Electron scattering studies of magnetic isovector transitions in 12C at high momentum transfer

Nuclear Physics A411 (1983) 337-356 @ North-Holland Publishing Company

ELECTRON SCATTERING STUDIES OF MAGNETIC ISOVECTOR TRANSITIONS IN ‘*C AT HIGH MOMENTUM TRANSFER U. DEUTSCHMANN,

G. LAHM and R. NEUHAUSEN

Institut fiirKernphysik, Universitiit Mainz, 6500 Mainz, Germany

and J.C. BERGSTROM Saskatchewan Accelerator Laboratory, University of Saskatchewan, Saskatoon, Canada S7N OWO

Received 8 April 1983 (Revised 23 June 1983) Abstract: The inelastic electron scattering cross sections for the Ml transition to the 15.11 MeV (l+, T = 1) level and for the M2 transition to the 16.58 MeV (2-, T = 1) level in “C have been measured in the momentum transfer region q = 0.4-3.0 fm-‘, with emphasis on precise data at high momentum transfers. Additionally, a broad state near 15.4 MeV excitation has been observed and its excitation energy and natural width have been established as 15.44rtO.04 MeV and 1.5 f 0.2 MeV, respectively. The Fourier-Bessel technique for determining the MA transition current density has been applied to the Ml and M2 transitions. Particular attention has been paid to the Coulomb corrections required to deduce the PWBA form factors. The Ml radiative width isr,=38.5*0.8eV.

E

NUCLEAR

REACTIONS “C(e, e’), E = SO-330 MeV; measured o(E, E,,, 0). ‘*C deduced levels, r, form factors, transition current density f,,,.

1. Introduction In the past considerable effort has been expended investigating the complex of states near 15 MeV excitation in 12C by inelastic electron scattering. While the strong Ml transition to the 15.11 MeV (l’, T = 1) level has often stood in the centre of interest, our knowledge of the form factor of this state, especially its high-q behaviour, has nevertheless been very limited [for references to earlier experiments see ref. ‘)I. The experimental situation was somewhat improved by the recent work of Flanz et al. ‘), but there remains a demand for data with higher precision and higher momentum transfers. Photo- and electro-production of pions from ‘*C have been investigated by several groups in recent years 3-1o). In the threshold region, total and differential cross sections to discrete states have been measured and the results compared with predictions based on the electron scattering form factors of the analogue states in ‘*C; for the most part the agreement between theory and experiment has been good 1*11*12). Such experiments present an opportunity to probe the pion-nucleus 337 December1983

338

U. Deutschmann et al. /Electron scattering

interaction in the energy region between pionic atoms and low-energy pion scattering, providing the nuclear structure aspects are well understood. A related question concerns the role played by the meson-exchange currents (MEC) in the (e, e’) and (y, rr) reactions. One might expect the MEC to have a larger relative influence on the isovector form factors than on the (7,~) cross sections since, roughly speaking, the MEC diagrams for the former are similar to the lowest-order photoproduction diagrams. However, recent accurate measurements *) of the 12C(y, v)i2B(g.s.) cross section are in very good agreement with predictions derived from the 15.11 MeV (l+, T = 1) form factor of “C, with no regard for two-body effects in the latter. On the other hand, Dubach and Haxton 13) have fitted the available (e, e’) data at this state with the MEC simultaneously taken into account, and find that the MEC contribute nearly 30% to the form factor (squared) on the first maximum. If the two-body effects are indeed this large, it is difficult to understand why there is such good agreement between the (e, e’) and ( y, r) reactions. Since the resolution of these questions will require reliable nuclear structure input, the quality of the (e, e’) data will correspondingly have to improve. Accordingly, we have made new measurements of the “C isovector form factors with greater precision and over a larger range of momentum transfers than previous experiments *,14). In this paper we report our results on the 15.11 MeV (l’, T = 1) and 16.58 MeV (2-, T = 1) levels for 4 = 0.4-3.0 fm-‘. Particular attention has been paid to the Coulomb corrections required to deduce plane wave Born approximation (PWBA) form factors. The extent of the present data permits this to be done in a relatively model-independent way through Fourier-Bessel analyses of the transition current densities which also yield improved determinations of the radiative widths I’,,. The present paper is organized as follows. In sect. 2 we describe the experimental details leading to the raw spectra, while conversion of the spectra into cross sections is discussed in sect. 3. The Coulomb corrections necessary to extract the form factors in the PWBA are considered in sect. 4, as are the associated transition current densities deduced from the Fourier-Bessel analyses. Finally, in sect. 5 we comment on recent theoretical developments concerning the 15.1 MeV form factor. 2. Experimental

details

This experiment was performed at the electron scattering facility of the Mainz 350 MeV linear accelerator is). The energy compressing system 16) following the accelerator reduced the primary energy spread to less than 0.2%. The dispersion matching system “) was operated in the energy loss mode permitting on-target currents up to 30 ~.LA,distributed over an 8 mm square. The incident beam energies were 80-331 MeV while the scattering angles ranged from 62” to 138”; the corresponding momentum transfers spanned q = 0.39-2.98 fm-*.

U. Deutschmann

et

al. / Electron scattering

339

A carbon target with a thickness of 33.2 f 0.2 mg/cm* was used at 80 MeV, while a thickness of 70.3 f 0.3 mg/cm* was used at all other energies. The targets were fabricated from pure reactor graphite and the natural 1.11% abundance of 13C was taken into account in the subsequent data analysis. The targets were always oriented in the so-called transmission mode to achieve optimal resolution, limited to SE/E L 6 x 10e4 by straggling effects in the target and by kinematic broadening due to the finite acceptance angle of the spectrometer collimator (2.5” at 80 MeV and 1.3’ at all other energies). Scattered electrons were detected by a 300 channel overlapping scintillator array I*) with an intrinsic resolution of about 2.5 x 10v4 located in the focal plane of the 180” double-focussing spectrometer. The elastic and inelastic peaks were measured by the same scintillators to minimize dependency on detector efficiencies, spectrometer dispersion constants, etc., and in most cases the data were normalized to the elastic cross sections. Data were accumulated to give a statistical accuracy of about 1% for the 15.11 MeV peak area at 80 MeV, a few percent at higher energies except of the region of the form-factor minimum, and about 8% at the maximum momentum transfer of 2.98 fm-‘. The beam charge was measured to an accuracy of 0.1% by a non-intercepting device of the ferrite-ring variety r9). As an added check on the system stability, the elastic peak was continuously monitored by a fixed-angle spectrometer at an angle of 28” with respect to the beam-line *‘). The data were charge-normalized and histogrammed (“binned”) in the usual manner. A typical back-angle spectrum in the 16 MeV excitation region is shown in fig. 1. The peaks corresponding to the 15.11 MeV (I+, T = 1) and 16.58 MeV (2-, T = 1) levels are seen to be clearly resolved from the strong 16.11 MeV (2+, T = 1) peak. Note that the large natural width of the 16.58 MeV is quite apparent. Critical inspection of the spectra measured at 80 MeV and 120 MeV, especially at the more forward scattering angles, reveals a very broad peak at about 15.4 MeV. Some evidence for this new state also appears in hadronic scattering experiments 21-25).The large width of this peak can strongly influence the determination of the form factors for neighbouring states and therefore must be taken into account in the analysis of the electron scattering spectra. Since the excitation energy and the total width of the new state, as quoted in the recent compilation of AjzenbergSelove and Busch 26), are not sufficiently accurate for this purpose, we have carried out additional measurements to improve the definition of these quantities. The spin and parity of the new state have been assigned **) as 2+, so the form factor is expected to be essentially longitudinal and to reach its maximum at around q = 1 fm-‘. Therefore, we chose an incident electron energy of 250 MeV, and three spectra with high statistical accuracy were obtained at scattering angles of 39”, 56.6” and 74.6”, corresponding to momentum transfers of 0.82,1.16 and 1.48 fm-‘, respectively. The spectrum obtained at 56.6” and around 15 MeV excitation is

340

U. Deutschmann et al. / Electron scattering

Fig. 1. Backward-angle spectrum showing the region around 16MeV excitation. The spectrum is decomposed into individual peaks by a line-shape fitting procedure superimposed on the baseline B. (a.u. = arbitrary units.)

shown in fig. 2. The 15.4 MeV state shows up very clearly, as does the strong 16.11 MeV (E2) transition. Weaker peaks are seen at 14.08 MeV (E4), 15.11 MeV (Ml) and 16.58 MeV (M2), the latter two being strongly suppressed at forward angles due to their transverse character. 3. Data analysis 3.1. EXCITATION

ENERGY

AND TOTAL WIDTH OF THE 15.4MeV

LEVEL

To determine the excitation energy and total width of the 15.4 MeV level, the three spectra at 250 MeV were decomposed into individual peaks by means of a line-shape fitting procedure similar to that described in ref. 14). The 15.11 MeV and 16.11 MeV peaks were fitted using the shape parameters of the elastic peak, since their natural widths are negligible compared to the system resolution. The broad levels, whose natural widths were clearly discernible in all spectra, required a different approach. Investigations of giant dipole resonances in (y, n) experiments have suggested their natural shapes are best described by Breit-Wigner resonance functions 27), so in the present analysis the broad peaks were represented by a convolution of the elastic line shape with Breit-Wigner functions. The validity of the convoluted shapes is attested to by the high quality of the fits as demonstrated, for example, in figs. 1 and 2.

U. Deutschmannet al. / Electronscattering

Oh, 230

I

t

I

231

232

233

341

em

E CMeVl

Fig. 2. Forward-angle spectrum showing the region around 15.4MeV excitation. The spectrum is decomposed by a fine-shape fitting procedure into individual peaks whose fitted line shapes are shown on top of the basline B. The FWHM of the corresponding elastic peak is 150 keV. (a.u. = arbitrary units.)

A polynomial “background” (baseline B in the figures) was included in the fitting program to accommodate contributions from the radiation tails of the elastic and lower-lying inelastic states, from the complex of several broad levels around 19 MeV, and from the proton continuum which starts at 15.9 MeV. The positions of all peaks were established relative to the well-known excitation energy of the 15.11 MeV state. The three spectra were repeated& fitted with different sets of initial parameters, and the resulting excitation energies and total widths from those fits which returned reasonable x2 were combined to yield the final values. We obtained 15.44~0.04MeV for the excitation energy and I.S-L~O,ZM~Vfor the total width of the state under consideration, where the quoted errors correspond to one standard deviation. The above quantities are compared with previous estimates in table I. The earlier values for the excitation energy agree with our result within their quoted errors; the error in the present measurement, however, is considerably smaller. The new value for the total width is approximately 25% smaller than the previous estimates. 3.2. CRQSS SECTIONS FOR THE TRANSITIONS

TO THE 15.11 MeV (Ml) AND i6SBMeV

(MZ) LEVEL

The line-shape ~tt~ngprocedures for all other spectra were performed in a manner similar to that described above. The areas were found by integrating the line shapes

342

U. Deutschmann et al. / Electron scattering TABLE 1 Excitation energy E and total width r of the 15.4 MeV level from various experiments Experiment

E (MeV)

f (MeV)

(P, P’) (3He, 3He’) 6% a ‘)

15.3ztO.2 15.2hO.3 15.5*0.1

2.ozto.2 1.8*0.3 2.1kO.3

adopted value

15.4ztO.l

2.0*0.2

15.44*0.04

1.5hO.2

(e,4

Ref. 22 21 i 22

)

26

)

this work

between 6r,,, above and lOr,,, below the peak maximum, where f,,, is the FWHM of the given peak. Schwinger, bremsstrahlung and ionization loss corrections were made as in ref. 2K) (eq. (4.1)), ref. 29) and ref. 30), respectively. The terms of order Z2 were omitted from the Schwinger correction of ref. 28) in view of the remarks by MO and Tsai 29). In most cases the inelastic peak areas were normalized to the elastic peaks, but in the vicinity of the elastic diffraction minimum and at the highest momentum transfers we relied on the overall system calibration deduced from elastic measurements at neighbouring momentum transfers. The elastic cross sections were evaluated by phase-shift analyses using the Fourier-Bessel charge distribution parameters as determined by Merle 31), with allowances made for the finite acceptance of the spectrometer. Subsequently, the elastic cross section of 12C was re-measured by Reuter et al. 20) and also analysed with a Fourier-Bessel parameterization. Since the slight deviations between the two data sets do not lead to a significant change of our normalization, we deemed it unnecessary to modify our inelastic cross sections. The experimental cross sections for the Ml transition to the 15.11 MeV state and the M2 transition to the 16.58 MeV state are listed in tables 2 and 3, respectively, where the deviations reflect statistical errors, uncertainties in the system calibration, beam energy and scattering angle, all quadratically added. An additional 5% error has been applied (quadratically) to the M2 cross sections in table 3 to allow for an uncertainty in the “background” previously mentioned. 4. Coulomb

4.1. FORMALISM

corrections,

AND ANALYSIS

PWBA form factors, and the transition current densities PROCEDURE

We now consider the Coulomb corrections required to convert the experimental data into PWBA cross sections. The correction factors are defined here by

U. Deutschmannet al. / Electron scattering TABLE

343

2

Tabulation of cross sections, Coulomb correction factors and the PWBA form factors for the Ml transition to the 15.11 MeV (l+, T = 1) level in “C as determined by the present experiment (the cross

sections are multiplied by lo”, where the power II is given in column 5)

8 80.5 80.5 80.5 80.6 80.5 80.5 80.5 119.9 120.0 119.9 119.9 119.9 119.9 149.9 149.9 149.9 149.9 169.8 169.8 200.0 200.0 200.0 199.9 199.9 239.8 239.8 239.8 239.8 239.9 300.2 300.2 300.1 300.1 171 .rl 331.1 331.0 331.0 331.1 331.1

(d-/da)

(deg)

(im4-1,

62.24 73.30 85.33 97.38 109.56 121.68 133.73 69.96 94.11 106.37 118.39 130.56 136.56 109.85 121.95 127.94 133.09 133.01 138.00 110.01 118.99 124.01 133.01 138.01 109.66 115.75 122.13 127.72 133.72 100.25 112.45 124.41 136.59 110.00 110.00 119.00 124.00 130.00 137.00

0.39 0.45 0.50 0.56 0.60 0.64 0.68 0.65 0.83 0.91 0.97 1.03 1.05 1.17 1.25 1.28 1.31 1.49 1.52 1.58 1.66 1.70 1.76 1.79 1.90 1.96 2.03 2.08 2.13 2.24 2.42 2.57 2.69 2.64 2.64 2.77 2.83 2.90 2.98

x 10”

(cm’/sr) 6.6340.13 4.48kO.12 3.14tkO.06 2.21 *to.04 1.73rto.04 1.31 *to.03 1.03 f 0.02 2.35*0.03 6.45zkO.11 3.43Lko.10 1.91 zkO.06 1.26~0.04 9.57 zt 0.26 2.89~kO.13 5.2650.90 3.29* 0.72 <8.6 4.511to.73 4.93zkO.66 7.321t0.86 1.05kO.12 1.122to.12 1.12kO.08 1.10*0,09 1.31kO.06 1.15rto.04 1.10*0*04 9.67 f 0.44 8.29AO.28 8.14ztO.37 4.52rt0.19 2.661tO.14 1.40*0.10 2.18k0.21 2.14kO.20 1.02*0*09 8.261t0.69 5.381tO.55 4.07*0.44

32 32 32 32 32 32 32 32 33 33 33 33 34 34 35 3s 36 35 35 3s 34 34 34 34 34 34 34 35 35 35 3s 3s 35 35 3s 35 36 36 36

1.070 1.067 1.062 1.0.55 1.045 1.034 1.023 1.026 0.980 0.946 0.905 0.865 0.846 0.765 0.604 0.496 2.247 1.926 1.455 1.310 1.264 1.211 1.188 1.116 1.092 1.072 1.056 1.042 1.027 0.989 0.954 0.925 0.950 0.950 0.919 0.909 0.896 0.890

24.9zt0.S 27.8*0.5 30.1 kO.6 30.1*0.5 31.liO.7 29.3zkO.7 27.3kO.5 29.1 kO.4 19.2kO.3 14.2k0.4 10.3 zto.3 8.52kO.27 7.09*0.19 2.48kO.11 0.71hO.12 0.59*0.13 CO.082 0.245 f 0.040 0.329 f 0.044 0.595~0.070 1.12*0.13 1.33*0.14 1.561tO.11 1.66*0.14 1.98*0.10 2.~~0.07 2.17kO.08 2.10*0.10 1.97*0.07 1.72kO.08 1.29kO.06 0.962 f 0.051 0.608 f 0.043 0.754 f 0.073 0.739 i 0.069 0.43lkO.038 0.381 kO.032 0.273 *0.028 0.227 f 0.025

where DWBA means distorted-wave Born approximation. Since the fC are closely tied to the MA transition current densities, we will discuss them together. The di~erential cross section in the PWBA is related to the magnetic form factor by (dfl/df&wnA

= a&+

tan* &FE

(4) ,

(2)

344

U. Deutschmann et al. / Electron scattering

G

‘*C

!

15.11 MeV

[l+, T-1)

0 NBS x BATES l MAINZ

IO‘=

I& 1 -6 g 10 7

n T I

IO7

I 1

0

, 2

1 3

c

q [frri’l

Fig. 3. Form factor for the Ml transition to the 15.11 MeV level and the Fourier-Bessel fit (solid curve) to the combined NBS 14) and Maim data. The Bates data were taken from ref. ‘). A11data have been corrected for Coulomb distortion.

where CM is the Mott cross section for a point charge 2. For a magnetic transition from the ground state with spin Ji to an excited state with Jf the form factor is connected to the reduced matrix element by the expression

(2Ji + l)-‘J(J~II~~”(q>IIJi)I’ *

R&A(4) =$

(3)

Finally, the reduced matrix element is given in terms of the transition current density:

(J$‘?” (s)llJi) = Jm x(qr)~~A(r>r2dr, 0

(4)

where J**(r) defines the radial distribution of the current density and contains the magnetization (or spin) current, the convection current and contributions from two-body effects. It is significant for our purpose that the nuclear structure aspects of the Coulomb distortion are completely determined by JAn(r) and the ground-state charge density. It is assumed that J**(r) may be expanded to sufficient accuracy in terms of a truncated Fourier-Bessel series, and that the current density beyond some radius R has a negligible effect on the resulting error band. Thus 32) JAA

(r)

=

j,

lo,

‘,iA br)

9

forrcR forr>R,

(5)

345

U. Deutschmannet al. / Electronscattering TABLE 3

Tabulation of cross sections, Coulomb correction factors and the PWBA form factors for the M2 transition to the 16.58 MeV (2-, T = 1) level in “C as determined by the present experiment (the cross sections are multiplied by lo”, where the power n is given in column 5)

(MZ)

80.5 80.5 119.9 119.9 149.9 149.9 149.9 149.9 169.9 169.9 169.9 169.8 200.0 200.0 200.0 199.9 199.9 239.8 239.8 239.8 239.8 239.9 300.2 300.2 300.1 300.1 331.1 331.1 331.1 331.1 331.1 331.1

122.00 149.40 130.56 136.56 109.84 121.95 127.94 133.99 119.00 124.00 133.01 138.00 110.01 118.99 124.01 133.01 138.01 109.66 115.75 122.13 127.72 133.72 100.25 112.45 124.41 136.59 110.00 110.00 119.00 124.00 130.00 137.00

0.64 0.71 1.02 1.04 1.17 1.24 1.28 1.31 1.40 1.43 1.48 1.51 1.57 1.65 1.69 1.76 1.79 1.89 1.96 2.02 2.07 2.12 2.24 2.42 2.56 2.69 2.63 2.63 2.76 2.82 2.90 2.97

(do/da) x 10” (cm’/sr)

n

<4.3 <2.6 3.32kO.78 4.81 f 1.00 1.13*0.10 1.50*0.09 1.67 f 0.09 1.72kO.10 2.59*0.15 2.48*0.14 2.25kO.15 2.27*0.14 2.48hO.18 2.22hO.19 2.05kO.15 1.91*0.14 1.85+zO.13 1.73*0.10 1.51kO.08 1.35*0.08 1.22*0.07 1.14ztO.06 9.88k0.56 5.59kO.32 3.65*0.24 1.89hO.13 3.46k0.27 3.281tO.27 1.7a*o.14 1.24zkO.10 7.38&O 72 5.34i0.48

34 34 34 34 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 34 34 34 34 34 34 34 34 35 35

fc

F&, x lo4

1.522 1.462 1.241 1.203 1.191 1.180 1.134 1.126 1.113 1.108 1.081 1.068 1.061 1.050 1.045 1.030 1.022 1.014 1.008 1.002 0.999 0.980 0.957 0.930 0.952 0.952 0.922 0.902 0.876 0.842

co.099 co.081 0.128~0.030 0.207 f 0.043 0.600*0.053 l.OlzkO.06 1.24hO.07 1.40*0.08 2.28kO.13 2.38zkO.14 2.46kO.17 2.64*0.16 2.71*0.20 2.89kO.25 2.90* 0.21 3.08kO.23 3.18kO.22 2.85ztO.16 2.82kO.15 2.81hO.17 2.77zt0.16 2.81 kO.15 2.15zt0.12 1.61 ho.09 1.32zko.09 0.823 kO.057 1.20*0.09 1.13*0.09 0.751rto.59 0.575 f 0.046 0.385kO.038 0.315rtO.028

where the constants qy are defined by jv(qYR) = 0; i.e. they are derived from the vth zero of the speherical Bessel function of order A. The expansion coefficients a, are determined by a least-squares fit to the data through eqs. (3) and (4). Instrumental in the analysis is the DWBA code HADES recently developed by Andresen et al. 33) at Mainz. This code has the advantage of reduced computing time and improved numerical accuracy, especially at large energies and scattering angles, compared to other DWBA programs. The transition current densities in HADES are described by Fourier-Bessel expansions, as in eq. (5), with similar expressions for the static charge density.

346

U. Deulschmann et al. /Electron scattering

The transition current densities and Coulomb correction factors were determined as follows. A preliminary set of a, coefficients was calculated from a Fourier-Bessel fit to the “experimental” form factors F&exp &R) = (d~/dJ2)),,,[(+&+tan2

~~)I-’,

where some allowance was made for Coulomb distortion momentum transfers qeff = 4ri + 3~~zn5~R~~i .

(6)

by using the effective (7)

Here, R,, is the radius of a uniform sphere having the same rms radius as 12C, and the other symbols have their usual meaning. Thus, eq. (6) represents the first generation of PWBA form factors. The PWBA and DWBA cross sections were subsequently calculated with this (I, set and the corresponding fc were applied to the form factors of eq. (6), but now using 4 instead of qeff to create the second generation of form factors. A new set of a, coefficients was obtained by a FourierBessel fit to the second generation form factors, and the cycle with HADES was repeated. This process converged quickly giving a final set of correction factors and PWBA form factors. The correction factors and PWBA form factors for the 15.11 MeV and 16.58 MeV transitions are given in tables 2 and 3, respectively. 4.2. THE

15.11 MeV

(I+, T= 1) LEVEL

For the DWBA analysis of the 15.11 MeV transition, the low-q measurements of Chertok et al. 14) made at NBS (Washington) were included as part of the total data set. The present fc factors are less model-dependent than those based on the incompressible liquid-drop model as used by Chertok et al. 14). Therefore we have re-analysed the NBS data, starting with the published inelastic/elastic ratios, and the results are summarized in table 4 where for comparison we also list the original TABLE

4

PWBA form factors for the Ml transition to the 15.11 MeV level [the original data were taken from Chertok et al. 14) and were re-analysed in the present work]

FL, x lo4 (M&J

(d:gI

35.695 40.638 45.705 55.719 35.538 40.599 50.609 55.624

75.2 75.2 75.2 75.2 110.64 110.64 110.64 110.64

(fl& 0.184 0.213 0.243 0.303 0.237 0.278 0.360 0.402

original 0.714 f 0.054 1.029+0.064 1.241 kO.067 1.775 f 0.083 1.067*0.061 1.520*0.064 2.047 f 0.076 2.442 f 0.095

FL, x lo4

fc re-analysed 1.116 1.114 1.110 1.105 1.137 1.130 1.112 1.100

0.719*0.054 1.034*0.064 1.250+0.068 1.783 f 0.083 1.071*0.061 1.534kO.065 2.073 f 0.077 2.480*0.096

U. Deutschmann etal. / Electron scattering

'2C

15.11 MeV

w,

347

7=1 )

Fig. 4. The Ml transition current density for the 15.11 MeV level from the Fourier-Bessel analysis of the experimental form factor. The shaded error band is two standard deviations wide.

values. Note that although the new analysis was based on a completely different method, the original form factors were modified by less than 2%. The 15.11 MeV Ml form factor is shown in fig. 3, together with the re-analysed NBS data. The agreement between the two data sets is excellent. The present data also compare favorably with the recent measurements of Flanz er al. ‘) made at the BATES facility, although there is some inconsistency between the two sets of Coulomb correction factors on the second lobe. The transition current density J**(r) for the 15.11 MeV level was deduced from a Fourier-Bessel analysis of the final PWBA form factors using the code XDEN, originally developed in Saskatoon and modified at Mainz. Some of the formalism, the method for choosing the cut-off radius R, etc., are presented in the appendix; that presentation also applies to the Fourier-Bessel analyses described in connection with the fc factors. Fig. 4 illustrates the Ml transition current density for the 15.11 MeV level based on the combined MAINZ-NBS (re-analysed) data, while the corresponding Fourier-Bessel coefficients are given in table 5. The error band represents a variation of *l standard deviation, reflecting the experimental statistics and the assumed behaviour of the form factor beyond the last experimental point. Finally, we consider the ground-state radiative width and the transition radius which derive from the Fourier-Bessel analysis (see the appendix for details). The radiative width for the 15.11 MeV level from the parameters in table 5 is r&Ml)

= 38.5 kO.8 eV .

This is about 4% larger than the currently accepted value 26), based on the original

348

U. Deutschmann et al. / Electron scattering TABLED

Fourier-Bessel coefficients of the Ml transition current density for the 15.11 MeV level (the number of coefficients is N, while N, designates the number in the region q.
4.

0.541 0.931 1.314 1.695 2.075 2.454 2.834

1

2 3 4 5 6 7 N=13

(fm-'1

N, = 7

a, X lo3 (fmm3) 2.181*0.007 4.131kO.021 1.182*0.053 -4.218*0.067 -7.962kO.071 -8.61*0.13 -6.30k0.15

Y

4. (fm-r)

a, X lo3 (fmW3)

8 9 10 11 12 13

3.213 3.592 3.971 4.349 4.728 5.107

-4.o* 1.3 -0.9rt3.3 0.7+3.2 -0.4zt2.9 0.3 f 2.6 -0.2k2.2

R = 8.3 fm

,y2/f=1.15(f=39)

NBS work 14), T,,(Ml) = 37.0* 1.1 eV. The slight disparity might be a function of the analysis procedure: the ground-state radiative width was determined here from the transition current density as an integral quantity, whereas in the original NBS work it was obtained by extrapolation to the so-called photon-point without any knowledge of the actual transition current density. In other words, we have benefited from a much wider range of momentum transfers. Historically, r,, has played a role in a test of the conserved vector current hypothesis i4), which implies a direct relation between r,, and the weak-magnetism components in the P-decay of the analogue nuclei “B and 12N. The new result improves slightly the comparison with the CVC prediction. The transition radius which obtains from the parameters in table 5 is R,, = 2.73 f 0.06 fm .

The ratio of R,, to the rms radius R, of the ground state, RJR,

= 1.11* 0.02 ,

is in excellent agreement with the average value RJR, = 1.09 deduced by Theissen 34) from a consideration of several Ml transitions in light nuclei. Again, we note that the present transition radius is derived directly from the transition current density, in contrast to the conventional method of deriving it from the slope of the form factor at low momentum transfers. 4.3. THE 16.58 MeV (2-, T = 1) LEVEL

The M2 form factor for the broad 16.58 MeV level is shown in fig. 5, along with the data of Flanz 35). The slight disagreement between the two experiments on the

U. Deutschmann et al. / Electron scattering 12C

349

16.58 MeV (2-,T=l)

I

3 q [fni’l Fig. 5. Form factor for the M2 transition to the 16.58 MeV level and the Fourier-Bessel fit (solid curve) to the Mainz data. The Bates data were taken from ref. 35) and DWBA corrected in the present work. The dashed curve shows the shell-model prediction from Donnelly 36) divided by a factor 2.

second maximum originates primarily in the different line-shape functions used for fitting the spectra; we employed the convoluted Breit-Wigner shape, while Flanz used a broadened version of the elastic shape. The Fourier-Bessel analyses of the M2 form factor suffer somewhat from the lack of experimental data in the momentum transfer region below 1 fm-‘. We have accumulated two spectra with very high statistical accuracy at 80.5 MeV and scattering angles of 122” and 149.4”, corresponding to momentum transfers of 0.64 and 0.71 fm-‘, but in both cases the 16.58 MeV peak was completely hidden in the background and only upper limits could be assigned to the form factor. Even the 180” scattering experiment 35), which considerably improves the ratio of transverse to longitudinal contributions in the spectra, could only establish upper limits in this region. In view of the systematic difference between the two data sets shown in fig. 5, the Fourier-Bessel analyses were applied to the present data only, excluding the upper limits in the low-q region. A second analysis included three “pseudo-data” points at low q, based on Donnelly’s theoretical form factor 36) (the dashed curve in fig. 5; note that the theoretical F2M2 has been renormalized downward by a factor of 2) at q = 0.33, 0.59 and 0.82 fm-’ and arbitrarily assigned a &60% error, to provide a mild constraint consistent with the upper limits. The results of both analyses are virtually indistinguishable. The M2 transition current density (fig. 6) is strongly peaked in the interior of the nucleus, but the small bumps at r = 3 and 5 fm are necessary for a correct description of the second lobe of the form factor.

3.50

U. Deutschmann et al. /Electron scattering

16.58 MeV

- 0.8

(2-,T=l )

1

Fig. 6. The M2 transition current density for the 16.58 MeV level from the Fourier-Bessel analysis of the experimental form factor. The shaded error band is two standard deviations wide.

The Fourier-Bessel radiative width:

coefficients

are listed in table 6 and yield the 16.58 MeV

f,,,,(M2) = 48 f 8 meV . The 16.58 MeV level is presumably the lowest member of a (2-, T = 1) triad built on the particle-hole states (2s1J(lp~&~, (ld5,2)(1p3,2)-1 and (ld3&(1p3&*, where according to Donnelly 36) the last term should have a very small amplitude in this level. The corresponding theoretical M2 form factor has a peculiar shape distinguished by a small first maximum, a large second maximum, and a minimum around q = 0.9 fm-‘. The strong suppression of the first lobe is caused in part by interference between the spin and orbital terms in the transition current, while the second maximum is characteristic of the spin current. Our measurements are TABLE 6 Fourier-Bessel

Y

coefficients of the transition current density for the 16.58 MeV level (analysis with the low-q pseudo-data included) qv (fm-9

a, X lo* (fmm3)

Y

4” (fm-‘)

a, x 10’ (fme3)

1.048 1.654 2.241 2.821 3.398

-0.331*0.013 -3.057Ito.031 -4.669 f 0.059 -3.788 f 0.066 -0.68 f 0.79

6 7 8 9

3.973 4.548 5.121 5.695

-0.4* 1.5 0.4* 1.6 -0.3* 1.5 0.2*1.4

1 2 3 4 5 N=9

N,=4

x2/f =0.95 (f = 29)

R = 5.5 fm

U. Deutschmann et al. /Electron scattering

351

essentially in agreement with these predictions. As a final comment, we note that the radiative pion capture reaction r*C(r-, y)l*B(2-, T = 1) is not affected by the interference since in this case the transition operator is totally spin dependent; hence comparison with (e, e’) at the first maximum (the pion point is q = 0.62 fm-‘) could serve to separate the spin and orbital pieces. On the second lobe the form factor is large and predominantly spin-flip in nature, therefore in the region q 3 1.5 fm-’ the (y, 7r*) reactions should strongly populate the corresponding analogue states in ‘*B and ‘*N. 5. Comparison with theoretical developments For a long time the 15.11 MeV level in ‘*C was the outstanding example in the discussion of pion-like states, states of unnatural parity (J” = O-, lc, 2-, . . .) which can be excited by AT = 1 transitions. Several theoretical groups have put considerable efforts into developing new theoretical approaches in an attempt to understand the enhancement of the Ml form factor at high momentum transfers. A complete discussion of these developments and a comparison with experiment was recently given by Weise 37), who came to the conclusion that there is no tendency towards critical enhancement of the pion field, if the short-range spin-isospin correlations are sufficiently repulsive. The experimental Ml form factor is very well described by the calculation of Toki and Weise 38) over the whole momentum transfer range and we refer the reader to the illustrations in refs. 37*38)for the comparison with the present work, The enhancement at high momentum transfers achieved by the Toki-Weise calculation is not unrelated to the quenching of other spin-isospin-dependent phenomena at low q, such as the Gamow-Teller transitions observed in the Indiana (p, n) reaction studies, so we will review briefly the salient features of the model. For reasons we discuss later, the success in describing the present data is also a bit disconcerting. According to the Toki-Weise model 37-39), the T = 1 spin-isospin-dependent part of the Ml operator is renormalized in a q-dependent manner which serves to suppress the low-q behaviour of the transition matrix element while enhancing the high-q behaviour, the convection-dependent part being relatively unaffected. For the most part, the renormalization originates from the excitation and destruction of virtual A(1232) particle-nucleon hole pairs in the surrounding nuclear medium, along what would normally be considered the virtual photon path. It is this “polarization” of the medium by the A-hole degrees of freedom that effectively modifies the response of the ordinary nuclear degrees of freedom to the virtual photon. Note that this has nothing to do with whether or not there are A(1232) components in the ground-state wave function. Excitation of the virtual pairs is mediated by the usual attractive one-pion exchange plus a repulsive short- (zero-) range spin-isospin interaction proportional

352

U. Deutschmann et al. /Electron scattering

to g’(ai - v~)(T~ - Q). Toki and Weise pointed out that by including two-pion exchange, one achieves a much more natural coupling to the transverse axq electron scattering interaction, and the 15.1 MeV Ml form factor can then be explained using a reasonable value for g’. Previous attempts to describe the form factor without consideration of the two-pion exchange in the particle hole interaction required such a small g’ that “pion condensation” became possible 37*39). The structure of the 15.1 MeV state was assumed to be accurately described by the Cohen-Kurath model. Of course, it has been known for some time that this model gives a very poor description of the Ml form factor, for example it underestimates the experimental FL1 (q) by more than a factor of 5 at the second maximum and produces a diffraction minimum considerably above the observed position [see e.g. ref. 37)]. It is therefore remarkable that the Toki-Weise renormalization overcomes both these obstacles and still avoids the necessity for pion condensates. However, as Weise remarks 37), the complicated structure of ‘*C does not make it an ideal test case for the theory. Actually, it would be disturbing if the polarization effects indeed produce such drastic modifications of the isovector magnetic form factors, since the nuclear structure information implicitly contained in measurements made over a wide range of momentum transfers would then be hopelessly entangled with the subtle details of operator renormalization. It would therefore be most interesting to see the calculation repeated for a simpler system than r*C, for example 6Li. Both the isoscalar and the isovector Ml form factors have been measured for that nucleus 40) and are found to have very similar shapes, perhaps indicating that renormalization effects are negligible in very light nuclei. While polarization phenomena in **C tend to shift the minimum of the isovector form factor to lower momentum transfers, the meson-exchange currents seem to have the opposite effect. The influence of MEC on the 15.1 MeV form factor was investigated by Dubach and Haxton 13) using the data available at that time, and according to their calculation the interference between the one- and two-body amplitudes shifts the diffraction minimum to higher momentum transfers than predicted by the one-body terms alone. The exchange currents cannot account for the enhancement at high momentum transfers but apparently they do amplify the form factor on the first lobe, for example FL1 (q) is increased by about 20% at the photon point and nearly 30% at the first maximum. Several discussions concerning the high-q behaviour of the 15.1 MeV Ml form factor have started with the familiar Cohen-Kurath shell-model wave functions in the belief that they give a reliable description, at least in the long-wavelength limit. The discrepancy at the high momentum transfers occurs in a region where strong cancellations occur in the form factor, and it is conceivable that some improvement could be gained by making small changes in the Cohen-Kurath matrix elements 13). With respect to the long-wavelength limit, we note that the present measurements of f,,(, (15.1) is 25% higher than the Cohen-Kurath prediction, but if the MEC are

353

U. Deutschmann et al. / Electron scattering

taken into account the difference is only about the wave functions. However, this completely polarization which presumably would tend to cancelling the MEC influence. Clearly, a correct both phenomena together.

5%) an impressive achievement for ignores the quenching due to core depress the Ml strength, partially treatment is needed which considers

The authors acknowledge the excellent cooperation with all members of the Institut fiir Kernphysik and are grateful for their generous help and overall support. The present work was supported in part by the Deutsche Forschungsgemeinschaft.

Appendix Since the Fourier-Bessel technique for deducing transition current densities from magnetic form factors is not a generally established method, we present here some of the formalism and discuss parts of the procedure. For further information the reader is referred to refs. 32*40). For a magnetic transition from the ground state with spin Ji to an excited state with Jf the reduced transition probability is given by the expression B(MA, 4, Ji+ Jr) =Uhq-**(Wi+ l)-‘((J~~~T~g~~Ji)~2 9

(A.11

with a, = A[(2A + 1)!!12/(A + 1) . The radiative width f,, for y-decay to the ground state is related to the reduced transition probability in the limit q = k, with k = E/~c, where E is the excitation energy. One finds

where 2Ji+l B(MA, k, Jt+Ji)=mB(B(MA,

ky Ji+Jf) e

f

(A.3)

Combining eqs. (A.l) through (A.3) leads to the expression ryO = 8raE (2 Jf+ l)-‘J(J~llT~’ (k)I(Ji)l* .

64.4)

We introduce the transition current density via eq. (4) and obtain r,, = 8mxe (2 Jr + 1)-r

II

j* (kr)J,* (r)r* dr * . I

Because kr cc 1, it is in general a good approximation to expand j* (kr) into a power series in kr and to keep only the leading term. However, in the case of the 15.11 MeV state in l*C this approximation leads to a deviation of about 2%. We therefore

354

U. Deutschmann et al. / Electron scattering

stay with the exact formula, eq. (AS), and insert the truncated series (eq. (5)) for J**(r):

Fourier-Bessel

R

I

jA

(kr)J,, (r)r* dr = E a, v=l

The integral is straightforward

I0

i* (krh (qvr)r2

dr .

64.6)

and one obtains

j*(kr)J,,(r)r*

dr =j*(kR)

qvR* f! -iA-x(qvR). v-1 “k*-q,

64.7)

Eqs. (AS) and (A.7) are used for deducing the radiative width from the parameters of the Fourier-Bessel expansion of the transition current density. The MA transition radius (squared) is related to the transition current density by the expression 34)

R2 =A + 1 (JrI(f”*IIJi) tr A +3 (Jfllr^llJi> ’

(A.8)

where the reduced matrix elements are given by

(A.9 with m = 0, 2. Again by inserting the Fourier-Bessel solving the integrals we obtain

series (eq. (5)) for J,,*(r) and

= -R”+*.i, qv,_yLq YR) 3 (J~llr”+*//Ji) = -R”‘4

jl

ff [ I- 2j42h,+;:)]h-1(q.R)

”

.

(A.ll)

Eqs. (A.8), (A.lO) and (A.ll) are used for deducing the transition radius from the Fourier-Bessel parameters. When applying the Fourier-Bessel technique to experimental form factors, one faces two difficulties: (i) the lack of experimental data beyond the maximum momentum transfer qmaxr (ii) the suitable choice of the cut-off radius R. We approached the first difficulty in the manner as described and discussed in ref. 32) (eqs. (19) and (20)). The form factor for q >qmax is assumed to lie with uniform probability within an envelope function of the form F,,,(q)

For the cases under consideration values for the parameters:

= A, epBnq” .

(A.12)

here we have chosen n = 1 and find the following

E = 15.11 MeV,

A

E = 16.58 MeV,

A I= 0.0274,

1

=

0.0297 ,

B1 = 0.86 fm

,

B1 = 0.54 fm

.

355

W.Deutschmann et al. / Electron scattering X2/f

!I& &

50

6

Le\if

8

10

c12

1G R[fm]

N=13

36 30 12 / 2L-j

I 41 , 4

1

I4

6

6

,

I

10

12

14

w

112

I

w

lb

11

R[fm]

R tfml

N -13

d

Fig. 7. Quality of the least-squares tit, radiative width and the transition radius versus the cut-off radius R for the Ml transition to the 15.11 MeV level. The arrows indicate the final choice of R.

To overcome the second difficulty we have determined the parameters of the transition current density (eq. (5)) as a function of the cut-off radius R by a least-squares fit to the experimental form factors. In fig. 7, we demonstrate, for the Ml transition to the 15,ll MeV level, how the quality of the fit (expressed by x2 per degree of freedom), the radiative width and the transition radius depend on R. For small cut-off radii the vafue for x2/f decreases very fast with increasing R and, then, stays practicalfy constant between R = 6-10 fm. In the same R-range, the radiative width and the transition radius are aIso nearly independent of R. Increasing R beyond 10 fm leads to a slight reduction of x2/f, This means that an increasing number of qy will lie inside the experimental region q
356

U. Deutschmann et al. / Electron scattering

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