Heitjans P., Karger J. (Eds.). Diffusion in Condensed Matter: Methods, Materials, Models

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3 Diﬀusion Studies of Solids by Quasielastic Neutron Scattering 141

0,0 0,5 1,0 1,5 2,0

0,0

0,2

0,4

0,6

0,8

1,0

0,0 0,5 1,0 1,5 2,0

0,0

0,2

0,4

0,6

0,8

1,0

2.18 Å

1.54 Å

1.45 Å

1.00 Å

H(SeO

)

T=500 K

NEAT

=6.2 Å

(

∆t)

= 0.4x10

-10

EISF

Q [ Å

-1

]

Fig. 3.28. Theoretical EISF curves for several examples of simple two-site and

three-site proton jump models (dashed, dotted and dashed-dotted lines) are com-

pared to the “apparent” EISF (open circles) obtained at medium energy resolution

with the time-of-ﬂight spectrometer NEAT (after [112]). The simple models are

not able to reproduce the experimentally observed data, whereas the successful

trigonal-asymmetric hydrogen bond (TAHB) model (solid line; see text) gives good

agreement [122].

The time-dependent site occupation probabilities for the 4 sites are de-

noted by W

(t). Their time-averages,

∞

(t) = lim

t→∞

(t), (3.67)

are used to introduce an order parameter η,

η =2W

∞

− 1, (3.68)

where full order (η = 1) means that there is no proton exchange with the ex-

ternal sites. The incoherent scattering function of the model is given by [122]:

inc

(Q,ω)=EISF(Q) ·δ(ω)+QISF

(Q) ·L

,ω)+QISF

(Q) ·L

,ω)

(3.69)

Here the elastic and quasielastic incoherent structure factors, EISF and

QISF

, depend on the jump vectors and on η. The two Lorentzians are given

by:

,ω)=H

/[(H

)

+ ω

]/π (3.70)

=(3/τ

+1/τ

)=6/(1 − η)/τ

; (3.71)

142 Tasso Springer and Ruep E. Lechner

Fig. 3.29. Local site arrangement corresponding to the trigonal-asymmetric hy-

drogen bond model. This model is compatible with the trigonal symmetry of

H(SeO

)

and with the measured EISF. The central site 1, with the average

proton residence time τ

, is connected to the three external sites, with residence

time τ

,bythejumpvectorsR

, R

and R

. The EISF analysis (Fig. 3.28)

suggests that most of the proton probability density is located in site 1. This is,

because the two proton exchange rates, 1/τ

and 1/τ

, are diﬀerent (after [122]).

=1/τ

=3(1+η)/(1 − η)/τ

. (3.72)

A comparison of this model to the QINS spectra obtained at 500 K gave η

= 0.746 and R

=1.785

A. The asymmetry represented by this value of η

is appreciable. Accordingly, the occupancy of site 1 is at least 5 times larger

than the sum of the occupation probabilities of the 3 external sites. From

this it also follows, that – at the temperature of the experiment – the average

lifetime of the dimer is about 20 times shorter than that of the monomers, a

consequence of the asymmetry of the potential. The meaning of these obser-

vations has been discussed in more detail in [112] and [122]. To summarize,

the important aspect of the obtained η-value is the result that the proton

essentially remains bonded to the original selenate top oxygen, i.e. to one

selenate group, for a period of the order of at least 10

−10

s. The value of the

jump distance R

may, however, be taken as an indication that the proton is

also exploring - on the 10

−11

s to 10

−10

s time scale - a path into the hydrogen

bridge, although with rather low probability. These observations have been

interpreted as a consequence of the presence of an intracrystalline chemical

equilibrium reaction: alternation between the association of the monomers

[HSeO

]

1−

and [SeO

]

2−

, resulting in the dimer [H(SeO

)

]

3−

(H-bond for-

3 Diﬀusion Studies of Solids by Quasielastic Neutron Scattering 143

mation), and the dissociation of the latter into the two monomers (H-bond

breaking) [112]. Reaction rates and potential barriers for association and dis-

sociation processes have been determined, as well as temperature-dependent

proton exchange rates between neighbouring selenate ions.

It may be concluded, that the dynamical disorder existing in the high-

temperature phase of Rb

H(SeO

)

can be described by the above-mentioned

chemical equilibrium reaction. The two distinct transient chemical states play

diﬀerent speciﬁc roles in the mechanism of protonic conductivity. While in the

monomer state the protonated monomer via reorientation makes the proton

available for three diﬀerent elongated (and therefore weak) hydrogen bonds,

the dimer state corresponding to the creation of a strong (but short-lived)

H-bond, with the proton close to the center of the bond, provides the oppor-

tunity for proton transfer across the hydrogen bridge.

3.11 Two-Dimensional Diﬀusion

In view of the important role played by low-dimensional ionic conduction

and diﬀusion in solid-state physics, electrochemistry and even in biology, we

present several results concerning this problem. These have been obtained,

for instance, in studies of layered structures exhibiting quasi two-dimensional

proton conduction and diﬀusion, such as CsOX · X

O, M

X(AO

)

,and

Zr(XBO

)

· X

O, (where M = NH

,Rb,Cs;A=S,Se;B=P,As;X

=H,D).

Obviously, the concept of low-dimensionality should not be understood in

the sense of strictly planar motions, since there will be local deviations from

this idealized picture. The essential condition is, that the diﬀusing protons

stay within the conducting layers of the crystal at least during the observa-

tion time deﬁned by the energy resolution of the experiment (see Sect. 3.2,

(3.21), and the related discussion). This also implies that the mean square

displacement parallel to the layer, the proton has acquired during this time,

is much larger than perpendicular to it.

In previous paragraphs, we have already mentioned the cases of the lay-

ered structures Ba

NH [88] (see Sect. 3.8), CsOH · H

O [97] (see Sect. 3.9)

and Rb

H(SeO

)

[122] (see Sect. 3.10). All these structures are pseudo-

hexagonal or trigonal. The distances between neighbouring (transient) sites

of mobile ions found in crystallographic studies are much shorter within the

layer planes than in directions parallel to the c-axes being perpendicular to

these planes. This leads to the assumption, that ion diﬀusion and ionic con-

ductivity are likely to be conﬁned to two dimensions. Such assumptions have

been made occasionally, in the absence of single crystals which would al-

low the direct determination of the dimensionality by a QENS experiment.

For CsOH · H

O, this assumption has been conﬁrmed by spin-lattice relax-

ation time (T

) measurements revealing a logarithmic frequency dependence

of the relaxation rate T

−1

[43], typical for two-dimensional diﬀusion. It is

144 Tasso Springer and Ruep E. Lechner

known from theory [123–125], that the corresponding behaviour of T

−1

as a

function of ln ν

is linear, whereas for three-dimensional diﬀusion T

−1

is inde-

pendent of the nuclear magnetic resonance frequency ν

at high frequencies

(for details see Sect. 9.2 in Chap. 9). Spin-lattice relaxation also led to the ex-

perimental conﬁrmation of 2D-diﬀusion of Li ions between graphite layers in

Li-graphite intercalation compounds via the observed logarithmic magnetic-

ﬁeld dependence of the diﬀusion-induced component of T

−1

[126]. The same

authors also employed quasielastic neutron scattering in order to determine

the mean residence time and the jump distance in the diﬀusion plane of Li

ions from the oscillatory Q-dependent behaviour of the measured quasielas-

tic linewidth. In the case of Rb

H(SeO

)

, the two-dimensional character of

proton diﬀusion was revealed by conductivity measurements on single crys-

tals [114,115]. Zr(HPO

)

,initsβ-phase, is another two-dimensional proton

conductor with a proton site lattice similar to that of Rb

H(SeO

)

[127].

The proton diﬀusion mechanism in this crystal comprises – in addition to the

two-dimensional long-range transport – a very fast localized diﬀusive motion

related to librations of protonated phosphate tetrahedra [121].

The incoherent neutron scattering function in the low-Q limit for long-

range translational diﬀusion (TD) was already considered in Sect. 3.3 (see

(3.31) to (3.33). Accordingly, in three dimensions, when the diﬀusive motion

is isotropic, with a diﬀusion coeﬃcient D

,wehaveforthisfunction

(Q,ω)=

)

+ ω

, (3.73)

Certain analytic results [128] concerning the anisotropic case, where the diﬀu-

sion process is restricted to the planes of a layered structure, will be discussed

in the following. For a single-crystalline sample we have then [129]

(Q,ω)=

(Q sin θ)

)

+ ω

, (3.74)

where D

is the coeﬃcient of self-diﬀusion in two dimensions, Q sin θ the

component of the the scattering vector in the diﬀusion plane, and θ the angle

between Q and the normal to this plane. If single-crystals are not available,

one has to resort to polycrystalline samples requiring orientational averaging

of the above expression. It is known, that the resulting integral over all orien-

tations exhibits a logarithmic singularity at zero energy transfer [129]. This

is caused by the fact that diﬀusion planes which are perpendicular (or close

to perpendicular) to the scattering vector Q contribute elastic (or almost

elastic) scattering to the QINS function. Experimentally, such a singularity

cannot easily be distinguished from truly elastic scattering, and the numeri-

cal convolution with a resolution function is rather tedious. The question is

to what extent the characteristic feature of low-dimensionality represented

by the above-mentioned singularity can still be exploited when the observed

3 Diﬀusion Studies of Solids by Quasielastic Neutron Scattering 145

scattering function is resolution-broadened. Evidently the logarithmic sin-

gularity is cancelled by ﬁnite resolution. It has been shown that for resolu-

tion functions which have the shape of a Lorentzian (with HWHM equal to

H) or of a sum of Lorentzians (which is typically the case in BSC experi-

ments) the orientationally averaged resolution-broadened QINS function for

2D-diﬀusion,

S

(Q,ω)

resol

orient

2π



2π

dΦ



sin θdθL(θ, ω) (3.75)

where

L(θ, ω)=

(Q sin θ)

+ H

(Q sin θ)

+ H]

+ ω

(3.76)

can be written in closed form [128]. The explicit expression is lengthy

and will not be given here. Instead, Fig. 3.30 shows the result of numer-

ical calculations of this function for a two-dimensional diﬀusion coeﬃcient

with D

=7.5 µeV and for four diﬀerent values of the resolution

∆(ω)=H. The curves are shown as full lines with circles. For compari-

son, the Lorentzian- shaped three-dimensional diﬀusion scattering functions

are also shown (simple full lines). Here D

has been set equal to 2D

/3,

because this leads to identical linewidths (and lineshapes) for two- and three-

dimensional diﬀusion in the low-resolution limit. The resolution functions are

shown together with the scattering functions, but on a reduced scale. Al-

though a diﬀerence between the lineshapes for diﬀerent dimensionality starts

to be visible, when D

∼ H (see Fig. 3.30b), it is seen that H must be

about 10 times smaller than D

in order to permit a clear distinction of

2D- and 3D-diﬀusion by the shapes of the respective scattering functions. The

expression given above is valid only at small Q, where Lorentzian linewidths

(for single-crystals) exhibit the D

-behaviour and are therefore very small

and diﬃcult to be measured. At large Q, i.e. outside of the D

-regime,

where much larger and correspondingly more easily measurable linewidths are

observed, lineshapes very similar to expression (3.75) are expected. But un-

fortunately, in this Q-region the shape typical for low-dimensionality cannot

be observed in its pure form, because of the simultaneous presence of addi-

tional quasielastic components due to fast localized diﬀusive motions usually

connected with the diﬀusion mechanism. These are probably the reasons, why

the lineshape predicted for polycrystalline samples by (3.75) has so far not

been observed experimentally.

The study of the anisotropy of translational diﬀusion in single-crystalline

material is straight-forward. As an example we discuss the transport of wa-

ter molecules on the surface of a membrane with a two-dimensional crys-

talline structure [130, 131]. In biological membranes (cf. Chap 12), proton

diﬀusion connected with conduction of protons provides an important mech-

anism of energy transduction in living organisms. The purple membrane

146 Tasso Springer and Ruep E. Lechner

Fig. 3.30. Assuming a two-dimensional diﬀusion coeﬃcient D

,withD

7.5 µeV, the resolution-broadened scattering function (3.75) has been calculated for

four diﬀerent energy resolutions H (HWHM): (a) 50 µeV; (b) 5 µeV; (c) 0.5 µeV;

(d) 0.05 µeV. The corresponding curves are shown as full lines with circles. For

comparison the Lorentzian-shaped three-dimensional diﬀusion scattering functions

for the same “eﬀective” diﬀusion coeﬃcient are also shown (simple full lines). The

resolution functions are given on a reduced scale (after [128]).

(PM) of Halobacterium salinarum, for instance, contains the protein bacteri-

orhodopsin (BR) which becomes a one-dimensional pulsed proton conductor

when activated by light. This is a light-driven proton pump generating an

electrochemical gradient across the membrane, which is employed by the bac-

terium as an energy source. After having been pumped from the cell-interior

to the membrane surface, the protons are transported by a mechanism of sur-

face conduction towards other proteins located within the same membrane:

The light-generated electrochemical potential across the cell membrane is

utilised by the halobacteria to furnish the driving force for ATP synthesis

by energizing the rotation of the turbine-like machinery in ATP synthase.

Water molecules near the surface are known to be relevant for this biological

function, since they have been shown to assist the proton conductivity [132].

Hydration water, its interaction with the surface of biological macromolecules

and macromolecular complexes, and its diﬀusion generally play an important

role in structure, dynamics and function of biological systems [133, 134]. It

is therefore of considerable interest to study the proton diﬀusion mechanism

within and close to the hydration layers of membranes. Similar to the case

of bulk water, it is expected that during the diﬀusion process protons are

3 Diﬀusion Studies of Solids by Quasielastic Neutron Scattering 147

exchanged between water molecules acting either as acceptors (forming for

instance (H

) or as donors (producing (OH)

−

ions). A “solid-like” Grot-

thuß feature (see Sect. 3.9) is added to the diﬀusion in the liquid water phase

by the presence of ﬁxed protonation sites on the surface of purple mem-

brane. It is worthwhile to note, that these sites are arranged in space in a

perfectly regular manner, since the bacteria use the most eﬃcient packing of

BR: trimers of BR molecules embedded in a lipid bilayer matrix are aligned

in a two-dimensional hexagonal single-crystalline structure [135]. However,

since the pH-value is not very diﬀerent from 7, the concentration of charged

particlesisverylow.Theprotonsspendmostofthetimeaspartofdif-

fusing neutral water molecules with only rare events of exchange between

diﬀerent “vehicles” (see Sect. 3.9). Therefore, for the purpose of analyzing

neutron scattering experiments, the whole mechanism of diﬀusion may to a

good approximation be classiﬁed as that of molecular diﬀusion.

The crystalline order of PM has been exploited in QINS [130] and PFG-

NMR [131] measurements in a study of the anisotropy of proton diﬀusion

relative to the membrane surface. For this purpose about 20.000 layers of

purple membranes were stacked approximately parallel to each other (mosaic

spread: about 12

◦

FWHM) at deﬁned relative humidities (r.h.). The mem-

brane stacks had been produced, starting from an aqueous suspension of

membrane pieces, by alignment of the membranes through evaporation of

water, using aluminium foils as a substrate. At 100 % r.h., with water layer

thicknesses of about 10

A between neighbouring membranes, the protons of

water molecules were found to participate in a process of two-dimensional

long-range translational diﬀusion parallel to the membrane plane, with a self-

diﬀusion coeﬃcient D

=4.4 · 10

−6

−1

at room temperature, i.e. about

ﬁve times smaller than the known bulk value of water. At the same time, they

are also participating in a fast localized diﬀusive motion which can - at least

partially - be asigned to the rotation of water molecules. This was observed to

be about six times slower than in bulk water. These motions are suﬃciently

fast to produce quasielastic broadenings clearly seen with an elastic energy

resolution of 16 µeV FWHM (see Fig. 3.31). Finally, a translational diﬀusion

jump distance of 4.1

A was derived from the Q-dependent behaviour of the

quasielastic linewidth at large scattering angles (see Fig. 3.32). This distance

is three times larger than the corresponding quantity of bulk water.

The relative slowness of the diﬀusion process may partly be due to

the restricted space available within the hydration layers. It might also be

caused, together with the large value of the jump distance, by the presence of

ﬁxed protonation sites on the membrane surface. These might have nearest-

neighbour distances similar to those of neighbouring lipid head groups, which

are of this order of magnitude. At these protonation sites, hydrogen bonds

are expected to be formed, with lifetimes exceeding those of bonds which

exist between neighbouring H

O molecules in bulk water.

148 Tasso Springer and Ruep E. Lechner

Fig. 3.31. Fit of a model for two-dimensional translational diﬀusion of water to

QINS spectra (data points shown as triangles) of hydrated purple membrane stacks

oriented with an angle α = 135

◦

with respect to the incident neutron beam. These

data were taken with the inverted time-of- ﬂight technique (IRIS, see Sect. 3.4) at

the scattering angle ϕ =90.4

◦

. The spectra were not corrected for multiple scat-

tering (MSC), but MSC was taken into account in the ﬁtted theoretical model:

The solid lines represent (from top to bottom) the ﬁt result for the total scatter-

ing function including all terms, the sum of all rotation-broadened components of

the scattering function (including the backgound B), and B. The dotted lines are

the two MSC contributions: a rotation-broadened MSC component sitting on the

background line, and a MSC component without rotation broadening on top of the

rotation-broadened single-scattering contribution; after [131].

An attempt to relate the measured proton diﬀusion coeﬃcient D

to the

protonic conductivity σ (see Sect. 3.8) is of interest for elucidating the trans-

port mechanism [91]. The surface conductivity is known from dielectric mea-

surements [136]. By extrapolation to the hydration level of H =0.28 em-

ployed in [130, 131] (H is given in units of g of water per g of PM), a value

of σ =2.15 · 10

−8

Ω

−1

was obtained. From this the charge diﬀusion co-

eﬃcient D

is found in the region 4.4 ·10

−6

−1

≤ D

≤ 58 ·10

−6

−1

if the (unknown) concentration of the positive charge carriers is varied from

pH = 5.9 to pH = 7.0, where the lower limit would correspond to D

= D

The Haven ratio, H

= D

, contains information on the proton transport

mechanism (see Sect. 3.8 and Chap. 1). Whereas H

= 1 would correspond to

the unrealistic case of a pure vehicle mechanism, the upper limit (H

=1/13)

probably is equally unrealistic, because the PM surface is known to be neg-

atively charged, which means that pH < 7.0 near the membrane surface.

An intermediate ratio D

appears to be the reality. It corresponds to a

mixed mechanism, where protons are transported by molecular ion diﬀusion

3 Diﬀusion Studies of Solids by Quasielastic Neutron Scattering 149

Fig. 3.32. Translational diﬀusion linewidths (exhibited by spectra as shown in

Fig. 3.31) of water on hydrated purple membrane, plotted as a function of Q

for

two sample orientations: α =45

◦

,opencircles;α = 135

◦

, full circles. The experi-

mental values are compared with the theoretical width corresponding to an isotropic

approximation of the Chudley-Elliott jump-diﬀusion model in two dimensions. Note

that for α = 135

◦

,theQ vector is exactly parallel to the membrane plane, when

the scattering angle is ϕ =90

◦

,whereasQ is perpendicular to the membrane for

the same scattering angle, when α =45

◦

. Therefore the linewidth curve for the

latter case approaches zero near Q

=1.784

−2

, in agreement with the values

observed experimentally. This means that diﬀusion parallel to the membrane plane

has clearly been seen in this experiment, whereas in the direction perpendicular

to it the diﬀusive motion is too slow to be observable with the QINS technique.

Dotted lines: behaviour of the width according to the D

-law, if this was valid

in the whole region of Q

shown in the ﬁgure (after [131]).

and exchanged from time to time between diﬀusing particles which encounter

each other.

3.12 Coherent Quasielastic Scattering

The diﬀusion of individual particles treated in the preceding experimental

sections is investigated by measuring S

inc

(Q, ω), i.e. by scattering without

the interference of waves that have been scattered by diﬀerent atoms. Now

we deal with aspects of collective motions which reﬂect correlations between

diﬀerent diﬀusing particles. Information on these correlations is obtained from

the coherent dynamic structure factor S

coh

(Q,ω). It yields more details on

the scattering system; however, the theoretical interpretation is a priori more

complex; so far the number of experimental studies on this matter has re-

mained comparatively small.

150 Tasso Springer and Ruep E. Lechner

The theoretical derivation of the coherent scattering function for transla-

tional diﬀusion requires the introduction of the mutual interaction between

diﬀerent particles. A very simple description is the “hard core” or exclusion

potential: The particles diﬀuse over a lattice of cells or sites which are acces-

sible or blocked. This leads to a modiﬁed Chudley-Elliott formula [137] and

the resulting coherent width is given by

coh

(c)=Γ

(Q,c→ 0) . (3.77)

which we call Γ

(Q), and where c is the average site occupancy. In this

case the width is equal to the result for incoherent scattering on the dilute

system, Γ

, as derived before (see (3.30)). This coincidence is due to the

fact that the thermodynamic coeﬃcient in the chemical diﬀusion coeﬃcient,

∂(µ/k

T )/∂ ln c (proportional to 1/(1 − c)) just cancels the blocking factor

in the single particle diﬀusion coeﬃcient, (1 − c).

Instead of the hard core model, a general potential of the mutual in-

teraction is introduced by the method of linear response in the mean-ﬁeld

approximation [138] which leads to the result

coh

(Q,ω)=

c(1 − c)Γ

(Q)/π

coh

(Q)+ω

(3.78)

with

coh

= Γ

(Q)



c(1 − c)V (Q)



. (3.79)

(Q) is again the value for the dilute case of incoherent scattering, c → 0.

V (Q) is the Fourier transform in space of the atomic interaction potential.

With the Clapp-Moss relation [139] the unknown potential can be directly ex-

pressed by the static structure factor for the short-range order of the diﬀusing

particles (corrected for contributions from the host lattice distortion)

coh

(Q)=

c(1 − c)

c(1−c)V (Q)

. (3.80)

This ﬁnally leads to

coh

(Q,ω)=S

coh

(Q)

coh

/π

coh

+ ω

(3.81)

with the coherent line width

coh

(Q)=

c(1 − c)Γ

(Q)

coh

(Q)

. (3.82)

For very small Q, S

coh

(Q)

−1

is proportional to the thermodynamic factor

∂(µ/k

T )/∂ ln c, the derivative of the chemical potential with respect to the