Nuclear Physics A 714 (2003) 21–43

www.elsevier.com/locate/npe

Beta-decay properties of the neutron-rich94–99Krand142–147Xe isotopes

U.C. Bergmanna, C.Aa. Digetb, K. Riisagerb, L. Weissmana,∗,G. Auböckc, J. Cederkälla, L.M. Frailea,1, H.O.U. Fynboa,

H. Gausemeld, H. Jeppesenb, U. Köstera, K.-L. Kratze, P. Möllerf,T. Nilssona, B. Pfeiffere, H. Simona,2 , K. Van de Velg, J. Äystoa,

ISOLDE Collaboration

a ISOLDE, CERN, CH-1211 Genève 23, Switzerlandb Institut for Fysik og Astronomi, Aarhus Universitet, DK-8000 Aarhus C, Denmark

c Technical University Graz, A-8010 Graz, Austriad Kjemisk Institutt, Universitetet i Oslo, Postboks 1033, Blindern, N-0315 Oslo, Norway

e Institut für Kernchemie, Universität Mainz, D-55128 Mainz, Germanyf Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

g IKS, Katholieke Universiteit Leuven, B-3001 Leuven, Belgium

Received 14 May 2002; received in revised form 15 October 2002; accepted 18 October 2002

Abstract

Beta-decay half-lives and delayed-neutron emission probabilities of the neutron-rich noble-gasisotopes94–99Kr and 142–147Xe have been measured at the PSB-ISOLDE facility at CERN. Theresults are compared to QRPA shell-model predictions and are used in dynamic calculations ofr-process abundances of Kr and Xe isotopes. 2002 Elsevier Science B.V. All rights reserved.

PACS: 27.60.+j; 23.40.-s; 07.05.Kf; 26.30.+k

Keywords: Half-lives;Pn values; On-line separation; QRPA calculations; r-process nuclei;94−99Kr; 142−147Xe

* Corresponding author. Present address: NSCL, Michigan State University, East Lansing, MI 48824-1321,USA.

E-mail address: weissman@nscl.msu.edu (L. Weissman).1 On leave from Dpto. de Física Atómica, Molecular y Nuclear, Univ. Complutense, E-28040 Madrid, Spain.2 Present address: Institut für Kernphysik, Technische Universität, D-64289 Darmstadt, Germany.

0375-9474/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0375-9474(02)01352-0

22 U.C. Bergmann et al. / Nuclear Physics A 714 (2003) 21–43

1. Introduction

Fifty years ago, the first isotope separation on-line (ISOL) experiment was carried outat the Institute for Theoretical Physics in Copenhagen, Denmark [1–3]. In that pioneeringwork the isotopes89−91Kr were produced and extracted from a uranium containing targetinstalled at a cyclotron, and theirβ-decay half-lives were measured for the first time “on-line” after mass separation. Since then there has been surprisingly little progress in the in-vestigation of neutron-rich Kr and Xe nuclei. A number of short-lived Kr and Xe isotopeswere measured at various ISOL facilities [4–11], however, these studies were limited to91−94Kr and139−143Xe. The heaviest noble-gas isotopes known so far,95Kr and144,145Xe[12], were in fact obtained in earlier experiments utilizing emanation, chemical separationand gas-jet techniques [13–17]. These indirect half-life determinations generally sufferedfrom systematic errors and were not always as reliable as the measurements with mass sep-aration. In contrast to the noble gases, the results from on-line measurements of extendedisotope sequences of other elements, such as the alkaline, are much more detailed. For ex-ample, the heaviest isotopes of Rb and Cs studied so far are102Rb and150Cs [12,18,19],8 and 7 mass units beyond the corresponding Kr and Xe nuclides. One reason for this situa-tion may be the more difficult analysis of half-lives and rather weak delayed-neutron emis-sion probabilities (Pn values) of the Kr and Xe nuclides in the presence of the much moreintense neutron branching ratios of their alkali daughters. Another complication in studiesof noble gas isotopes is associated with their fast diffusion from the implantation material.

On the other hand, the Kr and Xe isotopic chains are of considerable interest for theunderstanding of the development of nuclear structure in theN 56 andN 90 shape-transitional regions, respectively. Moreover, theN 58 andN 90 even-neutron isotopesof Kr and Xe are expected to be so-called “waiting-point” nuclei [20] in rapid neutron-capture (r-process) nucleosynthesis processes.

Noble gases are difficult to ionize, usually requiring chemically non-selective plasma-discharge ion sources that, for a selected mass, ionize the whole isobaric chain. However,the combination with a water-cooled transfer line between target and ion source adds thenecessary element selectivity by condensing and depositing all less volatile reaction prod-ucts. This rather simple target-ion-source system is ideal for production of pure radioactivebeams of noble-gas elements with high ionization efficiencies [21]. The high fission yieldsof Kr and Xe isotopes, their high ionization efficiency and isobaric selectivity, togetherwith the simple operation of this type of plasma ion source has made these pure noble-gas radioactive beams very attractive for use in various target tests. In this publication wereport on new results for heavy Kr and Xe nuclei obtained at the ISOLDE separator.

2. Experiments and results

2.1. Experimental techniques

In two different experiments, referred to as A and B, neutron-rich Kr and Xe isotopeswere produced in fission reactions induced by a pulsed beam of 1 GeV or 1.4 GeVprotons (3×1013 protons/pulse) from the CERN proton synchrotron booster (PSB) facility

U.C. Bergmann et al. / Nuclear Physics A 714 (2003) 21–43 23

impinging on a standard ISOLDE uranium-carbide/graphite target (with 50 g/cm2 of U)[22]. The reaction products diffused from the target heated to about 2000C and effusedthrough a low-temperature, water-cooled transfer line to an ion-source chamber, where theionization by plasma discharge took place [23]. The temperature of the transfer line waskept at about 50C, providing efficient condensation of all elements except volatile gases.The lighter noble-gas isotopes87−93Kr and137−142Xe were mass-separated and implantedinto an aluminized mylar tape and transported to a 4π β-detector setup. The implantedisotopes were identified via their half-lives. The counting rates for various delay times afterthe proton pulse were measured for these longer-lived isotopes to provide full informationon the release characteristics from the target. In the context of this paper the releasecurves for Kr and Xe were only used to determine the total yields of their neutron-richisotopes which were implanted during restricted time periods for half-life measurementsas explained below.

The heavier Kr and Xe isotopes were measured with a different detection system. Theradioactive beam was stopped in, (A) an aluminum foil, or (B) an aluminized mylar tape,situated in the center of a cylindrical 4π neutron long-counter. This detector consists of12 parallel-coupled3He proportional counters embedded in a paraffin moderator [24]. Theneutrons emitted afterβ decay were thus thermalized in a random walk process throughthe moderator before detection. The time delay introduced by the thermalized follows anexponential distribution with a mean residence time ofτ = 89± 1 µs [25]. The detectedrandom neutron background was only 0.3 Hz. The detection efficiency,εn, has beenmeasured previously to be 19± 1% [26] which was verified in the present experimentsusing a calibrated Am(Li) neutron source.

In experiment A theβ particles were measured with a detector consisting of two plasticscintillators of 50 mm diameter placed directly behind the aluminum foil. A Bicron BC-444 scintillator, of 1 mm thickness with a decay time of 160 ns, was optically cemented toa 25 mm thick Bicron BC0420 scintillator with a 2 ns decay time. A gated integrator, set tointegrate the signal between 50 and 150 ns, followed by a low-level discriminator detectedthe slow component of the light signal. This allowed to discriminateβ-rays fromγ -raysand neutrons which do not produce a signal in the thin scintillator. With this setup datawere collected for the isotopes8He, 94−98Kr and 142−147Xe. The former served as a testbeam. The efficiency of theβ detector,εβ , was determined in fits of the Kr and Xe decays(as explained in Section 2.2) and was always in the range of 20–25%, but varied from oneisotope to another due to slight changes in the position and shape of the implantation spot.A cross check was made by calculating the efficiency from the ratio of the number of timecorrelatedβn pairs to the number of detected neutrons [27], and agreement was found. Thedetection rate of long-livedβ background was 100–400 Hz. During these measurementsonly one proton pulse per the PSB cycle (14.4 s) was used, allowing detailed analysis ofthe daughter decays. The total number of proton pulses taken for each measurement wasin the range of 25–70.

For experiment B improvements were made to the detector setup which allowed bettersignal-to-background conditions. The isotopes94,96−98Kr were remeasured with longermeasuring times (100–500 proton pulses) and the data were extended to include also99Kr. Again data were taken with a beam of8He to make cross checks of the analysis.After each implantation and subsequent decay, and before the arrival of the next proton

24 U.C. Bergmann et al. / Nuclear Physics A 714 (2003) 21–43

pulse, long-lived daughter activities were removed from the detection region by moving thealuminized tape. Consequently, the detection rate of long-livedβ activity was always in therange of 3–10 Hz. The implantation point was situated close to a 125 µm mylar windowtransparent to high-energyβ particles. A 1.5 mm thick plastic detector was positionedclose to the mylar window inside the neutron long-counter for detection ofβ particles.The scintillation light from the plastic detectors was guided outside the neutron counterusing optical fibers to two photo-multipliers per detector. The signals from the photo-multipliers were amplified and fast time signals generated. Only coincidences betweentime signals from two photo-multipliers detector were considered as a true detection ofa β particle. This allows for a reduction of random electronic noise to a level of 1 Hz[28]. The detector efficiency was determined by the fits of the Kr decays and was always4–5%. Again this agreed with the results from studies ofβn time correlations. In thisexperiment the target was bombarded by the PSB proton pulses each 8.4 s. After themeasurements it was discovered that the aluminized tape had been installed upside downwhich meant that the ions were implanted into the mylar side of the tape. It is well knownthat a significant fraction of Kr ions implanted into mylar diffuses rapidly [29]. This fastdiffusion affects dramatically the measured half-lives for longer-lived isotopes, leaving thePn values unchanged to first order. Being very unfortunate this effect can, however, beanalyzed and proper half-lives extracted as shown in the following two subsections.

Using an electrostatic deflector (“beam-gate”) the ion beam was directed to theimplantation point only in designated time intervals shortly after the impact of the protonbeam. The collection time was chosen for each case to optimize the activity of the noble-gas isotope after implantation and at the same time to minimize the fraction of daughternuclei created during collection. For strong activities, the start of implantation was delayedin order to reduce dead-time effects. For the isobaric background (of Br, Rb and I, Csrespectively) possibly leaking through the water-cooled transfer line an upper limit of 1%was determined by measurements on masses where only stable Kr and Xe isotopes occur(80, 86, 130 and 136). Hence it is negligible for this analysis.

The detection times ofβ particles and thermalized neutrons were recorded by a precisetime-stamping module based on a 10 MHz clock with 32 bits time resolution. Eachregistered event consisted of the time relative to the proton pulse, and a pattern word toindicate whether the event was triggered by aβ particle or a neutron. This allowed toproduce both “singles” time spectra relative to proton beam impact of neutrons and betasindependently, referred to as “decay spectra” below, as well asβ-neutron time correlationson theµs scale.

2.2. Analysis procedure

A standard description of chain decay is given in text books (such as [30]). However,when considering nuclei with non-negligibleβ-delayed neutron branches one arrives ata situation where most isotopes involved in the chain decay are fed by more than oneprecursor and decay to several daughter nuclei. Below, an iterative method to solve thisproblem is outlined.

For all nuclei in question the multi-neutron emission probabilities can be neglected, i.e.,P1n = Pn andPxn = 0 for x > 1. To find theβ and neutron decay-activities at any timet

U.C. Bergmann et al. / Nuclear Physics A 714 (2003) 21–43 25

after proton impact on target, we need to know the concentration in the sample at timet

of all decaying isotopes. Let us assume that we know the isotopic concentrations for theelement with proton numberZ− 1 and we want to determine the amounts of nuclei at thelevelZ of proton number. Let us consider the nucleusA

ZX3 which is fed inβn andβ decayby the nucleiA+1

Z−1X1 and AZ−1X2. If these nuclei have decay constantsλ1, λ2 andλ3 and

Pn values ofP1, P2 andP3, the number ofX3 nuclei is given by the differential equation

dN3

dt= −λ3N3(t)+ P1λ1N1(t)+ (1− P2)λ2N2(t) (1)

which is valid after the implantation has finished, i.e., at timest > t0 + t0 where t0denotes the start of implantation andt0 the collection time. To simplify the notationlet us introduce the time variablet ′ = t − (t0 +t0). If N1 andN2 are of exponential form,that is

N1(t′) =

n1∑i=1

ai exp(−αi t ′), (2)

N2(t′) =

n2∑j=1

bj exp(−βj t ′), (3)

a complete solution to Eq. (1) is

N3(t′)=

n1∑i=1

Ai exp(−αi t ′)+n2∑j=1

Bj exp(−βj t ′)+C exp(−λ3t′), (4)

whereAi = aiP1λ1(λ3 − αi)−1, Bj = bj (1 − P2)λ2(λ3 − βj )

−1 andC = N3(t′=0) −∑n1

i=1Ai −∑n2j=1Bj . In Eqs. (2) and (3) the arbitrary parametersai , bj , αi , βj , n1 andn2

should be determined iteratively from the initial conditions (see below).We note thatN3 is again of exponential form and can be used, along with similar results

for the other isotopes with proton numberZ, as input equations in the next step of theiteration, to calculate the amounts of isotopes with proton numberZ + 1. This procedurecontinues until one reaches isotopes which are stable on the time scale of the measurement.

To start the iteration one needs to know the amount of mother nuclei in the sample,N(t ′). Since the mother nucleus is an isotope of a noble gas it is subject to diffusionand depending on its atomic size, the implantation depth, the implantation material,and its half-life it may leave the collection point before decaying and disappear fromthe detection region, through the vacuum tube of the beam line, with thermal velocity(∼ 270 A−1/2 cm/ms). From decay data on8He (T1/2 = 119 ms) it is apparent that asignificant part of the implanted8He atoms diffuse out of both aluminum and mylar with adiffusion “half-life” of around 32 ms. Further analysis shows that the diffusion process canbe described with an additional exponential component in the time evolution of implanted8He atoms

N(t ′)=Nndexp

(− ln 2

T1/2t ′)

+Nd exp

(− ln2

Tefft ′). (5)

Thus, at timet ′ = 0 the implanted sample consists ofNnd noble-gas atoms which are stuckat the implantation site and cannot diffuse (denotednon-diffusible) andNd atoms which are

26 U.C. Bergmann et al. / Nuclear Physics A 714 (2003) 21–43

Fig. 1. A general scheme presenting the relevant nuclei participating in the decay chain of a neutron-rich Kror Xe isotope. The nomenclature used for the various half-lives and delayed-neutron emission probabilities isintroduced.

subject to diffusion (denoteddiffusible). Both groups of atoms undergo radioactive decaywhich for the latter gives rise to an effective half-life ofTeff = T1/2Td/(T1/2+Td), Td beingthe diffusion half-life.

The relevant isotopes of the decay chain are shown in Fig. 1. On the experiment timescale of order of 10 s we need, apart from the mother decay, to consider the decays oftheβ , βn, ββ , ββn, andβββ daughters. For the nuclei studied here one can neglect thePn values of theββ , ββn, andβββ daughters. Having determined the concentration of allthese isotopes, using the procedure described above, the measuredβ and neutron activitiesare given as

Rβ(t′) = εβ ln2

(N(t ′)T1/2

+ Nβ(t′)

Tβ+ Nβn(t

′)Tβn

+ Nββ(t′)

Tββ+ Nββn(t

′)Tββn

+ Nβββ(t′)

Tβββ

)

+Rbackβ , (6)

Rn(t′) = εn ln2

(PnN(t

′)T1/2

+ PβNβ(t′)

Tβ+ PβnNβn(t

′)Tβn

)+Rback

n , (7)

where the amount of isotopes, half-life andPn value of theβx daughter are denotedNβx(t

′), Tβx andPβx . The parametersRbackβ andRback

n describe the long-lived backgroundcomponents which were not included in the decay chain. Due to the decay of mother nucleiduring the finite collection timet0, in principle allNβx(t ′ = 0) are non-zero. However,since short collection times were used (see Tables 1 and 2) onlyNβ(t

′ = 0) andNβn(t ′ = 0)

U.C

.Bergm

annetal./N

uclearP

hysicsA

714(2003)

21–4327

Table 1Results from the fitting ofβ and neutron decay-spectra for noble-gas isotopes implanted into aluminum during Experiment A. Different time intervals of implantationafter proton beam impact, [t0; t0 +t0], were used. The obtained mother half-lives,T1/2, andPn values, and alkali-daughter half-lives,Tβ , are compared to results fromprevious experiments [33]. The fit functions were constructed by using Eq. (4) iteratively with Eq. (5) as initial input, and inserting the results into Eqs. (6) and (7). In

this way, the total number of implanted noble-gas ions per pulse (still present when the implantation was finished),Nnd +Nd, the diffusible fraction, NdNnd+Nd

, and the

diffusion half-life, Td, were determined. (See text for more details)

Precursor t0 t0 Nnd +NdNd

Nnd+NdTd T1/2 Tβ Pn

Present Previous Present Previous Present Previous(ms) (ms) (atoms

pulse) (%) (ms) (ms) (ms) (ms) (ms) (%) (%)

8He 5 100 1.50(6)× 104 45(4) 32(6) 103(7) 119.0(15) 841(5) 838(6) 15(3) 16(1)

94Kr 100 5 5.2(2)× 104 20(4) 110(30) 212(5) 200(10) 2670(40) 2702(5) 1.26(10) 5.7(22)95Kr 100 10 1.75(8)× 104 13(4) 50(30) 114(3) 780(30) 384(4) 377.5(8) 2.87(18)96Kr 5 100 1.3(1)× 104 22(10) 70(30) 80(8) 189(5) 203(3) 3.8(4)97Kr 5 100 1050(50) 18(17) 61(5) 160(5) 169.9(7) 8.2(15)98Kr 5 100 36(13) 44(12) 130(17) 114(5) 7(4)

142Xe 100 2 5.15(15)× 104 10(2) 100(50) 1250(25) 1220(20) 1650(40) 1689(11) 0.21(6) 0.41(3)143Xe 100 5 4.4(2)× 104 9(2) 41(14) 511(6) 300(30) 1770(30) 1791(8) 1.00(15)144Xe 100 5 7400(500) 11(3) 31(13) 388(7) 1150(200) 983(20) 993(13) 3.0(3)145Xe 100 100 9000(200) < 3 188(4) 900(300) 558(9) 582(6) 5.0(6)146Xe 5 200 800(50) < 10 146(6) 300(20) 323(6) 6.9(15)147Xe 5 100 4.5(20) 100

(+100−50

)225(5) < 8

28U

.C.B

ergmann

etal./Nuclear

Physics

A714

(2003)21–43

Table 2As Table 1, but for noble-gas isotopes implanted into mylar during Experiment B

Precursor t0 t0 Nnd +NdNd

Nnd+NdTd T1/2 Tβ Pn

Present Previous Present Previous Present Previous(ms) (ms) (atoms

pulse) (%) (ms) (ms) (ms) (ms) (ms) (%) (%)

8He 30 30 1.11(5)× 104 78(2) 32(2) 100(7) 119.0(15) 834(10) 838(6) a 16(1)

94Kr 450 3 8300(1000) 84(3) 79(4) 210(20) 200(10) 2530(300) 2702(5) 0.98(9) 5.7(22)96Kr 200 7 2000(400) 75(13) 69(9) 80(10) 202(4) 203(3) 3.5(8)97Kr 5 80 6000(600) 62(13) 60(12) 68(7) 171.0(15) 169.9(7) 6.5(6)98Kr 5 80 450(100) 77(13) 48(10) 126(10) 114(5) 7.0(10)99Kr 5 80 9(3) 40(11) 59(12) 50.3(7) 11(7)

a Due to parameter correlations the fit did not converge when bothPn andεβ were varied. Theβ efficiency is not known for the8He run since changes had been madeto the detector setup. When fixingPn = 16% the fit gaveεβ = 10%.

U.C. Bergmann et al. / Nuclear Physics A 714 (2003) 21–43 29

were treated as non-zero. These numbers were estimated to lowest order from the in-growthof activity in the regiont0< t < t0+t0 of theβ decay-spectrum.

The fitting procedure was carried out as follows: theβ and neutron decay-spectra werefitted simultaneously, as histograms with 2 ms channel width, using the “Poisson likelihoodχ2” statistic3 with theoretical functions

Nbinβ =

∫bin

dt ′NpulseRβ(t′)[1− τeff

(Rβ(t

′)+Rn(t′))], (8)

Nbinn =

∫bin

dt ′NpulseRn(t′)[1− τeff

(Rβ(t

′)+Rn(t′))], (9)

respectively. HereNpulse is the number of proton pulses taken andτeff (∼ 2 µs) is aneffective dead time parameter for the electronics and data acquisition [27]. For the purposeof determining the half-life andPn value of the implanted nucleus, and their statisticaluncertainties, the following 9 parameters were varied freely while fitting:Nnd + Nd,Nβ(t

′ = 0), T1/2,Pn,Rbackβ ,Rback

n , εβ ,Nd/(Nnd+Nd), andTd. The parameter minimizationand error analysis were performed with MINUIT [32]. The decay parameters of the noble-gas daughters are well known in literature, they were fixed at the values given in [33].Varying these parameters within the uncertainty quoted in the literature had a negligibleeffect on the fit [34]. For the cases with low count numbers the 9-parameter fit showedstrong parameter correlations. This is an undesirable situation in which one can no longertrust the resulting error bars. Therefore, for such cases, the diffusion parameterTd wasfixed at an averaged value determined from fits of lighter Kr and Xe isotopes, respectively,which restored a normal fit situation. This procedure is justified by the fact that differentisotopes of the same noble-gas element were implanted at nearly the same depth and shouldtherefore have similar diffusion times. The decay spectra were fitted in the time region fromthe end of implantation to the onset of the next proton pulse. (For Experiment B the fit wasmade only as far as 7.5 s, the time where the tape was moved.)

2.3. Results

The measured decay spectra and the fitted decay components are shown in Figs. 2–4.The resulting fit parameters and their estimated uncertainties are given in Tables 1 and 2 forExperiment A and B, respectively. All resultingχ2 values are acceptable. Similar diffusionhalf-lives were obtained for aluminum and mylar, about 32 ms for8He and 75 ms for Kratoms, whereas an average diffusion half-life of 38 ms resulted for Xe which was onlyimplanted into aluminum. Thus, the atomic size and the implantation depth determine thediffusion time, rather than the material properties. As expected, a large fraction of the Kratoms, about 80%, were able to diffuse from mylar, whereas a smaller, but measurable,fraction of 20% could diffuse from aluminum. For Xe in aluminum a diffusible fractionabout 10% was obtained. Since the fraction of diffusible8He atoms was less dependent on

3 As discussed in [31] this method, which is based on the maximum-likelihood principle, gives more reliableresults than the classical chi-square method in the cases of poor statistics and histogram binning.

30 U.C. Bergmann et al. / Nuclear Physics A 714 (2003) 21–43

Fig. 2. Simultaneously fitted (using 2 ms channel width)β and neutron decay-spectra for Kr isotopes (and8Heas a test) implanted into aluminum during Experiment A. The resulting total fit functions (Eqs. (8) and (9)) areshown as thick dashed curves on top of the data. Below, the individual mother andβ-daughter activities are shownas thin dashed and dotted curves, respectively. In addition, theβ activity of ββ daughter and the neutron activityof theβn daughter are drawn, both as dashed-dotted curves. (See Fig. 1 and Eqs. (6) and (7) for nomenclature.)

U.C. Bergmann et al. / Nuclear Physics A 714 (2003) 21–43 31

Fig. 3. As Fig. 2, but presenting the Xe data.

32 U.C. Bergmann et al. / Nuclear Physics A 714 (2003) 21–43

Fig. 4. As Fig. 2, but for Kr (and8He) implanted into mylar during Experiment B.

U.C. Bergmann et al. / Nuclear Physics A 714 (2003) 21–43 33

Table 3Yields of Kr and Xe ions per µC of protons on target (1 pulse∼ 4.8 µC) at the exit of the mass separator. Theyields of94,96−99Kr are taken from Experiment B (generally a factor of two larger in Experiment A). Also, thecombined results from the two experiments of the Kr half-lives andPn values are given

Precursor Yield T1/2 Pn Precursor Yield(atoms/µC) (ms) (%) (atoms/µC)

94Kr 2.2×106 212(5) 1.11(7) 142Xe 1.7×107

95Kr 1.3×105 114(3) 2.87(18) 143Xe 3.2×106

96Kr 6.8×104 80(7) 3.7(4) 144Xe 4.9×105

97Kr 7200 63(4) 6.7(6) 145Xe 1.7×104

98Kr 520 46(8) 7.0(10) 146Xe 89099Kr 10 40(11) 11(7) 147Xe 7.5

the material we conclude that this parameter also depends on the size of the implanted ion.Consistent diffusion times and diffusible fractions were obtained for different Kr isotopesin the same material. The latter should decrease when the collection time is increased (theimportant parameter being the ratiot0/Td) since a part of the ions had already diffusedduring implantation. Due to the short collection times which did not exceed the diffusionhalf-life this effect was not detectable for Kr with the obtained precision. In contrast, forXe the diffusion was sufficiently fast that no measurable fraction of diffusible145,146,147Xeatoms was left after 100–200 ms collection (t0/Td 3–4, Table 1). Similarly, thedifference in diffusible fraction for8He in aluminum and mylar was enhanced by the useof different collection times. For some isotopes the diffusion time is not given in Tables 1and 2, although the diffusible fraction is quoted. In these casesTd was fixed in the fit inorder to avoid large parameter correlations. For the isotopes with poorest statistics neitherof the two diffusion parameters could be determined, i.e., the confidence interval for thediffusible fraction was larger than[0;1]. The results on diffusion of lighter isotopes werethen used to fix these parameters at realistic values in the fit, with their uncertainties takeninto account in the quoted half-lives andPn values.

The analyses of the Kr data from Experiments A and B give consistent results forT1/2 and Pn. The combined results for Kr are given in Table 3. At first one wouldexpect the latter experiment to give more precise estimates, thanks to the improvedsignal-to-background ratio and longer data taking, but this is only the case for the mostexotic isotopes since the diffusion component dominates the fit. This demonstrates theimportance of the right choice of implantation host in investigations of nuclear decaychains that include noble-gas isotopes. There is only partial agreement between the resultsof the present experiment and the existing data. In particular, the half-lives from theearlier indirect radiochemical measurements [13–17] (quoted by nuclear data evaluatorsfor 95Kr and 144,145Xe [33]) deviate considerably from our results, indicating that theseidentifications probably were not correct. The half-lives quoted by [33] for94Kr and142,143Xe were obtained at on-line separators [5–7,9,10] and are in reasonable agreementwith the present experiment, although theirPn values for94Kr and142Xe are a factor 3–5

34 U.C. Bergmann et al. / Nuclear Physics A 714 (2003) 21–43

larger4 [6,9]. The present data for147Xe allow only a rough estimate of the half-life andPnvalue. If the147Xe component is removed from the fitχ2 increases by 7. Since removingthe component effectively removes two fit parameters we conclude that147Xe is presentat a 95% confidence level. The results for8He implanted into both materials justify thepresent analysis. Even for this extreme case whereTd T1/2/4 the resulting mother anddaughter half-lives are reliable.

In order to test the data for systematic errors the half-life of the alkali daughter, thefit parameterTβ , was released after completion of the 9-parameter fit (which determinedT1/2 andPn for the mother nucleus as explained in Section 2.2). As can be seen fromTables 1 and 2 the fittedTβ values did not move further away from their starting points,taken from previous measurements [33], than could be expected according to their deduceduncertainties. Note that these uncertainties are in some cases comparable in size withthose in the literature. This is a very sensitive cross check for systematic errors whichbecomes evident if one tries to remove the diffusion component from the analysis: apartfrom 8He, the effects of diffusion are most pronounced for the nucleus94Kr which has thelongest half-life (of the Kr isotopes). When fixingNd = 0 one obtains for implantationin aluminumT1/2 = 192.7 ± 1.1 ms andTβ = 2580± 30 ms, and for mylarT1/2 =78.8 ± 1.5 ms andTβ = 1570± 60 ms. In contrast, the fittedPn value is not affectedby diffusion. Thus, in both cases the mother and daughter half-lives are underestimated(compare with Tables 1 and 2) and in the worse case, the implantation into mylar, the fittedT1/2 value gets close to the effective half-live of the diffusion component,Teff = 56 ms.Theχ2 values obtained in the fit of the94Kr decay were for both data sets decreased bymore than 100 by introducing the two additional diffusion parameters (notice the two-component nature, cf. Eq. (5), of the fittedβ and neutron contributions from94Kr inFig. 4). Generally, when diffusion is not included, the fittedT1/2 value will lie somewherein the interval[Teff;T1/2], depending on the actual size ofNd/(Nnd + Nd). For 99Kr thisinterval is[27 ms;41 ms] which shows that for the heavier Kr isotopes the error made byneglecting diffusion is less important. When using aluminum for implantation the fact thatthe diffusible fraction is small assures a small error for all Kr isotopes.

The fit parameterNβ(t ′ = 0), the amount of alkali-daughter nuclei present in the samplewhen the implantation was stopped, gives an additional cross check of the analysis. In allcases, when diffusion effects were included, this parameter agreed with the number ofdaughter nuclei expected from decay during implantation. Thus, no neutron-rich Rb or Csnuclei were present in the noble-gas beams.

The yields of Kr and Xe isotopes at the exit of the mass separator are presentedin Table 3. They were obtained from the fittedNnd + Nd values (Tables 1 and 2) bycorrecting for the beam transport efficiency and for the fraction of ions not arrivingwithin the implantation period. The latter was deduced from a release curve measuredfor both target/ion-source systems for long-lived isotopes (as explained in Section 2.1) andcorrected for decay losses for the case of short-lived isotopes. This yield information isimportant for the planning of future experiments.

4 It is interesting to note that the second method of analysis of the neutron spectra used in [9] provided aPnvalue of 1.6± 0.9% for 94Kr which is in agreement with our measurement. However, only thePn value from thefirst analysis method [9] (5.7± 2.2%) was accepted by nuclear data evaluators [33].

U.C. Bergmann et al. / Nuclear Physics A 714 (2003) 21–43 35

3. Discussion

3.1. Nuclear structure

The β-decay half-livesT1/2 and β-delayed neutron emission probabilitiesPn areintegral nuclear properties. Although one cannot expect to obtain detailed informationabout nuclear structure from them, taken together they often lead to some first insights.In general, while theβ-decay half-life is dominated by the lowest-energy part in theβ-strength function, theβ-delayed neutron-emission probability is sensitive to the feedingto the energy region just above the neutron separation energy (Sn). The feeding, or decay-rate, to the regiondE at excitation energyE is proportional to the product of theβ-strengthfunctionSβ(E) and the Fermi functionf (E) that is it is proportional toSβ(E)f (E)dE.Becausef (E) ∼ (Qβ − E)5 the contribution of transitions to low-lying energy states isdominant. SinceSβ(E) strongly depends on nuclear structure, the measuredT1/2 andPndo as well. It is because the half-lives and neutron emission probabilities are, roughly,determined by the structure of low-lying and high-lying parts of theβ-strength function,respectively, that we can draw some conclusions about nuclear structure from the relativemagnitude of these two functions.

The measurements carried out here for Kr and Xe isotopes extend over a range ofneutron numbers that take us from spherical or only weakly deformed nuclei just beyondβ-stability towards very deformed nuclei further into neutron-rich regions. Deformationhas a profound effect onSβ(E). For spherical nuclei the strength function is dominatedby a few strong peaks substantially separated in energy, whereas the strength function ofdeformed nuclei typically consists of more densely spaced, weaker peaks [35]. Despite theclear signature of deformation on the strength functions, the effect on the integral half-life and delayed-neutron emission probabilities is not always as dramatic. To see why thismay occur, consider the following case. Suppose that for a specific spherical nucleus, itsstrength function has no low-lying strength and the first strength occurs at, say 4 MeVwhere a strong peak is located. Now suppose we add a neutron to this nucleus and thisinduces deformation. The strength function now has a few small peaks near the groundstate. However, because the Fermi function is proportional to(Qβ −E)5 the transition rateto the ground state of the deformed nucleus will be the product of a large Fermi functionand a small strength function and may be approximately equal to the transition rate to the4 MeV state in the spherical isotope. That rate would be given by the product of a largestrength and a small Fermi function. On the other hand, in cases where both the deformedand the spherical nucleus haveβ-strength at zero excitation, the effect of deformationonset on the half-life would be dramatic. Since the Kr and Xe isotopes in this experimentextend from spherical towards deformed regions we compare measured data to two sets ofcalculations, one of which prescribes spherical nuclear shape and one in which we use theshape coordinates that were obtained in a microscopic-macroscopic calculation of nuclearground-state shapes and masses [36].

TheQβ andSn values used for these calculations were taken from experimental evalu-ations when available [33] otherwise from calculated masses [36,37]. The calculations oftheβ-decay half-livesT1/2 andβ-delayed neutron-emission probabilities are calculated ina microscopic quasi-particle random-phase approximation (QRPA), see Refs. [35,37,38]

36 U.C. Bergmann et al. / Nuclear Physics A 714 (2003) 21–43

(a) (b)

(c) (d)

Fig. 5. Comparison of experimental half-lives andPn values for Xe (a), (b) and Kr (c), (d) isotopes with QRPAcalculations performed for a spherical and deformed set of parameters. The experimental values for light92,93Krand 141Xe are taken from [33]. The staggering between isotopes observed in the calculation occurres due tostrong contribution of the odd neutron to the decay of even–odd isotope. In the case of93,95Kr isotopes themodel predicts the ground states non-correctly also leading to additional staggering in spherical calculations. Theresults of calculation for “correct” ground state (see text and Fig. 6) are also shown.

which has been enhanced to account for first-forbidden decays. These are calculated in thestatistical gross theory [39,40]. In Fig. 5 we show measuredT1/2 andPn values for Xe andKr isotopes, compared to calculated values. As mentioned earlier two sets of calculationswere performed, one for spherical shapes and one for shapes obtained in a calculation ofground-state shapes and masses.

For the range of Xe isotopes studied here, the calculated value of the quadrupoledeformation coordinateε2 evolves gradually and smoothly fromε2 = 0.125 for 141Xe toε2 = 0.208 for147Xe. There is good agreement between measured half-lives and the half-lives calculated for deformed shapes (solid-squares). There is also excellent agreementbetween the measuredβ-delayed neutron-emission probabilities and the calculated valuesbased on the deformed ground-state shapes, except possibly for the two lightest Xeisotopes. In these two cases the calculations based on spherical shape agree better withthe experimental data. However, thePn values for these two nuclei are only fractions

U.C. Bergmann et al. / Nuclear Physics A 714 (2003) 21–43 37

of one percent. In such cases the model calculations are quite uncertain, and the betteragreement with the spherical calculations are probably just accidental. The odd–evenisotope staggering observed in the calculation occurs because for an odd-neutron nucleusthe odd neutron contributes strongly to the decay. This contribution is absent in an even–even nucleus. The staggering effect is especially strong in the calculations for sphericalshapes due to low level density in these nuclei. For further discussion of this subject see[38].

For the Kr isotopes it is known thatN = 56 is a magic subshell, and this isotope andthe lighter ones are essentially spherical nuclei. Recent experimental data [41,42] indicatesthat no sizable deformation appears in the regionN 58. On another hand, an atomicspectroscopy experiment [43] showed that the96Kr (N = 60) is moderately deformed.BeyondN = 60 deformation is expected to set in fairly rapidly. We see in Fig. 5(c) thatthe half-lives calculated with deformed ground-state shapes atN = 60 and beyond arein reasonable agreement with the experimental measurements. On the other hand, theshape of the light93

36Kr57 is mainly spherical in character and one would expect betteragreement with the spherical calculation, rather than the more than one-order-of-magnitudedisagreement observed in Fig. 5(c). However, for this odd–even nucleus two somewhatdifferent types of decays occur: (a)v = 2 transitions to a one-quasiparticle phonon inthe daughter, where the initial odd particle acts only as a spectator (herev is the numberof unpaired neutrons), and (b) av = 0 transition from the oddv = 1 ground state toa v = 1 state in the daughter nucleus which directly involves the unpaired odd neutronin the neutron single-particle orbital corresponding toN = 57. Selection rules determinewhich final states are reachable from the initialN = 57 state. Our nuclear ground-stateshape and configuration calculations place the valenceN = 57 neutron in aΩν = 7/2+orbital. For this configuration we obtain theβ-strength function shown in Fig. 6(a).A strong peak in the strength function occurs near 1.5 MeV excitation energy. This low-lying peak results in a short calculated half-life. However, experimentally the spin/parityof the ground state in93,95Kr isotopes was found to beΩν = 1/2+ [43]. In our QRPAprogram we can calculate the strength function for this initial configuration. The result isshown in Fig. 6(b), and the half-life andβ-delayed neutron-emission probability are givenin Fig. 5(c) and (d) as open triangles. When we use the correct initial configuration wenow obtain excellent agreement with the experimental data for93Kr. For 95Kr the half-lifecalculated with spherical shapeand correctN = 59 configuration agrees less well with datathan does the deformed calculations. This isotope may be already deformed in character.We find that our discussion of the agreement between calculated and experimental decayhalf-lives of the Kr chain of isotopes carries over to the case of theβ-delayed neutron-emission probabilities. For the heavier deformed isotopes agreement between calculatedand experimental neutron-emission probabilities is good. For the basically spherical93Krisotope calculations based on a spherical shape agrees well with dataonly if we changethe configuration of theN = 57 neutron from the calculated spherical configuration tothe experimentally observed configuration. For the95Kr isotope the deformed calculationagrees best with data, just as is the case for the decay half-life.

We note that it is a well-known difficulty to reproduce theoretically theN = 56 subshellenhancement that occurs nearZ = 40. As far as we know, no general, global, unifiednuclear-structure model, such as ours, performs satisfactory in this respect. However,

38 U.C. Bergmann et al. / Nuclear Physics A 714 (2003) 21–43

(a)

(b)

Fig. 6. (a)β-strength function for93Kr decay for the calculated single-particle configuration with assumption ofspherical shape (7/2+ ground state). The strong transition at about 1.4 MeV leads to a short half-life. (b) Thesameβ-strength function calculated for a configuration with unpaired neutron placed in an orbital correspondingto the experimentally observed ground state (1/2+ ground state). The strong low-lying transition has disappearedresulting to a longer half-life. The two wide arrows indicate one and two neutron separation energy and the narrowone refers theQ-value.

calculations more specialized to this particular region do sometimes better describe theN = 56 subshell structure. For additional discussions of the structure of nuclei in thisregion see for example Refs. [38,44,45].

In summary we have good agreement between calculated and experimentalbeta-decay half-lives andβ-delayed neutron-emission probabilities when we use correct parentinitial configurations, that is correct nuclear deformationsand correct single-particle

U.C. Bergmann et al. / Nuclear Physics A 714 (2003) 21–43 39

level assignments. Normally we take the parent configuration from our macroscopic-microscopic nuclear ground-state mass calculation [36]. Our microscopic code willcalculate a default single-particle level configuration based on these deformations but itis possible to set the single-particle configuration differently. We showed that in the veryfew cases when deformation and/or level configuration did not agree with experimentallymeasured deformations and/or level structure, then we obtained very good agreement whenwe performed the half-life and delayed neutron-emission probabilities with the parentconfiguration given by these experimentally established configurations. We thereforeconclude that the structure of these Kr and Xe isotopes is well understood.

3.2. Astrophysics

Both the heaviest observed Kr (Z = 36) and Xe(Z = 54) isotopes lie already in ther-process path(s) in theA 100 andA 150 mass regions for moderate neutron-densityconditions. For the modeling of rapid neutron-capture nucleosynthesis (the astrophysicalr-process), several nuclear-physics quantities of very neutron-rich isotopes are required,the most important being nuclear masses (in particular neutron separation energies,Sn) andβ-decay properties (T1/2 andPn values). The majority of these nuclear data is inaccessibleto experiments at present and has to be taken from model predictions. Therefore anynew information about so-called r-process waiting-point nuclei is highly important forreplacing the theoretical model predictions, testing and improving the theoretical resultsand providing more reliable predictions for the more exotic nuclei close to the neutrondrip-line. The experimentalT1/2 andPn values of the Kr and Xe nuclei as well as improvedQRPA predictions of the next heavier isotopes can now be incorporated into dynamic (time-dependent) multicomponent r-process calculations to replace the theoretical predictions[37].

Fig. 7(a) and (b) show dynamic calculations of the relative r-abundances of neutron-rich even–even Kr and Xe isotopes under freeze-out conditions at a stellar temperatureof T9 = 1.35 (1.35× 109 K) as a function of the neutron density (nn). The abundancesare normalized to the abundance of Si equal to 106. In these calculations, the “quenched”ETFSI mass model has been used [46]. As odd-neutron isotopes have a smaller neutronseparation energy and, hence, higher neutron-capture cross-sections than the neighboringeven–even isotopes, they do not build high progenitor abundances before the “freeze out”.Therefore only the even–even Kr and Xe isotopes will act as classical “r-process nuclei”carrying the main r-abundances in both isotopic chains.

In the case of Kr isotopes the experimental half-lives of the important waiting points atA= 94,96,98 are more than a factor 2 shorter compared to the FRDM/QRPA predictions[37]. As can be seen from Fig. 7(a) at modest neutron densities in the range nn 3×1019–3 × 1022 n/cm3, representative of the r-matter flow between the classical iron seed andthe low wing of theA 130 solar-system r-abundance peak,96Kr and 98Kr are themain waiting-point nuclei with high relative isotopic abundances. For higher neutrondensities of nn 3 × 1022 n/cm3 corresponding to the conditions for building up andpassing theA≈ 130 r-abundance peak region, the still unknown isotopes100Kr and102Krbecome the main waiting-point nuclei. In the Kr chain the100Kr isotope has the highest r-abundance at neutron density of 1023 n/cm3. As the calculations presented in Fig. 7(a)

40 U.C. Bergmann et al. / Nuclear Physics A 714 (2003) 21–43

(a)

(b)

Fig. 7. Dynamic calculations for relative r-abundances of neutron-rich even–even Kr (a) and Xe (b) isotopes underfreeze-out conditions (T9 = 1.35) as a function of neutron density (nn). The abundances are presented in absoluteunits, following the convention of a Si abundance of 106.

suggests that the Kr isotopes are the main r-progenitors for the stable isotopes in theA≈ 100 mass region, it is interesting to compare the calculated r-abundances of the formerones with the r-abundances the latter ones. The r-abundances of the stable isotopes thathave mixed s- and r-origin are obtained by subtracting their measured solar abundancesand the calculated contribution from the s-process [47]. However, the obtained this wayr-abundances values have significant uncertainties. Only the abundances of the stableisotopes, which have exclusively r-process origins, provide the reliable low limit estimationof the r-process contribution in theA ≈ 100 region. The sum of the abundances of thestable “r-only”96Zr, 100Mo and104Ru isotopes is 0.89 [47] in units of Si= 106, whereasthe corresponding calculated sum for the Kr r-progenitors is 1.10. This confirms that thelater are indeed the dominant progenitors in this region. Use of the obtained experimental

U.C. Bergmann et al. / Nuclear Physics A 714 (2003) 21–43 41

data in the calculations significantly improved the agreement between the calculated andmeasured abundances.

The measured half-lives of the Xe isotopes agree within a 50% range to the theoreticalpredictions [37] (an exception is146Xe isotope which half-life is a factor 2 shorter thanthe prediction). The general good agreement allows with higher confidence use of thepredictions values for heavier isotopes in the r-process network. As it is seen from Fig.7(b) the main r-process progenitors in the Xe chain are the still unknown nuclei150−158Xe.They represent the waiting-point isotopes for neutron-density conditions in the rangenn ≈ 3 × 1023–3× 1026 n/cm3, where the r-matter flow has already passed theA≈ 130peak to form the rare-earth region and to start the “climb up the staircases” [20] towards theN = 126 neutron shell closure. The sum of abundances of the “r-only” stable148,150Nd and154Sm isotopes is 0.155 [47], while the corresponding sum of the calculated r-abundancesfor the Xe chain is equal to 0.215, indicating that the150−158Xe are the main progenitorsof these stable isotopes.

4. Conclusion

Gross decay properties of the neutron-rich noble-gas isotopes94−99Kr and 142−147Xehave been studied at the PSB-ISOLDE facility at CERN using isobaric selectivity achievedby combining a plasma ion-source with a water-cooled transfer line. Theβ-decay half-lives and delayed-neutron emission probabilities were measured with aβ-neutron detectorsetup.

The obtained experimental data can provide some insights in the nuclear structure of thestudied isotopes when compared to the QRPA shell-model predictions. The comparisonshows that results for Xe isotopes are consistent with a prolate deformation descriptionof these nuclei. For the Kr chain that covers the transitional region the obtained data areconsistent with the picture of spherical ground-states of Kr isotopes withN 58 andincreasing prolate deformation for the heavier isotopes.

The obtained data and improved model prediction for heavier nuclei were used indynamic calculations of r-process abundances of Kr and Xe isotopes. The investigated96,98Kr are shown to be r-process waiting-point nuclei under moderate neutron-densityconditions of nn 3× 1019–3× 1022 n/cm3. The waiting-points in the Xe chain lie in theregion of unknown152−158Xe isotopes.

The present data represent a significant step forward for our knowledge on decayproperties of the neutron-rich Kr and Xe isotopes. The half-lives for the most neutron-rich nuclei known previously were shown to be wrong by factors of 5, and 2–4 newisotopes were added for each case, thereby demonstrating the power of the ISOL technique.Although for the neutron-rich Kr nuclides we now cover a mass range twice as large asbefore [1,2] (99Kr lies 13 mass units away from the heaviest stable isotope86Kr), it is worthto remember that we still have to go another 20 mass units in order to reach the neutrondrip-line aroundN = 82. In the analysis it is of course important to take several steps in thedecay chains into account, but we stress that when noble-gas atoms are involved their fastdiffusion out of the experimental setup is a key feature that must be taken seriously. Wehave demonstrated this here for the case where noble-gas atoms were collected as primary

42 U.C. Bergmann et al. / Nuclear Physics A 714 (2003) 21–43

species, but similar effects will occur whenever they appear as part of the decay chain.Nevertheless, the technique used here is quite efficient and should in dedicated experimentsallow us to go one or two isotopes further out even at the present ISOLDE facility.

Acknowledgements

We would like to thank the crews of the CERN booster (PSB) and the ISOLDE facilityfor help in operation of the separator and ion source. We are grateful to Michael Hass(Weizmann Institute) for providing theβ detector for Experiment A and to Adrian Folley(CERN), Mark Huyse (KU Leuven) for their help with the construction of theβ detectorfor Experiment B. We are also grateful to Dr. E. Hagebø (University of Oslo) for fruitfuldiscussion of the obtained yield information. Two of us (B.P. and K.-L.K.) would liketo acknowledge the German BMBF (06MZ9631) for financial support. This work wasalso supported by the European Union Fifth Framework Programme “Improving HumanPotential—Access to Research Infrastructure”.

References

[1] O. Kofoed-Hansen, K.O. Nielsen, Phys. Rev. 82 (1951) 96.[2] O. Kofoed-Hansen, K.O. Nielsen, Mat. Fys. Medd. Vid. Selsk. 26 (7) (1951) 1.[3] O. Kofoed-Hansen, CERN Report, 1976-13, 65.[4] G. Holm, S. Borg, V. Fagerquist, F. Kompff, Ark. Fys. 34 (1967) 447.[5] W.L. Talbert, A.B. Tucker, G.M. Day, Phys. Rev. 177 (1969) 1805.[6] G.C. Carlson, W.C. Schick, W.L. Talbert, F.K. Wohn, Nucl. Phys. A 125 (1969) 267.[7] S. Amiel, H. Feldstein, M. Oron, E. Yellin, Phys. Rev. C 5 (1972) 270.[8] G. Rudstam, E. Lund, Phys. Rev. C 13 (1976) 321.[9] M. Asghar, et al., Nucl. Phys. A 247 (1975) 359.

[10] S.H. Faller, et al., Phys. Rev. C 38 (1988) 905.[11] T. Mehren, et al., Phys. Rev. Lett. 77 (1996) 458.[12] R.B. Firestone, V.S. Shirley, Table of Isotopes, Wiley, New York, 1996.[13] A.C. Wahl, R.L. Ferguson, D.R. Nethaway, D.E. Troutner, K. Wolfsberg, Phys. Rev. 126 (1962) 1112.[14] P. Patzelt, G. Herrmann, Physics and Chemistry of Fission, Vol. 2, 1965, p. 243.[15] O.L. Cordes, J.E. Cline, C.W. Reich, Nucl. Instrum. Methods 48 (1967) 125.[16] K. Wolfsberg, J. Inorg. Nucl. Chem. 33 (1971) 587.[17] H. Ahrens, P. Patzelt, G. Herrmann, J. Inorg. Nucl. Chem. 38 (1976) 191.[18] B. Pfeiffer, K.-L. Kratz, P. Möller, nucl-ex/0106020, Prog. Nucl. Energ., in press.[19] U. Köster, PhD thesis, Technische Universität München, Germany, 2000.[20] E.M. Burbidge, G.R. Burbidge, W.A. Fowler, F. Hoyle, Rev. Mod. Phys. 29 (1957) 54.[21] T. Bjørnstad, et al., Phys. Scripta 34 (1986) 578.[22] H.L. Ravn, Nucl. Instrum. Methods B 26 (1987) 72.[23] S. Sundell, H. Ravn, Nucl. Instrum. Methods B 70 (1992) 160.[24] O. Tengblad, Diploma thesis, Chalmers Tekniska Högskola, Göteborg, 1982, unpublished.[25] R.E. Azuma, et al., Phys. Lett. B 96 (1980) 31.[26] U.C. Bergmann, et al., Nucl. Phys. A 658 (1999) 129.[27] U.C. Bergmann, et al., Eur. Phys. J. A 11 (2001) 279.[28] L. Weissman, et al., Nucl. Instrum. Methods A 423 (1999) 328.[29] G.J. Beyer, H.L. Ravn, Y. Huang, Int. J. Appl. Radiat. Isot. 35 (1984) 1075.[30] K. Heyde, Basic Ideas and Concepts in Nuclear Physics, IOP, 1994.

U.C. Bergmann et al. / Nuclear Physics A 714 (2003) 21–43 43

[31] Y. Jading, K. Riisager, Nucl. Instrum. Methods A 372 (1996) 289.[32] F. James, M. Roos, MINUIT, Function Minimization and Error Analysis, CERN D506, Long Writeup.[33] G. Audi, O. Bersillon, J. Blachot, A.H. Wapstra, Nucl. Phys. A 624 (1997) 1.[34] C. Aa. Diget, BSc thesis, University of Aarhus, Denmark, 2001, unpublished.[35] J. Krumlinde, P. Möller, Nucl. Phys. A 417 (1984) 419.[36] P. Möller, J.R. Nix, W.D. Myers, W.J. Swiatecki, At. Data Nucl. Data Tables 59 (1995) 185.[37] P. Möller, J.R. Nix, K.-L. Kratz, At. Data Nucl. Data Tables 66 (1997) 131.[38] P. Möller, J. Randrup, Nucl. Phys. A 514 (1990) 1.[39] K. Takahashi, M. Yamada, T. Kondoh, At. Data Nucl. Data Tables 12 (1973) 101.[40] P. Möller, B. Pfeiffer, K.-L. Kratz, http://t16web.lanl.gov/Moller/publications/rspeed2002.html, Phys. Rev.

C, submitted for publication.[41] T. Rzaca-Urban, et al., Eur. Phys. J. A 9 (2000) 165.[42] G. Lhersonneau, A. Wöhr, B. Pfeiffer, K.-L. Kratz, Phys. Rev. C 63 (2001) 034316.[43] M. Keim, et al., Nucl. Phys. A 586 (1995) 219.[44] R.E. Azuma, G.L. Borchert, L.C. Carraz, P.G. Hansen, B. Jonson, S. Mattsson, O.B. Nielsen, G. Nyman,

I. Ragnarsson, H.L. Ravn, Phys. Lett. B 86 (1979) 5.[45] R. Bengtsson, P. Möller, J.R. Nix, J.-Y. Zhang, Phys. Scripta 29 (1984) 402.[46] J.M. Pearson, R.C. Nayak, S. Goriely, Phys. Lett. B 387 (1996) 455.[47] F. Käppeler, H. Beer, K. Wisshak, Rep. Prog. Phys. 52 (1989) 945.