Analysis of data from reactions of the form A+B→a1+a2+a3

NUCLEAR

INSTRUMENTS

AND

METHODS

47 (I964) 2 6 0 - 2 6 4 ; © N O R T H - H O L L A N D

ANALYSIS OF DATA F R O M REACTIONS OF THE F O R M A + B P.A. A S S I M A K O P O U L O S

PUBLISHING

--+ a t - r a 2 + a

CO.

3

and N . H . G A N G A S

Nuclear Research Center " D e m o c r i t u s " , Aghia Paraskevi, A ttikis, Greece Received 3 July 1966 T h e advantages and s h o r t c o m i n g s o f the various methods, currently in use for drawing i n f o r m a t i o n f r o m a n d presentation o f data o f " c o m p l e t e " experiments on nuclear reactions with

three particles in the final state, are briefly considered. A new m e t h o d for h a n d l i n g data from such experiments is presented.

1. Introduction The study of nuclear reactions with more than two particles in the final state has recently been the subject of a considerable amount of experimental and theoretical work1). This is mainly due to the relatively recent development of new experimental facilities in the field, such as the solid state detector and multi-dimensional analysers. The information which can be drawn from the study of such nuclear reactions has been extensively demonstrated in the literature 1'2); reaction mechanisms, phase shifts for processes in which either the scatterer or the scattering particle or even both are unstable and cluster configuration of nuclei are only a few examples. A complete measurement of the final state of a threebody reaction requires the determination of the momenta of the three outgoing particles i.e. the measurement of nine scalar parameters. The number of independent parameters in the final state, however, is immediately reduced to five if energy and momentum conservation are taken into account. In studying a three-body nuclear reaction one, therefore, is left with the choice of either performing an "incomplete" experiment, usually measuring the momentum of only one of the outgoing particles, or performing a "complete" experiment by measuring five independent parameters of the final state, e.g. the m o m e n t u m of one particle and the direction of emission of a second. In the first case all information will be drawn from the analysis of the energy spectrum of the measured particle although in most cases this is not sufficient for an unambiguous interpretation of the experimental results obtained3). The actual practice in conducting a three-body " c o m p l e t e " experiment is to measure six of the final state parameters, usually the momenta of two of the outgoing particles in coincidence and use this overdetermination of the final state for the elimination of background effects. It is the purpose of the present paper to consider first briefly the advantages and shortcomings of the

various methods, currently in use for drawing information from and presentation of data of "complete" experiments on nuclear reactions with three particles in the final state. A new method for handling data is presented in part 3 and the advantages of this procedure, in the opinion of the authors, over the procedures more often used, is discussed. A series of F O R T R A N II subroutines, processing the data from a three-body nuclear reaction following this new method is also available. The kinematics for "incomplete'' and " c o m p l e t e " experiments has been discussed elsewhere*) and, therefore, only the main results will be used in this paper, when necessary, without derivation.

260

2. Presentation and processing of data in a "complete" experiment 2.1. GENERAL CONSIDERATIONS In the typical experimental set up for the study of a reaction of the form A+B--*al+a2+a

3,

(t)

B is in the form of a target while the nucleus A is the bombarding particle. Two detectors D1 and D2 placed at polar angles 01, $1, 02 and ~b2, with respect to the beam measure in coincidence the kinetic energies T1 and T2 of particles a~ and a2, respectively. Energy and m o m e n t u m conservation predict that T1 and 7"2 will be the solutions of the equation Q = (1 +

rollins) Tj +(1 + melm3) T2-(1 -

2 cos 0, {(,,,

- nlAlm3)

TA

,,A/,n~) T,, T1}"

mA/m~) TA T2}~ + 2 cos 0t2 {(ml rn2/m~) T~ 7"2}~,

-- 2 cos 02 {(m2

(2)

where cos 0j2 = cos 0t cos 02 + sin 0 t sin 02 cos ( 4 1 - ~b2)'

ANALYSIS

OF DATA

FROM

xxxxxx

---I

261

REACTIONS

x

l x 1/2

x x

10

x

×

x

xXXx x x

×× x

×

×Xo x

x× x 5

o:

j I ,)

21 -

50

I

I I

! )(J 200 NO OF C O I N C I D E N C E S

/ 300

×

ox

I 400

2

5

10

T1

(MeV)

400

~°B~- d -

e.~- e.÷ a.

%= 0 . 4 0 MeV @ = 9 0 ° (I),=180 ° ®2=77 ° ¢~=0 o

3O0 ~n w

~

200

100

to

o

Fig. 1. Coincidence spectra for the reaction 10B÷d -+ ~÷~t F~ in a three-dimensional N-T1-Tg plot. The lab bombarding energy and the detection angles together with the various symbols for population densities are indicated. The axes T1 and T2 in the upper right hand three-dimensional plot represent the kinetic energy coordinates of the coincident co-particles received in detectors D1 and D2, respectively. The observed curves are projected onto the T-axes and presented as a histogram on the left and below. The calculated curves El, E2 and Elz relate the energy /'1 of an event to the excitation energy (on the right hand energy scale) of the intermediate nucleus SBe. mA, m j, m2 a n d m 3 are the rest masses of particle A, al a2 a n d a 3 respectively, TA is the b o m b a r d i n g energy a n d Q the Q-value o f the reaction. W i t h o u t lack of generality, q~t is usually taken equal to 180 °. F o r a particular nuclear reaction of the form in eq. (1) a n d a fixed set o f b o m b a r d i n g energy a n d angles o f detection, eq. (2) will give the geometrical locus of the expected c o n t r i b u t i o n s in a three d i m e n s i o n a l isometric representation of p o p u l a t i o n vs f l (T1, 7"2) a n d f2 (T1, 7"2) where.f1 a n d f2 is a set of functions of the

kinetic energies of the two detected particles. In particular the choice J] = T~ a n d f2 = T~ will give an ellipse on the f l , f 2 plane for TI a n d T 2 satisfying eq. (2). The more frequent choice, f~ = T I a n d J2 = T2, o n the other hand, will give a distorted ellipse. Still other choices have been used. The transition probability for the reaction in eq. (1) will be given by u, ~-~ IMI 2 pF(E),

(3)

262

P . A . A S S I M A K O P O U L O S AND N.H. GANGA.S

where M is a transition amplitude and pv(E) is the density of final states. For a direct three-body break-up, m 2 in eq. (3) is usually assumed constant, so that the decay probability along the kinematic locus will be directly proportional to the available phase space and may be easily calculated. The case of particular interest is that in which the three particle final state is reached through a two-body sequential decay rather than a direct three-body break-up implied in eq. (1) Such a decay is usually described by A + B ~ C~t2)+a3

I

(4)

a l -I-a2...

where C(~2) is an intermediate nucleus formed at some excitation level and the dots denote all possible permutations of the indices. Contributions from such a sequential decay will still be included in the direct reaction kinematic lccus, as a result of the overall energy-momentum conservation. However, the transition amplitude M in eq. (3) will no longer be constant and has to be evaluated. In performing a "complete" experiment on a threebody nuclear reaction the experimentally obtained contributions in any representation will be distributed over the kinematic locus predicted by eq. (2). All information concerning the particular reaction can be obtained by studying deviations of the experimental spectra from phase-space predictions. These deviatiens will be due to sequential decays of the form in eq. (4). 2.2. THE N - T I - T 2 REPRESENTATION

in fig. l the most usual form of plotting data is given. The two mutually perpendicular axes denote the measured kinetic energies of particle a~ and a2, detected at angles 01, ~b~ and 02, t~2 respectively. The population axis is perpendicular to both energy axes and for the plot in fig. 1 is directed out of the plane of the paper. The solid curve on the TI, T 2 plane is the calculated geometrical locus of contributions for the particular set of bombarding energy and angles of detection indicated in the figure. For the direct threebody reaction in eq. (1), i.e. a constant transition matrix M in eq. (3), the phase space consideration predicts a uniform population along this kinematic locus. The study of the mechanism of the nuclear reaction is therefore immediately reduced for this particular representation to the study of deviations from uniform population along the kinematic locus. Such a deviation of the experimental data from uniform population is obvious for the reaction l ° B + d - - ~

- - ~ + c~+ e in fig. 1 and a first hand qualitative picture of the reaction mechanism can be extracted from the three-dimensional plot. However, when quantitative analysis of the three-dimensional spectra is attempted, the calculations become highly complicated and one is inevitably forced to the study of two-dimensional projections of the spectra along the energy axes. For the particular case in fig. l the two energy-axis projections are given on the left of and under the three-dimensional plot. The nomograms E 1, E2 and E12 giving the excitation energy of the intermediate nucleus C~z) in eq. (4) are plotted in the lower figure containing the T 1-prOjectiOn histogram. The disadvantages of working with energy axis projections of the three-dimensional plot have already been pointed out3). They arise mainly from the sharp dependence of phase-space on kinetic energy of the observed particle and the required normalization o f the spectra with respect to it. Under actual experimental conditions, i.e. finite angular acceptance in the detectors dE/dx spread of the beam in the target, etc., a non-uniformly populated spread of the kinematic locus on the T1, T2 plane is introduced. The calculation of the phase space along each energy axis from such a population over the kinematic locus becomes highly complicated. At any rate, the result will depend very strongly - in particular at points above which there is a vertical tangency of the kinematic l o c u s - on the specific assumptions made in the calculations concerning the experimental conditions causing the spread of the events. Consequently, the phase space shape thus obtained cannot be often used with confidence for the normalization of the single-axis spectra. Further difficulties may arise from the distorted shape of the three-dimensional contribution in the case of a wide excitation level of an intermediate nucleus in the sequential decay. Such a distortion may introduce a considerable experimental error in the calculation of the excitation level width or the conjugate time delay involved. Still another point of difficulty may arise in certain cases from the double valuedness of the nomograms E 2 and E~ although in most cases this difficulty is resolved from the comparison of the two single axis energy spectra. Without overlooking the obvious advantages of this method in presenting a clear, straightforward physical picture of the reaction mechanism, the uncertainty introduced by the consideration of energy-axis projections has recently created a trend towards seeking new forms of data analysis. Still other attempts have been made in the direction of making all calculations directly from the three-dimensional spectra by building

ANALYSIS

OF

DATA

in the theoretically calculated spectra the experimental parameters, e.g. dE/dx loss, angular acceptances. Such a procedure, however, requires the employment of high-speed electronic computers and expensive display apparatus. 2.3. THE DALITZ PLOT When the experimental data contain a statistical distribution over all permissible angles of emission the application of some form of the Dalitz plot 6) is possible. In particular the Dalitz plot has been employed in the study of reactions with three identical particles in the final state7). The employment of this powerful method, however, has required the introduction of new experimental techniques such as position sensitive detectors. Still other experimental techniques borrowed directly from high energy physics might be more suitable for this representation. At any rate, the Dalitz plot with results obtained through the currently employed experimental techniques in low energy nuclear reactions should be used with considerable care. An interesting case of misleading results through the employment of the Dalitz plot in conjucntion with the usual two-detectors-in-coincidence technique has been reported by the Rice University groupS).

263

REACTIONS

where To = Tt(T2)I'r2=0|t should be noted that the usual case of the kinematic locus extending into the unphysical T~, T2 region for negative kinetic energy is treated here. In the case of the kinematic locus included entirely in the positive-positive quadrant, eq. (5) must be modified accordingly. The variable N along the vertical axis is the population around equal segments of the kinematic locus. Due to the nature of its construction the N-S plot is expected to be at most, smoothly varying for the direct three-body reaction. Any deviations from such population will then be attributed to sequential decay and all parameters may be calculated through T I and T z determined by eqs. (5) and (2) for a given S. With the help of F O R T R A N Code the experimental results in fig. are transformed into the histogram in fig. 2.

300

1,5

/

Ld 0 Z

\

w 200 £3

3. The N-S plot In analysing the experimental results from a "complete" experiment on the reaction I°B (d, 2c0 c~ a new method of obtaining all information directly from the N-T~-T2 three-dimensional plot has been attempted by "unfolding" the kinematic locus. This method is based on the fact that along the kinematic locus phase space is uniformly populated or at most smoothly varying if other effects such as Coulomb forces and angular momentum conservation are taken into account. The kinematic locus on the T1, T2 plane is divided into n equal parts and the three-dimensional presentation in fig. 1 is transformed into a two-dimensional histogram. The variable S along the horizontal axis of the new plot is the ratio of the length of the kinematic locus segment, measured from the 7"2 axis, to the entire length

S = f0"

FROM

{I + [dT2/'dT;]2} ~ dTi /

fo

°{l + [dT2/dT~]Z} ½d T ;

,

(5)

z O u

0z

4 0

I 5

~

w z 5 w

I

~3 1 0 0

I0

/ I 10

I 15

I 20

I 25

I 30

35

I 4O

S--CHANNELS

Fig. 2. A two-dimensional transformation of the coincident spectra in fig. 1, into an N-S plot. The nomograms E~, E2 and E12 have the same meaning as in fig. 1. Two advantages of this representation over one-axis projections are immediately evident: a. Phase-space varies smoothly along the S-axis so that no radical normalization of the experimental spectra is needed. b. The excitation energy of an intermediate state nucleus is a single-valued function with respect to S. Furthermore, it has in general, a much smoother variation over S than over either Tl o r T 2 (fig I), so that no large distortion of the Gaussian contribution of a wide level occurs. Finally, it should be mentioned that a similar

264

P.A. ASSIMAKOPOULOS AND N.H. GANGAS

m e t h o d o f analyzing results has been r e p o r t e d by Jones et al.9). This m e t h o d consists o f dividing the TI, T2 plane into equal a n g u l a r segments. The three dimensional a r r a y is thus t r a n s f o r m e d into a twodimensional h i s t o g r a m by plotting the p o p u l a t i o n on each a n g u l a r sector, m u l t i p l i e d by the reciprocal o f the radius to the kinematic locus, vs the p o l a r angle in the T1, 7"2 plane. A l t h o u g h the a b o v e t r a n s f o r m a t i o n also aims at a t w o - d i m e n s i o n a l p l o t o f the three-dimensional spectra, it does n o t offer a clear physical picture. M o r e o v e r , no satisfactory physical justification has been given for this m e t h o d . The results from the reaction I°B + d - + 3 ~ shown in figs. 1 a n d 2 for illustration p u r p o s e s are p a r t o f a " c o m p l e t e " experiment on the above reaction, still in progress. The a u t h o r s wish to a c k n o w l e d g e a series o f fruitful discussions with Mr. S. K o s s i o n i d e s during the first

p a r t o f the present work. T h a n k s are also extended to Mrs. C. Babili o f the Polytechnic School o f A t h e n s for m a k i n g available to the a u t h o r s the c o m p u t e r o f the a b o v e Institute and to Mr. S. V a l a m o n t e for the execution o f the drawings.

References 1) Conf. Correlations of particles emitted in nuclear reactions, Rev. Mod. Phys. 37 (1965) 327. ~) G.C. Phillips, Rev. Mod. Phys. 36 (1964) 1085. 3) G.C. Phillips, Lectures on few nuclear problems (Hercegnovi, July, 1964). 4) G.G. Ohlsen, Nucl. Instr. and Meth. 37 (1965) 240. 5) P.F. Donovan, Rev. Mod. Phys. 37 (1965) 501 and private communication. ~) CERN 64-13, Proc. 1964 Easter school for physicists, 2 (1964) 33. 7) E. Norbeck et al., Rev. Mod. Phys. 37 (1965) 455. 8) Comment by G.C. Phillips, Rev. Mod. Phys. 37 (1965) 457. 9) C.M. Jones et al., Rev. Mod. Phys. 37 (1965) 437.