Self-radiolysis of tritiated water. 4. The scavenging effect of azide ions (N3−) on the molecular hydrogen yield in the radiolysis of water by 60Co γ-rays and tritium β-particles at room temperature

The effect of the azide ion N3− on the yield of molecular hydrogen in water irradiated with 60Co γ-rays (∼1 MeV Compton electrons) and tritium β-electrons (mean electron energy of ∼7.8 keV) at 25 °C is investigated using Monte Carlo track chemistry simulations in conjunction with available experimental data. N3− is shown to interfere with the formation of H2 through its high reactivity towards hydrogen atoms and, but to a lesser extent, hydrated electrons, the two major radiolytic precursors of the H2 yield in the diffusing radiation tracks. Chemical changes are observed in the H2 scavengeability depending on the particular type of radiation considered. These changes can readily be explained on the basis of differences in the initial spatial distribution of primary radiolytic species (i.e., the structure of the electron tracks). In the “short-track” geometry of the higher “linear energy transfer” (LET) tritium β-electrons (mean LET ∼5.9 eV nm−1), radicals are formed locally in much higher initial concentration than in the isolated “spurs” of the energetic Compton electrons (LET ∼0.3 eV nm−1) generated by the cobalt-60 γ-rays. As a result, the short-track geometry favors radical–radical reactions involving hydrated electrons and hydrogen atoms, leading to a clear increase in the yield of H2 for tritium β-electrons compared to 60Co γ-rays. These changes in the scavengeability of H2 in passing from tritium β-radiolysis to γ-radiolysis are in good agreement with experimental data, lending strong support to the picture of tritium β-radiolysis mainly driven by the chemical action of short tracks of high local LET. At high N3− concentrations (>1 M), our H2 yield results for 60Co γ-radiolysis are also consistent with previous Monte Carlo simulations that suggested the necessity of including the capture of the precursors to the hydrated electrons (i.e., the short-lived “dry” electrons prior to hydration) by N3−. These processes tend to reduce significantly the yields of H2, as is observed experimentally. However, this dry electron scavenging at high azide concentrations is not seen in the higher-LET 3H β-radiolysis, leading us to conclude that the increased amount of intra-track chemistry intervening at early time under these conditions favors the recombination of these electrons with their parent water cations at the expense of their scavenging by N3−.


Introduction
A detailed understanding of the radiolysis of water and aqueous solutions is important both from a fundamental science point of view and for a variety of practical applications, 1-4 in particular, in the nuclear power industry and in radiation biology where living cells and tissues consist mainly of water ($70-85% by weight). Exposed to ionizing radiation, water is the site of ionizations and excitations: which result, within a few picoseconds, in a cascade of events leading to the formation of free radicals and molecular products along the track of the incident radiation. Ejected secondary electrons (also called "dry" electrons) have generally sufficient kinetic energy to cause further ionizations and excitations in close proximity to the original water positive ion. Aer slowing down to sub-excitation energies and thermalization, these electrons become trapped and hydrated. Under ordinary irradiation conditions (i.e., at modest dose rates so that no track overlap occurs), the initial products of radiolysis are generated in a highly nonhomogeneous "track structure" geometry. 5-11 They include 12,13 the hydrated electron (e aq À ), H 3 O + , OH À , H , H 2 , OH, H 2 O 2 , O 2 À (or its protonated form HO 2 ; pK a ¼ 4.8 at 25 C), O( 1 D), O ( 3 P), O À , etc. This early nonhomogeneous spatial distribution of radiolytic species is strongly dependent on the radiation quality, a measure of which is given by the "linear energy transfer" (LET) (also called "stopping power" by physicists and denoted by ÀdE/dx). For low-LET, sparsely ionizing radiation (such as g-rays from 60 Co or fast electrons; LET $0.3 eV nm À1 ), tracks are formed initially by widely spaced clusters of reactive species, commonly known as "spurs" (spherical in shape). 14,15 In this case, the predominant effect of radiolysis is radical production. In fact, when diffusion has brought about homogeneity in the system (i.e., within a few microseconds aer the initial energy deposition), relatively few radicals have combined in the spurs, resulting in an excess of radicals over molecular products. However, with increasing LET, the isolated spur structure changes to a situation in which the spurs eventually overlap and form (initially) a dense continuous column of species. This is actually the case for the low-energy b-electrons of tritium, which are involved in the "self-radiolysis" of tritiated water ( 3 HOH), [16][17][18] the subject matter of the present study. In the terminology of the Mozumder-Magee model of energy deposition, 6,19 while the Compton electrons ($1 MeV) produced by 60 Co g-radiolysis predominantly form spurs, these so, higher-LET tritium b-electrons predominantly deposit their energy as "short tracks". This leads to an increased local concentration of reactants and therefore an increased amount of intra-track chemistry that favors radical-radical reactions. Under these conditions, the radiation chemical yields (or G-values) 20 of the molecular products increase at the expense of the individual radicals. For the sake of illustration, Fig. 1 shows typical 2-D representations of the complete track of a 7.8 keV 3 H b-electron and the track segment of a 300 MeV proton (which mimics irradiation with 60 Co grays), calculated with our IONLYS Monte Carlo track structure simulation code (see below). In close connection with the LET and the relationship between track structure and chemistry, one critical area of research focuses on elucidating the basic radiation chemical mechanisms that operate in the "self-radiolysis" of tritiated water as compared with 60 Co g-radiolysis. [21][22][23][24][25][26] The present paper is the fourth in a series 18,27,28 dedicated to this subject. The Monte Carlo track-chemistry simulation work we reported previously revealed signicant differences between the chemical properties of short tracks and spurs using either g-rays/fast electrons or tritium b-particles. Overall, the results of our simulations provided strong support for the picture of tritium b-radiolysis mainly driven by the chemical action of short tracks of high local LET. In the present study, we now attempt to distinguish further the chemical properties of spur and short track geometries by examining the differences in the scavengeability of molecular H 2whose yields are relatively well-documented experimentally 21,22,25,26,[29][30][31][32] when passing from gto tritium b-electron radiolysis.
Molecular hydrogen is one of the most interesting radiolytic species, in part because of the questions it raises about the source of its formation. At very short times (<50-300 fs) aer the passage of the ionizing radiation, 33 H 2 can be formed by the following reactions: 34,35 ( (3) dissociation of excited water molecules (formed either by direct excitation or by geminate electron-hole recombination) where O( 1 D) is the oxygen atom in its singlet 1 D rst excited state.
In the low-LET g-irradiation case, this "initial"then described as "unscavengeable" (i.e., not removable by scavenger experiments) -H 2 yield was rst estimated by Schwarz 37 in 1969 to be $30% of the total "escape" yield 20 for molecular hydrogen [g(H 2 ) ¼ 0.45 molecule/100 eV]. 5,7,11,12 Recent scavenger studies 38 have shown, however, that Schwarz's initial estimate was undervalued. In fact, it was found that a major fraction ($75%, i.e., $0.34 molecule/100 eV) of the total H 2 produced was due to reactions of the dry/subexcitation electrons in the subpicosecond physicochemical stage of the radiolysis. In other words, these results suggest that only $0.11 molecule of H 2 per 100 eV remains to be formed during the subsequent nonhomogeneous chemical stage (i.e., in the radiation tracks as they expand by diffusion) on the picosecond-microsecond time scale. At this stage, three radical-radical combination reactions of the hydrated electron and H atom intervene in hydrogen formation. They are 18,34,39,40 e aq À + H (+H 2 O) / H 2 + OH À , k 1 ¼ 2.5 Â 10 10 M À1 s À1 (1) e aq À + e aq À (+2H 2 O) / H 2 + 2OH À , k 2 ¼ 6.2 Â 10 9 M À1 s À1 (2) and, but to a much lesser extent, with the corresponding rate constants (k) taken from the compilation of Elliot and Bartels. 12 Besides the mechanism of its formation, a better knowledge of the radiolytic production of molecular hydrogen is crucial in the "water chemistry" management of current water-cooled nuclear reactors to optimize plant performance and lifetime. As we know, 2 H 2 is currently added to the primary coolant water to suppress the formation of stable oxidizing products (H 2 O 2 and eventually its decomposition product O 2 ) from water radiolysis by a short chain reaction, thereby preventing corrosion and activity transport. The in situ radiolytic formation of H 2 in these reactors could, therefore, affect the minimum concentration of excess H 2 , referred to as the "critical hydrogen concentration", 41,42 required to suppress net radiolysis (no stable products formed) in the cores. Knowledge of this optimum H 2 level, which would minimize the damaging consequences of corrosion, is still a subject of debate in the chemical literature.
The anomalous increase in the escape yield of H 2 at high temperature is another key motivation for this study. In fact, although H 2 is a molecular product, g(H 2 ) increases with temperature under g/fast electron irradiation, 12,34,35 from $0.45 molecule/100 eV at 25 C to $0.76 molecule/100 eV at 350 C. This behavior is an exception to the generally accepted diffusion-kinetic model, 6,43 which predicts that, when the temperature increases, diffusion of free radicals out of spurs or tracks becomes more important than recombination, resulting in less molecular recombination products. At present, no denitive mechanism has yet been established to account for this anomalous radiolytic production of H 2 at high temperature. 34 For these different reasons, the escape yield of H 2 has attracted much attention from experimentalists and modelers in order to explore in more detail its formation under various irradiation conditions. In this work, we use Monte Carlo track chemistry simulations to examine further the chemical differences underlying the production of molecular hydrogen in tritium b-radiolysis as compared with cobalt g-radiolysis. No real-time studies on H 2 formation have been performed; its temporal dependence is usually probed by varying the concentration of appropriate scavengers for the hydrated electron and the hydrogen atom, which are the dominant free radical precursors of H 2 within the diffusing spurs or tracks. We here report data on the scavengeability by azide ions (N 3 À ) of the molecular H 2 yield produced by gand tritium b-radiolysis. This particular scavenger was chosen as it presents very different reactivities towards e aq À and H atoms, being highly unreactive towards the former but reacting very rapidly with the latter. Our aim is to study the different H 2 scavengeabilities found for the two types of irradiation considered and to examine how these differences reect the structure of the radiation track (i.e., spurs vs. short tracks) in both cases.

Monte-Carlo track chemistry simulations
Monte Carlo simulation methods are well suited to take into account the stochastic nature of the complex sequence of events that are generated in irradiated aqueous solutions containing reactive scavengers. In the case of interest here, the experimentally observed yield value for molecular hydrogen is a composite one to which each of the processes producing H 2 contributes. The addition of a scavenger that competes with these processes to different extents will change the relative amount that each process contributes to the total yield. The simulation allows the reconstruction of the intricate action of the radiation, thus providing a powerful tool for studying the relationship between the initial radiation track structure, the ensuing chemical processes, and the stable nal products formed. In this work, a full Monte Carlo track-chemistry computer code, called IONLYS-IRT, 11 has been used to simulate the radiolysis of water and aqueous solutions containing various concentrations of scavengers. This code rst models, in a 3D geometrical environment, the initial, highly nonhomogeneous radiation track structure ("IONLYS" program), and then the diffusion and chemical reactions of the various radical and molecular products formed by radiolysis with themselves or with solutes if present ("IRT" program). A detailed description of this code has been given previously. 11,18,28,34,[44][45][46] Only a brief overview of its most essential features is given below. The IONLYS code is a step-by-step simulation program that covers the early physical and physicochemical stages 47 of radiation action up to $1 ps in the track development. It is composed of two modules. One is for transporting the investigated incident charged particle (called either TRACEPR for an impacting primary electron or TRACPRO/TRACION for an incident proton/ion). The other (called TRACELE) is for transporting all of the energetic (or dry) electrons (collectively named "secondary electrons") resulting from the ionization of the water molecules until they become hydrated. In this study, we used the TRACEPR module of IONLYS to simulate the track structures of low-energy ($7.8 keV) tritium b-electrons. As for the TRACPRO module, it was used here to simulate track segments of 300 MeV incident protons (which, as mentioned before, mimic 60 Co g/fast electron irradiation) (see Fig. 1).
The complex, highly nonhomogeneous spatial distribution of reactants at the end of the physicochemical stage is provided as an output of the IONLYS (TRACELE) program. It is then used directly as the starting point for the subsequent nonhomogeneous chemical stage 47 (from $1 ps to $0.1-1 ms at 25 C, 48 i.e., until all tracks/spurs have dissipated). This stage, during which all different species diffuse (we assume $1 ps also marks the beginning of diffusion) randomly at rates determined by their diffusion coefficients and react with one another or with any added solutes present at the time of irradiation, is covered by our "independent reaction times" (IRT) program. This program employs the IRT method, 49,50 a computer-efficient stochastic simulation technique used to simulate reaction times without having to follow the trajectories of the diffusing species. Its implementation has been previously described in detail 45 and its ability to give accurate, time-dependent chemical yields has been well validated 51,52 by comparison with full random ight (or step-by-step) Monte Carlo simulations, which do follow the reactant trajectories in detail. Finally, this IRT program has also been used successfully to describe the evolution of radiationinduced yields in the homogeneous chemical stage 47 aer spur/track expansion is complete (i.e., when the radiolytic products become homogeneously distributed in the bulk solution), in the time domain typically beyond a few microseconds.
The reaction scheme and rate constants for the radiolysis of pure liquid water at 25 C employed in the current version of IONLYS-IRT are the same as used previously (see Table 1 of ref. 18). The values of the diffusion coefficients of the various intervening track species are listed in Table 6 of ref. 53. In order to simulate the radiolysis of the N 3 À solutions, we have supplemented the pure-water reaction scheme to include the primary e aq À and H atom scavenging reactions that occur in the system (vide infra). Under normal irradiation conditions, the concentrations of radiolytic products are low compared with the background concentrations of N 3 À ions considered, and their reactions could be modeled in the IRT program as pseudo rstorder reactions. In the computer simulations reported here, the diffusion coefficient used for N 3 À in liquid water at 25 C was 1.84 Â 10 À5 cm 2 s À1 . 54 This same value was also used for the diffusion coefficient of the azide radical N 3 . In addition, we have introduced in the IRT program the effect of the ionic strength of the solutions on all reactions between ions. 55 The correction to the reaction rate constants was made as described in ref. 56 and 57. Finally, for highly concentrated N 3 À solutions (some experimental data are available up to 5 M), we neglected complications due the "direct" action of ionizing radiation on the solute (which our Monte Carlo code does not take into account). This is certainly a very good approximation for $1-2 M N 3 À concentrations (in that case, $2-4% of the total energy is absorbed directly by the azide anions). For 5 M N 3 À solutions, the proportion of direct effects increases to about 11%, which remains relatively low and may reasonably still be ignored at least as a rst approximation.
To mimic the effects of 60 Co g/fast electron-radiolysis, we used short segments (typically, $150 mm) of $300 MeV irradiating proton tracks, over which the average LET of the proton remains nearly constant and equal to $0.3 eV nm À1 at 25 C. 2,45 Such model calculations thus gave "track segment" yields 8,45 at a well-dened LET. Briey, the simulations, performed with the TRACPRO module of IONLYS, consisted of following the transport and energy loss of an incident proton until it penetrated the chosen length of the track segment into the solution. As shown in Fig. 1, due to its large mass, the impacting proton is almost not deected by collisions with the target electrons. The number of individual proton "histories" (usually $150) was chosen to ensure only small statistical uctuations in the computed averages of chemical yields, while keeping acceptable computer time limits.
As indicated above, tritium-b primary electron track structures were simulated using the TRACEPR module of IONLYS. Each simulation typically involved 6000 different whole track histories. This number was chosen to permit averaging of results with acceptable statistical condence. In all the simulations, a single "effective" initial electron energy of $7.8 keV (mean LET in water: $5.9 eV nm À1 ) 16 was used to mimic the radiation chemical action of the tritium b-particles at 25 C (Fig. 1). This energy was found previously to be better suited to produce representative G-values when using tritium b-rays than the commonly used mean kinetic energy of $5.7 keV released by tritium decay. 18,58 Throughout this study, we assumed that tritiated water could simply be described as a "dilute" solution of 3 HOH in light water, with concentrations of low volumic activity (typically, less than $1 Ci per mL) so that dose-rate effects could be ignored. 25,59
Indeed, even in a 5 M N 3 À solution, the scavenging time 72 of e aq À by N 3 À is about the same order of magnitude as the lifetime of a spur ($0.2 ms) 48 in the 60 Co g-radiolysis of water at 25 C. Under these conditions, the molecular hydrogen yield was measured in irradiated aerated azide solutions. 22,25,60 Oxygen between $2.5 Â 10 À4 M (air-saturated conditions) and $3-5 Â 10 À5 M was used as e aq À scavenger on the $0.1-1 ms time scale.
Noteworthy, the azide radical is inert towards molecular oxygen, 66 but may react with the superoxide anion radical 62,66 While these low O 2 concentrations hardly affect g(H 2 ), they do prevent, at long times, the reactions of e aq À with itself and with water 62 in the bulk of the solutions. Fig. 2 (panels a and b) shows the effect of azide concentration on the kinetics of H 2 formation over the interval $1 ps to 10 ms, as obtained from our Monte Carlo simulations of the radiolysis of aerated neutral pH aqueous solutions of NaN 3 by $300 MeV incident protons and $7.8 keV tritium b-electrons at 25 C. Results are shown for six different concentrations of azide anions, ranging from 10 À4 to 5 M. As can be seen, for both types of radiation, the time proles of the H 2 yields are essentially similar although the magnitude of the G(H 2 ) values differs. In fact, the simulations show a clear increase in the absolute value of G(H 2 ) for 3 H b-electrons compared to 60 Co g-rays. As mentioned earlier, this increase in H 2 yields, when comparing the effects of higher-LET tritium b-radiolysis with the g/fast electron-radiolysis, is consistent with differences in the initial structure of electron tracks in the two cases. In the short-track geometry of the b-electrons (in contrast with spur geometry), Calculations were carried out using our Monte-Carlo track chemistry simulations over the time interval 1 ps to 10 ms. The blue, green, red, orange, cyan, and magenta lines correspond to six different concentrations of N 3 À anions: 10 À4 , 10 À3 , 10 À2 , 0.1, 1, and 5 M, respectively. For both types of radiation, the limiting plateau values of G(H 2 ) continuously decrease with increasing the concentration of N 3 À ions. For 60 Co g/fast electron irradiation, the arrow pointing downwards indicates the time s s $0.2 ms required for the changeover from nonhomogeneous spur kinetics to homogeneous kinetics in the bulk solutions, at 25 C. The black solid line in panels a and b show the kinetics of H 2 formation in azide-free aerated solutions (shown here for the sake of reference). Finally, the concentration of dissolved oxygen used in the simulations was 2.5 Â 10 À4 M.
the reactive intermediates are formed in much closer initial proximity, which is favorable to the additional formation of H 2 through the inter-radical combination reactions (1)-(3).
The decrease in the yield of H 2 with concentration of N 3 À ions for 300 MeV incident protons and 7.8 keV 3 H b-electrons in the radiolysis of aerated azide solutions is further illustrated in Fig. 3. The H 2 yields shown in this gure are the G(H 2 ) limiting plateau values corresponding to each considered N 3 À concentration, taken from Fig. 2. As can be seen, our simulated yields compare well with the experimental escape yields of Gagnon and Appleby, 22 Christman, 25 and Peled et al. 60 obtained for 60 Co g and tritium b-particle irradiations. In the case of g-radiolysis, this agreement is particularly good at low and moderated N 3 À concentrations. However, at concentrations higher than $0.5 M, there are signicant differences, the experimentally observed H 2 yields showing a very sharp decrease 73 compared to the simulation results. This efficiency in reducing the molecular hydrogen produced strongly suggests that the concentration of azide ions is now high enough to allow their reaction with the dry electron (e dry À ) prior to trapping and hydration (i.e., with the precursor to e aq À ), in the subpicosecond physicochemical stage. 33 Similar ndings about the N 3 À scavenging of the short-lived hydrated electron precursor were obtained by Harris and Pimblott 26 in recent Monte Carlo studies of the 60 Co g-radiolysis of azide solutions of concentration greater than 1 M. The present study clearly corroborates their results. Assuming the validity of  this hypothesis would imply a (e dry À + N 3 À ) reaction rate constant of $10 12 -10 13 M À1 s À1 at 25 C, in agreement with Harris and Pimblott 26,74 results. For the case of 3 H b-particle radiolysis, the effectiveness in lowering g(H 2 ) at high azide concentration differs considerably from the case of g-radiolysis. Despite a relatively large dispersion of experimental data, 75 we do not observe any sharp decrease at concentrations higher than $0.1-1 M as we do for g irradiation. There is only a slight continuous decrease of the yield of H 2 without any clear supporting evidence that, in this case, N 3 À ions scavenge the short-lived dry electrons. This is consistent with the enhanced contribution of short tracks for the higher LET tritium b-radiolysis as compared to g radiolysis. Indeed, in this case, the short-track geometry would be competitively more favorable to the subpicosecond recombination reaction of e dry À with its nearby parent water cation (H 2 O + ) than to its scavenging by the homogeneously distributed N 3 À ions.
A nal remark should be made here regarding the origin of the small reduction that is observed, for both types of radiation, in the yields of H 2 with increasing azide concentration from 10 À4 up to $0.1-1 M. In fact, as shown in Fig. 4 . Actually, it is indirectly because the hydrated electrons that have not reacted with H through reaction (1) become now available to participate to reaction (2). Overall, there is a kind of compensation between the two contributions involved in the H 2 production, the contribution from reaction (1) dominating slightly.

Conclusions
Monte Carlo track chemistry simulations have been employed to investigate the scavengeability by azide ions (N 3 À ) of the molecular hydrogen yield produced in water irradiated with 300 MeV protons (which mimic irradiation with 60 Co g rays or fast electrons) and tritium b-electrons at 25 C. From this study, we clearly show that the formation of H 2 from 3 H b-particles is higher than in the case of 60 Co g rays, a result that is easily explained by the difference of the structure of radiation tracks. The track structure in the case of 60 Co g irradiation is composed of well-separated (spherical) spurs, which contrasts with the short (roughly cylindrical) tracks observed in the case of higher-LET tritium b-electrons. The greater linear energy transfer of 3 H b-electrons leads to an increased local concentration of reactants. The distance between the primary events is thus much smaller than in the tracks of 60 Co g rays. Consequently, we nd more molecular products (H 2 in the case considered in this work) in tritium radiolysis than in g radiolysis. Our calculations of the H 2 yields from gand 3 H b-radiolysis of NaN 3 solutions show a very good agreement with experiment over a large range of N 3 À concentrations. For 60 Co g-radiolysis, however, our H 2 yields fail to reproduce the sharp decrease that is observed experimentally at high (>1 M) azide concentrations. These results are consistent with previous Monte Carlo simulations that suggested that such a decrease reected the possibility that low-energy (or "dry") secondary electrons could be scavenged by N 3 À prior to trapping and hydration in the subpicosecond physicochemical stage. Most interestingly, for 3 H bradiolysis, we do not observe any marked decrease in the molecular hydrogen yields at high N 3 À concentrations as we do for g irradiation. In other words, there is no clear evidence that, in this case, N 3 À ions scavenge the short-lived dry electrons. This is consistent with the enhanced contribution of short tracks for the higher LET 3 H b-radiolysis as compared to g radiolysis. Indeed, the short-track geometry is competitively more favorable to the geminate recombination of e dry À with their nearby parent water cations than their scavenging by the homogeneously distributed N 3 À ions. In order to further examine these results, we are currently working to introduce this ultra-fast (<1 ps) capture of the dry electron into our simulation models. In summary, this work, like our previous ones on the subject, provides a strong support for a picture of tritium b-radiolysis in terms of short tracks of high local LET.

Conflicts of interest
There are no conicts of interest to declare. 36 The secondary (or "dry") electron released in the ionization event can cause further ionization and excitation to occur if it has sufficient kinetic energy. Eventually, its energy falls below the rst electronic excitation threshold of water ($7.3 eV in amorphous ice at 14 K, see: M. Michaud, P. Cloutier and L. Sanche, Phys. Rev. A, 1991, 44, 5624), forming the so-called "sub-excitation electron" (R. L. Platzman, Radiat. Res., 1955, 2, 1). The latter loses the rest of its energy relatively slowly by exciting vibrational and rotational modes of water molecules. Once it is thermalized (e th À ), it can get localized or "trapped" (e tr À ) in a pre-existing potential energy well of appropriate depth in the liquid before it reaches a fully relaxed, hydrated state (e aq 73 For example, in the presence of 5 M N 3 À , the measured yield of H 2 is decreased to almost one-third of its value in the absence of azide (see ref. 60). 74 See also S. M. Pimblott and J. A. LaVerne, J. Phys. Chem., 1998, 102, 2967.
75 This relatively large dispersion of experimental data is probably explained by the fact that experiments are difficult to perform with 3 H as the source of b-particle radiation. Contrary to g-irradiation studies, the irradiation cannot be stopped and hence estimation of the dose given to the sample is susceptible to signicant error (see ref. 26).