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Systematics of Radii and Nuclear Charge Distributions Deduced from Elastic Electron Scattering, Muonic X-Ray

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Systematics of Radii and Nuclear Charge Distributions Deduced from Elastic Electron Scattering, Muonic X-Ray SYSTEMATICS OF RADII AND NUCLEAR CHARGE DISTRIBUTIONS DEDUCED FROM ELASTIC ELECTRON SCATTERING, MUONIC X-RAY AND OPTICAL ISOTOPE SHIFT MEASUREMENTS H.J. Emrich+), G. Fricke+^, M. Hoehn+++), K. Käser++), M. Mallot+), H. Miska+), B. Robert-Tissot++), D. Rychel+^, L. Schaller++), L. Schellenberg"^, H. Schneuwly++^, B. Shera+++), H.G. Sieberling+), R. Steffen++++), H.D. Wohlfahrt+++), Y. Yamazaki+++) +) Institut für Kernphysik, Universität Mainz, Mainz, Germany ++) Institut de Physique de l'Université de Fribourg, Pérolles, Fribourg, Suisse +++) Los Alamos Scientific Lab., University of California, Los Alamos N.M., U.S.A. ++++) Department of Physics, Purdue University, Lafayette, Indiana, U.S.A. Abstract nuclear charge distributions between neighbouring nuclei. The nuclear charge distribution and nuclear charge distribution differences have been investiga• These quantities are determined by high energy ted by 350 MeV elastic electron scattering at the elastic electron scattering measurements - up to Institut für Kernphysik, Universität Mainz. 350 MeV at the Mainz University(l>2,3,4) _ an(j ^ muonic X-ray studies: the region ^Ca to 70Zn in Muonic X-ray measurements yield complementary Los Alamos (5,6,7) and the region 69Ga to 100Mo at information to electron scattering results. Both S.I.N.(8). All the latter data are not yet pub• experimental data are analysed in an almost model lished and are to be regarded as preliminary. independent way. Muonic X-ray measurements have been performed for the region 40Ca up to IOOMO. Elastic electron scattering yields information about the charge distribution and charge distribu• Muonic X-ray transition energies allow the deter• tion differences. Electron scattering data essen• mination ofone radial parameter R^ - the Barrett tially determines the Fourier transform of the nucle• radius - with high precision. This Barrett radius ar charge distribution p(r) even in the framework combined with the charge distribution from elastic of a phase shift analysis. Therefore, these measure• electron scattering yields the following precise ments determine directly the charge distribution radial parameters <r2>l/2,<r4>l/4 and <r6>l/6. With and charge distribution differences, provided the the absolute values of these radial moments,it is measurements are accurate enough and extend over a possible to determine the two optical constants: the sufficiently large range of momentum transfer. By electron density at the nucleus and the specific using a Fourier-Besse! expansion technique^), rea• mass shift. Thus,the calibrated optical isotope shift listic error bands can be derived for the charge data from the off stability region yield the absolute distributions and their differences, which reflect values of these RMS radii. Therefore,it will be both the statistical error and the error due to the also possible to compare the RMS-radii of isobaric limited range of measured momentum transfer. As an nuclei with good accuracy. example Fig. 1 shows the charge distribution of 5&Ui with the correlated errors. The differences in RMS-radii between isotopes p(r) (102 e 1m3) and isotones show systematic tendencies with a strong shell effect. As one example: one observes that the differences in radii are largest at the beginning of a neutron or proton shell and decrease linearily to the end of a shell. Experimental re• sults will be compared with spherical Hartree-Fock calculations and the influence of the quadrupole deformation parameter ß. Data of odd even staggering effects on isotopes and isotones will be discussed in the N=28 and N=50 region. Some differences of charge distributions will also be presented. 1. Introduction Detailed investigation of the rearrangement of the nuclear charge distribution in response to the addition of neutrons or protons throughout a whole series of isotopes provides a challenging test of one's understanding of the structure of nuclear ground states. In this collaboration of the laboratories of Mainz, Los Alamos and S.I.N.-Fribourg a systematic survey is being performed of all stable isotopes and isotones in the region between 40ca to lO^Mo. It is the goal of these experiments to determine nucle• 0 2 A 6 8 r(fm) ar charge distributions and accurate differences of FIG. 1 Fourier-Bessel deduced charge distribution of 60Ni. The error band includes the statistical error of the cross sections, the model error due to the unmea• presented by G. Fricke sured momentum transfer range, and systematic uncer• tainties. - 33 - Muonic data, on the other hand, determine one where VyP(r) and VyS(r) are the potentials genera• integral quantity like the Barrett radial moment ted by the 2p and Is bound muons, respectively. <rke-ar> of the nuclear charge distribution(10). As an example for Ni, Z=28, k=2,12 and a=0,076 fnT1. To extract a "model independent" radius, the The precision of the muonic data is almost 5-10 procedure proposed by Barrett(lO) has been used. times better compared with elastic electron scatte• Barrett has shown that the potential difference in ring radii. To compare these results with optical eq. (3) can be approximated by isotope shift data,one needs the radial moments v2p_Yls = A+Brke-ar (3) <rn>l/n with n=2,4,6. A combination of electron scattering data with the muonic X-ray measurements y y v allows one to determine the even radial moments with almost the same precision as the Barrett ra• which demonstrates that the quantity dius. With the knowledge of these radial moments k ar k ar from three isotopes it is possible to extract the <r e' >=/p(r)r e" dT (4) two constants of the optical isotope shift measure• ments: the density of the electron at the nucleus is model independent within the experimental error. and the specific mass shift. The value of k in this equation is different for different transitions and also varies slowly as With an absolute calibration of these two op• a function of Z for a given transition. The value tical constants,the absolute values of the RMS-ra• of a varies linearly with Z and is usually kept dii of long chains of optical isotope shift measu• fixed for all transitions of one element (for exam• rements from the off stability region are avail• ple 70zn: a=0,079 fm"l, k=2.130). able: e.g. Hg, Ba, Rb, Cs a.s.o. An equivalent radius - the "Barrett" radius The results of these three methods can be in• RB - can be defined by equating this quantity with terpreted entirely on the basis of electromagnetic the corresponding quantity of an uniformly charged interaction and exhibit; therefore, the following sphere of radius Rß: advantages exist: B ar 2 k ar 3RB-3^ rV r dr = <r e" > (5) a) Since the electromagnetic interaction is exactly known, an accurate determination of The data in this paper are given without errors nuclear structure properties is possible. and are preliminary. After a careful analysis,the experimental error for Rß will be in the order of b) Compared with the nuclear forces the electro• ±l«10_3fm, whereas the uncertainties due to theore• magnetic interaction is weaker,therefore in tical corrections (for the most part nuclear polari• these experiments the distortion of the nu• zation) is ±5-10-3fm. The theoretical error for clei to be measured can be accounted for by radius differences reduces to ±l«10"3fm. only small correction terms. The older muonic atom data and elastic electron 3. High Energy Elastic Electron Scattering scattering data could be described in good approx• imation with a two parameter Fermi charge distribu• Though this method is well known and has been tion. By assuming a constant surface thickness and described before, it is worthwhile to summarize a constant central density, the equivalent radii of few important facts(12»13): a homogeneously charged sphere deduced from the RMS radii of the above Fermi distributions were re• The elastic electron scattering cross section presented by the following A-dependence: in first order Born approximation is given by 1/3 1/3 1 Req = 1.12 A + 0.009A" - 1.513A" (1) in good agreement with the experiments(11). In the present paper results are reported from -r- describes the point charge cross muonic X-ray and elastic electron scattering mea• a^Mott section. surements in the region Ca-Mo. The experiments clearly show systematic effects of the shell struc• For spherical, 1=0, nuclei the nuclear form factor ture, which will be discussed. is given by 2. Muonic X-Ray Measurements F(q2) =4^p(r)lM3HdT (7) The experimental apparatus and the procedure to measure and extract nuclear charge parameters where q stands for the momentum transfer from muonic isotope and isotone shifts are des• q=(2E0/nc)sin 0/2. cribed in detail in the paper of E.B. Shera(^). Nevertheless,some remarks and definitions may be A variation of the momentum transfer q can be useful. The muon binding energy of adjacent nuclei achieved by variation of either the incoming elec• differ slightly from each other due mainly to two trons EQ or of the scattering angle e. This results effects: the difference in the mass (which can be in a variation of the weighting function sin (qr)/qr accounted for by the reduced mass),and the size of for the charge distribution in the nuclear form the charge distribution ("field shift"). The "field factor. The electron scattering experiments covered shift" has been used to determine differences in the a range of q: 0.5fm"l<q<2.5fm-I, which allows one to nuclear radii. resolve structures of the nuclear charge distribu• tions in the order of the proton diameter. The "field shift" difference for spherical charge distribution differences Ap(r) is given by The evaluation of the nuclear ground state charge density from the scattering data has been s AE=Z/AP(r) I V^P(r)-V¡ (r)|dT, (2) performed by an almost model independent method - 54 - described in detail by B.
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