- Email: [email protected]
On radii of neutron distributions in nuclei
]
Nuclear Physics A306 (1978) 3 4 3 - 3 5 9 ; (~) North-HollamtPublishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
ON RADII OF NEUTRON DISTRIBUTIONS IN NUCLEI * GIRISH K. V A R M A
Seritt Physics Laharatorv, Rutgers Uidrersit)', Frelinghysen Road, Piscataway, New Jersey 08854, USA and LARRY Z A M I C K
Joseph Hem3' Lahoratorie.~', Princeton Universio', Jadwin Hall, Prhweton, New Jersey 08540, USA and Serin Physics Lahoratoo', Rutgers University, Frelin.qh)'sen Road, Piscataway, New Jersey 08854, USA *+ Received 14 March 1978 Abstract: We consider the analyses of the differences between rms radii (A = r - rD) of neutron and proton distributions in a wide variety of nuclei. We note that apart from its own intrinsic interest, the quantity zl is of importance for isotope shifts, core polarization contributions to the Coulomb energy differences of mirror pairs (Nolen-Schiffer anomaly) and the renormalization of the effective interaction. For example, if A were very small in 48Ca then the Nolen-Schiffer anomaly could be explained by a core polarization mechanism. We consider critically the various methods of determining A and conclude that at present probably the most reliable method is high energy ( ~ 1 GeV) proton-nucleus scattering. The different theoretical analyses based upon, e.g., the multiple diffraction theory (where Glauber amplitude is the leading term) or the optical potential (KMT) formalisms appear to be converging to essentially the same answer when analyzing the same data. High energy 0t-particles and medium energy pions can also become useful sources of information if higher order optical potentials are treated with care. We find that A is rather large in 48Ca, i.e. there is a neutron skin, so that the Nolen-Schiffer anomaly cannot be explained by a core polarization mechanism. The results of high energy proton-nucleus scattering are in excellent agreement with current density dependent Hartree-Fock calculations.
1. Introduction Among the most fundamental quantities in nuclear physics are the radii of nucleondensity distributions in nuclei. Most of the existing information on the subject comes from electron scattering and/~-mesic X-ray measurements which have yielded accurate radii for the proton distributions. At the same time, there have also been extensive theoretical efforts to understand nuclear structure in terms of nucleon-nucleon interactions. However, some of the best Hartree-Fock (HF) calculations at present employ density dependent effective interactions which are ultimately adjusted to fit charge densities as well as the binding energies and the single-particle energies of the closed shell nuclei. The HF results for radii of neutron distributions, on the other hand, are essentially predictions which can provide a test of the underlying theoretical approximations involved in such calculations. Unfortunately, the experimental information t Supported by the National Science Foundation. ** Permanent address. 343
344
G. K. VARMA A N D L. ZAMICK
on neutron radii comes mostly from the scattering of strongly interacting projectiles; the analyses are complicated and have often yielded conflicting results (see, for example, ref. 1)). However, high energy (~ 1 GeV) proton-nucleus scattering, if carefully analyzed, now appears to provide an accurate method for obtaining the rms radii (and other moments) of nuclear neutron density distributions. Other methods such as study of ~± elastic scattering angular distributions and high energy s-particle scattering may also prove to be useful in the near future. Whether the difference A = r , - r p between the radii of neutron and proton distributions in nuclei such as 4SCa and 2°spb is very small ~ 0.06 fm (as the Coulomb energy differences assuming no charge symmetry breaking appear to suggest) or is large ~ 0.2 fm as predicted by H F calculations, has important consequences. One implication of a small A in 4SCa is to have strong monopole core polarization. In that case the valence neutrons will push in the core neutrons and, in mirror nuclei, valence protons will shrink the core protons leading to an increase in the electrostatic energy. This would remove the so-called Nolen-Schiffer anomaly 3) in mirror pairs such as 41Sc-41Ca (this anomaly refers to a ~ 4 % - 1 0 % disagreement between experimental Coulomb displacement energies and theoretically calculated electromagnetic energy differences). This fact has led to considerable enthusiasm in reviving the idea of large core polarization 3). On the other hand, if A is large (as the high energy proton scattering results indicate), then the above core polarization mechanisms will be too small to explain the anomaly. In this paper we first consider, in sect. 2 the implications of large core polarization; it is shown that they lead to difficulties with quantities such as the renormalized effective interactions between identical nucleons. In sect. 3 we examine the values of A for closed shell nuclei extracted from proton scattering measurements at high energies where microscopic scattering theories are reliable. The results are in close agreement with current H F calculations and are consistent with almost no core polarization. Evidence from other strongly interacting projectiles is considered in sect. 4 and we summarize our results in sect. 5.
2. Possible implications of the magnitude of A Let us consider the change in mean square radius of a core when a valence neutron is added. In perturbation theory this is written as 2
2
or" • (j~FV - [ j v ( p h - i )Mr,,=o]~V)<(ph - 1)or"~r2½(1--Zz)O). phT"
(1)
~"-" ph
It is both more elegant and more convenient to insert as intermediate states only the monopole states IMT") = ~ <0r2z(ph - 1)or">(ph- ,)or" (~ i12)~ , (2) ph
NEUTRON
DISTRIBUTIONS
345
where z = 1 for T' = 0 and z = Zz for T' = 1. We note that the expression involves the following vertices:
V x = (j~ AET,,=o [j~M~°o]Jv), (3) V
Y = (Jr dEr"=l fJvM~"~==°]Jv). We insert for AE r'' the mean energies of the monopole states (for'example, from current somewhat imprecise empirical knowledge
AE ~ -2hto,
AE 1 ~ -4htn.
These vertices x and y also occur in other expressions. If we further assume harmonic oscillator radial wave functions with the oscillator length parameter b = ~/h/mo9 we find ( M ° ~ r 2 0 ) = x/~b 2, where X = sum ( 2 n + l + ~ ) over occupied states, e.g. _r = 36 for 160 and X = 120 for 4°Ca. We now summarize the dependence of various quantitites on x and y. (a) The change in mean square charge radius due to the addition of a valence neutron or proton, ti~.2 (neutron~ = "ch~,proton ;'
1
-
2 x/,Y, b2(x T- y).
(5)
(b) The Auerbach-Kahana-Weneser (AKW) diagram 4),
-4y(OVcMl).
(6)
The calculation of (OVc M1) is given in the appendix. This diagram was proposed by the authors as a possible explanation of the Nolen-Schiffer anomaly. It may be described as follows: the Coulomb interaction introduces an isospin impurity into the core. The nuclear interaction of a valence proton with this isospin impure core is different from that of a neutron. A priori, such a mechanism is quite a reasonable thing to consider. It is somewhat analogous to a calculation of an effective charge. These effective charges are known to be large. However, recent density dependent H F calculations by Negele, Vautherin and Brink and others have obtained very small values for this diagram. (c) The exchange of a monopole phonon between two neutrons (e.g. 42Ca = 4°Ca + 2f~ neutrons), tiEr= 1 = _ 2x2IAEO I_ 2y21AEll.
(7)
This is Of interest when considering the mass formula for Ca (and other isotopes) introduced by Talmi 5), E[Ca(40 + n)] = nC + [½n]fl + ½n(n- 1)~.
(8)
346
G. K. VARMA A N D L. ZAMICK
The empirical values for the Ca isotopes are: C = - 8.38,
fl = - 3.33,
e = 0.23.
The quantity 6E T= 1 can be identified as 6~, the change in ~ due to the exchange of a monopole phonon between two valence neutrons. Recall that using the bare G-matrix elements of all standard interactions, e.g. Kuo-Brown matrix elements 6) derived from the Hamada-Johnston interaction, the resulting ebare is negative. The exchange of a quadrupole bubble does yield a positive correction to e, and indeed, this quadrupole exchange may be the correct physical explanation of the repulsive pairing interaction. The AKW term can be written as 4a[AE r=lly where a 2 is the probability that isovector monopole state is mixed in the ground state from HF calculations. The value of a is about -0.08 for 4°Ca (i.e. ground-state impurity of 0.64 ~o). We now consider the general condition on the isotope shift. Let rp(40), rp(48), rn(40 ) and rn(48) be the rms radii of respectively, protons in 4°Ca and 48Ca and neutrons in 4°Ca and 4aCa. In first-order perturbation theory the condition on the neutrons reads
~/Z (x + y)b 2 = - I(2n + l + 3)- ~ l bZ + ~-[r~(40)- rZ.(40)] (10) + -~[r.Z(48) - r~(48)] + ~[%z(48)- r~(40]. The condition on the protons is
V ~ ( x - y)b 2 = -~[r~(48)- r~(40)].
(11)
To proceed further we need rpZ(40)- r,Z(40). Because the Coulomb interaction pushes out the protons this quantity should be positive. See the appendix for a discussion. We take rp(40)-r,(40) = 0.05 fm. This is consistent both with the calculations in the appendix, the H F calculations and the analysis of high energy proton-nucleus scattering. We also take rp(48) ,~ rp(40). This is also consistent with experiment. Recall that for a long time it was thought that %(48) was slightly smaller than %(40). But effects introduced by Bertozzi et al. v) reverse the situation. For our purposes, the assumption o f equality is more than adequate. We consider the different values of r.(48) and see what happens to 6e. Case 1: r.(48)- rp(48) = 0. This is an extreme case, but there is a belief among some people that r, = rp everywhere. Also, within error bars, this is consistent with 79 MeV a-scattering result. We find x = -0.057,
y = -0.057,
fie = -0.41 MeV, (12)
AKW diagram = + 824 keV.
NEUTRON DISTRIBUTIONS
347
We see that ~ is quite large and negative. This goes very strongly in the wrong way, and it is doubtful if there is any other mechanism (i.e. stronger quadrupole exchange 8)) that can compensate for this. On the other hand, the AKW diagram 4) does account for 824 keV, more than enough to explain the Nolen-Shiffer anomaly 2). Case 2: r,(48)-rp(48) = 0.2 fm. This is what proton scattering gives. We find x=0,
y=0,
fi~=0,
(13)
AKW diagram = 0. Now fi~ is negligibly small and the monopole exchange mechanism causes no trouble. But the AKW diagram 4) is also zero, so there is no explanation of the NolenSchiffer 2) anomaly. We thus see that a mechanism which helps to solve the Nolen-Schiffer anomaly 2), causes trouble with the repulsive pairing between identical particles. We feel therefore that at least the extreme possibility r,(48)- rp(48) = 0 can be ruled out. This argument is not so strong that one can rule out r,(48)-rp(48) ~> 0.1 fm, however.
104
, ~\
I I Proton Scattering at 1.05GeV
/
103 \i 102
48Ca
\\\~]"
~, ~.
c-~
\\
A = 0.19 fm
"~"
-
Io ~
,o o
A=I0 -I
10 .2
~
i
I
5
I0
15
C)C.M (deg)
20
Fig. 1. Elastic scattering of 1.05 GeV protons from ,o. ,8Ca" The data are from ref. 2 ~) and the theoretical curves are based upon the Glauber approximation including spin and correlation effects.
348
G. K. V A R M A A N D L, Z A M I C K
3. Neutron radii from high energy proton scattering Perhaps the most reliable tool at present for probing neutron density distributions is scattering of high energy protons from nuclei. The scattering data can be analyzed in terms of the multiple diffraction theory of Glauber 9) and its extensions 1o, 1~) or the optical potential formalism of K M T t 2) and its variants t 3). For example, the leading corrections 10) to the Glauber theory arising from non-eikonal propagation, from corrections to the frozen target approximation and from the kinematic transformations involved in relating bound nucleon t-matrices to free t-matrices, have been found to have little effect 11) on the extraction of neutron radii near 1 GeV. Wave function corrections such as those arising out of Pauli, c.m. and dynamical shortrange correlations as well as isospin effects have also been examined ~4). While detailed shapes of extracted densities are sensitive to some of these higher order effects, the rms radii do not change much under these improvements 15). 0 i05
5
I0
15
2O
I
I
r
I
,
Elastic
Scattering -
i04
-
103
_
102
-
i01
1.05 GeV
i05
A = 0.21 fm
\
¢o
..o E
-i05
p -2°spb
'~/-\
104
-!
i0 z 0.8GeV t~ = 0.21 fm
I0 i
I0 °
I0o
.
0.8GeV
t0 -i
1
0 0
i0 -I
0.3 P
iO-e
5
3 I0
~ 15
20
25
30
E~:.M (deg) Fig. 2. Proton-2°Spb elastic scattering and polarization. The data are from refs. 50. si) and the theoretical curves are based upon the Glauber approximation including spin and correlation effects. Note the different scales for 0.8 and 1.05 GeV,
N E U T R O N DISTRIBUTIONS
349
In figs. I and 2, we show typical fits to the high energy data on scattering of protons from 4°'4SCa and 2°spb. The theoretical curves are based on the Glauber theory and include spin dependent effects as well as the correlation effects described above. (The detailed expressions for including these effects are given in refs. 14, 16).) We should emphasize that in such microscopic calculations there are no free parameters (once the neutron and proton densities, and nucleon-nucleon (NN) scattering amplitudes are specified). In the above analyses the input NN amplitudes were taken from NN measurements, input proton densities were taken to be of the SOG forms 17) (determined from model independent analyses of electron scattering data) and neutron densities were obtained by fitting the data. One cannot hope to determine the densities in the interior of medium and heavy nuclei from proton scattering but both the
Nucleus
A = r, = % (fm) from proton scattering (GeV) .... 1.05 1.0 0.8
DDHF
DME, SKII
CS
~.0 b)
--0.02
--0.03
--0.02
160
- 0 . 0 2 a) --0.04 ')
4°Ca
0.04 4) . - 0 . 0 3 a) - 0 . 0 3 ')
- 0 . 0 7 b)
--0.04
tO.04
--0.05
4SCa
0.19 a) 0.16 a) 0.17 ')
0.21 b)
0.23
0.18
0.18
- -
9°Zr 20ap b
0.21 ~) 0.21 c)
0.13 ~)
0.08 4)
0.12
0.08
0.07
0.08 b)
0.21 n) 0.19 c)
0.20
0.20
0.21
As discussed in text, the analyses of refs. 18, 19) neglect all correlation effects. Ref. 16) includes c.m. correlations whereas ref. is) includes both c.m. and Pauli correlations. The typical error bars in A from proton scattering are ~ +0.05 fm. a) Ref. ,s). b) Ref. 16). ') Ref. ,9). d) Ref. is).
In table 1 we list the results for A = rn - rp for closed shell nuclei from the available theoretical analyses of the proton data t. As mentioned above, the analysis of ref. 15) includes spin dependent effects as well as all the correlation effects which contribute t The 1.00 GeV Leningrad data was analyzed earlier in ref. 16). The analysis of 2°SPb data in this paper was approximate in that an average matter distribution was extracted. The slightly different result given in table I for ref. 16) is due to an improved calculation using input SOG proton densities and treating the Pauli correlations.
350
G.K. VARMA AND L. ZAMICK
significantly. The analyses of refs. 18, 19) employ a first-order K M T potential; they include spin effects but neglect all correlation effects. Other analyses which also neglect spin effects are not listed in the table. [We, however, do not wish to imply that such analyses are meaningless. Some of the spin independent analyses employ effective N N amplitudes which mock up part of the spin effects in a restricted angular range. Thus, for example, Ahmed 20) obtains A = 0.24 fm for 48Ca and A = 0.26 fm for 2°8pb whereas Alkhazov e t al. 21) obtain A = 0.19 fm for 48Ca; these results are consistent with those from more sophisticated analysis 15).] From table 1 we notice that the results from different analyses are consistent with each other (within the error bars), and are also in excellent agreement with the Hartree-Fock calculations employing density dependent interactions 22 - 25). In 48Ca, as we shall see in sect. 5, they also appear to be consistent with a naive shell-model picture in which the change in radius is due solely to the addition of f~ neutrons. The smaller value of A in 2°8pb from 1.00 GeV data is due to disagreement of this data (0 < 15°) with the 1.05 GeV Saclay data (0 < 19°). The analyses of recent 0.8 GeV L A M P F data (0 ~< 30°) favour the 1.05 GeV data. We believe the 0.8 GeV analysis to be the most reliable one because the accompanying p-E°Spb polarization data are also well fitted by the theory and thus remove the small uncertainties associated with the spin parts of the N N amplitudes. For comparison, it should be noted that in a naive calculation in which one assumes the same oscillator parameter for the neutrons and protons the value of A in 2°8pb would be 0.4 fm as compared with the proton result of 0.2 fm. The smaller value of A is due in part to the fact that the Coulomb interaction pushes out the protons, and in part is due to the fact that the symmetry energy favors near equality of neutron and proton orbits. A rough calculation indicates that at least half of the effect comes from the Coulomb interaction pushing out the protons. This is shown in the appendix.
4. Neutron radii from other scattering processes In table 2 we list the values of A for 48Ca and 2°8pb from all available analyses 26- 3 8) (with respectable error bars) of scattering measurements. The pion reaction and total cross-section measurements appear to yield values of A which are generally small whereas detailed 130 MeV angular distribution measurements appear to be consistent with a large value of A. The analysis 36) of 10 ~ 0 GeV/c pion reaction cross sections ignores, for example, the existence of inelastic intermediate states. Such effects are large even in deuterium 41) and become larger in heavier nuclei. Since the neutron radii in this case are being extracted by fitting one number, this can lead to serious errors t. Similarly, at lower energies there are many higher order effects. Even effects such as double charge exchange and double spin flip are important 42) and will Similar analysis of proton reactioncross sectionsat 700 MeV lB. D. Anderson et al., Bull. Am. Phys. Soc. 22 (1977) 1005] leads to a matter radius differencebetween 48Ca and 4°Ca of 0.44 fm which is twice as large as the results from proton angular distributions.
N E U T R O N DISTRIBUTIONS
351
TABLE 2 The results for A = r . - rp in 4SCa and 2°8Pb from the available analyses of scattering measurements Nucleus
f 4SCa
A = r,-%
Ref.
0.19+0.05 0.21 -{-0.05 0.394-0.10
16) 26)
MeV MeV GeV
0.03+0.08 0.38 _+O. 12 0.20_+0.06
27) 2s) 29)
MeV
0.08_+0.05
32)
a)
33)
1.05 1.00 800 30
GeV GeV MeV MeV
0.21 _+0.05 0.08+0.05 0.21 -+ 0.05 0.36+0.20
l~) 16) ~5) 26)
104
MeV
f0.26+0.13
3,*)
104
MeV
140 166
MeV MeV
~.0.30_+0.07 0.0 -+0.07 0.42 _+0.20 0.25 ± 0.10
35) 36) 34)
Method
p elastic scattering
elastic scattering
n+ total cross sections
1.05 GeV 1.0 GeV 10.8-16.3 MeV 79 166 1.37 9C~240
nucleon-transfer reactions
2O8pb
" p elastic scattering
ct- elastic scattering
n - reaction cross sections 7r± reaction cross sections sub-Coulomb pick-up nucleon-transfer reactions
2~60 1-2
GeV GeV
0.0 -+0.1 0.0 +0.1 0.20 a)
15)
2s) 39) 39) 40)
33)
For comparison Coulomb energy differences imply A = 0.06 (0.07) fm for 4SCa (2°Spb). a) Consistent with D D H F 22).
occur through a second-order optical potential. True pion absorption (annihilation) also appears to have 4-3) significant effect on the cross sections. It is not obvious that these effects cancel even if ratios of reaction or total cross sections are used. Clearly, measurements involving detailed angular distributions of pions may provide much more reliable information. Recently, differential cross sections for elastic scattering of 130 MeV rc~: from 4o, 4SCa have been measured at SIN 30). A preliminary analysis of this data by Bethe and Johnson 31) indicates that the results for A are smaller than the predictions of DDHF. However, detailed analyses which include effects of higher order optical potentials, isolate the rather large Coulomb effects and, furthermore, treat the first-order pote'ntial accurately, must be carried out before the results become reliable. Different results obtained from lower energy or-particles are in conflict. These analyses are generally semiphenomenological. The real part of the potential is related to the nuclear density through a folding model while the imaginary potential is treated phenomenologically. In 4SCa the 166 MeV data 28) gave A = 0.38 ___0.12 fm but 79 MeV data 27) gave A = 0.03 --+0.08 fm. In the case of 2°apb, Bernstein et al. 34)
352
G . K . V A R M A A N D L. Z A M I C K
found A = 0.26_ 0.13 fm and Tatischeffet al. za) at 166 MeV obtained A = 0.25 +_0.10 fm. An analysis 35) of recent 104 MeV a-2°Spb data gave A -- 0.3_+0.07 fm but a subsequent analysis a6) of the same data using a slightly different imaginary potential (resulting in a somewhat lower chi square) led to A = 0_+0.1 fm. [The effects of different imaginary potentials on A are discussed inref. 3s).]If the imaginary potential is taken to be proportional to the real potential, as suggested in earlier works 34. aT), the resulting value of A is 0.32 fm, ref. 3s)]. That small changes in a completely phenomenological imaginary potential can cause such dramatic changes in extracted neutron radii is clearly worrisome and casts doubts on the reliability of such analyses in determining nuclear radii * As the energy goes up, however, the a-nucleus potential can be derived microscopically from either the Glauber or KMT theories. The "rigid projectile approximation" employed in ref. 29) to analyze the 1.37 GeV ~-4SCa data has been shown to be reasonable at small angles 44). The extracted value A = 0.20_+0.06 is consistent with the high energy proton scattering and HF results. A more careful analysis of this data should also treat the correlation effects 4s). One of the least ambiguous measurements of the spatial distribution of excess neutrons is provided by sub-Coulomb pick-up from 2°apb. A careful analysis 40) which treats the effects of the non-local exchange potential shows the results to be in agreement with HF calculations (A = 0.20 fm). The results obtained from analyses 33) of nucleon transfer reactions in Ca isotopes and in 2°spb also appear to be consistent with HF results and disagree with those obtained from Coulomb displacement energies. We briefly mention magnetic scattering of electrons, for example, the recent work 46) on 51V. In the single-particle picture all the magnetization is carried by a valence f~ proton. Thus one should be able to learn something about the radius of the valence proton (as opposed to Coulomb scattering, which gives the charge radius of the entire nucleus). It is observed that at high q the form factor [FM(q)[2 is larger than what is predicted by density dependent HF theories. In the oscillator model FM(q) is some polynomial in q times exp(-¼q2b:). A way of increasing FM(q) is to make b smaller, i.e. decrease the radius of the valence orbit relative to the HF value. If correct, this would afford an alternative explanation of the Nolen-Schiffer anomaly, indeed one proposed by Nolen and Schiffer themselves. Obviously, if the valence nucleon has a smaller radius its electrostatic interaction with the core will be stronger. It is, however, premature to assign the observed slow fall-off in q to a small radius. One must first consider effects of core polarization and exchange currents. This has not yet been done. By looking at the fits to the magnetization one might think that the largest M2 dominates at high q. But in fact, for given configurations nl, n'l' in • It is encouraging that attention is now being paid to microscopic calculations of the imaginary potential. See, for example, N. Vinh Mau, Phys. Lett. 71B (1977) 5, and also B. Sinha, Phys. Rev. C l l (1975) 1546. More recent calculations 61) of :t-Pb data give A consistent with HF results.
NEUTRON DISTRIBUTIONS
353
the limit of very large q, all M2 have the same q-dependence. Thus the asymptotic q-dependence depends on the configuration, not on 2. The M2 operator has terms j~_ l(qr)[Y~_ la] a and j~+ l(qr)(Yz+ la] a (also a can be replaced by I). For the f~-f~ configuration the largest term is j6(qr)Y6(t2) which yields an asymptotic term for FM(q)which goes as (qb)~ exp (-1q2b2) in the oscillator model. However, from core polarization, in particular the term f(f-th), one can get an asymptotic q-dependence which is higher (qb) 8 exp ( - ¼q2bZ). It is interesting to note that from core polarization one can induce a term of the formja(qr)Ys(f2 ) coming from the configuration f(f- lh)8. However, to get such a term one requires a tensor interaction between two nucleons. This is because the initial configuration f has l = 3 and the final configuration f(f- l h)S has angular momentum 13+81 ~ 3. Calculations of the core polarization effects have been carried out by Arita sT). For the highest multipole (M7) the results are force dependent. For a Serber force there is significant quenching of the M7 form factor, but for a Rosenfeld force there is hardly any quenching. Exchange currents are important as well. Indeed, in a recent conference proceedings, Arima 5s) notes that for 170 the effect of the exchange currents for large q is to increase :the transverse form factor of the magnetization by a significant amount. But this is precisely the effect that one gets by pushing the valence neutron in. Hence in x70 at least, one might erroneously conclude that the radius of the valence orbit was too large, when in fact exchange currents were responsible for the deviation * Clearly, from the calculated sizes of the effects of exchange currents and the force dependence of core polarization corrections, it is doubtful that one can make precise statements about the radius of the valence orbit.
5. Summary and conclusions We have seen that high energy proton scattering appears to yield results for differences in neutron and proton distribution radii in closed shell nuclei which are generally in agreement with the Hartree-Fock calculations employing currently popular interactions. They are consistent with results obtained from sub-Coulomb pick-up and 1.37 GeV a-particles, but appear to be larger than preliminary 130 MeV pion elastic scattering results. Some low energy a-scattering results are consistent with proton and HF results while others are not. We believe high energy proton results to be reliable because the analyses employ theories which are completely microscopic and where higher order corrections have been calculated. Proton scattering can, of course, give information about shapes of neutron densities as well (at least near the surface). The correlation effects, however, have to be treated much more carefully 14). The rms radii, on the other hand, appear to be remarkably stable ** * Recent calculations by J. Dubach of exchange effects in 51V appear to resolve the bulk of the discrepancy with HF calculations. ** This is only true if effective NN amplitudes are used and the data covers a small angular range. If, for example, the spin-dependent parts of the NN amplitude and correlations are simply switched off', significant errors can occur.
354
G . K . V A R M A A N D L. Z A M I C K
under theoretical improvements. We have therefore concentrated on rms radii (the mean square radius also happens to be one of the best determined moments in proton scattering). More careful calculations in future should be able to test the details of neutron distributions predicted by Hartree-Fock calculations. In the calcium region high energy proton scattering results are consistent with typical Hartree Fock calculations. There seems to be, however, some confusion about what is the "normal" behavior of the neutron radius in going from 4°Ca to 48Ca. For example, Chaumeaux et al. 13) interpret the proton scattering results in 48Ca as an anomaly assuming that "normal shell-model behavior" would have A = 0.4 fm. We should first remark that the shell model per se makes no statement whatsoever about radial charges. It instructs us to put particles in a well, and that is all. Let us then consider the simplest model where the 4°Ca core is described by the harmonic oscillator model (the oscillator parameter should then be b = 1.98 fm to fit the charge radius). Then putting eight neutrons in the f~ shell leads to r,(48)-rn(40) = 0.24 fm. Since rp(48) ~ rp(40) Fref. 7)] and if we include the Coulomb repulsion in 4°Ca so that rp(40)- r,(40) = 0.05 fm (see the appendix) then we obtain A = r. - rp = 0.19 fm in 4 8 C a . The clearest implication of the,closeness of this result with high energy proton scattering is that there is almost no monopole core polarization in 48Ca and that a resolution of Nolen-Schiffer anomaly in terms of such monopole core polarization is not possible. Thus it would appear that it is premature to criticize current nuclear structure theories solely on the basis of Coulomb energy anomaly. The problem is more likely tied with the interpretation of Coulomb energy shifts. Indeed, numerous possible corrections have been suggested * among them charge asymmetry. An early calculation indicated 48) that a charge symmetry breaking (CSB) potential could explain the anomaly in the 3He-3H system but not in 41Sc-4~Ca. However, a recent analysis by Sato 49) indicates that the magnitude and systematics of the Nolen-Schiffer anomaly (after taking into account admixtures of core excitation corrections) can be explained by a CSB force which is not inconsistent with the NN data. Whether a CSB force is the final explanation for this anomaly or not, it seems fair to say that in view of the current theoretical and experimental ignorance regarding charge asymmetry in nuclear forces, the role of Coulomb energy results in testing Hartree-Fock theories is seriously limited. We lastly cite our recent work 59) in which we correlate the neutron radius of 48Ca with the charge radius of 5°Ti using charge symmetry considerations. This analysis leads to a neutron skin in 48Ca more or less in agreement with the current Hartree-Fock calculations and with our analysis of the proton-nucleus scattering. [After this manuscript was submitted, we received a preprint by Auger and Lombard 6o) who show that the 1 GeV proton scattering data are reasonably well described by using nucleon density distributions from Hartree-Fock calculations as input.] • Some of these corrections are discussed in detail in ref. 4~).
NEUTRON DISTRIBUTIONS
355
It is a pleasure to thank Dr. L. C. Liu and Dr. H. Sato for a useful discussion. We are extremely grateful to A. M. Lane for some clarifying remarks about Coulomb energies. One of us (L.Z.) would like to thank Frank Calaprice and Rubby Sherr for their hospitality.
Appendix A.I. ISOSPIN IMPURITY DUE TO THE COULOMB INTERACTION AND ITS IMPLICATIONS To obtain the ground-state isospin impurity in the ground state of a nucleus, it is an excellent approximation to replace the two-body Coulomb interaction by a onebody field. For a uniform charge distribution this potential (field) is
I Ze2 Vc(r) =
r>R
t 3Ze2( r'
(A.1)
r2)
It is, furthermore, a very good approximation to use only the value of Vc(r) for
r
for all r.
(A.2)
This immediately establishes a relationship between the Coulomb field and the square radius operator. The states which mix into the ground state, are for the most part the monopole states, consisting of particle-hole states coupled to J = 0. We define them thus: (a) isoscalar Mo =
Q ~ r2(i)[0)
[(~r2(i) Q ~, r2(i))]½,
(A.3)
Q ~, r2(i)zz(i)[O) []½"
(A.4)
(b) isovector
M1=
Alternatively, we can talk about proton and neutron states: (c) proton
Qr2(i)lO> M
1t
~
g
[(~ r2O~r2>] ½'
* We are grateful to Dr. A. M. Lane for this suggestion.
(a.5)
356
G. K. VARMA AND L. ZAMICK
(d) neutron
Qr2(i)lO> M~=
~ [<~r2Q
~
v
(A.6)
r2>] & •
v
In the above, Q projects out the intermediate states orthogonal to the ground state. If [0> is a shell-model ground state with harmonic oscillator radial wave functions then the following matrix element is of interest: < M ° ~ r 2 0 > = [-~ I(ph-')J:° ~ ",r20> I2] ½ ph
= ~ (n+ 1)(n+ l+~)]~b 2 = x/-Zb 2,
(A.7)
where b is the oscillator length parameter, and ~ is the sum over occupied states of (2n+ l + a) (where we use the convention n starts from zero). Likewise ( M ' ~ r2Zz0> = ~ b 2, ( M~ E r20> = ~ ~ 2~b=,
(A.8).
;¢
( M ~ ~ r20)
z~ 2 x/~b~.
=
v
The matrix element for isospin impurity in the ground state is
CI=(MI~Vc(O0>~~
1 - ~r2 ) O> =
2R aZe2(½~/Zb2).
(A.9)
Also of interest is C = = =
Ze2 ~ 2
~
x/Z=b~.
(A.10)
i
We define a 1 = C1/]AE11 where IAE1] is the energy of the isovector breathing mode state. We take IAEI[ ,,~ 4hco. We can write R 2 = -~Z~b=/Z.S 2 Hence a I - Ze2
~/~
(A.11)
4b (~Z=/Z)glAEXI" The quantity a ~ is the isospin impurity amplitude. For example in 4°Ca is we take rico = 10.37 MeV we get a 1 ~ 0.035 or 0.7 ~o admixture of T = 1 in the ground state of 4°Ca. Another estimate of the isospin impurity amplitude can be made, following the method of Lane and Mekjian s 2) in which one considers the energy weighted sum rule EWSR = ~ ( E , -
Eo)ll2.
n
We obtain the previous estimate for the matrix element (n ~ r2ZzO>and setting E , - E o = 2hco. The effect of the potential energy is to change EWSR by a factor
NEUTRON DISTRIBUTIONS
357
(1 + r ) and to raise ( E , - E o ) to E ~- E o = IAEll . Thus we get the change (m E r2zzo) = ~/~b2 (!1
+ r)2hto'] ½
/"
The quantity r was thought for a long time to be about 0.4 fm the dipole case. But recently there hive been claims, both experimental and theoretical that r is closer to 1 [refs. 53-56)]. Note that if we t a k e r = 1 and AE 2 = 4h~, we will get no change in (n ~ r2zz0) from what we had previously. Therefore, we will stick with our previous calculation. But one. should keep in mind, that until the issue of what x is has been resolved unambiguously, there will be some uncertainty in (nSr2ZzO) and hence in the isospin impurity amplitude C 1. A.2. DIFFERENCE IN PROTON RADIUS AND NEUTRON RADIUS DUE TO THE COULOMB INTERACTION
The repulsion due to protons tends to push them out. We will here estimate the amount. We define the state IM ° ) = NO( ~ rp2+ ~ r2)10) where N O is a normalization. In the oscillator model
N O = (,Y,,~b~+ ,Y,~b~)-½.
(A. 12)
We define the state [M 1) to be orthogonal to IM°): [M 1) = NO[(X~b4/X.b~)½ ~, rp-(,Y.~bJX~b~) 2 44 4~ ~ rE].
(A.13)
We assume that the mean energy of the states IM °) znd IM ~) are, respectively, IAE°I = 2ho9 and IAEll = 4h~o. The coupling of the shell model ground state to those states via the Coulomb interaction Vc = (3Ze2/2R)(1- r2/3R 2) is Ko
K~
-
(OVcM ° ) o 4 Ze2 / - N Z.b~ ~ - ~ / [AE°[. AE o
(0VcM~)
4. vb~) 4 ½ ~Z e 2 N 0 (I,.b~I,
/IAEll.
(A.i4)
The ground state wave function is ~ = c[0)+ d°[M °) + d I IM 1). In lowest order perturbation theory c = 1, d o = K ° and d 1 = K 1. The difference in mean square radius of the protons and neutrons due to this Coulomb push is given by A(rp2 - - r nz) -----
rp _
r.
= 2cdONO
+ 2cdlN°x/Xvb~,~,b](A/NZ) + terms in d °2 and d xz.
(A.15)
358
G. K. VARMA AND L. ZAMICK
In perturbation theory we get
A(rZp_r2n)=2(No)2Ze22R 3 ]Az°ll ( Z ; ~
Z~bN4)Z~b4+2(NO)2 Ze 2 J
2R 3
1
4
4 A
lAX11Z~b~Z~b~ N Z (A.16)
Using first order perturbation theory in 4°Ca with b, = b v = 1.95 fm, Z~ = Z~ = 60 and hco = 10.5 MeV, 2 A(rp2 - r,) = (% + rn)A(r p - rn) = 0.322418 fm 2.
(A.17)
HereA(rp- rn) = 0.051 fm. In 2°spb Z~ = 393, Z v = 693. We take b, = b,, = 2.5 fm, hco = 7 MeV, rp ,~ r. = 5.5 fm. If we use perturbation theory, we find
A(rp-r,) = 0.19 fm.
(A.18)
However, perturbation theory is not justified in /°spb. The value of (0Vc M ° ) = 12.295 MeV. This is comparable with 2he) = 14 MeV, Likewise (0Vc M~) = 16.27 MeV. When the appropriate matrix diagonalization is done, the result is reduced to 0.10 fm. Such Coulomb calculations have been previously done by McDonald 6z) who used IAE°I = ]AE 11 = 2hco.
Note added in proof." The proton-nucleus analyses have thus far been carried out using charge densities rather than proton densities. One should use the latter. In 4 8 C a for example, it was shown by Bertozzi et al. v) that the proton rms radius is 0.021 fm larger than the charge radius. This suggests that the neutron rms radius be decreased by 0.015 fm in order to keep the matter radius the same. But then the difference A = r , - r p (with rp the proton radius) will decrease by 0.036 fm. Of course, more quantitative calculations concerning this should be carried out. It should be added that the effect in 2°Spb is much smaller. References 1) 2) 3) 4) 5) 6) 7) 8) 9)
D. F. Jackson, Rep. Prog. Phys. 37 (1974) 55 J. A. Nolen and J. P. Schiffer, Phys. Lett. 29B (1969) 396: Ann. Rev. Nucl. Sci. 19 (1969) 471 E. Friedman and S. Shlomo, Z. Phys. A283 (1977) 67: Phys. Rev. Lett. 39 (1977) 1180 E. H. Auerbach, S. Kahana and J. Weneser, Phys. Rev. Lett. 23 (1969) 1253 I. Talmi, Rev. Mod. Phys. 34 (1962) 704 T. T. S. Kuo and G. E. Brown, Nucl. Phys. A l l 4 (1968) 241 W. Bertozzi et al., Phys. Lett. 41B (1972) 408 L. Zamick, Phys. Lett. 20B (1966) 176 R.J. Glauber, Lectures in theoretical physics, vol. 1, ed. W. E. Britten and L. G. Dunham (lnterscience, NY, 1959) p. 359; V. Franco and R. J. Glauber, Phys. Rev. 142 (1966) 1195; D. R. Harrington, Phys. Rev. 184 (1969) 1745 10) S. J. Wallace, Phys. Rev. CI2 (1975) 179: C. W. Wong and S. K. Young, Phys. Rev. CI2 (1975) 1301, and references therein 11) D. R. Harrington and G. K. Varma, Phys. Lett. 74B (1978) 316 12) A. K. Kerman, H. McManus and R. M. Thaler, Ann. of Phys. 8 (1959) 557
NEUTRON DISTRIBUTIONS
359
13) H. Feshbach, A. Gal and J. Hufner, Ann. of Phys. 66 (1971) 20; E. Lambert and H. Feshbach, Ann. of Phys. 76 (1973) 80; E. Boridy and H. Feshbach, Phys. Lett. 50 (1974) 433 14) D. R. Harrington and G. K. Varna, Nucl. Phys., to be published 15) G. K. Varma, Bull. Am. Phys. Soc. 22 (1977) 1009 16) G. K. Varma and L. Zamick, Phys. Rev. C16 (1977) 308 17) 1. Sick, Nucl. Phys. A218 (1974) 509, and private communication 18) A. Chaumeaux, V. Layly and R. Schaeffer, Phys. Lett. 72B (1977) 33 19) L. Ray and W. Coker, preprint (1977): L. Ray et al., preprint (1978), and private communication 20) 1. Ahmed, Nucl. Phys. A247 (1975) 418 21) G. D. Alkhazov et al., Nucl. Phys. A274 (1976) 443 22) J. W. Negele, Phys. Rev. CI (1970) 1260 23) J. W. Negele and D. Vautherin, Phys. Rev. C5 (1972) 1472 24) D. Vautherin and D. M. Brink, Phys. Rev. C5 (1972) 626 25) X. Campi and D. W. Sprung, Nucl. Phys. A194 (1972) 401 26) J. C. Lombardi et al., Nucl. Phys. A188 (1972) 103; G. W. Greenlees et al., Phys. Rev. 20 (1970) 1063: Phys. Rev. C3 (1971) 1560 27) G. M. Lerner et al., Phys. Rev. C12 (1975) 778 28) B. Tatischeff et al., Phys. Rev. C5 (t972) 234; I. Brissaud et al., Nucl. Phys. A191 (1972) 145 29) G. D. Alkhazov et al., Nucl. Phys. A280 (1977) 365 30) J. P. Egger et al., Phys. Rev. Lett. 39 (1977) 1608 31) H. Bethe and M. Johnson, preprint 32) M. J. Jakobson et al., Phys. Rev. Lett. 38 (1977) 1201 33) E. Friedman, Phys. Rev. C8 (1973) 996; E. Freidman et al., Phys. Rev. C9 (1974) 2340 34) A. M. Bernstein and W. A. Seidler, Phys. Lett. 39B (1972) 583 35) H. J. Gils and H. Rebel, Phys: Rev. C13 (1976) 2159 36) E. Friedman and C. J. Batty, Phys. Rev. CI6 (1977) 1425 37) D. F. Jackson and V. K. Kembhavi, Phys. Rev. CI (1969) 1626 38) H. J. Gils et al., Z. Phys. A2"/9 (1976) 55 39) C. J. Batty and E. Friedman, Nucl. Phys. A179 (1972) 701, B. W. Allardyce et al., Nucl. Phys. A209 (1973) 1 40) J. W. Negele, Phys. Rev. C9 (1974) 1054; H. J. Korner and J. P. Schiffer, Phys. Rev. Lett. 27 (1971) 1457 41) D. P. Siddhu and C. Quigg, Phys. Rev. 1)7 (1973) 755 42) V. Franco, Phys. Rev. C9 (1974) 1690 43) L. C. Liu, Phys. Rev. C, to be published, and private communication 44) G. K. Varma, Nucl. Phys. A294 (1978) 465 45) G. K. Varma, Phys. Rev. C16 (1977) 308 46) P. K. A. De Qitt Huberts et al., Phys. Lett. 71B (1977) 317 47) J,W. Negele, Comm. Nucl. and Part. Phys. 6 (1974) 15 48) S. Shlomo, Phys. Lett. 42B (1972) 146: Ph.D. Thesis (1973) 49) H. Sato, Nucl. Phys. A269 (1976) 378 50) R. Bertini et al., Phys. Lett. 45B (1973) 119 51) G. S. Blanpied et al., Phys. Rev. Lett. 39 (1977) 1447; G. W. Hoffman et al., preprint (1978) 52) A. M. Lane and A. Z. Mekjian, Phys. Rev. C8 (1973) 1981 53) J. Ahrens, Nucl. Phys. A251 (1975) 479 54) A. Arima, G. E. Brown, H. Huyga and M. Ichimura, Nucl. Phys. A205 (1973) 27 55) D. Drechsel and Y. E. Kim, Phys. Rev. Lett. 40 (1978) 531 56) M. W. Kirson, preprint 57) K. Arita, Proc. of the Meeting on giant resonances and related topics, University of Tokyo Tanashi-shi (1977) 58) A. Arima, Hyperfine. Interactions 4 (1978) 151 59) G. K. Varma and L. Zamick, Phys. Lett. 73B (1978) 1 60) J. P. Auger and R. J. Lombard, preprint (1978) 61 ) A. M. Bernstein and C. N. Papanicolas, Bull. Am. Phys. Soc. 23 (1978) 630, and private communication 62) W. M. McDonal,:l, Ph.D. Thesis
Nuclear Physics A306 (1978) 3 4 3 - 3 5 9 ; (~) North-HollamtPublishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
ON RADII OF NEUTRON DISTRIBUTIONS IN NUCLEI * GIRISH K. V A R M A
Seritt Physics Laharatorv, Rutgers Uidrersit)', Frelinghysen Road, Piscataway, New Jersey 08854, USA and LARRY Z A M I C K
Joseph Hem3' Lahoratorie.~', Princeton Universio', Jadwin Hall, Prhweton, New Jersey 08540, USA and Serin Physics Lahoratoo', Rutgers University, Frelin.qh)'sen Road, Piscataway, New Jersey 08854, USA *+ Received 14 March 1978 Abstract: We consider the analyses of the differences between rms radii (A = r - rD) of neutron and proton distributions in a wide variety of nuclei. We note that apart from its own intrinsic interest, the quantity zl is of importance for isotope shifts, core polarization contributions to the Coulomb energy differences of mirror pairs (Nolen-Schiffer anomaly) and the renormalization of the effective interaction. For example, if A were very small in 48Ca then the Nolen-Schiffer anomaly could be explained by a core polarization mechanism. We consider critically the various methods of determining A and conclude that at present probably the most reliable method is high energy ( ~ 1 GeV) proton-nucleus scattering. The different theoretical analyses based upon, e.g., the multiple diffraction theory (where Glauber amplitude is the leading term) or the optical potential (KMT) formalisms appear to be converging to essentially the same answer when analyzing the same data. High energy 0t-particles and medium energy pions can also become useful sources of information if higher order optical potentials are treated with care. We find that A is rather large in 48Ca, i.e. there is a neutron skin, so that the Nolen-Schiffer anomaly cannot be explained by a core polarization mechanism. The results of high energy proton-nucleus scattering are in excellent agreement with current density dependent Hartree-Fock calculations.
1. Introduction Among the most fundamental quantities in nuclear physics are the radii of nucleondensity distributions in nuclei. Most of the existing information on the subject comes from electron scattering and/~-mesic X-ray measurements which have yielded accurate radii for the proton distributions. At the same time, there have also been extensive theoretical efforts to understand nuclear structure in terms of nucleon-nucleon interactions. However, some of the best Hartree-Fock (HF) calculations at present employ density dependent effective interactions which are ultimately adjusted to fit charge densities as well as the binding energies and the single-particle energies of the closed shell nuclei. The HF results for radii of neutron distributions, on the other hand, are essentially predictions which can provide a test of the underlying theoretical approximations involved in such calculations. Unfortunately, the experimental information t Supported by the National Science Foundation. ** Permanent address. 343
344
G. K. VARMA A N D L. ZAMICK
on neutron radii comes mostly from the scattering of strongly interacting projectiles; the analyses are complicated and have often yielded conflicting results (see, for example, ref. 1)). However, high energy (~ 1 GeV) proton-nucleus scattering, if carefully analyzed, now appears to provide an accurate method for obtaining the rms radii (and other moments) of nuclear neutron density distributions. Other methods such as study of ~± elastic scattering angular distributions and high energy s-particle scattering may also prove to be useful in the near future. Whether the difference A = r , - r p between the radii of neutron and proton distributions in nuclei such as 4SCa and 2°spb is very small ~ 0.06 fm (as the Coulomb energy differences assuming no charge symmetry breaking appear to suggest) or is large ~ 0.2 fm as predicted by H F calculations, has important consequences. One implication of a small A in 4SCa is to have strong monopole core polarization. In that case the valence neutrons will push in the core neutrons and, in mirror nuclei, valence protons will shrink the core protons leading to an increase in the electrostatic energy. This would remove the so-called Nolen-Schiffer anomaly 3) in mirror pairs such as 41Sc-41Ca (this anomaly refers to a ~ 4 % - 1 0 % disagreement between experimental Coulomb displacement energies and theoretically calculated electromagnetic energy differences). This fact has led to considerable enthusiasm in reviving the idea of large core polarization 3). On the other hand, if A is large (as the high energy proton scattering results indicate), then the above core polarization mechanisms will be too small to explain the anomaly. In this paper we first consider, in sect. 2 the implications of large core polarization; it is shown that they lead to difficulties with quantities such as the renormalized effective interactions between identical nucleons. In sect. 3 we examine the values of A for closed shell nuclei extracted from proton scattering measurements at high energies where microscopic scattering theories are reliable. The results are in close agreement with current H F calculations and are consistent with almost no core polarization. Evidence from other strongly interacting projectiles is considered in sect. 4 and we summarize our results in sect. 5.
2. Possible implications of the magnitude of A Let us consider the change in mean square radius of a core when a valence neutron is added. In perturbation theory this is written as 2
2
or" • (j~FV - [ j v ( p h - i )Mr,,=o]~V)<(ph - 1)or"~r2½(1--Zz)O). phT"
(1)
~"-" ph
It is both more elegant and more convenient to insert as intermediate states only the monopole states IMT") = ~ <0r2z(ph - 1)or">(ph- ,)or" (~ i
NEUTRON
DISTRIBUTIONS
345
where z = 1 for T' = 0 and z = Zz for T' = 1. We note that the expression involves the following vertices:
V x = (j~ AET,,=o [j~M~°o]Jv), (3) V
Y = (Jr dEr"=l fJvM~"~==°]Jv). We insert for AE r'' the mean energies of the monopole states (for'example, from current somewhat imprecise empirical knowledge
AE ~ -2hto,
AE 1 ~ -4htn.
These vertices x and y also occur in other expressions. If we further assume harmonic oscillator radial wave functions with the oscillator length parameter b = ~/h/mo9 we find ( M ° ~ r 2 0 ) = x/~b 2, where X = sum ( 2 n + l + ~ ) over occupied states, e.g. _r = 36 for 160 and X = 120 for 4°Ca. We now summarize the dependence of various quantitites on x and y. (a) The change in mean square charge radius due to the addition of a valence neutron or proton, ti~.2 (neutron~ = "ch~,proton ;'
1
-
2 x/,Y, b2(x T- y).
(5)
(b) The Auerbach-Kahana-Weneser (AKW) diagram 4),
-4y(OVcMl).
(6)
The calculation of (OVc M1) is given in the appendix. This diagram was proposed by the authors as a possible explanation of the Nolen-Schiffer anomaly. It may be described as follows: the Coulomb interaction introduces an isospin impurity into the core. The nuclear interaction of a valence proton with this isospin impure core is different from that of a neutron. A priori, such a mechanism is quite a reasonable thing to consider. It is somewhat analogous to a calculation of an effective charge. These effective charges are known to be large. However, recent density dependent H F calculations by Negele, Vautherin and Brink and others have obtained very small values for this diagram. (c) The exchange of a monopole phonon between two neutrons (e.g. 42Ca = 4°Ca + 2f~ neutrons), tiEr= 1 = _ 2x2IAEO I_ 2y21AEll.
(7)
This is Of interest when considering the mass formula for Ca (and other isotopes) introduced by Talmi 5), E[Ca(40 + n)] = nC + [½n]fl + ½n(n- 1)~.
(8)
346
G. K. VARMA A N D L. ZAMICK
The empirical values for the Ca isotopes are: C = - 8.38,
fl = - 3.33,
e = 0.23.
The quantity 6E T= 1 can be identified as 6~, the change in ~ due to the exchange of a monopole phonon between two valence neutrons. Recall that using the bare G-matrix elements of all standard interactions, e.g. Kuo-Brown matrix elements 6) derived from the Hamada-Johnston interaction, the resulting ebare is negative. The exchange of a quadrupole bubble does yield a positive correction to e, and indeed, this quadrupole exchange may be the correct physical explanation of the repulsive pairing interaction. The AKW term can be written as 4a[AE r=lly where a 2 is the probability that isovector monopole state is mixed in the ground state from HF calculations. The value of a is about -0.08 for 4°Ca (i.e. ground-state impurity of 0.64 ~o). We now consider the general condition on the isotope shift. Let rp(40), rp(48), rn(40 ) and rn(48) be the rms radii of respectively, protons in 4°Ca and 48Ca and neutrons in 4°Ca and 4aCa. In first-order perturbation theory the condition on the neutrons reads
~/Z (x + y)b 2 = - I(2n + l + 3)- ~ l bZ + ~-[r~(40)- rZ.(40)] (10) + -~[r.Z(48) - r~(48)] + ~[%z(48)- r~(40]. The condition on the protons is
V ~ ( x - y)b 2 = -~[r~(48)- r~(40)].
(11)
To proceed further we need rpZ(40)- r,Z(40). Because the Coulomb interaction pushes out the protons this quantity should be positive. See the appendix for a discussion. We take rp(40)-r,(40) = 0.05 fm. This is consistent both with the calculations in the appendix, the H F calculations and the analysis of high energy proton-nucleus scattering. We also take rp(48) ,~ rp(40). This is also consistent with experiment. Recall that for a long time it was thought that %(48) was slightly smaller than %(40). But effects introduced by Bertozzi et al. v) reverse the situation. For our purposes, the assumption o f equality is more than adequate. We consider the different values of r.(48) and see what happens to 6e. Case 1: r.(48)- rp(48) = 0. This is an extreme case, but there is a belief among some people that r, = rp everywhere. Also, within error bars, this is consistent with 79 MeV a-scattering result. We find x = -0.057,
y = -0.057,
fie = -0.41 MeV, (12)
AKW diagram = + 824 keV.
NEUTRON DISTRIBUTIONS
347
We see that ~ is quite large and negative. This goes very strongly in the wrong way, and it is doubtful if there is any other mechanism (i.e. stronger quadrupole exchange 8)) that can compensate for this. On the other hand, the AKW diagram 4) does account for 824 keV, more than enough to explain the Nolen-Shiffer anomaly 2). Case 2: r,(48)-rp(48) = 0.2 fm. This is what proton scattering gives. We find x=0,
y=0,
fi~=0,
(13)
AKW diagram = 0. Now fi~ is negligibly small and the monopole exchange mechanism causes no trouble. But the AKW diagram 4) is also zero, so there is no explanation of the NolenSchiffer 2) anomaly. We thus see that a mechanism which helps to solve the Nolen-Schiffer anomaly 2), causes trouble with the repulsive pairing between identical particles. We feel therefore that at least the extreme possibility r,(48)- rp(48) = 0 can be ruled out. This argument is not so strong that one can rule out r,(48)-rp(48) ~> 0.1 fm, however.
104
, ~\
I I Proton Scattering at 1.05GeV
/
103 \i 102
48Ca
\\\~]"
~, ~.
c-~
\\
A = 0.19 fm
"~"
-
Io ~
,o o
A=I0 -I
10 .2
~
i
I
5
I0
15
C)C.M (deg)
20
Fig. 1. Elastic scattering of 1.05 GeV protons from ,o. ,8Ca" The data are from ref. 2 ~) and the theoretical curves are based upon the Glauber approximation including spin and correlation effects.
348
G. K. V A R M A A N D L, Z A M I C K
3. Neutron radii from high energy proton scattering Perhaps the most reliable tool at present for probing neutron density distributions is scattering of high energy protons from nuclei. The scattering data can be analyzed in terms of the multiple diffraction theory of Glauber 9) and its extensions 1o, 1~) or the optical potential formalism of K M T t 2) and its variants t 3). For example, the leading corrections 10) to the Glauber theory arising from non-eikonal propagation, from corrections to the frozen target approximation and from the kinematic transformations involved in relating bound nucleon t-matrices to free t-matrices, have been found to have little effect 11) on the extraction of neutron radii near 1 GeV. Wave function corrections such as those arising out of Pauli, c.m. and dynamical shortrange correlations as well as isospin effects have also been examined ~4). While detailed shapes of extracted densities are sensitive to some of these higher order effects, the rms radii do not change much under these improvements 15). 0 i05
5
I0
15
2O
I
I
r
I
,
Elastic
Scattering -
i04
-
103
_
102
-
i01
1.05 GeV
i05
A = 0.21 fm
\
¢o
..o E
-i05
p -2°spb
'~/-\
104
-!
i0 z 0.8GeV t~ = 0.21 fm
I0 i
I0 °
I0o
.
0.8GeV
t0 -i
1
0 0
i0 -I
0.3 P
iO-e
5
3 I0
~ 15
20
25
30
E~:.M (deg) Fig. 2. Proton-2°Spb elastic scattering and polarization. The data are from refs. 50. si) and the theoretical curves are based upon the Glauber approximation including spin and correlation effects. Note the different scales for 0.8 and 1.05 GeV,
N E U T R O N DISTRIBUTIONS
349
In figs. I and 2, we show typical fits to the high energy data on scattering of protons from 4°'4SCa and 2°spb. The theoretical curves are based on the Glauber theory and include spin dependent effects as well as the correlation effects described above. (The detailed expressions for including these effects are given in refs. 14, 16).) We should emphasize that in such microscopic calculations there are no free parameters (once the neutron and proton densities, and nucleon-nucleon (NN) scattering amplitudes are specified). In the above analyses the input NN amplitudes were taken from NN measurements, input proton densities were taken to be of the SOG forms 17) (determined from model independent analyses of electron scattering data) and neutron densities were obtained by fitting the data. One cannot hope to determine the densities in the interior of medium and heavy nuclei from proton scattering but both the
Nucleus
A = r, = % (fm) from proton scattering (GeV) .... 1.05 1.0 0.8
DDHF
DME, SKII
CS
~.0 b)
--0.02
--0.03
--0.02
160
- 0 . 0 2 a) --0.04 ')
4°Ca
0.04 4) . - 0 . 0 3 a) - 0 . 0 3 ')
- 0 . 0 7 b)
--0.04
tO.04
--0.05
4SCa
0.19 a) 0.16 a) 0.17 ')
0.21 b)
0.23
0.18
0.18
- -
9°Zr 20ap b
0.21 ~) 0.21 c)
0.13 ~)
0.08 4)
0.12
0.08
0.07
0.08 b)
0.21 n) 0.19 c)
0.20
0.20
0.21
As discussed in text, the analyses of refs. 18, 19) neglect all correlation effects. Ref. 16) includes c.m. correlations whereas ref. is) includes both c.m. and Pauli correlations. The typical error bars in A from proton scattering are ~ +0.05 fm. a) Ref. ,s). b) Ref. 16). ') Ref. ,9). d) Ref. is).
In table 1 we list the results for A = rn - rp for closed shell nuclei from the available theoretical analyses of the proton data t. As mentioned above, the analysis of ref. 15) includes spin dependent effects as well as all the correlation effects which contribute t The 1.00 GeV Leningrad data was analyzed earlier in ref. 16). The analysis of 2°SPb data in this paper was approximate in that an average matter distribution was extracted. The slightly different result given in table I for ref. 16) is due to an improved calculation using input SOG proton densities and treating the Pauli correlations.
350
G.K. VARMA AND L. ZAMICK
significantly. The analyses of refs. 18, 19) employ a first-order K M T potential; they include spin effects but neglect all correlation effects. Other analyses which also neglect spin effects are not listed in the table. [We, however, do not wish to imply that such analyses are meaningless. Some of the spin independent analyses employ effective N N amplitudes which mock up part of the spin effects in a restricted angular range. Thus, for example, Ahmed 20) obtains A = 0.24 fm for 48Ca and A = 0.26 fm for 2°8pb whereas Alkhazov e t al. 21) obtain A = 0.19 fm for 48Ca; these results are consistent with those from more sophisticated analysis 15).] From table 1 we notice that the results from different analyses are consistent with each other (within the error bars), and are also in excellent agreement with the Hartree-Fock calculations employing density dependent interactions 22 - 25). In 48Ca, as we shall see in sect. 5, they also appear to be consistent with a naive shell-model picture in which the change in radius is due solely to the addition of f~ neutrons. The smaller value of A in 2°8pb from 1.00 GeV data is due to disagreement of this data (0 < 15°) with the 1.05 GeV Saclay data (0 < 19°). The analyses of recent 0.8 GeV L A M P F data (0 ~< 30°) favour the 1.05 GeV data. We believe the 0.8 GeV analysis to be the most reliable one because the accompanying p-E°Spb polarization data are also well fitted by the theory and thus remove the small uncertainties associated with the spin parts of the N N amplitudes. For comparison, it should be noted that in a naive calculation in which one assumes the same oscillator parameter for the neutrons and protons the value of A in 2°8pb would be 0.4 fm as compared with the proton result of 0.2 fm. The smaller value of A is due in part to the fact that the Coulomb interaction pushes out the protons, and in part is due to the fact that the symmetry energy favors near equality of neutron and proton orbits. A rough calculation indicates that at least half of the effect comes from the Coulomb interaction pushing out the protons. This is shown in the appendix.
4. Neutron radii from other scattering processes In table 2 we list the values of A for 48Ca and 2°8pb from all available analyses 26- 3 8) (with respectable error bars) of scattering measurements. The pion reaction and total cross-section measurements appear to yield values of A which are generally small whereas detailed 130 MeV angular distribution measurements appear to be consistent with a large value of A. The analysis 36) of 10 ~ 0 GeV/c pion reaction cross sections ignores, for example, the existence of inelastic intermediate states. Such effects are large even in deuterium 41) and become larger in heavier nuclei. Since the neutron radii in this case are being extracted by fitting one number, this can lead to serious errors t. Similarly, at lower energies there are many higher order effects. Even effects such as double charge exchange and double spin flip are important 42) and will Similar analysis of proton reactioncross sectionsat 700 MeV lB. D. Anderson et al., Bull. Am. Phys. Soc. 22 (1977) 1005] leads to a matter radius differencebetween 48Ca and 4°Ca of 0.44 fm which is twice as large as the results from proton angular distributions.
N E U T R O N DISTRIBUTIONS
351
TABLE 2 The results for A = r . - rp in 4SCa and 2°8Pb from the available analyses of scattering measurements Nucleus
f 4SCa
A = r,-%
Ref.
0.19+0.05 0.21 -{-0.05 0.394-0.10
16) 26)
MeV MeV GeV
0.03+0.08 0.38 _+O. 12 0.20_+0.06
27) 2s) 29)
MeV
0.08_+0.05
32)
a)
33)
1.05 1.00 800 30
GeV GeV MeV MeV
0.21 _+0.05 0.08+0.05 0.21 -+ 0.05 0.36+0.20
l~) 16) ~5) 26)
104
MeV
f0.26+0.13
3,*)
104
MeV
140 166
MeV MeV
~.0.30_+0.07 0.0 -+0.07 0.42 _+0.20 0.25 ± 0.10
35) 36) 34)
Method
p elastic scattering
elastic scattering
n+ total cross sections
1.05 GeV 1.0 GeV 10.8-16.3 MeV 79 166 1.37 9C~240
nucleon-transfer reactions
2O8pb
" p elastic scattering
ct- elastic scattering
n - reaction cross sections 7r± reaction cross sections sub-Coulomb pick-up nucleon-transfer reactions
2~60 1-2
GeV GeV
0.0 -+0.1 0.0 +0.1 0.20 a)
15)
2s) 39) 39) 40)
33)
For comparison Coulomb energy differences imply A = 0.06 (0.07) fm for 4SCa (2°Spb). a) Consistent with D D H F 22).
occur through a second-order optical potential. True pion absorption (annihilation) also appears to have 4-3) significant effect on the cross sections. It is not obvious that these effects cancel even if ratios of reaction or total cross sections are used. Clearly, measurements involving detailed angular distributions of pions may provide much more reliable information. Recently, differential cross sections for elastic scattering of 130 MeV rc~: from 4o, 4SCa have been measured at SIN 30). A preliminary analysis of this data by Bethe and Johnson 31) indicates that the results for A are smaller than the predictions of DDHF. However, detailed analyses which include effects of higher order optical potentials, isolate the rather large Coulomb effects and, furthermore, treat the first-order pote'ntial accurately, must be carried out before the results become reliable. Different results obtained from lower energy or-particles are in conflict. These analyses are generally semiphenomenological. The real part of the potential is related to the nuclear density through a folding model while the imaginary potential is treated phenomenologically. In 4SCa the 166 MeV data 28) gave A = 0.38 ___0.12 fm but 79 MeV data 27) gave A = 0.03 --+0.08 fm. In the case of 2°apb, Bernstein et al. 34)
352
G . K . V A R M A A N D L. Z A M I C K
found A = 0.26_ 0.13 fm and Tatischeffet al. za) at 166 MeV obtained A = 0.25 +_0.10 fm. An analysis 35) of recent 104 MeV a-2°Spb data gave A -- 0.3_+0.07 fm but a subsequent analysis a6) of the same data using a slightly different imaginary potential (resulting in a somewhat lower chi square) led to A = 0_+0.1 fm. [The effects of different imaginary potentials on A are discussed inref. 3s).]If the imaginary potential is taken to be proportional to the real potential, as suggested in earlier works 34. aT), the resulting value of A is 0.32 fm, ref. 3s)]. That small changes in a completely phenomenological imaginary potential can cause such dramatic changes in extracted neutron radii is clearly worrisome and casts doubts on the reliability of such analyses in determining nuclear radii * As the energy goes up, however, the a-nucleus potential can be derived microscopically from either the Glauber or KMT theories. The "rigid projectile approximation" employed in ref. 29) to analyze the 1.37 GeV ~-4SCa data has been shown to be reasonable at small angles 44). The extracted value A = 0.20_+0.06 is consistent with the high energy proton scattering and HF results. A more careful analysis of this data should also treat the correlation effects 4s). One of the least ambiguous measurements of the spatial distribution of excess neutrons is provided by sub-Coulomb pick-up from 2°apb. A careful analysis 40) which treats the effects of the non-local exchange potential shows the results to be in agreement with HF calculations (A = 0.20 fm). The results obtained from analyses 33) of nucleon transfer reactions in Ca isotopes and in 2°spb also appear to be consistent with HF results and disagree with those obtained from Coulomb displacement energies. We briefly mention magnetic scattering of electrons, for example, the recent work 46) on 51V. In the single-particle picture all the magnetization is carried by a valence f~ proton. Thus one should be able to learn something about the radius of the valence proton (as opposed to Coulomb scattering, which gives the charge radius of the entire nucleus). It is observed that at high q the form factor [FM(q)[2 is larger than what is predicted by density dependent HF theories. In the oscillator model FM(q) is some polynomial in q times exp(-¼q2b:). A way of increasing FM(q) is to make b smaller, i.e. decrease the radius of the valence orbit relative to the HF value. If correct, this would afford an alternative explanation of the Nolen-Schiffer anomaly, indeed one proposed by Nolen and Schiffer themselves. Obviously, if the valence nucleon has a smaller radius its electrostatic interaction with the core will be stronger. It is, however, premature to assign the observed slow fall-off in q to a small radius. One must first consider effects of core polarization and exchange currents. This has not yet been done. By looking at the fits to the magnetization one might think that the largest M2 dominates at high q. But in fact, for given configurations nl, n'l' in • It is encouraging that attention is now being paid to microscopic calculations of the imaginary potential. See, for example, N. Vinh Mau, Phys. Lett. 71B (1977) 5, and also B. Sinha, Phys. Rev. C l l (1975) 1546. More recent calculations 61) of :t-Pb data give A consistent with HF results.
NEUTRON DISTRIBUTIONS
353
the limit of very large q, all M2 have the same q-dependence. Thus the asymptotic q-dependence depends on the configuration, not on 2. The M2 operator has terms j~_ l(qr)[Y~_ la] a and j~+ l(qr)(Yz+ la] a (also a can be replaced by I). For the f~-f~ configuration the largest term is j6(qr)Y6(t2) which yields an asymptotic term for FM(q)which goes as (qb)~ exp (-1q2b2) in the oscillator model. However, from core polarization, in particular the term f(f-th), one can get an asymptotic q-dependence which is higher (qb) 8 exp ( - ¼q2bZ). It is interesting to note that from core polarization one can induce a term of the formja(qr)Ys(f2 ) coming from the configuration f(f- lh)8. However, to get such a term one requires a tensor interaction between two nucleons. This is because the initial configuration f has l = 3 and the final configuration f(f- l h)S has angular momentum 13+81 ~ 3. Calculations of the core polarization effects have been carried out by Arita sT). For the highest multipole (M7) the results are force dependent. For a Serber force there is significant quenching of the M7 form factor, but for a Rosenfeld force there is hardly any quenching. Exchange currents are important as well. Indeed, in a recent conference proceedings, Arima 5s) notes that for 170 the effect of the exchange currents for large q is to increase :the transverse form factor of the magnetization by a significant amount. But this is precisely the effect that one gets by pushing the valence neutron in. Hence in x70 at least, one might erroneously conclude that the radius of the valence orbit was too large, when in fact exchange currents were responsible for the deviation * Clearly, from the calculated sizes of the effects of exchange currents and the force dependence of core polarization corrections, it is doubtful that one can make precise statements about the radius of the valence orbit.
5. Summary and conclusions We have seen that high energy proton scattering appears to yield results for differences in neutron and proton distribution radii in closed shell nuclei which are generally in agreement with the Hartree-Fock calculations employing currently popular interactions. They are consistent with results obtained from sub-Coulomb pick-up and 1.37 GeV a-particles, but appear to be larger than preliminary 130 MeV pion elastic scattering results. Some low energy a-scattering results are consistent with proton and HF results while others are not. We believe high energy proton results to be reliable because the analyses employ theories which are completely microscopic and where higher order corrections have been calculated. Proton scattering can, of course, give information about shapes of neutron densities as well (at least near the surface). The correlation effects, however, have to be treated much more carefully 14). The rms radii, on the other hand, appear to be remarkably stable ** * Recent calculations by J. Dubach of exchange effects in 51V appear to resolve the bulk of the discrepancy with HF calculations. ** This is only true if effective NN amplitudes are used and the data covers a small angular range. If, for example, the spin-dependent parts of the NN amplitude and correlations are simply switched off', significant errors can occur.
354
G . K . V A R M A A N D L. Z A M I C K
under theoretical improvements. We have therefore concentrated on rms radii (the mean square radius also happens to be one of the best determined moments in proton scattering). More careful calculations in future should be able to test the details of neutron distributions predicted by Hartree-Fock calculations. In the calcium region high energy proton scattering results are consistent with typical Hartree Fock calculations. There seems to be, however, some confusion about what is the "normal" behavior of the neutron radius in going from 4°Ca to 48Ca. For example, Chaumeaux et al. 13) interpret the proton scattering results in 48Ca as an anomaly assuming that "normal shell-model behavior" would have A = 0.4 fm. We should first remark that the shell model per se makes no statement whatsoever about radial charges. It instructs us to put particles in a well, and that is all. Let us then consider the simplest model where the 4°Ca core is described by the harmonic oscillator model (the oscillator parameter should then be b = 1.98 fm to fit the charge radius). Then putting eight neutrons in the f~ shell leads to r,(48)-rn(40) = 0.24 fm. Since rp(48) ~ rp(40) Fref. 7)] and if we include the Coulomb repulsion in 4°Ca so that rp(40)- r,(40) = 0.05 fm (see the appendix) then we obtain A = r. - rp = 0.19 fm in 4 8 C a . The clearest implication of the,closeness of this result with high energy proton scattering is that there is almost no monopole core polarization in 48Ca and that a resolution of Nolen-Schiffer anomaly in terms of such monopole core polarization is not possible. Thus it would appear that it is premature to criticize current nuclear structure theories solely on the basis of Coulomb energy anomaly. The problem is more likely tied with the interpretation of Coulomb energy shifts. Indeed, numerous possible corrections have been suggested * among them charge asymmetry. An early calculation indicated 48) that a charge symmetry breaking (CSB) potential could explain the anomaly in the 3He-3H system but not in 41Sc-4~Ca. However, a recent analysis by Sato 49) indicates that the magnitude and systematics of the Nolen-Schiffer anomaly (after taking into account admixtures of core excitation corrections) can be explained by a CSB force which is not inconsistent with the NN data. Whether a CSB force is the final explanation for this anomaly or not, it seems fair to say that in view of the current theoretical and experimental ignorance regarding charge asymmetry in nuclear forces, the role of Coulomb energy results in testing Hartree-Fock theories is seriously limited. We lastly cite our recent work 59) in which we correlate the neutron radius of 48Ca with the charge radius of 5°Ti using charge symmetry considerations. This analysis leads to a neutron skin in 48Ca more or less in agreement with the current Hartree-Fock calculations and with our analysis of the proton-nucleus scattering. [After this manuscript was submitted, we received a preprint by Auger and Lombard 6o) who show that the 1 GeV proton scattering data are reasonably well described by using nucleon density distributions from Hartree-Fock calculations as input.] • Some of these corrections are discussed in detail in ref. 4~).
NEUTRON DISTRIBUTIONS
355
It is a pleasure to thank Dr. L. C. Liu and Dr. H. Sato for a useful discussion. We are extremely grateful to A. M. Lane for some clarifying remarks about Coulomb energies. One of us (L.Z.) would like to thank Frank Calaprice and Rubby Sherr for their hospitality.
Appendix A.I. ISOSPIN IMPURITY DUE TO THE COULOMB INTERACTION AND ITS IMPLICATIONS To obtain the ground-state isospin impurity in the ground state of a nucleus, it is an excellent approximation to replace the two-body Coulomb interaction by a onebody field. For a uniform charge distribution this potential (field) is
I Ze2 Vc(r) =
r>R
t 3Ze2( r'
(A.1)
r2)
It is, furthermore, a very good approximation to use only the value of Vc(r) for
r
for all r.
(A.2)
This immediately establishes a relationship between the Coulomb field and the square radius operator. The states which mix into the ground state, are for the most part the monopole states, consisting of particle-hole states coupled to J = 0. We define them thus: (a) isoscalar Mo =
Q ~ r2(i)[0)
[(~r2(i) Q ~, r2(i))]½,
(A.3)
Q ~, r2(i)zz(i)[O) [
(A.4)
(b) isovector
M1=
Alternatively, we can talk about proton and neutron states: (c) proton
Qr2(i)lO> M
1t
~
g
[(~ r2O~r2>] ½'
* We are grateful to Dr. A. M. Lane for this suggestion.
(a.5)
356
G. K. VARMA AND L. ZAMICK
(d) neutron
Qr2(i)lO> M~=
~ [<~r2Q
~
v
(A.6)
r2>] & •
v
In the above, Q projects out the intermediate states orthogonal to the ground state. If [0> is a shell-model ground state with harmonic oscillator radial wave functions then the following matrix element is of interest: < M ° ~ r 2 0 > = [-~ I(ph-')J:° ~ ",r20> I2] ½ ph
= ~ (n+ 1)(n+ l+~)]~b 2 = x/-Zb 2,
(A.7)
where b is the oscillator length parameter, and ~ is the sum over occupied states of (2n+ l + a) (where we use the convention n starts from zero). Likewise ( M ' ~ r2Zz0> = ~ b 2, ( M~ E r20> = ~ ~ 2~b=,
(A.8).
;¢
( M ~ ~ r20)
z~ 2 x/~b~.
=
v
The matrix element for isospin impurity in the ground state is
CI=(MI~Vc(O0>~~
1 - ~r2 ) O> =
2R aZe2(½~/Zb2).
(A.9)
Also of interest is C = =
Ze2 ~ 2
~
x/Z=b~.
(A.10)
i
We define a 1 = C1/]AE11 where IAE1] is the energy of the isovector breathing mode state. We take IAEI[ ,,~ 4hco. We can write R 2 = -~Z~b=/Z.S 2 Hence a I - Ze2
~/~
(A.11)
4b (~Z=/Z)glAEXI" The quantity a ~ is the isospin impurity amplitude. For example in 4°Ca is we take rico = 10.37 MeV we get a 1 ~ 0.035 or 0.7 ~o admixture of T = 1 in the ground state of 4°Ca. Another estimate of the isospin impurity amplitude can be made, following the method of Lane and Mekjian s 2) in which one considers the energy weighted sum rule EWSR = ~ ( E , -
Eo)l
n
We obtain the previous estimate for the matrix element (n ~ r2ZzO>and setting E , - E o = 2hco. The effect of the potential energy is to change EWSR by a factor
NEUTRON DISTRIBUTIONS
357
(1 + r ) and to raise ( E , - E o ) to E ~- E o = IAEll . Thus we get the change (m E r2zzo) = ~/~b2 (!1
+ r)2hto'] ½
/"
The quantity r was thought for a long time to be about 0.4 fm the dipole case. But recently there hive been claims, both experimental and theoretical that r is closer to 1 [refs. 53-56)]. Note that if we t a k e r = 1 and AE 2 = 4h~, we will get no change in (n ~ r2zz0) from what we had previously. Therefore, we will stick with our previous calculation. But one. should keep in mind, that until the issue of what x is has been resolved unambiguously, there will be some uncertainty in (nSr2ZzO) and hence in the isospin impurity amplitude C 1. A.2. DIFFERENCE IN PROTON RADIUS AND NEUTRON RADIUS DUE TO THE COULOMB INTERACTION
The repulsion due to protons tends to push them out. We will here estimate the amount. We define the state IM ° ) = NO( ~ rp2+ ~ r2)10) where N O is a normalization. In the oscillator model
N O = (,Y,,~b~+ ,Y,~b~)-½.
(A. 12)
We define the state [M 1) to be orthogonal to IM°): [M 1) = NO[(X~b4/X.b~)½ ~, rp-(,Y.~bJX~b~) 2 44 4~ ~ rE].
(A.13)
We assume that the mean energy of the states IM °) znd IM ~) are, respectively, IAE°I = 2ho9 and IAEll = 4h~o. The coupling of the shell model ground state to those states via the Coulomb interaction Vc = (3Ze2/2R)(1- r2/3R 2) is Ko
K~
-
(OVcM ° ) o 4 Ze2 / - N Z.b~ ~ - ~ / [AE°[. AE o
(0VcM~)
4. vb~) 4 ½ ~Z e 2 N 0 (I,.b~I,
/IAEll.
(A.i4)
The ground state wave function is ~ = c[0)+ d°[M °) + d I IM 1). In lowest order perturbation theory c = 1, d o = K ° and d 1 = K 1. The difference in mean square radius of the protons and neutrons due to this Coulomb push is given by A(rp2 - - r nz) -----
rp _
r.
= 2cdONO
+ 2cdlN°x/Xvb~,~,b](A/NZ) + terms in d °2 and d xz.
(A.15)
358
G. K. VARMA AND L. ZAMICK
In perturbation theory we get
A(rZp_r2n)=2(No)2Ze22R 3 ]Az°ll ( Z ; ~
Z~bN4)Z~b4+2(NO)2 Ze 2 J
2R 3
1
4
4 A
lAX11Z~b~Z~b~ N Z (A.16)
Using first order perturbation theory in 4°Ca with b, = b v = 1.95 fm, Z~ = Z~ = 60 and hco = 10.5 MeV, 2 A(rp2 - r,) = (% + rn)A(r p - rn) = 0.322418 fm 2.
(A.17)
HereA(rp- rn) = 0.051 fm. In 2°spb Z~ = 393, Z v = 693. We take b, = b,, = 2.5 fm, hco = 7 MeV, rp ,~ r. = 5.5 fm. If we use perturbation theory, we find
A(rp-r,) = 0.19 fm.
(A.18)
However, perturbation theory is not justified in /°spb. The value of (0Vc M ° ) = 12.295 MeV. This is comparable with 2he) = 14 MeV, Likewise (0Vc M~) = 16.27 MeV. When the appropriate matrix diagonalization is done, the result is reduced to 0.10 fm. Such Coulomb calculations have been previously done by McDonald 6z) who used IAE°I = ]AE 11 = 2hco.
Note added in proof." The proton-nucleus analyses have thus far been carried out using charge densities rather than proton densities. One should use the latter. In 4 8 C a for example, it was shown by Bertozzi et al. v) that the proton rms radius is 0.021 fm larger than the charge radius. This suggests that the neutron rms radius be decreased by 0.015 fm in order to keep the matter radius the same. But then the difference A = r , - r p (with rp the proton radius) will decrease by 0.036 fm. Of course, more quantitative calculations concerning this should be carried out. It should be added that the effect in 2°Spb is much smaller. References 1) 2) 3) 4) 5) 6) 7) 8) 9)
D. F. Jackson, Rep. Prog. Phys. 37 (1974) 55 J. A. Nolen and J. P. Schiffer, Phys. Lett. 29B (1969) 396: Ann. Rev. Nucl. Sci. 19 (1969) 471 E. Friedman and S. Shlomo, Z. Phys. A283 (1977) 67: Phys. Rev. Lett. 39 (1977) 1180 E. H. Auerbach, S. Kahana and J. Weneser, Phys. Rev. Lett. 23 (1969) 1253 I. Talmi, Rev. Mod. Phys. 34 (1962) 704 T. T. S. Kuo and G. E. Brown, Nucl. Phys. A l l 4 (1968) 241 W. Bertozzi et al., Phys. Lett. 41B (1972) 408 L. Zamick, Phys. Lett. 20B (1966) 176 R.J. Glauber, Lectures in theoretical physics, vol. 1, ed. W. E. Britten and L. G. Dunham (lnterscience, NY, 1959) p. 359; V. Franco and R. J. Glauber, Phys. Rev. 142 (1966) 1195; D. R. Harrington, Phys. Rev. 184 (1969) 1745 10) S. J. Wallace, Phys. Rev. CI2 (1975) 179: C. W. Wong and S. K. Young, Phys. Rev. CI2 (1975) 1301, and references therein 11) D. R. Harrington and G. K. Varma, Phys. Lett. 74B (1978) 316 12) A. K. Kerman, H. McManus and R. M. Thaler, Ann. of Phys. 8 (1959) 557
NEUTRON DISTRIBUTIONS
359
13) H. Feshbach, A. Gal and J. Hufner, Ann. of Phys. 66 (1971) 20; E. Lambert and H. Feshbach, Ann. of Phys. 76 (1973) 80; E. Boridy and H. Feshbach, Phys. Lett. 50 (1974) 433 14) D. R. Harrington and G. K. Varna, Nucl. Phys., to be published 15) G. K. Varma, Bull. Am. Phys. Soc. 22 (1977) 1009 16) G. K. Varma and L. Zamick, Phys. Rev. C16 (1977) 308 17) 1. Sick, Nucl. Phys. A218 (1974) 509, and private communication 18) A. Chaumeaux, V. Layly and R. Schaeffer, Phys. Lett. 72B (1977) 33 19) L. Ray and W. Coker, preprint (1977): L. Ray et al., preprint (1978), and private communication 20) 1. Ahmed, Nucl. Phys. A247 (1975) 418 21) G. D. Alkhazov et al., Nucl. Phys. A274 (1976) 443 22) J. W. Negele, Phys. Rev. CI (1970) 1260 23) J. W. Negele and D. Vautherin, Phys. Rev. C5 (1972) 1472 24) D. Vautherin and D. M. Brink, Phys. Rev. C5 (1972) 626 25) X. Campi and D. W. Sprung, Nucl. Phys. A194 (1972) 401 26) J. C. Lombardi et al., Nucl. Phys. A188 (1972) 103; G. W. Greenlees et al., Phys. Rev. 20 (1970) 1063: Phys. Rev. C3 (1971) 1560 27) G. M. Lerner et al., Phys. Rev. C12 (1975) 778 28) B. Tatischeff et al., Phys. Rev. C5 (t972) 234; I. Brissaud et al., Nucl. Phys. A191 (1972) 145 29) G. D. Alkhazov et al., Nucl. Phys. A280 (1977) 365 30) J. P. Egger et al., Phys. Rev. Lett. 39 (1977) 1608 31) H. Bethe and M. Johnson, preprint 32) M. J. Jakobson et al., Phys. Rev. Lett. 38 (1977) 1201 33) E. Friedman, Phys. Rev. C8 (1973) 996; E. Freidman et al., Phys. Rev. C9 (1974) 2340 34) A. M. Bernstein and W. A. Seidler, Phys. Lett. 39B (1972) 583 35) H. J. Gils and H. Rebel, Phys: Rev. C13 (1976) 2159 36) E. Friedman and C. J. Batty, Phys. Rev. CI6 (1977) 1425 37) D. F. Jackson and V. K. Kembhavi, Phys. Rev. CI (1969) 1626 38) H. J. Gils et al., Z. Phys. A2"/9 (1976) 55 39) C. J. Batty and E. Friedman, Nucl. Phys. A179 (1972) 701, B. W. Allardyce et al., Nucl. Phys. A209 (1973) 1 40) J. W. Negele, Phys. Rev. C9 (1974) 1054; H. J. Korner and J. P. Schiffer, Phys. Rev. Lett. 27 (1971) 1457 41) D. P. Siddhu and C. Quigg, Phys. Rev. 1)7 (1973) 755 42) V. Franco, Phys. Rev. C9 (1974) 1690 43) L. C. Liu, Phys. Rev. C, to be published, and private communication 44) G. K. Varma, Nucl. Phys. A294 (1978) 465 45) G. K. Varma, Phys. Rev. C16 (1977) 308 46) P. K. A. De Qitt Huberts et al., Phys. Lett. 71B (1977) 317 47) J,W. Negele, Comm. Nucl. and Part. Phys. 6 (1974) 15 48) S. Shlomo, Phys. Lett. 42B (1972) 146: Ph.D. Thesis (1973) 49) H. Sato, Nucl. Phys. A269 (1976) 378 50) R. Bertini et al., Phys. Lett. 45B (1973) 119 51) G. S. Blanpied et al., Phys. Rev. Lett. 39 (1977) 1447; G. W. Hoffman et al., preprint (1978) 52) A. M. Lane and A. Z. Mekjian, Phys. Rev. C8 (1973) 1981 53) J. Ahrens, Nucl. Phys. A251 (1975) 479 54) A. Arima, G. E. Brown, H. Huyga and M. Ichimura, Nucl. Phys. A205 (1973) 27 55) D. Drechsel and Y. E. Kim, Phys. Rev. Lett. 40 (1978) 531 56) M. W. Kirson, preprint 57) K. Arita, Proc. of the Meeting on giant resonances and related topics, University of Tokyo Tanashi-shi (1977) 58) A. Arima, Hyperfine. Interactions 4 (1978) 151 59) G. K. Varma and L. Zamick, Phys. Lett. 73B (1978) 1 60) J. P. Auger and R. J. Lombard, preprint (1978) 61 ) A. M. Bernstein and C. N. Papanicolas, Bull. Am. Phys. Soc. 23 (1978) 630, and private communication 62) W. M. McDonal,:l, Ph.D. Thesis