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Analysis of elastic scattering of 0.8 GeV polarized protons from 116Sn and 124Sn
Volume 76B, number 4
PHYSICS LETTERS
19 June 1978
ANALYSIS OF ELASTIC SCATTERING OF 0.8 GeV POLARIZED PROTONS FROM l l 6 S n AND 124Sn G.W. H O F F M A N N University of Texas, Austin, TX 78712, USA and Los Alamos Scientific Laboratory, Los Alamos, NM 87545, USA G.S. BLANPIED l , W.R. COKER and R.P. LILJESTRAND University o f Texas, Austin, TX 78712, USA L. RAY, J.E. SPENCER and H.A. THIESSEN Los Alamos Scientific Laboratory, Los Alamos, NM 87545, USA N.M. HINTZ and M.A. OOTHOUDT 2 University of Minnesota, Minneapolis, MN 55455, USA T.S. BAUER, G. IGO, G. PAULETTA, J. SOUKUP and C.A. WHITTEN Jr. University of California, Los Angeles, CA 90024, USA H. NANN and K.K. SETH Northwestern University, Evanston, IL 60201, USA C. GLASHAUSSER Rutgers University, New Brunswick, NJ 08903, USA D.K. McDANIELS, J. TINSLEY and P. VARGHESE University of Oregon, Eugene, OR 97403, USA Received 6 March 1978
Differential cross section and analyzing power data for elastic scattering of 0.8 GeV polarized protons from ll6Sn and 1~Sn are analyzed in terms of a spin-dependent Kerman-McManus-Thaler formalism. Neutron matter densities and rms radii are deduced with careful attention to sources of error, and found to be in good agreement with Hartree-Fock predictions.
A number o f detailed predictions have been made for the ground state matter densities of neutrons and protons in a wide range o f stable nuclei, using the density-dependent Hartree-Fock approach or the densitymatrix-expansion approximation [ 1 - 4 ] . The predictions for the proton densities can, in many cases, be 1 Present address, New Mexico State University, Las Cruces, NM 88001, USA. 2 Present address, Los Alamos Scientific Laboratory, Los Alamos, New Mexico 87545, USA.
compared to empirical densities extracted from electron scattering data extending to high enough momentum transfer to be sensitive to the nuclear interior. Empirical neutron densities are desirable for comparison to both the empirical proton densities and to the Hartree-Fock predictions. In recent years, hadron-nucleus elastic scattering experiments [ 5 - 1 2 ] at medium energies have provided data through which neutron matter densities can be explored. In this letter we present new data, both angular distributions and analyzing powers, for 0.8 GeV polarized 383
Volume 76B, number 4
PHYSICS LETTERS 0.9
19 June 1978
,
,
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,
i
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Fig. 1. Proton elastic angular distributions and analyzing powers at 800 MeV for targets 116,124Sn. The solid curves are results of analysis using either three-parameter Fermi or gaussian densities as discussed in the text. The 116Sn data and calculation have been reduced by a factor of 100 for clarity. proton elastic scattering from ll6Sn and 124Sn, and the details and results of a theoretical analysis made in an effort to learn more about neutron matter density distributions. The data, shown in fig. 1, were obtained using the High Resolution Spectrometer (HRS) at the Clinton P. Anderson Meson Physics Facility (LAMPF) of the Los Alamos Scientific Laboratory. The experimental arrangements have been described elsewhere [12,13]. The overall normalization of the cross sections is believed accurate to +10%, the center-of-mass angle to +0.05 ° . The analysis of the data was done by solving the Schr6dinger equation with relativistic kinematics and a microscopic, first-order optical potential obtained from the Kerman-McManus-Thaler (KMT) expansion, as modified by Feshbach et al. [14,15]. In this approach, the nucleon-nucleus optical potential in me384
mentum-transfer space depends directly on the free nucleon-nucleon scattering amplitudes and on the Fourier transforms of assumed point-nucleon matter densities of the target nucleus. When all important contributions which are first-order in the nuclear densities are carefully treated, including Coulomb effects and the spin-orbit interaction, relatively unambiguous information on at least the mean square radius of the neutron matter density can be obtained by simultaneously fitting the elastic cross section and analyzing power [12,13,16,17] when the proton matter density is fixed empirically from electron scattering studies [ 18]. The parametrization chosen for the general spin-dependent nucleon-nucleon scattering amplitudes is based upon the form [12]
tp](q 2) = tpj(q 2) + i t~](q2)(~p +~j)"t~,
(1)
Volume 76B, number 4
PHYSICS LETTERS
where] stands for p or n, q = k i - kf is the momentum transfer, and ri = (k i × kf)/Ik i X kfl. The specific parametrization adopted is, with M the nucleon mass,
tp](q 2) = (ikooT//4n)(1 -
iapj)
exp(-Bpiq2),
(2)
tp/(q 2) = (iko Opi /47r)(q2 /4M 2) l/2 X (1 -- iasp])
exp(-Bsp/q2).
If nucleon-nucleon elastic scattering and polarization data at 0.8 GeV were available, one could determine the twelve parameters of eq. (2) from a direct fit to these data. Unfortunately, only elastic cross section data are available at present, and the p + n data are of rather poor quality in the region of momentum transfer most important for our application [ 19]. Keeping the empirical total cross sections fixed at appT, Op nT = 4.73 -+ 0.05 fm 2, 3.8 -+ 0.l fm 2 [19], we obtained good fits to the p + p and p + n cross section data with values app, apn = 0.056 -+ 0.006, - 0 . 2 + 0.02, and Bpp, Bpn = 0.09 -+ 0.005 fm 2, 0.12 -+ 0.008 fm 2. With no complete nucleon-nucleon polarization data at 0.8 GeV, but a relative abundance of 0.8 GeV proton-nucleus polarization data (on targets 12C, 58Ni, 90Zr, 116,124Sn and 208pb) [13] we defined isospin-averaged spin-dependent parameters ~Tp,asp and Bsp and fixed/~sp to a value of 0.2 fm 2 on the basis of preliminary proton-proton polarization data at 0.8 GeV [20]. The parameters {Tp and ~sp were then obtained for each target nucleus by a free search on the available proton-nucleus elastic polarization data. The results for the six nuclei considered have a smooth dependence on N/Z, as expected, namely gp = (23.6 - 9.2(N/Z) + 2) fm 2 and ~sp = 0.17 + 0.28(N/Z) + 0.1. The final values adopted for ll6Sn (124Sn) were 0p = 10.6 fm2 (10.2 fm 2) and ~sp = 0.57(0.55). It should be noted that a direct fit to two-nucleon scattering and polarization data, using eqs. (1) and (2), determines only the product Opi(asp] - ap]). Triplescattering experiments, for instance [21 ], are required to fix Op] and asp ] separately - experiments not yet done at 0.8 GeV. However, while the proton-nucleus analyzing power is also primarily sensitive to the product 0p~sp, such details as peak-to-valley ratios do depend on the separate parameters, and this allowed the determinations quoted in the previous paragraph. Roughly half of the quoted error is due to the ambiguity just discussed, the rest being due to experimental uncertainties, as discussed below.
19 June 1978
Table 1 Point density parameters and rms radii of the density distribu/~2xl/2 tions used in this analysis. ,, 'CH is the rms radius of the charge distribution obtained from electron scattering. The three-parameter Fermi and three-parameter Gaussian charge d,:nsities used for each nucleus differ in rms by about 0.04 fm arid result from two different analyses of two different sets of electron scattering data; see re]. [181. The model dependence of the neutron radii is 0.02 fin (see text). Nucleus w
R z k @2)1/2 (fm) (fro) (fro)
1/2 ~r2,~CH (fro)
116Sn p n p n
0.0 0.0 0.23 0.32
5.36 0.53 1 4.596 5.44 0.56 1 4.701 5.14 2.50 2 4.549 5.02 2.70 2 4.679
4.665
124Sn p n p n
0.0 5.49 0.50 0.0 5.63 0.56 0.26 5.22 2.49 0.42 5.29 2.67
1 1 2 2
4.641 4.826 4.602 4.823
,
4.619 4.709 4.671
We chose for the pohat-neutron and-proton matter densities the three-parameter Fermi (or gaussian) form:
1 + wi 21R2 P/(r) = Poi
1 + exp((r k -
R~)/zlk. )
,
(3)
where ] = p or n, k = 1 (or 2). The point-proton density parameters were obtained numerically by unfolding the nuclear charge density as determined via electron scattering [18]. These parameters (see table 1) were held fixed in the Subsequent analysis, as were all nucleon-nucleon parameters. The Coulomb potential for proton-nucleus scattering was also determined microscopically from the empirical charge density. The procedure was then to search freely on the three neutron matter density parameters, in order to optimize the fit to the experimental data. The usual chi-squared criterion was used to judge the excellence of the fit; a simultaneous fit to cross section and analyzing power was required. The results of the calculations are shown as solid curves in fig. 1. Indistinguishably good fits can be obtained with either the three-parameter Fermi or gaussian forms. The final results for the neutron rms matter radii using each of these forms disagree by about 0.02 fm out of 4.7--4.8 fm, which suggests the degree of model-dependence of our results. (See also caption for table 1, and discussion below.) 385
Volume 76B, number 4
I00|
I
[
PHYSICS LETTERS
t
I
pn(lZ4Sn) -pn("SSn)
',
]
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8oi
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Experimental- 3pF Experimental- 3pG
/" I I
i 2
I 3
4
I
5
I 6
I 7
I 8
I 9
r (fro)
Fig. 2. Folded neutron density differences for l16Sn-124Sn (see text) from a density-matrix-expansion calculation (solid curve), the three-parameter Fermi model result (dashed curve), and the three-parameter gaussian model result (dash-dot curve).
A comparison of the difference Apn(r ) = pln24(r) pnll6(r), predicted by the density matrix expansion approximation to Hartree-Fock [4], with that found empirically in our analysis, is shown in fig. 2. Here the densities shown are not the point-nucleon densities of table 1, but have the finite nucleon radius folded in. It is seen that the trend of both the three-parameter Fermi and gaussian results is in reasonably good agreement with the Hartree-Fock prediction for r ~> 4 fro. Our fits to the data are relatively insensitive to the region 0 < r < 3 fm, and the behavior of the dashed and dot-dash curves in fig. 2 in this region has little physical significance. The difference between the neutron and proton root mean square radii, Amp, for -
-
116 Sn (124 Sn) is, according to the density-matrix expansion approach, 0.14 fm (0.21 fm) [4] whereas the
386
19 June 1978
result of our analysis is, for the three-parameter Fermi density, 0.10 fm (0.18 fm) and for the three-parameter gaussian density, 0.13 fm (0.22 fm). The importance of a simultaneous analysis of angular distribution and polarization data can be seen if one simply leaves out the spin-dependent term and repeats the analysis. So doing reduces Arnp by 0.03 fm. The constraint introduced by inclusion of polarization data is significant in light of the overall error in Arnp, which we now discuss. Taking into account all the sources of error in our analysis and the data, we have tried to assign a realistic overall error to our determination of Arnp. We include the systematic experimental errors in the incident beam energy (-+2 MeV), overall data normalization, angle determination [13] and the empirical proton size (-+0.007 fm) [18], as well as those uncertainties in the KMT optical potential due to the uncertainty in the spin-dependent and independent nucleon-nucleon amplitudes [19]. We also include an estimate of the error in our analysis due to the omission of Pauli-correlation effects [22,23]. If we fix all the quantities just discussed at their upper and lower error bounds, and compare the final result for the neutron matter densities, we arrive at a total uncertainty in Arnp due to these systematic errors of -+0.05 fro, or -+1% of (rn2)1/2. Experimental uncertainties contribute 36% of this error. The statistical error in the data, and the finite maximum momentum transfer attained, also introduce an rdependent error in the extracted form of Pn(r) ~,d consequently an additional error in (r2n)1/2.This model dependence" is most usefully studied by plotting, for the derived densities, an "error envelope" like those of Brissaud et al. [24] or of Friar and Negele [25]. Ways in which this can be done will be the subject of a forthcoming paper. In the present work, we have been content to estimate the extent of model-dependent error simply by comparing the three-parameter gaussian and Fermi results. In conclusion, we have simultaneously analyzed elastic scattering and analyzing power data for 0.8 GeV polarized protons incident on 116,124 Sn to obtain and compare the neutron density distributions of these two nuclei. Our final result for Arn_, combining the systematic and statistical errors for 11~ Sn (124 Sn) is O. 12 + 0.06 fm (0.20 -+ 0.06 fm), in good agreement with predictions. Equally good agreement between theory and experiment is found for differences between
Volume 76B, number 4
PHYSICS LETTERS
p124(r) and p116(r) in the nuclear surface region. Comparing Arn-vf°r 124Sn and l l 6 S n , we obtain Arnp(124 Sn) - ' A r n l , ( l l 6 S n) = 0.08 + 0.02 fm (systematic errors cancel leaving only the statistical error), as compared to 0.07 fan, the Hartree-Fock result; again the good agreement between theory and experiment is impressive. This research was supported in part by the U.S. Department of Energy, The Robert A. Welch Foundation, and the U.S. National Science Foundation.
References [1] [2] [3] [4]
J.W. Negele, Phys. Rev. C1 (1970) 1260. J.W. Negele, Phys. Rev. C9 (1974) 1054. D, Vautherin and D.M. Brink, Phys. Rev. C5 (1972) 626. J.W. Negele and D. Vautherin, Phys. Rev. C5 (1972) 1472, and priv. comm. [5] G.D. Alkhazov et al., Phys. Lett. 42B (1972) 121. [6] J. Thirion, Proc. fifth Intern. Conf. on High-energy physics and nuclear structure, ed. G. Tibell (NorthHolland, 1974) p, 168.
[7] [8] [9] [10] [11 ] [12] [13] [14] [15] [16] [17] [ 18] [19] [20] [21] [22] [23] [24] [25]
19 June 1978
R. Bertini et al., Phys. Lett. 45B (1973) 119. G.D. Alkhazov et al., Phys. Lett. 67B (1977) 402. G.D. Alkhazov et al., Nucl. Phys. A274 (1976) 443. G.D. Alkhazov et al., Phys. Lett. 57B (1975) 47. G.D. Alkhazov et al., report No. LINP-244, Leningrad, 1976 (unpublished). G.S. Blanpied et al., Phys. Rev. Lett. 39 (1977) 1447. G.W. Hoffmann et al., to be published. A.K. Kerman et al., Ann. Phys. (N.Y.) 8 (1959) 551. H. Feshbach et al., Ann. Phys. (N.Y.) 66 (1971) 20. R.M. Lombard, Nuovo Cimento Lett. 18 (1977) 415. A. Chaumeaux et al., Phys. Lett. 72B (1977) 33. C.W. de Jager et al., At. Data and Nucl. Data Tab. 14 (1974) 479. J. Bystricky et al., CEA-N-1547 (E)-Saclay-Aout 1972. M. McNaughton, private communication. M.J. Moravcsik, The two nucleon interaction (Clarendon Press, Oxford, 1963). E. Boridy and H. Feshbacb, Ann. Phys. (N.Y.) 109 (1977) 468. D.R. Harrington and G.K. Varma, preprint. I. Brissaud and M.K, Brussel, Phys. Rev. C15 (1977) 452. J.L. Friar and J.W. Negele, in: Advances in nuclear physics eds. M. Baranger and E. Vogt (Plenum Press, New York, 1975) vol. 8, p. 219.
387
PHYSICS LETTERS
19 June 1978
ANALYSIS OF ELASTIC SCATTERING OF 0.8 GeV POLARIZED PROTONS FROM l l 6 S n AND 124Sn G.W. H O F F M A N N University of Texas, Austin, TX 78712, USA and Los Alamos Scientific Laboratory, Los Alamos, NM 87545, USA G.S. BLANPIED l , W.R. COKER and R.P. LILJESTRAND University o f Texas, Austin, TX 78712, USA L. RAY, J.E. SPENCER and H.A. THIESSEN Los Alamos Scientific Laboratory, Los Alamos, NM 87545, USA N.M. HINTZ and M.A. OOTHOUDT 2 University of Minnesota, Minneapolis, MN 55455, USA T.S. BAUER, G. IGO, G. PAULETTA, J. SOUKUP and C.A. WHITTEN Jr. University of California, Los Angeles, CA 90024, USA H. NANN and K.K. SETH Northwestern University, Evanston, IL 60201, USA C. GLASHAUSSER Rutgers University, New Brunswick, NJ 08903, USA D.K. McDANIELS, J. TINSLEY and P. VARGHESE University of Oregon, Eugene, OR 97403, USA Received 6 March 1978
Differential cross section and analyzing power data for elastic scattering of 0.8 GeV polarized protons from ll6Sn and 1~Sn are analyzed in terms of a spin-dependent Kerman-McManus-Thaler formalism. Neutron matter densities and rms radii are deduced with careful attention to sources of error, and found to be in good agreement with Hartree-Fock predictions.
A number o f detailed predictions have been made for the ground state matter densities of neutrons and protons in a wide range o f stable nuclei, using the density-dependent Hartree-Fock approach or the densitymatrix-expansion approximation [ 1 - 4 ] . The predictions for the proton densities can, in many cases, be 1 Present address, New Mexico State University, Las Cruces, NM 88001, USA. 2 Present address, Los Alamos Scientific Laboratory, Los Alamos, New Mexico 87545, USA.
compared to empirical densities extracted from electron scattering data extending to high enough momentum transfer to be sensitive to the nuclear interior. Empirical neutron densities are desirable for comparison to both the empirical proton densities and to the Hartree-Fock predictions. In recent years, hadron-nucleus elastic scattering experiments [ 5 - 1 2 ] at medium energies have provided data through which neutron matter densities can be explored. In this letter we present new data, both angular distributions and analyzing powers, for 0.8 GeV polarized 383
Volume 76B, number 4
PHYSICS LETTERS 0.9
19 June 1978
,
,
~
o.e,
,
i
,
,
,
,
i
,
ELASTIC 0.8 GeM
e
,
,
,
i
,
,
,
,
i
~
~
e
ELASTIC 0.8 GeV
0
IZ4S n
~.
~SN
to: .~t. v
Id
(,,
"
,-I,,,,
I,,,,
I,,,,
t
I
"°C
cl
o,
~J6Snx
..
o..-
N I
°"
/t% r
io-I
02- ~ al 0
5
I0
15
{:
20
t'
" j
I
,
I
5
. . . .
I , , ' i I , , , i I ,
I0
15
20
ee.m. (deg)
Fig. 1. Proton elastic angular distributions and analyzing powers at 800 MeV for targets 116,124Sn. The solid curves are results of analysis using either three-parameter Fermi or gaussian densities as discussed in the text. The 116Sn data and calculation have been reduced by a factor of 100 for clarity. proton elastic scattering from ll6Sn and 124Sn, and the details and results of a theoretical analysis made in an effort to learn more about neutron matter density distributions. The data, shown in fig. 1, were obtained using the High Resolution Spectrometer (HRS) at the Clinton P. Anderson Meson Physics Facility (LAMPF) of the Los Alamos Scientific Laboratory. The experimental arrangements have been described elsewhere [12,13]. The overall normalization of the cross sections is believed accurate to +10%, the center-of-mass angle to +0.05 ° . The analysis of the data was done by solving the Schr6dinger equation with relativistic kinematics and a microscopic, first-order optical potential obtained from the Kerman-McManus-Thaler (KMT) expansion, as modified by Feshbach et al. [14,15]. In this approach, the nucleon-nucleus optical potential in me384
mentum-transfer space depends directly on the free nucleon-nucleon scattering amplitudes and on the Fourier transforms of assumed point-nucleon matter densities of the target nucleus. When all important contributions which are first-order in the nuclear densities are carefully treated, including Coulomb effects and the spin-orbit interaction, relatively unambiguous information on at least the mean square radius of the neutron matter density can be obtained by simultaneously fitting the elastic cross section and analyzing power [12,13,16,17] when the proton matter density is fixed empirically from electron scattering studies [ 18]. The parametrization chosen for the general spin-dependent nucleon-nucleon scattering amplitudes is based upon the form [12]
tp](q 2) = tpj(q 2) + i t~](q2)(~p +~j)"t~,
(1)
Volume 76B, number 4
PHYSICS LETTERS
where] stands for p or n, q = k i - kf is the momentum transfer, and ri = (k i × kf)/Ik i X kfl. The specific parametrization adopted is, with M the nucleon mass,
tp](q 2) = (ikooT//4n)(1 -
iapj)
exp(-Bpiq2),
(2)
tp/(q 2) = (iko Opi /47r)(q2 /4M 2) l/2 X (1 -- iasp])
exp(-Bsp/q2).
If nucleon-nucleon elastic scattering and polarization data at 0.8 GeV were available, one could determine the twelve parameters of eq. (2) from a direct fit to these data. Unfortunately, only elastic cross section data are available at present, and the p + n data are of rather poor quality in the region of momentum transfer most important for our application [ 19]. Keeping the empirical total cross sections fixed at appT, Op nT = 4.73 -+ 0.05 fm 2, 3.8 -+ 0.l fm 2 [19], we obtained good fits to the p + p and p + n cross section data with values app, apn = 0.056 -+ 0.006, - 0 . 2 + 0.02, and Bpp, Bpn = 0.09 -+ 0.005 fm 2, 0.12 -+ 0.008 fm 2. With no complete nucleon-nucleon polarization data at 0.8 GeV, but a relative abundance of 0.8 GeV proton-nucleus polarization data (on targets 12C, 58Ni, 90Zr, 116,124Sn and 208pb) [13] we defined isospin-averaged spin-dependent parameters ~Tp,asp and Bsp and fixed/~sp to a value of 0.2 fm 2 on the basis of preliminary proton-proton polarization data at 0.8 GeV [20]. The parameters {Tp and ~sp were then obtained for each target nucleus by a free search on the available proton-nucleus elastic polarization data. The results for the six nuclei considered have a smooth dependence on N/Z, as expected, namely gp = (23.6 - 9.2(N/Z) + 2) fm 2 and ~sp = 0.17 + 0.28(N/Z) + 0.1. The final values adopted for ll6Sn (124Sn) were 0p = 10.6 fm2 (10.2 fm 2) and ~sp = 0.57(0.55). It should be noted that a direct fit to two-nucleon scattering and polarization data, using eqs. (1) and (2), determines only the product Opi(asp] - ap]). Triplescattering experiments, for instance [21 ], are required to fix Op] and asp ] separately - experiments not yet done at 0.8 GeV. However, while the proton-nucleus analyzing power is also primarily sensitive to the product 0p~sp, such details as peak-to-valley ratios do depend on the separate parameters, and this allowed the determinations quoted in the previous paragraph. Roughly half of the quoted error is due to the ambiguity just discussed, the rest being due to experimental uncertainties, as discussed below.
19 June 1978
Table 1 Point density parameters and rms radii of the density distribu/~2xl/2 tions used in this analysis. ,, 'CH is the rms radius of the charge distribution obtained from electron scattering. The three-parameter Fermi and three-parameter Gaussian charge d,:nsities used for each nucleus differ in rms by about 0.04 fm arid result from two different analyses of two different sets of electron scattering data; see re]. [181. The model dependence of the neutron radii is 0.02 fin (see text). Nucleus w
R z k @2)1/2 (fm) (fro) (fro)
1/2 ~r2,~CH (fro)
116Sn p n p n
0.0 0.0 0.23 0.32
5.36 0.53 1 4.596 5.44 0.56 1 4.701 5.14 2.50 2 4.549 5.02 2.70 2 4.679
4.665
124Sn p n p n
0.0 5.49 0.50 0.0 5.63 0.56 0.26 5.22 2.49 0.42 5.29 2.67
1 1 2 2
4.641 4.826 4.602 4.823
,
4.619 4.709 4.671
We chose for the pohat-neutron and-proton matter densities the three-parameter Fermi (or gaussian) form:
1 + wi 21R2 P/(r) = Poi
1 + exp((r k -
R~)/zlk. )
,
(3)
where ] = p or n, k = 1 (or 2). The point-proton density parameters were obtained numerically by unfolding the nuclear charge density as determined via electron scattering [18]. These parameters (see table 1) were held fixed in the Subsequent analysis, as were all nucleon-nucleon parameters. The Coulomb potential for proton-nucleus scattering was also determined microscopically from the empirical charge density. The procedure was then to search freely on the three neutron matter density parameters, in order to optimize the fit to the experimental data. The usual chi-squared criterion was used to judge the excellence of the fit; a simultaneous fit to cross section and analyzing power was required. The results of the calculations are shown as solid curves in fig. 1. Indistinguishably good fits can be obtained with either the three-parameter Fermi or gaussian forms. The final results for the neutron rms matter radii using each of these forms disagree by about 0.02 fm out of 4.7--4.8 fm, which suggests the degree of model-dependence of our results. (See also caption for table 1, and discussion below.) 385
Volume 76B, number 4
I00|
I
[
PHYSICS LETTERS
t
I
pn(lZ4Sn) -pn("SSn)
',
]
I
I
/~\
8oi
// ,,"\\\
/ /
\[
i
oc
4" 0 x
i ,'/ I //
\
\\!
4C
% •-
2C
#o
!
/
/ -20
/
/ -4.0 0
./
--
..... ....
/
/
DUE
Experimental- 3pF Experimental- 3pG
/" I I
i 2
I 3
4
I
5
I 6
I 7
I 8
I 9
r (fro)
Fig. 2. Folded neutron density differences for l16Sn-124Sn (see text) from a density-matrix-expansion calculation (solid curve), the three-parameter Fermi model result (dashed curve), and the three-parameter gaussian model result (dash-dot curve).
A comparison of the difference Apn(r ) = pln24(r) pnll6(r), predicted by the density matrix expansion approximation to Hartree-Fock [4], with that found empirically in our analysis, is shown in fig. 2. Here the densities shown are not the point-nucleon densities of table 1, but have the finite nucleon radius folded in. It is seen that the trend of both the three-parameter Fermi and gaussian results is in reasonably good agreement with the Hartree-Fock prediction for r ~> 4 fro. Our fits to the data are relatively insensitive to the region 0 < r < 3 fm, and the behavior of the dashed and dot-dash curves in fig. 2 in this region has little physical significance. The difference between the neutron and proton root mean square radii, Amp, for -
-
116 Sn (124 Sn) is, according to the density-matrix expansion approach, 0.14 fm (0.21 fm) [4] whereas the
386
19 June 1978
result of our analysis is, for the three-parameter Fermi density, 0.10 fm (0.18 fm) and for the three-parameter gaussian density, 0.13 fm (0.22 fm). The importance of a simultaneous analysis of angular distribution and polarization data can be seen if one simply leaves out the spin-dependent term and repeats the analysis. So doing reduces Arnp by 0.03 fm. The constraint introduced by inclusion of polarization data is significant in light of the overall error in Arnp, which we now discuss. Taking into account all the sources of error in our analysis and the data, we have tried to assign a realistic overall error to our determination of Arnp. We include the systematic experimental errors in the incident beam energy (-+2 MeV), overall data normalization, angle determination [13] and the empirical proton size (-+0.007 fm) [18], as well as those uncertainties in the KMT optical potential due to the uncertainty in the spin-dependent and independent nucleon-nucleon amplitudes [19]. We also include an estimate of the error in our analysis due to the omission of Pauli-correlation effects [22,23]. If we fix all the quantities just discussed at their upper and lower error bounds, and compare the final result for the neutron matter densities, we arrive at a total uncertainty in Arnp due to these systematic errors of -+0.05 fro, or -+1% of (rn2)1/2. Experimental uncertainties contribute 36% of this error. The statistical error in the data, and the finite maximum momentum transfer attained, also introduce an rdependent error in the extracted form of Pn(r) ~,d consequently an additional error in (r2n)1/2.This model dependence" is most usefully studied by plotting, for the derived densities, an "error envelope" like those of Brissaud et al. [24] or of Friar and Negele [25]. Ways in which this can be done will be the subject of a forthcoming paper. In the present work, we have been content to estimate the extent of model-dependent error simply by comparing the three-parameter gaussian and Fermi results. In conclusion, we have simultaneously analyzed elastic scattering and analyzing power data for 0.8 GeV polarized protons incident on 116,124 Sn to obtain and compare the neutron density distributions of these two nuclei. Our final result for Arn_, combining the systematic and statistical errors for 11~ Sn (124 Sn) is O. 12 + 0.06 fm (0.20 -+ 0.06 fm), in good agreement with predictions. Equally good agreement between theory and experiment is found for differences between
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p124(r) and p116(r) in the nuclear surface region. Comparing Arn-vf°r 124Sn and l l 6 S n , we obtain Arnp(124 Sn) - ' A r n l , ( l l 6 S n) = 0.08 + 0.02 fm (systematic errors cancel leaving only the statistical error), as compared to 0.07 fan, the Hartree-Fock result; again the good agreement between theory and experiment is impressive. This research was supported in part by the U.S. Department of Energy, The Robert A. Welch Foundation, and the U.S. National Science Foundation.
References [1] [2] [3] [4]
J.W. Negele, Phys. Rev. C1 (1970) 1260. J.W. Negele, Phys. Rev. C9 (1974) 1054. D, Vautherin and D.M. Brink, Phys. Rev. C5 (1972) 626. J.W. Negele and D. Vautherin, Phys. Rev. C5 (1972) 1472, and priv. comm. [5] G.D. Alkhazov et al., Phys. Lett. 42B (1972) 121. [6] J. Thirion, Proc. fifth Intern. Conf. on High-energy physics and nuclear structure, ed. G. Tibell (NorthHolland, 1974) p, 168.
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R. Bertini et al., Phys. Lett. 45B (1973) 119. G.D. Alkhazov et al., Phys. Lett. 67B (1977) 402. G.D. Alkhazov et al., Nucl. Phys. A274 (1976) 443. G.D. Alkhazov et al., Phys. Lett. 57B (1975) 47. G.D. Alkhazov et al., report No. LINP-244, Leningrad, 1976 (unpublished). G.S. Blanpied et al., Phys. Rev. Lett. 39 (1977) 1447. G.W. Hoffmann et al., to be published. A.K. Kerman et al., Ann. Phys. (N.Y.) 8 (1959) 551. H. Feshbach et al., Ann. Phys. (N.Y.) 66 (1971) 20. R.M. Lombard, Nuovo Cimento Lett. 18 (1977) 415. A. Chaumeaux et al., Phys. Lett. 72B (1977) 33. C.W. de Jager et al., At. Data and Nucl. Data Tab. 14 (1974) 479. J. Bystricky et al., CEA-N-1547 (E)-Saclay-Aout 1972. M. McNaughton, private communication. M.J. Moravcsik, The two nucleon interaction (Clarendon Press, Oxford, 1963). E. Boridy and H. Feshbacb, Ann. Phys. (N.Y.) 109 (1977) 468. D.R. Harrington and G.K. Varma, preprint. I. Brissaud and M.K, Brussel, Phys. Rev. C15 (1977) 452. J.L. Friar and J.W. Negele, in: Advances in nuclear physics eds. M. Baranger and E. Vogt (Plenum Press, New York, 1975) vol. 8, p. 219.
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