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The difference between neutron and proton radii in the Ca isotopes
Volume 83B, number 1
PHYSICS LETTERS
23 April 1979
THE DIFFERENCE BETWEEN NEUTRON AND PROTON RADII IN THE Ca ISOTOPES S. SHLOMO and R. SCHAEFFER CEN-Saclay, 91190 Gif sur Yvette, France
Received 9 November 1978 Revised manuscript received 16 February 1979
We take a close look at the recent analysis of 1 GeV proton scattering on Ca isotopes. Considering the more accurate values of relevant differences of matter and proton rms radii, a value of r n - rp = 0.10 *~0.03 fm is obtained for 48Ca. This value is much smaller than those quoted recently and is consistent with Coulomb displacement energies.
High-energy (1 GeV) proton scattering has recently become an important tool for determining matter or neutron density distributions in nuclei (see refs. [1 ] and [2] as well as references therein). In particular, root mean square (rms) radii can be obtained for the neutron distribution. Table 1 summarizes the results for 40Ca and 48Ca obtained by the Saclay group [1] using the impulse approximation and the Gatchina group [2] using the Glauber model. Here, rm(A), rp(A) and rn(A ) are, respectively, the rms radii of the mass, proton and neutron point density distributions of a nucleus with mass A. The analysis of these experiments is motivated by the extraction of the difference A(A) = rn(A ) - rp(A) from the data. This quantity is of considerable interest, mainly because the values obtained from Coulomb energy shift measurements and H a r t r e e - F o c k calculations differ significantly. We would like to emphasize here two more or less obvious, but important points: (i) high-energy protons do not determine A(A) directly, but merely r m and (ii) the error on the difference rm(A ) - rm(40 ) between neighboring isotopes is much smaller than the error on r m itself. As discussed in refs. [1,2], the cross sections are sensitive to the optical potential which is a linear combination of the proton COp) and neutron (Pn) densities with respectively the p r o t o n - p r o t o n (tpp) and the p r o t o n - n e u t r o n (tpn) scattering amplitude, Fop t = tpppp + tpnPn = ½(tpp + tpn)P m + ½(tpn - tpp)(Pn -- p p ) .
(1)
The scattering amplitudes tpp and tpn being nearly identical [1] and pp - Pn being a factor ( N - Z ) / A smaller than the matter density Pm = Pp + Pn, the term proportional to the latter is nearly two orders of magnitude larger than the term proportional to On pp. So, although the results are usually given in terms of rn, or A = r n _ rp, high-energy protons lead actually to information about Pm or the matter rms radius r m. The neutron density, hence r n and r n - rp, is always obtained indirectly by combining the information on pp taken from electron scattering with the proton scattering information. Any change in the size parameters 6rp of pp will induce an inverse change 6r n ~ - 6 r p in the parameters of On when the latter is fitted to a given experiment, so as to keep the optical potential and Pm as close as possible to their initial values. This induces a change of about - 2 8 r p in A. Since A is usually a small number, such modifications are far from being negligible. The second point, which is also emphasized in ref. [3], is that most of the uncertainties in the radii cancel out when relative values between isotopes measured consistently are considered (as, for instance, a systematic angular shift). A correction currently discussed among the authors of refs. [1 ] and [2], which does not affect rm(A ) - rm(40), but changes A(A), is the discrepancy between various values given for the proton form factor. The compilation of ref. [4] leads to a proton radius of 0.81 -+ 0.01 fm. This value was raised [5] up to 0.88 + 0.03 fm but very recently the
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PHYSICS LETTERS
Table 1 Rms radii (in fro) of 4°Ca and 48Ca obtained in previous analyses of 1 GeV proton scattering. rms (point)
4°Ca r m rp rn r n - rp 48Ca r m rp rn rn-r p
rm(48)- rm(40)
rms (folded)
(a)
(b)
(c)
(d)
(e)
3.375 3.39 3.36 -0.03
3.40 3.39 3.41 +0.02
3.49 3.49 3.49 0.00
3.49 3.48 3.50 +0.02
3.47
3.474 3.38 3.54 0.16
3.50 3.38 3.59 0.21
3.58 3.48 3.65 0.17
3.59 3.47 3.66 0.19
3.57
0.10
0.10
0.10
0.10
0.10
(a) Ref. [1 ], table 9. Results obtained using the Saclay data [8] and the impulse approximation including the spin-orbit nucleon-nucleon amplitude. (b) Ref. [1 ], table 9. Same calculation as in column (a) except that the correlations were included. (c) Ref. [2], table 6.1. Results obtained using the Gatchina data [7] and the Glauber approximation without spin-orbit or correlations. (d) Re/'. [2], table 6.1. Same calculation as in column (c), but using the Saclay data [8 ]. (e) Ref. [2], table 6.2. Results fitting directly the matter density to both the Gatchina [71 and the Saclay [8] data. data of ref. [5] were reanalyzed [61, leading to 0.86 or 0.84 fro. In ref. [1], the value of 0.88 was retained, in ref. [2], on the other hand, 0.81 was used with no correction for the neutron form factor, which amounts to using 0.88 and to correct for the nuetrons. A decrease of 0.04 fm of the proton radius diminishes A(A) by 0.02 fro, but leaves rm(A ) - rm(40) invariant. The same kind of changes arise from possible uncertainties in the spin-orbit interaction, the use of the eikonal approximation, the calculation of the correlation correction, the model dependence o f the densities used in the analysis, but can be cancelled out by considering rm(A ) - r m (40), since all these effects amount to shifting the values of all matter radii by a nearly constant amount. A striking illustration is provided by the compilation o f table 1, which summarizes the values obtained in refs. [1,2]. The first two columns show the effect of including the correlations: A(48) changes by 0.05 fm, whereas rm(48 ) - rm(40) does not. The comparison of columns (a) and (c) shows a similar effect: in column (a) the results are obtained using the impulse approximation including the spin-orbit inter-
23 April 1979
action, and in column (c)with the Glauber approximation and no spin-orbit. Finally, the comparison of columns (c) and (d) shows the results using the Gatchina [7] and the Saclay [8] data, respectively: only A(48) is different. We thus conclude that the quantity obtained by 1 GeV proton scattering which is the most free of systematic errors is the difference rm(A ) - rm(40 ). It remains constant, and equal to 0.10 fm whatever model is used (table 1), whereas A(48) varies from 0.16 to 0.21 fro. In the following we consider this value of rm(48 ) - rm(40 ) = 0.10 + 0.02 fm the most accurate (and relevant) quantity which is determined from 1 GeV proton scattering and the starting point for our determination of A(48). It is easy to see that for the Ca isotopes we have A(A)= rn(A ) - rp(A) = ( A / N ) [ ( r m ( A ) - rm(40)) - (rp(A) - rp(40)) + ½A(40)] ,
(2)
except for corrections which are negligible since the difference of the rms radii is small. This equality relates A(A) to two very well-known quantities, namely the difference in matter rms radii and the difference in protons rms radii. It involves, however, a quantity which is not well known, A(40), and its choice has to be discussed. Let us consider 48Ca in more detail. To determine A(48) we use: (i) From table 1, rm(48 ) rm(40 ) = 0.10 -+ 0.02 fro, the error being mainly due to experimental accuracy. (ii) For rp(48) - rp(40) we make use of the very accurate and recent muonic Xray measurements of ref. [9]. Although these results agree with previous measurements [101 (within their experimental error of lO-2fm), they are much more accurate (with an error of 10 - 3 fro) and also agree with recent results obtained using both laser [11 ] and atomic beam absorption optical spectroscopy [12]. The charge rms radius of 48Ca is found to be equal to that of 4°Ca within 10 -3 fm. Thus, taking into account the effect of the valence neutrons [13] on the charge rms radius of 48Ca we find that rp(48) - rp(40) = 0.021 + 0.001 fm. (iii) A(40) = rn(40 ) -- rp(40), which is practically unknown experimentally. If one neglects the Coulomb interaction among the protons, then A(40) = 0. With the Coulomb interaction one expects (intuitively) that A(40) < 0. In the independent-particle model, using a Woods-Saxon potential, we find A(40) = - 0 . 0 7 fro. Due to self-consistency, H a r t r e e - F o c k results are somewhat smaller in abso-
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PHYSICS LETTERS
lute value. However, practically all existing HF calculations lead [1 ] to A(40) ~ --0.04 fm with a very few exceptions predicting A(40) = --0.05 fm. We adopt the value A(40) = --0.04 fro. Substituting in eq. (2) we find that A(48) = 0.10 e 0.03 fm. This is significantly smaller than the values given in table 1. This is basically due to: (i)! rp(48) - rp(40) is taken to be +0.02 fm instead o f - 0 . 0 1 fm in table 1, diminishing £x(48) by 0.05 fm and (ii) the choice of A(40). Indeed those analyses which resulted [1] in A(48) = 0.20 fm also give A(40) = + 0.02 fm (instead of - 0 . 0 4 fro). A similar analysis, using eq. (2), was carried out for the other Ca isotopes. The results are summarized in table 2. The values obtained for r n - r_ are much smaller than those previously claimed (1,2]. Note that for 44Ca we find A(44) = 0 e 0.03 fm. Although it re. lies on the choice of A(40), the present determination of ~(A) (i.e. using eq. (2)) is more accurate than those of refs. [1] and [2] (based on an absolute determination of rm). Even with the extreme values A(40) = 0.00 or --0.06 fm we get A(48) = 0.14 ± 0.03 fm or 0.08 + 0.03 fro, which are significantly smaller than the values given in table 1. Relation (2) was used in previous analyses of 79 MeV a scattering [14] and 1 GeV proton scattering [7] to deduce A(48), leading also to small values for this number. It has been shown recently [15] that the calculated value of the Coulomb displacement energy (2~Ec) of 48Ca agrees with experiment if A(48) = 0.08 fm, i.e. the Nolen-Schiffer anomaly [16] is in this case absent. The value of 0.10 +- 0.03 fm obtained here agrees with that obtained from 2rEc. It may indicate the need for a contribution of 70 +- 100 keV to ZXEc from charge symmetry breaking interaction. (A decrease of A(48) of 0.1 fm increases zXEc o f 48Ca by 350 keV.) Even for A(40) = 0, one needs only 210 ± 100 keV
23 April 1979
charge symmetry breaking. These values are in agreement with the conclusions of ref. [15], but based on more reliable data. Well-known Hartree-Fock calculations [17] which give A(48) > 0.2 fm disagree with the present result and with the experimental value of zXEc. Some more recent Hartree-Fock calculations [18] lead to a value o f ~(48) = 0.14 fm (and 2x(40) = - 0 . 0 4 fm) which is closer to our result. It is also worthwhile to note that the difference rn(A ) - rn(40 ) can be extracted quite accurately from the 1 GeV proton and electron scattering data, since it is almost independent of A(40): rn(A ) - rn(40 ) = (A /N) [rm (A ) - rm(40)] (3)
- (Z/N)[rp(A) - rp(40)] - ( ( N -
Z)/2N)A(40).
The results are also shown in table 2. We have, for instance rn(48 ) - rn(40 ) = 0.16 ± 0.03 fm whatever choice of A(40) is made and thus independent of any assumption. Finally, let us make a short comment on the most recent measurements [19] o f the f7/2 valence particle radius rex. They also allow to deduce a value for rn(A ) - rn(40), although this is still somewhat preliminary since the model dependence of rex as well as the exchange current corrections have not yet been thoroughly examined. Nevertheless, using the average value [19] of 4.01 ± 0.04 for rex and assuming the rms radius of the 20 neutron core of 48Ca is equal to that of 4°Ca (rn(40) = 3.36 • 0.02 fm), we find that rn(48 ) - rn(40 ) -- 0.20 ± 0.02 fm. This may indicate a small (0.05 ± 0.05) compression in the core neutron rms radius when adding eight neutrons. The exchange current corrections are likely [20] to increase rex by 1 or 2% leading to a somewhat larger core compression. Unfortunately, the present theoretical and experimental uncertainties do not permit a firm conclusion.
Table 2 Results of present analysis using eqs. (2) and (3). A
rm(A ) _ rm(40) a) rp(A) - rp(40) b) rn(A ) - rp(A) rn(A) - rn(40) a) Refs. [1,2].
b) Ref. [9].
40
42
44
48
_ _ -0.04 c) -
0.055 ± 0.02 0.036 0.0 ± 0.03 0.07 _+0.03
0.07 ± 0.02 0.053 0.0 ± 0.03 0.09 ± 0.03
0.10 +-0.02 0.021 0.10 ± 0.03 0.16 ± 0.03
c) Assumed.
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PHYSICS LETTERS
References [1] A. Chaumeaux, V. Layly and R. Schaeffer, Ann. Phys. 116 (1978) 247. [2] G.D. Alkhazov, S.L. Belostotsky and A.A. Vorobyov, Phys. Rep. 42C (1978) 89. [3] L. Ray et al., Phys. Rev. C18 (1978) 2641. [4] L.N. Hand et al., Rev. Mod. Phys. 35 (1963) 335. [5] F. Borkowski et al., Nucl. Phys. B93 (1975) 461. [6] B. Frois, private communication; V. Walter, Lecture given at Frascati, Italy (March 1978). [7] G.D. Alkhazov et al., Phys. Lett. 57B (1975) 47. [8] G.D. Alkhazov et al., Nucl. Phys. A274 (1976) 443. [9] H.D. Wohlfahrt et al., Phys. Lett. 7313 (1978) 131. [10] R.D. Ehrlich et al., Phys. Rev. Lett. 18 (1967) 959; E.R. Macagno et al., Phys. Rev. C1 (1970) 1202. [11] R. Neumann et al., Z. Phys. A279 (1976) 249. [12] H.W. Brandt et al,, Z. Phys. A288 (1978) 241.
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[13] W. Bertozzi et al., Phys. Lett. 41B (1972) 408. [14] G.M. Lerner et al., Phys. Rev. C12 (1975) 778. [15] E. Friedman and S. Shlomo, Z. Phys. A283 (1977) 283; S. Shlomo and E. Friedman, Phys. Rev, Lett. 19 (1977) 1180; S. Shlomo, Rep. Prog. Phys. 41 (1978) 957. [16] J.A. Nolen and J.P. Schiffer, Ann. Rev. Nucl. Sci. 19 (1969) 527. [17] J.W. Negele, Phys. Rev. C1 (1970) 1260; D. Vautherin and D. Brink, Phys. Rev. C5 (1972) 626. [18] H. Beiner et al., Nucl. Phys. A238 (1975) 29; D. Gogny, private communication. [19] P.K.A. de Witt Huberts et al., Phys. Lett. 7113 (1977) 317; S.K. Platchkov et al., Saclay preprint (1978); and private communication. [20] A. Arima et al., Phys. Rev. Lett. 40 (1978) 1001; and private communication to S.K. Platchkov et al., quoted in ref. [19].
PHYSICS LETTERS
23 April 1979
THE DIFFERENCE BETWEEN NEUTRON AND PROTON RADII IN THE Ca ISOTOPES S. SHLOMO and R. SCHAEFFER CEN-Saclay, 91190 Gif sur Yvette, France
Received 9 November 1978 Revised manuscript received 16 February 1979
We take a close look at the recent analysis of 1 GeV proton scattering on Ca isotopes. Considering the more accurate values of relevant differences of matter and proton rms radii, a value of r n - rp = 0.10 *~0.03 fm is obtained for 48Ca. This value is much smaller than those quoted recently and is consistent with Coulomb displacement energies.
High-energy (1 GeV) proton scattering has recently become an important tool for determining matter or neutron density distributions in nuclei (see refs. [1 ] and [2] as well as references therein). In particular, root mean square (rms) radii can be obtained for the neutron distribution. Table 1 summarizes the results for 40Ca and 48Ca obtained by the Saclay group [1] using the impulse approximation and the Gatchina group [2] using the Glauber model. Here, rm(A), rp(A) and rn(A ) are, respectively, the rms radii of the mass, proton and neutron point density distributions of a nucleus with mass A. The analysis of these experiments is motivated by the extraction of the difference A(A) = rn(A ) - rp(A) from the data. This quantity is of considerable interest, mainly because the values obtained from Coulomb energy shift measurements and H a r t r e e - F o c k calculations differ significantly. We would like to emphasize here two more or less obvious, but important points: (i) high-energy protons do not determine A(A) directly, but merely r m and (ii) the error on the difference rm(A ) - rm(40 ) between neighboring isotopes is much smaller than the error on r m itself. As discussed in refs. [1,2], the cross sections are sensitive to the optical potential which is a linear combination of the proton COp) and neutron (Pn) densities with respectively the p r o t o n - p r o t o n (tpp) and the p r o t o n - n e u t r o n (tpn) scattering amplitude, Fop t = tpppp + tpnPn = ½(tpp + tpn)P m + ½(tpn - tpp)(Pn -- p p ) .
(1)
The scattering amplitudes tpp and tpn being nearly identical [1] and pp - Pn being a factor ( N - Z ) / A smaller than the matter density Pm = Pp + Pn, the term proportional to the latter is nearly two orders of magnitude larger than the term proportional to On pp. So, although the results are usually given in terms of rn, or A = r n _ rp, high-energy protons lead actually to information about Pm or the matter rms radius r m. The neutron density, hence r n and r n - rp, is always obtained indirectly by combining the information on pp taken from electron scattering with the proton scattering information. Any change in the size parameters 6rp of pp will induce an inverse change 6r n ~ - 6 r p in the parameters of On when the latter is fitted to a given experiment, so as to keep the optical potential and Pm as close as possible to their initial values. This induces a change of about - 2 8 r p in A. Since A is usually a small number, such modifications are far from being negligible. The second point, which is also emphasized in ref. [3], is that most of the uncertainties in the radii cancel out when relative values between isotopes measured consistently are considered (as, for instance, a systematic angular shift). A correction currently discussed among the authors of refs. [1 ] and [2], which does not affect rm(A ) - rm(40), but changes A(A), is the discrepancy between various values given for the proton form factor. The compilation of ref. [4] leads to a proton radius of 0.81 -+ 0.01 fm. This value was raised [5] up to 0.88 + 0.03 fm but very recently the
Volume 83B, number 1
PHYSICS LETTERS
Table 1 Rms radii (in fro) of 4°Ca and 48Ca obtained in previous analyses of 1 GeV proton scattering. rms (point)
4°Ca r m rp rn r n - rp 48Ca r m rp rn rn-r p
rm(48)- rm(40)
rms (folded)
(a)
(b)
(c)
(d)
(e)
3.375 3.39 3.36 -0.03
3.40 3.39 3.41 +0.02
3.49 3.49 3.49 0.00
3.49 3.48 3.50 +0.02
3.47
3.474 3.38 3.54 0.16
3.50 3.38 3.59 0.21
3.58 3.48 3.65 0.17
3.59 3.47 3.66 0.19
3.57
0.10
0.10
0.10
0.10
0.10
(a) Ref. [1 ], table 9. Results obtained using the Saclay data [8] and the impulse approximation including the spin-orbit nucleon-nucleon amplitude. (b) Ref. [1 ], table 9. Same calculation as in column (a) except that the correlations were included. (c) Ref. [2], table 6.1. Results obtained using the Gatchina data [7] and the Glauber approximation without spin-orbit or correlations. (d) Re/'. [2], table 6.1. Same calculation as in column (c), but using the Saclay data [8 ]. (e) Ref. [2], table 6.2. Results fitting directly the matter density to both the Gatchina [71 and the Saclay [8] data. data of ref. [5] were reanalyzed [61, leading to 0.86 or 0.84 fro. In ref. [1], the value of 0.88 was retained, in ref. [2], on the other hand, 0.81 was used with no correction for the neutron form factor, which amounts to using 0.88 and to correct for the nuetrons. A decrease of 0.04 fm of the proton radius diminishes A(A) by 0.02 fro, but leaves rm(A ) - rm(40) invariant. The same kind of changes arise from possible uncertainties in the spin-orbit interaction, the use of the eikonal approximation, the calculation of the correlation correction, the model dependence o f the densities used in the analysis, but can be cancelled out by considering rm(A ) - r m (40), since all these effects amount to shifting the values of all matter radii by a nearly constant amount. A striking illustration is provided by the compilation o f table 1, which summarizes the values obtained in refs. [1,2]. The first two columns show the effect of including the correlations: A(48) changes by 0.05 fm, whereas rm(48 ) - rm(40) does not. The comparison of columns (a) and (c) shows a similar effect: in column (a) the results are obtained using the impulse approximation including the spin-orbit inter-
23 April 1979
action, and in column (c)with the Glauber approximation and no spin-orbit. Finally, the comparison of columns (c) and (d) shows the results using the Gatchina [7] and the Saclay [8] data, respectively: only A(48) is different. We thus conclude that the quantity obtained by 1 GeV proton scattering which is the most free of systematic errors is the difference rm(A ) - rm(40 ). It remains constant, and equal to 0.10 fm whatever model is used (table 1), whereas A(48) varies from 0.16 to 0.21 fro. In the following we consider this value of rm(48 ) - rm(40 ) = 0.10 + 0.02 fm the most accurate (and relevant) quantity which is determined from 1 GeV proton scattering and the starting point for our determination of A(48). It is easy to see that for the Ca isotopes we have A(A)= rn(A ) - rp(A) = ( A / N ) [ ( r m ( A ) - rm(40)) - (rp(A) - rp(40)) + ½A(40)] ,
(2)
except for corrections which are negligible since the difference of the rms radii is small. This equality relates A(A) to two very well-known quantities, namely the difference in matter rms radii and the difference in protons rms radii. It involves, however, a quantity which is not well known, A(40), and its choice has to be discussed. Let us consider 48Ca in more detail. To determine A(48) we use: (i) From table 1, rm(48 ) rm(40 ) = 0.10 -+ 0.02 fro, the error being mainly due to experimental accuracy. (ii) For rp(48) - rp(40) we make use of the very accurate and recent muonic Xray measurements of ref. [9]. Although these results agree with previous measurements [101 (within their experimental error of lO-2fm), they are much more accurate (with an error of 10 - 3 fro) and also agree with recent results obtained using both laser [11 ] and atomic beam absorption optical spectroscopy [12]. The charge rms radius of 48Ca is found to be equal to that of 4°Ca within 10 -3 fm. Thus, taking into account the effect of the valence neutrons [13] on the charge rms radius of 48Ca we find that rp(48) - rp(40) = 0.021 + 0.001 fm. (iii) A(40) = rn(40 ) -- rp(40), which is practically unknown experimentally. If one neglects the Coulomb interaction among the protons, then A(40) = 0. With the Coulomb interaction one expects (intuitively) that A(40) < 0. In the independent-particle model, using a Woods-Saxon potential, we find A(40) = - 0 . 0 7 fro. Due to self-consistency, H a r t r e e - F o c k results are somewhat smaller in abso-
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lute value. However, practically all existing HF calculations lead [1 ] to A(40) ~ --0.04 fm with a very few exceptions predicting A(40) = --0.05 fm. We adopt the value A(40) = --0.04 fro. Substituting in eq. (2) we find that A(48) = 0.10 e 0.03 fm. This is significantly smaller than the values given in table 1. This is basically due to: (i)! rp(48) - rp(40) is taken to be +0.02 fm instead o f - 0 . 0 1 fm in table 1, diminishing £x(48) by 0.05 fm and (ii) the choice of A(40). Indeed those analyses which resulted [1] in A(48) = 0.20 fm also give A(40) = + 0.02 fm (instead of - 0 . 0 4 fro). A similar analysis, using eq. (2), was carried out for the other Ca isotopes. The results are summarized in table 2. The values obtained for r n - r_ are much smaller than those previously claimed (1,2]. Note that for 44Ca we find A(44) = 0 e 0.03 fm. Although it re. lies on the choice of A(40), the present determination of ~(A) (i.e. using eq. (2)) is more accurate than those of refs. [1] and [2] (based on an absolute determination of rm). Even with the extreme values A(40) = 0.00 or --0.06 fm we get A(48) = 0.14 ± 0.03 fm or 0.08 + 0.03 fro, which are significantly smaller than the values given in table 1. Relation (2) was used in previous analyses of 79 MeV a scattering [14] and 1 GeV proton scattering [7] to deduce A(48), leading also to small values for this number. It has been shown recently [15] that the calculated value of the Coulomb displacement energy (2~Ec) of 48Ca agrees with experiment if A(48) = 0.08 fm, i.e. the Nolen-Schiffer anomaly [16] is in this case absent. The value of 0.10 +- 0.03 fm obtained here agrees with that obtained from 2rEc. It may indicate the need for a contribution of 70 +- 100 keV to ZXEc from charge symmetry breaking interaction. (A decrease of A(48) of 0.1 fm increases zXEc o f 48Ca by 350 keV.) Even for A(40) = 0, one needs only 210 ± 100 keV
23 April 1979
charge symmetry breaking. These values are in agreement with the conclusions of ref. [15], but based on more reliable data. Well-known Hartree-Fock calculations [17] which give A(48) > 0.2 fm disagree with the present result and with the experimental value of zXEc. Some more recent Hartree-Fock calculations [18] lead to a value o f ~(48) = 0.14 fm (and 2x(40) = - 0 . 0 4 fm) which is closer to our result. It is also worthwhile to note that the difference rn(A ) - rn(40 ) can be extracted quite accurately from the 1 GeV proton and electron scattering data, since it is almost independent of A(40): rn(A ) - rn(40 ) = (A /N) [rm (A ) - rm(40)] (3)
- (Z/N)[rp(A) - rp(40)] - ( ( N -
Z)/2N)A(40).
The results are also shown in table 2. We have, for instance rn(48 ) - rn(40 ) = 0.16 ± 0.03 fm whatever choice of A(40) is made and thus independent of any assumption. Finally, let us make a short comment on the most recent measurements [19] o f the f7/2 valence particle radius rex. They also allow to deduce a value for rn(A ) - rn(40), although this is still somewhat preliminary since the model dependence of rex as well as the exchange current corrections have not yet been thoroughly examined. Nevertheless, using the average value [19] of 4.01 ± 0.04 for rex and assuming the rms radius of the 20 neutron core of 48Ca is equal to that of 4°Ca (rn(40) = 3.36 • 0.02 fm), we find that rn(48 ) - rn(40 ) -- 0.20 ± 0.02 fm. This may indicate a small (0.05 ± 0.05) compression in the core neutron rms radius when adding eight neutrons. The exchange current corrections are likely [20] to increase rex by 1 or 2% leading to a somewhat larger core compression. Unfortunately, the present theoretical and experimental uncertainties do not permit a firm conclusion.
Table 2 Results of present analysis using eqs. (2) and (3). A
rm(A ) _ rm(40) a) rp(A) - rp(40) b) rn(A ) - rp(A) rn(A) - rn(40) a) Refs. [1,2].
b) Ref. [9].
40
42
44
48
_ _ -0.04 c) -
0.055 ± 0.02 0.036 0.0 ± 0.03 0.07 _+0.03
0.07 ± 0.02 0.053 0.0 ± 0.03 0.09 ± 0.03
0.10 +-0.02 0.021 0.10 ± 0.03 0.16 ± 0.03
c) Assumed.
Volume 83B, number 1
PHYSICS LETTERS
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