Applications of kinematics to nuclear reactions using radioactive ion beams

Nuclear Instruments and Methods in Physics Research A316 (1992) 143-146 North-Holland

&METHOD6 IN I g g Y ~ Secl,onA

Applications of kinematics to nuclear reactions using radioactive ion beams N.R. Fletcher Physics Department, Florida State UniL'ersity, Tallahassee, FL 32306, USA Received 20 November 1991

The observation of nuclear reactions by use of radioactive ion beams, as non-accelerated secondary beams, has been limited by relatively poor energy resolution due to the large energy spread in the secondary beam, its large angular spread and the large beam spot size. In addition there is often an uncertain beam composition. This work shows that under certain conditions all of these problems can be minimized by employing kinematic coincidence for complete determination of the secondary reaction final state, from which secondary beam particle, energy, and angle, can be determined. A partial cancellation can lead to even better energy resolution in the Q-spectrum than in the beam energy spectrum.

SLi at 0.981 MeV are extracted only with great difficulty under such conditions. In nuclear and particle physics it has been a common practice for some time to use complete kinematic determination of multiparticle final states for the determination of mass and momentum of a missing particle. Those applications depend on knowing the initial momentum accurately. In the case of radioactive secondary beam experiments, it is the beam particle which has an uncertain momentum. If we perform experiments in which the total final state momentum for all of the particles is measured, then the momentum of the beam particle which produced the reaction can easily be calculated. We describe in the following paragraphs the resolutions expected for beam energy and

1. Introduction One of the most successful radioactive beam facilities to date is the University of Notre D a m e / U n i v e r sity of Michigan collaboration at the U N D FN-Tandem accelerator laboratory, which has produced ~ 107 pps of SLi ions [1,2]. This secondary beam typically has an energy of ~ 14 MeV, energy widths of ~ 0.5 MeV FWHM and ~ 1.0 MeV FWTM, an angular divergence cone half angle of 3 deg, and a beam spot diameter of ~ 5 mm. These properties combined to produced scattered 8Li energy spectra with energy resolution of approximately 1 MeV in a recent report of t2C+SLi elastic and inelastic scattering [2]. The data for inelastic scattering to the first excited state of det 3

z

\

det 4

det 3 ~p \

:~ It I '

det 4

det 3

det4 ~\

I~31

P~ ~'~

P'I (a)

1-P

(b)

(c)

Fig. 1. Illustration of particle detection geometry for (a) well defined beam momentum, (b) large angular divergence, and (c) large target illumination area. 0168-9002/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved

144

N.R. Fletcher / Applications o f kinematics

angle, and for energies in the final state from kinematically complete measurements. The range of application and the limitations of the method are also discussed.

2. Kinematic correction method

In a two body final state reaction we will number the particles in a standard notation by 2(1, 3)4. The momenta of the final state particles can be determined by use of xy-position sensitive, E-AE counter telescopes. The momentum of the beam is of course given by p~ = P3 + P4, as illustrated in fig. I a. In terms of the measured quantities, the beam energy and the scattering angle are then given by the equations, 1

El = -m.~ - [ E3m3 + Eam4 + 2]/E~E4m~m4cos 0341, " "

(1) 013 = arctanlsin O~l(liE3m31E4m,l + cos 034)1, (2) and the Q-value for the reaction defined by Q = E 3 + E 4 - E l becomes, Q=E3

l-~

+E4 l-~ m I

In 1

2 -- - - d E 3 E 4 m 3 m 4 m !

cos

034.

(3)

The angle, 034 , is the angular separation for the coincidence detection of the two final state particles, 3 and 4. For each event in which particles 3 and 4 are detected in coincidence, the energies E 3 and E4, and the angle between the particles, 0~, are recorded, and from these events, spectra for El, 0 l, and Q can be generated. While the spectral widths in spectra for E~ and 0~ would reflect the initial beam properties, in the case referenced above [2] values of FWHM of ~ 500 keV and 3 deg respectively, the FWHM in the Q-spectrum and the spectrum of 013 values can be an order of magnitude less as we shall soon illustrate. Note that 013 is the true scattering angle, while 0~ is the angle of the secondary beam particle relative to the z-axis. In fig. lb, we illustrate the effect of an angular divergent beam particle. Particles 3 and 4 have their momentum measured by detectors 3 and 4 respectively, which are at the same angular positions as in fig. l a. The magnitude of these measured momenta are however different from the case in fig. la, and that difference leads to different calculated values of 0~3. The angle of the incoming beam particle, 01, would be the difference between 0~3 and the angular position relative to the z-axis of particle 3 as determined by the position sensitive detector number 3. The effect of a

finite beam spot size is illustrated in fig. lc. The case illustrated is nearly equivalent to the case in fig. lb, since the magnitude of the detected momenta are the same and the calculated value of 013 is also the same. It is clear then that the effects of beam divergence and the lateral translation of a beam particle are essentially equivalent, however the difference between 0L~ and the angular position of the point of detection of 3 relative to the z-axis, is representative of the beam phase space (d01 d X l) rather than the angular divergence alone. An additional difference arises since 034 is approximately equal to 0~4 only when particles 3 and 4 are detected at nearly the same angle but on opposite sides relative to the z-axis. This effect is discussed later. The errors expected in the determination of values for E l, 013, and Q for each event are calculated from the partial derivatives of eqs. (1), (2) and (3), with respect to the measured parameters, E 3, E4, and 034, times the errors in determining those parameters and adding them in quadrature. Methods for measuring position resolution and for calibrating detectors in energy as a function of position and in position as a function of energy have been discussed elsewhere in detail [3]. Illustrations of the errors in determining 0~3, El, and Q for the case of 8Li elastic scattering from ~:'C at 32 MeV and 14 MeV bombarding energy are given in fig. 2. These calculations are based on assumed constant detector resolutions of 100 keV for the energy measurement of particles 3 and 4, and an angular resolution in 034 of 0.002 rad, representing the combined position resolutions readily available [3] with 1 cm diameter double position sensitive, E-AE detectors at l0 cm from the target. It is interesting to note that at forward angles the resolution in the Q-spectrum would exceed even that of a single detector experiment (100 keV) with an incident beam of perfect energy and angular resolution. This is because more information than just the energy of one particle type is going into the determination of the Q-value. Of course energy resolution of detectors is in reality also energy dependent instead of a constant 100 keV. If the minimum energy dependent energy resolution were used, from for example, Bass et al. [4], the calcu!aled values A E ! and AQ would be reduced, especially at back angles.

3. Discussion of advantages and limitations of the method

There are several factors which have not thus far entered the discussion which must be considered. Some of these factors will severely limit the applicability of the method. Other factors, which are characteristic of ordinary particle spectroscopy with detectors, are of

145

N.R. Fletcher / Applications of kinematics little importance with this method. Below we discuss firstly the advantages, followed by the limitations and disadvantages. We have already illustrated in fig. 2 that the poor beam quality factors of energy resolution and angular divergence are no longer effects which need concern us, since we see that application of the method would improve observed final state energy resolution by a factor of ten [2] as well as identifying the reaction angle more accurately by a factor of ten. Because of this a great advantage is realized in that the thickness of the primary target can be substantially increased, thus increasing the radioactive beam luminosity while introducing an additional energy spread which is now much less important. The increased energy spread in the secondary beam may slightly increase the beam spot size, however the effect is only minor compared to the general problem of beam spot size effects. There is another very important advantage, which must be appreciated especially if facilities similar to UND are to operate at higher energies. Due to complete identification of the final state and hence the initial state, the possible ambiguity arising from secondary beam con-

0.6 0.5

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0.1 El <3 t~ ~5 o.o

//

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,,

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,r~ Mostlav~le energyrange

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(b) \

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,'='"

,,o ,,

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"', if.,

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u.

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0.2 0.1 0.0

I

0

20

AEI(MeV)

2 eV) 40 I

610

I

I

I

I

80 100 120 140 013 (deg)

160180 I

Fig. 2. Expected full width at half maximum values in 0~3, El, and Q. when measuring E 3, E4, and 034 with resolutions of 100 keV, 100 keV and 0.002 rad respectively. These calculations do not include effects of secondary target thickness or secondary beam spot size. (a) E I = 32 MeV: (b) E= = 14 MeV.

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i

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i

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i

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, , , , I 80 100 120 140 160 180 013 (deg)

Fig. 3. Effect of various secondary beam spot sizes (FWHM) on observed reaction Q-value.

tamination due to unwanted primary target reactions is no longer a problem provided their yield does not totally dominate the desired secondary beam particles. The effect of finite beam spot size on the calculated Q-value energy resolution has been calculated by an analytical account of the effect of the difference between 0'34, the actual value of the angle between patticles 3 and 4, and the measured value, 0 N (see fig. lc). The result of a sample calculation, using the same detector geometry and resolutions mentioned earlier, are shown in fig. 3 for a variety of A X] values, the FWHM in beam spot illumination. The value AX~ (FWHM) = 3 mm might be representative of the 5 mm beam diameter of the UND facility [2]. This figure illustrates clearly that it would be an advantage to use a shorter focal length for the focusing of the secondary beam to decrease A X~ while increasing the angular spread of the secondary beam. Note that at approximately 50 deg there is no effect of beam spot size. This is for the case when 0]3 and 0~4 are equal and the resolution defect cancels for A X=/(detector distance) <<1. A very important limiting factor in obtaining greatly improved energy resolution by this method is the secondary target thickness, since both of the reaction products must escape the target with an energy which is still a fair representation of their energies at the reaction site in the target. The experimenter no longer has the luxury of using a relatively thick secondary target to match the poor beam energy resolution, however the target angle can be adjusted to insure that the combined energy loss of the two finai state particles is minimized. A target thickness a factor of three less than normally used in these radioactive beam experiments would provide an estimated energy resolution of ~ 200 to 300 keV. This need not reduce count rate since the primary target thickness may be considerably

146

N.R. Fletcher / Applications of kinematics

increased when this coincidence detection method is used. The stability of the reaction products is also of prime interest for the application of this method. Any particle instability in particles three or four would almost surely destroy all advantages unless the particle decay energy were less than or the order of 100 keV. For details on the effects of multiparticle final states the reader is referred to ref. [3]. The effects of ~/-ray decay will also be detrimental to the resolutions projected by the method, but the effect is rather small. The energy change of a reaction product due to its ~/-ray decay is extremely small, but the angular deviation of the nuclides momentum can have a significant effect. If one assumes isotropic decay by ~/-ray emission, the average angular spread introduced in the measured angular position of the reaction product will be d0(radians) ~ 0.074Ev/MvrM-E, where Ev is the ~/-ray energy in MeV, and M and E are the mass and energy of the decaying nuclide in amu and MeV respectively. This loss in resolution of the measured angular position of a final state particle would be of little importance for Ev ~ 1 MeV and the masses and energies of particles typical of current UND experiments, but ~/decay energies of 5 to 10 MeV would have significant effects. It is interesting to note that for life times the order of the time of flight to the detector, a broadened and a narrow line would appear superimposed in the resultant Q-spectrum and the yield ratio in the two width components would be characteristic of that lifetime. This is similar to the doppler shifted ~/-ray energy technique for measuring lifetime, but here the line shape alteration is a result of a direction spread in the detected particle, not an energy spread. In general one would obtain a cross section for a particular excited state of a reaction product by directly converting the yield in a group in the Q-spectrum in the usual way. This would still be the procedure if one of the double position sensitive detectors, say detector 4, were sufficiently large, and properly placed, that the solid angle of the other, detector 3, would be the defining solid angle for the cross section calculation. This means that the size of detector 4 would have to be large enough to accommodate the expanded kinematic space caused by the considerable angular and ~patial spread in the secondary, beam. If this size accommodation is not made, then it would be necessary to make a simulation calculation for the detection efficiency which would require considerable knowledge of the secondary beam.

A different kind of limiting factor is that the two final state particles should be comparable in mass, probably within a factor of two or three, since both particles must be sufficiently far in angle from the diverging secondary beam that they can be cleanly detected. This of course also limits the angular range of application, but a very useful range of 40 to 80 deg seems to be practical in the scattering experiment which has been considered in these calculations, however it is clear that increased secondary beam energy would be advantageous. Although there are number of factors limiting the application of this method, the net effect would be greatly improved energy resolution without a tremendous loss in count rate. In summary one would use a thicker primary target, a thinner secondary target, a shorter solenoid focal length to decrease beam spot size and increased secondary beam angular divergence, and one should also keep in mind that the position sensitive detectors would allow an increased count rate without loss of resolution due to kinematic spread.

Acknowledgements The author wishes to thank F. Becchetti, D. Caussyn, K. Kemper, J. Kolata and R. Warner for many discussions of the UND facility and helpful comments on this manuscript.

References [1] J.J. Kolata, A. Morsad, X.J. Kong, R.E. Warner, F.D. Becchetti, W.Z. Liu, D.A. Roberts and J.W. Janecke, Nucl. Instr. and Meth. B40/41 (1989)503; F.D. Becchetti, W.Z. Liu, D.A. Roberts, J.W. Janecke, J.J. Kolata, A. Morsad, X.J. Kong and R.E. Warner, Phys. Rev. C40 (1989) R1104; F.D. Becchetti et ai., Nucl. Instr. and Meth. B56/57 (1991) 554. [2] R.J. Smith, J.J. Kolata, K. Lamkin, A. Morsad, F.D. Becchetti, J.A. Brown, W.Z. Liu, J.W. Janecke and D.A. Roberts, Phys. Rev. (243 (1991) 2346. [3] D.D. Ca~sy,, G L G¢l~il~,A. Toppic~ -..~ " ~ "" ~ Fletcher, ,~.,,. Nucl. Instr. and Meth. A286 (1990) 348; D.D. Caussyn, G.L. Gentry, J.A. Liendo and N.R. Fletcher, Phys. Rev. C43 (1991) 205. [4] R. Bass et al., Nucl. Instr. and Meth. 130 (1975) 125.