3.2.1.  General considerations of ideal solution counting samples

Most of the advances in LS counters over the past twenty years — at least those that resulted in greater counting efficiency — have been due to improvements in the phototubes and in the optical and electronic design. Scintillator formulations, by comparison, have changed very little. Now, as twenty years ago, an example of an unquenched counting sample is ³H-toluene in toluene containing 3 g/l PPO and 0.5 g/l POPOP⁴.

Different formulations can be compared using the figure of merit given by Birks [69]. The figure of merit is the number of photoelectrons arriving at the first dynode of the phototube per kiloelectronvolt of energy deposited in the scintillator. Its use in radionuclide metrology is discussed in Chapter 4 and Chapter 5 of this Monograph. More recently Birks [238] has isolated those terms from the figure of merit over which the chemist has some control and defined the product of those terms as the scintillator figure of merit which depends upon

1. the choice of the solvent,

2. the solvent-solute energy transfer efficiency,

3. the solute quantum yield and concentration, and

4. the matching between solute fluorescence and phototube absorption.

Table 2.  Abbreviations, chemical formulae and wavelengths ($\lambda$) at fluorescence-emission maxima for organic scintillators referred to in this Monograph⁵

Following procedures given in Birks [238], one can arrive at scintillator combinations which give slightly higher scintillator figures of merit than the toluene solution of PPO and POPOP. Tables comparing the most frequently used LS solvents are given in Birks [69]. Simple aromatic compounds such as benzene, toluene and p-xylene have long been recognized as the most efficient solvents, for reasons described in Fuchs et al. [170] and Laustriat et al. [158].

The choice of scintillators is often governed by the spectral response of the phototube. Response curves for the current bialkali phototubes, such as the RCA 8850, Philips 56 DUVP, and EMI 9813-B, are given in the manufacturers’ literature, and fluorescence spectra and quantum yields for a large number of scintillators are” given in Berlman [166]. Although PPO is still the most common primary solute, terphenyl, PBD and tert-butyl-PBD are also used for some applications [209]. For many of these solutes there is an optimum concentration beyond which the scintillation yield decreases due to “self quenching”. Optimum concentrations for primary solutes are given in Birks [69] and Horrocks [223]. Details of quenching caused by eximers (excited complexes containing two solute molecules) are discussed by Horrocks [176], [223].

In the latest bialkali tubes, the absorption spectra sufficiently overlap the PPO fluorescence spectrum that there is little need for a wavelength shifter. Nevertheless, most workers continue to add the secondary solute, particularly if color quenching due to the sample is anticipated. The three most often used are POPOP, dimethyl-POPOP and bis-MSB. Bis-MSB is slightly more soluble in the common solvents than the oxazoles and solubility is often the determining factor.

3.2.2.  Uses of ideal solution counting samples

Nobles gases — As may be seen from the periodic table in Figure 1, noble gas radionuclides are among the few which have been assayed in unquenched or ideal counting samples. At one symposium on the noble gases [247], several papers were devoted to LS counting. For accurate work the solubility of the noble gas in the scintillator must be determined at the temperature at which measurements will be made [79]. The solubility of these gases in toluene increases with atomic weight, so the proportion of activity in the gas phase is relatively larger for lighter gases, such as ³⁷Ar and ⁸⁵Kr, than it is for heavier members of the group, such as ¹³³Xe and ²²²Rn.

Group VII A — The halides — The halides, such as ³⁶Cl and ¹²⁵I, are normally assayed as salts in alkaline solution. If it is necessary to prepare unquenched samples, however, they may be converted to organic halides. An often cited example is that of Yerick and Ross [67], who reacted molecular iodine, as both ¹²⁵I and ¹²⁹I, with an olefin by the reaction

The alkyl halide obtained is only a mild quenching agent and is quite stable except in alkaline solution, where the iodide groups may be replaced by -OH groups.

Ronzani and Tamers [105] report that unquenched samples containing ³⁶Cl may be prepared by synthesizing silicon tetrachloride (SiCl₄) which is then dissolved in a toluene-PBD liquid scintillator.

Groups II B and III A — VI A — If it is necessary to dissolve radionuclides from these groups in unquenched counting samples, there are a number of organometallic compounds available. Dimethyl-Hg, for instance, could be used for ²⁰³Hg. Many organometallic compounds of gallium, indium, and selenium are available commercially because of their usefulness in semiconductor research [250]. Organometallic compounds of heavy elements, such as tetrabutyltin and tetraethyl lead, are often used to increase the detection efficiency for $\mathrm{\gamma }$ rays [69].

Most of these organometallic compounds are synthesized by Grignard reactions. A useful example is the work of Brodzinski et al. [71] with ³²Si. They prepared a nonquenching tetralkyl silicon compound by the reaction

where $\text{X}=\text{Br}$ or $\text{F}$.

The magnesium compound on the left is called a Grignard reagent. Such agents are widely used for attaching alkyl groups to metals and some non-metals in groups III A-VI A. In the work cited above, the ³²Si-tetraethyl silane was introduced into a toluene solution scintillator.

3.2.3.  Preparation of ideal solution counting samples

Pure solvents and scintillators must be used to avoid quenching caused by impurities [229]. One can buy scintillators individually, as premixed blends, or as concentrates, which are diluted with additional solvent to the optimum concentration. These optimum concentrations are typically 5 to 10 g/l for the primary solute and 1 to 2 g/l for the secondary solute.

In metrology studies two preparation methods are used for these types of samples. The most prevalent method is to pipet the scintillator solution into the counting vial and then to dispense an aliquot of the sample material into the vial. There are two drawbacks to this straightforward technique. First, it is desirable to remove oxygen from the counting sample, to minimize quenching, and one cannot easily purge the vial after a volatile sample has been added. Second, errors in dispensing volatile samples will be greater than those incurred in dispensing aqueous samples in the same mass range.

Merritt [207] and BIPM [237] describe in detail the inaccuracies involved in dispensing aqueous samples in the range $20$ to $70\mathrm{mg}$. It is exceedingly difficult to dispense ³H-toluene with comparable accuracy.

The more accurate method is to spike a large quantity of the solvent (of the order of one litre) with a small sample ($1$ to $5\mathrm{g}$). Aliquots of the large batch are then dispensed into the cells. See, for example, Garfinkel et al. [90]. In this procedure weighing errors are much less than 0.1% because of the relatively large masses being transferred. Deoxygenating however is still a formidable problem.

The oxygen may be removed by purging with a nonquenching gas, usually nitrogen or argon, or by vacuum techniques. Despite the simple principles involved in these two operations, a great deal of care is necessary to insure that part of the sample is not removed with the oxygen. An example of a vacuum system used for LS sample preparation is given by Johns [241]. In this system, an LS vial with a Luer fitting is placed on a vacuum rack, the gaseous sample (⁸⁵Kr, ¹³³Xe, ³H-methane) is cryogenically transferred to the vial, and then de gassed scintillator is introduced.

In another method described by Shuping et al. [143], the solvent is refluxed for about fifteen minutes to remove oxygen. Subsequently, the system is enclosed and the solvent permitted to cool. The solvent may then be transferred either in a manifold or by using an inert gas atmosphere.

3.3.1.  General considerations of real solution counting samples

Most measurements in LS metrology before 1970 were made using what we have called real solutions. That is, some other agent was added to insure the solubility of the sample radionuclide in the solvent. The five predominant types of real solutions are

1. those containing labelled cation salts of organic acids, such as ⁸⁷Rb-octoate [36],

2. those containing labelled cations complexed by alkyl phosphate chelating agents, such as ¹⁴⁷Pm-diethyl hexyl phosphoric acid [82],

3. binary liquid solutions, such as 8% toluene-15% ethanol, which will tolerate small quantities of water containing the sample radionuclide as an inorganic salt,

4. dioxane-naphthalene solutions, which may also be used with small aqueous samples, and lastly

5. those scintillator solutions in which organic amines are used as trapping agents for labelled anions.

Item 5) will not be dealt with here, as the use of amines for trapping ¹⁴CO₂ and ³⁵SO₂ is a widely used technique and thorough reviews are available [224], [242], [275].

For most radionuclides present as cations, the choice among the first four types of solutions is somewhat arbitrary. As a general rule the counting efficiency would be slightly less for types 3) and 4) as ethanol is a mild quenching agent [223] and dioxane-naphthalene is a poor solvent. relative to xylene or toluene [284]. The quenching in solutions prepared using the organic acids (types 1) and 2) will be dependent for the most part on the purity of the scintillators and reagents used.

Horrocks [223] notes that the most often used organic acids (solution type 1) are octoic and naphthenic acids. A method of preparing cadmium, bismuth, and lead octoates suitable for use in liquid scintillators is described by Ronzio [40]. Many metal salts of these acids are commercially available as they are often used as catalysts in the petrochemical industry.

The most often used chelating agent is “diethylhexyl phosphoric acid, normally abbreviated as HDEHP. Horrocks [174], [221], for example, has used HDEHP for such diverse nuclides as ⁵⁵Fe, ¹¹³Sn, and natural uranium, while Erdtmann [75] and Erdtmann and Herrmann [88] have used the same agent for ²²Na, ⁶⁰Co, ⁹⁵Zr-⁹⁵Nb, ¹²⁴Sb, ¹⁴⁴Ce-⁴⁴Pr and ¹⁹⁸Au. HDEHP is also widely used by inorganic and nuclear chemists for sol vent extraction studies [126], [180], [183]. Mason et al. [261], for example, used HDEHP to extract ⁹¹Y, ¹⁴⁷Pm, ¹⁵²Eu, ¹⁷⁰Tm, and ¹⁷⁷Lu from acid solutions into benzene. After equilibration of the two phases, aliquots were taken from each, evaporated to dryness, and counted with a proportional counter. Mason [274] reports that in more recent work, they have introduced the sample aliquot from each phase directly into an emulsion scintillator for LS counting.

Many other alkyl phosphates have, of course, been used for extracting radionuclides into liguid scintillators. Ludwick [45] used dibutyl phosphate to introduce ⁹⁵Zr-⁹⁰Nb into a xylene scintillator, while Horrocks and Harkness [59] used ⁶³Ni labelled dioctyl phosphate, also in a xylene scintillator. Reference to other useful complexing agents may be found in [193] and [242].

Too few articles in the literature on LS metrology give detailed descriptions of the scintillator formulation used. Six papers which do contain a wealth of detail are listed here, along with the type of solution scintillator employed and the number of radionuclides studied (in parentheses):

Flynn et al. [77]

toluene-ethanol-octoic acid: (10)

Dyer et al. [74]

toluene-ethanol: (4)

Vaninbroukx and Spernol [95]

toluene-ethanol: (15)
dioxane-naphthalene: (13)

Erdtmann [75], Erdtmann and Herrmann [88]

toluene-ethanol-HDEHP: (6)
dioxane-naphthalene: (6)

Steyn [122]

xylene-ethanol: (5)

The relative amounts of primary and secondary solvents differ significantly in these works. The solvent used by Vaninbroukx and Spernol contained 89% toluene and 11% ethanol by volume while some of those used by Dyer et al. contain over 50% ethanol. It is also interesting to compare in this regard the types of solution scintillators used by the participants in the 1963 BIPM international comparison of ²⁴¹Am [84]. This report contains details of the scintillators used by the six laboratories who used LS counting methods (Toluene, xylene and dioxane solution scintillators were used).

The real key to stability in these solution scintillators probably lies in selecting the proper chemical form of the radionuclide and concentration of carrier.

Dioxane-naphthalene scintillators were developed by Furst et al. [17] over twenty years ago, as a means of dissolving aqueous samples in an organic solvent. In addition to the precision metrology studies cited above, these scintillators have also been widely used for environmental measurements [141] and for biological samples [148]. There seems, however, to be little reason for their continued use. Dioxane is more toxic than toluene and p-xylene has a lower flash point, and its naphthalene solutions are less efficient solvents. For practical applications, the dioxane-naphthalene systems have been largely replaced by emulsion scintillators.

3.3.2.  Uses of real solution counting samples

It is evident from Figure 1 that most radionuclides can be incorporated into real-solution counting samples. For noble gas radionuclides, however, there is no particular advantage to adding solubilizing agents, as the noble gases are most soluble in pure aromatic hydrocarbons [151].

Sufficient examples have already been given for real-solution scintillators used for radionuclides present as cations (²²Na, ⁸⁷Rb, ⁶⁰Co, etc.). Some radionuclides normally counted as anions are ¹⁴C (${\text{CO}}_{\stackrel{=}{3}}$), ³⁵S (${\text{SO}}_{\stackrel{=}{3}}$), ³²P (${\text{HPO}}_{\stackrel{=}{4}}$), and the halides (Cl^-, I^-, etc.). Such solutions are usually made basic (with KOH, for example) to prevent escape of volatile labelled material. Solution scintillators used for radioiodines are described by Rhodes [92].

3.3.3.  Preparation of real solution counting samples

In most experiments, the counting samples are prepared by first pipetting the scintillator solution into the vial and then transferring an aliquot of an aqueous solution into the vial. For accurate work, gravimetric techniques are used, such as those described by Merritt [207] and Vaninbroukx and Spernol [95].

The procedures described in those works are too lengthy to be reproduced here in full. We should like to mention, however, one technique described there [95] for insuring that the sample and solution scintillator are thoroughly mixed. First, about $0.1\mathrm{ml}$ of the aqueous sample (containing the radionuclide and appropriate carriers) is weighed into an open $100-\mathrm{ml}$ bottle. Then the sample is dissolved in $6\mathrm{ml}$ of ethanol, $44\mathrm{ml}$ of toluene-based scintillator is added, and the bottle is closed and shaken mechanically for 10 minutes. Individual counting samples are then weighed into counting cells.

The main limitation to the use of toluene-ethanol scintillators is that the counting samples are often unstable and the radionuclide may precipitate or become adsorbed on the surface of the vial. The first point to consider when instability is recognized is the homogeneity of the solution. If the sample is indeed a true solution (one in which the relative proportions of constituents are the same in any differential volume element as they are in the total sample), then several steps may be taken to prevent the radioactive cation from being adsorbed on the container walls. The most effective means of keeping the sample in solution is to add sufficient inactive carrier to insure that the reactive sites on the glass surface are occupied by inactive ions. Several examples are given by Erdtmann [75], in which the stability is studied as a function of carrier concentration. The amount of carrier necessary to prevent adsorption will depend on the temperature, sample pH, and concentrations of complexing agents.

In other work Vaninbroukx and Spernol [95] added ethylene glycol with solutions of SrCl₂ to prevent adsorption while Smith and Stuart [245] reported adding a drop of concentrated acid to the sample, a technique also mentioned in Monographie BIPM-l [237]. One should exercise caution, however, in adding strong acid to these solutions. Nitric and perchloric acids, in particular, may produce explosive compounds. Concentrated acids may also accelerate oxidation and decomposition of the fluors. A reasonable limit to the acid concentration for these solutions would be that obtained on adding $150\mathrm{mg}$ of an aqueous solution of $4\mathrm{mol}$ HCl per ${\mathrm{dm}}^3$ to $10\mathrm{ml}$ of solution (90% toluene-10% ethanol, by volume).

Smith and Stuart [245] have also mentioned using a silicone treatment on the surface of the glass vials. Physical and chemical models for these wall-surface treatments are given in recent work by Harding and Dixon [267] and techniques for applying such treatments are described in Monographie BIPM-l [237].

3.4.1.  General considerations of emulsion counting samples

The commercial emulsion scintillators in use today have evolved from the work of many investigators over the past fifteen years. The history of the development of these scintillators and their practical applications were discussed recently by Benson [251]. As Fox has demonstrated in a series of papers [216], [217], [254], the chief advantage of emulsion scintillators is that one may incorporate relatively large proportions of aqueous sample, and still maintain reasonable counting efficiencies. His figure of merit for low-level ³H counting is proportional to the product of counting efficiency and volume fraction of water.

The present treatment, however, will be limited to high-efficiency emulsion counting samples, which contain less than 5% water by volume. In such systems, water micelles are dispersed in the organic phase consisting of a solvent and a dispersing agent. Dispersing agents are also called emulsifiers, detergents, and surfactants. The tenn surfactant is used here since “surface-active” agent best describes their function in these ternary mixtures.

Reference works on emulsion technology [127], [226] reveal that a large number of surfactants can be used to form stable emulsions. Lieberman and Moghissi [159] evaluated many of these and found certain non-ionic surfactants best suited for use in emulsion scintillators. The most popular of these, which are known by the Röhm and Haas trademark Triton, are formed by the reaction of an alkyl phenol with ethylene oxide and have the general formula

Generally $m$, the number of ethylene oxide groups, is about the same as $n$, the number of methyl groups. For Triton-X 100, $m=9$ and $n=8$, while for Triton N101, $m=10$ and $n=9$. Another useful agent is BBS-3, a proprietary compound distributed by Beckman [256]. When a small amount of water is dispersed in a large volume of solvent, the surfactant molecules stabilize the two-phase system by orienting themselves with the alkyl chain, or hydrophobic end, in the organic phase and the hydrophilic end (containing the ether groups with the unbound electrons on oxygen) in the aqueous phase.

The proportions of solvent, surfactant, and water necessary to form a stable emulsion will depend on variables such as temperature, pH and salt concentrations [118], [188], [216], [217]. For the small samples used inm etrology studies, it is only necessary to establish the minimum amount of surfactant required to form a stable system. Excess surfactant will serve only to lower the scintillation yield (although recent work [231] has shown that the aromatic surfactants do behave as scintillators, to a limited extent). Few references are available on emulsion scintillators which are specifically formulated to give high counting efficiency for small samples. One such system, reported by Horrocks [256], consists of 16% BBS-3 and 84% toluene by volume. This seems to be a reasonable lower limit to the amount of surfactant required to prevent phase separation. Most of the commercial products on the market contain 20 to 33% surfactant by volume.

An intriguing question about these emulsion systems is whether the micelle diameters are small compared to the range of particles emitted in the nuclear decay (or transition) process. As early as 1967, van der Laarse [118] showed that there was a small but significant difference in $\mathrm{\beta }$-particle counting efficiency depending upon whether the activity was in the organic scintillator phase (³H-toluene) or inside the micelles (³H-water).

In a recent work devoted to this topic, Horrocks [256] determined counting efficiency as a function of water concentration for samples of ³H, ¹⁰⁹Cd-¹⁰⁹Ag^m, and ²³³U. Gibson [255] at the same symposium suggested that (at least for systems described by Horrocks [256]) the micelle diameters are of the order of $10\mathrm{nm}$ or less and, therefore, energy loss within the micelles will only be significant for a few radionuclides such as ³H and ⁵⁵Fe. Gibson has suggested more recently [291] that emulsion scintillators should not be used in metrology studies until the question of energy loss within the micelles has been resolved.

It is usually assumed that when samples containing dissolved salts are dispersed in these two-phase systems, the ionic species will be in the aqueous phase. The extent to which radioactive species are extracted into the organic phase has received little attention. It is clear however that when one adds an extractant such as HDEHP to these emulsion systems, the radioactive sample is extracted to some extent frOm the water micelles. This is similar in fact to McDowell’s work [183] with actinides, in which aqueous samples are introduced into a toluene-HDEHP “extractive scintillator”. With plutonium the HDEHP serves two purposes. First it extracts plutonium from the acidic aqueous phase and second it prevents Pu⁺⁴ from polymerizing in the weak acid environment of the organic phase [63], [183].

In recent work at NBS, ²³⁹Pu solution samples ($100$ to $300\mathrm{mg}$ in an aqueous solution of $5\mathrm{mol}{\mathrm{l}}^{-1}$ HNO₃ were introduced directly into $10\mathrm{ml}$ of a commercial emulsion scintillator and a rapid loss in count rate was observed (5% per day). Stable samples were obtained however when the emulsion was made 0.05 molar in HDEHP.

Another “extractive scintillator” for curium, californium and americium has been described by Miglio [283].

3.4.2.  Uses of emulsion counting samples

Radionuclides of at least 22 of the elements shown in Figure 1 have already been incorporated into emulsion scintillators. Several extensive works are listed here, together with the brandname-emulsion scintillator employed and the number of radionuclides examined (in parentheses):

Horrocks [221], [222], [256], [257]

Ready-Solv VI: (6)

Handler and Romberger [202]

Insta-Gel: (7)

Ishikawa and Takiue [205]

Insta-Gel: (4)

Bransome and Sharpe [213]

BBS-3 — toluene: (4)

Only recently have precise radionuclide metrology results been reported in which emulsion scintillators were used. In the international comparison on ¹³⁹Ce [276], sponsored by the Bureau International des Poids et Mesures (BIPM), the three laboratories who used LS counting (IBJ, NAC, and NPL) ⁶ all used commercial emulsion scintillators. Several of the participants in the ¹³⁴Cs international comparison (the results of which are presented in Chapter 8 of this Monograph) also used commercial emulsion scintillators.

3.4.3.  Preparation of emulsion counting samples

Many of the procedures given in previous sections, for preparing solution counting samples, are also applicable for preparing emulsions. When using commercial products, however, one should heed suggestions given by the supplier as many of these products are formulated for specific types of samples. Some of the emulsions may appear cloudy at first and then clear after standing for several hours. If samples are to be counted at reduced temperature, they should be stored at that temperature and examined before counting to insure that phase separation has not occurred.

Chemiluminescence and phosphorescence are often observed with emulsion systems [251]. Presumably these problems are associated with impurities in the surfactants. Some workers report reducing the chemiluminescence by pretreating the surfactant [251], while others find that warming the sample to $40\mathrm{°C}$ reduces this component of the background [159]. Many workers also advise doing as much of the sample preparation as possible under red light to avoid unnecessary photoactivation.

Figure 2 — Loss in count rate observed for some ⁸⁹Sr counting samples prepared using a commercial emulsion scintillator. Measurements were made at $6\mathrm{°C}$ using a Packard TriCarb, with a window fixed by a single low-level discriminator. The counting efficiency in this window was 0.995. The counting samples consisted of $\approx 75\mathrm{mg}$ of aqueous solution (SrCl₂ in an aqueous solution of $0.1\mathrm{mol}$ HCl per ${\mathrm{dm}}^3)\in stem:\left[15\mathrm{ml}\right.$ of emulsion scintillator in borosilicate-glass vials (from measurements made at NBS).

Figure 3 — Relative loss in count rate due to sample adsorption for ten counting samples containing different amounts of strontium carrier and chelating agent: ◦ strontium only, • strontium $+0.2\mathrm{mmol}$ HDEHP. Each vial contained $\approx 75\mathrm{ml}$ of aqueous sample in $7\mathrm{ml}$ of a commercial emulsion scintillator (from measurements made at NBS).

The stability of counting samples can still be a problem with emulsion systems. Figure 2, for instance, shows the loss in count rate for some ⁸⁹Sr samples dispersed in a commercial emulsion scintillator. Note that the addition of chelating agent alone was not sufficient to prevent the sample from being adsorbed on the wall of the glass vial. A series of vials having different carrier concentrations was then prepared and $1\mathrm{mg}$ of strontium per $7\mathrm{ml}$ of emulsion was found sufficient to give a stable sample. Figure 3 shows the response for this series of samples as a function of added carrier after 4.5 days.

Volatility of the sample is another problem that still has to be considered when using emulsion scintillators. The commercial formulations are usually supplied in acid form in order to inhibit chemiluminescence reactions [150], which may occur when strongly alkaline clinical samples are mixed with the scintillator. A problem occurs if a slightly alkaline sample, such as ¹⁴C-labelled Na₂CO₃, is added to a commercial product which is acidic. In this case volatile ¹⁴CO₂ may escape into the void space above the scintillator, as shown in recent work by Herbland [268] and Huskisson and Ward [280]. This problem can be avoided for carbonate samples by first adding sufficient basic carrier solution to neutralize the scintillator emulsion, before adding the sample.

where

$S_{\text{E}}$

scintillation yield, the fraction of electron energy which is converted to light energy [223],

$h\overline{\nu }$

average energy of emitted photons,

$G$

light-collection efficiency, that fraction of the photons generated in step Section 4.3, 1) that arrive at the photocathode,

$m{C}_{\text{pe}}{g}_{\text{c}}$

phototube response, the fraction of photons arriving at the photocathode, which result in photoelectrons arriving at the first dynode,

4.3.1.  The scintillation yield

The scintillation yield is about 4.5% for 5-keV electrons in the best liquid scintillators. Some of the problems that limit direct measurements of $S_{\text{E}}$ are:

1. the yield depends on the nature of the ionizing particle; for example, the yield for an $\mathrm{\alpha }$ particle is only about one-tenth that of a $\mathrm{\beta }$ particle of the same energy, due to differences in ionization density along the particle track [69],

2. the yield depends on the energy of the particle; for $\mathrm{\alpha }$ and $\mathrm{\beta }$ particles, the energy loss per unit length of path is inversely proportional to the square of the particle velocity [9] (see also Section 5.4.1, ionization quenching),

3. finally the yield depends on the chemical composition of the sample [238].

Given the difficulties involved it is perhaps not too surprising that there are few data available on scintillation yields. Most workers rely on the absolute-scintillation- yield measurements of Hastings and Weber [65] with ³H and ¹⁴C in a liquid scintillator com.f.osed of PPO and POPOP in toluene, and those of Skarstad et al. [137] with ¹⁴C in a solution of terphenyl in benzene.

Malcolm and Stanley [259] have combined these yield data with Horrocks’ [80] relative yield vs. electron-energy data to arrive at an empirical expression giving $S_{\text{E}}$ as a function of $E_{\text{e}}$. Using their value of $S_{\text{E}}=0.04$ for $5\mathrm{keV}$, and comparative data for different solvents [69], [238] one would expect $S_{\text{e}}\approx 0.045$ for p-xylene with an efficient scintillator such as PPO or t-butyl PBD. Using $3.2\mathrm{eV}$ as the average photon energy ($385\mathrm{nm}$), one finds that a 5-keV electron will produce 71 primary photons. This is the first term in brackets in Equation (1).

4.3.2.  Light-collection efficiency

Estimates of the fraction $G$ of these 71 primary photons that reach the phototube have been very crude and range from as low as 10% to as high as 100%. Before addressing the problem of estimating $G$ we will discuss some of the design factors that must be considered in optimizing light-collection efficiency, namely:

• average photon light-path length,

• shape of sample container,

• interfaces,

• reflecting surfaces, and

• light guides.

Photon light-path length — The further the photons have to travel through the scintillator, the greater the attenuation on the photon pulse before it reaches the boundaries of the container. The two major processes in this attenuation are

• absorption of the photons by other scintillator molecules, and

• absorption of the photons by colored impurities in the solution (referred to as color quenching).

Sipp and Miehe [232] have studied time resolution and fluorescence — absorption in liquid-scintillator cells between two RCA 8850 phototubes. Their results clearly show the dependence of these two parameters on the path length in the cell, which was varied from $0.2$ to $5\mathrm{cm}$.

Shape of the container — It has long been recognized that certain geometrical shapes, rectangular cells [2] and spherical vials [120], [259], for example, are better suited for photon measurements than the cylindrical vials used in most commercial instruments. Further evidence of light losses in the cylindrical geometry is presented by Stanley [234]. Figure 6 in that reference is a photograph of the bottom of a vial, in which the exposure is due only to the fluorescence from the cell. The cylindrical symmetry there is shown to have a collimating effect (a bright ring near the edge is observed) which results in excessive losses at the bottom of the cylinder.

Interfaces — Clearly every interface between the scintillator and the photocathode represents a potential reflection for the photons [134]. There are two usual ways to reduce losses at these interfaces. First, one can sandblast the outer surface of the container [106], to make a more diffuse surface and reduce internal reflection. Second, one can employ optical coupling greases or fluids which have a refractive index close to that of the cell and phototube face. The refractive index of the liquid scintillators is about 1.50, while that for borosilicate glass, which is used for many counting vials and is also used in the envelope for the RCA 8850, is 1.47. One useful silicone coupling grease is “Dow Corning Optical Couplant Q2-3067” for which $n_{\text{D}}=\mathrm{1.464 8}$.

Reflective surfaces — Materials most often used for reflecting surfaces are polished aluminum, TiO₂ (such as NE 560 from Nuclear Enterprises), MgO and BaO. In some very early work Hayes et al. [21] looked at the efficacy of Al and TiO₂ reflectors for scintillators which emit at different wavelengths. Some of the results rearranged by Birks [69] are given here.

It appears then that polished aluminum is superior if only the primary scintillator is used. If wavelength shifters are employed however, (see Section 3.2.1, TiO₂ would be better due to its better reflecting properties at longer wavelengths.

Light guides — Materials most often employed in light guides [32], [69] are cylindrical rods of polymethylmethacrylate (Perspex, Lucite, Plexiglas), although quartz rods or fiber optics [296] also exhibit very high transmission. In the close geometries employed in LS counters, however, the light guide most often is a hollow Lucite container filled with a light-weight silicone oil of matching refractive index, for example, Dow Corning 550 Silicone Fluid, which has a $n_{\text{D}}=1.50$ and a kinematic viscosity of $1.15{\mathrm{cm}}^2{\mathrm{s}}^{-1}$ [290].

Monte-Carlo methods for calculating light-collection efficiency — While the details presented in the last five subsections are useful in optimizing the light-collection efficiency, we have still not addressed the problem of how one makes a quantitative estimate of $G$ to use in Equation (1).

One could obtain estimates of $G$ using the Monte-Carlo calculations of Malcolm and Stanley [271]. Their computer program allows them to simulate photon emission, attenuation and collection processes for a given geometrical arrangement of vial, reflector and phototubes. Thus far they have reported results for only two arrangements, namely, for the IEC standard vial containing $20\mathrm{ml}$ of liquid scintillator, and for a spherical vial containing, for example, $5\mathrm{ml}$ of scintillator.

Rather than use their estimates here we have elected to estimate all other parameters in Equation (1) and then to solve for $G$ as the unknown. With this objective in mind we consider now the phototube response.

4.3.3.  Phototube response

The bracket term in Equation (1), [ $m{C}_{\text{pe}}{g}_{\text{c}}$ ] which we have called the phototube response, gives the fraction of photons arriving at the photocathode which result in photoelectrons arriving at the first dynode, and is about 0.25 for the typical scintillator/phototube combinations used in radionuclide metrology. We consider now in a little more detail the three terms that make up this response.

The spectral matching factor $m$ is given by

where $I\left(\lambda \right)$ is the fluorescence spectrum of the scintillator, normalized such that ${\int }_0^{\infty }I\left(\lambda \right)\text{d}\lambda =1$, and $\Phi \left(\lambda \right)$ is the photocathode response function, normalized such that the maximum value of $\Phi \left(\lambda \right)=1$.

According to Birks [238] spectral matching factors are quite high for the usual primary scintillators with bialkali photocathodes. For instance, $m=0.97$ for PPO with a bialkali tube.

The second term $C_{\text{pe}}$ is the actual maximum photoelectric efficiency, or “quantum efficiency” reported by the tube manufacturer, and does not include $g_{\text{c}}$, the photoelectron collection efficiency at the first dynode [196].

Details of the construction of phototubes often used in LS counting are given by Coates [149], Krall and Persyk [198] and Persyk and lewis [230]. Typical values for $m$, $C_{\text{pe}}$ and $g$ are given in Table 3. Of these three the $g_{\text{c}}$ value is the most difficult to measure and it causes the greatest uncertainty in the phototube response [162], [292].

Table 3.  Typical figures of merit for commercial counting system⁹ and for high-efficiency lS counting system for $E_{\text{e}}=5\mathrm{keV}$

4.3.4.  Comparison of figures of merit for two liquid-scintillation counting systems

In Table 3 we give some typical values for the parameters in Equation (1), which allow an interesting comparison to be made between the commercial system and a high-efficiency LS system. To arrive at estimates of the light-collection efficiency, Equation (1) was solved for $G$, using the values for other parameters shown in the table. This result indicates that about 11% of the primary photons emitted are incident on each phototube in an arrangement such as the one shown in Figure 4, which compares well with about 14% given by the Monte-Carlo program for a similar geometry [271], Table 2b).

4.4.1.  Cylindrical vials perpendicular to phototube

Sensitivity can be optimized in this geometry by using small-diameter tubing for the container. An efficient light guide is then used between the sample container and the phototube face [45], [79]. McDowell and Weiss [262], in a system designed for assaying $\mathrm{\alpha }$ particles, use a silicone oil-filled container as the light guide. The inner surfaces of the light guide are painted with reflecting paint.

Figure 6-1 — Cylinder with axis perpendicular to axis of phototubes¹¹

Figure 6-2 — Cylinder with axis perpendicular to axis of phototube¹²

Figure 6-3 — Rectilinear cell with single phototube¹³

Figure 6-4 — Cylinder with axis parallel to axis of phototube¹⁴

Figure 6-5 — Cylinder with axis paralel to axis of phototubes¹⁵

Figure 6-6 — Spherical cell between two phototubes¹⁶

Figure 6 — Geometrical arrangements of sample containers and phototubes often used in high-efficiency LS systems.

The light-collection efficiency will be better for small-radius cylinders [81] and this results in higher counting efficiency for low-energy $\mathrm{\beta }$ particles and better energy resolution for $\mathrm{\alpha }$ particles. The resolution for $\mathrm{\alpha }$ particles, expressed as FWHM/$E_{\mathrm{\alpha }}$)( (full width at half maximum divided by $\mathrm{\alpha }$-particle energy), is 20 to 25% for a 25-mm diameter cylinder, while Horrocks reported 10.7% using vials constructed from 6-mm diameter thin-walled Pyrex tubing.

4.4.2.  Cylindrical vials with axis of cylinder parallel to that of phototube

The single phototube arrangements in Figure 6-4 have been used extensively in metrology work and current systems used at NPL and at NBS are of this design. Light-collection efficiency here depends markedly on the dimensions of the scintillator. The NPL samples, for instance, consist of $5\mathrm{ml}$ of scintillator in a cylindrical vial, while the NBS sources consist of $3\mathrm{ml}$ in a hemispherical vial.

The light-collection efficiency is, as expected, inversely proportional to the height of scintillator in the vial. This was demonstrated by Vaninbroukx and Spernol [95] by observing the width of the $\mathrm{\alpha }$-particle peak from ²¹⁰Po as a function of the height of scintillator in the vial.

The second very useful arrangement Figure 6-5 allows one to couple the flat ends of a cylinder directly to the faces of two phototubes [122]. Kolarov et al. [157] used a quartz-spectrophotometer cell with two RCA 8850 phototubes and obtained $\epsilon$(³H) = 73% for the tubes in coincidence, and 89% in summation. These cells are available in both glass and quartz, with specified light paths of from $0.1\mathrm{cm}$ to $5\mathrm{cm}$.

Although there are no data on the $\epsilon$(³H) as a function of path length in these spectrophotometer cells, there are some useful results published by Kaufman and coworkers 20 years ago [44], [61], using a cylindrical container made by connecting the faces of the two phototubes with an aluminum tube, which was polished on the inner surface. The curves shown in Figure 7 give efficiency as a function of separation distance of the phototubes. The fact that the three curves have different slopes is evidence of differences in the attenuation of light in the scintillators. As we mentioned before,different concentrations of solute molecules or colored impurities will give rise to different attenuation characteristics [266].

4.4.3.  Spherical vials

Malcolm and Stanley [273], using the Monte-Carlo method, derived a design for a system employing a spherical 5-ml vial, in which the $\epsilon$(³H) for an unquenched sample was as high as 86%.

There is some early experimental work for this geometry, but since the work was published in Russian it appears to have received little notice in reviews on this subject. Two of the most interesting drawings from this work by Paligorić et al. [120] are included here in Figure 8. They did not report $\epsilon$(³H) directly, so we cannot compare their results with those derived by the Monte-Carlo method. The figures of merit ${\eta }_{49}\mathrm{keV}$ of about 0.3 photoelectron/keV for each tube suggest, however, that even though the light-collection efficiency was high, the scintillation yield and phototube response were less than optimum. According to the English translation, the scintillator used was composed of PPO and POPOP in dioxane. In the usual formulations, this scintillator contains 60 grams of naphthalene per liter of dioxane. If naphthalene is omitted from this formulation, the scintillation yield is considerably less than that found with PPO in toluene.

It can be seen in Figure 8-1 that the efficiency begins to decrease for ³²P in the small vials due to the wall effect. This means that in designing a system for general use in metrology one has to consider which radionuclides are to be assayed, so that the dimensions of the container will be large compared to the range of the $\mathrm{\alpha }$ or $\mathrm{\beta }$ particles. For the spherical vials it appears the $5\mathrm{ml}$ vial suggested by Malcolm and Stanley [273] is close to optimum for low-energy $\mathrm{\beta }$ emitters.

Figure 7 — ³H counting efficiency as a function of separation distance between the phototubes, after Kaufman et al. [44].

In this system (see also Figure 6-5, the DuMont phototube face plates formed the ends of a cylindrical cavity containing the scintillator. The three curves are for different scintillator formulations, and demonstrate the effect of color quenching at longer light-path lengths.

Figure 8-1 — Relative counting efficiency as a function of mass of scintillator for system employing spherical vials.

Figure 8-2 — Schematic arrangement of system employing spherical vial.

Nine vials of different diameters were used: $10$ to $42\mathrm{mm}$ corresponding to $0.3$ to $35\mathrm{g}$ of scintillator.

1. phototubes,

2. scintillator solution $7\mathrm{g}$ PPO and $50\mathrm{mg}$ POPOP in 1 l dioxane,

3. silicone oil,

4. rubber seal,

5. counter casing with TiO₂ reflector.

Figures are adapted from [120], originally in Russian, and are based on an English translation provided by the authors (courtesy IAEA, Vienna).

4.4.4.  Other systems

Another rather unique design of an LS counting system for use in radionuclide metrology has been reported by Pochwalski and Radoszewski [293]. In this system, three phototubes are arranged symmetrically about the axis of a cylindrical glass vial, and triple coincidence events, as well as single and dual coincidence events, may be recorded.

Several systems are also in use which employ optical filters between the cell and phototube to allow one to vary the counting efficiency for a single sample [190], [294], [299].

5.2.1.  The statistics of the scintillation counter

The pulse-height distribution from a scintillation counter is not a simple function of the input energy except at high energies ($>500\mathrm{keV}$) where a simple Gaussian distribution gives an adequate representation of the output. In considering radiations from radionuclides such as ³H, ⁵⁵Fe, ⁶³Ni and ¹⁴C, such an approach is not correct. Neither can the output be considered as a Poisson distribution at low energies because the phototube modifies this distribution at the input and produces a variance in excess of that expected from the Poisson distribution. This additional distortion is small for the modern high-gain phototubes.

Gale and Gibson [101] reviewed the mathematical methods used to describe the pulse-height distribution for a single-electron input to the first dynode and produced generating functions for any number of electrons and also for an input with a Poisson distribution. The formulae developed in an earlier paper [89] are as follows. The probability of obtaining $n$ electrons from the $k$th stage of the phototube from an input to the first dynode of exactly $r$ electrons is

and

is the zero-detection probability. Now

where $m_k=\text{gain of the }k\text{th stage }\left(k=1,2,3,4,\dots \right)$.

It will be noted that the probability distribution is calculated from the single-electron output from the $\left(k-l\right)$th stage and can be calculated directly for any particular value or $r$, independent of a calculation of any other value of $r$.

$P_k\left(1,0\right)$ and $P_k\left(r,0\right)$ are the zero-detection probabilities and the generating functions are based upon the Poisson distribution Equation (3). The pulse-height distribution from an input to the first dynode distributed with a Poisson mean of $\overline{r}$ electrons is

and

The output is again derived from the single-electron input, but this time at the $k$th stage. Similar generating functions have been derived by Prescott [104] to take account of non-uniformity in the dynodes of the phototube and they contain the Poisson distribution as a special case. However, as has been shown by Coates [149], this additional complication is unnecessary if a high-gain phototube is used.

The major problem with this exact model is the amount of computer time required to produce a pulse-height distribution and Gale and Gibson [101] proposed an algebraic approximation in the form of a Gamma function to reduce the computing time. However the introduction of the RCA phototube with a first stage gain of between 30 and 35 has reduced the necessity for such an approach because the distortion introduced by the dynodes is small. For example the mean and variance calculated using the generating functions above are:

Mean

$\mu =\overline{r}\cdot m_1\cdot m_2^{k-1},$

Variance

${\sigma }^2=\overline{r}\cdot m_1^2\cdot m_2^{2\left(k-1\right)}\left\{1+\frac{m_2}{m_1\left(m_2-1\right)}\right\}.$

An initial stage with a gain of $m_l$ and subsequent stages of gain $m_2=m_3=m_4=\dots =m_k$ are assumed. The relative variance is

At the input of the first stage, the relative variance is $1/\overline{r}$ so that the term in the brackets in Equation (9) represents the additional distortion introduced by the dynodes. Now if $m_1=35$ and $m_2=4$, the relative variance is $1.038/\overline{r}$ and the relative standard deviation is $1.019/\sqrt{\overline{r}}$. This means that the width of the peak from a single (mono) or multielectron input is increased by less than 2% over that expected for a Poisson distribution. The effects on the zero-detection probability are not detectable ($\approx {10}^{-14}$) and so it is reasonable to assume that a Poisson distribution can be used to represent the pulse-height distribution from this type of phototube. Other tubes with a lower first-stage gain require the more exact approach discussed above.

5.2.2.  Pulse-height distributions

An excellent review of the output from the RCA 8850 phototube was prepared by Houtermans [204] and he shows the pulse-height distribution from single photons for this phototube Figure 9. The relative peak width at half maximum (FWHM) is 41%, which is a function of the relative variance such that, for a Gaussian distribution,

Figure 9 — Mono spectrum for an electron-multiplier phototube of type RCA 8850 [204] (courtesy North-Holland Publishing Company, Amsterdam).

Thus $\overline{r}=\left(8\text{ln}2\right)/{0.41}^2=33$, which is equal to the first stage gain ($m_1$) as given in the manufacturers’ specification. The distribution below the peak is due to edge effects [149] but, by careful choice of the tube, these can be reduced to a minimum (peak to valley > 5). The dark noise should also be minimised and it is important to have a high quantum efficiency for the photocathode.

In the ideal situation, the distribution would be purely Poissonian and the result of having $r=2,3,4,\dots$ electrons incident at the same time would produce a series of distributions with means of 66, 99, 132, and so on. If an input to the first stage is itself distributed according to the Poisson distribution, then it is necessary to add together the contributions such that the final distribution may be approximated by the distribution at the output of the first stage, viz,

where $m_1$ is the gain of the first stage and $P_0$ is the Poisson distribution as defined by Equation (3). Thus

Figure 10 — Calculated pulse-height distributions for an input to the first dynode distributed according to Poisson statistics with means of $\overline{r}$ from 1 to 8 [204] (courtesy North-Holland Publishing Company, Amsterdam).

For a more exact calculation, to allow for the edge effects on the dynodes, the generating functions produced in Section 5.2.1 can be used. The spectrum of monos that is measured can be used instead of the Poisson distribution. The zero-detection probability is unchanged because the major contribution (>99.99%) is due to the input distribution to the first dynode.

Figure 11 — Measured pulse-height distribution of ⁵⁵Fe [204] (courtesy North-Holland Publishing Company, Amsterdam).

Houtermans [204] used the measured mono-electron distribution to calculate the spectra expected and these are shown in Figure 10. The actual observed spectrum for ⁵⁵Fe given in Figure 11 shows the peaks from 1, 2, 3 and 4 monos as expected by the theory. The value of $\overline{r}$ in this figure is between 2 and 3 [204].

This distribution demonstrates the difficulty of defining an equivalent energy threshold. At higher energies the input will be a Gaussian distribution and the resolution into quantised peaks will not be observed. An example is also shown by Houtermans for ¹¹³Sn+ ¹¹³In^m Figure 12, with an input energy of about $380\mathrm{keV}$. The FWHM is 15.5%, which corresponds to a mean number of monos of 231. In the actual determination of $\overline{r}$, it is necessary to allow for the double peak at $368$ and $392\mathrm{keV}$ and this partially accounts for the difference from a value of $\overline{r}=264$ calculated from the peak position.

Figure 12 — Measured pulse-height distribution of ¹¹³Sn + ¹¹³In^m [204] (courtesy North-Holland Publishing Company, Amsterdam).

Thus it is possible to calculate the pulse-height distribution from a single phototube for the measured parameters in the system. However, in order to do this, the non-linear effects must be included and these are considered in Section 5.4.

5.2.3.  The pulse-height distribution for a coincidence system

Most modern systems are arranged so that the outputs of the two phototubes are added when a pulse is observed in both tubes (see for example Gibson and Lally [173]). Thus since both tubes are required to produce a pulse, then there will be the equivalent of at least two photoelectrons in each detected event. For an input of exactly $i$ electrons shared randomly between the two phototubes the output is reduced by a factor corresponding to the probability of all $i$ electrons being detected by one of the phototubes. The factor is $\left(1-2^{1-i}\right)$ as quoted by Swank [32], but it can be deduced simply from the binomial distribution.

Thus, in calculating the spectrum from a Poisson-distributed input to the first dynode Equation (10), this factor must be included, i.e.

As discussed by Jordan [178], $\overline{r}$ can be less than 2 or less than 1 and there is still a finite probability of detection with a coincidence system.

5.2.4.  The pulse-height distribution for a $\mathrm{\beta }$-ray spectrum

The distribution for a spectrum of electrons can be calculated by dividing it up into a sufficient number of discrete energy bands of mean energy, $E_j$, which, when multiplied by the appropriate figure of merit ($\eta$ electrons/keV, give a value of $\overline{r}$ such that

$\overline{r}$ can be used with Equation (9) or Equation (10) and weighted according to the proportion of the spectrum in the energy band $E_j-\Delta E/2$ to $E_j+\Delta E/2$. The weighting factors, $H\left(j\right)$, when integrated over the spectrum, are equal to unity. Thus the pulse-height distribution is obtained from

where $J$, the number of elements in the spectrum, is normally about twenty depending upon the degree of precision required. There is little point in using element widths, $\Delta E$, much less than the resolution of the system. It should be noted that $\eta$ is not independent of $E$ at low energies (see Section 5.4.1.

5.2.5.  Electron capture and internal conversion

X rays and electrons from electron-capturing (EC) nuclides and internal-conversion (IC) transitions can usefully be used to measure the figure of merit, $\eta$, for the system and as was shown in Figure 11, they also are used to demonstrate the validity of the theoretical calculation of the pulse-height distributions. However to calculate the complete distribution it is necessary to consider all possible modes of decay involving the production of cascades of Auger electrons and X rays (from the L, M, N, …​ shells) which may escape from the scintillator without interacting. In some cases, fifteen or more alternative modes of decay are possible and Figure 13 indicates the sort of analysis required for the calculation. IC transitions often do not occur singly either because of the ejection of both K and L electrons with appropriate probabilities or because two or more transitions produce electrons of similar energy. Consideration of these effects is important in assessing the peak position and vital in using the resolution of the peak to determine the figure of merit. This point was discussed above for the decay of ¹¹³Sn + ¹³¹In^m Figure 12.

Having decided upon the input spectrum, the same approach can be adopted as for $\mathrm{\beta }$-ray emitters and values of $H\left(j\right)$, based upon this assumed spectrum, inserted into Equation (13).

Figure 13 — Selected interactions for EC nuclides and a calculation of the efficiency for ⁵⁵Fe in a coincidence system at figures or merit of ${\eta }_0=1.0$ and $0.5$ photoelectron/keV [195] (courtesy Pergamon Press Ltd., Oxford).

5.2.6.  The Compton edge

It is necessary to measure the response of the system to a wide range of energies in order to determine the extent of the non-linear response of the scintillator. There are insufficient nuclides decaying by isomeric transitions, with the ejection of monoenergetic electrons, so that it is necessary to use Compton-recoil electrons of maximum energy (the Compton edge) from a series of external $\mathrm{\gamma }$-ray sources. The energy for the maximum transferrance of energy to the scintillator is given by

where $E_{\mathrm{\gamma }}$ equals the incident $\mathrm{\gamma }$-ray energy and $m_0{c}^2$ is the rest mass of the electron in appropriate energy units, such that $E_{\mathrm{\gamma }}/m_0{c}^2$ is dimensionless. The shape of the Compton spectrum is a function of the energy, and the appropriate function is given in various sources (e.g. [130]). This theoretical spectrum can also be divided up into energy bands, weighted by $H\left(j\right)$, and the output distribution calculated using Equation (13).

It is conventional to take the point corresponding to half the maximum height as the position of the Compton edge. This may lead to small errors in calibration but this should not affect the calculation of the overall counting efficiency.

The position of the Compton edge is also used by Horrocks [256] as a means of determining the degree of quenching in a liquid scintillator. His method is relative, but it is very useful for checking that a particular sample conforms to the operating parameters of the system.

5.3.1.  The counting efficiency for a single phototube

Starting with a monoenergetic source of electrons in the scintillator, the number of electrons at the input to the first dynode will follow a Poisson distribution of mean $\overline{r}$. The zero-detection probability, $Z$, is $e^{-\overline{r}}$ and the efficiency

With the RCA tube, there is no further reduction in efficiency in the phototube. The presence of the low-energy tail on the mono distribution has less than a 0.01% effect. Unfortunately no radionuclide decays with a single electron energy, so it is always necessary to determine the counting efficiency by means of an integral, or a summation. It is also necessary in the practical case to use a summation as discussed in Section 5.2.4, Equation (13). The efficiency ${\epsilon }_1\left(\mathrm{\beta }\right)$ for a single phototube is then given by

where $\eta$ is the figure of merit (photoelectrons/keV) and $E$ is the energy in $\mathrm{keV}$.

The spectral characteristics of most $\mathrm{\beta }$ emitters are known and calculation methods based upon the Fermi theory can be used [16] to evaluate $H\left(j\right)$. A similar approach can be made for EC nuclides where the contributions from the Auger electrons and X rays need to be included [195].

5.3.2.  The counting efficiency for two phototubes in coincidence

Again starting with a monoenergetic source of electrons and two phototubes in coincidence, if the zero-detection probability for phototube $A$ is $Z\left(a\right)$ and for photofube $B$ is $Z\left(b\right)$, then the efficiency is

The efficiency for counting a $\mathrm{\beta }$-ray emitter is then obtained by summation,

where $\eta \left(a\right)$ and $\eta \left(b\right)$ are the figures of merit for phototubes $A$ and $B$ respectively.

The counts from the two phototubes separately and then in coincidence can be expressed as three independent simultaneous equations

where $F\left(y\right)=\sum _{j=1}^{\mathrm{J}}H\left(j\right)\text{exp}\left[\eta \left(y\right)E\left(j\right)\right]$, with $y=a,b$ or $a+b$.

Thus with the measured values $N_1$, $N_2$ and $N_{\text{c}}$ and three unknowns, $a$, $b$ and $N_0$, Tissen [33] and, later, Kolarov et al. [157] suggested that $N_0$ could be determined. However it is necessary to know the form of the summation $F$ since for a $\mathrm{\beta }$-ray spectrum $F\left(a\right)\cdot F\left(b\right)\ne F\left(a+b\right)$.

In general, it will be better to compare the calculated and measured distributions before calculating the counting efficiency by one of the above methods.

5.4.1.  Ionization quenching

As mentioned in the introduction to this chapter, the energy transfer from the initial energy deposition in the scintillator to light output is very inefficient (1 to 5%). If the ionizing particle deposits energy in a very short track length (high $\text{d}E/\text{d}x$) as is observed for $\mathrm{\alpha }$ particles and low-energy electrons then, as Birks [69] has shown, there is a further reduction in the scintillation efficiency. Birks used the term “ionization quenching” to describe this loss and produced an equation which can be derived from the Stern-Volmer [1] theory of quenching, for a first-order effect, as the rate of light output per unit track length,

where $\text{d}E/\text{d}x$ is the specific energy loss (normally expressed in $\mathrm{MeV}$ ${\mathrm{cm}}^{-2}{\mathrm{g}}^{-1}$), $k$ is a rate constant and $B\text{d}E/\text{d}x$ is the concentration of ionizing events. Other workers have included a further term $C{\left(\text{d}E/\text{d}x\right)}^2$ in the denominator and Voltz [144] proposed an equation of the form

Equation (15) due to Birks [69] is to be preferred but the use of alternatives may be necessary if an appropriate fit to the observed data cannot be obtained. The meaning of the terms in Equation (15) will now be explained. The value of the constant $kB$ is about $9\mathrm{mg}{\mathrm{cm}}^{-2}{\mathrm{MeV}}^{-1}$ and thus for $\mathrm{\beta }$ particles and electrons with energies greater than about $1\mathrm{keV}$, where $\text{d}E/\text{d}x=100\mathrm{MeV}{\mathrm{cm}}^{-2}{\mathrm{g}}^{-1}$, the Equation (15) and Equation (16) give similar results ($53{\eta }_s$ and $40{\eta }_s$, respectively). Here ${\eta }_s$ is defined as the figure of merit for the scintillator when $\text{d}E/\text{d}x$ tends to zero (i.e. high $\mathrm{\beta }$-ray energies). If $kB\text{ d}E/\text{d}x$ is small compared to 1, then the light output from an electron of energy $E$ is

and ${\eta }_s=L/E$ is the light output per unit energy input. A second figure of merit for the complete system is ${\eta }_0=\lambda L/E$, where $\lambda$ is the photocathode efficiency. In the general case when $kB\text{ d}E/\text{d}x$ is not small compared to 1, then

where $Q\left(E\right)$ is obtained by numerical integration. Thus in general

The shape of $Q\left(E\right)$ is dependent upon the value chosen for $kB$ and an example with $kB=9\mathrm{mg}{\mathrm{cm}}^{-2}{\mathrm{MeV}}^{-1}$ is given in Figure 14, which was calculated using a modified form of the Bethe-Bloch theory for $\text{d}E/\text{d}x$ [22], with $\text{d}E/\text{d}x$ proportional to $E^{-0.5}$ below $400\mathrm{eV}$ [160].

Figure 14 — Variations of scintillation efficiency with electron energy due to ionization quenching and wall effects [133] (courtesy Institute of Physics, London).

It should be noted that for $\mathrm{\alpha }$ and other charged particles, when $kB\text{ d}E/\text{d}x>>1$, Equation (15) reduces to

and

and the light output is proportional to the range of the particle, not to its energy.

5.4.2.  Boundary losses at high electron energies

One major difficulty in determining ${\eta }_0$ at high electron energies is that a significant proportion of the energy may be lost into the walls of the scintillator cell and out of the surface of the scintillator. This will distort the spectrum and may give misleading results. Benjamin et al. [55] derived a formula for the fraction of energy deposited within a cylinder of radius $r$ and height $h$, such that

where $R\left(E\right)$ is the range of an electron of energy $E$ [22]. As far as possible the energy lost to the boundaries should be reduced to a minimum so that corrections to the observed spectrum are unnecessary. Normally this boundary loss is not sufficient to reduce the counting efficiency as enough energy will be deposited in the scintillator to produce a detectable output from the phototube and this effect is only important in setting up a system. Illustrative values of $W\left(E\right)$ are shown in Figure 14.

5.4.3.  Loss of photon radiation from the scintillator without interaction

This problem arises in three cases:

1. in the loss of X rays after electron capture,

2. where a nuclide decays by an isomeric transition without emitting a $\mathrm{\beta }$ particle or an electron and only $\mathrm{\gamma }$ rays are produced (e.g. ¹³⁷Ba^m) and

3. where the liquid scintillator is used as the $\mathrm{\beta }$-ray detector in the $4\mathrm{\pi }\mathrm{\beta }-\mathrm{\gamma }$ coincidence method (e.g. [88]).

Gibson and Marshall [195] calculated the probability of escape for a right circular cylinder of radius $a=10.5\mathrm{mm}$ and half length, $l=27.75\mathrm{mm}$, based upon the following integral for the total probability of escape

where the solid angle $\text{d}\Omega =\text{sin}\theta \text{d}\theta \text{d}\Psi$ with the integral of $\theta$ from $0$ to $\mathrm{\pi }$ and $\Psi$ from $0$ to $\mathrm{\pi }$ as well. The volume element $\text{d}V=x\text{d}x\text{d}z$ Figure 15 where the radial term, $x$, is integrated from $0$ to $a$ and the axial term, $z$, from $0$ to $1$, the half length of the scintillator. The divisor can be integrated analytically to give $\mathrm{\pi }{a}^{2}l\left(=V/2\right)$ and the numerator integral must be solved numerically using tabulated values of the mass attenuation coefficient, ${\mu }_{\text{a}}$, for different photon energies (e. g. [130]. The distance, $d$, from the point of origin of the X ray to the boundaries of the scintillator is complicated to calculate. Thus the escape is considered in three parts:

1. top of the cylinder,

   ${\mu }_{\text{a}}d={\mu }_{\text{a}}\left(l-z\right)\text{cos}\theta$,
   with integration limits for $\theta$ from $\theta =0$ to $\theta ={\theta }_1={\text{tan}}^{-1}\left\{\frac{-x\text{cos}\Psi \pm f}{1-z}\right\}$,

2. curved section,

   ${\mu }_{\text{a}}d=-{\mu }_{\text{a}}\left(x\text{cos}\Psi \pm f\right)/\text{sin}\theta$,
   with integration limits for $\theta$ from $\theta ={\theta }_1$ to $\theta ={\theta }_2={\text{tan}}^{-1}\left\{\frac{-x\text{cos}\Psi \pm f}{-1-z}\right\}$,

3. bottom of the cylinder,

   ${\mu }_{\text{a}}d=-{\mu }_{\text{a}}\left(1+z\right)/\text{cos}\theta$,
   with integration limits for $\theta$ from $\theta ={\theta }_2$ to $\theta =\mathrm{\pi }$.

Figure 15 — Geometry used in calculating the escape of photon radiation from the scintillator cell [195] (courtesy Pergamon Press Ltd., Oxford).

The limits for $\Psi$ are $0$ to $\mathrm{\pi }$, for $x$ from $0$ to $a$ and for $z$ from $0$ to $1$. The function $f$ above is given by

The value of $1-J\left(E\right)$ for a $15{\mathrm{cm}}^3$ volume with $a=10.5\mathrm{mm}$ and $l=21.75\mathrm{mm}$ is plotted as function of energy in Figure 16 and shows that this correction is important above about $2\mathrm{keV}$. The efficiency for counting Auger electrons is also shown in this figure for two values of the figure of merit, ${\eta }_0=0.5$ and 1.0 photoelectron/keV (${\eta }_0/2$ for each phototube).

Figure 16 — Counting efficiency for X rays and Auger electrons, and X-ray capture probabilities [195] (courtesy Pergamon Press Ltd., Oxford).

5.4.4.  Other effects

There are many minor effects which can distort the spectrum and hence lead to a loss of counting efficiency. Starting with the scintillator and vial, it is important to ensure complete solubility with no plating out on to the vial walls. Colour and chemical quenching distort the spectra in different ways. In the case of chemical quenching the effect is equivalent to a gain change such that

where $k$ is the rate constant in the Stern-Volmer [1] equation and $w$ is the concentration of the quenching agent per unit volume of the scintillator [113]. Thus the figure of merit, ${\eta }_0$, is reduced to $g{\eta }_0$. Colour quenching introduces a more subtle distortion because light produced close to the photocathode is not absorbed butt hat from deeper in the scintillator may be lost completely. It is better to reduce quenching to a minimum in direct counting, but checks must still be made to make sure this objective is achieved. The vials should be as uniform as possible so that the light escapes equally to both phototubes in a coincidence system. The vial should be well coupled optically to the phototubes to avoid internal reflections.

The photocathode may be non-uniform as shown by Stanley and Malcolm [277] and it may be necessary to limit the area used to avoid distortions in its response. The collection efficiency of the first dynode needs to be as good as possible and inhomogeneities in all dynodes should be minimised. Coates [149] has demonstrated that edge effects in the dynodes are the major source of spectral distortion with the RCA tube and must be allowed for in calculating the pulse-height distribution. Photoemission from the first dynode is unimportant with a phototube of the RCA type. Instrumental effects due to a poor high-frequency response in the electronics or to poor layout of phototube on other connectors giving rise to ringing or reflections must be avoided. Dead times in excess of $100\mathrm{ns}$ may be required to avoid this problem.

Afterpulses in the phototube can cause problems but this will be discussed in a later chapter. Equally, long decay times in the scintillator may produce similar effects and should be avoided. In general a careful survey of the system is required on setting up and at regular intervals to ensure that ageing effects do not change any of the parameters or introduce interfering effects.

6.2.1.  Experimental arrangement

1. The cell and collection of the light — The object is to collect the maximum of the light created by the scintillator. Various authors have used one or more electron-multiplier phototubes. In arrangements using a single phototube the cell, or vial, containing the scintillator is arranged vertically above the photocathode of the phototube with an optical joint of silicone grease and a reflecting or diffusing cover which directs the light towards the photocathode [13], [19], [23], [36], [46], [49], [95], [114]. Amongst other advantages, the arrangement with two coaxial phototubes allows better collection of the light. Many types of cells have been used with two phototubes. The authors who use standard vials generally improve the collection of the light by using a silicone bath or by a light duct with a silicone joint [36], [46], [49], [95], [114], [185]. Certain authors have used a sphere in a silicone oil bath [120]. Others have employed glass cylindrical cells, the axis of which coincides with that of the phototube [26], [123]; see also Section 4.4.

2. The electron-multiplier phototubes — Much progress has been made in the development of phototubes, particularly in the quantum efficiency which is nearly 0.32 photoelectron per photon at $385\mathrm{nm}$ for a bialkali photocathode. The photoelectron collection efficiency has also been improved by using highly sophisticated electron optics. Dynodes activated with gall ium phosphide (GaP), which are used in the first stage of certain recent phototubes (RCA 8850), have a very high gain which can reach 40, and this has the effect of greatly reducing fluctuations in low-amplitude pulses. Improved construction has also eliminated ion build-up on the photocathode, so reducing one of the principal sources of parasitic pulses¹⁹. The new photocathodes, with increased sensitivity, have a reduced dark current, and this means that it is no longer necessary to work at low temperatures to achieve low levels of noise.

6.2.2.  The scintillating solution

1. The solvent — The solvent most widely used in lS metrology is still toluene. For diluting inorganic salts in aqueous solution, as is the normal case when measuring standards, ethanol or methanol are most frequently used as the intermediate solvent. Dioxane is little used in metrology because of the chemiluminescence that it exhibits when it comes into contact with water, as has been pointed out by many authors, and also because of its lower luminous output. Furthermore the standards to be measured generally have such high radioactivity concentrations that it is possible to introduce only very small quantities of the radioactive solution into the scintillator (less than 1%). Few authors carry out systematic degassing of the solvent.

2. The scintillating solution — Most laboratories still use the PPO-POPOP system, even with the bialkaline photocathode. The reason for this is that the properties of this scintillator are perfectly known, and that it remains very effective even in the presence of quenching elements. Certain workers use PBD or t-butyl-PBD. When measuring toluene labelled with ¹⁴C or ³H it is advantageous to use BIBUQ [157]. Chemical formulae for these and other scintillators are given in Table 2.

3. Preparation of the sources — Two source-preparation procedures are possible. In certain laboratories a small aliquot of the active solution is placed in each vial to be used in the measurement. Other authors prefer to dissolve a larger aliquot in a large quantity of scintillator ($50$ to $100{\mathrm{cm}}^3$) and then to fill the measuring vials with the ready-prepared solution. Obviously, the measurement of the aliquots is carried out by weighing, usually by difference weighing of a polythene pycnometer or using a cal ibrated micropipette [284]. The first method involves the measurement of small masses, usually a few milligrams, the second procedure allows easier and more accurate weighing since the masses are greater, generally of the order of a few hundred milligrams. Furthermore, homogenisation of the active-solution — scintillator mixture is more easily achieved in a flask of larger dimensions than the measuring cell. We should note that the weighing of volatile solutions does present special difficulties [90].

6.2.3.  The counting electronics

At the present time charge-sensitive preamplifiers using field-effect transistors completely solve the problem of background noise in the amplification chain. Nevertheless care must be taken not to allow saturation from pile-up in the preamplifier before the first differentiation. In the same way transistor amplifiers are easily protected against overloads so that there is no difficulty in working at very high gains and high-overload levels in count-rate measurements, for example, of pure $\mathrm{\beta }$-particle emitters. Differentiating amplifiers with a single delay line with base-line restoration give particularly good performance in this field. The discrimination and dead-time circuits should be carefully chosen; for example, the dead time should be non-extending, in order to reduce the dead-time correction uncertainty. No comment needs to be made about scalers, except that provision be made for obtaining the data in a form that is convenient for subsequent computer processing. The electronic clock which controls the counting interval should, in general, have an accuracy of 1 in 10⁵. When it is necessary to carry out extrapolation as a function of the discriminator level it is possible, in the case of $\mathrm{\beta }$-particle emitters of reasonably high maximum energy, to repeat the measurements by varying the discriminator voltage setting each time [157], or by recording count rates at several levels of discrimination simultaneously [123], [246].

6.2.4.  Measurements using $\mathrm{\beta }-\mathrm{\gamma }$ coincidences

In the case where coincident $\mathrm{\gamma }$ rays are detected many arrangements of the Nal(Tl) crystal or crystals are possible [26], [49], [95], [114], [120], [123], [185]. In order to obtain a high $\mathrm{\gamma }$-ray efficiency it is possible to use Nal(Tl) crystals which completely end ose the sample vial and the $\mathrm{\beta }$-detecting phototube. The $\mathrm{\gamma }$-channel amplifiers are of the conventional type. The single-channel analyzer should be designed to give precise time information and to impose a well-defined dead time so as to make exact corrections possible.

6.3.1.  Measuring radionuclides which disintegrate by pure $\mathrm{\beta }$-particle emission

The $\mathrm{\beta }$-ray spectra show a non-zero emission probability density down to and including zero energy. Part of the spectrum is situated in the region of the background noise of the detector and the amplifier. It is therefore necessary to have recourse to extrapolation of the rate of integral counting to zero value of the energy. The shapes of the $\mathrm{\beta }$-ray spectra are very different, and the detection procedure considerably modifies them. For those of high maximum energy it is possible [19], [23], [36], [46], [49], [95], using adequate amplification, to obtain a flat region of the spectrum; this justifies linear extrapolation, although no theoretical reason can be advanced to explain this fact. Many workers have also observed a build-up of the differential spectrum towards the lower energies without being able to attribute this to the background noise of the amplifier or the detector background [8], [13], [19]. This phenomenon, attributed to multiple pulses arising from the detector or the amplifier, can be eliminated by a dead-time circuit preceding the counting scaler [8], [19].

Comparison of these results with those obtained by other methods shows, in the case of radionuclides of maximum energy lower than $200\mathrm{keV}$, a detection efficiency which is less than 100% [10], [29], [91] (¹⁴C, ³⁵S, ⁶³Ni, ¹⁰⁶Ru, ²⁴¹Pu and, in particular, ³H). This is attributed to statistical fluctuations in the number of photoelectrons emitted by the cathode of the phototube, and the non-vanishing probability that a low-energy event results in no photoelectron being emitted (see Chapter 5).

Basing his work on the hypothesis of the Poisson distribution of the electronic emission of the photocathode Wright [14] calculated the zero-detection probability (ZDP) In the case of monoenergetic ionising particles. Tissen [33] extended this calculation to continuous $\mathrm{\beta }$-ray spectra in the cases of detection by a single phototube, and by two phototubes in coincidence. He characterized the quality of the detection assembly — scintillator and photocathode — by the mean number of photoelectrons obtained for unit loss of energy of the primary particle in the scintillator, and showed that, approximating to the allowed form of the decay of radionuclides of low atomic number, and on the hypothesis of linear response of the scintillator, the efficiency of detection depends only on the quality parameter and the maximum energy of the continuous spectrum. Values calculated in this way for ³H and ¹⁴C are found to be in good agreement with current experimental results.

Horrocks and Studier [53] showed that, for a given spectrum, the ratio of the detection efficiencies with a single phototube and two in coincidence depends only on the quality parameter (figure of merit). They obtained excellent agreement between their experimental measurements and theoretical values calculated on the basis of an average of one photoelectron per $1.5\mathrm{keV}$ of energy dissipated in the scintillator.

The high ionization densities created by low-energy electrons reduce the luminous output of organic scintillators. This results in a non-linear response for energies less than about $100\mathrm{keV}$. Birks [3] has given a formula describing the phenomenon where the luminous output varies with the loss of specific energy of the particle. In their article Gibson and Gale [133] take this effect into account by the introduction of a term dependent on the energy, $Q\left(E\right)$, when calculating the detection efficiency (see Chapter 5. According to them the uncertainty in respect to the coefficients does not introduce an error of more than 3% in the value of $Q\left(E\right)$ for an energy of $5\mathrm{keV}$. The quality parameter measured on their installation is 0.45 photoelectron/keV. From this they deduce a detection efficiency with a single phototube of 62.9% for ³H and, by comparison with a Radiochemical Centre standard, they observe a difference of — 2.4%. With the phototubes in coincidence, they calculate an efficiency of 43% with a difference, in respect to an NBS standard, of +2.7%. For ¹⁴C the calculated values are 96.4% and 93.7%, respectively (1971 revised values) [171], with differences of +3.5% and +2.5% in respect of the standards.

These calculations assume that the quality parameter is known. In the method of Kolarov et al. [157] the quality parameter is determined at the time of the measurement itself by solving the system of non-linear equations which link the efficiency of detection in coincidence, summed and for each phototube separately, with the experimental count rates. In these equations the quality parameter occurs as an unknown in the same way as the activity. When applied to ³H this method gives differences of about 4% in excess when compared with an NBS standard (unpublished results).

The accuracy of all these methods obviously depends on the accuracy of the theories and of the values of the parameters occurring in them, since they all lead to an a priori calculation of the detection efficiencies. The principal sources of systematic error are due to the theory of the energy loss of the electrons in matter (Bethe), to the theory of ionization quenching (Birks, Voltz), to the theory of the form of spectra (Fermi), and to the calculation of the deformation of the spectra by wall effects.

Let us examine these various points:

1. The Bethe formula is not very accurate and is almost certainly inaccurate below $0.5\mathrm{keV}$ for organic solvents. This results in an error which is difficult to quantify but which is certainly less than 0.1% in the case of ¹⁴C. At the present time there is no accurate theory in this field of energy [76].

2. Two different theories [3], [76], [96], [133], [171] giving numerically very similar results make it possible to calculate the yield of organic scintillators [122], [174]. Gibson [133] has estimated that the uncertainty in respect of the coefficient in the Birks formula was likely to introduce an error of $\pm 2\%$ in the detection efficiency in the case of ³H, but less than 0.1% in the case of ¹⁴C. Experimental determination of the luminous response of organic scintillators [80] is inaccurate, particularly below $20\text{unitmsl(keV)}$. There are, in fact, few references to energies in this range and, furthermore, resolution of the peaks is poor.

3. A formula due to Benjamin [55] has been used up to the present time to take account of the wall effect. Actually, this only gives the proportion of particles which encounter the wall. It is however preferable to calculate precisely the deformed spectrum. The method of Snidov and Warren [121] can be used. This effect is considerable in the case of high-energy emitters, since it causes the appearance of a large number of small pulses for which the zero-detection probability is considerable.

4. The exact form of the $\mathrm{\beta }$-ray spectra themselves is not always well known. Only rarely does theory make it possible to forecast the nature of the transition, and few factors relating to the experimental shape have been published. Furthermore theoreticians have introduced new correction terms (radiative, polarization, etc.) [37], [116], [165], the practical calculation of which has been found to be difficult.

Methods not requiring a priori calculations — The guiding idea here is to make the non-detection probability increase and to carry out extrapolation for a zero non-detection probability. Several procedures are possible: inhibition of the luminescence of the scintillator, modification of the optical coupling by interposing optical absorbing filters [169], defocussing of the phototube [172]. It is enough to know simply the attenuation ratios, and this can easily be achieved by using an external $\mathrm{\gamma }$-ray source or, better, using an internal monoenergetic source (e.g. of conversion electrons or $\mathrm{\alpha }$ particles). The only source of systematic error is then the uncertainty of the extrapolation. In the communication already cited, Flynn and his co-authors carry out a linear extrapolation. They estimate their maximum error as $\pm 2\%$ for ³H. It should be possible to improve the method by increasing the number of points, and this is easily done by defocussing and by carrying out a non-linear extrapolation.

6.3.2.  Measurements on $\mathrm{\beta }-\mathrm{\gamma }$ emitters

In the method of activity measurement using $\mathrm{\beta }-\mathrm{\gamma }$ coincidences it is very advantageous if at least one of the detectors has a high detection efficiency. Detectors using liquid scintillators for the detection of $\mathrm{\beta }$ radiation satisfactorily meet this condition, but in addition to this the intimate mixing of the radionuclide and the scintillator means that the sensitivity is more or less constant throughout the whole volume of the source (if there is no adsorption at the boundaries), this property also being very favorable to the quality of measuring by coincidence.

The elementary theory of the method of $\mathrm{\beta }-\mathrm{\gamma }$ coincidences assumes that the detectors have zero detection efficiency for the type of radiation for which they are not intended.

This is only true as a first approximation, particularly for the $\mathrm{\beta }$ detector. Whilst for a proportional counter this sensitivity ${\epsilon }_{{\mathrm{\beta }}_{\mathrm{\gamma }}}$ is several parts in a thousand, it reaches several percent and varies with the level of filling of the cell in an LS counter. Under these conditions it is preferable to use an extrapolation method [256]. However with a liquid scintillator, which preserves the general form of the $\mathrm{\beta }$-ray spectrum reasonably well, one can easily make the efficiency of detection vary by simple modification of the counting threshold. For the same reasons the method is equally suitable for measuring the activities of emitters with a complex decay scheme [60], [97].

It has been shown that the method of extrapolation in coincidence involves an error if certain conditions are not satisfied [87], [125], [108], [138]. The systematic error arises from the difference in the forms of the spectra of the various $\mathrm{\beta }$ branches, from the lack of homogeneity of the source, and the escape of part of the radiation. Obviously this error becomes smaller as the detection efficiencies which are achieved become greater. With LS the detection efficiency is greater than 95% for $\mathrm{\beta }$ radiation of energy greater than about $5\mathrm{keV}$, and this is a very distinct advantage when compared with measurements with counters using a solid source. One way of detecting the systematic error in extrapolation is to repeat the measurement with different $\mathrm{\gamma }$-ray gates [108]. Spahr and Binkert [187] have recently reported results on measuring ¹³⁴Cs with different $\mathrm{\gamma }$-ray gates: ${\mathrm{\gamma }}_9$ ($\mathrm{1 365}\mathrm{keV}$) in coincidence with $\mathrm{\beta }$‘s of lower energy, ${\mathrm{\gamma }}_5$ ($796\mathrm{keV}$) in coincidence with $\mathrm{\beta }$‘s of high energy. They obtained in these cases extrapolated values 0.04% higher and 0.33% lower in relation to the value found with ${\mathrm{\gamma }}_4$ ($604.7\mathrm{keV}$), in coincidence with three $\mathrm{\beta }$ branches. The same authors have carried out a comparison of the results obtained with a proportional counter and with a liquid scintillator in the case of ⁶⁰Co. They found a deviation of 0.14% lower with the liquid scintillator, their statistical accuracy being 0.14% (the confidence interval is not given). For more details on ¹³⁴Cs measurements, the reader may consult Chapter 8 and [301].

It is necessary in all these very important questions to pursue, in a systematic manner, the theoretical and experimental study of the accuracy of the methods obtained by the extrapolation method.

6.3.3.  Measurement of pure $\mathrm{\beta }$-particle emitters with a $\mathrm{\beta }-\mathrm{\gamma }$ tracer

The method of $\mathrm{\beta }-\mathrm{\gamma }$ tracing previously used in the assay of pure $\mathrm{\beta }$-particle emitters with proportional counters [35], [43], [68] has subsequently been used with a LS detector [66]. This can, in fact, be regarded as a variant of the method for measuring $\mathrm{\beta }-\mathrm{\gamma }$ emitters with a complex decay scheme, and the general theory [97] can be fully applied. LS detectors allow good mixing of the two radionuclides and high homogeneity of the source. As a result they are not subject to the principal cause of systematic error in detectors with a solid source. Extrapolation can conveniently be carried out by variation of the counting threshold. The systematic error arising from the difference in the $\mathrm{\beta }$ spectra still, obviously, continues to exist. It can be detected and estimated by repeating the measurement with different $\mathrm{\beta }-\mathrm{\gamma }$ tracers. In this way Spahr and Binkert [187] have assayed ³⁵S using ⁶⁰Co, ¹³⁴Cs and ²⁰³Hg as tracers. Their results expressed as ratios to the result obtained with ¹³⁴Cs ($\mathrm{\gamma }$-ray gate unspecified) were

and

It can be seen that the difference between these ratios (0.50%) is higher than the random errors in the ratios. It is interesting to note that, in a measurement of ³⁵S by tracers with a proportional counter, Williams and Goodier [125] found a larger difference in the opposite direction.

The complementarity of these two methods can be clearly seen, and it is reasonable to assume that systematic comparisons would make it possible to detect and reduce the errors.

An accurate extrapolation could be made if it were known how the ratio of the counting efficiencies for the two radionuclides varied as the counting threshold is varied. A novel solution to this problem was developed by Bryant et al. [109], [156] in which an opto-mechanical device simulated the scintillations produced by $\mathrm{\beta }$ particles in the scintillator taking into account the theoretical $\mathrm{\beta }$ spectra and the variation of scintillator efficiency with $\mathrm{\beta }$-particle energy. Tests showed that this system accurately reproduced the shapes of known efficiency functions. A modification of this method in which the opto-mechanical device has been replaced by a computer simulation is routinely used for the standardization of pure $\mathrm{\beta }$-particle emitters [298].

6.3.4.  Measurement of radionuclides disintegrating by electron capture

In a liquid scintillator the Auger electrons are detected with high efficiency and good reproducibility [123]. X rays of low energy have a short path in the liquid scintiilator, and escape is not of great importance with the volumes of scintillator normally used (a few milliliters). Cascades of X and Auger events make it possible to collect all the energy of the vacancy created by a capture. All these reasons lead to high detection efficiency — 80% for ⁵⁴Mn — for radio-nuclides which disintegrate by electron capture.

Steyn [122] has applied the extrapolation method in coincidence to the measurement of the activity of emitters which disintegrate by electron capture followed by $\mathrm{\gamma }$ emission. In this way he measured ⁵¹Cr, ⁵⁴Mn, ⁶⁵Zn and ⁵⁷Co with small differences with respect to measurements made with a proportional counter, except for ⁵¹Cr where he observed a difference of -2.8%. The maximum efficiency obtained with ⁵⁷Co was ($89.1\pm 0.1$)% with the liquid scintillator, whilst it varied from 39% to 62% according to the sources when using a proportional counter. This high reproducibility of liquid-scintillator efficiency makes it possible to envisage the measurement of radionuclides which disintegrate by electron capture without $\mathrm{\gamma }$-ray emission, the detection efficiency being known by interpolation of the efficiencies obtained with other radionuclides (it must obviously be assumed that the quenching remains small). With radionuclides of high atomic number, that part of the efficiency resulting from the escape of part of the X radiation constitutes a geometrical effect which may involve a systematic error in the extrapolation by variation of the counting threshold [138].

One can therefore envisage the measurement of emitters disintegrating by pure electron capture by using, as the tracer, a radionuclide which disintegrates with electron capture followed by $\mathrm{\gamma }$-ray emission. If the atomic numbers of the two elements do not differ greatly, the method of extrapolation should not introduce an excessive error.

If the electron capture is in competition with positron emission the situation becomes more complex. A report dealing with this subject is by Steyn and de Jesus [107] who, however, used a plastic instead of a liquid scintillator.

6.3.5.  Measurement of $\mathrm{\alpha }$ emitters

The resolution of $\mathrm{\alpha }$-particle spectra obtained by LS systems is of the order of 20 to 30% (in terms of full width of the peak at half maximum divided by $\mathrm{\alpha }$-particle energy). This dispersion arises from the location of light emission in the scintillator, rather than from statistical fluctuations in the number of photoelectrons in the phototube. The boundary effect results in the formation of a “tail” between the peak and zero energy. This tail normally represents less than 1% of all the pulses. Therefore, under these conditions, together with low-noise electronics, the only source of error being that inherent in the extrapolation, it is possible to obtain high accuracy. Ihle [114] has shown that underlight-collecting conditions which had deliberately been made poor, the resultant error remains small ($\approx 0.1\%$).