inelastic neutron-scattering study of free proton dynamics in /-mno 2

E L S E V I E R Chemical Physics 209 (1996) 111-125

Chemical Physics

Inelastic neutron-scattering study of free proton dynamics in /-MnO 2

F. Fillaux "'*, S.M. Bennington b j. Tomkinson b, L.T. YU c a Laboratoire de Spectrochimie lnfrarouge et Raman, Centre National de la Recherche Scientifique, 2 rue Henry-Dunant,

94320 Thiais, France b Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, 0 X l l OQX, UK

c Laboratoire d'Electrochimie et Synth~se Organique Centre National de la Recherche Scientifique, 2 rue Henry-Dunant, 94320 Thiais, France

Received 31 July 1995; in final form 9 April 1996

Abstract

Inelastic neutron scattering measurements of "/-MnO 2 at 30, 100 and 200 K give S(Q, to) maps of intensity over large o 1

energy and momentum transfer ranges (from 0 to 1 eV and from 0 to 40 A - , respectively). They reveal a ridge of intensity due to recoil of free particles with effective mass of -- 1 amu. The width of the recoil line is one order of magnitude greater than that anticipated from the kinetic momentum distribution of an isolated gas of protons at the temperature of the sample. Several models are considered: ideal gas obeying Maxwell-Boltzmann or Fermi statistics, multiple scattering, phonon wings, protons trapped in shallow potentials and quantum correlations between free and bound protons. None of them is satisfactory. Finally, broadening by zero-point motions of the lattice provides a satisfactory theoretical framework to represent the shape and intensity of the recoil spectra. New aspects of proton dynamics in 3,-MnO 2 are highlighted.

1. Introduction

Manganese dioxides are extremely complex materials of great importance for basic research and practical use for battery electrodes [1-16]. The electrochemical and chemical reduction in aqueous me- dia proceed via proton insertion in the lattice. There- fore, the rate of proton diffusion contributes to the electrochemical reactivity relevant to high rate battery discharge performances. Recent advances in inelastic neutron-scattering spectroscopy (INS) have shed new light on proton dynamics and localization in 3,-MnO 2. The existence of free protons has been

* Corresponding author.

revealed [17-20]. These free protons could be charge carriers of great impact to the conductivity and electrochemical activity.

The y-MnO 2 lattice is an intergrowth of ramsdellite and pyrolusite structures (Fig. 1) [21] which contains various types of protons. Some of these protons are associated with Mn 3+ defects. The loca- tion of protons due to structural water molecules has given rise to much controversy [22,23]. In the model proposed by Ruetschi [24], a fraction of Mn 4÷ ions is missing and the corresponding vacancies are coor- dinated to four charge compensating protons. Each vacancy is thus equivalent to 2 H 2 0 molecules. These protons are supposed to be mobile. They could interconnect the 1-D channels of the ideal structure, allowing thus for proton mobility in 3-D. Additional

0301-0104/96/$15.00 Copyright © 1996 Elsevier Science B.V. All fights reserved PI! S0301-0 104(96)00140-1

112 F. Fillaux et al. / Chemical Physics 209 (1996) 111-125

A.,\ tRuannmSdellit \ e PtYro/us ite

>b

Double chain of octahedra © Mn(IV)

J Single chain of octahedra • Mn(lll)

Fig. t. Schematic view of a cation-deficient intergrowth structure of ramsdellite and pyrolusite, projected onto the [001] plane.

protons can be chemically or electrochemically inserted into the structural channels. Their mobility along the channels and through the vacancies is supposed to be the key mechanism for the high electrochemical reactivity of these materials [25-34].

INS is uniquely suited to observation of hydrogenous species in non-hydrogenous matrices. The scattering cross-section of the hydrogen atom is more than one order of magnitude greater than those of the other atoms (Mn and O). Even for concentrations in hydrogenous species as low as = 1 wt.% (weight percent) the total cross-section for the matrix is such that it can be regarded as effectively transparent.

Previous INS spectra from - -20 c m - l to 4000 cm-~ were obtained with the TFXA spectrometer at the ISIS pulsed neutron source (Rutherford Appleton Laboratory, UK) [17-20]. Bands due to vibrations of localized protons could be distinguished: water-like

OH at - -500 cm-1 on the one hand, and protons inserted in the structural channel between 700 and 1150 cm -1, on the other. Samples with various reduction degrees gave detailed information on inserted protons: single protons surrounded by empty sites at = 750 cm- I, protons in sites nearest neighbor to vacancies at - -890 cm-~ or next nearest neighbor at -- 1010 c m - l and protons further away at -- 1120 c m - l [20]. For the most reduced samples, overtones of the band at 1120 cm-1 revealed that inserted protons behave as isotropic oscillators with effective mass = 1 amu [19]. It was thus concluded that these protons are sitting at the center of the oxygen octahedra forming the structural channels.

Beside these inserted protons in the channels, charge compensating protons in Mn 4+ vacancies were observed between 100 and 300 K as freely rotating entities represented as (H+)4 at the center of the vacancies [18]. These entities were no longer observed at 20 K. It was proposed that charge compensating protons are outside the vacancies at low temperature and the formation of freely rotating species is a thermally activated process. The energy difference between the two configurations is --- 90 c m - i.

All the bands are superimposed to a continuum with almost constant intensity over the whole spectral-range. This was attributed to recoil of particles with mass -- 1 amu. It was suggested that a significant fraction of the protonic species are quasi-free delocalized proton entities [17]. It is necessary to emphasize that INS gives no information on the electrical charges of these particles. They are referred to as 'protons' in the sense of scattering nuclei. The dynamics of these protons is the central problem which is addressed in this present work.

Evidence for free protons in solids is dramatically lacking. Such protons cannot be characterized with techniques commonly used (infrared, Raman, eSR, NMR, etc.). Only INS may provide a clear cut signature for free protons. However, previous measurements [17-20] did not fully exploit the great advantage of the INS technique, that spectra can be measured over a range of kinetic momentum transfer (Q). This is the vector difference between the incident and scattered wave-vectors (] k i ] = 2"rr/A i and [ k j [ = 2"tr//Aj, where h i and Aj are the incident and scattered wave-lengths, respectively). The variation

F. Fillaux et al. / Chemical Physics 209 (1996) 111-125 113

a ) -.,~'"? ~:"J': '

20

10 r.~

)

J (b)

800

14

~,-MnO 2

20 K

;>

E

e-

#

e.. u.]

600

400

200

-200

17.5 -- 20.0

l l 15.0 -- 17.5

l l 12.5 -- 15.0

l l i0.0 -- 12.5

1 1 7 . 5 0 -- 10.0

1 1 5 . 0 0 -- 7.50

l l 2.50 -- 5.00

l l 0.00 -- 2.50

10 20 30 40 Momentum transfer (A-l)

Fig. 2. INS spectrum of y-MnO 2 at 30 K. (A) landscape view; (B) iso-intensity contours equally spaced on a linear scale in arbitrary units. Recoil lines for H, O, AI and Mn atoms are superimposed.

114 F. Fillaux et a l . / Chemical Physics 209 (1996) 111-125

of the INS spectral intensity of a transition with the angle at which the scattered neutron is measured contains spatial information on the proton wave- function. Some spectrometers allow the momentum transfer and the energy transfer (7~to) to be varied independently and yield the full scattering function S(Q, to). This is not the case of the TFXA spectrometer for which the kinetic momentum transfer and the energy transfer are correlated according to: h to-- 2Q 2 (with the energy and momentum transfer in meV and ~,-~ units, respectively). This trajectory in (Q, to) space, corresponds to the maximum in the scattering function for particles with mass --- 1 amu, either oscillating (bound) or recoiling (free) and is often referred to as the 'proton recoil line'. This narrow slice of S(Q, to) gives no information on the Q dependence of the proton signal at constant energy transfer and information on free proton dynamics (namely the distribution of kinetic momentum) is consequently absent. Recent works on coals have highlighted the relevance of full S(Q, to) maps to proton recoil dynamics [35,36].

In the present paper we report totally new INS measurements of S(Q, to) maps for a y-MnO 2 sample at 30, 100 and 200 K. Recoil of free protons is observed over large energy and momentum transfer ranges. The spectra are consistent with free bare protons with rather large kinetic energy, incompatible with an isolated gas at the temperature of the sample. Several mechanisms are considered and we conclude that the observed kinetic energy distribution is due to Doppler broadening by the zero-point motions of the lattice.

2. Experimental

The 3~-MnO2 sample is identical to that used in previous INS experiments [17-20], This is an elec- trodeposited manganese dioxide (EMD), purchased from Tekkosha. The average valency corresponds to MnOL96 and the amount of manganese is --59.5 wt.% (compared to 63.2 wt.% for the ideal MnO 2 stoichiometry). This is equivalent to 7 wt.% of water molecules. Thermogravimetric measurements reveal that -- 7% of the Mn sites are vacant and -- 4% are occupied by Mn 3+ ions. Therefore, there is --- 0.32 wt.% H + in vacancies and = 0.05 wt.% H + associ-

ated to Mn 3÷ ions. According to previous INS stud- ies, apart from water-like protons (band at = 500 cm-~), most of the protons are either free (underlying continuum) or single (band at 750 cm - l ) or in sites next neighbor to vacancies (band at - -890 c m - l ) .

Spectra were obtained with the MARI spectrometer at the ISIS pulsed neutron-source, Rutherford Appleton Laboratory, Chilton, UK. The incident neutron energy was 1 eV and the spectrometer resolution was Ato/to -- 2%. Spectra were converted from counts per channel and angle to S(Q, to) per energy and momentum transfer by standard programs. About 10 g of sample were wrapped in aluminum foils and loaded into a closed cycle refrigerator. Since less than 5% of the incident neutrons were scattered, multiple inelastic scattering events are negligible.

3. Results

3.1. Bound and free protons

The ridge of intensity in the S(Q, to) map measured for the sample at 30 K (Fig. 2) is due to proton recoil. OH stretching modes give a rather weak inelastic peak, at -- 400 meV/3200 cm -~, superimposed on this ridge. Bands previously observed between 60 and 150 meV (480-1200 cm -~) with the TFXA spectrometer [17,20] are not distinguished from the elastic peak centered at the origin.

The ridge of weaker intensity observed below 100 meV/800 c m - ~ and yet at large momentum transfer corresponds to the vibrational density of states of heavy atoms such as O and Mn atoms in the sample and A1 in the container and cryostat.

S(Q, to) maps for the same sample at 100 and 200 K are very similar in shape. The recoil signals are slightly broadened. All these data are quite similar to those obtained previously for a coal sample [361.

4. Theory

4.1. The S(Q, to) map

In this section, different models are considered in order to account for the observed S(Q, to) maps:

F. F illaux et a l . / Chemical Physics 209 (1996) 111-125 115

protons bound to heavy atoms, gas of free protons, quantum correlations and Doppler broadening by lattice oscillations.

4.1.1. Bound protons For the sake of simplicity, the dynamics of bound

protons are represented as harmonic oscillators. The scattering law for the 0 ~ n transition of an isolated isotropic_ oscillator at low temperature with frequency hto 0 and mass m is [37]:

S( Q, to)= n------]i~---, exp(-O2u2)6( to - ntoo),

(1)

where u 2 =-h/2mto o is the mean square amplitude in the ground state. These transitions give peaks of intensity with maxima at E. = nhto 0 and Q2 = nu-2 = 2mE.-h -2. Therefore, the maximum intensity for profiles in Q at constant energy transfer nhto o provides a direct estimate of the effective oscillator mass: m* -- 2 Q2/E., with m* in atomic mass units,

o 1 E. in meV and Q. in A - . The observed S(Q, to) map for the OH stretching mode at = 400 meV/3200 c m - l is thus consistent with an effective mass of 1 amu, as anticipated for a proton bound to a heavy rigid lattice.

Because a uniform distribution of vibrational frequencies extending over the whole energy transfer range is very unlikely, the continuum of intensity cannot be represented with discrete transitions of

1000

8O0

'~_~ 600

t, 400

200

0 -0,5 0 ( A ) o,5

Fig. 3. Shallow potential function with a single bound state, n = 0: V(x) = 1000[1 - 1 /cosh2(6x)] . V and x are in c m - i and ,~ units,

respectively. The bound state is at = 550 c m - ~ and E D = 450 c m - ~ is the dissociation energy.

isolated oscillators. However, weakly bound protons with a dissociation threshold at low energy could give a continuum of intensity. Moreover, coupling to the lattice modes might give manifolds of phonon wings which could merge into a quasi continuum.

4.1.2. Shallow potential with a single bound state Recoiling protons are supposed to be trapped in a

shallow local potential with only one bound state below the dissociation threshold (schematically represented in Fig. 3). The scattering-law for transitions from the ground state to the continuum, h to > E D ,

can be calculated within the impulse approximation which applies provided the final state is free [37,38]:

S(Q, to)

( h 2 h 2 ) __p2_ p+Q)2 = fn(p)a hto+ 2m -~m ( dp;

hto > E o. (2)

The 6 function expresses the conservation of the kinetic energy which applies to the collision between the nucleus with mass m and the neutron [37]. The kinetic-momentum distribution in the ground state is n(p). In the isotropic harmonic case:

S(Q, to)

[ m - 4,rr ER u2 exp 4ERU 2 ;

hto>ED, (3) h 2 Q 2

w h e r e : E R = h t o R = 2 m (4)

E R is the recoil energy. The corresponding S(Q, to) map is a ridge of intensity along the recoil line defined by Eq. (4). The solid line calculated with m = 1 amu (labeled H in Fig. 2B) coincides with the maximum of the observed ridge of intensity. There- fore, the recoiling particles are bare protons. Dynam- ical coupling with heavy atoms (lattice modes) or electrons is negligible.

At sufficiently high energy transfer, a cut of S(Q, to) at constant Q has a Gaussian profile in energy, Eqs. (3) and (4), with maximum of intensity proportional to Q-~ and standard deviation proportional to the momentum transfer value. The cut at 17 A -~ (Fig. 4) gives the best view of the recoil spectrum

116 F. Fillaux et aL / Chemical Physics 209 (1996) 111-125

. . . . . . . . . ' . . . . . . . . When the scattering sample is isotropi, 30 shown [39]:

m ,-, S(Q, to) = J ( y ) ,

, ~ 20 m

"~ I where y = ~-~ ( h to - E R )

10 ~= ~ / and J ( y) = 2"trfl : n ( p ) dp

t " ~ y

o . . . . . . . . . . . . . . . . . . .

0 200 400 600 800 ---- 2--~u2] e x p - . Energy Transfer (meV)

Fig. 4. Intensity profile for a cut at 174- 1 ,~-~. Solid line: measured at 30 K: broken lines: Gaussian fit (Table 1).

and minimizes contributions from the vibrations of bound protons. These contributions are thus negligible. For the sample at 30 K, the cut reveals two Gaussian like profiles centered at = 21.4 meV/171 cm - l and 510 meV/4150 cm - l (Table 1). Using Eq. (4), these peaks correspond to effective masses of 27 amu (A1) and 1.1 amu (H), respectively. This confirms that recoiling particles are virtually bare protons. The width of the Gaussian fit for protons gives the 'harmonic' frequency in the bound ground state: hto 0 = 135 meV/1080 cm -1.

The width of the recoil spectrum for any cut at constant energy transfer is also related to the distribution of kinetic momentum in the ground state.

)ic it can be

(5)

(6)

(7)

J ( y ) is the probability that a particle has a compo- nent of momentum along Q with magnitude y. This is the well-known Compton profile which is independent of the position of the cut on the energy scale. It has a Gaussian form with variance cr 2 = u 2. Using Eqs. (5) and (6) transforms cuts of S(Q, to) at constant energy transfer into the Compton profile.

Compton profiles calculated for cuts at constant energy-transfer from 600 to 700 meV/4800 to 5600 cm- i (see Fig. 5) or from 650 to 750 meV/5200 to 6000 cm- i are Gaussian-like in shape (see Table 1), as anticipated within the impulse approximation. These profiles are centered at y = 0 for the appropri- ate value of the effective mass: 1.19 amu. The dispersion for the estimated effective masses and temperatures could stem from experimental errors (background definition, statistics and resolution) and small contributions from bound protons and phonon wings (see below). Nevertheless, the estimated effec-

Table 1 Analysis of the Q cut and Compton profiles

Temperature Cut at constant Q Compton profiles (K)

1 6 - 1 8 / ~ - l 600-700 meV 650-750 meV

v A m* ~Ito o T * m* o" hto o T* m* o" hto o T*

(meV) (meV) (ainu) (cm - l ) (K) (ainu) ( ,~- I ) ( c m - l ) (K) (ainu) ( /~ - t ) ( c m - t ) (K) + 10 ::t:2 4-0.1 4-70 +50 +0.05 4-0.2 4- 130 4- 100 4-0.05 -I-0.2 4- 130 4- 100

30 540 318 1.07 1080 810 1.19 4.08 1120 840 1.19 4.24 1200 900 100 521 320 1.11 1130 850 1.19 4.14 1150 860 1.19 4.27 1220 914 200 518 328 1.12 1200 900 1.19 4.29 1230 920 1.19 4.38 1280 962

m* and T * are the estimated effective mass and temperature, respectively, for the proton gas; v and A are the position of the maximum and the half width at half maximum of the Gaussian fit to the recoil spectrum; o" is the variance of the Gaussian fit to the Compton profile. h w o is the corresponding harmonic frequency.

F. F illaux et a l . / Chemical Physics 209 (1996) 111-125 117

1000

500

_=

,, , , . , /~ ~A . ~ - - ~ 0

-t0 -5 0 5 10 (A "l)

Fig. 5. Compton profile for the cut at 650 4-50 meV. Solid line: measured at 30 K; broken line: Gaussian fit (Table 1).

tive masses confirm that the recoiling entities are virtually bare protons. Averaged estimates of the frequency for the ground state (hco0 = 1130, 1170 and 1240 + 70 cm- ~ for sample temperatures of 30, 100 and 200 K, respectively) are consistent with those derived from cuts at constant momentum transfer. They reveal a significant increase with temperature, which should not occur for ideal isolated oscillators.

For weakly bound protons, a cutoff of intensity is anticipated at energy transfer h co < E D. Moreover, Eq. (3) does not apply rigorously for h co = E D since near the dissociation threshold, quasi-bound states cannot be treated within the impulse approximation (final state effects) [38]. In the present experiments, the energy resolution is not sufficient to discard the existence of a cutoff of intensity at low energy. Bound protons in a local potential with a dissociation threshold lower than = 200 meV would give sub- stantially the same spectrum. However, previous measurements with the TFXA spectrometer have demonstrated that there is no visible cut-off for the continuum down to a few meV [17]. A potential function with hco0 = 1100 cm - l and a single bound state very close to the dissociation threshold is extremely narrow and of questionable physical relevance. Therefore, it is concluded that the existence of any bound-state for recoiling protons is unlikely.

Furthermore, for protons in shallow potentials the intensity at maximum for the recoil spectrum is proportional to Q- l in Eq. (3). It should decrease at high energy and momentum transfer. In contrast to

this, the intensity of the continuum on the S(Q, to) map (Fig. 2), or previously measured with the TFXA spectrometer [17], is virtually a constant over the whole energy transfer range. Therefore, this model is not adequate.

4.1.3. Phonon wings Dynamical coupling of protons with lattice modes

occurs because the proton mass is very small compared to the effective oscillator mass for the lattice modes. Therefore, protons can 'follow' the lattice vibrations and a significant part of their oscillator mode spectral intensity is shifted into combination with those lattice vibrations. These 'phonon wings' mimic the external density of states spectrum and attenuate the oscillator mode intensity by a Debye- Waller factor: e x p ( - Q 2 ( U ~ ) ) ; where (U~) is the powder-averaged mean-square displacement of lattice vibrations. The oscillator mode intensity lost in this way is distributed between multiphonon wings. The intensity transferred to the nth wing is:

(8) nt

In the S(Q, co) map, phonon-wings should manifest themselves as convolution of the oscillator mode transitions (parent band) with the lattice density of states observed at low frequency. For a sample at low temperature, they appear as wings exclusively on the high frequency and large momentum transfer side of the parent band. For large (U~) , the parent band may become invisible and the spectrum looks like recoil, along the line corresponding to the effective mass of the lattice modes. Therefore, phonon wings cannot account for proton recoil with effective mass of 1 amu. On the contrary, this mass means that coupling to the lattice modes is negligible.

4.1.4. Gas of free protons For a gas of free particles (e.g., hydrogen atoms)

with the Maxwell-Boltzmann distribution of kinetic momentum, the scattering law calculated within the impulse approximation, Eq. (2), is [37,38]:

S(Q, co) ' '

k 4~rff~RkT ] 4ERkT (9)


This equation is similar to Eq. (3) with u 2= mkT, where T is the temperature of the gas. Averaged estimates for T obtained from the analysis of the Compton profiles (Table l) give 850, 875 and 930 _ 50 K for sample temperatures 30, 100 and 200 K, respectively. Consequently, free protons cannot be represented as an isolated Boltzmann gas at the temperature of the sample. For a sample at 30 K, the variance of the Compton profile should be only

o 2 -~ 0.6 A- , instead of = 17 ,~-2 observed.

For a gas of free protons (H + with spin 1/2) at low temperature, Fermi statistics should be of importance and give significant deviations from the Maxwell-Boltzmann distribution. At T = 0 K, n(p) is a solid sphere: n(p) = p for p --<PF and n(p) = 0 for p > PF, where PF is the modulus of the kinetic momentum corresponding to the Fermi energy E F. For large momentum transfer values (Q > 2PF), the scattering law is [37]:

3 1 S(Q, t o ) = 8yEF [ --( x -y2

( y 2 _ 2y) < x < (y2 + 2y) , (10)

with: x = h t o / E F and y = Q/PF- S(Q, to) also gives a ridge of intensity along the recoil line. Cuts at constant momentum transfer are parabola centered at E R and vanishing intensity at E R +__ 2(EREF) 1/2. At high temperature (kT>> E F) the profile is broadened. The Fermi and Maxwell- Boltzmann distributions are indistinguishable.

In the present experiments, it is not straightforward to distinguish Gaussian profiles from parabola within experimental errors. Nevertheless, only a rather large Fermi energy would account for the width of the recoil spectrum. However, this Fermi energy is directly related to the amount of free protons in the sample under investigation [40]:

h2 (3"ff2N)2/3, (II) E F = 2 m ~ V

where N/V is the number of protons per units of volume. The average valency (MnOt.96) gives 0.08 protons per MnO 2 entity. The volume of the unit cell containing four MnO 2 entities is --120 ~3 [21]. Therefore, E F = 0.4 meV/3 cm-' and PF ---- 0.4 ,~-'. The recoil spectrum should be extremely nar-

row and the observed S(Q, to) map is not amenable to a Fermi gas of free protons at the temperature of the sample.

4.1.5. Multiple scattering Multiple scattering may contribute to the width of

the recoil spectrum. The amount of sample in the beam was such that less than 5% of the incident neutrons were scattered. Therefore, multiple inelastic scattering events are negligible. However, inelastic scattering followed by elastic scattering is more probable. This should correspond to convolution of the recoil spectrum of the gas with the elastic peak, which is Gaussian in profile [Eq. (1) with n = 0]:

S(Q, to)= f dqf dpexp(-(Q-q)2(U2>)n(p)

- - p 2 - . × 6 h t o + 2m

(12 )

n(p) is the kinetic momentum distribution for free protons. Since the width of the recoil spectrum due to a cold, Fermi or Maxwell-Boltzmann, gas is negligible with regard to that of the elastic peak, the spectrum should look like the 'recoil' of the elastic peak:

S(Q, to) ( h2q2 )

=f dqexp(-(Q-q)Z(ut2))8 hto- 2----m

or s ( a , to)

8 ( h t o - e R ) . -- exp - Q h

(13)

In this case, cuts at constant energy transfer should be Gaussian profiles with width and intensity at maximum virtually independent of the energy transfer value. This is in agreement with the observation (Fig. 2). Cuts of the S(Q, to) map at constant energy transfer are amenable to Gaussian profiles whose widths and intensities at maximum are independent of the energy transfer value, within experimental errors (Fig. 6 and Table 2). However, the width of the elastic peak, S(Q, 0), estimated from the experi-

F. Fillaux et al./ Chemical Physics 209 (1996) 111-125 119

. . . . I r , , ~ I . . . . I . . . .

15

"~ lO

0 ~"2"~J ' ' '

0 10 20 30 40 Momentum Transfer (A ~)

Fig. 6. Gaussian fit to the cut at 650_+ 25 meV. Solid line with error bars: measured at 30 K; broken line: Gaussian fit (Table 2).

250 ' ~ I . . . . I . . . . I . . . . . . . I . . . .

,-. 200

1oo i

50

o ' 0 5 10 15 20 25 30

Momentum Transfer (A -l)

Fig. 7. Gaussian fit to the cut at 0_+ 5 meV. Solid line with error bars: measured at 30 K; broken line: Gaussian fit. Bragg peaks are represented with triangular functions.

ments (Fig. 7), is about twice that o f the recoil line

(Table 2). This d iscrepancy cannot be expla ined with

mul t ip le scattering events . It is conf i rmed that mult i-

ple scattering effects are largely negl igible .

4.1.6. Gas o f free protons interacting with bound protons

Free protons in the structural channels are in

contact with those protons inserted in the structural

channels which behave like isotropic osci l lators with

f requencies be tween 95 m e V / 7 5 0 c m - ~ and 140

m e V / l l 2 0 cm - I [18,20]. These va lues compare

very favorably to those der ived f rom the width o f the

recoil spectrum (Table 2). Water - l ike protons, on the

other hand, g ive the stretching m o d e at ~ 400

m e V / 3 2 0 0 c m - ~ and l ibrat ional modes around 70

m e V / 5 6 0 cm -~. The mean f requency o f these pro-

tons is much greater, about 200 m e V / 1 6 0 0 c m - ~ .

These protons are l ikely to be at the surface o f the

part icles and should have no interact ion with free

protons inside the structure.

Because free protons are delocal ized , there is a

non-vanish ing probabi l i ty for dis tances be tween free

and bound protons to be small enough that e lectro-

static interactions should be considered. The total

Hami l ton ian is then:

H = H B + H F + HBF ( 1 4 )

with H B = ~.,Pi2 / 2 m + Vi( riR); H F = Y ' ~ p 2 j 2 m i j

and HBF = ~_,cteZ/[ rib -- rjF I. i j

(15)

H B corresponds to bound protons exper ienc ing local

potentials Vi(rB). The wave funct ions are qt,(riB).

Table 2 Gaussian fit to cuts of the recoil spectra at constant energy transfer values

Temperature Cut at constant E

(K) - 5- + 5 meV 625-675 meV 675-725 meV 725-775 meV

(UI2) m* (u 2 ) -hto m* (u 2) 7/to m* (u 2) 7tto

(10- 2 ,~. 2 ) (amu) (10- 2 ,~, 2 ) (cm -I ) (amu) (10- 2 ,~ 2 ) (cm - j ) (amu) (10- 2,~ 2 ) (cm -1 ) + 0.05 + 0.02 + 0.02 + 20 _+ 0.02 + 0.02 + 20 + 0.02 + 0.02 + 20

30 1.08 1.03 2.08 810 1.03 2.04 820 1.03 2.08 810 100 0.96 1.04 1.98 850 1.04 1.94 865 1.04 1.92 870 200 0.98 1.04 1.89 890 1.04 1.93 870 1.04 1.91 880

m* is the estimated effective mass for the proton gas. (u 2) = ( 2 0 " 2 ) - l, where 0" is the variance of the Gaussian profile, hto is the harmonic frequency for mean-square displacement ( u 2 ).


H F corresponds to free protons with planar wave functions (])k(rjF) and wavevector k. HFB is the electrostatic interaction between bound and free protons. The density of free protons is sufficiently low that interactions between free protons are negligible. The eigenvalues of this Hamiltonian are:

l Elk + = ~( HBii "at- HFkk)

"~-+¢(OBii--OFkk) 2 +4H2aik (16)

where HBi i = (~i* I HB IV/>;

HFk * = (qbk* [HF I@, )

2 =1<%*1 IvI i)<rI i I I%>1. and HFBik HFB * HFB (17)

2 However, since the coupling term HFBik contains terms which diverge:

[(~k* l ae2/ ( l rB- rF [ ) l e x p - - a2r2,)

×(exp-a2r2alo~e2/(IrB-rFI)le~)l (18)

the Hamiltonian, Eq. (14), cannot be treated simply with these basis sets of wave-functions. Therefore, recoiling protons cannot be represented as planar waves. An alternative approach is to consider that 'free' protons are experiencing a distribution of in- finitely deep potential wells whose walls correspond to bound protons. The corresponding mean energy levels ( E , ) are [40]:

~2h2 ( E n ) = n 2 2re(a2) , n > 1, (19)

where ( a z) is the mean square distance between bound protons. Our estimate for a uniform distribution of free and bound protons is ( E . ) -- n 2 × 10 -3 meV. The distribution of kinetic momentum should be very narrow ( ( p 2 ) = 2 × 10 -2 ,~- 1). Therefore, the spectrum should be a quasi-continuum similar to that anticipated for a cold gas of totally free protons. This is again incompatible with the observation.

4.1.7. Doppler broadening The kinetic energy and momentum of free protons

are defined with respect to the host lattice. In this referential, free protons are represented as planar- waves with kinetic momentum distribution nH(PH) (e.g., Maxwell-Boltzmann or Fermi). At very low temperature free protons are at rest but the lattice is

still oscillating with respect to a 'fixed' laboratory referential. This corresponds essentially to the zero- point energy of the lattice. These collective oscillations are represented as phonons, i.e., planar-waves, with distribution of kinetic momentum:

= ~ e x p ( - 2p2 • (UL2)).

(20)

All together, nL(PL) and nH(PH) form the total density-of-states which is probed by neutrons. The scattering function is the convolution of the recoil spectrum for the gas of protons with the kinetic momentum distribution of the lattice modes:

S(Q, to)

=f dp,f dpnnL(PL)nH(PH)8 ( h2 )

x + 7-£m - + Q .

( 2 1 )

For mkT.~ ((U~))-~, the scattering law is:

S(Q, ,o)

[< ct - exp - 2 Q h

(22)

Therefore, at very low temperature, a cut at constant energy has a Gaussian profile with variance ( 4 ( U 2 ) ) -1, which is twice that of the elastic peak [compare Eqs. (13) and (22)]. Moreover, the intensity at maximum is largely independent of the energy transfer values. This model is in perfect agreement with the observation (see Table 2).

This broadening mechanism can be compared to the Doppler effect. The wave-vector for the incident neutrons is defined with respect to a 'fixed' laboratory frame. For a given value eL for the lattice kinetic momentum, the incoming wave-vector expe- rienced by free protons (k i - -pL ) is either blue or red shifted, depending on the mutual orientation of the two vectors. The momentum transferred to the recoiling protons (QH = k i - - e L - I f ) is different from that transferred to the whole sample: Q = k i - kf.

F. Fillaux et aL / Chemical Physics 209 (1996) 111-125 121

4.1.8. Quantum correlations Here we consider a less common approach in

terms of quantum correlations related to the Ein- stein-Podolsky-Rosen (EPR) paradox [41]. These quantum effects were further, but almost simultane- ously, described by SchriSdinger in straightforward words [42]: 'When two systems of which we know the states by their respective representations enter a temporary physical interaction due to known forces between them and then, after a time of mutual influence, the systems separate again, then they can no longer be described in the same way as before viz by endowing each of them with a representative state vector. I would not call that one but rather the characteristics of quantum mechanics.'

Quantum correlations due to nonlocal quantum entanglement, or EPR correlations, have been, and are still, subjected to controversy. This is because fundamental concepts of quantum mechanics regarding physical reality and separability are involved [43]. These quantum effects are studied in particle physics and quantum optic [44], but, their role in the dynamics of condensed matter is questionable. EPR correlations are not commonly used in solid state physics, due to the lack of clear cut experimental evidences. Because they depend on the size of the de Broglie wave-length (or delocalization) of the quantum objects under consideration, they should be best observed for light entities like protons. Therefore, protonic systems containing delocalized (mobile) entities could be the best opportunity to observe these correlations. Quantum correlations have been in- voked, but in the somewhat different context of coherent dissipative structures, to account for 'anomalous' mobilities of protonic species in HEO/D20 mixtures [49] and in metal hydrides [50].

Applied to the case of free protons interacting with bound protons, quantum correlations would mean that once interactions have occurred, the final state cannot be represented with a product of two wave functions. It corresponds to a non-factorizable EPR state: I ~ ' j (r))= I~i(r)clgk(r)). Consequently, the two particles are indistinguishable with respect to neutron scattering events and the scattering law is:

S(Q, t o ) = I(~'y* lexp iQ.rl ~i)l 2

or S(Q, to)

= If dq(aPr* ( r ) [ exp i ( Q - q ) , r Ig',~(r))

× (q~k)( r ) ] exp iq. r lCPkr ( r ) ) [ 28

× (hto - Ei, f - E¢,f). (23)

For a scattering event corresponding to inelastic scattering by free proton and elastic scattering by bound protons, it becomes:

S(Q, to)=f dqf d k i e x p ( - ( Q - q ) 2 u 2 ) n ( k i ) ~

( ) × hto+ 2 m k ~ - ~ m ( k i + q ) 2

(24)

This is very similar to Eq. (13), but, here, bound protons are harmonic oscillators with mean square amplitude u 2. At low temperature (mkT ,~ u2), cuts at constant energy transfer are Gaussian profiles with variance -- (2u2) -1 . Numerical values obtained from the measured S(Q, to) maps (see Table 2) and the almost constant intensity at maximum of the recoil spectra are in agreement with this model. Therefore, the dynamics can be represented with quantum correlations between a gas of free protons at thermal equilibrium and those bound protons located in the structural channels which are accessible to free protons.

Although this approach is in reasonably good agreement with the observation, it suffers from cer- tain weaknesses. First, interactions between free and bound protons which yields EPR states are difficult to rationalize. Second, according to Eq. (23), the spectrum of the bound protons should be totally recoiled for the sake of symmetry. Therefore, the bound protons which are observed do not form EPR states. Any coincidence between the width of the recoil spectrum and the frequency of bound protons is fortuitous and does provide any support to quantum correlations.

5. Discussion

Among the models presented above to account for the width of the recoil spectrum, phonon wings,

122 F. Fillaux et a l . / Chemical Physics 209 (1996) 111-125

isolated free gas at thermal equilibrium and electrostatic interactions give distributions of kinetic momentum narrower than observed by at least one order of magnitude. Multiple scattering, on the other hand, gives a width about twice that observed. All these effects are likely to be of secondary importance.

The shallow potential with a single bound state and a small dissociation threshold cannot be rejected so easily since the observed cuts at constant energy and Compton profiles are both amenable to Gaussian profiles, within experimental errors (see Figs. 5 and 6). The absence of any cut-off of intensity for the continuum in previous INS measurements with the TFXA spectrometer [17] imposes that the dissociation threshold should be very close to, and no more than a few meV above the ground state. The physical meaning of such a potential is certainly questionable but this model cannot be rejected definitely on this basis. Another drawback is that the shallow potential model does not give the correct temperature effect. At low temperature, kT << E D, the kinetic momentum distribution in the ground state should be temperature independent since there is virtually no coupling to the lattice. Alternatively, at higher temperature, protons should escape from the bound state to become free, and slow. The width of the recoil line should decrease as the temperature increases, but the opposite trend is observed. However, the weakness of the observed temperature effects and the many possible contributions do not allow to reject the shallow potential safely. Finally, the main discrep- ancy of this model is that the maximum of intensity along the recoil line should be proportional to E~ 1/2, Eq. (9), whilst it is observed to be largely independent of the energy transfer value. Therefore, alternative models providing better agreement with the observation are preferred.

Quantum correlations between free and bound protons provide an interesting framework to account for the observations. However, this approach raises controversial questions regarding the nature of the interactions yielding EPR states. Further experiments should be worthwhile to rationalize this mechanism.

Finally, Doppler broadening appears as the best and most straightforward explanation for the recoil spectra. Because free protons are delocalized and behave as planar waves they probe the lattice oscillations. The whole density-of-states for the sample

include phonons and free protons which cannot be distinguished. The wave-vector of the incoming neutrons is modulated (Doppler effect) by the lattice vibrations. The broadening of the recoil line is determined by the distribution of kinetic momentum for the lattice density-of-state. This yields a straightforward proportionality between widths for the elastic peak and for the recoil line, that is effectively observed. This superimposition of lattice and proton planar-waves is totally different in nature from dynamical coupling to the lattice modes (riding effect) which is negligible since the effective mass for free protons is very close to 1 amu.

A consequence of this model is that free protons contribute to the intensity of the elastic peak. Within the harmonic approximation, the width of the elastic peak is determined by the mean-square amplitudes for atom vibrations averaged over the whole density- of-states: ( U 2 ) a ( F . i ( [d~ilJi )-1 >, where/x i and v i are the effective masses and frequencies for all modes. Bound protons are very few and can be neglected. The width of the elastic peak is essentially due to acoustic modes. For translational modes of Mn atoms with effective mass of 55 amu, the mean square amplitude estimated from the elastic peak ( (U 2 ) -- 1 x 10 -2/~2, Table 2) gives a harmonic frequency of ---30 cm-1 which corresponds to the maximum of the density-of-states [17]. Optical lattice modes are at much higher frequencies ( - 6 0 0 c m - l ) and their mean-square amplitudes are negligible. It transpires that the proportionality observed for mean square amplitudes (2(UL 2) ---- (U 2) ----- 2 × 10 -2 ,~2, see Table 2) could be fortuitous.

As anticipated for Doppler broadening, temperature effects on the elastic peak and on the width of the recoil spectrum are similar (see Table 2). Be- cause Eq. (21) is a convolution of two gaussian like functions, the variance of the Gaussian-like cuts at constant energy transfer is -- mkT + (4(U2(T))) - l At increasing temperature, the contribution of the gas of free protons broadens as mkT. On the other hand, thermal excitation of anharmonic lattice modes increases the mean square amplitude ( U 2 ( T ) ) , and thus decreases the width of the elastic peak. The broadening with increasing temperature observed for both the elastic peak and recoil for T = 30 and 100 K (see Table 2) are essentially due to thermal effects on the gas of free protons. At higher temperature,

F. Fillaux et a l . / Chemical Physics 209 (1996) 111-125 123

further broadening is partially canceled by thermal excitation of the anharmonic lattice.

In principle, INS can provide estimates for the relative concentrations of free and bound protons. The measured intensities for different types of protons are proportional to their relative amounts in the sample, provided atomic cross-sections for non hydrogen atoms are negligible. However, exact analyti- cal formula for bound, Eq. (1), and free protons, Eqs. (20) and (21), are not available and approximations are necessary. Within the Doppler broadening mechanism, free protons can be treated as harmonic oscillators with mean square amplitude (U 2 ) ~--- 0.02 .~2, very similar to that for protons inserted in the structural channels, (u~) . Therefore, relative intensities for free and bound protons are approximately proportional to their relative amounts in the sample.

Unfortunately, only the stretching mode of water- like protons is observable on the S(Q, to) maps of intensity presented above. This signal is very weak and quantitative analysis is not worthwhile. With a spectrometer of the TFXA type, it is possible to compare intensities along the recoil line which corresponds to the maximum of intensity for free and bound protons. Previous measurements and band decomposition (see Fig. 5 and Table II in Ref. [20]) have revealed four components at 747, 891, 1003 and 1100 cm- l, respectively, with very similar integrated intensities, proportional to --5, 6, 6 and 3, respectively, in arbitrary units. These bands are superimposed to a continuum of intensity. The ratios of intensity for the underlying continuum compared to each band are: ---30, 25, 25 and 50, respectively. The ratio for total intensity for free and bound protons is thus --7.5. This value compares very favorably to the ratio derived from chemical analysis ( - -6 .5) for the amounts of protons in vacancies (0.32 wt.%) and those inserted in the channels (0.05 wt.%). Therefore, it transpires that free protons could be essentially those charge compensating protons associated to the Mn 4+ vacancies.

It is worth noticing that here we compare intensities which are proportional to the nuclear scattering cross-sections. This is totally different from total incoherent scattering cross sections (i.e., integrated over energy and momentum transfer) for free and bound protons, which axe in a ratio of 1 / 4 [37].

The great similarity of the recoil spectra for coals

[35,36] and y-MnO 2 suggest that Doppler broadening mechanisms apply to both systems. It should be worthwhile to re-analyze coal data in this context.

6. Conclusion

S(Q, to) maps of y-MnO 2 confirm the existence of a gas of free particles with effective mass very close to 1 amu, which are referred to as 'protons'. The mean kinetic energy distribution measured at 30 K is about one order of magnitude greater than that anticipated for an isolated gas at thermal equilibrium. Phonon wings, electrostatic interactions, multiple scattering and shallow local potentials cannot account for the observed recoil spectra. Quantum correlations between free and bound protons in the channels are questionable. Finally, the spectra are well represented with a superimposition of planar waves due to the lattice density-of-states and free protons. The Doppler broadening mechanism provides a satisfactory theoretical framework for a detailed analysis of the recoil spectra.

A dramatic consequence is that Doppler-free dynamics cannot be observed with INS. Consequently, it is not possible to decide whether free particles are bosons (for example, hydrogen atoms) or fermions (for example, protons, H+). Throughout this paper (and in previous works [17-20]), we have called 'protons' the recoiling entities. This is justified because neutrons are scattered by nuclei. However, INS provides no information on the electrical charges and it is beyond the limits of this work to conclude whether these entities are protons or hydrogen atoms or any possible intermediate entity.

The S(Q, to) maps presented in this paper shed a new light on proton dynamics in manganese dioxides. The picture which emerges is that various proton dynamics are closely related to defects in the T-MnO 2 structure. In the non reduced sample, apart from protons at the surface which do not recoil, two types of protons can be distinguished: vacancy charge compensating protons which recoil and protons associated to Mn 3+ defects which are trapped in the structural channels. This is in line with previous INS works [18] which have revealed that charge compensating protons are outside vacancies at low temperature ( - 2 0 K) and enter vacancies to form freely

124 F. Fillaux et al . / Chemical Physics 209 (1996) 111-125

rotating (H+)4 entities at higher temperature (e.g., above 100 K). The present work suggests that charge compensating protons outside the vacancies, which were previously supposed to remain near the vacancies [18], are delocalized in the structural channels and form a cold gas of free particles nearly at rest with respect to the lattice. This is further confirmed by the relative intensities for free and bound protons at - -20K.

At higher temperature, (H+)4 entities replace the delocalized protons. However, the binding energy is very weak ( -- 90 cm- 1 ) and recoil is still observed at large energy and momentum transfer. These entities have an internal vibration which corresponds to the tetrahedron inversion. The zero-point energy is that of a harmonic oscillator with hto 0 ~ 1500 cm- [18]. Therefore, the recoil of protons forming (H+)4 entities corresponds to the shallow potential case [see Eq. (2)]. Although the shape of the recoil lines for these protons [Eq. (3)] and for free protons outside vacancies [see Eq. (22)] are formally different, they have similar widths and cannot be distinguished in the S(Q, to) maps presented in this paper.

During the reduction process, i.e., simultaneous insertion of protons and electrons, INS band intensi- fies due to bound protons are well correlated to the reduction degree whilst the continuum remains virtually unchanged [20]. Consequently, free protons are essentially related to the structure and largely independent of the reduction state. This is in accord with the INS spectrum of manganite (MnOOH) [19] which gives no evidence for proton recoil.

In spite of the remarkable insight onto proton dynamics provided by INS, outstanding questions remain with no answer. The mechanism of the inter- conversion of protons delocalized in the channels into freely rotating entities inside vacancies is not understood at a microscopic level. It is logical to suppose that localization/delocalization of protons is favored by simultaneous localization/delocalization of electrons, y-MnO 2 is a semiconductor with an energy gap smaller than 0.5 eV [51]. Localized electronic states associated to vacancies and located about 90 cm-l above the valence band are suggested by the thermally activated proton inter-conversion process.

Besides, electrical charges attached to free particles are largely unknown. It is tempting to speculate

that such particles should be almost neutral (e.g., hydrogen atoms) to be free in the MnO 2 lattice with ionic character. Similarly, it was observed that rotating entities in vacancies obey Bose statistics [18], but it should not be concluded that this is definitely against bare protons. This information could simply mean that there is no spin correlation between the four particles. A naive view is that hydrogen atoms, either free or confined in vacancies, should react rapidly to form hydrogen molecules. Such molecules are not observed despite the great sensitivity of INS, specifically to this molecule. It is thus necessary to suppose that the MnO 2 lattice prevents the formation of H 2 molecules. Such coupling of hydrogen atom electronic orbitals to the lattice should be weak since the observed effective mass is ~. 1 amu. The physical origin of these remarkable properties and their possible contributions to the electrochemical activity of y-MnO 2 are strong incentive for further investiga- tions.

Acknowledgments

We are indebted to C.A. Chatzidimitriou-Dreis- mann for stimulating discussions stressing the importance of quantum correlations.

References

[1] K. Sasaki and A. Kozawa, J. Electrochem. Soc. Jpn. 25 (1957) 115.

[2] K. Sasaki and A. Kozawa, J. Electrochem. Soc. Jpn. 25 (1957) 273.

[3] A. Kozawa, J. Electrochem. Soc, 106 (1959) 79. [4] W.C. Worsburgh, J. Electrochem. So=, 106 (1959) 839. [5] M. Fukuda, J. Electrochem. Soc. Jpn., 28 (1960) 67. [6] S. Niwagi, Y. Miyak¢ and E. Prcisler, J. Electtochem. Soc.

Jpn., 30 (1962) 727. [7] J. Brenet, G. Coeffier and J. Gabano, Electtochim. Acta, 8

(1963) 273. [8] A. Travusko, J. Electrochem. Soc, 111 (1964) 125. [9] R.F. Amelie and A. Travusko, J. Eiectrochem. Soc, 111

0964) 496. [10] J. Brenet, P. Chartier, M-T. Dott, M. Gross, K. Le Tran and

K. Traore, Electtochira. Acta, 13 (1968) 2167. [11] D.S. Freeman, P.F. Pelter, F.L. Tye and L.L. Wood, J. Appl.

Electrochem., 1 (1971) 127. [12] J.A. Lee, C.E. Newnham, F.S. Stone and F.L. "lye, J. Colloid

Interface Sci., 42 (1973) 372.

F. Fillaux et a l . / Chemical Physics 209 (1996) 111-125 125

[13] K. Miyasaki, in Proceedings of Manganese Dioxide Sympo- sium, Vol. 2, B. Schumm Jr., H.J. Joseph and A. Kosawa, Editors (Sample Office Publisher, Cleveland, OH 1975), p. 111.

[14] E. Preissler, in Proceedings of Manganese Dioxide Sympo- sium, Vol. 2, B. Schumm Jr., H.J. Joseph and A. Kosawa, Editors (Sample Office Publisher, Cleveland, OH 1980), p. 184.

[15] T. Yoshimori, M. Kato, K. Nippashi, J. Murayama, S. Turner and P.R. Busek, in Proceedings of Manganese Dioxide Sym- posium, Vol. 2, B. Schumm Jr., H.J. Joseph and A. Kosawa, Editors (Sample Office Publisher, Cleveland, OH 1980), p. 369.

[16] Progress in Batteries & Battery materials, edited by ITE-JEC Press Inc. and IBA Inc., 13 (1994).

[17] F. Fillaux, H. Ouboumour, J. Tomkinson and L.T. Yu, Chem. Phys., 149 (1991) 459.

[18] F. Fillaux, H. Ouboumour, C. Cachet, J. Tomkinson, G.J. Kearley and L.T. Yu, Chem. Phys., 164 (1992) 311.

[19] F. Fillaux, C. Cachet, H. Ouboumour, J. Tomkinson, C. L6vy-Cl6ment and L.T. Yu, J. Electrochem. Soc., 140 (1993) 585.

[20] F. Fillaux, C. Cachet, H. Ouboumour, J. Tomkinson, C. l.,6vy-Cl6ment and L.T. Yu, J. Electrochem. Soc., 140 (1993) 592.

[21] P.M. De Wolff, Acta Cryst., 12 (1959) 341. [22] F. Freund, E. Konen and E. Preisler, Acta Cryst., t2 (1959)

14. [23] K.M. Parida, S.B. Kanungo and B.R. Sant, Electrochim.

Acta, 26 (1985) 435. [24] P. Ruetschi, J. Electrochem. Soc., 131 (1984) 2737. [25] J. Coleman, Trans. Electrochem. Soc., 90 (1946) 545. [26] S. Atlung and T. Jacobsen, Electrochim. Acta, 26

(1981) 1147. [27] W.C. Maskell, J.E.A. Shaw and F.L. Tye, Electrochim. Acta,

28 (1983) 225. [28] W.C. Maskell, J.E.A. Shaw and F.L. Tye, Electrochim. Acta,

28 (1983) 231. [29] F.L. Tye, Electrochim. Acta, 30 (1985) 17. [30] D.T. Ferrel and W.C. Worsburgh, J. Electrochem. Soc., 98

(1951) 334.

[31] D.T. Ferrel and W.C. Worsburgh, J. Electrochem. Soc., 100 (1953) 471.

[32] W. Feiknecht, H.R. Oswald and U. Feiknecht-Steinhmarm, Helv. Chim. Acta, 50 (1967) 1947.

[33] P. Ruetschi, J. Electrochem. Soc., 135 (1988) 2567. [34] P. Ruetschi and R. Giovanoli, J. Electrochem. Soc., 135

(1988) 2663. [35] F. Fillaux, R. Papoular, A. Lauti6 and J. Tomkinson, Fuel, 74

(1995) 865. [36] F. Fillanx, R. Papoular, J. Tomkinson and S.M. Bennington,

J. Noncryst. Solids 188 (1995) 161. [37] S.W. Lovesey, International Series of Monographs on Physics

72. Theory of Neutron Scattering from Condensed Matter Vol. I. Oxford Science Publications (Claxedon Press, Oxford, 1987).

[38] J. Mayers, C. Andreani and G. Bacioco, Phys. Rev., B 39 (1989) 2022.

[39] V.F. Sears, Phys. Rev., B 30 (1984) 44. [40] C. Cohen-Tannoudji, B. Diu and F. Lalo~, in: M6canique

Quantique (Hermann, Paris, 1977). [41] A. Einstein, B. Podolsky and N. Rosen, Phys. Rev., 47

(1935) 777. [42] E. Schr&linger, Proc. Cambridge Phil. Soc., 31 (1935) 555. [43] F. Selleri, in: Quantum Paradoxes and Physical Reality, ed.

A. van der Merwe, Fundamental Theories of Physics, Vol. 35 (Kluwer Accademic Publishers, Dordrecht, 1990).

[44] D.I. Fivel, Phys. Rev. Lett., 67 (1991) 286. [45] J. Maddox, Nature, 352 (1991) 277. [46] R. Omnbs, Rev. Modem Phys., 64 (1992) 339. [47] B. d'Espagnat, Scient. Am., 241 (1979) 128. [48] H. Primas, in: Chemistry, Quantum Mechanics and Reduc-

tionism, (Springer, Berlin, 1983). [49] C.A. Chatzidimitriou-Dreismann and E.J. BP, indas, Ber. Bun-

senges. Phys. Chem., 95 (1991) 263. [50] C.A. Chatzidimitriou-Dreismann, E.J. Bffindas and E. Kar-

losson, Phys. Rev. B, 42 (1990) 2704. [51] K.J. Euler and H. Mueller-Helsa, J. Power Sources, 4

(1979) 77.

inelastic neutron-scattering study of free proton dynamics in /-mno 2

Documents