- Email: [email protected]
Pionic fusion study of the 6He halo
Nuclear Physics A 779 (2006) 47–62
Pionic fusion study of the 6He halo M. Andersson a , Chr. Bargholtz a , Kj. Fransson a , E. Fumero a , L. Gerén a,∗ , L. Holmberg a , K. Lindh a , L. Mårtensson a , I. Sitnikova a , P.-E. Tegnér a , G. Weiss a , K. Wilhelmsen b a Department of Physics, Stockholm University, SE-106 91 Stockholm, Sweden b Department of Physics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden
Received 20 September 2005; received in revised form 7 September 2006; accepted 7 September 2006 Available online 25 September 2006
Abstract The halo nucleus 6 He has been studied in a pionic fusion experiment at the CELSIUS ring in Uppsala. The aim of the experiment was to investigate, in particular, the high-momentum part of the halo wave function by measuring the differential cross section for the 4 He(d, 6 He)π + reaction 0.6, 1.2 and 5.0 MeV above threshold in the centre-of-mass frame. The 6 He ions were detected in a E − E solid-state detector telescope inserted into the ring vacuum. The result for the total cross section is, respectively, 22(1), 38(1) and 57(9) nb with a common systematic uncertainty of ±35%. The differential cross section is clearly dσ (0◦ ) − dσ (180◦ )]/[ dσ (0◦ ) + dσ (180◦ )] = −0.25(5), −0.29(5) and −0.35(15) at the anisotropic: [ dΩ dΩ dΩ dΩ same three energies. The preferred direction of emission for the pion is parallel to the momentum of the 4 He ion in the initial state. In terms of a simple model for the reaction the results depend sensitively on the state of motion of the centre of mass of the halo relative to that of the core. © 2006 Elsevier B.V. All rights reserved. PACS: 25.10.+s; 25.45.-z; 27.20.+n; 21.60.Gx Keywords: N UCLEAR REACTIONS d + 4 He, E = 217.3, 218.2, 224.1 MeV; measured particle spectra, σ , σ (θ), anisotropies. 6 He deduced halo features
* Corresponding author.
E-mail address: [email protected] (L. Gerén). 0375-9474/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2006.09.002
48
M. Andersson et al. / Nuclear Physics A 779 (2006) 47–62
1. Introduction The T = 1, I = 0 ground state of 6 He is the lightest two-neutron halo state. The two valence neutrons are bound by a little less than one MeV and their wave function extends far beyond that of the four tightly bound nucleons of the core. The point-nucleon matter radius was first derived by Tanihata et al. [1,2] to be rm = 2.73(4) fm from measurements of the total interaction cross section in collisions with Be, C and Al. These same data have been repeatedly analysed to yield significantly differing results for the matter radius depending on the method of analysis. In the latest of these analyses known to us Tostevin and Al-Khalili [3] arrived at rm = 2.54(4) fm. Later Alkhazov et al. and Neumaier et al. [4–6] made a thorough investigation of matter distributions describing 6 He and 8 He based on measurements of proton elastic scattering in inverse kinematics. At a beam energy of 0.7 GeV per nucleon the differential cross section, dσ /dt, was measured for |t| = 0.05 (GeV/c)2 . Their final result for 6 He, rm = 2.45(10) fm, included a tail to the density distribution taken from theory. The authors conclude from their data that the 6 He nucleus may, to first approximation, be treated as a three-body system consisting of an inert alpha particle and two valence neutrons forming an extended halo. Recently Wang et al. published the results of a laser-spectroscopic measurement of the 6 He charge radius [7], determining the point-proton charge radius to be, rp = 1.912(18) fm. This high-precision determination of the charge radius provides a crucial test of any theoretical description of the 6 He nucleus. In a three-body description, although insensitive to the relative distance between the halo neutrons, the charge radius provides a measure of the displacement of the alpha-particle core relative to the nuclear centre of mass [8]. Mass and charge radii provide information primarily regarding the dominant low-momentum parts of the nuclear wave function. In order to probe the high-momentum components of the ∗ halo wave function the differential cross section of the d(4 He, 6 Li )π 0 reaction to the isobaric analogue state of the 6 He ground state was measured [9]. At beam energies of 417.95 and 420.30 MeV the pion was emitted with a kinetic energy of 1.2 and 1.9 MeV in the centre-of-mass (c.m.) system respectively. In the pionic fusion process essentially all of the kinetic energy available in the entrance channel is transferred to the meson field. It is to be expected that such a highly coherent process is particularly sensitive to correlations among the nucleons in the initial and final state and theoretical treatments in the literature [10–16] stress the sensitivity to the detailed structure of the nuclei involved. Kajino et al. [14] obtained an order of magnitude increase ∗ in the cross section for the 4 He(3 He, π + )7 Li reaction using a correlated cluster wave function for 7 Li as compared to the result using a shell model wave function without correlations. Here we report the results of measurements of the differential cross section at three energies close to threshold for the pionic fusion of deuterons and 4 He, forming 6 He in its ground state and a positively charged pion. A preliminary account of the experiment has been reported earlier [17]. The details of the experiment are described in Section 2 followed, in Section 3 by an account of data reduction and analysis. The results are presented and discussed in the light of simple theoretical considerations in Section 4. Finally, in Section 5, our main conclusions are summarised.
M. Andersson et al. / Nuclear Physics A 779 (2006) 47–62
49
2. Experiment The 4 He(d, 6 He)π + reaction was studied in the CELSIUS accelerator and storage ring at The Svedberg Laboratory in Uppsala. Simultaneously 3 He-ions from the 4 He(d, 3 He)t reaction were detected for monitoring purposes. A storage ring with a cooled ion beam and an internal target provides ideal conditions for the study of reactions very close to threshold. Low-energy pions are difficult to detect and so in pionic fusion close to threshold it is advantageous to detect the fused nuclear system instead [18–20]. Following pionic fusion reactions close to threshold the heavy reaction products are emitted in a narrow cone around zero degrees and with essentially all of the beam momentum. Hence, in a storage ring, they stay close to the circulating beam and are difficult to detect. Given that the charge of the recoiling nuclei is different to that of the beam particles the recoils may be spatially separated from the beam in a magnetic field and then detected. 2.1. Apparatus An electron cooled deuteron beam was used in combination with a windowless 4 He gas target in the fourth straight section of CELSIUS. The heavy reaction products were detected in the zero-degree spectrometer [21] 6.1 m away from the normal target position at the centre of the straight section (cf. Fig. 1). In this experiment the cluster-jet target station was operated with 4 He providing a windowless, however spread out, gas target with an integrated thickness of approximately 6 × 1013 atoms cm−2 . In normal operation, with clustering gasses, the target is concentrated to within approximately one centimeter along the beam at the centre of the scattering chamber of the cluster-jet target station. In the present case, the extension along the beam was of the order of meters (cf. Section 3.1). The quadrupole and dipole magnets in the fourth quadrant of the ring focused and deflected the heavy reaction products onto a solid-state detector telescope inside the ring vacuum in the second dipole magnet 1.5 m into the bend following the gas target (cf. Fig. 1). The detector telescope in this experiment comprised a 1.0 mm thick silicon transmission (E) detector and a 1.7 mm thick stopping (E) detector made of high-purity germanium. The stopping detector was position sensitive with contacts divided into 18 horizontal strips of 2 mm width on one side and 66 vertical, 1 mm wide, strips on the other. The dead layers of the silicon detector corresponded to less than 1 µm Si + 0.3 µm Al (specified by the manufacturer). The contacts on the germanium detector are made by ion implantation creating a dead layer of less than 0.5 µm germanium equivalent. The detectors were cooled to liquid nitrogen temperature by circulating
Fig. 1. Schematic drawing of the experimental apparatus (not to scale) showing the zero-degree spectrometer and the luminosity monitor telescope aiming at the normal target position.
50
M. Andersson et al. / Nuclear Physics A 779 (2006) 47–62
liquid nitrogen through the cold head on which the detectors were mounted. For an event to be accepted a coincidence was required between the two detectors. Although the solid angle of the zero-degree spectrometer for ions emanating from the normal target position is not more than 0.7 msr in the present experiment the effective acceptance for 6 He ions from the 4 He(d, 6 He)π + reaction varied from approximately 56% at the lowest energy to 7% at the highest. ∗ In the measurement of the differential cross section of the d(4 He, 6 Li )π 0 reaction [9] there were difficulties associated with the determination of the luminosity which in turn rendered the determination of the cross section (although not the asymmetry) somewhat uncertain. In order to avoid this difficulty a scintillator detector telescope comprising two 2 mm thick, 40 × 40 mm2 , plastic detectors and one larger, 100 mm thick, conically shaped BGO detector was placed outside of the CELSIUS ring at an angle of 50◦ from the direction of the beam and pointing towards the normal target position at the centre of the scattering chamber of the cluster-jet target station. An accepted event required a triple coincidence between the two plastic scintillators, placed 440 mm apart, and the BGO detector. By means of this so-called monitor telescope a direct measure of the slightly varying luminosity was provided. The data acquisition system used in the experiment was based on the SVEDAQ [22] system with NIM, CAMAC and VME electronics. The signals from the silicon detector and from the horizontal contacts of the germanium detector were fed to shaping amplifiers and then to peaksensing analogue-to-digital converters as were the signals from the three detectors of the monitor telescope. The preamplifier signals from the vertical contacts on the germanium detector were shaped and fed to discriminators the output of which were recorded in the SVEDAQ system. The leakage current of the solid state detectors was monitored continuously and stored as was information regarding the detector position and a time stamp.
2.2. Measurements
Measurements were made at four different beam energies, Ebeam = 214.6, 217.3, 218.2, and 224.1 MeV. The lowest beam energy lies below the threshold for the 4 He(d, 6 He)π + reaction and was included in order to provide information concerning the background. The three higher energies correspond to c.m. energies, Q, 0.6, 1.2, and 5.0 MeV above threshold. CELSIUS was operated in 900 s cycles and the detector telescope was moved into measuring position 40 s after injection, when the beam had been accelerated to its final energy and then cooled. After 800 s the telescope was removed to the parking position before the beam was dumped. For most of the measurements the detector telescope was positioned with its centre 170 mm away from the circulating beam. In order to determine the density of the target gas along the straight section the distribution of 3 He ions in the detector plane from the 4 He(d, 3 He)t reaction was measured for distances of 113 to 237 mm from the circulating beam. Energy calibration of both detectors was done by means of alpha particles from 241 Am and 232 U sources mounted inside the vacuum chamber. The energy resolution (FWHM) for alpha particles at 8 MeV was measured to 350 keV for the silicon detector and to between 300 and 400 keV for the strips of the position sensitive germanium detector (these values include effects of dead layers and oblique incidence). For 73 MeV 6 He ions the energy resolution of the telescope was determined to be approximately 0.9 MeV (below).
M. Andersson et al. / Nuclear Physics A 779 (2006) 47–62
51
3. Analysis Coincident spectra from the silicon and germanium detectors were used in E − E technique in order to identify particles resulting from d + 4 He reactions. The stopping power of the thin silicon detector depends on the charge-to-mass ratio and the energy of the ions. In Fig. 2 ridges are seen corresponding to protons, deuterons, tritons (weakly), 3 He, 4 He, and 6 He. The concave part of the ridges correspond to ions stopped in the germanium detector whereas particles passing through both detectors, depositing less energy with increasing velocity, give rise to the convex ridges. The 3 He ions from the 4 He(d, 3 He)t reaction indicated in Fig. 2 have an energy of approximately 200 MeV and pass through both detectors. 3.1. Determination of target density The 4 He density distribution along the target straight section could be determined from the distribution in the plane of detection of 3 He ions from the 4 He(d, 3 He)t reaction. The acceptance decreases rapidly with emission angle in the c.m. system so that at an angle of 5◦ it is a tenth of its maximum value at 0◦ . These ions, thus, form an almost mono-energetic distribution appearing as a peak in E − E spectra (cf. Fig. 2). Their distribution in the plane of detection depends strongly on the extension of the target along the straight section. This is illustrated in Fig. 3. These experimental data were collected during a total of more than 40 hours with the detector telescope in 4 different positions. Simulated data were produced using a Monte Carlo code with ray tracing that has been developed for the zero-degree spectrometer. The code uses the actual geometry of CELSIUS and measured field maps. The angular distribution of 3 He ions was assumed constant over the narrow range of emission angles and their point of emission was sampled according to a target density
Fig. 2. E − E spectrum for Ebeam = 217.3 MeV where E(E) is the energy deposit in the silicon (germanium) detector (cf. text).
52
M. Andersson et al. / Nuclear Physics A 779 (2006) 47–62
Fig. 3. Horizontal distribution of 3 He ions in the detector plane. Results of simulations assuming a localized target at the normal target position (top) and a distributed target according to Eq. (1) for b1 = 0.69 m−1 and b2 = 0.186 m−1 (bottom, grey) and experimental data (bottom, line).
ρ(x) determined by three parameters, the central density ρ0 and two different slope parameters, b1 and b2 , in the backward and forward direction respectively: 1 − b1 |x| , − b11 < x < 0, (1) ρ(x) = ρ0 1 − b2 |x| , 0 < x < b12 , where x is the distance along the beam from the normal target position. The values of the slope parameters were obtained in a maximum likelihood fit of simulated to experimental twodimensional position distributions. The horisontal position distribution corresponding to the best fit, obtained for b1 = 0.69(7) m−1 and b2 = 0.186(1) m−1 , is shown in Fig. 3. From Eq. (1) the integrated target thickness, d, is obtained as 1/b 2
d= −1/b1
1 ρ(x) dx = ρ0 b1−1 + b2 −1 . 2
(2)
The target thickness, d, was determined in a measurement of the stopping power of the target gas. For this purpose the electron cooler was switched off and the change of orbital frequency was determined from Schottky spectra. The result for the target thickness, d = 5.9(1) × 1013 cm−2 , fixes the value of the density at the centre of the target chamber, ρ0 = 1.7(2) × 1011 cm−3 . 3.2. Luminosity Given the density distribution in (1) we may define a luminosity per unit length l(x) = Ibeam ρ(x)
(3)
such that the number of reactions per second per unit length of target is given by r(x) = σ l(x).
(4)
M. Andersson et al. / Nuclear Physics A 779 (2006) 47–62
53
Fig. 4. The acceptance for 3 He ions from the 4 He(d, 3 He)t reaction (assuming isotropy in the c.m.) as a function of where along the beam line the reaction occurs. The data are simulated assuming the detector telescope to be in the main measuring position with its centre 170 mm away from the circulating beam.
In these expressions Ibeam denotes the beam current (particles per unit time) and σ the total cross section. Immediately preceding the measurement of the target thickness the beam current was measured while at the same time registering the rate of events in the monitor telescope, nmon . Since the monitor telescope accepts particles due to reactions in the vicinity of x = 0 only, a constant of proportionality, C = Ibeam ρ0 /nmon , could be determined in terms of which the luminosity per unit length is given by 1 − b1 |x| , − b11 < x < 0, l(x) = Cnmon (5) 1 − b2 |x| , 0 < x < b12 . The possible variation of the slope parameter b2 was monitored during the entire measurement by the number of 3 He-ions from the 4 He(d, 3 He)t reaction reaching the detector. The acceptance for these ions in the main measuring position is shown in Fig. 4. The variation of the average value of the slope parameter was smaller than 10% for all beam energies. For each beam energy the integrated luminosity was determined from (5) by substituting for the event rate the total number of events registered in the monitor telescope at the corresponding energy. 3.3. Selection of events due to 6 He The reaction 4 He(d, 6 He)π + was studied at three different beam energies, Ebeam = 217.3, 218.2, and 224.1 MeV. Events corresponding to production of 6 He ions are easily identified in E − E spectra like the one shown in Fig. 2. The separation from the 4 He distribution is excellent. By selecting events in a narrow gate around the expected 6 He distribution in the E − E spectrum samples with only a small background contribution were obtained. Events were approved of if the signal from the position-sensitive detector emanated from one or two neighbouring horizontal strips and from one to three neighbouring vertical strips. This division of charge among neighbouring strips is at least partly due to the strong magnetic field of the CELSIUS bending magnets perpendicular to the electric field of the detector. The loss of efficiency caused by multiple hits was less than 1%. An upper limit to the efficiency loss due to nuclear reactions in the detectors, estimated from [9], is 1.4%. Energy spectra of events selected in this way at the four different beam energies are shown in Fig. 5. At each beam energy above the threshold for the 4 He(d, 6 He)π + reaction the spectra exhibit two peaks corresponding to 6 He ions emitted at 0 and 180 degrees in the c.m. system.
54
M. Andersson et al. / Nuclear Physics A 779 (2006) 47–62
Fig. 5. Kinetic energy spectra of selected events, measured at four different beam energies (from top to bottom) 214.6, 217.3, 218.2, and 224.1 MeV respectively. The spectrum for the highest beam energy is the sum of spectra taken at two different detector positions (130 and 175 mm from the beam). The integrated luminosities during the measurement of these spectra were approximately 1.9, 8.0, 7.3 and 3.7 in units of 1034 cm−2 respectively.
The energy difference is a sensitive measure of the beam energy. From the spectrum at the lowest beam energy, which is below threshold, the magnitude of the background can be estimated. This background is mainly due to misidentified events, i.e. events due to nuclear reactions or due to sum-up from particles hitting neighbouring strips. 3.4. Fitting to experimental data Simulations of 4 He(d, 6 He)π + reactions were done again using the MC code developed for the zero-degree spectrometer. The reaction points along the target straight section were sampled according to the luminosity distribution (5) determined from the number of monitor events and 3 He ions detected in parallel to the detection of 6 He. The differential cross section, at each beam energy, of the 4 He(d, 6 He)π + reaction was expanded in Legendre polynomials dσ = ak Pk (cos θc.m. ), (6) dΩ c.m. k0
M. Andersson et al. / Nuclear Physics A 779 (2006) 47–62
55
where θc.m. denotes the c.m. angle between the pion and deuteron momenta. Each simulated event was characterized by the energy and position of detection of the 6 He ions reaching the detector, the same parameters as those recorded for the experimental data. Results from the MC simulations were fitted simultaneously to the experimental data for all three beam energies above threshold. A maximum-likelihood algorithm was used to determine the coefficients in Eq. (6). Additional parameters determined in the fit were the magnitude of an assumed flat background of misidentified events (for each beam energy), one independent beam energy and the energy resolution of the detector telescope. The steps in beam energy between the three measurements, 0.90 and 5.90 MeV respectively, were assumed precisely determined by the change in high-voltage of the electron cooler. 4. Results and discussion 4.1. Experimental results The result of the fit is summarised in Table 1, assuming ak = 0 for k 2 in Eq. (6). In Fig. 6, as an illustration of the quality of the fit, the simulated distribution for Ebeam = 218.2 MeV is compared to the measured one. The systematic uncertainty in the Legendre coefficients in Table 1 are due to two sources. One is the uncertainty in the beam current calibration, which we estimate to approximately 5%. The second, and larger, is the uncertainty in the shape of the density distribution of the target gas, i.e. where along the beam line the gas of known thickness d is located. The fitted triangular distribution, Eq. (1), gives a good description of the results for 3 He. In order to assess the systematic uncertainty in the results for 6 He we have investigated two extreme cases for the gas distribution. In the first case the target gas was assumed to be confined to the target chamber with no gas in the beam pipes outside the chamber. This provided a lower limit to the Legendre coefficients since the acceptance for the reaction leading to 6 He was largest in this region. In the second case we used the vacuum calculation code VAKTRAK [23] in order to estimate the limit to the extension of the target gas in the backward direction where the acceptance was low, thus obtaining an upper limit to the Legendre coefficients. This calculation was done for the actual geometry of the target straight section and readings of the vacuum gauges and assuming the ion pumps along the straight section to have been all switched off. The total systematic uncertainty in the Legendre coefficients, from both sources, amounts to approximately 35%. The uncertainty, however, amounts to a common multiplicative factor, leaving the asymmetry and the energy dependence of the cross section unaffected. The results in Table 1 differ from the preliminary results reported previously [17] due to erroneous assumptions in the early simulations. Table 1 Results for beam energy, Legendre coefficients and total cross section obtained in the fit to experimental data. The energy resolution (FWHM) of the detector telescope was determined to be 0.89(2) MeV Results from the fit Beam energy (MeV)
a0 (nb/sr)
a1 (nb/sr)
σ (nb)
217.28 ± 0.01
+0.6 1.78 ± 0.07−0.6 3.03 ± 0.10+1 −1 4.5 ± 0.7+1 −1
+0.2 −0.44 ± 0.10−0.2 +0.3 −0.89 ± 0.16−0.3 +0.5 −1.6 ± 0.8−0.5
22.4 ± 0.9+8 −8
218.18 ± 0.01 224.08 ± 0.01
38.1 ± 1.3+14 −14 57 ± 9+18 −17
56
M. Andersson et al. / Nuclear Physics A 779 (2006) 47–62
Fig. 6. Experimental results (full drawn line) of the 4 He(d, 6 He)π + reaction at Ebeam = 218.2 MeV and fit (grey). Energy distribution of 6 He ions (top), vertical distribution (middle) and horizontal distribution (bottom). One horizontal strip was disconnected which explains the dip in the vertical distribution. Table 2 ∗ Comparison between parameters for the 4 He(d, 6 He)π + and the d(4 He, 6 Li3.56 )π 0 [9] reactions. The total kinetic energy in the c.m. system is denoted by Q. In the last column the cross section for the reaction leading to 6 He and a positive pion has been corrected for the Coulomb interaction in the final state (cf. text). The statistical uncertainties are indicated. Systematic uncertainties in the 6 He cross sections are of the order of ±35% . Q (MeV) Asymmetry a1 /a0 σ (nb) σCc (nb) 4 He(d, 6 He)π + reaction
0.61 1.20 5.02
−0.25 ± 0.05 −0.29 ± 0.05 −0.35 ± 0.15
22.4 ± 0.9 38.1 ± 1.3 57 ± 9
−0.11 ± 0.04 −0.43 ± 0.10
228 ± 6 141 ± 12
38 ± 2 55 ± 2 68 ± 10
∗
d(4 He, 6 Li3.56 )π 0 reaction 1.2 1.9
∗
Unlike in the d(4 He, 6 Li )π 0 reaction, the charged pion in the 4 He(d, 6 He)π + reaction experiences Coulomb interactions in the final state suppressing the cross section. In order to facilitate a comparison of the results for the two reactions the total cross section in the absence of Coulomb interaction in the final state, σCc , has been calculated and is presented in Table 2 together with dσ dσ dσ dσ (0◦ ) − dΩ (180◦ )]/[ dΩ (0◦ ) + dΩ (180◦ )] in the c.m. frame). The results for the asymmetry ([ dΩ Gamow factor suppressing the cross section in case of the charged pion has been calculated according to the method for describing the cross section difference between np → dπ 0 and pp → dπ + reactions described by Fäldt and Wilkin [24].
M. Andersson et al. / Nuclear Physics A 779 (2006) 47–62
57
From the results compiled in Table 2 we see that the differential cross section for the two analogue pionic fusion reactions is clearly anisotropic. In both reactions the preferred direction of emission of the pion is parallel to the momentum of 4 He in the initial state. 4.2. A simple model for the reaction For the discussion of our results we apply essentially the same model of the reaction as that developed in [9] (cf. Fig. 7) save for the introduction of Jacobi-type coordinates in order to guarantee translational invariance. Inspired by the treatment of Hiller and Pirner [26] of coherent pion production we picture the deuteron in the initial state as two nucleons surrounded by a cloud of virtual pions static in the rest frame of the deuteron. For positively charged pions the pseudoscalar pion current connects the deuteron ground state to the T = 1, I = 0 state of two neutrons in a relative L = 0 state. The virtual pions are brought on shell by scattering on the rapidly approaching alpha particle. In the intermediate state, following pion emission, the now quasi deuteron carries essentially all of its initial momentum irrespective of the direction of emission of the pion that only affects the momentum of the alpha particle. Further, we assume pion scattering on the alpha particle to be only weakly dependent on the final pion momentum within the narrow range of momentum transfer probed in the present experiment. In addition the pion mean free path is long compared to the dimensions of the final state nucleus at the present energies. The measured differential cross section, therefore, should be only moderately affected by interactions in the final state other than the Coulomb interaction. In this approximation the Coulomb corrected differential cross section, (dσ/dΩ)Cc , is proportional to a form factor times a phase space factor, the c.m. momentum of the pion. The form factor is proportional to the square of the overlap of the nuclear wave functions following pion emission and that of the final A = 6 bound state. Following pion emission, the c.m. motion of the quasi deuteron and the alpha particle are described by delta functions of momentum, Φd ∗ (p) = δ(p − p∗d ) and Φα (p) = δ(p + p∗d + pπ ) d respectively, where we take p∗d = mdm+m pd , pd and pπ referring to the c.m. momentum of the π deuteron in the initial state and the pion in the final state. The final A = 6, T = 1 state we describe by a cluster wave function composed of two halo nucleons in a relative s-state and an alpha-particle like core. Then in the Hartree approximation the internal wave functions of the core and the halo enter only into the over-all normalization of the cross section. The momentum dependence is fully determined by the wave function describing the relative motion of the core and halo systems. In the (translationally non-invariant) limit, where the neutron—core relative motion is described by single-particle 1p oscillator wave functions, the lowest L = 0 state of the halo is given by
Fig. 7. Model of pionic fusion reaction.
58
M. Andersson et al. / Nuclear Physics A 779 (2006) 47–62
√ 2 − 1 (k 2 +k 2 ) √ (α π )−3 (k1 · k2 )e 2α2 1 2 3
− 12 t 2 − 12 s 2 2 2 1 2 2 − 32 2αt = √ t − 3 − s − 3 (α α π) e e 2αs t s αs2 2 3 αt2
Ψ=
α2
(7)
where
√ α t = k1 + k2 , αs = √ , αt = α 2 2 and α is the single particle oscillator constant. This wave function corresponds to a linear combination of a 1s state of internal motion of the halo (momentum coordinate s) multiplied by a 2s wave function describing the motion of the halo relative to that of the stationary core (momentum coordinate t) and a 2s internal state of the halo multiplied by a 1s state of motion of the halo relative to the core. The latter component would give only a small contribution to the reaction cross section in our model since the internal state of the halo is in this case almost orthogonal to the internal wave function of the deuteron. In our model calculation we therefore take the wave function for the momentum conjugate to the coordinate describing the relative motion of the core and halo to be a 2s harmonic oscillator wave function q2 1 √ − 32 2 2 (8) q − 3 exp − 2 , ν(q) = √ αq π αq2 2αq 6 1 s = (k1 − k2 ), 2
where q = (2kh − kc )/3. Here kh and kc denote the total c.m. momentum of the halo and of the core respectively. The differential cross section is then calculated to be
dσ 1 pπ2 = Npπ λ2 exp − 2 + pd∗2 dΩ αq 9
1 ∗2 2 1 1 1 × 1 + pd pπ + − P0 (cos θc.m. ) 27 4λ2 2αq4 αq2 λ 1 1 1 − 2 P1 (cos θc.m. ) + pd∗ pπ 3 λ αq 1 1 2 ∗2 2 1 P2 (cos θc.m. ) + − + pd pπ 27 4λ2 2αq4 αq2 λ
(9)
to second order in Legendre polynomials, where θc.m. is the angle between the momentum of the pion, pπ , and the incoming deuteron, pd , in the c.m. frame, N is a normalization constant not determined by the model and 1 ∗2 1 2 3 2 (10) pd + pπ − αq . λ= 4 9 2 Our choice of a harmonic oscillator wave function for the relative motion of core and halo is convenient as it leaves only one parameter, the harmonic oscillator constant αq , to be determined (besides the normalization). However, expression (9), appropriately modified, could be applied also in the case of other wave functions approximated by a series of wave functions of the form (8) with different harmonic oscillator constants.
M. Andersson et al. / Nuclear Physics A 779 (2006) 47–62
59
4.3. Discussion In Figs. 8 and 9 the energy dependence of the cross section, Eq. (9), and of the asymmetry is compared to the experimental results for the reaction leading to the 6 He ground state and to the analogue state in 6 Li. Also included are the results of a calculation by Fäldt and Wilkin [25] based on amplitudes for the p(d, 3 He)π 0 reaction. Fitting our model (9) to the experimental results for the cross section and asymmetry for 6 He alone leads to the result αq = 0.55(6) fm−1 . This in turn translates to a point-proton radius of rp = 1.85(8) fm for the ground state of 6 He assuming an alpha-particle core, a result that is consistent with that of Wang et al. [7]. The experiment of Wang et al. and the present one probe the wave function at highly different momenta. This consistency may be interpreted as an indication of the adequacy of the chosen form, (8), for the wave function. Before drawing any firm conclusion, however, we believe a more thorough theoretical treatment would be needed. Already at this point, however, it is clear that the present data provide information on the highmomentum parts of the wave function describing the relative motion of the halo nucleons relative to that of the core. For comparison we have included in Figs. 8 and 9 the model predictions assuming for the final state a single particle harmonic-oscillator wave function characterized by the same oscillator constant as in the alpha-particle. In particular the asymmetries are systematically underpredicted with such a wave function. The systematic uncertainty in the distribution of the target gas discussed above affects the over-all normalization of the cross sections. In our model this corresponds to a change of the normalization constant, N , leaving the result for the harmonic oscillator constant, αq , and the prediction for the asymmetries unchanged. This could lead to a marginally improved agreement
Fig. 8. Present experimental results (circles) and from [9] (squares) for the (Coulomb corrected) total cross section as a function of the kinetic energy, Q, in the c.m. frame compared to results obtained with the model (9) described in the text for αq = 0.55 fm−1 (full line), 0.49 fm−1 (long dash) and 0.61 fm−1 (short dash) respectively. Also shown are the ∗ results for the d(4 He, 6 Li )π 0 reaction calculated by Fäldt and Wilkin [25] (open triangles). For comparison we have included also the result predicted assuming for 6 He a single particle harmonic-oscillator wave function with the same oscillator constant as for the alpha particle (thin line).
60
M. Andersson et al. / Nuclear Physics A 779 (2006) 47–62
Fig. 9. Present experimental results (circles) and from [9] (squares) for the asymmetry as a function of the kinetic energy, Q, in the c.m. frame compared to results obtained with the model (9) described in the text for αq = 0.55 fm−1 (full ∗ line), 0.49 fm−1 (long dash) and 0.61 fm−1 (short dash) respectively. Also shown are the results for the d(4 He, 6 Li )π 0 reaction calculated by Fäldt and Wilkin [25] (open triangles). For comparison we have included also the result predicted assuming for 6 He a single particle harmonic-oscillator wave function with the same oscillator constant as for the alpha particle (thin line).
with the measured cross section for the reaction leading to the analogue state in 6 Li [9]. The prediction, however, would still fall far below the cross section measured for the production of 6 Li 1.1 MeV above the threshold for the d(4 He, 6 Li∗ )π 0 reaction. The large cross section in conjunction with the anomalously small asymmetry found in that same measurement could be interpreted as indicating a second source of 6 Li ions with (or mimicking) a more symmetric distribution in the c.m. frame. However, we have not been able to identify any candidate reaction for such a second source. The clear identification of 6 Li ions left no room for any substantial background due to other ions. The situation regarding the measured results for the reaction to the analogue state in 6 Li remaining unchanged we note, however, that including in our fit the results from [9] measured 1.9 MeV above threshold leads to a marginal 5 per cent (0.4σ ) decrease of the harmonic oscillator constant, αq . 5. Conclusions We have measured the differential cross section for the 4 He(d, 6 He)π + reaction at three centre-of-mass energies, 0.61, 1.20 and 5.02 MeV above the absolute threshold. The result for the total cross section is 22.4(9), 38.1(13) and 57(9) nb respectively with a common systematic uncertainty of ±35%. The differential cross section is clearly anisotropic at all three energies. The pion is preferentially emitted parallel to the momentum of the 4 He particle in the initial state. By means of a simple model for the reaction the results are tied, in a semi-quantitative way, to earlier results for the analogue reaction to the 3.56 MeV state in 6 Li. In this picture the differential cross section depends sensitively on the wave function describing the relative motion of the core
M. Andersson et al. / Nuclear Physics A 779 (2006) 47–62
61
and the centre of mass of the halo. Assuming an alpha-particle core our result for this wave function is consistent with a recent and precise measurement of the charge radius of 6 He. In order to provide more data on the T = 1 six-nucleon state, in particular the importance of triton–triton or triton–3 He clustering we have made an experiment studying the 3 He(3 He, 6 Li∗ )π + reaction close to the absolute threshold. The data from this experiment are in the process of analysis. Acknowledgements We are deeply indebted to the operating crew and the technicians at The Svedberg Laboratory. Their contribution to the successful completion of the experiment is gratefully acknowledged. This work was supported in part by the Swedish Science Research Council (VR). References [1] I. Tanihata, H. Hamagaki, O. Hashimoto, S. Nagamiya, Y. Shida, N. Yoshikawa, O. Yamakawa, K. Sugimoto, T. Kobayashi, D.E. Greiner, N. Takahashi, Y. Nojiri, Phys. Lett. B 160 (1985) 380. [2] I. Tanihata, H. Hamagaki, O. Hashimoto, Y. Shida, N. Yoshikawa, K. Sugimoto, O. Yamakawa, T. Kobayashi, N. Takahashi, Phys. Rev. Lett. 55 (1985) 2676. [3] J.A. Tostevin, J.S. Al-Khalili, Nucl. Phys. A 616 (1997) 418c. [4] G.D. Alkhazov, M.N. Andronenko, A.V. Dobrovolsky, P. Egelhof, G.E. Gavrilov, H. Geissel, H. Irnich, A.V. Khanzadeev, G.A. Korolev, A.A. Lobodenko, G. Münzenberg, M. Mutterer, S.R. Neumaier, F. Nickel, W. Schwab, D.M. Seliverstov, T. Suzuki, J.P. Theobald, N.A. Timofeev, A.A. Vorobyov, V.I. Yatsoura, Phys. Rev. Lett. 78 (1997) 2313. [5] S.R. Neumaier, G.D. Alkhazov, M.N. Andronenko, A.V. Dobrovolsky, P. Egelhof, G.E. Gavrilov, H. Geissel, H. Irnich, A.V. Khanzadeev, G.A. Korolev, A.A. Lobodenko, G. Münzenberg, M. Mutterer, W. Schwab, D.M. Seliverstov, T. Suzuki, N.A. Timofeev, A.A. Vorobyov, V.I. Yatsoura, Nucl. Phys. A 712 (2002) 247. [6] G.D. Alkhazov, A.V. Dobrovolsky, P. Egelhof, H. Geissel, H. Irnich, A.V. Khanzadeev, G.A. Korolev, A.A. Lobodenko, G. Münzenberg, M. Mutterer, S.R. Neumaier, W. Schwab, D.M. Seliverstov, T. Suzuki, A.A. Vorobyov, Nucl. Phys. A 712 (2002) 269. [7] L.-B. Wang, P. Mueller, K. Bailey, G.W.F. Drake, J.P. Greene, D. Henderson, R.J. Holt, R.V.F. Janssens, C.L. Jiang, Z.-T. Lu, T.P. O’Connor, R.C. Pardo, K.E. Rehm, J.P. Schiffer, X.D. Tang, Phys. Rev. Lett. 93 (2004) 142501. [8] B.V. Danilin, S.N. Ershov, J.S. Vaagen, Phys. Rev. C 71 (2005) 057301. [9] M. Andersson, Chr. Bargholtz, K. Fransson, L. Holmberg, K. Lindh, L. Mårtensson, I. Sitnikova, P.-E. Tegnér, P. Thörngren Engblom, G. Weiss, K. Wilhelmsen Rolander, Phys. Lett. B 481 (2000) 165. [10] J.-F. Germond, C. Wilkin, Phys. Lett. B 106 (1981) 449. [11] K. Klingenbeck, M. Dillig, M. Huber, Phys. Rev. Lett. 47 (1981) 1654. [12] M. Huber, K. Klingenbeck, R. Hupke, Nucl. Phys. A 396 (1983) 191c. [13] J.-F. Germond, C. Wilkin, J. Phys. G 10 (1984) 745. [14] T. Kajino, H. Toki, K. Kubo, Phys. Rev. C 35 (1987) 1370. [15] Harzheim, M.G. Huber, B.C. Metsch, Z. Phys. A 340 (1991) 399. [16] A. Volya, S. Pratt, V. Zelevinsky, Phys. Rev. C 59 (1999) 305. [17] M. Andersson, Chr. Bargholtz, Kj. Fransson, E. Fumero, L. Gerén, L. Holmberg, K. Lindh, L. Mårtensson, I. Sitnikova, P.-E. Tegnér, G. Weiss, K. Wilhelmsen Rolander, Phys. Scr. T 104 (2003) 96; M. Andersson, Chr. Bargholtz, Kj. Fransson, E. Fumero, L. Gerén, L. Holmberg, K. Lindberg, K. Lindh, L. Mårtensson, I. Sitnikova, P.-E. Tegnér, G. Weiss, K. Wilhelmsen Rolander, in: Proceedings of the International Workshop MESON 2002, 24–28 May 2002, Cracow, Poland, ISBN 981-238-160-0, World Scientific, Singapore, 2003, p. 481. [18] W. Schott, W. Wagner, P. Kienle, R. Pollock, R. Bent, M. Fatyga, J. Kehayias, M. Green, K. Rehm, Phys. Rev. C 34 (1986) 1406. [19] J. Homolka, W. Schott, W. Wagner, W. Wilhelm, R.D. Bent, M. Fatyga, R.E. Pollock, M. Saber, R.E. Segel, P. Kienle, Phys. Rev. C 38 (1988) 2686. [20] D. Horn, G.C. Ball, D.R. Bowman, W.G. Davies, D. Fox, A. Galindo-Uribarri, A.C. Hayes, G. Savard, L. Beaulieu, Y. Larochelle, C. St-Pierre, Phys. Rev. Lett. 77 (1996) 2408.
62
M. Andersson et al. / Nuclear Physics A 779 (2006) 47–62
[21] Chr. Bargholtz, K. Lindh, D. Protic, N. Ruus, P.-E. Tegnér, P. Thörngren Engblom, K. Wihelmsen Rolander, Nucl. Instrum. Methods A 390 (1997) 160. [22] T. Sundqvist, J. Nyberg, in: A. Ingemarsson (Ed.), TSL Progress Report, 1996–1997, p. 21, http://www3.tsl.uu. se/~nyberg/tsl/svedaq/. [23] V. Ziemann, Stanford Linear Accelerator Center Publication SLAC-PUB-5962 (1992). [24] G. Fäldt, C. Wilkin, Phys. Rev. C 56 (1997) 2067. [25] G. Fäldt, C. Wilkin, Phys. Lett. B 486 (2000) 67. [26] B. Hiller, H.J. Pirner, Phys. Lett. B 109 (1982) 338.
Pionic fusion study of the 6He halo M. Andersson a , Chr. Bargholtz a , Kj. Fransson a , E. Fumero a , L. Gerén a,∗ , L. Holmberg a , K. Lindh a , L. Mårtensson a , I. Sitnikova a , P.-E. Tegnér a , G. Weiss a , K. Wilhelmsen b a Department of Physics, Stockholm University, SE-106 91 Stockholm, Sweden b Department of Physics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden
Received 20 September 2005; received in revised form 7 September 2006; accepted 7 September 2006 Available online 25 September 2006
Abstract The halo nucleus 6 He has been studied in a pionic fusion experiment at the CELSIUS ring in Uppsala. The aim of the experiment was to investigate, in particular, the high-momentum part of the halo wave function by measuring the differential cross section for the 4 He(d, 6 He)π + reaction 0.6, 1.2 and 5.0 MeV above threshold in the centre-of-mass frame. The 6 He ions were detected in a E − E solid-state detector telescope inserted into the ring vacuum. The result for the total cross section is, respectively, 22(1), 38(1) and 57(9) nb with a common systematic uncertainty of ±35%. The differential cross section is clearly dσ (0◦ ) − dσ (180◦ )]/[ dσ (0◦ ) + dσ (180◦ )] = −0.25(5), −0.29(5) and −0.35(15) at the anisotropic: [ dΩ dΩ dΩ dΩ same three energies. The preferred direction of emission for the pion is parallel to the momentum of the 4 He ion in the initial state. In terms of a simple model for the reaction the results depend sensitively on the state of motion of the centre of mass of the halo relative to that of the core. © 2006 Elsevier B.V. All rights reserved. PACS: 25.10.+s; 25.45.-z; 27.20.+n; 21.60.Gx Keywords: N UCLEAR REACTIONS d + 4 He, E = 217.3, 218.2, 224.1 MeV; measured particle spectra, σ , σ (θ), anisotropies. 6 He deduced halo features
* Corresponding author.
E-mail address: [email protected] (L. Gerén). 0375-9474/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2006.09.002
48
M. Andersson et al. / Nuclear Physics A 779 (2006) 47–62
1. Introduction The T = 1, I = 0 ground state of 6 He is the lightest two-neutron halo state. The two valence neutrons are bound by a little less than one MeV and their wave function extends far beyond that of the four tightly bound nucleons of the core. The point-nucleon matter radius was first derived by Tanihata et al. [1,2] to be rm = 2.73(4) fm from measurements of the total interaction cross section in collisions with Be, C and Al. These same data have been repeatedly analysed to yield significantly differing results for the matter radius depending on the method of analysis. In the latest of these analyses known to us Tostevin and Al-Khalili [3] arrived at rm = 2.54(4) fm. Later Alkhazov et al. and Neumaier et al. [4–6] made a thorough investigation of matter distributions describing 6 He and 8 He based on measurements of proton elastic scattering in inverse kinematics. At a beam energy of 0.7 GeV per nucleon the differential cross section, dσ /dt, was measured for |t| = 0.05 (GeV/c)2 . Their final result for 6 He, rm = 2.45(10) fm, included a tail to the density distribution taken from theory. The authors conclude from their data that the 6 He nucleus may, to first approximation, be treated as a three-body system consisting of an inert alpha particle and two valence neutrons forming an extended halo. Recently Wang et al. published the results of a laser-spectroscopic measurement of the 6 He charge radius [7], determining the point-proton charge radius to be, rp = 1.912(18) fm. This high-precision determination of the charge radius provides a crucial test of any theoretical description of the 6 He nucleus. In a three-body description, although insensitive to the relative distance between the halo neutrons, the charge radius provides a measure of the displacement of the alpha-particle core relative to the nuclear centre of mass [8]. Mass and charge radii provide information primarily regarding the dominant low-momentum parts of the nuclear wave function. In order to probe the high-momentum components of the ∗ halo wave function the differential cross section of the d(4 He, 6 Li )π 0 reaction to the isobaric analogue state of the 6 He ground state was measured [9]. At beam energies of 417.95 and 420.30 MeV the pion was emitted with a kinetic energy of 1.2 and 1.9 MeV in the centre-of-mass (c.m.) system respectively. In the pionic fusion process essentially all of the kinetic energy available in the entrance channel is transferred to the meson field. It is to be expected that such a highly coherent process is particularly sensitive to correlations among the nucleons in the initial and final state and theoretical treatments in the literature [10–16] stress the sensitivity to the detailed structure of the nuclei involved. Kajino et al. [14] obtained an order of magnitude increase ∗ in the cross section for the 4 He(3 He, π + )7 Li reaction using a correlated cluster wave function for 7 Li as compared to the result using a shell model wave function without correlations. Here we report the results of measurements of the differential cross section at three energies close to threshold for the pionic fusion of deuterons and 4 He, forming 6 He in its ground state and a positively charged pion. A preliminary account of the experiment has been reported earlier [17]. The details of the experiment are described in Section 2 followed, in Section 3 by an account of data reduction and analysis. The results are presented and discussed in the light of simple theoretical considerations in Section 4. Finally, in Section 5, our main conclusions are summarised.
M. Andersson et al. / Nuclear Physics A 779 (2006) 47–62
49
2. Experiment The 4 He(d, 6 He)π + reaction was studied in the CELSIUS accelerator and storage ring at The Svedberg Laboratory in Uppsala. Simultaneously 3 He-ions from the 4 He(d, 3 He)t reaction were detected for monitoring purposes. A storage ring with a cooled ion beam and an internal target provides ideal conditions for the study of reactions very close to threshold. Low-energy pions are difficult to detect and so in pionic fusion close to threshold it is advantageous to detect the fused nuclear system instead [18–20]. Following pionic fusion reactions close to threshold the heavy reaction products are emitted in a narrow cone around zero degrees and with essentially all of the beam momentum. Hence, in a storage ring, they stay close to the circulating beam and are difficult to detect. Given that the charge of the recoiling nuclei is different to that of the beam particles the recoils may be spatially separated from the beam in a magnetic field and then detected. 2.1. Apparatus An electron cooled deuteron beam was used in combination with a windowless 4 He gas target in the fourth straight section of CELSIUS. The heavy reaction products were detected in the zero-degree spectrometer [21] 6.1 m away from the normal target position at the centre of the straight section (cf. Fig. 1). In this experiment the cluster-jet target station was operated with 4 He providing a windowless, however spread out, gas target with an integrated thickness of approximately 6 × 1013 atoms cm−2 . In normal operation, with clustering gasses, the target is concentrated to within approximately one centimeter along the beam at the centre of the scattering chamber of the cluster-jet target station. In the present case, the extension along the beam was of the order of meters (cf. Section 3.1). The quadrupole and dipole magnets in the fourth quadrant of the ring focused and deflected the heavy reaction products onto a solid-state detector telescope inside the ring vacuum in the second dipole magnet 1.5 m into the bend following the gas target (cf. Fig. 1). The detector telescope in this experiment comprised a 1.0 mm thick silicon transmission (E) detector and a 1.7 mm thick stopping (E) detector made of high-purity germanium. The stopping detector was position sensitive with contacts divided into 18 horizontal strips of 2 mm width on one side and 66 vertical, 1 mm wide, strips on the other. The dead layers of the silicon detector corresponded to less than 1 µm Si + 0.3 µm Al (specified by the manufacturer). The contacts on the germanium detector are made by ion implantation creating a dead layer of less than 0.5 µm germanium equivalent. The detectors were cooled to liquid nitrogen temperature by circulating
Fig. 1. Schematic drawing of the experimental apparatus (not to scale) showing the zero-degree spectrometer and the luminosity monitor telescope aiming at the normal target position.
50
M. Andersson et al. / Nuclear Physics A 779 (2006) 47–62
liquid nitrogen through the cold head on which the detectors were mounted. For an event to be accepted a coincidence was required between the two detectors. Although the solid angle of the zero-degree spectrometer for ions emanating from the normal target position is not more than 0.7 msr in the present experiment the effective acceptance for 6 He ions from the 4 He(d, 6 He)π + reaction varied from approximately 56% at the lowest energy to 7% at the highest. ∗ In the measurement of the differential cross section of the d(4 He, 6 Li )π 0 reaction [9] there were difficulties associated with the determination of the luminosity which in turn rendered the determination of the cross section (although not the asymmetry) somewhat uncertain. In order to avoid this difficulty a scintillator detector telescope comprising two 2 mm thick, 40 × 40 mm2 , plastic detectors and one larger, 100 mm thick, conically shaped BGO detector was placed outside of the CELSIUS ring at an angle of 50◦ from the direction of the beam and pointing towards the normal target position at the centre of the scattering chamber of the cluster-jet target station. An accepted event required a triple coincidence between the two plastic scintillators, placed 440 mm apart, and the BGO detector. By means of this so-called monitor telescope a direct measure of the slightly varying luminosity was provided. The data acquisition system used in the experiment was based on the SVEDAQ [22] system with NIM, CAMAC and VME electronics. The signals from the silicon detector and from the horizontal contacts of the germanium detector were fed to shaping amplifiers and then to peaksensing analogue-to-digital converters as were the signals from the three detectors of the monitor telescope. The preamplifier signals from the vertical contacts on the germanium detector were shaped and fed to discriminators the output of which were recorded in the SVEDAQ system. The leakage current of the solid state detectors was monitored continuously and stored as was information regarding the detector position and a time stamp.
2.2. Measurements
Measurements were made at four different beam energies, Ebeam = 214.6, 217.3, 218.2, and 224.1 MeV. The lowest beam energy lies below the threshold for the 4 He(d, 6 He)π + reaction and was included in order to provide information concerning the background. The three higher energies correspond to c.m. energies, Q, 0.6, 1.2, and 5.0 MeV above threshold. CELSIUS was operated in 900 s cycles and the detector telescope was moved into measuring position 40 s after injection, when the beam had been accelerated to its final energy and then cooled. After 800 s the telescope was removed to the parking position before the beam was dumped. For most of the measurements the detector telescope was positioned with its centre 170 mm away from the circulating beam. In order to determine the density of the target gas along the straight section the distribution of 3 He ions in the detector plane from the 4 He(d, 3 He)t reaction was measured for distances of 113 to 237 mm from the circulating beam. Energy calibration of both detectors was done by means of alpha particles from 241 Am and 232 U sources mounted inside the vacuum chamber. The energy resolution (FWHM) for alpha particles at 8 MeV was measured to 350 keV for the silicon detector and to between 300 and 400 keV for the strips of the position sensitive germanium detector (these values include effects of dead layers and oblique incidence). For 73 MeV 6 He ions the energy resolution of the telescope was determined to be approximately 0.9 MeV (below).
M. Andersson et al. / Nuclear Physics A 779 (2006) 47–62
51
3. Analysis Coincident spectra from the silicon and germanium detectors were used in E − E technique in order to identify particles resulting from d + 4 He reactions. The stopping power of the thin silicon detector depends on the charge-to-mass ratio and the energy of the ions. In Fig. 2 ridges are seen corresponding to protons, deuterons, tritons (weakly), 3 He, 4 He, and 6 He. The concave part of the ridges correspond to ions stopped in the germanium detector whereas particles passing through both detectors, depositing less energy with increasing velocity, give rise to the convex ridges. The 3 He ions from the 4 He(d, 3 He)t reaction indicated in Fig. 2 have an energy of approximately 200 MeV and pass through both detectors. 3.1. Determination of target density The 4 He density distribution along the target straight section could be determined from the distribution in the plane of detection of 3 He ions from the 4 He(d, 3 He)t reaction. The acceptance decreases rapidly with emission angle in the c.m. system so that at an angle of 5◦ it is a tenth of its maximum value at 0◦ . These ions, thus, form an almost mono-energetic distribution appearing as a peak in E − E spectra (cf. Fig. 2). Their distribution in the plane of detection depends strongly on the extension of the target along the straight section. This is illustrated in Fig. 3. These experimental data were collected during a total of more than 40 hours with the detector telescope in 4 different positions. Simulated data were produced using a Monte Carlo code with ray tracing that has been developed for the zero-degree spectrometer. The code uses the actual geometry of CELSIUS and measured field maps. The angular distribution of 3 He ions was assumed constant over the narrow range of emission angles and their point of emission was sampled according to a target density
Fig. 2. E − E spectrum for Ebeam = 217.3 MeV where E(E) is the energy deposit in the silicon (germanium) detector (cf. text).
52
M. Andersson et al. / Nuclear Physics A 779 (2006) 47–62
Fig. 3. Horizontal distribution of 3 He ions in the detector plane. Results of simulations assuming a localized target at the normal target position (top) and a distributed target according to Eq. (1) for b1 = 0.69 m−1 and b2 = 0.186 m−1 (bottom, grey) and experimental data (bottom, line).
ρ(x) determined by three parameters, the central density ρ0 and two different slope parameters, b1 and b2 , in the backward and forward direction respectively: 1 − b1 |x| , − b11 < x < 0, (1) ρ(x) = ρ0 1 − b2 |x| , 0 < x < b12 , where x is the distance along the beam from the normal target position. The values of the slope parameters were obtained in a maximum likelihood fit of simulated to experimental twodimensional position distributions. The horisontal position distribution corresponding to the best fit, obtained for b1 = 0.69(7) m−1 and b2 = 0.186(1) m−1 , is shown in Fig. 3. From Eq. (1) the integrated target thickness, d, is obtained as 1/b 2
d= −1/b1
1 ρ(x) dx = ρ0 b1−1 + b2 −1 . 2
(2)
The target thickness, d, was determined in a measurement of the stopping power of the target gas. For this purpose the electron cooler was switched off and the change of orbital frequency was determined from Schottky spectra. The result for the target thickness, d = 5.9(1) × 1013 cm−2 , fixes the value of the density at the centre of the target chamber, ρ0 = 1.7(2) × 1011 cm−3 . 3.2. Luminosity Given the density distribution in (1) we may define a luminosity per unit length l(x) = Ibeam ρ(x)
(3)
such that the number of reactions per second per unit length of target is given by r(x) = σ l(x).
(4)
M. Andersson et al. / Nuclear Physics A 779 (2006) 47–62
53
Fig. 4. The acceptance for 3 He ions from the 4 He(d, 3 He)t reaction (assuming isotropy in the c.m.) as a function of where along the beam line the reaction occurs. The data are simulated assuming the detector telescope to be in the main measuring position with its centre 170 mm away from the circulating beam.
In these expressions Ibeam denotes the beam current (particles per unit time) and σ the total cross section. Immediately preceding the measurement of the target thickness the beam current was measured while at the same time registering the rate of events in the monitor telescope, nmon . Since the monitor telescope accepts particles due to reactions in the vicinity of x = 0 only, a constant of proportionality, C = Ibeam ρ0 /nmon , could be determined in terms of which the luminosity per unit length is given by 1 − b1 |x| , − b11 < x < 0, l(x) = Cnmon (5) 1 − b2 |x| , 0 < x < b12 . The possible variation of the slope parameter b2 was monitored during the entire measurement by the number of 3 He-ions from the 4 He(d, 3 He)t reaction reaching the detector. The acceptance for these ions in the main measuring position is shown in Fig. 4. The variation of the average value of the slope parameter was smaller than 10% for all beam energies. For each beam energy the integrated luminosity was determined from (5) by substituting for the event rate the total number of events registered in the monitor telescope at the corresponding energy. 3.3. Selection of events due to 6 He The reaction 4 He(d, 6 He)π + was studied at three different beam energies, Ebeam = 217.3, 218.2, and 224.1 MeV. Events corresponding to production of 6 He ions are easily identified in E − E spectra like the one shown in Fig. 2. The separation from the 4 He distribution is excellent. By selecting events in a narrow gate around the expected 6 He distribution in the E − E spectrum samples with only a small background contribution were obtained. Events were approved of if the signal from the position-sensitive detector emanated from one or two neighbouring horizontal strips and from one to three neighbouring vertical strips. This division of charge among neighbouring strips is at least partly due to the strong magnetic field of the CELSIUS bending magnets perpendicular to the electric field of the detector. The loss of efficiency caused by multiple hits was less than 1%. An upper limit to the efficiency loss due to nuclear reactions in the detectors, estimated from [9], is 1.4%. Energy spectra of events selected in this way at the four different beam energies are shown in Fig. 5. At each beam energy above the threshold for the 4 He(d, 6 He)π + reaction the spectra exhibit two peaks corresponding to 6 He ions emitted at 0 and 180 degrees in the c.m. system.
54
M. Andersson et al. / Nuclear Physics A 779 (2006) 47–62
Fig. 5. Kinetic energy spectra of selected events, measured at four different beam energies (from top to bottom) 214.6, 217.3, 218.2, and 224.1 MeV respectively. The spectrum for the highest beam energy is the sum of spectra taken at two different detector positions (130 and 175 mm from the beam). The integrated luminosities during the measurement of these spectra were approximately 1.9, 8.0, 7.3 and 3.7 in units of 1034 cm−2 respectively.
The energy difference is a sensitive measure of the beam energy. From the spectrum at the lowest beam energy, which is below threshold, the magnitude of the background can be estimated. This background is mainly due to misidentified events, i.e. events due to nuclear reactions or due to sum-up from particles hitting neighbouring strips. 3.4. Fitting to experimental data Simulations of 4 He(d, 6 He)π + reactions were done again using the MC code developed for the zero-degree spectrometer. The reaction points along the target straight section were sampled according to the luminosity distribution (5) determined from the number of monitor events and 3 He ions detected in parallel to the detection of 6 He. The differential cross section, at each beam energy, of the 4 He(d, 6 He)π + reaction was expanded in Legendre polynomials dσ = ak Pk (cos θc.m. ), (6) dΩ c.m. k0
M. Andersson et al. / Nuclear Physics A 779 (2006) 47–62
55
where θc.m. denotes the c.m. angle between the pion and deuteron momenta. Each simulated event was characterized by the energy and position of detection of the 6 He ions reaching the detector, the same parameters as those recorded for the experimental data. Results from the MC simulations were fitted simultaneously to the experimental data for all three beam energies above threshold. A maximum-likelihood algorithm was used to determine the coefficients in Eq. (6). Additional parameters determined in the fit were the magnitude of an assumed flat background of misidentified events (for each beam energy), one independent beam energy and the energy resolution of the detector telescope. The steps in beam energy between the three measurements, 0.90 and 5.90 MeV respectively, were assumed precisely determined by the change in high-voltage of the electron cooler. 4. Results and discussion 4.1. Experimental results The result of the fit is summarised in Table 1, assuming ak = 0 for k 2 in Eq. (6). In Fig. 6, as an illustration of the quality of the fit, the simulated distribution for Ebeam = 218.2 MeV is compared to the measured one. The systematic uncertainty in the Legendre coefficients in Table 1 are due to two sources. One is the uncertainty in the beam current calibration, which we estimate to approximately 5%. The second, and larger, is the uncertainty in the shape of the density distribution of the target gas, i.e. where along the beam line the gas of known thickness d is located. The fitted triangular distribution, Eq. (1), gives a good description of the results for 3 He. In order to assess the systematic uncertainty in the results for 6 He we have investigated two extreme cases for the gas distribution. In the first case the target gas was assumed to be confined to the target chamber with no gas in the beam pipes outside the chamber. This provided a lower limit to the Legendre coefficients since the acceptance for the reaction leading to 6 He was largest in this region. In the second case we used the vacuum calculation code VAKTRAK [23] in order to estimate the limit to the extension of the target gas in the backward direction where the acceptance was low, thus obtaining an upper limit to the Legendre coefficients. This calculation was done for the actual geometry of the target straight section and readings of the vacuum gauges and assuming the ion pumps along the straight section to have been all switched off. The total systematic uncertainty in the Legendre coefficients, from both sources, amounts to approximately 35%. The uncertainty, however, amounts to a common multiplicative factor, leaving the asymmetry and the energy dependence of the cross section unaffected. The results in Table 1 differ from the preliminary results reported previously [17] due to erroneous assumptions in the early simulations. Table 1 Results for beam energy, Legendre coefficients and total cross section obtained in the fit to experimental data. The energy resolution (FWHM) of the detector telescope was determined to be 0.89(2) MeV Results from the fit Beam energy (MeV)
a0 (nb/sr)
a1 (nb/sr)
σ (nb)
217.28 ± 0.01
+0.6 1.78 ± 0.07−0.6 3.03 ± 0.10+1 −1 4.5 ± 0.7+1 −1
+0.2 −0.44 ± 0.10−0.2 +0.3 −0.89 ± 0.16−0.3 +0.5 −1.6 ± 0.8−0.5
22.4 ± 0.9+8 −8
218.18 ± 0.01 224.08 ± 0.01
38.1 ± 1.3+14 −14 57 ± 9+18 −17
56
M. Andersson et al. / Nuclear Physics A 779 (2006) 47–62
Fig. 6. Experimental results (full drawn line) of the 4 He(d, 6 He)π + reaction at Ebeam = 218.2 MeV and fit (grey). Energy distribution of 6 He ions (top), vertical distribution (middle) and horizontal distribution (bottom). One horizontal strip was disconnected which explains the dip in the vertical distribution. Table 2 ∗ Comparison between parameters for the 4 He(d, 6 He)π + and the d(4 He, 6 Li3.56 )π 0 [9] reactions. The total kinetic energy in the c.m. system is denoted by Q. In the last column the cross section for the reaction leading to 6 He and a positive pion has been corrected for the Coulomb interaction in the final state (cf. text). The statistical uncertainties are indicated. Systematic uncertainties in the 6 He cross sections are of the order of ±35% . Q (MeV) Asymmetry a1 /a0 σ (nb) σCc (nb) 4 He(d, 6 He)π + reaction
0.61 1.20 5.02
−0.25 ± 0.05 −0.29 ± 0.05 −0.35 ± 0.15
22.4 ± 0.9 38.1 ± 1.3 57 ± 9
−0.11 ± 0.04 −0.43 ± 0.10
228 ± 6 141 ± 12
38 ± 2 55 ± 2 68 ± 10
∗
d(4 He, 6 Li3.56 )π 0 reaction 1.2 1.9
∗
Unlike in the d(4 He, 6 Li )π 0 reaction, the charged pion in the 4 He(d, 6 He)π + reaction experiences Coulomb interactions in the final state suppressing the cross section. In order to facilitate a comparison of the results for the two reactions the total cross section in the absence of Coulomb interaction in the final state, σCc , has been calculated and is presented in Table 2 together with dσ dσ dσ dσ (0◦ ) − dΩ (180◦ )]/[ dΩ (0◦ ) + dΩ (180◦ )] in the c.m. frame). The results for the asymmetry ([ dΩ Gamow factor suppressing the cross section in case of the charged pion has been calculated according to the method for describing the cross section difference between np → dπ 0 and pp → dπ + reactions described by Fäldt and Wilkin [24].
M. Andersson et al. / Nuclear Physics A 779 (2006) 47–62
57
From the results compiled in Table 2 we see that the differential cross section for the two analogue pionic fusion reactions is clearly anisotropic. In both reactions the preferred direction of emission of the pion is parallel to the momentum of 4 He in the initial state. 4.2. A simple model for the reaction For the discussion of our results we apply essentially the same model of the reaction as that developed in [9] (cf. Fig. 7) save for the introduction of Jacobi-type coordinates in order to guarantee translational invariance. Inspired by the treatment of Hiller and Pirner [26] of coherent pion production we picture the deuteron in the initial state as two nucleons surrounded by a cloud of virtual pions static in the rest frame of the deuteron. For positively charged pions the pseudoscalar pion current connects the deuteron ground state to the T = 1, I = 0 state of two neutrons in a relative L = 0 state. The virtual pions are brought on shell by scattering on the rapidly approaching alpha particle. In the intermediate state, following pion emission, the now quasi deuteron carries essentially all of its initial momentum irrespective of the direction of emission of the pion that only affects the momentum of the alpha particle. Further, we assume pion scattering on the alpha particle to be only weakly dependent on the final pion momentum within the narrow range of momentum transfer probed in the present experiment. In addition the pion mean free path is long compared to the dimensions of the final state nucleus at the present energies. The measured differential cross section, therefore, should be only moderately affected by interactions in the final state other than the Coulomb interaction. In this approximation the Coulomb corrected differential cross section, (dσ/dΩ)Cc , is proportional to a form factor times a phase space factor, the c.m. momentum of the pion. The form factor is proportional to the square of the overlap of the nuclear wave functions following pion emission and that of the final A = 6 bound state. Following pion emission, the c.m. motion of the quasi deuteron and the alpha particle are described by delta functions of momentum, Φd ∗ (p) = δ(p − p∗d ) and Φα (p) = δ(p + p∗d + pπ ) d respectively, where we take p∗d = mdm+m pd , pd and pπ referring to the c.m. momentum of the π deuteron in the initial state and the pion in the final state. The final A = 6, T = 1 state we describe by a cluster wave function composed of two halo nucleons in a relative s-state and an alpha-particle like core. Then in the Hartree approximation the internal wave functions of the core and the halo enter only into the over-all normalization of the cross section. The momentum dependence is fully determined by the wave function describing the relative motion of the core and halo systems. In the (translationally non-invariant) limit, where the neutron—core relative motion is described by single-particle 1p oscillator wave functions, the lowest L = 0 state of the halo is given by
Fig. 7. Model of pionic fusion reaction.
58
M. Andersson et al. / Nuclear Physics A 779 (2006) 47–62
√ 2 − 1 (k 2 +k 2 ) √ (α π )−3 (k1 · k2 )e 2α2 1 2 3
− 12 t 2 − 12 s 2 2 2 1 2 2 − 32 2αt = √ t − 3 − s − 3 (α α π) e e 2αs t s αs2 2 3 αt2
Ψ=
α2
(7)
where
√ α t = k1 + k2 , αs = √ , αt = α 2 2 and α is the single particle oscillator constant. This wave function corresponds to a linear combination of a 1s state of internal motion of the halo (momentum coordinate s) multiplied by a 2s wave function describing the motion of the halo relative to that of the stationary core (momentum coordinate t) and a 2s internal state of the halo multiplied by a 1s state of motion of the halo relative to the core. The latter component would give only a small contribution to the reaction cross section in our model since the internal state of the halo is in this case almost orthogonal to the internal wave function of the deuteron. In our model calculation we therefore take the wave function for the momentum conjugate to the coordinate describing the relative motion of the core and halo to be a 2s harmonic oscillator wave function q2 1 √ − 32 2 2 (8) q − 3 exp − 2 , ν(q) = √ αq π αq2 2αq 6 1 s = (k1 − k2 ), 2
where q = (2kh − kc )/3. Here kh and kc denote the total c.m. momentum of the halo and of the core respectively. The differential cross section is then calculated to be
dσ 1 pπ2 = Npπ λ2 exp − 2 + pd∗2 dΩ αq 9
1 ∗2 2 1 1 1 × 1 + pd pπ + − P0 (cos θc.m. ) 27 4λ2 2αq4 αq2 λ 1 1 1 − 2 P1 (cos θc.m. ) + pd∗ pπ 3 λ αq 1 1 2 ∗2 2 1 P2 (cos θc.m. ) + − + pd pπ 27 4λ2 2αq4 αq2 λ
(9)
to second order in Legendre polynomials, where θc.m. is the angle between the momentum of the pion, pπ , and the incoming deuteron, pd , in the c.m. frame, N is a normalization constant not determined by the model and 1 ∗2 1 2 3 2 (10) pd + pπ − αq . λ= 4 9 2 Our choice of a harmonic oscillator wave function for the relative motion of core and halo is convenient as it leaves only one parameter, the harmonic oscillator constant αq , to be determined (besides the normalization). However, expression (9), appropriately modified, could be applied also in the case of other wave functions approximated by a series of wave functions of the form (8) with different harmonic oscillator constants.
M. Andersson et al. / Nuclear Physics A 779 (2006) 47–62
59
4.3. Discussion In Figs. 8 and 9 the energy dependence of the cross section, Eq. (9), and of the asymmetry is compared to the experimental results for the reaction leading to the 6 He ground state and to the analogue state in 6 Li. Also included are the results of a calculation by Fäldt and Wilkin [25] based on amplitudes for the p(d, 3 He)π 0 reaction. Fitting our model (9) to the experimental results for the cross section and asymmetry for 6 He alone leads to the result αq = 0.55(6) fm−1 . This in turn translates to a point-proton radius of rp = 1.85(8) fm for the ground state of 6 He assuming an alpha-particle core, a result that is consistent with that of Wang et al. [7]. The experiment of Wang et al. and the present one probe the wave function at highly different momenta. This consistency may be interpreted as an indication of the adequacy of the chosen form, (8), for the wave function. Before drawing any firm conclusion, however, we believe a more thorough theoretical treatment would be needed. Already at this point, however, it is clear that the present data provide information on the highmomentum parts of the wave function describing the relative motion of the halo nucleons relative to that of the core. For comparison we have included in Figs. 8 and 9 the model predictions assuming for the final state a single particle harmonic-oscillator wave function characterized by the same oscillator constant as in the alpha-particle. In particular the asymmetries are systematically underpredicted with such a wave function. The systematic uncertainty in the distribution of the target gas discussed above affects the over-all normalization of the cross sections. In our model this corresponds to a change of the normalization constant, N , leaving the result for the harmonic oscillator constant, αq , and the prediction for the asymmetries unchanged. This could lead to a marginally improved agreement
Fig. 8. Present experimental results (circles) and from [9] (squares) for the (Coulomb corrected) total cross section as a function of the kinetic energy, Q, in the c.m. frame compared to results obtained with the model (9) described in the text for αq = 0.55 fm−1 (full line), 0.49 fm−1 (long dash) and 0.61 fm−1 (short dash) respectively. Also shown are the ∗ results for the d(4 He, 6 Li )π 0 reaction calculated by Fäldt and Wilkin [25] (open triangles). For comparison we have included also the result predicted assuming for 6 He a single particle harmonic-oscillator wave function with the same oscillator constant as for the alpha particle (thin line).
60
M. Andersson et al. / Nuclear Physics A 779 (2006) 47–62
Fig. 9. Present experimental results (circles) and from [9] (squares) for the asymmetry as a function of the kinetic energy, Q, in the c.m. frame compared to results obtained with the model (9) described in the text for αq = 0.55 fm−1 (full ∗ line), 0.49 fm−1 (long dash) and 0.61 fm−1 (short dash) respectively. Also shown are the results for the d(4 He, 6 Li )π 0 reaction calculated by Fäldt and Wilkin [25] (open triangles). For comparison we have included also the result predicted assuming for 6 He a single particle harmonic-oscillator wave function with the same oscillator constant as for the alpha particle (thin line).
with the measured cross section for the reaction leading to the analogue state in 6 Li [9]. The prediction, however, would still fall far below the cross section measured for the production of 6 Li 1.1 MeV above the threshold for the d(4 He, 6 Li∗ )π 0 reaction. The large cross section in conjunction with the anomalously small asymmetry found in that same measurement could be interpreted as indicating a second source of 6 Li ions with (or mimicking) a more symmetric distribution in the c.m. frame. However, we have not been able to identify any candidate reaction for such a second source. The clear identification of 6 Li ions left no room for any substantial background due to other ions. The situation regarding the measured results for the reaction to the analogue state in 6 Li remaining unchanged we note, however, that including in our fit the results from [9] measured 1.9 MeV above threshold leads to a marginal 5 per cent (0.4σ ) decrease of the harmonic oscillator constant, αq . 5. Conclusions We have measured the differential cross section for the 4 He(d, 6 He)π + reaction at three centre-of-mass energies, 0.61, 1.20 and 5.02 MeV above the absolute threshold. The result for the total cross section is 22.4(9), 38.1(13) and 57(9) nb respectively with a common systematic uncertainty of ±35%. The differential cross section is clearly anisotropic at all three energies. The pion is preferentially emitted parallel to the momentum of the 4 He particle in the initial state. By means of a simple model for the reaction the results are tied, in a semi-quantitative way, to earlier results for the analogue reaction to the 3.56 MeV state in 6 Li. In this picture the differential cross section depends sensitively on the wave function describing the relative motion of the core
M. Andersson et al. / Nuclear Physics A 779 (2006) 47–62
61
and the centre of mass of the halo. Assuming an alpha-particle core our result for this wave function is consistent with a recent and precise measurement of the charge radius of 6 He. In order to provide more data on the T = 1 six-nucleon state, in particular the importance of triton–triton or triton–3 He clustering we have made an experiment studying the 3 He(3 He, 6 Li∗ )π + reaction close to the absolute threshold. The data from this experiment are in the process of analysis. Acknowledgements We are deeply indebted to the operating crew and the technicians at The Svedberg Laboratory. Their contribution to the successful completion of the experiment is gratefully acknowledged. This work was supported in part by the Swedish Science Research Council (VR). References [1] I. Tanihata, H. Hamagaki, O. Hashimoto, S. Nagamiya, Y. Shida, N. Yoshikawa, O. Yamakawa, K. Sugimoto, T. Kobayashi, D.E. Greiner, N. Takahashi, Y. Nojiri, Phys. Lett. B 160 (1985) 380. [2] I. Tanihata, H. Hamagaki, O. Hashimoto, Y. Shida, N. Yoshikawa, K. Sugimoto, O. Yamakawa, T. Kobayashi, N. Takahashi, Phys. Rev. Lett. 55 (1985) 2676. [3] J.A. Tostevin, J.S. Al-Khalili, Nucl. Phys. A 616 (1997) 418c. [4] G.D. Alkhazov, M.N. Andronenko, A.V. Dobrovolsky, P. Egelhof, G.E. Gavrilov, H. Geissel, H. Irnich, A.V. Khanzadeev, G.A. Korolev, A.A. Lobodenko, G. Münzenberg, M. Mutterer, S.R. Neumaier, F. Nickel, W. Schwab, D.M. Seliverstov, T. Suzuki, J.P. Theobald, N.A. Timofeev, A.A. Vorobyov, V.I. Yatsoura, Phys. Rev. Lett. 78 (1997) 2313. [5] S.R. Neumaier, G.D. Alkhazov, M.N. Andronenko, A.V. Dobrovolsky, P. Egelhof, G.E. Gavrilov, H. Geissel, H. Irnich, A.V. Khanzadeev, G.A. Korolev, A.A. Lobodenko, G. Münzenberg, M. Mutterer, W. Schwab, D.M. Seliverstov, T. Suzuki, N.A. Timofeev, A.A. Vorobyov, V.I. Yatsoura, Nucl. Phys. A 712 (2002) 247. [6] G.D. Alkhazov, A.V. Dobrovolsky, P. Egelhof, H. Geissel, H. Irnich, A.V. Khanzadeev, G.A. Korolev, A.A. Lobodenko, G. Münzenberg, M. Mutterer, S.R. Neumaier, W. Schwab, D.M. Seliverstov, T. Suzuki, A.A. Vorobyov, Nucl. Phys. A 712 (2002) 269. [7] L.-B. Wang, P. Mueller, K. Bailey, G.W.F. Drake, J.P. Greene, D. Henderson, R.J. Holt, R.V.F. Janssens, C.L. Jiang, Z.-T. Lu, T.P. O’Connor, R.C. Pardo, K.E. Rehm, J.P. Schiffer, X.D. Tang, Phys. Rev. Lett. 93 (2004) 142501. [8] B.V. Danilin, S.N. Ershov, J.S. Vaagen, Phys. Rev. C 71 (2005) 057301. [9] M. Andersson, Chr. Bargholtz, K. Fransson, L. Holmberg, K. Lindh, L. Mårtensson, I. Sitnikova, P.-E. Tegnér, P. Thörngren Engblom, G. Weiss, K. Wilhelmsen Rolander, Phys. Lett. B 481 (2000) 165. [10] J.-F. Germond, C. Wilkin, Phys. Lett. B 106 (1981) 449. [11] K. Klingenbeck, M. Dillig, M. Huber, Phys. Rev. Lett. 47 (1981) 1654. [12] M. Huber, K. Klingenbeck, R. Hupke, Nucl. Phys. A 396 (1983) 191c. [13] J.-F. Germond, C. Wilkin, J. Phys. G 10 (1984) 745. [14] T. Kajino, H. Toki, K. Kubo, Phys. Rev. C 35 (1987) 1370. [15] Harzheim, M.G. Huber, B.C. Metsch, Z. Phys. A 340 (1991) 399. [16] A. Volya, S. Pratt, V. Zelevinsky, Phys. Rev. C 59 (1999) 305. [17] M. Andersson, Chr. Bargholtz, Kj. Fransson, E. Fumero, L. Gerén, L. Holmberg, K. Lindh, L. Mårtensson, I. Sitnikova, P.-E. Tegnér, G. Weiss, K. Wilhelmsen Rolander, Phys. Scr. T 104 (2003) 96; M. Andersson, Chr. Bargholtz, Kj. Fransson, E. Fumero, L. Gerén, L. Holmberg, K. Lindberg, K. Lindh, L. Mårtensson, I. Sitnikova, P.-E. Tegnér, G. Weiss, K. Wilhelmsen Rolander, in: Proceedings of the International Workshop MESON 2002, 24–28 May 2002, Cracow, Poland, ISBN 981-238-160-0, World Scientific, Singapore, 2003, p. 481. [18] W. Schott, W. Wagner, P. Kienle, R. Pollock, R. Bent, M. Fatyga, J. Kehayias, M. Green, K. Rehm, Phys. Rev. C 34 (1986) 1406. [19] J. Homolka, W. Schott, W. Wagner, W. Wilhelm, R.D. Bent, M. Fatyga, R.E. Pollock, M. Saber, R.E. Segel, P. Kienle, Phys. Rev. C 38 (1988) 2686. [20] D. Horn, G.C. Ball, D.R. Bowman, W.G. Davies, D. Fox, A. Galindo-Uribarri, A.C. Hayes, G. Savard, L. Beaulieu, Y. Larochelle, C. St-Pierre, Phys. Rev. Lett. 77 (1996) 2408.
62
M. Andersson et al. / Nuclear Physics A 779 (2006) 47–62
[21] Chr. Bargholtz, K. Lindh, D. Protic, N. Ruus, P.-E. Tegnér, P. Thörngren Engblom, K. Wihelmsen Rolander, Nucl. Instrum. Methods A 390 (1997) 160. [22] T. Sundqvist, J. Nyberg, in: A. Ingemarsson (Ed.), TSL Progress Report, 1996–1997, p. 21, http://www3.tsl.uu. se/~nyberg/tsl/svedaq/. [23] V. Ziemann, Stanford Linear Accelerator Center Publication SLAC-PUB-5962 (1992). [24] G. Fäldt, C. Wilkin, Phys. Rev. C 56 (1997) 2067. [25] G. Fäldt, C. Wilkin, Phys. Lett. B 486 (2000) 67. [26] B. Hiller, H.J. Pirner, Phys. Lett. B 109 (1982) 338.