energy of the transition and M is the nucleon mass, and provided one is not at backward angles, where 2 cos 2 ½0 << 1 + sin 2 ½0. For single particle transitions, as was originally pointed out by Schiff tl), the situation is no longer so simple. An extensive analysis of the contributions of the various terms in eq. (1) to single particle transitions has been carried out by one of us is) and a specific example is discussed in sect. 4. The general conclusions can be summarized as follows. Coulomb scattering dominates the single particle transitions if i) The lowest multipole which contributes to the transition is electric, that is, the transition is parity-favoured. ii) One stays close ("close" depending on the transition involved, see sect. 4) to the forward direction. We shall henceforth keep only the Coulomb term with the above restrictions in mind and calculate tile reduced transition probability for some of the models discussed previously to show how the results will differ. 2. Discussion of the Models
2.1. THE POLARIZATION MODEL We discuss first the polarization model. We imagine that the transition is that of a single particle going between two levels, but the particle now has a complex structure as it is a nucleon which has surrounded itself with some positive charge through the action of the nuclear force. We therefore rewrite eq. (2) as •J}'~/(q) =
4-~/sf
dO~ Y*M(K2,)e' ' ' p
fe~'t'-'.)p(lx-xpl)d(x-xO,
(4)
where xp is the position of the odd nucleon and p(lx-xpl) is the single particle charge density referred to that point. If we now define
f(q2) _ f e,~.j,p(y) dy,
(5)
. M ~ (q) = f(q2)js(qXp) YsM(f2p).
(6)
we have
This is just the singe particle transition operator back again, only now it is multiplied by the form factor of the effective charge. We can write in general f(q2) = e * [ l _ { < r 2 > q 2 + . . . ] . e
(7)
ELECTRON SCATTERING AND EFFECTIVE CHARGE
285
Evidently f(0) = e*/e and if the effective charge were a point charge, f(q2) would be independent of q~-. If the odd particle is a proton, we must include in p(y) the spatial distribution of the intrinsic proton charge ~s). 2.2. WEAK COUPLING TO SURFACE OSCILLATIONS We next ask for the transition multipole operator in the case where the odd nucleon is weakly coupled to surface oscillations of the core. If the nuclear potential of the core is assumed to have the same shape as the core, the interaction between the odd particle and the core is
H' = - aVo
0 Z Q," r,, l
(8)
where a is the equilibrium radius of the core, Vo is the depth of the potential and Q~ • Y~is the tensor scalar product of Y,,,(t2,) referring to the particle and Q,. = (-1)mq,_. = ( - 1)r" ( ~ f
r [ a , _ ' + ( - 1)a , .+] "
referring to the core 16). We now write our states as [jLJM), where j represents the particle quantum numbers, L those of the surface vibration quanta, and J is the total angular momentum. If H' is weak enough to be treated as a perturbation, we can write for our initial and final states
~],m~ = [jiOji mi) + ~ [j"Ljl mi)(jaLji milH'lJiOjt mi), [%, + hCOL] ~[Jjfmf ~
]jfOjfmf)+ ~ ]jnLjfmf)(jnLjfmflH'ljfOjfmf) jaL
(9)
8 j r - - [ S J a "/ff hOJL]
We have used the fact that H ' is a scalar under rotations and that it can only create one surface oscillation quantum from the vacuum. The multipole moments of the charge distribution are 3Z . "~JM(q) = G Jj(qa)qzM-b ½(1 -[-'r3)jz(qXp) Yj~(~p),
(10)
where the first term is from the core 16) and the second term is from the odd particle. Now taking matrix elements of this operator between the states given above (to first order in tt') and noting the selection rules on the matrix dements involved we find
(Ttdflld[s(q)]lT j,) = (jfOjf[[,..,~'jl[jiOji) + (jfOjfii'gs[ljfJji)(jfJjimilH']jiOjimi) 8ji -- ejf-- h~s q- (jr 0jr mdH'lj i Jjf mf)(j i Jjfll,/gjl[ji 0ji ) % - % - hoj s
(11)
~86
$. D. WALECKA AND R. S. WILLEY
Making use o f the rules for the matrix elements of tensor operators in coupled schemes ~7) we find that (~j~ IIdt's(q)ll wj~) = (Jr] Id[)'P'(q) + ~')'P'(q)ll Ji),
(12)
where ~'~'~t" + ~'~'~/ = ½(1 + za)js(qxp)YsM(12p)
(3z)(,,~) + Vo -~
2
2h~o~
.
,~ (ncoj),2----7"- ( e j f - e i l ,2 ) Js(qa)ab(xv- a) YjM(~2p)
(13)
and the matrix element is taken between the singleparticle states t. Thus mixing in the collective states and allowing for the collective transitions effectively modifies the single particle operator. Since the entire angular dependence of the operator is contained in YsM(Op), we have all the selection rules of single particle transitions (/-forbiddenhess for example) even when the transition is really taking place by another mechanism. If we concentrate on odd neutrons, the form factor is characteristicallyjs(qa) and the effective charge is proportional to [(hcos) 2 - (ejf-ejf)2] - 1 so that it will depend on the process, being in general somewhat less for static moments than for transitions. (We have tacitly assumed hoga > lejf-e~,l in making a perturbation expansion in H'.) The odd nucleon likes to pull the surface along with it and hence acquires a charge by the polarization of the core. The form factor for transitions is then characteristically the form factor for core transitions. 2.3. COUPLING TO CORE OF ELLIPSOIDAL DEFORMATION A trivial extension o f the above calculation is the case when the odd particle is coupled to a core which has a small permanent ellipsoidal deformation. The nuclear potential is t h e n
V~p,(rOc~) = - VoO[a(1 + 2 E ~2'O(--~--fl--~)Y2M'(O, ~b))--r]
(14)
M'
O(x) = {1, O,
x > 0, x
The core-particle perturbation is therefore
U' = -- 21Io aft(r-- a ) ~ .
Y2,
(15)
where 2 is the deformation parameter and ~ o2 • II2 is the scalar tensor product o f (-1)M~o(-a-fl-7) referring to the core and Y2u'(0q0 referring to the particle. The unperturbed states are again labelled by [jLJM) where L refers to the rotational t The observation that the result of a configuration mixing calculation such as this can be cast in the form of a modified single particle operator is due to de-Shalit 7).
ELECTRON SCATTERING AND EFFECTIVE CHARGE
287
state of the deformed core. In this case the calculation yields eq. (12) with ½(1 +
+6j2
,3)jj(qxp) rj (Op)
Vo \ 4 r c / \ 5 ] [3h2~2_(ei_sj,)2J2(qa)a6(Xp-a)Y2M(g2P)
"
(16)
The only difference from the previous case is that there is only an extra term for J = 2 if we have ellipsoidal deformations. The energy of the core is given by [h2L(L + 1)]/2I where I is the moment of inertia and we have again made the assumption that the deformation is small enough so that 3h2/I > [eJf-zji[. 2.4. THE DE-SHALIT C A L C U L A T I O N F O R T H E MULTIPOLE O P E R A T O R
Finally, we discuss the de-Shalit calculation for the multipole operator 7) (we shall actually oversimplify things a little for convenience). We restrict the discussion to single odd neutrons outside of closed proton shells (noloJo). We assume a residual neutron-proton interaction of the Wigner type Vnp = -- Z K
fK(rnrp)CK(n)" CK(P),
(17)
[Ckq = ~/4zc/(2k+ 1 ) Y j . Using the odd-group modelwe write the states as ]jndpJM) where j~ refers to the neutron and Jp to the protons. We now start with the states [L0jimj), [jfOjfmO and mix in all the states reached by treating g = Z Vnp np
(18)
as a perturbation. The transition multipole operator is that of the protons
~ s u = Z Js(qxi)YsM(Oi) • protons
The calculation proceeds just as in subsects. 2.2 and 2.3. The result can again be written in the form of eq. (12) with -s.. _
I
~ s M - 2J +-~1
Z
2~p
nlj>nolojo 8p x (no loJolljj(qx)Yj(Ox)llnlj)(no loJollfj(r,y)Cj(f2y)llnlj),
(19)
where 8p is the proton excitation energy to the higher shell (nlj). (Again note that because &the Cj~(On), all the single particle selection rules hold.) We cannot immediately read off the q dependence because the admixture of states depends on thefs (or on the force). We can, however, say that the q dependence is that characteristic of a
proton transition to a higher shell.
288
$. D. WALECKA AND R. S. WILLEY
3. Discussion of the Calculations Knowing the operator dt'~/~/(q) we immediately obtain the transition probability for real ? emission, if the transitions are electric, (_Dfi ~--- 8~0~C
(-2 J-+ 1)!!
~
[(Jf[[Qs[[Ji)]
(20) QSM = Lim ( 2 J + 1)[!
~o
qS
dts~(q).
We see, therefore, that according to the four models discussed the odd particle acquires an effective charge for electric emission, the value of the charge depending on the model. Each model, however, predicts a different q dependence of the inelastic electron scattering cross section. In the weak coupling to the collective motion of the core, the q dependence is given exactly byjs(qa ) where a is the equilibrium radius of the core. In the fourth model, where proton states are admixed by a neutron-proton interaction, we see that the q dependence is that characteristic of a proton transition between different proton shells. In the polarization model, the q dependence is that of the single particle operator js(qx)Ysu(f2x) taken between the single particle states and then multiplied by the form factor of the effective charge. In these last two models one must, of course, know the radial wave functions well before the q dependence can be calculated t. 4. Numerical Example In order to get an estimate of the magnitude of the effects discussed above we have carried out a numerical calculation for 017. The shell model is expected to have validity here 4) and the configuration of the ground state should be 016 + ld~ neutron. The first excited state is thought to be 016 +2s½ neutron, as indicated in fig. 1. The mean life of the first excited state is zo) ¥ = 2.55 x 10 - l ° see. Assuming harmonic oscillator radial wave functions with an oscillator parameter ,t 1
/~
h
-
- 3.2 fm 2
Mo~o
and using a single particle operator
Q2M = e*x2y2M(Ox),
(21)
t H. Kendall and L Talmi la) have tried to fit the data on scattering from Vzl. They work with the reduced 7 ray transition probabilities (eq. (20)) obtained by extrapolating the cross section to zero momentum transfer. The different models discussed above give different values for these transition probabilities (and actually slightly different procedures for extracting them from the data). Our proposal is really complementary to this, as we advocate looking at the shape of the cross sections themselves to distinguish between the models. *t This result is obtained by interpolating the results lo, z2) of elastic electron scattering from Oin to 0 is.
ELECTRON
SCATTERING
AND
289
EFFECTIVE CHARGE
A ~
LEVEL SCHEME FOR MeV 3.06
I/2-
u
O. 87 "{'=2"55x10
0
-io
secI/2+
II
6 512+
Fig. 1. Level scheme for 80~ 1T. T h e data are taken f r o m refs. ~0,~1).
.io-~0
L
I
1
I
~
I
*co= 197 MeV
Lu--
~
Io--0.51el
--
\ I0"" (0 °}
I g (fff~')l~
P 0.2 0.4 (11-5"} (2:5.1°)
0.6 0.8 [55.0"} (47.2")
1
l
1.0
1.2 (T3.9")
[60 °)
1.4 (89 °}
Fig. 2. Inelastic electron scattering cross section to the first excited state of e l l T h e incident electron energy is 197 MeV. The C o u l o m b cross section (d
--1.91(eh/2Mc).
I
I
I
I
Eo= 197 MeV
0"87T2$ o-_=
T io-_==
~m
~-,, (0'=l
I
~
r q(fr~)7
,
0.2 0.4 0.6 0.8 (11.5") (23.1") (35.0")(47.2")
f
I
1.0 1.2 1.4 {60 =) (73.9")(89*)
Fig. 3. Inelastic electron scattering cross section for a single p r o t o n o f charge e m a k i n g a ld i -+ 2s½ transition. T h e incident electron energy is 197 MoV. T h e c o n t r i b u t i o n o f t h e transverse multipoles (daE2, daM 3) was c o m p u t e d for a charge e a n d m o m e n t / ~ p = +2.79(eti/2Mc).
I
1
I
I
I
I
I
Eo - 197 MeV
G87
2SII=
0 " ~ - - " Ids/= Coulomb Cross Section
io-_=
I \,x
~X 2p, Ip protont m n ~ t ~ n ~ "vocuum p o l o r i z a t i o l l ~ \ \ \
io-_~
io"* {0°}
I
0.2 04 0.6 0.8 (11.5") (23,1") (35.0"} H72")
1.0
1.2
1.4
(600)
(73.9 °)
(89")
Fig. 4. T h e C o u l o m b cross section for inelastic electron scattering to t h e first excited state of O z7 c o m p u t e d for various m o d e l s o f t h e effective charge. T h e m o d e l s a n d p a r a m e t e r s u s e d are discussed in sect. 4 o f the text.
ELECTRON SCATTERING AND EFFECTIVE CHARGE
291
one finds
The q u a ~ u p o l e moment of the ground state is Q / e = - 2 . 7 1 fm 2 and a similar calculation gives = 0.42.
--
e
Q
(Note that this is consistent with our assertion that e* should be less for static moments.) We now turn to the cross section for inelastic electron scattering. In fig. 2, we show the cross section for scattering to the s~ level assuming that the extra neutron has a point e* = 0.51e and including the contributions of the allowed transverse multipoles E2 and M3. (We have taken an electron energy of 197 MeV [K = lfm -1 ] and have used e = 0 for the transverse multipoles.) For purposes of comparison, we have plotted in fig. 3 the cross sections calculated for a single proton of charge e making the ld t --* 2s½ transition. We see that if one stays at angles < 75 °, the cross section is dominated by the Coulomb scattering. We have calculated the Coulomb scattering cross section using harmonic oscillator wave functions and the value of fl given above for the various models we have discussed and the results are given in fig. 4. The curves have all been normalized to fit the experimental lifetime (or equivalently, to give the same e*). For the polarization model, curve I, we have usedf(q 2) = e -*~2~" with b = 1.5 fin, a reasonable correlation distance. In curve II, the result of coupling to the collective modes of the core, we have taken a = 3.55 fm. For the results o f t b e de-Shalit calculation (eq. (19)) we have plotted two simple extremes. In the first we assume the force is such that the protons only make a l p ~ 2p transition, and in the second, only a l p ~ i f transition. The curves are again normalized to give the same e*. The result is that there are significant differences in the shapes of the various curves. These differences are greater than the (calculable) background contribution of the transverse photons as can be seen from the figures. Since the cross sections are large enough to be seen, it would be interesting to have some experimental data bearing on this question.
References
1) 2) 3) 4) 5) 6) 7) 8)
J. H. Fregeau and R. Hofstadter, Phys. Rev. 99 (1955) 1503 H. Crannel, R. Helm, H. Kendall, J. Oeser and M. Yearian, Phys. Rev. 123 (1960) 923 W. C. Barber, F. Berthold, G. Fricke and F. E. Gudden, Phys. Key. 120 (1960) 2081 I. Talmi and I. Unna, Ann. Rev. Nucl. Sci. 10 (1960) 353 R. Blieden and L. Wilets, Bull. Am. Phys. Soc. 5 (1958) 244 J. P. Elliot and A. M. Lane, Handbuch der Physik (Springer-Verlag, Berlin, 1957) p. 364 A. de-Shalit, Phys. Rev. 113 (1958) 547 G. Barton, D. M. Brink and L M. Delves, Nuclear Physics 14 (1959) 256
9.9~ 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22)
J. D. WALECKAAND R. S. WILLEY R. Hofstadter, Rev. Mod. Phys. 28 (1956) 214 L. R. B. Elton, Nuclear sizes (Oxford University Press, London, 1961) L. I. Schiff, Phys. Rev. 96 (1954) 765 G. Morpttrgu, Nuovo Cim. 3 (1956) 430 K. Aider, A. Bohr, T. Huns, B. Mottelson and A. Winther, Rev. Mod. Phys. 28 (1956) 475 L. J. Tassie, Nuovo Cim. 5 (1957) 1 4 9 7 K. W. McVoy and L. Van Hove, Phys. Roy. 125 (1962) 1034 J. D. Walecka, Phys. Rev. 126 (1962) 653 A. R. Edmonds, Angular momentum in quantum mechanics (Princeton University Press, Princeton, New Jersey, 1957) R. S. Willey, to be published H. Kendall and I. Talmi, to be published F. Ajzenberg-Selove and T. Lauritsen, Nuclear Physics 11 (1959) 1 Nuclear spectroscopy, Part B., ed. by F. Ajzenberg-Selove (Academic Press, New York, 1960) F. Lacoste and G. R. Bishop, Nuclear Physics 26 (1961) 511