A unified approach to the liquid scintillation counting process—I: A comprehensive stochastic model

International Journal of Applied Radiation and ~

1976, Vol. 27, pp. 397.-414.. I ' ~ o n

Prms. Printed in Northern Ireltnd

A Unified Approach to the Liquid Scintillation Counting Process--I: A Comprehensive Stochastic Model PHILIP J. MALCOLM* Biometry Section, Waite Agricultural Research Institute, University of Adelaide, Glen Osmond, South Australia 5064 and PI-HLIP E. STANLEY t Department of Clinical Pharmacology, The Queen Elizabeth Hospital, Woodville, South Australia 5011

(Received 8 September 1975)

A mathematical model of the liquid scintillation counting (I~C) process is described in detail. It numerically unifies the essential energetic and optical aspects of this process, and the results it produces are consistent with those observed in laboratory practice. The model has the following uses: firstly, as an aid to understanding the dynamical relationships constituting the LSC process; secondly, to determine relationships otherwise very difficult to ascertain; and thirdly, as presented in the second part of this paper, in improving the optical design of the LSC instrument. INTRODUCTION WHINE aspects of the liquid scintillation counting (henceforth LSC) process have been described in varying degrees of detail, there has as yet been no comprehensive report in the literature in which existing knowledge has been consolidated into a numerically unified whole. Such a description, a stochastic computer model of the whole liquid scintillation process, is presented in this the first part of this communication; optimization of instrument design is discussed in the second part of this communication, where the clear superiority of

spherically over cylindrically symmetrical systems is demonstrated. The model includes events occurring within the sample vial and in the detector assembly. The results it generates are comparable to observations made in real life, which indicates that most of the important features of the liquid scintillation process have been included. By unifying the details of the liquid scintillation process, the model enables otherwise inaccessible dynamical relationships to be ascertained (e.g. proportion of primary photons escaping the vial); secondly, the improvement or even optimizing of instrument design; and *Present address: Ahmadu Bello University, thirdly, a background for improving the utilization of the instrument is provided. Zaria, Nigeria. The extensive results generated by this t To whom reprint requests should be sent. model have been presented elsewhere °-3) as Copyright reserved by the Authors. 397

398

P.J. Malcolm and P. E. Stanley

space herein precludes their inclusion; similarly it was not possible to include the rigorous detail of the model at these Symposia. As discussed previously~4) several other workers have proposed fairly complete models of the LS process (TEN HAAF (5'6), NEARY and BUDD (7), KACZMARCZYK(8), GIBSON and GALE(9'10)), but none yet appear to have developed a detailed model. NON-MATHEMATICAL TREATMENT OF THE MODEL Many components of the LS process behave in a random fashion; a nuclear disintegration for example occurs at a random position within the volume of the binary scintillator solution (PPO, toluene and quenching agents, henceforth solution) used in this work. Similarly, an emitted /3-particle has an energy in the range appropriate to the isotope, and moreover, while the energies of any two /3particles are mutually independent, the /3spectrum produced is constrained to a particular shape, the so-called Fermi distribution. Such behaviour can be modelled by programming the equation for the/3-spectrum so that it can be driven by a pseudo-random number generator. Independent "/3energies"* mimicking the real/3-spectrum can thus be obtained. Successive/3-disintegrations are then simulated individually by successive calls to the program. This is the stochastic, probabilistic, or Monte Carlo approach to modelling. This program is included together with others in an event simulation loop which forms the heart of the model, a different event being simulated each time the loop is executed. Figure 1 is a simplified flowchart of this loop, components of which will be referred to by the italic numerals in the bottom right hand corner of the boxes. Thus the loop is entered at box 1 and left at box 5. The broad design of the whole model will be presented first, then followed by the mathematical details of the components. A/3-event is characterized firstly by its position in 3-dimensions within the solution, and * Quotations will where necessary be used to distinguish a simulated from a real entity.

FIG. 1. secondly by the energy of the emitted /3particle (see box 1). A pseudo-random number generator was used to produce uniformly random co-ordinates of the "event" within the solution. The /3-spectrum of the particular radioisotope being simulated is treated as a probability distribution, and random "/3energies" are generated following that distribution. Only a small fraction (2-5%) of the

399

Model of LSC process

"g-energy" is available for primary photon production, and this portion is calculated (see below) using the experimentally determined seintiUation efficiency.(m Thus the energy available for primary photon production Ep is determined from the "g-energy," E and the scintillation efficiency at the "g-energy," S~ Ep = SsE.

The effect of impurity or chemical quenching is to lower still further the energy available for primary photon production. This is simulated by multiplying F_~ by a fraction q, where ( 1 - q) is the proportion of energy lost by impurity quenching. The energy conversion equation now becomes F_~ = qSEE.

Note that while it is known that SE depends upon E, (11) it is assumed that q is independent of E. Thus, the g-disintegration has been characterised by two attributes, its random position within the solution and the energy available for primary photon production. The remainder of Fig. 1 fails into two loops, the first one entering between boxes 1 and 2 signifying a new primary emission by PPO, and the second entering between boxes 2 and 3 indicating a secondary (or higher order) emission by PPO. Note that any secondary emission occurs at the point of absorption of the primary photon by PPO, and not at the position of the event. New "photons" are emitted (box 3), and are characterized by escape direction, which is random in 4ir geometry, and energy. The corrected PPO emission spectrum is treated as a probability distribution, and random "energies" following that distribution are produced. "Energies" of "photons" produced by primary emission by PPO are subtracted from the energy Ep (available for primary photon production) until exhaustion completes the simulation of the event as shown in boxes 4 and 5. The colour absorption spectrum of the solu-. tion being simulated was experimentally determined and included in the model. The absorbance of the solution at that photon's

"energy" is then calculated (box 6), and used to generate a random distance of travel of the "photon." Note that this distance may exceed that necessary to leave the vial. Since the geometry of the vial and solution now becomes complicated, the remainder of the flowchart has been simplified to facilitate description. The asymmetric nature and irregular reflective properties of the vial bottom and elevator platform, and also the vial top and cap, are very difficult to describe mathematically in an accurate fashion. It is assumed therefore that the solution is a circular cylinder enclosed by a circular-cylindrical pipe with a flat bottom, and with perfect absorbers closing the ends; there are small air spaces between the upper absorber and the top of the solution, and also between the lower absorber and the bottom of the vial. Comparable experimental data are then obtained using a vial with black photographic paper on the top of the elevator and in the cap of the vial (Fig. 2). This method is practical with both instruments used experimentally, a Packard 3375 (with a light pipe) and a Searle Analytic Isocap 300, as the detector in each case subtends only the solution and not the absorbers. It is therefore assumed, both experimentally and in the model, that any photons leaving the / Block looper in vial cap

4.0 cm

i Elevator

FIG. 2.

Black paper on elevatOr

400

P. J. Malcolm and P. E. Stanley

bottom of the vial are absorbed, as are photons entering the air space above the solution; and also that only those photons leaving the curved sides of t h e vial enter the light pipe. The decision in box 7 as to whether the "photon" leaves the vial is therefore readily made: it must travel a distance sufficient to reach the curved wall of the vial at an angle (in 3-dimensions) less than the critical angle (~43"); note that it may however be totally internally reflected at the top of the solution or at the bottom of the vial. If the "photon" does not leave the vial then its fate is determined in boxes 8-19. Should it reach the top of the solution (box 9), then it is either totally internally reflected (box 11), or it is lost in the absorber above the solution (box 10). Should it reach the b o t t o m of the solution, perhaps via a reflection at the top, then it is either totally internally reflected (box 13), or absorbed (box 14). The "photon" may avoid absorption (boxes 10 or 14) and is in this ease absorbed within the solution itself, either by PPO or by some coloured agent (note that we are therefore assuming that no photons are absorbed in the vial walls). The measured absorbances of both this coloured agent and PPO at the "energy" of this photon are then used to randomly determine which agent was the absorber (box 16). Thus colour quenching arises (box 17), or, if PPO was the absorber, higher order emission may occur according to the fluorescence quantum efficiency of PPO (box 20); if not then PPO has acted as a colour quencher (box 19). Any "photon" negotiating a path to the point of re-emission certainly deserves a second chance, and it is indeed propitious that the fluorescence quantum efficiency of PPO is 0.83. (12) In the case of higher order emissions (box 20), the position (3-dimensionally within the solution) of the new emission is calculated (taking into account the number of total internal reflections and the thickness of the vial wall). A new "photon" is then generated at this point (box 3). Note that the law of conservation of energy is not applicable in higher order emission, and thus the full PPO emission spectrum is utilized. °3) We turn now to the case where the

"photon" leaves the vial (box 21), and therefore (see earlier) enters the detector. A point source of light was used to investigate the directional response of the detector in both instruments used, and the data collected indicated wide variation in this response (Table 1, and Appendix). Detector response for a partieular "photon" is ascertained from its point of exit from the "vial," treated as a probability, and stochastically modelled in the usual fashion. It is assumed that all photons not lost in the detector reach the appropriate photocathode (box 23) without re-entering the vial, and the next step is therefore to simulate photocathode response. This is achieved by calculating the photocathode quantum efficiency from the photon's "energy," and then treating the efficiency as a probability in the usual fashion. Any "photoelectrons" generated are counted (box 24), and a new primary excitation of PPO is then simulated (box 2). 1HATHEMATICAL T R E A T M E N T OF THE MODEL The heart of the spectrometer is two matched photo-multipliers axially opposed against a light pipe or mirror system enclosing the vial. For example the light pipe of the Packard 3375 instrument (see Figs. 3 and 4) is a rectilinear parallel pipe of perspex containing a vertical cylindrical hole. An elevator moves down this hole bringing with it a cylindrical glass vial containing the solution to be counted. A cylindrical coordinate system was used, with the origin at the centre of the vial as shown in Fig. 3. Note that the origin of the vertical axis is at the mid-point up the light pipe. The position of a radioactive disintegration in the vial is given by the coordinates (p, a, z) where p is the horizontal radius, ~ is the horizontal angle, and Z is the height. The internal radius of the vial was used as a unit of distance throughout this work, with the top and bottom of the solution being Zm~ and Zmin (negative) respectively. It is assumed that Zmax--Zmin is not greater than 4.4cm, the height of the light pipe. Then a volume element d V centred at the point (p, a, z) is given

Model of I.,$C process

401

TAm_~ l(a). Directional response of the detector chamber of a model 3375 Packard Tricarb fitted with a light pipe (normalized data) Source eight

Left hand photomultiplier

m

0 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300 315 330 345 360

0.05

0.5

2.0

3.0

3.95

0.5537 0.5085 0.4220 0.3750 0.3044 0.2144 0.0784 0.0094 0.0026 0.0004 0.0009 0.0007 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0704 0.2066 0.3092 0.3796 0.4152 0.4799 0.5537

0.8494 0.8825 0.7583 0.6215 0.5193 0.3282 0.1213 0.0305 0.0007 0.0021 0.0035 0.0019 0.0081 0.0053 0.0055 0.0052 0.0073 0.0106 0.0372 0.2755 0.4900 0.5896 0.6644 0.7695 0.8494

0.8793 0.8396 0.7951 0.7540 0.6547 0.4708 0.1582 0.0275 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0003 0.0000 0.0000 0.0620 0.2831 0.5589 0.7631 0.8405 0.8755 0.8793

0.8942 0.9079 0.8335 0.8430 0.6866 0.5502 0.2415 0.0362 0.0012 0.0022 0.0113 0.0023 0.0098 0.0054 0.0071 0.0066 0.0117 0.0022 0.0545 0.3455 0.6354 0.8035 0.9737 1.0000 0.8942

1.0000 0.9864 0.8944 0.7554 0.6138 0.3011 0.1382 0.0144 0.0011 0.0020 0.0005 0.0007 0.0000 0.0034 0.0046 0.0049 0.0033 0.0074 0.0380 0.2366 0.6028 0.6575 0.7238 0.9674 1.0000

Left hand photomultiplier 0 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 255

0.0000 0.0023 0,0031 0.0003 0,0014 0.0019 0.0142 0.0815 0.2335 0.3166 0.3543 0.4325 0.5029 0.5203 0.4280 0.3696 0.3137 0.21.90

0.0060 0.0042 0.0089 0.0037 0.0056 0.0042 0.0338 0.2033 0.3733 0.5322 0.6064 0.7613 0.8154 0.8572 0.7662 0.6387 0.5320 0.3170

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0613 0.1633 0.4690 0.6643 0.7961 0,8001 0.7997 0.7985 0.8041 0.7928 0.6684 0.4695

0.0067 0.0000 0.0000 0.0000 0.0000 0.0000 0.0059 0.1535 0.4191 0.7213 0.8041 0.8390 0.8386 0.8600 0.9095 0.8844 0.7613 0.4520

0.0036 0.0012 0.0025 0.0058 0.0039 0.0019 0.0233 0.2045 0.4820 0.5998 0.6437 0.8497 0.9188 0.9659 0.8875 0.7088 0.6196 0.4049

P. Z Malcolm and P. E. Stanley

402 TABLE l(a). (Continued) Source eight

270 285 300 315 330 345 360

Right hand photomultiplier 0.05

0.5

2.0

3.0

3.95

0.0642 0.0112 0.0003 0.0004 0.0000 0.0017 0.0000

0.1290 0.0136 0.0012 0.0066 0.0072 0.0035 0.0060

0.2220 0.0073 0.0000 0.0000 0.0000 0.0000 0.0000

0.1436 0.0097 0.0000 0.0000 0.0000 0.0000 0.0067

0.1390 0.0027 0.0033 0.0028 0.0011 0.0031 0.0036

* Angles expressed in degrees relative to the line joining the centre of the vial with the centre of the face of the right photomultiplier.

TABLE l(b). Directional response of the detector chamber of a Searle Analytic Isocap 300 fitted with reflectors (normalized data) Source , ~ height radial ~ m angle ~ 0 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300 315 330 345 360

Rear photomultiplier 0.05

0.5

2.0

3.0

3.95

0.4460 0.3323 0.2639 0.1587 0.1373 0.1468 0.0614 0.0000 0.0000 0.0100 0.0139 0.0107 0.0000 0.0108 0.0000 0.0000 0.0000 0.0000 0.1381 0.2091 0.2468 0.2200 0.2238 0.3651 0.4460

0.9003 0.8025 0.6938 0.4602 0.3582 0.3333 0.3251 0.0252 0.0000 0.0030 0.0177 0.0128 0.0077 0.0164 0.0089 0.6710 0.4041 0.0045 0.1237 0.4906 0.4851 0.4489 0.7404 0.8563 0.9003

0.6018 0.5723 0.5610 0.5444 0.5108 0.4640 0.3050 0.0598 0.0000 0.0037 0.0069 0.0051 0.0000 0.0000 0.0000 0.0005 0.0034 0.0000 0.0497 0.5259 0.5860 0.6496 0.6669 0.6585 0.6018

0.6291 0.6430 0.6404 0.6102 0.5349 0.4779 0.2076 0.0387 0.0277 0.0463 0.0536 0.0330 0.0515 0.1034 0.1181 0.0745 0.0387 0.0312 0.2042 0.5860 0.6443 0.6668 0.7000 0.7136 0.6291

0.8010 0.7611 0.6551 0.4390 0.2731 0.3129 0.2879 0.0000 0.0000 0.0000 0.0033 0.0054 0.0000 0.0123 0.0077 0.0060 0.0053 0.0161 0.0788 0.4612 0.3799 0.3670 0.6174 0.7491 0.8010

Model of LSC process

403

TABLE l(b). (Continued) Source eight

0 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300 315 330 345 360

Frontphotomultiplier 0.05

0.5

2.0

3.0

3.95

0.0027 0.0033 0.0040 0.0139 0.0102 0.0058 0.1335 0.2632 0.1923 0.1928 0.1682 0.2172 0.1994 0.2074 0.1954 0.1540 0.2012 0.2077 0.0792 0.0045 0.0011 0.0142 0.0104 0.0009 0.0027

0.0022 0.0023 0.0059 0.0160 0.0189 0.0076 0.0674 0.4736 0.5602 0.4773 0.5587 0.7461 0.6940 0.8179 0.7583 0.4914 0.4181 0.5326 0.3188 0.0100 0.0017 0.0173 0.0162 0.0153 0.0022

0.0013 0.0038 0.0010 0.0030 0.0053 0.0035 0.0447 0.2977 0.5587 0.6655 0.7713 0.8358 0.8903 0.8966 0.8317 0.7421 0.6888 0.5305 0.3769 0.0055 0.0045 0.0030 0.0044 0.0023 0.0013

0.0175 0.0000 0.0190 0.0263 0.0173 0.0108 0.1905 0.6278 0.6809 0.7191 0.8397 0.9247 1.0000 0.9767 0.9246 0.7917 0.6978 0.6533 0.1791 0.0163 0.0132 0.0179 0.0186 0.0145 0.0175

0.0000 0.0005 0.0038 0.0130 0.0029 0.0032 0.1083 0.5271 0.5740 0.4672 0.6756 0.7344 0.9591 0.8746 0.6917 0.3511 0.3054 0.6175 0.4421 0.0001 0.0018 0.0040 0.0045 0.0032 0.0000

* Angles expressed in degrees relative to the line joining the centre of the vial with the centre of the face of the rear photomultiplier. by d V = p dp d a dz. Radioactive disintegrations a r e assumed to be distributed uniformly within the volume of the solution. R a n d o m event positions within the solution are therefore generated by

(p •

x/U[O,1), aeU(-~r, lr], z

•

U[Zmin,Zmax])

where U is the uniform distribution, " • " means "is drawn at r a n d o m f r o m , " [0, 1) denotes an interval 0-< x < 1, (-~r, ~'] denotes an interval - ~ r < x -- w etc. Excited fluor molecules are assumed to emit photons in r a n d o m directions from a point source. T h e coordinates (/3, tb) can therefore be used to define the direction of emission. /3 is the horizontal emission angle relative to a

(Fig. 5) and th is the angle of inclination of the emission to t h e horizontal plane (Fig. 6). A n element d A of the surface area of the unit sphere centred at the point of emission is given by d A = cos ~b d/3 d~b. R a n d o m escape directions for photons are therefore generated by (/3 • U(-Tr, ~r], 4~ • sin -1 ( U ( - 1 , 1])). The remainder of this description is formatted according to the box numbers given in the flow diagram (Fig. 1).

Box 1 Event position: see above. Event energy: the theoretical Fermi spectrum for an allowed beta transition, modified

404

P. J. Malcolm and P. E. Stanley

Perspex

I Vial cap

I Glass wall of

f = 0.007297 (fine structure constant) m = 511 (rest mass of electron in keV) z = atomic number of product nucleus (2 for 3H, 7 for t4C) E,, = maximum energy of beta emission (18.6 keV for 3H, 156 keV for ~4C). J. B. B I R K S (13) has noted the suitability of this distribution to describe observed 3H and t4C /3-spectra. A random drawing technique was therefore used to produce random energies following F ( E ) . It has been particularly difficult to obtain monoenergetie data relating primary scintillation efficiency (i.e. efficiency of production of

t

El evator I movement C't F I O . 3.

by the Coulomb attraction term, is used to give the energy distribution of electron emission. The (non-normalized) distribution function for emission of an electron of energy E in keV is therefore a

F ( E ) = 1 - e ------~( E + m ) ( E ~ - E) 2 ~/E 2 + 2mE

where l_e-=

is the Coulomb attraction term

and a =

OF VIAL

2 ' n ' f z ( E + m)

FIG. 5. Radius OB is of length 1; radius OC is

~/E2+2mE Reflective surface W o l l ~ ,J Glass/

P

C

Perspex light pipe Air space between vicII add

Fio. 4.

light pipe

of length 1 + d. primary photons) to g-energy. Horrocks' data(IS)have been normalized to the SE value of 0.052 (obtained by Hastings and Weber (14) using PPO and POPOP as scintillators) for the average energy ( - 5 0 k e V ) of the 14C /3spectrum. (~s) In respect of determining primary scintillation efficiencies it is clear that Hastings' ex-' perimental determination of SE must include secondary photons, and must therefore depend upon the geometrical light trapping characteristics of his special counting system

Model of LSC process

405

X TANGENTIAL PLANE TO OUTER WALL AT POINT I'

HORI ZON TAL PLANE

F,G. 6.

which is quite different in design to the normal LS instnnnent. This means firstly that their estimates are likely to be low, since the fluorescence quantum effieieneies of PPO (=,0.83) and POPOP (==0.85) are <1, and these efficieneies are critical to the production of secondary photons; secondly, Hastings' estimates do not necessarily apply to the normal LS instrument (as modelled) which has quite different light trapping characteristics. Note that geometrical light trapping can be avoided by using the spherically symmetric counting system proposed by the present authors (see (~ and part 2 of this paper). A spherically symmetric system could therefore be used to obtain machine-independent data; in addition an easily established formula (see part 2 of this paper) could then be used to correct for emission of secondary photons and thereby obtain data for primary scintillation efficiency. The second difficulty arising with data relating scintillation efficiency to g-energy is that of requiring effieiencies for a monoenergetic source of, say, 50 keV, rather than for /3-energies averaged to 50 keV over the broad range of 0-156 keV for 14C. Horrocks' data °x) were plotted on a linear-log scale and a roughly linear relationship was

found. We therefore assumed that the relationship SE=a+b lnE could be used, and obtained estimates for a and b in terms of Horrocks' data of $5 = 0.042 and $5o = 0.052 for the average g-energies of the aH and 14C g-spectra. Let Fn(e) and Fc(e) be the Fermi distributions for ~H and 14C respectively; then it can be seen that for average scintillation efliciencies Sr~ and Sc obtained respectively from 3H and 14C sourc~s~

Sn l eFn( e ) de = a l eFH( e ) de + bf e In eFn( e ) de or, more simply, SHAH = a AH + b BH and also S c A c = a A C + bBc whereupon a = ( S c A c B r r - SrIA~Bc)I(AcBM- ArrBc) and b = A n A c ( S c - Sn)/(AnBc - AcBM).

406

P. Z Malcolm and 1>. E. Stanley

Estimates of a = 0.033 and b = 0.0046 were thereby obtained and used to determine monoenergetic scintillation efficiencies in the model. Box 3

similarly, 3' is given by sin 3'

produce random wavelengths followexperimentally determined and corfluorescence emission spectrum for toluene.

Boxes 6, 7 The B e e r - B o u g e r absorption law states that the probability u of transmission over a distance x in a medium of absorption coefficient aa at wavelength )t is tl :

1/31= sin O l+d

whence

Escape direction: see above. Energy: a random drawing technique was

used to ing an rected PPO in

_ p sin l+d

= sin-~(sin O~ 7

\1 + d /

The photon must travel the distance BE

cos 0 - p cos/3

cos 4'

cos 4'

to leave the solution; we require -lnu a~

cos O - p cos /3

cos 4'

e -a~,x.

The absorption coefficient for the wavelength X (generated in box 3 above) is determined from an experimentally determined absorption spectrum, and if u ~ U(0, 1], then x = - I n u/ax generates a randomized distance travelled by the photon. The projection of the distance x onto the horizontal plane is given by y = x cos 4,. Whether the photon escapes from the vial into the light pipe depends firstly upon the distance x travelled before absorption. Escape depends secondly upon whether the photon approaches the curved outside of the vial at an angle less than the critical. The distance to be travelled for the photon to reach the inside wall of the vial is found as follows (see Fig. 5): the horizontal projection of the distance to escape the solution is E B now

A E = - p cos/3

and A B = x/1 - p2 sin 2/3 = c o s 0

where O = sin -1 (p sin

1/31)

The photon, to escape, must also approach the curved outer wall of the vial at an angle less than the critical. Referring to Fig. 6, let E be the point of emission, let E I be the photon's trajectory to the vial's outer surface at point I (note that the glass used in Packard low background vials and the solution are assumed to be optically continuous), and let B A I be the tangential plane to the outer wall at point I (actually B A / i s tangential to the curved side along the line A / o f contact). The B A E plane is perpendicular to the B A / plane. Then ~/, is the angle of incidence (to the normal) to the vial's outer surface at p o i n t / . Now

EB cos ~ = - ~ - = cos v cos 4'

whence sin t~ = x/1 - c o s 2 3, cos 2 4, and we require sin @
cos 0 - p cos/3 c o s 4,

407

Model of LSC process

1~

AIR

/

Glass w a i l of vial

AIR

"£~-'~

inl;..,~h

of photon

Boxes 8-15 Consider now the case where the photon does not leave the solution via the curved side of the vial. Let the photon be emitted at height z in the solution. The height to which the photon would travel in an infinite cylinder is z ' = z + sin 4, +Ah

Final path

of photon

Fxo. 7. Let C = 42*< 45* be the critical angle of the solution and also the vial glass. Then I < C < 45* for the photon to escape. Therefore 90"-I=J>45">42" and the photon must be totally internally reflected at the solution-air interface. and

where Ah is the vertical distance travelled in the glass walls. Ah will be specified in a later section. If z ' > Zm~, then the photon reaches the top of the solution whereupon it is totally internally reflected (if cos 4,--> 1/1.5018), or alternatively escapes to the air pocket above the solution (cos 4, < 1/1.5018) where it is lost in the perfect absorber (see above). If the photon is totally internally reflected then the height at which it is absorbed by the solution is

1

z" = 2Zmax- z'.

~/1-cos 2 T cos2 4 , < - .

r

The photon may encounter the top of the solution or the bottom of the vial in transit. Any photon reaching the top of the solution and potentially satisfying the condition sin ¢ < 1/r at the curved side, must be totally internally reflected at the solution-air interface at the top of the solution. This can be easily seen from Fig. 7 since the critical angle of the solution is about 1

sin-1 ( ~ ) - - ~ 4 2

.

Reaching the top of the solution is therefore immaterial to the photon's potential escape, and can only influence the height at which the photon leaves the vial. An analogous argument applies to photons reaching the bottom of the vial and potentially satisfying sin qJ< 1/r. Consider a photon emission at height z in the solution. Then it can be seen that the height of possible exit of the photon from the solution is z ' = z+(cos O - p cos 13) tan 4' where the calculation of z' must be performed in steps (see below).

Similarly, if z ' < Zminthen the photon is either totally internally reflected (if cos 0 > 1/r), or lost in the perfect absorber beneath the vial. A totally internally reflected photon will be absorbed at a height

Z" = 2Zmin -- Z' in the solution.

Box 16 A photon reaching box 16 is absorbed in the solution by either the fluor or another coloured material. A random number u U[0, 1) is generated and compared to absorbance of fluor at A absorbance of solution at A to determine the absorbing agent. Should the photon be absorbed by the fluor then another (new) photon may be emitted by the excited fluor molecule. This case is easily determined by generating a random number u ~ U[0, 1) and comparing it with the quantum efficiency of emission by the fluor (=0.83 for PPO). Box 20 Finding the point of emission of a new photon is more tedious than the preceding

408

P. Z Malcolm and P. E. Stanley

calculations. The height z' of the new emission is already known (but only approximately if the photon travels through the vial walls; see boxes 8-15 above), but its other two coordinates (p, a) are not, and must therefore be determined (recall that y is the horizontal projection of the distance x travelled by the photon through the solution). Consider first the case where the photon is absorbed before reaching the wall of the vial. It can be seen from Fig. 8 that O B = p sin1/31 and whence

B E = - p cos/3

¥

c

/

S A

/

p' = O A = x/p 2 sin2/3 + (y + p cos/3)2

= x/p2+ 2py cos/3+y2 and

ven by

et'-ot = [ A O E is

IIAOEI=c°s-'x[p

2 + p,2 --

and the sign of ] A O E is the sign of /3, thus giving the coordinates (p', a', z') of the new emission.

/

¥J

FIG. 9. The photon follows a path EB, DF, HA through the solution, and a path BCD, FGH through the vial walls. Consider now the case where the photon is totally internally reflected at the curved outside wall of the vial. The photon will, because of the vial's circular symmetry, be successively reflected around the vial until absorbed (it may also be totally internally reflected at the top of the solution or the bottom of the vial, but this will not alter its horizontal trajectory). Referring to Fig. 9, it can be seen that the photon travels a distance D F through the solution between successive reflections. While we assume in Fig. 9 that the photon is reflected at points C and G before finally being absorbed at point A, we will obtain results generalized to any number of reflec, tions. Let n be the number of reflections off the curved sides. Then n , 1 = integer value of

= l.~J

Distance

AE • y FIG. 8.

=

2 ~.o~ 0 "

i j"

(The foreshadowed correction to allow for the vertical height travelled by the photon inside the walls can now be made. It can be easily seen that the photon travels a vertical distance

409

Model of LSC process

through the glass walls of

a ' = a + (I/31 + 0 - 2 7 + ( n - 1)('a" -

2n((1+ d) cos 3 , - c o s 0) tan 4, =2n

+ cos-l(.1 - h cos 0), x the sign of/$. p'

tan 4,('~/cos 2 0 + 2 d + d 2 - c o s

0),

giving z ' = z + 2 n tan 4 , ( x / c o s 2 0 + 2 d + d 2 cos O)+x sin 4, for boxes 8-15 above.) The final distance H A travelled to the point

of absorption is given in the general case by H A = h = y - c o s O+ p cos/3 - 2 ( n -

= y+p cos/3-(2n-1)

27)

1) cos 0

cos 0

The coordinates (p', a', z') of the new emission have now been calculated for all cases. Box 21

From Figs. 5 and 6 it can be shown that the photon leaves the vial at the point a ' = ot + ( i g l - 7) x t h e sign of/3 z ' = z +(cos 7 - P cos/3) tan 4,

From Fig. 10 it can be easily seen that

on the outside wall (note that the calculation of z' is performed in steps, with the sign of p' = x/h z sin 2 0 + ( h cos 0 - 1 ) 2 tan 4, being negated whenever the photon is reflected at the top of the solution or the = x / h 2 - 2 h cos 0 + 1 bottom of the vial). Consider Fig. 11. The photon is emitted at The final coordinate a' of the new emission the point E in the solution, and takes a path is calculated as follows (refer to Fig. 9): E1 to its point of exit I where it is refracted along the new path / J outside the vial. We I/Eocl=l l-3' wish to determine the horizontal and vertical I / C O G [ = ( n - 1)(w- 23") in the general case components (3" and 4,' respectively) of the II OOHI = o- 3" angle of refraction q,' in order to determine ! - h cos the photon's path outside the vial. It can be seen intuitively (or from the principle of least action, or from the full statement of Snell's and the sign of [ E O A is the sign of gwhence law) that the points E, B, I, J, L will always lie in the same plane as the line E A rotates about the axis A / K with angle of rotation 3'. It is however not so clear that Snell's law for the vertical component angle 4, is preserved during rotation, that is that

[/HOAi=cos-1

P' O)

sin4, sin 4,'

1 = -

r

as well as

sinqJ

1

. = sin ~' r

as 3" varies (r is the refractive index of the

glass). We willtherefore prove that sin.4, = 1. That Sln 4,'

B I L is a straight line means that tan/BIA =

H~rm h cese 03

R h cos e

-!

Fio. 10.

AB IA

=

sin tan

3, 4,

KL = tan/KI.L = KI

sin 7' t a n 4,'

r

410

P. Z Malcolm and P. E. Stanley

.N

tangential Plane at Point Z on vial's o u t e r Surface

FIG. 11. Planes B A E , NIM, J K L are parallel and perpendicular to planes A B I K L and O A I N K P . The points E, B , / , J, L all lie in the same plane; so do the points K, J , / , M. The plane E B I J L is perpendicular to the plane A B I K L . ~ is the angle of incidence, and ~' is the angle of refraction at point I. ~/' = / N I M = / P K J =/KJL. ~b' = ~JIM = / K J L 0/' = / J I N =/LJL proved above, that

whence sin ~ / t a n 4)'

sin ~/' =

1 - cos 2 ~b' = 1 - cos 2 4~' cos E ~/'

t a n 4)

= 1 - cos E ~ ' (cosE ¢~' sin2 t~ -- sinE ¢~' COS2 t~ sinE ~ ) X ~b' COSE sin E ~b

and sin E ~ t a n z q~' tan E cos2 6 ' s inE 4) - s i n 2 th' cos 2 th s i n 2 ~{

cos2 7 ' -- 1

cos 2 6 ' sinE

b u t S n e l l ' s law s t a t e s t h a t sin ~

1

sin ~b'

r

then, 1 - c o s 2 ~b'= r 2 ( 1 - c o s E ~b)

using

c o s 2 6 ' sin 2 tk - sin E ~b' c o s E 6 sin 2 7 = sin E th - r 2 sin 2 th(1 - c o s E ~b) then, sin E 4 ) - s i n E ~b' sin 2 ~b - s i n E th' cos 2 ~b sin E ~/ = sin E 4 ) - r 2 sin 2 4)(1 - c o s 2 $ )

whence

and,

hence,

the

result

cos $ = cos tk cos

sin E th'(sin 2 4~ + cos E th sin E ~') = r 2 sin 2 t h ( 1 - c o s E ~b)

Model of LSC process

411

(note that the sign of sin 4, may have to be negated; see above) relative to the normal at the point (a', z').

but, sin2 4, + COS24, sin2 7 = 1 -- COS2 4,

"~-C082 4 , - COS24, COS27 ~-~1 - c o s 2 ~b hence, sin 2 4,' = r 2 sin 2 4, and sin 4, sin 4"

1

r

or $' = sin -1 (r sin 4,) where the sign of sin 4, is negated whenever the photon is reflected at the top of the solution or the bottom of the vial. We now need an expression for 7'. It can be seen from the above that sin2 T' = sin2 4,' cos2 4, sin2 7 COS24,' sin2 7 r 2 sin 2 4,(1 - s i n 2 4,) sin 2 7 (1 -- r 2 sin 2 4,) sin 2 4,

or _ ( 1 --sin 2 4, sin 2 \ ( l / r ) 2 - sin 2 4~] 7 whence [ / 1 - s i n z4, sin 3') 7' = sin-1 ~ r ~ l : 7 s-~n2~b

[./- !-sin2 4,

= sin -1 ~ ~ / ( 1 / r ) 2 _ s i n 2 4} sin

7).

The photon therefore leaves the outside surface of the vial at the point or' = ot + (I/31- T) x the sign of/3 z ' = z +(cos 7 - P cos/3) tan 4, (this last calculation being performed in steps; see above) travelling in the direction 7' = sin-1 ( ~ /

1 - sin24, 7) (1]r)2_sin 2 4, sin

4,' = sin -1 (r sin 4,)

Boxes 21-25 The directional response of the detector is determined by bivariate interpolation of Table 1 according to the coordinates (a', z') of the point of escape of the photon. Note that there are considerable losses (---9%; see part 2 of this paper) due to photons escaping from the detector chamber via the air gap between the vial and detector. These losses are included in our experimental data; note also that, during the calibration of the detectors, any photons reflected back towards the vial were absorbed by the black walls of the vial (see Appendix), and are therefore treated as losses (a fuller discussion of behaviour in the detector chamber can be found in~2)). The directional response was treated as a probability and modelled in the usual fashion. It was assumed that all photons which were not lost in the detector reached the photocathode without re-entering the vial and further that photocathode response was uniform across the entire face and followed the manufacturers' specifications. ~16> The photocathode quantum efficiency for a photon of that wavelength is calculated and treated as a probability (in the usual fashion; see above) to determine whether a photoelectron was generated. Since suitable data describing first dynode losses and pulse spread are as yet not available, "photoelectron counts" in the left and right "photomultipliers" are the main output of the current model. "Coincidence" is established when a " g - e v e n t " gives rise to at least one "photoelectron" in each "photomultiplier." All "coincidences" are counted and used in the overall "efficiency" calculation. A digital "pulse height spectrum" is produced using the summed "photoelectrons" from many "events." Pulse height discriminators are mimicked by assessing "counting efficiency" based on any number of "photons" of interest. Similar calculations could be made using only the lesser "pulse" of "photoelectrons.,,(17,18)

412

P. Z Malcolm and P. E. Stanley

T h e model was p r o g r a m m e d in (nonUSASI) F O R T R A N and run on a Control D a t a 6400 computer. Although its m e m o r y requirements (45K of 60 bit words) are quite reasonable the model exhibits a considerable appetite for processor time, each event requiring up to several seconds to simulate, with runs of several thousand events being necessary for behaviour to stabilize. Its widespread use is therefore precluded. CONCLUSION The model has enabled a comprehensive picture of the LSC process to be gained by logically structuring rigorous descriptions of its details. It has enabled us to ascertain dynamical relationships otherwise inaccessible, to validate existing knowledge, and to point out weaknesses in it as well as in instrum e n t design. T h e r e are for example several areas where further information is necessary if observed behaviour is to be m o r e accurately modelled. These include details of the contours of the photocathode response and its reflectivity; perturbations due to pulse spreading and losses along the dynode chain; details of multiple reflections within counting c h a m b e r (especially for those photons reentering the vial); and monoenergetic scintillation efliciencies corrected for secondary emissions. T h e model has been tested in detail, and found to ~ r o v i d e extensive results, reported elsewhere ~ -- 3) which are comparable to those observed in the laboratory. U n e x p e c t e d results have also b e e n obtained. For example the model indicates that a disproportionately large portion of light (---9%) escapes through the annular air gap between the vial and the top and b o t t o m of the detector chamber. The mathematical description of this behaviour is presented in part 2 of this communication, as is the design of a spherically symmetric LSC system in which this and other losses are minimized.

Acknowledgements---Grateful thanks are given to the staff of the Computing Centre of the University of Adelaide, without whose assistance this study would have been impossible. We are also indebted

to Dr. Jom~ BIRKS,Dr. DONALD HORROCKS,Mr. PHILIP KmaVELD, Mr. DONALD MOORE and Mr. EDWARD POLIC for providing invaluable numerical data and advice. Finally we thank Mrs. ERMIONI MOURTZIOS and Mrs. HELEN SIMPSONfor preparing the manuscripts.

APPENDIX A modified vial was used to provide the point source of light needed to investigate the directional response of the detector chamber of a model 3375 Packard Tricarb and Searle Analytic Isocap 300 spectrometers. The vial was fitted with a black lid and filled with solution (4.0 cra column) which had been spiked with several microcuries of 14Ctoluene. It was surrounded by a closely fitting cylinder of black photographic paper (6 cm high), and its base was covered permanently with similar paper. A single 0.1 cm diameter hole was punched in the paper cylinder at a height of 2.0 cm from the base of the solution and the vial then loaded in the detector chamber at a known angle and the photomultipliers switched out of coincidence. The response for each photomultiplier was then measured in an integral window and this was followed by measurements at radial intervals of 15° for total of 360 °. The black paper cylinder was then replaced, in turn, by one with a hole at 0.05, 0.5, 3.0, 3.95 cm. Thus the response for each photomultiplier tube was determined for each of 5 x 24 = 120 positions. Heights were selected in order to approximately linearise the total loss curve (see Fig. 4 of Part II of this contribution).

REFERENCES 1. MALCOLMP. J. and STANLEY P. E. Systematically understanding the liquid scintillation counting process: a stochastic computer model. To be published in Liquid Scintillation Counting, Vol. 4. Heyden and Son, London (1976)~ 2. STANLEY P. E. and MALCOLM P. J. Practical liquid scintillation spectrometry: organizing a methodology. To be published in Liquid Scintillation Counting, Vol. 4. Heyden and Son, London (1976). 3. MALCOLM P. J. and STANLBV P. E. Low level counting using liquid scintillation spectrometry: optimizing optical design. To be published in the Proceedings of an International Conference on Low Radioactivity Measurements and Applications, High Tatras, Czechoslovakia, October 1975, Comenius University Press, Bratislava.

Model of LSC process 4. MALCOLM P. J. and STANLEY P. E. Liquid

Scintillation Counting: Recent Developments (Edited by STANLEY P. E. and ScoooINS B. A.), pp 77-90. Academic Press, New York (1974).

11.

5. TEN HAAF F. E. L. Liquid Scintillation Counting, Vol. 2. (Edited by CROOK M. A., JOHNSON P. and SCALESB.), pp 39--48. Heyden and

12.

Son, London (1972). 6. TEN HAAF F. E. L. Liquid Scintillation Counting, Vol. 3 (Edited by CROOK M. A. and JOHNSON P.), pp 41-46. Heyden and Son, London (1974). 7. NEARY M. P. and BUDD A. L. The Current Status of Liquid Scintillation Counting (Edited by BRANSOME E. D.), pp 273-282. Grune Stratton, New York (1970). 8. K A C ~ C Z V K N. Organic Scintillators and Liquid Scintillation Counting (Edited by HORROCKS D. L. and PENO C-T), pp 977-990. Academic Press, New York (1971). 9. GXBSONJ. A. B. and GALE H. J. J. Sci. Instr. Ser. 2 1, 99-106 (1968). 10. GIBSONJ. A. B. Liquid Scintillation Counting,

13. 14. 15. 16. 17.

18.

413

Vol 2, (Edited by CROOK, M. A., JOHNSONP. and SCALES B.), pp 23-27. Heyden and Son, London (1972). HORROCKSD. L. Liquid Scintillation Counting, Vol. 3 (Edited by CROOK M. A. and JOHNSON P.) pp 1-2Q. Heyden and Son, London (1974). Hom~ocKs D. L. Applications of Liquid Scintillation Counting. Academic Press, New York (1974). BmKs J. B., personal communication. HASTINGS J. W. and WEBER G..7- Opt. Soc. Amer. 53, 1410-1415 (1963). HORROCKSD. L., personal communication. EMI specifications for 9635QB and 9805/A photomultipliers. LANEVB. Liquid Scintillation Counting: Recent Developments (Edited by STANLEY P. E. and ScoooiNS B. A.), pp 455-464. Academic Press, New York (1974). EDISS C., NOUJAIM A. A. and WmBE L. I.

Liquid Scintillation Counting: Recent Developments (Edited by STANLEYP. E. and Scoc_a31NS B. A.), pp 91-103. Academic Press, New York (1974).

ETUDE D'ENSEMBLE DU PROCESSUS DE COMPTAGE PAR SCINTILLATION LIQUIDE 1. M O D E L E STOCHASTIQUE COMPLET Un module math~matique du processus de comptage par scintillation llquide (CSL) est d&crit en d&tail. I1 r&unit de facon num&rique les aspects essentiels ~nerg&tiques et optiques de ce processus, et les r&sultats obtenus sont en bon accord avec ceux obtenus dans la pratique # de laboratoire. Le module sert ~ (1) faciliter la comprehension du rapport dynamlque qui constitue le processus CSL, (2) determiner des rapports qu'il serait " " • • et (5) - comme o~ le autreme n t dlfflclle de verifier, verra A la deuxleme partie de ce memolre - am&liorer la construction optique de l'instrument de CSL.

EINE EINHEITLICHE STUDIE DES FLOSSIGKEITS - SZINTILLATIONSPROZESSES I. EIN UMFASSENDES

STOCHASTISCHES MODELL

Ein mathematisches Modell des FIGssigkeits-Szintillatlons-Z~hhlprozesses (FSZ) ist ausfithrllch beschrieben.

414

P. 3. Malcolm and P. E. Stanley

Es vereint zahlenm~ssig die wesen~lichen energetischen und optischen Aspekte dieses Prozesses, und die errechneten Resultate stlmmen mit den in der Laborpraxis beobachteten ~berein. Das Modell hat die folgenden Anwendungen:- I. als Hilfe zum Verst~ndnls der den FSZ Prozess bestimmenden dynamischen Beziehung; 2. Beziehungen, die anderweitig sehr schwer bestimmbar sind, aufzudecken und 3. die Konstruktlon, bezogen auf die Optik des FSZ-Instrumentes, zu verbessern, wie dies im zweiten Tell dieser Arbeit pr~sentiert wird.

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HO~XO~ K F~O~ECC¥ CTOXACT~qECKAH

CqETA X ~ K O C T H O ~

MO~E~

0nzc~,saeTc~ n o = p o ~ o M a T e M a T z q e c x a x uo=e=~ npot~ecca cqeTa XMZKOCTHOII C ~ X H T M ~ M M (OXC). Mozez~ CqMczeHHO o6'e~*gH~eT c y ~ e c T s e H ~ e ~ H e p r e T ~ e c K H e M O n T ~ q e c K x e acneKT~ 9TOFO npouecca, H no~yqeHH~e p e 3 y ~ T a ? ~ B XOpOme~ coP~acgg c p e a y ~ T a T a M M H a 0 ~ a e ~ u ~ M g B ~a6opaTOpHO~ npaKT~xe. M o £ e ~ ~MeeT c~e~y~mge up~MeHeHH~: BIIepBHX,

B IIO,HMMaHHH £ H H a M M q e C K O F O O T H O m e H M ~ COOTaBJI~-cqeTa XPI~KOOTHOR C ~ H H T H ~ H H ; BO BTOpHX, B onpe~e.eHHI~ O T H O m e H ~ H , K O T O p H e H H a q e T p y £ H O 6 H " O 6 H yOTaHaB~HBaT%; B TpeT%I~, - - K a K OII~CMBaeTCH BO B T O p O H q a C T H ~TOrO £ox~a~a - B y~ly~eHH~ OIITHqecKoH K O H C T p y H ~

m,~ero npo~ecc

np~0opa C~KC.