Irradiated ice

Lifetime measurements were done on both poly- and monocrystalline samples. They were irradiated at 77 K in a ⁶⁰Co irradiation facility up to a dose of 11 Megarad (Mrad). Measurements were carried out as function of dose and as function of temperature for doses of 4.1 and 11 Mrad.

Angular correlation measurements as function of temperature between 4 and 160 K were done on a monocrystal which was irradiated by the positron source itself to an average dose of approximately 2 Mrad per day.

HF-doped ice

Monocrystals were grown by the Czochralski method doped to different HF concentrations (Bilgram, 1973). The concentrations were measured before and after measurements with an Orion specific-ion electrode on small pieces of the crystals which were melted. Samples were cut by lathe out of larger crystals, ground on emery paper, and polished with a silk cloth. Orientation of the crystals for the angular-correlation measurements was determined by polarized light and by etch pits or by X-ray diffraction. Handling of the crystals took place in a cold room at — 18^°C. Some HF certainly diffused out of the crystals during the handling period (one to a few hours). This gave rise to some uncertainty in the HF concentration of the crystals.

Both lifetime and angular-correlation measurements were made on the doped crystals. The parameters which were varied were (1) HF concentration, (2) measuring temperature, (3) cooling rate and the temperature from which cooling started, and (4) the time for which the sample was kept at a certain temperature, in other words the thermal history of die sample before a measurement.

Data analysis

The lifetimes and their relative intensities were extracted from the measured positron lifetime spectra by use of the computer programme for least-squares fitting Positronfit (Kirkegaard and Eldrup, 1974). Similarly the various components in the angular correlation curves were extracted by use of the least-squares filling programme Paacfit (Kirkegaard and Mogensen, unpublished). Λ more detailed discussion of the data analysis is given by Mogensen and Eldrup (1977th]) and Eldrup (1976).

Irradiated ice

The main effect of γ-irradiation at 77 K in the lifetime spectra is the appearance of two long-lived components with lifetimes of τ ₃ = 1.2 ns and τ ₄ = 2.3 ns. They are associated with ortho-Ps trapped in vacancies and divacancies, respectively. Their intensities, I ₃ and I ₄, increase with dose showing an increase of the defect concentration. Simultaneously the intensity, I ₁, of a short-lived component mainly associated with para-Ps decreases from about 22% to about 3% at 11 Mrad. This is explained in part as a decrease in the Ps yield brought about by irradiation-created OH radicals (Eldrup, 1976). The results of heating crystals irradiated with 4.1 and 11 Mrad are shown in Figure 1. The measuring lime at each temperature was 3–4 d. There is a clear transition at 93 K. I ₁ increases sharply to its pre-irradiation value and I ₃ and I ₄ increase somewhat. τ ₃ is constrained to its average value 1.2 ns at all temperatures (to reduce the scatter of the fitted parameters (Eldrup, 1976)), but above approximately 105 K, τ ₄ increases strongly. This shows that at these temperatures some ortho-Ps becomes trapped in larger holes (vacancy clusters) where ortho-Ps lives longer. Simultaneously, the intensities I ₃ and I ₄ drop.

Angular-correlation measurements on ice irradiated at 4 K showed a strong decrease of the intensities of the narrow para-Ps peaks and the appearance of a “middle-broad” (3–4 mrad) component associated with para-Ps trapped in vacancies. On heating, the middle-broad component disappeared and the narrow component intensities increased strongly at about 100 K (paper by M. Eldrup, R. J. Douglas, and L. Lupton, in preparation). This is evidence that the main annealing occurs at about 100 K and that vacancies created at lower temperatures disappear in this stage.

Fig. 1. Lifetime results fir pure ice γ-irradiated at 77 K and measured as a function of increasing temperature. Circles are for polycrystalline samples (11 Mrad). Triangles are for one set of monocrystals (4.1 Mrad). Squares are for another set of monocrystals (II Mrad). The full and the dashed curves far I ₃ and I₄ are drawn for the 11 and 4.1 Mrad points, respectively. The changes at 93 K are associated with vacancy migration, the increase of τ₄ with the growth of the size of voids.

HF-doped ice

As briefly mentioned above, doping of ice with HF gives rise to the appearance of a 1.2 ns component in lifetime spectra and an angular correlation component 3–4 mrad broad. The intensities of these components increase with increasing HF concentration. Mogensen and Eldrup (1973) showed I ₃ (the intensity of the τ ₃ = 1.2 ns component) as a function of HF concentration for polycrystalline ice (frozen aqueous solutions). The results for doped monocrystals follow the same curves, but the scatter of the points is much larger than for polycrystals. Probably, this is partly due to the uncertainty in HF concentration (because of out-diffusion during sample handling) and partly due to uncertainty in cooling rate (see below). Mogensen and Eldrup (1973) also showed angular correlation curves illustrating the effect of HF doping.

The measurements carried out on crystals doped to 25 p.p.m. HF were particularly detailed. First the samples were quenched from —18^°C to liquid-nitrogen temperature (cooled in less than 1 min), and measurements were then made as a function of increasing temperature. The measuring time at each temperature was about 3 d. In Figure 2 the most important lifetime parameters for two different sets of samples are shown (τ ₃ was constrained to 1.20 ns). At low temperatures I ₃ ≈ 54%, i.e. all ortho-Ps becomes trapped in vacancies.

Fig. 2. Annealing of 25 p.p.m. HF-doped crystals after quenching from — 20^°C to liquid-nitrogen temperature. Lifetimes and intensities of the two long-lived components (trapped ortho-Ps) are shown. Open and closed circles ere for two different sets of samples. The increase of I ₄ is associated with migration of vacancies which form divacancies, the increase of τ₄ with the growth of the size of voids.

At about 100 K the intensity I ₄ of a longer-lived component (τ ₄ ≈ 2.5 ns) starts to increase. Its lifetime stays constant up to about 115 K above which it increases strongly, while the intensity drops to zero at 160 K. This increase of τ ₄ is very similar to that observed for γ-irradiated ice (Fig. 1) and the appearance of such a long-lived component shows again that some ortho-Ps becomes trapped in clusters of vacancies at these temperatures.

After the above-mentioned annealing up to 160 K, one set of samples was cooled slowly from 160 K. Measurements were made as a function of decreasing temperature (Fig. 3, open circles). Each measurement took 3–4 d and the cooling rate in between was 1 deg/min. Later on the sample was cooled to 95 K from 173 K with a rate of 1 deg/min (squares) and as fast as possible (triangles). Figure 3 shows that the lifetime τ ₃ is essentially constant at 1.2 ns, but its intensity I ₃ (and hence the concentration of vacancies) measured at low temperature is very dependent on cooling rate. This observation also holds true for other HF concentrations, for which quenching from temperatures above 170 K results in a larger I ₃ at low temperature than does cooling by 1 deg/min.

Finally, the set of samples was heated to —26^°C and measurements were made as a function of time. The intensity I ₃ of the τ ₃ = 1.2 ns component is shown in Figure 4. Within a week the intensity has dropped to that in pure ice (the hatched area indicates I ₃ (1.2 ns) for pure ice), but stays slightly below this value for the rest of the measuring period. The sample was melted afterwards and a concentration of about 1 p.p.m. HF was found. Similar results were obtained by measuring a 4 p.p.m. HF sample as function of temperature, viz, that at temperatures above c. 220 K I ₃ fell below the pure ice values, and the measured HF concentration after several days at higher temperatures was about 0.6 p.p.m. HF.

Fig. 3. Effect of cooling rate from about 170 K of a 25 p.p.m. HF-doped crystal. The “1.2 ns” lifetime and its intensity are shown. Cooling rates tire: Circles, 1 deg/min between points, 3–4 d measuring time at each point; average cooling rate. = 2 × 10^-3 deg/min. Squares, 1 deg/min. Triangles, quenched, i.e. about 100 deg/min.

Fig. 4. The intensity I₃ of the “1.2 ns” component in ice initially doped to 25 p.p.m. HF as function of time, measured at — 26^°C. The hatched area indicates results for pure ice. The decrease of I₃ is associated with out-diffusion of HF.

Vacancy migration

First let us discuss the changes in parameters taking place around 100 K in both irradiated and highly HF-doped crystals (Figs 1 and 2).

In the field of radiation chemistry, a number of investigations of irradiated ice have been carried out, in particular to study trapped electrons and OH and H radicals which are created during the irradiation. Irradiation at liquid-nitrogen temperature creates H and OH radicals, most likely by dissociation of water molecules by which H and OH are ejected into the crystal lattice leaving a vacancy behind. The interstitial OH radicals (identified by an absorption at 2 800 ?) are stable at liquid-nitrogen temperature but about 120 K (for a heating rate of a few degrees per minute) these radicals disappear (Ghormley and Hochanadel, 1971; Kawabata, 1976). The yield of trapped electrons (absorption at 6 200 ?) peaks at 120 K and decreases strongly at higher temperatures. Kawabata (1976) suggests that the electrons are trapped in radiation-created vacancies which anneal out around 120 K. This interpretation agrees very well with the picture obtained from the results of Figure 1.

The decrease of the intensities I ₁, and the narrow angular-correlation para-Ps peaks I _N with increasing dose is interpreted as a result of OH-radical formation. The presence of this radical with a high electron affinity is likely to decrease the Ps formation probability (Mogensen, 1974) which is seen as a reduction of I ₁ and I _N. With the slow heating rates used in ?Λ? measurements the OH radicals disappear at about 100 K causing the Ps yield, and hence I ₁ and I _N, to go up. Also I ₃ and I ₄ increase showing that more Ps is now trapped in vacancies and divacancies (since more Ps is formed). However, the increase of I ₃ is rather small, indicating a decrease of the vacancy concentration simultaneously with the disappearance of the interstitial OH radicals. These concentration changes can be brought about by the migration of either the radicals or the vacancies (or maybe both). In cither case vacancies will trap the OH radicals into substitutional positions and the vacancy (and interstitial OH) concentration will thus decrease in agreement with observations. We find there is convincing evidence to believe that at least the vacancies migrate at about 100 K. The increase of τ ₄ (Figs 1 and 2) above approximately no K shows that holes larger than mono- and divacancies (i.e. vacancy clusters) are formed. This requires the migration of mono- and/or multivacancies in this temperature region. Migration of vacancies much below 100 K seems unlikely, as an appreciable non-equilibrium vacancy concentration is present al 100 K. The angular correlation measurements also support this. The strongest evidence for vacancy migration, however, is the appearance of the 2.5 ns component (τ ₄, ? ₄) above 100 K in HF-doped ice (cf. the increase of I ₄ from o to 4% in Figure 2). It shows that divacancies (or maybe trivacancies) are formed at this temperature. There seems to be no other way for this to happen than by migration and pairing of vacancies. τ ₄ does not start to increase until 10–15 deg higher, suggesting that divacancies become mobile (or unstable) and cluster at and above this temperature into multivacancies.

Thus we can conclude from the above discussion, supported by the results of Kawabata (1976) that an appreciable migration of vacancies in ice take place over a time period of days at about 100 K. If the time scale is minutes the temperature has to be raised to about 120 K to give significant migration.

From the vacancy migration temperature it is possible to calculate the activation energy for migration. Experimentally we observe the temperature at which the concentration of vacancies (a supersaturation of which has been frozen-in at low temperature) decreases or at which divacancies are formed. This happens when vacancies migrate and reach sinks such as interstitials, other vacancies, and dislocations. In order to reach a sink a vacancy must, on the average, jump a certain number of times determined by the sink concentration. The jump frequency increases with increasing temperature as determined by the activation energy for vacancy migration. Hence, at a certain temperature the jump frequency will be large enough for vacancies to reach sinks within the time spent at that temperature (i.e. the measuring time), and a change in vacancy concentration or a divacancy signal will be measured. The migration energy can therefore be deduced from the migration temperature and the estimated sink concentration as follows (Eldrup, 1976).

The average number of jumps n of a vacancy during a time t is

(1)

where v is the average jump frequency. It is given by (Shewmon, 1963) :

(2)

where v ₀ is a frequency close to lattice vibration frequencies, k is the Boltzmann constant, T the absolute temperature, S _m the entropy change, and E _m the activation energy for vacancy migration. Ramseier (1967) finds that an expression by Zener for the total entropy change of diffusion is in good agreement with experients. Therefore we will use Zener’s expression for S _m:

(3)

where β is a constant (0.33 according to Ramseier (1967)) and T _mp is the ice melting temperature. By inserting the expressions for v and S _m (Equations (1) and (3)) into Equation (2) we get

(4)

E _m is given by the temperature T at which a vacancy on the average jumps n times during the time t. For vacancies of a concentration of about 10 p.p.m. to cluster, they have to jump n ≈ 10⁴–10⁵ times during the typical measuring time of t = 2 × 10⁵ s. This clustering sets in at a temperature of about T = 98 K. With these numbers and v ₀ = 4.7 × 10¹² s^-1 (Ramseier, 1967) we obtain E _m = 0.3 eV. If the concentration of sinks for vacancies is taken 100 times higher (i.e. n 100 times smaller) E _m will increase by 0.04 eV. Other uncertainties are of course present in this simplified determination of E _m, probably amounting to a total of about ±20%. Hence, the activation energy for vacancy migration is estimated at E _m = 0.34±0.07 eV. The activation energy for self-diffusion, which is generally accepted to take place by a vacancy mechanism, is well determined at Q = 0.62 eV (Ramseier, 1967). Since Q = E _m + E _f, where E _f is the vacancy formation energy, we get E _f = 0.28±0.07 eV in agreement with the estimates from pure ice (Mogensen and Eldrup, 1977[b], 1978). These new experimentally determined values for E _m and E _f are strikingly different from older estimates by Granicher (1958) and Fletcher (1970) of E _f ≈ 0.5 eV. A very important effect of this is that the equilibrium vacancy concentration close to 0^°C is at least 10⁴ times higher than previously assumed (Mogensen and Eldrup, 1977[a], 1978).

Void formation

In both irradiated and HF-doped ice the vacancy migration is followed at higher temperatures by a strong increase of τ ₄ and a simultaneous decrease of its intensity I ₄ (Figs 1 and 2). As mentioned earlier, the long lifetimes of several nanoseconds can only be explained as the lifetimes of ortho-Ps trapped in holes in the lattice, i.e. vacancy clusters or voids. The size of these voids (indicated by τ ₄) increases with temperature, at least up to 145 K. A rough estimate of their size suggests that a 10 ns lifetime is equivalent to a 15 ? diameter (Eldrup, 1976). The growth of small voids into larger ones probably takes place by a net emission of vacancies from the smaller voids and a net absorption of vacancies by larger voids (Mae, 1968) analogous to the way by which void coarsening lakes place in metals. During the void coarsening some vacancies are lost to other vacancy sinks, and the total void concentration decreases. Both effects contribute to the decrease of the intensity I ₄ of the void component.

Cavities in ice, much larger than the voids observed here, have been seen by optical and electron microscopes (Mae, 1968; Unwin and Muguruma, 1972), under, however, quite different experimental conditions. The advantage of ?A? measurements clearly is that it is possible to follow the formation and growth of the voids from the starting mono- (and in some cases di-) vacancy configuration. This has so far not been possible by other experimental methods.

Vacancy creation by HF doping

An important problem, of course, concerns the mechanism by which doping with HF creates vacancies in ice in excess of the pure-ice thermal-equilibrium concentration. Two fundamentally different possibilities should be considered. First, that the vacancies observed at low temperatures are created during cooling down of the crystal as a result of the stress in the crystal induced by thermal gradients. Secondly, that HF influences the equilibrium properties of vacancies at a given temperature, such as the vacancy concentration and jump frequency, with the effect that the equilibrium concentration increases and it becomes easier to quench-in a high concentration at low temperatures.

Let us first consider the first possibility. Jones and Gilra (1973) have shown that doping with HF strongly increases the dislocation density. The thermal gradients in the samples in our case are probably much larger than in their crystals. Therefore, we would expect the dislocation densities in our samples to be appreciably larger than those observed by Jones and Gilra, whether the role of HF is to ease or provoke dislocation formation (Jones and Gilra, 1973). One would expect that the creation and movement of dislocations would be accompanied by formation of some vacancies. Fast cooling to low temperature would then produce larger stresses and higher dislocation and hence vacancy concentrations for a given HF concentration than slow cooling would, in agreement with observations (Fig. 3). Since vacancies are mobile above about 100 K they will be in thermal equilibrium above this temperature. To explain the high I ₃ values at — 100^°C (Figs 2 and 3) and at —26^°C (Fig. 4) we would therefore have to assume that Ps can be trapped by dislocations, where ortho-Ps would have a lifetime close to its lifetime in vacancies. Against this interpretation is the fact that in pure ice no trapping has been observed at low temperature after quenching from about — 20^°C. Jones and Gilra (1973) observed a fivefold increase in the initial 10⁴–10⁵ cm^-2 dislocation density for a 5 p.p.m. HF concentration. To account for the amount of Ps trapping at this HF concentration (Mogensen and Eldrup, 1973) a concentration of trapping sites equivalent to about 1 p.p.m. of vacancies is probably required. If each molecular position along a dislocation line is assumed to be a trapping site equivalent to a vacancy, a dislocation density of about 10⁹ cm^-2 is needed. The concentration of vacancies created during the formation of the dislocations would probably be much less than 1 p.p.m. Hence, to account for the void formation a dislocation density of 10¹⁰–10¹¹ cm^-2 would probably be necessary. It seems unlikely that such high densities could be obtained even by the combined action of the present quenching and HF doping (Muguruma and Higashi, 1963).

Let us consider the second possibility, viz. that doping with HF increases the equilibrium vacancy concentration and makes it possible to quench-in at least part of the high concentration present at higher temperatures. Bilgram and others (1976) have previously discussed the creation of vacancies by Frenkel-pair formation as a result of HF doping to interpret NMR spin-lattice relaxation data. We shall discuss another possibility here. It seems very probable that there will be a certain binding energy between vacancies and one or more other defects in the ice lattice (e.g. F, ionic, or Bjerrum defects). In that case, the total concentration of vacancies will consist of the free vacancies in the bulk plus vacancies bound to defects. If for instance there is an attraction between vacancies and substitutional F, and/or L, and/or H₃O⁺ defects (or bound states of these) one would expect that HF doping would increase the total vacancy concentration. The maximum increase would be about the same order of magnitude as the HF concentration. Vacancies present at low temperatures will (at least partly) be quenched-in vacancies. A larger cooling rate is expected to result in a larger quenched-in concentration of vacancies. This is in qualitative agreement with observations (Fig. 3). Very slow cooling (≈ 2 × 10^-3 deg/min) results in a relatively low vacancy concentration at 95 K, giving rise to I ₃ = 20% ; a cooling rate of 1 deg/min results in a higher concentration (I ₃ = 49%), while quenching gives I ₃ = 54%. (Thus, quenching gives a higher I ₃ (54%) at 95 K than the one observed at 160 K (41%, Fig. 3), although the quenched-in concentration of vacancies at 95 K cannot be higher than the concentration at 160 K. This difference in I ₃ is associated with a decrease of the Ps trapping rate per vacancy with increasing temperature (paper by M. Eldrup, O. E. Mogensen, and J. H. Bilgram, in preparation) and will not be discussed further here.) With a higher equilibrium vacancy concentration than in pure ice, it is of course easier to quench-in a certain vacancy concentration at low temperatures. But the results illustrated in Figure 3, viz. that even a rather slow cooling rate (1 deg/ min) is sufficient to quench-in an appreciable vacancy concentration, show that the equilibrium vacancy concentration at these temperatures is attained rather slowly. Only for very low cooling rates (circles in Fig. 3) do we see a clear decrease of the vacancy concentration with decreasing temperature, maybe approaching an equilibrium situation. If a binding exists between the vacancies and other defects it is reasonable to expect that this will reduce the vacancy mobility. Also the mobility of vacancies not bound to other defects might be influenced by the presence of defects. Actually, the migration of vacancies in 25 p.p.m. HF-doped ice as evidenced by the increase of I ₄ (Fig. 2) seems to lake place 10–15 deg above the migration in pure, irradiated ice (Fig. 1). Hence, it is not unreasonable to expect that vacancies in thermal equilibrium at higher temperatures can be quenched-in at lower temperatures in HF-doped ice.

We have discussed two fundamentally different mechanisms which might explain the increase in vacancy concentration observed in HF-doped ice. It seems that the more plausible is the latter mechanism by which the vacancy equilibrium properties are influenced by the doping, resulting in an increased equilibrium vacancy concentration by vacancy-defect binding, and resulting in a decrease of the average vacancy jump frequency, making it easier to quench-in vacancies at low temperature.

Our earlier, less detailed, measurements on ice doped with other impurities than HF showed that in all cases trapping of Ps (probably in vacancies) took place as a result of doping (Milošević-Kvajić and others, 1972; Eldrup and others, 1972). Thus, the increase in vacancy concentration at low temperatures seems to be a general effect of doping. The reason that HF apparently gives the largest effect may be its larger solubility. However, we shall not go into further details here. To obtain a more quantitative understanding of the mechanisms of vacancy formation and migration in ice doped with HF (and other impurities), more experimental as well as theoretical work is needed, the latter taking into account the interaction of vacancies with other defects.

Diffusion of HF

The decrease of I ₃ with time for an initially 25 p.p.m. HF doped crystal at —26^°C (Fig. 4) (and also for an initial 4 p.p.m. HF doping) is ascribed to out-diffusion of HF, as evidenced also by the decrease of the HF concentration to the 1 p.p.m. level found in the crystal after the measurements. Using a simple trapping model which relates I ₃ to the vacancy concentration (West, 1973), and assuming HF and vacancy concentrations to be proportional, we find that during the first week HF diffuses with a diffusion constant that is in agreement with the results of Haltenorth and Klinger (1969) within the uncertainty of about a factor of two. For longer times than a week, I ₃ falls below the pure-ice value. This might be a result of a decrease in trapping rate per vacancy due to the presence of HF, or it might be that trapping in vacancies bound to other defects gives a slightly shorter ortho-Ps lifetime than trapping in free vacancies in pure ice, which will lead to the same experimental result.

The HF concentration of 1 p.p.m. measured after 30 d is much larger than expected from the diffusion constant given by Haltenorth and Klinger (1969). This could be because an equilibrium has been established between the HF in the crystal and the out-diffused gaseous HF surrounding the sample in the sample container.

Another explanation could be the one proposed by Bilgram and others (1976) which postulates a fast interstitial diffusion at high HF concentrations and a slow substitutional diffusion at low HF concentrations. Haltenorth and Klinger (1969, 1977) have proposed that the fast HF diffusion might be along dislocations or small-angle grain boundaries.

Effects at 100 K

A number of different effects have been observed to take place in ice at temperatures close to 100K such as in electrical polarization and depolarization (Dengel and others, 1964; Sixou and Jeneveau, 1973), in fluorescence and thermoluminescencc (Grossweiner and Matheson, 1954) in heat capacity (Pick, 1969; Matsuo and others, 1973), in elastic constants (Helmreich, 1969), in yield of trapped excess electrons (Kawabata, 1976), and in annealing of damage created by protons in channelling experiments (Golecki and others, 1976; Golecki and Jaccard, 1978). With the results of the present PAT investigations on ice in mind, it seems worthwhile to consider the possible role of vacancy migration for the explanation of these effects. Only Kawabata (1976) seems to have done so, as mentioned earlier.