Nuclear Physics A563 (1993) 225-246North-Holland

The54,56Fe(n, p)54,56Mn reactions at En = 97 MeV

T. R6nngvist, H. Condd, N. Olsson, E. Ramstr6m, R. ZorroDepartment of Neutron Research, Uppsala University, Box 535, S-75121 Uppsala, Sweden

J. Blomgren', A. Hdkansson, A. Ringbom, G. TibellDepartment of Radiation Science, Uppsala University, Box 535, S-75121 Uppsala, Sweden

O. Jonsson, L. Nilsson, P.U. RenbergThe Svedberg Laboratory, Uppsale University, Pox 533, S-75121 Uppsala, Sweden

S.Y. van der WerfKernfysisch Versneller Instituut, Zemikefaan 25, 9747AA Groningen, The Netherlands

W. UnkelbachIndiana University Nuclear Theory Center, Bloomington, IN 47408, USA

F.P. BradyDepartment of Physics, University ofCalifornia, Davis, CA 95616, USA

Received 21 January 1993

AbstractDouble-differential cross sections of the ss .sa Fe(n, p) reactions have been measured at 97

MeV in the angular range 0-30° for excitation energies up to 40 MeV. The spectra have beendecomposed into different muttipolarities by a technique based on the use of sample angulardistributions calculated within the distorted-wave Born approximation. From the identifiedGamow-Teller strength, Sit . values were obtained for "Fe and "Fe. Comparisons withavailable shell-model calculations of the GT strength were made . The results are important formodels of supernova explosions since electron-capture rates, which are proportional to Sp., inlf2p-shell nuclei affect the dynamics of the star. At higher excitation energies, the spectra aredominated by L =1 strength in broad distributions with a maximum at about 12 MeV.Microscopic calculations based on the random-phase approximation were performed and com-pared with the experimental data .

Key words: NUCLEAR REACTIONS 54,56Fe(n, p), E=97 MeV; measured o(Ep , 0) DeducedGamow-Teller and dipole strengths Statistical multistep direct DWBA calculations multipoledecomposition, comparison with RPA calculations.

' Present address : Indiana University Cyclotron Facility, Bloomington, Indiana 47408, USA.

0375-9474/93/$06.00 0 1993 - Elsevier Science Publishers B.V. All rights reserved

NUCLEARPHYSICS A

226

T. Räruzgcèst et al. / s'."Fe(n, p)

1. Introduction

The charge-exchange reaction (p, n) and (n, p) select isovector excitations innuclei and are therefore ideal probes for studies of the isovector part of thenucleon-nucleus interaction .

The 0° Gamow-Teller (GT) cross sections (AT= 1, AS= 1, AL = 0, NO)excitations) are proportional to the analogous ß-decay strengths. Charge-exchangereactions at small momentum transfer can therefore be used to study /3-decaystrength distributions when /3-decay is not energetically possible. The (p, n)reaction probes the Sß - strength, corresponding to ß--decay, and the (n, p)reaction gives the strength for ß+-decay, Sß.. These strengths are related to eachother by the model-independent Ikeda sum rule [1] (summed over all states)

Sß-- Sß.= 3(N- Z) .

The GT(ß+) strengths are proportional to the electron-capture strengths whichare of interest in astrophysics since the electron-capture rates influence theelectron-to-baryon ratio and entropy in presupernova stars. At the beginning of thestellar collapse, the center of the star is made up of nuclei in the Fe region. Slowerelectron capture gives a collapse at a higher entropy per baryon and with a largerinitial core mass. Both of these effects decrease the probability for a viablesupernova explosion. Electron-capture rates in If2p-shell nuclei are thereforeImportant for the understanding of the evolution of the star . Discussions on thistopic can be found in refs . [2-4].

In this paper we present double-differential cross sections for the 54 Fe(n,p)"Mn and 56Fe(n, p)"Mn reactions at 97 MeV. Cross sections for the 54 Fe(n,p)"Mn and 54Fe(p, 054Co reactions at 300 MeV have been measured [5] atTRIUMF. Measurements [6] of the cross sections for the 54Fe(p, 054Co and55 Fe(p, n)"Co reactions at 160 MeV have been made at IUCF.

The GT((3+) strengths of 54 Fe and 56 Fe have also been measured in the56 Fe( 1ZC, 12 N)56Mn [7] and 54.s6Fe(t, 3 He)54.56Mn [8] reactions . Due to the compar-atively good energy resolution in these experiments it is possible to resolveindividual states at low excitation energies . Above a few MeV, however, thecontinuum contribution makes the extraction of GT strength very difficult. An-other problem is that the effective interactions in these reactions are not as wellknown as the effective nucleon-nucleon interaction in the (n, p) reaction . Theheavy-ion measurements could, however, be used to determine the excitationenergies and strengths of individual low-lying GT states .

The experimental apparatus and procedure used in the present experiment aredescribed in sect . 2, while the data reduction and the experimental results arepresented in sect. 3. Sect . 4 deals with the analysis of the experimental protonspectra in terms of multipole distributions, which were obtained by a decomposi-

COLD TRAP

PROTONDEFIECTING MAGNETS

T. Rönngcist et al. / 54"saFe(n, p)

Fig. 1 . The Uppsala (n, p) facility.

tion using calculated sample angular distributions . Finally, a summary and theconclusions are given in sect. 5.

2. Experimental apparatus and procedure

227

The present experiment was performed at the (n, p) facility of the The Svedberglaboratory in Uppsala, Sweden. The equipment has been described in detail in arecent publication [9] and only a brief summary will be given here.A general overview of the experimental equipment is shown in fig. 1 . Protons

from the cyclotron impinged on the neutron production target from the left in thefigure . The 97.0±0.5 MeV neutrons were produced by the 7Li(p, 07 Be reaction,using 150-200 mg/cm' thick discs of lithium, enriched to 99.98% in 7Li. After thetarget the proton beam was deflected into a well-shielded beam dump. A narrowneutron beam in the forward direction was defined by a system of three collima-tors. The vacuum system was terminated with a 40 p,m Kapton foil after the firstcollimator . Charged particles produced in this foil and along the collimatorchannel were deflected by a clearing magnet . The diameter of the neutron beamwas about 7 cm at the (n, p) target position, 8 m from the neutron productiontarget . With a 5 .0 )t,A proton beam incident on a 200 mg/cm2 thick 7 Li target, theneutron yield was about1.0X106 s- ' within the solid angle defined by the

228

T. Rïinngtisr er ai. / se,sspe(n, pi

collimators. The charged-particle contamination of the beam was about five ordersof magnitude lower.

To improve the (n, p)-reaction rate without impairing the energy resolution, asandwiched multi-target system was used. It consisted of a stack of thin (n, p)target foils interspaced by multi-wire proportional chambers (MWPC). In this wayit was possible to determine in which foil the reaction occurred, so that correctionsfor the energy loss in the subsequent targets could be applied. The efficiency perchamber plane was in general >, 99%. The first two MWPCs provided veto signalsfor rejection of charged particles contaminating the neutron beam .Two 135 mg/cm2 target foils of 'Fe, with 99% enrichment, were mounted in

the multi-target box. Two 113 mg/cm2 natural iron foils, with a chemical purity of99.0%, were used to extract the "Fe(n, p) data. The isotopic composition ofnatural iron is : 5.8% 5°Fe, 91.8% 56 Fe, 2.1% 57Fe and 0.3% 58Fe. In addition, a 50mg/cm2 CH Z reference foil was placed in the last target position. This foil wasused to normalize the 54,56 Fe(n, p) cross sections to np scattering . To determinethe instrumental background, spectra were recorded at all experimental settingswith only the CH Z foil mounted in the multi-target box.

The momentum determination of the charged particles emitted from the reac-tion target was performed with a spectrometer consisting of a dipole magnet andfour drift chambers, two in front of and two behind the magnet . The scatteringangle was determined by the trajectory through the first two drift chambers. Thedetection efficiency of a single drift-chamber plane was about 98%. To reducedthe multiple scattering of charged particles in air, the space between the first twodrift chambers and the volume in the pole gap were filled with helium gas.The data were taken with the spectrometer magnet at two different positions,

covering the scattering angular ranges 0-15° and 15-30°, respectively. In the firstposition magnetic fields of 0.8 and 1.0 T were used for the angular ranges 0-8° and8-15°, respectively, whereas a single magnetic field of 0.7 T was sufficient to coverthe angular range 15-30°. The energy resolution (FWHM) was about 2.8 MeV inthe angular range 0-15° and 3.7 MeV for the larger angles.A trigger signal for the data acquisition system was generated by a triple

coincidence between two large plastic scintillators, located behind the last driftchamber, and a thin scintillator, positioned immediately after the multi-target box.The neutron time-of-flight (t.o.0 was measured using pulses from the thin scintil-lator and the cyclotron RF as start and stop signals, respectively . This informationwas used to reject events from low-energy neutrons in the tail of the neutronspectrum . Together with information on the particle momentum, the pulse heightsfrom the two large scintillators were used for particle identification, enablingseparation of protons from other charged particles.

The data acquisition system was based on a vME-bus configuration in conjunc-tion with CAMAC and NIM electronic modules. For each event the time informa-tion from the drift chambers and the MWPCs of the multi-target box, together

with the linear signals from the scintillators and the neutron t.o.f., were stored onmagnetic tape .

3. Data reduction and experimental results

The data were analyzed off line on an event-by-event basis. The accepted "'Feand S6 Fe events were stored as relative double-differential cross sections inmatrices with angular and energy bin widths of 1° and 1 MeV, respectively. Thedata from the CH 2 foil were treated in a similar way. The contribution from thet2C(n, p) reaction [101, which interferes with the np scattering data at laboratoryangles larger than 15° was subtracted and a gaussian distribution was fitted to theproton peak in the energy spectrum for each angular bin. Differential crosssections were obtained by normalizing to up scattering data of a previous measure-ment [111 .The background spectra were treated in the same way as the iron data and

subtracted after proper flux normalization. The instrumental background is domi-nated by np scattering in the MWPC foils, as illustrated in Fig. 2. Due to anefficiency loss in one of the MWPCs, a small number of events from the naturaliron targets contaminated the "Fe(n, p) data. Corrections for this contributionwere made and are also included in the background from the target box, as shownin Fig. 2a .

To obtain the S6Fe(n, p)"Mn cross sections from the natural iron data, theisotopic contribution (5.ß%) from 54 Fe(n, p) reactions (right-hatched area in Fig.2b) was subtracted from the natural iron data . After this subtractions, the 56Fe(n,

T. Rlnnquist et a(. /sas*Fe(n, p)

229

E, (MeV)

Fig . 2 . (a) Uncorrected spectrum from the 54Fe(n, p)54 Mn reaction at E � =97 MeV in the angularinterval 0-2°, plotted versus the excitation energy in the residual nucleus (filled circles). The back-ground from the target box is shown by the histogram. The correction for the low-energy tail of theneutron beam is represented by the cross-hatched area . (b) As in (a) but for the 56Fe(n, P)56Mnreaction, obtained using a natural iron target . The right-hatched area represents the contribution from

the 54Fe(n, p)54Mn reaction .

230

T. Rönngt "ist et ai. /s'"s°Fe(n, p)

p) data can be considered as originating from an iron target, enriched to 96% in5'Fe . Due to the finite width of the micropulses from the cyclotron (3-4 ns) thet.o.f. rejection off low-energy neutrons was not complete . The remaining tail oflow-energy neutrons was therefore subtracted from the iron spectra using anunfolding technique . The tail below the peak in the proton spectrum was dividedinto bins where the contents were compared to that of the total hydrogenspectrum . Scaled by this ratio, a shifted iron spectrum was subtracted from the fullspectrum for each bin . As can be seen in Fig . 2, the effect of this correction wassmall (cross-hatched area).

Acceptance corrections were determined experimentally by making narrowsoftware cuts in the vertical scattering angle to allow only trajectories almostparallel to the median plane, i.e., unperturbed by the pole faces of the magnet. Bycomparing the energy spectra and angular distributions from this analysis with thetotal spectra, the required corrections factors could be extracted .

Experimentally determined double-differential cross sections in the center-of-mass (c.m.) system, grouped in 2° bins, are shown as filled circles in Fig . 3 . Theerror bars represent statistical uncertainties . In addition, the cross-section scalehas a systematic or normalization uncertainty of 5% which is dominated by the 4%uncertainty in the np scattering cross sections [11] and target thickness uncertain-ties . The data set, spanning the angular range 0-30°, is complete up to about 30MeV excitation energy (in 54,56 Mn), but for most angles the data extend up to 40MeV.

4. Data analyses and discussion

The experimental cross-section data are expected to be due to direct reactionsof different multipolarities as well as to multistep reactions. The analysis methodsused in this experiment are similar to the ones used in a 9°Zr(n, p)9°Y measure-ment which has recently been published (12], where a more detailed description ofthe methods is given .

4.1. Statistical multistep calculations

For the calculation of the double-differential cross sections for multistep reac-tions the compute code of Bonetti and îMesa [13], based on the statistical theoryof multistep direct reactions (SMDE) of Feshbach et al . [14], was used . Similarcalculations have been made previously for the 9°Zr(n, p)9° Y reaction [12]. In thesecalculations, the effective nucleon-nucleon (NN) interaction is described by aYukawa potential . The depth of the potential (Vo) used in the 9°Zr(n, p)9°Y workwas 17.3 MeV, which was obtained from the energy dependence reported byScobel et al . [15]. Using this interaction strength in the SMDE calculations for

LilaCa

Éb-0

3 0... .=2 -4°44"

2

14+#t

+

4"

"

1 +

0

1

0

3

2

1

0

3

2

1

0

0��=10-12°

0,-=18-20'

Fig. 3. Experimental data at selected angles together with the calculated contribution from multistepprocesses (dotted line). The calculated contribution from quasifree scattering, using values of the

energy cutoff parameter T of 12 MeV(solid line) and 70 MeV(dashed line) is also shown.

s4 Fe(n, p), the calculated full cross sections (single step + multistep) are almost afactor of two smaller than the experimental data at the largest angles of thisexperiment . Therefore, a Va value of 21 MeV was used, which gives betteragreement (Fig . 4) . As can be seen, the contribution from multistep processes issmall, in particular at lower excitation energies. The calculated multistep contribu-tions vary smoothly with mass number (A). The multistep calculation for the54 Fe(n, p) reaction was therefore used as an estimate for the "Fe(n, p) reaction .

4.2 Calculation of angular distributions

T. Rönnquist et al. / -"--"Fe(n, p)

231

"Fe(n,p) s6Fe(n,p)

_ 0,,=2-4*

0,.�=10-12°

E, (MeV)0 10 20 30 0 10 20 30

To decompose the experimental data into different multipolarities, sampleangular distributions are needed . Angular distributions for several J~-values were

232

T. Rdnngrist et at. / sa.seFe(n, p)

aEwcv

Èe 0ô 0 10 20 30

E, (MeV)Fig . 4 . Double-differential cross sections for the 54Fe(n, p)14Min reaction at E� =97 MeV in theangular range 28-311° (filled circles). The SMDE calculation of the cross sections (solid line) is shown;

the dashed line represents the contribution from multistep processes.

calculated using a simple model of lplh excitations in the framework of thedistorted-wave Born approximation (DWBA). As described for the "Zr(n, p)9°Yanalysis [12], normalized ph amplitudes are obtained from calculations based onnormal mode excitations [16], which are the response to a simple tensor multipoleoperator acting on the ground state.

The single-particle energies used to determine the wave functions were ob-tained from experimental data in studies of the 54,56Fe(d, 3He) [17], 54 Fe(d, p) [18]and 56 Fe(d, p) [19] reactions. Unbound single-particle states were assigned a smallbinding energy to simplify the calculations . The wave functions were generatedwith a Woods-Saxon potential (ro = 1.25 fm, a = 0.65 fm).

The reaction calculations were performed using the code DW81 [20] where thecontributions from the different particle-hole configurations are added coherently .The effective NN interaction was represented by the density-dependent G-matrixat 100 MeV as given by Nakayama and Love [21]. For comparison, calculationswere also made using the free t-matrix of Franey and Love [22] and the density-de-pendent G-matrix derived by von Geramb [23] . The in- and outgoing neutron andNroton waves were obtained with the optical model, using the potential derived bySchwandt et al . [24], with proper adjustments for the neutron channel .

Differential cross sections were calculated for L 5 4 in steps of 10 MeV up to40 MeV excitation energy in the final nuclei 54 Mn and "Mn. By interpolatingbetwee- the..e distributions, cross sections could be obtained at intermediateexcitation energies .

As an example, the angular distributions for spin-flip transitions with L = 0-4in the 54Fe(n, p)54 Mn reaction, calculated at an excitation energy of 10 MeV, areshown in Fig . 5 . The cross sections for the different J~-values of each multipolarityL were added and these distributions (solid lines in Fig . 5) were used as one set ofsample angular distributions in the multipole decomposition . Since calculationsshowed that the non-spin-flip contributions to the cross sections for each L-valueare more than a factor of five smaller than the spin-flip ones, and also very similar

10

Cv

-110

T. R6nnquist et at / sd.s6FO(; D)

233

0 10 20 300 10 20 30

0e,. (de9 .)Fig . 5 . Angular distributions from DWBA calculations, using the Nakayama-Love G-matrix [21], forL=0-4 of the 5"Fe(n, p) reaction at E� =97 MeV and at an excitation energy of 10 MeV. The

contributions of different J°'s to each multipolarity L are shown, as well as their sum .

in angular dependence, they were neglected in the multipole decompositions. ForL = 0 we used GT distributions up to 10 MeV excitation energy and spin isovectormonopole (SIVM) above that energy. This is justified because the GT strength,being of Oh(o character, is expected to be located at low excitation energy. Theshapes of the different angular distributions become more similar to each other athigher excitation energies, as illustrated in Fig. 6. This might cause ambiguities inthe multipole decomposition at excitation energies above 20 MeV.

E.=0.0 MeV E=--20.0 MeV

SUM (solid)l' (dotted

_ 2' domed)3' dot-dashed)

0 10 20 300 10 20 300 10 20 30

G,.m . (de9.)

Fig . 6. Angular distributions from DWBA calculations, using the Nakayama-Love G-matrix [21], forL=0-3 of the: `^Fe(n, p) reaction at E � =97 MeV and at excitation energies of 0, 20 and 40 MeV.

Each multipolarity L is summed over the contributing J"s .

G

ÉL=O. GT (solid) - L=O. SIVM (solid)

b " L=1 (dotted)L=2 dashed) "

L=1 (dotted)L=2 (dashed)

101 - L=3 (dot-dashed)1 1

= L-3 (dal-dashed)

The L-grouping of the J'-values described above is one out of several possibili-ties. A simplified grouping, choosing the J=L angular distribution components,was also used in the multipole decomposition . The parameter space was reducedto four by combining the L = 3 and L = 4 distributions . This set of sample angulardistributions is closer to the one used in the 54 Fe(n, p) work performed atTRIUMF [5]. These authors used a simple configuration where, e.g., the angulardistributions of a single 1 - transition represented the L = 1 (dipole) strength . Theway the dipole strength is represented influences the multipole decompositionsince the angular shapes for the different JR components of L = 1 are different(see Fig. 5).

Angular distributions were also calculated for the "Fe(n, p)"Mn reaction .These distributions were found to be very similar to the ones for the 54 Fe(n,p)"Mn reaction.

4.3. Multipole decomposition of experiniental data

Multipole decompositions of the entire 54Fe(n, p),94Mn and 56Fe(n, p)56Mnspectra were made after subtraction off the small multistep contribution . For each 1MeV energy bin the experimental angular distributions were described by

M) cxp _

dor da~A't 12

T. Riintngrisi et al. / s4.seFe(n, p)

where the is represent the different multipolarities and (do,/d1D; denotes thecalculated sample angular distributions (subsect . 4.2), normalized to unity at thepeak values . The coefficients A ; are determined by a least-squares fit to the data .These coefficients were set to zero whenever they became negative in the fitting .The results of the multipole decomposition of the 54 Fe(n, p) data, using angulardistributions from the L-grouping is shown in Fig. 7, where the strength for eachmultipolarity L = 0-4 is displayed separately . As can be seen the spectrum isdominated by L = 0 and L = 1 strength. The L = 0 strength is mainly located atexcitation energies below 9 meV (GT strength), but a tail extends to higherexcitation energies . A broad distribution of dipole strength is peaked at about 12MeV.The results from the decomposition using the simplified J=L grouping are

shown in Fig. 8, where different effective NN interactions [22], [21], [23] have beenused . It is obvious that the choice of interaction has a very small effect on theresult . The error bars, which represent the uncertainties due to the statistics andthe multipole decomposition are displayed for the Franey-Love interaction. Dueto the similarities in the shapes of the different angular distributions for higherexcitation energies (see Fig. 6), the uncertainties in the multipole decompositionbecome large at excitation energies above 20 MeV. The L=0 strength above 9

>N

N

i=

1=

C41

In

2

0

2

00

2

0

2

0

2

0

L=3

L, AmLlr"

JI'~1414F,a}l

0

N

20 0 20E, (MeV)

Fig . 7. Strength distributions for L = 0-4 of the s4Fe(n, p) reaction at E� = 97 MeV as obtained fromthe multipole decomposition, in which sample angular distributions were obtained from DWBAcalculations, using the Nakayama-Love G-matrix [21) . The strengths are shown at the angles where theangular distributions of the different multipolarities have their maxima. The contribution from multi-

step processes was subtracted from the experimental data .

Froney-Love

Nakayama-Love

von Geramb

T. Rönnquist et ai. / 5a,56Fe(n, p)

E, (MeV)

3-+4'

235

Fig . S . Strength distributions for the ' 4Fe(n, p) reaction at E � = 97 MeV as obtained from the multipoledecomposition, using the free Froney-Love t-matrix [22], the Nakayama-Love G-matrix [21] and thevon Geramb G-matrix [23]. Angular distributions from the J = L grouping (subsect . 4.2) were used . Theuncertainties are shown for the decomposition where the Froney-Love t-matrix was used . Thestrengths are shown at the respective angles where the angular distributions of the multipolarities havetheir maxima . The contribution from multistep processes was subtracted from the experimental data .

2

0

T. Riinngrist et aL / s4 "saFe(n, p)

4.4. Quasifree scattering contribution

0 10 20 30

0 10 20 30

E, (MeV)Fig. 9. Strength distributions for the 56Fe(n, p) reaction at E� =97 MeV as obtained from the multipoledecomposition, using the Nakayama-Love G-matrix [21] . Angular distributions from the J= L group-ing (subsect . 4.2) were used . The strengths are shown at the respective angles where the angulardistributions of the multipolarities have their maxima. The contribution from multistep processes was

subtracted from the experimental data .

MeV is decreased compared to the decomposition using the L-grouping. ThisL = 0 strength could be the result of misidentified dipole strength . No definiteconclusions can therefore be made concerning the L = 0 strength above 9 MeV.

The result from the multipole decomposition of the 5 Fe(n, p) data, usingangular distributions from the simplified J = L grouping, is shown in Fig. 9. Thestrength distributions are similar to the ones obtained for 5°Fe(n, p), although theGT and dipole resonances are less pronounced . The L = 0 strength above 7 MeVis sensitive to the choice of sample angular distributions . No definite conclusionscan therefore be drawn about the L = 0 strength at higher excitation energies .

The giant-resonance strengths are most likely overestimated in the multipoledecomposition discus5md i ;1 the previous section, because reaction mechanisms,such as quasifree scattering (QFS), should also be considered, especially at higherexcitation energies .

The QFS was estimated following the technique used by Erell et al . [25] in astudy of pion charge-exchange reactions, where a simple phenomenological expres-sion was fitted to the continuum. The double-differential cross section is given by

d2o, =NI- exp[ -(E-Eo)/T]Z ,

E>Eo, (3)dE dfl

1 + [(E-EOF)IWL]

where N is a momentum-transfer-dependent normalization factor, Eo is theseparation energy of the least bound neutron in 5° Mn ("Mn), T is an energycutoff parameter and EOF and WL are the energy and width of the QFS peak,

T. Rönngi ist et at. / 54.5"Fe(n, p)

237

respectively. EQF is given by the kinematics of the reaction 'H(n, p)n*, wherem�. = m� + B. Within this model, EQF is the difference in kinetic energy betweenthe incoming neutron and the outgoing proton and B is a parameter representingthe average binding energy of struck protons. The peak width WL is expressed as

WL = WL0LI +a(glkF) 21,

(4)

where q is the momentum transfer at EQF, kF is the Fermi momentum and WLand a are adjustable parameters.

This procedure has previously been used in the analysis of "Zr(n, p) data[26,12], where the following parameter values were used : a/k F = 0.36 fm', B = 11.9MeV, WL, = 22.7 MeV and T= 12 MeV. Since the present data did not extendhigh enough in excitation energy to allow a good fit of the parameters, these valueswere used in the present work. Different values of a/k2 were tested but nosignificant difference in the shape of the QFS contribution was seen . No A-depen-dence of WL , was found in the work by Erell et al . [25]. A value of the energycutoff parameter T of 70 MeV was used in that work. Calculations of the QFScontribution to the present data were made also with Lhat value. After subtractionof the SMDE contribution, the QFS calculations were fitted to the data above 28MeV. The results are shown in Fig. 3 (sect. 3) .

Multipole decompositions were also made after subtraction of the calculatedQFS contribution . In the excitation energy region where the contribution from theQFS is fitted to t".e ôta, n:, giant-resonance strengths are assumed. The contribu-tion from QFS is therefore certainly overestimated.

4.5. Gamow-Teller strength

From the

L=0 strength deduced in the multipole decomposition, theGamow-Teller strength B(GT) can be obtained . A factorized expression for theGT cross section at low momentum transfer, has been derived [27],

Q(q, w) = (~GT(E, A)F(q, (o )B(GT),

where the factor F(q, w), which gives the dependence of the cross section onmomentum transfer q and energy loss w, goes to unity in the limit of q =w = 0. Incases where /3-decay between the initial and final nuclear states is energeticallypossible, the corresponding GT strength can be determined [27]. Subsequently, theunit cross section &GT(E, A) can be determined by extrapolating the GT crosssection to q = w = 0.A theoretical value for the unit cross section can be calculated using a DWBA

code such as DW81 [20] . In a simple shell model, the GT strength originates fromar1f7/2 ~ v1f5/2 transitions for both the "Fe(n, p) and "Fe(n, p) reactions. For a

238

T. Rövnagi"ist et at

sa..snFr(a, p)

f7,, r shell with six protons and an empty f5/2 neutron shell, this transitioncorresponds to 10.29 sum-rule units. Microscopic DWBA calculations of thistransition, under the same assumption, gave q = 0 cross sections of 37.0 mb/sr and36.1 mb/sr and corresponding unit cross sections of 3.6 mb/sr and 3.5 mb/sr for54Fe and S'Fe, respectively . The DWBA calculations were made with the opticalpotentials of Schwandt et al . [24] and the effective NN interaction of Nakayamaand Love [21] . Taddeucci et al . [27] have pointed out that ambiguities in the opticalpotential could give uncertainties of 25-30% in the inelastic distorted-wave crosssections, which can be taken as the uncertainty in the calculations.

The 64Ni(n, p)64Co reaction, in the energy range 90-240 MeV, has recentlybeen studied [28] at Los Alamos . The first experimental values o1 the Gamow-Teller unit cross section from the (n, p) reaction for an 112p-shell nucleus wereobtained from that measurement which gave a unit cross section of 3.6 ± 0.4mb/sr at 105.5 MeV. By estimating the A- and E-dependence of the unit crosssections from DWBA calculations, the experimental "Ni value could be used as areference. In this way, unit cross sections for the 54 Fe(n, p) and "Fe(n, p)reactions of 3.6 ± 0.4 mb/sr and 3.5 ± 0.4 mb/sr, respectively, were obtained .These values, which are equal to the ones obtained with the DWBA calculation,were used in the analysis of the present experiment .

The GT strength was extracted from the low-lying L = 0 strength in themultipole decompositions, using the effective NN interaction of Nakayama-Loveand the ! = L grouping of the angular distributions (subsect . 4.2). To extrapolateto q =W = 0, the L = 0 cross sections at 0° were divided by F(q, (0) . This factorwas calculated in steps of 1 MeV, using the expression

F(q, w) =

ffDwal(q , W)

(YDws1(q = 0, W=0),

where ODwai is a cross section calculated with the code DW81 .The GT strength obtained for the "Fe(n, p) and "Fe(n, p) reactions is shown in

Tables 1 and 2, respectively. The ratios of the L = 0 cross sections to the full crosssections are also shown in the tables . For reasons discussed in subsect. 4.3, L = 0strength above 9 MeV (for 54 Fe) or 7 MeV ("Fe) has not been considered . Theintegrated GT strength (S,,) for 54 Fe(n, p) and "Fe(n, p) is 3.5 ± 0.3 ± 0.4 and2.3 ± 0.2 ± 0.4, respectively . The first uncertainties are due to the statistics and themultipole decomposition, and the second ones to the unit cross sections.

In the 54 Fe(n, p) measurement at 300 MeV, Vetterli et al . [5] obtained an Sß .value of 3.1 ± 0.6, using a unit cross section of 5.1 ± 0.8 mb/sr. Considering theuncertainties in the unit cross sections, this is in agreement with the present result .

Shell-model calculations for the GT(,ß+) strength in "Fe and 56Fe have beenperformed by Bloom and Fuller [3]. For 54 Fe they obtained Sß.= 10.29 in thesimplest (Op2h -> lp3h) model. The strength was reduced to 9.12 when lplh

T. Rörtttgt-ist et al. / sa.-'ÔFe(n, p)

239

Table 1Zero-degree cross sections and Gamow-Teller strength (column 5), given in 1 MeV bins, for thes°Fe(n, p)s°Mn reaction at 97 MeV. Column 2 gives the L =0 cross sections and column 3 the ratiobetween the L = 0 and the total cross sections. The L = 0 cross sections are extrapolated to q =w =0in column 4

excitations were included in both the parent and daughter nuclei . A correspondingvalue of 10.00 was obtained for 56Fe, using a (2p2h ~ 3p3h) model. For bothnuclei, the strength is distributed in the excitation energy region 0-10 MeV, ingood agreement with the present results. However, the strengths are greatlyoverestimated in the calculations . Furthermore, these calculations fail to predictthe smaller amount of GT strength seen in the "Fe(n, p) reaction compared to the" Fe(n, p) reaction . Muto [29] extended the model space to include 2p2h excita-tions in the parent nuclei and obtained Sp.= 7.4 for 54 Fe. Thus, it is obvious that alarger configuration space reduces the theoretical predictions of the GT strength .

Table 2Zero-degree cross sections and Gamow-Teller strength (column 5), given in 1 MeV bins, for thes6Fe(n, p)-"Mn reaction at 97 MeV. Column 2 gives the L=0 cross sections and column 3 the ratiobetween the L = 0 and the total cross sections . The L = 0 cross sections are extrapolated to q=w =0in column 4

E�(MeV)

OL =0(mb/sr)

UL =a/°iut oq-.=o(mb/sr)

B(GT*)

-24-1) 0.18 t 0.03 0.86 0.18f 0.03 0.05 ±0.01-1-0 0.42 t 0.05 0.96 0.42±0.05 0.12 ±0.010-1 0.89+0.06 0.95 0.90±0.06 0.25 ±0.021-2 1 .56+0.07 0.98 1.63 ±0.07 0.45 ±0.022-3 1 .92 ± 0.06 0.99 2.06t0.06 0.57+0.023-4 1 .75 f 0.09 0.95 1.94 t0.10 0.54 ± 0.034-5 1 .59+0.10 0.92 1.82 ± 0.12 0.51+0.035-6 1 .15 F0.10 0.77 1.37+0.12 0.38 ± 0.036-7 1 .18 ± 0.09 0.78 1,47+0.12 0.41 ± 0.037-8 0.52±0.10 0.42 0.68±0.12 0.19±0.038-9 0.12±0.10 0.10 0.17±0.15 0.05±0.04

E�(MeV)

OL =0(mb/sr)

QL=o/fftot oq -. -o(mb/sr)

B(GT')

-2-(-1) 0.16±0.05 0.78 0.16±0.03 0.05±0.01-1-0 0.48±0.05 0.91 0.52±0.05 0.15±0.010-1 0.91 ± 0.04 0.94 1 .01+0.06 0.29+0.011-2 1.22+0.040.04 0.94 1 .39+0.070.07 0.40+0.010.012-3 1.29+0.040.04 0.93 1 .52 +_ 0.06 0.44+0.020.023-4 0.99+0.090.09 0.77 1 .21+0.100.l0 0.35+0.030.034-5 0.84 +_ 0.10 0.68 1 .07+0.12 0.31+0.045-6 0.60+0.09 0.50 0.80+0.12 0.23+0.046-7 0.27+0.10 0.22 0.38+0.12 0.11+0.04

240

T. Riinngi ist et nL / s+.snFeln, p)

Auerbach et al . [30] have shown that RPA correlations can reduce the Sß . valueof 'Ni by a factor of 2. The quenching of the GT(ß`) strength in 'Mg using a fullld2s shell-model space relative to a 2p2h model space has been calculated byAuerbach et al. [31] . By exploring the connection between the quenching and thequadrupole collectivity, these authors obtained an estimate of the GT(/3+) strengthin "Fe which would be obtained in a full lf2p shell-model calculation . Thesecalculations, together with the global quenching factor of 0.6, gave a Sß . value of3.8 for "Fe which is in good agreement with the present experimental result .Furthermore, the calculations give a larger quenching with a larger quadrupolemoment. Experimental B(E2) values connecting the ground state and the first 2*state are 10.6 W.u. [32] and 16.8 W.u . [33] for 54Fe and 5°Fe, respectively . Thecalculations are supported by the present measurement since the product of theS,, . and B(E2) values are nearly equal for the two isotopes.

Comparisons with the Ikeda sum rule were made with the present resultstogether with the previously measured [6] values of Sß - of 7.8 ± 1 .9 and 9.9 ± 2.4for the 54pe(p, 054Co and 5''Fe(p, n)5'Co reactions, respectively. For 5° Fe we get:Sß -- Sß .= (7.8 ± 1.9) - (3.5 ± 0.3 ± 0.4) = 4.3 ± 2.0, which is 72% of the Ikedasum-rule strength of 6. Similarly, we get for 5'Fe: Sß -- Sß .= (9.9 ± 2.4) - (2.3 ±0.2 ± 0.4) = 7.6 ± 2.5, which is 63% of the sum-rule strength of 12. The Sy . andSß - values are determined in excitation energy regions which are different in the(n, p) and (p, n) measurements . Goodman et al . [34] have pointed out that undersuch circumstances, comparisons with the Ikeda sum rule have no simple inter-pretable meaning.

4.6. Dipole strength

The total strength for each multipole operator can be obtained as the incoher-ent sum of the strengths for the individual particle-hole excitations, calculated onthe basis of normal mode excitations . A more detailed description of this proce-dure can be found in ref. [l2] . Since the spectra are dominated by L = 1 strengthand the uncertainties in the multipole decomposition for the higher multipolaritiesare quite large, only the dipole strength will be discussed here . The total spin-di-pole strength from the normal mode calculations is 195 fm' and 181 fMZ for the54Fe(n, p) and 56Fe(n, p) reactions, respectively. The strengths for all three J"components were added. These values were obtained using harmonic-oscillatorwave functions with the oscillator constant b = 1 .95 fm .

The cross sections of the sample angular distributions calculated in DWBA alsocorrespond to the total strength . The fraction of the strength exhausted in theexperiment was estimatcd by summing the coefficients A ; of the 1 MeV bins fromthe multipole decomposition up to an excitation energy of 30 MeV. This limit waschosen because of ambiguities in the multipole decomposition at higher excitationenergies . The L-grouping (subsect . 4.2) of the sample angular distributions and the

_E

CN

(n

T. Rönngt-ist et al. / "."Fe(rt, p)

241

b)

n

E, (MeV)

Fig. 10. Strength distributions at an angle of 8° for L = I of the 54Fein, p) reaction as obtained from themultipole decomposition, in which angular distributions from the L-grouping (subsect . 4.2) and theeffective NN interaction of Nakayama and Love [21] were used . The strength was determined (a)without subtraction of the QFS contribution and with subtraction of this contribution using an energycutoff parameter T of (b) 70 MeV and (c) 12 MeV. The contribution from multistep processes was

subtracted from the experimental data .

NN interaction by Nakayama and Love [21] were used in the decompositiondiscussed here .

The effect of subtracting the calculated QFS contribution (subsect . 4.4) on thedipole strength was also investigated. As can be seen in Fig. 10, this subtractionconcentrates the strength to one broad peak. More strength is removed if a smallervalue of the energy cutoff parameter T is chosen. The corresponding decomposi-tion was made for "Fe(n, p). The results are summarized in Table 3, where thefractions of the spin-dipole strength are shown. The energy-integrated crosssections at an angle of about 8° are also given for the multipole decompositionswithout subtraction of the QFS part . A smaller part of the strength is exhausted inthe "Fe(n, p) reaction than in the 54 Fe(n, p) reaction. The uncertainties gives forthe strengths and the cross sections are due to statistics and to the decompositionprocedure. The given fractions of strength must be regarded as upper limits, sincethe contributions from non-spin-flip transitions were neglected. These contributeless than 20% of the spin-flip cross sections if both strengths are fully exhausted.Less than half the predicted dipole strength is found in the broad peak at about 12

Table 3Experimental energy-integrated L = I cross sections up to 30 MeV at an angle of 8° (column 2) and theratio between this strength and the spin-dipole strength calculated for normal modes. The experimentalstrength was deduced with and without subtraction of the calculated contribution from QFS. Twovalues of the energy cutoff parameter T were used in the QFS calculations. The contribution frommultistep processes was subtracted from the experimental data

Reaction (do/dfl) (do/dfl)/(mb/sr) (do/dfl)NM

No QFS No QFS QFS, T= 70 MeV QFS, T= 12 MeV

54 Fe(n, p) 36.9+1 .3 0.84+0.04 0.40 ± 0.01 0.29+0.01"Fe(n,p) 27.1±1 .2 0.67±0.04 0.30±O.Oi 0.22±0.01

242

T. Riittegrist et ai. J s+.5'Fetn, pl

MeV. If the total non-spin-flip strength is contained in this peak, it is possible thata substantial part off the peak is due to non-spin-flip transitions .

4.7. Comparison of data with RPA calculations

RPA calculations, similar to the ones made in the "Zr(n, p) work [12], were alsomade for the "'Fe(n, p) reactions. The single-particle states were determined byWoods-Saxon calculations where the parameters of the potentials were chosen toreproduce the single-particle spectra near the Fermi level . The single-particleWoods-Saxon solutions were expanded in terms of harmonic-oscillator wavefunctions. In this way a discretized continuum was obtained . For the RPAcalculation all single-particle states up to 50 MeV in the continuum were takeninto account . Near the Fermi level single-particle energies were taken fromexperimental data . For the residual interaction î'rC� we used a Landau-Migdalinteraction

P..(r , r') = CO,S(r - r)[fii+gtia'Q']T'T I

with the parameters

fti = 1 .5,

gt; = 1 .0

and

Ca = 320 MeV - fm' .

(8)

The parameter values were chosen according to a systematic (p, p') work [35] .The t-matrix of Frane; and Love [22] was chosen as the effective interaction .

Distorted-wave calculations were performed with a code based on DWUCK4 [36],where the incoming and outgoing distorted waves were calculated using theoptical-model potential of Schwandt et al . [24] . In the calculations, particle-holeexcitations with 1 < 7 were taken into account .A simple method was used to account for continuum effects and 2p2h excita-

tions . The discrete spectra were folded with a Breit`vvigner distribution with anenergy-dependent width as described by Smith and Wambach [37]. The experimen-tal energy resolution was included in the folding procedure .

Examples of the RPA calculations for the 54 Fe(n, p) reaction are shown in Fig .11 . A simple shell structure with a filled If7/2 neutron shell and a If7/. protonshell with two holes was used to describe the target nucleus in the RPA calcula-tions . At low excitation energies, these calculations give a few states which aredominated by a single ph transition. For instance, the whole GT strength is foundin one state originating from a alf7/2 -> vlf.12 transition. The shell-model treat-ment in the present RPA calculations is obviously too simple to describe thespectra at low excitation energies . Moreover, the calculations underestimate thestrength above 10 MeV with about a factor of two. The results of the RPAcalculations for the 5'Fe(n, p) reaction are almost identical to the ones for the54 Fe(n, p) reaction .

243

Fig. 11 . Results from total RPA calculations (solid line) show together with experimental data from the54Fe(n, p) reaction . The calculated multistep contribution (dashed line) has been added to the RPA

calculations to get the solid line.

The RPA cross sections for J' = 0- , 1 - and 2 - are shown in Fig. 12 togetherwith the L = 1 strength, which was extracted in the multipole decompositionsusing the L-grouping (subsect . 4.2). One peak at 11 MeV, being mainly of 1 -character, can be seen in the calculated spectra. The shapes of the calculateddipole strength distributions are in agreement with the experimental data althoughthe calculations underestimate the cross sections with approximately a factor oftwo. With the simple shell structure on which the RPA calculations are based, theonly difference between SaFe and "Fe is two neutrons in the 2paiz shell. Transi-tions to this shell are not of particular importance for the calculated dipole peaks.Therefore, the dipole calculations for the 54 Fe(n, p) and "Fe(n, p) reactions givevery similar results and do not reproduce the experimentally observed difference inthe dipole strength for these reactions .

The RPA calculations together with multistep calculations for the "'Zr(n,p)reaction at 100 MeV [121 show good agreement with the total energy spectra . Itshould, however, be noted that the contributions from multistep processes arelarger for the 9°Zr(n, p) reaction than for the 54 ."Fe(n, p) reactions . Thus, atexcitation energies above 25 MeV, half of the total strength or more werepredicted to originate from multistep processes.

N\d_EWUC

bâ E° (MeV)

Fig. 12 . Results from RPA calculations (solid lines) and experimental strength distributions for L = 1 at

an angle of S° for the (a) "Fe(n, p) and (b) 56 Fe(n, p) reactions. Angular distributions from the

L-grouping (subsect . 4.2) and the effective NN interaction of Nakayama and Love [21] were used in the

multipole decomposition of the experimental data .

T. R6nnquist et aL / s' "S6Fe(n, p)

3 o°-= to-t2° O,m=26-28°

2

OC

t 0 20 0 20 20t>ô E° (MeV)

244

â. süir~®®®a .-y U®®û cUMlü3io®®s

T. Riirvtgrist et at / "_s"Fete6 P)

Double-differential cross sections of the "Fc(n, p)s°Mn and 56Fe(n, p)"Mn

reactions at E�=97 MeV have been measured in the excitation energy andangular regions 0-40 MeV and 0-30°, respectively, using a magnetic spectrometerwith position-sensitive detectors for the determination of the proton energy andemission angle. Contributions from multistep processes, predicted to be small,were subtracted from the en=rgy spectra . From multipole decompositions of thespectra, strength distributions for different multipolarities were obtained .GT strength distributions were obtained from the L = 0 strength up to excita-

tion energies of 9 and 7 MeV for the "Fe(n, p) and "Fe(n, p) reactions,respectively . Possible GT strength at higher excitation energies could not beextracted due to uncertainties in the multipole decomposition . The deduced Sß .values for "Fe and "Fe are 3.5 ± 0.3 ± 0.4 and 2.3 ± 0.2 ± 0.4, respectively, wherethe first uncertainties are due to statistics and to the multipole decomposition andthe second ones are related to the unit cross sections . The result of 54 Fe(n, p) is inagreement with a previous measurement at 300 MeV [5]. Shell-model calculationsreproduce the shapes of the GT strength distributions quite well although thestrengths are strongly overestimated . Available calculations show that a largerconfiguration space and RPA correlations reduce the theoretical predictions of thestrength . Auerbach et al . [31] estimated the B(GT+) strength in 54Fe which wouldbe obtained in a full If2p shell-model calculation . With the global quenching factorof 0.6, they obtained a Sß . value in agreement with the present experimentalresult . The fact that a smaller amount of GT strength is observed in the 16Fe(n, p)reaction than in the 54 Fe(n, p) reaction is not in accordance with the shell-modelcalculations by Bloom and Fuller [3], which predict an almost equal amount ofstrength . The calculations by Auerbach et al . [31], however, give a larger reductionof the strength with a larger quadrupole moment. This is supported by the presentresults since the product of the S. . and B(E2) values are nearly equal for 56Feand 5°Fe.

The measured distribution o¬ GT(,ß') strength can be used to estimate elec-tron-capture rates which are important in order to understand the dynamics ofsupernovae . Aufderheide [2] has estimated the uncertainties in stellar electroncapture . From the "Fe(n, p) measurement by Vetterli et al . [5], Aufderheideestimated experimental electron-capture uncertainties of 20-30%, which originatefrom the uncertainty in the SO . value and the uncertainties in the strengthdistribution, caused by the limited energy resolution . The largest source of uncer-tainty in the electron-capture calculations is, however, the theoretical uncertaintiesin the treatment of the strength functions of the excited parent states . The presents'Fe(n, p) results give new experimental information for the electron-capture ratecalculations and the "Fe(n, p) results support those from the previous measure-ment [5] .

T. Rdnnquisr ef at / s' " s 6Fe(n, p)

245

The spectra measured in this work are dominated by L = 1 strength, with broadpeaks around 12 MeV excitation energy, and in addition, tails reaching up to atleast 30 MeV. These tails were removed when the contribution from quasifreescattering (QFS) was subtracted from the spectra. The phenomenologic QFScalculations are uncertain and it is therefore difficult to decide if the strength athigher excitation energies originates from direct multipole reactions or from QFS.A smaller part of the L = 1 strength is exhausted in the 56Fe(n, p) reaction than inthe "Fe(n, p) reaction .

The spectra were compared with RPA calculations . Neutron and proton shellswith sharp Fermi surfaces were used to describe the target nucleus in the RPAcalculations . At low excitation energies, the calculations therefore give few statesand do not describe the data well . For excitation energies above 10 MeV, thecalculations underestimate the cross sections with about a factor of two. Thepositions of the dipole peaks, however, are reproduced by the calculations .

The authors wish to thank the The Svedberg laboratory staff for significantcontributions in the construction of the equipmAnt and for assistance during themeasurements. Thanks are also due to Prof. J. Wambach for help with the RPAcalculations and due to Prof. J. Rapaport for valuable discussions . This work wasfinancially supported by the Swedish Natural Science Research Council. Supportwas also given by the Alexander von Humboldt Foundation and the U.S . Depart-ment of Energy .

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