Giant resonance spectroscopy of 40Ca with the (e,e′x) reaction (III): Direct versus statistical decay

Nuclear Physics A 696 (2001) 317–336 www.elsevier.com/locate/npe

Giant resonance spectroscopy of 40Ca with the (e, e
x) reaction (III): Direct versus statistical decay ✩ J. Carter a , H. Diesener b,1 , U. Helm b,2 , G. Herbert b,3 , P. von Neumann-Cosel b,∗ , A. Richter b , G. Schrieder b , S. Strauch b,4 a School of Physics, University of the Witwatersrand, PO Wits, Johannesburg, 2050 South Africa b Institut für Kernphysik, Technische Universität Darmstadt, D-64289 Darmstadt, Germany

Received 21 August 2000; revised 7 May 2001; accepted 14 June 2001

Abstract The present article is the third out of three on a study of the 40 Ca(e, e
x) reaction discussing the role of direct and statistical contributions to the decay of the observed giant resonance strengths. The proton and α decay modes leading to low-lying final states in 36 Ar and 39 K were investigated. The branching ratios for the p0 , p123 , α0 and α1 channels are compared to statistical model calculations. In the excitation region of dominant isoscalar E2 strength (Ex = 12–18 MeV) good agreement is observed. Model predictions of direct E2 decay for the (α0 + α1 )/(p0 + p1 ) ratio describe the data poorly. In the isovector E1 excitation region large excess strength is found in the population of lowlying states in 39 K. A fluctuation analysis shows the direct contributions to the p0 , p1 channels to be
85%. The presence of preequilibrium components is indicated by the significant nonstatistical decay to the p3 level which has a dominant ‘phonon ⊗ hole’ structure. Cross correlations reveal no significant branching between the different channels. The correlations between different electron scattering angles in the p0 , p1 and p3 decay result in an interaction radius compatible with the whole nucleus acting as an emitting source.  2001 Elsevier Science B.V. All rights reserved. Keywords: N UCLEAR REACTION 40 Ca(e, e
p), (e, e
α), E0 = 78, 183.5 MeV; analysed proton and α branching ratios; deduced direct contributions, emitter size, related reaction mechanism features. Statistical model calculations; fluctuation analysis.

✩

Work supported by the DFG under contract FOR 272/2-1 and through SFB 201.

* Corresponding author.

E-mail address: [email protected] (P. von Neumann-Cosel). 1 Present address: EDS, D-63037 Offenbach, Germany. 2 Deceased. 3 Present address: Sun Microsystems, D-63235 Langen, Germany. 4 Present address: Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854, USA.

0375-9474/01/$ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 4 7 4 ( 0 1 ) 0 1 2 0 8 - 8

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1. Introduction One of the basic problems which has to be solved for a full understanding of giant resonances is the interpretation of the width and the damping in microscopic theories. As a first approximation the width Γ can be written as [1] Γ = Γ + Γ ↑ + Γ ↓.

(1)

Here, Γ stands for the Landau damping due to the fragmentation of the initial oneparticle–one-hole (1p–1h) doorway states. The 1p–1h states couple to the continuum and acquire an escape width Γ ↑ which gives rise to a direct decay contribution (some authors prefer the term ‘semidirect’ [2]) into dominant hole states of the daughter nucleus. Lastly, Γ ↓ describes a spreading width resulting from mixing into more complex 2p–2h and further into np–nh configurations until an equilibrated compound nucleus is reached. The total spreading width can be split further into two parts: Γ ↓ = Γ ↓↑ + Γ ↓↓, where Γ ↓↑ represents a preequilibrium partial width and Γ ↓↓ stands for the fully-statistical decay component. A primary goal of all giant resonance decay experiments is to determine the relative contributions of the partial widths Γ ↑ and Γ ↓ to the total width Γ . The experimental signature of direct nucleon decay is the enhanced population of hole states in the daughter nucleus A − 1. Statistical decay can be identified by comparison of the measured spectrum with Hauser–Feshbach model calculations. If deviations are observed for the cross sections or branching ratios of the different decay channels, these contributions are assigned to nonstatistical processes and, in the case of population of hole states, to direct decay. A distinction between the preequilibrium component Γ ↓↑ and the fully statistical part Γ ↓↓ is very difficult in most cases. An exception is the case of damping through surface vibration excitations which have been shown to contribute significantly to the spreading width of giant resonances [3] as well as to the fragmentation of single-particle strength [4]. Experimentally, these contributions can be inferred from decay to final states with a dominant ‘phonon ⊗ hole’ structure. Most calculations and experiments agree that Γ ↓ dominates in heavy nuclei (A
60) for isoscalar E2 (giant–quadrupole resonance, GQR) and isoscalar E0 (giant–monopole resonance, GMR) as well as for isovector E1 (giant–dipole resonance, GDR) excitation [2,5]. However, Dolbilkin et al. [6] observed a strong population of low-lying hole states in the 58 Ni, 64,66Zn(e, e
p) reactions which could not be described by statistical model calculations. In light nuclei (A < 40) predominant direct decay has been observed [2,7,8]. In 40 Ca, the results of an (α, α
x; x = p, α) study of Zwarts et al. [9] are compatible with a statistical decay of the GQR while photoproton studies [7,10] indicate the importance of direct decay in the GDR excitation region. In the following, two methods used to extract the different decay contributions are discussed for the 40 Ca(e, e
x; x = p, α) data described in part I. Section 2 presents a statistical model calculation of the decay into 36 Ar and 39 K. In case of the data taken at MAMI [11] at larger momentum transfers q, branching ratios of the decay into resolved final states are discussed, while particle emission spectra are investigated from

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the measurement at low q at the S-DALINAC [12]. In Section 3 the different reaction contributions are extracted from a fluctuation analysis of the Ericson type [13]. The resulting direct and compound reaction cross section parts are identified with the direct and statistical decay discussed above. A short conclusion is given in Section 4. The results presented here have been discussed partially in [14–16]. Also, further details can be found in [17,18].

2. Statistical model description of decay branching ratios and particle emission spectra 2.1. Experimental data The measured coincidence spectra for the reaction 40 Ca(e, e
x; x = p, α) have been discussed in detail in part I. In each case, proton decay to the ground state of 39 K, J π = 3/2+ (p0 ), and to the group of the first three excited states (p123), Ex = 2.523 MeV, J π = 1/2+ , Ex = 2.814 MeV, J π = 7/2− , Ex = 3.019 MeV, J π = 3/2− , could be resolved from decays to higher-lying states. Also, α-decays to the ground state, J π = 0+ (α0 ) and in most cases to the first excited state, Ex = 1.97 MeV, J π = 2+ (α1 ) of 36 Ar were separated. The resulting angle-integrated energy spectra are shown in Figs.13–15 of part I. Branching ratios Γx /Γ for the various decay channels were obtained by dividing binby-bin the angle-integrated energy spectra for the individual channels by the corresponding spectrum for all decay channels (pall +αall ) at each momentum transfer. Since the following analysis seeks to explain the gross structures seen in the branching ratios, the original 25 keV bins have been enlarged to 250 keV. By way of example and to afford a comparison to statistical model calculations, Fig. 1 shows the resulting branching ratios for E0 = 183.5 MeV, Θe = 22.0◦ . These are plotted as a function of excitation energy in 40 Ca. The particle decay of 40 Ca starts with the p0 channel which opens at Ex 9 MeV and rapidly becomes weaker after the α0 channel opens at Ex 11 MeV and higher-lying states in 39 K and 36 Ar become populated. As can be seen, the p123, α0 and α1 channels rise to a maximum at Ex 14 MeV. The α0 and α1 channels become very weak at Ex
20 MeV while the p0 and p123 channels remain populated strongly due to the isospin allowed GDR decay. The data measured at q = 0.26 fm−1 at the S-DALINAC are treated differently. Proton and α-particle emission spectra summed over an excitation energy region Ex = 11–21 MeV in 40 Ca are extracted for the different final channels. The main advantage of such a comparison with statistical model results lies in the inclusion of the decay to higher-lying states (pres , αres ) which contain an appreciable part of the total cross sections. 2.2. Hauser–Feshbach calculations In the spirit of the statistical model it is necessary for the initial 1p–1h excitations to be mixed strongly with more complex configurations. Thus, specific nuclear structure effects become smeared out. Statistical decay is determined completely by level densities in the

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Fig. 1. Relative contribution of different spins J π = 0+ , 1− , 2+ (solid, dashed and dotted lines, respectively) and isospins T = 0, 1 in the description of the 40 Ca giant resonance decay branching ratios (histograms) from Hauser–Feshbach calculations.

daughter nuclei produced together with the transmission coefficients for the corresponding light particles emitted. If, however, large differences are observed between the calculated and measured branching ratios or particle emission spectra then it indicates that nuclear structure effects are present. It can be expected, therefore, that such an analysis could determine the direct component of giant resonance decay. In the present experiment, 0+ , 1− and 2+ states are excited strongly in the range 8  Ex  26 MeV where considerable isospin mixing can be expected. Consequently, both T = 0 and T = 1 states are excited in 40 Ca in this excitation energy region. In order to investigate the influence of the different spins and isospins, statistical model calculations were performed for the various pure initial conditions. The Hauser–Feshbach calculations were based on the code CASCADE of Pühlhofer [19] modified by Harakeh [20] to include parity and isospin conservation. Examples of these results are shown in Fig. 1. It should be noted that the decay of pure T = 1 strength by α-particle emission to T = 0 final states is isospin forbidden and, therefore, there is no contribution to the α0 and α1 channels. Irrespective of the spin and parity of the initial decaying state, the predicted branching ratios follow the general trend of the measured data. However, the magnitudes differ appreciably for the different J π , T combinations. A realistic calculation has to include

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as a starting point a population of the available phase space that reflects the conditions of the experimental measurements. Statistical decay probabilities for the emission of nucleons and light clusters from a fully equilibrated nucleus can be calculated as follows. The rate for emitting particle x from nucleus 1 at excitation energy E1 , with angular momentum J1 and parity π1 , to nucleus 2 at E2 , with J2 and π2 , is given by Rx d x = Γx (x )

1 ρ2 (E2 , J2 , π2 ) 2π h¯ ρ1 (E1 , J1 , π1 )

J
2 +Sx

J
1 +S

Tx (x ) dx ,

(2)

S=|J2 −Sx | =|J1 −S|

where – x = kinetic energy of particle x = E1 − E2 (separation energy), – Sx = spin of particle x, –  = orbital angular momentum, – S = J 2 + S x , – ρ1,2 = level densities and – Tx = transmission coefficients. In addition, isospin Clebsch–Gordon coefficients coupling excited levels to decay channels [21] have to be introduced into the formalism. Fig. 2 shows schematically the excitation and deexcitation of 40 Ca. The lowest decay channels 1α, 1p, 2p and 1n have ground-state Q values of 7.04, 8.33, 14.71 and 15.64 MeV, respectively. Angular momentum, parity and isospin of low-lying levels of the various decay daughter nuclei [22] were entered explicitly into the calculation. In the region of high excitation shown by the cross hatching in Fig. 2, level densities were calculated using the back-shifted Fermi-gas model [23], where the density of states of either parity is given by

Fig. 2. Excitation and decay channels included in the statistical model calculations of 40 Ca. The low-lying levels are entered explicitly into the CASCADE programme [19]. The level densities in the cross-hatched regions are determined from the back-shifted Fermi-gas model [23].

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ρ(Ex , J ) = ρ(Ex )

  (2J + 1) J (J + 1) exp − , 2σ 2 2σ 2

(3)

and the spin cut-off parameter σ is determined as a function of Ex [23]. Here, ρ(Ex ) takes the following form:   1 exp 2[a(Ex − ∆)]1/2 1 , (4) ρ(Ex ) = √ (Ex − ∆ + t)5/4 12 2 σ a 1/4 with the thermodynamic temperature t obtained from Ex − ∆ = at 2 − t.

(5)

Values for the fictive ground-state position ∆ and the level density parameter a for 40 Ca were taken from [24]. Thus, the quantity ρ2 (E2 , J2 , π2 ) in Eq. 2 is determined completely with parameters from experiment and has no freedom to be varied. The transmission coefficients Tlx (x ) were calculated from standard optical model potentials implemented in CASCADE [19]. Decay only by n, p and α emission was included. Alternative modes of decay by d, t and 3 He with higher separation energies were found not to affect the final results significantly and thus were omitted. Finally, the relative initial population of states ρ1 (E1 , J1 , π1 ) excited in 40 Ca has to be considered. This depends very strongly on the electron momentum transfer. The relative strengths for E2(E0) and E1 excitation obtained from the multipole analysis described in Section 2 and shown in Fig. 1 of part II were used. It should be noted that for excitation energies below the main peak of the GDR at Ex ≈ 19 MeV E2(E0) cross sections dominate that of E1. Since the analysis described in II does not distinguish between E2 and E0 strength because of the similarity of the form factors, the relative contributions to the cross sections were taken to be in proportion to the corresponding energy-weighted sum-rule strengths, giving E2 : E0 = 2.0 : 1.0. Further, the isospin dependence of the level densities has to be considered. Following the prescription of Jensen [25] the relative amount of T = 1 states can be determined as a function of excitation energy. Generally, the density of states with T = T0 + 1 can be written as [25] ρ(Ex , T = T0 + 1) ρ(Ex − E2 , T0 ),

(6)

where E2 is the energy of the volume symmetry term and has a value of 2.75 MeV for 40 Ca. Fig. 3 shows the fraction of T = 1 states as a function of E . It rises from about 10% x at Ex = 10 MeV to roughly 20% at Ex = 25 MeV. 2.3. Branching ratios and particle emission spectra The calculated branching ratios of the MAMI data are compared in Fig. 4 to the various decay channels measured. Reasonable to good agreement is observed. The calculated Ex dependence provides overall a satisfactory description except for the p123 channel, where the calculated maximum is somewhat above the data and shifted about 1 MeV to higher energies. A shift of several hundred keV is also observed for the α1 channel. For the

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Fig. 3. Variation of the 40 Ca, T = 1 level density fraction in the excitation energy region 10–25 MeV as calculated using [25].

Fig. 4. Branching ratios of the decay of giant resonance strength excited in the 40 Ca(e, e
x) reaction into the p0 , p123 , α0 and α1 channels. The solid lines give the results of the statistical model calculations described in the text.

p channels the calculations tend to increase relative to the data towards larger Θe . The description of the the α decay channels is clearly superior. In order to gain more insight as to how well the statistical model predicts the overall structure of the measured data, plotted in Fig. 5 is the ratio (Γexp − Γth )/(Γexp + Γth ) versus excitation energy. The α0 and α1 decay for Ex 13–25 MeV is explained well by the statistical model. One should note that the deviations at about 25 MeV result most

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Fig. 5. Deviations between measured and calculated branching ratios of the 40 Ca(e, e
x) reaction for decay into the p0 , p123 , α0 and α1 channels.

probably only from the very poor statistics. At lower Ex , deviations are partly due to the decay of strongly excited discrete levels with a large α decay width; similar cases are observed in the p123 channel [22,26]. The proton channels show a large excess strength for Ex > 18 MeV, where the Hauser–Feshbach predictions drop to zero, indicating the presence of nonstatistical contributions. Because of the dominant hole structure of the final 39 K states this additional strength is attributed to direct reactions. An alternative possibility to extract information on the decay is provided by the particle emission spectra. Such an analysis has been performed for the measurement performed at the S-DALINAC [27]. The proton emission spectra reflect the behaviour already visible in the branching ratios: at the highest proton energies, where the statistical model predicts negligible contributions, excess strength is seen corresponding to decay into the p0 and p1 channels. A quantitative analysis of direct contributions in these decay channels is presented in the next section. As an example, the α-particle emission spectra separated into the α0 , α1 and αres channels are presented in Fig. 6. The lines are statistical model results using the E1 and E2(E0) cross sections from the multipole decomposition described in part II. Since the mixture of E0 and E2 contributions is unknown, calculations for the two extremes assuming pure E2 (dashed line) and pure E0 (dotted line) are shown. The calculated spectra have been shifted by 500 keV to lower energies Eα to correct for the average energy loss in the target. For Eα = 5–7 MeV good correspondence is found with the Hauser–Feshbach calculations. The enhancement of the experimental spectra at lower Eα is related probably to a strong angle dependence of the α particle energy loss because of the variation of the effective target thickness as a function of the emission angle relative to the beam axis. Some dependence on the initial multipole strength composition is found for the α0 channel

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Fig. 6. Alpha emission spectra of the 40 Ca(e, e
α) reaction summed over an excitation energy range 11–21 MeV. The dashed and dotted lines are statistical model calculations described in the text assuming mixtures of E1 + E2 and E1 + E0, respectively.

at low energies when comparing the two extremes for the E2/E0 ratio. However, all the data with Eα > 6 MeV in all channels are described well by the calculations. The good correspondence indicates again that α decay from the giant resonance region in 40 Ca can be described within the statistical model. The small excess at higher Eα for the αres channel should be considered as an upper limit only. The normalization of the calculation to the data depends on the assumptions about the low-energy strength discussed above. Because E1 strength is suppressed by isospin rules it can be concluded that the character of the E2 (and E0) decay by α-particle emission is statistical predominantly. 2.4. Discussion The analysis allows two major conclusions. On one hand, significant nonstatistical parts can be identified in the proton decay at excitation energies above about 18 MeV. This should be largely attributed to the GDR because the dipole cross sections dominate for the lower momentum transfers measured. Significant fractions of direct proton decay from the GDR are in agreement with findings in sd-shell [7] and fp-shell nuclei [6] and have also been observed in neutron emission from 48 Ca [28]. On the other hand, α decay is in agreement with the statistical model predictions. Because of isospin selection rules the corresponding cross sections stem from decay of the GQR and GMR. At lower excitation energies also proton emission seems compatible with the Hauser–Feshbach results. However, the presence of nonstatistical contributions on the level of 30% cannot be excluded and, for example, have been observed in a 40 Ca(40 Ca, 40 Ca
p) experiment exciting the GQR [29]. The excitation energy dependence is governed mainly by the transmission coefficients which also enter into a description of direct decay. The presence of nonstatistical decay modes can thus be distinguished clearly only at excitation energies where the statistical decay becomes very small.

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Fig. 7. Experimental ratio of the (α0 + α1 )/(p0 + p1 ) branching ratios for decay of E2 (+ E0) giant resonance strength excited in the 40 Ca(e, e
x) reaction. The solid line describes a shell-model calculation of the ratio for direct E2 decay [30].

An additional test is possible through direct charged particle decay predictions for the isoscalar GQR in 40 Ca available from a SU(3) shell-model calculation [30]. In Fig. 2 of [30] the (α0 + α1 )/(p0 + p1 ) ratio is given for excitation energies Ex = 15–21 MeV. Experimentally, for the α0 and α1 channels the E2 strength extracted from the angular correlation analysis described in part II is used. The proton decay channels are taken from a form factor multipole decomposition of the 4π integrated spectra described in part II. The resulting spectra might contain some unresolved E0 strength. However, comparison with (α, α
p) and (p, p
p) experiments under kinematics where λ = 2 strength dominates indicates that E0 contributions are small in the p0 and p1 decay [9,15,31]. The results are compared to those of [30] in Fig. 7. The ratio is underestimated by a factor of about four over the whole energy range. Also, the decrease predicted towards higher excitation energies is not seen in the data. Even allowing for as much as 50% E0 strength in the α decay channels the ratio would still be underestimated severely. While the direct decay model provides a good description of the GQR decay in light nuclei like 16 O [30], the data of the present work are described poorly. However, it may be noted that the role of proton emission from the GQR still remains unclear. One cannot distinguish whether the model failure is due to a completely statistical character of the GQR decay or due to the statistical nature of α emission alone. At present, no theoretical explanations for the different role of direct decay from the GDR and the GQR exist. Indeed, microscopic calculations of the role of statistical and direct decay contributions would be of considerable interest, not only for 40 Ca but systematically from light to heavy nuclei. Although not providing a full explanation of the experimental observations two remarks may be in order. Firstly, the level density of isovector excitations is smaller inferring a reduced mixing with the underlying complicated np–nh states. However, a self-conjugate nucleus like 40 Ca is a special case. For targets with nonzero g.s. isospin T< , the effect should then be particularly pronounced for T> = T< + 1 parts of the GDR cross sections. Secondly, the direct α-decay from giant resonances

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observed in light nuclei is intimately linked with α clustering. The present results indicate that α clustering plays no significant role for 40 Ca in the giant resonance excitation region.

3. Fluctuation analysis The strong deviations from the statistical model predictions for decay from the GDR region into low-lying states of 39 K shown in the previous section require a quantitative description. A natural way to do this is by means of an Ericson fluctuation analysis since in the region of interest there are strongly overlapping levels where the coherence width Γ is large compared to the average level spacing D. In this case, the correlations observed can be related directly to the compound and direct reaction contributions (used in the sense as described in the introduction). The correlation method can be extended to cross correlations between decay channels or different electron scattering angles, thus providing information on the size of the emitting source. An extensive review of the fluctuation analysis method can be found in [32]. 3.1. Autocorrelation function The principles of the autocorrelation function are described in Fig. 8. In a first step, the original spectrum is folded with a Gaussian of FWHM = 800 keV (l.h.s., upper part). In this way the fluctuations induced by intermediate structures are removed. Division of

Fig. 8. Principles of the fluctuation analysis. L.h.s., top: original and smoothed (Gaussian, FWHM = 800 keV) spectrum to remove intermediate structures. L.h.s., bottom: stationary cross section σ/σ . R.h.s.: autocorrelation function. The solid line corresponds to a fit with Eq. (8).

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the original spectrum by the smoothed cross sections leads to a stationary cross section spectrum (l.h.s., lower part) reflecting the fluctuating cross section part only. Then the autocorrelation function is calculated using C() =

σ (E) · σ (E + )I − 1, σ (E)I σ (E + )I

(7)

where  is the energy increment which was taken equal to the bin-width of 25 keV. The averaging interval I was restricted to the region 16–23 MeV. Here, the upper bound of this interval was determined by the need for reasonable statistics, while the lower bound results from the condition of overlapping levels. Based on the level density calculations described above and using Γ = 15 keV as suggested in [33], a value Γ /D ≈ 2 can be estimated at Ex = 16 MeV. It was shown by Moldauer [34] that this relaxed condition is still sufficient as compared to the strict Ericson formulation of Γ /D  1. The resulting autocorrelation function is depicted in the r.h.s. of Fig. 8, where the variation of C() is described by the Lorentzian Γ2 (8) Γ 2 + 2 displayed as a solid line. Note that the fluctuations of the experimental points at large  are result from the finite data interval [32]. The value of Eq. (7) obtained at  = 0 is related to the direct part of the cross section by  1  1 − yD2 . (9) C(0) = nN Here, N corresponds to the effective number of spin channels contributing to the reaction and n describes a damping factor due to the experimental energy resolution E. It has been shown [35] that the damping depends on E and Γ only, and for cases where E  Γ can be approximated by C() = C(0) ·

E . (10) πΓ + 1 However, for an experimental resolution of E 120 keV and typical coherence energies of a few tens of keV it seems more appropriate to use the exact solution for a square resolution function [36] n=

n=

( E/Γ )2 . 2( E/Γ ) tan−1 ( E/Γ ) − ln[1 + ( E/Γ )2 ]

(11)

The procedure described to extract C(0) values is of course sensitive to the choice of the width in the smoothing Gaussian function. The variation of C(0) with the chosen FWHM is displayed in Fig. 9. For values of 600–1000 keV the variation is found to be minimal and therefore a value FWHM = 800 keV was adopted for all further calculations. 3.2. Probability distribution Another independent method to evaluate the direct part of the cross section is given by the probability distribution. The principles are demonstrated in Fig. 10. One determines the

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Fig. 9. Dependence of the normalized variance C(0) of the 40 Ca(e, e
p0 ) autocorrelation function on the width of the smoothing Gaussian function.

frequency distribution of the variation of the stationary cross sections y = σ/σ  around the mean value y = 1. The probability distribution is a function of the direct reaction part yD and the number of open channels N and can be written as   N  N −N(y − yD ) JN−1 (2ix)
P (y) = · y N · exp with (12)
1 − yD 1 − yD (ix)N √ N yyD and N
= N − 1. x = 1 − yD Here, Jn is the complex Bessel function of order n. The complex argument in Eq. (12) can be removed utilizing the relation In (x) = i−n Jn x,

(13)

where In (x) represents the modified real Bessel function. Since independent variation leads to almost identical results for different combinations of N and yD , the number of open channels was kept fixed at the same values as used in the autocorrelation function and with the same damping factor due to the finite energy resolution. Fig. 11 presents results for

Fig. 10. Distribution of fluctuations around the mean value σ/σ  = 1 for the example of the 40 Ca(e, e
p ) reaction channel. 0

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Fig. 11. Probability distribution of the 40 Ca(e, e
p0 ) channel. The solid line corresponds to a fit of Eq. (12) with N = 23 and yD = 0.908.

the p0 channel as an example. The solid line corresponds to a fit using Eq. (12) with the parameters indicated in the figure caption. 3.3. Discussion of results The normalized variances of the autocorrelation function obtained for the p0 , p1 , p3 and pres channels are presented in Table 1. In order to convert C(0) to a direct reaction crosssection fraction, independently determined values for N and n must be inserted in Eq. (9). The maximum number of spin channels R can be calculated from [13,37] R=

(2I + 1)(2i + 1)(2I
+ 1)(2i
+ 1) . 2

(14)

Here, I , i, I
, i
stand for the spin of projectile, target, residual nucleus and ejectile, respectively, and angular momentum conservation for the z-components, Table 1 Results of the fluctuation analysis of the 4π -integrated 40 Ca(e, e
p) spectra Channel N

n

p0

6.90

p1

3.45

p3

6.90

2.26 3.37 2.26 3.37 2.26 3.37

pres a Ref. [10] b Ref. [38]

Autocorrelation function Γ (keV) C(0) 26a 15b 26 15 26 15

1.09 × 10−2 1.75 × 10−2 2.83 × 10−2 0.32 × 10−2

yD (%) 86.4 91.1 89.2 92.9 58.6 74.7

Probability distribution N yD (%) 16 23 8 12 16 23

85.9 90.8 88.9 92.9 58.1 67.9

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M + m = M
+ m
,

331

(15)

can be used as an additional constraint. It has been shown in [37] that Eq. (14) can serve as a good approximation for emission angles 40◦ –140◦. The remaining angular ranges were interpolated graphically from Fig. 10 of [37]. The numbers obtained at the measured particle angles are multiplied with the weighting factors used in the 4π integration (see Eq. (3) in part I). The resulting N values are listed in the second column of Table 1. Because of an experimental situation where the energy resolution E 120 keV is large compared to Γ , the coherence energy cannot be determined independently and must be taken from the literature. Diener et al. [10] extracted two values, Γ = 26 keV and 13 keV, from a 39 K(p, γ0 ) study depending on the method of averaging of the original spectrum. In [38] a value of Γ = 15 keV was deduced from the 39 K(p, α)36Ar reaction. This value is in excellent agreement with the model calculations of [33]. Table 1 presents results of the autocorrelation function assuming Γ = 15 keV or 26 keV, respectively. Also included are results from the probability distributions. Here, N was assumed to be the product of the N value deduced from the spin coupling, Eq. (14), multiplied with the damping factor n due to the experimental energy resolution, Eq. (11). Since Eq. (13) is defined for integer values only, the resulting numbers were rounded to the nearest integer. One should note that the N values are derived under the assumption of statistical decay and therefore represent upper bounds. Accordingly, the extracted direct cross-section parts given in Table 1 are lower limits only. Despite some variations due to the choice of Γ both methods consistently demonstrate a direct reaction cross-section part of about 85– 95% for the p0 and p1 decay. This finding is in qualitative agreement with results obtained in the 58 Ni, 64,66Zn(e, e
p) reactions populating low-lying states in the corresponding daughter nuclei [6] which show predominantly direct decay from the GDR excitation region. For light nuclei (sd-shell) the direct nature of the GDR decay into the ground state has been demonstrated in investigations of the time-reversed (p, γ0 ) reaction [39–42] on various nuclei. The results of the fluctuation analyses depend on assumptions about N , but in all cases direct reactions dominated (yD > 90%). Diener et al. [10] investigated the 39 K(p, γ0 )40Ca reaction and concluded that assuming N = 1 one obtains yD ≈ 99%, which is in qualitative agreement with the value given in Table 1 for the 40 Ca(e, e
p0 )39 Kg.s. reaction. In the case of the p3 channel a substantially larger compound reaction part is found. A possible explanation for the difference might be as follows. The ground and first-excited −1 −1 states of 39 K are of dominant 1d3/2 and 2s1/2 character, respectively, as can be inferred from nucleon pick-up reactions [43]. The p3 level is excited only weakly in the transfer reaction and has a more complex wave function. It has been interpreted to belong to −1 −2 ] particle–phonon multiplet with some admixture from the (1f7/21d3/2 ) the [3− ⊗ 1d3/2 configuration [44]. While a pure single hole structure favours decay from the initial 1p–1h excitations, the overlap with the strongly mixed parts of the giant resonance wave function should be enhanced for the more complex p3 state. The coupling to phonon excitations plays a major role in the damping of the initial 1p–1h strength [3]. It has been suggested, therefore, that nonstatistical decay into dominant ‘phonon ⊗ hole’ states can be interpreted as a sign of preequilibrium emission from the

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2p–2h mixed states. Thus, the large nonstatistical decay fraction of the p3 channel indicates the presence of a preequilibrium contribution. The corresponding cross section presents a lower limit of Γ ↓↑ only, since further strength might be hidden in the pres channel which constitutes many unresolved states. In fact, the strong nonstatistical population of 39 K levels around E = 4 MeV observed in the particle emission spectra might well be x −1 explained by a decay to the J π = 5/2− and 7/2− members of the [3− ⊗ 1d3/2 ] multiplet which are found experimentally [22] at 3.88 and 4.13 MeV, respectively. The importance of direct reactions in the proton decay of the GDR in sd-shell nuclei has been emphasized by Eramzhyan et al. [7]. They estimate the total direct cross-section part in 40 Ca to be 50%. In the present work the major part of the cross section leads to residual states in 39 K at higher excitation energy. The autocorrelation function yields a very small C(0) value. However, the number of contributing channels is unknown and might be very large, so no conclusion from this method is possible. In [7] a close correlation was also presented between the integrated GDR cross section to the ground state and spectroscopic factors of nucleon pick-up reactions leading to the A − 1 nucleus. On the basis of the same arguments, it should be possible to compare results for different hole states like p0 , p1 in one residual nucleus. With a cross section ratio σ (p0 ) : σ (p1 ) = 2.5 : 1, good correspondence is observed with respect to the 40 Ca(d, 3 He)39 K results of Doll et al. [43], who find a ratio of 2.2 : 1, and an (e, e
p) study [45] giving 2.6 : 1. If one adopts instead the (d, 3 He) results of Devins et al. [46] of 1.4 : 1 the agreement would be worse. The difference of the two (d, 3 He) experiments is not clear. It might depend on the difference in incident energy and the choice of 3 He optical potentials. 3.4. Cross correlations Another important application of the correlation method is the study of cross correlations. The cross correlation function C1,2 () is defined as  

 σ2 (E) − σ2 (E) σ1 (E + ) − σ1 (E + ) 1 . (16) C1,2 () = √ σ1 (E + ) σ2 (E) C1 (0)C2 (0) The function is normalized to the geometrical mean of the variances C1 (0) and C2 (0). Three examples will be discussed, viz. correlations between the 39 K(p, γ0 )40 Ca and the 40 Ca(e, e
p )39 K 0 g.s. reactions, between different decay channels and between different electron scattering angles. The close relation between photo-proton reactions and the p0 decay channel investigated in the previous section is demonstrated impressively in Fig. 12. The comparison of the 39 K(p, γ )40 Ca spectrum [10] with the 40 Ca(e, e
p )39 K channel shows close agreement 0 0 even in fine structure. The correlation coefficient C1,2 (0) = 0.53 is limited mainly by the additional GQR strength in the Ex = 16–18 MeV region which is excited only in the (e, e
) reaction. This strong correlation is further proof of the predominant direct nature of the p0 decay as discussed above. The correlations between different decay channels like p0 , p1 , α0 , etc., show values C(0) ≈ 0.1–0.4. However, at large  comparable correlation values are obtained which

J. Carter et al. / Nuclear Physics A 696 (2001) 317–336

333

Fig. 12. Proton emission angle-integrated 40 Ca(e, e
p0 ) spectrum compared to the result from the 39 K(p, γ )40 Ca reaction [10]. 0

indicate that the cross correlations at  = 0 result from ‘finite range of data’ errors [32] and poor statistics. Overall, the results indicate no significant branching into different emission channels in the excitation energy region 16–20 MeV. In contrast, such cases are well known for levels at lower Ex [22]. The cross correlation method was applied also to spectra of the same reaction channel belonging to different electron scattering angles. If the difference of the scattering angles Θ1 and Θ2 is smaller than the coherence angle α, one can expect [47] J1 (kRα) 2 , CΘ1 ,Θ2 (0) = kRα

(17)

where k is the emitted-particle wave number. The correlation interval is limited by the data to I = 16–20 MeV and the interaction radius R can be approximated by R=

1 . kα

(18)

The method of obtaining the coherence angle is demonstrated in Fig. 13 for the example of the p1 channel. The solid line is a least-squares fit of Eq. (17) to the data and the coherence angle is defined as the width at half maximum. The results for different proton channels are summarized in Table 2. Considering the relatively large error, all channels evaluated are consistent with the 40 Ca equivalent radius Re 4.4 fm [48] indicating that the whole nucleus acts as an emitting source.

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Fig. 13. Cross correlations between different electron scattering angles for the 40 Ca(e, e
p1 ) reaction as an example. The solid line corresponds to a fit of Eq. (17). The coherence angle obtained at C( Θ) = 0.5 is indicated also. Table 2 Coherence angle α and interaction radius R from cross correlations between different electron scattering angles Decay channel p0 p1 p3

α (deg)

R (fm)

17(6) 14(5) 12(4)

4.9(1.7) 5.9(1.9) 6.9(2.3)

4. Conclusions The relative importance of direct and statistical contributions to the charged particle decay in the 40 Ca(e, e
x) reaction at Ex 10–25 MeV was investigated with different methods. A Hauser–Feshbach calculation of the branching ratios into the resolved final channels shows reasonable to good agreement with the data in the excitation energy region 12–18 MeV, where the major part of the GQR strength is found. In this energy region the transmission coefficients are the decisive component determining the relative strength rather than nuclear structure details. Nonstatistical strength observed in the proton emission to low-lying states in 39 K comes from the region around 20 MeV and is most likely connected to the GDR. These findings are confirmed by the analysis of the particle emission spectra. Therefore, it is concluded that there is a dominant statistical character for the GQR decay in 40 Ca. However, a claim of a 30% direct contribution in the GQR decay from a 40 Ca(40 Ca, 40 Ca
p) study [29] cannot be excluded on the basis of the present results. On the other hand, direct model predictions for the decay of the GQR show little correspondence to the data.

J. Carter et al. / Nuclear Physics A 696 (2001) 317–336

335

The direct nature of the decay of the GDR excitation region (Ex ≈ 16–23 MeV) into low-lying states of 39 K is proven by a fluctuation analysis. This method gives at least 85–95% contribution to the decay into the ground state and first excited state in 39 K, in qualitative agreement with previous results in a study of the 39 K(p, γ0 )40 Ca reaction. The ratio of cross sections in the direct decay channels was shown to correspond to the ratio of the single-nucleon transfer spectroscopic factors. The relatively strong (> 50%) nonstatistical population of the ‘phonon ⊗ hole’ type p3 indicates the presence of significant preequilibrium emission through mixing with 2p–2h states. Cross correlations between different decay channels indicate no significant branching in the excitation energy region investigated except for well known strongly excited levels at Ex ≈ 9–12 MeV. From the correlations of spectra of one decay channel for different electron scattering angles a coherence angle can be extracted. It is related to the size of the emitter. Within error bars, all investigated channels are compatible with the assumption that the whole 40 Ca nucleus acts as the emitting source.

Acknowledgements Discussions with N. Frascaria, M.N. Harakeh and J. Wambach are gratefully acknowledged.

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