capture reactions at astrophysically relevant energies: extended gas target experiments and geant...

Nuclear Instruments and Methods in Physics Research A 431 (1999) 160}176

Capture reactions at astrophysically relevant energies: extendedgas target experiments and GEANT simulations

V. KoK lle!, U. KoK lle!, S.E. Braitmayer!, P. Mohr!,*, S. Wilmes!, G. Staudt!,J.W. Hammer", M. Jaeger", H. Knee", R. Kunz", A. Mayer"

!Physikalisches Institut, Universita( t Tu( bingen, Auf der Morgenstelle 14, D-72076 Tu( bingen, Germany"Institut fu( r Strahlenphysik, Universita( t Stuttgart, Allmandring 3, D-70569 Stuttgart, Germany

Received 29 January 1999

Abstract

Several resonances of the capture reaction 20Ne(a, c)24Mg were measured using an extended windowless gas targetsystem. Detailed GEANT simulations were performed to derive the strength and the total width of the resonances fromthe measured yield curve. The crucial experimental parameters, which are mainly the density pro"le in the gas target andthe e$ciency of the c-ray detector, were analyzed by a comparison between the measured data and the correspondingsimulation calculations. The excellent agreement between the experimental data and the simulations gives detailedinsight into these parameters. ( 1999 Elsevier Science B.V. All rights reserved.

PACS: 29.30.Kv; 29.85.#c; 29.25.Pj

Keywords: Capture reactions; Extended gas target; GEANT simulations

1. Introduction

Capture reactions play a leading role in nucleo-synthesis of elements. The experimental deter-mination of resonance strengths and astrophysicalS-factors and thereby the exact knowledge ofreaction rates is necessary for the modelling ofstellar nucleosynthesis. Since the cross sections tobe determined are very small at astrophysicallyrelevant energies, a high-current particle acceler-ator combined with a gas target is needed for the

*Corresponding author. Present address: Institut fuK r Kern-physik, Technische UniversitaK t Darmstadt, Schlossgartenstr. 9,D-64289 Darmstadt, Germany.

E-mail address: [email protected] (P. Mohr)

experiments because many of the target substancesare gaseous. Additionally, a windowless gas targetsystem is required because of the high currents andthe low projectile energies. Both components areavailable at the Institut fuK r Strahlenphysik at theUniversity of Stuttgart [1,2].

The use of an extended gas target in combinationwith a c detector arranged in close geometry to thetarget chamber demands some considerations inthe analysis of the measured yield curves. Up tonow, in most cases, resonance strengths weregained from the analysis of experimental data in theyield maximum [3]. That means that only c raysemitted from the vicinity of the center of the cham-ber come into consideration. With this restrictionthe e$ciency of the c detector in most cases can be

0168-9002/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved.PII: S 0 1 6 8 - 9 0 0 2 ( 9 9 ) 0 0 2 4 7 - 8

determined straightforward using pointlike calib-ration sources.

The use of large volume HPGe detectors leads toa large e$ciency for the detection of c rays. BGOdetectors arranged around the HPGe detector andused as an active shield reduce the backgrounddrastically. This detector combination used for themeasurement of capture reactions results in a verylarge dynamical range of the yield curves. Thevalues found in the maxima of the yield resonancesand those found in the minima between them di!erup to six orders of magnitude [5]. This result opensthe possibility to gain more and better informationby an analysis of the full yield curve instead only ananalysis of the peak maximum. In many cases, notonly the strength, but also the total width of a res-onance can be determined.

In order to simulate the shape of peaks in theyield curves data for the gas pressure along thebeam line in the interior and in either side ofthe extended gas target have to be known. Enteringthe resulting values for the mean energies and theenergy straggling of the incoming particles in de-pendence of the position of c ray emission, and thevalue of the resonance width C, the yield curve of anisolated single resonance can be computed overmany orders of magnitude if the e$ciency of thec detector for the emitted c rays is well known. TheMonte Carlo program GEANT [6] developed forapplications in high energy physics facilitates thecalculation of the detector e$ciency using all geo-metrical data of the experimental set up and takinginto consideration the energy and the angular dis-tribution of the c ray emission. Comparing thesimulated and the experimental data over a widedynamical range yields values of the branchingratios, of the resonance strengths, and of the widthsof the resonances.

In Section 2 of this paper a short description ofthe experimental set up is given and, as an example,a yield curve of the reaction 20Ne(a, c)24Mg isdiscussed which has been measured in this arrange-ment. In Section 3 some results of simulationscalculated with the program GEANT and theexperimental veri"cation of the c detector e$ciencyare displayed. In Section 4 the simulation of yieldcurves is described together with the representationof the di!erent components whose knowledge is

necessary for these calculations. Finally, some re-sults are shown and a conclusion is given.

Note that Ea

is the a energy in the laboratorysystem in this paper.

2. Experimental set-up and procedures

2.1. Accelerator and gas target system

The 4 MV Dynamitron accelerator of the InstitutfuK r Strahlenphysik, University of Stuttgart, pro-vides beams of 4He ions with currents up to 250 lAat energies between about 0.45 and 3.50 MeV. De-tails of the accelerator, the beam handling system,and the beam characteristics have been describedpreviously [1].

For the measurement of capture reactions thewindowless and recirculating gas target systemRHINOCEROS is used [2]. A schematic diagramof the relevant parts of the experimental set-up isshown in Fig. 1. The reaction chamber is a #at cellwith four ports radiating from the center of thechamber. These ports are used for gas inlet and forthe supply of a manometer and two particle de-tectors. The beam enters the chamber through ca-nal A and leaves it through canal B. The length ofeach canal is 15 mm, the diameters are 5 mm each.The length of the beam axis in the interior of thetarget chamber is 60 mm, the width of the chamber is15 mm. The beam alignment can be controlled op-tically through a plastic window because the targetgas is excited by the beam and emits characteristiclight. To avoid background reactions all inner surfa-ces are plated with a layer of gold of about 10 lmthickness. The particle beam is dumped into a Fara-day cup about 150 cm behind the target chamber.

The gas pressure in the chamber is measuredwith a Baratron capacitance manometer. Fluctu-ations of the gas pressure are compensated by anelectronically controlled valve installed in the gascirculation system. Thus, #uctuations could be re-duced to *p/p)1%. The pressure in the gas targetwas about 1}2 mbar. GoK rres et al. [7] have ob-served a linear decrease in the actual target densityas the dissipated power in the gas target increasesdue to the energy loss of the projectiles along thebeam axis. It was found that this e!ect is negligible

V. Ko( lle et al. / Nuclear Instruments and Methods in Physics Research A 431 (1999) 160}176 161

Fig. 1. Schematic view of the target chamber of the windowless gas target system RHINOCEROS [2].

to the extent of 10% if the dissipated power isbelow 20 mW/mm. For ion currents of about 100lA as used in our experiments, the dissipatedpower in the target gas is below 20 mW/mm.

Up to now we have measured alpha-capture re-actions on 2H,15N,16O, and 20Ne [4,5,8}10] andthe proton-capture reaction 16O(p, c)17F [11]. Inthe following, mainly results of the reaction20Ne(a, c)24Mg will be described as an example.

2.2. Gamma-ray detection

For the detection of the c rays, several high-purity Germanium detectors (HPGe) with relativee$ciencies between 85% and 100% have beenused. In the experiments two detectors were ar-ranged in close geometry perpendicular to thebeam axis and symmetrical to the center of thetarget chamber covering c-ray angles from about603 to 1203 and 403 to 1403, respectively. The dis-tance to the beam axis was as small as possible. Forbackground reduction, one of the HPGe detectorswas surrounded by an active BGO shield (Fig. 2).This detector arrangement as well as the secondHPGe detector (not shown in Fig. 2) was embedded

in a lead block which served as a passive back-ground shield. The BGO detector is made up ofeight segments, each segment having its own photo-multiplier. The eight outputs build a stop signal fora time-to-amplitude converter (TAC), whereas thestart signal is generated via the timing output of theHPGe detector. The achieved time resolution isbetter than 15 ns (FWHM). All signals from theTAC output within a 60 ns window around thepeak maximum are de"ned as coincidental. Theyare selected with a single-channel-analyzer and givean anticoincidence gate for the HPGe output. Inthis way, a simultaneous output from at least onesegment of the BGO and from the HPGe detectoris rejected.

The application of the BGO shield leads to a re-duction of the background by a factor of about 8 inthe energy region near E

c"5 MeV and by a factor

of about 10 in the region near Ec"10 MeV, as can

be seen in Fig. 3.

2.3. Particle detection

The detected gamma-ray spectra are normalizedusing the number of projectiles elastically scattered

162 V. Ko( lle et al. / Nuclear Instruments and Methods in Physics Research A 431 (1999) 160}176

Fig. 2. Schematic view of the c-ray detector. The HPGe detectoris placed perpendicular to the beam axis at a distance of about6 cm, and it is actively shielded by a BGO detector.

Fig. 3. c-ray spectra of the HPGe detector with (full line) andwithout (dashed line) active BGO shielding. The background isreduced by about one order of magnitude at E

c+10 MeV.

Table 1Geometry of the particle detectors mounted at 0"293 and0"603. These detectors were used for the normalization of thec-ray yield

0 s r f d la

ladX

( 3 ) (mm) (mm) (mm) (mm) (mm) (10~5 mmsr)

29 0.35 0.80 328.9 353.5 0.78 1.2560 0.50 1.10 304.5 334.5 0.63 2.15

Fig. 4. Energy spectrum of the particle detector mounted at0"293. Note that no contamination of the target gas can beseen with the exception of 4He.

by the target nuclei. The detection was done usinglight-tight silicon surface barrier detectors, 100 lmthick, which were placed at angles 293 and 603relative to the beam direction. These detectors aremounted in separate vacuum-pumped chambers atthe end of tubes branching away from the mainreaction chamber (see Fig. 1). The use of a slit(width s) perpendicular to the beam axis assures themeasurement of particles scattered within an e!ec-tive target length l

1. An aperture of the detector

chamber inlet (diameter 2r) determines the solid

angle for detection. Assuming l1;d, where d is the

distance from the scattering center to the apertureof the detector and f the distance between the en-trance slit and the detector aperture, one gets the`geometry factora [3]

l1dX"

nsin0

sr2

fd. (2.1)

The exact detection angles result from the geometryof the set up. The used entities are listed in Table 1.Fig. 4 shows a spectrum taken during the experi-ment of the capture reaction 20Ne(a, c)24Mg.Clearly visible in the spectrum are the peak of theelastically scattered a particles, the peak of therecoiled 20Ne nuclei, and a small peak of elastica-4He scattering. 4He was used for #ushing theequipment before starting the experiment. It has tobe pointed out that the normalization of the c-rayyield curve cannot be done by the integrated beam


Fig. 5. c-ray yield curve for the ground state transition of thereaction 20Ne(a, c)24Mg. Note that the c-ray yield changes bysix orders of magnitude between the maximum in the resonancesand the upper limits measured in between.

Fig. 6. Partial level scheme of 24Mg (taken from Ref. [12]). Allenergies are given in keV.

current because of charge exchange reactionsbetween the ionized 4He` beam and the neutraltarget gas.

The projectiles experience a small angularstraggling after colliding with the target gas atoms.While the mean beam direction remains un-changed, the mean scattering angle is shifted toa smaller value due to the strong angular depend-ence of the Rutherford cross section. For an invest-igation of this e!ect, measurements were taken witha particles as projectiles at various target gas pres-sures. From the number of scattered a particles andrecoil nuclei, the mean scattering angles were deter-mined. It resulted that they change by about0.23/mbar. Furthermore, the measurements con-clude the independence of the angular stragglingfrom the incident beam energy in the range 1.0MeV )E

a)2.7 MeV.

2.4. An example of a measured yield curve

The yield >%91

of a capture reaction is de"ned as

>%91

"

Ic

I1

(2.2)

with Ic

and I1

being the number of counts in thec detector and the particle detector, respectively. InFig. 5, as an example, a part of a measured yieldcurve of the reaction 20Ne(a, c)24Mg is shown [5],and in Fig. 6 a simpli"ed level scheme of the A"24system. The yield curve represents the primarytransition RPg.s. (0`) in the energy range E

a"

1.8}2.7 MeV, which corresponds to excitation ener-gies E

xin 24Mg and therefore to c-transition ener-

gies Ecof E

x"E

c"10 800}11 600 keV. The lower

tail of the Ex"11 390(1~) resonance can be ob-

served down to Ea+2.38 MeV and, adding the

measured upper limits in the range betweenE

a+2.18 and 2.35 MeV, down to E

a+2.18 MeV.

The yield values appertaining to this range covernearly six orders of magnitude. The two resonancesbelonging to the states at E

x"11 162(3~) and the

doublet Ex"11 208/11 217(2`/4`) are sum peaks

of the cascade transitions RP1369(2`) plus1369Pg.s. The upper limit of the measured yield atE

a+2.18 MeV corresponds to a cross section of

pac+300 pb.

3. Simulation and experimental veri5cationof the c-detector e7ciency

3.1. The simulation program GEANT

The e$ciency of the HPGe detectors was cal-culated using the Monte Carlo program GEANT,


Fig. 7. E$ciency of the HGe detector for the 1332.5 keV c-ray ofa 60Co source. The e$ciency depends only weakly on the posi-tion of the 60Co source as long as the source is in the chamber.Outside the chamber the e$ciency drops sharply because of thelead shielding.

version 3.21 [6]. This program was developed foruse in high-energy physics. It allows to pursue allpossible interactions between incoming photonswith the atoms of both the detector and the shield-ing material. All its secondary products are in-cluded in the e$ciency calculation. The simulationof the destiny of one individual photon which en-ters the detector building is completed if either allsecondary particles or photons have left the de-tector building, or if the energy of these particles islower than a given threshold energy. Obviously, thesuccessful application of the program requiresthe exact knowledge of all geometric details of thearrangement. It is stated in the instruction manualof the program [6] that all possible processes in theenergy range between 10 keV and 10 GeV havebeen parametrized in the code with an error rangebelow 10%.

3.2. Experimental determination of ezciency curves

The applicability of the program to energies inthe MeV range was tested by comparing simulatedresults with experimental data.

In a "rst experiment a calibrated and nearlypoint-like 60Co source was displaced along thebeam axis. Via b decay of the 60Co nucleus the2505.7 keV level of 60Ni is populated. Subsequentlytwo photons are emitted with energies E

c1"1173.2

keV and Ec2"1332.5 keV. Therefore in the HPGe

detector two c peaks are observed at these energies,furthermore a sum peak at E

4"E

c1#E

c2.

Combining the number of photons emitted bythe 60Co source, N

0, and the number of counts

N1(E

c1) and N

2(E

c2) at the respective photon ener-

gies, an e!ective e$ciency e@ can be de"ned

N1(E

c1)"e@

1)N

0,

N2(E

c2)"e@

2)N

0.

(3.1)

In order to extract e@1

and e@2

from the simulationseveral e!ects have to be considered based on thecoincidental emission of two photons. First, thedetector e$ciencies e

1and e

2are determined for the

photon energies Ec1

and Ec2

, respectively, presum-ing the existence of only one photon and neglectingthe other coincidental photon. Supposing that both

coincidental photons are detected at their full en-ergy values, the simulation yields for the sum peak

N4(E

c1#E

c2)"e

1) e

2)N

0. (3.2)

The counts of the sum peak are absent at the singlephoton peaks. Furthermore, the single photonpeaks are missing those events in which the secondphoton in a sum event is not detected at its fullenergy and is therefore detected with another e$-ciency e6 . Finally, the single event peaks are alsomissing those counts in which the second photon isdetected in one of the BGO counters with an e$-ciency e6

Acreating an anticoincidence signal. All

these cases and therefore the entities e, e6 and e6A

canbe simulated with the Monte Carlo programGEANT. As a summary, we get

e@1"e

1(1!e

2!e6

2!e6

A2),

e@2"e

2(1!e

1!e6

1!e6

A1).

(3.3)

Fig. 7 shows a comparison between the measuredyield curve for photons at E

c2"1332.5 keV and

the simulated one. An excellent agreement can beobserved. This good agreement further indicatesthe correct input of the geometric parameters of theexperimental set up into the simulation program.

Over the range of the target chamber, the e$-ciency shows a #at plateau that falls o! very rapidlywith increasing distance of the source from thecenter of the chamber. This is due to the lead shield.


Fig. 8. Energy dependence of the e$ciency of the HPGe de-tector. The experimental results agree very well with a GEANTsimulation for energies 100 keV(E

c(12 MeV. Note that

a di!erent geometry was used for the measurements in the inset;therefore the absolute values are di!erent in the diagram and inthe inset.

The consistency of the results can be examinedcalculating the number of photons emitted by thesource, N

0, using Eq. (3.2). From both the experi-

mental value of N4

and from the value of e1

ande2

gained by the simulation, we get a value ofN

0that di!ers from the experimental value by less

than 1%.In a second experiment the energy dependence

of the c detector e$ciency was measured. Botha calibrated 226Ra source and the capture reson-ance 27Al(p, c)28Si at E

P"992 keV were used. For

this reaction, the energies and branching ratios ofthe gamma transitions are well-known [12], andthe angular distributions are close to isotropic [13].The measurements were taken in a set-up simularto the one described in Section 2.2. The 226Ra sourceand the Al foil were placed in such a way that thesource position coincided with the center of the gastarget chamber. The results of the measurementsand of the simulation are shown in Fig. 8. We cansee from this "gure that the experimental results inthe overlap region correspond extremely well as doall the measured data with the simulated ones.

As seen from Fig. 8, the detector e$ciency de-creases by approximately one order of magnitudein the energy range 500 keV )E

c)12 MeV. The

consistency between the experimentally gained andthe calculated values for the c detector e$ciencyshows that the Monte Carlo program GEANT canbe used successfully in the energy range 500 keV)E

c)12 MeV.

Additionally, the low-energy e$ciency of aHPGe detector was determined in a separate ex-periment using di!erent radioactive sources atlarge distances. As can be seen from the inset of Fig. 8,the GEANT simulation works well down to thelowest measured energies of about E

c+100 keV.

3.3. Ezciency for non-isotropic gamma-emission

Gamma rays emitted following alpha-capturegenerally have a non-isotropic angular distributionthat can be represented by

=(H)"+k

akPk(cosH), (3.4)

where Pk(cosH) are the Legendre polynomials and

akthe coe$cients that must be determined for each

gamma transition. Values of k up to 4 are required.If the state with total angular momentum J

1is

excited by alpha-capture and if the excited statedecays via c emission into a state with angularmomentum J

2, the coe$cients a

kcan be written as

a product of two factors [14]: the "rst factor de-pends only on the relative probabilities of popula-tion, w(M

1), of the substates M

1of the excited state

J1. The second factor describes the transition

J1PJ

2. For non-polarized target nuclei, we have

w(M1)"w(!M

1), so all Legendre coe$cients

with odd k-values vanish, the angular distributionis symmetric to 903. For a-particle projectiles, thechannel spin S is given by the spin of the targetnucleus. Denoting ¸ for the relative angular mo-mentum between the a particle and the target nu-cleus, we get J

1"L#S. Choosing the beam axis

as the axis of quantization, we get ML"0 and

hence M1"M

S. In the case of 20Ne we have S"0

and J"L.Generally several multipoles contribute to

gamma transitions, but in the most common case inpractice only two multipoles have to be taken intoconsideration. In this case, for the descriptionof a transition J

1PJ

2, the mixing ratio d has to


Fig. 9. Calculated e$ciency of the HPGe detector fortransitions with di!erent angular distributions. In the upperpart a pure E2 transition (2`P0`) is shown, whereas in thelower part an almost isotropic E2/M1 transition (2`P2`) isshown which is similar to the result from an isotropic source (seeFig. 7).

be known. Usually the value of d is determinedexperimentally.

The computer program GEANT was extendedin such a way that the angular distribution ofthe emitted photons was accounted for in theMonte Carlo simulation. Calculations carriedout for a number of point-like emission-centersalong the beam axis (see Fig. 2) yield resultsas partly shown in Fig. 9 for the transitionsR(10 917)Pg.s. and R(10 917)P1369 of thereaction 20Ne(a,c)24Mg.

For the transition R(10 917;2`)Pg.s.(0`) (upperpart of Fig. 9) the e$ciency curve re#ects the char-acteristic angular distribution of an E2 transitionwith *J"2, *M

J"0. The e$ciency strongly de-

pends on the center of emission, x. On the otherhand, two multipolarities (E2, M1) with a mixingratio d"2.0$1.4 [15] contribute to the transitionR(10 917; 2`)P1369(2`) (lower part of Fig. 9).In this case the angular distribution is nearly iso-tropic in the detection range of the c detector.Therefore the e$ciency curve is very similar to thatshown in Fig. 7 as measured with a radioactive60Co source.

3.4. Line proxles of c spectra

The line pro"les of c spectra o!er another possi-bility to test the simulated e$ciency values. We getthe pro"le of a c line by sorting the simulatedc events (see Fig. 9) of a "xed center of emission x,with regard to the emission angle H and using theDoppler shift formula

Ec"E

c0A1#v

ccosHB. (3.5)

In Fig. 10 the results of a simulation for two c linescorresponding to the transitions R(10 917)Pg.s.(left-hand side) and R(10 917)P1369 (right-handside) of the reaction 20Ne(a, c)24Mg are shown. Thecalculations are carried out for three centers ofemission: (a) the emission takes place near the beaminlet, (b) in the center, and (c) near the beam outletof the target chamber.

For the simulated transition R(10 917)Pg.s.,both the shape of the simulated gamma curve andthe total number of counts strongly depend on thecenter of emission. If the emission takes place in thecenter of the target chamber (Fig. 10b), the doubleline which re#ects the angular distribution of thistransition is (nearly) symmetrical. Due to Dopplershift the simulation yields a distance between thetwo maxima of *E4*.

c+25 keV.

On the other hand, for the transitionR(10 917)P1369 (right-hand side of Fig. 10) thesimulated gamma lines show a regular shape due tothe nearly isotropic distribution of the emittedphotons. A Doppler shift of the emitted photonswhich depends on the center of emission, x, can beobserved: the photon energy decreases with in-creasing x: *E4*.

c/*x+3 keV/cm using Eq. (3.5).

The simulated gamma lines can be comparedwith experimental ones (Fig. 11). It is worth noting,however, that the simulations used point-likecenters of emission along the beam axis while in thereal experiment the centers of emission are spreadout due to straggling e!ects (see Section 4) at a "xedincident beam energy. In analogy to the simulatedresult (Fig. 10) the doublet structure of the experi-mental line R(10 917)P g.s. (Fig. 11, left-hand side)can be observed. It was also clearly visible in Fig. 3.It is evident that the shape of the double line is very


Fig. 10. Simulated c-ray spectra for the transitionsR(10 917; 2`)Pg.s. (left) and R(10917)P1369; 2` (right) atdi!erent emission positions: (a) near the beam inlet, (b) inthe center, (c) near the beam outlet. Because of the anisotropyof the pure E2 (2`P0`) angular distribution a doublet canbe seen if the emission occurs in the center of the chamber.This doublet does not appear for the almost isotropic transi-tion 2`P2`.

Fig. 11. Experimental c-ray spectra for the transitionsR(10 917; 2`)Pg.s. (left) and R(10 917)P1369; 2` (right) at dif-ferent emission positions: (a) near the beam inlet, (b) in thecenter, (c) near the beam outlet. The experimental spectra com-pare very well to the simulated ones (see Fig. 10). Especially, thedoublet structure is clearly visible in the experimental spectrum(b) for the anisotropic 2`P0` transition.

sensitive to variations of the projectile energy, thatmeans to variations of the center of emission. Thesymmetry of the double line (Fig. 11b) indicatesthat the center of gamma emission can be localizedwithin the target chamber: this shape gives evidencethat the projectile energy in the center of the targetchamber is identical with the resonance energy andcan be used as indicator for some further analyses(see Section 4). Due to straggling e!ects, the shapeof the experimental double line is smeared outcompared to the simulated result (Fig. 10). The

distance between the two maxima is somewhatlarger: *E%91

c+29 keV.

For the transition R(10 917)P1369 a Dopplershift of the center of the gamma peak can be ob-served in the experimental data (Fig. 11, right-handside) as has been seen in the simulation (Fig. 10,right-hand side): the center of emission moves withincreasing projectile energy from the beam inletthrough the center to the beam outlet of the targetchamber. Therefore the gamma energy of the peakaverage decreases with increasing beam energy.


Due to energy loss, a variation of the incidentenergy of about 4 keV corresponds to a shift of thecenter of emission, x6 , of about 3 cm (see Section 4).The detected Doppler shift is similar to thesimulated value: *E%91

c/*x+3 keV/cm. This Dop-

pler shift can be used for the analysis of c spectra inorder to separate closely neighbouring resonances[10,11].

In the simulation it was assumed that the emit-ting nuclei move at their full speed, so they did notsu!er any deceleration before the gamma quantaare emitted. An estimate shows that the mean lifetime (&fs) is small compared to the mean collisiontime (&ns) at the gas pressures used in the experi-ment (1.5 mbar). The validity of this assumption isdemonstrated by the similarity of the *E

c/*x

values.

4. Simulation of yield curves

4.1. The yield function

The experimental yield curves which describea de"ned c transition triggered by a capture reac-tion (e.g.: 20Ne(a, c)24Mg (g.s.): see Fig. 5) can becompared to a simulated yield function. The simu-lation has to take into consideration the "nite ex-tension of both the gas target and the c detector. Inorder to calculate this function the following termshave to be well known: (i) the density of the targetgas, o(x), (ii) the mean particle energy, EM

x, together

with the energy distribution of the projectiles,A

x(E), both given for all positions x along the beam

axis, (iii) the energy dependence of the capture crosssection, p(E) (including the branching ratio of thefollowing c transition), (iv) the detector e$ciencye(x,E

c), and (v) the number of elastically scattered

projectiles, I1(E), detected in the particle detector.

Due to both the interaction of the projectileswith the atoms of the target gas and the ripple ofthe accelerator voltage [1], for a given positionx along the beam axis an energy distribution A

x(E)

of the projectiles is observed. Multiplying this dis-tribution function by the capture cross section p(E)and integrating over the energy E, a term is ob-tained which is proportional to the number ofc rays emitted from the position point x.

In order to simulate one point of the yield curve,characterized by the energy of the projectiles, E!##

1,

the contributions from all positions x have to betaken into account. The density of the target gas atthese positions x is given by o(x). In the c detectorthese events are detected with the e$ciency e(x, E

c).

The number of c rays registered in the Ge detectortherefore can be written

Ic"N

1PxAP

E

Ax(E)p(E) dEBo(x)e(x,E

c) dx. (4.1)

The number of particles, elastically scattered anddetected in the particle detector is given as

I1"N

1o0

dpdX

(E0, 0)l

1dX. (4.2)

Here N1is the number of incident particles, $p

$X (E0,0)

the di!erential cross section for elastic scattering,E0is the mean energy of the projectiles in the center

of the target chamber, 0 the scattering angle, o0

thedensity of the target gas in the chamber, and l

1dX

the geometry factor (Eq. (2.1)) of the particledetector.

Combining Eqs. (2.2), (4.1) and (4.2) one obtainsthe yield for one point of the yield curve

>4*.

(E!##1

)

,AI

cI1BE

!##1

"

1

dpd X

(E0, 0)l

1dX

]P`=

x/~=AP

E

Ax(E)p(E) dEB

o(x)

o0

e(x, Ec) dx. (4.3)

In the following sections the terms o(x),Ax(E), and

p(E), are discussed. The cross section $p$X(E0

,0) has tobe determined in a separate experiment. In thesimplest case this cross section is given by theRutherford formula.

4.2. Gas-density proxle and mean energy loss

The gas density distribution o(x) along the beamaxis is determined by the target chamber geometry(see Fig. 1) and by the pumping speed of the recir-culation pumps in front and in the back of thewindowless gas target chambers. Fig. 12 (upper


Fig. 12. Cross section of the target chamber (upper part) and the calculated gas density pro"le in the chamber (lower part).

part) illustrates the cross section of the target cham-ber. In the lower part of Fig. 12, the gas densitydistribution is shown according to both the laws ofgas dynamics in a well-de"ned chamber geometryand the known pumping speed of the pump [2,8].

Knowing the gas density distribution, o(x) andusing stopping power values as given by Ziegler[16], the mean energy loss of projectiles along thebeam line from the accelerator up to position

x,*Ex, can be calculated. The knowledge of *E

xis

required for the computation of the energy straggl-

ing (see Section 4.3). The entity *Ex

can be deter-

mined from Fig. 12 if its value *Ex/0

, i.e. in thecenter of the chamber, is known.

For the two projectile energies corresponding toEx"10 917 keV and E

x"11 453 keV in the reac-

tion 20Ne(a, c)24Mg, the mean energy loss *Ex/0

was determined experimentally for "ve di!erent gaspressures in the target chamber. The symmetricdouble peak in the c-spectrum for the transition2`P0`(*¸"2,*M

L"0) was used as an indi-

cator for the alpha-particle energy being at theresonance energy in the center of the target cham-ber (see Section 3.4). The result is shown in Fig. 13.

From these data and using values for the stop-ping power from [16], the relevant values of themean energy loss can be determined. Table 2lists values for a gas pressure of 1.5 mbar in thetarget chamber. From Fig. 13, furthermore theenergy o!set of the accelerator can be extracted;depending on the ion source condition, the o!setranges from 20 to 30 keV for the two series ofmeasurements.

4.3. Energy straggling and accelerator ripple

The interaction of charged projectiles with elec-trons of the target atoms is a statistical process.Hence, energy straggling occurs. When penetratingthrough matter of a thickness *R"x!x

0, or

n2*R (atoms/cm2), the projectiles su!er energy loss

described by a distribution Ax(E). The shape of this

distribution depends on the areal density n2*R,

and therefore on the mean energy loss *Ex.

For very thin targets, Ax(E) takes on the shape of

the (strongly asymmetric) Landau distribution[17]; for su$ciently thick targets it takes ona Gaussian. The distribution given by Symon


Fig. 13. Energy loss of the projectiles in the target chamber at di!erent pressures. At high gas pressures the beam energy has to beincreased to shift the resonance into the center of the chamber. The position is determined accurately by the doublet structure of the2`P0` transition.

Table 2

Mean energy loss *E at a pressure p"1.5 mbar

Mean energy loss *E (keV)

Ea

Ex(24Mg) to the entrance of to the center of in the

(keV) (keV) the chamber the chamber chamber

1926 10917 6.4$0.5 11.0$1.0 9.3$1.02570 11453 6.0$0.5 10.0$1.0 8.0$1.0

[18,19] and independently by Vavilov [20] con-tains both the Landau and the Gauss distributionas the two extremes. This distribution takes on

a Gaussian [20] if the mean energy loss *Ex

islarger than the energy E

.!9transferred in a colli-

sion of the projectile with an electron of the targetatom. In our case (a!20Ne system, p"1.5 mbar,E

a+2 MeV), this condition is satis"ed: the max-

imum energy transferred per collision is E.!9

+

1 keV, the mean energy loss of the projectiles up to

the target chamber is *E+6 keV (see Table 2).Hence, the energy distribution is almost a Gaus-sian. According to Bohr [21], the variance of thisdistribution is given by

X2B"4pZ2

1Z

2e4n

2*R

"2.6]10~19Z21Z

2n2*RkeV2 (4.4)

where Z1

and Z2

are the atomic numbers of theprojectile and the target, and n

2*R is the thickness

of the target material in atoms per cm2. Taking thevalues mentioned above, we get X2

B+2 keV2.

In the Bohr formula, the velocity distribution ofthe electrons in the target atoms is neglected. Forprojectile velocities close to electron velocities onthe Bohr orbits, these e!ects become signi"cant.They determine the shape of the Bragg curve at andbelow its maximum. For projectile energies close to2 MeV in the a!20Ne system, we are just abovethe maximum. In the model of Lindhard andSchar! [22], these e!ects and their in#uence on theenergy straggling was accounted for. Using thismodel, Besenbacher et al. [23] performed calcu-lations on the a!20Ne system and comparedthem with experimental data. The authors "nd that


Bohr's formula (Eq. (4.4)) renders values for X2B

invery good agreement with measured data forE

a+2 MeV.A signi"cant contribution to the energy distribu-

tion Ax(E) of the projectiles is given by the ripple of

the Dynamitron accelerator. These variations inthe accelerator voltage ;!## come about by meansof a non-ideal DC conversion in the machine. Ina calculation, these variations can be approximatedby taking two e!ective accelerator voltages;

B";!##$d

r/e

0whose separation was measured

by Hammer et al. [1] on an ideal machine con"g-uration: 2d

3+1.7 keV.

The energy distribution Ax(E) at position x can

therefore be written as a sum of two Gaussians

centered around E#d3

and Ex!d

3, respectively.

Their widths are given by the variance X2B

causedby energy straggling, and their separation is givenby the accelerator ripple 2d

3/e

0

Ax(E)"

1

2J2pXBGexpC!

(E!Ex#d

3)2

2X2B

D

#expC!E!E

x!d

32X2

BDH. (4.5)

This energy distribution Ax(E) can be evaluated for

any position x along the beam line. Once the mean

energy loss of the projectiles, *Ex, is known at any

given x, both the mean projectile energy, Ex, and

the variance X2B

can be determined using tabulatedvalues by Ziegler et al. [16] and using Eq. (4.4),respectively.

4.4. Cross section and resonance strength

The cross section of a capture reaction fora single-level resonance is described by theBreit}Wigner formula

p(E)"p1

k2u

C1C

c(E!E

3%4)2#(C/2)2

. (4.6)

The statistical factor u is given by the angularmomentum J of the excited state in the compoundnucleus and the spins of the projectiles J

1and the

target nucleus JT, respectively

u"

2J#1

(2J1#1)(2J

T#1)

. (4.7)

E3%4

is the resonance energy, C1

and Cc

are thepartial widths, and C"C

1#C

cis the total width

of the resonance if no further decay channels areopen. All energies and widths are in the center-of-mass system.

The partial width C1

is energy dependent sincethe incident particle must penetrate through boththe Coulomb and the centrifugal barrier. In thesimplest model, assuming a rectangular nuclear po-tential with radius R

N, the particle width C

1is given

[3] by

C1"

3+2kkR

N

Pl(E,R

N)H2

1. (4.8)

In this equation k is the reduced mass and Pl(E,R

N)

is the penetration factor, which can be describedby the regular and irregular Coulomb functionsFl(E,R

N) and G

l(E,R

N), respectively

Pl(E,R

N)"(F2

l(E,R

N)#G2

l(E,R

N))~1. (4.9)

H21

is the reduced particle width of the resonance. Itindicates the probability to "nd the con"gurationof the incoming channel t

T?t

Pin the resonance

state t3%4

. The numerical value of H21

is model-dependent.

The energy-dependent particle width C1(E) can

be expressed by its width at the resonance energy

C1(E)"

k

k3%4

Pl(E,R

N)

Pl(E

3%4,R

N)C

1(E

3%4). (4.10)

Then the expression for the simulated yield resultsfrom Eqs. (4.3) and (4.6) under consideration of theenergy dependence of C

1

>4*.

(E!##1

)

"

1

dpdX

(E0, 0)l

1dXP

=

~=

o(x)

o0

e(x,Ec)

]APE

p

k2A

x(E)u

C1(E)C

c

(E!E3%4

)2#AC

1(E)#C

c2 B

2dEBdx

(4.11)


Fig. 14. Simulated yield curves for R(10917)Pg.s., R(10 917)P1369 and (1369)Pg.s. compared to experimental data points measuredat a pressure of 1.5 mbar. The perfect agreement can only be obtained for an accelerator ripple of 2d

r"3.4 keV.

where Ax(E) is calculated by using Eq. (4.5), C

1(E)

by using Eqs. (4.8) and (4.9), and e(x,Ec) by

using the code GEANT (see Section 3). For narrowresonances (C;E

3%4), the energy dependence

of C1

and hence of C can be neglected. However,if the condition (C;E

3%4) is not satis"ed, the

energy dependence of C1(E) has to be accounted

for in both the numerator and denumerator ofEq. (4.11).

The simulations were computed by means ofa program that determines Eq. (4.11) except for thevalue of the resonance strength

uc"uC1(E

3%4)C

cC(E

3%4)

. (4.12)

So, the program YIELD [24] calculates the expres-sion >@

4*.(E!##

1) in units of (eV)~1

>4*.

(E!##1

)"uc>@4*.

(E!##1

). (4.13)

The resonance strength uc is determined indepen-dently by "tting the simulated to the experimentalyield curve. For broad resonances, uc can be ap-proximated by uc"uC

c.

Examples of simulations and some results arediscussed in the following subsection.

4.5. Examples for yield simulations

The following examples were chosen from theinvestigations of the reaction 20Ne(a,c)24Mg [5,25](see Fig. 5). First, we consider the resonance atEx"10 917keV. For this resonance, all widths are

known: C"7.5 eV and Cc"0.48 eV [12], hence

Ca"7.0 eV. This means that the observed width of

the yield resonance is determined by the mean

energy loss *E in the target, by the width of theenergy distribution due to straggling, and by theripple of the accelerator, d

3. The mean energy loss

*E is well known using the results described inSection 4.2 (see Fig. 13); also, energy straggling isknown using results of Besenbacher et al. [23]. Ananalysis of the shape of the yield curve for the g.s.transition reveals an accelerator ripple valuegreater than found by Hammer et al. [1] underideal conditions. In Fig. 14 we show the perfectagreement between the simulated and the experi-mental yield curve. This agreement can only beobtained using values of 3 keV (2d

3(4 keV.

A detailed analysis [25] performed on several targetgas pressures results in 2d

3"3.4 keV. Fig. 14 shows

the experimental yield curves R(10 917)Pg.s.,R(10 917)P1369 and (1369)P g.s. at p"1.5 mbar


Fig. 15. Yield curves for the transition R(11 016; 2`)Pg.s.The calculated curves agree well with the experimental datapoints only for a total width 5 eV(C(20 eV. A similarresult was obtained from the analysis of the transitionR(11 016; 2`)P1369;2`.

with their simulations in very good agreementusing 2d

3"3.4 keV. The g.s. transition shows an

asymmetrical double peak. The reason for thisfeature found by simulation is due to the character-istic angular distribution of the transition 2`P0`

(see Section 3.3); the asymmetry is caused by theenergy straggling since the width of the distributionfunction A

x(E) increases with x. The resonance

strength was determined to uc"(2.67$0.11) eVcompared to uc"(2.2$0.3) eV [15] and uc"(2.5$0.8) eV [26].

In the following two examples, resonance widthsC will be determined using simulations of experi-mental yield resonances measured over a verybroad dynamic range.

The transition R(11 016; 2`)Pg.s. is shown inFig. 15. Two experimental yield points of this curveat E

a+2.03MeV and 2.06 MeV di!er in their

dynamic range compared with the point at the yieldmaximum by almost four orders of magnitude. Forthis resonance, only a lower limit C'1.4 eV hadbeen known [12]. The simulation reveals a value5 eV(C(20 eV. A similar result was obtainedfrom the analysis of the transition R(11 016; 2`)P1369; 2`. The resonance strength was determinedto uc"(1.77$0.07) eV in comparison to uc"(1.5$0.2) eV [15] and uc"(1.6$0.5) eV [26].

For the resonance at Ex"11 390 keV (1~) the

value C"(0.5 keV) was found in the literature [12].

In this experiment, the low energy tail of the reson-ance could be measured over "ve orders of magni-tude (see Fig. 5). Comparison of the data and thesimulation reveals the width C"(0.4$0.1) keVand the resonance strength uc"(0.72$0.03) eV.Older values are uc"(0.46$0.10) eV [15] anduc"(0.6$0.2) eV [26].

It has to be pointed out that the analysis of theyield curve over several orders of magnitude allowsthe determination of the total width of a resonanceeven in a region of the total width 1 eV)C)

1 keV, where other experimental techniques cannotbe used without di$culties. A direct determinationof C from the measured p(E) curve becomes di$cultfor widths below C+1 keV, and lifetime measure-ments like the Doppler shift attenuation methodbecome problematic for very short lifetimes corre-sponding to widths above C+1 eV.

The simulation also shows the contributions tothe yield curve from individual points x of thetransition R(11390; 1~)Pg.s. along the beam axis.The results of this calculation are illustrated inFig. 16 for a number of projectile energies. Theintegral over each curve in Fig. 16 corresponds toa point on the yield curve. For the resonance local-ized in the center of the gas target chamber (Fig.16h), the yield distribution curve is symmetric withrespect to the center of the chamber and the valuesare very large. When lowering the projectile energy,the maximum of the curve moves towards the inletof the target chamber (Fig. 16f, g). Lowering theenergy even more, the resonance can no longer befound within the chamber, its virtual position isthen in front of the target chamber. Since the gasdensity o(x) and the counter e$ciency e(x,E

c) only

show signi"cant values between !3 cm )x)3cm, i.e. across the chamber dimensions, and due tosteep slopes of the Breit}Wigner cross section, themaximum of the emission distribution stays localiz-ed at the target chamber inlet, while the yield valuesdecrease rapidly with decreasing projectile energy(Fig. 16c}e). For Ea+2300 keV (Fig. 16b ), the yieldis 5}6 orders of magnitude smaller than in themaximum of the yield curves. The distributioncurve is then almost symmetric with respect to thecenter of the chamber since the low energy tail ofthe cross section is almost constant across the tar-get chamber at this projectile energy. The curve is


Fig. 16. Contributions to the yield in arbitrary units (diagrams a}i) at di!erent projectile energies for the transitionR(11390;1~)P1369; 2`. The vertical lines indicate the position of the target chamber where both the gas pressure and the detectione$ciency are high. Lowering the projectile energy (indicated in the diagrams in keV) shifts the main contribution from the center (h) tothe inlet (f ) of the chamber. At energies far below the resonance, the cross section becomes almost constant in the chamber, and the maincontribution comes from the chamber again (a,b). Note that the scale of the diagrams covers several orders of magnitude!

then shaped by the detector e$ciency e(x, Ec), and

the maximum returns back to the center of thetarget chamber.

5. Summary

The analysis of capture reactions with an ex-tended gas target was discussed in detail. Manyingredients have to be well-understood for a re-

liable data analysis: (i) the e$ciency of the HPGedetector; (ii) the density pro"le of the gas in thetarget chamber; (iii) the energy loss of the projec-tiles; (iv) the energy straggling of the projectiles;(v) the energy distribution of the beam energy fromthe accelerator.

The in#uence of all these ingredients was ana-lyzed experimentally by comparison of simulatedand experimental results for the c-ray spectraand the c-ray yield of the 20Ne(a, c)24Mg reaction.


Especially the Doppler shift and Doppler broaden-ing together with the angular distribution of E2(2`P0`) transitions gave detailed insight into theexperimental parameters listed above.

Because all the parameters are well understood,now an analysis of the shape of the yield curve ispossible over several orders of magnitude. Fromthis analysis it is possible to derive not only thestrength, but also the total width of resonances ina wide range. This can be done even in the region ofthe total width around several eV where othermethods cannot be used easily.

We have focused on the analysis of resonantcapture in this work. Of course, an analysis ofnon-resonant capture is also possible: one has toreplace the Breit}Wigner cross section in Eq. (4.6)by a slowly varying S factor curve which usuallymakes the analysis easier. One example is theanalysis of the non-resonant capture reaction16O(p, c)17F. The experimental low-energy datadown to about 200 keV proved the halo propertiesof the "rst excited 1/2` state in 17F [11].

The achieved detection limit with our set up is inthe order of uc+1l eV or p+300 pb. Many cap-ture reactions of astrophysical relevance have notyet been analyzed at this level of sensitivity.A closer approach to the Gamow window willbecome possible with the presented experimentaltechnique and the careful data analysis.

Acknowledgements

We thank U. Kneissl for the support of the ex-periments at the Institut fuK r Strahlenphysik inStuttgart, and we want to thank the Dynamitrongroup for the high-current beam. This work wassupported by Deutsche Forschungsgemeinschaft(DFG) under contracts Sta290/5 and Graduierten-kolleg Mu705/3.

References

[1] J.W. Hammer, B. Fischer, H. Hollick, H.P. Trautvetter,K.U. Kettner, C. Rolfs, M. Wiescher, Nucl. Instr. andMeth. 161 (1979) 189.

[2] J.W. Hammer et al., in preparation.[3] C. Rolfs, W. Rodney, Cauldrons in the Cosmos, The

University of Chicago Press, Chicago, 1988.[4] P. Mohr, V. KoK lle, S. Wilmes, U. Atzrott, G. Staudt, J.W.

Hammer, H. Krauss, H. Oberhummer, Phys. Rev. C 50(1994) 1543.

[5] V. KoK lle, Ph.D. Thesis, Univ. TuK bingen, 1997.[6] R. Brun, F. Carminati, GEANT Detector Description and

Simulation Tool, CERN Program Library, Long WriteupW 5013 ed., CERN, Geneva, 1993.

[7] J. GoK rres, K.V. Kettner, H. KraK winkel, C. Rolfs, Nucl.Instr. and Meth. 177 (1980) 295.

[8] H. Knee, Ph.D. Thesis, Univ. Stuttgart, 1994.[9] S. Wilmes, P. Mohr, U. Atzrott, V. KoK lle, G. Staudt,

A. Mayer, J.W. Hammer, Phys. Rev. C 52 (1995) R 2823.[10] S. Wilmes, Ph.D. Thesis, Univ. TuK bingen, 1996.[11] R. Morlock, R. Kunz, A. Mayer, M. Jaeger, A. MuK ller,

J.W. Hammer, P. Mohr, H. Oberhummer, G. Staudt,V. KoK lle, Phys. Rev. Lett. 79 (1997) 3837.

[12] P.M. Endt, C. van der Leun, Nucl. Phys. A 521 (1990) 1.[13] A. Anttila, J. Keinonen, M. Hautala, I. Forsblom, Nucl.

Instr. and Meth. 147 (1977) 501.[14] H.J. Rose, D.M. Brink, Rev. Mod. Phys. 39 (1967) 306.[15] P. Schmalbrock, H.W. Becker, L. Buchmann, J. GoK rres,

K.U. Kettner, W.E. Kieser, H. KraK winkel, C. Rolfs, H.P.Trautvetter, J.W. Hammer, Nucl. Phys. A 398 (1983) 279.

[16] J.F. Ziegler, Helium Stopping Powers and Ranges in allElements, Pergamon, New York, 1978.

[17] R. Landau, J. Phys. USSR 8 (1944) 201.[18] W.R. Symon, Ph.D. Thesis, Harvard University, 1948.[19] B. Rossi, High-Energy Particles, Prentice Hall, New York,

1952.[20] P.V. Vavilov, Sov. Phys. JETP 5 (1957) 749.[21] N. Bohr, Mat. Fys. Medd. Dan. Vid. Selsk. 18 (8) 1948.[22] J. Lindhard, M. Schar!, Mat. Fys. Medd. Dan. Vid. Selsk.

27 (15) (1953).[23] F. Besenbacher, J.U. Andersen, E. Bonderup, Nucl. Instr.

and Meth. 168 (1980) 1.[24] U. KoK lle, Computer program YIELD, Univ. TuK bingen,

unpublished.[25] U. KoK lle, Diploma Thesis, Univ. TuK bingen, 1996.[26] P.J.M. Smulders, Physica 31 (1956) 973.


capture reactions at astrophysically relevant energies: extended gas target experiments and geant...

Documents