On the reverse micelle effect in liquid scintillation counting

Applied Radiation and Isotopes 125 (2017) 94–107

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On the reverse micelle effect in liquid scintillation counting ⁎

Youcef Nedjadi , Jean-Pascal Laedermann, François Bochud, Claude Bailat

MARK

Institut de Radiophysique, Lausanne, Switzerland

A R T I C L E I N F O

A B S T R A C T

Keywords: Liquid scintillation Ultima Gold Reverse micelle Geant4-DNA Ionisation quenching 3 H 63 Ni 54 Mn 55 Fe TDCR

This work looks into the tracks of electrons in nanoemulsive scintillating media using the Monte Carlo Geant4DNA code which simulates event-by-event interactions of electrons in liquid water down to the eV, without resorting to the condensed history method. It demonstrates that the average number of micelles in which electrons deposit energy is quite large, increasing with their emission energy, decreasing with micelle size, and rising with micelle concentration. The probability of an electron ending its track in a micelle is found to be rather large and micelle size-dependent below 1 keV, and approximating the aqueous fraction at higher energies. Analyses of the Monte Carlo estimated energy depositions in the aqueous phase and in the scintillant tell of a micelle quenching effect, with the micelle size shaping the quenching at low energy and the micelle concentration governing it at higher energies. The micelle effect on the 3H and 63Ni beta spectra is discussed for a range of micelle sizes and concentrations. This paper also computes the ionisation quenching function using Birk's law whilst considering the full energy losses in the micelles bisecting the electron pathway, and not just that incurred in the primary micelle enclosing the decaying nuclide. The ionisation quenching function is then used to calculate the detection efficiencies for 3H, 63Ni, 54Mn and 55Fe. The effect of the micelle size is found to be small for beta emitters but significant for the electron capture nuclides. TDCR measurements of 63Ni samples covering 8 aqueous fractions are analysed with and without explicit treatment of the micelle effect. Activities in the two representations agree within 0.02%. The ratios of the corresponding figures of merit are found to coincide with the scintillant fractions.

1. Introduction

commercial scintillants and found them to be smaller than the then widely used 8 nm diameter (Bergeron, 2012). In addition, Bergeron studied the effects of changing the aqueous and acid contents on the micelle sizes. We also measured more recently the diameters of micelles in two scintillants, Ultima Gold and Ultima Gold AB, with the same technique. For Ultima Gold (UG), the micelle diameters were measured to be about 2 nm and slightly dependent of the aqueous fraction, while for Ultima Gold AB (UGAB) they were found to vary strongly with the aqueous fraction and range between 1 and 4 nm. These results agree only partly with the measurements of Bergeron for these two scintillants (Nedjadi et al., 2016). Knowing the size of micelles in a given sample for liquid scintillation analysis has been deemed important because the electrons, ejected by radioactive decays within a micelle, will lose part of their energy in the aqueous phase before reaching the organic scintillating medium, thus reducing the detection efficiency. Grau Carles initiated the work to account for this efficiency loss, using the results of NOREC Monte Carlo computations of the electron energy depositions inside nanospheres of liquid water (Grau Carles, 2006; Semenenko et al., 2003). He argued that the efficiency loss caused by the micelle size is significant only for

Sample preparation for liquid scintillation analysis often involves dissolving radioactive solutions into organic liquid scintillation cocktails. Since most radioisotopes are available in aqueous form, which is not miscible with the aromatic solvents of the cocktails, surfactants are used to increase the interface between the aqueous phase and the organic one. This stable dispersion of aqueous droplets – reverse micelles of nanometric size – in the continuous organic medium may be regarded as a nanoemulsion (Anton and Vandamme, 2011). The nanoemulsive nature of samples in liquid scintillation counting has been the focus of increased interest in recent years. Several works were published on the effects of the reverse micelle size on the liquid scintillation detection efficiency (Grau Carles, 2007a, 2007b; Kossert and Grau Carles, 2010; Bergeron and Laureano-Perez, 2014) and efforts were made to determine experimentally the sizes of reverse micelles in commonly used commercial liquid scintillation cocktails (Kaushik et al., 2006; Bergeron, 2012; Nedjadi et al., 2016). Using dynamic light scattering, Bergeron measured the sizes of the reverse micelles – henceforth denoted as micelles, for short – in eight

⁎

Corresponding author. E-mail address: [email protected] (Y. Nedjadi).

http://dx.doi.org/10.1016/j.apradiso.2017.04.020 Received 2 December 2016; Received in revised form 13 March 2017; Accepted 11 April 2017 Available online 12 April 2017 0969-8043/ © 2017 Elsevier Ltd. All rights reserved.

Applied Radiation and Isotopes 125 (2017) 94–107

Y. Nedjadi et al.

This code was used to study the number of micelles crossed as well as the energy depositions in the micelles and in the scintillants, in the case of two cocktails, UG and UGAB, for a range of aqueous fractions used in liquid scintillation radionuclide metrology and for a range of micelle sizes. The code was also used to examine the micelle effects on the beta spectra of tritium and 63Ni. In addition, the electron energy track records into and out of the micelles and the scintillating medium produced by the code were utilised to calculate ionisation quenching functions which take into account all electron energy losses to the aqueous phase. The micelle effect on the detection efficiency in the case of 3H, 63Ni, 54Mn and 55Fe were also estimated. TDCR liquid scintillation measurements of a 63Ni solution were performed as well to probe the micelle effect on the activity concentrations. Analyses of these measurements provide interesting insights into how one may incorporate the aqueous fraction in the traditional efficiency calculation models in liquid scintillation counting. This work and its main results are described in the remainder of this paper.

electrons in the 0.1–1 keV energy range, as ionisation quenching renders the effect undetectable for energies below 0.1 keV, while electrons with energies higher than 1 keV ‘do not deposit appreciable amount of energy in targets of liquid water of a few nanometers’ (Grau Carles, 2007a, 2007b). Thus the prescription he used for accounting for the micelle size effect in the detection efficiency involves correcting the energy available for scintillation, or reduced energy,

EQ(E) =

∫0

E

dE′ 1 + kB⋅

dE ′ dx

(1)

by replacing E with E–ΔE, where ΔE is the energy deposited within the micelle housing the radioactive decay. ΔE is evaluated as a function of the initial energy of the emitted electron with the NOREC electron track Monte Carlo simulation code (Grau Carles, 2006). The same approach is used in the stochastic models for computing the efficiency (Grau Carles, 2007a, 2007b; Kossert and Grau Carles, 2010). One may call this approach the ‘first-micelle correction model’ because it accounts only for the electron energy deposited in the original micelle, and it overlooks any subsequent energy depositions in other micelles bisecting the electron pathway. The starting point of this paper is to question the representation of reality underlying this model. It runs counter to the physical intuition that electrons traverse more micelles than the one from which they originate. Consider a 20 mL vial, filled with 15 mL of UG into which 45 μL (typically 3 pycnometer drops) of aqueous radioactive solution are deposited. The aqueous fraction, defined here as the ratio between the aqueous sample volume and the scintillant volume, is then 0.3%. Assuming the micelle size distribution is single peaked at 2 nm means the micelle concentration is about 7.2 1017 micelles per mL. If we now suppose the micelles are spread uniformly in the organic phase, and that they sit on vertices of elementary cubic cells, then the average intermicellar distance is about 11 nm. Given that the range of say a 10 keV electron is about 2500 nm, i.e. around 250 micelle separations, it is unlikely that such an electron would evade crossing micelles on its track after it leaves the first one. Furthermore, close to the end of its pathway, the energy of the electron drops significantly, below the keV say, and the electron may then traverse micelles or even end its course in one. Such energy depositions are not accounted for by the firstmicelle correction model. Another issue with this model pertains to a basic geometric requirement. Since the aqueous phase is dispersed uniformly in the organic phase, and both media have close densities, one would expect the partitioning of the electron energy between the two phases to reflect on average and to some extent their respective fractions. This should be the case for sufficiently energetic electrons. But in the first-micelle correction model all the energy goes to the organic phase except the bit deposited in the first micelle, regardless of the aqueous fraction involved. The omission or here incomplete treatment of the electron energy losses in the aqueous phase in the liquid scintillation detection efficiency models seems all the more puzzling because in plastic scintillation with microspheres – where the water-scintillant configuration is reversed – the models require a full account of the energy dissipations in the aqueous phase to obtain meaningful results. (Sanz and Kossert, 2011; Tarancón et al., 2015). These issues were the motivations to explore quantitatively the reverse micelle effect in liquid scintillation counting using a Monte Carlo simulation of the electron interactions at the nanoscale level. Among the several codes that are available (PARTRAC, NOREC, TRIOL, PITS etc.), we chose the open-source, general purpose Geant4-DNA Monte Carlo simulation toolkit to study the electron tracks structures at the molecular level. This track structure code simulates all the elementary interactions on an event-by-event basis, with no use of condensed history approximations.

2. Method The Geant4 toolkit (version 4.10.0.1) with the Geant4-DNA lowenergy extension was used for this study. This code – indicated as G4DNA hereafter – simulates explicitly the discrete interactions of particles with liquid water without resorting to the condensed history method (Chauvie et al., 2006; Incerti et al., 2010a, 2010b; Bernal et al., 2015). For electrons, the interactions include elastic scattering, ionisation, electronic excitation, vibrational excitations and molecular attachment (Ivanchenko et al., 2012). Electrons are transported down to 7.4 eV, and below this threshold their kinetic energy is deposited locally. For the validation of G4DNA physical processes see Incerti et al. (2010a, 2010b, 2014), Francis et al. (2011), Champion et al. (2014). The micelles were modelled as nanospheres consisting of just liquid water, neglecting the surfactant molecules which interface the aqueous and organic phases. Furthermore, no distinction is made between the bulk-like water molecules in the core of the micelle and those trapped interfacing with the surfactant molecules (Fayer and Levinger, 2010; Fayer, 2011). The micelles were also assumed to be monodisperse, with their diameter a parameter in the simulations. Assuming a uniform nanoemulsion, each nanodroplet sits within a 3D cubic simulation box filled with liquid scintillation cocktail (UG or UGAB). Densities of 0.98 and 0.96 g/cm3 are taken for the waterequivalent UG and UGAB media respectively. The size of this elementary box is determined by the selection of the aqueous fraction and the choice of the micelle size. Simulations were performed with the micelles located randomly within the cubic cells as well as with the micelles in the centres of the lattice, but no significant differences between the two geometries were found. The results reported here are for nanospheres randomly positioned within the simulation box. Although micelles are in a state of Brownian motion, with some of them redistributing their contents through fusions and/or fissions, a static perspective is assumed here, with no micelle collisions and a single size distribution. Each simulation starts with the water nanosphere randomly placed within a cubic cell. The electron begins its pathway from a random position around the centre of the micelle, and then it is tracked until it reaches the boundary of the box. Periodical boundary conditions are applied in the 3 directions as the box may multiply in any of them. Electron kinematic continuity is enforced at cell boundaries. Tallies are made for the electron energies deposited within micelles and outside them, for the number of micelles crossed, and for the sequences of micelle entrance and exit energies. Most simulations were carried out for 143 initial electron energies between a few eV and 500 keV. Around 106 events were simulated for each of these energies. The computations for the micelle effect on the 3H and 63Ni spectra were performed with 95

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103 beta emission energies between zero and the maximum; 5×107 decay events were simulated for each disintegration energy. For internal consistency checks, two tracking methods were used. In the first, which emphasises the cell, the primary electron and all the secondary ones are tracked in turn, until they either stop within the cell or reach its boundaries. They are stacked regardless of their degree of branching, and the tracking proceeds till the stack is empty. In the alternative method, the focus is on the degree of branching, as the tracking proceeds with the primary electron from its original micelle in the first cube, through the cells, until it stops, then with secondary, tertiary etc. electrons, and so on. The observables (energy depositions, numbers of traversed micelles etc.) computed by the two methods agree well within the statistical uncertainties. These calculations were performed over several months on three computer clusters with 24, 96 and 192 cores.

The penetrating electron encounters randomly distributed micelles within its range, and passes through a number of them before it ends its course in the scintillating medium or in a micelle. Fig. 1a pictures how the average number of micelles traversed by the electron varies with its emission energy expressed in keV units. The number reported is a tally of the micelles bisected without energy loss as well as those in which the electron deposits some energy. It is given for UG, faq=10%, and the micelle diameters listed in Table A1. The electron clearly passes through a large number of micelles. For instance, for Ømic=2 nm which is close to the UG empirical value, a 0.5 keV electron tracks through ~25 micelles, while it crosses ~50 nanodroplets at 1 keV, ~630 micelles at 10 keV, and ~14800 nanospheres at 100 keV. This number scales as a power law, with an exponent of 1.43, with respect to the electron energy (E≥1 keV), whereas it decreases with increasing micelle diameter, reflecting obviously the same trend for the micelle concentration. The same average number is now reported, in Fig. 1b, when the micelle diameter is kept constant at 2 nm while faq is varied between 0.1% and 15%. For an aqueous fraction of 0.5%, which corresponds approximately to 3–4 aqueous pycnometer drops blended with 15 mL of UG, the electron traverses ~3 micelles at 0.5 keV, ~4 at 1 keV, ~36 at 10 keV, ~810 at 100 keV and ~8300 at 500 keV. The same power law in terms of the electron energy obtains. The number of traversed micelles increases with aqueous fraction as does the micelle concentration. What matters for the micelle effect on the detection efficiency is not these total numbers, but the number of micelles in which the electron actually deposits some of its energy. This number is shown in Fig. 2a as a function of the electron energy for UG, faq=10%, and the nanodroplet sizes given in Table A1. The number of energy dissipating micelles is large, though one order of magnitude smaller than the overall number of traversed micelles. This reflects the dominance of the elastic crosssection at low-energies and its considerable magnitude at higher energies (Incerti et al., 2014). For example, in the case of Ømic=2 nm, an electron of 0.5 keV deposits energy in ~9 micelles, while it dissipates some of it in ~18 nanospheres at 1 keV, in ~180 micelles at 10 keV, and in ~1820 micelles at 100 keV. This number increases linearly with energy for E≥1 keV. It gets smaller in emulsions with

3. Results and discussion A first set of computations were carried out for the micelle sizes and aqueous fractions of UG and UGAB listed in Table 1 of Nedjadi et al. (2016). These are typical sample conditions encountered in liquid scintillation counting metrology. An analysis of the results showed subtle interplays between the effect of the micelle size and that of the aqueous fraction. This work will not be reported in this paper. In order to disentangle these two effects, we made a second set of simulations for the twenty micelle sizes and aqueous fractions listed in Table A1. 15 mL scintillant volume is assumed in all cases. Ten conditions are for a fixed aqueous fraction and varying water droplet diameters (Ømic), and the rest have the micelle size unchanging while the aqueous fraction (faq) is variable. Some of the diameters or fractions shown in the table may not refer to realistic samples but they help to clarify the underlying physics. Results reported below refer only to simulations performed with the parameters of Table A1. 3.1. Number of crossed micelles Once an electron escapes the micelle enclosing the nucleus which emitted it, it follows a stochastic path inside the scintillating emulsion.

Number of crossed micelles

10

10

10

10

10

Ømic Ømic Ømic Ømic Ømic Ømic Ømic Ømic Ømic Ømic

5

4

3

= 0.5 nm = 1.0 nm = 2.0 nm = 3.0 nm = 4.0 nm = 5.0 nm = 6.0 nm = 7.0 nm = 8.0 nm = 10.0 nm

10

faq = 0.1 % faq = 0.5 % faq = 1.0 % faq = 2.0 % faq = 4.0 % faq = 6.0 % faq = 8.0 % faq = 10.0 % faq= 12.0 % faq = 15.0 %

10

10

2

10

4

3

2

1

10

1

b)

a) 10

5

0

0.01

0.1

1

10

100

0.01

Energy (keV)

0.1

1

10

100

Energy (keV)

Fig. 1. Variation of the number of micelles crossed with electron emission energy (in keV units) for various a) micelle sizes (faq=10%), and b) aqueous fractions (Ømic=2 nm). Data is for UG.

96

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Number of effective micelles

10

10

10

10

10

4

10

Ømic Ømic Ømic Ømic Ømic Ømic Ømic Ømic Ømic Ømic

3

2

= 0.5 nm = 1.0 nm = 2.0 nm = 3.0 nm = 4.0 nm = 5.0 nm = 6.0 nm = 7.0 nm = 8.0 nm = 10.0 nm

faq = 0.1 % faq = 0.5 % faq = 1.0 % faq = 2.0 % faq = 4.0 % faq = 6.0 % faq = 8.0 % faq = 10.0 % faq = 12.0 % faq = 15.0 %

10

10

1

10

a)

0

0.01

0.1

1

10

b)

100

0.01

0.1

1

10

10

4

3

2

1

0

100

Energy (keV)

Energy (keV)

Fig. 2. Variation of the number of energy dissipating micelles with electron emission energy for various a) micelle sizes (faq=10%), and b) aqueous fractions (Ømic=2 nm). Data is for UG.

not more significant than the energy losses in subsequent micelles. In fact, the maximum deposition in the second micelle is larger, and the maxima of the subsequent depositions drop gradually. This may be due to the fact that the electron path in the second micelle is on average larger than in the first one because in the latter it starts randomly around its centre. Note that as the ordinal number of the traversed micelle increases, the spectrum of the energy depositions shifts to higher energies as electrons need larger energies to reach high ordinal number micelles. Summing the energies the electron deposits in the successive micelles yields the total energy loss. The average of this quantity is now represented in Fig. 6a as a function of the electron emission energy. For electrons ejected with less than 1 keV, the total average electron energy dissipated in micelles rises with emission energy, but above this threshold it grows proportionately with the emission energy. Notice how the micelle size shapes the low energy behaviour but is not relevant at high energy. Fig. 6b displays the same quantity for a fixed micelle size (2 nm) and a range of aqueous fractions. In this case one observes that this quantity is not sensitive to the aqueous fraction below ~0.5 keV but varies significantly with it at higher energy. Otherwise stated, the total energy deposited by the electron in the aqueous phase increases with its emission energy, the magnitude of this increment proceeding from the micelle size at low energy and from the micelle concentration at high energy. The total average energy dissipated in micelles can be represented more revealingly as a percentage of the electron emission energy. This is shown in Fig. 7 for different micelle sizes and aqueous fractions. They illustrate well the statement that the micelle size regulates the energy loss at low energy while the micelle concentration determines it at high energy. Note that, at high energy, the energy ratio dissipated in micelles tends to the value of the aqueous fraction, construed as the ratio between the aqueous volume and the total sample volume. This observation has important consequences which are discussed in Section 3.7.

larger micelle sizes, but at quite low energy (E≤0.5 keV) this number does not depend much on Ømic. Fig. 2b now displays the same average quantity for a micelle diameter fixed at 2 nm and an aqueous fraction incremented from 0.1% to 15%. In the case of the 0.5% aqueous fraction, the electron loses energy in about one micelle between 0.5 and 1 keV, and in ~10 nanospheres at 10 keV, ~100 at 100 keV, and ~500 at 500 keV. A quasi-linear law relates the number of energy dissipating micelles with the electron energy above 1 keV, and it clearly grows with the aqueous fraction. Because the simulations record the sequences of micelle entrance and exit energies, it is also possible to compute the frequentist probabilities of electrons ending their track in a micelle. These are shown in percentages in Fig. 3a for a fixed aqueous fraction and a range of micelle diameters. This probability is quite large and micelle sizedependent below 500 eV, and it levels out at the value of the aqueous fraction above that energy. Fig. 3b reports the same quantity for a fixed micelle diameter and various aqueous fractions. The aqueous fraction clearly determines this probability at sufficiently high energy, as should be expected from the ratio of the compounds in the composite medium it traverses. 3.2. Energy deposited in micelles We now turn to quantifying the electron energy depositions. Fig. 4 displays the average energy deposited in the original micelle enclosing the ejected electron as a function of the emission energy, for three micelle diameters. These energy losses peak below 1 keV and increase with micelle size. The figure also compares the energy depositions in the first micelle computed with G4DNA with those predicted by NOREC for three micelle diameters (Kossert and Grau Carles, 2010). The Monte Carlo codes predict the same profiles but G4DNA energy depositions are larger below 1 keV. Above, they agree within a few percent. Since the simulations tally the micelle ingress and egress energies, it is possible to calculate the average energy depositions in any of the N micelles an electron may pass through. For example, Fig. 5 shows the average electron energy losses in the initial ten successive micelles as a function of the electron emission energy. The energy dissipation in the first micelle accommodating the ejected electron is important, but it is

3.3. Energy depositions in the scintillant Using the tallies of the energy depositions in the scintillant, one can estimate the total average energy deposited in it as a ratio of the 97

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Probability of ending track in a micelle (%)

70

25

Ømic Ømic Ømic Ømic Ømic Ømic Ømic Ømic Ømic Ømic

60

50

40

faq = faq = faq = faq = faq =

= 0.5 nm = 1.0 nm = 2.0 nm = 3.0 nm = 4.0 nm = 5.0 nm = 6.0 nm = 7.0 nm = 8.0 nm = 10.0 nm

0.1 % 0.5 % 1.0 % 2.0 % 4.0 %

faq = 6.0 % faq = 8.0 % faq = 10.0 % faq = 12.0 % faq = 15.0 %

20

b) 15

10

30

a) 20 5

10 0 0.01

0.1

1

10

100

0.01

0.1

Energy (keV)

1

10

100

Energy (keV)

Fig. 3. Probability of an electron finishing its track inside a micelle for a range of a) micelle sizes (faq=10%), and b) aqueous fractions (Ømic=2 nm). Data is for UG. The abscissa is the electron emission energy.

Energy deposited in first micelle (eV)

70

NOREC

60

50

Ø = 2 nm Ø = 4 nm Ø = 8 nm

40

G4DNA

30

20

10

0 0.01

0.1

1

10

100

Energy (keV) Fig. 4. Comparison between G4DNA and NOREC predictions for the electron energy deposition in the first micelle. NOREC results are shown with markers while G4DNA depositions are pictured with lines. The blue, green and red colours refer to 2 nm, 4 nm and 8 nm micelles respectively. The x-axis refers to the electron emission energy. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

Fig. 5. Electron energy deposition in successive micelles in the case of UG, Ømic=3 nm and faq=10%, as a function of the electron emission energy.

3.4. 3H and

63

Ni beta spectra

Models for the calculation of the detection efficiency in liquid scintillation counting involve the beta emission spectra. But the electrons emitted in the radioactive decays may lose a sizeable portion of their energy in the aqueous phase as they emerge from the primary micelle and then track through a number of micelles interspersed in the scintillating medium. It would therefore be interesting to compare the beta emission spectrum with the energy distribution of the electrons in the scintillant. The G4DNA Monte Carlo calculations discussed above were performed with mono-energetic electrons. In order to quantify the previously described effects in a situation closer to liquid scintillation counting, 3H and 63Ni beta emission spectra were used to generate the electrons in the simulation. In each case the emission spectrum, ranging between zero and the maximum beta energy, was discretized into 1000 bins. First, Fig. 8a shows the average number of micelles in which the

electron emission energy. Appendix A displays this quantity for a range of micelle diameters and for different aqueous fractions. Here the energy ratio deposited in the scintillator approaches the scintillant fraction (fLSC), defined as the ratio between the organic volume and the total sample volume, at high energy. The departure of this ratio from one clearly shows that not all the electron energy is available for the scintillation process as a significant portion of it is reduced in the numerous micelles it crosses. This reduction may be described as a micelle quenching effect, with the micelle size shaping the quenching at low energy and the micelle concentration governing it at higher energies.

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100

Total energy lost in micelles (keV)

100

Ømi Ømi Ømi Ømi Ømi Ømi Ømi Ømi Ømi Ømi

10

1

= 0.5 nm = 1.0 nm = 2.0 nm = 3.0 nm = 4.0 nm = 5.0 nm = 6.0 nm = 7.0 nm = 8.0 nm = 10.0 nm

faq = 0.1 % faq = 0.5 % faq = 1.0 % faq = 2.0 % faq = 4.0 % faq = 6.0 % faq = 8.0 % faq = 10.0 % faq = 12.0 % faq = 15.0 %

10

1

0.1 0.1

0.01 0.01

a)

b)

0.001

0.001 0.01

0.1

1

10

100

0.01

0.1

1

10

100

Energy (keV)

Energy (keV)

Fig. 6. Variation of the total energy deposition in micelles with electron emission energy for varying a) micelle sizes (faq=10%), and b) aqueous fractions (Ømic=2 nm). Data is for UG.

beta particles from 3H and 63Ni deposit energy for a range of micelle diameters, while Fig. 8b reports the same numbers for different aqueous fractions. Clearly, the electrons ejected from these two nuclides lose energy in tens of micelles. This number decreases with swelling micelle size and increases with larger aqueous fractions. The energy distribution of the 3H electrons in UG is shown in Fig. 9a for different aqueous fractions. The corresponding spectrum for 63Ni is given in Fig. 9b. For both nuclides, the spectra shift distinctly towards lower energies as the aqueous fraction or micelle concentration increases. The mean energies of the distributions provided in Table

B1 underscore this observation. These results thus agree with the experimental fact that increasing aqueous quenching displaces energy spectra towards lower energies. Fig. 9c and 9d represent now the reduced energy spectra of 3H and 63 Ni in UG, respectively, for various micelle diameters and a fixed aqueous fraction. A log scale is used for the energy to zoom on the low energy region. The micelle size does not matter visibly above 1 keV but it does affect appreciably the spectrum below that. The larger the micelle size the more the spectrum gets reduced at low energies. Inspecting the averages and higher moments of the energy distributions

Ratio of total energy lost in micelles (%)

80

30

a)

Ømic Ømic Ømic Ømic Ømic Ømic Ømic Ømic Ømic Ømic

60

b)

= 0.5 nm = 1.0 nm = 2.0 nm = 3.0 nm = 4.0 nm = 5.0 nm = 6.0 nm = 7.0 nm = 8.0 nm = 10.0 nm

faq = 0.1 % faq = 0.5 % faq = 1.0 % 25 faq = 2.0 % faq = 4.0 % faq = 6.0 % faq = 8.0 % 20 faq = 10.0 % faq = 12.0 % faq = 15.0 %

40

15

10 20 5

0 0.01

0 0.1

1

10

100

0.01

Energy (keV)

0.1

1

10

100

Energy (keV)

Fig. 7. Ratio of the total energy deposited in micelles against the electron disintegration energy for a range of a) micelle sizes (faq=10%), and b) aqueous fractions (Ømic=2 nm). Data is for UG.

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Number of energy dissipating micelles

5

5

3

4

H

63 3

3

4

H

63

Ni

3 2

Ni

100 7 6 5

2

4 3 2

a)

100

b)

9

10

8

7 6 5

7 6

4 3

5

2

4

1

3

0

2

4

6

8

10 0

Fig. 8. Average numbers of micelles in which beta particles from H and Ømic=2 nm).

63

4

6

8

10

12

14

Aqueous fraction (%)

Micelle diameter (nm) 3

2

Ni deposit energy, for a range of a) micelle diameters (UG, and faq=10%) and b) aqueous fractions (UG and

lating the effect of the micelle size on the spectrum would require a liquid scintillation counter with very high energy resolution below 1 keV, but no such counter is available.

does not reveal marked trends with the micelle size. This reflects the opposing effects the increase in micelle size (at constant aqueous fraction) induces in the detection efficiency. Larger diameters means increased energy depositions in the primary and subsequent aqueous nanospheres traversed, but, on the other hand, it implies a smaller micelle concentration and hence longer electron pathways in the scintillant before encountering the next micelle. Experimentally, iso-

3.5. Ionisation quenching function In organic scintillation at low electron energies, there is a fluores-

Fig. 9. Normalised 3H (a) and 63Ni (b) reduced spectra for different aqueous fractions (Ømic=2 nm). Tritium (c) and nickel (d) spectra for different micelle sizes (faq=10%); the energy is in log-scale to highlight the low energy region. The x-axis refers to the electron emission energy.

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where E is the energy, kB is a parameter and dE/dx is the electron stopping power of the liquid scintillator. The actual reduced energy available for scintillation, when all electron depositions in micelles are taken into account, is therefore:

cence loss because primary molecular excitations are quenched by the high density of ionised and excited molecules (Birks, 1964). G4DNA does not have yet a library of cross-sections (elastic scattering, ionisation, electronic and vibrational excitation, dissociation etc.) for the solvent and fluor molecules in commercial liquid scintillants which would be required to simulate the scintillation and ionisation quenching processes at the molecular level. We therefore make do here with a realistic implementation of Birk's semi-empirical law for the ionisation quenching function. As explained in the introduction, the first-micelle correction model involves altering Birk's law through replacing the emission energy E with E–ΔE, where ΔE is the energy lost in the micelle housing the decay. In other words, the integration is carried out only when the electron is in the organic scintillating medium, the implicit premise being that the bit of energy dissipated in the aqueous micelle does not produce light. We retain here the latter simplifying premise, thereby explicitly assuming that none of the energy deposited in the aqueous matter may produce light, either directly through a Cerenkov process for sufficiently energetic electrons (E≥265 keV), or indirectly by some supposed process propagating and coupling it to the fluor molecules. Relaxing this simplification would not alter the main conclusions discussed below, since one can include Čerenkov and/or coupling coefficients to the quench function (Grau Carles and Grau Malonda, 2001a, 2001b; Kossert, 2010), but this intricacy is beyond the scope of this paper. Now since the electron deposits energy in a larger number of micelles, the adequate correction of the ionisation quenching function should account for the full energy losses in the aqueous phase bisecting the electron pathway. Let f(E) be the integrand of the ionisation quenching function, i.e.

EQ(E) =

+

dE

1 + k B⋅ dx

E1out

f(E′)dE′ +

2in

∫0

E Nout

E 2out

f(E′)dE′+…+

3in

∫E

E j−1out

f(E′)dE′+…

jin

f(E′)dE′

(3)

(4)

Q(E) < Q(E),

i.e. correcting the ionisation quenching function for the micelle effect makes it smaller, because as we saw in Section 3.3 the energy available for scintillation is smaller. G4DNA makes it possible to compute Q(E) since the Monte Carlo simulation generates as many {E1out, E2in, E2out, E3in, E3out, …} tallies of the ingress and egress energies as random samplings used. One can thus compute an average corrected ionisation quenching function Q(E) which accounts for the micelle effect. The numerical integrations in Eq. (3) are carried out with the Romberg algorithm in double precision. Fig. 10a displays the ionisation quenching functions Q(E) taking into account the micelle effect, using estar stopping powers (Berger, 1992) and kB=0.012 cm/Mev, for various micelle sizes. They were

(2)

c)

b)

0.02

Ionisation quenching function

0.00 -0.02

Residuals

1.0

a)

∫E

where {E1out, E2out, …, ENout} are the electron exit energies from the first, second, …, Nth micelles respectively, while {E2in, E3in, etc.} are the corresponding entrance energies. Of course the last term integral in this equation refers to a case where the electron ends its pathway in the scintillant; if it were to terminate its track within a micelle, then its boundaries would be ENin and EN-1out. The integration is carried out only when the electron is in the organic scintillating medium as Birk's law applies only when the electron is in the scintillating medium. One can clearly write

1

f(E) =

∫E

0.8 1.0

0.8 0.6

0.4

0.2

0.01

0.6

0.5 nm 1.0 nm 2.0 nm 3.0 nm 4.0 nm 5.0 nm 6.0 nm 7.0 nm 8.0 nm 10.0 nm UG, no H2O 0.1

1

10

100

0.1 % 0.5 % 1.0 % 2.0 % 4.0 % 6.0 % 8.0 % 10.0 % 12.0 % 15.0 %

0.4

Q(E) Q(E) S(E) Q(E) • S(E)

UG, no H2O 0.01

0.1

1

10

100

0.01

0.1

1

10

0.2

100

Energy (keV) Fig. 10. Ionisation quenching functions taking into account the micelle effect for various a) micelle sizes (UG, faq=10%) and b) aqueous fractions (UG, Ømic=2 nm). Plot c) shows a comparison between Q(E)·S(E) (triangle) and Q(E) (red dot-dashed line), for UGAB, faq=6.5%, Ømic=4 nm and kB=0.012 cm/MeV. The abscissa is the electron emission energy. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

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For comparison purposes, the efficiencies are constrained here to the same λ and kB values, even though in realistic activity calculations they are phenomenological parameters determined by minimising the deviations between the theoretical and experimental TDCR values whilst requiring the inferred activities to remain independent of the efficiency variation used to alter the TDCRs (Broda et al., 2007). This exercise serves to quantify the consequence of the ionisation quenching function change on the efficiency ceteris paribus. Our calculations of the tritium double detection efficiencies for a range of micelle diameters and a fixed aqueous fraction show that increasing the micelle size does not alter significantly the detection efficiency. The same observation holds for the triple detection efficiency, and hence for their ratio, the TDCR. The 63Ni efficiencies present similar trends, but the case of 55Fe displays different features. Here the micelle size does have a significant impact on the efficiencies. Fig. 11a reports the relative differences between the 55Fe double detection efficiencies for different micelle sizes and that for Ømic=0.5 nm, with faq=10%. We evaluate this quantity, rather than differences with the efficiency when no micelles are present, in order to offset the effect of the aqueous fraction which is much larger as will be shown below. Clearly the micelle diameter does matter. The double detection efficiency decreases appreciably with increasing diameter. The same conclusion holds for the triple detection efficiency. Fig. 11c displays the relative differences between the triple detection efficiencies for Ømic=3 nm and 0.5 nm for the four nuclides of interest. For λ=2, a 2.5 nm micelle size effect reduces the 3H and 63Ni triple detection efficiencies by less than 0.2% but it decreases those of the electron capture nuclides by 1–2%. Table C1 gives the relatives differences when the micelle size growth is 3.5 nm, to approximate the case of UGAB for moderate aqueous fractions, also for λ=2. In the case of 55Fe, the double and triple efficiencies can drop by 1.5% and 2.8% respectively. The effect of the micelle concentration is much larger than that of the micelle size. Incrementing the micelle concentration while maintaining a fixed micelle diameter of 2 nm was found to decreases appreciably the tritium double detection efficiency. This is also true for the triple detection efficiency and TDCR. The same features are shown by the 63Ni efficiencies. Fig. 11b depicts the relative differences between the 55Fe double detection efficiencies for different aqueous fractions and that when no aqueous phase is present, for a set micelle size (2 nm). The effect increases significantly with micelle concentration, ranging from about a percent efficiency drop for very low aqueous fractions to several percent for moderate fractions, when λ=2. We report in Fig. 11d the relative differences between the triple detection efficiencies for an aqueous fraction of 6% – our usual practice at IRA-METAS – and that for UG only. For λ=2, the triple detection efficiencies of 63Ni and 3H decrease by 3.5% and 9.9% respectively, while those of 54Mn and 55Fe see a reduction of 6.3% and 14.5%, in that order. Table D1 provides the relative differences when the aqueous fraction is 0.5%, which corresponds to a few drops of radioactive solution in 10 mL scintillant, also for λ=2. In the case of 55Fe, the double and triple efficiencies fall by 1.4% and 2.6% respectively.

calculated for UG with faq=10%. For reference, the black solid line shows Q(E) when no micelle effect is included. As expected Q(E) is smaller than Q(E). The figure also shows that the micelle size is mainly relevant below about 1 keV. Above this energy, it is the aqueous fraction which determines the correction to the ionisation quenching function. One sees that the various profiles of Q(E) coincide when the same aqueous fraction is used. These observations are confirmed by Fig. 10b which reports the ionisation quenching functions including the micelle effect for various aqueous fractions in the case of UG but a fixed micelle size (2 nm). Estar stopping powers and kB=0.012 cm/Mev are used. Note that the higher the aqueous fraction the lower Q(E) gets. The micelle concentration affects the ionisation quenching function at all energies. We saw in Section 3.3 that only part of the electron energy is available for the scintillation process due to micelle quenching. Let S(E) be the electron energy ratio deposited in the scintillant; typical profiles of S(E) are shown in the Appendix A. One can then compare Q(E)·S(E) – the product of the energy fraction usable for scintillation and the ionisation quenching function uncorrected for the micelle effect – with Q(E) given by Eq. (3). The comparison is shown in Fig. 10c, in the case of UGAB for kB=0.012 cm/MeV. The Q(E)·S(E) product approximates quite well the computationally exacting Q(E) function as evidenced by their small difference shown as residuals. Systematic comparisons for both scintillants and a wide spectrum of conditions (faq and Ømic) show that this is a good approximation except for low kB values where it breaks down. Note that, since S(E) tends to the scintillant fraction (flsc) at high energy (see Section 3.3), one can further approximate the micelle corrected ionisation quenching function by Q(E)·flsc in such energy regime. An explicit account of the water content of a liquid scintillation sample, or rather its concomitant scintillant fraction, obtains in this case. 3.6. Micelle effect on the detection efficiency We now seek to evaluate the micelle effect on the TDCR detection efficiencies for beta emitters (3H and 63Ni) and low energy electron capture nuclides (54Mn and 55Fe). Assuming identical PMTs, isotropic light emission and Poisson statistics for the photoelectrons at the photo-cathodes, the double and triple detection efficiencies for a pure beta emitter are modelled as:

εD =

∫0

Eβ max

N(E)⋅[3(1 − e−m(E) )2− 2(1 − e−m(E) )3]dE,

(5)

and

εT =

∫0

Eβ max

N(E)⋅(1 − e−m(E) )3dE

(6)

respectively (Grau Malonda and Coursey, 1988). N(E) is the normalised beta spectrum density function while the average number of photoelectrons is

m(E) =

EQ(E) , 3λ

(7)

where λ is a free parameter – the figure of merit – which quantifies the average energy deposition required to produce one photoelectron (Grau Carles and Grau Malonda, 2001a, 2001b). The micelle effect on the 3H and 63Ni efficiencies was computed using Eqs. (5)–(7). For 54Mn and 55Fe, this was done by means of the MICELLE2 code (Kossert and Grau Carles, 2010) which was used with its micelle option switched off and our own set of ionisation quenching functions described in the previous section. This effect was explored for an ionisation quenching parameter kB set at 0.010 cm/MeV and a figure of merit varying between 0.1 and 4 keV/photoelectron, although the latter typically ranges from 1.5 to 3.3 keV/photoelectron for our first TDCR system (Nedjadi et al., 2015).

3.7. Micelle effect on the activity concentration of

63

Ni

The previous section looked at the translation of the micelle corrected ionisation quenching function to the efficiency. But how does it translate to the activity concentration? Thus far we studied only the effect on the activity of the pure betaemitter 63Ni (Emax=66.98 keV). The reference solution was in the form of 50 μg/g NiCl2 in HCl 0.5 M, and had a concentration of about 114 kBq/g at the reference time. Twenty four samples covering eight aqueous fractions – 0.1%, 0.5%, 1%, 2%, 4%, 6%, 8% and 10% – in triplicates were prepared. These fractions are meant as the ratio 102

Applied Radiation and Isotopes 125 (2017) 94–107

ΔεD (

55

Fe ) [ % ]

Y. Nedjadi et al. 0

0

-1

-5

a)

b)

-10

-2

1.0 nm 2.0 nm 3.0 nm 4.0 nm 5.0 nm 6.0 nm

-3

-4

7.0 nm 8.0 nm 10.0 nm 1

0.0

0.1 % 0.5 % 1.0 % 2.0 % 4.0 % 6.0 % 8.0 %

-15

10.0 % 12.0 % 15.0 %

-20

-25 2

3

4

1

2

3

4

0

-0.5

ΔεT

[%]

-5 55

-1.0

54 3

-1.5

Fe Mn

-10

H

63

Ni -15

-2.0

d)

c) -2.5

-20 1

2

3

4

1

2

λ

3

4

λ

Fig. 11. a) Relative difference between εD( Fe) for a) various micelle sizes and that for Ømic=0.5 nm (faq=10%), and b) for different aqueous fractions and that for UG with no aqueous phase (Ømic=2 nm). Relative difference between εT for Ømic=3 and 0.5 nm (faq=10%) (c), and between εT for faq=6% and that for no aqueous fraction (d), in the cases of 3H, 63Ni, 54Mn and 55Fe. A value of kB=0.010 cm/MeV is used in all cases. 55

discussed in Section 3.5. λ and λ denote the related figures of merit. Table E1 reports the activities of the 24 samples. The table gives the inferred activities taking into account the micelle effect along with the corresponding activities neglecting the aqueous phase. The uncertainties stated include only the standard deviations of the counting points. Clearly the two treatments agree quite well, the deviations between them ranging between 0.01% and 0.02%. The same table provides the kB values which minimise the slopes

between the aqueous sample volume and the scintillator volume. Aliquots of the radioactive solution were added gravimetrically to 14.4 mL Ultima Gold in low potassium high performance borosilicate vials. Variable amounts of ultrapure water top-ups were supplemented, as appropriate, to realise the eight aqueous fractions, which were determined gravimetrically by weighing the vials before and after dispensing the scintillant and the water top-ups. Each vial was agitated for 2 min with a vortex shaker, and then centrifuged at 15 revolutions per second for 3 min in order to settle down the liquid on the cap and walls. The vials were measured on our second TDCR counter with Hamamatsu R331-05 photomultiplier tubes (PMTs). They were operated at 2.1 kV anode voltage and a focusing voltage changing between 280 and 560 V in 20 V steps. A resolving time of 60 ns was used. Each of the 15 counting points was measured for at least 600 s. The TDCR values ranged between 0.77 and 0.87. For the activity calculations, the PMTs are assumed to be asymmetric. IRA's Fortran code is used. The nickel spectrum includes the screening and atomic exchange corrections (Angrave et al., 1998) as well as finite size, recoil and radiative corrections. Two parallel analyses – with and without the micelle effect – of the measured rates are made. The average numbers of photo-electrons for the three PMTs is

mj(E) =

EQ(E) , j= A, B, C, 3λ j

λmic

1.4

EQ(E) , 3λ j

1.2

1.0

0.8

(8) 0.8

when taking into account the micelle effect, and

mj(E) =

0.1 % 1.0 % 2.0 % 4.0 % 6.0 % 8.0 % 10.0 %

1.2

1.0

1.4

λ¬mic Fig. 12. Comparison of the figures of merit obtained when including micelles with the corresponding numbers when ignoring the aqueous matter. The data is for PMT A, for the first vial of each aqueous fraction, and for the 15 counting points. The dashed line is the bisector.

(9)

when the aqueous matter is overlooked. Q(E) and Q(E) refer to the ionisation quenching functions, computed with the estar stopping powers, including and omitting the micelle effect respectively, as 103

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Residuals (%)

Y. Nedjadi et al.

the aqueous matter. This explains why the same activities were found, and it is not surprising since the TDCR activity calculation procedure aims at reproducing the average number of photoelectrons (Broda et al., 2007). For 63Ni and nuclides with higher beta energies, Eq. (11) suggests three possible postures in dealing with the aqueous matter. It can either be treated explicitly using the micelle-corrected ionisation quenching function Q (E ), or else approximately through scaling down the usual ionisation quenching function by the scintillant fraction. In both cases the average energy required to produce a photoelectron is lower and retains its usual interpretation as reflecting chemical and colour quenching and the quantum efficiency of the photocathode. The third alternative would be to use the standard reduced energy ignoring micelles but then the figure of merit is higher and must be reinterpreted as including micelle quenching. Of course, for lower energy beta emitters and electron capture nuclides, this may not be the case because the micelle size becomes then as relevant as the micelle concentration. This is what we intend to investigate next.

0.08 0.04 0.00 -0.04 -0.08

100

98

96 λmic (%) λ¬mic 94

92

90 90

92

94

96

98

100

Scintillant fraction (%) Fig. 13. Ratios of the figures of merit, with and without micelle quenching, against the corresponding scintillant fraction. Data is for PMT A, the 560 V focusing voltage counting point, and the first vial of each aqueous fraction.

4. Conclusion This study showed that the average number of micelles in which electrons deposit energy is quite large, increases with their emission energy, decreases with micelle size, and grows with micelle concentration. Statistics of the electron tracks also found that the probability of an electron ending its track in a micelle is very large and micelle size dependent below 1 keV, and that it approximates closely the aqueous fraction at higher energies. These findings run counter to representing the electron as losing energy solely in the primary micelle housing the decaying nuclide. The Monte Carlo simulations ascertained that the total energy deposited by the electron in the aqueous phase increases with its emission energy, the micelle size regulating the energy loss at low energy and the micelle concentration or aqueous fraction determining it at high energy. The simulations also established that the electron energy ratio deposited in the scintillator is much lower than one and micelle size dependent at low energy whereas it tends strictly to the scintillant fraction at high energy. This may be described as a micelle quenching effect, with the micelle size shaping the quenching at low energy and the micelle concentration governing it at higher energies. This effect may be exploited in the choice of surfactant in the design of scintillants. Since micelle quenching is mostly dependent on micelle size at low energy and on micelle concentration at higher energies, selecting the right combination of surfactant and aqueous fraction for a given radionuclide should provide a way to predictably optimise scintillant efficiency. The micelle effect on the 3H and 63Ni beta spectra were discussed for a range of micelle sizes and concentrations. The effect of the aqueous matter is to shift the spectra towards lower energies. The ionisation quenching function was computed on the premise that the electron energy deposited in the micelles produces no light. Birk's law was used in such a way as to account for the full energy losses in the aqueous droplets bisecting the electron pathway, and not just that incurred in the primary micelle enclosing the disintegrating nuclide. This was done for various micelle sizes, aqueous fractions, and ionisation quenching parameters (kB). The micelle size alters the ionisation quenching function below 1 keV but the micelle concentration affects it at all energies. The computationally exacting micellecorrected ionisation quenching function was found to be well approximated by the product of the ionisation quenching function ignoring the aqueous phase with the electron energy ratio deposited in the scintillant. Since the latter tends to the scintillant fraction above a few keV, one may use this approximation to represent explicitly the aqueous fraction in the efficiency model. These ionisation quenching functions were used to calculate the

when treating fully the micelles next to the reciprocal kB values which minimise the slopes when neglecting micelles. The kB values agree within 0.1–0.4%. Since the activities and the kB values are almost equal with rather different ionisation quenching functions, one needs to look closely at the figures of merit. This is done in Fig. 12 for PMT A. The ordinates are the figures of merit obtained when including micelles (λA ) whilst the abscissae are the related numbers when ignoring the aqueous matter (λA). The figures of merit are shown for the first vial of each aqueous fraction, and for the 15 counting points. One clearly sees that the figures of merit when micelle quenching is included are smaller than, and proportional to, the corresponding figures when omitting micelles. The other two PMTs show analogous profiles. The relationship between the figures of merit, with and without micelles, is made clearer in Fig. 13. It presents the ratios of the figures of merit, with and without micelles, against the corresponding scintillant fraction (fLSC). This is done for PMT A, the 560 V focusing voltage counting point, and the first vial of each aqueous fraction. The eight points lay on the diagonal. The same plot obtains if one uses the triplication averages rather than just the first vial of each aqueous fraction. We did verify that one gets the same profile for all the fifteen counting points. These features are likewise reproduced in the other two PMTs. One can therefore write:

λ = flsc. λ

(10)

This means that if one treats explicitly and fully the micelle effect, the ionisation quenching function is lowered but then one needs less energy to produce a photoelectron. If however the aqueous phase is omitted, then the reduced energy is higher but in this case one needs more energy to produce a photoelectron. In other words, efficiency models which do not incorporate explicitly the aqueous matter get its effects through their phenomenological figure of merit. Using this relation, and since the micelle size has a negligible effect for 63Ni (Section 3.6) and it was pointed out in Section 3.5 that Q (E ) can be approximated by Q(E)·flsc at high energy, it follows that the average number of photo-electrons treating the micelle effect explicitly

m(E) =

EQ(E) EQ(E)⋅flsc EQ(E) EQ(E) ≃ = = = m(E) λ 3λ 3λ 3λ 3f lsc

(11)

is equivalent to the average number of photoelectrons when ignoring 104

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detection efficiencies for 3H, 63Ni, 54Mn and 55Fe. The effect of the micelle size was found to be small except for the electron capture nuclides where it is significant. TDCR measurements of 63Ni samples representing eight aqueous fractions were analysed with and without explicit treatment of the micelle effect. Activities in the two approaches agreed within 0.02%. The ratios of the reciprocal figures of merit were found to coincide with the scintillant fractions. It was argued that if the aqueous phase is not explicitly treated in the efficiency model, then one gets its effect through the figure of merit which must in that case be re-interpreted

as including micelle quenching. We plan to study the micelle effect on the liquid scintillation activities of 3H and 55Fe solutions next.

Acknowledgements The authors are grateful to one of the referees for pointing out the practical implications of this work on surfactant selection in the design of liquid scintillation cocktails.

Appendix A Graph a) shows how the percentage of the total average energy deposited in the scintillant varies with the electron decay energy for a range of micelle sizes, keeping the aqueous fraction at 10%. The plot b) gives the corresponding energy ratio for different aqueous fractions with Ømic fixed at 2 nm.

(See Tables A1–E1).

Table A1 Twenty liquid scintillation sample conditions used for the simulations. An UG volume of 15 mL is assumed in all cases. The aqueous fraction is denoted faq while the micelle diameter is indicated as Ømic. faq (%)

Ømic (nm)

||

faq (%)

Ømic (nm)

0.1 0.5 1.0 2.0 4.0 6.0 8.0 10.0 12.0 15.0

2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0

|| || || || || || || || || ||

10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0

0.5 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 10.0

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Table B1 Average energies in keV units of the reduced spectra of tritium and 63Ni for ten aqueous fractions and the same micelle diameter of 2 nm. The mean energy of the beta emission spectrum is 5.680 keV for 3H and 17.434 keV for 63Ni. faq (%)

3

63

0.1% 0.5% 1.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 15.0%

5.664 5.637 5.609 5.552 5.444 5.340 5.238 5.142 5.045 4.916

17.108 17.034 16.949 16.780 16.447 16.135 15.826 15.531 15.250 14.852

H

Table C1 Relative difference between εD, εT and the TDCR for Ømic=4 and 0.5 nm, faq=10%, for kB=0.010 and λ=2, in the cases of 3H,

63

Ni,

54

Mn and

55

Ni

Fe.

Nuclide

ΔεD (%)

ΔεT (%)

ΔTDCR (%)

55

−1.5 −1.3 −0.1 −0.03

−2.8 −1.1 −0.2 −0.06

−1.2 +0.2 −0.1 −0.03

Fe Mn 3 H 63 Ni 54

Table D1 Relative difference between εD, εT and the TDCR for faq=0.5% and that for aqueous phase, with Ømic=2 nm, kB=0.010 and λ=2, in the cases of 3H, 63Ni, 54Mn and 55Fe. Nuclide

ΔεD (%)

ΔεT (%)

ΔTDCR (%)

55

−1.4 −1.3 −0.5 −0.1

−2.6 −1.2 −0.9 −0.3

−1.2 −0.1 −0.4 −0.2

Fe Mn 3 H 63 Ni 54

Table E1 Comparisons of the activity concentrations and kB values of the 24 samples when treating fully the micelles against their reciprocal values when omitting micelles. kB values are obtained by minimising the activity slopes. Source

C mic (kBq/g)

Cmic

ΔC %

|| ||

kBmic (cm/MeV)

kBmic

ΔkB %

63Ni01 63Ni02 63Ni03 63Ni04 63Ni05 63Ni06 63Ni07 63Ni08 63Ni09 63Ni10 63Ni11 63Ni12 63Ni13 63Ni14 63Ni15 63Ni16 63Ni17 63Ni18 63Ni19 63Ni20 63Ni21 63Ni22 63Ni23 63Ni24

115.06(7) 114.67(10) 114.44(5) 114.82(10) 115.2(8) 114.95(11) 114.69(12) 115.07(9) 114.84(9) 114.73(10) 115.00(10) 114.66(10) 115.26(7) 114.62(7) 115.36(8) 114.45(8) 114.74(11) 114.73(9) 115.00(10) 115.39(12) 114.53(7) 115.02(12) 114.97(6) 115.01(8)

115.04(7) 114.64(10) 114.42(5) 114.80(10) 115.18(8) 114.93(11) 114.67(12) 115.05(9) 114.82(9) 114.71(10) 114.98(10) 114.64(10) 115.24(8) 114.60(7) 115.34(8) 114.44(8) 114.73(11) 114.71(9) 114.98(10) 115.37(12) 114.51(7) 115.01(12) 114.96(6) 115.00(8)

0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01

|| || || || || || || || || || || || || || || || || || || || || || || ||

0.00976 0.00913 0.00856 0.00911 0.01073 0.00934 0.00919 0.00994 0.00955 0.00879 0.00998 0.00878 0.01045 0.00871 0.01076 0.00800 0.00866 0.00909 0.00954 0.01101 0.00907 0.00980 0.00966 0.00945

0.00972 0.00909 0.00852 0.00907 0.01069 0.00930 0.00916 0.00991 0.00951 0.00876 0.00994 0.00874 0.01042 0.00868 0.01073 0.00798 0.00863 0.00907 0.00951 0.01099 0.00905 0.00979 0.00965 0.00944

0.40 0.42 0.44 0.37 0.34 0.37 0.35 0.33 0.34 0.38 0.35 0.38 0.28 0.31 0.27 0.31 0.29 0.28 0.25 0.23 0.26 0.14 0.15 0.15

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