This paper investigates the possibility of distinguishing between the effects  of radiation coming from two or more radioactive isotopes, by using  methods of statistical mathematics. The procedure uses a mixed distribution  of an additive type. Mathematical treatment is demonstrated herein on an  example of analysis of composite radiation from two radioactive sources

Dolićanin-Đekić Diana

Pućić Dženis F.

Stanković Koviljka Đ.

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AP  PLI  CA  TION  OF  AN  AD  DI  TIVE-TYPE  MIXED  PROB  A  BIL  ITYDIS  TRI  BU  TION  TO  THE  ANAL  Y  SIS  OF  RA  DI  A  TION  FROM  AMIX  TURE  OF  RA  DIO  AC  TIVE  SOURCESbyDi  ana ]. DOLI]ANIN-DJEKI] 1, Dženis F. PU]I] 2, and Koviljka Dj. STANKOVI] 3*1Fac  ulty of Tech  ni  cal Sci  ences, Uni  ver  sity of Kosovska Mitrovica, Serbia2State Uni  ver  sity of Novi Pazar, Novi Pazar, Ser  bia 3Fac ulty  of  Elec tri cal  En gi neer ing,  Uni ver sity  of  Bel grade,  Bel grade,  Ser biaShort pa  perDOI: 10.2298/NTRP1302191DThis  pa per  in ves ti gates  the  pos si bil ity  of  dis tin guish ing  be tween  the  ef fects  of  ra di a tion  com -ing from two or more ra  dio  ac  tive iso  topes, by us  ing meth  ods of sta  tis  ti  cal math  e  mat  ics. Thepro ce dure  uses  a  mixed  dis tri bu tion  of  an  ad di tive  type.  Math e mat i cal  treat ment  is  dem on -strated herein on an ex am ple of anal y sis of com pos ite ra di a tion from two ra dio ac tive sources.Key words: mix ture  of  ra dio ac tive  iso topes,  ad di tive-type  mixed  prob a bil ity  dis tri bu tion,  pa ram e tersfor  in di vid ual  dis tri bu tion  uni versesIN TRO DUC TIONDe cay time of an in di vid ual nu cleus fluc tu ates ir -reg  u  larly from one nu  cleus to an  other, so that, at thepres  ent level of de  vel  op  ment of sci  ence, it is not pos  si  -ble to pre  dict whether a spe  cific nu  cleus will de  cay to  -mor row, next week, or in a mil lion years. An in di vid ualnu cleus  de cay  is  con sid ered  com pletely  ran dom,  with -out any  thing to de  ter  mine the time when it hap  pens,there be ing no law that would re late it to any thing [1, 2].For this rea  son, when a mix  ture of ra  dio  ac  tive iso  topesis  con sid ered,  ex per i men tally  ob tained  dis tri bu tionfunc  tions for the ran  dom num  ber of de  cays (i. e. thenum ber of pulses in a de tec tor) rep re sent a com bi na tionof  two or  more the o ret i cal  dis tri bu tion func tions.  Com -bi na tions  of  dis tri bu tions  re sult  in  mixed  dis tri bu tions.These can be of ad di tive or multi pli ca tive type, with thelet ter  ones  gen er ally  per tain ing  to  prob a bil ity  en large -ment laws [3-5]. Be  fore one can reach the stage of sta  -tis ti cal  for mu la tion  of  a  ran dom  (sto chas tic)  pro cess,  itneeds to be “phenomenologically known”. One there  -fore has to start with ex  per  i  ments whose re  sults (num  -ber of pulses in a de  tec  tor) vary within cer  tain ran  domlim its.  The  causes  of  vari a tions  of  ex per i men tal  re sultscan be in  her  ent to the pro  cess of de  cay of ra  dio  ac  tiveiso  topes, aris  ing from its bound  ary con  di  tions, or mayre  side in ran  dom mea  sur  ing er  rors. This last cat  e  goryshould be min i mized and be de ter mined as ac cu rately as pos  si  ble, since it can not be cov  ered by the al  go  rithmpro  posed in this pa  per [6, 7].The aim of this pa  per is to de  velop an al  go  rithmwhich en  ables ra  di  a  tion from a mix  ture of ra  dio  iso  -topes to be an  a  lyzed by a sta  tis  ti  cal method.THE  EX PER I MENTEm pir i cal  dis tri bu tions  of  the  ran dom  vari abledes  ig  nated as “the num  ber of pulses in the de  tec  tor”for a mix  ture of sources were ob  tained by means of aGM coun ter and two ra  dio  ac  tive iso topes, Cs-137 andAm-241. The sources were spe cif i cally po si tioned rel -a  tive to the GM coun  ter's tube, so that the Am sourcewould pro  vide a pulse count higher by ap  prox  i  matelyone third than that from the Cs source. The GM coun  -ter  pro vided  with  anti-co in ci dence  pro tec tion  wasplaced within a shielded booth (shield  ing it from elec  -tro mag netic  ra di a tion)  with  a  sup pres sion  level  ofmore than 100 dB. Mea  sure  ments were per  formed atfive min  ute in  ter  vals, first with the Cs source only,then with the Am source, and fi  nally with bothsources.  Geo met ri cal  pa ram e ters  of  the  sys tem,  in -clud ing the po si tion of the sources and the de tec tor, re -mained  un changed  dur ing  mea sure ments.  Re sultswere cor  rected for back  ground ra  di  a  tion, which wasalso mea  sured, al  though it proved to be neg  li  gi  ble.The com  bined mea  sure  ment un  cer  tainty of the wholeex per i men tal  pro ce dure  was  be low  3%  [8,  9].RE SULTS  AND  DIS CUS SIONNum ber of pulses in the de tec tor is a sto chas tic quan  tity  which  fol  lows the  nor  mal  dis  tri  bu  tionD. ]. Doli}anin-Djeki}, et al.: Ap  pli  ca  tion of an Ad  di  tive-Type Mixed ...Nu  clear Tech  nol  ogy & Ra  di  a  tion Pro  tec  tion: Year 2013, Vol. 28, No. 2, pp. 191-194 191*  Cor re spond ing  au thor;  e-mail:  kstankovic@etf.rs[10, 11]. Fig  ure 1 shows the ex  per  i  men  tally ob  -tained dis  tri  bu  tion func  tions for the Cs source,F(Cs), for the Am source, F(Am), and for the su per -po si tion of Cs and Am sources, F(Cs + Am). Re sultsin di cate that the last of these three dis tri bu tions rep -re  sents a mixed dis  tri  bu  tion of the ad  di  tive type.The phys  i  cal causes of such additively com  -posed  mixed  dis tri bu tion  are  dif fer ent  mech a nisms,dis trib uted  ac cord ing  to  dis tri bu tion  func tions  F1(x) in ac  cor  dance with which the en  tire pro  cess (dis  trib  -uted ac cord ing to dis tri bu tion func tion Fi(x) can takeplaceF x F x i iijiij( ) ( ), = == =å å a a1 11 where (1)Multi pli ca tive  mixed  dis tri bu tions  to  be  con sid -ered here are math  e  mat  i  cally very dif  fi  cult to pro  cess,be  cause of the six pa  ram  e  ters in  volved (mCs, sCs, aCs,mAm,  sAm,  and aAm), which ac cord ing to (1) and aCs ++.aAm = 1  can be re duced to five pa ram e ters (mCs, sCs,aCs, mAm,  sAm,).  Be cause  of  this,  math e mat i cal anal y -sis of a mixed dis  tri  bu  tion con  sist  ing of two nor  maldis  tri  bu  tions is rather com  plex and sub  ject to a greatun cer tainty.  There  con se quently  re main  em pir i calmeth ods, which can be em ployed ei ther us ing com put -ers,  or  graph i cally,  us ing  a  prob a bil ity  pa per.  Graph i -cal meth  ods are based on the em  pir  i  cal prob  a  bil  ityden sity  func tion  f *(x), which is di  vided into den  sityfunc tions  f x i*( )  of a par  tic  u  lar type of dis  tri  bu  tion, insuch a way that the ini tial den sity f *(x)  is ob tained as asu per po si tion of  f x i*( ) func tions. This pro ce dure is fa -cil  i  tated by the fact that, when nor  mal dis  tri  bu  tionprob a bil ity pa per is used, the branches of the em pir i cal den sity func tion  f *(x)  are al most nor mally dis trib utedif the com po nent den si ties  f x i*( ) are also nor mally dis -trib uted.In  a  graph i cal  ap prox i ma tion,  these  straightbranches should be the start  ing point. The mixed dis  -tri bu tion from fig. 1 is de com posed into its con stit u entparts in fig. 2.The char  ac  ter  is  tic of the  mixed  dis  tri  bu  tionF(Cs  + Am) and the modes de  scribed sug  gest thatF(Cs + Am)  is com  posed of two nor  mal dis  tri  bu  -tions. Since the start  ing point has to be the em  pir  i  -cal den  sity func  tion, the mea  sured val  ues are re  -corded in a fre  quency ta  ble [12, 13] and pro  cessed(tab. 1). The em  pir  i  cal den  sity func  tion is graph  i  -cally repre sented (fig. 2) and di vided into two “uni -verse”  (pop u la tions).  The  rel a tive  fre quency  ofeach com po nent uni verse is en tered into tab. 1. Fre -quency  sums  yield  com po nent  pa ram e ters  as  aCs == 0.42 and aAm = 0.58.Cu mu la tive  fre quen cies  are  then  formed  foreach  uni verse,  ac cord ing  toh h k iikS ==å11 acs,Am(2)(see tab. 1) and rep re sented in or der to de ter mine to the pa ram e ters  graph i cally  as  quantiles    n50Cs,  s50Cs,  and n50Am, s50Cs. The re  sult ob  tained is a mixed dis  tri  bu  -tion,  con sist ing  of  two  nor mal  dis tri bu tions  (markedwith  F), with the dis  tri  bu  tion func  tion ex  pressed asFn n( ). ( ; ; ) . ( ; ; ) ( )Cs AmCs Am+ == + 042 125 7 058 149 6 32 2 F FThe  pa ram e ters  of  the  com po nent  dis tri bu tionsare in good agree  ment with the dis  tri  bu  tion func  tionfor in  di  vid  ual sources (figs. 1 and 2).D. ]. Doli}anin-Djeki}, et al.: Ap  pli  ca  tion of an Ad  di  tive-Type Mixed ...192 Nu  clear Tech  nol  ogy & Ra  di  a  tion Pro  tec  tion: Year 2013, Vol. 28, No. 2, pp. 191-194Fig ure  2.  Anal y sis  of  mixed  dis tri bu tion  F(Cs + Am)from fig. 1Fig  ure 1. Mixed dis  tri  bu  tion of the “num  ber of pulses”ran dom  vari ableF(Cs) – dis  tri  bu  tion func  tion for the num  ber of pulses fromthe Cs source; F(Am) – dis  tri  bu  tion func tion for the num  berof pulses from the Am source; F(Cs + Am) – dis  tri  bu  tionfunc  tion for the num  ber of pulses from a mixed Cs and AmsourceCON CLU SIONSThe method pre  sented in the pa  per en  ables in  di  -vid  ual ra  dio  iso  topes in a mix  ture to be sep  a  rately de  -tected and char ac ter ized. This method also makes pos -si  ble that a GM coun  ter's dead time be de  ter  minedac  cord  ing to a sin  gle mea  sure  ment. Be  cause of themath e mat i cal dif fi cul ties as so ci ated with mixed dis tri -bu  tions, they should only be used when that is phys  i  -cally ne ces si tated,  i. e. when the phys i  cal model of theran dom  pro cess  be ing  in ves ti gated  pro duces  a  mixeddis  tri  bu  tion. One should on no ac  count in  ter  pret anem pir i cally-found  re la tion ship  from  the  out set  as  amixed dis  tri  bu  tion with  out such a model. Many re  la  -tion ships in na ture and tech nol ogy in trin si cally fol lowmixed dis tri bu tions, but spe cific in di vid ual in flu encesusu ally dom i nate, so that mixed com po nents can be ig -nored.AC KNOWL EDGE MENTThe Min is try of Ed u ca tion, Sci ence and Tech no -log i cal  De vel op ment  of  the  Re pub lic  of  Ser bia  sup -ported this work un  der con  tract 171007.AU THOR  CON TRI BU TIONSThe o ret i cal  anal y sis  was  car ried  out  by  D.  ].Doli}anin-Djeki} and Dž. F. Pu}i}. Ex  per  i  mentswere car ried out by Dž. F. Pu}i} and K. Dj. Stankovi}. All of the au thors have ana lysed and dis cussed the re -sults. The manu  script was writ  ten by Dž. F. Pu}i}.The fig  ures and ta  bles were pre  pared by K. Dj.Stankovi}.REF ER ENCES[1] Stankovi}, K., In  flu  ence of the Plain-Par  al  lel Elec  -trode Sur  face Di  men  sions on the Type A Mea  sure  -ment Un  cer  tainty of GM Coun  ter, Nucl TechnolRadiat, 26 (2011), 1, pp. 39-44[2] Pejovi}, M. M., et al., Suc ces sive Gamma-Ray Ir ra di -a tion  and  Cor re spond ing  Post-Ir ra di a tion  An neal ingof  pMOS  Do sim e ters, Nucl Technol Radiat, 27(2012), 4, pp. 341-345[3] Stankovi}, K., et al., Sur  face Time En  large  ment Lawfor Gas Pulse Break  down, IEEE Trans  ac  tions on Di  -elec trics  and  Elec tri cal  In su la tion,  15  (2008), 4, pp.994-1005[4] Stankovi}, K., et al., Time En  large  ment Law for GasPulse Break down, Plasma Sources Sci ence and Tech -nol ogy,  18 (2009), 025028, p. 12[5] Osmokrovi}, P., et al., Va lid ity of the Space-Time En -large  ment Law for Vac  uum Break  down, Vac uum,  85(2010), 2, pp. 221-230[6] Hauschild, W., Mosch, W., Sta  tis  ti  cal Tech  niques forHigh-Volt age En gi neer ing, Pe ter Peregrinus Ltd, 1992[7] Stankovi}, S., et al., MSV Sig  nal Pro  cess  ing Sys  temfor Neu tron-Gamma Dis crim i na tion in a Mixed Field, Nucl Technol Radiat., 27 (2012), 2, pp. 165-170[8] Vujisi}, M., Stankovi}, K., Osmokrovi}, P., A Sta tis ti -cal  Anal y sis  of  Mea sure ment  Re sults  Ob tained  fromNon lin ear  Phys i cal  Laws,  Ap plied  Math e mat i calMod el ing,  35 (2011), 7, pp. 3128-3135[9] Stankovi}, K., Vujisi}, M., Deli}, Lj., In  flu  ence ofTube Vol  ume on Mea  sure  ment Un  cer  tainty of GMCoun ters, Nucl Technol Radiat, 25 (2010), 1, pp. 46-50[10] Doli}anin, ].,  et al., Statistisal Treat  ment of Nu  clearCount ing  Re sults,  Nucl Technol Radiat, 26 (2011), 2,pp. 164-170[11] Lazarevi}, Dj., et al.,  Ra di a tion  Hard ness  of  In diumOx  ide Films in the Coo  per-Pair In  su  la  tor State, NuclTechnol Radiat, 27 (2012), 1, pp. 40-43[12] Hill,  T.,  Sta tis ti cal  Me chan ics,  Prin ci ples  and  Se -lected Ap  pli  ca  tions, McGraw-Hill Book Com  pany,Inc, 1956[13] Doli}anin, D., et al., Con struc tion of an AutogeneratorDy namic  Model  Ap pli ca ble  to  Nu clear  Pro cesses,Nucl Technol Radiat, 26 (2011), 1, pp. 74-77Re  ceived on March 30, 2013Ac  cepted on April 30, 2013  D. ]. Doli}anin-Djeki}, et al.: Ap  pli  ca  tion of an Ad  di  tive-Type Mixed ...Nu  clear Tech  nol  ogy & Ra  di  a  tion Pro  tec  tion: Year 2013, Vol. 28, No. 2, pp. 191-194 193Ta  ble 1. Eval  u  a  tion of the mixed dis  tri  bu  tion (fig. 2) with a sam  ple size of n = 199Classnumber Class limits AbsolutefrequencyRelativecumulativefrequencyRelativefrequencyRelativefrequencyUniverse CsRelativefrequencyUniverse AmRelative cumulative frequencyUniverse CsRelative cumulative frequencyUniverse Amk nCs nAm hmk hSk hk hkCs hkAm hSkCs hSkAm1 >107 112 2 0.010 0.010 0.010 – 0.024 –2 >112 117 4 0.050 0.040 0.040 – 0.119 –3 >117 122 12 0.120 0.070 0.070 – 0.286 –4 >122 127 20 0.220 0.100 0.100 – 0.525 –5 >127 132 19 0.315 0.095 0.093 0.002 0.747 0.0036 >132 137 15 0.390 0.075 0.062 0.013 0.895 0.0267 >137 142 20 0.490 0.100 0.032 0.068 0.971 0.1448 >142 147 28 0.630 0.140 0.010 0.130 0.995 0.3709 >147 152 34 0.800 0.170 0.020 0.168 0.999 0.66210 >152 157 26 0.930 0.130 – 0.130 – 0.88811 >157 162 11 0.985 0.055 – 0.055 – 0.98312 >162 167 2 0.995 0.010 – 0.010 – 0.9990.995 aCs = 0.419a » 0.42aAm = 0.5761 – a » 0.58D. ]. Doli}anin-Djeki}, et al.: Ap  pli  ca  tion of an Ad  di  tive-Type Mixed ...194 Nu  clear Tech  nol  ogy & Ra  di  a  tion Pro  tec  tion: Year 2013, Vol. 28, No. 2, pp. 191-194Di  ana ]. DOLI]ANIN-\EKI], Xenis F. PU]I], Koviqka \. STANKOVI]PRIMENA  ME[OVITE  STATISTI^KE  RASPODELE  ADITIVNOG  TIPA  NA ANALIZU  ZRA^EWA  SME[E  RADIOAKTIVNIH  IZOTOPAU radu se razmatra mogu}nost razdvajawa efekata zra~ewa iz dva ili vi{e radioaktivnih izotopa primenom metoda statisti~ke matematike. Postupak se svodi na primenu me{oviteraspodele aditivnog tipa. Matemati~ki postupak prikazan je na primeru analize slo`enogzra~ewa iz dva radioaktivna izvora.Kqu~ne re~i: zra~ewe sme{e izotopa, me{ovita statisti~ka raspodela aditivnog tipa,.........................parametri pojedina~nih univerzuma raspodele

Nuclear Technology and Radiation Protection

Application of an additive-type mixed probability distribution to the analysis of radiation from a mixture of radioactive sources

This paper investigates the possibility of distinguishing between the effects
   of radiation coming from two or more radioactive isotopes, by using
   methods of statistical mathematics. The procedure uses a mixed distribution
   of an additive type. Mathematical treatment is demonstrated herein on an
   example of analysis of composite radiation from two radioactive sources.</jats:p

Application of an additive-type mixed probability distribution to the analysis of radiation from a mixture of radioactive sources

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