Diffusion and CIDEP of H and D atoms in solid H2O, D2O and isotopic mixtures

Chemical Physics North-Holland

164 (1992)

421-437

Diffusion and CIDEP of H and D atoms in solid H20, D20 and isotopic mixtures * David

M. Bartels,

Ping Han

Chemistry Dwu~on, Argonne Natlonal Laboratory, 9700 South Cass Avenue, Argonne, IL 60439, USA

and Paul W. Percival Department of Chemistry and TRIUMF, Simon Fraser University, Burnaby, Britrsh Columbra, Canada V5A lS6 Received

17 December

199 1; in final form 6 April 1992

Hydrogen and deuterium atoms have been studied by pulsed EPR spectroscopy in polycrystalline samples of HZ0 ice, D20 ice, and various isotopic mixtures. At high temperature ( - 10°C) the pattern and the time-dependence of the EPR line intenstties are similar to previous results for H and D m liquid water. Chemically induced dynamic electron polarization (CIDEP) is generated in second-order atom combination reactions. The CIDEP behavior was found to change over the temperature range studied ( - 5°C to 130”(Z), consistent with additional contributions from spur reactions, and at lower temperatures, geminate recombination of (D...OD) radical pairs. Transverse spin relaxation times were measured by the spin-echo technique, and interpreted in terms of translational motion of free atoms diffusing through the ice lattice. One surprising result is that D atoms diffuse faster than H atoms below 200 K. This 1s explained as a vibrational zero point energy effect, by applying transition state theory to a model in which the diffusing atom must pass through a tight “bottleneck” in the electronic potential surface, as it passes from one minimum energy site in the lattice to the next. The H and D spin relaxation rates were successfully simulated by means of a semiclassical potential which was constructed by pairwise addition of atom-atom contnbutions represented by modified Buckmgham potential functions. Extension of the model to include tunneling resulted in little change to the tit of the H and D data. Although predictions of muonium diffusion rates using the same potential do not give quantitative agreement with published results from spin relaxation measurements, they do serve to illustrate the dominant effect of tunneling over a wide temperature range for that light atom.

1. Introduction

Although the radiation chemistry of liquid water has been intensively studied over the past four decades [ l-4 1, comparatively little work has been done on pure ice [ 5- 12 1. The action of ionizing radiation on ice crystals has recently taken on some importance, because this chemistry may impact the “ozone hole” problem in atmospheric chemistry [ 13 1. Ra* Work at Argonne performed

under the auspices of the Office of Basic Energy Sciences, Division of Chemical Sciences, US-DOE under contract number W-31-109-ENG-38. The work of P.W. Percrval is supported by the Natural Sciences and Engineering Research Counctl of Canada. 0301-0104/92/%

05.00 0 1992 Elsevier Science Publishers

diolysis of ice results in the same primary products as water, H30+, e-, H and OH, but the time-scales of the recombination reactions differ. In particular, recombination of the ions proceeds one to two orders of magnitude faster in ice due to the much higher ion mobilities in the crystal [ lo- 12 ] . On the other hand, OH radicals move orders of magnitude slower in ice than in water [ 9,141. Surprisingly, very little seems to have been published on the behavior of hydrogen atoms in pure ice [ 57,141. The work described in this paper represents a large step in tilling this information gap. EPR detection was first reported in 1959 for hydrogen atoms trapped in irradiated pure ice [ 5 1. The hyperfine constant is very close to that for H in vac-

B.V. All rights reserved.

422

D.M. Bartels et al. / Diffuuon and CIDEP of H and D atoms

uum, implying that the atom resides in an interstitial site in the crystal, with essentially zero net interaction with the surrounding lattice atoms. The signal is long-lived at 4.2 K, but decays rapidly above 50 K due to the onset of H atom mobility [ 6 1. By using continuous electron irradiation Shiraishi et al. [7] were able to study H in neutral ice at high temperatures, up to the melting point. However, it was only possible to detect EPR signals above 160 K, where motional line narrowing and chemically induced dynamic electron spin polarization (CIDEP) [ 15,161 serve to enhance signal intensities. Thus, direct study of H atoms in ice has not been possible for the temperature range x 50- 160 K (except for highly acidic ice, an entirely different medium [ 8 ] ). In contrast to H, the exotic light isotope muonium (Mu = u+e- ) has been studied in ice over the whole temperature range from 4 K to the melting point [ 1723 ]. A small axially symmetric hyperfine anisotropy was discovered, suggesting that Mu (and by analogy H) diffusion is preferentially along the c axis channels of the ice lattice [ 221. The focus of the most recent work [ 171 was the study of muonium diffusion via measurements of spin relaxation rates. One of the conclusions is that muonium still diffuses rapidly through the ice lattice at temperatures below 10 K, in contrast to H atoms, which are considered trapped below 50 K [ 6 1. There is also disagreement between muonium diffusion rates at higher temperatures and those reported for H atoms by Shiraishi et al. [ 71. Our desire to resolve this discrepancy was the main motivation for the work described in this paper. An important consideration is that both the muonium and the hydrogen atom diffusion rates are determined indirectly. The methods are very different from a practical point of view, but they are both based on the spin relaxation of the unpaired electron in the atom, caused by modulation of the dipolar interaction with magnetic nuclei (H, D as appropriate) in the lattice as the Mu/H atoms jump from one interstitial site to another. The muonium spin relaxation rates were measured directly in the time domain, but Shiraishi’s data for H were deduced from measurements of EPR linewidths [ 71. A major problem with this procedure is the assumption that the measured linewidths are due entirely to the relaxation mechanism in question. In particular, an anisotropic hyperfine interaction gives rise to inhomogeneous broad-

ening [ 241. The muonium study avoided this by using single-crystal samples [ 171. The H and D atom work reported here involved polycrystalline samples, but avoided inhomogeneity effects by means of the spinecho technique [ 25 1. The pulsed EPR kinetics techniques used in this study have previously been applied to several other radiolytically generated transient radicals [ 26-301 including H atoms in water [30 ] and fused silica [28]. A common feature of the EPR spectra is the existence of anomalous line intensities at short times, due to the CIDEP phenomenon [ 15,161. The spectra of H and D atoms in water are always observed with low field line in emission and high field line with enhanced absorption (an E/A pattern). The EPR line intensities, pumped by the spin-dependent radical recombination reactions, typically reach maximum amplitude within roughly a microsecond (depending on initial radical concentrations generated), and then decay away more slowly (primarily by bimolecular spin exchange) [ 301. Study of the small initial spin polarization immediately (within z 30-50 ns) after the radiolysis pulse reveals an E/A pattern as well [ 301, which must arise in random cross-recombination reactions in the radiolysis spurs, because geminate recombination of singlet H...OH pairs should give rise to the opposite A/E polarization pattern [ 15,161. (No evidence for radical pair mechanism spin polarization in OH recombination reactions has ever been reported in any case [ 301, presumably because the (subnanosecond [ 3 1 ] ) spin relaxation of OH is too fast to maintain the critical spin coherence in an encounter pair.) Comparison of the CIDEP kinetics in ice with those in liquid water is of great interest to gain insight into the important geminate and random free radical recombination pathways in the solid. Following a description of the experimental procedure (section 2 ), this paper contains a short report (section 3) on the CIDEP results and their implication for the chemistry of hydrogen atoms in ice. However, the main emphasis is on the interpretation of the spin relaxation results (section 4) in terms of hydrogen atom diffusion, and in particular, the isotope effects evident in that diffusion. In sections 5 and 6 a simple transition state theory model is developed and evaluated to correlate the observations on Mu, H and D diffusion.

D.M. Bartels et al. / D~ffuslon and CIDEP of H and D atoms

2. Experimental Samples of triply-distilled HZ0 and D20 were purged of oxygen by bubbling with N2 or Ar and then degassed by a pump/shake method. The degassed samples were transferred under helium into 2 mm i.d., 3 mm o.d. Pyrex sample tubes. The samples were again degassed on a vacuum line and the tubes were sealed off. Immediately prior to use, the water was frozen by immersion in liquid nitrogen, yielding a random polycrystalline ice sample. The tube was then transfemed to the pre-cooled cooling jacket in the EPR cavity. During irradiation, the samples were maintained at a constant desired temperature using the low temperature flow system shown in fig. 1. Several liters of isopentane were kept in a large well-stirred dewar flask. The reservoir temperature was regulated to ? 0.5”C by controlling the flow of cold nitrogen gas through a heat exchange coil. The isopentane was circulated through the Pyrex jacket and around the sample tube in the EPR cavity. Testing showed that the sample temperature, as measured with a thermocouple embedded in the ice, remained within 1C of the isopentane coolant for electron beam currents of up to 1 PA. All of the experiments reported here were performed at approximately this (average) current level. Thus the basic uncertainty in sample tempera-

Cold isopentane out

Fig. I. Schematic outline of expenmental arrangement. Isopentane coolant temperature was controlled to f 1 “C to provide the variable sample temperature.

423

ture is + 1 ‘C. Dry air was used to flush the EPR cavity and prevent water condensation on the cold walls of the cooling jacket. Tests with a solid Pyrex rod in place of the ice revealed no detectable H atom signal from the Pyrex or isopentane over the 130” C to - 5 oC temperature range investigated. The pulsed EPR kinetics techniques used in this study have been reported in several previous publications [ 26-301. Briefly, the sample is irradiated with a 3 MeV electron pulse of duration lo- 100 ns, creating the free radical species of interest (H and D atoms). After some programmed time delay t, a 25 ns 7c/2 microwave pulse is applied, and a free induction decay (FID) is observed. After a period r on the order of some microseconds ( 20.5 us) a second microwave (A) pulse is applied, which generates an electron spin echo at time 25 after the initial x/2 pulse. The initial amplitude of the FID and the amplitude of the spin echo are measured in separate gated integrators (Stanford Research Model SR50). Both sigcnls are digitized and recorded by a PDP 11/23 computer which also controls the digital delays (t and T). In the present study, integration gate widths were 150 ns for the FID amplitude and 200 ns for the spin echo. H and D atom Tzs were measured by the standard spin echo envelope decay method [ 25 1. The general procedure is to record a baseline scan in a time window just prior to the electron pulse, and then to record several z scans with the n/2 pulse placed at different times t after the electron pulse ( t = 20,60, and 100 ps was most often used). A systematic trend toward longer effective T, at longer times t is a good indication that bimolecular spin exchange has affected the Tz measurement. The experiment was repeated at lower and lower radiation doses until the T2 result was independent of the dose/radical concentration. ac coupling of the EPR signal into the gated integrator with a 10 kHz high-pass filter was found to produce a very significant S/N improvement over dc coupling, but only at the cost of baseline linearity as the R pulse was scanned in time relative to the x/2 pulse. By inverting the phase of the n/ 2 pulse, the sign of the spin echo signal is changed without changing the baseline. Calculating the difference of two sets of alternate-phase echo data collected back-to-back was found to both improve the S/N and remove the baseline artifacts. For the deuterium lines, it was also necessary to subtract aback-

D.M. Bartels et al. / D~ffislon and CIDEP of H and D atoms

424

on the radiation dose in the pulse, i.e., on the initial radical concentrations.) Given the very fast recombination of ionic species in ice [ lo- 121, only H + H or possibly H+OH reactions can be responsible for the polarization generation on a microsecond timescale. The high field (ml= - 1) and low field lines are antisymmetrically (m I= + 1) deuterium polarized to roughly the same extent, which is typical of dominant ST0 mixing in the radical pair CIDEP mechanism. The hydrogen low field (ml= + l/2) line becomes polarized much more strongly than the high field (m,= - l/2) line. This indicates that ST_ mixing is important in the H atom CIDEP, thanks to the very large hypertine coupling of the H proton. One very important aspect of the kinetics at z - 10°C is the presence of an initial signal immediately after the electron pulse, which already shows the E/A pattern characteristic of random radical pair CIDEP. Just as in liquid water, this prompt signal must arise from radical-radical cross-reactions in radiolysis spurs [ 301. As the temperature is lowered, the kinetics become slower, the point of the maximum CIDEP signal moves to longer times, and the maximum signal becomes generally weaker. These

ground echo signal due to the paramagnetic centers in the irradiated Pyrex tubing. Experiments were performed on samples of HzO, DzO, 50% H,0/50% DzO, and 10% H20/90% D20.

3. Spin polarization and chemical reactions Typical results of the FID kinetics experiments are shown for H in HZ0 ice in fig. 2 and for D in DzO ice in fig. 3. At the high temperature limit ( = - lO’C), the signals are very similar to those found in liquid water [ 301. The kinetics are dominated by chemically induced dynamic electron polarization (CIDEP) generated in random (F pair) spin-dependent chemical reactions [ 15,16 1. This gives rise to the characteristic low field emission/high field absorption (E/A) pattern in the spectra of both H and D. The polarization grows in over a period of z 20 us as the second-order recombination reactions proceed, until a maximum is reached when the polarization rate (proportional to recombination rate) equals the rate of spin relaxation. (Although not shown in these figures, the positions of the maxima in time depend

0.8 (-1/2)line 0.4 -

-67 “C

t

-1.2 -

E

a I

0.8 0.4 -

(-1/2)line F

(+112)line

-38 “C

-96

"C

r'

0.0-0.4 -

1,

-0.8 .,.2j

(+1/2)line , 0

,

,

,

,

-0.8 ,

400

,

,

)

,

,

3

(+1/2)line

800

Time (psec) Fig. 2. Kintetics of the H atom magnetization in HZ0 ice at several different temperatures, following irradiation with a 55 ns electron pulse. The CIDEP phenomenon is readily observed, and the time-scale mdxates fast motion of the H atoms through the lattice.

425

DM. Bartels et al. / Diffusion and CIDEP ofH and D atoms

E

0.5

3 0

0

400

0

800

Time

400

800

(psec)

with a 55 ns electron in D20 ice at several different temperatures, following irradiation Fig. 3. Kinetics of the D atom magnetization pulse. The CIDEP effect is readily apparent, and the time-scale indicates fast motion of the D atoms through the lattice.

effects are all consistent with slower diffusion of the H and D atoms. However, a dramatic change in initial spectra is observed as shown in the insets of figs. 2 and 3 for the lower temperatures. The initial line amplitudes are plotted versus temperature in figs. 4 and 5 for H in Hz0 ice and D in D20 ice, respectively. By -30°C for D, and by -70°C for H, both

(-1)line

\

2.0 q 5 E

(-112)line

1.0

9 C E I

I .t: 5

-100

\

-40

/

\

(+1/2) line

-2.0 -100

-80

-60

Temperature

-40

-20

0

(“C)

Fig. 4. Initial amplitude of the H magnetlzatlon temperature, following a 55 ns lrradlation pulse.

signals versus

-20

0

(“C)

Fig. 5. Initial amplitude of the D magnetization temperature, followmg a 55 ns irradiation pulse.

-1.0

-120

-60

Temperature

0.0

I

-80

signals versus

the high and low field lines are initially in emission. By -5O”C, the high field line of D is even more strongly polarized in emission than the low field line. In the parlance of chemically induced magnetic resonance, this is an A/E multiplet pattern superimposed upon overall net emissive polarization [ l&l6 1. The initial A/E multiplet pattern is characteristic of

426

D.M. Bartels et al. / Diffusion and CIDEP of H and D atoms

geminate radical pairs produced with singlet spin phasing [ 15,16 1. This is in fact the polarization pattern one should expect from geminate recombination of singlet (D...OD) radical pairs. Presumably the initial H spectrum is also strongly influenced by this geminate reaction, but the presence of strong ST_ emissive polarization masks the A/E multiplet polarization from the ST0 mixing. The geminate polarization is presumably absent at higher temperatures because the spin dephasing time of OH (OD) is very short, probably subnanosecond, as in liquid water [ 3 11. When the T, of OH becomes longer than one or two nanoseconds, the geminate spin coherence persists long enough [ 15,161 for CIDEP effects to become manifest. The temperature dependence of the prompt signals was found to be qualitatively. but not quantitatively reproducible from run to run. The temperature at which the high field lines cross over into emission varied by ? 10°C and the ratio of line intensities could change by 50% at a given temperature. We believe that this variation resulted from the irreproducible polycrystalline nature of our ice samples. The CIDEP phenomenon depends upon the difference in hyperline and Zeeman energies of the interacting radicals, and their diffusion rates, which together determine the amount of singlet triplet mixing which occurs between (re-)encounters [ 15,161. In an ice crystal, one expects that both quantities will be anisotropic. The magnitude of geminate CIDEP from the (H...OH) and (D...OD) radical pairs should be sensitive to crystal orientation because of the large anisotropy in the OH (OD) EPR spectrum in ice [ 91. The temperature at which the geminate (A/E) CIDEP dominates the (E/A) prompt spur cross-reaction contribution would then depend upon the distribution of crystal axes in the sample under study. We expect that this effect will prove a fascinating aspect of future single-crystal studies. A large isotope effect favoring H over D formation was found in experiments on isotopically mixed ice of 50%/50% and 10%/90% H20/DZ0 composition. In the 50/50 mixture, the H atom signal was decreased relative to pure H20, but only a very small D atom signal was detected. Signal amplitudes of H and D were comparable in the 10%/90% mixture. The kinetics observed for H and D in the 10%H~0/90% DzO system was qualitatively very similar to the iso-

topically pure ices. A previous study [ 32 ] of prompt CIDEP in isotopically mixed alkaline liquid water led to the conclusion that H formation was favored over D, both by the branching ratio of HDO dissociation (on the order of 2.5 : 1) and in the recombination of electrons with hydronium ion. The CIDEP effect further enhances the H EPR signal relative to D, because of the larger H hyperfine splitting [ 15,161. The similarity of the isotope effects in mixed HzO/D20 ice to those in the liquid suggests that the same mechanisms may be operative. Our observations of the CIDEP kinetics substantially confirm those of Shiraishi et al. [ 7 1, who first reported the phenomenon in Hz0 ice. Just as in our experiments, the H atom signal could not be detected below x 110°C due to a combination of line-broadening and smaller CIDEP signal enhancements. The ST_ polarization in the H atom spectrum was only postulated by Shiraishi et al. in pure ice, whereas in our time-resolved experiments the effect is clearly evident. The steady-state irradiation technique used in the previous study was incapable of separating the different effects of geminate and F pair polarization. As we demonstrate below, the CW EPR detection method used by Shiraishi et al. unfortunately also led to significant error in the spin-relaxation rate estimates which were based on the observed H atom linewidths.

4. Spin relaxation The primary aim of our study was the measurement of spin relaxation rates to elucidate the mechanism of H and D diffusion through the ice lattice. A few measurements of spin-lattice relaxation times, T,, were attempted by the saturation-recovery method, but the results are not reported here as the relaxation was found to be dominated by Heisenberg spin exchange. Studies with lower radiation doses are planned for the future. Transverse relaxation times, T,, were determined over a wide temperature range by the spin-echo method. Measurements were made on H atoms in pure Hz0 ice, 50% Hz0/50% D,O and 10% H20/90% D20, and for D atoms in pure D20 and 10% H,0/90% D20. A spin-echo envelope decay curve is shown in fig. 6, which is representative of the difficult (small S/N) experiments in the 10%

D.M. Bartels et al. / Diffislon

10%

D (+l) -61

T,

:

g P

:

6.45

D,O

H,0/90% Line c

psec

100

E

a z s

427

and CIDEP qfH and D atoms Table 1 Arrhenms

parameters

for H, D transverse

Atom

LatticeH/D

composrtion

D D H H H

0.4%/99.6% I O%/YO% 10%/90% 50%;50% natural abundance

spin relaxation

A (s-l)

-E,

201 482 153 473 753

10.01 9.90 11.86 11.61 12.00

in ice

(kJ mol-‘)

50

c 'E 0-J

0

t

I

1 10

0

I 30

I 20

2’Tau

I 50

I 40

(psec)

Fig. 6. Spin-echo envelope decay of D atom ( + 1) line at - 6 1 “C in 10% H,O/YO% D20 ice. Each point represents an average of 6400 shots. The S/N ratio of each individual shot was about 0.1 in the first channel (2 T= 1 us). Larger signals could be generated with higher radiation dose, but then Heisenberg spin exchange in diffusive radical-radical encounters contributes to the spin relaxation.

2.-

t_.+-T

O.Ol-

4.0

I

I

I

I

I

4.5

5.0

5.5

6.0

65

1lTemp

(K-h1

O3 )

Fig. 7. Arrhenms plot of H and D transverse spin relaxation rate versus mverse temperature for several tsotopically mixed ices.

H,0/90% D20 lattice. The results are collected in fig. 7 in the form of an Arrhenius plot, and best-fit Arrhenius parameters are collected in table 1. Our data for H in Hz0 can be compared with the findings of Shiraishi et al. [ 7 1, who measured EPR linewidths over a similar temperature range. They found a threefold increase in linewidth as the temperature was reduced from 250 to 160 K, a much weaker temperature dependence than that shown in fig. 7. Furthermore, our value of l/T, at 25 K

(2.4 x 1OC5 s- ’ ) corresponds to a homogeneous linewidth of only 15 mG, an order of magnitude smaller than the width measured by Shiraishi. The conclusion is that at high temperatures the EPR signal is inhomogeneously broadened, as previously suggested [ 221 following the discovery of the small hyperfine anisotropy for muonium in ice. The temperature and isotope dependence of the relaxation times evident in fig. 7 are both qualitatively consistent with the spin relaxation mechanism suggested by Shiraishi et al. [ 7 1. This is modulation of the electron-nuclear dipolar interaction between the free atom and the lattice nuclei by diffusional motion of the atom through the lattice. As will be shown later, the activation energies extracted from fig. 7 may be identified with the activation energies for diffusion, and are slightly different for H and D atoms. For one isotopic mixture ( 10% H,0/90% D20) it was possible to measure both H and D relaxation times. A crossover in their magnitudes occurs at about 200 K, leading to the surprising and interesting inference that D atoms diffuse faster than H atoms below 200 K. As we discuss in section 5, this behavior is entirely consistent with a semi-classical model for diffusion through a “bottleneck”. For electron-nuclear dipole spin relaxation caused by random translational motion of the atom through the ice lattice, the longitudinal and transverse relaxation rates should be given by

(1) (2) where wt is the second moment of the random magnetic field created by the lattice nuclei, w. is the Larmor frequency of the electron spin (2rr x 9.3 GHz),

D.M. Bartels et al. / Dgffus~onand CIDEP ofH and D atoms

428

and T is the motional correlation time. In the system under study, the correlation time can be identified with the inverse frequency of “hops” between adjacent sites in the lattice. In the limit wo7~ 1, which applies over most of the temperature range investigated here, eq. (2) reduces to the simple expression 1/T,=w:r.

(3)

The transverse relaxation rate is seen to be inversely proportional to the hopping rate, which accounts for the negative activation energies of the Arrhenius plots in fig. 7. The pre-exponential factors depend upon the proton/deuteron ratio in the lattice through the wt factor, which can be written (for a randomly oriented powder spectrum) [ 24 ]

o_$=$:fi*C yiZk(Zk+ 1 )Rk6, k

4.0

4.5

D,O

5.0

l/Temp

5.5

(K“

6.0

6.5

~10~)

Fig. 8. Arrhenius plots of H and D transverse spin relaxation after correction for the lattice isotope effect according to eq. (5).

(4)

where l)e and & are electron and nuclear gyromagnetic ratios, Zkis the nuclear spin of the lattice nuclei, and Rk is the distance from the interstitial atom to the kth nucleus in the lattice. Given the gyromagnetic ratios and nuclear spins of the two lattice isotopes, one can readily calculate that protons are 15.9 1 times more effective than deuterons in causing spin relaxation of the diffusing atoms. Assuming random isotopic distribution of the protons and deuterons throughout the ice lattice, the 0:. factor in a mixed H20/D20 ice can be related to pure Hz0 by the formula ot( mix) d(H20)

SO% H,O/SO%

=Xu+&

where X, and X, are the respective mole fractions of proton and deuteron in the mixture. In fig. 8 the correction formula is applied to the least-square fit representation of the data shown in fig. 7. The simple correction works very well, bringing all of the H data and all of the D data for the several samples to within 20% of common values at a given temperature. Thus there is little or no dependence of the atom diffusion rate on the isotopic composition of the lattice. The remaining differences could easily be residual Heisenberg spin exchange in the measurement, because the signal-to-noise ratio of the experiment was insufficient to test for second-order kinetic effects by a further reduction in dose-per-

pulse. There is, however, a clear difference in the Arrhenius activation energies and pre-exponential factors in the diffusion of H and D, with the activation energy for H being greater than that of D. A crossover in diffusion rates apparently occurs, such that the diffusion of D is faster than that of H below 200 K. (This is most clear in the 10% H,0/90% DzO ice where relaxation of both atoms could be measured in the same lattice environment. )

5. Diffusion of H and D The isotope effect on the diffusion of H and D atoms deduced from the spin relaxation is unusual, but can be readily explained with standard transition state theory (TST) arguments [ 33-361. The basic idea is that H and D atoms are trapped or caged for the most part in minimum energy sites (MES) of the ice lattice. In order to “hop” into an adjacent MES, the atom must pass over a barrier in the electronic potential surface, through a tight restriction, or “bottleneck” at the transition state. The unusual isotope effect is a consequence of the large change in zero-point vibrational energy as the atom moves from the MES to the transition state. The transition state theory rate expression for a diffusive jump can be written [ 33 ] K

k(T) = ph

rL Qcn,

e: a

ew( -PO) ,

D.M. Bartels et al. / Difislon

where p= l/kT, h is the Planck constant, V0 is the electronic barrier, Qc is the partition function for vibrational motion of the atom at the MES in the direction of the diffusion, and the Q, represent vibrational partition functions for all other modes in the MES and the transition state (I). K is the transmission coefficient, set equal to 1 in typical TST calculations. In the most general case, the product of partition functions should be taken over all vibrational modes of the lattice as well as the diffusing atom. The near coincidence of the data sets in fig. 8 suggests that the vibrations of the lattice are of secondary importance for H and D diffusion in ice above 150 K, and it is reasonable to assume that to first-order there is no change of the lattice vibrational frequencies in the transition state. The product over partition functions will therefore be restricted to just the vibrational motions of the diffusing atom in our subsequent discussion. Assuming harmonic vibrations of the H or D atom in its MES and for motion normal to the diffusion pathway at the transition state, the vibrational partition functions may be written [ 33 ] Q

=

’

ew( -Phv,D) 1 -exp(

-/3hv,)

(7)

The effective activation energy for the diffusion, including the vibrational zero point energies (which enter in the exponential numerator of the vibrational partition functions), is E,=V,-jhv,+$hAv,+jhAv,,

429

point effect on the (c axis) “reaction coordinate”. In many chemical problems the transverse vibrations change little in the transition state, so that the reaction coordinate term dominates, and the activation energy for the deuterated system is larger than for the protonated system [ 371. In the present case, the transverse vibrational frequencies must increase very significantly at a transition state “bottleneck”, such that the !~Av,,~ terms dominate the isotope effect. In order to investigate the properties of the H atom MES and the postulated diffusional “bottleneck”, we constructed a simple model for the H atom in ice, and optimized the parameters to give the best “tit” of the transition state theory hopping rate to the H and D spin relaxation data. The electronic potential surface for a free hydrogen atom diffusing in ice was constructed by pairwise addition of atom-atom contributions V( X-H) and V( X-O), where X ( = H or D) labels the interstitial (diffusing) atom. V( X-H) and V(X-0) were presented by three-parameter Buckingham potential functions, as used in an earlier calculation for muonium [ 17 1, but modified to include a damping function [ 381 to correct the short-range behavior of the dispersion energy term [ 39 ] V(R) =A exp( -bR)

-f(R)C/R6,

(10)

where f(R)=exp[-(1.28R,/R-l)‘], =l,

forR<1.28R,, for R > 1.28R,.

(8)

where Av,=v;-v, The frequencies of the harmonic motions will always be & times larger for H than for D because of the factor of two difference in atomic masses. The difference in the vibrationally adiabatic activation energies for the two isotopes can therefore be written (in terms of the frequencies for H) G(H)-&(D) =0.293(-fhv,+jhAv,+jhAv,,) .

and CIDEP ofH and D atoms

(9)

A “normal” H/D isotope effect comes from the - 1h v, term of eq. (9 ), which is the vibrational zero

Since R, is implicitly defined as the position of the potential minimum, there remain only three independent parameters. The exponent of the repulsion term was calculated with the combination rule [ 401 L=f(L+&)

9

(11)

using the Bohm and Ahlrichs values for b,, and boo [ 40 1, and a value of 2.83 A- ’ for b,,. The latter parameter was determined by fitting eq. ( 10) to the 3Z,, Hz potential [ 4 1,42 1. This fit also provided a value for the dispersion coefficient C,, (5.74 eV A6). C,, and Co, were calculated from the Slater-Kirkwood expression [ 43 ] and combined with Cx, using Tang’s rule [44]

430

D.M. Bartels et al. / D&ion

aA

ffB CA,

CBB

CAB= Cf:C~fj +CY~CAA ’ where the (Yare the static polarizabilities of a free H atom (X ) and the H or 0 atom in an OH bond [ 45 1. R, was chosen as the third parameter required to define each potential, since it can be represented by the sum of van der Waals radii. Commonly accepted values of the latter are 1.20 and 1.52 8, for lattice H and 0, respectively [ 461. However, it is not at all clear what value is appropriate for the interstitial atom [ 17 1. An upper limit is given by one-half of the minimum distance of the ‘&, state in Hz (2.08 A) but a preliminary investigation showed that this gives too high a potential barrier between interstitial sites, based on the activation energy for spin relaxation. Accordingly, the van der Waals radius of X was left as a free parameter, to be optimized by reproducing the experimental temperature dependence of the relaxation rates (see below). A summary of the atomatom potential parameters is given in table 2. Although they are not independent parameters of the potential, the pre-exponential factor A and the welldepth E are included in the table. To calculate the total interaction of the H atoms with the ice, a sum of atom-atom potentials was taken over 156 lattice protons and 56 lattice oxygen atoms in the vicinity of the starting position of X (the MES) and the paths for diffusion to next-neighbor interstitial cavities. For a rigorous calculation on a perfect ice crystal, the protons of the lattice need to be distributed according to the Bernal-Fowler rules, such that each oxygen has two close (sigma-bonded) and two farther (hydrogen-bonded) nearest neighbor protons (i.e. a single proton between each pair of oxygen atoms) [ 471. In fact, the defect concentration (zero or two protons between 0 atoms) may be quite Table 2 Parameters

of model potential Potential

V(X-H) V( X-O)

for X

and CIDEP of H and D atoms

high, particularly under the harsh conditions of our experiment. As the X atom moves it will sample all the statistically distributed possibilities. The ,Y, V( XH,) part of the potential will therefore have small variations from period to period. We represented the average of this potential by summing over all of the possible lattice proton sites, and dividing by two to get an effective total interaction. A representation of the ice lattice is shown in fig. 9, showing the position of the H atom MES and the “bottlenecks’* for diffusion along the c axis, and in the u-b plane. Two equivalent pathways into adjacent MES locations are available along the c axis, and three equivalent exits are available in the u-b plane (only one is indicated for the sake of clarity). Each u-b exit channel splits into two equivalent pathways, thus connecting every MES with six nearest neighbors in the u-b crystal plane. In fig. 10 we plot the optimized atom-atom potential along the c axis (upper plot ) and two axes (x, y) normal to this path at the MES and at the transition state. The minimum energy path along the c axis is closely sinusoidal with period 3.67 8, and barrier amplitude 94.5 meV. The lower plot clearly indicates the narrower electronic potential “bottleneck” at the c axis transition state relative to the MES. By symmetry both the purely electronic barrier and the zero point energy of transverse vibrations reach their maxima at 1.835 A from the MES. The (H atom) transverse vibrational frequencies are both approximately 254 cm-’ (3 1.5 meV) at the MES and increase to 565 cm- ’ (70.0 meV) at the transition state bottleneck. The ground state vibrational frequency for motion along the c axis is 209 cm-’ (25.9 meV). The u-b exits are complicated by the bifurcation of the reaction pathway, so that the separate electronic and zero point vibration contributions to the activation energy do not reach maximum at the same location.

( = H, D) m ice-Ih A ‘)

b

ta)

(A-‘)

c (eVA6)

R, b’

(ev)

(A)

(meV)

27.7 704.3

3.09 3.96

4.05 6.72

1.20+R, 1.52+R,

2.89 3.67

a) Parameter values imphcitly defined by the others. b, The optimum X atom van der Waals radius Rx= 1.63 A.

D.M. Bartels et al. / Dlffuusmnand CIDEP of H and D atoms

431

C axis

I’

0

I

I

I

1

2

3

4

Distance along c-axis (Angstroms)

B c 5

g B Fig. 9. Diagram of the ice-Ih lattice showing the positions of the oxygen atoms (spheres) and the axes (connectmg lines) for location of lattice protons between nearest-neighbor oxygens. LocatIons of the H atom minimum energy sites (MES) are mdicated by the large spheres. One hexagonal “cage” surrounding an H atom MES 1s outlined m bold. Positions of the transition states for c axis motion (above and below) and for diffusion in the ab plane (out the sides) are indicated with “bull’s eye” symbols.

Roughly speaking, the transition state for a-b diffusional jumps is located at 2 8, from the MES. The electronic barrier (not plotted here) is 50% higher, and the bottleneck is slightly narrower for this motion, making it a much less important component of the diffusion. To optimize parameters of the atom-atom potential just described, the semi-classical transition state theory expression (6) was evaluated at a number of temperatures between 150 and 250 K, and fit by least squares to the H and D relaxation data in the 10% H,0/90% DzO ice. The total relaxation rate was calculated from eq. (2) with the correlation time 1 ‘= 2kc+3kah ’

(13)

where kc and kab are the TST rates for motion through the c axis and a-b barriers, respectively. The trans-

i

; I.,. -:py‘_&

:

:

100

: 8’

‘,

‘\

,’

‘,

50

‘\

‘\

‘\.\

0

--_____/-

\

-0.75

-0.50

-0.25

0.00

I’

I’

,’

81’ I

0.25

0.50

0.75

Distance from C-BUS (Angstroms) Fig. 10. Electronic potential for H atoms along the c axis (upper plot) and orthogonal to the c axis at the MES and the c axis transition state (lower plot).

mission coefficient K was taken to be unity. To calculate the effect of transverse vibrational motions, the potential was sampled at 0.1 8, away from the minimum energy path to estimate an effective harmonic vibrational frequency from the curvature. Eq. (7) was used for the partition functions. The partition function for c axis vibration near the MES was corrected for anharmonicity using the Pitzer-Gwinn approximation for sinusoidal potentials [ 48 1, as given by Hill [33]: Qc=QHo(2xlT)L’Zexp(-U)Zo(U)1

(14)

where U= jj3Vo and Z, is the zero-order Bessel function of the first kind. QHo is the harmonic oscillator partition function given in eq. (7). For any reasonable barrier height the correction factor is between 1.2 and unity in the temperature range investigated. In fig. 11 we show the results of an optimization in

D.M. Bartels et al. /Diffusion and CIDEP ofH and D atoms

432

2 I

160

160

I

I

I

200

220

240

Temperature (K) Fig. 11. Simulation of the H and D spin relaxation rates m 10% H,0/90% D,O, using normal transition state theory and the model atom-atom potential summarized in table 2.

which the X atom van der Waals radius, the exponen-

tial b parameters, and the ol factor have been adjusted to the values listed in table 2. The lit is reasonably good, and displays the experimentally observed crossover in H and D relaxation rates, although the isotope effect is somewhat too small at higher temperatures. Reduction of the van der Waals radius from the initial estimate of 2.08 A to an optimum value in the range 1.6-1.8 8, was essential to obtain reasonably correct activation energies. Further improvement of the tit was obtained from a slight ( z 10%) increase in the b parameters (i.e. to produce steeper exponential repulsions) over the initial estimates. A similar lit could be obtained by adjustment of the C parameters, but this produced unrealistic values. No combination of parameters (even unreasonable) was found which could produce a “perfect” match to the data. The second moment of the random lattice field obtained in the fit displayed in fig. 11 is 5.12 x 1015 se2 (normalized for a pure HZ0 lattice). This is about live times larger than the value employed by Shiraishi et al. [ 7 1, which was based on the limiting EPR linewidth for H in 0.5 M sulphuric acid ice. (For static dipolar line broadening, wt in magnetic field units = D2/4, where D is the peak-to-peak width of the Gaussian first derivative lineshape of a powder spectrum [ 241. ) A corresponding measurement on pure ice at 4 K [49] is equivalent to a second moment, wf of 8.1 x 1OL5se2. This value should be considered an upper limit, since additional line broadening

factors were probably active. Another useful comparison is with the second moments derived for muonium in Hz0 and D20 ice [ 17 1. The muonium experiments were carried out in low magnetic fields, for which the detected muonium precession frequency is approximately 1.4 MHz per Gauss of applied field. This effective gyromagnetic ratio is just one half that of H in the high fields used in EPR. Consequently, if the second moment of local fields (measured in Gauss) is expressed in radian frequency units, the muonium value must be corrected by a factor of 4 to make it comparable to the wt values discussed for H above. Thus, the wt values derived from muonium correspondto3.0~10’~s-*and4.4X10’~s-’inH~0 and D20, respectively (the D20 value is scaled by 15.9 1). However, these values were determined from experiments on single crystals oriented with their c axes parallel to the applied magnetic field. Since 0:. is orientation-dependent, these values should be considered lower limits for our EPR experiments, where a powder average would be more appropriate. The computed second moment for Mu in H20 [ 501, averaged over all orientations, has the value 4.3 x 10t5 ss2. The agreement of the muonium data with the present tit for H and D is excellent. It is interesting to estimate the actual rate of diffusion in ice near the melting point based on this model fit, and compare it with diffusion in liquid water. A diffusion coefficient may be calculated from the formula [34] D(T)=

gk(7.J ,

(15)

where k(T) is the rate for jumping from site-tosite, d is the jump distance, and y is the dimensionality of the diffusion ( 1, 2, or 3). The diffusional motion in the ice crystal is highly anisotropic. The ratio 3kab/2kc estimated from the TST calculation is only 0.07 at 273 K, and decreases significantly at lower temperatures. Thus it is reasonable to assume essentially one-dimensional motion along the c axis, and set y= 1, d= 3.67 A. The jump frequency for H at 273 K is approximately 2.5 x 10” s- ‘, and from eq. ( 15 ) we arrive at a diffusion coefficient D,,, = 1.7 x 10 -’ m* s-l. In liquid water at room temperature, the H atom diffusion coefficient is approximately [51 8 x 10e9 m2 s-‘, and given twice the viscosity at 273 K, we can estimate D,,,,,=~x 1O-9 m2 s-l. These

]

433

D.M. Bartels et al. / D~ffiuon and CIDEP of H and D atoms

calculations demonstrate the near equivalence of H atom diffusion rates in ice and water, which was already apparent from the very similar timescale of the CIDEP recombination kinetics. Little significance should be attached to this observation, however, because the modes of motion in the two phases must be very different (hopping versus hydrodynamic diffusion ). The calculation of a diffusion coefficient allows a cross-check on one further assumption made in the analysis, which is that the diffusion occurs in the bulk of the material and not along grain boundaries in the polycrystalline sample. Near the melting point, the spin relaxation time in the DZO ice is z 50 us. In this time (which is a worst case in our experiments), the average distance traveled by an X atom is [ ( 1.7X10-9m2s-‘) (5x 10-5s)]‘~2=300nmalong the c axis of the crystal. The average size of ice crystals in our sample is certainly much larger than this, so that the calculation provides ex post facto conlirmation of the basic assumption in the data analysis.

6. Tunneling effects: the case of muonium The semi-classical transition state theory calculation of the previous section is in excellent qualitative and reasonable quantitative agreement with the H and D spin relaxation data. The remaining deviations could quite plausibly be ascribed to quantum mechanical tunneling effects, which are ignored in the initial approximation. Spin relaxation data for muonium in ice is also available for the temperature range 8-263 K [ 171. These data are not described at all well by the semi-classical model; the experimental muonium activation energy is lower than that of H and D, whereas the semi-classical TST model predicts it should be higher because of the zero point energy “bottleneck” effect. This discrepancy is certainly a manifestation of the tunneling phenomenon in the light muonium isotope. Thus in an attempt to model the muonium data and improve the description of the H and D spin relaxation, we have extended the semi-classical model to include tunneling effects. The method adopted is the semi-classical adiabatic ground state (SAG) treatment of the transmission coefficient [ 36 1, as described in some detail by Lau-

derdale and Truhlar [ 341, who applied it to variational transition state (VTST) calculations of surface diffusion of H on copper. The VTST/SAG method has also been applied to H diffusion in bulk metals by Garrett and coworkers [ 351. Briefly, the “reaction coordinate” or diffusion pathway of the atom is treated separately from all of the other vibrational modes of the system, which are taken to be independent quantum harmonic oscillators with zero point energy thv,. The frequencies v,(s) are taken as functions of the position of the system along the reaction coordinates s. The multi-dimensional problem is reduced to one dimension with effective Hamiltonian H(p,,s;n)=

$

+v,(S,n)

7

(16)

where the adiabatic potential V, is given by the sum of the electronic and vibrational zero point energies V,=V,(s)+C J

hv,(s)(n,+i)

.

(17)

For the present we will continue to assume that lattice motions are unimportant, so that the sum over vibrational modes is restricted to just the motions of the atom perpendicular to the reaction path. In the low temperature range of interest to us, we need only consider the lowest transverse vibrational states, n,= 0. In the general case, the reaction coordinate, as defined by the least energy pathway between adjacent equilibrium (MES) positions, is not linear. The momentum, ps, and effective mass, m, of the diffusing particle are then functions of the curvature of the reaction pathway [ 341. In the ice c axis diffusion problem this complication does not arise, and m is just the mass of the diffusing atom. The transition rate can now be written

(18) where

D.M. Bartels et al. / Diffusron and CIDEP of H and D atoms

434 cc

+

-I

ew( -PE) ~

K(T)=

s

dEP(E)

>

>

exp(-BE)

.

I’$ a

(19)

P(E) is the quantum mechanical probability for the particle of mass m with kinetic energy E to tunnel through or pass over the adiabatic potential l’s(s). The E, are eigenenergies of the particle in the onedimensional adiabatic potential well centered at the MES. The first term in the numerator of K(T) represents the probability for tunneling through the adiabatic barrier from bound states of the potential while the second (continuum) term represents the probability for passing over the barrier. In the one-dimensional problem, the Q:.QJQXQ,, ratio for the transverse vibrations are degeneracy factors for the adiabatic potential. The adiabatic potential at the MES is taken as zero. The new definition for the zero of energy requires

Qx,, =

1 1 -exp(

(20)

-W,,,)

Qs is evaluated directly from the Boltzmann-weighted sum over bound states

Qs=C

n

ew(-PE,)

(21)

.

Both the energies E, and the frequency factors dE/ dn can be calculated numerically on V,(s) using the WKB approximation [ 341. In the present case where Va(s) is nearly sinusoidal, we found it convenient to obtain the energy levels from numerical diagonalization of the Hamiltonian matrix formed with eq. ( 16 ) and a periodic basis set of the form ( - d/ 2 < s < d/ 2 ) Om(~) = -$exp(im m=O,

+l,

-+2, ... .

7)

lated from the (inverse) period of the classical trajectory of the particle on V,(s). Optimization of the potential parameters was carried out in this treatment just as for the semi-classical TST without tunneling. The minor contribution of diffusion in the a-b plane was estimated semi-classically as before. The optimum van der Waals radius for H increased marginally from 1.63 to 1.64 A, but the agreement with experimental relaxation rates became slightly worse overall. Once again, no combination of potential parameters could be found which gave “perfect” agreement with the data. The transmission coefficient K was several percent larger than unity for the H atom at 200 K, which confirms that the tunneling effect should not be ignored in the temperature range of the H and D spin relaxation data. Application of the same model potential to c axis diffusion of muonium (m = 0.11 u) proved to be most revealing. The predicted rates for barrier crossing of H, D, and Mu are plotted together in fig. 12. In the low temperature limit, the rate reaches a constant which is purely a function of the tunneling rate from the lowest vibrational state. The H atom reaches a low temperature limit of 0.6 s- ’ at roughly 25 K, which is entirely reasonable based on the low temperature EPR measurements of Floumoy et al. [ 6 1. For the same potential, the low temperature limit for the muonium rate is predicted to be lo9 s-i, which according to fig. 12 should obtain for all temperatures below 120 K. This prediction is completely at odds with the spin relaxation measurements of Leung et al.

1 O'O ___...._________________~ 1 o*

I

; (22)

Nineteen basis functions proved more than adequate to define the bound energy levels for all of the isotopes. For each eigenenergy, dE/dn was calcu-

Temperature

(K)

Fig. 12. Simulation of H, D and muonium c axis barrier crossing rates versus temperature, using the SAG method for tunneling corrections.

D.M. Bartels et al. / Dlffuslon and CIDEP of H and D atoms

[ 17 1, who deduce a rate of at most several times 10’ s-l at 8 K in D20 ice, with indications that the rate is still decreasing with temperature. One possible flaw in application of the H, D model potential to muonium is the anharmonicity of vibrations in the u-b plane, which should be more significant for the higher-frequency Mu vibrations. The seriousness of this problem was investigated by solution of the one-dimensional vibrational Schrodinger equation for H and for Mu, on the x and y axes of the potential plotted in fig. 10. Despite the visible (asymmetric) anharmonicity of the x axis potential around the MES, the ground state frequency for muonium is very nearly three times the H frequency as expected from the mass ratio for a harmonic oscillator. It is unlikely that a rigorous treatment of the anharmonicity can change the Mu tunneling rate by more than a factor of two. The model calculation which has been carried out here has assumed that low frequency lattice motions are unimportant. Inclusion of these modes is most likely to increase the rate of hopping, especially in the limit of low temperature. Therefore the calculation we have done should represent (apart from anharmonicity corrections) a lower limit for the true c axis diffusion rate. The experimental finding that the muonium tunneling rate is at least two orders of magnitude smaller than calculated, can only mean that our model atom-atom potential is seriously flawed. In order to have a much lower Mu tunneling rate, the true adiabatic potential barrier must be significantly higher. At the same time, a higher electronic barrier will make the activation energy too large for agreement with the H and D measurements. These requirements might be reconciled if a larger fraction of the adiabatic barrier is zero point energy, so that the effective muonium barrier becomes much larger than that for H and D. A lower electronic threshold, but narrower “bottleneck” is qualitatively indicated. Nevertheless, the two order of magnitude gap between theory and experiment cannot be reconciled just in terms of the adiabatic barrier height . To investigate this point we carried out a tunneling calculation based on a model c axis potential of the form I’,(s) = ( V, + fhAy, + #Av,) Xf[l-cos(2ns/d)],

(23)

435

where dz3.67 A. Both the electronic potential I’,(s) and the adiabatic potential V,(s) proved to be nearly sinusoidal in the atom-atom potential, so this model is not unreasonable. The vibrational frequencies Y_, are assumed equal at all positions s, and scale as the square root of the isotope mass. Using parameters h~,,~, h&_ and V,, the model was optimized to tit the high temperature H and D spin relaxation as well as the low temperature muonium limit. The lowest muonium hopping rate that could be calculated which still agreed even qualitatively with the H and D measurements was 10’ s-l, still an order of magnitude faster than indicated by experiment. Several factors might conceivably contribute to the failure of the calculations to extend to the muonium data. First among these would be a distortion of the ice lattice around the free atom, which might have important consequences on the tunneling factor. A static distortion would constitute an asymmetric potential, which is less permeable than the symmetric case. If the lattice distortion accompanies the diffusing atom, then that is tantamount to a heavier “effective” mass for the diffusing particle. Such an effect would be more severe for the light muonium atom, which samples more of the potential surface by virtue of its larger amplitude vibrations. A related, and perhaps more important factor is the neglect of lattice vibrations in our calculation. The low temperature muonium diffusion is certainly characteristic of tunneling through the rigid lattice, and the data seems to demand a larger barrier than our model for H and D will allow. This suggests that lattice vibrations (of the oxygen atoms, particularly) are important in letting the H and D atoms “squeeze through” the c axis barrier in the higher temperature regime. Our simple TST model may be able to mimic the activation energy of the H and D diffusion by choice of an “effective” rigid barrier. The muonium data lends further support for this idea, in that there is some difference between the motional correlation times found in HZ0 and D20 ice [ 17 1. Thanks again to the larger-amplitude motions of Mu, its diffusion should be more sensitive to the proton/deuteron lattice vibrations. Finally, one should more carefully investigate the validity of the VTST/SAG prescription for calculating rate constants within the context of this problem. The separation of the degrees of freedom into semi-classical

436

D.M. Bartels et al. / Diffusion and CIDEP of H and D atoms

reaction path and uncoupled quantized transverse motions may not be a good approximation, particularly for muonium. Thus, the limitations of our current model can be put into two categories: the assumption of a rigid ice lattice, and the semi-classical treatment of quantum particles. There is a clear need for a more sophisticated model than we used for our calculations; our combined data on H, D, and Mu should prove valuable in testing such further work.

Acknowledgement The authors wish to thank Dr. David Werst and Mr. Bob Lowers for operation of the Van de Graaff accelerator and for help in constructing the isopentane flow system. A special thanks goes to Dr. Stephen Gray for his advice in the VTST/SAG calculations.

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