Time-interval analysis of β decay. V. Horvat and J. C. Hardy

Transcription

1 Tme-nerval analyss of β decay V. Horva and J. C. Hardy Work on he even analyss of β decay [1] connued and resuled n he developmen of a novel mehod of bea-decay me-nerval analyss ha produces hghly accurae resuls for he half-lfe, decay rae, and background rae, regardless of he even rae, naure of he deecon-sysem dead me, and/or exen of he dead me, whle beng concepually smple, free of noable numercal challenges, fas and exac. The mehod apples when even arrval mes are measured and recorded ndvdually, whch can be accomplshed easly usng exsng elecronc modules, such as me-o-dgal converer (TDC) or waveform dgzer (WFD). These modules are conrolled by means of a personal compuer, usng cusomzed sofware developed n our lab. Ths repor descrbes he prncples behnd he me-nerval analyss mehod and demonsraes s robusness based on an example nvolvng smulaed evens. The half-lfe of a β-decayng nuclde s deermned from he known me-dependence of he even rae expeced under deal condons (.e., n he absence of he deecon sysem s dead me), whch s descrbed by a funcon we denoed by ρ and call he deal rae funcon. For example, n he case of a sngle-componen decay n he presence of a consan background B, ρ = A exp(-λ) + B, (1) where A s he nal deal rae (a me = 0) n he absence of background, and λ = ln(2) / T 1/2 (2) s he nuclde-specfc decay consan, whch s relaed o he nuclde s half-lfe T 1/2. Here s assumed ha he sysem s deecon effcency does no depend on ρ. The me-nerval analyss mehod was derved based on Eqs. (9) and (10) of Ref. [1], whch express he probably dp ha an even occurrng a me zero wll be followed by he nex even n he me nerval [, + d), provded ha a known deecon-sysem dead me d follows he even deeced a me zero: dp = Θ( - d ) exp[- <ρ> d ( - d )] ρ d, (3) where 1 d d d ( ) d ; (4) ρ s he value of ρ a me ; and Θ( - d ) s he Heavsde (un-sep) funcon. The Heavsde funcon reflecs he fac ha he probably of deecng an even n he me nerval [0, d ) equals zero. IV-38

2 Consequenly, dp on he lef-hand sde of Eq. (3) s also he probably of deecng an even n me nerval [, + d) followng he deecon of no evens n he me nerval [ d, ). In fac, replacng d n Eqs. (3) and (4) wh l, where l s he deecon-sysem lve me precedng he deecon of he even a me, yelds dp = Θ( l ) exp[- <ρ> l l ] ρ d, (5) where 1 l l l ( ) d. (6) The key elemen n he me-nerval analyss mehod, whch ensures ha he mehod s exac (.e., no nvolvng any approxmaons n s concep), s o know d exacly. Unforunaely, n realy, he acual value of d canno be deermned exacly for each measured even. However, f he mng of each measured even s recorded, s possble o mpose, by means of sofware, a known fxed exendable dead me τ e ha follows each measured even. The measured (prmary) evens ha are no elmnaed by he mposed dead me can hen be used o form a secondary even se. If τ e s se o be equal o (or greaer han) he larges acual dead me d n he orgnal (prmary) even se, hen he acual dead me d, as well as he acual lve me l, for each even n he secondary se can be deermned exacly. Consequenly, an exac analyss can be performed on he secondary even se, even hough he naure and/or he exen of he deecon-sysem dead me may no be known exacly for each even n he prmary se. To ensure ha he dead me τ e mposed on he evens recorded n a real measuremen s suffcenly large bu no oo large (o avod removng oo many evens), analyss of he prmary-even se mus be performed several mes, each me wh a dfferen value of τ e. By plong he resuls of he analyses as a funcon of τ e, should be sragh-forward o deermne he bes value of τ e (here denoed by τ m ), below whch he resuls show a rend, and above whch he resuls vary randomly. The exen of he random varaons for τ e > τ m mus be smaller han hose normally expeced based on he number of evens analyzed. Ths s because he secondary even ses obaned from he same prmary even se by mposng dfferen values of τ e are no sascally ndependen. The goal of daa analyss s o deermne he bes esmaes (or mos-lkely values) of he parameers of ρ and her unceranes. In he me-nerval analyss mehod, as appled o a sngle measuremen of bea decay ha sared a me = 0, hs s accomplshed by evaluang quany Z, whch s proporonal o he probably of obanng, n a repeaed measuremen under he same condons, he acual me sequence of evens ha survved afer he dead me τ m had been mposed. Then Z exp( f z d) N 1 ( dp / d), (7) IV-39

3 where ( dp / d) [ ( )]exp[ l ( ) l ( ) d] ; (8) N s he oal number of such evens; ( = 1, 2,, N) s he nsan when he -h even occurred; l () s he deecon-sysem lve me perod precedng even ; f s he nsan when he measuremen ended; ρ s he value of he deal even rae a me ; and z = mn( f, N + τ m ). (9) The exponenal funcon n Eq. (7) represens he probably of measurng no evens n he me nerval ( N, f ). Fnally, he parameers of ρ [.e., A, T 1/2, and B, f ρ s assumed o be gven by Eqs.(1) and (2)] are vared eravely n order o fnd he se of her mos-lkely values, whch are aken o be hose ha maxmze he value of Z. For praccal reasons, hs s done by mnmzng he value of quany E gven by E 2ln Z. (10) If several measuremens are analyzed smulaneously so ha, for example, a common value of T 1/2 can be deermned, he parameers of ρ are vared eravely n order o maxmze he produc of Z-values or o mnmze he sum of E-values obaned for each measuremen. The uncerany of any parameer of ρ s obaned as he square roo of he correspondng dagonal elemen of he nverse of he Hessan marx of E. Accuracy and sascal conssency of he resuls can be bes assessed by applyng he menerval analyss mehod o smulaed even ses ha have been consruced based on mposed values for he parameers of ρ. The smulaed even ses used o es he me-nerval analyss mehod were made o mmc hose obaned n he acual measuremens of he 26m Al half-lfe, whch used he K-500 superconducng cycloron, he Momenum Achromac Recol Separaor, and he Precson On-Lne Decay Facly a Texas A&M Unversy [2]. Specfcally, was assumed ha ρ s gven by Eqs. (1) and (2), wh B =1 s -1 and T 1/2 = s [3], whle A ranged from 10 2 s -1 o 10 5 s -1. The orgnal ses of smulaed evens were made assumng ha here s no dead me, bu a dead me per even (τ m ) of up o 512 µs, as needed, was mposed on he daa by he sofware before he begnnng of he me-nerval analyss. The oal number of evens n each prmary se was abou 60 mllon, whch corresponds o a sascal precson slghly above 0.01%. The number of ndvdual decay measuremens n each smulaed even se ranged from 66 a A = 10 5 s -1 o 57,693 a A = 10 2 s -1. Each smulaed measuremen was assumed o las 125 s, whch corresponds o abou 20 half-lves. IV-40

4 In order o dsngush beween he values of B, T 1/2, and A, on whch he even smulaon was based, and he correspondng values obaned n he analyss of he smulaed even ses, lowercase symbols a, 1/2, and b wll be used for he laer. To assess he meanngfulness and qualy of resuls from he me-nerval analyss mehod, a 500-channel decay specrum was consruced for each smulaed measuremen, along wh he correspondng specrum of predced values. These predced values were obaned for each channel by negraon of he mos-lkely deal even rae ρ (as obaned n he analyss) over me, from he channel lower lm o he channel upper lm, whle skppng he me nervals whn he channel ha were covered by he dead me. The correspondng channel conens of he ndvdual specra from he same se were hen combned o consruc a sngle specrum n order o presen sascally more meanngful resuls and o amplfy and expose any sysemac errors ha mgh have occurred n he daa analyss. An example of a specrum and he resuls obaned hs way are shown n Fg. 1. Fg. 1 demonsraes ha he me-nerval analyss mehod produces accurae resuls, n parcular FIG. 1. Resuls of he me-nerval analyss presened n he form of he combned decay specrum and he correspondng specrum of he resduals for he case of a smulaed deal even se obaned assumng A = 10 5 s -1, B = 1 s -1, and T 1/2 = s, on whch an exendable dead me of 64 µs was mposed. In he decay specrum, he daa pons represen he number of evens n each (0.25 s wde) channel, he hck sold (red) lnes represen he expeced values calculaed based on he bes esmaes of he deal rae parameers obaned n he analyss (and on he mposed dead me). Lkewse, he hck dashed (gray) lnes represen he background, whle he hn sold (blue) lnes represen he decay componen. The hn dashed (black) lnes represen he expeced resuls under deal condons (.e., no dead me). The resduals are shown as a funcon of me n a separae graph locaed above he correspondng graph of he decay specrum, whle her hsograms are shown as nsers n he decay specrum graph, usng grey bars. Each hsogram of he resduals was fed by a Gaussan funcon. The bes f s shown by he sold (red) lne and he bes-f sandard devaon (σ) s ndcaed n he graph IV-41

5 for 1/2, even n he case n whch he decay specrum s drascally dsored due o he presence of an exendable dead me. Noe ha he example shown n Fg. 1 s raher exreme and was chosen only o demonsrae he robusness of he me-nerval analyss mehod. [1] V. Horva and J. C. Hardy, Progress n Research, Cycloron Insue, Texas A&M Unversy ( ), p. V-28. [2] hp://cycloron.amu.edu/ [3] J.C. Hardy and I. S. Towner, Phys. Rev. C 79, (2009). [4] V. Horva and J.C. Hardy, Nucl. Insrum. Mehods Phys. Res. A713, 19 (2013). IV-42