Physica B 289}290 (2000) 476}481
Hyper"ne coupling constants of muonium in sub and supercritical water Khashayar Ghandi, Jean-Claude Brodovitch, Brenda Addison-Jones, Paul W. Percival*, Joachim SchuK th TRIUMF and Department of Chemistry, Simon Fraser University, Burnaby, B.C., Canada V5A 1S6
Abstract Muonium, like the hydrogen atom, is a hydrophobic solute in water under standard conditions. Molecular dynamics simulations suggest that the free atom exists in a transient clathrate-like cage of hydrogen-bonded water molecules. The hyper"ne constants of Mu and H are very close to their vacuum values, supporting the picture of an atom `rattlinga around in a hole in the liquid. Muonium has now been studied in water over a wide range of temperatures and pressures, from standard conditions to over 4003C and 400 bar (the critical point is at 3743C, 221 bar). Drastic changes occur in the properties of water over this range of conditions, so large changes in the muonium hyper"ne constant might well be expected. Surprisingly, the changes are small. The hyper"ne coupling constant goes through a minimum in the subcritical region, and then increases toward the vacuum value under supercritical conditions. 2000 Elsevier Science B.V. All rights reserved. PACS: 31.30.Gs; 31.70.DK; 36.10.Dr; 76.75.#i Keywords: Muonium; Supercritical water; Hyper"ne coupling
1. Introduction Under standard conditions water shows many anomalies when compared with other solvents. The unusual properties are usually ascribed to extensive hydrogen bonding , so it is reasonable to expect that the drastic changes which occur when water is heated under pressure are due to changes in local structure (intermolecular bonding). The changes in the physicochemical properties of water as temper-
* Corresponding author. Fax: 1-604-291-3765. E-mail address: [email protected]
(P.W. Percival). Current address: Institut fuK r Strahlen- und Kernphysik, UniversitaK t Bonn, Nussallee 14-16, 53115 Bonn, Germany.
ature and pressure are varied make it an adjustable solvent for industrial use. Indeed, above its critical point (3743C, 221 bar) water has many characteristics of an organic solvent. Applications range from a medium for toxic waste destruction  to cooling of nuclear-power reactors . Despite the fast growing industrial interest, and in spite of increased theoretical study, there is a lack of explicit knowledge of the microscopic properties of supercritical aqueous solutions. Muonium, like the hydrogen atom, is a hydrophobic solute in water under standard conditions. Molecular dynamics simulation suggests  that the free Mu atom exists in a transient clathrate-like cage of hydrogen-bonded molecules in water. The hyper"ne constants of Mu and H in water are very
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 0 ) 0 0 4 4 0 - 3
K. Ghandi et al. / Physica B 289}290 (2000) 476}481
close to their vacuum values , supporting the picture of an atom &rattling' around in a cavity in the liquid. However, interactions with the water molecules comprising the cage should induce changes in the hyper"ne constant, and thus muonium could be used as a probe to study sub- and supercritical water at the molecular level. The high energy of muons and their decay positrons permits relatively easy access to the sample in a high pressure vessel, and this is a considerable advantage over optical spectroscopic methods. Previous measurements on Mu in water were limited to room temperature; H and D atoms were studied as a function of temperature up to about 903C . As part of our investigations of muonium chemistry in supercritical water  we have determined Mu hyper"ne constants over a wide range of temperatures and pressures, from standard conditions to over 4003C and 400 bar.
2. Experimental and results The experiments used `backwarda muons from the M9 beam line at TRIUMF in Vancouver. Spectra were collected with the standard transverse "eld muon-spin rotation technique [7,8]. The hyper"ne constant (hfc) was determined from the splitting of the muonium precession frequencies  in an intermediate "eld, typically 200 G. The high-pressure vessel used for measurements on H O is cylindrical with about 110 cm internal volume. It is made of nonmagnetic stainless steel and has a titanium window. Later measurements on D O used a sample cell with the same geometry but made of titanium. The pressure system includes an expansion vessel to accommodate changes in sample density, so that temperature and pressure can be adjusted independently. Details of our experimental set up are described elsewhere . The hyper"ne constant was determined for Mu in both H O and D O under similar thermodyn amic conditions, mostly at pressures around 250 bar. The results are displayed in Fig. 1. As indicated by the curve, the hfc values for Mu in H O appear to go through a shallow minimum as the temperature is increased. Although the error bars are too large to show any signi"cant temperature variation
Fig. 1. Hyper"ne constants determined for Mu in H O (䊐) and D O (E) as a function of temperature at roughly constant pressure (&250 bar). Literature data  for Mu in standard conditions are denoted by the open (H O) and "lled (D O) diamonds. The dotted line represents the vacuum hyper"ne value (4463 MHz). Table 1 Hyper"ne constants (A ) for Mu in H O at di!erent pressures l (P), temperatures (¹), and densities (o) P/bar
A /(MHz) l
127 250 345 361 350 350 352 335 350 250 244 382 250 127 250 250 128 250 126 249 77
350 400 420 425 420 420 410 400 400 375 357 375 325 285 285 250 200 200 150 150 93
0.063 0.166 0.311 0.317 0.324 0.324 0.408 0.453 0.475 0.505 0.598 0.602 0.692 0.751 0.769 0.821 0.873 0.881 0.924 0.930 0.967
4470(7) 4456(3) 4431(7) 4431(8) 4442(6) 4437(4) 4438(8) 4447(11) 4441(10) 4443(3) 4442(5) 4438(5) 4425(5) 4433(9) 4434(8) 4434(7) 4413(7) 4412(6) 4417(7) 4425(9) 4432(8)
of the D O data, there seems to be a consistent trend to higher hfcs in D O than in H O under similar conditions. This "nding is consistent with the isotope e!ects reported by Roduner et al. 
K. Ghandi et al. / Physica B 289}290 (2000) 476}481
Fig. 2. Hyper"ne constants of Mu in H O as a function of water density. Data are at 127 bar (䊐), 250 bar (*), and 350}370 bar (*). The value for standard conditions (䉫) is from Ref. . The dotted line represents predictions based on the model and parameters given by Roduner et al. . A similar model but with density-dependent coe$cients, de"ned by Eqs. (3) and (4), results in the predictions denoted by the dashed (250 bar) and solid (350 bar) curves.
for Mu in water at room temperature, and also for H in H O and D in D O. The temperature dependence evident in the H O results is also consistent with the results  for H and D atoms in liquid water at low temperatures (a linear trend with negative-temperature coe$cient). At higher temperatures our Mu data pass through a minimum and rise towards the vacuum value (4463 MHz) in the supercritical region (Fig. 1). In some experiments pressure was varied but temperature was held constant; it was found that in regions where water has low compressibility the pressure does not signi"cantly a!ect the hfc. The data for Mu in H O is collected in Table 1 and displayed as a function of density in Fig. 2. Compared to the drastic changes in the properties of water over this range of conditions, the changes in hfc are small.
3. Discussion The isotropic hyper"ne constant measured in the experiments reported here arises from the Fermi contact interaction and is proportional to the unpaired electron spin density at the nucleus. Interac-
tion with the solvent molecules perturbs the electronic wave functions of solvated atoms and hence the hfc di!ers from the vacuum value. Roduner et al. analyzed their experimental data on the hfc of hydrogen isotopes in terms of a perturbation theory for the atoms in a spherical parabolic model potential . The conclusion was that only spin delocalization onto the surrounding water molecules can describe isotope e!ects of the magnitude observed. Such an interaction reduces the hfc of a solvated atom from the vacuum value (A ). The degree of spin delocalization depends on the average distance of the atom from the molecules of the solvent cage. The relative hfc can be expressed as an expansion in terms of the displacement from the centre of the cage. The temperature e!ect comes from the quadratic term after applying the Boltzmann distribution to the population of the energy levels (E ): T A/A "1#C #C (M K)\ + (#v #v #v )exp(!E /k¹) V W X T ; T . (1) exp(!E /k¹) T T Roduner et al.  "tted Eq. (1) to their experimental data with C , C and K as adjustable para meters. This approach is consistent with the highdensity part of our data (0.85}1.0 g cm\) using the same values for these parameters. However, it is obvious from the dotted line in Fig. 2 that this model does not "t the entire range of our experimental results. This is not surprising, since at higher temperatures (supercritical region) the behaviour of water would be more gas-like and a cage model would not be appropriate at the low-density extreme. An alternative approach is to base a model on collision theory. A transient transfer of spin polarization occurs every time Mu collides with a water molecule. Therefore, we expect the shift in the hyper"ne frequency to be proportional to the average collision rate (Z) between Mu and water molecules. Thus, from standard kinetic theory:
8k¹ (A !A)JZ"opp , (2) Mp P where p represents the interaction distance between Mu and H O, and M is their reduced mass. As can P
K. Ghandi et al. / Physica B 289}290 (2000) 476}481
Fig. 3. Collision frequency (relative to standard conditions) of Mu in water as a function of temperature at 250 bar.
Fig. 4. Experimental values and model predictions for the deviation of the hfc from its vacuum value as a function of density. The curves represent predictions of the collision model at 127 bar (dashed), 250 bar (solid) and 350 bar (dotted). The vertical dotted line denotes the density at the critical point (0.322 g cm\).
be seen from Fig. 3, the collision frequency "rst increases with temperature, due to the (¹) factor, but then falls as the density (o) is reduced at elevated temperatures. These competing e!ects result in a broad maximum in the range of temperatures between 1503C and 2003C, the same region where the hyper"ne coupling constant has its maximum deviation from the vacuum value. This supports the idea of a correlation between the hfc and the collision frequency. Fig. 4 shows plots (at di!erent pres-
sures) of the relative collision frequency scaled to "t the relative deviation of the hfc, (A !A)/A . As can be seen, both the model and the experimental results pass through a maximum in the high density region. It is remarkable that the simple collision model appears to "t the data as well as it does. This is due in part to the relatively poor quality of the data, which does not allow precise determination of the maximum in Fig. 4. Similarly, it is not clear if the apparent jump in hfc between densities 0.82 and 0.87 g cm\ is signi"cant. Our data seem to change rapidly with density above 0.90 g cm\, where the collision model predicts only a mild dependence * according to Eq. (2) the collision frequency depends mainly on (¹) at densities close to 1 g cm\, where the liquid is relatively incompressible. On the other hand, Roduner et al.  found a direct dependence on temperature for the hfcs of H and D atoms. Another region where the data may deviate from the collision model predictions is in the vicinity of the critical point of water (0.32 g cm\), where a local maximum can be seen in the hfc deviation (Fig. 4). This can be explained if water molecules are clustered around Mu. If the water density in the vicinity of Mu is greater than the bulk water density, Mu is subject to an increased collision frequency, resulting in a larger deviation of hyper"ne coupling constant from the vacuum value. Such a local solvent density enhancement around Mu is consistent with other data on supercritical #uids . It is important because the hydrophobic nature of Mu under standard conditions is similar to nonpolar organic compounds and gases used in supercritical water oxidation reactors. Interpretation of the local-density enhancement calls for deeper analysis of the Mu}H O interactions, for example by means of molecular dynamics simulations. Most likely it will also take such powerful tools to adequately model the behaviour of the hfc over the whole density range, from a dense liquid with structure to a low-density gas. However, it is possible to extend the cavity model (Eq. (1)) in a purely empirical manner, by allowing C , C and K to be polynomial functions of density. In preliminary investigations we found that K, the force constant for harmonic vibration of Mu in the solvent cage,
K. Ghandi et al. / Physica B 289}290 (2000) 476}481
Table 2 Coe$cients for the polynomial expansion of C and C (Eqs. (3) and (4)) used to "t Eq. (1) to the data (Fig. 2) C /10\ x x x x C /nm\ y y
5.01 20.8 !72.9 46.1 0.0089 !1.059
a bubble in water. To sustain the bubble the outward pressure exerted by the kinetic motion of Mu (proportional to K) must match that of the surface tension forces which would otherwise shrink the cavity. The e!ective surface tension of the cavity can be attributed to hydrogen bonding between the water molecules. Thus, K depends on the extent of hydrogen-bonding, and the proportionality with density parallels the "nding of Guillot and Guissani  that hydrogen-bonding scales with density for liquid water near the liquid gas coexistence curve. The dependence of the coe$cients C and C on density are not so easily interpreted. As shown in Fig. 5, C becomes positive at low densities, which would mean that compression of the Mu wave function, which increases the hfc, is the dominant interaction for static Mu under such conditions. At higher densities spin delocalization and attractive interactions result in an overall negative value for C . Coe$cient C characterizes the dynamic ef fects, dominated by spin delocalization. It seems reasonable that the time-averaged interaction becomes smaller as the density falls, and thus larger displacements of Mu from the cavity centre are required to produce an e!ect.
Fig. 5. Best "t values of C and C as a function of density.
scales with density with an exponent 0.97, i.e. K is nearly proportional to density. Therefore, in subsequent "ts we assumed that K is indeed proportional to the density, with a proportionality constant of 3.8 N m\, the value obtained by Roduner et al. for standard density. C and C were assumed to vary with density o according to C "x #x (o/o )#x (o/o )#x (o/o ), (3) C /nm\"y #y (o/o ), (4) where o is the density at standard temperature and pressure. The best "t parameters are given in Table 2, and the coe$cients C and C are plotted in Fig. 5. Predictions using these parameters are shown by the solid and broken curves in Fig. 2, for 350 and 250 bar, respectively. The dependence of K on density can be rationalized by viewing the cavity in which Mu resides as
Acknowledgements We thank Syd Kreitzman and the sta! of the TRIUMF lSR Facility for technical support. This research was "nancially supported by the Natural Sciences and Engineering Research Council of Canada and, through TRIUMF, by the National Research Council of Canada. J. SchuK th was the recipient of a NATO postdoctoral fellowship administered by DAAD (Deutscher Akademischer Austauschdienst).
References  F. Franks, Water, The Royal Society of Chemistry, London, 1984.  Th. Hirth, E.U. Franck, Ber. Bunsenges. Phys. Chem. 97 (1993) 1091.
K. Ghandi et al. / Physica B 289}290 (2000) 476}481  Y. Oka, S. Koshizuka, T. Yamasaki, J. Nucl. Sci. Technol. 29 (1992) 585.  B. De Raedt, M. Sprik, M.L. Klein, J. Chem. Phys. 80 (1984) 5719.  E. Roduner, P.W. Percival, P. Han, D.M. Bartels, J. Chem. Phys. 102 (1995) 5989.
 P.W. Percival, J.-C. Brodovitch, K. Ghandi, B. AddisonJones, J. SchuK th, Phys. Chem. Chem. Phys. 1 (2000) 4999.  B.D. Patterson, Rev. Mod. Phys. 60 (1988) 69.  E. Roduner, Appl. Magn. Reson. 13 (1988) 1.  S.C. Tucker, Chem. Rev. 99 (1999) 391.  Y. Guissani, B. Guillot, J. Chem. Phys. 98 (1993) 8221.