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NMR studies of superionic β -aluminas
J.L. Bjorkstam, M. Villa
To cite this version:
J.L. Bjorkstam, M. Villa. NMR studies of superionic β-aluminas. Journal de Physique, 1981, 42 (2),
pp.345-351. �10.1051/jphys:01981004202034500�. �jpa-00209016�
NMR studies of superionic 03B2-aluminas
J. L. Bjorkstam
Department of Electrical Engineering, University of Washington, Seattle, Washington 98195, U.S.A.
and M. Villa
Istituto di Fisica « A. Volta » e Gruppo Nazionale di Struttura della Materia del C.N.R., 27100 Pavia, Italy (Reçu le 8 juillet 1980, révisé le 29 septembre, accepté le 6 octobre 1980)
Résumé.
2014Les temps de relaxation nucléaire spin-réseau (T1) pour 23Na et 27Al dans l’électrolyte solide Na 03B2-alumine sont interprétés avec un modèle de diffusion continue à deux dimensions. Nous montrons que, avec
ce modèle, les données expérimentales de la Résonance Magnétique Nucléaire sont en bon accord avec celles
de la diffusion. Nous montrons aussi que l’on peut étudier, par le T1 de 27Al, la dynamique dans les plans de
conduction des 03B2-alumines partiellement substituées.
Abstract.
201423Na and 27Al nuclear spin-lattice relaxation data in superionic Na 03B2-alumina are interpreted with
a two-dimensional, continuum-diffusion (2D-cd) model. This model is shown to bring agreement between Nuclear
Magnetic Resonance and diffusion results. Also, the immobile 27Al relaxation is shown to be a valid probe for investigating the mixed-cation conduction dynamics.
Classification Physics Abstracts
66.30H - 76-60E
1. Introduction.
-The recent substantial interest in superionic conductors is motivated by both their potential importance as the ion conduction medium in high energy density electrical storage systems and the unusual physics associated with restricted-dimen-
sionality, liquid-like transport in solids. We will focus attention upon nuclear magnetic resonance
studies of the latter question in the planar conductor, fi-alumina. In particular, we will address two topics :
-
interpretation of 23Na and 27 Al spin-lattice
relaxation times ( T1 ) of Na p-alumina with a two-
dimensional continuum-diffusion (2D-cd) model and
-
use of the 27 Al T1 to study mixed alkali and co-ionic effects in partially substituted fl-aluminas [1].
NMR studies of diffusion processes in solids are
often interpreted in terms of a hopping model in
which the particle resides an average time i in a
lattice position before performing a jump. The NMR spectral densities have the form j (m) - -r(1 +W2 Ir 2)-l.
This assumption implies that individual jumps cause
environmental fluctuations responsible for relaxa-
tion. A minimum in T1 is expected when mo i - 1
(where mo is the NMR measuring frequency). In this
case, T 1 vs. T measurements in solid electrolytes provide a way of determining the attempt frequency
and testing the absolute rate theory of transport in solids [2]. However, for Na p-alumina there is a
substantial disagreement between the attempt fre- quencies determined with 23Na NMR [3] and with
other techniques [4]. Furthermore, in the hopping scheme, an interpretation of the asymmetric shape
of the 23Na T 1 vs. T -1 curve requires a substantial distribution of jump activation energies [3]. Such a
distribution is difficult to reconcile with the simple
Arrhenius behaviour of ionic conductivity over the
entire 120-800 K range [5].
Another point of view may be associated with the continuum model of motion, often applied to liquids [6, 8]. The relevant NMR time is then
with D the diffusion coefficient and do the mobile
particle displacement which causes the environmental fluctuations responsible for relaxation. When D is
known, the Tl-minimum gives a distance rather than
a time. As a function of temperature, the maximum value of the local field spectral densities is reached when CorD is of the order of ten [7, 8]. In addition,
the relaxation rates for continuum diffusion in the
high frequency regime (mo TD » 1) are approximated by :
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01981004202034500
346
with fl = 1.5 for some analytically soluble models [6]
or 1.2 Z fi Z 1.8 near the highest calculated fre-
quencies for models which have been solved numeri-
cally [7, 8]. Since fi = 2 for Lorentzian densities,
continuum models predict a weaker frequency and temperature dependence for the low-temperature Tl’s
than do hopping models.
We will show that the NMR data above 150 K
are compatible with macroscopic diffusion results for Na fi-alumina when the 2D-cd spectral densi-
ties [8] are used. Applicability of this model for temperatures below the T1-minima implies that many individual jumps take place in traversing the dis-
tance do ; i.e., the time scale to which the NMR results are sensitive is not short enough to detect
the individual hops of Na+ diffusion in this tempe-
rature range.
Wolf [9] suggests that the difference in slope of
the NMR T 1 data above and below the T 1-minimum
is associated with actual differences in the controlling energies in the two temperature régimes ; i.e., at high temperatures where the associated regions around
an excess compensating 02 - become small the measured activation energy is related to the long
range diffusive motion while at low temperatures the motion is controlled by the energy necessary to free mobile interstitials from the larger associated regions. Wolf is careful to point out that the agree- ment so obtained with the Tl vs. T slopes above
and below the Tl minimum may be accidental, because of the lack of a detailed microscopic theory,
for the Na relaxation [9]. While relaxation and spectral
measurements below - 150 K allow NMR to be used as a tool for studying further details of the local cation motion [10], we suggest here that the NMR data above this temperature are not so dis- criminatory. In fact, both the 23Na and 27 Al Ti-data
above 150 K will be shown to give a single activation
energy, typical of long range diffusion, when the
2D-cd model is applied. The relationship between
ion transport and NMR response will be further demonstrated through a 27 Al Tl study of partially
substituted fi-aluminas.
2. General information and experimental détails.
-The formula for stoichiometric sodium si-alumina
is Na20, 11 A’203. Sodium lies in a mirror plane
between spinel blocks containing Al. In our melt-
grown crystal from Union Carbide there was 22 %
excess Na+. The most generally accepted mechanism
for charge compensation of the excess cation is the presence of excess oxygen (O2-) in the conducting plane.
We will report measurements on 7Li, 23Na and
2’AI central lines in single crystals with the magnetic
field Ho perpendicular to the conduction planes
At this orientation, the Am = ± 2 23Na transition
probability W2 is much higher than W 1 [3]. There-
fore it is easy to subtract the slow recovery due to the Am
=+ 1 transition and to construct a recovery
plot for only that portion of the magnetization which
relaxes for Am
=± 2 transitions. Recovery of the 2’AI magnetization after a pulse which saturates the
central transition is not exponential. The same often
occurs for the ’Li magnetization, even though in
this case the whole spectrum (central and satellite
lines) can be irradiated and detected. Therefore, we
define T 1 as the inverse relaxation rate at the time
origin and T* as the time for the magnetization to
reach (1- e-1) of its equilibrium value.
Complete or partial substitution of the conducting
ions in fi-aluminas has been performed by immersing
a crystal in an appropriate molten salt bath [11]. The
extent of the ionic substitution was obtained within
an estimated ± 5 % with gravimetric and neutron
activation analyses.
Thermogravimetric analyses of the fi-aluminas
used in this study have been performed in crystals exposed to various gases : He, N2, 02, CO2 and
air with a 55 % relative humidity at 20 °C [12]. For
small crystals (size 63-88 ym) in air the following
amounts of water are reversibly absorbed/desorbed during cooling/heating cycles covering the 300 - 900 K interval : 0.9 % (in weight) in Na fi-alumina, 2.5 %
in Li (50%)-Na (50%) and 3.3 % in crystals with
nominal [13] composition Li (85 %)-Ag (15%). The desorption process begins in the 360 + 450 K interval and is apparently completed below 600 K. Weight changes less than 0.08 % occur in dry atmospheres
and for fl-aluminas in air having Na+ totally or partially substituted by K+, Rb+, Ag+ or Tl+. During
the first heating, non-annealed single crystals weight- ing - 0.1 g lost only a fraction of the water exchanged by the crushed crystals and did not have a noticeable
intake of water when kept in air at RT for 2 3 days
after annealing. This indicates a slower and/or
reduced reactivity with water for the large-size crystals.
All the NMR measurements reported here have
been performed in single crystals (typical size
15 x 6 x 3 mm) in air. When the crystals were
annealed above their desorption temperature, they
were left for months in air before taking new data
below 350 K. With these precautions, the NMR
results were reproduced well within the experimental uncertainty (up to ± 15 % for the 23Na data and less than + 10 % for the other measurements).
Preliminary NMR measurements have been made in crystals immediately after annealing for 1 h at
800 K. For Na fi-alumina, neither the 23Na nor the 2’Al response below 300 K appears to be sensitive
to the annealing [14]. At RT, we found nearly equal
values for 7Li longitudinal and transverse relaxation times in our Li+ fl-alumina crystals as exchanged
and just annealed. At the present time we cannot say
if this similarity is coincidental, as the data of walstedt
et al. [15] in flux-grown Li-Na fi-aluminas would suggest, or is due to a modest intake of water in large single crystals, as indicated by the thermogravimetric
analyses.
_ ’3. Results and discussion. - Figure 1 presents spin-
lattice relaxation data for 23Na and 2’AI in Na fi-
alumina at 21 MHz with ce = 0°. The (2 W2)-1
values for 23Na are represented by circles and triangles
above and below 150 K, respectively, to signify that
the law of recovery for the magnetization is different
in the two temperature regions. While the slow part of the recovery, which is due to the Am = ± 1 tran- sitions, is always described by a single exponential,
the part due to the Am = ± 2 transitions acquires a markedly multi-exponential character below - 150 K.
In this temperature region, we define 2 W2 as the
relaxation rate at the time origin for the fast decaying portion of the magnetization.
Figure 2 illustrates how, according to Wolf [9], Na+ diffusion takes place through correlated motion of doubly occupied cells. In terms of the attempt frequency vo21 for the m0-m0 - aBR-BR interstitial
jumps (a - b in figure 2), the diffusion coefficient
(in cm2 . s-1) can be written as [9]
where the numerical values have been determined by comparing parameters of a complex theoretical expres- sion with experimental results. Consistency of the
Wolf theory is supported by the agreement between the vo21 1 value estimated from conductivity and that directly given by Raman results (vo21 = 1.7(1012) s-1).
As anticipated, the 2D-cd model gives the dis-
tances do for 23Na and 27 Al relaxation by comparing
diffusion data and the temperatures where the T 1-
minima of figure 1 occur. Assuming that the Am= + 2
process dominates the relaxation for 2’Al, as well as
for 23Na, and that the Da value given by equation (3)
is applicable over the temperature range of our Tl results, we find dô’ 5.7 À and dô a 12 À. Since
widely different conductivity data have been reported
for the Union Carbide Na p-alumina [5, 9], the values
of do have a large ( N 50 %) uncertainty. However,
the values quoted above represent an average of various estimates which gave do AI and do Na -values
in the - 4-8 A and - 9-18 A range, respectively. It
is thus clear that dô is very close to the lattice spacing
of figure 2. It seems quite reasonable that the cation motion over one unit cell would most effectively
rattle the spinel cage [1] thus modulating the electric
field gradient (efg) tensor at the 2’Al sites. The fact that do -= 2 dô 1 two lattice spacings is suggestive
since it is approximately half the mean distance
between the afore-mentioned excess compensating 02- ions in the conducting plane. This fact leads to
Fig. 1.
-Circles and crosses represent, respectively, (2 W2)-1 of
23Na and T1* of 2’Al in Na p-alumina at 21 MHz and a
=0°.
Triangles represent (2 W2)-1 at the time origin when the relaxation due to the Am
=± 2 transitions displays a multi exponential
behaviour (see text). Solid lines are calculations using the 2D-cd
model while the dashed lines are given by Lorentzian spectral
densities.
Fig. 2.
-Transport sequence of a Na+ ion from cell 1 to cell II.
Open symbols indicate lattice positions not occupied by the cation.
Dashed lines include the two ions of a doubly occupied cell.
the hypothesis that 23Na relaxation above 150 K is driven by efg inhomogeneities caused by the compen-
sating oxygens.
In the elementary model for uncorrelated hopping,
the relationship between correlation time i and attempt frequency vo would be
With 2 Wo i - 1 at the temperature of the 2’Al T1-
minimum one finds vo éé 1.3(1012) s-1, a value consis-
348
tent with the Raman result. This agreement is for- tuitous since the hopping model presumes i to be the time for individual jumps rather than that for the a -+ b transition of figure 2. Since many individual correlated hops must occur for the cation to move over the distance of a lattice spacing, a simple hopping
model should give an attempt frequency much higher than vo21. Instead, precisely the opposite
result follows when the hopping picture is applied to
the 23Na data. The solid curves in figure 1 represent theoretical predictions of the 2D-cd model with the
previously obtained values of do and D, while the
dashed lines refer to the hopping model, as discussed
below equation (4). In drawing these curves, the only requirement was that they agree with the values of Tl
at the minima. The 2D-cd model gives agreement with the following experimental findings :
i) The Tl-minima are much broader than predict-
ed by Lorentzian spectral densities (see Fig. 1) and ii) at temperatures below the minima, T1 has a
much weaker temperature and frequency [3] depen-
dence than given by the hopping model.
Figure 3 shows 27 Al T* results in some partially
substituted p-alumina crystals. The relaxation rates of 27 Al are very sensitive to the cation present with larger, less mobile ions yielding T1*-minima at higher temperatures. Arrows indicate the positions of T1-
minima as calculated with the 2D-cd model by taking
o = 5.6 A and the conductivity data for pure Na, K, Rb and Tl p-alumina [11]. For K-Na and Rb-Na crystals the temperatures of the Tl-minima are consi-
Fig. 3.
-27 Al Ti in Na fl-alumina and in some mixed crystals :
K (90 %)-Na (10 %), Tl (60 %)-Na (40 %), and Rb (55 %)-Na (45 %).
These percentages were determined through weight measurements.
Solid lines have no theoretical significance. For the meaning of
the arrows, see text.
derably higher than those predicted theoretically.
Therefore, in these mixed crystals the mobilities of both ions are low compared with the mobilities in the pure crystals (mixed alkali effect) [16].
For mixed fi-aluminas containing Li+ a different
situation may occur. At the present time, we have analysed only crystals with two different contents
of Li+ : Li (50 %)-Na and Li (85 %)-Ag [13], hereafter
indicated as Li-Na p and Li-Ag fi, respectively. For
both crystals we have found two "AI Tl-minima.
It was suggested [1] that the double 7B-minima are
due to the fact that Li+ has a mobility considerably
different than that of the other ion. In the following,
we summarize the NMR evidence which leads to such
an interpretation. For the Li-Na fl crystal, the 2’Al Tl-minima occur approximately at 175 K and 310 K.
Also the spin-lattice relaxation time of 7Li undergoes
two minima with changing temperature, one near 180 K and another, much more pronounced at 400 K (see Fig. 4). Figure 5 reports, as function of temperature,
the peak-to-peak separation (b vpp) of the absorption
derivative for the 23Na central transition in Na fil
and Li-Na fi and for the 7Li spectrum in Li-Na fl.
In both crystals, the 23Na signal at a = 0° undergoes
a first narrowing process in the 80 : 140 K range.
In Na fi, this process causes the disappearance of a
some structure of the spectrum [17], disappearancç
that so far, has not been interpreted in detail. Instead,
the 23Na line narrowing process above 240 K has been explained in terms of a T1-effect and related to
the diffusive motion of Na+ [17]. The’Li line narrow-
Fig. 4.
-Relaxation times at the time origin (Tl) for ’Li and 2’AI in Li-Na fi at 21 MHz and a
=()o. The solid curve is a fit
of the 2’Al Tl data in Na fi-alumina made with the 2D-cd model.
Fig. 5.
-Absorption derivative peak-to-peak separation for 23Na
central line in Na fi and Na-Li fi and for the ’Li spectrum in Na-Li fi.
The working frequency was 15 MHz and a
=00. The quantities wres and ÔVRL used in equation (5) are indicated. The inset shows the Arrhenius plot for the correlation frequencies deduced from the ’ Li data.
ing in Li-Na fl is apparently well described by the phenomenological equation [18]
where c5vres is the residual linewidth at high tempera-
tures due to the inhomogeneity of the magnetic field, c5VRL is the rigid lattice peak-to-peak separation and a
is a constant of the order of one. In addition, an activat-
ed behaviour
is assumed for the correlation time (i) of the motion
responsible for the narrowing. With a = 1, the ’Li data of figure 5 would give -rOl = 3(106) s-’ and
E = 0.063 eV.
It appears, from the 23Na lineshape data, that no major differences exist between Na+ dynamics in Na
and Li-Na fi. Since the low-temperature ’Li and 27 Al T1-minima in Li-Na fi occur in the same tempera-
ture region as the 27 Al Tl -minimum in Na fi (see Fig. 4), we conclude that these minima are mostly
caused by efg fluctuations related to the Na+ diffu- sion process. By exclusion the other 2’AI T1-minimum
should be related to motion of Li+--Li+ or Li+--Na+
pairs. Notice that, at 200 K the Li+ motion, as charac-
terized through the 7Li lineshape, should have fre-
quencies in the 105 s-1 range, i.e. three order of magni-
tude smaller than that required to have a T1-minimum.
Above 250 K, the’Li and 27 Al relaxation behaviours in Li-Na fi resemble those of 23Na and 27 Al in Na fl
discussed above. The Arrhenius plot for ’Li T1 gives
an asymmetric minimum with the energy of the low- temperature side which is a fraction of that of the high- temperature side. The Tl-minimum for the nucleus of the mobile species (Li+) occurs at higher tempera-
ture than that of the non-mobile species (27 Al).
Obviously, an explanation of this fact may be more
involved than that given for Na fi.
Interpretation of the 27 Al Tl in Li-Na fi with the
Wolf model would require consideration of four diffe- rent transfer sequences (see Fig. 2) for both Na+
and Li+ and an impractical number of adjustable parameters. However, something may be learned by comparing the available NMR data in Li-Ag fl and Li-Na fi (see Figs. 4-6). The fact that both the 27 Al
and ’Li high temperature Tl-minima in Li-Ag fi
occur - 50 K higher than the corresponding minima
in Li-Na fi, is a strong indication that 28 Al and 7 Li relaxation processes are due to the same cause. The
high temperature 17 AI Tl-minimum is much more pronounced in the crystal with high Li+ concentra-
tion (Li-Ag fi) as it can be expected if this minimum is due to motion of Li+ pairs. With equations (5)
and (6) we foundr 1 = 2.9(108) s-1 and E = 0.168 eV
for the’Li narrowing process in Li-Ag fi. Again, order
of magnitude considerations rule out that the motion that causes the 7Li line to narrow is responsible for
the low temperature 27 Al and 7Li ri-minima. Below
Fig. 6.
-Relaxation time at the time origin (T1) for ’Li and "AI in Li-Na fi at 21 MHz and a
=0°. The solid curve fits the 27 Al T1-
data in Ag p-alumina.
350
200 K, the 27 Al relaxation times in Li-Ag f3 are un- usually short. Since Ag+ motion proved a very effi- cient 27 Al relaxation mechanism in Ag fi-alumina, (see Fig. 6) we suggest that the low temperature T1-
minima in Li-Ag fl are primarily due to Ag+ motion.
Due to the uncertainty in the Li-Ag fl composition [13]
and to the complex nature of the 27 Al relaxation pro-
cess in mixed fi-alumina, more experimental evi-
dence is needed to confirm this hypothesis.
4. Conclusions.
-We have interpreted NMR data
in Na ¡J-alumina in terms of a 2D-cd model which
was previously applied to spins coupled by the magne- tic dipolar interaction [8]. The correlation functions
describing the relaxation in quadrupole-perturbed spin systems may be different compared to the pair
correlation of distinguishable particles entering the dipolar spin problem. However, for some models, the dipolar and quadrupolar relaxation have essentially
the same type of correlation function. We refer to a recent review article for discussion of this point [19].
Recent molecular dynamics [20] and mechanical [21]
simulation experiments have already indicated that
cation diffusion in fi-aluminas has a continuum, liquidlike character. The success of the 2D-cd model shows that, for the temperature region above 150 K, the NMR time scale is not sufficiently short to unravel
details of the hopping motion associated with ion transport in the fi-aluminas. Otherwise stated, many steps are necessary to produce the fluctuations in nuclear environment responsible for the Tl-minima
of figure 1. Thus, on the NMR time scale, the motion
is well described by a continuum process.
A question may arise as to whether NMR can detect correlated aspects of motion or whether Tl-data
should be compared with DT (tracer diffusion coeffi-
cient) rather than D,,. Unfortunately, the experimental uncertainty of D,, is even larger than differences bet-
ween the DT and D,,, values reported by the Argonne investigators [22]. Moreover, we have calculated
T1-curves according to the 2D-cd model by using the
two most different expressions for Do reported in the
literature. The theoretical curves agree with experi-
ment nearly as well as those reported in figure 1. There- fore, at the present stage of accuracy, the above ques- tion cannot be answered experimentally.
We have shown that the 27 Al Tl in fi-aluminas
is very sensitive to the diffusion processes of cations and that, in some cases, a simple relationship exists
between NMR and ionic conductivity. However,
when ionic species, or ionic pairs, have widely diffe-
rent jump rates, percolative aspects may influence the conductivity more than the microscopic dynamics sampled by NMR. For example, the tracer diffusion coefficient of Na+ in our Li-Na fi crystal is described
by the expression [23]
in the 450-700 K interval. At 300 K this expression
would give DT = 2.8(10-8) cm2 . s-1 which is 15 times smaller than the Na+ tracer diffusion coefficient in
melt-grown Na fi [22]. Instead, similar Na+ mobilities would have been predicted in Na fl and Li-Na fi by
the NMR results. According to the model calculations
of Wang et al. [24], when Na+ forms a pair with Li+,
the Na+ jump out of the doubly occupied cell is highly improbable since it requires an energy of 0.65 eV. Therefore, we suggest that, while the hopping
of Na+ in a Na+--Li+ pair affects the transport pro-
perties of Na+, it is the Na+ motion in cells not occupi-
ed by Li+ which determines the 27 Al and 23Na relaxa- tion of Na-Li near 200 K. A preliminary 23Na Tl
measurement in Li-Na fi at RT showed a distinct multi-
exponential character in that part of the recovery due to the Am = ± 2 transitions. According to theory
of relaxation in the presence of exchanging phases,
this can happen when the Na+ hopping time in Na--Li
is longer than (2 W2)-1 at the time origin (350 ys at RT).
It has been noted that for solid electrolytes NMR
leads often to attempt frequencies different by orders
of magnitude from 1012 +- 1013 s-1, typical of ionic
diffusion [2, 25, 26]. The ’Li line-narrowing processes in Li-Na fi and Li-Ag fl offer another example of this phenomenon. It has been suggested that low dimen-
sionality effects cause the NMR quantities to display
activation energies substantially smaller than those for ionic motion, thus leading to prefactor anomalies [26]. Another explanation calls for the breakdown
of the absolute rate theory upon which the concept of attempt frequency is based [2, 25]. Altematively,
it may occur that NMR is sensitive to local motion of low energy which do not contribute to diffusion.
For the Li+-containing samples, we cannot definiti- vely rule out any of the above interpretations. However,
we believe that in both Li-Na fi and Li-Ag fl the ’Li
line narrowing process, as well as the ’Li and 2’Al
T1-minima at high temperature, are due to the Li+
diffusion. In fact, a qualitative argument suggests that the correlation time i is of the order of (2 n £5VRJ-l
when the linewidth becomes ÔVRL/2. By putting
ro = 2(1012) s-1 in equation (6) and by choosing E equal to the activation energy of ’Li at high tempera- ture, the temperature when the linewidth becomes
ÔVRL/2 is correctly predicted for both Li-Na fi and Li-Ag fi. In addition, with the previous assumptions
about T. 1 and E we would have Wo ’L = 1 at tempera-
tures between those of the 2’Al and ’Li Tl-minima
at high temperature. At the present time, we cannot explain why the phenomenological BPP equation (Eq. (5)) fails so badly to yield the expected values for To and E while providing an accurate description for
the line narrowing process. Among mixed aluminas, Li+ containing fi-aluminas were known to have some unique properties. For Li-Na fl-aluminas some evi-
dence was presented for a conductivity which was due
essentially to Li+ mobility and was higher than that
of pure Li fl-alumina [27]. The term « co-ionic conduc-
tivity » has been coined to signify that Li+ ions can
be electrochemically passed through Li-Na p-alu-
minas without significantly altering the Li+/Na+
ratio [27]. Instead, our results can be qualitatively
summarized as follows :
-
In Li-Na fi, Na+ has a mobility comparable to
that of Na + in y Na fi. This is a major difference in
comparison with K-Na fi, Rb-Na fl and Tl-Na fl
which show the ordinary mixed alkali effect.
-
