Electroexcitation of giant resonances in 181Ta

2.B : 2.C

Nrrclaar PAgyala A278 (1977) 261-284 ; © North-Holland Prtbliddnp Co., Mutadern Not to be repzoduced by photoprint or micro5lm without wrlüm pxmiwion !}oat the pabliahis

>~~OEXCTTATION OF GIANT RESONANCFS IN "'Ta t R. S . HICKS rr, I. P. AUER rrr , J, C . BERGSTROM and H . S. CAPLAN Saskatchewan Accelerator Laboratory, Untoeraity ojSaakatchewan, Saskatoon, Canada S7N OWO Received 24 Artgast 1976 Abrtrnct : The giant resonance region of tst Ta has been investigated by means of inelastic eloctron scattering, with primary rolectron energies of 79 .1 to 118.3 MeV. A peak-fitting procedure was employed to separate the measured spxtrum into nine ditï'erent resonance components . Multi polarity and strength assignments were deduced using DWHA analysis with the Goldhaber-Teller and Steinwedel-Jer>sen models. In addition to the well-known gust dipole structure, other troeronances were identified at 23.2 t 0.3 MeV (E2), 9.5 f 0.2 and 11 .5 t0.2 MeV (E2 or EO), 19 .5 f 0.8 MeV (E3), 3 .70 f 0.14 MeV (E3 or E4~ and 5.40 f 0.15 MeV (E4 or E~ . The model dependence of the analysis is discussed.

E

NUCLEAR REACTIONS '°tTa(e, e~, E = 79-120 MeV ; measured a(E, deduced giant multipole resonance parameters, B(F.x) .

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'°'Ta

1. Introduction During the decades of the fifties and sixties "real" photon reactions were utilized to amass a large body of information describing the giant dipole resonance (GDR) . Although proposals were occasionally made for giant resonance modes other than El, there was little direct experimental evidence to support these suggestions, principally because in real photon reactions such modes are strongly suppressed relative to the dominant El component.. In 1965 Drechsel t) drew attention to the potential ofinelastic electron scattering : If the momentum transfer were judiciously chosen to depress the GDR, thepredicted E2 resonanoes should be readily observable . On the other hand, for a given incident electron energy the E2 cross section is similar to the EO in its dependence on scattering angle, so the identification of quadrupole structure might not be unambiguous. Six years later, at Darmstadt, evidence was found for a giant E2 or FA resonance in (e, e~ spectra of N = 82 nuclei s ), . and this discovery was quickly supported by a r Work supported by the Atomic Energy Control Board of Canada and the National Research Council of Canada . tt prexnt address : Instituut voor Kernphysisch Onderzoek, Poatbus 4395, Amsterdam 1006, The Netherlander. ttt present address : High Energy Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA . 261

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re-evaluation of earlier (p, p') data from Oak Ridge 3). The position of this newfound resonance, x 63 A -~ MeV, lay in good agreement with Bohr and Mottelson's calculation of the excitation energy for the isoscalar (i.e., dT = 0) E2 giant resonance °), and it became commonly identified as such. The isoscalar nature was subsequently confirmed by its appearance in spectra from inelastic a-particle S " 6 } and deuteron'} scattering measurements, with the (d, d') results also indicating a preference for the E2 assignment . This interpretation had also been hinted at by {p, p') data s " 9}, and by comparisons of the observed strength of the resonance with the energy-weighted sum rule (EWSR) values for isoscalar EO and E2 excitation '°). It exhausts 25-95 % ofthe E2 EWSR, but is often several times the monopole EWSR. Later electron scattering work ' 1 " 1 z) has also indicated further giant resonance structure at z lOSA - } MeV (E3) and .. 132A - } MeV (EO or E2) . The latter resonance is usually interpreted to be the isovector (d T = 1) branch of the giant quadrupole resonance, and its strength is generally seen to exhaust a substantial part (60-95 ~) of the corresponding EWSR. This paper reports measurements of the spectrum of la'Ta using inelastic electron scattering with particular emphasis on the giant resonance region. The nucleus isiTa has been relatively neglected in the field of electron scattering. In an early measurement performed at Stanford 13), a peak observed at 3.6 MeV excitation seemed to have an E4 angular dependence, and a value of 9.3±4.6 Weisskopf single-particle units was extracted for the reduced transition probability, B(E4j). The existence of a weak state at 5.6 MeV was also indicated . To date little is known about the new-found resonances in this nucleus . Early inelastic a-particle and 3He scattering measurements'¢) revealed a broad peak at 11 MeV (~ 63A - } MeV) which has since been re-examined at Orsay 9) using the (p, p~ reaction. An E2 assignment again seems preferred, and the resonance is estimated to account for (42 f9)~ of the isovector E2 EWSR. The prime motivation for the present experiment was to complement a remeasurement at Melbourne of the tantalum photoneutron cross section ls) in which broad structure observed at 23 MeV was tentatively identified as the isovector E2 resonance . We have sought to examine this identification further, since the structure has been seen in the (y, n) reaction an EO assignment is precluded . It will also be instructive to note whether the large deformation of this nucleus causes the new resonances to fragment or broaden, as occurs, for example, in the GDR. In an excited, dynamically triaxial nucleus the 2L+ 1 eigenstetes of a giant resonance ofmultipolarity Lwill separate, and provided the separation-to-width ratio ofthe eigenstetes is sufficiently large, this splitting may be observable experimentally . 2. Experimental apparatne This experiment was performed using the electron scattering facility at the University of Saskatchewan linear accelerator laboratory. Details ofthe accelerator

'°'Ta GIANT RESONANCES

26 3

and the electron scattering apparatus have been given elsewhere 16 .1~), so only a brief description is needed here . Because the giant resonance structures of 191Ta are expected to have large natural widths, high resolution was not of primary concern, and the energy spread of the beam incident on the target was set at values between 0.2 and 0.4 ~. Natural foil _ targets, with thicknesses ranging from 7.6 to 500 mg/cm z were used . The beam current was measured by a non-intercepting torroidal feinte monitor la) whose response was periodically calibrated against a Faraday cup. Electrons scattered from the target were detected by a 45~hannel array of plastic scintillators located in the focal plane of a 127°, 50 cm radius, double-focusing magnetic spectrometer . The momentum acceptance of each channel is 0.09 ~. Two long back-up detectors were positioned behind the array, allowing the broad background "ghost peaks", produced by rescattering of electrons within the spectrometer system, to be determined and subtracted l '). With an incident electron energy of 118.3 MeV, spectra were accumulated at scattering angles of 37 .9°, 54.0°, 65.0°, 78.0°, 92.0°, 108.9° and 127° . Data were also taken at 102.0 MeV, 62° and 79.5 MeV, 149°. These kinematic conditions result in elastic momentum transfers ranging from 0.39 to 1 .07 fm -1. Except for the most backward angle, the targets were oriented in the transmission mode with respect to the incident and scattered electrons, thereby minimizing the broadening of resonance structures due to straggling . 3. Aealysîs of data For each incident beam energy and scattering angle the raw experimental data were corrected for dead-time losses, non-linear spectrometer dispersion, and the differing counting efficiencies of the 45 channels in the detector array, and then combined to give an energy spectrum of scattered electrons. A typical result, fig. 1, shows the giant resonance structures as bumps on top of a large, continuous background . The (e, e~ cross sections of the observed resonances were deduced by comparing the radiative- and ionization-corrected areas of the inelastic and elastic Peaks, Ago and Ae~ Cref. '9)] ~~/io

A~ ~~/ei~

with the 191Ta elastic cross section being determined by calibration against the known elastic cross sections of 12C and 9Be [ref. s°)] . 3.1 . SUBTRACTION OF THE ELASTIC PEAK TAIL

Probably the major obstacle in the use of electron scattering as a means of investigating the giant resonance region is the problem of separating the resonance

Fig. 1 . Spectrum of electrons inelastically scattered from a '°'Ta target . The continuous line is the theoretical radiation tail, calculated using the Maximon-Isabege =') formula and including target thickness contributions =_). Above an excitation of 5 MeV it falls well below the measurod spectrum. The hatched area designates the range of variation for "reasonable" lifted backgrounds .

structures from the underlying background components . The first step in this process is usually the subtraction of the radiative tail of the elastic peak. The large Coulomb potential of t e'Ta introduces a significant distortion into the incoming and outgoing electron waves, so that a plausible theoretical calculation of the radiative tail should be made without the simplification of the plane-wave Born approximation. However, since no convenient method of executing such calculations is presently available, we make a first approximation to it using planewave techniques . This calculation followed the peaking approximation of Maximon and Isabelle 21), with target thickness effects being accounted for using the formulae of Butcher and Messel zs) . A representative comparison of a measured spectrum and its associated theoretical tail is shown in fig. 1 . Except for the region within a few MeV of the elastic peak, the calculated tail falls well below the observed spectrum . The excess cross section may consist of at least four different components : (i) the calculated elastic peak tail may be underestimated, for example, by the use of the Born approximation, or by deficiencies in the model accounting for target thickness effects; (ü) the contribution of processes such as quasi-elastic scattering Za) ; (iii) unaecounted-for experimental background ; and (iv) giant resonance and other inelastic structures . In this work, no attempt is made to determine specifically all the various contributions to the measured spectra. Instead, we mathematically separate the cross section into two components ; a resonant part, which is assumod to be the giant resonance structure, and a smooth, slowly varying part, which we take to be the sum

'°'Ta GIANT RFSONANCFS

26 5

of all other contributing processes. In effect, our approach to the problem of the background subtraction follows the method developed at Darmstadt za) : A set of Lorentz resonance shapes and an empirical tail function are simultaneously fitted to the experimental spectrum . From the best-fit parameters the giant resonance cross sections are deduced. We presume that this method of analysis results in the separation of the giant resonances from the quasi-elastic cross section, since the latter is expected to have a very broad energy dependence Zs) . On the other hand, the nuclear processes that are involved with the giant resonance excitation also influence the quasi-elastic scattering ss . 26), so the two effects should probably be considered together . However, in this work no attempt was made at the difficult problem of isolating the quasi-elastic contributions to the measured spectra ; we deal only with the resonant Parts. The acceptability of the fitted tail rests on several factors. Firstly, it must be broad and smooth over the region of interest . Above and below the giant resonance structures we also demand that it have the same asymptotic shape as the measured spectrum . Since we usually fit over a wide range of excitation, approximately 1 to 50 MeV, the asymptotic behaviour of the measured spectrum and the fitted tail can be closely compared . In the l e'Ta spectra the relatively unstructured regions above 40 MeV excitation and in the range 1-3 MeV play a decisive role in defining the form of the fitted tail . An important check on the reliability of this method is that the cross sections derivedfor the two known GDR peaks, at 12.4 and 15.3 MeV, should conform to the expected El dependence on energy and scattering angle. The errors in the deduced giant resonance cross sections are mainly determined by the background uncertainty. We estimate this by fitting the data with a variety of different tail functions : T(co) = To(a~xa+bw)+(c+dw+ecoZ),

T(w) = a+bz+czz +d/z+e/zz ;

z = f -co,

(1)

(3)

where To (m) is the calculated Born approximation tail at excitation co, and a, b, c, d, e and fare free parameters. Forms similar to eqs. (1) and (3) have been used by others to fit (e, e~ giant resonance spectra 2a . z,). In each case we used the smallest number of free parameters required to give a satisfactory fit to the data, with chisquare values of 1 .~5 p.d.f. generally being attained . Fig. 1 shows the spread of "reasonable" tails fitted to the 118.3 MeV, 65° data.

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3.2 . DETERMINATION OF INELASTIC PEAK AREAS ; RADIATION AND IONIZATION CORRECTIONS

As stated previously, we have followed the procedure offitting the giant resonance spectrum with a set of Lorentz line-shapes . However, note that the observed spectrum does not directly describe the actual spreading of the giant resonance strength, since distortion is introduced, not only by radiation and ionization effects, but also as a result of the systematic change in momentum transfer across the resonance width s °). In particular, owing to the large "backgrounds" and overlapped nature ofthe giant resonance structures, the aptness of any line-shape funotion becomes more and more uncertain as we move away from the peak maximum . We assume the Lorentz line shape function provides a reasonable representation for the region within a few half-widths of the maximum . The contribution to the total peak area from beyond this region is rather uncertain but, on the other hand is expected to be small. With these considerations in mind we elected to compute the peak areas by integrating the fitted line-shapes between set limits . For example, a peak centred at energy Ej was integrated between the limits E~+d~ and E~-d~, where d~ is comfortably greater than the resonance half-width (HWHM) . For most peaks in our spectra we set d~ equal to 10 MeV. If we consider the integrated area to be concentrated at E1, we may then apply the usual radiation and ionization corrections as ifthe broad resonance were a narrow level. This method is simple and convenient, but lacks the rigour ofmore detailed techniques, such as that proposed by Tsai 28). The first step in Tsai's approach is the calculation and subtraction of the radiative tails of the elastic and discrete inelastic levels. The continuum region is then corrected by dividing the spectrum into small bins, each of which is treated as an inelastic level . A Schwinger-type correction is applied to the bin at the continuum threshold and, using an assumed or measured q-dependence, its appropriate radiation tail is calculated and subtracted from the remaining spectrum . This process is repeated for successive bins until the whole spectrum has been "stripped" of radiative effects . Remaining then are presumably the "radiationless" giant resonance structures superimposed upon a monotonically increasing background, which in large part is considered to be the quasi-elastic cross section s9). When Tsai's technique was applied to one of our thicker target runs at backward angle it was found that the deduced cross sections agreed closely with the values obtained using the cruder method . In view of this, and the fact that the accuracy of our data was not sufficient to warrant the computational effort associated with a more detailed technique, it was decided that the simple radiative correction procedure was adequate for our purposes. 3.3. DISTORTED-WAVE PHASE-SHIFT CALCULATIONS

The inelastic electron scattering cross sections are presented in units of the Mott

'°'Ta GIANT RESONANCES

26 7

cross section, which describes the elastic scattering through an angle B of an electron of energy E, from a spinless, infinitely massive, point charge Ze : dQ ~~~~«~

aZ cos 2B 2 - C2E~ sinz 2B) . _

Not only does this make the nuclear contribution to the cross section more evident, but it also allows for a more convenient comparison with theoretical predictions. It is this comparison between experiment and theory which enables us to estimate the multipolarities and strengths of the observed giant resonance structures. The interpretation of these resonace was made using the inelastic distorted-wave code DUELS 3°), which requires a model for the nuclear transition charge and current densities'` One frequently used model is that of Tassie 31}, where the excited nuclear state is represented as a surface vibration of an irrotatianal, incompressible, liquid drop with transition charge density pL(r} given by

Although the charge density p i(r) has often been assumed to have the same parameterization as the ground-state charge density, po(r), it has been established that the parameters in the Tassie model must be changed considerably if it is to fit successfully the electron scattering data for many well-known transitions' Z). The charge densities p ° (r) and p=(r) are presumed to be described by the two-parameter Fermi distribution where c is the half-density radius and t = 4z1n3 is the skin thickness . In this model the rms radius is given by ~rZi~ _ ~~ 2+~ZZ2]~. For the tantalum ground-state parameters we use values that are averaged over the orientation of the nuclear axis : c° = 6 .38 and t° = 2.80 fm [ref. 33 }]. The cross sections computed with DUELS are for a reduced transition probability B(XL, E= j) = 1 .OeZ ~ MeV- Z~, so that the experimental transition probability is estimated by normalizing the calculated curve to the measured points . That it is appropriate to analyse the data presented here with the Tassie model is indicated by noting that the model's transition charge density, given by eq. (6), is the same as that derived from the Goldhaber-Teller (GT) representation of giant resonance vibrations zs). In the original GT picture of the GDR'4), the neutron and proton distributions were considered to be separately rigid, and the frequency of the harmonic oscillations of these two distributions against one another gave the giant dipole energy . This model was subsequently generalized by Ùberall zs) so that other multipolarities and "spin" vibrations could be considered within its

268

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framework. Nevertheless, both the Tassie and GT models have been formulated assuming a spherically symmetric nucleus, so that some care is demanded in their application to the deformed nucleus, 181Ta. This shortcoming should be most serious for the El oscillations, which are expected to be strongly polarized along the semi-major and semi-minor axes . Accordingly, we treat these El vibrations separately as de-0oupled resonances with different charge density parameters . On the other hand, quadrupole and higher multipolarity resonances are expected to be more evenly distributed over the nucleus ss), so that the approximation of spherical symmetry may be less detrimental. Another representation of the giant resonance that has been used in the present analysis is the Steinwedel-Jensen model ae), in which interpenetrating neutron and proton "fluids" vibrate against each other within a rigid nuclear surface. The separate neutron and proton distributions are not in themselves rigid, but the vibration is constrained to preserve a constant matter density throughout the nucleus. Thusfar, the SJ model has been applied to isovector modes only, and in this respect the dynamic collective model, a refined extension of the SJ model in which the giant resonance oscillations are coupled to the vibrational and rotational degrees 3s . ") . of freedom of the nuclear surface, has been developed considerably In the SJ model the transition charge density is given by a spherical Bessel function Pt:(r) a .la(kt:r~ PL(r) = 0,

r 5 R o, r > Ro,

where Ro is the uniform density radius and the eigenvalues kL are given by the boundary condition d 0. dr ~a(kcr)L=RO =

A comparison of the Tassie (or GT) and SJ transition charge densities for multipolarities El and E2 is shown in fig. 2. In the Tassie model the transition density is concentrated at the nuclear surface, whereas the interior of the nucleus participates more in a SJ oscillation. To illustrate the sensitivity of the DWBA cross sections to the transition multie polarity and model parameters, we present a few examples applicable to 1 1Ta. Fig. 3 shows E1, E2, E3 and E4 cross sections, in the Tassie model, for 118.3 MeV electrons exciting a hypothetical state at 16 MeV excitation . Two dipole curves are shown, one for an oscillation along the long nuclear symmetry axis, the other for an oscillation perpendicular to it. By arbitrarily normalizing the curves to peak at the same cross-section value, their relative displacements are more easily seen . As the transition multipolarity increases the peak of the cross section is shifted towards more backward angles . The one exception to this is the EO cross section, which cannot be computed with the unmodified DUELS code . In this work it was calculated

'°'Ta GIANT R.ESONANCES

269

Fig. 2 . Comparison of the transition charge densities given by the Goldhaber-Teller ") [or Tassie a')] and Steinwedel-Jensen 3s) models .

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Fig. 4. This figure shows the dependence of the computed DWBA cross section on j, the ratio of the excited and ground-state radii . All curves are normaliaod to a reduced transition probability of B(E2) - 1 .0 e~ ~ fm`.

270

R . S . HICKS et aJ.

using the monopole DWBA program of Rosenfelder 3a) . For the monopole transition charge density we take the expression for the hydrodynamic breathing mode 3~ : PL=o(r) °~

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(12)

Although a strong dependence on Jis evident in this figure, it is also known that this considerable model dependency does not extend to separate variations of c= and zi [ref. a°)] . DecreasingJpushes the computed cross sections to more backward angles, an effect which we have seen can also be achieved by increasing the transition multipolarity . Consequently, an E2 cross section calculated with a particular radius ~, will peak at the same angle as an E3 cross section computed using a radius somewhat larger . Seldom are the giant resonance data from electron scattering measurements of sufficient quality to permit the discrimination between multipolarities merely on the basis of the finer details of cross-section shape. The number of possible multipolarities is usually restricted by demanding that ~ . In cases where well-known levels of similar multipolarity can be identified in the discrete inelastic spectrum, the (e, e') cross sections of these states can be used to guide the multipolarity assignments for the giant resonances . From fig. 4 we also see the sensitivity of the cross-section magnitude to changes in the radius
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ieiTa GIANT RESONANNCS

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table 1 . Fig. 5 shows three examples of the giant resonance spectra after subtraction of the fitted tails. At low momentum transfer the dominance of the known El components at 12.40 and 15.32 MeV is noted. However, beginning with the resonances centred at 9.54, 11.47 and 23.2 MeV, the relative importance ofthe other peaks increases as we move to higher q. A more detailed discussion of the nine resonances will now follow . 4.1 . THE GIANT DIPOLE RESONANCE

Virtually all prior information relevant to resonace observed in this experiment pertains to the electric dipole structure. The splitting of the tantalum GDR into two components has been well established by photoreaction work, and in the present case we use the mean of three separate measurements of the ie iTa(y, n) cross section's " a') to fix the positions and widths of the El resonances . In the hydrodynamic model the splitting of the GDR in heavy deformed nuclei is interpreted as originating from differences in the resonant energies of oscillations directed along the semimajor and semi-minor radii, a and b, of the nuclear spheroid as). These dipole vibrations are treated separately, with model parameters determined from the ratio of the giant dipole energies, Ea and E6. In the uniform density model the radii a and b are given by az) b - (E,

0.089 1I0 .911,

(13)

with the volume constraint where Ra is the mean uniform density radius of the ground state. Eq. (8) relates the values a and b to the radial parameters of the equivalent Fermi charge distributions. Assuming equal skin thicknesses, i.e. z, = zb = za = 0.64 frn, we obtain ce = 5 .87 fm and cb = 7.49 fm. With the GT transition charge density given directly by these values, (i .e.,J = 1 .0), DUELS yields the solid curves shown in fig. 6, where a least-squares fitting procedure was employed to determine the normalization of the curves to the data. In order to display the measurements taken at 102.0 MeV, 62° and 79.1 MeV, 149° we have plotted these points at angles which give the same inelastic momentum transfers for an incident electron energy of 118.3 MeV. To correct for the failure of the Born approximation the magnitudes of these cross sections have been multiplied by the ratio of the DUELS predictions for incident energies of 118.3 MeV, and for 102.0 or 79.1 MeV as the case may be. Particular attention is drawn to the fact that when the 79.1 MeV, 149° data is plotted in this manner, the points lie close to the trend of the data taken at more forward angles. This will be seen to be true not only for the dipole structure, but for the other resonace as well. The fact that these 149° points do not lie above the forward-angle

Fig . 6. Differential cross sections for l 18 .3 MeV electrons inelastically scattering from the 12 .4 and 15 .3 MeV components ofthe GDR of' s 'Ta. The data points from measurements with 79 .1 and 102 .0 MeV electrons have been repositioned as described in the text. The continuous and dashed curves were computed in DWBA using the Goldhaber-Teller (f = 1 .00) and Steinwedel-Jensen (j = l .I5) models .

data indicates that cross sections measured in the present experiment are predominantly determined by longitudinal matrix elements. The normalization ofthe DWBA curves to the data yields reduced transition probabilities B(E1,12.4 McVj) = 26.Ot 13.1 ez ~ fmz, B(E1,15.3 McVj) = 61.0 f 18.7 ez ~ frnz. These figures are somewhat larger than the respective values derived from photoreaction measurements, 19.9 and 36.7 (f 10 ~) ez ~ fmz, through the relationship zs)

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275

'°'Ta GIANT RESONANCES

transverse term, B(EL, E,~, that is measured in y-absorption . However, if the electric spin-flip contributions are neglected, the wntinuity equation shows that these two quantities are approximately equal 23) . If we repeat the above (f = 1 .0) calculation using the SJ model we obtain cross sections that fall off rather more slowly with increasing scattering angle compared to what is observed experimentally . Better agreement results when the excited-state radial parameters are multiplied by a factor f = 1 .15, and this calculation is also shown in fig. 6. Perhaps the preference for a larger radius is not entirely unexpected, for as Danos 3') has pointed out, the rigid boundary assumption of the SJ model is erroneous in view of the well-known softness of the nuclear surface. The radial velocity of the neutron and proton fluids will not vanish exactly at the uniform-density radius Ro . Danos has estimated that a more realistic radius would be obtained by replacing Ro ( z 1 .2A - } fm) with 1 . l A -} + 1.4 fm. In the present case this would correspond to a multiplicative factor of about 1 .12. The analysis with the SJ model yields dipole transition probabilities B(E1, 12.4 McVj) = 22.0 f 13.2 e2 ~ fmZ, B(E1,15.3 McVj) = 54.0 f 25.1 eZ ~ fmZ.

Fig. 6 shows that, although the cross sections computed with the SJ model do not fit the measurements as well as the Tassie model, the trends of the data and both sets ofmodel calculations lie in moderately good agreement. This gives confidence in the technique employed for the "background" subtraction. Support for the use of the polarized vibration model described above is seen to come from a plot of the ratio of the two E1 cross sections. If the radial parameters o.~ N M N

w

0.6 QS

vw 0A Q2

40

SO

60

70

9

80

90

(degrees)

100

Ip

120

130

Fig. 7. Ratio of the electroexcitation cross sections for the 12.4 and 15.3 MeV dipole resonances of '°'Ta. The continuous (GT, j= 1 .0) and dashed (SJ,j= 1 .15) curves were computed in DWBA, assuming polarized charge vibrations along the major and minor nuclear axes . The nord~alization of these curves is given by the ratio of the photonuclear transition probabilities (dot~ash curve).

R . S . HICKS et a/.

276

of the two modes were similar, this ratio should be independent of scattering angle, and close to the ratio of transition probabilities that is deduced from photoreaction data . On the other hand, a somewhat different behavior might be anticipated ifthe transition radii were not comparable . Normalizing the DWBA predicted ratios by the ratio of the photonuclear transition probabilities B(E1,12.4 MeV)/B(E1,15 .3 MeV) = 0.5410.06, we obtain the curves shown in fig. 7. Clearly, the overall magnitude of the experimental results is more consistent with these curves than it is with the constant photonuclear ratio, and therefore offers some support for the concept of the giant dipole oscillations being concentrated along the major and minor axes in deformed nuclei . However, it is noted that the shapes of the DWBA predictions are rather exaggerated in comparison with the measured ratio. This may well originate from shortcomings in the simple GT and SJ models. Moreover, we have considered the two dipole modes to be completely uncoupled, whereas experiments with tesgo IeITo

(e,é)

E x " 23.2 MeV

Ex =9.54 MsV ' E E3 ~ `,~

Ex " I I . 47 MsV

E3 0

c ~ a

EI " 118.31 MeV EI " 101 .97MaV,9 " 82' E I " 7~ 10 MeV, B " 149' t 40

60

90

100

120

40

B (degrees)

60

80

100

120

140

Fig . 8 . I)iffetential cross sections for 118 .3 MeV electrons scattering from giant resonancea of `e'Ta. Resonanees fitted with E2 curves are equally well described by FA calculations.

ieiTa GIANT RESONANCES

277

have shown that 10-20 ~ of the absorption cross section is independent of nuclear orientation `s. "). Further reservations concern the use of an inelastic HWBA code which does not treat the nuclear distortion rigorously. 4.2 . E2 (OR EO) STRUCTURE

We first turn to the broad structure observed in the excitation range 9-12 MeV, just below the GDR. In fitting the measured spectra, consistently better results were obtained if this strength was fitted with two Lorentz line-shapes centred at 9.54 and 11 .47 MeV, rather than a single resonance shape. From fig. 8 it is seen that the shapes of the derived differential cross sections for these two resonances are quite similar and, compared to the E1 cross sections shown in fig. 6, are displaced towards more backward angles . These cross sections are equally well fitted by either an E2 or an EO calculation. The peak at 11 .47 (64.9 A - }) MeV lies close to the generally accepted location of the isoscalar E2 giant resonance. The best E2 Tassie-model fit to this peak was obtained with an excited state radius specified by f = 0.8, and indicates a reduced transition probability B(E2,11.47 MeV) = 1340 t 500 e2 ~ fm`.

This corresponds to (22 i8)% of the isoscalar E2 EWSR, given by `s) _ e 2~2 L(2L+ 1)~ ZZ

t where M is the nucleon mass and (rzL- zi the radial moment of the target ground state. Although the errors quoted above include a contribution from the uncertainty in f, it is difficult to specify a completely model-independent error. The 9.54 MeV peak is well described by E_ = 53 A - } MeV, previously established for E2 or EO resonances in s°sPb and 19'Au [refs .'2 .3e)] . The DWBA analysis of the present data gives B(E2, 9.54 MeV) = 580 t320 eZ ~ fm`, or (8 f4)~ of the isoscalar EWSR . Together, the 9.54 and 11 .47 MeV resonances can therefore account for (30 f9)% of this sum rule . Monopole DWBA analysis results in essentially the same angular dependence as the E2 curves shown in fig. 8. If an EO assignment is considered for the 9.54 and 11 .47 MeV resonances, then they respectively exhaust (13 f 7)~ and (40t 13)~ of the isoscalar mopole EWSR `e):

where M,t is the monopole matrix element.

27 8

R. S . HICKS et a/.

Our results are consistent with the Orsay 1 s 1Ta(P, P7 measurements 9), which showed a broad peak closely resembling the combined strength of the presently proposed 9.54 and 11 .47 MeV resonances . Again a multipolarity of EO or E2 was assigned and the structure was estimated to exhaust (42 f 10)~ of the E2 isoscalar EWSR . It might be speculated that the preference for two resonances in the 9-12 MeV structure is a manifestation of the broadening of the isoscalar E2 giant resonance due to the large nuclear deformation . However, the fact that Pitthan t z) has observed a similar distribution of strength in the spherical nucleus s°sPb seems to rule against this possibility. Moreover, it is interesting to note that the measured widths of the 9.54 and 11 .47 MeV resonace do not appear to be significantly larger than the observed widths of the equivalent z°ePb states . Above the GDR, the broad resonance observed at 23.2 MeV is also consistent with an E2 or EO transition, as fig. 8 shows. However, the position of this peak agrees well with structure observed in the Ta(y, n) cross section 1 s), and since a longitudinal monopole state cannot be excited by real photons, an EO assignment would seem to be precluded. Furthermore, the distribution of this strength lies in excellent agreement with theoretical predictions for the isovector giant quadrupole resonance 3s .4'), and we identify it as such. In the Melbourne measurement of the Ta(y, n) cross section 1 s), this structure was assessed an integrated cross section (5±1)% of the integrated dipole strength . With eq. (14) we deduce that it therefore exhausts (75 f 19) ~ of the isovector E2 EWSR, which is given by as): - eZitZL(2L+ 1)ZNZ ~rsL

- i, (17) L > 1 ~ 8~cM ,q t This is several times greater than the (18 f 5) ~ indicated by the best (f = 0.7) Tassie-model fit to the present data . The basis for this poor agreement may lie in the following: In the representation of the dynamic collective model, Ligensa, Greiner and Danos a') point out that "there exist five main E2 giant resonances containing most of the quadrupole strengths, which correspond to the excitation of the proton-neutron fluids along the long nuclear axis (1 mode), along the short nuclear axis (2 modes), and along the directions 8 = ~, r¢ = 0, ~n (2 modes)". With reference to fig. 4 it is seen that, for equal photonuclear transition probabilities, the electroexcitation cross sections for vibrations along the minor nuclear axes will be considerably enhanced over those concentrated along the longer axes . Conceivably, the strength of these longer-axis modes may be relatively neglected in the extraction of the B(E2j) value from (e, e~ data. This hypothesis was examined in a manner similar to our treatment of the two GDR components . The differential cross sections of the five E2 modes were computed separately, and, assuming equal photonuclear transition probabilities, the mean of these calculations taken to represent the overall DWBA cross section. The angular dependence of this more detailed calculation was found to be somewhat E=~~L, E~

ieiTa GIANT RESONANCES

279

broader at backward scattering angles, but otherwise the effect was not great. The best fit to the data was obtained with a radial scale factor off = 0 .73, and gave B(E2, 23.2 MeV) = 920f 250 e2 ~ fm4, or 20 f5 ~ ofthe isovector EWSR. Because of its assumed isovector character, the 23.2 MeV resonance was also analysed using the SJ model, which yielded B(E2, 23.2 MeV) = 850f 230 eZ

~

&n4.

The discrepancy with the photonuclear data is seen to persist despite the use of several different computational approaches. It is also apparent (table 1) that the deduced radii of the giant E2 states, both isovector and isoscalar, remain unexpectedly small in comparison with the ground state and giant E1 values . In the dynamic collective model, the distribution of the isovector quadrupole strength is almost completely determined by the observed properties of the GDR and the low-energy spectra of neighbouring doubly even nuclei ; only the E2 widths remain as free parameters . For 18'Ta, the dT = 1 E2 strength is calculated to be distributed throughout the excitation range 21-26 MeV [ref. a')] . The quality of the data presented here prevents us from confirming such fragmentation, which may also be obscured by the apparently large E2 resonance width. Numerous other approaches have been made towards a theoretical understanding of giant E2 modes, although little of this work is specific to deformed nuclei . For example, Bohr and Mottelson 4), Hamamoto ae) and Suzuki a~ have employed harmonic oscillator models for collective vibrations including a virtually-excited core, whereas Krewald and Speth s °) and Bertsch and Tsai si) have performed microscopic random-phase-approximation calculations using Skyrme interactions. All these methods predict the E2 giant resonance to have dT = 0 and dT = 1 branches near 63 A - } and 130 A - } MeV respectively . They are therefore in accord with the results presented here and elsewhere. The possible existence of giant electric monopole resonances has also received considerable theoretical attention a9-ss) . For example, the "breathing mode" has been calculated to lie at excitations ranging from 56 A - } MeV [ref. to ; ., 120 A - ~ MeV [ref. si)], and is expected to have large natural width. To date, no unambiguous experimental evidence has been presented to substantiate the predictions of giant monopole strength . s2)]

4.3. RESONANCES OF HIGHER MULTIPOLARITY

As we progress to larger momentum transfers, a broad peak rises in the electroexcitation spectrum to fill in the trough between the GDR and the assumed isovector E2 resonance at 23 .2 MeV. Fig. 8 shows that the differential cross section of this structure is consistent with an E3 Tassie-model calculation normal-

28 0

R. S. HICKS et al.

ized to

B(E3,19.5 MeV) _ (5.2 f 3.3) x 10` e2 ~ fm6. The 110 A - } MeV excitation of this resonance is in good agreement with similar structures observed in (e, e~ measurements on z °8Pb and t 9'Au, and also identified as E3 [refs. t t " ts)] . However, for those nuclei the percentage exhaustion of the sum rule is approximately four times greater than the (12 f 8)% deduced from the present measurements. Although the inclusion of a very broad resonance around 30 MeV improves the quality of the fit at the high energy end of the giant resonance spectrum, the great width of this state, combined with the uncertainty in the tail subtraction, renders the derived cross section somewhat uncertain. An E4 calculation provides a moderate fit to the slowly-varying angular dependence, but there may also be evidence for a small E2 (or EO) contribution . However, this interpretation is only speculative. At the low excitation end of the spectrum, the conspicuous 3.7 MeV peak has been previously observed at Stanford ' 3), where, on the basis of plane-wave Born approximation analysis, a tentative E4 assignment was made. Despite the improved

b-~20

40

60

a0

8 (degrees)

120

Fig. 9. Differential dectroexcitation , cross seàions for peab observed at 3.7 and 5 .4 MeV, and oom perisona with DWBA calcu]adons. The data are better fitted if allowance is made for an 1?1(or FA) oomponent-in the transitions .

ieiTa GIANT RESONANCES

28 1

statistical quality of the present measurement, and the use of the more rigorous distorted-wave treatment, a definite multipolarity still cannot be assigned to the peak, mainly on account of the strong model dependence of the theoretical analysis. As shown in fig. 9, there is little to choose between fits obtained by either an E3 (f = 0.94) or E4 (f ~ 0.80) calculation. In both cases, the cross section is better fitted if the additional contribution of an E2 (or EO) component is allowed. We are also able to confirm speculation by the Stanford workers on the existence of a state near 5 .6 MeV. The derived cross section for this state, preséntly observed at an excitation of 5.40 f0.15 MeV, is shown in fig. 9. Its angular dependence is seen to be rather broad, indicating a higher multipolarity than that of the 3.7 MeV state, possibly E4 or E5, again mixed with E2. Like the other resonances observed in this experiment, the 3.7 and 5.4 MeV states appear to be strongly collective in character. Both the Bohr-Mottelson model's) and microscopic formalisms so .si) have been employed to compute the distribution of E3 and higher multipolarity strength in 9°Zr and Z°sPb. If the results of these calculations can be extended to the deformed nucleus t s iTa, isoscalar E3 strength may be expected near excitation energies of 3 and 20.5 MeV. The tempting identincation of these states with the observed resonances at 3.7 and 19.5 MeV should, however, await the availability ofmore definitive experimental evidence. According to both the Bohr-Mottelson as) and dynamic collective ss) models the bulk of the isovector E3 strength would not be expected until an excitation of approximately 32 MeV. As we move to multipolarities higher than E3, theoretical calculations so. si) indicate that the giant resonance strength becomes more and more fragmented, and in fact is widely distributed throughout the giant resonance spectrum . S. Condoling remake and satnmary

The technique of inelastic electron scattering has been employed to determine the excitation, width, multipolarity and parity of giant resonance structures in the continuum of 1 a 1 Ta. The reduced transition probabilities of the identified structures have also been deduced, but are somewhat model dependent. There are two aspects to this model dependency, the first of which concerns the reduction of the data into differential cross sections while the second involves the comparison with DWBA calculations. In order to determine the cross section of a particular giant resonance component, it is not only necessary to subtract off a considerable and uncertain background, but also to isolate the desired cross section from those of other overlapping resonance structures . We have followed the course of simultaneously fitting the measured spectra with an empirical tail and a set of Lorentz line-shapes, which are taken to represent the giant resonances . The interpretation of the deduced cross sections rests on comparisons with theoretical DWBA calculations . The importance of the model dependency in this

282

R. S. HICKS et al.

regard has been demonstrated and cannot be too strongly emphasized. We not only have the freedom of choosing between different forms ofthe transition charge density (e.g., GT or SJ), but also of varying the parameters within a particular model (e.g., by changing the parameters. At this point it should be restated that although we have used a simple method to account for the effects of the nuclear deformation with existing DWBA codes for spherical nuclei, there is clearly need for a more rigorous approach to this problem. In the electroëxcitation of discrete inelastic levels, the validity of various models can often be checked by extending the data to higher momentum transfer sa). However, due to their great width, it is doubtful that this would be very informative in the case of giant resonances . Theoretical calculations so's~) have suggested that . considerable E4 and higher multipolarity strength is widely distributed throughout the giant resonance spectrum . At high q it would be difficult to separate this satisfactorily from the measured, lower multipolarity, cross sections. Indeed, it may well be that the high-q data points derived from the present work are inflated by such contributions, in which case both the excited state radii and the transition strengths (note relative cross-section magnitudes in fig. 4) will have been underestimated. Particularly in the case of the E2 and E3 resonances, this may at least partially explain the unexpectedly low values obtained for these parameters. An extension of the measurements to lower momentum transfer may permit more model-independent estimates of the reduced transition probabilities, particularly of the E2 components ' 9). In this region the contribution of E3 and higher multipolarity structure should be negligible (see fig. 3), and the El cross section can be reliably estimated from photoabsorption data . We conclude with a brief summary of some of the more important points to emerge from this work. 5.1 . El RESONANCES

The positions and shapes of the two GDR components were taken from photoreaction data; only the electrcexcitation cross section magnitudes were determined from the present measurements . The results lie in moderate agreement with the predictions of both the GT (or Tassie) and SJ models . The concept of the polarization of the GDR vibrations is favoured. 5.2. E2 AND EO STRUCTURE

Considerable electric quadrupole (or monopole) strength was identified in the excitation range 8-12 MeV. In order to fit this structure, it was necessary to use two resonance line-shapes, rather than one. A broad resonance centred at 23.2 MeV was also observed to have an E2 or EO character, however, the fact that it has also been seen in the (y, n) cross section would seem to rule out a monopole assignment .

isiTa GIANT RESONANCES

28 3

Unfortunately, the photoreaction data cannot similarly resolve the ambiguous multipolarity assignment for the 8-12 MeV strength . Even if this structure were to exhaust fully the E2 isoscalar EWSR, eq . (14) shows that the corresponding peak in the photoneutron cross section would have an amplitude of no more than 4-5 mb. Broad peaks of this magnitude would be difficult to observe experimentally . 5.3 . RESONANCES OF HIGHER MULTIPOLARITY

Resonances of higher multipolarity have been observed at 19.5±0.8 MeV (E3), 3.70±0.14 MeV (E3 or E4), and at 5.40±0.15 MeV (E4 or ES), with the possibility of the 3.7 and 5.4 MeV peaks being mixed with E2 strength . There may also be evidence of high multipolarity strength around 30 MeV. Table 1 forms a quantitative statement of our results. The authors gratefully acknowledge the support given to this project by former members of the electron scattering group at the University of Saskatchewan, in particular Dr. J. L. Groh, B. Norum, Dr. K. Itoh and Dr. F. J. Kline. Dr. A. Schwierczinski is thanked for generously providing the EO computer code . One of the authors (R.S.H .) wishes to express his appreciation for the hospitality extended to him by the laboratory staff under Drs. L. Katz and Y. M. Shin during his stay at Saskatoon. References

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