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Lifetime analysis using the Doppler-shift attenuation method with a gate on feeding transition
Nuclear Instruments and Methods in Physics Research A 437 (1999) 274}281
Lifetime analysis using the Doppler-shift attenuation method with a gate on feeding transition P. Petkov!,",*, D. Tonev!,#, J. Gableske!, A. Dewald!, P. von Brentano! !Institut fu( r Kernphysik der Universita( t zu Ko( ln, 50937 Ko( ln, Germany "Bulgarian Academy of Sciences, Institute for Nuclear Research and Nuclear Energy, 1784 Soxa, Bulgaria #Faculty of Physics, University of Soxa, 1164 Soxa, Bulgaria Received 2 June 1999; accepted 13 July 1999
Abstract A method for the description of the c-ray line-shapes observed in coincidence lifetime measurements using the Doppler-shift attenuation method is proposed. The case where a gate is set on a transition which feeds directly the level of interest is considered in detail. The formalism for lifetime determination in the general framework of the di!erential decay-curve method is developed. The method is illustrated by an application to data simulated for the case of the 116Cd(16O, 4n)128Ba reaction at a beam energy of 72 MeV observed with a multi-detector set-up. ( 1999 Elsevier Science B.V. All rights reserved. PACS: 21.10.Tg; 29.85.#c; 29.30#r Keywords: Line-shapes in coincidence DSAM measurements; Lifetime determination
1. Introduction Experiments performed according to the Doppler-shift attenuation method (DSAM) are commonly used for the measurement of sub-picosecond lifetimes of excited nuclear states. More details about that method can be found e.g. in Ref. [1] and the references quoted therein. The method is based on the fact that in a solid material a moving excited nucleus obtained in e.g. a heavy-ion-induced nuclear reaction does not come immediately to rest
* Corresponding author. Tel.: #49-221-470-5732; fax: #49221-470-5168. E-mail address: [email protected] (P. Petkov)
but after a "nite time, typically of the order of 1 ps. The emission of c-rays during this time interval leads to the observation of a line-shape whose details are sensitive to the time evolution of the population of the level of interest and in particular, to its lifetime. For the analysis of such data, the slowingdown history of the recoiling ions must be known, indeed. The present work is dedicated to the analysis of coincidence DSAM data where a gate is set on a transition feeding the level of interest. Nowadays, this is feasible in experiments when e$cient multidetector arrays are used. Our approach therefore di!ers from the recently developed procedure [2] where `thina gates are set on the stopped portion of the peak corresponding to a transition
0168-9002/99/$ - 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 7 7 1 - 8
P. Petkov et al. / Nuclear Instruments and Methods in Physics Research A 437 (1999) 274}281
depopulating a level directly fed by the level of interest (NGTB). Setting a gate on a feeding transition also eliminates the long-standing problem of the unknown (side-) feeding which is inherent to the singles measurements or when gating from below the level of interest is used. Such gates can be set on many feeding transitions in the cascade above the level of interest to increase the statistics of the data to be analysed. However, for this analysis a precise description of the gated spectra must be achieved where both the time dependence of the population of the level and the slowing-down history have to be taken into account. As a matter of fact, the problem becomes more complicated when the gate is set only on the `shifteda portion of the peak which corresponds to emission during the slowing-down and selects only a speci"c fraction of the possible slowing-down histories. Nevertheless, this is of course the very desired case because a gating on the fully stopped component of the feeding transition does not generate data containing information on the lifetime of the level which is fed by it. The development of a method for the description of spectra obtained with gates set only on the shifted component of the feeding transition is one of the aims of our work. The second aim is to derive from the "t of the line-shape quantities which allow a simple determination of the lifetime of the level of interest. In this way, the involvement of a complicated level scheme information for taking into account the feeding history in the analysis can be avoided. We remind that the usual approach for treatment of DSAM data needs many assumptions on unknown feeding intensities and level lifetimes. The present approach is a further development within the general framework of the di!erential decay-curve method (DDCM) [3,4]. It should be mentioned that a gate on the full c-ray peak (line-shape) of the feeding transition, generated by emissions during both the slowingdown (shifted part) and at rest (unshifted part), solves in principle the problem with the side-feeding but generates a spectrum which is analogous to a singles one. Hence a knowledge of the feeding history is again necessary for the analysis of such data. The useful timing information originates only from the gate on the shifted part. In addition, the
275
spectra generated by many gates of the latter type on di!erent portions of the shifted part should be consistently described by the analysis. Therefore one may hope that a successful solution of this problem could lead to an increased sensitivity with respect to the stopping powers necessary for the simulation of the slowing-down history and thus to a more precise knowledge of these quantities. The paper is built as follows. The presentation starts with the quantitative description of the lineshape observed in coincidence DSAM experiments where a gate is set on a transition directly feeding the level of interest and the procedure for the derivation of the lifetime q in this case. Further, an application of the procedure to DSAM data simulated for conditions found after a 116Cd(16O,4n)128Ba reaction at a beam energy of 72 MeV and an exemplary c-ray cascade is presented. In the last section, the analysis of data obtained with gates set on highly lying feeding transitions is shortly considered.
2. DSA line-shapes and lifetime determination with gate set on a direct feeder In a DSAM experiment, the measured spectrum (line-shape) at a given direction of observation, e.g. at an angle h with respect to beam axis, is determined by: (i)
(ii)
(iii)
The energy of the investigated c-ray transition E corresponding to an emission at 0 rest The time-dependence of the velocity projections v on this direction i.e. of the Dopplerh shifted energies ES)"E (1#v /c) where c is h 0 h the velocity of light The time-dependence of the population n (t) i of the level of interest i.
For every investigated line shape corresponding to a c-ray depopulating the level of interest, the last two factors are correlated i.e. the slowing-down history of the ion and the time evolution of the population of the level are coupled. To illustrate how this mechanism works in a c}c coincidence measurement, let us consider the level scheme presented in Fig. 1. The transitions A and B are
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P. Petkov et al. / Nuclear Instruments and Methods in Physics Research A 437 (1999) 274}281
Fig. 1. Level scheme used for the simulation of DSA line-shapes in the present work. The lifetime of the level a has to be determined from the analysis of the line-shapes of the transition A generated by the gates set on the shifted part of the directly feeding transition B. The singles line shape of the latter transition and the three gates are shown in the insert. The energy calibration is given by E "Channel/2. c
registered by two detectors e.g. belonging to a multi-detector set-up (the detectors are also labeled by A and B). In the insert of the "gure, the `singlesa line-shape of the transition B which depopulates the level b and directly feeds the level a is shown. Di!erent gates can be set on the shifted portion of transition B which is `measureda in the case displayed at a forward direction with respect to the beam axis. The line-shape corresponding to the transition A has to be analysed. This line-shape is obtained by folding the spectrum of the shifted energies ES) (i.e. the spectrum of the velocity projecA tions r(v )) with the detector response function A U(E , ES)). The latter function describes the probA A ability for measuring in the full-energy peak an energy E at incident c-ray energy ES). Since the A A gate is set on a folded spectrum, di!erent energies ES), respectively di!erent projections v , contribute B B to the generation of the gated spectrum. The probability to contribute i.e. the unfolding weight is
P
= (v )" B B
B E.!9 dE U (E , ES)(v )) B B B B B B E.*/
(1)
where E.*/ and E.!9 indicate the lower and upper B B gate limits, correspondingly. Under these condi-
tions, the spectrum of the velocity projections r(v ) A for a particular slowing-down (velocity) history is given by r(v )"b A af
P P =
ta
dt = (vi (t ))d(v , vi (t )) A A a b B B b 0 0 ]j j n (t ) e~ja (ta ~tb )F. (2) a b b b In Eq. (2), i labels the velocity history, b is the af branching ratio of the transition A(aPf ) while j and j are the decay constants of the two levels. a b The d-function serves to increment the content of the point in the velocity spectrum with a coordinate v which corresponds to the projection of the ion A velocity at the emission time t of the transition A. a The feeding transition B occurs at t"t and b = (vi (t )) is the unfolding weight of the velocity B B b projection vi (t ) on the axis of the detector registerB b ing the transition B at its emission time. The best approach to calculate the time evolution of the velocity distribution of the recoils consists in performing a Monte-Carlo (MC) simulation of the process of creation of the recoils followed by a slowing-down in the target and the stopper. To apply in practice Eq. (2), it is necessary to sum-up over many MC-histories and therefore to know, dt
a
P. Petkov et al. / Nuclear Instruments and Methods in Physics Research A 437 (1999) 274}281
individually for each history, the velocity (i.e. its three projections) as a function of the time. For the simulation of the histories we have used a modi"ed version of the computer code DESASTOP [5,6], which follows in three dimensions the evolution of the velocity of the recoils. Further details on the description of the slowing-down process in this code can be found in Ref. [7]. The modi"ed version of DESASTOP and the randomisation of the slowing-down (velocity) histories with respect to the multi-detector set-up used are described in more detail in a recent work [8] dedicated to the treatment of DSA e!ects in recoil distance Doppler-shift measurements. Here, we mention only that the "nal result of the simulation is a four-dimensional array of velocity projections at a set of times t of interest (v (t), v (t), v (t), t). The third projection v represA B Z Z ents the component of the recoil on the beam axis while the time of interest covers the range from the creation of the recoil to the moment where it comes to rest. The numerical simulation of DSAM data using Eq. (2) reveals that a summation over several thousand velocity histories is su$cient to stabilise statistically the line-shape. On the basis of the comparison of the calculated line-shape with the data, it is possible in principle to "t the unknown properties of the investigated c-ray cascade and in particular, the lifetime of the level of interest. Below, we present a di!erent approach which relies on the decomposition of the DSA line-shape into different components. These components are associated with the possible emission times of the subsequent transitions of interest B and A (Fig. 1). The transitions can occur during the slowing-down (shifted in slowing down, labeled by SS) and at rest (unshifted, labeled by U). Therefore, three contributions to the line-shapes are possible, namely MSS, SSN, MSS, UN and MU, UN, where the letters on the l.h.s. indicate the emission of the feeding transition B while these on the r.h.s. indicate the emission of the transition of interest A. We shall denote the areas of these contributions (components) by MB , A N, MB , A N and MB , A N, correspondSS SS SS U U U ingly. It should be mentioned that the emissions occurring in the target and in the stopper during the slowing-down are not distinguished here and the same label SS is used for them. The time t at which s
277
the recoil comes to rest can be used in a natural way to split the integration in Eq. (2) over speci"c regions in the (t , t ) plane. Thus, one obtains a b r(v )"rM (v )#rM N (v "0)#rM N (v "0) A SS,SSN A SS,U A U,U A (3) where (v )"b rM SS,SSN A af
P P ts
ta
dt = (vi (t )) b B B b 0 0 ]d(v , vi (t ))j j n (t ) e~ja (ta ~tb ) (4) A A a a b b b = ts dt dt = (vi (t )) rM N (v "0)"b SS,U A af a b B B b ts 0 ]j j n (t ) e~ja (ta ~tb ) a b b b ts "b dt = (vi (t ))j n (t ) e~ja (ts ~tb ) af b B B b b b b 0 (5) dt a
P P
P
and
P P
= ta dt dt = (0)j j n (t ) a b B a b b b ts ts ]e~ja (ta ~tb )
rM N (v "0)"b af U,U A
"b = (0) (S (R)!S (t )) af B b b s "b = (0)R (t )). (6) af B b s In the latter equation, R (t ) denotes the decay b s function of the level b i.e. the number of decays of that level occurring after the time t while S (t) is s b the shifted decay function
P
t
j n (t@) dt@. (7) b b 0 It is easy to show that the derivative of the quantity rM with respect to t is simply given by SS,SSN s ts drM N /dt "b dt = (vi (t ))j j n (t ) e~ja (ts ~tb ). SS,SS s af b B B b a b b b 0 (8) S (t)" b
P
Then, dividing the quantities on both sides of Eq. (5) by those participating in Eq. (8) yields the lifetime of the level a: /dt ). q "1/j "rM N /(drM SS,U SS,SSN s a a
(9)
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P. Petkov et al. / Nuclear Instruments and Methods in Physics Research A 437 (1999) 274}281
This equation holds, of course, also for the folded integrals of the velocity distributions rM N and SS,U rM i.e. for the areas of the corresponding comSS,SSN ponents of the DSA line-shape: q "MB , A N/(dMB , A N/dt ). (10) a SS U SS SS s In this way, it becomes clear that with a convenient choice of a gate set on a direct feeder, the lifetime of the level of interest can be determined according to the above procedure. A necessary condition is that the gate does not generate any MU, UN component in the gated spectrum. Then, the area of MSS, UN component can be "tted with the response function of the detector while the MSS, SSN component can be "tted using Eq. (4). We have developed a computer code which "ts simultaneously the shifted decay curve S (t) of the level of interest and the a DSA c-ray spectrum. The approach is quite similar to the one described in Ref. [8] dealing with the description of DSA e!ects in RDDS measurements as well as to the procedure [4,7] for the analysis of DSA line-shapes in the singles case. We note only that on the time scale the S (t) function is representa ed by second-order polynomials over separate intervals. These polynomials are continuously interconnected at the interval borders. Therefore the "tting problem is linear with respect to the independent polynomial parameters and the area of the unshifted peak (MB , A N). After careful SS U background subtraction, the "tting procedure consists in changing the position of the interval borders until the best "t of the data is obtained. For every new set of borders, a summation over all MChistories of the simulation is performed to generate the quantities necessary for the reproduction of the spectrum and calculation of the derivative in the denominator of Eq. (9). Here, an important point has to be mentioned. The use of Eqs. (4) and (8) implies the knowledge of the lifetime q of the level a of interest and of the decay-function j n (t) of the b b feeding level. The latter can be expressed using the equation of Bateman: dn (t) dS (t) d2S (t) a . j n (t)"j n (t)# a " a #q b b a a a dt dt dt2 (11) Thus, an initial guess for the value of q enters as an a additional parameter in the "tting procedure. How-
ever, this value must be the same as the value deduced from the equation for the derivation of the lifetime (Eq. (9)). Hence the internal consistency of the procedure is not hindered by the necessity to involve a `guesseda value for q as a parameter. a 3. Practical application In order to verify and illustrate the procedure described in the previous section, we have simulated coincidence DSAM data using the level scheme presented in Fig. 1. Di!erent gates (shown in the insert of the "gure) were set on the shifted portion of the transition B which feeds directly the level a. The velocity histories were generated for the case of the reaction 116Cd(16O, 4nc)128Ba at a beam energy of 72 MeV. Coincidences were `recordeda between every pair of six detectors positioned nearly symmetrically at a mean angle of 34.63 with respect to the beam axis. An 1.1 mg/cm2 thick target and a 6.2 mg/cm2 thick gold stopper were employed in the calculation. A set of standard stopping powers for the A+130 mass region were used (see e.g. Ref. [7]). For instance, the electron stopping power was deduced from the semi-empirical tables of Northcli!e and Schilling [9] with taking into account e!ects of the medium atomic structure [10,11]. Under these circumstances, the mean velocity of the recoils leaving the target was calculated to be 0.925% of the velocity of light, c. We do not present more details on this topic here since the same stopping powers, respectively, slowing-down histories, were used for the simulation of the data using Eq. (2) and for their analysis ("tting) according to the procedure described in Section 2. We note also that a constant background was added to the data obtained using Eq. (2) and normally distributed statistical uncertainties were additionally introduced. The results of the "ts of the line-shapes of the investigated transition A as generated by the three gates and the derived lifetimes are presented in Fig. 2B}D. The "ts were accepted when they were characterised by a small value of s2 and consistent values of the assumed and deduced according to Eq. (10) lifetime q (c.f. previous section). As it can a be seen, the lifetime of the level a incorporated in
P. Petkov et al. / Nuclear Instruments and Methods in Physics Research A 437 (1999) 274}281
279
Fig. 2. (A) Line-shape of the 1300 keV transition generated by a gate set on the full line-shape of the feeding transition of 1314.5 keV. (B}D) Fits of the line shapes of the investigated transition generated by three di!erent gates and derived lifetimes with their statistical errors. The data points are randomized by assuming that they are normally distributed around their expectation values after the addition of a constant background. The dotted line represents the shifted part (the MSS, SSN contribution) while the dashed line represents the unshifted part (the MSS, UN contribution). The full line is the sum of both contributions. See also text and Fig. 1.
the simulation (q "0.312 ps) is very close to the a derived lifetime values from the three gated spectra within their statistical errors. For comparison, the line-shape of the investigated transition generated by a gate set on the full line shape of the feeding transition is also presented in part A of the "gure. Obviously, the useful timing information in the coincidence spectra is increased when the gate is set on the shifted portion of the peak corresponding to the feeding transition. Furthermore, the line-shapes generated by di!erent gates of that type di!er from each other. Therefore, the requirement for an agreement between the extracted lifetime values should lead in real experimental cases to an increased sensitivity with respect to the precision of the stopping powers used in the data analysis.
4. Gates on higher feeders In principle, the generalisation of Eq. (2) for the case of a gate set on any high-lying feeding transition is straightforward. For simplicity, let us consider the case where only one feeding path goes to the investigated level and every level in the cascade is fed only from one level lying above it. Let us label the level of interest with 1, the level which directly feeds this level with 2 and set the gate on the shifted part of level number k in the cascade. The analysed transition is recorded by a detector A and the gate is set on the shifted portion of the transition from the level k recorded by a detector B. It is easy to show that the spectra of velocity projections (c.f. Eq. (2)) generated by the transitions from level 1 and level 2 will
280
P. Petkov et al. / Nuclear Instruments and Methods in Physics Research A 437 (1999) 274}281
be described by the following expressions:
P
= dt d(v , v (t ))j e~j2 t2 2 A A 2 2 0 t2 ] dt = (v (t ))j n (t ) k B B k k k k 0 ]f(t , t D j 2j ) 2 k 2 k~1
Then, the following relation is obtained for the lifetime of the level 1
r (v )" 2 A
and
P
/dt ) q "(r M N !r M N )/(dr M 2 SS,U 1 SS,SSN s 1 1 SS,U
P
(12)
= dt d(v , v (t ))j e~j1 t1 1 A A 1 1 0 t1 ] dt = (v (t ))j n (t ) k B B k k k k 0 1 t ) ] dt j e~(j2 ~j1 )t2 f (t , t D j 2j 2 2 2 k 2 k~1 k t (13)
r (v )" 1 A
P P
where the function f (t , t D j 2j ) is the same 2 k 2 k~1 in both cases and represents the result of the integration over the emission times of the intermediate transitions (from level 3 to level k!1). The term r M can be obtained but setting the upper limit 1 SS,SSN of the integration over t in Eq. (13) to t . The 1 s derivative of this term with respect to t equals to s ts dr M dt = (v (t ))j n (t ) /dt "j e~j1 ts 1 SS,SSN s k B B k k k k 1 0 ts ] dt j e~(j2 ~j1 )t2 f (t , t D j 2j ). (14) 2 2 2 k 2 k~1 tk Correspondingly, the terms r M N and r M N are 1 SS,U 2 SS,U given by the expressions
P
P
r M N "e~j1 ts 1 SS,U
P P
ts
P
ts
0
dt = (v (t )) j n (t ) k B B k k k k
dt j e~(j2 ~j1 )t2 f (t , t D j 2j ) 2 2 2 k 2 k~1 tk = ts # dt j e~j2 t2 dt = (v (t ))j n (t ) 2 2 k B B k k k k ts 0 ]f (t , t D j 2j ). (15) 2 k 2 k~1 ]
P
and
P
r M N" 2 SS,U
=
dt j e~j2 t2 2 2
P
ts
dt = (v (t ))j n (t ) k B B k k k k ts 0 ]f (t , t D j 2j ). (16) 2 k 2 k~1
(17)
which holds also for the coincident events (areas) obtained after folding with the detector response functions. Obviously, this equation can also be used for lifetime determination. However, the problem of what approach is optimal for its use requires further investigations which are out of the scope of the present work. These investigations have to clarify if the part of the function to be integrated for the calculation of the terms r M and r M and 1 SS,SSN 2 SS,SSN which re#ects the time-dependence of the population of the levels involved can be conveniently approximated with elementary functions for the case of an arbitrary cascade. If so, then an analysing procedure analogous to the one for the case of a gate set on a directly feeding transition is feasible. If not, one has to rely on the simulation of gated spectra and deducing the lifetimes of the cascade levels from the comparison with the experimental spectra.
5. Summary and conclusions A new method for the description of the c-ray line-shapes observed in coincidence lifetime measurements using the Doppler-shift attenuation method is proposed for the case where a gate is set on the shifted portion of a transition which feeds the level of interest. A procedure for lifetime determination in the general framework of the di!erential decay-curve method is developed for the case where the gate is set on a transition which feeds directly the level of interest. The procedure is illustrated by an application to data simulated for the case of the 116Cd(16O, 4n)128Ba reaction at a beam energy of 72 MeV observed with a multi-detector set-up. It can be expected that the simultaneous description of spectra obtained with di!erent gates on the same feeding transition will increase the sensitivity with respect to the stopping-powers employed in the analysis. The case where the gate is set higher in the feeding cascade is also brie#y considered.
P. Petkov et al. / Nuclear Instruments and Methods in Physics Research A 437 (1999) 274}281
Acknowledgements Two of us (P.P. and D.T.) are grateful for the kind hospitality of our colleagues at the Cologne University. D.T. would like to acknowledge the support of the Deutsche Akademische Austauschdienst (DAAD). This work was funded by the BMBF under contract 06 OK862 I(0). The support of the Bulgarian Science Foundation under contract Ph.801 is also appreciated. References [1] T.K. Alexander, J.S. Forster, Adv. Nucl. Phys. 10 (1978) 197. [2] F. Brandolini, R.V. Ribas, Nucl. Instr. Meth. Phys. Res. A 417 (1998) 150.
281
[3] A. Dewald, S. Harissopulos, P. von Brentano, Z. Phys. A 334 (1989) 163. [4] G. BoK hm, A. Dewald, P. Petkov, P. von Brentano, Nucl. Instr. Meth. Phys. Res. A 329 (1993) 248. [5] G. Winter, ZfK Rossendorf Report ZfK-497, 1983. [6] G. Winter, Nucl. Instr and Meth. 214 (1983) 537. [7] P. Petkov, J. Gableske, O. Vogel, A. Dewald, P. von Brentano, R. KruK cken, R. Peusquens, N. Nicolay, A. Gizon, J. Gizon, D. Bazzacco, C. Rossi-Alvarez, S. Lunardi, P. Pavan, D.R. Napoli, W. Andrejtsche!, R.V. Jolos, Nucl. Phys. A 640 (1998) 293. [8] P. Petkov, D. Tonev, J. Gableske, A. Dewald, T. Klemme, P. von Brentano, Nucl. Instr. and Meth. A 431 (1999) 208. [9] L.C. Northcli!e, R.F. Schilling, Nucl. Data Tables A7 (1970) 233. [10] J.F. Ziegler, W.K. Chu, Atomic Data Nucl. Data Tables 13 (1974) 463. [11] J.F. Ziegler, J.P. Biersack, in: D.A. Bromley (Ed.), Treatise on Heavy-Ion Science, Vol. 6, Plenum Press, New York, 1985, p. 95.
Lifetime analysis using the Doppler-shift attenuation method with a gate on feeding transition P. Petkov!,",*, D. Tonev!,#, J. Gableske!, A. Dewald!, P. von Brentano! !Institut fu( r Kernphysik der Universita( t zu Ko( ln, 50937 Ko( ln, Germany "Bulgarian Academy of Sciences, Institute for Nuclear Research and Nuclear Energy, 1784 Soxa, Bulgaria #Faculty of Physics, University of Soxa, 1164 Soxa, Bulgaria Received 2 June 1999; accepted 13 July 1999
Abstract A method for the description of the c-ray line-shapes observed in coincidence lifetime measurements using the Doppler-shift attenuation method is proposed. The case where a gate is set on a transition which feeds directly the level of interest is considered in detail. The formalism for lifetime determination in the general framework of the di!erential decay-curve method is developed. The method is illustrated by an application to data simulated for the case of the 116Cd(16O, 4n)128Ba reaction at a beam energy of 72 MeV observed with a multi-detector set-up. ( 1999 Elsevier Science B.V. All rights reserved. PACS: 21.10.Tg; 29.85.#c; 29.30#r Keywords: Line-shapes in coincidence DSAM measurements; Lifetime determination
1. Introduction Experiments performed according to the Doppler-shift attenuation method (DSAM) are commonly used for the measurement of sub-picosecond lifetimes of excited nuclear states. More details about that method can be found e.g. in Ref. [1] and the references quoted therein. The method is based on the fact that in a solid material a moving excited nucleus obtained in e.g. a heavy-ion-induced nuclear reaction does not come immediately to rest
* Corresponding author. Tel.: #49-221-470-5732; fax: #49221-470-5168. E-mail address: [email protected] (P. Petkov)
but after a "nite time, typically of the order of 1 ps. The emission of c-rays during this time interval leads to the observation of a line-shape whose details are sensitive to the time evolution of the population of the level of interest and in particular, to its lifetime. For the analysis of such data, the slowingdown history of the recoiling ions must be known, indeed. The present work is dedicated to the analysis of coincidence DSAM data where a gate is set on a transition feeding the level of interest. Nowadays, this is feasible in experiments when e$cient multidetector arrays are used. Our approach therefore di!ers from the recently developed procedure [2] where `thina gates are set on the stopped portion of the peak corresponding to a transition
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 7 7 1 - 8
P. Petkov et al. / Nuclear Instruments and Methods in Physics Research A 437 (1999) 274}281
depopulating a level directly fed by the level of interest (NGTB). Setting a gate on a feeding transition also eliminates the long-standing problem of the unknown (side-) feeding which is inherent to the singles measurements or when gating from below the level of interest is used. Such gates can be set on many feeding transitions in the cascade above the level of interest to increase the statistics of the data to be analysed. However, for this analysis a precise description of the gated spectra must be achieved where both the time dependence of the population of the level and the slowing-down history have to be taken into account. As a matter of fact, the problem becomes more complicated when the gate is set only on the `shifteda portion of the peak which corresponds to emission during the slowing-down and selects only a speci"c fraction of the possible slowing-down histories. Nevertheless, this is of course the very desired case because a gating on the fully stopped component of the feeding transition does not generate data containing information on the lifetime of the level which is fed by it. The development of a method for the description of spectra obtained with gates set only on the shifted component of the feeding transition is one of the aims of our work. The second aim is to derive from the "t of the line-shape quantities which allow a simple determination of the lifetime of the level of interest. In this way, the involvement of a complicated level scheme information for taking into account the feeding history in the analysis can be avoided. We remind that the usual approach for treatment of DSAM data needs many assumptions on unknown feeding intensities and level lifetimes. The present approach is a further development within the general framework of the di!erential decay-curve method (DDCM) [3,4]. It should be mentioned that a gate on the full c-ray peak (line-shape) of the feeding transition, generated by emissions during both the slowingdown (shifted part) and at rest (unshifted part), solves in principle the problem with the side-feeding but generates a spectrum which is analogous to a singles one. Hence a knowledge of the feeding history is again necessary for the analysis of such data. The useful timing information originates only from the gate on the shifted part. In addition, the
275
spectra generated by many gates of the latter type on di!erent portions of the shifted part should be consistently described by the analysis. Therefore one may hope that a successful solution of this problem could lead to an increased sensitivity with respect to the stopping powers necessary for the simulation of the slowing-down history and thus to a more precise knowledge of these quantities. The paper is built as follows. The presentation starts with the quantitative description of the lineshape observed in coincidence DSAM experiments where a gate is set on a transition directly feeding the level of interest and the procedure for the derivation of the lifetime q in this case. Further, an application of the procedure to DSAM data simulated for conditions found after a 116Cd(16O,4n)128Ba reaction at a beam energy of 72 MeV and an exemplary c-ray cascade is presented. In the last section, the analysis of data obtained with gates set on highly lying feeding transitions is shortly considered.
2. DSA line-shapes and lifetime determination with gate set on a direct feeder In a DSAM experiment, the measured spectrum (line-shape) at a given direction of observation, e.g. at an angle h with respect to beam axis, is determined by: (i)
(ii)
(iii)
The energy of the investigated c-ray transition E corresponding to an emission at 0 rest The time-dependence of the velocity projections v on this direction i.e. of the Dopplerh shifted energies ES)"E (1#v /c) where c is h 0 h the velocity of light The time-dependence of the population n (t) i of the level of interest i.
For every investigated line shape corresponding to a c-ray depopulating the level of interest, the last two factors are correlated i.e. the slowing-down history of the ion and the time evolution of the population of the level are coupled. To illustrate how this mechanism works in a c}c coincidence measurement, let us consider the level scheme presented in Fig. 1. The transitions A and B are
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Fig. 1. Level scheme used for the simulation of DSA line-shapes in the present work. The lifetime of the level a has to be determined from the analysis of the line-shapes of the transition A generated by the gates set on the shifted part of the directly feeding transition B. The singles line shape of the latter transition and the three gates are shown in the insert. The energy calibration is given by E "Channel/2. c
registered by two detectors e.g. belonging to a multi-detector set-up (the detectors are also labeled by A and B). In the insert of the "gure, the `singlesa line-shape of the transition B which depopulates the level b and directly feeds the level a is shown. Di!erent gates can be set on the shifted portion of transition B which is `measureda in the case displayed at a forward direction with respect to the beam axis. The line-shape corresponding to the transition A has to be analysed. This line-shape is obtained by folding the spectrum of the shifted energies ES) (i.e. the spectrum of the velocity projecA tions r(v )) with the detector response function A U(E , ES)). The latter function describes the probA A ability for measuring in the full-energy peak an energy E at incident c-ray energy ES). Since the A A gate is set on a folded spectrum, di!erent energies ES), respectively di!erent projections v , contribute B B to the generation of the gated spectrum. The probability to contribute i.e. the unfolding weight is
P
= (v )" B B
B E.!9 dE U (E , ES)(v )) B B B B B B E.*/
(1)
where E.*/ and E.!9 indicate the lower and upper B B gate limits, correspondingly. Under these condi-
tions, the spectrum of the velocity projections r(v ) A for a particular slowing-down (velocity) history is given by r(v )"b A af
P P =
ta
dt = (vi (t ))d(v , vi (t )) A A a b B B b 0 0 ]j j n (t ) e~ja (ta ~tb )F. (2) a b b b In Eq. (2), i labels the velocity history, b is the af branching ratio of the transition A(aPf ) while j and j are the decay constants of the two levels. a b The d-function serves to increment the content of the point in the velocity spectrum with a coordinate v which corresponds to the projection of the ion A velocity at the emission time t of the transition A. a The feeding transition B occurs at t"t and b = (vi (t )) is the unfolding weight of the velocity B B b projection vi (t ) on the axis of the detector registerB b ing the transition B at its emission time. The best approach to calculate the time evolution of the velocity distribution of the recoils consists in performing a Monte-Carlo (MC) simulation of the process of creation of the recoils followed by a slowing-down in the target and the stopper. To apply in practice Eq. (2), it is necessary to sum-up over many MC-histories and therefore to know, dt
a
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individually for each history, the velocity (i.e. its three projections) as a function of the time. For the simulation of the histories we have used a modi"ed version of the computer code DESASTOP [5,6], which follows in three dimensions the evolution of the velocity of the recoils. Further details on the description of the slowing-down process in this code can be found in Ref. [7]. The modi"ed version of DESASTOP and the randomisation of the slowing-down (velocity) histories with respect to the multi-detector set-up used are described in more detail in a recent work [8] dedicated to the treatment of DSA e!ects in recoil distance Doppler-shift measurements. Here, we mention only that the "nal result of the simulation is a four-dimensional array of velocity projections at a set of times t of interest (v (t), v (t), v (t), t). The third projection v represA B Z Z ents the component of the recoil on the beam axis while the time of interest covers the range from the creation of the recoil to the moment where it comes to rest. The numerical simulation of DSAM data using Eq. (2) reveals that a summation over several thousand velocity histories is su$cient to stabilise statistically the line-shape. On the basis of the comparison of the calculated line-shape with the data, it is possible in principle to "t the unknown properties of the investigated c-ray cascade and in particular, the lifetime of the level of interest. Below, we present a di!erent approach which relies on the decomposition of the DSA line-shape into different components. These components are associated with the possible emission times of the subsequent transitions of interest B and A (Fig. 1). The transitions can occur during the slowing-down (shifted in slowing down, labeled by SS) and at rest (unshifted, labeled by U). Therefore, three contributions to the line-shapes are possible, namely MSS, SSN, MSS, UN and MU, UN, where the letters on the l.h.s. indicate the emission of the feeding transition B while these on the r.h.s. indicate the emission of the transition of interest A. We shall denote the areas of these contributions (components) by MB , A N, MB , A N and MB , A N, correspondSS SS SS U U U ingly. It should be mentioned that the emissions occurring in the target and in the stopper during the slowing-down are not distinguished here and the same label SS is used for them. The time t at which s
277
the recoil comes to rest can be used in a natural way to split the integration in Eq. (2) over speci"c regions in the (t , t ) plane. Thus, one obtains a b r(v )"rM (v )#rM N (v "0)#rM N (v "0) A SS,SSN A SS,U A U,U A (3) where (v )"b rM SS,SSN A af
P P ts
ta
dt = (vi (t )) b B B b 0 0 ]d(v , vi (t ))j j n (t ) e~ja (ta ~tb ) (4) A A a a b b b = ts dt dt = (vi (t )) rM N (v "0)"b SS,U A af a b B B b ts 0 ]j j n (t ) e~ja (ta ~tb ) a b b b ts "b dt = (vi (t ))j n (t ) e~ja (ts ~tb ) af b B B b b b b 0 (5) dt a
P P
P
and
P P
= ta dt dt = (0)j j n (t ) a b B a b b b ts ts ]e~ja (ta ~tb )
rM N (v "0)"b af U,U A
"b = (0) (S (R)!S (t )) af B b b s "b = (0)R (t )). (6) af B b s In the latter equation, R (t ) denotes the decay b s function of the level b i.e. the number of decays of that level occurring after the time t while S (t) is s b the shifted decay function
P
t
j n (t@) dt@. (7) b b 0 It is easy to show that the derivative of the quantity rM with respect to t is simply given by SS,SSN s ts drM N /dt "b dt = (vi (t ))j j n (t ) e~ja (ts ~tb ). SS,SS s af b B B b a b b b 0 (8) S (t)" b
P
Then, dividing the quantities on both sides of Eq. (5) by those participating in Eq. (8) yields the lifetime of the level a: /dt ). q "1/j "rM N /(drM SS,U SS,SSN s a a
(9)
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This equation holds, of course, also for the folded integrals of the velocity distributions rM N and SS,U rM i.e. for the areas of the corresponding comSS,SSN ponents of the DSA line-shape: q "MB , A N/(dMB , A N/dt ). (10) a SS U SS SS s In this way, it becomes clear that with a convenient choice of a gate set on a direct feeder, the lifetime of the level of interest can be determined according to the above procedure. A necessary condition is that the gate does not generate any MU, UN component in the gated spectrum. Then, the area of MSS, UN component can be "tted with the response function of the detector while the MSS, SSN component can be "tted using Eq. (4). We have developed a computer code which "ts simultaneously the shifted decay curve S (t) of the level of interest and the a DSA c-ray spectrum. The approach is quite similar to the one described in Ref. [8] dealing with the description of DSA e!ects in RDDS measurements as well as to the procedure [4,7] for the analysis of DSA line-shapes in the singles case. We note only that on the time scale the S (t) function is representa ed by second-order polynomials over separate intervals. These polynomials are continuously interconnected at the interval borders. Therefore the "tting problem is linear with respect to the independent polynomial parameters and the area of the unshifted peak (MB , A N). After careful SS U background subtraction, the "tting procedure consists in changing the position of the interval borders until the best "t of the data is obtained. For every new set of borders, a summation over all MChistories of the simulation is performed to generate the quantities necessary for the reproduction of the spectrum and calculation of the derivative in the denominator of Eq. (9). Here, an important point has to be mentioned. The use of Eqs. (4) and (8) implies the knowledge of the lifetime q of the level a of interest and of the decay-function j n (t) of the b b feeding level. The latter can be expressed using the equation of Bateman: dn (t) dS (t) d2S (t) a . j n (t)"j n (t)# a " a #q b b a a a dt dt dt2 (11) Thus, an initial guess for the value of q enters as an a additional parameter in the "tting procedure. How-
ever, this value must be the same as the value deduced from the equation for the derivation of the lifetime (Eq. (9)). Hence the internal consistency of the procedure is not hindered by the necessity to involve a `guesseda value for q as a parameter. a 3. Practical application In order to verify and illustrate the procedure described in the previous section, we have simulated coincidence DSAM data using the level scheme presented in Fig. 1. Di!erent gates (shown in the insert of the "gure) were set on the shifted portion of the transition B which feeds directly the level a. The velocity histories were generated for the case of the reaction 116Cd(16O, 4nc)128Ba at a beam energy of 72 MeV. Coincidences were `recordeda between every pair of six detectors positioned nearly symmetrically at a mean angle of 34.63 with respect to the beam axis. An 1.1 mg/cm2 thick target and a 6.2 mg/cm2 thick gold stopper were employed in the calculation. A set of standard stopping powers for the A+130 mass region were used (see e.g. Ref. [7]). For instance, the electron stopping power was deduced from the semi-empirical tables of Northcli!e and Schilling [9] with taking into account e!ects of the medium atomic structure [10,11]. Under these circumstances, the mean velocity of the recoils leaving the target was calculated to be 0.925% of the velocity of light, c. We do not present more details on this topic here since the same stopping powers, respectively, slowing-down histories, were used for the simulation of the data using Eq. (2) and for their analysis ("tting) according to the procedure described in Section 2. We note also that a constant background was added to the data obtained using Eq. (2) and normally distributed statistical uncertainties were additionally introduced. The results of the "ts of the line-shapes of the investigated transition A as generated by the three gates and the derived lifetimes are presented in Fig. 2B}D. The "ts were accepted when they were characterised by a small value of s2 and consistent values of the assumed and deduced according to Eq. (10) lifetime q (c.f. previous section). As it can a be seen, the lifetime of the level a incorporated in
P. Petkov et al. / Nuclear Instruments and Methods in Physics Research A 437 (1999) 274}281
279
Fig. 2. (A) Line-shape of the 1300 keV transition generated by a gate set on the full line-shape of the feeding transition of 1314.5 keV. (B}D) Fits of the line shapes of the investigated transition generated by three di!erent gates and derived lifetimes with their statistical errors. The data points are randomized by assuming that they are normally distributed around their expectation values after the addition of a constant background. The dotted line represents the shifted part (the MSS, SSN contribution) while the dashed line represents the unshifted part (the MSS, UN contribution). The full line is the sum of both contributions. See also text and Fig. 1.
the simulation (q "0.312 ps) is very close to the a derived lifetime values from the three gated spectra within their statistical errors. For comparison, the line-shape of the investigated transition generated by a gate set on the full line shape of the feeding transition is also presented in part A of the "gure. Obviously, the useful timing information in the coincidence spectra is increased when the gate is set on the shifted portion of the peak corresponding to the feeding transition. Furthermore, the line-shapes generated by di!erent gates of that type di!er from each other. Therefore, the requirement for an agreement between the extracted lifetime values should lead in real experimental cases to an increased sensitivity with respect to the precision of the stopping powers used in the data analysis.
4. Gates on higher feeders In principle, the generalisation of Eq. (2) for the case of a gate set on any high-lying feeding transition is straightforward. For simplicity, let us consider the case where only one feeding path goes to the investigated level and every level in the cascade is fed only from one level lying above it. Let us label the level of interest with 1, the level which directly feeds this level with 2 and set the gate on the shifted part of level number k in the cascade. The analysed transition is recorded by a detector A and the gate is set on the shifted portion of the transition from the level k recorded by a detector B. It is easy to show that the spectra of velocity projections (c.f. Eq. (2)) generated by the transitions from level 1 and level 2 will
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P. Petkov et al. / Nuclear Instruments and Methods in Physics Research A 437 (1999) 274}281
be described by the following expressions:
P
= dt d(v , v (t ))j e~j2 t2 2 A A 2 2 0 t2 ] dt = (v (t ))j n (t ) k B B k k k k 0 ]f(t , t D j 2j ) 2 k 2 k~1
Then, the following relation is obtained for the lifetime of the level 1
r (v )" 2 A
and
P
/dt ) q "(r M N !r M N )/(dr M 2 SS,U 1 SS,SSN s 1 1 SS,U
P
(12)
= dt d(v , v (t ))j e~j1 t1 1 A A 1 1 0 t1 ] dt = (v (t ))j n (t ) k B B k k k k 0 1 t ) ] dt j e~(j2 ~j1 )t2 f (t , t D j 2j 2 2 2 k 2 k~1 k t (13)
r (v )" 1 A
P P
where the function f (t , t D j 2j ) is the same 2 k 2 k~1 in both cases and represents the result of the integration over the emission times of the intermediate transitions (from level 3 to level k!1). The term r M can be obtained but setting the upper limit 1 SS,SSN of the integration over t in Eq. (13) to t . The 1 s derivative of this term with respect to t equals to s ts dr M dt = (v (t ))j n (t ) /dt "j e~j1 ts 1 SS,SSN s k B B k k k k 1 0 ts ] dt j e~(j2 ~j1 )t2 f (t , t D j 2j ). (14) 2 2 2 k 2 k~1 tk Correspondingly, the terms r M N and r M N are 1 SS,U 2 SS,U given by the expressions
P
P
r M N "e~j1 ts 1 SS,U
P P
ts
P
ts
0
dt = (v (t )) j n (t ) k B B k k k k
dt j e~(j2 ~j1 )t2 f (t , t D j 2j ) 2 2 2 k 2 k~1 tk = ts # dt j e~j2 t2 dt = (v (t ))j n (t ) 2 2 k B B k k k k ts 0 ]f (t , t D j 2j ). (15) 2 k 2 k~1 ]
P
and
P
r M N" 2 SS,U
=
dt j e~j2 t2 2 2
P
ts
dt = (v (t ))j n (t ) k B B k k k k ts 0 ]f (t , t D j 2j ). (16) 2 k 2 k~1
(17)
which holds also for the coincident events (areas) obtained after folding with the detector response functions. Obviously, this equation can also be used for lifetime determination. However, the problem of what approach is optimal for its use requires further investigations which are out of the scope of the present work. These investigations have to clarify if the part of the function to be integrated for the calculation of the terms r M and r M and 1 SS,SSN 2 SS,SSN which re#ects the time-dependence of the population of the levels involved can be conveniently approximated with elementary functions for the case of an arbitrary cascade. If so, then an analysing procedure analogous to the one for the case of a gate set on a directly feeding transition is feasible. If not, one has to rely on the simulation of gated spectra and deducing the lifetimes of the cascade levels from the comparison with the experimental spectra.
5. Summary and conclusions A new method for the description of the c-ray line-shapes observed in coincidence lifetime measurements using the Doppler-shift attenuation method is proposed for the case where a gate is set on the shifted portion of a transition which feeds the level of interest. A procedure for lifetime determination in the general framework of the di!erential decay-curve method is developed for the case where the gate is set on a transition which feeds directly the level of interest. The procedure is illustrated by an application to data simulated for the case of the 116Cd(16O, 4n)128Ba reaction at a beam energy of 72 MeV observed with a multi-detector set-up. It can be expected that the simultaneous description of spectra obtained with di!erent gates on the same feeding transition will increase the sensitivity with respect to the stopping-powers employed in the analysis. The case where the gate is set higher in the feeding cascade is also brie#y considered.
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Acknowledgements Two of us (P.P. and D.T.) are grateful for the kind hospitality of our colleagues at the Cologne University. D.T. would like to acknowledge the support of the Deutsche Akademische Austauschdienst (DAAD). This work was funded by the BMBF under contract 06 OK862 I(0). The support of the Bulgarian Science Foundation under contract Ph.801 is also appreciated. References [1] T.K. Alexander, J.S. Forster, Adv. Nucl. Phys. 10 (1978) 197. [2] F. Brandolini, R.V. Ribas, Nucl. Instr. Meth. Phys. Res. A 417 (1998) 150.
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[3] A. Dewald, S. Harissopulos, P. von Brentano, Z. Phys. A 334 (1989) 163. [4] G. BoK hm, A. Dewald, P. Petkov, P. von Brentano, Nucl. Instr. Meth. Phys. Res. A 329 (1993) 248. [5] G. Winter, ZfK Rossendorf Report ZfK-497, 1983. [6] G. Winter, Nucl. Instr and Meth. 214 (1983) 537. [7] P. Petkov, J. Gableske, O. Vogel, A. Dewald, P. von Brentano, R. KruK cken, R. Peusquens, N. Nicolay, A. Gizon, J. Gizon, D. Bazzacco, C. Rossi-Alvarez, S. Lunardi, P. Pavan, D.R. Napoli, W. Andrejtsche!, R.V. Jolos, Nucl. Phys. A 640 (1998) 293. [8] P. Petkov, D. Tonev, J. Gableske, A. Dewald, T. Klemme, P. von Brentano, Nucl. Instr. and Meth. A 431 (1999) 208. [9] L.C. Northcli!e, R.F. Schilling, Nucl. Data Tables A7 (1970) 233. [10] J.F. Ziegler, W.K. Chu, Atomic Data Nucl. Data Tables 13 (1974) 463. [11] J.F. Ziegler, J.P. Biersack, in: D.A. Bromley (Ed.), Treatise on Heavy-Ion Science, Vol. 6, Plenum Press, New York, 1985, p. 95.