Determination of Qβ values from endpoint energies of beta spectra

N U C L E A R I N S T R U M E N T S AND METHODS

166 (1979) 5 0 7 - 5 1 4 ;

(~) N O R T H - H O L L A N D

P U B L I S H I N G CO.

D E T E R M I N A T I O N OF QB VALUES F R O M E N D P O I N T ENERGIES OF BETA SPECTRA* H. OTTO, P. PEUSER

lnstitut fiir Kernchemie der Universit6t Mainz, Mainz, Germany and G. NYMAN +, E. ROECKL

Gesellschctfi fiir SchwerionenJbrschung, Darmstadt, Germany Received 10 May 1979 and in revised form 16 July 1979 A plastic scintillator telescope is described for determining beta endpoint energies up to 15 MeV. The response function of the telescope was measured using monoenergetic electrons from a betatron. A nonlinear procedure for unfolding the measured beta spectra on the basis of the experimental response function was tested successfully.

1. Introduction In studying nuclei far off stability the mass excess of nuclear ground states is of particular interest, because it contains basic information on nuclear binding and allows extrapolation into unknown regions of the mass surface. A valuable method for the determination of mass excesses is the measurement of mass differences as total beta decay energies Qi~" Establishing a Q/~ value depends upon knowledge of the corresponding gamma decay scheme as well as of the endpoint energies of gamma-coincident beta spectra. We restrict ourselves here to discussing the determination of endpoint energies and assume that the decay scheme is well known. Due to the short half-lives and the small yields (or formation cross sections) of far unstable nuclides, the beta spectra ranging up to several MeV will in general be characterized by low statistics, in particular close to the endpoint. Data evaluation methods should therefore aim at including the part of the beta spectrum, which contains the main part of the intensity. In the present paper a beta telescope is described, which has been constructed for use with on-line mass separators. Its response function was determined to compensate the distortions in beta spectra introduced by the detector system. The compensation is performed by an evaluation program utilizing a nonlinear unfolding procedure. The applicabil-

ity of this procedure is tested by a real spectrum measured for 144pr (endpoint energy 2996 keV).

* This work comprises part of the Doctoral Dissertation of H. Otto, to be submitted to Johannes Gutenberg-Universit~it, Mainz. t On leave from Department of Physics, Chalmers University of Technology, S-41296 G6teborg, Sweden.

Fig. 1. Beta gamma detector system: (1) NUPLEX plastic scintillator system; (2) NE 102A plastic scintillator close to tape or foil; (3) Ge(Li) detector; (4a, b) RCA 8850 photoraultipliers; (5a, b) photomultiplier bases; (6) vacuum chamber of the tape transport system; (7) position of radioactive sources.

2. The beta telescope The beta telescope (fig. 1) consists of a 3 E - E detector arrangement mountable in the vacuum chamber of the mass separator. The radioactive sources are positioned either by a moving-tape collector or by implanting the mass-separated beam directly onto a foil at the end of a beam line [in the latter case the Ge(Li) detector shown in fig. 1 is removed]. The AE detector, consisting of a thin disc of NE 102A plastic scintillator (8 m m diameter, 5 m m thickness), discriminates against gamma ra-

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diation and furthermore defines an aperture of approximately 68 ° for the E detector. The AE scintillator is coupled via a light guide to a photomultiplier tube of type RCA 8850. The E detector consists of a cylindrical NUPLEX plastic scintillator (75 m m diameter, 75 mm length) connected to a photomultiplier-base of the same type as above. 3. Measurement of the response function A detector, exposed to particles with an energy spectrum S(E'), will show at its output a spectrum SIn(E) according to

Sn,(E) =

R(E,E')S(E')dE'.

(1)

~0

The response function R(E,E') describing the detection properties of the telescope can be measured by means of monoenergetic electrons represented here by a delta function 5(E'-E~)

f ~ R(E,E')6(E'-E'o)dE' = R(E,E'o).

The efficiency of the system was assumed to be independent of energy and angle of incidence in the energy region under consideration. The spectra of the E detector were measured using a fast-slow coincidence technique, recorded in a multichannel analyzer, and written on magnetic tape for subsequent computer analysis. In each spectrum approximately 104 events were collected. The long term stability of the system was checked by regularly taking spectra of the known conversion electron lines of ~37Cs, 2°7Bi, 2°8T1. Shape and peak position of these lines were independent of the operating state of the betatron and the quadrupole magnets of the beam transport system. Additionally, the stability of the electronic part of the measuring system was supervised with a precision pulser. An example of these spectra is given in fig. 2a. In order to determine the point source response function, the spectra for a given beam energy were integrated over the four measured angles of inci-

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In measuring R(E, Eo) one has to ensure that the energy spread of the beam is smaller than the resolution of the measuring system. Moreover, since the response to an isotropically-radiating point source differs significantly from that of a parallel beam, the point source has to be simulated by variation of the angle of incidence of the parallel beam. The response function was measured at the betatron of the Physikalisches Institut der Universit~it Wfirzburg for electrons in the energy range from 4 - 1 5 MeV in steps of 1 MeV and angles of incidence of 0 °, 10 °, 20 °, and 30 ° with respect to the detector axis. The mean energy E ' of the electron beam was known with an uncertainty of 1% and its relative energy width AE'/E' was known with an uncertainty of 0.1%. The energy of the incident electrons was calculated from the frequency of a nuclear resonance device measuring the magnetic field in a sector magnet used for definition of the beam energy. The spectra in the E detector were taken in coincidence with events in the AE detector. For pulses of the AE detector, a lower discriminator level was set corresponding to the valley separating the noise distribution and the meanenergy-loss peak. In order to keep the pile-up in the coincidence spectra below 3 %, the AE-rate was held at 10 cps by adjusting the beam intensity of the betatron.

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DETERMINATION

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E'[MeV] Fig. 3. Parameters of the response function for a point source as a function of incident energy E'. Points represent fit parameter values of the response function or values derived from these parameters. Solid lines give fits to these values, dashed lines are to guide the eye. (a) centre of the gaussian peak: (b) width a of the gaussian peak: (c) square-root dependence of F W H M o f gaussian peak:, (d) relative peak resolution AE/E; (e) area ,6 of low-energy tail: (f) onset E 1 of the low-energy tail; (g) ratio P/T of gaussian peak to low energy tail for scintillator and Ge(Li) telescopes.

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dence. A polynomial of second degree was fitted to the data points of each energy channel as a function of the incident angle and weighted with the surface element integrated over the aperture of the E detector. Using the covariance matrix of the fit, the corresponding statistical errors were calculated. An example of these point source spectra is given in fig. 2b. A function R consisting of a gaussian full-energy peak and a low-energy tail function h, which increases with energy and decreases below the fullenergy peak, was fitted to these spectra as a function of energy E. (The low-energy part h can be understood qualitatively as being due to electron escape and to incomplete light collection) I |-- / ~ R(E o, ~r,~,E1, c;E) = c [ ~

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This fit contains the following five parameters: mean energy of the gaussian full-energy peak; width of the gaussian peak; area of the low-energy tail; onset of the low-energy tail; total area below the function R (normalization factor). Polynomials up to the 4th degree were fitted to the parameters E0, e, fl, and E~ as functions of incident energy E'. With the parameters of these polynomials the response function R can be represented as a function of E and E'; R ( E , E ' ) = R ( E , p ( E ' ) ) , w i t h p containing the parameters E0, cr,/3 and E~. The centre of the gaussian peak E0 (fig. 3a) shows a linear dependence on the incident energy E' with a m a x i m u m integral nonlinearity of 0.9%. The maximum integral nonlinearity is defined as the maximum ratio of the residual between a measured point and its corresponding fit value to this fit value. For the width of the gaussian peak (fig. 3b) one expects from statistical reasons a proportionality to rE. For a detector of similar shape this dependence was shown ~) to be valid up to 8 MeV. In our measurements significant deviations occur at higher energies (fig. 3c) being obviously due to the finite size of the detector. This results in variation of the resolution of the gaussian peak (fig. 3d) between 5% and 8%. The area of the low energy tail (fig. 3e) is nearly constant between 4 and 9 MeV, but increases linearly with increasing energy. As a constraint of this fit,/3 at 0 MeV was approximated

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by the average /3 resulting from the energy range between 4 and 9 MeV. The point of onset E~ of the low-energy tail (fig. 3f) increases linearly with energy up to 8 MeV, but remains constant for higher energies. As a constraint of this fit the low energy tail was assumed to be 0 MeV at E ' = 0 MeV. In valuing the resolving power of a beta telescope one should not only consider the resolution of the gaussian peak which is correct in the case of gamma spectroscopy, where the events due to a certain gamma transition are regarded as being mapped to a unique peak. Since its area is proportional to the number of events divided by the photopeak efficiency, only the peak portion of the gamma response function is important. For beta decay, however, events from a specific transition are mapped to a continuous spectrum Sm according to eq. (1). Considering the whole response function as a distribution, one has to calculate the effective F W H M from its variance. Comparing our system with one using a Ge(Li) or intrinsic Ge as the E detector, it was shown 2) that the advantages of good peak resolution (fig. 3d) were overcompensated by the disadvantages of a small ratio of peak to low-energy tail (fig. 3g) resulting in a better system resolution of the plastic scintillator telescope (fig. 4). Despite some advantages in the energy calibration of a solid state system, its good peak resolution alone is not necessarily a strong argument. 50 Ge(Li)SYSTEM 40

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511

DETERMINATION OF Q# VALUES

4. Principle of the unfolding procedure Writing eq. (1) as a matrix equation with an energy interval A E = 1 results in (4)

sm = R S .

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According to the Banach principle of contracting mapping, the limit S* of the sequence Sn is a unique fixed point of ~ which is obtained independent of the choice of the starting value So. This is obvious from considering the first term of eqs. (6) and (10). The second term of eqs. (6) and (10) imply that S*

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which has to be compensated for. Due to the subtraction in eq. (5) the statistical fluctuations tend to increase with the number of steps in the unfolding procedure. This is physically irrelevant because the hypothesis S~ has to be a smooth function overlaid by the statistical fluctuations of the measured spectrum only. Therefore at the end of each iteration step, the spectrum S, used as input for the next iteration step is smoothed by a procedure reducing the fluctuations by the means of a x 2 testS). As with all smoothing procedures this one, too, introduces some ambiguity in the hypotheses which is compensated partly by the convergence of the algorithm (5). The output spectra of this modified procedure for real spectra are the hypotheses overlaid with the statistical fluctuations of the measured spectrum.

5. Test of the unfolding procedure In order to test the unfolding procedure with respect to convergence and systematical errors, the following cases were investigated: A "simulated" beta spectrum with an endpoint energy of 8 MeV was calculated according to the Fermi theory for 8 × 104 events and folded with the response function of the beta telescope. A real spectrum was measured using a ~44pr source with an endpoint energy of 2996.0_+2.9 keV 9). The simulated spectrum, which corresponds to a r

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512

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real one except for statistical fluctuations was unfolded iteratively in three steps. The distortions in the folded spectrum display the structures mentioned by Cramer et al.7). Its behaviour at the endpoint is shown in fig. 5. The largest increments towards the calculated spectrum are performed in the first two or three steps. Due to the low statistics and possible pile-up in real spectra, the information content will be small in the region around the endpoint compared with data points in the large region below the endpoint. The difference between the unfolded spectra and the calculated spectrum (fig. 6) display an oscillatory behaviour about the calculated spectrum with little gain between the second and third step. Due to the detector response function they result 7) in shifts of the fitted endpoint energies and in their dependence upon the fit range. In fig. 6, Fermi-Kurie plots are shown of the measured ~44pr spectrum and of the unfolded spectrum (3 steps) together with linear fits for both

et al.

cases. The deviation from a straight line of the Fermi-Kurie plots leads to an uncertainty in endpoint determination over a wide range, the degree of uncertainty being dependent upon the position and the width of the fit interval. A crucial test for the validity of the unfolding procedure will be to show in contour plots of fitted endpoint versus both these parameters how far this dependence can be reduced. Although complete Mean position of fit intervaL[MeV] 2.0 2.5

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350 400 450 500 Neon position 0f tit interval [channels] Fig. 7. Contour plots of fitted endpoint energies as a function of fit-interval position and fit-interval width. The fits were made to a measured spectrum of 144pr in Fermi-Kurie plot, Contour plots of constant endpoint energies are drawn for ]0 keV intervals, for some contour lines of interest the endpoint energies are given in MeV. The region around the known endpoint energy of 2996 keV is hatched within ± 2 0 keV (E0-band). (a) Based on the measured spectrum. (b) Based on the spectrum resulting from three steps of the unfolding procedure applied to the measured spectrum.

D E T E R M I N A T I O N OF QB V A L U E S

independence cannot be reached because of the finite number of steps in the procedure, one hopes to decrease this systematic error to a level below the minimum statistical error introduced by the energy calibration. (We assumed 20 keV for the calibration uncertainty in the range 3-8 MeV). The test consists in fitting Fermi-Kurie analysed beta spectra before and after the application of a three-step unfolding procedure, varying the parameters "fit-interval position" % and "fit-interval width" C,w. The dependence of the fitted endpoint energy Eof upon C~ and c~w Eof = Eof(Cip, ci~ )

(13)

is represented in contour plots by lines of constant endpoint energies. Comparing the contourplots of the measured and unfolded spectra (see fig. 7) gives detailed information concerning the convergence of the procedure as well as the amount of systematical error reduction. (For contour plots of the simulated spectrum see ref. 8.) For the interpretation of these contour plots it is useful to consider a simplified model for the fit assuming that the fit of a straight line to a spectrum can be represented by a tangent or a secant to the spectrum over the fit interval. An expression for Eof is then easily obtained and with the additional information about the derivatives of the spectrum, Eof can be expanded with respect to the parameters % and c~,. The salient features in the contour plots are summarized here only briefly: Be/bre applying the unfolding procedure the contour plot displays the following structures (see fig. 7a): (1) The values of Eof depend essentially upon the fit-interval position and to a lesser extent upon its width. (2) Due to the curvature in the spectra a minimum (UoT~°0 exists for Eof. (3) An energy band with a width of twice the statistical error of energy calibration including the correct endpoint energy E0 are situated on both sides of the ,~ofr'lmm~valley (E0-bands). Mostly the lower E0-band is below the region of interest. The upper E0-band seems to be in the middle between Elm~n~ and the maximum possible Eof. of (4) The lines Eof = constant diverge with increasing fit-interval width in the range of the upper E0 -band. A.Ber unfolding, the structures in the contour plot change (see fig. 7b): (5) the whole landscape has become shallower.

513

(6) ~o~ in/ increases towards E0, the maximum of Eof decreases. (7) Both E0-bands approach Uo~i°l. (8) The area of both E0-bands increases remarkably. (9) Both E0-bands flow together. All the above features are observed in the simulated spectrum 8) as well as in the measured one. The additional fine-structures in the contour plots might be interpreted (with this simple model for the fit) in terms of local variations in the spectrum or its statistical variations over a certain region are taken into account. On the other hand,each contour plot is calculated from about 100equally spaced Eof-values with a scan width of 20 channels. Structures with a spatial expansion below this width may be due to the contourplot algorithm and should be omitted from interpretation. Because of the complexity of the problem, we want to restrict ourselves to the following qualitative conclusions concerning the use of the unfolding procedure: (1) The unfolding procedure is convergent within the statistical error of the energy calibration (20 keV) to the correct value E0. (2) The probability to hit the value E0 within the calibration error has increased remarkably after application of the unfolding procedure, i.e. Eof has become more independent of the choice of fit-interval position and width. The several MeV wide plateau around the "correct" endpoint value allows indeed a meaningful endpoint determination. (3) The representation of the response function is reasonable within the limits of the attainable accuracy. 6. Conclusions We have described a telescope for measuring beta endpoint energies up to 15 MeV. As was shown in two test cases, the unfolding procedure based on the experimental response functions enables a reliable determination of endpoint energies. It is evident that the test cases are extremely simple ones. Q-value measurements of interest today concern however far-unstable nuclei, the decay schemes of which are rather complicated (and badly known), and the beta spectra of which contain several sometimes unresolvable components. Moreover, such cases underly serious limitations with respect to counting statistics, which often reaches only 10 3 events in a gama-coincident beta spectrum compared with 10 4 events in the 144pr

514

H. O T T O et al.

spectrum discussed above. Q/~-value measurements have been extended recently, for example, to very neutron-rich isotopes of rubidium, strontium, niobium, molybdenum and technetium in the mass region 96 to 102~°), as well as to very neutrondeficient isotopes of rubidiumS). We would ',iKe to thank Dr. H. Roos for valuable help during the measurements, Mrs. E. Rocholz and Mrs. G. W611 for assistance in data processing. The financial support of the Bundesministerium ftir Forschung und Technologie is gratefully acknowledged. References l) E. Beck, Nucl. Instr. and Meth. 76 (1969) 77.

2) W. Lauppe, G. Nyman, H. Otto, P. Peuser and E. Roeckl, Annual Report (1975) lnstitut for Kernchemie, Mainz, p. 72. 3) D. D. Slavinskas, T.J. Kennett and W. V. Prestwich, Nucl. Instr. and Meth. 37 (1965) 36. 4) E. E. Habib and J.P. Labrie, Nucl. Instr. and Meth. 130 (1975) 199. 5) T.J. Kennett, W.V. Prestwich and A. Robertson, Nucl. Instr. and Meth. 151 (1978) 285. ~) T.J. Kennett, W.V. Prestwich and A. Robertson, Nucl. Instr. and Meth. 151 (1978) 293. 7) j. C. Cramer Jr., B.J. Farmer and C. M. Class, Nucl. Instr. and Meth. 16 (1962) 289. 8) It. Otto, Doctoral Dissertation, Universit/:it Mainz, to be submitted. 9) A. H. Wapstra and K Bos, Atomic Nucl. Data Tables 19 (1977) 175. 10) p. Peuser, H. Otto, N. Kaffrell, G. Nyman and E. Roeckl, Nucl. Phys. A (in print).