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AgI-type solid electrolytes
Progress in Solid SVate Chemistry, 1976, Vol ii, pp. 345-402
PergamonPress
Printed in Great Britain
AgI-TYPE SOLID ELECTROLYTES K. Funke Institut pdr Phys~kahsche Chemie der Umversit~t G~ttingen and Sonderforschungsbereich 126 Gotnngen, Germany
CON TEN TS
I
Introduction
345
II
General S u r v e y
348
III
Ionic C o n d u c t i v i t y
350
IV
Structural
V
Heat C a p a c i t i e s
Properties
357
VI
Tracer Diffusion
VII
Complex Conductivity
VIII
F a r - I n f r a r e d and R a m a n - S c a t t e r i n g E x p e r i m e n t s
385
IX
quasielastic Neutron Scattering
390
and D i s o r d e r i n g
Processes
367 373
in the M i c r o w a v e Range
377
I INTRODUCTION
Most
ionic
defect
crystals have degrees of d i s o r d e r well below
concentrations
being small,
at a time and carry charge vities of these "normal" peratures. ca.
Typical
through the crystal lattice.
ionic
crystals are therefore
values are ca.
exceeds
the energy r e q u i r e d the thermal energy,
ent and unusual behaviour. trical p r o p e r t i e s
pair greatly
(2,9). The c o n c e n t r a t i o n s
ionic crystal
therefore
of acfects
strcngiy depenu on
over a t e m p e r a t u r e
Flg.l
shows T u b a n d t ' s
AgCi,
AgBr,
crystal ~ith a quite differ-
they o b s e r v e d
and Lorenz's
has a h i g h l y
- stable b e t w e e n
that solid silver
range of more than 400 °C, has ionic conductivi-
to those of the best c o n d u c t i n g
and AgI versus
AgCI and AgBr,
an ionic
In the course of their i n v e s t i g a t i o n of the eitc-
of the silver halldes,
~ comparable ties o n ~y
-phase
tem-
(~) and
and the same is true of the ionic conductivity.
In 1919, T~bandt and Lorenz e n c o u n t e r e d
iodide,
conducti-
(j), both at 200 °C. In these "normal"
for the formation of a defect
respectively
ions in a typical
temperature,
The ionic
low at m o d e r a t e
kBT. At 200 °C the factors are ca. 95 and 50 in
the cases of AgCI and NaCI, and mobile
(i). The
iO -4 ( ~ cm) -I in the case of igCl
I0 -~ ( 9 em) -I in the case of NaCI
crystals
10-2,
only a limited fraction of ions can move
liquid e l e c t r o l y t e s
o r i g i n a l plot of the c o n d u c t i v i t i e s
temperature.
It is seen that AgI,
(4)
of
in contrast
to
c o n d u c t i n g solid phase - now known as the a-
147 and 555 °C. At the 2 - ~
phase transition,
the
c o n d u c t i v i t y of AgI increases by more than three orders of m a g n i t u d e up to
345
346
K. Funke
,/
I
f
r
I
~o
,/
/
/
I
I t I I
/
1i
ir"
I ! I t
;
Ill
/ _~
i r"
~o, ~ ~Oo
SO0 o
600 °
700~
C °
FIG.
i. Conductivity of the silver halides according
to Tubandt and Lorenz
~ 1.3 (~ cm) -l, cf. Fig 4. Within the ly up to
a -phase,
(4) it increases only slight-
~ ~ 2.6 ( ~ cm) -I and then drops on melting.
value of the ionic conductivity of
a-AgI
The abnormally high
and its weak temperature dependence
are typical of the class of ionic crystals to be reviewed in this paper. The measurement
of the conductivity,
~ , of silver iodide may be regarded as
the starting point for the investigation of AgI-type their early work, Tubandt et al. identified various
ionic crystals as belonging
these were
g- and ~-CuBr,
From their measurements experiments
to the same class of materials. a-Ag2S,
a-Ag2Se,
and
the charge is being carried by the cations
phases were found to be mixed ionic and electronic
The highly mobile
(5). The conductors
cations were supposed to move in a "liquid-like"
as will be seen in section II, the principal the same as early in this century.
for new solid electrolytes
~)The high electrical by Faraday in 1634.
(6)
manner
In the years to come further
support for this concept was provided from X-ray structure analysis
essentially
Among
a_Ag2T e (5)~
of transference numbers and from their inter-dlffusion
within the rigid framework of the anion lattice.
Today,
In
phases of
they concluded that in the highly conducting phases of the silver
and cuprous halides chalcogenide
a-Cul,
solid electrolytes.
the high-temperature
(7)
lines of development
On the one hand,
is of great current interest;
are
the search
on the other hand,
conductivity of solid Ag2S had already been observed
Agl-Type solid electrolytes
we are these
still
trying
to get
a better
features
common
347
understanding
of the ion].c m o t i o n
in
materials.
The most
prominent
to A g I - t y p e
solid
electrolytes
are
the
following. (i)
The
cations
of voids the
cations
less
at random.
regions (ii)
The
these
differences
a large
the a p p a r e n t
In o r d e r
iodide
ions
deduced,
the
two s i l v e r
two s i l v e r
(9,10,11)
the p o s s i b l e quately
number
(i)
potentials
energy
are voids.
- as far as
- are of the o r d e r
to more
extended
The
it is still
data
sites
more
However,
appear
or less
pointed
out
still
arising
sites
do not
data ~2, i~,14) more
provided
with
regard
adeby the the
to the
as follows.
is really
in c h a r a c t e r
was
of these
by far e x c e e d s
ions may be s u m m a r i z e d cations
it is
by d i f f e r e n t
scattering
of space
The
statistically
to be d e s c r i b e d
regions
diffusive
are
of iodide
structure"
the o c c u p a n c y
regions
of the
cube
as such
an "average
local
the m o t i o n
(zz),
a-AgI.
Geometrically,
of these
of the s i l v e r
the
(7,5).
questions
of
plots
and experi-
consider
in a b o d y - c e n t r e d
recent n e u t r o n
principal
that
briefly
as has been
ions
Because
from A r r h e n i u s
at
thus
energy.
distributed (7,8).
compounds to void,
process.
on two p r e f e r r e d
literally,
The n u m b e r
from void
and the c o n c e p t u a l
lattice
diffraction
Is it p o s s i b l e that
situation
of the s i l v e r
cations.
sense
of
or
neighbouring
- or in certain
obtained
contained
of sites too
of r a t h e r
and m o t i o n s
more
can find vacant
the local
cation-diffusion
ions b e i n g
According
framework.
of the
positions
that
- can move
cubic
ions
X-ray
(42)
positons
in terms
iodide-ion
cation
interconnect
let us once more
precisely
must not be taken
authors
cations
the m i c r o s c o p i c
a large n u m b e r
sites
the n u m b e r
the n u m b e r
occupied
by the anions
of a c t i v a t i o n
involved,
the o b s e r v e d
the
are
in potential
about
of them
in a general
be l o c a t e d
From
a typical
lines w h i c h
all of the
fraction
form a b o d y - c e n t r e d
that
cannot
voids
that
exceeds
neighbourhood.
the d i f f e r e n c e s
energies
to e x p l a i n
difficulties
exist.
these
in such a way
are b r o u g h t
lattice
to be of the o r d e r of the thermal
mental
evident
in the sense
energy.
As a consequence,
found
situation
certain
lines,
thermal
that
in its i m m e d i a t e
along
participating
over
In this
these
least
and
are a r r a n g e d
flat
disordered
for them by the anion
themselves~
Along
of the
ions
structurally
of space
anions
rather
(iii)
are
provided
liquid-like
on a scale
in
of roughly
1 ~ or less? (ii)
On the o t h e r cept
that
hand,
the
metal-hydrogen "residence (iii) W h a t
kind
ordered
answers
systems?
times"
and
with
and
reliable
times"
as has
been
moving
in favour
e.g.
should
of the
like h y d r o g e n
be able
conin
to d i s t i n g u i s h
of the ions.
between
rapidly
jumps,
case one
exists
theoretical
Questions.
arguments
discrete
In this
"jump
is it true,
domains
experimental to these
there perform
of c o r r e l a t i o n
For example,
Recent
are
cations
the motions
proposed
of d i f f e r e n t
(15,16)
that
there
boundaries?
research
has
just b e g u n
to provide
ions? are
348
K. Funke
II GENERAL SURVEY In Table
i, a llst of AgI-type
their respective lattice
temperature
parameters.
obtained
solid electrolytes
Furthermore,
from these compounds.
references
Table
ting compounds
which contain halides
valent
However,
copper.
has considerably complicated copper salts
(17)
beyond
or chalcogenides
of silver or of mono-
composition.
systems
like Li2S04,
many silver-ion
conducting
These latter are mostly of more
By restricting
ourselves
to silver and
on which much work has been done
nor will they be reviewed
(20) and the anlon-conducting
in this article.
(21) solid electrolytes
(18, The
are well
the scope of this paper.
The experimental
data referred
to in Table
a) ionic conductivity
in the low-frequency
b) structure
from X-ray,
analysis
c) heat capacity
at constant
d) tracer-dlffusion
limit,
electron,
pressure,
coefficient
1 include ~ (T),
or neutron
ions, D~(T),
and permittivity
in the microwave
f) conductivity
and permittivity
in the far infrared
g) Raman scattering h) quasielastic
1 certainly
data.
In particular
phase diagrams
ments are not quoted. (30)
etc
the results
scattering
Furthermore,
(FIR),
(26,27)
and thermodynamic
ultrasonic
In this review,
spectrum of available
attenuation
however•
experimental
data from e.m.f,
data exist of thermoelectric (28,29)
electronic
we will essentially
measure-
power
''~22-25) proper-
concentrate
on
listed in the table.
It can already be seen from the general AgI-type
(MW),
(C.NS).
does not cover the whole
ionic Hall effect
range
(R),
neutron
Table
diffraction,
Cp(T),
of the mobile
e) conductivity
ties
data
to those cation-conduc-
in the last few years,
19), have not been included, 2-alumina-type
are given of experimental
and
i is restricted
have had to be omitted.
chemical
along with
anion arrangements,
as the number of phases known to belong to this class
increased
solid electrolytes
is presented
ranges of stability,
solid electrolytes
shape of Table
i that in the field of
we can at present distinguish
two different
major
llnes.of development. ~) A systematic search has begun for ionic crystals which exhibit high ionic, These
but negligible compounds
construction
electronic
are of considerable
of solid-state
current
batteries.
at ambient
interest
temperature.
in view o£ the
Such solid-state
systems have
become much in demand in recent years because
of their advantages
respect
large temperature
to conventional
of operation,
battery-systems,
e.g.
long shelf life, and the possibility
Suitable
materials
species.
In many cases not even the structure
usually
various solid silver-lon shi (31). (ii)
conductivities
New experimental understand
techniques
of the main requirements
of miniaturization.
contain AgI with silver ions as the mobile
conductors
the microscopic
with
ranges
is known.
has recently
A review of
been given by Takaha-
are being used in order to analyze and dynamical
behaviour
of these techniques
of the mobile
ions. One
is that the information
they yield should be on the time scale of the ionic motion,
i.e. rough-
b.c.c.
~430 ?
Ag~SBr
?
50,49 50,49
43O 48O
a:ii.97 c: 7.41 a:12.77 c:26.54 I
25
b.c.c. 4 90 ordered b.c.c. 4 99 disordered " b.c.c. 4.81 ordered -30 25
25
300
60
6.38
f.e.c.
170
5.84
see section IV see section IV
6,56
81,65 82
65 66
80
80
63,64 64
80
77-79
74-76
63,64
61,62
60
74,58
74
6,54,55 74,75
-)5.66 e~(581 57-59
250
250
72,73
72,73
71,72
6,51-54 74,75
48,49
250
3truct.
cp(¢)
)4
)2
)2
)2,93
89,91,73
89,91,73
89-91,7]
54
54,98
83, 4,43,44 7,8,12 l~a67~6L 8,84-8? 95,96 88 45-47 69,70 97
~(T)
References:
43O
25
250
[°4
(T)
f.c.c.
f. C.C.
6.57
thermodynamically unstable at T < 27 °C
((CH3)4N)2AgI3115
(C5H5NH)Ag5I 6
235-
a-Ag~Sl
<235
~-Ag~SI
i~110-1113
a-Cu2Se
51- 164
~430-I127
a-Cu2S
a-Ag2Hgl 4
~140-~790
a-Ag2Te
f.c.c.
b.c.c.
~13o- 897
a-Ag2Se
5.00
4.88
b.c.e.
~177-~600
a-Ag2S
4.53
469- 487
~-CuBr
h.c.p,
6.13 a:4.04 c:6.~
11.24
5.06
b.c.c.
386- 469
ee see~)27- 232 stion IV 408- 602 f.c.c.
147- 555
Eo4
Stability Anion L~ttice Range Structure Const.
B-CuBr
~-Cul
a-RbAg415
a-AgI
Solid Electrolyte
TABLE i. GENERAL SURVEY ON AgI-TYPE SOLID ELECTROLYTES
#4,99
33
32
36
35
32,37
FIR
I
38
R
14,40
o~s
~o
W
O
rv
tD
T
350
K. Funke
ly between
109 and 1013 Hz. Therefore,
(32,35-37),
and Raman experiments
microwave
(32-34)
far-infrared
(38) have been undertaken.
A similar
frequency range is covered by the quasielastic and inelastic scattering of cold neutrons
(39,14,40),
neous spacial resolution.
with the additional
advantage of simulta-
In the lO 9 to 1013 Hz frequency range,
diffe-
rent kinds of behaviour of the mobile ions have been predicted by different models
(37•41,42).
These models are now being tested by comparison
with experiment.
III IONIC CONDUCTIVITY The most prominent property of AgI-type ly high ionic conductivity. -ionic conductors"
solid electrolytes
is their unusual-
They have therefore sometimes been called "super
or even "ionic superconductors".
However,
these terms seem
to be misleading• (lO0) ty
especially because there is no relation to superconductivi-
For the graphical
representation of ionic conductivities
systems•
it is convenient
to choose a plot in which the data give straight
lines. Let us consider two extreme (i)
in solid or liquid
cases.
As is well known•
in a "normal"
linear dependence
is observed in a plot of log(~ T) vs I/T. In the
intrinsic
temperature
equilibrium,
region,
ionic crystal•
like NaCI or AgCI,
a
where the lattice defects are in thermal
the slope of this line is determined by the defect-forma-
tion enthalpy and by the migration enthalpy which both clearly exceed the thermal energy. (ii)
For liquid systems• proposed by Reynik
a special "itinerant
oscillator"
model has been
(i01). This model predicts a linear dependence
the coefficient of self-diffusion D on temperature, indeed observed in many liquids,
of
D = bT-a, which is
including molten salts. Let us assume
that in a given system the motion of the ions obeys this relation. Secondly,
let us assume that the Haven ratio• HR~•
depend on temperature•
does not noticeably
that is, that D is proportional
stability range of the system under consideration.
to ~ T
in the
In a ~ vs 1/T dia-
gram, we then simply find a straight line with a negative slope. In the case of AgI-type solid electrolytes,
a specific difficulty arises.
Plots suggested by (i) and (ii) yield nearly straight lines and the same is true of a number of other plots which may be made. Therefore,
an interpre-
tation may be dangerous as long as no independent information is a~ailable about the microscopic behaviour of the mobile ions. As an example• conductivity of ~ - A g I may be considered. by Tubandt and Lorenz
In Figs. 2 and 3, the data obtained
"~4~ marked "u", and by Kvist and "
'°o"• are plotted in three different ways,
the ionic
Josefson ( 4 3 )
i.e. 5 vs i/T, l o g g
marked
vs l/T• and
log (gT) vs I/T. ~he
Haven ratio is the factor in the Nernst-Einstein
as the ratio of tracer to charge diffusion see section VI.
equation and is defined
coefficient•
H R = D~/D~
(102)
Agl-Type solid electrolytes
T [°C] - - ~ 300 400
200 i
i
351
500
1
r
2.6
2.4 2.2 6 [ ( a c m ) -I ] 2,0 1.8 1.6
o/ / /
1.4
/,
2~ =
FIG.
2.Electrical
21o
,:0
conductivity
,I~
,2
of
~-Agl
in a
~ vs I/T r e p r e s e n t a t i o n .
T [oc] --~ 400 500
200
t
,16
T -1 [tO 3 K-1 ]
300
e5
2.0
~1
[ ( n c m ) "1 ] 2.5 11"5
( points below stroight line)
i
( points obove
straight llne)
n
2.0
1.5
oe • II
/.
./.-
1.0
0.5 I
I
Z2
2.0 •" - -
FIG.
3.Electrical log
i
1
i
1.8
1.6
1.4
1.2
T - ' [10 3 K - ' ]
conductivity
of
a-Agl
(G T) vs I/T representations.
in the l o g ~
vs I/T and
352
K. Funke
In the ~ vs 1/T representation, found at temperatures
Fig. 2, an accurately linear dependence
above 250 °C. In the high-temperature
ductivity extrapolates
limit,
is
the con-
to a value similar to those abtained from correspon-
ding plots of molten salts. According
to Reynik's
theory,
this would give ca.
0.5 ~ for the mean displacement of a centre of oscillation during one oscillation period,
cf. 0'Keeffe
(103). However,
below 250 °C, the linear fit becomes
as the temperature
is decreased
increasingly worse. Moreover,
known from microwave and neutron-scatterlng
experiments,
and IX, that the diffusive motion of the silver ions in
it is now
see sections VII g-AgI
is not proper-
ly described by the itinerant oscillator model.
Instead,
that distinct
with mean jump-lengths
steps of Jump-diffusion do occur,
the order of a lattice constant. activation
the conclusion is
In the concept of distinct
of
jumps, a thermal
for the ionic motion across the maxima of local potential has to
be considered.
Thus,
the logarithmic
plots,
rather than the linear one. In the l o g ~ of the experimental
Fig ), might be appropriate
vs I/T plot,
curve is slightly negative,
the second derivative
while it is slightly positive
in the log (~ T) vs I/T representation. Biermann and Jost
(i04) have considered
the f r a c t i o n m
of cations which have
a certain migration enthalpy H m of the order of kBT. For simplicity, assume
they
just two sets of cations with enthalpies differing by H m. Then, -Hm/kBT
(1).
e
a
=
-H m /
1+e
kB~
'
the second term in the partition sum is not negligible Interestingly, onal to
a,
the experimental
giving the observed
~-values log G
compared to unity.
are found to be directly proporti-
vs 1/T dependence,
when this approach
is used. The corresponding "value of H m is 64 meV in the case of On the other hand,
in
g-AgI.
assuming a continuous distribution of possible energy
states, we find that the fraction of ions with enthalpies H • H m is given simply by a = exp(-Hm/kBT). rate is proportional hence of
to a
Furthermore,
one should expect that the jump
and, therefore,
that the same is true of D and
g T rather than of 6 . The upward bend of the experimental
curve in
the log (e T) vs I/T plot might then indicate increasing values of H m with increasing temperature.
As a tentative interpretation,
the following con-
sideration might be helpful. Because of the motions of the neighbouring cf. sections VII-IX,
ca. I0 -12s, with amplitudes
comparable
to the thermal energy. Therefore,
ions in different momentary microscopic enthalpies defined.
of activation
cat-
situations will need different
for a Jump, and a unique value of H m cannot be
At relatively high temperatures,
larger enthalpies
ions,
the local potentials will fluctuate on a time scale of
the fraction of jumps requiring
of activation will be larger than at relatively low
temperatures and hence a temperature variation of the apparent overall value of H m should be expected. In Figs.4-6,
the ionic conductivities
are represented have
of various AgI-type
solid electrolytes
in log (~ T) vs I/T plots. As indicated above,
precautions
to be taken in the interpretation of the slopes. In particular,
and meaningful enthalpies
of migration cannot be extracted.
unique
Agl-Type solid electrolytes
T[ oc] - ~ - - - . -
353
T[oC]
200
=
2,,o0
soo
soo
)
~a- Cu Br
a-AgZ
103 !T 103
I)-CuBr
• 1 [(~ c m)"~1
loZ[~crn] 102
~j/--AgCl
10
10
1
1
1l~1
ld 1
16 2
16'I
I
I
I
3
2
1
3
1031T[K "1] FIG.
4. Ionic conductivities
I
FIG.
,
1
5. Ionic conductivities
of
highly conducting silver and
highly cation-conducting
cuprous halides.
silver chalcogenides.
For clarity, classified (i)
of
I
2 •..~.--.-1031 T[K 4]
the ionic conductors quoted below and in Figs. 4-6 have been
into three groups of compounds:
the halides of silver and monovalent
(ii)
the chalcogenides
(iii)
compounds with more complicated
As a reference, also in Figs.
copper,
of silver and monovalent
copper,
chemical compositions.
the conductivity of Agl is recorded not only in Fig. 4, but
5 and 6.
(i) The highly cation conducting solid phases of the silver and cuprous halides are
a-Agl,
a-Cul,
a-CuBr,
and ~-CuBr ~. The data presented in Fig. 4 were
obtained
from resistance measurements
see also
(105)
ly carefully
between platinum electrodes
"'~4'5"43)
The cuprous halide samples had to be prepared under extreme-
control]ed
conditions and measured in an atmosphere
halogen and oxygen in order to obtain reproducible results. While
free of the early
data of Tubandt et al. were later on reproduced exactly in the case of a-Agl
(43), Wagner and Wagner,
using copper electrodes,
-values than the Tubandt group in the cases of (49). However, CuBr
recent microwave
g-CuI,
conductivity data of
observed lower' a-CuBr,
a-CuI
and B-CuBr
(33) and ~ -
(34) _ both in equilibrium with copper - are in good agreement with the
results given by Tubandt and coworkers. The cuprous halides,
An
our notation,
which transform into their cation-disordered
the phases are termed a-,
phases at
$-, T- etc. in the order of
their stability with decreasing temperature.
354
K. Funke
relatively vities
high temperatures,
already display
below these phase transitions,
at their transition conductivity duction
considerable
in
solid electrolyte
a-CuBr.
cationic
conducti-
(106), and the discontinuities
points are less pronounced
of a Agl-type
is observed
cf.
with negligible
electronic
The transport number of the electrons
known to be less than 10 -7 in a-Agl
"'~i07~, of the order
and of the order of 10 -4 in
"'~49).
a-CuI
of
than in AgI. The highest con-
is
of 10 -5 in ~-CuBr~49!~
(ii) The silver and cuprous ductors.
chalcogenides
The electronic
ionic conductivities In contrast
conduction
are mixed cationic
predominates
of the order of 1 ( ~ c m ) -I are observed
to the ionic conductivities,
strongly depend on deviations be described
and electronic
the electronic
from the stoichiometric
by Ag2+ ~ X and Cu2_ ~ X, respectively.
perties
and effects
associated
treated
in this paper.
The ionic conductivity
with deviations
of a mixed silver-ion
con-
even in those phases where (6,51,56-58,60)
contributions
compositions
However,
which may
electronic
from stoichiometry
and electron
to pro-
will not be
conductor
can be
analyzed with the arrangement Ag / a-AgI / specimen / a-AgI / Ag , a-AgI is used as an electron blocking layer (51,
where
may be performed and measuring
by sending a current of constant
the voltage drop across
the specimen.
switch the current on and off and observe -up and decay of the potential proposed by Yokota species. passage
Here,
of cations.
conducting
see Miyatani
voltage,
In this kind of experiment,
the steady state voltage,
Highly
cation-conducting
Ag2S ,
a-Ag2Se,
a-Ag2S
and
a-Ag2Te,
exist different The
a-phase
phases belonging
a-Ag2Te,
a-Cu2S,
for the build
(51). Another method,
are used to block the
a constant
current is
can be deduced
and the residual
to the chalcogenide
a-Cu2Se,
and ~-Cu2S.
from the
voltage. group are
is used in spite of the fact that in these cases there (Ii0)
(59), stable above ca. 430 °C (58), and the "digenite"-
phase CUl.8 S, stable above ca. 90 °C, have been shown to be identical Ionic conductivity
g-
The usual notation,
solid phases above 600 °C and 800 °C, respectively
of Cu2S
this cell one can
to silver ions as the mobile ionic
electrodes
switched on and off, and the ionic conductivity initial
through
Alternatively,
the time constants
difference,
(109) is not restricted
electronically
108). The experiment
intensity
data on
a_Cu2S
scarce and have not been included
(57,59) in Fig.
and
a-Cu2Se
5. According
Kawai (57), the ionic conductivity of B-Cu2S increases at ii0 °C to ca. 0.5 ( ~ c m ) -I at 220 °C, and Wehefritz
(58)
(60) are relatively to 0kamoto
and
from ca. 0.i ( ~ c m ) -I (58) reports an ionic
conductivity of CUl.8 S of ca. 1.6 ( ~ c m ) -I at 400 °C. In the case of a-Cu2Se, ~elustka and Ogorelec obtained values increasing from 3 ( ~ c m ) -I to 4 ( ~ c m ) -1 in the temperature
range from 580 °C to 750 °C (60)
(iii) A large number of highly cation conducting conductance
can be derived
the iodide ions, with different
from AgI by partial
or even both,
structures
high ionic conductivities
solids with negligible
by different
substitution
electronic
of the silver or
kinds of ions. Thus new compounds
are obtained which in some cases exhibit unusually even at room temperature
and below.
In these com-
Agl-Type solid electrolytes
pounds,
except
exclusively
in CuiHgl 4 and its alloys
the silver
ions.
are CuiHgl 4 and Ag3SBr: Agl are replaced As the first Ketelaar
by different
example
in 1994
At a later date,
Exceptions
in these
with AgiHgl4,
all
of the silver
of this group of materials, numbers
and Pond proved
Reuter
two phases
and Hardel
they both structurally different
(63).
1967, Bradley
discovered
where M represents bit the largest
ionic
a -Agl, Ag3SBr
and Greene
a group
~ - A g i H g l 4 was described
similar
conductivity see section
is similar (45,111)
of solid ionic
interesting IV, although
to ~-Ag3SI
and Owens
electrolytes
conductivities
of being
into Agl and MiAglg, In order
thermodynamically especially
to find Agl-type
unstable
solid electrolytes
(64)
and Argue these
(66,8?)
and ((CH~)4N)iAgI~II5 ductors, see Fig. 6. The electrical
conductivities
of various
MAg415,
at present,
exhi-
e.g.
appears to be compounds have
the
at 298 K. They decompose of moisture
which
have been
(46) inde-
compounds
(i12)~
are stable
at room
temperature, compounds with partial substitution of the silver organic groups have been investigated. Thus, among others (31),
(65)
The
because having
of composition
known
in the presence
(62)
data of Ag]SI.
0.?7(~cm) -I in the case of RbAg415 (47). Although RbAg415 inert at room temperature £or long periods of time, these disadvantage
by
as the eutectoid
properties
Rb, K, or NH 4. At room temperature,
solid-state
ions in
are tag+ ~ 0.94 and t H g i + ~ 0 . 0 6 .
and B, are particularly
resemble
conductivities
In 1966 and pendently
reported
of this compound, ~
substitution
or iodide
that CuiHgl 4 as well
of the AgiHgl 4 - CuiHgl 4 system have quite In !961,
ions are
species.
(61). The transport Suchow
the mobile
to the rule of partial
cases
ionic
355
identified
systems
MX-#gI
ions in Agl by
as good
(CsHsNH)Ag516 ionic
con-
have been determined
as a function of composition. The result obtained by Owens and Argue in the case of RbI-AgI at 22 °C (iI~) is shown in Fig. 7. It may be regarded as typical
in the sense
that in a number
maximum
in the ionic
conductivity
cent AgI. g~is04
T h i s i ~ i s also
- #gl
example, the in Fig. 6.
~}See
(45) and
~l16j,
of systems,
has been observed
true of thei~systems
Ag~V04
_ Agl
~(T)
curve
(ili)
for the phase
at room temperature,
\l16j
KCN
and AgiW04
of the compound
diagrams.
at, or near,
- ~gI
(AgiW04)
(114)
Ag~PO 4~
a
~0 mole per - AgI
(I15)
- Agl (I17)_ As an Ag414 has been
included
356
K. Funke
T[K] 125
150
I
v
200
300
I
I
I
500 A9- ,I~
10 3
I I;
(AgzWO4)~14 RbAg#~j
10-z
eT
6
[I~ cm) -I] ([CHzI,-N)zAg~Im
10-1
{ Csl'.lsNH) AgsI 5
Ag2 H9 I~
10-3
d6
I
I
I
I
I
I
I
I
I
9
8
?
6
5 1031 T[K "1]
4
3
2
1
•~
FIG. 6. Ionic conductivities
i
I
I
IC
of various AgI-type
i
!
|
I
I
solid electrolytes.
I
"T " " 0.25 E U
0.20 > 0.15
g
o.;o
U U
0.05
U
0 Rbl
FIG. 7.
2O
40 60 mole % Agl
80
100 AgI
Conductivity of the system Rbl - Agl at 22 °C according to Owens and Argue (I08).
Agl-Type solid electrolytes IV STRUCTUKAL The main structural arrangement
357
PROPERTIES
property of Agl-type solid electrolytes
is the disordered
of the mobile cations within a relatively rigid anion lattice.
In the case of the simpler compounds, anion structures,
we can distinguish
three different
namely
(i)
body-centred
cubic
(ii)
face-centred
cubic
(iii)
hexagonal
(b.c.c.), (f.c.c.),
close-packed
(h.c.p.).
The anion structures of those AgI-type
solid electrolytes which do not fit
into one of these categories are generally much more complicated. be discussed
in paragraph
They will
(iv).
(i) The structure of
~-Agl was first determined by Stroek in 1934
iodide ions form a b,c.c, to the silver ions, available
lattice,
see section I. Strock proposed a large number of sites
for the two silver ions within a cube of iodide ions, namely 6
octahedral
(b),
i2 tetrahedral
(d), and 24 trigonal bipyramidal
see Fig. 3. In order to explain the X-ray patterns, Hoshino
(?). The
while no definite sites can be assigned
(~), who confirmed Strock's
results,
Strock,
(h) positions,
and later on
assumed that the two silver
ions are statistically distributed over these 42 sites. At the same time, was pointed out by Strock to a "quasi-molten"
that the observed structural
or "liquid-like"
properties
state of the silver ions,
of.
interpretation has sometimes been regarded as self-contradictory clearly,
to X-ray diffraetlon,
has the advantage
This
structural analysis by neutron
diffraction
of yielding information about the loci of the nuclei and
not of the electron on
(4)
(i0).,
Strock himself did not take his 42 sites too literally.
In comparison
Fender
it
correspond
clouds.
Recently,
B~hrer and H~ig
(12) and Wright and
(13) have independently undertaken neutron diffraction experiments
a-Agl 6. The neutron data indicate that the silver ions are preferentially
found in oblong ellipsoidal and extending
regions of space centred at the tetrahedral
in the directions
Fig. 9. This result suggests completely
liquid-like
channel-like results,
of the nelghbourlng octahedral directions
should be regarded as
This point of view is supported by microwave
see section VII.
From the structural
refinements
nents are
for both the silver and the iodide ions. According
obtained
Debye-Waller
theory,
(8,12,1j)
they correspond(at
the ions from their mean positions Here,
see
(13) that the motion of the silver ions is not
and that the (100>
diffusion paths.
sites,
sites
unusually
large Debye-Waller expoto
200 °C) to r.m.s, displacements
of
of ca. 0.4 ~ and ca. 0.3 ~, respectivity.
the mean positions of the silver ions have been taken to be ± 0,3
away from the tetrahedral
sites,
along the ~100>
view of the large cationic displacements,
passageways.
However,
in
the actual distribution of the
silver ions is more properly described by the above-mentioned
ellipsoids
than by pairs of sites. ~Although
single crystals of
determination
a-AgI
can be produced
(i18,119)
on a single crystal has so far been reported.
no structural
358
K. Funke
FIG.
8. Crystal
structure
Hoshino marked
(8) ~
constant
of
a-AgI
according
The 6b sites are marked
to Strock •
, and the 24h sites are marked is 5.06 ~ at 250 °C
(7) and
the 12d sites are o
. The lattice
(8)
I
I
I ,i,S FIG.
9. Crystal
structure
The figure gives nuclei
at 200 °C.
of
l_uX g-AgI
according
the positons
to Wright
and r.m.s,
and Fender
displacements
(13)
of the
Agl-Type solid electrolytes
359
In this connection it must be emphasized that the evaluation of the temperature
factors in terms of simple Debye-Waller theory becomes questionable
AgI-type
solid electrolytes.
The large-amplitude vibrations
are generally considered anharmonic
and anisotropic
in
of the cations
(120,79).
For a further
discussion of the local motion of the cations see sections VIII and IX. In the case of the iodide ions,
the large Debye-Waller exponents
can perhaps
be partially explained by quasistatic displacements due to the cation disorder
(121). Fig. 9 gives the positons and r.m.s, displacements
according
of the nuclei
to (13). The structural disorder of the silver ions and the large
temperature
factors are also reflected by the intense diffuse
scattering
a-AgI (8,68) which has recently been measured with thermal neutrons A peculiarity of and M~ller
~-AgI
is the so-called "memory-effect",
(122) and by Burley
heating of the hexagonal
(67). Samples of
low-temperature
from
(121)
observed by Bloch
~-AgI may be obtained by
~-phase or the metastable
f.c.c,
r-
-phase. Samples arising from these two different sources seem to have slightly different
cation-arrangements
170 °C is not surpassed.
as long as a temperature of approx.
According to (122) and (67), cooling below the
transition point at 147 °C results in regeneration of the original phase. This "memory effect" 170 °C (67)
becomes irreversibly erased at temperatures above ca.
The motion of cations of different by the b.c.c,
anion lattice of
Flygare and Huggins
(123)
sizes along the ¢I00)
a-AgI has been examined theoretically by
In this calculation,
actions of the anions with a cation in a as well as overlap repulsion effects. simplicity,
tunnels provided
the electrostatic
inter-
tunnel have been considered
On the other hand,
in order to retain
the authors have ignored the influence of the positions and
velocities of the neighbouring shows the loci of
cations on the local potentials
(124). Fig.
minimum energy paths for cations of different
sizes.
i0
They
do not follow the centreline of the turmel but deviate periodically with both direction and magnitude depending on the cationic size. Small ions are attracted
towards pairs of iodide ions, while the paths of larger ions are
orthogonal
to the anion-anion axis. Flygare and Huggins have
periodic variations
minimum energy path during translation through the tunnel. have opposite phases for small and large cations. barriers
calculated the
of the local potentials as the cations follow their These variations
The respective
energy
are found either near the tetrahedral or near the octahedral
and originate
either from the electrostatic or from the repulsive
potential energy. At an optimum intermediate
sites
term in the
cationic size, with cationic
radius r ~ 0.83 ~, these effects largely balance each other, energy variations well below the thermal energy,
see Fig.
yielding small
ii. The actual
radius of the silver ion is somewhere between that of the lithium ion, 0.60 ~, and the sodium ion, 0.95 ~. Thus,
the predictions of the Flygare-Huggins
theory are in good agreement with the low potential b a r r i e r s ties
actually observed in
volume change of approx. AgI
(125-127)
~-AgI.
and high mobili-
In the light of this theory,
4.5 % associated with the 8 ---+Q
the negative
transition of
appears to be related to the fact that in ~ -AgI there is a
particular ratio of lattice constant to cationic radius which is highly
360
K. Funke
J Y
FIG.
y
10. Loci of minimum energy paths for cations of different sizes along a tunnel in a b.c.c, anion lattice, after Flygare and Huggins (123). On the left, small ions are attracted towards pairs of I'-ions
(indicated by circles), on the right, paths of
larger ions are orthogonal to the anion-anion axis.
12S
loo
,oi 2! 0 O~
l 0.7
I 0.8
I 0.9
1.0
r 1,;,1 FIG.
II. Activation energy as a function of cationic radius, as calculated by the Flygare-Huggins theory for the motion of cations along
tunnels in the AgI-anion-lattice
favourable for fast cation diffusion,
(123).
cf. (127).
Besides g-AgI, the phases ~-CuBr, g-Ag3SI, ~-Ag2S, and ~-Ag2Se have b.c.c, anion structures. The number of cations per b.c.c, unit cell is two in g-AgI and m-CuBr, three in g-Ag)SI, and four in g-Ag2S and ~-Ag2Se.
Agl-Type solid electrolytes
In 1952, Hoshino ~-CuBr
(73) and Krug and Sieg
is isostructural
phase diagrams ~-AgI-like According
with
of CuC1 and CuBr,
it is concluded
to Rahlfs,
1936,
(74),
a-Ag2S, unit
(ii) proposed
of
~-Ag3SI
besides
I- ions are s t a t i s t i c a l l y On the other hand, ~-AgjSI
the anions
and also in Ag3SBr;
composed
of oimple
the silver ions, R e u t e r and Hardel the anion structure. (x,[/2,0)~ Ag3SBr.
in the
in the low-temperature
lattice
proposed
-~(80). In
the S 2- and lattice. phase
can now be regarded
sublattices.
as being
For the sites of
the largest voids available
or Br
in the corners
these voids are situated
in B-Ag~SI
unit
cell,
for geometrical
can be occupied a unit
cell,
therefore
corresponding
±
and 0.3~5 in
to the tetrahedral
reasons,
regions
at a time. Most silver ions are thus confined
readily explained. for squeezing
along
a-Agl
and
The activation
a-Ag3Sl,
than in
enthalpies
to one face of ~-Ag3SI
necessary
the silver ions past the large anions,
directions,
are close to i/4 eV,
see. Fig.
in In
only one site out of a group of four
and the fact that they are less mobile
and Ag3SBr the
in
and the smaller
at the 12 positions
9, and there are as many groups of voids as there are silver ions.
most cases,
of
voids
there is a group of four voids in the vicinity of each face
centre of the b.c.c, Fig.
are
In the
by Reuter and Hardel
and ± (x, 0 , 1 / 2 ) ~ , where x is 0.390
Thus,
~-Ag2Se.
that the six octahedral
(or Br-)
In a cube with I
S 2- ion in the centre,
~-Ag2Se
by the four silver ions.
are ordered and I
p
with a pre-
over the sites of a b.c.c,
the b.c.c,
cubic S
~-Ag2S and
disorder of the silver ions,
distributed
that
vs
(126)
over the 42 Strock-sites
has been established
the structural
ascertained of the T
that there also exists an
sites in the case of
cell are randomly occupied
The structure this case,
Rickert
~ 2 kbar
the silver ions in
distributed
for the larger tetrahedral
case of b.c.c,
(72) independently
From the similarity
phase of CuC1 at pressures
more or less statistically ference
a-AgI.
361
is
in ~-AgjSI probably
clearly exceeding
those
6.
(ii) In c o n t r a d i s t i n c t i o n
to
the anion arrangement
a-Agl
in
a-Cul
and
a-CuBr,
in view of the ratios of the cationic compounds, structure
is observed
a-Ag2S
a-Cu2S,
and
and
still exhibit f.c.c,
paths
have been discussed
the cations
consist
tetrahedron
by Az~roff
of alternating
through
(74).
a-Ag2Te,
i.e.f.e.c,
In an f.c.c,
The diffusion
unlike voids:
unit
and
cell,
there
paths available
for
tetrahedron ~ o c t a h e d r o n
shares
faces with 4 octahedra,
Thus there is a large variety
the anion lattice. 12.
than
chalcogenides
structure,
structures,
(i29).
voids.
with 6 tetrahedra.
is shown in Fig.
are smaller
of the anion
arrangements.
etc.. Each anion tetrahedron
passageways
linear tunnel
variation
the b.c.e,
in close-packed
and 4 octahedrai
and each octahedron possible
have
diffusion
are o tetrahedral
radii of these three
anion lattice
The same systematic
anion lattices,
This is not unexpected
in the case of the silver and cuprous
a-Ag2Se
~-Cu2Se
The possible h.c.p.,
lattice.
cubic.
to anionic
as the voids provided by an f.c.e,
those in a b.e.c, While
which have b.c.c,
is face-centred
For visual
clarity,
of an almost
362
K. Funke
U FIG.
12. An almost
linear p a s s a g e w a y
in an f.c.c,
anion lattice~
The d i s t r i b u t i o n of the cations in the c h a l c o g e n i d e phases ~-Cu2Se,
and
In all of these phases, (i/4,
I/4,
~-Ag2Te ,
~ - C u I 8S was c a r e f u l l y studied by Rahlfs as early as 1935 (74) roughly one half of the cations occupy the sites
i/4; plus f.c.c.) w h i c h
correspond
to a z i n c b l e n d e
lattice.
The
r e m a i n i n g 4 cations w i t h i n the f.c.c, unit cell are more or less r a n d o m l y distributed.
Besides
the o c t a h e d r a l
c o n s i d e r e d 32 sites called
and t e t r a h e d r a l
16a(I/3)
and
positions,
Rahlfs also
16a(2/3). E a c h of these is s i t u a t e d
on a
4111>
site,
at the i n t e r s e c t i o n w i t h the face common to b o t h anion polyhedra.
16a(I/3)
p a s s a g e w a y b e t w e e n a t e t r a h e d r a l and a n e i g h b o u r i n g o c t a h e d r a l
positions
occupied zincblende
form sets of 4 s u b s i d i a r y sites around each of the 4 sites,
while
the 16a(2/3)
a s s o c i a t e d w i t h the 4 t e t r a h e d r a l results, various
sites
positions are c o r r e s p o n d i n g l y
(3/4, 3/4,
3/4;
plus
f.c.c.).
in p a r t i c u l a r his e s t i m a t e of the fractional o c c u p a n c i e s possible p o s i t i o n s
c o n f i r m e d by others
in the f.c.c,
(75,76). However,
chalcogenide
phases,
tials are known to be flat. W i t h the help of neutrons, 42 sites in a unit cell of
a-AgI
since the potenconcept of
could be r e p l a c e d by the more r e a l i s t i c n e u t r o n s might
also be useful for a s t r u c t u r e d e t e r m i n a t i o n in c a t i o n - d i s o r d e r e d
~-CuI,
of the
have since been
Strock's
v i e w of several r a t h e r e x t e n d e d regions of space• Similarly, e l e c t r o l y t e s w i t h f.c.c,
Rahlfs'
the d i s t r i b u t i o n of the cations over a
large n u m b e r of d i s t i n c t sites is n e c e s s a r i l y an artifact,
In the case of
The
solid
lattices.
Miyake et al.
X - r a y data by the a s s u m p t i o n
(71) o b t a i n e d the best fit to their
that the 4 cuprous ions w i t h i n the unit cell are
r a n d o m l y d i s t r i b u t e d over the 4 zincblende
sites plus the
16a(i/3)
sites. An
Agl-Type solid electrolytes
alternative
363
and probably more realistic explanation of the data, which has
also been given by Krug and Sieg
(72) and by Matsubara
presence of a location in the zincblende-type vibrational
amplitudes.
1 ~ for the r.m.s, displacement of the
cuprous ions. However,
confined to the zincbiende-type cationic mobilities,
voids with unusually large
Applying an anharmonic oscillator model, Matsubara
(120) derives values of approx. oscillating
in these models,
voids,
the cations remain
and the prerequisite
view of the ionic conductivity of
~-CuI,
is still unfulfilled.
structural disorder in the sense that a fraction of the
~-Ag2HgI 4 and
~-Cu2HgI4,
cell. On the average,
alike.
every forth one of these voids is thus unoccupied. As pointed out by Hoshino
in the X-ray line intensities of In fact,
vibrations
voids in the f.c.c, unit
very high ionic conductivities were already predicted
prior to their measurement. arities
(i0)
first analyzed by Ketelaar in 19J4 (77), the
cations statistically occupy the 4 zincblende-type this reason,
In
it is evident that there must be
cuprous ions should have left the zincblende-typevoids In
for very high
namely that the number of available and partially
occupied voids should exceed the number of cations, some additional
(120), is the possible
in
~-Ag2HgI4,
of the cations
as in
~-CuI and
a-CuI,
have been reported
For
(77)
(78), certain peculi~-Ag2HgI # are much
large-amplitude
anharmonic
(79)
(iii) The only known Agl-type solid electrolyte with an h.c.p, ~-CuBr. However,
the above-mentioned
of CuCI and CuBr
(i28) suggests
that
in the relatively narrow temperature have properties
anion structure
is
similarity of the T vs p phase diagrams a-CuCl,
which is hexagonal and stable
interval from 407 °C to 422 °C, might
similar to those of ~-CuBr. Very careful conductivity
measurements on CuCI are needed in order to decide this question. The hexagonal anion lattice provides One half of the tetrahedral the other hand, interconnect see Fig.
tetrahedral as well as octahedral
voids are occupied in the wurtzite
the most probable diffusion paths available
the face-sharing octahedra along parallel
voids.
structure.
On
for the cations
straight lines
(i29, i0),
i3. The conductivity and the self-diffusion of the cuprous ions in
a single crystal of ~-CuBr should therefore be markedly anisotropic. Structural
analyses of 8-CuBr have been performed by Hoshino
Krug and Sieg
(72)
the wurtzlte-type
According to Hoshino, tetrahedral voids.
and following Matsubara's tain r.m.s,
Using Hoshino's Debye-Waller factors
anharmonic-oscillator
amplitudes of almost
(79) and by
the cations are situated within treatment,
we once more ob-
i ~ for the cuprous ions. Unfortunately,
the high mobility of the cations is not reflected by Hoshlno's model. and Sieg proposed octahedral different
that a fraction of roughly
sites. However,
Krug
I0 ~ of the cations occupy the
they did not preclude the possibility of still
cation distributions
in B-CuBr.
364
K. Funke
FIG.
13. A linear diffusion
path in an h.c.p,
anion lattice.
(iv) The structure determined
of ~-RbAg4I 5 and its isomorphs a-KAg4I 5 and ~-NH4Ag4I 5 was
by Bradley
single-crystal
and Greene
analysis ~ ) ~ .
spect to the arrangement describe
the structure
(70) and by Geller
The results
(69), who performed
of the silver ions. In the following,
according
to Oeller
in a cubic unit cell. The structure
is characterized
the unit cell is similar to that of the manganese 56 tetrahedral
octahedra.
fractional
occupancies.
ding tetrahedra
share
the 4 Rb + ions are
The 56 tetrahedral
three sets of crystallographically
nonequivalent
These sets are ~-, 24-, l, 0, and 2 faces,
by the enantiomorphic of the 20 iodide ions in
atoms in B-Mn and provides
voids for the 16 silver ions, while
in distorted
we will
(69). There are 4 formula units
space groups P413(07 ) and P413(06 ). The arrangement
situated
a
of these authors differ with re-
positions
consist of
sites with different and 24-fold;
respectively,
the correspon-
with neighbouring
anion octahedra containing Rb + ions. The other faces are shared with tetraheda, and thus a large number of possible passageways is available for the silver ions. Interestingly,
the alternating
face-sharing
of tetrahedra
~)
The production of single crystals of RbAg415 has been described by Fullmer and Hiller (i~0) and by Manning, Venuto, and Boden (131)
~)
Ladd and Lee compounds
(132) have investigated
the crystal energies
MAg415 using the data of (70) and (69)
of the
be-
Agl-Type solid electrolytes
365
longing to the two 24-fold sets results in the formation of channels through the anion lattice. cell,
One of these channels is shown in Fig.
there are two parallel
channels
in each
14. Within a unit
direction.
Perpendicular
channels are cross-linked by the face-sharing of two tetrahedra belonging to either of these channels. RbAg4I 5 undergoes
phase transitions
at 209 K and at 122 K (88)
The one at
209 K is of the lambda type, see section V. As seen from Fig. 6, it is associated with a discontinuity
in d 4 / d T ,
while S is continuous.
sition at 122 K is a first order one, with a discontinuous two orders of magnitude.
Both transitions
distortion of the iodide-rubidium a-phase are still identified neither of the low-temperature
The tran-
change in ~ of
seem to involve relatively
lattice,
as the diffraction
(69,133). However,
little
lines of the
birefringence
shows that
phases is really cubic. According to Geller
(69), domains are visible under a microscope within the 8-phase. The crystal structure of ((CH3)4N)2AgI3II5 was determined by Geller and Lind
(82), cf.
(134). It is more complicated
characterized by similar properties,
than that of
a-RbAg415,
but is
namely the existence of nonequivalent
sets of tetrahedral voids and of channels formed by face-sharing tetrahedra. The
structure of (C5H5NH)Ag5I 6 belongs to space group P6/mcc
formula
units in the hexagonal unit cell
sets of tetrahedral (4c) are available
FIG.
positions
(D~h)'u with two
(81,65). Within the unit cell,
(6f and 24 m) and one set of octahedral
two
sites
for the silver ions. The structure at -JO °C is shown in
14. View along one of the channels available
for the
diffusing Ag + ions in RDAg4I 5. Iodide and rubidium ions, but no silver ions are shown in the figure.
366
K. Funke
Fig.
15. As the temperature
is increased,
6f sites into the 24 tetrahedral site occupancies (65)
as functions
of temperature
At ca. 50 °C, there appears
from a phase of relatively
silver ions move from the 4c and
(m) sites previously
from the log
(~T)
In the (C5H5NH)I electrolyte, conductivity
to a more disordered
vs I/T plot,
- AgI system,
At the same temperature,
of the formal activation
energy obtained
see Fig. 6 (65). there also exists
,-r~~135,1Do). Its structure
(135) and by Geller,
in particular
a "two-dlmensional"
solid
has been describedt~ by Geller and
Skarstad,
and Wilber
~136j
These authors
show
that the total volume of the crystal occupied by the possible
passageways
available
dimensional
solid electrolytes
for the silver ions is lower than in any of the threediscussed
above.
O
O.
FIG.
phase in which
(CsH£NH)5AglSI23 ~ However, at room temperature, its average of 0 . 0 0 8 ~ i ( ~ c m ) - is an order of magnitude smaller than that of
(C5H=NH)Ag516~ Skarstad
decrease
The fractional
to be a higher than first order transition
low disorder
the silver ions can move more freely than before. there is a discontinuous
vacant.
are given by Geller and Owens
15. Plan view of the crystal -30 °C, after Geller
=I"
Ag+
structure
of (C5H5NH)Ag5I 6 at
(81)
Let us finally mention two general structural models. The first, introduced by Geller (69), cf. (134) and (137), has already been used in the present discussion. It is based on the idea that networks of passageways are formed by the face-sharing of anion polyhedra. This model appears to be very useful except in the case of b.c.c, anion structures. In ~-AgI, e.g., it predicts direct passages from one tetrahedral site to a neighbouring one, along one of
Agl-Type solid electrolytes
the
4110>
directions,
through the face shared by both anion tetrahedra.
m o t i o n along the ever,
367
The
channels would be i n c o m p a t i b l e with this model. How-
these channels are formed by a l t e r n a t i n g anion t e t r a h e d r a and anion
o c t a h e d r a w h i c h do not share s i t u a t i o n is p a r t i c u l a r l y octahedral cf. (12))
voids
faces but in fact overlap. Evidently,
favourable
this
for the cationic m o t i o n as long as the
cannot be e x c l u d e d because of interionlc
repulsive
forces,
The second model i s due to van Gool (10,15) and van Gool and Bottelberghs (16) This model suggests
that the structure of a typical A g I - t y p e e l e c t r o l y t e
c o n s i s t s of domains
in w h i c h the cations occupy specific r e g u l a r and symme-
tric sets out of the more sites are o c c u p i e d
than e q u i v a l e n t n u m b e r of sites. D i f f e r e n t sets of
in d i f f e r e n t domains.
ries would therefore
A fast motion of the d o m a i n bounda-
imply a high cationic mobility.
It has been p r o p o s e d
that the domains o b s e r v e d f ~ b y G e l l e r in ~-RbAg4I 5 (69) should be i n t e r p r e t e d in terms of this model
tlO, 13~). In most cases,
however,
d o m a i n - m o d e l will not be easy to prove or disprove
the v a l i d i t y of the
experimentally.
V H E A T C A P A C I T I E S AND D I S O R D E R I N G PROCESSES
The
d i s o r d e r i n g process
solid e l e c t r o l y t e s capacities
and latent heats have been d e t e r m i n e d
Historically, interest,
leading to the high ionic c o n d u c t i v i t y of A g I - t y p e
is r e f l e c t e d by their t h e r m o d y n a m i c
)~
change of the anion arrangement, Thus the excess heat effects,
cations.
in a n u m b e r of cases.
phase t r a n s i t i o n there is only a slight from a tetragonal
a-Ag2HgI4,
to an f.c.c,
lattice
a t t r i b u t e d to the d i s o r d e r i n g of the
a c c o r d i n g to K e t e l a a r
(77) ,
the three cations per
unit cell are s t a t i s t i c a l l y d i s t r i b u t e d over the four z i n c b l e n d e - t y p e while in the ~-phase one vacant.
three of these voids are r e g u l a r l y occupied,
The heat c a p a c i t y Cp of Ag2HgI 4 n e a r its ~
is g i v e n in Fig. 40 to 50 °C, C
l Cp [ ~
400 /~g 2Hg
] •K
P
c o n t i n u o u s l y by a factor of approx.
I I i
14
t
Cp
/I,.,'
300, 200
12 R
!
I
100
t
324 K 300
310
320
330
T [K]
FIG.
)a
16. At first, w i t h i n the n a r r o w t e m p e r a t u r e increases
(139)
o b s e r v e d at and in the v i c i n i t y of the transi-
can be almost e x c l u s i v e l y
In
Heat
the first example was Ag2HgI 4 (94). It is of p a r t i c u l a r
because at the ~
tion point,
properties.
16. Heat c a p a c i t y of Ag2Hgl 4, after K e t e l a a r
(94).
voids,
leaving
phase t r a n s i t i o n interval
from
3, and finally
368
K. Funke
a latent heat is found at the first-order lambda-type order hence
behaviour
phase
of C
in alloys.
of the d i s o r d e r i n g formation,
ca.
second-order analogy
change is followed Moreover,
lattice becomes
From the relatively a-phase,
"domain becomes
structure",
in contrast
requires
the d i s p l a c e m e n t s
of these
anion structures
~-phase.
in
Nevertheless,
of the anions at
small amount of energy.
(2 meV) phonon branch
Long-wavelength
of the iodide
transition
by the following
a) B e l o w
the transition
observed
heat
capacity
point, exceeds
data by an amount ACp(T). disordering
This
in B-Agl,
ions necessary
within
a temperature
the linearly
which is (6.2 i 0.3) kJ/mole
heat capacity
17. The
(83) is given
in Fig.
range of ca. 50 °C,
extrapolated
lb~ )
to Perrott
the
low-temperature reflects
the integrated
an value of
to the latent heat at the phase
(b3). Thus,
of Agl had previously According
However,
in comparison
to ~-Ag2Hgl 4, one can assume a relatively
(84,8,85,86,87).
for their re-
in Fig.
The excess heat capacity ACp(T) e.g.
of this mode
features:
of the cation lattice.
is rather small,
phonons
(141) as illustrated
curve of Agl as obtained by N S l t i n g and Rein
It is characterized
crease
in
rather than a
a differentiation
and its b.c.c.
only a relatively
(142,38,40,141).
at the phase
~he
(140) that,
structure"
when the mean size of the domains
of a very low lying
can provide
transition
associated
constants.
B-phase
arrangement
A Cp(T)
on
to the lower one of two doubly degenerate E 2 modes observed
at the zone centre
initial
kJ/mole
(141) that in Agl the r e o r g a n i z a t i o n
is due to the existence
Cp(T)
heat effect of 6.0
meaningless
hexagonal
transition
there is some
of particles
disordered.
see section IV. Of course,
it can be estimated
corresponding
transition,
the arrangement
to Ag2Hgl 4, has two rather different
its w u r t z i t e - t y p e
part of the trans-
i0 ~ of the "forbidden"
in Ag2Hgl 4, it has been concluded
becomes
character
In view of the fact that the
by a first-order
as small as a few lattice
the phase
(140).
there should be a true "averaged
two possibilities
cooperative
and roughly
in both cases,
large overall
with the t r a n s f o r m a t i o n
the highly
At the end of the continuous are consumed
sites are occupied
to Cu3Au.
an f.o.c,
process.
1.2 kJ/mole
zincblende-type
Agl,
at 50.7 °C. The
is similar to that of second-order order-disP However, the increase of C with temperature, and P the decrease of the degree of long-range order, appears to be more changes
rapid in the case of Ag2Hgl 4, indicating
the
transition
small
in ~-Agl,
in contrast
degree of disorder
been determined
and Fletcher
(~7),
by various
and hence
authors
there is an in-
of C (T) in the a - p h a s e until 430 °C where an o r d e r - d i s o r d e r transP is reported. The authors claim that this behaviour is typical
formation
only for exactly metry result a-phase. reproduced
stoichiometric
in the "usual"
These results by others.
Agl and that small deviations
from stoichio-
C (T) curves which are relatively flat in the P which are not easy to understand (9) have not been
AgL-Type solid electrolytes
369
I¢
~ A
Ig+
a '~- -
FIG.
17.
-- -----"~---~
Q
l-
----~b
J
On the left: Ionic displacements associated with the lowenergy Q=0 E 2 mode in B-Agl. On the right: This mode is assumed to support the rearrangement of the anions at the transition from the wurtzite-type B-phase to the b.c.c. ~-phase of Agl. ~fter B~hrer and Br~esch (141)
I
70
I I I I
I Cp
Ag !
I
65
/,
[m .K ] 55
14 t Cp 1.3
6R
,.2
420 K 300
I)28E
500 T
1.1
700 [K]
•
FIG. 16. Heat capacity of Agl, after NUlting and Rein (03) a negligible temperature dependence of the defect-formation enthalpy. Denoting the formation([~thalpy of Frenkel pairs by Hf and the degree of disorder by a, we have d
ACp(T) = ~T (Hr. a (T)) : ~-T (Hf" ~o and t h e r e f o r e Hf in( ACp. T 2) = const.- 2RT
.e-Hf/2RT
)
(s) T
and
~(T) = HfI- f A C p ( T ' ) d T '
(9)
O N~'!ting and Rein thus obtained Hf = (if0 ± i0) kJ/mole and ~(147 °C) : 0.~[.i0-3 (O3). The disordered cations in B-AgI most probably occupy the octahedral interstices, causing a local contraction of the iodide lattice in the vicinity of each defect (14~). As a result, ~-AgI has a negative thermsl expansion coefficient with an increasing value as the ~ )a transition is approached (126,127). Furthermore, according to a theory by
370
K. Funke
Rice, Str~ssler,
and Toombs
(144)
the interaction of the cation defects
with the strain field they induce is considered primarily responsible phase transition b) Within the indicating
to the cation-disordered
a-phase,
for the
staten!
the heat capacity of AgI is essentially
that a further cation-disordering
constant,
which might give further contri-
butions to C
is no longer possible. The value of C exceeds 6R only by ca. P P 15 %, in fair agreement with the Neumann-Kopp rule. The slight decrease of Cp
just above the phase transition has been attributed residual short-range order
to the destruction Of
(91). This is particularly
destruction of short-range order in this temperature by the observation
that the "memory effect"
interesting,
since a
range is also implied
(67) in a-AgI vanishes near ca.
440 K, see section IV. The slight increase of C (T) at temperatures above P approx. 700 K is due to the formation of defects in the anio~ lattice. The concentration of anion-defects
remains however low up to the melting point;
this is confirmed by the results of I--tracer diffusion experiments (146"147'
95) Inspection of Cp(T) curves of various AgI-type
solid electrolytes
shows that
excess heat capacities due to an increasing degree of cation-disorder nerally found before the transition the highly conducting phases,
to the cation-disordered
the heat capacities
are ge-
state. Within
are either found to be
essentially
constant or to decrease. Decreasing values of C , with relatively constant slopes, are co-se rv ed in ~- A g2 S (92) ' B - Cu 2 S (92) ' ~ - CuBr (89,91) '
and a-CuI
(89,91),
see Figs.19 and 20.
The origin of the negative
coefficients of Cp is still unknown. Short-range order effects, can at most explain
I0 % of the decrease
(89)
N~iting,
Rein,
temperature
as in
}~I,
and Troe
have tentatively proposed an interpretation which is directly based on the high cationic mobility in the disordered phases. potential barriers in AgI-type solid electrolytes thermal energy,
In view of the fact that the are of the order of the
these authors conceive a gradual transition of the cationic
motion from the oscillatory
to the translational
state.
number of degrees of freedom and thus the heat capacity. analogous
This would lower the The treatment is
to that of a hindered rotator.
From the heat capacities of CuBr and CuI in their ~-phases
(90,148,149,89,91)
it is seen that B-CuBr is already structurally cation-disordered,
while B-CuI
is not. The increase of Cp(T) already observed in ~-CuI is continued in ~-CuI, indicating further disordering of the cation lattice, attained.
On the other hand,
until the a-phase
the heat capacity of ~-CuBr already displays the
typical decreasing behaviour,
which is also found in
a-CuI.
different model for the phase transition to the cation-disordered has been proposed by Huberman
site and a vacancy. However,
interaction is incorporated into the expression of the low-temperature
state
(145). This model is based on the interaction
between an ion on an intersticial energy
is
the way this
for the Helmholtz free
phase appears difficult
to justify.
Agl-Type solid electrolytes
1
/
150
Cp
Cu Br 125
mole. K
100
400
500
capacity
of CuBr,
3
:: I I
Cp
II If
I,
2,5
6R
I I I I
"
20
"
1.5
II
I
700 •
after N~iting,
I
T Cp
t
i t
i
600 T [K]
l~s. Heat
I
659 K -742 K-
50
FIG.
I
i
1
75
371
Rein,
and Troe
(59)
I
150 I
Cul
] ,K
Cp
125
2.5
6R
I
100
75
I i I I
I I
I •
2,0 1.5
681 6/,2 K
_
50 300
I I I I
500
700 T [K]
FIG. The heat
20. Heat capacities
ston, Wiedersich, a first-order Figs.
capacity
transition
experimental
Cv(T ) curve
served
about
above
silver
IV. Excess heat
capacities,
from the calculated
is taking place
two phase
this transition
(89)
are unknown
are probably
mined by Graham and Chang
~ from single-crystal (29)
Unfortunately,
from the fact the
for the
from X-ray data, elastic
the
of ~-RbAg4I 5 to
~ needed
to
of the
closely related,
coefficient
of the
are ob-
at 122 K seems
process
- apart
distortions
of Cv(T ) = C p ( T ) - T V ~ 2 / ~ were obtained
at 209 K, cf.
contributions,
disordering
by John-
transitions,
deviations
transformation
appear to represent
A l t h o u g h both structures
compressibility
i.e.
in the ~ - and ~-phases.
~)The molar volume V and the cubic expansion isothermal
lattice
50 K. The crystallographic
above and below
calculation
and Troe
at 122 K and one of the lambda-type
that the R b + - I - - f r a m e w o r k s lower symmetry.
Rein,
. RbAg415 undergoes
onto a more or less continuous
ions which
structures
after N~iting,
C (T~ ~nd C~T(T)~)of RbAg4I 5 have been determined
and Lindberg
6 and 21 and section
be s u p e r i m p o s e d
of Cul,
constants
and the deter-
3 72
K. Funke
~
~ t r a n s i t i o n largely affects
type phase transition,
the ionic conductivity.
there is no d i s c o n t i n u i t y of ~,
is not a usual o r d e r - d i s o r d e r
change.
J o h n s t o n et.al.
At the lambda-
i n d i c a t i n g that this (88) interpret the
209 K t r a n s i t i o n as the result of the interplay between changes in siteenergy differences, On the other hand, ordered domains
in site occupations, Pardee and Mahan
and concarrent lattice distortions.
(138) propose that the d i s i n t e g r a t i o n of
- domains w h i c h have p o s s i b l y been o b s e r v e d in B - R b A g 4 1 5 by
G e l l e r - might be r e s p o n s i b l e
for the transition.
similar to that in R b D g 4 1 5 p r o b a b l y exists
A h i g h e r - o r d e r transition
in (CsHsNH)AgsI 6 at ~2J K, of.
section IV. Above the lambda-point,
the heat capacity C V of R b A g 4 1 5 is found to decrease,
at least up to 310 K (88), while Cp(T) ature range. W i e d e r s i c h and J o h n s t o n heat capacity in this t e m p e r a t u r e
is r e l a t i v e l y constant in this temper-
(133) have compared the e x p e r i m e n t a l
range to the results of a model
calculation
w h i c h is based on the n o n - r a n d o m d i s t r i b u t i o n of the silver ions on the three nonequivalent
sets of t e t r a h e d r a l
to the q u a s i e h e m i e a l transformations,
sites in RbAg41 ~. A p p l y i n g a method similar
approximation,
used in the theory of o r d e r - d i s o r d e r
they obtain site energy d i f f e r e n c e s of ca. 0.027 and ca.
0.042 eV between the d i f f e r e n t types of silver sites and a mutual r e p u l s i o n e n e r g y b e t w e e n silver ions on n e a r e s t - n e i g h b o u r these parameters,
sites of ca. O . 0 J 5 eV. W i t h
the p r e d i c t i o n s of the c a l c u l a t i o n are in fair a g r e e m e n t
with the e x p e r i m e n t a l data. Thermodynamic etc.
properties
are very useful
a whole,
but,
like excess heat capacities,
for the study of the d i s o r d e r i n g process
of course•
in a system as
digression,
Appropriate
(42) and
(37)
#
(41,42,j7,150)
I
Rice and Roth
(41) have pre-
1.5
!
Cv
Cv I
30 R
! Cv,exP,
250
Cv , Debye
125
0.5
100
200 T
FIG.
21. Heat
[K]
300 D-
capacity C~7 of RbAg;~l 5, 8fter Johnston,
and L i n d b e r g ($~}.
have been
and theoreti-
For a brief d i s c u s s i o n of mo-
see sections VII and VIII.
375
we briefly
experiments
w h i c h will be d e s c r i b e d in the f o l l o w i n g sections,
cal models have been p r o p o s e d dels
state. As a
the efforts made in this direction.
undertaken,
entropies
cannot give any i n f o r m a t i o n about the details of
cationic motion in the c a t i o n - d i s o r d e r e d mention
latent heats,
Wiedersich,
Agl-Type solid electrolytes
sented
a free-ion-like
basic
idea
lixe
state,
~ecording where
consists
states.
energy
to the theory,
M is the
ionic
However,
conventional
hopping
free-ion-like
model,
theory
which
evident with
so that
motion
correlated
in terms
there
the
ion itself
electrostatic a diffusion the
same
Thus also
are
each those
did not
coefficient
n is the n u m o e r side
D~
of the
be derived
from a
no need
has
we will
steps~{
of" the
local
These
comparable
exist,
D ~ would
density
of e q u a t i o n
In the absence
ease
of cubic
Einstein
relation
for invoking
a
Each
to be r e g a r d e d
describe single
potentials
changes,
to the average only
influence
the c o n d u c t i v i t y ~
be simply
connected
2 ' nq and q the charge
(4) is also
9 and
structures
~ denote
the
called
of correlations,
this
diffusion seen
(i) by
mainly
due
energy
needed
the next
to the for
step of
and
the tracer-
by the N e r n s t - E i n s t e i n
:
jump
14) of the mobile the charge
we would
(152) , both D ~ and D~
D~ where
predictions
~ kB T
right-hand
the
states,
of these
of its neighbours.
D~
cient,
those
electrolytes
diffusion
step will not
equation
where
energy
by Vm=(2Em/M)9/2t
free-ion-lixe
can also
following,
(ii) by its neighbours.
correlations
-diffusion
in a free-ion-
threshold
the life-times
is a c t u a l l y
solid
in a change
interactions,
step.
of the
all
are
The
DIFFUSION
In the
unique
result
and
ion but
If these
in A g l - t y p e
process.
of single,
step of an ion will
ions
then given
contains
that
electrolytes.
(151)
of the cations
as a h i g h l y
are
experiment
Vl TRACER
The m o t i o n
the mobile
the o c c u p a t i o n s
become
in agreement
solid
a characteristic
velocitzes
is postulated,
it has
~re
For
conducting
that
~m exceeds
the ionic
mass.
equation
which
for highly
of the a s s u m p t i o n
if their
a Boltzmann
theory
model
373
o
:
ions.
diffusion
thus have would
and
coeffi-
D ~ = D~
be given
t2
frequency
The
In
by the
(5)
the
(fixed)
jump
length
of the
ions. On the other
hand,
one
electrolytes.
The
ratio
Possible H H will
generally D~/D6
D ~ % D~
come
correlation
effects
w h i c h might
be d i s c u s s e d
in this
section.
H R is composed
of two d i f f e r e n t
associated
wlth D ~ and D~
The
diffusion
tracer
~;trong
finds
has
{'vidence
and q u a s i e l a z t [ c
in h i g h l y
to be called explain
First,
correlation
conducting
the Haven
the e x p e r i m e n t a l
however, factors,
we will f and
fl'
solid
ratio
HR • values of
poznt
out
which
that
are
, respectively.
coefficient
for this view
of a cubic
has
neutron-scattering
solid
been obtained experiments,
electrolyte
from recent
can
oe w r l t t e n
microwave-
see sections
VII
and
IX.
374
K. Funke
The fact that the directions correlated
only if these
correlations
the directions other.
of two
do not exist. mobile
may also be imagined
For example,
particles,
moving
f; f equals unity
f is smaller
result
the efficiency
of the jumps in c o n t r i b u t i n g
In formulating
D~ in terms of ~ and
~
than
opposite
1 if
to each
in a crystallographic
to jump over several
This process would
between
factor
jumps are p r e f e r e n t i a l l y
length {(153)
which correlations
jumps of the tracer ion are
by the correlation
consecutive
On the other hand,
channel,
of consecutive
is taken into account
elementary
distances
of
in f > i. The factor f thus gives to D ~.
we first
consider
the jumps of different
a cubic crystal
ions do not exist.
in
We then
find
There
is no correlation
concerned
~2.
(7)
factor f in equation
(7), because we are now not
with the directions
of consecutive
rather with the net motion of charge process
as long as the jumps of different
(102,154).
Of course,
the case of Agl-type
as pointed
The meaning
2
of the " c o n d u c t i v i t y
demonstrated is correlated
in Fig.
22: e.g.
the Haven ratio
These authors strated
in p a r t i c u l a r resulting
" fI "
fI (io3,154,155)
factor"
= f/fI
method"
and equation (9)
However,
•
fl
= 0
• ;
-0
• fl
processes
are influenced
are a priori
!
-0
• fl
=
2
between different factor"
ions, and the
fI"
to n e a r e s t - n e i g h b o u r
in the S a t o - K i k u c h i
the theory predicts equivalent.
by
D~/Ds no longer depends
-0
= 1
correlation
not considered
of
They demon-
is obtained.
-0
O"
(155)
for the calculation
jump frequencies
interactions.
of the r e s t r i c t i o n
(103). Instead,
carrier
in the opposite
interactions.
like those shown on the left- and right-hand
unfortunately
is
(9)
only n e a r e s t - n e i g h b o u r
how the actual
22. C o r r e l a t i o n
As a consequence
by
(8)
the " p a t h - p r o b a b i l i t y
"conductivity
cf.
(7) is replaced
(9) has first been used by Sato and Kikuchi
•
-sites
this premise does not hold in
fI is 0 or 2 if each jump of a charge
from these
on jump frequencies,
processes
are u n c o r r e l a t e d
respectively.
considering
effects
but
is given by
equation
applied
D M, D@, and H H,
FIG.
carriers
and equation
correlation
HR = D~/D" In this n o t a t i o n
ion,
this is a random
with a jump of another one which is performed
or the same direction, Thus,
4
However
charge
out above,
solid electrolytes, D~ = ~
jumps of one specific
carriers.
interactions,
side of Fig.
fI = i if all available
An e x t e n s i o n
22 are
theory in its present
of the theory,
state,
cation
which should
Agl-Type solid electrolytes
,o'o
~
T [°C }~ 3o0
200
[.f~'] 8
500-~o
~oo
"-Agl
375
T [% I
1O91oD
250
200 i
' ~
l "~'2s
"" 300
T
.
IO5D
X
(o)..
2
.~
HR
o
HR
HR
L4
HR 4
o (k,t) • (j,p)
l 2t
I
2.&
I
2.0
1.6
0.2
I
1.2
i
2.2
•"qP~--1031T [ K "1]
25
I06D
~
T I°C] ~ 50 100
150
[~']
I
2.0
Iog.~)
1.8
1031T { K "11
T[oCl
300
150
-5.0 T -4.25
200
!
!
al'Ag2Se
(o
10s D
4
-5.25 -/,.5 2 1
-5.0
":~
2
0.4 HR
01
~2 I
32
i
3.0 2.6 1031T [K4I
FIG. 2). Tracer diffusion
2'.,~ ~ coefficient
I
20 ll0"lfTIK41
D ~, charge diffusion
and Haven ratio H R for ~-AgI, a-RbAg4Is, (j,p) : Jordan and Pochon (95) (k,t) (o)
: Kvist and T~rneberg : 0kazaki (54)
(b,m) (a,m)
: Bartkowicz and Mrowec : Allen and Moore (98)
(o,a) (b)
: Owens and Argue : Bentle (97)
(46)
~
(96) (~2)
,'8 coeffi2ient
~-Ag2S , and a-Ag2ge.
D~
376
K. Funke
allow for a proper treatment of the most relevant might become relatively better understanding
involved.
of diffusion
correlation
processes,
However it would be of much value for a and conductivity
data of highly disordered
solid electrolytes. Tracer diffusion coefficients of the mobile cations ~ h a v e been experimentally determined in a-Agl (95,96), ~_Ag2S ( 9 8 , 5 4 ) ~_Ag2S e (54), and a_RbAg415
(97). The data are presented
and the resulting smaller
than one. However different
~HR/~T is slightly and negative
positive
in
temperature
~-Ag2S
and
dependences
~-Ag2Se,
are observed:
negligible
in
a-Agl,
in a-RbAg415.
Can one decide whether consequence
in Fig. 2~ along with those of D e
values of H R . In all of these cases, H R is found to be
the experimental
result HR< i is essentially
of f < l or of fl>l or of both inequalities?
possibilities
have already been proposed
a
All of these three
in the particular
case of
a-Ag2S:
(i) Rickert
(ll) pointed out that,
in a-Ag2S,
an octahedral
site plus the neigh-
bouring Strock sites can at most be occupied by one silver ion at a time. He therefore
introduced
the concept of six octahedral
available
for the four silver ions per unit cell,
In this view, most voids are occupied, scribe
the cation motion
estimate
in ~-Ag2S
Yokota
regions"
cf. Fig. 35 in section IX.
and hence Rickert proposed
to de-
in terms of a vacancy mechanism.
gave f ~ 0.5, not too different
(ii) On the other hand,
"elementary
from the experimental
(157) and 0kazaki
(54) by putting H R = i/f I. They proposed
(54) tried to explain
a cooperative
ions along a line, known as the "caterpillar
A rough
H R values.
their data
motion of several
mechanism",
see Fig. 24. It is
assumed
that an ion on a site is able to jump not only into a vacant neigh-
bouring
site,
but also into an occupied one,
inducing
site to perform a Jump in the same direction. one of the cations cations -AgI,
the
< i00>
fI
occupied find r ~ 8
that a void is occupied.
in a-Ag2S
In
The ratio of the jump frequencies and r ~ 1 6
if it is assumed
paths for expression
g-Ag2S,
p is rough-
into vacant and into
(10) with H R : i/f I, one would
(54). Smaller values of r are ob-
that the directions
identical
For tracer diffusion
in a-Ag2Se
and
(I0)
sites is denoted by r. From equation
are not always
the following
(r'I)(I-P2)+(I+p)(I-p)-I (r-l)(l-p)~+l
see Rickert.
until
~-Ag2Se
as appropriate
For fI' Yokota derived
p is the probability
ly 2/3,
tained
:
as well as in
channels have to be regarded
mechanism.
continues
to Jump into. The average number of
involved will then equal fI" In m-Ag2S,
the caterpillar (157).
Here,
finds a vacancy
the ion on the latter
This process
of inducing
jump and induced
(54)
coefficients
in the case of ~-Agl and
of the anions,
(156) in the case of
see e.g.
a-Cul.
(146,147,95)
jump
Agl-Type solid electrolytes
FIG.
24.
377
Caterpillar mechanism.
(iii) Bartkowicz
and Mrowec
HR= f/fI in
m-Ag2S
(52) have proposed to interpret the Haven ratio
by the collinear interstitialcy mechanism.
ment is based on a comparison of the H~ values found in obtained from calculations
for AgCI
(lwS)
process
g-Ag2S with those
but the structural differences
between ~-Ag2S and AgCI are not taken into account. interstitialcy
Their argu-
Of course,
the collinear
can be regarded as a particular kind of caterpillar
mechanism. It is seen from (i), physical reasons
(ii),
(iii) that, in
g-Ag2S and in
a-Ag2Se,
there are
for f < 1 as well as for fi>i. Thus the observed values of
H R are probably due to both effects.
Certainly,
the caterpillar mechanism is
not the only conceivable process yielding fi) l: jumps into vacant sites might also induce
Jumps of neighbouring
ions in the same or at least in a
positively correlated direction. In g-AgI and
~-RbAg4I 5, the number density of the mobile cations is much
lower than in the chalcogenide inappropriate
phases.
for the description of the ionic motion.
appears to be much less important, proximation.
A vacancy mechanism is therefore Hence the role of f
and HR~I/f I might be a reasonable
ap-
The caterpillar mechanism has been applied to g-AgI by Okazaki
(54). Assuming r = 1 or r = 10, one finds i/f I = 0.5 or i/f I = 2/3, ively. The actual value of H R is ca. 0.6 (95,
respect-
96).
Vll COMPLEX CONDUCTIVITY IN THE MICROWAVE RANGE We now return to our introductory questions
concerning
the microscopic
aspects of the diffusive motion of the cations in AgI-type One of the principal questions motion is continuous
is the following.
solid electrolytes
Can we decide whether this
or discontinuous on scales of
~ 10-1~ and
~i0 -8 cm?
The two extreme possibilities are presented on the left- and right-hand of Fig. 25. The continuous simple monoatomic
liquid.
They obey the laws of simple diffusion,
sented by the Langevin-equation given above
(39,159).
sides
type of motion is displayed by the atoms in a as repre-
and Fick's second law, down to the limits
On the other hand, hydrogen in certain hydrogen-metal
systems as H-Pd and H-Nb definitely perform~ a jump diffusion from one well -defined site to another mean time of flight, (160) TO Atoms in liquids,
(39)
In this case it is generally assumed that the
Tl, is negligible
compared to the mean residence
hydrogen in hydrogen-metal
solid electrolytes have comparable
systems,
in these systems are quite
from each other. From the present microwave,
and quasielastic neutron-scattering
and cations in AgI-type
coefficients of self-dlffusion of the or-
der of lO-ScmRs -I. Yet the diffusion mechanisms different
time,
data,
far-infrared,
Raman,
it is concluded that the diffusion
378
K. Funke
in A g I - t y p e simple
solid e l e c t r o l y t e s
is in a way i n t e r m e d i a t e between that in
liquids and that in h y d r o g e n - m e t a l
shown in the f o l l o w i n g sections, k i n d of motion,
c h a r a c t e r i z e d by large a m p l i t u d e s
to be a good candidate
for a first a p p r o a c h .
in c o n t r a s t to h y d r o g e n - m e t a l be c o m p a r a b l e
systems,
systems,
in A g I - t y p e
seems
systems, ~i" seem to
~ . o were i n i t i a t e d by the follow-
If a j u m p - d i f f u s i o n model is valid in the highly cation-
c o n d u c t i n g solid electrolytes,
6o(~ ° +~)
~e can use the e q u a t i o n
~2,
=
where 4 is the m e a n jump distance. that the c h a r a c t e r i s t i c
(i~1
Inserting
for ~ values of a few ~, we find
times ~o" ~i might be of the order of 10-11s.
a d i s p e r s i o n of the e l e c t r i c a l
served in the m i c r o w a v e range. a long time before various
and d i e l e c t r i c
properties
In this
should be ob-
This was pointed out by W. Jost in 1967
t h e o r e t i c a l models
cerning the m o t i o n of the cations
Interestingly,
the o b s e r v e d m i c r o w a v e data.
the i n f o r m a t i o n d e r i v e d from these data on a time scale of
should be r e g a r d e d as s i g n i f i c a n t
(161),
(41,42,37) were d e v e l o p e d con-
in these systems.
these models has been able to predict words,
and strong damping,
However,
The m i c r o w a v e m e a s u r e m e n t s now to be discussed,
case,
25. As will be
the mean times of flight,
to the mean r e s i d e n c e times,
ing consideration.
see Fig.
a s u p e r p o s i t i o n of j u m p - d i f f u s i o n and a local
none of In other ~ i 0 -II s
for a proper d e s c r i p t i o n of the actual
diffusion mechanism.
Systems
with
simple monoatomic liquids
I,
~
25.
hydrogen in metals, e. g. H - P d , H-Nb
I
local motion ~- jump diffusion "Co ~ "I~1 "~ 10-11s
I
jump diffusion "~1 <<~'o ~ 10-11 s
D i f f u s i o n in d i f f e r e n t systems.
In the m i c r o w a v e experiments, appropriate
o<- Ag 1 c<- Cu | - Cu Br etc.
I
simple diffusion To= 0
FIG.
D P- 10-5 cm 2 s -1
a special technique
(162) was used which is
for the m e a s u r e m e n t of h i g h - l o s s materials.
under investigation
- s e c t i o n is shown in Fig. d u c t i n g waveguide,
The solid e l e c t r o l y t e
forms the s l d e - w a l l s of a r e c t a n g u l a r waveguide.
Fig.
26b. In c o m p a r i s o n to a normal,
26a,
two effects
can be observed.
A cross
n e a r l y ideally conFirstly,
a
Agl-Type solid electrolytes
TEl0-wave~)propagating the side-walls, these data,
i~o~
are calculated
tions of the w a v e - e q u a t i o n s 26,
of losses
£'- i~" and the complex
the ~aveguide
in
From
conduc-
are given along with solutions
component E . At low frequencies, Y short-circuit technique was used.
electrolytes
have been performed
with different
(b.c.c.),
a-CuI
continuous
structures
(f.c.c.),
(32,33,34)
(162)
for the non
i.e. at 2 - ~ GHz,
on three A g l - t y p e
of their anion
and B-CuBr
solu-
and within the walls
field
So far, m e a s u r e m e n t s a-AgI
E=
because
change of wavelength.
by means of the proper
within
the w a v e - e q u a t i o n s
zero electrical a coaxial
is attenuated
is a small
A
~=
In Fig.
there
the complex p e r m i t t i v i t y
A
tivity
in the waveguide
and secondly,
379
lattices,
(h.c.p.).
solid
namely
on
The results,
A
~(W)
= Re ~ (V) and
£'(~) = Re
~ (w),
W=
~/2~,
are plotted
in Figs.
27,2b,
and 29. In the case of m-AgI,
the far-infrared
spectrum
is also included
the figure,
The curves drawn
for ~(V)
and
to the criterion
of the K r a m e r s - K r o n i g
mutually
see section VIII.
consistent
according
y
in
£' (v) are rela-
Y
t,x
px
o FIG.
b
26. TEIo waves
in r e c t a n g u l a r
a) perfectly
conducting
b) two side-walls tions
(169). As a common
increasing
frequency
low frequencies, @ Correspondingly, should become tive values of exist
feature,
~A TElo-wave field vector
the @ ( 9 ) - s p e c t r a
to approach
according
constant
and negative
values of
is defined
the numbers
Fig.
26, are
exhibit
in
Recently,
~-Cul
and Taylor
properties:
half-waves
i and 0, respectively.
value
At
~(0).
relations,
limit.
and in $-CuBr,
Armstrong
to the direction
with
in the 20 - p0 GHz range.
a' in the case of ~-RbAg415,
of standing
dielectric.
a decrease
low-frequency
in the low-frequency
by the following
is transverse
(ii)
maximum
its known
conducting
to the data and the K r a m e r s - K r o n i g
~' have been observed
negative
from highly
and an additional
seems
in the 2 - 8 GHz range.
reported 10 8 Hz.
made
waveguides,
metal wails;
E'
Indeed nega-
where data (164) have also
at frequencies
below
(i) the electrical
of p r o p a g a t i o n
of the wave;
in the x- and y-directions,
see
380
K. Funke
In search of an explanation of the decrease disregard
the conductivity maximum.
much like the
~(~)-curve
in conductivity let us first
The remaining part of the spectrum looks
predicted by the Drude model
has actually been proposed by Huberman and Sen by Armstrong and Taylor ~ois
charge,
definded by
(42,166)
fractionmof
~(~o)
= 6(0)/2,
moving ions to be
g(0)'oJo" q-2n-l.
m and q are the ionic mass and
and n is the number density of the cations.
the result would be
and~more tentatively,
(164). Applying the Drude model as a first attempt,
we would find the momentary Here,
(165), and this model
~ ~ 2"i0 -3. On the other hand,
In the case of B-CuBr, the Drude model would
give a mean time of flight T l ~ 2 / ~ o, the • sign being valid if there is friction during each individual
flight. However,
such friction has to be ex-
pected because of the fluctuating local potentials due to the motions of the neighbouring From and
~=~/(~ ~
ions. Thus we would find ~l ~ 4 • 10-11s in the case of B-CuBr. 2 + ~ ) and 6D (~o + ~i ) ~ ~ we would then infer "Co • 2"10-Ss
200 ~ which appears to be quite unrealistic
for an AgI-type
solid
electrolyte.
9 l'~m-' ] 6 [(ncm)-' ]
1
10
!
I
100 I
II II
3
® (2SO°C)~; 2
~,(o) ---.~ 1 I
6 (25 °C~ I
10
R~9 i;10 I011 1012
FIG. 27. Conductivity ~ of
1013
g-AgI at 250 °C and of B-AgI at
25 °C in a broad frequency range. The difficulties
in applying the Drude model are at once resolved if we in-
troduce one simple additional jump-lengths
assumption
(34,99).
It is assumed that the
of the ions are fixed, being given by the geometry of the anion
lattice and not being affected by the electrical
field.
In this case,
the
effect of acceleration or retardation of the moving ions by the electrical field changes their times of flight, but not their jump rates. cannot give any contribution
to
explain the observed conductivity
~(0).
However,
Therefore
as will be shown,
it
it can
peaks at 20 - 30 GHz. This effect will be
called the "acceleration effect". Assuming fixed Jump lengths, ty and its decrease
we have to explain the low-frequency conductivi-
in the microwave-range
by another model.
This can be done
by a simple consideration which is familiar in the case of "normal" crystals:
the electrical
ionic
field slightly supports or hinders the initiation
Agl-Type solid electrolytes
of jumps in or against a constant value of
the instantaneously
~(~)
including the "start effect"
dependence
~(~)
will now be qualitaE'(0) will be given.
according to our physical model,
and the "acceleration effect".
Finally the
from this model will be compared to the observed data.
V[GHz] 2 4
10 20 40 I ~-CuBr,/,10
e [(~cm) "1]
e (o)
@(w)
thus causing
limit. The consequence of
and a simple relation between ~l and
We will then sketch a calculation of predictions
preferred direction,
in the low-frequency
this "start effect" with regard to the tively described,
381
I °C
v [GHz ] 0.2 0.4 1 2 4 10 2040 100 1 I t [ 1 t I I TI I - - - ~.-Cu 1,450 ° C ~ 1 . 2
IL
-
+
4
0.5
1
1
0
0
100
~
t
0 ¢'
O
~
E'
-100 - 200 '
-2001~[-~E~[~ 2 4 10 20 40 v[GHz] -'= FIG.
450 °C F
0.2 (14
I 1 2 4
-300
I
l 10 2 0 4 0
v [GHz] FIG. 29. C o n d u c t i v i t y ~
2~. C o n d u c t i v i t y ~ and permittivi-
mittivity
ty E' of ~-CuBr at 410 °C.
and per-
E' of ~-CuI
at 450 °C. The low-frequency
limit may be defined by ~i ~
T = 2~/~. In this limit the
phase of the field is practically constant while an ion is performing a j u m p When the frequency is increased the following situation will occur. A cation which has started
for an "extra"
jump with the help of the electrical
field
is still on its way while the field is already changing its sign. After this change of sign the ion will move against and will give a negative
contribution
will result in a decrease of ~.
the momentarily preferred direction
to the overall
In the high-frequency
conductivity, limit
~ ~
which
T, the
field will change its sign many times during the flight of an ion and the overall
conductivity will be zero.
An interesting aspect of this can be demonstrated
in a complex plane rota-
382
K. Funke
ting at the angular
frequency ~ of the applied
Fig. 30, let E be fixed in the vertical phase difference
betweer
field.
direction.
to Fig. 30, this phase difference
given by (~/2 + ~) where 2 ~ / 2 ~ = ~I/T = ~ low-frequency limit, we have ~ ~ 2 ~ and
~ ~"T
m E'
i.e.
~= ~ / 2 .
to the actual
value of ~i from
G(0) ~ 1 . 5 5 ( ~ c m )
For the calculation (~) = q
of "
E' (0). Inserting -I and ~^( ~ )
~(0)
and
g'(0),
effect"
the data from 8-CuBr,
will also Fig.
(34,99) , we start with the equation (13)
,
flux in the direction
of the electric
field E e -i~t. o let us introduce
In order to formulate the problem in only one dimension, a the flux IID which we would expect, if all jumps were being performed directly direction
in or against
the field direction.
and the jump direction
If in three dimensions
of an ion form an a n g l e ~ ,
are proportional
=sphere
I ^ID=
to c o s ~
~ID/3
phase angle
the field
of this motion
we obtain (14)
.
coc . E.
. Therefore
either
then the influence
of the field on the motion of the ion as well as the component in the field direction
28,
~' (0) ~ -225, we find %'1 = 2"5"i0-iis"
A I(w,t)/ (Eoe_i~t)
where I is the complex
is
In the
(12)
it has to be kept in mind that the "acceleration
contribute namely
/2~,
E'(o)
We can thus obtain a first approximate although
We are now asking for the
E and D = go.~.E in the case of a jump diffusion
with a mean time of flight ~l" According
AW
In this phase diagram,
/
/
/
~c= ~"> O"Z, I
I
FIG. 30. Complex phase diagram,
demonstrating
the phase difference
tween E and D = Eo ~ ~ in the case of jump-dlffusion. 2W corresponds
be-
The angle
to the period T, and the angle 2~ corresponds
the mean duration
of a Jump,
~l
"
to
Agl-Type
In our one dimensional
solid electrolytes
calculation,
densities of ions starting
jumps
in the time
Secondly,
interval
dt'.
v+(t',t)
of its
flight,
or the negative
for its jump of fixed
let us consider
are the number respectively,
value
at time t has the positive
v (t',t).
~ is
Thlrdlj,
"g+(t') or
T
with-
an ion which has started
on its way at time t > t'. Depending
its velocity
length
and n_(t')dt'
in the + and - direction,
a jump st time t' and is still direction
h+(t')dt'
383
on the
value
the time thls ion needs (t'),
respectively.
Then
the
A
flux IID is given by
t ^
=
lid
J
fl (t')v+(t',t)@(t'+~+(t')-t)dt'
(IX)
t +
J
~_(t')v_(t',t)~(t'+Z'_(t')-t)dt'
~ith
e(x)
:
for x ~ 0
The recuisite
~±(t') where
field.
start
= ~o± ~ f i ( t ' )
rates and velocities
and v+ ( t ' , t )
no and ± V o ( t ' , t Here,
simplifying
: ± Vo(t',t
) denote
their
Vo(t',t ) is considered assumption
From equation
(16),
is made
~(t')
jump-length
~ . Considering t
IiD
in t h e
that one function
a r ~ expressed
: 2
values
) + ~v(t',t),
as a function
~+(t') and v±(t',t)
jump-times
are ~ritten
only
absence only of
(!6) o f an a p p l i e d (t-t'),
Vo(t-t' ) holds
are inserted
by v±(t',t)
under
first-order
terms,
and the
for all
into equation
the constraint
jumps.
(15).
The
of fixed
we then obtain
from
(15):
a~(t')Vo(t',t)dt' t-T I (i7) t
t ~v(t_~l , t")dt" }
t- ~'1
t-~- 1
A
:
In equation
(liD)start
(17),
the first
+
(~ID) accel.
term involves
"start effect", while the second the "acceleration effect". For the calculation proportional
of the "start
to the electric
~ [ and therefore
term involves
effect",
~v
describes
and therefore
the
describes
it is supposed that ~ ( t ' ) is • ~ e -i~t' field at time t' i.e. ~o , and, for sim-
384
K. Funke
plicity,
that the cations
Introduction of
have a constant
of different
~start(~)
functions
more,
influence
occur in a correlated
s of these two jumps,
In order to calculate
to the field at time t,
friction-like (negative)
= 0, A x ( t ' • t ' ) effect
the arrangement The d e r i v a t i o n
is obtained
Av(t',t')
of rapidly
changing
microwave
(34). The curves
spectra
of
to formulate
the Haven
can assume H R ~
for
the change
in position
~-AgI
(i~],__,
smoothed
for the
and V" is the
potential
^ ~(~)
mainly due to
27
straightforward (microwave
They closely
TI, determines
the position
The location
to be ca. 0.6,
similar
This result
on
The height
of the c o n d u c t i v i t y
TI'
decrease
~-AgI
s = I/4 have been
and B-CuBr.
to that of ~-CuBr,
the decrease
of
as an indication
jumps occur in a correlated
manner
~-CuI.
T 1 / ( T ° + TI).
peak depends
on the f r i c t i o n - t e r m
Therefore,
~o is roughly
the value of b cannot i0 ps (14),
be fixed,
see part IX. Then,
b as well as
instant of time, unless
~o is known.
b becomes
15 ps -1 and V"• which e s s e n t i a l l y
determines
high-frequency
to be ca. -15 meV/~ 2. In ~-CuBr,
side,
is estimated
be similar or somewhat
larger.
Depending
on
the shape of the ~o'
curves are possible
these
b and V" are of the same order of magnitude
given
for
with different
various
experimental
jump distances
and ca. one unit-cell
values
close
approx.
~-peak
on its
~o might
fits to the
of b and V".
In all of
as the corresponding
a-AgI.
The above values of ZI' together with residence imply
In ~-CuI,
should be checked by a t r a c e r - d i f f u s l o n
on the fraction of cations w h i c h are in flight at a given
parameters
of
fI' and s. In
see section VI, and we
This might be u n d e r s t o o d
that there even more than two s u c c e e d i n g
of the conducti-
of the
by the parameters
of the spectra of both
experiment
part) and in
fit the experi-
15 ps in the case of
fI = 1/0.6 and a time overlap
lower frequencies.
and
and ~-CuBr.
ratio H R is known
i/f I. Indeed•
in the same direction.
is now
scale and is found to be ca.
T1 should have a value
cases,
due
from
shown in Fig.
scale is d e t e r m i n e d
for the calculation
g-AgI,
the
of a moving ion.
= 0. The term b allows
and ca. 25 ps in the case of 8-CuBr.
on the frequency
is at somewhat
we have
and velocity
from this equation.
of the model,
vity peak on the frequency
~-AgI,
section VI. The
paramete~
local potentials,
of an overall
28 have been calculated
~-AgI
=
jumps of two
cf.
= qEoe-i~t
of the final equation
The main parameter
In
curves
of the anions.
is given elsewhere mental
effect",
a jump at time t'
Ax(t',t),
second derivative
manner,
< s < i, is another
field on the position
mA~(t' • t ) + m b A £ ( t ' , t ) + V " ' a x ( t ' , t ) with A x ( t ' , t ' )
0
the " a c c e l e r a t i o n
of the applied
For an ion which has started
where
the resulting
for the fact that to a certain extent
ions in the same d i r e c t i o n
used
Vo(t-t' ) can modify
their flight.
and
~e have allowed
time overlap
6
v o during
E' start (~) ' but cannot change their general shapes. F u r t h e r by i n t r o d u c t i o n of the " c o n d u c t i v i t y correlation factor" fI as a para-
meter,
Fig.
velocity
of a p p r o x i m a t e l y c-axis,
~6.7
times of roughly
one lattice
~, in ~-CuBr.
constant,
~5
10 ps, would ~, in
~-AgI
These rather large values
Agl-Type solid electrolytes
would favour the m o t i o n along channels would then move along the dral sites.
In B-CuBr,
in both cases.
tunnels
In ~-Agl,
!
local motion which and 36. This is
tron-scattering permit
it is c o n c l u d e d
is s u p e r i m p o s e d
ioc91 motion,
Lnde~d o b s e r v e d
from V"
•
their jump di?f'asion,
Raman scattering,
25
and q u a s i e l a s t i c neu-
The term b itself is p o s s i b l y net too large
to
from site to site. b a r r i e r s b e t w e e n these sites may be roughly e s t i m a t e d
They are found to be close to the thermal energy,
the formal a c t i v a t i o n e n t h a l p i e s derived meter
cf. Figs.
w h i c h has been d e s c r i b e d by the f r i c t i o n - t e r m b,
experiments.
The a v e r a g e d potential
from one p o s i t i o n
that the cations perform an 9ddit~cnal
onto
in far-infrared,
s cetion to move
sites
l0 larger than those one would ex-
pect if the ions moved with their thermel v e l o c i t y d i r e c t l y Therefore
and oct~he-
the o c t ~ h e d r a l
d i f f u s i o n paths.
T~ are by a factor of approx,
to another.
the ions
connecting tetrahedral
the channels w h i c h i n t e r c o n n e c t
would be the a p p r o p r i a t e The times
385
from L r r h e n i u s
is the time o v e r l a p s of two successive
in agree~nent
plots.
correlated
~
W ~ n
The last para-
jumps, whose value
of i/4 might a p p r o x i m a t e l y have been expected• It is n-)t easy to e s t i m a t e
the influence of Lorentz
They have not been c o n s i d e r e d
in this calculation.
local
field effects.
Nor has any statistics
been i n t r o d u c e d w h i c h might have allowed for a d i s t r i b u t i o n of jump times. In this model
the jump d i f f u s i o n has been treated in a s i m i l a r l y simple man-
ner as in the G i s s l e r - S t u m p model
(167), which is used for the i n t e r p r e t a t i o n
of q u a s i e l a s t i c n e u t r o n - s c a t t e r i n g
results,
see section IX.
VIII F A R - I N F R A R E D AND R A M A N - S C A T T E R I N G EXPERIFHENTS
It is a common feature of A g l - t y p e
solid e l e c t r o l y t e s
Debye-Waller exponents
of the mobile
and n e u t r o n s t r u c t u r a l
refinements,
u s u a l l y high a m p l i t u d e s not only w i t h i n
the cations
section IV. They correspond
p r o v i d e d by the anions,
but also during
in these materials.
context,
it is once more stressed that the i n t e r a c t i o n s of a
tentatively describe
that of h i g h l y damped,
of
section.
cause
fluctuations of its local potential
of the o r d e r of the thermal energy and on a time scale of 10-12s.
view seems
flight from
This had to be taken into account by the
cation w i t h its m o v i n g n e i g h b o u r s
one might
to un-
However,
some local kind of m o t i o n seems to be c h a r a c t e r i s t i c
i n t r o d u c t i o n of the f r i c t i o n - t e r m b in the p r e c e d i n g In the present
large
cations are f o r m a l l y d e r i v e d from X - r a y of.
of a local m o t i o n w i t h i n flat potentials•
the voids
one void to another,
that u n u s u a l l y
Therefore,
the s h o r t - t i m e b e h a v i o u r of the cations as
more or less s t o c h a s t i c a l l y driven oscillators.
This
to be c o n f i r m e d by the f a r - i n f r a r e d and R a m a n - s c a t t e r i n g experi-
ments so far p e r f o r m e d on A g l - t y p e By the use of F o u r i e r - s p e c t r o s c o p y , far been studied in
solid electrolytes.
~-Agl (32'37) and
exist in the case of B-CuBr
A
the f a r - i n f r a r e d d i s p e r s i o n of [ has so ~_RbAg415
(36). Br~esch,
(35). P r e l i m i n a r y results
Str~ssler,
m i n e d the r e f l e c t i v i t y of single crystals of
~-Agl,
and Z e l l e r while
(37) deter~
t r a n s m i t t a n c e as
386
K. Funke
well as reflectivity measurements by the other authors. either a Michelson
on polycrystalline
The spectrometers
interferometer
grating interferometer able at frequencies
(Beckman FS 720)
(Beckmann LR i00)
below ca.
from
m-AgI is almost opaque in the broad frequency range 1012 Hz,
the reflectivity increases
frequency until the limit of the experimentally accessible
frequency range is attained at ca. 2 " 1 0 1 1 H z . by (97) and
or a lamellar
the latter being prefer-
samples of g-AgI are presented in
% 2 " i 0 II Hz to 5"1012 Hz. Below ca.
with decreasing
(32,35-]7)
(92,35,~6),
contain
i012 Hz.
The results obtained on polycrystaliine Fig. 91. Remarkably,
samples were carried out
used in these experiments
The reflectivity data obtained
(32) are in good agreement with each other.
^
A
g and hence
6 = i~g~
In order to calculate
A
from the transmittance
T and the reflectivity R, the
p
following two scalar equations R = Irl 2
^
;
1/2
T =
are used:
l(l+r)exp(_i~d~/c).(l_r) 1 2
Here, n =
~
reflection
factor at the vacuum-sample
sample.
is the complex refractive
In the equation for T
(19)
index,
r = (I-~)/(I+~)
interface,
multiple
dispersion of the complex permittivity
and d the thickness of the
reflections ~=
the complex
could be neglected.
e' - i g" of
~-AgI at
The
250 °C
is included in Fig. 31. It is recognized
that R and
a similar manner.
~', g" also attains its largest
Interestingly,
values at the lowest accessible data has the advantage
besides
frequencies.
that the maxima of
A
6(~)
~(~),
g' depend on frequency in representation of the
in contrast
to those of ~"(~),
f
£'£"
T,R
E
t.O "1o
20 °1,
i
,
f
16
12
/ |
2.1011
20
l
5.1011 1012 2.1012
5.1012 1013
I
2.1011
1012
1013
[Hz]
[Hz]
FIG. 31. Transmittance T, reflectivity R, real and imaginary parts of permittlvity
~=
g' - ia" of polycrystalline
g-Agl at
250 °C. Thicknes~ of sample used for measurement of T : 7 3 ~ m . indicate the transverse mode frequencies of a crystal for any degree of damping
(168). Therefore,
6 (~) has been calculated
from
g"(~),
included in Fig. 27, section VII. The mutual consistency of
and this curve is ~'(~) and
6 (~)
Agl-Type solid electrolytes
was
checked
by a K r a m e r s - K r o n i g
seen that
~-Agl
exhibits
absorption
s p e c t r u m above
analysis.
From the
a r e l a t i v e l y normal,
~(~)
dependence
though highly damped,
it is lattice
lO 12 Hz. This part of the spectrum does not differ
much from the one at room temperature. an intense,
387
B e l o w this frequency,
broad and rather s t r u c t u r e l e s s
however,
there
is
a b s o r p t i o n band, which is unusual
in ionic crystals. Before ding
e m b a r k i n g on a d i s c u s s i o n of these results, ~(~)-date
from
are o b s e r v e d as in
m-RbAg415 ~-Agl.
is lo~er than that of far-infrared tWO
in Fig.
we Lresent
]2. Cualitatively,
the same features
A l t h o u g h the dc c o n d u c t i v i t y of
~-Agl
the correspon-
~ - R b A g 4 1 5 at 20 °C
at 250 °C by about one order of magnitude,
the
c o n d u c t i v i t y maxima n e a r 5"i0 ii Hz d i f f e r only by a factor of
•
Ecru-I] [(~ c~m)-I ]
10
20
50
I
I
I
100 200 I
I
125 °C. ,A
20 ° C . ~ o
I
I
1
k
I
01011 2.1011 5.10111012 2.~12 5.~12 1013
v CHz] FIG.
)2.
Far-infrared
c o n d u c t i v i t y of'
~-RbAg415
at d i f f e r e n t
temperatures. In ionic
crystals,
phonon f r e q u e n c i e s However,
a b s o r p t i o n bands beyond
the transverse optical processes (169)
as the intensity of d i f f e r e n c e bands is rather temperature
their c o n t r i b u t i o n the absorptzon. of
far-infrared
are usually due to m u l t i p h o n o n d i f f e r e n c e
can be roughly e s t i m a t e d
Besides
room temperature measurements,
of
f a r - i n f r a r e d spectra
~ - R b A g 4 1 5 have been o b s e r v e d at -40 °C and at + 125 °C. In these spectra,
the shape and the s t r e n g t h of the band are hardly altered. range
from -40 °C to + 125 °C,
tor of seven, 15 per
cent.
low-frequency cesses.
in w h i c h the de c o n d u c t i v i t y
the c o n d u c t i v i t y m a x i m u m n e a r 5 - 1 0 1 1 H z Therefore
in
~-RbAg415
to a n a r r o w e r
be even more intense
changes by a fac-
rises by not more
to the
~-Agl where the e x p e r i m e n t s
I/T range.
at 400 °C, the f a r - i n f r a r e d a b s o r p t i o n
than in
than
is not p r o v i d e d by m u l t i p h o n o n pro-
The same seems to hold in the case of
In the case of S-CuBr,
In the temperature
it is c o n c l u d e d that the major c o n t r i b u t i o n
absorption
were r e s t r i c t e d
~-Agl,
band seems to
and a d i s t i n c t i o n b e t w e e n
the lattice a b s o r p t i o n s p e c t r u m e x p e c t e d n e a r 9-1012 Hz appears
(96)
sensitive,
from the temperature d e p e n d e n c e
this band and to be impossible
388
K. Funke
For an interpretation of the broad low-frequency bands in m-RbAg4I 5, one has to consider the fact that Bloch's a crystal with structural disorder.
Therefore
S-CuBr,
and
the usual optical selection
rule Q ~ 0 is broken in the sense that all phonon-mode observable
g-AgI,
theorem is not valid in frequencies become
regardless of the values of their wave-vectors ~. So far, the
situation is similar to that in polar liquids• where a similar far-infrared behaviour has indeed been found terms of "smeared-out
(170,171). However•
an interpretation
in
liquid lattice bands ''(170"171) would imply that the
phonon concept is applicable
for the type of motion which is responsible
the far-infrared absorption.
In the limiting case of very high damping one
for
would rather prefer the description given at the beginning of this section, treating the cations as an assembly of stochastically driven oscillators. Actually,
quasielastic neutron-scattering
experiments on
g-AgI demonstrate
that the local motion of the silver ions is of an overdamped
type,
see
section IX. In the case of overdamping, + 2 r
~ +
the solution of a homogeneous equation of motion, = O,
(20)
which now contains a damping-term resulting
E" (~) ~ 2 t ~ l and (172)
~w
o, is no longer periodic.
6 (~)
=
~o ~
( ( ~ - ~ ) 2+4 2 ~ )
the
(21)
a"(~),
• still exhibit maxima on the frequency scale.
spectrum of frequencies a very broad band of The Raman scattering Hanson,
However,
functions
Fjeldly,
~oj• ~(~)
the contributions
If there is a whole
~j(~) might well add up to
as it is observed.
from a single crystal of
and Hochheimer
(38)
In their
~-AgI has been examined by 90°-scattering
experiment•
they used the 6471 ~ line of a Krypton laser. The Raman spectrum shown in Fig.33 does not noticeably depend on the crystal orientation and changes very little with temperature.
The scattered radiation is polarized
in the same
direction as the incident beam, while the Raleigh component in the light scattered with crossed polarization is believed to be an artifact of the experiment.
Very strong Raman scattering is found in the low-frequency region
from ca. 5 cm -I~
1.5 "i0 II Hz to ca. 50 cm -I ~ 1.5"1012 Hz. A lattice mode•
cf. Fig. 27, is observed at ca. i00 cm-l~ 3"1012 Hz. It is noted that this spectrum closely resembles frequencies•
however,
that of
E"
(~) presented in Fig. 31. At very low
there seems to be an indication of a peak in the ~"(~)
spectrum while no such peak is observed in the Raman spectrum. We will now briefly compare the actual far-infrared response of AgI-type solid electrolytes diffusion-model
to the predictions
from various models and theories.
The
given in the previous section is of course not intended to
predict any realistic
infrared spectra,
since the "local motion" has been
taken into account only by a frlction-term•
i.e. in a highly unspecific man-
ner. The "free-ion-like"
model presented by Rice and Roth
(41) yields a Drude-
Agl-Type solid electrolytes
like b e h a v i o u r like
lifetime"
expected optical Fig.
~(~)
~f. In this model,
to occur phonon
27 for
6(O)/(l-i~f)
=
w i t h the " c h a r a c t e r i s t i c
the D r u d e - l i k e d e c r e a s e of
in the f a r - i n f r a r e d region,
frequencies.
~-Agl,
The e x p e r i m e n t a l
is o b v i o u s l y more
frequency
Singwi,
and S j ~ l a n d e r
w e is introduced,
present at f r e q u e n c i e s ~ > ~ . " d i f f u s i v e modes", ~ < ~ ¢ , states
is then assumed
ductivity
in the v i c i n i t y of the
~(~)
dependence,
(42)
a cutoff forces are only
the motion of the cations is spllt into
and v i b r a t i o n a l
modes, ~ > ~ .
for both types of modes.
to a d i s t r i b u t i o n of damped oscillators.
demonstrated
as shown in
which is based on an "Ansatz"
''L173). In this model,
A Debye denszty of
The e x p r e s s i o n
is simply w r i t t e n as a sum of a Drude d i f ? u s i o n
the m i c r o w a v e
~ (~) was
i.e.
and it is assumed that r e s t o r i n g Thus,
free-ion-
complex and quite different.
H u b e r m a n n and Sen have proposed a model given by Rahman,
389
for the con-
term and a term due
That it is impossible
to describe
spectra with the help of a Drude d i f f u s i o n term has been in the p r e v i o u s
the o b s e r v e d broad
section.
far-infrared
However,
as has been pointed out above,
c o n d u c t i v i t y bands are not inconsistent with
the a s s u m p t i o n of damped or o v e r d a m p e d oscillators.
m - Agl
>.
T
-- 3 4 0
P
=
°C
1bar
C C C
0
E 0
n-
50
100 A9
FIG.
33 • R a m a n s c a t t e r i n g and H o c h h e i m e r
Br~esch,
Str~ssler,
151
[ c m -1 ]
from
~-Agl,
. after Hanson,
and Z e l l e r have r e c e n t l y tried to calculate
c o n d u c t i v i t y of h i g h l y c o n d u c t i n g solid e l e c t r o l y t e s cies from only one d i f f e r e n t i a l
equation
vin e q u a t i o n i n c l u d i n g a m e m o r y function. vin e q u a t i o n oscillator model
the complex
for arbitrary
frequen-
(37). This is a g e n e r a l i z e d LangeIt transforms into the usual Lange-
for long m e m o r y times and into the e q u a t i o n of a damped har~nonic
for short m e m o r y times.
It seems to be a typical feature oi" tais
that it can give a good fit to the e x p e r i m e n t a l
but that it cannot e a s i l y predict any c h a r a c t e r i s t i c wave range.
Fjeldiy,
(3~)
f a r - i n f r a r e d data, d i s p e r s i o n in the micro-
It is felt that this is due to the fact that the process of
J u m p - d i f f u s i o n should not be d e s c r i b e d by a L a n g e v i n - t y p e
equation.
390
K. Funke
IX QUASIELASTIC NEUTRON SCATTERING
The quasielastic
scattering of cold neutrons provides a powerful tool
for
the study of the diffusive motion of highly mobile atoms or ions. The fundamental advantage
in comparison
to the microwave
and far-infrared
described above is the possibility of simultaneous times,
but also of the distances
characterizing
experiments
resolution not only of the
the elementary diffusion
steps. This is a consequence of the fact that the energy transfer ~ the
momentum transfer
neously,
~Q
and
of the scattered neutrons are measured simulta-
which is practically impossible
in the case of electromagnetic
radiation. Although
for example the diffusion of hydrogen in metals has been examined
in various quasielastic neutron-scattering was not applied to Agl-type
solid electrolytes until
have been due to the additional silver and copper,
experiments
complications
in contrast to hydrogen,
(39), this technique
1973. This may partly
arising from the fact that
scatter neutrons essentially
coherently. Recently, quasielastic neutron-scattering samples of
~-Agl have been undertaken
time-of-flight
spectrometer of the Institut Laue-Langevin,
these two experiments, resolutions were and 0.177 meV,
respectively.
(~,68,121)
energy transfer
The temperature of the sample was 250 °C. spectra are plotted against energy transfer
~=0.
the origin of the diffuse scattering
There is intense low-energy scattering, As Q increases,
tail extending to higher values o f ~ Basically,
In
the ranges of wavevector transfer and the energy
in Fig. ]4. The spectra readily explain m-AgI
Grenoble.
1.4 ~-i( Q<2.7 ~-i and 0.4 meV, and 0.56 ~-i~ Q • 2.16 ~-i
Typical examples of corrected in
experiments on polycrystalline
(40,14) at the "IN5 multichopper"
the quasielastic
centred around
its broad and almost structureless becomes more and more pronounced.
scattering reflects the structural disorder,
because it is mostly coherent.
Its shape gives information on the diffusion
of the silver ions. Its tail at higher energies
corresponds
to the broad
low-frequency absorption band found in the infrared experiments. Spectra obtained by Axe and Hoshino at the Brookhaven ter qualitatively display the same features. tals of
~-AgI.
In their experiment,
however,
triple-axis
spectrome-
These workers used single crysboth the scattered intensity
and the energy resolution were comparatively unfavourable evaluation (174)
to a numerical
In order to extract information on the diffusive motion of the silver ions, we have to analyze energy width
the Q-dependence
A E of the quasielastic
of the shape and, to begin with,
the width of the incoherent part of the quasielastic
scattering is known to
be A E = 2~DQ 2, where D is the coefficient of self-diffusion. be the case for sufficiently small Q, corresponding In
~-AgI,
of the
peak. As long as Fick's law is fulfilled,
the validity of this simple translational
be verified in the Q-range below approx,
~)This should
to sufficiently
large r.
diffusion model could
i ~-i. At larger Q, the experimental
Agl-Type solid electrolytes
widths
are found to be smaller.
most of the s c a t t e r i n g
391
This cannot be e x p l a i n e d by the feet that
is coherent.
Rather,
it is an i n d i c a t i o n that the
simple d i f f u s i o n model no longer applies. Closer i n s p e c t i o n of the spectra shows component
that they seem to contain a narrow
- w h i c h is h o w e v e r b r o a d e r than the r e s o l u t i o n
of a m u c h b r o a d e r d i s t r i b u t i o n . dels for t r a n s l a t i o n a l tr~l shapes
Moreover,
it is found that none of the mo-
d i f f u s i o n known to us
and their Q - d e p e n d e n c e .
function - on top
(177) can explain
these spec-
In order to derive a physical
moael
ap-
p l y i n g to the d i f f u s i v e motion of the silver ions, we make the following assumptions. a) The spectra are free of elastic Por the f o l l o w i n g reasons.
scattering.
This a s s u m p t i o n
There is no coherent
is justified
elastic scattering,
B r a g g peaks have been e x c l u d e d by the choice of the e x p e r i m e n t a l On the other hand, incoherent
since
condltiors.
there is no i n c o h e r e n t elastic s c a t t e r i n g because
cross section of iodine is p r o b a b l y n e g l i g i b l e
the
and because
the
silver ions have no fixed lattice sites. b) The i n e l a s t i c
contributions
to our spectra are n e g l i g i b l e The inelastic
in c o m p a r i s o n
to the q u a s i e l a s t i c
scattering.
does not exceed ca.
i0 g even at our largest s c a t t e r i n g angles.
c) The q u a s i e l a s t i c
scattering
is e x c l u s i v e l y due to the silver ions.
particular,
its coherent
correlation
f u n c t i o n GAg-Ag(~,t),
butions
are o b s e r v e d
It is n o t e d spectra
while
the elastic and q u a s i e i a s t i c
functions GAg-l(~,t),
contri-
Gl-~g(~,t),
and
in the B r a g g peaks.
that there is a s u r p r i s i n g s i m i l a r i t y between the q u a s i e l a s t i c
found in
~-Agl
and in some m o l e c u l a r systems w h i c h exhibit d i f f u s i v e
m o t i o n s of atoms w i t h i n r e s t r i c t e d regions of space. statistical
This motion is due to
f l u c t u a t i o n s of the m o l e c u l a r o r i e n t a t i o n
(!78). The spectra <)~
such systems e s s e n t i a l l y
consist of a n a r r o w u n b r o a d e n e d
broad q u a s i e l a s t i c
It is also noted
oscillations this kind,
In
part is the F o u r i e r t r a n s f o r m of the s i l v e r - s i l v e r
a r i s i n g from the c o r r e l a t i o n
Gl-l(~,t)
fraction of the s c a t t e r i n ~
peak.
yield s i m i l a r spectral
except
that o v e r d a m p e d
shapes.
leads us to assume
~)In terms of the Van Hove s e l f - c o r r e l a t i o n
The i n c o h e r e n t Sinc
large-amplitude
spectra are also of This
that
function Gs(~,t ) which gives
of finding an atom at (~,t),
the s o l u t i o n of Fick's Gs(~,t)
~-Agl
that the n a r r o w component is slightly broadened.
pheno:nenological r e s e m b l e n c e
the p r o b a b i l i t y
The
line on top of a
if it was at
(2,0)
(175),
law is
: (4~Dt)-~/2exp(-r2/4Dt). scattering
(~, ~ )
function is the F o u r i e r t r a n s f o r m of Gs(~,t):
= 2DQ2/((DQ2)2
The e n e r g y w i d t h of Sin c (Q, ~ )
is
+ ~2). AE
= 2~DQ 2, cf.
(176)
392
K. Funke
st°t- (~p, ¢1)) 10 5 0 0 1: o = 73 ps 1:1 = 1 6 . 5 p s
7000
r
f
= 41.5 °
= 1~
Dr =
1 ps -1
3500
0
3
. . . . . . . . .
4
' ; - - 6 )
....
• "l'im [meV]
st°t'(f,w) 16 5 0 0
I
=5~
I: 0 = 7.5 ps
%1 =16.5ps
11000
~' =
105 °
r = 1~ Dr =
1 ps -1
5500
o
o
.
.
.
.
.
.
.
.
.
.
-3
FIG. 34. Experimental
i
-2 -1
0
quasielastic
at two different
2
3
4 •
neutron
scattering
The solid lines result is sketched
1
5 6 ? "l'iuJ [meV]
scattering
angles
(o).
from the model
calculation
which
in the text.
Wavelength
of incoming neutrons:
Scattering
angles:
The experimental
5.343
41.5 ° and i05 °.
spectra are corrected
in the usual way
for detector efficiency, sample-container sorption, and self-shielding. The spectra
spectra
are markedly
asymmetric
scattering,
because
(and calculated) at constant scattering constant momentum transfer ~ Q .
ab-
they are taken
angle ~ and not at
Agl-Type solid electrolytes
d) the silver ion is carrying different
time scales.
sive motion within iodide
lattice.
bution
plus a
lational
-diffusion would
~(~)-peak
type.
observe
broaden
over
region of space,
crystal
line.
An example
of the a b o v e - m e n t i o n e d with the elementary
translational
diffusion
35. Local
in a void of the b.c.c
voids
follows n a t u r a l l y
distri-
kind of motion
is a trans-
the spectrum which one
~lone;
in particular
of a possible
regions
or diffu-
volume which might be of the Jump
from the first kind of moticm
of cations between
e.g.
on top of it. The second the whole
on two
an o v e r d a m p e d
could cause a broad quasiel~stic
are identical
FIG.
types of motion
it performs
It might only slightly broaden
the elastic
arrangement
a restricted
Such a motion
diffusion
out two different
On the one hand
393
it would
three-dimensional
is shown in Fig.
35. These regions
proposed by Rickert from this picture
(ii)~)
The
as an exch3nge
these voids.
regions
of space available
for the silver
ions
in
g-AgI.
~)Fig.
35 gives a rough illustrat~on
accessible of B ~ h r e r
to the silver ions. and H~lg
relatively
(12) and of Wright
large value of
the following
~I'
interpretation:
dence
ti~es,
of Fig. ~o'
(13)
located within during
the results
In view of the
one can, however,
their motion
35 might be occupied
of space which are
to contradict
and Fender
see section VII, during
the silver ions are e s s e n t i a l l y the regions
of those regions
It is not meant
along the
the regions
of Fig.
the relatively
conceive channels, 9, while
short resi-
394
K.
e) We furthermore simultaneously assumption
assume
that the silver ions perform both kinds of motion
and that these are dynamically
permits
and the scattering elestie
Funke
scattering
independent
of each other.
us to write the silver self-correlation
This
function GsAg(~,t )
function S~ g (C,~) for the incoherent part of the quasiinc -from the silver ions as the convolutions of the individual
functions:
G~g
(r,t)_ =
$ G~ ( r ' ' t ) . G L ( ~S- ~ ' ' t ) d r ' -
-
/oT --
where
~inc
the superscripts
respectively.
models
~L (q,~), ~'inc
for both types of motion,
a n d ~inc(,{,~) ~,.o ~ ,
S t°t
sectionsof ~cattering
•
(q,~)
and
silver, respectively, function sAg~Ag(Q,~).
motion,
in Fig. 36. By the
see below,
for the total quesielastic
and ~inc are the coherent
local"
the functions
can be approximated.
~Ag-Ag ~coh~coh (C,~) +
=
(22)
"~inc (~' ~ -
T and L stand for "translatlonal
In order to derive expressions
wherev~oh
--
The meaning of eouations (22) is illustrated
use of simplifying s Tznc ( q , ~ ) ,
,
As 6inc S.±nc
ana incoherent
scattering
(q,,~)
~,e still need the coherent It is assumed that
(2))
'
scattering
function
cross
quasielastic
con
f) oAg-agfc ~) can be estimated by using a phenomenological model ~hieh ~coh ~-' was proposed a few years ago for liquid argon~179 jI~ and which yields the relation sAg-Ag (q,~) cob The required mentel
=
~Bg (Q. (sAg(c) )-I/2, ~ ) .s#g (Q) ~inc
function sAg(Q):
and theoretical
=Is"g_Ag(c,~
results
con
of ref.
112])
) d~ is taken from the experi-
f+l':r'f : }:{+} sL FIG.
36.
Illustration
to equations
(24)
(22), cf. Fig. 25.
Agl-Type solid electrolytes
In principle, derived
the total Q u a s i e l a s t i c
and compared
mathematically movements The
tractable
models
jump-diffusion
by Rowe et el.
does not appear needed which 71,
to occur
jump length,
in a crystal assume
g) the G i s s l e r - S t u m p
model
of S ~) n c.( <_, ~
dence
time ~
(ii) The length, more,
~.
provides
At present,
without
from either
appreach
scattering
results
h) the solution an adequate
tetrahedral
sites,
approximated nately,
motion
weighting
i) we have
point
~re:
of still
Furtherv = ~/~.
for ~he ouasi-
is useG in two and
ions,
factors
tha numerical
the two possibilities oscillatory
by prolate
results
affecting
to approximate
menticned
motion,
are
if one takes into
the frequency
spectra ~hen
that
equation
in a restricted
provided
region of space
ions.
of Fig.
Possible
candidates
centred at the
35 which may be roughly
centred at the octahedral
an ellipsoid
is
for the local motion of the
by the iodide
9, or the regions
within
~o' ~i" and
tecb_niques.
ellipsoids
too complicated.
ob-
not only with the neutron
of space might be the oblong ellipsoids see Fig.
results
On the other
of the parameters
for setting up a model
the voids
the d i f f u s i o n
matically
model
indistingulsnable.
and overdamped
we assume
of the d i f f u s i o n
for these regions
are concerned,
They are both compatible
convenience,
silver ions within
expression
choicc- of jump directions,
but also with the infrared
starting
for the
the mean resi-
their jumps is constant,
az'e p r a c t i c a l l y
with the two different
For m a t h e m a t i c a l
Thus
TI, of the jumps are constants.
depend on the choice
local diffusive
much alike.
measured
treat~
and Stump
made in this moae]
as exp(-t/~o)
the G i s s i e r - S t u m p
any :pecific
strongly
the different
jump
with jumps along the ~i00> tunnels.
physically account
is
nameIj
As for the local motion of the silver namely
~ model
To, a mean
good a p p r o x i m a t i o n
assumptions
0, the C h u d l e y - E l l i o t t
as far as the final spectre
the spectra
Rather,
time,
T o. In
therefore
is recovered.
of S~nc(¢:,~),
modifications,
(i) isotropic,
~I ~
model
the only model which
a sufficiently
of the atoms during
function
(ii) polycrystalline,
~-Agl.
however,
that
case ~i ~
For the calculation
In this model,
in this way is the one of Gissler
is introduced. o ~, and the duration,
scattering
different
above,
and local
a given site at time t = 0, the probability
the velocity
In the limiting
tained
ar.d
used in the litera-
in the sense of
a mean residence
~' d "ing it there at time t>0 decreases ~in
hand,
can now be
adecuate
It has been generalized
the Chudiey-Elliott
The simplifying
occupies
(i$2).
to the case of
as parameters,
therefore
(i) If an a t o m
results,
to be applicable
and a mean
ca] culation
frequently
and Elliott (I~0).
instantaneously
microwave
contains
the j u m p - d i f f u s i o n (!67). We
by Chudley
(181) and by Gissler and Rother
the jumps are assumed
However,
if p h y e i c a i ! y
can be found for the translationa~
model which has been most
view of the existing
elastic
function st°t'(Q,~)
spectra,
of the ions.
ture is the one proposed
time,
scattering
to the experimental
395
and even within
sites.
a sphere
Unfortuis mathe-
Therefore, the local motion by a d i f f u s i o n
on the surface
of a
396
K. Funke
sphere, mation
the only case which Is mathematically is not too far from a diffusion
ion which is allowed
to diffuse
found near the surface -diffusion" sphere,
model within a full sphere since an
than near the centre of the sphere.
approximation
contains
two parameters,
certainly
of physical
simplifications
Making
However,
in order to retain conceptual
the assumptions
st°t'(Q,~)
namely
coefficient,
Some of the above assumptions reality.
(183). This approxi-
in the interior of a sphere is more probably
r, and the "rotational-diffusion"
necessary
tractable
Dr .
imply rather drastic of this
for various
simplifications
kind appear to be
clarity and mathematical
a) to i), the total quasielastic
are calculated
The "rotational the radius of the
feasibility.
scattering
sets of parameter values.
functions
The functions
st°t'(Q,~) are then transformed into corresponding functions which describe scattering at fixed scattering angles ~ . Finally, convolution of these functions
with the resolution
(~,~) which may be compared In Fig.
34, calculated
-dependence
the parameter
the broad component
scattering
close
shapes
angles
are presented
D r is essentially
of the quaslelastic
angles.
given by the energy widths arising
from the
fractions
of neutrons
scattered
broadened
~(~) peaks.
The best fits arc achieved with D r ~ ips -I and r ~ l
It should,
however,
be emphasized
ions and that the values
values of ~ into the slightly
that our present
is only a very rough first approach
to the actual
of its parameters
of
from the local
r, is then determined
at different
The Q
described.
scattering,
motion of the ions. The second parameter,
along
(~ = 41.5 ° and i05°).
fits at all scattering is correctly
spectra S t°t"
ones. functions
spectra at two scattering
surprisingly
of the spectral
In our model,
the calculated
to the experimental
quasielastic
with the experimental Our model yields
function yields
the
rotational
~.
diffusion
model
local motion of the silver
should hence not be taken too
literally. The width of the narrow quasielastic transfer
essentla]ly
determines
It is found to be somewhat On the other hand,
component
the value of the mean residence
(2.5 ~, 0 ps) to more than
Fortunately,
and vice versa.
to extract
component
independent
Thus close ranging
from
this uncertain-
the energy shapes and
at different
information
~o"
A best value
~i)- combinations
(6 ~, 22 ps). In order to overcome
it would be necessary
widths of the narrow quasielastic precision.
(~
time,
than I0 ps.
to be less certain.
as soon as ~ is known,
fits are possible wlth several different ty experimentally,
large momentum
larger than 5 ps, but still smaller
the value of ~i appears
of ~i can be determined
at relatively
Q wlth much higher
about the Jump-time
T1 is
available from the microwave data which give ~i(250 °C)~15 ps, see section VII. The spectra st°t'(~,~) shown in Fig. 34 have been calculated with ~o = 7.5 ps, consistent
TI = 16.5 ps, and
~=
5 ~, a set of parameter values which is
both with the neutron-scatterlng
of the implications
and the microwave
results.
of these values have already been discussed
Some
at the end of
section VII. The parameters
of the translational
diffusion
model are connected with the
coefficient of self-diffuslon of the silver ions, D, by equation (ll): 6D(T ° +~i ) =~2. The a b o v e parameter values give D ~ 1 . 7 5 • 10 -5 cm2s -I, in
Agl-Type solid electrolytes
397
good agreement with the experimentally determined tracer-diffusion coefficient which is roughly 2'10 -5 cm2s -I at 250 °C (96). Thus, in the case of
~-AgI, our present physical model for the cation motion,
cf. Figs. 25 end 36, appears to be consistent with the experimental results given in the last four sections. A direct determination of the starting points, directions, individual
and lengths of the
jump-diffusion steps will probably require a single-crystal neu-
tron-scattering experiment with an energy resolution better than 0.1 meV.
A CKN 0W LE DGE ME~ TS The author is much indebted to Prof. W. Jost for his advice and encouraging support throughout this work. Moreover, I wish to thank Prof. H. Schmalzried who also critically r~ad the manuscript, in wording. Finally,
and Dr. D. Field for his suggestions
thanks are due to the authors and publishers who gave
their permission to reproduce Figs. 7,9-ii,15,17-21,
and 33.
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PergamonPress
Printed in Great Britain
AgI-TYPE SOLID ELECTROLYTES K. Funke Institut pdr Phys~kahsche Chemie der Umversit~t G~ttingen and Sonderforschungsbereich 126 Gotnngen, Germany
CON TEN TS
I
Introduction
345
II
General S u r v e y
348
III
Ionic C o n d u c t i v i t y
350
IV
Structural
V
Heat C a p a c i t i e s
Properties
357
VI
Tracer Diffusion
VII
Complex Conductivity
VIII
F a r - I n f r a r e d and R a m a n - S c a t t e r i n g E x p e r i m e n t s
385
IX
quasielastic Neutron Scattering
390
and D i s o r d e r i n g
Processes
367 373
in the M i c r o w a v e Range
377
I INTRODUCTION
Most
ionic
defect
crystals have degrees of d i s o r d e r well below
concentrations
being small,
at a time and carry charge vities of these "normal" peratures. ca.
Typical
through the crystal lattice.
ionic
crystals are therefore
values are ca.
exceeds
the energy r e q u i r e d the thermal energy,
ent and unusual behaviour. trical p r o p e r t i e s
pair greatly
(2,9). The c o n c e n t r a t i o n s
ionic crystal
therefore
of acfects
strcngiy depenu on
over a t e m p e r a t u r e
Flg.l
shows T u b a n d t ' s
AgCi,
AgBr,
crystal ~ith a quite differ-
they o b s e r v e d
and Lorenz's
has a h i g h l y
- stable b e t w e e n
that solid silver
range of more than 400 °C, has ionic conductivi-
to those of the best c o n d u c t i n g
and AgI versus
AgCI and AgBr,
an ionic
In the course of their i n v e s t i g a t i o n of the eitc-
of the silver halldes,
~ comparable ties o n ~y
-phase
tem-
(~) and
and the same is true of the ionic conductivity.
In 1919, T~bandt and Lorenz e n c o u n t e r e d
iodide,
conducti-
(j), both at 200 °C. In these "normal"
for the formation of a defect
respectively
ions in a typical
temperature,
The ionic
low at m o d e r a t e
kBT. At 200 °C the factors are ca. 95 and 50 in
the cases of AgCI and NaCI, and mobile
(i). The
iO -4 ( ~ cm) -I in the case of igCl
I0 -~ ( 9 em) -I in the case of NaCI
crystals
10-2,
only a limited fraction of ions can move
liquid e l e c t r o l y t e s
o r i g i n a l plot of the c o n d u c t i v i t i e s
temperature.
It is seen that AgI,
(4)
of
in contrast
to
c o n d u c t i n g solid phase - now known as the a-
147 and 555 °C. At the 2 - ~
phase transition,
the
c o n d u c t i v i t y of AgI increases by more than three orders of m a g n i t u d e up to
345
346
K. Funke
,/
I
f
r
I
~o
,/
/
/
I
I t I I
/
1i
ir"
I ! I t
;
Ill
/ _~
i r"
~o, ~ ~Oo
SO0 o
600 °
700~
C °
FIG.
i. Conductivity of the silver halides according
to Tubandt and Lorenz
~ 1.3 (~ cm) -l, cf. Fig 4. Within the ly up to
a -phase,
(4) it increases only slight-
~ ~ 2.6 ( ~ cm) -I and then drops on melting.
value of the ionic conductivity of
a-AgI
The abnormally high
and its weak temperature dependence
are typical of the class of ionic crystals to be reviewed in this paper. The measurement
of the conductivity,
~ , of silver iodide may be regarded as
the starting point for the investigation of AgI-type their early work, Tubandt et al. identified various
ionic crystals as belonging
these were
g- and ~-CuBr,
From their measurements experiments
to the same class of materials. a-Ag2S,
a-Ag2Se,
and
the charge is being carried by the cations
phases were found to be mixed ionic and electronic
The highly mobile
(5). The conductors
cations were supposed to move in a "liquid-like"
as will be seen in section II, the principal the same as early in this century.
for new solid electrolytes
~)The high electrical by Faraday in 1634.
(6)
manner
In the years to come further
support for this concept was provided from X-ray structure analysis
essentially
Among
a_Ag2T e (5)~
of transference numbers and from their inter-dlffusion
within the rigid framework of the anion lattice.
Today,
In
phases of
they concluded that in the highly conducting phases of the silver
and cuprous halides chalcogenide
a-Cul,
solid electrolytes.
the high-temperature
(7)
lines of development
On the one hand,
is of great current interest;
are
the search
on the other hand,
conductivity of solid Ag2S had already been observed
Agl-Type solid electrolytes
we are these
still
trying
to get
a better
features
common
347
understanding
of the ion].c m o t i o n
in
materials.
The most
prominent
to A g I - t y p e
solid
electrolytes
are
the
following. (i)
The
cations
of voids the
cations
less
at random.
regions (ii)
The
these
differences
a large
the a p p a r e n t
In o r d e r
iodide
ions
deduced,
the
two s i l v e r
two s i l v e r
(9,10,11)
the p o s s i b l e quately
number
(i)
potentials
energy
are voids.
- as far as
- are of the o r d e r
to more
extended
The
it is still
data
sites
more
However,
appear
or less
pointed
out
still
arising
sites
do not
data ~2, i~,14) more
provided
with
regard
adeby the the
to the
as follows.
is really
in c h a r a c t e r
was
of these
by far e x c e e d s
ions may be s u m m a r i z e d cations
it is
by d i f f e r e n t
scattering
of space
The
statistically
to be d e s c r i b e d
regions
diffusive
are
of iodide
structure"
the o c c u p a n c y
regions
of the
cube
as such
an "average
local
the m o t i o n
(zz),
a-AgI.
Geometrically,
of these
of the s i l v e r
the
(7,5).
questions
of
plots
and experi-
consider
in a b o d y - c e n t r e d
recent n e u t r o n
principal
that
briefly
as has been
ions
Because
from A r r h e n i u s
at
thus
energy.
distributed (7,8).
compounds to void,
process.
on two p r e f e r r e d
literally,
The n u m b e r
from void
and the c o n c e p t u a l
lattice
diffraction
Is it p o s s i b l e that
situation
of the s i l v e r
cations.
sense
of
or
neighbouring
- or in certain
obtained
contained
of sites too
of r a t h e r
and m o t i o n s
more
can find vacant
the local
cation-diffusion
ions b e i n g
According
framework.
of the
positions
that
- can move
cubic
ions
X-ray
(42)
positons
in terms
iodide-ion
cation
interconnect
let us once more
precisely
must not be taken
authors
cations
the m i c r o s c o p i c
a large n u m b e r
sites
the n u m b e r
the n u m b e r
occupied
by the anions
of a c t i v a t i o n
involved,
the o b s e r v e d
the
are
in potential
about
of them
in a general
be l o c a t e d
From
a typical
lines w h i c h
all of the
fraction
form a b o d y - c e n t r e d
that
cannot
voids
that
exceeds
neighbourhood.
the d i f f e r e n c e s
energies
to e x p l a i n
difficulties
exist.
these
in such a way
are b r o u g h t
lattice
to be of the o r d e r of the thermal
mental
evident
in the sense
energy.
As a consequence,
found
situation
certain
lines,
thermal
that
in its i m m e d i a t e
along
participating
over
In this
these
least
and
are a r r a n g e d
flat
disordered
for them by the anion
themselves~
Along
of the
ions
structurally
of space
anions
rather
(iii)
are
provided
liquid-like
on a scale
in
of roughly
1 ~ or less? (ii)
On the o t h e r cept
that
hand,
the
metal-hydrogen "residence (iii) W h a t
kind
ordered
answers
systems?
times"
and
with
and
reliable
times"
as has
been
moving
in favour
e.g.
should
of the
like h y d r o g e n
be able
conin
to d i s t i n g u i s h
of the ions.
between
rapidly
jumps,
case one
exists
theoretical
Questions.
arguments
discrete
In this
"jump
is it true,
domains
experimental to these
there perform
of c o r r e l a t i o n
For example,
Recent
are
cations
the motions
proposed
of d i f f e r e n t
(15,16)
that
there
boundaries?
research
has
just b e g u n
to provide
ions? are
348
K. Funke
II GENERAL SURVEY In Table
i, a llst of AgI-type
their respective lattice
temperature
parameters.
obtained
solid electrolytes
Furthermore,
from these compounds.
references
Table
ting compounds
which contain halides
valent
However,
copper.
has considerably complicated copper salts
(17)
beyond
or chalcogenides
of silver or of mono-
composition.
systems
like Li2S04,
many silver-ion
conducting
These latter are mostly of more
By restricting
ourselves
to silver and
on which much work has been done
nor will they be reviewed
(20) and the anlon-conducting
in this article.
(21) solid electrolytes
(18, The
are well
the scope of this paper.
The experimental
data referred
to in Table
a) ionic conductivity
in the low-frequency
b) structure
from X-ray,
analysis
c) heat capacity
at constant
d) tracer-dlffusion
limit,
electron,
pressure,
coefficient
1 include ~ (T),
or neutron
ions, D~(T),
and permittivity
in the microwave
f) conductivity
and permittivity
in the far infrared
g) Raman scattering h) quasielastic
1 certainly
data.
In particular
phase diagrams
ments are not quoted. (30)
etc
the results
scattering
Furthermore,
(FIR),
(26,27)
and thermodynamic
ultrasonic
In this review,
spectrum of available
attenuation
however•
experimental
data from e.m.f,
data exist of thermoelectric (28,29)
electronic
we will essentially
measure-
power
''~22-25) proper-
concentrate
on
listed in the table.
It can already be seen from the general AgI-type
(MW),
(C.NS).
does not cover the whole
ionic Hall effect
range
(R),
neutron
Table
diffraction,
Cp(T),
of the mobile
e) conductivity
ties
data
to those cation-conduc-
in the last few years,
19), have not been included, 2-alumina-type
are given of experimental
and
i is restricted
have had to be omitted.
chemical
along with
anion arrangements,
as the number of phases known to belong to this class
increased
solid electrolytes
is presented
ranges of stability,
solid electrolytes
shape of Table
i that in the field of
we can at present distinguish
two different
major
llnes.of development. ~) A systematic search has begun for ionic crystals which exhibit high ionic, These
but negligible compounds
construction
electronic
are of considerable
of solid-state
current
batteries.
at ambient
interest
temperature.
in view o£ the
Such solid-state
systems have
become much in demand in recent years because
of their advantages
respect
large temperature
to conventional
of operation,
battery-systems,
e.g.
long shelf life, and the possibility
Suitable
materials
species.
In many cases not even the structure
usually
various solid silver-lon shi (31). (ii)
conductivities
New experimental understand
techniques
of the main requirements
of miniaturization.
contain AgI with silver ions as the mobile
conductors
the microscopic
with
ranges
is known.
has recently
A review of
been given by Takaha-
are being used in order to analyze and dynamical
behaviour
of these techniques
of the mobile
ions. One
is that the information
they yield should be on the time scale of the ionic motion,
i.e. rough-
b.c.c.
~430 ?
Ag~SBr
?
50,49 50,49
43O 48O
a:ii.97 c: 7.41 a:12.77 c:26.54 I
25
b.c.c. 4 90 ordered b.c.c. 4 99 disordered " b.c.c. 4.81 ordered -30 25
25
300
60
6.38
f.e.c.
170
5.84
see section IV see section IV
6,56
81,65 82
65 66
80
80
63,64 64
80
77-79
74-76
63,64
61,62
60
74,58
74
6,54,55 74,75
-)5.66 e~(581 57-59
250
250
72,73
72,73
71,72
6,51-54 74,75
48,49
250
3truct.
cp(¢)
)4
)2
)2
)2,93
89,91,73
89,91,73
89-91,7]
54
54,98
83, 4,43,44 7,8,12 l~a67~6L 8,84-8? 95,96 88 45-47 69,70 97
~(T)
References:
43O
25
250
[°4
(T)
f.c.c.
f. C.C.
6.57
thermodynamically unstable at T < 27 °C
((CH3)4N)2AgI3115
(C5H5NH)Ag5I 6
235-
a-Ag~Sl
<235
~-Ag~SI
i~110-1113
a-Cu2Se
51- 164
~430-I127
a-Cu2S
a-Ag2Hgl 4
~140-~790
a-Ag2Te
f.c.c.
b.c.c.
~13o- 897
a-Ag2Se
5.00
4.88
b.c.e.
~177-~600
a-Ag2S
4.53
469- 487
~-CuBr
h.c.p,
6.13 a:4.04 c:6.~
11.24
5.06
b.c.c.
386- 469
ee see~)27- 232 stion IV 408- 602 f.c.c.
147- 555
Eo4
Stability Anion L~ttice Range Structure Const.
B-CuBr
~-Cul
a-RbAg415
a-AgI
Solid Electrolyte
TABLE i. GENERAL SURVEY ON AgI-TYPE SOLID ELECTROLYTES
#4,99
33
32
36
35
32,37
FIR
I
38
R
14,40
o~s
~o
W
O
rv
tD
T
350
K. Funke
ly between
109 and 1013 Hz. Therefore,
(32,35-37),
and Raman experiments
microwave
(32-34)
far-infrared
(38) have been undertaken.
A similar
frequency range is covered by the quasielastic and inelastic scattering of cold neutrons
(39,14,40),
neous spacial resolution.
with the additional
advantage of simulta-
In the lO 9 to 1013 Hz frequency range,
diffe-
rent kinds of behaviour of the mobile ions have been predicted by different models
(37•41,42).
These models are now being tested by comparison
with experiment.
III IONIC CONDUCTIVITY The most prominent property of AgI-type ly high ionic conductivity. -ionic conductors"
solid electrolytes
is their unusual-
They have therefore sometimes been called "super
or even "ionic superconductors".
However,
these terms seem
to be misleading• (lO0) ty
especially because there is no relation to superconductivi-
For the graphical
representation of ionic conductivities
systems•
it is convenient
to choose a plot in which the data give straight
lines. Let us consider two extreme (i)
in solid or liquid
cases.
As is well known•
in a "normal"
linear dependence
is observed in a plot of log(~ T) vs I/T. In the
intrinsic
temperature
equilibrium,
region,
ionic crystal•
like NaCI or AgCI,
a
where the lattice defects are in thermal
the slope of this line is determined by the defect-forma-
tion enthalpy and by the migration enthalpy which both clearly exceed the thermal energy. (ii)
For liquid systems• proposed by Reynik
a special "itinerant
oscillator"
model has been
(i01). This model predicts a linear dependence
the coefficient of self-diffusion D on temperature, indeed observed in many liquids,
of
D = bT-a, which is
including molten salts. Let us assume
that in a given system the motion of the ions obeys this relation. Secondly,
let us assume that the Haven ratio• HR~•
depend on temperature•
does not noticeably
that is, that D is proportional
stability range of the system under consideration.
to ~ T
in the
In a ~ vs 1/T dia-
gram, we then simply find a straight line with a negative slope. In the case of AgI-type solid electrolytes,
a specific difficulty arises.
Plots suggested by (i) and (ii) yield nearly straight lines and the same is true of a number of other plots which may be made. Therefore,
an interpre-
tation may be dangerous as long as no independent information is a~ailable about the microscopic behaviour of the mobile ions. As an example• conductivity of ~ - A g I may be considered. by Tubandt and Lorenz
In Figs. 2 and 3, the data obtained
"~4~ marked "u", and by Kvist and "
'°o"• are plotted in three different ways,
the ionic
Josefson ( 4 3 )
i.e. 5 vs i/T, l o g g
marked
vs l/T• and
log (gT) vs I/T. ~he
Haven ratio is the factor in the Nernst-Einstein
as the ratio of tracer to charge diffusion see section VI.
equation and is defined
coefficient•
H R = D~/D~
(102)
Agl-Type solid electrolytes
T [°C] - - ~ 300 400
200 i
i
351
500
1
r
2.6
2.4 2.2 6 [ ( a c m ) -I ] 2,0 1.8 1.6
o/ / /
1.4
/,
2~ =
FIG.
2.Electrical
21o
,:0
conductivity
,I~
,2
of
~-Agl
in a
~ vs I/T r e p r e s e n t a t i o n .
T [oc] --~ 400 500
200
t
,16
T -1 [tO 3 K-1 ]
300
e5
2.0
~1
[ ( n c m ) "1 ] 2.5 11"5
( points below stroight line)
i
( points obove
straight llne)
n
2.0
1.5
oe • II
/.
./.-
1.0
0.5 I
I
Z2
2.0 •" - -
FIG.
3.Electrical log
i
1
i
1.8
1.6
1.4
1.2
T - ' [10 3 K - ' ]
conductivity
of
a-Agl
(G T) vs I/T representations.
in the l o g ~
vs I/T and
352
K. Funke
In the ~ vs 1/T representation, found at temperatures
Fig. 2, an accurately linear dependence
above 250 °C. In the high-temperature
ductivity extrapolates
limit,
is
the con-
to a value similar to those abtained from correspon-
ding plots of molten salts. According
to Reynik's
theory,
this would give ca.
0.5 ~ for the mean displacement of a centre of oscillation during one oscillation period,
cf. 0'Keeffe
(103). However,
below 250 °C, the linear fit becomes
as the temperature
is decreased
increasingly worse. Moreover,
known from microwave and neutron-scatterlng
experiments,
and IX, that the diffusive motion of the silver ions in
it is now
see sections VII g-AgI
is not proper-
ly described by the itinerant oscillator model.
Instead,
that distinct
with mean jump-lengths
steps of Jump-diffusion do occur,
the order of a lattice constant. activation
the conclusion is
In the concept of distinct
of
jumps, a thermal
for the ionic motion across the maxima of local potential has to
be considered.
Thus,
the logarithmic
plots,
rather than the linear one. In the l o g ~ of the experimental
Fig ), might be appropriate
vs I/T plot,
curve is slightly negative,
the second derivative
while it is slightly positive
in the log (~ T) vs I/T representation. Biermann and Jost
(i04) have considered
the f r a c t i o n m
of cations which have
a certain migration enthalpy H m of the order of kBT. For simplicity, assume
they
just two sets of cations with enthalpies differing by H m. Then, -Hm/kBT
(1).
e
a
=
-H m /
1+e
kB~
'
the second term in the partition sum is not negligible Interestingly, onal to
a,
the experimental
giving the observed
~-values log G
compared to unity.
are found to be directly proporti-
vs 1/T dependence,
when this approach
is used. The corresponding "value of H m is 64 meV in the case of On the other hand,
in
g-AgI.
assuming a continuous distribution of possible energy
states, we find that the fraction of ions with enthalpies H • H m is given simply by a = exp(-Hm/kBT). rate is proportional hence of
to a
Furthermore,
one should expect that the jump
and, therefore,
that the same is true of D and
g T rather than of 6 . The upward bend of the experimental
curve in
the log (e T) vs I/T plot might then indicate increasing values of H m with increasing temperature.
As a tentative interpretation,
the following con-
sideration might be helpful. Because of the motions of the neighbouring cf. sections VII-IX,
ca. I0 -12s, with amplitudes
comparable
to the thermal energy. Therefore,
ions in different momentary microscopic enthalpies defined.
of activation
cat-
situations will need different
for a Jump, and a unique value of H m cannot be
At relatively high temperatures,
larger enthalpies
ions,
the local potentials will fluctuate on a time scale of
the fraction of jumps requiring
of activation will be larger than at relatively low
temperatures and hence a temperature variation of the apparent overall value of H m should be expected. In Figs.4-6,
the ionic conductivities
are represented have
of various AgI-type
solid electrolytes
in log (~ T) vs I/T plots. As indicated above,
precautions
to be taken in the interpretation of the slopes. In particular,
and meaningful enthalpies
of migration cannot be extracted.
unique
Agl-Type solid electrolytes
T[ oc] - ~ - - - . -
353
T[oC]
200
=
2,,o0
soo
soo
)
~a- Cu Br
a-AgZ
103 !T 103
I)-CuBr
• 1 [(~ c m)"~1
loZ[~crn] 102
~j/--AgCl
10
10
1
1
1l~1
ld 1
16 2
16'I
I
I
I
3
2
1
3
1031T[K "1] FIG.
4. Ionic conductivities
I
FIG.
,
1
5. Ionic conductivities
of
highly conducting silver and
highly cation-conducting
cuprous halides.
silver chalcogenides.
For clarity, classified (i)
of
I
2 •..~.--.-1031 T[K 4]
the ionic conductors quoted below and in Figs. 4-6 have been
into three groups of compounds:
the halides of silver and monovalent
(ii)
the chalcogenides
(iii)
compounds with more complicated
As a reference, also in Figs.
copper,
of silver and monovalent
copper,
chemical compositions.
the conductivity of Agl is recorded not only in Fig. 4, but
5 and 6.
(i) The highly cation conducting solid phases of the silver and cuprous halides are
a-Agl,
a-Cul,
a-CuBr,
and ~-CuBr ~. The data presented in Fig. 4 were
obtained
from resistance measurements
see also
(105)
ly carefully
between platinum electrodes
"'~4'5"43)
The cuprous halide samples had to be prepared under extreme-
control]ed
conditions and measured in an atmosphere
halogen and oxygen in order to obtain reproducible results. While
free of the early
data of Tubandt et al. were later on reproduced exactly in the case of a-Agl
(43), Wagner and Wagner,
using copper electrodes,
-values than the Tubandt group in the cases of (49). However, CuBr
recent microwave
g-CuI,
conductivity data of
observed lower' a-CuBr,
a-CuI
and B-CuBr
(33) and ~ -
(34) _ both in equilibrium with copper - are in good agreement with the
results given by Tubandt and coworkers. The cuprous halides,
An
our notation,
which transform into their cation-disordered
the phases are termed a-,
phases at
$-, T- etc. in the order of
their stability with decreasing temperature.
354
K. Funke
relatively vities
high temperatures,
already display
below these phase transitions,
at their transition conductivity duction
considerable
in
solid electrolyte
a-CuBr.
cationic
conducti-
(106), and the discontinuities
points are less pronounced
of a Agl-type
is observed
cf.
with negligible
electronic
The transport number of the electrons
known to be less than 10 -7 in a-Agl
"'~i07~, of the order
and of the order of 10 -4 in
"'~49).
a-CuI
of
than in AgI. The highest con-
is
of 10 -5 in ~-CuBr~49!~
(ii) The silver and cuprous ductors.
chalcogenides
The electronic
ionic conductivities In contrast
conduction
are mixed cationic
predominates
of the order of 1 ( ~ c m ) -I are observed
to the ionic conductivities,
strongly depend on deviations be described
and electronic
the electronic
from the stoichiometric
by Ag2+ ~ X and Cu2_ ~ X, respectively.
perties
and effects
associated
treated
in this paper.
The ionic conductivity
with deviations
of a mixed silver-ion
con-
even in those phases where (6,51,56-58,60)
contributions
compositions
However,
which may
electronic
from stoichiometry
and electron
to pro-
will not be
conductor
can be
analyzed with the arrangement Ag / a-AgI / specimen / a-AgI / Ag , a-AgI is used as an electron blocking layer (51,
where
may be performed and measuring
by sending a current of constant
the voltage drop across
the specimen.
switch the current on and off and observe -up and decay of the potential proposed by Yokota species. passage
Here,
of cations.
conducting
see Miyatani
voltage,
In this kind of experiment,
the steady state voltage,
Highly
cation-conducting
Ag2S ,
a-Ag2Se,
a-Ag2S
and
a-Ag2Te,
exist different The
a-phase
phases belonging
a-Ag2Te,
a-Cu2S,
for the build
(51). Another method,
are used to block the
a constant
current is
can be deduced
and the residual
to the chalcogenide
a-Cu2Se,
and ~-Cu2S.
from the
voltage. group are
is used in spite of the fact that in these cases there (Ii0)
(59), stable above ca. 430 °C (58), and the "digenite"-
phase CUl.8 S, stable above ca. 90 °C, have been shown to be identical Ionic conductivity
g-
The usual notation,
solid phases above 600 °C and 800 °C, respectively
of Cu2S
this cell one can
to silver ions as the mobile ionic
electrodes
switched on and off, and the ionic conductivity initial
through
Alternatively,
the time constants
difference,
(109) is not restricted
electronically
108). The experiment
intensity
data on
a_Cu2S
scarce and have not been included
(57,59) in Fig.
and
a-Cu2Se
5. According
Kawai (57), the ionic conductivity of B-Cu2S increases at ii0 °C to ca. 0.5 ( ~ c m ) -I at 220 °C, and Wehefritz
(58)
(60) are relatively to 0kamoto
and
from ca. 0.i ( ~ c m ) -I (58) reports an ionic
conductivity of CUl.8 S of ca. 1.6 ( ~ c m ) -I at 400 °C. In the case of a-Cu2Se, ~elustka and Ogorelec obtained values increasing from 3 ( ~ c m ) -I to 4 ( ~ c m ) -1 in the temperature
range from 580 °C to 750 °C (60)
(iii) A large number of highly cation conducting conductance
can be derived
the iodide ions, with different
from AgI by partial
or even both,
structures
high ionic conductivities
solids with negligible
by different
substitution
electronic
of the silver or
kinds of ions. Thus new compounds
are obtained which in some cases exhibit unusually even at room temperature
and below.
In these com-
Agl-Type solid electrolytes
pounds,
except
exclusively
in CuiHgl 4 and its alloys
the silver
ions.
are CuiHgl 4 and Ag3SBr: Agl are replaced As the first Ketelaar
by different
example
in 1994
At a later date,
Exceptions
in these
with AgiHgl4,
all
of the silver
of this group of materials, numbers
and Pond proved
Reuter
two phases
and Hardel
they both structurally different
(63).
1967, Bradley
discovered
where M represents bit the largest
ionic
a -Agl, Ag3SBr
and Greene
a group
~ - A g i H g l 4 was described
similar
conductivity see section
is similar (45,111)
of solid ionic
interesting IV, although
to ~-Ag3SI
and Owens
electrolytes
conductivities
of being
into Agl and MiAglg, In order
thermodynamically especially
to find Agl-type
unstable
solid electrolytes
(64)
and Argue these
(66,8?)
and ((CH~)4N)iAgI~II5 ductors, see Fig. 6. The electrical
conductivities
of various
MAg415,
at present,
exhi-
e.g.
appears to be compounds have
the
at 298 K. They decompose of moisture
which
have been
(46) inde-
compounds
(i12)~
are stable
at room
temperature, compounds with partial substitution of the silver organic groups have been investigated. Thus, among others (31),
(65)
The
because having
of composition
known
in the presence
(62)
data of Ag]SI.
0.?7(~cm) -I in the case of RbAg415 (47). Although RbAg415 inert at room temperature £or long periods of time, these disadvantage
by
as the eutectoid
properties
Rb, K, or NH 4. At room temperature,
solid-state
ions in
are tag+ ~ 0.94 and t H g i + ~ 0 . 0 6 .
and B, are particularly
resemble
conductivities
In 1966 and pendently
reported
of this compound, ~
substitution
or iodide
that CuiHgl 4 as well
of the AgiHgl 4 - CuiHgl 4 system have quite In !961,
ions are
species.
(61). The transport Suchow
the mobile
to the rule of partial
cases
ionic
355
identified
systems
MX-#gI
ions in Agl by
as good
(CsHsNH)Ag516 ionic
con-
have been determined
as a function of composition. The result obtained by Owens and Argue in the case of RbI-AgI at 22 °C (iI~) is shown in Fig. 7. It may be regarded as typical
in the sense
that in a number
maximum
in the ionic
conductivity
cent AgI. g~is04
T h i s i ~ i s also
- #gl
example, the in Fig. 6.
~}See
(45) and
~l16j,
of systems,
has been observed
true of thei~systems
Ag~V04
_ Agl
~(T)
curve
(ili)
for the phase
at room temperature,
\l16j
KCN
and AgiW04
of the compound
diagrams.
at, or near,
- ~gI
(AgiW04)
(114)
Ag~PO 4~
a
~0 mole per - AgI
(I15)
- Agl (I17)_ As an Ag414 has been
included
356
K. Funke
T[K] 125
150
I
v
200
300
I
I
I
500 A9- ,I~
10 3
I I;
(AgzWO4)~14 RbAg#~j
10-z
eT
6
[I~ cm) -I] ([CHzI,-N)zAg~Im
10-1
{ Csl'.lsNH) AgsI 5
Ag2 H9 I~
10-3
d6
I
I
I
I
I
I
I
I
I
9
8
?
6
5 1031 T[K "1]
4
3
2
1
•~
FIG. 6. Ionic conductivities
i
I
I
IC
of various AgI-type
i
!
|
I
I
solid electrolytes.
I
"T " " 0.25 E U
0.20 > 0.15
g
o.;o
U U
0.05
U
0 Rbl
FIG. 7.
2O
40 60 mole % Agl
80
100 AgI
Conductivity of the system Rbl - Agl at 22 °C according to Owens and Argue (I08).
Agl-Type solid electrolytes IV STRUCTUKAL The main structural arrangement
357
PROPERTIES
property of Agl-type solid electrolytes
is the disordered
of the mobile cations within a relatively rigid anion lattice.
In the case of the simpler compounds, anion structures,
we can distinguish
three different
namely
(i)
body-centred
cubic
(ii)
face-centred
cubic
(iii)
hexagonal
(b.c.c.), (f.c.c.),
close-packed
(h.c.p.).
The anion structures of those AgI-type
solid electrolytes which do not fit
into one of these categories are generally much more complicated. be discussed
in paragraph
They will
(iv).
(i) The structure of
~-Agl was first determined by Stroek in 1934
iodide ions form a b,c.c, to the silver ions, available
lattice,
see section I. Strock proposed a large number of sites
for the two silver ions within a cube of iodide ions, namely 6
octahedral
(b),
i2 tetrahedral
(d), and 24 trigonal bipyramidal
see Fig. 3. In order to explain the X-ray patterns, Hoshino
(?). The
while no definite sites can be assigned
(~), who confirmed Strock's
results,
Strock,
(h) positions,
and later on
assumed that the two silver
ions are statistically distributed over these 42 sites. At the same time, was pointed out by Strock to a "quasi-molten"
that the observed structural
or "liquid-like"
properties
state of the silver ions,
of.
interpretation has sometimes been regarded as self-contradictory clearly,
to X-ray diffraetlon,
has the advantage
This
structural analysis by neutron
diffraction
of yielding information about the loci of the nuclei and
not of the electron on
(4)
(i0).,
Strock himself did not take his 42 sites too literally.
In comparison
Fender
it
correspond
clouds.
Recently,
B~hrer and H~ig
(12) and Wright and
(13) have independently undertaken neutron diffraction experiments
a-Agl 6. The neutron data indicate that the silver ions are preferentially
found in oblong ellipsoidal and extending
regions of space centred at the tetrahedral
in the directions
Fig. 9. This result suggests completely
liquid-like
channel-like results,
of the nelghbourlng octahedral directions
should be regarded as
This point of view is supported by microwave
see section VII.
From the structural
refinements
nents are
for both the silver and the iodide ions. According
obtained
Debye-Waller
theory,
(8,12,1j)
they correspond(at
the ions from their mean positions Here,
see
(13) that the motion of the silver ions is not
and that the (100>
diffusion paths.
sites,
sites
unusually
large Debye-Waller expoto
200 °C) to r.m.s, displacements
of
of ca. 0.4 ~ and ca. 0.3 ~, respectivity.
the mean positions of the silver ions have been taken to be ± 0,3
away from the tetrahedral
sites,
along the ~100>
view of the large cationic displacements,
passageways.
However,
in
the actual distribution of the
silver ions is more properly described by the above-mentioned
ellipsoids
than by pairs of sites. ~Although
single crystals of
determination
a-AgI
can be produced
(i18,119)
on a single crystal has so far been reported.
no structural
358
K. Funke
FIG.
8. Crystal
structure
Hoshino marked
(8) ~
constant
of
a-AgI
according
The 6b sites are marked
to Strock •
, and the 24h sites are marked is 5.06 ~ at 250 °C
(7) and
the 12d sites are o
. The lattice
(8)
I
I
I ,i,S FIG.
9. Crystal
structure
The figure gives nuclei
at 200 °C.
of
l_uX g-AgI
according
the positons
to Wright
and r.m.s,
and Fender
displacements
(13)
of the
Agl-Type solid electrolytes
359
In this connection it must be emphasized that the evaluation of the temperature
factors in terms of simple Debye-Waller theory becomes questionable
AgI-type
solid electrolytes.
The large-amplitude vibrations
are generally considered anharmonic
and anisotropic
in
of the cations
(120,79).
For a further
discussion of the local motion of the cations see sections VIII and IX. In the case of the iodide ions,
the large Debye-Waller exponents
can perhaps
be partially explained by quasistatic displacements due to the cation disorder
(121). Fig. 9 gives the positons and r.m.s, displacements
according
of the nuclei
to (13). The structural disorder of the silver ions and the large
temperature
factors are also reflected by the intense diffuse
scattering
a-AgI (8,68) which has recently been measured with thermal neutrons A peculiarity of and M~ller
~-AgI
is the so-called "memory-effect",
(122) and by Burley
heating of the hexagonal
(67). Samples of
low-temperature
from
(121)
observed by Bloch
~-AgI may be obtained by
~-phase or the metastable
f.c.c,
r-
-phase. Samples arising from these two different sources seem to have slightly different
cation-arrangements
170 °C is not surpassed.
as long as a temperature of approx.
According to (122) and (67), cooling below the
transition point at 147 °C results in regeneration of the original phase. This "memory effect" 170 °C (67)
becomes irreversibly erased at temperatures above ca.
The motion of cations of different by the b.c.c,
anion lattice of
Flygare and Huggins
(123)
sizes along the ¢I00)
a-AgI has been examined theoretically by
In this calculation,
actions of the anions with a cation in a as well as overlap repulsion effects. simplicity,
tunnels provided
the electrostatic
inter-
tunnel have been considered
On the other hand,
in order to retain
the authors have ignored the influence of the positions and
velocities of the neighbouring shows the loci of
cations on the local potentials
(124). Fig.
minimum energy paths for cations of different
sizes.
i0
They
do not follow the centreline of the turmel but deviate periodically with both direction and magnitude depending on the cationic size. Small ions are attracted
towards pairs of iodide ions, while the paths of larger ions are
orthogonal
to the anion-anion axis. Flygare and Huggins have
periodic variations
minimum energy path during translation through the tunnel. have opposite phases for small and large cations. barriers
calculated the
of the local potentials as the cations follow their These variations
The respective
energy
are found either near the tetrahedral or near the octahedral
and originate
either from the electrostatic or from the repulsive
potential energy. At an optimum intermediate
sites
term in the
cationic size, with cationic
radius r ~ 0.83 ~, these effects largely balance each other, energy variations well below the thermal energy,
see Fig.
yielding small
ii. The actual
radius of the silver ion is somewhere between that of the lithium ion, 0.60 ~, and the sodium ion, 0.95 ~. Thus,
the predictions of the Flygare-Huggins
theory are in good agreement with the low potential b a r r i e r s ties
actually observed in
volume change of approx. AgI
(125-127)
~-AgI.
and high mobili-
In the light of this theory,
4.5 % associated with the 8 ---+Q
the negative
transition of
appears to be related to the fact that in ~ -AgI there is a
particular ratio of lattice constant to cationic radius which is highly
360
K. Funke
J Y
FIG.
y
10. Loci of minimum energy paths for cations of different sizes along a
(indicated by circles), on the right, paths of
larger ions are orthogonal to the anion-anion axis.
12S
loo
,oi 2! 0 O~
l 0.7
I 0.8
I 0.9
1.0
r 1,;,1 FIG.
II. Activation energy as a function of cationic radius, as calculated by the Flygare-Huggins theory for the motion of cations along
favourable for fast cation diffusion,
(123).
cf. (127).
Besides g-AgI, the phases ~-CuBr, g-Ag3SI, ~-Ag2S, and ~-Ag2Se have b.c.c, anion structures. The number of cations per b.c.c, unit cell is two in g-AgI and m-CuBr, three in g-Ag)SI, and four in g-Ag2S and ~-Ag2Se.
Agl-Type solid electrolytes
In 1952, Hoshino ~-CuBr
(73) and Krug and Sieg
is isostructural
phase diagrams ~-AgI-like According
with
of CuC1 and CuBr,
it is concluded
to Rahlfs,
1936,
(74),
a-Ag2S, unit
(ii) proposed
of
~-Ag3SI
besides
I- ions are s t a t i s t i c a l l y On the other hand, ~-AgjSI
the anions
and also in Ag3SBr;
composed
of oimple
the silver ions, R e u t e r and Hardel the anion structure. (x,[/2,0)~ Ag3SBr.
in the
in the low-temperature
lattice
proposed
-~(80). In
the S 2- and lattice. phase
can now be regarded
sublattices.
as being
For the sites of
the largest voids available
or Br
in the corners
these voids are situated
in B-Ag~SI
unit
cell,
for geometrical
can be occupied a unit
cell,
therefore
corresponding
±
and 0.3~5 in
to the tetrahedral
reasons,
regions
at a time. Most silver ions are thus confined
readily explained. for squeezing
along
a-Agl
and
The activation
a-Ag3Sl,
than in
enthalpies
to one face of ~-Ag3SI
necessary
the silver ions past the large anions,
directions,
are close to i/4 eV,
see. Fig.
in In
only one site out of a group of four
and the fact that they are less mobile
and Ag3SBr the
in
and the smaller
at the 12 positions
9, and there are as many groups of voids as there are silver ions.
most cases,
of
voids
there is a group of four voids in the vicinity of each face
centre of the b.c.c, Fig.
are
In the
by Reuter and Hardel
and ± (x, 0 , 1 / 2 ) ~ , where x is 0.390
Thus,
~-Ag2Se.
that the six octahedral
(or Br-)
In a cube with I
S 2- ion in the centre,
~-Ag2Se
by the four silver ions.
are ordered and I
p
with a pre-
over the sites of a b.c.c,
the b.c.c,
cubic S
~-Ag2S and
disorder of the silver ions,
distributed
that
vs
(126)
over the 42 Strock-sites
has been established
the structural
ascertained of the T
that there also exists an
sites in the case of
cell are randomly occupied
The structure this case,
Rickert
~ 2 kbar
the silver ions in
distributed
for the larger tetrahedral
case of b.c.c,
(72) independently
From the similarity
phase of CuC1 at pressures
more or less statistically ference
a-AgI.
361
is
in ~-AgjSI probably
clearly exceeding
those
6.
(ii) In c o n t r a d i s t i n c t i o n
to
the anion arrangement
a-Agl
in
a-Cul
and
a-CuBr,
in view of the ratios of the cationic compounds, structure
is observed
a-Ag2S
a-Cu2S,
and
and
still exhibit f.c.c,
paths
have been discussed
the cations
consist
tetrahedron
by Az~roff
of alternating
through
(74).
a-Ag2Te,
i.e.f.e.c,
In an f.c.c,
The diffusion
unlike voids:
unit
and
cell,
there
paths available
for
tetrahedron ~ o c t a h e d r o n
shares
faces with 4 octahedra,
Thus there is a large variety
the anion lattice. 12.
than
chalcogenides
structure,
structures,
(i29).
voids.
with 6 tetrahedra.
is shown in Fig.
are smaller
of the anion
arrangements.
etc.. Each anion tetrahedron
passageways
linear tunnel
variation
the b.c.e,
in close-packed
and 4 octahedrai
and each octahedron possible
have
diffusion
are o tetrahedral
radii of these three
anion lattice
The same systematic
anion lattices,
This is not unexpected
in the case of the silver and cuprous
a-Ag2Se
~-Cu2Se
The possible h.c.p.,
lattice.
cubic.
to anionic
as the voids provided by an f.c.e,
those in a b.e.c, While
which have b.c.c,
is face-centred
For visual
clarity,
of an almost
362
K. Funke
U FIG.
12. An almost
linear p a s s a g e w a y
in an f.c.c,
anion lattice~
The d i s t r i b u t i o n of the cations in the c h a l c o g e n i d e phases ~-Cu2Se,
and
In all of these phases, (i/4,
I/4,
~-Ag2Te ,
~ - C u I 8S was c a r e f u l l y studied by Rahlfs as early as 1935 (74) roughly one half of the cations occupy the sites
i/4; plus f.c.c.) w h i c h
correspond
to a z i n c b l e n d e
lattice.
The
r e m a i n i n g 4 cations w i t h i n the f.c.c, unit cell are more or less r a n d o m l y distributed.
Besides
the o c t a h e d r a l
c o n s i d e r e d 32 sites called
and t e t r a h e d r a l
16a(I/3)
and
positions,
Rahlfs also
16a(2/3). E a c h of these is s i t u a t e d
on a
4111>
site,
at the i n t e r s e c t i o n w i t h the face common to b o t h anion polyhedra.
16a(I/3)
p a s s a g e w a y b e t w e e n a t e t r a h e d r a l and a n e i g h b o u r i n g o c t a h e d r a l
positions
occupied zincblende
form sets of 4 s u b s i d i a r y sites around each of the 4 sites,
while
the 16a(2/3)
a s s o c i a t e d w i t h the 4 t e t r a h e d r a l results, various
sites
positions are c o r r e s p o n d i n g l y
(3/4, 3/4,
3/4;
plus
f.c.c.).
in p a r t i c u l a r his e s t i m a t e of the fractional o c c u p a n c i e s possible p o s i t i o n s
c o n f i r m e d by others
in the f.c.c,
(75,76). However,
chalcogenide
phases,
tials are known to be flat. W i t h the help of neutrons, 42 sites in a unit cell of
a-AgI
since the potenconcept of
could be r e p l a c e d by the more r e a l i s t i c n e u t r o n s might
also be useful for a s t r u c t u r e d e t e r m i n a t i o n in c a t i o n - d i s o r d e r e d
~-CuI,
of the
have since been
Strock's
v i e w of several r a t h e r e x t e n d e d regions of space• Similarly, e l e c t r o l y t e s w i t h f.c.c,
Rahlfs'
the d i s t r i b u t i o n of the cations over a
large n u m b e r of d i s t i n c t sites is n e c e s s a r i l y an artifact,
In the case of
The
solid
lattices.
Miyake et al.
X - r a y data by the a s s u m p t i o n
(71) o b t a i n e d the best fit to their
that the 4 cuprous ions w i t h i n the unit cell are
r a n d o m l y d i s t r i b u t e d over the 4 zincblende
sites plus the
16a(i/3)
sites. An
Agl-Type solid electrolytes
alternative
363
and probably more realistic explanation of the data, which has
also been given by Krug and Sieg
(72) and by Matsubara
presence of a location in the zincblende-type vibrational
amplitudes.
1 ~ for the r.m.s, displacement of the
cuprous ions. However,
confined to the zincbiende-type cationic mobilities,
voids with unusually large
Applying an anharmonic oscillator model, Matsubara
(120) derives values of approx. oscillating
in these models,
voids,
the cations remain
and the prerequisite
view of the ionic conductivity of
~-CuI,
is still unfulfilled.
structural disorder in the sense that a fraction of the
~-Ag2HgI 4 and
~-Cu2HgI4,
cell. On the average,
alike.
every forth one of these voids is thus unoccupied. As pointed out by Hoshino
in the X-ray line intensities of In fact,
vibrations
voids in the f.c.c, unit
very high ionic conductivities were already predicted
prior to their measurement. arities
(i0)
first analyzed by Ketelaar in 19J4 (77), the
cations statistically occupy the 4 zincblende-type this reason,
In
it is evident that there must be
cuprous ions should have left the zincblende-typevoids In
for very high
namely that the number of available and partially
occupied voids should exceed the number of cations, some additional
(120), is the possible
in
~-Ag2HgI4,
of the cations
as in
~-CuI and
a-CuI,
have been reported
For
(77)
(78), certain peculi~-Ag2HgI # are much
large-amplitude
anharmonic
(79)
(iii) The only known Agl-type solid electrolyte with an h.c.p, ~-CuBr. However,
the above-mentioned
of CuCI and CuBr
(i28) suggests
that
in the relatively narrow temperature have properties
anion structure
is
similarity of the T vs p phase diagrams a-CuCl,
which is hexagonal and stable
interval from 407 °C to 422 °C, might
similar to those of ~-CuBr. Very careful conductivity
measurements on CuCI are needed in order to decide this question. The hexagonal anion lattice provides One half of the tetrahedral the other hand, interconnect see Fig.
tetrahedral as well as octahedral
voids are occupied in the wurtzite
the most probable diffusion paths available
the face-sharing octahedra along parallel
voids.
structure.
On
for the cations
straight lines
(i29, i0),
i3. The conductivity and the self-diffusion of the cuprous ions in
a single crystal of ~-CuBr should therefore be markedly anisotropic. Structural
analyses of 8-CuBr have been performed by Hoshino
Krug and Sieg
(72)
the wurtzlte-type
According to Hoshino, tetrahedral voids.
and following Matsubara's tain r.m.s,
Using Hoshino's Debye-Waller factors
anharmonic-oscillator
amplitudes of almost
(79) and by
the cations are situated within treatment,
we once more ob-
i ~ for the cuprous ions. Unfortunately,
the high mobility of the cations is not reflected by Hoshlno's model. and Sieg proposed octahedral different
that a fraction of roughly
sites. However,
Krug
I0 ~ of the cations occupy the
they did not preclude the possibility of still
cation distributions
in B-CuBr.
364
K. Funke
FIG.
13. A linear diffusion
path in an h.c.p,
anion lattice.
(iv) The structure determined
of ~-RbAg4I 5 and its isomorphs a-KAg4I 5 and ~-NH4Ag4I 5 was
by Bradley
single-crystal
and Greene
analysis ~ ) ~ .
spect to the arrangement describe
the structure
(70) and by Geller
The results
(69), who performed
of the silver ions. In the following,
according
to Oeller
in a cubic unit cell. The structure
is characterized
the unit cell is similar to that of the manganese 56 tetrahedral
octahedra.
fractional
occupancies.
ding tetrahedra
share
the 4 Rb + ions are
The 56 tetrahedral
three sets of crystallographically
nonequivalent
These sets are ~-, 24-, l, 0, and 2 faces,
by the enantiomorphic of the 20 iodide ions in
atoms in B-Mn and provides
voids for the 16 silver ions, while
in distorted
we will
(69). There are 4 formula units
space groups P413(07 ) and P413(06 ). The arrangement
situated
a
of these authors differ with re-
positions
consist of
sites with different and 24-fold;
respectively,
the correspon-
with neighbouring
anion octahedra containing Rb + ions. The other faces are shared with tetraheda, and thus a large number of possible passageways is available for the silver ions. Interestingly,
the alternating
face-sharing
of tetrahedra
~)
The production of single crystals of RbAg415 has been described by Fullmer and Hiller (i~0) and by Manning, Venuto, and Boden (131)
~)
Ladd and Lee compounds
(132) have investigated
the crystal energies
MAg415 using the data of (70) and (69)
of the
be-
Agl-Type solid electrolytes
365
longing to the two 24-fold sets results in the formation of channels through the anion lattice. cell,
One of these channels is shown in Fig.
there are two parallel
channels
in each
14. Within a unit
direction.
Perpendicular
channels are cross-linked by the face-sharing of two tetrahedra belonging to either of these channels. RbAg4I 5 undergoes
phase transitions
at 209 K and at 122 K (88)
The one at
209 K is of the lambda type, see section V. As seen from Fig. 6, it is associated with a discontinuity
in d 4 / d T ,
while S is continuous.
sition at 122 K is a first order one, with a discontinuous two orders of magnitude.
Both transitions
distortion of the iodide-rubidium a-phase are still identified neither of the low-temperature
The tran-
change in ~ of
seem to involve relatively
lattice,
as the diffraction
(69,133). However,
little
lines of the
birefringence
shows that
phases is really cubic. According to Geller
(69), domains are visible under a microscope within the 8-phase. The crystal structure of ((CH3)4N)2AgI3II5 was determined by Geller and Lind
(82), cf.
(134). It is more complicated
characterized by similar properties,
than that of
a-RbAg415,
but is
namely the existence of nonequivalent
sets of tetrahedral voids and of channels formed by face-sharing tetrahedra. The
structure of (C5H5NH)Ag5I 6 belongs to space group P6/mcc
formula
units in the hexagonal unit cell
sets of tetrahedral (4c) are available
FIG.
positions
(D~h)'u with two
(81,65). Within the unit cell,
(6f and 24 m) and one set of octahedral
two
sites
for the silver ions. The structure at -JO °C is shown in
14. View along one of the
for the
diffusing Ag + ions in RDAg4I 5. Iodide and rubidium ions, but no silver ions are shown in the figure.
366
K. Funke
Fig.
15. As the temperature
is increased,
6f sites into the 24 tetrahedral site occupancies (65)
as functions
of temperature
At ca. 50 °C, there appears
from a phase of relatively
silver ions move from the 4c and
(m) sites previously
from the log
(~T)
In the (C5H5NH)I electrolyte, conductivity
to a more disordered
vs I/T plot,
- AgI system,
At the same temperature,
of the formal activation
energy obtained
see Fig. 6 (65). there also exists
,-r~~135,1Do). Its structure
(135) and by Geller,
in particular
a "two-dlmensional"
solid
has been describedt~ by Geller and
Skarstad,
and Wilber
~136j
These authors
show
that the total volume of the crystal occupied by the possible
passageways
available
dimensional
solid electrolytes
for the silver ions is lower than in any of the threediscussed
above.
O
O.
FIG.
phase in which
(CsH£NH)5AglSI23 ~ However, at room temperature, its average of 0 . 0 0 8 ~ i ( ~ c m ) - is an order of magnitude smaller than that of
(C5H=NH)Ag516~ Skarstad
decrease
The fractional
to be a higher than first order transition
low disorder
the silver ions can move more freely than before. there is a discontinuous
vacant.
are given by Geller and Owens
15. Plan view of the crystal -30 °C, after Geller
=I"
Ag+
structure
of (C5H5NH)Ag5I 6 at
(81)
Let us finally mention two general structural models. The first, introduced by Geller (69), cf. (134) and (137), has already been used in the present discussion. It is based on the idea that networks of passageways are formed by the face-sharing of anion polyhedra. This model appears to be very useful except in the case of b.c.c, anion structures. In ~-AgI, e.g., it predicts direct passages from one tetrahedral site to a neighbouring one, along one of
Agl-Type solid electrolytes
the
4110>
directions,
through the face shared by both anion tetrahedra.
m o t i o n along the
367
The
channels would be i n c o m p a t i b l e with this model. How-
these channels are formed by a l t e r n a t i n g anion t e t r a h e d r a and anion
o c t a h e d r a w h i c h do not share s i t u a t i o n is p a r t i c u l a r l y octahedral cf. (12))
voids
faces but in fact overlap. Evidently,
favourable
this
for the cationic m o t i o n as long as the
cannot be e x c l u d e d because of interionlc
repulsive
forces,
The second model i s due to van Gool (10,15) and van Gool and Bottelberghs (16) This model suggests
that the structure of a typical A g I - t y p e e l e c t r o l y t e
c o n s i s t s of domains
in w h i c h the cations occupy specific r e g u l a r and symme-
tric sets out of the more sites are o c c u p i e d
than e q u i v a l e n t n u m b e r of sites. D i f f e r e n t sets of
in d i f f e r e n t domains.
ries would therefore
A fast motion of the d o m a i n bounda-
imply a high cationic mobility.
It has been p r o p o s e d
that the domains o b s e r v e d f ~ b y G e l l e r in ~-RbAg4I 5 (69) should be i n t e r p r e t e d in terms of this model
tlO, 13~). In most cases,
however,
d o m a i n - m o d e l will not be easy to prove or disprove
the v a l i d i t y of the
experimentally.
V H E A T C A P A C I T I E S AND D I S O R D E R I N G PROCESSES
The
d i s o r d e r i n g process
solid e l e c t r o l y t e s capacities
and latent heats have been d e t e r m i n e d
Historically, interest,
leading to the high ionic c o n d u c t i v i t y of A g I - t y p e
is r e f l e c t e d by their t h e r m o d y n a m i c
)~
change of the anion arrangement, Thus the excess heat effects,
cations.
in a n u m b e r of cases.
phase t r a n s i t i o n there is only a slight from a tetragonal
a-Ag2HgI4,
to an f.c.c,
lattice
a t t r i b u t e d to the d i s o r d e r i n g of the
a c c o r d i n g to K e t e l a a r
(77) ,
the three cations per
unit cell are s t a t i s t i c a l l y d i s t r i b u t e d over the four z i n c b l e n d e - t y p e while in the ~-phase one vacant.
three of these voids are r e g u l a r l y occupied,
The heat c a p a c i t y Cp of Ag2HgI 4 n e a r its ~
is g i v e n in Fig. 40 to 50 °C, C
l Cp [ ~
400 /~g 2Hg
] •K
P
c o n t i n u o u s l y by a factor of approx.
I I i
14
t
Cp
/I,.,'
300, 200
12 R
!
I
100
t
324 K 300
310
320
330
T [K]
FIG.
)a
16. At first, w i t h i n the n a r r o w t e m p e r a t u r e increases
(139)
o b s e r v e d at and in the v i c i n i t y of the transi-
can be almost e x c l u s i v e l y
In
Heat
the first example was Ag2HgI 4 (94). It is of p a r t i c u l a r
because at the ~
tion point,
properties.
16. Heat c a p a c i t y of Ag2Hgl 4, after K e t e l a a r
(94).
voids,
leaving
phase t r a n s i t i o n interval
from
3, and finally
368
K. Funke
a latent heat is found at the first-order lambda-type order hence
behaviour
phase
of C
in alloys.
of the d i s o r d e r i n g formation,
ca.
second-order analogy
change is followed Moreover,
lattice becomes
From the relatively a-phase,
"domain becomes
structure",
in contrast
requires
the d i s p l a c e m e n t s
of these
anion structures
~-phase.
in
Nevertheless,
of the anions at
small amount of energy.
(2 meV) phonon branch
Long-wavelength
of the iodide
transition
by the following
a) B e l o w
the transition
observed
heat
capacity
point, exceeds
data by an amount ACp(T). disordering
This
in B-Agl,
ions necessary
within
a temperature
the linearly
which is (6.2 i 0.3) kJ/mole
heat capacity
17. The
(83) is given
in Fig.
range of ca. 50 °C,
extrapolated
lb~ )
to Perrott
the
low-temperature reflects
the integrated
an value of
to the latent heat at the phase
(b3). Thus,
of Agl had previously According
However,
in comparison
to ~-Ag2Hgl 4, one can assume a relatively
(84,8,85,86,87).
for their re-
in Fig.
The excess heat capacity ACp(T) e.g.
of this mode
features:
of the cation lattice.
is rather small,
phonons
(141) as illustrated
curve of Agl as obtained by N S l t i n g and Rein
It is characterized
crease
in
rather than a
a differentiation
and its b.c.c.
only a relatively
(142,38,40,141).
at the phase
~he
(140) that,
structure"
when the mean size of the domains
of a very low lying
can provide
transition
associated
constants.
B-phase
arrangement
A Cp(T)
on
to the lower one of two doubly degenerate E 2 modes observed
at the zone centre
initial
kJ/mole
(141) that in Agl the r e o r g a n i z a t i o n
is due to the existence
Cp(T)
heat effect of 6.0
meaningless
hexagonal
transition
there is some
of particles
disordered.
see section IV. Of course,
it can be estimated
corresponding
transition,
the arrangement
to Ag2Hgl 4, has two rather different
its w u r t z i t e - t y p e
part of the trans-
i0 ~ of the "forbidden"
in Ag2Hgl 4, it has been concluded
becomes
character
In view of the fact that the
by a first-order
as small as a few lattice
the phase
(140).
there should be a true "averaged
two possibilities
cooperative
and roughly
in both cases,
large overall
with the t r a n s f o r m a t i o n
the highly
At the end of the continuous are consumed
sites are occupied
to Cu3Au.
an f.o.c,
process.
1.2 kJ/mole
zincblende-type
Agl,
at 50.7 °C. The
is similar to that of second-order order-disP However, the increase of C with temperature, and P the decrease of the degree of long-range order, appears to be more changes
rapid in the case of Ag2Hgl 4, indicating
the
transition
small
in ~-Agl,
in contrast
degree of disorder
been determined
and Fletcher
(~7),
by various
and hence
authors
there is an in-
of C (T) in the a - p h a s e until 430 °C where an o r d e r - d i s o r d e r transP is reported. The authors claim that this behaviour is typical
formation
only for exactly metry result a-phase. reproduced
stoichiometric
in the "usual"
These results by others.
Agl and that small deviations
from stoichio-
C (T) curves which are relatively flat in the P which are not easy to understand (9) have not been
AgL-Type solid electrolytes
369
I¢
~ A
Ig+
a '~- -
FIG.
17.
-- -----"~---~
Q
l-
----~b
J
On the left: Ionic displacements associated with the lowenergy Q=0 E 2 mode in B-Agl. On the right: This mode is assumed to support the rearrangement of the anions at the transition from the wurtzite-type B-phase to the b.c.c. ~-phase of Agl. ~fter B~hrer and Br~esch (141)
I
70
I I I I
I Cp
Ag !
I
65
/,
[m .K ] 55
14 t Cp 1.3
6R
,.2
420 K 300
I)28E
500 T
1.1
700 [K]
•
FIG. 16. Heat capacity of Agl, after NUlting and Rein (03) a negligible temperature dependence of the defect-formation enthalpy. Denoting the formation([~thalpy of Frenkel pairs by Hf and the degree of disorder by a, we have d
ACp(T) = ~T (Hr. a (T)) : ~-T (Hf" ~o and t h e r e f o r e Hf in( ACp. T 2) = const.- 2RT
.e-Hf/2RT
)
(s) T
and
~(T) = HfI- f A C p ( T ' ) d T '
(9)
O N~'!ting and Rein thus obtained Hf = (if0 ± i0) kJ/mole and ~(147 °C) : 0.~[.i0-3 (O3). The disordered cations in B-AgI most probably occupy the octahedral interstices, causing a local contraction of the iodide lattice in the vicinity of each defect (14~). As a result, ~-AgI has a negative thermsl expansion coefficient with an increasing value as the ~ )a transition is approached (126,127). Furthermore, according to a theory by
370
K. Funke
Rice, Str~ssler,
and Toombs
(144)
the interaction of the cation defects
with the strain field they induce is considered primarily responsible phase transition b) Within the indicating
to the cation-disordered
a-phase,
for the
staten!
the heat capacity of AgI is essentially
that a further cation-disordering
constant,
which might give further contri-
butions to C
is no longer possible. The value of C exceeds 6R only by ca. P P 15 %, in fair agreement with the Neumann-Kopp rule. The slight decrease of Cp
just above the phase transition has been attributed residual short-range order
to the destruction Of
(91). This is particularly
destruction of short-range order in this temperature by the observation
that the "memory effect"
interesting,
since a
range is also implied
(67) in a-AgI vanishes near ca.
440 K, see section IV. The slight increase of C (T) at temperatures above P approx. 700 K is due to the formation of defects in the anio~ lattice. The concentration of anion-defects
remains however low up to the melting point;
this is confirmed by the results of I--tracer diffusion experiments (146"147'
95) Inspection of Cp(T) curves of various AgI-type
solid electrolytes
shows that
excess heat capacities due to an increasing degree of cation-disorder nerally found before the transition the highly conducting phases,
to the cation-disordered
the heat capacities
are ge-
state. Within
are either found to be
essentially
constant or to decrease. Decreasing values of C , with relatively constant slopes, are co-se rv ed in ~- A g2 S (92) ' B - Cu 2 S (92) ' ~ - CuBr (89,91) '
and a-CuI
(89,91),
see Figs.19 and 20.
The origin of the negative
coefficients of Cp is still unknown. Short-range order effects, can at most explain
I0 % of the decrease
(89)
N~iting,
Rein,
temperature
as in
}~I,
and Troe
have tentatively proposed an interpretation which is directly based on the high cationic mobility in the disordered phases. potential barriers in AgI-type solid electrolytes thermal energy,
In view of the fact that the are of the order of the
these authors conceive a gradual transition of the cationic
motion from the oscillatory
to the translational
state.
number of degrees of freedom and thus the heat capacity. analogous
This would lower the The treatment is
to that of a hindered rotator.
From the heat capacities of CuBr and CuI in their ~-phases
(90,148,149,89,91)
it is seen that B-CuBr is already structurally cation-disordered,
while B-CuI
is not. The increase of Cp(T) already observed in ~-CuI is continued in ~-CuI, indicating further disordering of the cation lattice, attained.
On the other hand,
until the a-phase
the heat capacity of ~-CuBr already displays the
typical decreasing behaviour,
which is also found in
a-CuI.
different model for the phase transition to the cation-disordered has been proposed by Huberman
site and a vacancy. However,
interaction is incorporated into the expression of the low-temperature
state
(145). This model is based on the interaction
between an ion on an intersticial energy
is
the way this
for the Helmholtz free
phase appears difficult
to justify.
Agl-Type solid electrolytes
1
/
150
Cp
Cu Br 125
mole. K
100
400
500
capacity
of CuBr,
3
:: I I
Cp
II If
I,
2,5
6R
I I I I
"
20
"
1.5
II
I
700 •
after N~iting,
I
T Cp
t
i t
i
600 T [K]
l~s. Heat
I
659 K -742 K-
50
FIG.
I
i
1
75
371
Rein,
and Troe
(59)
I
150 I
Cul
] ,K
Cp
125
2.5
6R
I
100
75
I i I I
I I
I •
2,0 1.5
681 6/,2 K
_
50 300
I I I I
500
700 T [K]
FIG. The heat
20. Heat capacities
ston, Wiedersich, a first-order Figs.
capacity
transition
experimental
Cv(T ) curve
served
about
above
silver
IV. Excess heat
capacities,
from the calculated
is taking place
two phase
this transition
(89)
are unknown
are probably
mined by Graham and Chang
~ from single-crystal (29)
Unfortunately,
from the fact the
for the
from X-ray data, elastic
the
of ~-RbAg4I 5 to
~ needed
to
of the
closely related,
coefficient
of the
are ob-
at 122 K seems
process
- apart
distortions
of Cv(T ) = C p ( T ) - T V ~ 2 / ~ were obtained
at 209 K, cf.
contributions,
disordering
by John-
transitions,
deviations
transformation
appear to represent
A l t h o u g h both structures
compressibility
i.e.
in the ~ - and ~-phases.
~)The molar volume V and the cubic expansion isothermal
lattice
50 K. The crystallographic
above and below
calculation
and Troe
at 122 K and one of the lambda-type
that the R b + - I - - f r a m e w o r k s lower symmetry.
Rein,
. RbAg415 undergoes
onto a more or less continuous
ions which
structures
after N~iting,
C (T~ ~nd C~T(T)~)of RbAg4I 5 have been determined
and Lindberg
6 and 21 and section
be s u p e r i m p o s e d
of Cul,
constants
and the deter-
3 72
K. Funke
~
~ t r a n s i t i o n largely affects
type phase transition,
the ionic conductivity.
there is no d i s c o n t i n u i t y of ~,
is not a usual o r d e r - d i s o r d e r
change.
J o h n s t o n et.al.
At the lambda-
i n d i c a t i n g that this (88) interpret the
209 K t r a n s i t i o n as the result of the interplay between changes in siteenergy differences, On the other hand, ordered domains
in site occupations, Pardee and Mahan
and concarrent lattice distortions.
(138) propose that the d i s i n t e g r a t i o n of
- domains w h i c h have p o s s i b l y been o b s e r v e d in B - R b A g 4 1 5 by
G e l l e r - might be r e s p o n s i b l e
for the transition.
similar to that in R b D g 4 1 5 p r o b a b l y exists
A h i g h e r - o r d e r transition
in (CsHsNH)AgsI 6 at ~2J K, of.
section IV. Above the lambda-point,
the heat capacity C V of R b A g 4 1 5 is found to decrease,
at least up to 310 K (88), while Cp(T) ature range. W i e d e r s i c h and J o h n s t o n heat capacity in this t e m p e r a t u r e
is r e l a t i v e l y constant in this temper-
(133) have compared the e x p e r i m e n t a l
range to the results of a model
calculation
w h i c h is based on the n o n - r a n d o m d i s t r i b u t i o n of the silver ions on the three nonequivalent
sets of t e t r a h e d r a l
to the q u a s i e h e m i e a l transformations,
sites in RbAg41 ~. A p p l y i n g a method similar
approximation,
used in the theory of o r d e r - d i s o r d e r
they obtain site energy d i f f e r e n c e s of ca. 0.027 and ca.
0.042 eV between the d i f f e r e n t types of silver sites and a mutual r e p u l s i o n e n e r g y b e t w e e n silver ions on n e a r e s t - n e i g h b o u r these parameters,
sites of ca. O . 0 J 5 eV. W i t h
the p r e d i c t i o n s of the c a l c u l a t i o n are in fair a g r e e m e n t
with the e x p e r i m e n t a l data. Thermodynamic etc.
properties
are very useful
a whole,
but,
like excess heat capacities,
for the study of the d i s o r d e r i n g process
of course•
in a system as
digression,
Appropriate
(42) and
(37)
#
(41,42,j7,150)
I
Rice and Roth
(41) have pre-
1.5
!
Cv
Cv I
30 R
! Cv,exP,
250
Cv , Debye
125
0.5
100
200 T
FIG.
21. Heat
[K]
300 D-
capacity C~7 of RbAg;~l 5, 8fter Johnston,
and L i n d b e r g ($~}.
have been
and theoreti-
For a brief d i s c u s s i o n of mo-
see sections VII and VIII.
375
we briefly
experiments
w h i c h will be d e s c r i b e d in the f o l l o w i n g sections,
cal models have been p r o p o s e d dels
state. As a
the efforts made in this direction.
undertaken,
entropies
cannot give any i n f o r m a t i o n about the details of
cationic motion in the c a t i o n - d i s o r d e r e d mention
latent heats,
Wiedersich,
Agl-Type solid electrolytes
sented
a free-ion-like
basic
idea
lixe
state,
~ecording where
consists
states.
energy
to the theory,
M is the
ionic
However,
conventional
hopping
free-ion-like
model,
theory
which
evident with
so that
motion
correlated
in terms
there
the
ion itself
electrostatic a diffusion the
same
Thus also
are
each those
did not
coefficient
n is the n u m o e r side
D~
of the
be derived
from a
no need
has
we will
steps~{
of" the
local
These
comparable
exist,
D ~ would
density
of e q u a t i o n
In the absence
ease
of cubic
Einstein
relation
for invoking
a
Each
to be r e g a r d e d
describe single
potentials
changes,
to the average only
influence
the c o n d u c t i v i t y ~
be simply
connected
2 ' nq and q the charge
(4) is also
9 and
structures
~ denote
the
called
of correlations,
this
diffusion seen
(i) by
mainly
due
energy
needed
the next
to the for
step of
and
the tracer-
by the N e r n s t - E i n s t e i n
:
jump
14) of the mobile the charge
we would
(152) , both D ~ and D~
D~ where
predictions
~ kB T
right-hand
the
states,
of these
of its neighbours.
D~
cient,
those
electrolytes
diffusion
step will not
equation
where
energy
by Vm=(2Em/M)9/2t
free-ion-lixe
can also
following,
(ii) by its neighbours.
correlations
-diffusion
in a free-ion-
threshold
the life-times
is a c t u a l l y
solid
in a change
interactions,
step.
of the
all
are
The
DIFFUSION
In the
unique
result
and
ion but
If these
in A g l - t y p e
process.
of single,
step of an ion will
ions
then given
contains
that
electrolytes.
(151)
of the cations
as a h i g h l y
are
experiment
Vl TRACER
The m o t i o n
the mobile
the o c c u p a t i o n s
become
in agreement
solid
a characteristic
velocitzes
is postulated,
it has
~re
For
conducting
that
~m exceeds
the ionic
mass.
equation
which
for highly
of the a s s u m p t i o n
if their
a Boltzmann
theory
model
373
o
:
ions.
diffusion
thus have would
and
coeffi-
D ~ = D~
be given
t2
frequency
The
In
by the
(5)
the
(fixed)
jump
length
of the
ions. On the other
hand,
one
electrolytes.
The
ratio
Possible H H will
generally D~/D6
D ~ % D~
come
correlation
effects
w h i c h might
be d i s c u s s e d
in this
section.
H R is composed
of two d i f f e r e n t
associated
wlth D ~ and D~
The
diffusion
tracer
~;trong
finds
has
{'vidence
and q u a s i e l a z t [ c
in h i g h l y
to be called explain
First,
correlation
conducting
the Haven
the e x p e r i m e n t a l
however, factors,
we will f and
fl'
solid
ratio
HR • values of
poznt
out
which
that
are
, respectively.
coefficient
for this view
of a cubic
has
neutron-scattering
solid
been obtained experiments,
electrolyte
from recent
can
oe w r l t t e n
microwave-
see sections
VII
and
IX.
374
K. Funke
The fact that the directions correlated
only if these
correlations
the directions other.
of two
do not exist. mobile
may also be imagined
For example,
particles,
moving
f; f equals unity
f is smaller
result
the efficiency
of the jumps in c o n t r i b u t i n g
In formulating
D~ in terms of ~ and
~
than
opposite
1 if
to each
in a crystallographic
to jump over several
This process would
between
factor
jumps are p r e f e r e n t i a l l y
length {(153)
which correlations
jumps of the tracer ion are
by the correlation
consecutive
On the other hand,
channel,
of consecutive
is taken into account
elementary
distances
of
in f > i. The factor f thus gives to D ~.
we first
consider
the jumps of different
a cubic crystal
ions do not exist.
in
We then
find
There
is no correlation
concerned
~2.
(7)
factor f in equation
(7), because we are now not
with the directions
of consecutive
rather with the net motion of charge process
as long as the jumps of different
(102,154).
Of course,
the case of Agl-type
as pointed
The meaning
2
of the " c o n d u c t i v i t y
demonstrated is correlated
in Fig.
22: e.g.
the Haven ratio
These authors strated
in p a r t i c u l a r resulting
" fI "
fI (io3,154,155)
factor"
= f/fI
method"
and equation (9)
However,
•
fl
= 0
• ;
-0
• fl
processes
are influenced
are a priori
!
-0
• fl
=
2
between different factor"
ions, and the
fI"
to n e a r e s t - n e i g h b o u r
in the S a t o - K i k u c h i
the theory predicts equivalent.
by
D~/Ds no longer depends
-0
= 1
correlation
not considered
of
They demon-
is obtained.
-0
O"
(155)
for the calculation
jump frequencies
interactions.
of the r e s t r i c t i o n
(103). Instead,
carrier
in the opposite
interactions.
like those shown on the left- and right-hand
unfortunately
is
(9)
only n e a r e s t - n e i g h b o u r
how the actual
22. C o r r e l a t i o n
As a consequence
by
(8)
the " p a t h - p r o b a b i l i t y
"conductivity
cf.
(7) is replaced
(9) has first been used by Sato and Kikuchi
•
-sites
this premise does not hold in
fI is 0 or 2 if each jump of a charge
from these
on jump frequencies,
processes
are u n c o r r e l a t e d
respectively.
considering
effects
but
is given by
equation
applied
D M, D@, and H H,
FIG.
carriers
and equation
correlation
HR = D~/D" In this n o t a t i o n
ion,
this is a random
with a jump of another one which is performed
or the same direction, Thus,
4
However
charge
out above,
solid electrolytes, D~ = ~
jumps of one specific
carriers.
interactions,
side of Fig.
fI = i if all available
An e x t e n s i o n
22 are
theory in its present
of the theory,
state,
cation
which should
Agl-Type solid electrolytes
,o'o
~
T [°C }~ 3o0
200
[.f~'] 8
500-~o
~oo
"-Agl
375
T [% I
1O91oD
250
200 i
' ~
l "~'2s
"" 300
T
.
IO5D
X
(o)..
2
.~
HR
o
HR
HR
L4
HR 4
o (k,t) • (j,p)
l 2t
I
2.&
I
2.0
1.6
0.2
I
1.2
i
2.2
•"qP~--1031T [ K "1]
25
I06D
~
T I°C] ~ 50 100
150
[~']
I
2.0
Iog.~)
1.8
1031T { K "11
T[oCl
300
150
-5.0 T -4.25
200
!
!
al'Ag2Se
(o
10s D
4
-5.25 -/,.5 2 1
-5.0
":~
2
0.4 HR
01
~2 I
32
i
3.0 2.6 1031T [K4I
FIG. 2). Tracer diffusion
2'.,~ ~ coefficient
I
20 ll0"lfTIK41
D ~, charge diffusion
and Haven ratio H R for ~-AgI, a-RbAg4Is, (j,p) : Jordan and Pochon (95) (k,t) (o)
: Kvist and T~rneberg : 0kazaki (54)
(b,m) (a,m)
: Bartkowicz and Mrowec : Allen and Moore (98)
(o,a) (b)
: Owens and Argue : Bentle (97)
(46)
~
(96) (~2)
,'8 coeffi2ient
~-Ag2S , and a-Ag2ge.
D~
376
K. Funke
allow for a proper treatment of the most relevant might become relatively better understanding
involved.
of diffusion
correlation
processes,
However it would be of much value for a and conductivity
data of highly disordered
solid electrolytes. Tracer diffusion coefficients of the mobile cations ~ h a v e been experimentally determined in a-Agl (95,96), ~_Ag2S ( 9 8 , 5 4 ) ~_Ag2S e (54), and a_RbAg415
(97). The data are presented
and the resulting smaller
than one. However different
~HR/~T is slightly and negative
positive
in
temperature
~-Ag2S
and
dependences
~-Ag2Se,
are observed:
negligible
in
a-Agl,
in a-RbAg415.
Can one decide whether consequence
in Fig. 2~ along with those of D e
values of H R . In all of these cases, H R is found to be
the experimental
result HR< i is essentially
of f < l or of fl>l or of both inequalities?
possibilities
have already been proposed
a
All of these three
in the particular
case of
a-Ag2S:
(i) Rickert
(ll) pointed out that,
in a-Ag2S,
an octahedral
site plus the neigh-
bouring Strock sites can at most be occupied by one silver ion at a time. He therefore
introduced
the concept of six octahedral
available
for the four silver ions per unit cell,
In this view, most voids are occupied, scribe
the cation motion
estimate
in ~-Ag2S
Yokota
regions"
cf. Fig. 35 in section IX.
and hence Rickert proposed
to de-
in terms of a vacancy mechanism.
gave f ~ 0.5, not too different
(ii) On the other hand,
"elementary
from the experimental
(157) and 0kazaki
(54) by putting H R = i/f I. They proposed
(54) tried to explain
a cooperative
ions along a line, known as the "caterpillar
A rough
H R values.
their data
motion of several
mechanism",
see Fig. 24. It is
assumed
that an ion on a site is able to jump not only into a vacant neigh-
bouring
site,
but also into an occupied one,
inducing
site to perform a Jump in the same direction. one of the cations cations -AgI,
the
< i00>
fI
occupied find r ~ 8
that a void is occupied.
in a-Ag2S
In
The ratio of the jump frequencies and r ~ 1 6
if it is assumed
paths for expression
g-Ag2S,
p is rough-
into vacant and into
(10) with H R : i/f I, one would
(54). Smaller values of r are ob-
that the directions
identical
For tracer diffusion
in a-Ag2Se
and
(I0)
sites is denoted by r. From equation
are not always
the following
(r'I)(I-P2)+(I+p)(I-p)-I (r-l)(l-p)~+l
see Rickert.
until
~-Ag2Se
as appropriate
For fI' Yokota derived
p is the probability
ly 2/3,
tained
:
as well as in
channels have to be regarded
mechanism.
continues
to Jump into. The average number of
involved will then equal fI" In m-Ag2S,
the caterpillar (157).
Here,
finds a vacancy
the ion on the latter
This process
of inducing
jump and induced
(54)
coefficients
in the case of ~-Agl and
of the anions,
(156) in the case of
see e.g.
a-Cul.
(146,147,95)
jump
Agl-Type solid electrolytes
FIG.
24.
377
Caterpillar mechanism.
(iii) Bartkowicz
and Mrowec
HR= f/fI in
m-Ag2S
(52) have proposed to interpret the Haven ratio
by the collinear interstitialcy mechanism.
ment is based on a comparison of the H~ values found in obtained from calculations
for AgCI
(lwS)
process
g-Ag2S with those
but the structural differences
between ~-Ag2S and AgCI are not taken into account. interstitialcy
Their argu-
Of course,
the collinear
can be regarded as a particular kind of caterpillar
mechanism. It is seen from (i), physical reasons
(ii),
(iii) that, in
g-Ag2S and in
a-Ag2Se,
there are
for f < 1 as well as for fi>i. Thus the observed values of
H R are probably due to both effects.
Certainly,
the caterpillar mechanism is
not the only conceivable process yielding fi) l: jumps into vacant sites might also induce
Jumps of neighbouring
ions in the same or at least in a
positively correlated direction. In g-AgI and
~-RbAg4I 5, the number density of the mobile cations is much
lower than in the chalcogenide inappropriate
phases.
for the description of the ionic motion.
appears to be much less important, proximation.
A vacancy mechanism is therefore Hence the role of f
and HR~I/f I might be a reasonable
ap-
The caterpillar mechanism has been applied to g-AgI by Okazaki
(54). Assuming r = 1 or r = 10, one finds i/f I = 0.5 or i/f I = 2/3, ively. The actual value of H R is ca. 0.6 (95,
respect-
96).
Vll COMPLEX CONDUCTIVITY IN THE MICROWAVE RANGE We now return to our introductory questions
concerning
the microscopic
aspects of the diffusive motion of the cations in AgI-type One of the principal questions motion is continuous
is the following.
solid electrolytes
Can we decide whether this
or discontinuous on scales of
~ 10-1~ and
~i0 -8 cm?
The two extreme possibilities are presented on the left- and right-hand of Fig. 25. The continuous simple monoatomic
liquid.
They obey the laws of simple diffusion,
sented by the Langevin-equation given above
(39,159).
sides
type of motion is displayed by the atoms in a as repre-
and Fick's second law, down to the limits
On the other hand, hydrogen in certain hydrogen-metal
systems as H-Pd and H-Nb definitely perform~ a jump diffusion from one well -defined site to another mean time of flight, (160) TO Atoms in liquids,
(39)
In this case it is generally assumed that the
Tl, is negligible
compared to the mean residence
hydrogen in hydrogen-metal
solid electrolytes have comparable
systems,
in these systems are quite
from each other. From the present microwave,
and quasielastic neutron-scattering
and cations in AgI-type
coefficients of self-dlffusion of the or-
der of lO-ScmRs -I. Yet the diffusion mechanisms different
time,
data,
far-infrared,
Raman,
it is concluded that the diffusion
378
K. Funke
in A g I - t y p e simple
solid e l e c t r o l y t e s
is in a way i n t e r m e d i a t e between that in
liquids and that in h y d r o g e n - m e t a l
shown in the f o l l o w i n g sections, k i n d of motion,
c h a r a c t e r i z e d by large a m p l i t u d e s
to be a good candidate
for a first a p p r o a c h .
in c o n t r a s t to h y d r o g e n - m e t a l be c o m p a r a b l e
systems,
systems,
in A g I - t y p e
seems
systems, ~i" seem to
~ . o were i n i t i a t e d by the follow-
If a j u m p - d i f f u s i o n model is valid in the highly cation-
c o n d u c t i n g solid electrolytes,
6o(~ ° +~)
~e can use the e q u a t i o n
~2,
=
where 4 is the m e a n jump distance. that the c h a r a c t e r i s t i c
(i~1
Inserting
for ~ values of a few ~, we find
times ~o" ~i might be of the order of 10-11s.
a d i s p e r s i o n of the e l e c t r i c a l
served in the m i c r o w a v e range. a long time before various
and d i e l e c t r i c
properties
In this
should be ob-
This was pointed out by W. Jost in 1967
t h e o r e t i c a l models
cerning the m o t i o n of the cations
Interestingly,
the o b s e r v e d m i c r o w a v e data.
the i n f o r m a t i o n d e r i v e d from these data on a time scale of
should be r e g a r d e d as s i g n i f i c a n t
(161),
(41,42,37) were d e v e l o p e d con-
in these systems.
these models has been able to predict words,
and strong damping,
However,
The m i c r o w a v e m e a s u r e m e n t s now to be discussed,
case,
25. As will be
the mean times of flight,
to the mean r e s i d e n c e times,
ing consideration.
see Fig.
a s u p e r p o s i t i o n of j u m p - d i f f u s i o n and a local
none of In other ~ i 0 -II s
for a proper d e s c r i p t i o n of the actual
diffusion mechanism.
Systems
with
simple monoatomic liquids
I,
~
25.
hydrogen in metals, e. g. H - P d , H-Nb
I
local motion ~- jump diffusion "Co ~ "I~1 "~ 10-11s
I
jump diffusion "~1 <<~'o ~ 10-11 s
D i f f u s i o n in d i f f e r e n t systems.
In the m i c r o w a v e experiments, appropriate
o<- Ag 1 c<- Cu | - Cu Br etc.
I
simple diffusion To= 0
FIG.
D P- 10-5 cm 2 s -1
a special technique
(162) was used which is
for the m e a s u r e m e n t of h i g h - l o s s materials.
under investigation
- s e c t i o n is shown in Fig. d u c t i n g waveguide,
The solid e l e c t r o l y t e
forms the s l d e - w a l l s of a r e c t a n g u l a r waveguide.
Fig.
26b. In c o m p a r i s o n to a normal,
26a,
two effects
can be observed.
A cross
n e a r l y ideally conFirstly,
a
Agl-Type solid electrolytes
TEl0-wave~)propagating the side-walls, these data,
i~o~
are calculated
tions of the w a v e - e q u a t i o n s 26,
of losses
£'- i~" and the complex
the ~aveguide
in
From
conduc-
are given along with solutions
component E . At low frequencies, Y short-circuit technique was used.
electrolytes
have been performed
with different
(b.c.c.),
a-CuI
continuous
structures
(f.c.c.),
(32,33,34)
(162)
for the non
i.e. at 2 - ~ GHz,
on three A g l - t y p e
of their anion
and B-CuBr
solu-
and within the walls
field
So far, m e a s u r e m e n t s a-AgI
E=
because
change of wavelength.
by means of the proper
within
the w a v e - e q u a t i o n s
zero electrical a coaxial
is attenuated
is a small
A
~=
In Fig.
there
the complex p e r m i t t i v i t y
A
tivity
in the waveguide
and secondly,
379
lattices,
(h.c.p.).
solid
namely
on
The results,
A
~(W)
= Re ~ (V) and
£'(~) = Re
~ (w),
W=
~/2~,
are plotted
in Figs.
27,2b,
and 29. In the case of m-AgI,
the far-infrared
spectrum
is also included
the figure,
The curves drawn
for ~(V)
and
to the criterion
of the K r a m e r s - K r o n i g
mutually
see section VIII.
consistent
according
y
in
£' (v) are rela-
Y
t,x
px
o FIG.
b
26. TEIo waves
in r e c t a n g u l a r
a) perfectly
conducting
b) two side-walls tions
(169). As a common
increasing
frequency
low frequencies, @ Correspondingly, should become tive values of exist
feature,
~A TElo-wave field vector
the @ ( 9 ) - s p e c t r a
to approach
according
constant
and negative
values of
is defined
the numbers
Fig.
26, are
exhibit
in
Recently,
~-Cul
and Taylor
properties:
half-waves
i and 0, respectively.
value
At
~(0).
relations,
limit.
and in $-CuBr,
Armstrong
to the direction
with
in the 20 - p0 GHz range.
a' in the case of ~-RbAg415,
of standing
dielectric.
a decrease
low-frequency
in the low-frequency
by the following
is transverse
(ii)
maximum
its known
conducting
to the data and the K r a m e r s - K r o n i g
~' have been observed
negative
from highly
and an additional
seems
in the 2 - 8 GHz range.
reported 10 8 Hz.
made
waveguides,
metal wails;
E'
Indeed nega-
where data (164) have also
at frequencies
below
(i) the electrical
of p r o p a g a t i o n
of the wave;
in the x- and y-directions,
see
380
K. Funke
In search of an explanation of the decrease disregard
the conductivity maximum.
much like the
~(~)-curve
in conductivity let us first
The remaining part of the spectrum looks
predicted by the Drude model
has actually been proposed by Huberman and Sen by Armstrong and Taylor ~ois
charge,
definded by
(42,166)
fractionmof
~(~o)
= 6(0)/2,
moving ions to be
g(0)'oJo" q-2n-l.
m and q are the ionic mass and
and n is the number density of the cations.
the result would be
and~more tentatively,
(164). Applying the Drude model as a first attempt,
we would find the momentary Here,
(165), and this model
~ ~ 2"i0 -3. On the other hand,
In the case of B-CuBr, the Drude model would
give a mean time of flight T l ~ 2 / ~ o, the • sign being valid if there is friction during each individual
flight. However,
such friction has to be ex-
pected because of the fluctuating local potentials due to the motions of the neighbouring From and
~=~/(~ ~
ions. Thus we would find ~l ~ 4 • 10-11s in the case of B-CuBr. 2 + ~ ) and 6D (~o + ~i ) ~ ~ we would then infer "Co • 2"10-Ss
200 ~ which appears to be quite unrealistic
for an AgI-type
solid
electrolyte.
9 l'~m-' ] 6 [(ncm)-' ]
1
10
!
I
100 I
II II
3
® (2SO°C)~; 2
~,(o) ---.~ 1 I
6 (25 °C~ I
10
R~9 i;10 I011 1012
FIG. 27. Conductivity ~ of
1013
g-AgI at 250 °C and of B-AgI at
25 °C in a broad frequency range. The difficulties
in applying the Drude model are at once resolved if we in-
troduce one simple additional jump-lengths
assumption
(34,99).
It is assumed that the
of the ions are fixed, being given by the geometry of the anion
lattice and not being affected by the electrical
field.
In this case,
the
effect of acceleration or retardation of the moving ions by the electrical field changes their times of flight, but not their jump rates. cannot give any contribution
to
explain the observed conductivity
~(0).
However,
Therefore
as will be shown,
it
it can
peaks at 20 - 30 GHz. This effect will be
called the "acceleration effect". Assuming fixed Jump lengths, ty and its decrease
we have to explain the low-frequency conductivi-
in the microwave-range
by another model.
This can be done
by a simple consideration which is familiar in the case of "normal" crystals:
the electrical
ionic
field slightly supports or hinders the initiation
Agl-Type solid electrolytes
of jumps in or against a constant value of
the instantaneously
~(~)
including the "start effect"
dependence
~(~)
will now be qualitaE'(0) will be given.
according to our physical model,
and the "acceleration effect".
Finally the
from this model will be compared to the observed data.
V[GHz] 2 4
10 20 40 I ~-CuBr,/,10
e [(~cm) "1]
e (o)
@(w)
thus causing
limit. The consequence of
and a simple relation between ~l and
We will then sketch a calculation of predictions
preferred direction,
in the low-frequency
this "start effect" with regard to the tively described,
381
I °C
v [GHz ] 0.2 0.4 1 2 4 10 2040 100 1 I t [ 1 t I I TI I - - - ~.-Cu 1,450 ° C ~ 1 . 2
IL
-
+
4
0.5
1
1
0
0
100
~
t
0 ¢'
O
~
E'
-100 - 200 '
-2001~[-~E~[~ 2 4 10 20 40 v[GHz] -'= FIG.
450 °C F
0.2 (14
I 1 2 4
-300
I
l 10 2 0 4 0
v [GHz] FIG. 29. C o n d u c t i v i t y ~
2~. C o n d u c t i v i t y ~ and permittivi-
mittivity
ty E' of ~-CuBr at 410 °C.
and per-
E' of ~-CuI
at 450 °C. The low-frequency
limit may be defined by ~i ~
T = 2~/~. In this limit the
phase of the field is practically constant while an ion is performing a j u m p When the frequency is increased the following situation will occur. A cation which has started
for an "extra"
jump with the help of the electrical
field
is still on its way while the field is already changing its sign. After this change of sign the ion will move against and will give a negative
contribution
will result in a decrease of ~.
the momentarily preferred direction
to the overall
In the high-frequency
conductivity, limit
~ ~
which
T, the
field will change its sign many times during the flight of an ion and the overall
conductivity will be zero.
An interesting aspect of this can be demonstrated
in a complex plane rota-
382
K. Funke
ting at the angular
frequency ~ of the applied
Fig. 30, let E be fixed in the vertical phase difference
betweer
field.
direction.
to Fig. 30, this phase difference
given by (~/2 + ~) where 2 ~ / 2 ~ = ~I/T = ~ low-frequency limit, we have ~ ~ 2 ~ and
~ ~"T
m E'
i.e.
~= ~ / 2 .
to the actual
value of ~i from
G(0) ~ 1 . 5 5 ( ~ c m )
For the calculation (~) = q
of "
E' (0). Inserting -I and ~^( ~ )
~(0)
and
g'(0),
effect"
the data from 8-CuBr,
will also Fig.
(34,99) , we start with the equation (13)
,
flux in the direction
of the electric
field E e -i~t. o let us introduce
In order to formulate the problem in only one dimension, a the flux IID which we would expect, if all jumps were being performed directly direction
in or against
the field direction.
and the jump direction
If in three dimensions
of an ion form an a n g l e ~ ,
are proportional
=
I ^ID=
to c o s ~
~ID/3
phase angle
the field
of this motion
we obtain (14)
.
coc . E.
. Therefore
either
then the influence
of the field on the motion of the ion as well as the component in the field direction
28,
~' (0) ~ -225, we find %'1 = 2"5"i0-iis"
A I(w,t)/ (Eoe_i~t)
where I is the complex
is
In the
(12)
it has to be kept in mind that the "acceleration
contribute namely
/2~,
E'(o)
We can thus obtain a first approximate although
We are now asking for the
E and D = go.~.E in the case of a jump diffusion
with a mean time of flight ~l" According
AW
In this phase diagram,
/
/
/
~c= ~"> O"Z, I
I
FIG. 30. Complex phase diagram,
demonstrating
the phase difference
tween E and D = Eo ~ ~ in the case of jump-dlffusion. 2W corresponds
be-
The angle
to the period T, and the angle 2~ corresponds
the mean duration
of a Jump,
~l
"
to
Agl-Type
In our one dimensional
solid electrolytes
calculation,
densities of ions starting
jumps
in the time
Secondly,
interval
dt'.
v+(t',t)
of its
flight,
or the negative
for its jump of fixed
let us consider
are the number respectively,
value
at time t has the positive
v (t',t).
~ is
Thlrdlj,
"g+(t') or
T
with-
an ion which has started
on its way at time t > t'. Depending
its velocity
length
and n_(t')dt'
in the + and - direction,
a jump st time t' and is still direction
h+(t')dt'
383
on the
value
the time thls ion needs (t'),
respectively.
Then
the
A
flux IID is given by
t ^
=
lid
J
fl (t')v+(t',t)@(t'+~+(t')-t)dt'
(IX)
t +
J
~_(t')v_(t',t)~(t'+Z'_(t')-t)dt'
~ith
e(x)
:
for x ~ 0
The recuisite
~±(t') where
field.
start
= ~o± ~ f i ( t ' )
rates and velocities
and v+ ( t ' , t )
no and ± V o ( t ' , t Here,
simplifying
: ± Vo(t',t
) denote
their
Vo(t',t ) is considered assumption
From equation
(16),
is made
~(t')
jump-length
~ . Considering t
IiD
in t h e
that one function
a r ~ expressed
: 2
values
) + ~v(t',t),
as a function
~+(t') and v±(t',t)
jump-times
are ~ritten
only
absence only of
(!6) o f an a p p l i e d (t-t'),
Vo(t-t' ) holds
are inserted
by v±(t',t)
under
first-order
terms,
and the
for all
into equation
the constraint
jumps.
(15).
The
of fixed
we then obtain
from
(15):
a~(t')Vo(t',t)dt' t-T I (i7) t
t ~v(t_~l , t")dt" }
t- ~'1
t-~- 1
A
:
In equation
(liD)start
(17),
the first
+
(~ID) accel.
term involves
"start effect", while the second the "acceleration effect". For the calculation proportional
of the "start
to the electric
~ [ and therefore
term involves
effect",
~v
describes
and therefore
the
describes
it is supposed that ~ ( t ' ) is • ~ e -i~t' field at time t' i.e. ~o , and, for sim-
384
K. Funke
plicity,
that the cations
Introduction of
have a constant
of different
~start(~)
functions
more,
influence
occur in a correlated
s of these two jumps,
In order to calculate
to the field at time t,
friction-like (negative)
= 0, A x ( t ' • t ' ) effect
the arrangement The d e r i v a t i o n
is obtained
Av(t',t')
of rapidly
changing
microwave
(34). The curves
spectra
of
to formulate
the Haven
can assume H R ~
for
the change
in position
~-AgI
(i~],__,
smoothed
for the
and V" is the
potential
^ ~(~)
mainly due to
27
straightforward (microwave
They closely
TI, determines
the position
The location
to be ca. 0.6,
similar
This result
on
The height
of the c o n d u c t i v i t y
TI'
decrease
~-AgI
s = I/4 have been
and B-CuBr.
to that of ~-CuBr,
the decrease
of
as an indication
jumps occur in a correlated
manner
~-CuI.
T 1 / ( T ° + TI).
peak depends
on the f r i c t i o n - t e r m
Therefore,
~o is roughly
the value of b cannot i0 ps (14),
be fixed,
see part IX. Then,
b as well as
instant of time, unless
~o is known.
b becomes
15 ps -1 and V"• which e s s e n t i a l l y
determines
high-frequency
to be ca. -15 meV/~ 2. In ~-CuBr,
side,
is estimated
be similar or somewhat
larger.
Depending
on
the shape of the ~o'
curves are possible
these
b and V" are of the same order of magnitude
given
for
with different
various
experimental
jump distances
and ca. one unit-cell
values
close
approx.
~-peak
on its
~o might
fits to the
of b and V".
In all of
as the corresponding
a-AgI.
The above values of ZI' together with residence imply
In ~-CuI,
should be checked by a t r a c e r - d i f f u s l o n
on the fraction of cations w h i c h are in flight at a given
parameters
of
fI' and s. In
see section VI, and we
This might be u n d e r s t o o d
that there even more than two s u c c e e d i n g
of the conducti-
of the
by the parameters
of the spectra of both
experiment
part) and in
fit the experi-
15 ps in the case of
fI = 1/0.6 and a time overlap
lower frequencies.
and
and ~-CuBr.
ratio H R is known
i/f I. Indeed•
in the same direction.
is now
scale and is found to be ca.
T1 should have a value
cases,
due
from
shown in Fig.
scale is d e t e r m i n e d
for the calculation
g-AgI,
the
of a moving ion.
= 0. The term b allows
and ca. 25 ps in the case of 8-CuBr.
on the frequency
is at somewhat
we have
and velocity
from this equation.
of the model,
vity peak on the frequency
~-AgI,
section VI. The
paramete~
local potentials,
of an overall
28 have been calculated
~-AgI
=
jumps of two
cf.
= qEoe-i~t
of the final equation
The main parameter
In
curves
of the anions.
is given elsewhere mental
effect",
a jump at time t'
Ax(t',t),
second derivative
manner,
< s < i, is another
field on the position
mA~(t' • t ) + m b A £ ( t ' , t ) + V " ' a x ( t ' , t ) with A x ( t ' , t ' )
0
the " a c c e l e r a t i o n
of the applied
For an ion which has started
where
the resulting
for the fact that to a certain extent
ions in the same d i r e c t i o n
used
Vo(t-t' ) can modify
their flight.
and
~e have allowed
time overlap
6
v o during
E' start (~) ' but cannot change their general shapes. F u r t h e r by i n t r o d u c t i o n of the " c o n d u c t i v i t y correlation factor" fI as a para-
meter,
Fig.
velocity
of a p p r o x i m a t e l y c-axis,
~6.7
times of roughly
one lattice
~, in ~-CuBr.
constant,
~5
10 ps, would ~, in
~-AgI
These rather large values
Agl-Type solid electrolytes
would favour the m o t i o n along channels would then move along the dral sites.
In B-CuBr,
in both cases.
tunnels
In ~-Agl,
!
local motion which and 36. This is
tron-scattering permit
it is c o n c l u d e d
is s u p e r i m p o s e d
ioc91 motion,
Lnde~d o b s e r v e d
from V"
•
their jump di?f'asion,
Raman scattering,
25
and q u a s i e l a s t i c neu-
The term b itself is p o s s i b l y net too large
to
from site to site. b a r r i e r s b e t w e e n these sites may be roughly e s t i m a t e d
They are found to be close to the thermal energy,
the formal a c t i v a t i o n e n t h a l p i e s derived meter
cf. Figs.
w h i c h has been d e s c r i b e d by the f r i c t i o n - t e r m b,
experiments.
The a v e r a g e d potential
from one p o s i t i o n
that the cations perform an 9ddit~cnal
onto
in far-infrared,
s cetion to move
sites
l0 larger than those one would ex-
pect if the ions moved with their thermel v e l o c i t y d i r e c t l y Therefore
and oct~he-
the o c t ~ h e d r a l
d i f f u s i o n paths.
T~ are by a factor of approx,
to another.
the ions
connecting tetrahedral
the channels w h i c h i n t e r c o n n e c t
would be the a p p r o p r i a t e The times
385
from L r r h e n i u s
is the time o v e r l a p s of two successive
in agree~nent
plots.
correlated
~
W ~ n
The last para-
jumps, whose value
of i/4 might a p p r o x i m a t e l y have been expected• It is n-)t easy to e s t i m a t e
the influence of Lorentz
They have not been c o n s i d e r e d
in this calculation.
local
field effects.
Nor has any statistics
been i n t r o d u c e d w h i c h might have allowed for a d i s t r i b u t i o n of jump times. In this model
the jump d i f f u s i o n has been treated in a s i m i l a r l y simple man-
ner as in the G i s s l e r - S t u m p model
(167), which is used for the i n t e r p r e t a t i o n
of q u a s i e l a s t i c n e u t r o n - s c a t t e r i n g
results,
see section IX.
VIII F A R - I N F R A R E D AND R A M A N - S C A T T E R I N G EXPERIFHENTS
It is a common feature of A g l - t y p e
solid e l e c t r o l y t e s
Debye-Waller exponents
of the mobile
and n e u t r o n s t r u c t u r a l
refinements,
u s u a l l y high a m p l i t u d e s not only w i t h i n
the cations
section IV. They correspond
p r o v i d e d by the anions,
but also during
in these materials.
context,
it is once more stressed that the i n t e r a c t i o n s of a
tentatively describe
that of h i g h l y damped,
of
section.
cause
fluctuations of its local potential
of the o r d e r of the thermal energy and on a time scale of 10-12s.
view seems
flight from
This had to be taken into account by the
cation w i t h its m o v i n g n e i g h b o u r s
one might
to un-
However,
some local kind of m o t i o n seems to be c h a r a c t e r i s t i c
i n t r o d u c t i o n of the f r i c t i o n - t e r m b in the p r e c e d i n g In the present
large
cations are f o r m a l l y d e r i v e d from X - r a y of.
of a local m o t i o n w i t h i n flat potentials•
the voids
one void to another,
that u n u s u a l l y
Therefore,
the s h o r t - t i m e b e h a v i o u r of the cations as
more or less s t o c h a s t i c a l l y driven oscillators.
This
to be c o n f i r m e d by the f a r - i n f r a r e d and R a m a n - s c a t t e r i n g experi-
ments so far p e r f o r m e d on A g l - t y p e By the use of F o u r i e r - s p e c t r o s c o p y , far been studied in
solid electrolytes.
~-Agl (32'37) and
exist in the case of B-CuBr
A
the f a r - i n f r a r e d d i s p e r s i o n of [ has so ~_RbAg415
(36). Br~esch,
(35). P r e l i m i n a r y results
Str~ssler,
m i n e d the r e f l e c t i v i t y of single crystals of
~-Agl,
and Z e l l e r while
(37) deter~
t r a n s m i t t a n c e as
386
K. Funke
well as reflectivity measurements by the other authors. either a Michelson
on polycrystalline
The spectrometers
interferometer
grating interferometer able at frequencies
(Beckman FS 720)
(Beckmann LR i00)
below ca.
from
m-AgI is almost opaque in the broad frequency range 1012 Hz,
the reflectivity increases
frequency until the limit of the experimentally accessible
frequency range is attained at ca. 2 " 1 0 1 1 H z . by (97) and
or a lamellar
the latter being prefer-
samples of g-AgI are presented in
% 2 " i 0 II Hz to 5"1012 Hz. Below ca.
with decreasing
(32,35-]7)
(92,35,~6),
contain
i012 Hz.
The results obtained on polycrystaliine Fig. 91. Remarkably,
samples were carried out
used in these experiments
The reflectivity data obtained
(32) are in good agreement with each other.
^
A
g and hence
6 = i~g~
In order to calculate
A
from the transmittance
T and the reflectivity R, the
p
following two scalar equations R = Irl 2
^
;
1/2
T =
are used:
l(l+r)exp(_i~d~/c).(l_r) 1 2
Here, n =
~
reflection
factor at the vacuum-sample
sample.
is the complex refractive
In the equation for T
(19)
index,
r = (I-~)/(I+~)
interface,
multiple
dispersion of the complex permittivity
and d the thickness of the
reflections ~=
the complex
could be neglected.
e' - i g" of
~-AgI at
The
250 °C
is included in Fig. 31. It is recognized
that R and
a similar manner.
~', g" also attains its largest
Interestingly,
values at the lowest accessible data has the advantage
besides
frequencies.
that the maxima of
A
6(~)
~(~),
g' depend on frequency in representation of the
in contrast
to those of ~"(~),
f
£'£"
T,R
E
t.O "1o
20 °1,
i
,
f
16
12
/ |
2.1011
20
l
5.1011 1012 2.1012
5.1012 1013
I
2.1011
1012
1013
[Hz]
[Hz]
FIG. 31. Transmittance T, reflectivity R, real and imaginary parts of permittlvity
~=
g' - ia" of polycrystalline
g-Agl at
250 °C. Thicknes~ of sample used for measurement of T : 7 3 ~ m . indicate the transverse mode frequencies of a crystal for any degree of damping
(168). Therefore,
6 (~) has been calculated
from
g"(~),
included in Fig. 27, section VII. The mutual consistency of
and this curve is ~'(~) and
6 (~)
Agl-Type solid electrolytes
was
checked
by a K r a m e r s - K r o n i g
seen that
~-Agl
exhibits
absorption
s p e c t r u m above
analysis.
From the
a r e l a t i v e l y normal,
~(~)
dependence
though highly damped,
it is lattice
lO 12 Hz. This part of the spectrum does not differ
much from the one at room temperature. an intense,
387
B e l o w this frequency,
broad and rather s t r u c t u r e l e s s
however,
there
is
a b s o r p t i o n band, which is unusual
in ionic crystals. Before ding
e m b a r k i n g on a d i s c u s s i o n of these results, ~(~)-date
from
are o b s e r v e d as in
m-RbAg415 ~-Agl.
is lo~er than that of far-infrared tWO
in Fig.
we Lresent
]2. Cualitatively,
the same features
A l t h o u g h the dc c o n d u c t i v i t y of
~-Agl
the correspon-
~ - R b A g 4 1 5 at 20 °C
at 250 °C by about one order of magnitude,
the
c o n d u c t i v i t y maxima n e a r 5"i0 ii Hz d i f f e r only by a factor of
•
Ecru-I] [(~ c~m)-I ]
10
20
50
I
I
I
100 200 I
I
125 °C. ,A
20 ° C . ~ o
I
I
1
k
I
01011 2.1011 5.10111012 2.~12 5.~12 1013
v CHz] FIG.
)2.
Far-infrared
c o n d u c t i v i t y of'
~-RbAg415
at d i f f e r e n t
temperatures. In ionic
crystals,
phonon f r e q u e n c i e s However,
a b s o r p t i o n bands beyond
the transverse optical processes (169)
as the intensity of d i f f e r e n c e bands is rather temperature
their c o n t r i b u t i o n the absorptzon. of
far-infrared
are usually due to m u l t i p h o n o n d i f f e r e n c e
can be roughly e s t i m a t e d
Besides
room temperature measurements,
of
f a r - i n f r a r e d spectra
~ - R b A g 4 1 5 have been o b s e r v e d at -40 °C and at + 125 °C. In these spectra,
the shape and the s t r e n g t h of the band are hardly altered. range
from -40 °C to + 125 °C,
tor of seven, 15 per
cent.
low-frequency cesses.
in w h i c h the de c o n d u c t i v i t y
the c o n d u c t i v i t y m a x i m u m n e a r 5 - 1 0 1 1 H z Therefore
in
~-RbAg415
to a n a r r o w e r
be even more intense
changes by a fac-
rises by not more
to the
~-Agl where the e x p e r i m e n t s
I/T range.
at 400 °C, the f a r - i n f r a r e d a b s o r p t i o n
than in
than
is not p r o v i d e d by m u l t i p h o n o n pro-
The same seems to hold in the case of
In the case of S-CuBr,
In the temperature
it is c o n c l u d e d that the major c o n t r i b u t i o n
absorption
were r e s t r i c t e d
~-Agl,
band seems to
and a d i s t i n c t i o n b e t w e e n
the lattice a b s o r p t i o n s p e c t r u m e x p e c t e d n e a r 9-1012 Hz appears
(96)
sensitive,
from the temperature d e p e n d e n c e
this band and to be impossible
388
K. Funke
For an interpretation of the broad low-frequency bands in m-RbAg4I 5, one has to consider the fact that Bloch's a crystal with structural disorder.
Therefore
S-CuBr,
and
the usual optical selection
rule Q ~ 0 is broken in the sense that all phonon-mode observable
g-AgI,
theorem is not valid in frequencies become
regardless of the values of their wave-vectors ~. So far, the
situation is similar to that in polar liquids• where a similar far-infrared behaviour has indeed been found terms of "smeared-out
(170,171). However•
an interpretation
in
liquid lattice bands ''(170"171) would imply that the
phonon concept is applicable
for the type of motion which is responsible
the far-infrared absorption.
In the limiting case of very high damping one
for
would rather prefer the description given at the beginning of this section, treating the cations as an assembly of stochastically driven oscillators. Actually,
quasielastic neutron-scattering
experiments on
g-AgI demonstrate
that the local motion of the silver ions is of an overdamped
type,
see
section IX. In the case of overdamping, + 2 r
~ +
the solution of a homogeneous equation of motion, = O,
(20)
which now contains a damping-term resulting
E" (~) ~ 2 t ~ l and (172)
~w
o, is no longer periodic.
6 (~)
=
~o ~
( ( ~ - ~ ) 2+4 2 ~ )
the
(21)
a"(~),
• still exhibit maxima on the frequency scale.
spectrum of frequencies a very broad band of The Raman scattering Hanson,
However,
functions
Fjeldly,
~oj• ~(~)
the contributions
If there is a whole
~j(~) might well add up to
as it is observed.
from a single crystal of
and Hochheimer
(38)
In their
~-AgI has been examined by 90°-scattering
experiment•
they used the 6471 ~ line of a Krypton laser. The Raman spectrum shown in Fig.33 does not noticeably depend on the crystal orientation and changes very little with temperature.
The scattered radiation is polarized
in the same
direction as the incident beam, while the Raleigh component in the light scattered with crossed polarization is believed to be an artifact of the experiment.
Very strong Raman scattering is found in the low-frequency region
from ca. 5 cm -I~
1.5 "i0 II Hz to ca. 50 cm -I ~ 1.5"1012 Hz. A lattice mode•
cf. Fig. 27, is observed at ca. i00 cm-l~ 3"1012 Hz. It is noted that this spectrum closely resembles frequencies•
however,
that of
E"
(~) presented in Fig. 31. At very low
there seems to be an indication of a peak in the ~"(~)
spectrum while no such peak is observed in the Raman spectrum. We will now briefly compare the actual far-infrared response of AgI-type solid electrolytes diffusion-model
to the predictions
from various models and theories.
The
given in the previous section is of course not intended to
predict any realistic
infrared spectra,
since the "local motion" has been
taken into account only by a frlction-term•
i.e. in a highly unspecific man-
ner. The "free-ion-like"
model presented by Rice and Roth
(41) yields a Drude-
Agl-Type solid electrolytes
like b e h a v i o u r like
lifetime"
expected optical Fig.
~(~)
~f. In this model,
to occur phonon
27 for
6(O)/(l-i~f)
=
w i t h the " c h a r a c t e r i s t i c
the D r u d e - l i k e d e c r e a s e of
in the f a r - i n f r a r e d region,
frequencies.
~-Agl,
The e x p e r i m e n t a l
is o b v i o u s l y more
frequency
Singwi,
and S j ~ l a n d e r
w e is introduced,
present at f r e q u e n c i e s ~ > ~ . " d i f f u s i v e modes", ~ < ~ ¢ , states
is then assumed
ductivity
in the v i c i n i t y of the
~(~)
dependence,
(42)
a cutoff forces are only
the motion of the cations is spllt into
and v i b r a t i o n a l
modes, ~ > ~ .
for both types of modes.
to a d i s t r i b u t i o n of damped oscillators.
demonstrated
as shown in
which is based on an "Ansatz"
''L173). In this model,
A Debye denszty of
The e x p r e s s i o n
is simply w r i t t e n as a sum of a Drude d i f ? u s i o n
the m i c r o w a v e
~ (~) was
i.e.
and it is assumed that r e s t o r i n g Thus,
free-ion-
complex and quite different.
H u b e r m a n n and Sen have proposed a model given by Rahman,
389
for the con-
term and a term due
That it is impossible
to describe
spectra with the help of a Drude d i f f u s i o n term has been in the p r e v i o u s
the o b s e r v e d broad
section.
far-infrared
However,
as has been pointed out above,
c o n d u c t i v i t y bands are not inconsistent with
the a s s u m p t i o n of damped or o v e r d a m p e d oscillators.
m - Agl
>.
T
-- 3 4 0
P
=
°C
1bar
C C C
0
E 0
n-
50
100 A9
FIG.
33 • R a m a n s c a t t e r i n g and H o c h h e i m e r
Br~esch,
Str~ssler,
151
[ c m -1 ]
from
~-Agl,
. after Hanson,
and Z e l l e r have r e c e n t l y tried to calculate
c o n d u c t i v i t y of h i g h l y c o n d u c t i n g solid e l e c t r o l y t e s cies from only one d i f f e r e n t i a l
equation
vin e q u a t i o n i n c l u d i n g a m e m o r y function. vin e q u a t i o n oscillator model
the complex
for arbitrary
frequen-
(37). This is a g e n e r a l i z e d LangeIt transforms into the usual Lange-
for long m e m o r y times and into the e q u a t i o n of a damped har~nonic
for short m e m o r y times.
It seems to be a typical feature oi" tais
that it can give a good fit to the e x p e r i m e n t a l
but that it cannot e a s i l y predict any c h a r a c t e r i s t i c wave range.
Fjeldiy,
(3~)
f a r - i n f r a r e d data, d i s p e r s i o n in the micro-
It is felt that this is due to the fact that the process of
J u m p - d i f f u s i o n should not be d e s c r i b e d by a L a n g e v i n - t y p e
equation.
390
K. Funke
IX QUASIELASTIC NEUTRON SCATTERING
The quasielastic
scattering of cold neutrons provides a powerful tool
for
the study of the diffusive motion of highly mobile atoms or ions. The fundamental advantage
in comparison
to the microwave
and far-infrared
described above is the possibility of simultaneous times,
but also of the distances
characterizing
experiments
resolution not only of the
the elementary diffusion
steps. This is a consequence of the fact that the energy transfer ~ the
momentum transfer
neously,
~Q
and
of the scattered neutrons are measured simulta-
which is practically impossible
in the case of electromagnetic
radiation. Although
for example the diffusion of hydrogen in metals has been examined
in various quasielastic neutron-scattering was not applied to Agl-type
solid electrolytes until
have been due to the additional silver and copper,
experiments
complications
in contrast to hydrogen,
(39), this technique
1973. This may partly
arising from the fact that
scatter neutrons essentially
coherently. Recently, quasielastic neutron-scattering samples of
~-Agl have been undertaken
time-of-flight
spectrometer of the Institut Laue-Langevin,
these two experiments, resolutions were and 0.177 meV,
respectively.
(~,68,121)
energy transfer
The temperature of the sample was 250 °C. spectra are plotted against energy transfer
~=0.
the origin of the diffuse scattering
There is intense low-energy scattering, As Q increases,
tail extending to higher values o f ~ Basically,
In
the ranges of wavevector transfer and the energy
in Fig. ]4. The spectra readily explain m-AgI
Grenoble.
1.4 ~-i( Q<2.7 ~-i and 0.4 meV, and 0.56 ~-i~ Q • 2.16 ~-i
Typical examples of corrected in
experiments on polycrystalline
(40,14) at the "IN5 multichopper"
the quasielastic
centred around
its broad and almost structureless becomes more and more pronounced.
scattering reflects the structural disorder,
because it is mostly coherent.
Its shape gives information on the diffusion
of the silver ions. Its tail at higher energies
corresponds
to the broad
low-frequency absorption band found in the infrared experiments. Spectra obtained by Axe and Hoshino at the Brookhaven ter qualitatively display the same features. tals of
~-AgI.
In their experiment,
however,
triple-axis
spectrome-
These workers used single crysboth the scattered intensity
and the energy resolution were comparatively unfavourable evaluation (174)
to a numerical
In order to extract information on the diffusive motion of the silver ions, we have to analyze energy width
the Q-dependence
A E of the quasielastic
of the shape and, to begin with,
the width of the incoherent part of the quasielastic
scattering is known to
be A E = 2~DQ 2, where D is the coefficient of self-diffusion. be the case for sufficiently small Q, corresponding In
~-AgI,
of the
peak. As long as Fick's law is fulfilled,
the validity of this simple translational
be verified in the Q-range below approx,
~)This should
to sufficiently
large r.
diffusion model could
i ~-i. At larger Q, the experimental
Agl-Type solid electrolytes
widths
are found to be smaller.
most of the s c a t t e r i n g
391
This cannot be e x p l a i n e d by the feet that
is coherent.
Rather,
it is an i n d i c a t i o n that the
simple d i f f u s i o n model no longer applies. Closer i n s p e c t i o n of the spectra shows component
that they seem to contain a narrow
- w h i c h is h o w e v e r b r o a d e r than the r e s o l u t i o n
of a m u c h b r o a d e r d i s t r i b u t i o n . dels for t r a n s l a t i o n a l tr~l shapes
Moreover,
it is found that none of the mo-
d i f f u s i o n known to us
and their Q - d e p e n d e n c e .
function - on top
(177) can explain
these spec-
In order to derive a physical
moael
ap-
p l y i n g to the d i f f u s i v e motion of the silver ions, we make the following assumptions. a) The spectra are free of elastic Por the f o l l o w i n g reasons.
scattering.
This a s s u m p t i o n
There is no coherent
is justified
elastic scattering,
B r a g g peaks have been e x c l u d e d by the choice of the e x p e r i m e n t a l On the other hand, incoherent
since
condltiors.
there is no i n c o h e r e n t elastic s c a t t e r i n g because
cross section of iodine is p r o b a b l y n e g l i g i b l e
the
and because
the
silver ions have no fixed lattice sites. b) The i n e l a s t i c
contributions
to our spectra are n e g l i g i b l e The inelastic
in c o m p a r i s o n
to the q u a s i e l a s t i c
scattering.
does not exceed ca.
i0 g even at our largest s c a t t e r i n g angles.
c) The q u a s i e l a s t i c
scattering
is e x c l u s i v e l y due to the silver ions.
particular,
its coherent
correlation
f u n c t i o n GAg-Ag(~,t),
butions
are o b s e r v e d
It is n o t e d spectra
while
the elastic and q u a s i e i a s t i c
functions GAg-l(~,t),
contri-
Gl-~g(~,t),
and
in the B r a g g peaks.
that there is a s u r p r i s i n g s i m i l a r i t y between the q u a s i e l a s t i c
found in
~-Agl
and in some m o l e c u l a r systems w h i c h exhibit d i f f u s i v e
m o t i o n s of atoms w i t h i n r e s t r i c t e d regions of space. statistical
This motion is due to
f l u c t u a t i o n s of the m o l e c u l a r o r i e n t a t i o n
(!78). The spectra <)~
such systems e s s e n t i a l l y
consist of a n a r r o w u n b r o a d e n e d
broad q u a s i e l a s t i c
It is also noted
oscillations this kind,
In
part is the F o u r i e r t r a n s f o r m of the s i l v e r - s i l v e r
a r i s i n g from the c o r r e l a t i o n
Gl-l(~,t)
fraction of the s c a t t e r i n ~
peak.
yield s i m i l a r spectral
except
that o v e r d a m p e d
shapes.
leads us to assume
~)In terms of the Van Hove s e l f - c o r r e l a t i o n
The i n c o h e r e n t Sinc
large-amplitude
spectra are also of This
that
function Gs(~,t ) which gives
of finding an atom at (~,t),
the s o l u t i o n of Fick's Gs(~,t)
~-Agl
that the n a r r o w component is slightly broadened.
pheno:nenological r e s e m b l e n c e
the p r o b a b i l i t y
The
line on top of a
if it was at
(2,0)
(175),
law is
: (4~Dt)-~/2exp(-r2/4Dt). scattering
(~, ~ )
function is the F o u r i e r t r a n s f o r m of Gs(~,t):
= 2DQ2/((DQ2)2
The e n e r g y w i d t h of Sin c (Q, ~ )
is
+ ~2). AE
= 2~DQ 2, cf.
(176)
392
K. Funke
st°t- (~p, ¢1)) 10 5 0 0 1: o = 73 ps 1:1 = 1 6 . 5 p s
7000
r
f
= 41.5 °
= 1~
Dr =
1 ps -1
3500
0
3
. . . . . . . . .
4
' ; - - 6 )
....
• "l'im [meV]
st°t'(f,w) 16 5 0 0
I
=5~
I: 0 = 7.5 ps
%1 =16.5ps
11000
~' =
105 °
r = 1~ Dr =
1 ps -1
5500
o
o
.
.
.
.
.
.
.
.
.
.
-3
FIG. 34. Experimental
i
-2 -1
0
quasielastic
at two different
2
3
4 •
neutron
scattering
The solid lines result is sketched
1
5 6 ? "l'iuJ [meV]
scattering
angles
(o).
from the model
calculation
which
in the text.
Wavelength
of incoming neutrons:
Scattering
angles:
The experimental
5.343
41.5 ° and i05 °.
spectra are corrected
in the usual way
for detector efficiency, sample-container sorption, and self-shielding. The spectra
spectra
are markedly
asymmetric
scattering,
because
(and calculated) at constant scattering constant momentum transfer ~ Q .
ab-
they are taken
angle ~ and not at
Agl-Type solid electrolytes
d) the silver ion is carrying different
time scales.
sive motion within iodide
lattice.
bution
plus a
lational
-diffusion would
~(~)-peak
type.
observe
broaden
over
region of space,
crystal
line.
An example
of the a b o v e - m e n t i o n e d with the elementary
translational
diffusion
35. Local
in a void of the b.c.c
voids
follows n a t u r a l l y
distri-
kind of motion
is a trans-
the spectrum which one
~lone;
in particular
of a possible
regions
or diffu-
volume which might be of the Jump
from the first kind of moticm
of cations between
e.g.
on top of it. The second the whole
on two
an o v e r d a m p e d
could cause a broad quasiel~stic
are identical
FIG.
types of motion
it performs
It might only slightly broaden
the elastic
arrangement
a restricted
Such a motion
diffusion
out two different
On the one hand
393
it would
three-dimensional
is shown in Fig.
35. These regions
proposed by Rickert from this picture
(ii)~)
The
as an exch3nge
these voids.
regions
of space available
for the silver
ions
in
g-AgI.
~)Fig.
35 gives a rough illustrat~on
accessible of B ~ h r e r
to the silver ions. and H~lg
relatively
(12) and of Wright
large value of
the following
~I'
interpretation:
dence
ti~es,
of Fig. ~o'
(13)
located within during
the results
In view of the
one can, however,
their motion
35 might be occupied
of space which are
to contradict
and Fender
see section VII, during
the silver ions are e s s e n t i a l l y the regions
of those regions
It is not meant
along the
the regions
the relatively
conceive channels, 9, while
short resi-
394
K.
e) We furthermore simultaneously assumption
assume
that the silver ions perform both kinds of motion
and that these are dynamically
permits
and the scattering elestie
Funke
scattering
independent
of each other.
us to write the silver self-correlation
This
function GsAg(~,t )
function S~ g (C,~) for the incoherent part of the quasiinc -from the silver ions as the convolutions of the individual
functions:
G~g
(r,t)_ =
$ G~ ( r ' ' t ) . G L ( ~S- ~ ' ' t ) d r ' -
-
/oT --
where
~inc
the superscripts
respectively.
models
~L (q,~), ~'inc
for both types of motion,
a n d ~inc(,{,~) ~,.o ~ ,
S t°t
sectionsof ~cattering
•
(q,~)
and
silver, respectively, function sAg~Ag(Q,~).
motion,
in Fig. 36. By the
see below,
for the total quesielastic
and ~inc are the coherent
local"
the functions
can be approximated.
~Ag-Ag ~coh~coh (C,~) +
=
(22)
"~inc (~' ~ -
T and L stand for "translatlonal
In order to derive expressions
wherev~oh
--
The meaning of eouations (22) is illustrated
use of simplifying s Tznc ( q , ~ ) ,
,
As 6inc S.±nc
ana incoherent
scattering
(q,,~)
~,e still need the coherent It is assumed that
(2))
'
scattering
function
cross
quasielastic
con
f) oAg-agfc ~) can be estimated by using a phenomenological model ~hieh ~coh ~-' was proposed a few years ago for liquid argon~179 jI~ and which yields the relation sAg-Ag (q,~) cob The required mentel
=
~Bg (Q. (sAg(c) )-I/2, ~ ) .s#g (Q) ~inc
function sAg(Q):
and theoretical
=Is"g_Ag(c,~
results
con
of ref.
112])
) d~ is taken from the experi-
f+l':r'f : }:{+} sL FIG.
36.
Illustration
to equations
(24)
(22), cf. Fig. 25.
Agl-Type solid electrolytes
In principle, derived
the total Q u a s i e l a s t i c
and compared
mathematically movements The
tractable
models
jump-diffusion
by Rowe et el.
does not appear needed which 71,
to occur
jump length,
in a crystal assume
g) the G i s s l e r - S t u m p
model
of S ~) n c.( <_, ~
dence
time ~
(ii) The length, more,
~.
provides
At present,
without
from either
appreach
scattering
results
h) the solution an adequate
tetrahedral
sites,
approximated nately,
motion
weighting
i) we have
point
~re:
of still
Furtherv = ~/~.
for ~he ouasi-
is useG in two and
ions,
factors
tha numerical
the two possibilities oscillatory
by prolate
results
affecting
to approximate
menticned
motion,
are
if one takes into
the frequency
spectra ~hen
that
equation
in a restricted
provided
region of space
ions.
of Fig.
Possible
candidates
centred at the
35 which may be roughly
centred at the octahedral
an ellipsoid
is
for the local motion of the
by the iodide
9, or the regions
within
~o' ~i" and
tecb_niques.
ellipsoids
too complicated.
ob-
not only with the neutron
of space might be the oblong ellipsoids see Fig.
results
On the other
of the parameters
for setting up a model
the voids
the d i f f u s i o n
matically
model
indistingulsnable.
and overdamped
we assume
of the d i f f u s i o n
for these regions
are concerned,
They are both compatible
convenience,
silver ions within
expression
choicc- of jump directions,
but also with the infrared
starting
for the
the mean resi-
their jumps is constant,
az'e p r a c t i c a l l y
with the two different
For m a t h e m a t i c a l
Thus
TI, of the jumps are constants.
depend on the choice
local diffusive
much alike.
measured
treat~
and Stump
made in this moae]
as exp(-t/~o)
the G i s s i e r - S t u m p
any :pecific
strongly
the different
jump
with jumps along the ~i00> tunnels.
physically account
is
nameIj
As for the local motion of the silver namely
~ model
To, a mean
good a p p r o x i m a t i o n
assumptions
0, the C h u d l e y - E l l i o t t
as far as the final spectre
the spectra
Rather,
time,
T o. In
therefore
is recovered.
of S~nc(¢:,~),
modifications,
(i) isotropic,
~I ~
model
the only model which
a sufficiently
of the atoms during
function
(ii) polycrystalline,
~-Agl.
however,
that
case ~i ~
For the calculation
In this model,
in this way is the one of Gissler
is introduced. o ~, and the duration,
scattering
different
above,
and local
a given site at time t = 0, the probability
the velocity
In the limiting
tained
ar.d
used in the litera-
in the sense of
a mean residence
~' d "ing it there at time t>0 decreases ~in
hand,
can now be
adecuate
It has been generalized
the Chudiey-Elliott
The simplifying
occupies
(i$2).
to the case of
as parameters,
therefore
(i) If an a t o m
results,
to be applicable
and a mean
ca] culation
frequently
and Elliott (I~0).
instantaneously
microwave
contains
the j u m p - d i f f u s i o n (!67). We
by Chudley
(181) and by Gissler and Rother
the jumps are assumed
However,
if p h y e i c a i ! y
can be found for the translationa~
model which has been most
view of the existing
elastic
function st°t'(Q,~)
spectra,
of the ions.
ture is the one proposed
time,
scattering
to the experimental
395
and even within
sites.
a sphere
Unfortuis mathe-
Therefore, the local motion by a d i f f u s i o n
on the surface
of a
396
K. Funke
sphere, mation
the only case which Is mathematically is not too far from a diffusion
ion which is allowed
to diffuse
found near the surface -diffusion" sphere,
model within a full sphere since an
than near the centre of the sphere.
approximation
contains
two parameters,
certainly
of physical
simplifications
Making
However,
in order to retain conceptual
the assumptions
st°t'(Q,~)
namely
coefficient,
Some of the above assumptions reality.
(183). This approxi-
in the interior of a sphere is more probably
r, and the "rotational-diffusion"
necessary
tractable
Dr .
imply rather drastic of this
for various
simplifications
kind appear to be
clarity and mathematical
a) to i), the total quasielastic
are calculated
The "rotational the radius of the
feasibility.
scattering
sets of parameter values.
functions
The functions
st°t'(Q,~) are then transformed into corresponding functions which describe scattering at fixed scattering angles ~ . Finally, convolution of these functions
with the resolution
(~,~) which may be compared In Fig.
34, calculated
-dependence
the parameter
the broad component
scattering
close
shapes
angles
are presented
D r is essentially
of the quaslelastic
angles.
given by the energy widths arising
from the
fractions
of neutrons
scattered
broadened
~(~) peaks.
The best fits arc achieved with D r ~ ips -I and r ~ l
It should,
however,
be emphasized
ions and that the values
values of ~ into the slightly
that our present
is only a very rough first approach
to the actual
of its parameters
of
from the local
r, is then determined
at different
The Q
described.
scattering,
motion of the ions. The second parameter,
along
(~ = 41.5 ° and i05°).
fits at all scattering is correctly
spectra S t°t"
ones. functions
spectra at two scattering
surprisingly
of the spectral
In our model,
the calculated
to the experimental
quasielastic
with the experimental Our model yields
function yields
the
rotational
~.
diffusion
model
local motion of the silver
should hence not be taken too
literally. The width of the narrow quasielastic transfer
essentla]ly
determines
It is found to be somewhat On the other hand,
component
the value of the mean residence
(2.5 ~, 0 ps) to more than
Fortunately,
and vice versa.
to extract
component
independent
Thus close ranging
from
this uncertain-
the energy shapes and
at different
information
~o"
A best value
~i)- combinations
(6 ~, 22 ps). In order to overcome
it would be necessary
widths of the narrow quasielastic precision.
(~
time,
than I0 ps.
to be less certain.
as soon as ~ is known,
fits are possible wlth several different ty experimentally,
large momentum
larger than 5 ps, but still smaller
the value of ~i appears
of ~i can be determined
at relatively
Q wlth much higher
about the Jump-time
T1 is
available from the microwave data which give ~i(250 °C)~15 ps, see section VII. The spectra st°t'(~,~) shown in Fig. 34 have been calculated with ~o = 7.5 ps, consistent
TI = 16.5 ps, and
~=
5 ~, a set of parameter values which is
both with the neutron-scatterlng
of the implications
and the microwave
results.
of these values have already been discussed
Some
at the end of
section VII. The parameters
of the translational
diffusion
model are connected with the
coefficient of self-diffuslon of the silver ions, D, by equation (ll): 6D(T ° +~i ) =~2. The a b o v e parameter values give D ~ 1 . 7 5 • 10 -5 cm2s -I, in
Agl-Type solid electrolytes
397
good agreement with the experimentally determined tracer-diffusion coefficient which is roughly 2'10 -5 cm2s -I at 250 °C (96). Thus, in the case of
~-AgI, our present physical model for the cation motion,
cf. Figs. 25 end 36, appears to be consistent with the experimental results given in the last four sections. A direct determination of the starting points, directions, individual
and lengths of the
jump-diffusion steps will probably require a single-crystal neu-
tron-scattering experiment with an energy resolution better than 0.1 meV.
A CKN 0W LE DGE ME~ TS The author is much indebted to Prof. W. Jost for his advice and encouraging support throughout this work. Moreover, I wish to thank Prof. H. Schmalzried who also critically r~ad the manuscript, in wording. Finally,
and Dr. D. Field for his suggestions
thanks are due to the authors and publishers who gave
their permission to reproduce Figs. 7,9-ii,15,17-21,
and 33.
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