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3 Diﬀusion Studies of Solids by Quasielastic Neutron Scattering 151

Fig. 3.33. Diﬀusion of deuterium in Nb: D

(Q)=Γ

coh

(Q)/Q

- generalized chem-

ical diﬀusion coeﬃcient as a function of Q; D

= tracer or single particle diﬀusion

coeﬃcient obtained from incoherent quasielastic scattering. S(Q)=experimen-

tal structure factor for deuterium dissolved in Nb. The ratio D

/S(Q)iscloseto

(Q) [140].

concentration. This allows one to introduce the chemical diﬀusion coeﬃcient

,namely

coh

(Q → 0) =

c(1 − c)Q

(c → 0)

coh

(0)

= Q

. (3.83)

Like in diﬀusion measurements by light scattering (see Chap. 15), the

coherent quasielastic width can be intuitively understood by the decay rate

of concentration ﬂuctuations with a wave number Q. In (3.82) a maximum

of the static structure factor S(Q) causes a minimum of the quasielastic

width which means: density ﬂuctuations corresponding to strong short-range

correlations decay with a very small relaxation rate. For the ﬁrst time, this

“line narrowing” was predicted by De Gennes [141] and observed for liquid

argon [142].

In a deuterium loaded bcc niobium crystal the relation between the struc-

ture factor S(Q) and the coherent width, see (3.82), was investigated (deu-

terium has a strong coherent scattering contribution). Γ

coh

and Γ

are com-

pared in Fig. 3.33 [140] deﬁning D

(Q)=Γ

coh

(Q)/Q

, whereas the structure

factor was calculated with a site blocking model (dashed line) and also exper-

imentally determined, after correction for the strain ﬁeld contributions from

the host lattice.

152 Tasso Springer and Ruep E. Lechner

D [cm /s]

T [10K]

-3 -1-1

Fig. 3.34. Lower curves: Chemical diﬀusion coeﬃcient D (called D

in the text)

from mechanical relaxation experiments. Slowing down for T → T

near the critical

point at the coexistence curve of the NbH

phase diagram. Open circles: H/Nb

= 34.4 at % (critical concentration); open squares: 40.4 at %. D

∗

=self-diﬀusion

coeﬃcient calculated from the chemical diﬀusion coeﬃcient using the measured

factor (∂µ/∂ ln c). Fair agreement of this quantity is found with D

∗

from quasielastic

scattering with Γ = Q

∗

for 33 % (full triangle-up) and for 3.2 % H/Nb (full circle

and full triangle-down: upper curve). From [143].

Fig. 3.34 compares the diﬀusion coeﬃcient as obtained from the width

of incoherent quasielastic scattering for NbH

with the chemical diﬀusion

coeﬃcient; this was measured by the long-range mechanical relaxation on

such samples (Gorski eﬀect) [59,143]. The results are consistent with (3.83).

The “critical slowing down” of the chemical diﬀusion coeﬃcient D

is caused

by the singularity of the thermodynamic factor ∂(µ/k

T )/∂ ln c near the

critical temperature T

on the top of the miscibility gap in the Nb-H phase

diagram. This eﬀect does not show up in the self-diﬀusion coeﬃcient D

For hydrogen or deuterium atoms the incoherent and the coherent struc-

ture factor can be separated in the same scattering experiment, using polar-

ized neutrons and spin analysis after scattering [49].

An entirely diﬀerent aspect of quasielastic coherent scattering is related

to the ﬂuctuations of the host lattice, caused by the diﬀusing particle [144].

3 Diﬀusion Studies of Solids by Quasielastic Neutron Scattering 153

For instance, a hydrogen atom dissolved in niobium is surrounded by atomic

displacements. The resulting long-range strain ﬁeld induces diﬀuse scattering

between the Bragg reﬂections, which is especially strong close to the Bragg

lines, the so-called Huang scattering. The strain ﬁeld can be described by mu-

tually interpenetrating “distortion clouds”, moving along with the protons.

Consequently, the coherent diﬀuse scattering on the host lattice is quasielas-

tic due to these density ﬂuctuations. Neglecting retardation and assuming a

linear superposition of the weak distortion ﬁelds, the resulting coherent dy-

namic structure factor can be factorized into two terms: the static structure

factor caused by the strain ﬁelds of uncorrelated single particles, and the in-

coherent dynamic structure factor following the Chudley-Elliott formula for

the diﬀusing particles. First quasielastic measurements on the distortion ﬁeld

were performed for hydrogen in niobium [55] (see Sect. 3.5). The coherent

quasielastic scattering yields interesting information on the diﬀusion in the

lattice; however, its observation is diﬃcult. Although the distortion scatter-

ing is strong close to the Bragg reﬂections and therefore its integral is easy to

observe, the width of the quasielastic spectra is very small in these regions,

becauseitgoesas(Q − G)

,whereG is the reciprocal lattice vector cor-

responding to the Bragg line. So, in the experiment, a suitable compromise

must be found between the intensity of this eﬀect, the resolution, and the

closest distance from the reciprocal lattice point, where measurements are

feasible with the available energy and momentum resolutions.

A third type of quasielastic coherent scattering concerns the observation

of molecular structure factors in the context of translational diﬀusion. This

problem occurs, for instance, in studies of the crystalline high-temperature

fast-ion conductor modiﬁcations of alkali metal (M) salts, M

and M

of sulfuric and phosphoric acid, respectively. These phases with the poly-

atomic anions SO

2−

and PO

3−

, are not only cation conductors, but also

plastic crystals, since the anions exhibit dynamic rotational disorder. Part

of the motivation for quasielastic neutron scattering studies of these com-

pounds [145–147], has arisen from a dispute about a possible cation mobility

enhancement by dynamical coupling of cation migration and anion reorienta-

tion. Proponents of this coupling model have coined the term “paddle-wheel”

mechanism [148–150].

Obviously, the reorientation of the polyatomic tetrahedral sulfate and

phosphate ions implies a fast and rigidly coupled motion of the four oxygen

atoms bound in each anion, whereas the sulfur and phosphorus atoms may

be assumed to stay immobile if coupling between translation and rotation

of these ions is neglected. Since oxygen is a purely coherent scatterer, the

scattering function S

coh

(Q,ω) is required. If correlations between diﬀerent

rotating ions are neglected, the calculation is straight-forward, employing the

same (well-known) models as used for rotational motions in the incoherent

scattering case. Details can be found in the literature [145,151,152]. There are

154 Tasso Springer and Ruep E. Lechner

two essential diﬀerences between the incoherent and the coherent scattering

functions for rotational (localized) motions:

i) In the case of S

coh

(Q,ω) , the elastic scattering of the rotating molecules

is entirely concentrated in crystallographic Bragg peaks: the neutron waves

scattered from the (inﬁnite-time) probability density distributions in diﬀerent

unit-cells are capable of interference with each other. In other words, there is

no EISF that could be measured between the Bragg peaks, and therefore the

possibility of the self-consistent normalization based on a well-known integral

property

of S

inc

(Q,ω), customary in incoherent scattering, is not available

here.

ii) In the absence of elastic and quasielastic incoherent structure fac-

tors (EISF and QISF), the spatial information concerning the geometry of

the rotational motion can only be extracted from the integral of S

coh

(Q,ω)

(excluding Bragg peaks), i.e. from the quasielastic coherent structure factor

(QCSF). The normalization problem can be solved, if the compound stud-

ied contains an incoherently scattering component. This is the case, e.g. for

the sodium salts of sulfuric and phosphoric acid. The incoherent scattering

of sodium, determined at small angles, where the intensity of the coherent

component is negligible, has been used for normalization [145].

Earlier QENS measurements of Na

taken in the temperature range

from 300 K to 973 K have been successfully interpreted in terms of a uniaxial

reorientation model with a single time-constant in the 1 to 0.3 ps range (ac-

tivation energy: 184 meV) for the observed localized diﬀusive motion [145].

There is, however, a discrepancy between experiment and theory at low Q,

up to about Q =1.5

−1

, where the measured QCSFs are larger than the

calculated ones (see Figs. 6 and 7 in [145]). In a subsequent paper [147], the

discrepancy was explained by a modiﬁcation of the molecular structure fac-

tor. It is caused by a sodium ion attached temporarily at a distance of 2.15

from the central atom, above the center of one of the faces of a tetrahe-

dral anion involved in reorientation. Such transient “binding” increases the

eﬀective molecular “radius” and, therefore, leads to additional quasielastic

intensity for low Q. The same interpretation was also found to be consis-

tent with similar measurements on solid solutions of sodium phosphate and

sodium sulfate (Figs. 5 to 7 in [147]). The translational diﬀusion of sodium

ions in Na

was studied with high-resolution backscattering techniques;

it was concluded that sodium ion jumps between tetrahedral sites in the cu-

bic high-temperature phase appear to be most important [146]. The diﬀusion

coeﬃcient shows an Arrhenius-type behaviour,

(T )=1.3 × 10

−3

−1

exp(−630 meV/k

T ).

This value of the activation energy is considerably higher than that of the

reorientational motion, and the time scales of the two types of motion diﬀer

by roughly three orders of magnitude: While characteristic times of the an-

The integral of the incoherent scattering function over energy transfer is equal

to1[3].

3 Diﬀusion Studies of Solids by Quasielastic Neutron Scattering 155

ion rotation are in the picosecond range, average residence times of diﬀusing

sodium ions are of the order of nanoseconds. This means that cation transport

via a rotational step does not automatically lead to a step of translational

diﬀusion. It should rather be seen as a thermally activated phenomenon re-

lated with local potential ﬂuctuations which favour translational diﬀusion,

and which – from time to time – bring a cation into a position favorable for

jumping away to a more distant site. These results clearly suggest a certain

dynamical correlation between the diﬀusion of cations and anions in these

compounds. But, because of the very diﬀerent time-scales, a paddle-wheel

mechanism in the sense of a synchronous action of rotatory and translatory

motions can be excluded for Na

3.13 Conclusion

Quasielastic scattering of neutrons is an excellent tool to evaluate the el-

ementary steps of diﬀusion in solids on scales from atomic to mesoscopic

dimensions and times. Good examples are the understanding of correlation

eﬀects of diﬀusion in metals with vacancies or of hydrogen diﬀusion in met-

als. The latter problem is still exciting regarding, for instance, the elastic

displacements and the long-range strain ﬁelds produced by rapidly moving

protons, as well as the question, to what extent quantum eﬀects related to

the protonic motion might exist. So far no such eﬀects were observed, ex-

cept for the localized modes of a trapped proton. Many questions are still

unsolved and need more experiments, for instance quasielastic Huang scat-

tering. Another class of problems deals with diﬀusion in disordered solids.

While the theoretical treatment of systems with highly dilute or extremely

concentrated impurities is relatively simple, intermediate concentrations are

much more diﬃcult to handle. The behaviour of disordered alloys has not yet

been systematically treated.

In the case of ionic crystals, ion diﬀusion, and especially proton diﬀusion

has been studied in solid-state protonic conductors, e.g. in hydroxide hy-

drates, hydrogen sulfates and selenates of alkali metals, and in the perovskite

structures of acceptor-doped barium or strontium rare earth metal oxides.

The speciﬁc dynamical features which generally distinguish protonic conduc-

tors from other ionic conductors are connected with the fact that, in addition

to the ionic character of these crystals, hydrogen bonding plays an important

role. In particular, it is the process of formation and breaking of H-bonds,

which represents an important ingredient of the proton diﬀusion mechanisms.

The relation between this solid-state chemical reaction and the diﬀusion of

protons has been established. Suitable approximations have been developed

which allow a comparatively simple treatment also of systems with interme-

diate concentrations of mobile ions. When Coulomb interactions, e.g. in silver

iodide crystals, play a major role, ion-ion correlations become important and

must be taken into account; diﬀusing ions carry a “correlation cloud” along

156 Tasso Springer and Ruep E. Lechner

with themselves (similar to the Debye-H¨uckel cloud in ionic solutions). A

new theoretical model has been developed along these lines and contributes

to the understanding of the relation between ionic diﬀusion and conductivity

in terms of backward-correlated ion jumps. Also non-stoichiometric metallic

oxides are interesting with respect to diﬀusion and electrical conductivity.

An interesting special case are solid rotator phases with ionic conduc-

tivity, where the rotating polyatomic anions contribute quasielastic coherent

scattering, whereas the translational diﬀusion of incoherently scattering sim-

ple cations is easily isolated if the time-scales are suﬃciently diﬀerent. A

certain dynamical correlation between the diﬀusive cation and anion motions

in these compounds has been shown to exist. But, the renowned picture of a

“paddle-wheel” mechanism in the sense of a synchronous action of rotatory

and translatory motions has so far not been conﬁrmed by QENS experiments.

The interpretation of coherent quasielastic scattering in order to investi-

gate collective eﬀects of diﬀusion is a new ﬁeld with a promising future. The

high intensities available at modern neutron scattering instruments and es-

pecially at the high-power spallation neutron sources now under construction

will allow the systematic experimental separation of coherent and incoherent

scattering by spin analysis.

In general, future eﬀorts aiming at the synthesis of the results of quasi-

elastic neutron scattering, M¨oßbauer spectroscopy, nuclear resonance scatter-

ing of synchrotron radiation, nuclear magnetic resonance studies, frequency-

dependent conductivity spectroscopy using electromagnetic waves from the

microwave to the (far) infrared region, and ultrasonic measurements, com-

puter simulation and the development of new analytical approaches will fur-

ther contribute to a more comprehensive and sophisticated understanding of

the diﬀusion processes in solids.

Notation

a lattice parameter

b scattering length

c site occupancy probability

isotope concentration of species i

concentration of traps

d jump vector

collective or chemical diﬀusion coeﬃcient

diﬀusion coeﬃcient for free diﬀusion

rotational diﬀusion coeﬃcient

self-diﬀusion coeﬃcient (usually in 3 dimensions)

two-dimensional self-diﬀusion coeﬃcient

three-dimensional self-diﬀusion coeﬃcient

ionic charge or conductivity diﬀusion coeﬃcient

energy of neutron before and after scattering, respectively

3 Diﬀusion Studies of Solids by Quasielastic Neutron Scattering 157

act

activation energy

A(Q),F(Q) incoherent elastic structure factor (EISF)

G reciprocal lattice vector

g(r) pair-correlation function

G(r,t) space and time dependent pair-correlation function

(r,t) space and time dependent self-correlation function

H hydration level (given in units of g of water per g of biological

material)

h Planck’s constant

 Planck’s constant / 2π

Haven ratio

I(Q,t) coherent intermediate scattering function

(Q,t) incoherent intermediate scattering function

incident neutron current density

∆I

scattered intensity

, k

wave vector of neutron before and after scattering, respectively

wave numbers of neutron before and after scattering, respectively

∆k

div

wave number spread caused by divergence

l, d jump distances

distance between elements x and y of a TOF spectrometer

∆L uncertainty in the length of the ﬂight path

L(H

,ω) Lorentzian in energy transfer with HWHM = H

M atomic mass

number density of scattering atoms

number of identical atoms in the scattering specimen

P (r,t) time dependent probability of occupancy

P (Q,t) Fourier transform of P

Q scattering vector

eﬀ

trapping radius

S(Q,ω) scattering function, or dynamic structure factor

inc

(Q,ω) incoherent scattering function or dynamic structure factor

coh

(Q,ω) coherent scattering function or dynamic structure factor

(Q,ω) incoherent scattering function for two-dimensional translational

diﬀusion

(Q,ω) incoherent scattering function for three-dimensional translational

diﬀusion

∆t experimental observation time

V sample volume

v velocity

thermal velocity

velocity of incident neutrons

velocity of scattered neutrons

V (Q) Fourier transform of interaction potential

Γ quasielastic widths

158 Tasso Springer and Ruep E. Lechner

quasielastic widths for rotation

,Γ

widths following Chudley-Elliott model

coh

widths of coherent scattering

η order parameter

θ polar angle of spherical coordinates

Θ angle of Bragg reﬂection

∆Θ beam divergence

λ neutron wave length

ν repetition rate of chopper

σ ionic conductivity

coh

coherent bound scattering cross-section

inc

incoherent bound scattering cross-section

τ mean residence (or rest) time a particle spends on a single

site, when diﬀusing on an undisturbed lattice (according to the

Chudley-Elliott model)

time required for diﬀusive jump

mean time elapsing between two consecutive encounters, when a

particle undergoes diﬀusion according to the encounter model

MTOF spectrometer pulse length of pulsing chopper

MTOF spectrometer pulse length of monochromating chopper

relaxation time for the restoration of a displacement ﬁeld of elas-

tic distortion

relaxation time of (rotational) diﬀusion on a spherical surface

rpm

duration of the (spectral) period of a TOF spectrometer

mean residence time of a diﬀusing vacancy

residence (or rest) time of a particle in a trapped state (two-state

model of translational diﬀusion)

mean duration of free diﬀusion in the undisturbed lattice (two-

state model of translational diﬀusion)

,τ

relaxation times of the TAHB model

relaxation time of the two-site jump model

∆τ

chopper pulse length of XTL-TOF spectrometer

φ scattering angle

Φ azimuthal angle of spherical coordinates

Ω solid angle

ω energy transfer

∆(ω) half-width at half maximum (HWHM) of the energy transfer

resolution function.

BSC back scattering

CE Chudley-Elliott (jump-diﬀusion model)

CHx chopper number x

EISF elastic incoherent structure factor

FWHM full-width-at-half-maximum

HWHM half-width-at-half-maximum

3 Diﬀusion Studies of Solids by Quasielastic Neutron Scattering 159

MSC multiple scattering

MTOF multi-disk chopper time-of-ﬂight

nn nearest neighbour

nnn next-nearest neighbour

NSE neutron spin-echo

PDD probability density distribution

PWR pulse-width ratio (optimization method)

QCNS quasielastic coherent neutron scattering

QCSF quasielastic coherent structure factor

QENS quasielastic neutron scattering

QINS quasielastic incoherent neutron scattering

QISF quasielastic incoherent structure factor

sc simple cubic

SSPC solid-state protonic conductor

TAHB trigonal-asymmetric hydrogen bond

TD translational diﬀusion

TOF time-of-ﬂight

XTL (monochromator or analyzer) crystal

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