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
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.
v2p_Yls = A+Brke-ar (3)
B ar 2 k ar 3RB-3^ rV r dr =
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 - 54 - described in detail by B. Dreher et al.(9). The relative contribution of the higher radial moments The intensity of the incident electron beam is monitored by a ferrite-transformer(15,16) and a Faraday-cup. At the same time,a second fixed angle spectrometer mom* torsthe product of electron current and target thickness. The resulting accuracy for charge distribution differences can be seen in Fig. 12. From the charge distribution determined by elastic electron scattering*it is possible to de• duce the ratio of the RMS-radius to the Barrett- radius with a precision comparable of that of the Barrett-radius, measured by muonic X-ray experi• 0 . . r~~ ments. This is caused by the fact that the two 20 Wo 60 80 Z radial moments are quite similar. In this way it Fig. 2 is possible to extract the radial moments from For example,the mercury field shift measurement con• elastic electron scattering and muonic X-ray data tains a contribution of about 7.5 % from the higher with almost the same precision with which the radial moments. Fig. 3 shows a King-plot of optical Barrett radius is known. We will use these precise isotope shift measurements against muonic X-ray data. radial moments to calibrate the optical isotope shift data. 4. Optical isotope shifts and 6 The isotope shift ôveXpA,A* between two iso• topes with mass number A ana A' observed in the wave number v is the sum of the field shift and the mass shift, according to the following formula: A A1 2 ? 4 fi 6v ' = cA|i|;(o)¿le(|:o + WfV ' Mmass shift The density of the electron at the nucleus given by A|^(0)|2 is only approximately known. Also, the mass shift contains a contribution from the specific mass shift,which is not well known either. If one knows the absolute radial moments - 35 - Differences in Nuclear Charge Radii upon Addition of two Neutrons l\RMS 4 [NT'/H 1Î 22 ^4 26 \2&\3 0 32 34 36 38 40 42 „ 4>-. 46 48 52 54 1561 58 60 \ 9z| 37 \t*\ 32 fâë] 24 [Us] 38 36 («j 26 0 21 0 0 Zr ' * Zn 0 *J0 '« 0 '7 0 0-100 -7[m| J 7 2 9 1 8 5 o Ni 0 0 0 0 0 - © RH < û » FC Ñ « fr] « 0 0 -6 [So] -5 ¡MJ -4 [m] -4 0 A> Cr 0-2.^47 0 0-2 0 yyro 0 2 0 0 0 10 Se * 9 :o 16 i? a £ß • Ca-0 3°0" E-'3- è 0 0 0 0 Fig. 4 5. Results, Systematics and Interpretation shell,one observes that the radius becomes smaller upfoan addition of two neutrons e.g. of Differences of RMS Radii 48ca-46ca the RMS radius becomes 20-10-3fm smaller. The same can be seen for the Sr, Rb and Kr-isoto- pes. By adding four times in sequence two neutrons 5.1 Isotope shifts to fill the lgg^shel 1 ,the RMS radius of the Kr- isotopes is becoming always smaller. In the 2d5/2 The aim of this experiment has been to study subshell a decrease of the radius by adding the the systematics of differences of RMS-radii for last two neutrons does not occur. isotopes and isotones. Results for Fe, Ni, Cu and Zn isotopes have been already pub!ished(5,6). These measurements were extended to the lfy/o shell c) A sudden increase in the isotope shifts occurs at nuclei(7), the nuclei near the closed N=50 shell N = 20,28,50 and 56. and the N=56 subshell(8). The results are shown in Fig. 4 where the differences in the RMS-radii of d) At the beginning of a new neutron shell,the inde• isotopes from Ca to Mo for aN=2 are plotted. pendence of the proton configuration is stronger compared to the end of the neutron shell. Each isotope is In the following figures (5-9) represented by a square with the mass-number A. The figures between the squares indicate the difference in the RMS-radii between neighbouring AN=2 isotopes. The results are plotted against the neutron numbers. Differences in Nziclear Charge Radii The radii differences between even isotopes upon Addition of two Neutrons AN=2 due to the proton core polarisation by the L\RMS added two neutrons indicate strong systematic [io-Vm] shell effects (Fig. 4): a) The addition of the first two neutrons in the 40 If7/2 neutron shell e.g. 42ca ~ Ca or 40Ar-38Ar show a strong increase of the radius. The same is seen after the closed neutron shell N=28 between 56pe - 54Fe or 54r,r - 52o. This occurs also after N=50 or N=56. A sequential addition of neutron pairs in the different shells shown in Fig.4 results in an almost linear decrease in the successive isotope shifts. 42 44 46 48 ¡50} 52 54 jj/j 58 60 b) Furthermore, it is apparent that the isotope N shifts are independent of the proton configu• ration of the nucleus involved,(e.g. the 58Fe-56Fe and 60ní-58ní isotope shifts are identical within the experimental uncertainties).We find this be- havious quite remarkable,particularly in the view * Mo [92] » [94] [ii] z* fiäj » jTooj that the proton configuration of Ni is magic Ü¡]-@-3[?¡]-[¡3-5[^J Kr - Fig. 5 Z=28. At the end of the lf7/2Shell and the lgg/2 - 36 - Differences in Nxtclear Charge Radii Differences in Nuclear Charge Radii upon Addition of txuo Neutrons up on Addition of two Neutrons LRMS &RMS • 3 e o [i 38 40 42 44 46 48 (jo] N 3C 32 34 36 38 40 42 O N [84]-ÍO[86] -7J88J ST • .«[TÍ] 17 0 « 0 Fiq> 6 Ge » Fe [ii]«[¡i]»[sä] Fig. 8 Differences in Nuclear Charge Radii upon Addition of two Neutrons we compare the experimental results with spherical Hartree-Fock-calculations from Reinhard and Drechsel kRMS taking into account experimental B(E2) values to the ' [l0"3/m] lowest 2+ states. The H.F. term makes a smoothly varying and almost constant contribution to the dif• H.F. ^ ferences of the RMS-radii. The B(E2)-related term, which is identical to the corresponding term in the usual incompressible fluid model formulation, intro• duces a strong variation in the predictions for different isotopes. The deviation seems to be within a range of 15*10-3fm, except for the Ca-isotopes, where the inclusion of octupole deformations impro• ves the result. The agreement between experiment and théorie is, of course, limited by the error of the • measured B(E2) values. -0- TV The differences of the RMS-radii for the iso• topes of 32Ge up to 38sr at the end of the lgg/2 neutron shell in the region N<50 is given in Fig.10. Fig. 7 0 2 [78] O [so] I [¡2] 0 Differences in Nuclear Charge Radii upon Addition of two Neutrons ÙJUfS Differences in Nxtclcar Charge Radii [LO-'/m] upon Addition of tu o Nexttrons àRMS 3i bt 5 " [irr /m] o «-»•?- • Ê V».P. :fk. 48 (sol 3V *>«- • 35 %r . IT KV . 8 ¡2$ « A 76 [Tsj 30 32 [B41-U>|SC] -T[as] Sr ° ' Rb ° 0ME3_S[I3"40"4È Kr A 0-0 Br * # CT [5Ô]-'[52J^0 2 0 9 0 • [£2¡ Se * c &« 017 0 • E3 Ge 4**2. * A Ca .OFTTJ-^JTËJ-ÎOJTI] Fig. 9 Fig. 10 - 37 - Here the Z-independence of the change in the RMS-radii seems to be less pronounced compared to the beginning of the neutron shell. The average value for each element - as indicated at the left side of Fig. 10 - shows a systematic tendency that the ARMS-radius becomes smaller with increasing Z. 5.2 Isotone shifts The sequential addition of proton pairs after Ca (Z=20) and Ni (Z=28), shown in Fig. 11 results in an almost linear decrease in the successive isotone shifts. Differences in Nuclear Charge Radii upon Addition of two Protons MUÍS A [10"1 /m] 100- Vig. 12? 8 rtH 90- 80- 70 \ 60 50- \8 Isotone àZ=2 Radii Differences around N-50 40- 30- [92| |94 [96J lui 20- Mo 51 10 I Si Fe Ni Zn Ce Kr Sr 7.T Mo [90 0- Ar Ca Ti Cr Zr 0 ra ra O ¡96] 33 0 SV 0 0 Ö a [¡7] 93 [Ts] * S » (H3 O [Tz] 83 [Tt] * 0 38 0 O 0 46 46 [50] 33 0 0 0 [se] [sTJ 51 [5*J 38 0 O 0 43 0 4, 0 , 0 94 0 fiÔ) 53 [v] 47 [Tt] 46 48 [sol C- [76] 6) [Ta] 53 [To] O [Të] 93 [TU] 69 ¡SO] Fiq. 13 /V O 0 90[J¿] û [Tl] 66 [Tê] 66 0 0 6« [«J El77 @ 0«0 Fig. 11 Furthermore, it is apparent Isotone àZ=2 Radii Differences around N~28 that the isotone shifts are independent at the neu• tron configuration of the nuclei involved, es• pecially for the If7/2 proton shell nuclei Ca to Ni |58J [Boj Ni. For example, the 46Ti-44Ca, 48Ti-46ca and 50ji-48ca isotone shifts are all identical and not 38 dependent on,e.g. ,double magic nucleus like f§Ca28 ra or the not magic nucleus ^Ti26- A sudden increase Fe in the isotone shifts more than a factor of two 5I occurs at Z=28, while neutron-configuration inde• pendence is maintained. In this case, the sudden Cr [50] ra increase reflects the beginning of the 2P3/2 proton 61 shell. In Fig. 12 the differences in the cnarge distribution - measured by elastic electron scatte• Ti @ ring - show clearly the further radial extension of the 2p3/2 proton wave functions compared to the If7/2 proton wave functions. Ca 0 ra The isotone AZ=2 RMS-radii differences before and after the magic neutron number N=50,(see Fig.13) seems to be in a systematic way larger than at N=50. Fig. 14 N In Fig. 14, around N=28, this effect cannot be observed. It is interesting to see the similarity of the systematic behaviour between isotope and isotone shifts. - 38 - 5.3 Comparison of isotone and isotope shifts and Isotone and Isotope Odd-Even Staggering core polarisation effects "normal" Effect Isotone shifts, in contrast to isotope shifts, are dominated by the charge of the added protons. Thus, before isotope- and isotone-shift data can be compared on an equivalent basis, the charge dis• tribution of the added protons must first be con• sidered. Elastic electron scattering experiments -Fig. 12- have shown that the radial dependence of Ap(r) between lfy^-shell isotones is well descri• bed by If7/2-she11 harmonic oscillator wave func• tions. As a zeroth-order approximation, we there• fore calculated a set of lf7/2-shell isotone shifts considering only the charge distribution of the added protons In Fig. 15 the experimental isotone shifts are compared with a harmonic Oszillator shell model calculation. ..Mi. This may be called the "normal" odd-even staggering effect.Reehal and Sorenson predict smaller defor• mations for odd then for even isotopes, because ground state correlations are blocked by the unpaired nucleon(IS). The arrow in Fig.16 indicates the direction from smaller to larger increase of the RMS-radius.In Fig.17,the isotope odd-even staggering around N=28 shows for N>28 a "normal" effect. The isotone effect is "normal" below Z=28,but for Z>28 the first proton (63Ca-62Ni) yields almost the same increase of the RMS-radius as the second proton (64zn-63cu) in the new 2P3/2 proton shell. The Ca isotope shift in the first half of the If7/2 shell is "normal", whereas in the second half of the If7/2 shell the Ca and Ti shifts for the first added neutron (45ca-44ca; 47Ti-46Ti) is larger in the amount of the difference of the RMS-radius com• pared to the effect of the second neutron (46ca-45ca; 48jí_47tí). The "normal" odd-even staggering effect in the first half of the If7/2 neutron shell may be "Protons explained by the increasing deformation and the "not normal" effect in the second explained by the decreasing deformation. The details of this be• haviour, however, may not be easy to explain. The actual isotone shifts decrease more rapidly with increasing Z than the calculation would suggest. Isotone and Isotope Odd-Even Staggering It is evident that the effect of the added LRMS [10-An] nucléons on the proton core must be included when Zn considering either isotone or isotope shifts. In 0 V5 the case of isotope shifts, this "core polarisation" Cu is the only effect between N=20 and N=28, (see Ni' S0@ Fig.4).The difference between the measured isotone 15 shifts and the contribution of the proton charge Co 4Hi 0) should be the proton produced core polarisation, (see Fig. 15).This proton core polarisation is posi• Fe . tiv in the first half of the If7/2 proton shell Mn and negative in the second half. The core polari• sation caused by the added neutrons, as seen in Cr Fig.4, is in the average about two times stronger V than the proton produced core polarisation. We con• §5 clude that the proton - and neutron - core polarisa• Ti tion reveal qualitatively a similar behaviour in Sc the lf7/2Shel1. Ca H 22 2* ^ ' 26 5.4 Isotone and Isotope odd-even staggering N Around N=50 Fig.16 displays the odd-even staggering for isotones and isotopes. The increase of the RMS-radius due to the first unpaired nucleón is smaller compared to the effect of the second nucleón; this is shown for neutrons and protons. - 39 - 5.5. Differences of charge distributions and de- !_! ^ formation effects From the Fe, Ni and Zn charge distributions measured by elastic electron scattering one can deduce the change in the half-density radius Ac and the change in the nuclear skin thickness At. The result in Fig. 18 shows that AC is almost con• stant whereas At is decreasing if the neutron shell is more and more filled with neutrons. Û^FB [im] 0.05; f 4 T i— 23 30 32 34 35 38 0.0 1.0 2.0 3.0 4. 0 5. 0 6. 0 7. 0 8.0 9 Fe_ r Lfm] 0.10- Fig. 20 * Zn 0.05 0 i ^ v> • 1 p 1 r References: 0.05 (1) H.D. Wohlfahrt, 0. Schwentker, G. Fricke, Fig. 18 H.G. Andresen and E.B. Shera; P.R.C. 22(1980)264 (2) B. Dreher; Phys.Rev.Lett. 35(1975)716 J AtFB [fm] . (3) H. Rothhaas; Thesis, Universität Mainz, 1976 • Fe t " O Ni. 0.10 (4) H. von der Schmitt; KPH-Report 24/74, Univer• • Zn sität Mainz, 1974 0.05- (5) E.B. Shera, E.T. Ritter, R.B. Perkins, 0- 30 32 34 G.A. Rinker, L.K. Wagner, H.D. Wohlfahrt, G. Fricke and R.M. Steffen; - 0.05- Phys.Rev. 014(1976)731 Aß' (6) H.D. Wohlfahrt, E.B. Shera, M.V. Hoehn, • A» from experiment 6 (E2)-values Y. Yamazaki, G. Fricke and R.M. Steffen; 0.02- • A - Theorie : D.O.H.F. (Ko! 76) Phys.Lett. Vol 73B, No. 2(1978)131 0.01- (7) H.D. Wohlfahrt, E.B. Shera, M.V. Hoehn, Y. Yamazaki and R.M. Steffen; 0- 32 34 ^35^ • 38 Phys.Rev. 023(1981)533 - 0.01 - (8) L. Schellenberg, B. Robert-Tissot, K. Käser, L.A. Schauer, H. Schneuwly, G. Fricke, -0.02 Fig. 19 S. Gluckert, G. Mallot and E.B. Shera; Nucl.Phys. A333(1980)333 This tendency is compared in Fig. 19 (9) B. Dreher, J. Friedrich, K. Merle, H. Rothhaas with the quadrupole deformation changes A32. The and G. Lührs; similar behaviour of At in respect to Aß2 is ob• Nucl.Phys. A235(1974)219 vious and supports the explanation of the isotope (10) R.C. Barrett; Phys.Lett. 33B(1970)388 shifts by quadrupole deformations, see chapter 5.1. (11) H.R. Collard, L.B.R. Elton and R. Hofstadter; A new electron scattering measurement of the Nucl.Radi i in Landolt-Börnstein, Group I, Vol.2 difference of the 48ca-40ca isotopes shows -Fig.20- (12) R. Hofstadter; Rev.mod.Phys. 28(1956)214 that some charge is shifted from the inner and outer part of the nucleus to the surface region. This (13) T. de Forest, Jr. and J.D. Walecka; complex structure may be the reason why the isotone Advances in Phys. 15(1966)1 shifts in the Ca isotopes can not only be explained by quadrupole deformations and that octupole de• (14) H. Ehrenberg, H. Averdung, B. Dreher, G.Fricke, formations have also to be considered. H. Herminghaus, R. Herr, H. Hultzsch, G. Lührs, K. Merle, R. Neuhausen, G. Nöldeke, H.M. Stolz, V. Walther and H.D. Wohlfahrt; Nucl. Instr. and Meth. 105(1972)253 (15) G. Stephan; KPH-Report 71/3, Universität Mainz, 1971 (16) R. Steiner, K. Merle and H.G. Andresen; Nucl.Instr. and Meth. 127(1975)11 (17) J. Bonn et al.; Zeitschr.f.Physik A 276(1976)203 (18) B.S. Reehai and R.A. Sorensen; Nucl.Phys. A161(1971)385 - 40 - DISCUSSION masses in East Lansing rather convincing evidence was presented that as far as masses are concerned, G. Alkhazov: A comment on neutron distributions. the "magicity" of one particle supports or enhances The charge and correspondingly the proton distri• the "magicity" of the other particle. Can you butions, as was discussed before, are studied comment on this apparent discrepancy between your fairly well for a number of nuclei. As for the radius-based conclusions and those from masses. neutron distributions and the neutron r.m.s. radii, they are studied more poorly. The reason for this G. Fricke: Evidence for correlations between the is that for investigation of matter and neutron dis• nuclear charge radii and the binding energies for tributions one uses strongly interacting particles. the calcium isotopes has been found by F. Träger, Then, the analysis of the data is a difficult pro• Z. Phys. A299, 33-39 (1981). For the krypton iso• blem and it has only approximate solutions. Up to tope, pairs close beneath the magic neutron number now the most reliable results on neutron distribu• N = 50, H. Gerhardt et al., Z. Phys. A292, 7-14 tions have been obtained using elastic scattering (1979) give a further example for such a correlation. of high-energy (y 1 GeV) protons from nuclei. The On the other hand, for the long chain of Rb iso• experiments of this kind were performed in Gatchina topes the relation between radii and the two neutron (Ep = 1 GeV), Saclay (Ep = 1 GeV), and Los Alamos separation energies seems to be not so evident. (Ep = 800 MeV). The advantage of using high-energy protons as probing particles is due to the fact I cannot answer your second question, but I regard that at high energy the mechanism of scattering is your comment as an interesting hint to look into rather simple and there exists a rather accurate the systematics of the "magicity". theory of multiple scattering which allows the E.W. Otten: I like to comment on the recaí i brati on connection of measured cross-sections with the free of optical isotope shift data, which amount to about proton-nucleon amplitudes and the nuclear densities 80% in the case of Cd, e.g. It would have really under investigation. The main conclusions of these drastic consequences on the evaluation of our data studies are as follows. The differences between and destroy the consistency, especially in the sys• the neutron and the proton distributions for the tematic of isotonic shifts, where Cd was already one nuclei in the central part of the valley of nuclear of the cornerstones. We would have to change the stability are rather small. The largest difference 2 102 - 41 -