SIMPLEX OPTIMIZATION BY ADVANCE PREDICTION IN INSTRUMENTAL NEUTRON ACTIVATION ANALYSIS.

Transcription

1 SIMPLEX OPTIMIZATION BY ADVANE PREDITION IN INSTRUMENTAL NEUTRON ATIVATION ANALYSIS.

2 SIMPLEX OPTIMIZATION BY ADVANE PREDITION IN INSTRUMENTAL NEUTRON ATIVATION ANALYSIS By Patrik Hayumbu, B.S. A Thesis Submitted to the Faulty of Graduate Studies in Partial Fulfilment of the Requirements for the degree Masters MMaster University Deember, 1983

3 MASTERS (1983) (hemistry) MMASTER UNIVERSITY Hamilton, Ontario TITLE: AUTHOR: Simplex Optimization by Advane Predition in Instrumental Neutron Ativation Analysis. Patrik Hayumbu SUPERVISOR: Dr. D.O. Burgess NUMBER OF PAGES: x, 173

4 ABSTRAT Though Neutron Ativation Analysis is one of the most sensitive multielement analysis methods, ompton interferene in omplex sample matries usually presents a problem when hoosing irradiation and deay times for analysis. While various sientists have attempted to solve the problem, the aprroahes taken to date have the drawbaks of either requiring standard spetra of the sample omponents or not giving the optimum times automatially and simultaneously. The purpose of this thesis was to find a method to automatially and simultaneously obtain the optimum times for Instrumental Neutron Ativation Analysis using the Modified Simplex Method to evaluate the best figure of merit alulated from an advane predited spetrum of the sample. iii

5 AKNOWLEDGEMENTS I would like to thank Dr. D.D. Burgess for his assistane and patiene during the ourse of preparation of this thesis. Finanial support from the anadian ommonwealth Sholarship and Fellowship ommittee is gratefully aknowledged. I would also like to express my sinere thanks to Mrs. Gayle Griffin for typing this thesis.

6 TABLE OF ONTENTS ABSTRAT AKNOWLEDGEMENTS LIST OF OF TABLES LIST OF FIGURES HAPTER l iii iv viii X l l.l Objetive of Study l l.2 Need of Finding the Optimum onditions 1.3 Theory of Instrumental Neutron Ativation Analysis 5 (I) Introdution 5 (II) Sample Ativation 6 (III) Detetion of Gamma Rays (IV) Quantitative Determination in INAA 13 (V) alulation of Errors in INAA and Detetion Limit General Approah to Optimizing Irradiation and Deay Time Review of Previous Work Involving Optimizing Irradiation 18 and Deay Time in INAA (i) Introdution 18 (ii) Examples Simplex Optimization of Irradiation and Deay Time 21 HAPTER 2 Advane Predition and Simplex Algorithms Advane Predition 22 v 7

7 Page 2.2 Simplex Algorithm 26 HAPTER 3 Detetor alibration l Experiments 3.2 Data Redution 39 (I) Energy alibration 39 (I I) orretion Required on INAA Spetra 40 (I II) Photopeak Effiieny 41 (IV) Detetor Effiieny 42 (V) Photofration 42 (VI) Full Width at Half-Maximum Results 44 HAPTER Simplex Optimization by Advane Predition Program Strategy of Evaluating the Program Results Disussion of Results onlusion 71 APPENDIX A Determination of Photofration and Total Detetor 72 Effiieny Using a Nulide Emitting Two Gamma Rays APPENDIX B Desription of Programs and Their Flowharts 76 SAOMP 77 SIMPLEX 86 GRID 114 vi

8 Page APPENDIX Listing of Programs 117 l SAOMP SIMPLEX GRID 156 vii

9 LIST OF TABLES Number Title Page 1 Summary of the Interation of Photons with ~1atter. 8 2 Table showing Photon Esape and Origin of Peaks in a Pulse Height Spetrum. 9 3 Table of Standard Isotopes used to Find Detetor harateristis Table of Energies used for Energy alibration in Ra Table of ounting System omponents Table of Photoeffiieny of Several Gamma Energies Table of Total Detetor Effiieny for Several Gamma Energies Table of Photofration of Several Gamma Energies Table of Full Width Half-Maximum for Several Gamma Energies Table Showing the Derived Funtion of eah Detetor harateristi Table of Sample omposition used to Examine the Simplex Program Table of Elements Optimized for using the Simplex Program Table of Boundary onditions used for Optimization Table of Optimum Times for Several Elements a Table of the Initial and Final Simplexes of 76 As b Table of Initial and Final Simplexes of 52 v Table of the Initial and Final Simplex of 27 Mg Table of PF and PFe. 67 viii

10 LIST OF FIGURES Number 2b 3 4a 4b Title Detetion Limits of 79 elements irradiated for at a thermal neutron flux of 1.0 x lol3 m2-s-1 by gamma ray spetrosopy using a 40m3 Ge(Li) an hour followed detetor. Variation of Photoeletri, ompton and Pair Prodution reation ross setion as a funtion of energy. Flowhart giving a general outline of the subroutines neessary for Advane Predition of a given Sample's Spetrum. Possible moves in the Modified Simplex Method. Flowhart of the Modified Simplex Algorithm. A diagram of the Gamma Spetrosopy System. A diagram showing Soure-Detetor onfiguration. Graph of Photopeak effiieny as funtion of energy. Graph of Total Detetor Effiieny as a funtion of energy. Graph of Photofration as a funtion of energy. Graph of the square of Full-Width at Half-Maximum as a funtion of Energy. Grid plot of the response funtion of 76 As. Grid Plot of the Response funtion for 52 v. Grid Plot of the Response funtion of 27 Mg Flowhart of Program SAOMP. Flowhart of Subroutine SAMDAT. Flowhart of Subroutine PARAM Flowhart of Subroutine ENERGY Flowhart of Program SIMPLEX ix

11 Number Title Page 19 Flowhart of Subroutine REDAT Flowhart of Subroutine SELET Flowhart of Subroutine LIMIT Flowhart of Subroutine STIME Flowhart of Subroutine RESPON Flowhart of Subroutine REFLEK Flowhart of Subroutine EXPAND Flowhart of Subroutine ONTR Flowhart of Subroutine PEAK Flowhart of Subroutine SPE Flowhart of Subroutine SORT Flowhart of Funtion STDEV Flowhart of Program GRID a Part of a Spetrum of 88 v showing ompton 11 bakground. X

12 HAPTER OBJETIVE OF STUDY: In quantitative analytial praties the general objetive is the determination of some speifi element or omponent. The method of analysis is hosen on the basis of the quality of the results desired. Some of the measures of the quality of the results are preision, auray, seletivity, sensitivity, detetion limit and throughput of the method as well as the ost at whih the results are obtained (44). The objetive of this study was to find a method of determining the optimum irradiation and deay times simultaneously so that instrumental thermal neutron ativation analysis (INAA) an be arried out at onditions whih give minimum matrix interferene and hene optimum detetion limit for any element of interest. These optimum onditions were to be available automatially one the required inputs of the optimization program have been speified. 1.2 NEED OF FINDING THE OPTIMUM ONDITIONS: INAA is one of the most sensitive and nondestrutive methods of single or multielement determination (see Figure 1). When Ge or Ge(Li) detetors with high resolution are used its seletivity is also very high. In analyses where ativation of elements other than those to be determined ours, the detetion limit, auray and preision depend - 1

13 Figure 1. Shows the detetion limits of 79 elements irradiated for hour at a thermal neutron flux of 10 em s followed by gamma-ray spetrosopy using a 40 m 3 Ge(Li) detetor (24).

14 - 2 f- Polarography u ( Spetrosopy Yb Ho Sm Nd Hg Pt w Th Tm Tb Gd Pr e :< olorimetry &Atomi Absorption Gravimetri & Volumetri Methods Hf La s I Sb Rh Sr Br Ba Xe Te d Ag Pd Ru Mo Os Ta Sn Lu As Kr Er Ga Ge Dy Au u Zn Eu Ir Ar o Nb Zr Ni S Rb Se In Re Al Ti Mn v Na l K r Fe Mg a Si s l.ox E-10 E-8 E-6 E-5 E-4 E-3 Detetion limits for INAA{g/g)

15 - 3 on interfering gamma rays to a great extent (1-5, 12-16, 41). Three types of possible interferenes in instrumental ativation analysis are: (i) interferenes resulting from formation of the same radionulide as that formed by an (n,y) reation of the element of interest. (Examples of this are fast neutron (n,p) and (n,a) reations on an element one or two units higher in atomi number than the element of interest); (ii) interferenes resulting from the formation of one or more radionulides that emit gamma rays whih overlap with the photopeak of interest and (iii) interferenes resulting from ompton levels in the pulse height spetra aused by interations of gamma rays of higher energy than the peak of interest upon whih the peak of interest is superimposed. (See Figure 2a.) Analyses arried out at optimum detetion limit using INAA require that the effets of these interferenes are redued to a minimum. The first type of interferene is brought under ontrol by hoosing irradiation positions where almost all fast neutrons are thermalized. The seond problem is not often enountered when Ge or Ge(Li) detetors are used beause of their high resolution, but when it does our, the peaks an be resolved by mathematial methods implemented in the form of a subroutine of a spetral analysis program (30). The third problem is rather diffiult to solve beause it requires the proper hoie of irradiation and deay times. This is the problem whih alls for optimization of irradiation and deay time in INAA (l-18). An alternative approah to redue ompton interferene may be the use of hemial separation but in ases where a high throughput is required, separations onsume time and may be too labourious.

16 - 4 There is also a general optimization usually arried out in neutron ativation analysis to ensure that signifiant ounts giving reasonable statistis for the peak of interest are obtained at reasonable ount rates with minimum deadtime losses, pulse pile-up and spetral distortion. For these requirements, one optimizes the ounting time, geometrial effiieny, the ounting equipment and the data redution method (22). An example stressing the need for optimum deay and irradiation time is the analysis of atmospheri pollutants {27). In order to trae the pollution soures of aerosols and to determine how muh the anthropogeni ontribution is, it is neessary to determine the amount and elemental omposition of atmospheri partiulates and preipitation. For studies of soures, harateristis, atmoshperi transport proesses and removal rates sensitive and multielemental analyses are required. Moreover, the method of analysis should be fast and easy beause of the large masses of information required. Aurate analysis of airborne dust, dry deposition and rain water is diffiult beause of the omplex nature of the samples and the low onentrations involved. Analytial tehniques proposed for this purpose suh as olorimetry, emission spetrometry, atomi absorption, flame photometry and polarography may or may not involve a dissolution step of the sample and subsequent hemial and instrumental steps. Sine atmospheri pa~tiulate material is often to a great extent arbonaeous, it is largely insoluble in water. Ashing, in order to obtain omplete dissolution, without loss of volatile elements is very diffiult. Also the sensitivity and seletivity of most of these tehniques is inadequate to ope with the omplexity of environ

17 - 5 mental samples. Suh analyses have been readily arried out by INAA using Ge (Li) or Ge gamma spetrometry and omputer assisted data redution (27, 28). The irradiation and deay time shedules whih were used sometimes required several reirradiations and experiene of handling the matries involved (29). A method of easily finding these times for eah element at optimum detetion limit without having to do trial experiments is desirable THEORY OF INSTRUMENTAL NEUTRON ATIVATION ANALYSIS: (I) Introdution: The first part of INAA results in the prodution of radioative nulides formed when stable nulides apture thermal neutrons. The ompletion of the analysis involves the detetion and measurement of gamma rays emitted by desired nulides among all those produed in the irradiated matrix. The radionulides are haraterised by both their deay shemes and their deay kinetis. The deay sheme summarizes the modes of disintegration and the energies of the radiations released. The kinetis desribe the rates of disintegration of the radioative nulei. The deay sheme is important in radioativity measurements sine it relates the number of radiations of a given kind and energy to the atual number of disintegrations of the partiular nulide. Energy values in gamma spetrometry provide qualitative information while the emission intensity of the gamma rays together with the deay kinetis provide both qualitative and quantitative information.

18 - 6 (II) Sample Ativation: If one onsiders a partiular stable nulide of an element that is apable of apturing a neutron, the number of radionulides produed in a thin sample is given by the equation: N = No r <P S/1. 1. where No - is the initial number of target nulei, r- is the reation ross setion (square em), <P-is the neutron flux (per square em per seond), S - is the saturation fator and, >. - is the deay onstant (per seond) Sometimes the deay onstant is expressed as: A. = 0.693/T 2. where T- is the half-life of the radionulide (in seonds). The number of target nulides for any partiular isotope is given by the equation: No = Na m J/Ma 3. where Na - is the Avogadro number, m- is the mass of element in irradiated sample {in grams), J - is the isotopi abundane of the nulide and Ma - is the atomi weight of the element (in grams). The saturation fator is given by the equation: 4.

19 - 7 where Ti - is the irradiation time (in seonds). The ativity or rate of disintegration of the radionulides just at the end of irradiation is given by the equation: Ao = N \ 5. (III) Detetion of gamma rays: Typially, the ativated sample is ounted at some Td, after the end of an irradiation period lasting forti. after a deay time Td is: deay time, The ativity A = Ao D 6. where D - is the deay fator and is given by the equation: -Aid D = e 7. One seldom, in pratie determines the atual disintegration rate of an indued ativity, bu. rather one measures its ounting rate under a given set of ounting onditions. The ounting rate depends upon the type of detetor used, the size of the detetor, the deay sheme of the radionulide and the sample-to-detetor geometry (solid angle subtended by the detetor). If the ounting period is T and the detetor properties are haraterised by the total detetor effiieny (d(e)), the observed orreted total ounts for a given gamma ray deteted by GeorGe (Li) detetor is obtained from the equation: TTS = I A d(e) (l -e-\t) /\ 8. where I - is the emission intensity of the gamma ray of interest with energy E.

20 - 8 If the ounting time is muh less than the halflife, then equation 8 an be reast in a different form: TTSl = T A d(e) I 8a. The ounts arising from gamma rays as observed in the Multihannel Analyser (MA) onstitute a gamma spetrum. The spetral distribution of these ounts is understood from onsidering the interation of eletromagneti radiation with matter (see Table 1 and 2). TABLE 1. Summary of the Interation of Photons with Matter. (Ref. 22, Page 101) Partile ~1atter Interation Elasti Inelasti Absorption alli si on ollision Gamma NULEUS Negligible Nulear Photodisintegration Photon Resonane (e.g. Moss bauer Effet) ORBITAL Negligible ompton Photoeletri ELETRONS Sattering Effet Pair Prodution FIELD Negligible Negligible for gamma energies > 1.02 Mev.

21 - 9 TABLE 2. Photon Esape and Origin of Peaks in Pulse-Height Spetra Photon Spetrum Spetrum Esape Peak Peak Energy Origin Energy Name Mev 1.02Mev E 11 =E/(1+2a) 11 E to E 0 to E Total absorption E Photopeak Pair prodution and esape of one annihilation photon Pair prodution and esape of both annihilation photons ompton 180 sattering Single ompton sattering Multiple ompton sattering E-0.51 First pair esape peak E-1.02 Seond pair esape peak E =E/(l+a/2) e E to 0 ompton edge Single ompton distribution Multiple ompton distribution. where a m = E/m 2 =.511 MeV., i.e., eletron mass energy and = 180 degrees A gamma ray entering a Ge or Ge(Li) detetor an lose its energy through photoeletri interation, or ompton sattering, or pair prodution (36-37). In photoeletri interation, the inident photon loses all its energy to an eletron of an atom whih thus beomes ionized. The ejeted eletron an aus.e seondary ionization. The net harge indued in the detetor is deteted as a pulse whose magnitude is a funtion

22 - 10 of energy o~ hannel number in the MA. The rate at whih the pulses are produed depends on the disintegration rate of the sample. ompton sattering, only part of the energy is imparted to the ejeted eletron. The sattered photon either undergoes further interation or esapes from the detetor. In pair prodution, photons with more than 1.02 Mev produe eletron-positron pairs. This results in esape and.511 Mev peaks. The probability of photoeletri, ompton sattering and pair prodution are different and are funtions of energy of the photon. Photoeletri interation is predominant up to energies of about 1 Mev whereas ompton sattering is predominant up to energies of about 4 Mev. Pair prodution beomes the dominant form of interation after 4 Mev (see Figure 2). The detetor harateristis (suh as detetor effiieny) I depend on energy beause all the interations leading to registering the ourrene of an event are energy dependent. spetrometry in INAA uses energies up to about 3 r~ev In Usually, gamma ray so that the spetrum for a given gamma ray mainly onsists of ompton events and total absorption events. This point of view (15) leads to the onept that for any samp1e, the spetrum is mainly made up of the sum of ompton events and photopeak events for all the gamma rays emitted by the samp1e. The photopeak events approximately desribe a gaussian peak and the ompton events approximately desribe a retangular funtion starting at the ompton edge. photon an have. The ompton edge is the maximum energy a sattered If MP is the total ompton ounts deteted and PK is the photopeak ounts deteted for a gamma ray of energy E whose emission rate is A, for a sample ounted for T seonds, the total detetor effii

23 - 11 ~ J 1 2 2a... "' 0 ::... 0 u """'+---~---.--~ ~--.---~---.--~--~ ~00 1SOO hannel Number rjj z a: < al ~ z 0 i= (.) UJ <1.1 (/) (/) 0 r: (.) z 0 i= L r: 0 (/) al < 10' 1o I 1o ENERGY IN Kn Figure 2... Variation of Photoeletri, ompton and Pair Prodution reation ross setion as a funtion of Energy. Figure 2a. Part of a Spetrum of v 88 showing ompton interferene. = 1460 Kev peak of 40 ~ 2 = 1836 Kev peak of 88 v Peak 1 sits on the ompton bakground of Peak 2.

24 - 12 eny is defined by: d(e) = (PK+MP)/A T 9. The photoeffiieny (or photopeak effiieny), that is, the probability of deteting photopeak events during the ounting interval is defined as: \ Pf = PK/(A T) l 0. If we substitute Pf for d(e) in equation 8, we obtain the photopeak ount for a given gamma ray: PK = T No r ~ S D I Pf 11. The photofration, that is, the fration of events deteted in the photopeak is: f = PK/(~1P + PK) 12. An alternative expression for the photopeak ounts using equation 8 and 12 is: PK = TTS f or PK = T No r ~ S D I d(e) f 12a. and the ompton fration is f = 1-f 13. Using equation 8 and 13, the number of ompton ounts for a given gamma ray then beomes:

25 - 13 MP = TTS f 14. or MP = T No a ~ f.s D I 14a. From equations 9, 10 and 12, an alternative expression for photopeak effiieny is: Pf = d(e) f 15. (IV) Quantitative determination in INAA Three ways of determining the amount of an element in INAA use: (i) the absolute method or, (ii) omparator method or (iii) single omparator method. The absolute method (28) is based on the determination of the weight of the element by means of the ativation relationship 11. Rearranging equation 11 gives: No= PK/(T a ~ S D I Pf) 16. whih is the number of radionulides. Knowing the isotopi abundane and the atomi weight, this an be onverted to the amount of element deteted. However this method is rather rude beause it assumes a steady flux of neutrons. The omparator method is muh more aurate. In the omparator method (28), the sample and a standard of the element of interest are irradiated under similar onditions. This avoids the need to diretly use nulear onstants in finding the amount of the element. If Au is the ativity of the unknown sample and As is the ativity of the standard weighing Ws grams, then the amount of the unknown

26 - 14 sample is: W = (A W )/A 17. u u s s When multielement analysis involving shortlived nulides or high throughput is required this method is inonvenient beause it requires a standard for every element in the sample. is to use a single omparator method (28). A more onvenient approah In the Single omparator method, the standard onsists of a single element whih if possible should have both longlived and shortlived ativities (e.g. if Zin is used we have the isotopes 69 mzn (halflife 13.7 hr, y-ray at Kev) and 65 zn halflife (243.7 d, y-ray at Kev) ) An unknown onentration xy in a sample is alulated from the following equation: in whih -A t ) ~ e ( x y S A s = x s xy s 1s 18. xy -A t I (e s s) S A s. 1 X X X X y where x,y,s = subsripts for element, sample and standard, respetively

27 - 15 A = photopeak ount rate 1 = weight M= Atomi weight A = deay onstant S = Saturation fator t = deay time P = abundane of stable isotope a = effetive ross-setion for the irradiation position used y = abundane of seleted gamma ray. The $x values depend only on the neutron spetrum in the irradiation position used. These an be determined experimentally for the isotope to be used. For isotopes with more than one prominent y-ray, an average $x value has to be obtained. (V) alulation of errors in INAA and detetion limit The error in the peak is usually obtained by assuming Poisson distribution of the ounting rate (31). This assumption requires that the number of deaying nulei of interest in the sample is very large and the probability of deaying in a given period is very small. The ounting interval is hosen suh that the total number of ounts obtained (TTS) is large. Under these onditions, if a radionulide of interest emitting a harateristi gamma ray whih is superimposed on ompton and room bakground, the standard deviation (31) of the peak ounts for one run is: SDEV = IMP + PK + BK 21.

28 - 16 where BK - is the room bakground, MP - is the ompton bakground under the peak and PK - is the peak area. The% relative error (41) is given by: S% = SDEV.lOO/PK 22. The dependene of the detetion limit (DL) on the height of the ompton ontinuum an be learly seen from the formula introdued by urrie (37): DL = ( (BA) )/PA 23. where BA- is the area on whih the peak is situated (MP + BK), PA- is the peak area (PK). The base area is basially a funtion of the elemental omposition of the matrix and its most important omponent is the ompton ontinuum. 1.4 GENERAL APPROAH TO OPTIMIZING IRRADIATION AND DEAY TIME The general approah of optimizing irradiation and deay time is to define an analytial funtion whih varies with both irradiation and deay time in suh a way that the funtion is optimum when this type of interferene is lowest for some set of irradiation and deay time. A nonzero optimum deay time only exists if the half-life of the interfering nulide is less than half the halflife of the nulide of interest

29 - 17 (7). When this requirement is not met, the optimum deay time is zero sine waiting allows interfering ativity to inrease relative to the ativity of interest. The proedure is to optimize the funtion and obtain the optimum irradiation and deay times as a set of times at whih the funtion is optimum. From a mathematial point of view, the problem beomes one of searhing for extrema in a multidimensional spae (45). This an be arried out by numerial or differential methods. It is preferable that the method hosen is easily programmed on a small omputer. Generally, differential methods find the stationary points of a ontinuous funtion at whih the optimum is expeted to be by evaluating its partial derivatives. For an unimodal funtion, the required stationary point is a maximum or a minimum. When the onvexity of the funtion is known, the first partial derivatives adequately identify the extremum. The first derivatives are set equal to zero and solutions of the resulting equations when solved for the pertinent variables give the oordinates of the extremum. Methods for multimodal funtions evaluate derivatives higher than the seond so that stationary points suh as points of inflextion and saddle points whih are not extrema are distinguished. When it is not onvenient to use differential methods the alternative is to use numerial methods. (In INAA all optimizations whih have so far proved pratial are numerial (5)). One starts with an initial approximation of the funtion and proeeds by generating a sequene of its approximations so that eah of them is suessively better than the previous one and nearer to the solution. This group of methods an be subdivided

30 - 18 into arithmeti methods and logial methods. Arithmeti methods use an initial set of measurements to alulate the oordinates of the seond set of measurements. This is ontinued until an optimum set is approahed suffiiently well. The optimization usually proeeds by optimizing one variable at a time. An example of an arithmeti method is the Newton Raphson method (2, 45). Logial methods selet the positions of the seond set of measurements by logial deisions. The seond set of oordinates is aepted on the basis of the results of a logial omparison with the previous set of measurements. An example of a logial method is the simplex method (53). 1.5 REVIEW OF PREVIOUS WORK INVOLVING OPTIMIZING IRRADIATION AND DEAY TIME IN INAA (i) Introdution: The analytial funtions used to find the onditions at whih ompton interferene is minimum should naturally be funtions of ompton bakground. Some examples of suh funtions are detetion limit (10, 12-16, 18), relative error (equation 22), and standard deviation (equation 21). In the optimization algorithm, the value aounting for ompton ounts under the peak of interest an sometimes give problems if it is obtained from library spetra of interfering nulides (8). Only sample matries whose spetra are available an be analysed and samples must be measured under onditions whih are idential to those under whih the standard library spetra were measured. It is therefore preferable to use a 1 gori thms \'I hi h do not need 1 i brary spetra. A few examp1 es of

31 - 19 approahes used in previous attempts follow. (ii) Examples: 1. Okada (1) theoretially onsidered optimizing the irradiation and deay time to improve the error and detetion limit of the results of neutron ativation analysis. His approah did not inlude information on ompton interferene evaluation. He used an objetive funtion proportional to the peak ounts of interest, namely: Y= S D 24. where S - is the saturation fator and D - is the deay fator. The optimal onditions were obtained by using differential methods. This approah is rather limited beause it ignores inluding ompton bakground a feature whih prompted the optimization. Furthermore, searhing for optimum onditions using differential methods is rather diffiult when there are several interferenes as was pointed out by Isenhour and Morrison (2). 2. Isenhour and Morrison used an objetive funtion defined as a ratio of the peak to the sum of the ompton bakground in the region of the peak and the peak ounts, that is, R = PK/(PK + r MP) 25.

32 - 20 The fration of the ompton events in the region of interest was obtained from standard spetra of eah interfering nulide. The Newton-Raphson method was used to find the optimum ratio with respet to one variable at a time. The starting point for the optimization was obtained by alulating a matrix of ratio values over the entire range of irradiation and deay time. The highest ratio was taken as the starting point. 3. Quittner and Montvai (3) used relative error as the objetive funtion. They used differential methods to find the optimum onditions. They show results of sample alulations using the proedure for one interfering nulide. It is likely that the method would have similar problems as those enountered by Isenhour and Morrison if tried for a more omplex matrix. 4. Watterson (5) used the variane as the objetive funtion. The optimization gave graphial solutions obtained from a graph of % relative standard deviation as a funtion of irradiation time only and another graph of% relative standard deviation as funtion of deay time only. Library spetra were used to find the fration of ompton events in the region of the peak of interest. 5. In the method adopted by Guinn, a spetrum of the sample to be analysed is simulated for any given set of irradiation, deay and ounting time on the basis of the fat that it is mainly omposed of ompton events and photopeak events. Other peaks suh as the esape or 511 peaks whih

33 - 21 our in the spetrum are also inluded. Then the %relative error and the detetion limit for the peak of interest are alulated at these onditions. Several sets of onditions are tried and the one whih gives the optimum %relative error to a level aeptable to the analyst is aepted as the optimum set of onditions. This method has been tested and used to determine the detetion limits of several elements in INAA. 1.6 SIMPLEX OPTIMIZATION OF IRRADIATION AND DEAY TIMES Only the algorithm adopted by Guinn meets both requirements of not needing library spetra and using appropriate funtions. The others use rather restrited funtions in terms of the number of interfering nulides whih an be onsidered (Ref. 1,3) or they need library spetra (Ref. 2,5). However the Guinn algorithm (alled Advane Predition) does not give optimum results automatially. The searh for the optimum irradiation, deay and ounting times requires seleting trial sets whose results the analyst has to examine. If the results are not satisfatory more trial sets have to be entered. In this work, the advane predition algorithm is used to generate the spetrum expeted for the sample to be analysed, a figure of merit is defined,(that is, an analytial funtion of the property whih must be optimized) and the modified simplex searh method is used to simultaneously and automatially find the optimum irradiation and deay time.

34 HAPTER 2 ADVANE PREDITION AND SIMPLEX ALGORITHMS: 2.1 ADVANE PREDITION: The advane predition algorithm uses a simple model of gamma ray pulse height spetra (41). This allows the alulation of the umulative ompton ontinuum levels upon whih the various photopeaks are superimposed. The method assumes that either the major onstituents (major from the standpoint of levels of gamma ray emitting radionulides produed) of the matrix are known, or that they have been established by means of a simple and rapid exploratory ativation and gamma ray spetrometry measurement. In the original form of the Advane Predition algorithm one starts by entering the sample omposition followed by the entry of physial onstants (e.g. atomi weight, Avogadro s number and isotopi abundanes - to define the number of target nulides) and the nulear data onstants (e.g. halflife, reation ross setion, emitted energies and their intensities). Then the detetor harateristis are speified (in the form of equations of detetor property as a funtion of energy). Finally the onditions (neutron flux, soure-detetor geometry, and deay, irradiation and ounting times) of analysis are entered. Equation lla is used to alulate the photopeak ounts and equation 14a is used to alulate the total ompton ounts of eah gamma ray. The ompton ounts divided by the ompton edge gives the ompton distribution (ompton ounts per unit of energy or ompton ounts per - 22

35 - 23 hannel) of a gamma ray. The umulative ompton ounts per hannel is taken to be the sum of eah gamma ray's ompton ontribution in that hannel. In the original advane predition, the main objetive was to find the detetion limit for any given set of onditions though one ould obtain optimum times as well as optimum sample size. Hene, one the spetrum had been onstruted, the detetion limit or any other figure of merit (also alled objetive funtion or analytial funtion) to be evaluated is alulated. When optimization is the objetive, several sets of times are entered and the one yielding the best results is hosen. In this study, to searh for optimum times, three sets of irradiation and deay time are entered after all the neessary parameters outlined above have been entered. A figure of merit is evaluated for eah set. These sets of times ~ogether with their figures of merit define the initial simplex in three dimensions. The simplex algorithm is then used to find the optimum set of the figure of merit, irradiation time and deay time. Figure 3 gives a general outline of the Advane Predition subroutines used in this study. The detetor properties required in the Advane Predition algorithm depend on the quality of the simulated spetrum required. For a omplete spetrum, one alibrates the detetor for; (i) total detetor effiieny, (ii) photofration, (iii) Full Width at Half Maximum (FWHM), (iv) esape and.511 Mev peak effiienies~ and (v) energy (espeially if a plot of the spetrum is desired). Total detetor effiieny and photofration are neessary for photopeak and ompton ount alulations as shown in equations 11 and 14

36 - 24 Figure 3. General outline of the subroutines neessary to predit a sample s spetrum in this study. SAMDAT, PARAM and ENERGY are subroutines to enter sample omposition, nulear and physial onstants (see Appendix B) 1 Store emitting nulide identifying ode, halflife of the gamma emitting nulidi state, the emitted energies and its saturation ativity. 2 Exit and enter the program or subroutine whih needs a predited spetrum (e.g. SIMPLEX OM or GRID OM). REDAT - Subroutine to read data stored at 1. SELET - Subroutine to selet element and gamma energy of interest PEAK and SPE - Subroutines to alulate the spetrum. 3 Return to program needing the spetrum.

37 - 25 Start 0 I v I SAMDAT REDAT \'7 SELET I I ENERGY Sample size, Tr,Td&T I y I, PEAK SPE

38 - 26 respetively. The FWHM desribes the number of hannels a peak spans. This number is neessary when alulating the bakground under the peak. The esape (single and double) and.511 Mev peak effiienies failitate onstrution of realisti spetra when high energy gamma rays and positron annihilation are present. Energy alibration allows relating gamma ray energies to the peak positions (i.e. hannel number) in the spetrum. 2.2 SIMPLEX ALGORITHM: The simplex searh method allows many fators to be varied at the same time to arrive at and trak the optimum using rather simple deisions and oordinate alulations. This makes the method attrative for automated optimization. The method was first proposed by Spendley et al. (_47). Its first appliation to analytial hemistry was by Ernst (49) who used it for- instrumental optimization. Sine then, there has been inreased use of the method with various modifiations aimed at improved onvergene rates and preision. The Modified Simplex Method (MSM), an improved version of the original simplex algorithm (alled the Basi Simplex Method) was used in this study. MSM was introdued by Nelder and Mead (48). In the original algorithm (BSM), the step size is fixed. When the step size is too small, it takes many moves to find the optimum whereas when it is too large, the optimum is determined with insuffiient preision. In MSM, translation of the simplex is ahieved by refletion, expansion, ontration and massive ontration. The step size is variable throughout the whole proedure. In this study, rules for the simplex moves are those to attain the minimum of the detetion limit of the element of interest. The rules of performing these simplex moves

39 - 27 an be understood by referring to Figures 4a and 4b. RULES OF MODIFIED SIMPLEX METHOD. A simplex is a geometrial figure defined by a number of points {n+l) equal to one more than the number of dimensions of the fator spae(n). For the simplest multifator problem, namely an optimization of two parameters, the simplex is a triangle. In the initial simplex BNW shown in Figure 4a,vertex B has the best response (that is, the lowest value of the objetive funtion), W has the worst response and N has the next-to-worst response. P is the entroid of the line segment BN. It is logial to onlude that the response will probably be lower in the diretion opposite to the point with the worst response (W). Therefore, the triangle is refleted through the point P to the point R, the refleted point. The oordinates of R are obtained aording to the equation: R = P + (P-W) 26. The response at R an be lower than that at B, higher than that at B but lower than at N or higher than at N. Depending on whih of the three possibilities is found to be true, the following steps are undertaken; (i) If the response at R fs lower than the response at B the simplex seems to move in a favourable diretion. An expansion is therefore attempted by generating vertex E (the expanded point) by the equation: E = P + y (P-W) 27.

40

41 Start } Figure 4b Flowhart of the Modified Simplex Algorithm (48) I I Sort verties to form --lanw Determine P & R I y ~?!Yes I! ' Determine E Determine r Determine w r 0 t ~New l ~- Simplex 1. r< R? Massive ~/ i BNR is BNE '------<J ~ ~~~-es--~ t New Simplex I Yes ontrat ~ Exit 'V'Yes I ----/

42 - 30 where y is an expansion oeffiient whose value is usually 2. If the response at E is also better than that at B, then E is retained and the new simplex beomes BNE. If not, the expansion has failed and the new simplex is BNR. One then proeeds to heking whether the optimum onditions have been attained. If the optimum has not been attained, one reflets the worst point and tries to rejet it aording to the prevailing simplex rules. (ii) If the response at R is lower than the response at N but is greater than the response at B, the new simplex is BNR. No expansion or ontration is envisaged for the present simplex. One uses this new simplex to try to rejet the worst point using rule (i) if the optimum has not been attained. (iii) If the response at R is greater than that at N, the simplex has moved too far and should be ontrated. The ontration an be around either the refleted point or the worst point. The deision of whih point to ontrat around is based on the results of the omparison of the refleted point to the worst point. l. If the response at R is lower than that at W, the new vertex r is best situated nearer to R. the ontrated point is given by the equation: r = P + S (P-W) 28.

43 - 31 where s is the ontration oeffiient whih is usually If the response at R is greater than that at W, the new vertex w is nearer to W and is given by the equation: w = P-S (P-W) 29. The new simplex is BNr or BNw and one proeeds in the usual way of trying to rejet the worst point if the optimum has not been reahed. (iv) The failed ontration results if the response at r is greater than that at R or the response at w is 0reater than the response at W. Nelder and Mead reommend a massive ontration in whih the size of the simplex is diminshed even further. This simplex move is sometimes referred to as shrinking the simplex. This is done by replaing N in the original simplex with P and retaining the urrent worst point (w or r). If for onveniene we denote the point P as M, then the new simplex is BMr or BMw. (v) If a vertex lies outside the boundaries of one or more of the fators, a very undesirable response is assigned to that vertex. The simplex will then be fored bak inside the boundaries. The simplex is halted when the standard deviation of the response at the three verties beomes less than some predetermined value. Assessing that one is onverging at the global optimum is done by arrying out several trials

44 - 32 using different initial simplexes. In this study, the simplex rules are used to alulate the values of irradiation and deay time for the different simplex translations. For eah set of these times, defining a prospetive simplex vertex, the advane predition algorithm is used to generate a spetrum. This spetrum is used to evaluate the figure of merit. The figure of merit is used in the logial omparisons whih lead to aepting or rejeting the point as a valid vertex of the new simplex.

45 HAPTER 3 DETETOR ALIBRATION 3. 1 EXPERIMENTS Several standard soures (see Table 3 and 4) were ounted using a 45 m 3 ommerial ylindrial intrinsi Ge detetor. The ounting system onsisted of a preamplifier, pulse pileup rejetor, ratemeter, a spetrosopy amplifier with a 4 miroseond time onstant and a multihannel analyser (see Table 5 and Figure 5). The soure detetor distane was defined by a rigid plasti shelfing struture with slots for a tray to hold the sample at various distanes from the detetor asing (see Figure 6). The soures were plaed at the entre of the tray so that they were diretly above the axis of the detetor asing. All the soures were ounted at a nominal distane of 3.7 em from the detetor asing. This distane was used beause it was the lowest shelf slot and had therefore the maximum ounting rate. Three spetra were olleted for eah soure. Three room bakground spetra were also olleted for eah ounting period after ounting the standard. These spetra were stored on a magneti tape. A data routing system assembled by Burgess (38) was used to transfer spetra from the hannel analyser to either the tape reorder for spetra storage or an Osborne 1 miroomputer. The miroomputer was used for spetral analysis. Spetral analysis was arried out using the program NAASYS (NEUTRON ATIVATION ANALYSIS SYSTEM- Ref. 39). - 33

46 - 34 TABLE 3. Standard Isotopes Used to Find Detetor harateristis ISOTOPE ISOTOPE ATIVITY WHEN BOUGHT (in Mirouries) o ±4.4% s ±3. 7% Na ±3. 7% Mn ±3.7% o ±1. 9% Ba ±4.8% Y :!5 0% GAMMA ENERGY OF INTEREST (in Mev) l

47 - 35 TABLE 4. Energies used for energy alibration in Ra-226 PEAK NUMBER ENERGY IN MEV

48 - 36 TABLE 5. OUNTING SYSTEM OMPONENTS INSTRUMENT MODEL SUPPLIER Passivated Hyperpure Germanium Detetor Spetrosopy Amplifier S-A31 (1.8) Apte/NRD 2010 ANBERRA Linear Ratemeter 1481 ANBERRA Livetime orretor/ Pileup Rejetor 1468 ANBERRA Multihannel Analyser 8100 ANBERRA Referene Pulse 1407 ANBERRA Generator Data assette 8410 TEHTRAN

49 Figure 5. Gamma Spetrometry System A : Tape, B : Miroomputer & : Teletype 0 SOURE Ge Detetor, 1 Pileup L_r~[?>-t> MA I Rejetor.JI I ~ '"---o----r--' Pulser w '-I High Voltage Soure Aatemeter J Data Routing Swithes r 9 l tl? A 8

50 - 3~ Figure 6. Soure-Detetor onfiguration

51 DATA REDUTION: (I) ENERGY ALIBRATION: When a sample s spetrum is obtained, a spetral analysis program in INAA essentially alulates two parameters from the spetrum; viz.; the energies and the areas of the peaks. Peaks in the pulse height hannel analyser are assigned to hannels rather than energies. It is therefore neessary to find the relationship between hannel numbers and peak energies before performing spetral analysis on the sample s spetrum. Establishing this relationship is alled energy alibration. (It should be mentioned that energy alibration is not limited to the ase of using a spetral analysis program. Rather, it is part of the proedure of obtaining qualitative information in INAA.) The relationship between energy and hannel number is defined by a polynomial funtion (27) of the form: where E.. is the peak energy, 2 E = ao + a 1 + a 2 + a is the hannel number at the peak entroid and a. - stands for alibration onstants. 1 The alibration onstants are the values whih are sought for during energy alibration. This is performed by using standard soures with known energies. The hannel numbers are obtained from the spetra of the standards. The two parameters (energy and hannel number) are then used in a least squares program to find the required onstants. The polynomial nature of the peak energy - hannel number

52 - 40 relationship is due to the non-linearity of the spetrosopy amplifier. If the amplifier response an be defined by a linear funtion, only two gamma energies are needed for alibration. Otherwise more energies are needed. The soures an be: (i) sever~l single energy emitting nulides in the form (ii) standards of single nulides, or several single or multiple energy nulides in the form of a mixture of nulides (e.g. Ra-226), or (iii) a single nulide emitting at least two gamma ray energies (e.g. o-60). From the experimental point of view it is more onvenient to use a soure for whih only one spetrum is needed to obtain several peaks. This favours type (ii) and (iii) soures. The NAASYS program has an energy alibration subroutine whih makes available the onstants to the spetral analysis subroutine.... standard of Ra-226 was used for energy alibration. A (II) ORRETIONS REQUIRED ON INAA SPETRA 1. Pulse pile-up: This happens when a pulse ours in the detetion system suffiiently soon after a preeding one to ombine their heights. This is prevalent at high ounting rates. Instead of two pulses, the system registers one pulse with an energy somewhere between the individual omponents and their sum. The effet is to remove ounts from the peaks and hange the shape of the spetrum. The pulse pile-up rejetor was used to rejet suh peaks automatially in order to redue spetral distortion.

53 System deadtime losses: System deadtime refers to the fat that a multihannel analyser requires a period of time to proess a signal due to a deteted event. During this time, other signals are lost. The fration of ounts lost in all the peaks of signifiant intensity are the same. To orret for system dead time, the referene pulse generator was used to generate pulses at a predetermined frequeny of 60Hz and these pulses (of a hosen amplitude and therefore hosen hannel) were sent to the preamplifier. A ertain fration of these pulses were lost through the dead time phenomenon so that the pulser peak ounts observed were less than those determined by the produt of the pulser frequeny and ounting interval. The dead-time orretion fator was obtained by using the equation: r = Pex/Pob 31. where Pex - is the expeted pulser ounts and Pob- is the observed pulser ounts. Pex = 60 T Room Bakground: The determination of TTS (equation 8) for a given gamma ray requires the integration of all hannel ounts up to the high energy peak boundary hannel (obtained from NAASYS). Sine there is always ontribution from room bakground for a spetrum, this was subtrated from the obtained integral. (III) PHOTOPEAK EFFIIENY (Pf(E)): After energy alibration, NAASYS was used to alulate the

54 - 42 peak areas of spetra of the standard soures. The peak areas were orreted for dead-time losses and used to alulate the photopeak effiienies for eah gamma energy aording to equation 10. The photopeak effiieny as shown in equation 16 is an important quantity for quantitative analysis. In this study it was used to hek the validity of the relationships of photofration versus energy and total detetor effiieny versus energy by using equation 10. It was neessary to hek these relationships beause of the inherent approximations used to obtain them (see below). (IV) DETETOR EFFIIENY (d(e)): The total detetor effiieny was obtained from standard soures with one or two gamma rays by using equation 9. For soures with two gamma rays, either the effiieny was assigned to the average energy (o-57 and 0-60) or the gross ount (TTS) was deomposed to obtain a value at one energy (Y-88 and Na-22) (see Appendix A). (V) PHOTOFRATION (f(e)): The photofration was obtained by using equation 12 if there was only one gamma ray emitted by the nulide (e.g. s-137). For nulides emitting two gamma rays lose together, the average total detetor effiieny and the average photofration were used (see Appendix A). In the ase where the two gamma rays were too far apart as in Y-88, a regression program {43) was used to find the photofration flanked by photofrations whih ould diretly be obtained by equation 12. For example the photo

55 - 43 frations of s-137 (.662 Mev), Mn-54 (.835 Mev) and o-60 (1.253 Mev) were used in the regression program to find the photofration of the.898 Mev gamma ray of Y-88. One this was found, sine its peak ounts ould be obtained from the spetrum, use of equation 12 gave the total ounts due to the Mev gamma ray. The photofration of the other gamma ray (1.836 Mev) was found by first determining the total ounts (as in Appendix A, ase 2) followed by appliation of equation 12 sine the peak ounts are known from the use of NAASYS. Pl78). and energy: (VI) FULL WIDTH AT HALF-~1AXIMUM (FWHM): The peak width in gamma spetrometry is expressed as FWHM (27, Theoretially the following relationship exists between resolution 2 2 (FWHM) = k f s E + (p) 32. where k - is a proportionality onstant; f - is the Fano fator - is the energy required to produe one eletron-hole pair (2.98 ev forge); E - is the energy of the gamma photon and p - is the ontribution to the resolution by eletroni fators (noise from detetor leakage urrent, preamplifier, et.). The FWHM will beome pratially onstant below a ertain energy, depending on the quality of the detetor and assoiated eletronis. The method of Zimmerman (64) was used to determine FWHM at various gamma energies. In this method, the peak is onsidered as a

56 - 44 gaussian peak. A linearised form of the peak fs obtained by plotting the natural logarithm of the ratio Y(x-1)/Y(x+l) against x; where x is the hannel number and Y(x) is the number of ounts due to the peak (i.e. bakground orreted ounts) in hannel number x. The slope of the linear part of the graph is given b.y the equation: slope = 2 /(r) The FWHM is given by the equation: FWHM = r 34. Linear regression was used to obtain the linear part eah linearised gaussian urve (peak) for several standard soures. 3.3 RESULTS Tables 6-9 show the photopeak effiieny, total detetor effiieny, photofration and FWHM respetively. Sine these parameters are funtions of energy, their proper haraterisation required finding their funtional relationships with energy. The relationships were sought for using a regression program (43). The funtion giving the best orrelation for eah harateristi was hosen. These funtions together with their experimental values used to generate them were plotted (see Figures 7-10). Table 10 shows the funtion of eah harateristi.