Elastic electron scattering from 40Ar

Nuclear Physics A379 (1982) 396-406 © North-Holland Publishing Company

ELASTIC ELECTRON SCATTERING FROM °°Ar

C.R. OTTERMANN*, CH . SCHMITT**, G.G . SIMON", F. BORKOWSKI** and V.H . WALTHER Institut für Kernphysik, Johannes Gutenberg-Universität, D-6500 Mainz, Federal Republic of Germany

Received 16 October 1981 Abelrret : Cross sections for elastic electron scattering from 4°Ar have been measured for the momentum transfer range from 0.59 to 1.31 fm ~. We have analyzed with the Fourier-Bessel snaslz our data es well as the data of former experiments. The rms charge radius we have found is 3.423(14) fm. The results are in excellent agreement with latest muonic data. Furthermore, we have reanalyzed former °°Ca data and have discussed the °° Ca 4°Ar charge distribution.

1. Introduction In this paper we want to report on our extensive investigations of the 4°Ar nucleus. For these investigations we have measured elastic electron scattering cross sections with high accuracy. We have reanalyzed all former data and have compared our results with muonic atom data . There are many reasons for carrying out this experiment and the overall analysis : (i) The published argon data measured in different laboratories differ in part of our kinematic region by up to 20% [refs.')]. (ü) The determined charge distributions and the derived rms radii are not reliable and are in disagreement with all muonic data. a°Ar is a nucleus with the same mass as the closed shell nucleus a° Ca, which can be described by shell-model calculations very well. Therefore, a comparison of the charge distributions and of the charge rms radius of a°Ar with those of a°Ca is of high interest in respect to the nucleon distribution and to the shell effects. Another point, which we want to clarify in this experiment is the discrepancy between the electron scattering and the muonic atom data, which seems to be systematic . These data show remarkable deviations, not only in the case of argon and, thus, one has assumed that real physical effects could lead to these disagreements. In this paper we discuss, after a short description of the experimental set up, our elastic electron scattering data of argon. Then, we examine the published data of other laboratories and include them after some corrections in the overall analysis. This leads to a new charge distribution and rms value for argon. The calcium data are treated in the same way and the di$erence between the charge distributions of a°Ca and a°Ar is discussed. * Supported by the Studienstiftung des Deutschen Volkes . *'Supported by the Deutsche Forschungs Gemeinschaft. 396

C.R. Ottermann et ai. / Elastic electron scattering

39 7

2. Experimental set ap The experiment was performed on the Mainz 400 MeV electron linear accelerator. The details of the experimental procedure have been reported s.e) earlier . Only those details, which are important to the data analysis and the determination of the normalization error are discussed here . The electron beam with an energy spread of 0.1 % was directed onto the target within a typical area of 1 x 2 mm Z. The position of the spot was fixed on the target to t 1 mm by a split foil monitor. The incident charge was measured by a ferrite induction current monitoring system . The absolute calibration error was 0.1% for the beam currents, which were between 0.1 and 3 WA. For measurement of the cross section we made use of two magnetic spectrometers . The first one was a conventional double focussing 180° spectrometer, which could

be set at different scattering angles . The angle of the spectrometer with respect to the incoming beam direction was known with an error of 0.02° and could be reproduced with an uncertainty of 0.004°. The scattered electrons were recorded by a 300 channel detector with overlapping scintillators arranged in a ladder configuration and with ~erenkov backing counters. The intrinsic resolution was 0.025% . With this spectrometer the absolute energy calibration was determined to 10 .07% . The second spectrometer consisted of two quadrupole and a 13° bending magnet mounted at a fixed scattering angle of 28°. The electrons were detected by a ~erenkov counter. It was used for monitoring the stability of the experimental set up by measuring the product of target thickness times number of incident electrons. We used a high-pressure gas target system S) with 3 èells, one filled with argon, one with hydrogen and one cell was evacuated (dummy target). To suppress electrons scattered in the material of the walls of the cells, we used a special collimator system . The accepted scattering volume in the target cell was defined by the collimator system and the incoming electron beam. Therefore, no errors arise by the disposition of the target cells. The collimator system was aligned by an optical method with an accuracy of 0.1 mm . The target cells were filled with a pressure up to 12 bar at the TH Darmstadt. After filling the cells, the temperature and the pressure of the gas were measured and then the targets were sealed . The absolute gas density - after corrections with virial coefficients - was known with an uncertainty of less than 0.05 % . The gas diffusion rate for the gases used was measured to be less than 0.1 % per year . The target cells were filled at the beginning of the measurement and, therefore, we can neglect this error. The gases used contained a negligible amount of impurities (5 ppm), which were measured with a mass spectrometer during the filling procedure . Changes of the target densities by heating effects were measured by varying the beam intensity S). An additional control during the measurements was carried out by the fixed-angle spectrometer . The uncertainty in the ratio of the densities was of the order of 0.04% .

39 8

C.R . Ottermann et al. / Elastic electron scattering

To obtain absolute cross sections we used electron scattering at hydrogen for normalization. The absolute electron proton cross section in a four-momentum transfer up to 1 .9 fm z is known with an overall normalization error of only 0.5% [ref, e)]. By recoil effects the elasric peak from electrons scattered on hydrogen is wider than that from electrons scattered on argon. To avoid different efficiencies in the detector system, the magnetic field was changed run by run in such a way that the calculations for the ratio of e-p to e-Ar scattering could be made for each counter individually . Thus, the systematic errors caused by the relative efficiency of each counter were eliminated (method of individual counter channel weighting) . The radiative corrections were applied according to the procedure of Mo and Tsai') . Also, the corrections for real bremsstrahlung') and for Landau-straggling s) were taken into account. The overall normalization error in the cross section, linearly added single shares, was 0.8% and the systematic errors were in the order of 0.8% .This is an improvement of the absolute normalization error of one order of magnitude compared to all former argon measurements . The measurements were made at incident energies of 115.75 MeV and 249.03 MeV at scattering angles between 30° and 95°. This covers a range of effective momentum transfer from 0.59 fm' up to 1 .31 fm- 1 . The choice of these conditions led to cross sections in a kinematic region, where former experiments had shown the biggest discrepancies. In table 1 we have listed the measured cross sections . TABLE 1

Experimental results for 4°Ar E1 (Mew

B (deg)

115.75

55 .017 60 .012 65 .008 70 .005 75 .021 80 .015 85 .020 90 .026 95 .040

249.03

30.019 35 .010 40.010 45 .009 50 .012 55 .017 60 .012

dl1

Stat . error (%)

Syst . error (°k)

Norm . error (%)

6302 3417 1867 1027 553.3 295.1 158.6 82 .81 42 .92

0.8 0.6 0.7 0.6 0.5 0.7 0.7 0.7 0.8

9129 2310 536.7 100.0 13 .68 2.963 3.041

0.7 0.5 0.7 0.7 0.7 0.8 0.8

0.7 0.7 0.7 0.7 0.8 0.8 0.8 0.9 1 .0

0.8 0 .8 0.8 0.8 0.8 0.8 0.8 0.8 0.8

fm ( x 10-' sr~

0.7 0.8 0.9 1.1 1 .4 0.6 0.8

0 .8 0.8 0.8 0.8 0.8 0.8 0.8

C.R. Ottermann et al. / Elastic electron scattering

399

3. Data analysis We extracted the charge distribution from the measured cross sections by means of the Fourier-Bessel ansatz 9). In all former analyses a two- or three-parameterc Ferrai distribution was fitted to the argon cross section. But this method is inflexible and strongly model-dependent as the parameters depend on the measured q-range. In the Fourier-Bessel ansatz, the ground-state charge distribution is described by P(r)=~0~ 1

a~1o(gJ)~

r
(1)

R (cut-off radius) and vm ~ are model parameters, a are adjustable parameters of the fitting routine, q,, is detenmined by the condition l0(4~)=0-i4~=R v . As the cross sections are measured only in a limited q-range, we need a boundary condition for the asymptotic behaviour of the nuclear form factor dependence with growing momentum transfer. This is given by F(q) _ ~ 4

e-gza~ie

with (rô)1~2 = 0.862 fm. The constant c is usually adjusted at the last measured form factor maximum. The value of vm~ is directly correlated to the accepted momentum transfer range of the data analysis, R The Fourier-Bessel fit to the measured cross sections is influenced by the parameters R, v~ and c. In principle, these parameters are correlated. The influence of v~ and c on R is small and, therefore, the choice of vm~ and c is not critical . They were fixed at v~ =15 and c = 0.171 fm-4. Then the value of R was optimised in a XZ test until the influence of this parameter on the Fourier-Bessel fit vanished . The optimal parameter was found to be R = 9 fm. Fig. 1 shows some results of the (rz)'iz fit routine for Xz/point, and p(r=0) for different values for R. 4. Results In the first step of the data analysis with the Fourier-Bessel fit, we only used the data of this experiment. Fig. 2 shows the relative deviation of the measured cross sections to the fit values ~~,/~~. In the whole q-range, including the first diffraction z minimum of the form factor (at 1 .2 fm-'), the X /point of the fit is 0.71. This

C.R . Otttrmann tt al . / Elastic electron scattering

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9

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C.R . Ottennann tt al. / Elastic electron scattering

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x% point " 0.79 " ~"99s7s~v ~~s " a7o ~ E,"2kAaiMrV ~h " a79

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Fig . 2 . The relative deviation of the measured cross section to the fit values .

agreement is valid for both data sets at 116 and 249 MeV. The derived root mean square radius is (rZ)tiZ =

3.424(S) fm .

The total error is found by the quadratic addition of the statistical error (7 x 10 -a fm) and the model uncertainty (3 x 10 -s fm).

10

717

0

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Fig . 3 . The relative deviations of the former cross sections since 1971 with their quoted best fits to our fit values .

402

C.R. Ostermann et al. / Elastic electron scattering

In fig. 3 we have plotted the values of all former measurements since 1971 with their quoted best fits . Disagreements between the former measurements up to 25% are seen . The data of Finn et al. a) show a systematic deviation from our fit with increasing momentum transfer. The best fit is a two-parametric Fermi distribution . The biggest discrepancy is seen for point Pt in fig. 3 . The large value arises from a too large cut-off energy for the integration of the elastic peak . Subsequent discussions with Finn have confirmed this fact . The deviation of the remaining points to our fit can be eliminated by changing the total normalization by -4% . Wendling et al. 2) used a three-parametric gaussian distribution for their fit. The data were measured with a liquid-argon target and, therefore, fluctuations in the target thickness caused by heating effects and bubbles can not be excluded . By changing the normalization of all points by 8%,the data are in agreement with our best fit. The data of Groh et al . s) show big fluctuations and big error bars . The data of Schütz et al. t) at small momentum transfer remained unchanged in normalization. The final result is shown in fig. 4. The solid line shows the best fit including all data with a X2 /p =1 .09. Table 2 gives the Fourier-Bessel coefficients together with the statistical errors and the total uncertainty. The charge distribution is shown in fig. 5 . The statistic error band and the total error band are plotted. The rms radius of a° Ar with the total error is (r2)1~2

= 3 .423(14) fm .

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0

DA T7 I Ref SAS Tl IRef Mï74 IRef NBS76 IRef

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QS

i 1.0

1.S

?0

q."e

Fig. 4. The relative deviation of all cross sections since 1971 to the final fit.

(fm'7

C.R. Ottermanri et al. / Elastic electron scattering

403

TABLE 2 The Fourier-Beseel coefficients for ~ °Ar v

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

a (fm-3 )

Stat . error (fm - ')

Total error (fm-3 )

1 .69158E-3 3 .0740E-3 1 .1223E-3 -9 .313E-4 -7 .543E-4 -2 .4E-6 5 .11E-5 -2 .289E-5 6 .65E-6 -1 .10E-6 -2 .4E-7 3 .4E-7 -2 .1E-7 1 .0E-7 -4 .3E-8

1 .01E-6 2.81E-6 3 .54E-6 6.98E-6 1 .17E-5 6.34E-5 1 .01E-4 7 .26E-5 4 .56E-5 2 .83E-5 1 .76E-5 1 .09E-5 6 .55E-6 3 .8E-6 2 .2E-6

2 .95E-6 1 .497E-5 1 .113E-5 7 .11E-6 1 .32E-5 6 .51E-5 1 .049E-4 7 .28E-5 4 .56E-5 2 .85E-5 1 .77E-5 1 .09E-5 6 .59E-6 3 .8E-6 2 .2E-6

5 . Electron ecatterbig and muonic atom data The expression directly correlated to the muonic atom data is the Barren moment (r k e-ai)

= 4Tl

J

P(r)t'k+z

~ ar dr'

(S)

where a and k are specific nuclear constants (for a°Ar a = 0 .0622 fm-1 , k = 2 .116) [ref, 11 )]. To compare our electron scattering data with the muonic data, we have derived tote Barrett moment from our charge distribution . Table 3 compares the ao~, muonic data' Z) with the electron scattering data of this experiment . Excellent agreement is seen in the Barrett moment . The latest rms from muonic data 13) is 3 .425(1) fm . The old statement that muonic and electron scattering data always disagree is, at least in the case of argon, no longer true .

6 . The charge dieMbation of ~Ca and'°Ar For the Calcium analysis we have used the three latest published data sets' °-' 6 ) in the kinematic region up to 3 .64 fm -1 . The Fourier-Bessel analysis was the same as in the case of argon . As the absolute normalization of the electron scattering data of `°Ca from different laboratories were not known well enough, each data set was given a free normalization parameter in the fit routine . Therefore, the overall normalization was determined by the `°Ca Barren moment from the latest

404

CR . Otternsann et al. / Elastic electron scattering plrl

rf~ p -

rit --- statislicd srror band = total error band

p(rl+lE=a~j+lA,rl ~0

r
4rrJp(rli dr=1

7

Fig. 5. The resulting charge distribution of 4°Ar .

TABLE 3

Barrett moments for `°Ar

this experiment ref. ") ref .' 2)

(r k e_o . )

Total error

10.70 fmk 10.74 fm k 10 .71 fmk

0.08 fm k 0.03 fm k 0.03 fm k

rCfmJ

C R. Ottermann et al. / Elastic electron scattering Q10

405

pl .l Clm- ~7

409 408 0.07 QO6 QOS 0.04 403 OA2 0.01 0 Fig. 6a . The charge distribution of °°Ca and °°Ar .

4

5

r [fm]

Fig . 6b . The r2 -weighted charge difference

°°Ca-°°Ar .

muonic data set "). The rms radius we found for a°Ca is (~z)t~z

-

3,477(5) fm ,

which is bigger by 0.054(19) fm than the argon rms radius . Figs. 6a and b show the charge distributions of a°Ca and a°Ar and the rz-weighted charge difference .

406

C.R . Ottennann et al. / Elastic electron scattering

The °°Ar nucleus needs two protons for the closure of the ld3iZ shell. Then, the magic number 20 is achieved . The main peak in fig. 6b shows clearly the influence of the two added ld3iz protons and, hence directly the shell effect at 3.5 fm. A more detailed examination of the charge difference between °°Ar and a°Ca requires additional measurements at higher q. It is a pleasure to thank the entire laboratory staff of the Institut für Kernphysik, Universität Mainz, for overall help. We take this opportunity to express our thanks to Prof . H. Frank and his co-workers (TH Darmstadt) for their support by filling the gas targets. References 1) W. Schütz and H. Frank, Z. Phys. 243 (1971) 132 R.D . Wendung and V.H . Welther, Nucl. Phys. A219 (1974) 450 J.L. Groh, R.P . Singhal and H.S . Caplan, Can. J. Phys . 49 (1971) 2073 J.M. Finn, H. Crannell, P.L. Hallowell, J.T. O'Brien and S. Penner, Nucl . Phys. A274 (1976) 28 G.G . Simon, Ch. Schnitt, F. Borkowski, C. Ostermann, V.H . Welther, D. Bender and A. von Gunten, Nucl. matt . 158 (1979) 185 6) G.G . Simon, Ch . Schnitt, F. Borkowski and V.H . Welther, Nucl . Phys . A333 (1980) 381 7) L.W . Mo and Y.S . Tsai, Rev. Mod. Phys . 41 (1969) 205 ; L.C. Maxinom, Rev . Mod. Phys . 41 (1969) 193 8) L. Landau, J. Phys. (USSR) 8 (1944) 201 9) B. Dreher, J. Friedrich, K. Merle, H. Rothhaas and G. Lïilus, Nucl . Phys . A235 (1974) 219 ; H. Euteneuer, J. Friedrich end N. Voegler, Phys. Rev . Lett . 36 (1976) 129 10) R.C . Bettelt, Phys . Lett . 33B (1970) 388 11) R. Engfer, H. Schneuwly, J.L . Vuilleumier, H.K . Walter and A. Zehnder, Atomic Data and Nucl. Data Tables 14 (1974) 509 12) H. Daniel, H.-J. Pfeiffer, K. Springer, P. Stceckel, G. Backenstoss and L. Tauschet, Phys. Lett. 48B (1974) 109 13) G. Fricke, G. Mallot, T.Q . Phan, G. Piper, A. Rüetschi, L.A . Schellet, L. Schellenberg and H. Schneuwly, Schweiz. Phys . Ges., Spring Meeting, April 1981, Neuchgtel, p. 50 14) B.B .P . Sinha, G.A . Peterson, R.R . Whitney, I. Sick and J.S. McCarthy, Phys. Rev. C7 (1973) 1930 15) J.B . Bellicard, P. Bounin, R.F. Frosch, R. Hofstadter, J.S. McCarthy, F.J . Uhrtrane, B.C. Clark and R. Herman, Phys . Rev. C16 (1977) 1262 . 16) B. Frais, private communication 17) H .-D . Wohlfahrt, E.B . Shere, M.V . Hcehn, Y. Yamazaki, G. Fricke and R.M . Steffen, Phys . Lets. 37B (1978) 131 2) 3) 4) 5)