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Mechanisms of Reactions in the Solid State.

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[I661 H . Pape, unpublished results.
[167] R . Ukazaki, 7: Okazaki, J . L. Strominger, and A . M . Michelson,
J. Biol. Chem. 237,3014 (1962); H . Zarkowsky, E. Lipkin, and L. Glaser,
ibid. 245, 6599 (1970); S . F . Wang and 0.Gabriel, ibid. 245, 8 (1970).
[I681 G . Wober and 0.Hofmann-Ustenhof, Europ. J. Biochem. 17,393
(1970).
[I741 W Kelleher and H . Grisebach, Europ. J. Blochem. 23,136 (1971);
cf. H . Grisebach and U Dobereiner, Z. Naturforsch. 21 b, 429 (1966).
[168a] G . Wiiber and 0.Hoffman-Ustenhof, Monatsh. Chem. 101,
1861 (1970).
[I691 H . Sandermann j r . and H . Grisebach, Biochim. Biophys. Acta
208, 173 (1970).
[I701 E . Wellmann and H . Grisebach, Biochim. Biophys. Acta 235,
389 (1971)
[I711 D. Baron, E. Wellmann, and H . Grisebach, Biochim. Biophys.
Acta 258, 310 (1972).
[I771 R . Ortmann, H . Sandermann jr., and H . Grisebach, FEBS Lett. 7,
164 (1970).
[172] K . Hahlbrock, J . Ebel, R . Urtmann, A . Sutter, E. Wellmann, and
H . Grisebach, Biochim. Biophys. Acta 244,7 (1971).
[I731 E . Wellmann, D. Baron, and H . Grisebach, Biochim. Biophys.
Acta 244, 1 (1971).
[175] H . Grisebach and U . Dobereiner, Biochem. Biophys. Res. Comrnun. 17, 737 (1964).
[176] J . M . Picken and J . Mendicino, J. Biol. Chem. 242, 1629 (1967).
[178] A . Sutter, R. Urtmann, and H . Grisebach, Biochim. Biophys.
Acta 258, 71 (1972).
[I791 R. Schmid, unpublished results.
[I801 H . 7: Shigeura and S . D. Sampson, Nature 215,419 (1967).
[181] E . Walton, S . R . Jenkins, R . F . Nutt, and F. W Holly, J. Med.
Chem. 12, 306 (1969).
[I821 See e . g . J . M . Wilhefm,N . L. Uleinick, and J . W Corcoran: Antimicrobial Agents and Chemotherapy, 1970. American Society for
Microbiology, Bethesda, Md. 1971, p. 236.
Mechanisms of Reactions in the Solid State
By Kurt Hardel"'
The reaction of two solid phases X and Y to give a solid compound XI', is taken as an example
for the discussion of transport mechanisms and reaction steps. Thefollowing methods of investigation are discussed: determination of the rate law, marker experiments, and calculation of the
reaction rate. It is pointed out that the investigation of powder reactions leads to problematic
conclusions.
1. Introduction
4. Double reaction, e.g. PbS,,,+CdO,,,-tPbO,,,+CdS,,,"';
5 . Superposition of the solid-state reaction by an additional
decomposition of one reactant with formation of a gas, e. g.
If a solid phase X is in contact with a solid phase Y at a
temperature at which the mobilities of the particles in the
solid state are sufficiently high, X and Y can react to form
one or more solid reaction productsXY,,, XY,,, etc. which
separate the reactants from each other. A process of this
kind is a solid-state reaction. The reactants X and Y may
be elements or compounds. Generally the r e a c t p is
exothermic, since the entropy of reaction of solids is low.
Similar features are valid for reactions between several
solid phases, X,, X,, etc. and several solid phases Y,, Y,,
etc. to form one or more solid reaction products.
Only the relatively well-studied reaction type 1 will be
discussed, specifically the formation of a single compound
XY, with a small homogeneity range. X and Y will be binary
ionic compounds having the same anion, and X Y , will be a
ternary ionic compound. The reaction mechanisms and the
principal methods of investigation are discussed for this
simple case. Examples may be found in the literature cited.
If one considers only the reaction of a single solid phase X
with a single solid phase Y ,the following reaction types can
be distinguished :
2. Reaction Mechanisms
1. Formation of one or more compounds ;
2. Formation of a continuous series of solid solutions between the reactants;
3. Formation ofa series of solid solutions with a miscibility
gap ;
p]
Prof. Dr.-Ing. K. Hardel
Institut fur Anorganische und Analytische Chemie der
Technischen Universitat
IBerlin 12, Strasse des 17. Juni 135 (Germany)
Angew. Chem. internat. Edit.
Vol. I I (1972) 1 No. 3
At the beginning of the reaction, the following reaction
step ma!' be assumed :
(I) Exchange processes of particles at the interface between
the two reactants and formation of the reaction product
(phase boundary reaction).
If the reaction product is a ternary compound and if the
contact between the solids is ideal, there are three conceivable transport mechanisms after the formation of a homogeneous, pore-free reaction layer ;these mechanisms, which
173
should be regarded as limiting cases, are shown schematically in Figure 1 for spinel formation from the oxides. A
distinction is to be made between the migration of cations
and anions through the reaction product in the same direction andcountertransport ofcations[2?For thecasein which
the cations and anions of the substance A 0 migrate in the
same direction through the reaction product to the reactant
B,O, (Fig. I),
further reaction steps are as follows:
(11) Transfer of the particles through the interface 1 between the reactant A 0 and the reaction product (phase
boundary reaction).
(111) Diffusion of the cations and anions of the substance
A 0 through the reaction product, specifically volume
diffusion or diffusion via grain boundaries and dislocations
of the crystallites of the reaction product.
the interface at two-dimensional and one-dimensional
defects, at vacancies, or at impuritiest4’. In the spine1
formation from the oxides, the following mechanism has
been proposed for the case of countertransport of cations
and for the structural relationship between the reaction
product and one of the reactantsc5]: After transfer through
the interface between the reaction product and the reactant
(Fig. I)the
, small cations diffuse into the reactant and
occupy the sites of the cation lattice of the reaction product,
while the anion lattice of this reactant is transformed into
the anion lattice of the reaction product by alteration of
the stacking sequence of the lattice planes. This occurs by
migration of partial dislocations.
(IV) Reaction of the cations and anions of the substance
A 0 at the interface 2 with the reactant B,O, to form the
reaction product (phase boundary reaction).
Analogous reaction steps may be assumed for migration
of the cations and anions of the substance B,O, through
the reaction product in the same direction and for countertransport of cations (Fig. 1).
11861.21
I
6e-
I
Fig. 2. Transport mechanisms for the case in which contact between the
solids is inadequate and the partial pressure of the nonmetallic component is sufficiently high.
Fig. 1. Transport mechanisms with ideal contact between the solids.
If the contact between the reactants or between one of the
reactants and the reaction product is insufficient, or if the
reaction layer contains pores, other transport mechanisms
are possible; these are the transport of one reactant via the
gas phase 01’
the transport of the nonmetallic component
tlia the gas phase with simultaneous migration of cations
and of electrons or defect electrons through the reaction
layer (Fig. 2).
Exchange processes of particles at the interface at the
beginning of the reaction have been confirmed by LEED
investigation~[~].
The mechanism of the formation of the reaction product in
steps (I) and (IV) has not yet been sufficiently elucidated. It
is to be supposed that because of the limited solubility of
the reaction product in the reactant, supersaturation of the
reactant with the reaction product can occur in the phase
boundary reaction (I) or (IV), with consequent heterogeneous nucleation and crystal growth. The heterogeneous
nucleation is possible at the interface and in the vicinity of
I74
Reaction step (II), i. e. the finite rate of the transfer of the
particles through the interface between one reactant and
the reaction product (Fig. I),
can be detectedc6! Deviations
from thermodynamic equilibrium at the interface may
inhibit the transfer of the particles through the interface,
with the result that this becomes the rate-determining step
of the reaction.
The volume diffusion through the reaction product, which
depends only on the concentration of point defects, has
been extensively studied. Reaction layers that are at least
several pm thick, and in which the volume diffusion is often
the rate-determining step of the reaction at sufficiently high
reaction temperatures, are relatively easy to investigate.
If the reaction product is a ternary compound and if it is
assumed that the particles migrate independently of one
another and that the ion currents are related only by the
electroneutrality condition, it is possible to distinguish the
three diffusion mechanisms shown schematically in Figure 1.
Since the component diffusion coefficients of the various
types of ions are generally different, the rate of the solidstate reaction is determined by the type of ion whose
mobility lies between the mobilities of the other two types.
With counterdiffusion of cations, for example, the mobility
of the type of cation that diffuses more slowly is ratedetermining[’!
If the reaction temperature is not SO high that volume diffusion predominates, the diffusion of the particles via the
Angew. Chem. internal. Edit. Voi. 11 (1972) No. 3
grain boundaries and dislocations (“dislocation pipes”) of
the crystallites of the product can become the rate-determining step of the reaction. In this case, the reaction rate
depends on the relative orientation of neighboring crystallites of the product[*]and on the size of the coherent crystal
regions. The reaction rate increases with decreasing average
crystallite size or with decreasing average size of the coherent regions of the
If the product is a
ternary compound, it is possible, in analogy with volume
diffusion, to discuss diffusion of cations and anions in the
same direction or counterdiffusion of cations.
With good contact between the solid substances, if the
partial pressure of the nonmetallic component is sufficiently
high, one may find the special case that the reaction product
is formed not only between the reactants but also on part
of the surface of one of these sub~tances~’’~.
It is then possible that counterdiffusion of cations occurs in the interior
of the reaction layer (Fig. I), whereas in the zones near
the surface,cations and electrons or defect electrons migrate
through the reaction product, and the nonmetallic component is simultaneously transported via the gas phase
(Fig. 2).
If it is established that the formation of the reaction product
in step (IV) is coupled throughout the reaction with nucleation processes at the interface between the product and the
reactant (Fig. I),
it is possible that the average crystallite
size of the reaction product depends on the rate of nucleation of the product at this interface. If grain boundary
diffusion through the reaction product is rate-determining,
it is thus necessary to check whether factors that affect the
rate of nucleation of the reaction product at the interface
between the reactant and the product lead to a change in
the average crystallite size of the product, and hence to a
change in the reaction rate. Such factors would be the
crystal structure, the crystallographic orientation, and the
defect structure of the reactant, as well as contamination of
the reactant with impurities[” - 14]. The mechanism has
not yet been proved beyond doubt[”.
The rate-determining step of these reactions depends
mainly on the experimental conditions under which the
solid-state reaction takes place. The transport mechanism
is mainly determined by the experimental conditions, the
mobilities of the various types of ions in the reaction product, and the electronic properties of the product.
Where contact between the solid substances is inadequate,
the influence of the gas phase on the solid-state reaction
must be taken into account. If gaps in which a certain partial
pressure of the reactant is produced remain between one of
the reactants and the reaction layer during the reaction,
and if the reaction temperature is so high that this partial
pressure is not negligible, the mass transport takes place
via the gas phase. For sufficiently thin reaction layers, the
evaporation of the reactant, the transport in the gas phase,
and the condensation on the reaction product may become
rate-determir~ing[~].
This can also be observed[’5! With
thick layers of reaction product, however, diffusion through
the product is the rate-determining step of the reaction.
If partial conductivity of electrons or defect electrons in
the reaction product is not negligible, and if the partial
pressure of the nonmetallic component is sufficiently high,
the nonmetallic component may be transported via the gas
phase in cases of insufficient contact between the solid substances or in cases where the reaction layer contains pores,
while electroneutrality is maintained by an electron or
defect electron current through the reaction product. For
the example of spinel formation from the oxides, Figure 2
shows schematically the transport of oxygen through the
gas phase with simultaneous transport of the cation A z +
or B3+ and of electrons through the spinel layer. It is assumed that the particles migrate independently of one another
and that the ion and electron or defect electron currents
are coupled only by the condition of electroneutrality. The
rate-determining step is the diffusion of the cations A’+
or B 3 + through the reaction product. If the oxygen partial
pressure is decreased to such an extent that there are no
longer sufficient oxygen molecules available for the reaction
at the interface, another transport mechanism must come
into effect[l61.
Angew. Chem. internat. Edit.
1 Vol. 11
(1972)
1 No. 3
3. Methods of Investigation
It is assumed in the following discussion that the reaction
is isothermic. This condition is satisfied if the mass fraction
of the reaction product formed per unit time is relatively
small. This assumption is generally permissible in solidstate reactions of the type specified in Section 1.
The principal methods used to investigate the mechanisms
of such solid-state reactions are the following :
1. determination of the rate law to establish the ratedetermining step of the reaction ;
2. marking the interface between the reactants before the
beginning of the reaction to establish the transport mechanism ;
3. calculation of the reaction rate if the volume diffusion
through the product is the rate-determining step. It is
possible in this way to decide which of the diffusing types
of ions determines the rate of the reaction.
3.1. Determination of the Rate Law
If the phase boundary reaction (11) or (IV) is the rate-determining step, the reaction rate is independent of the thickness A x of the reaction layer; one obtains a linear rate law,
which can be expressed in the form
A X = l’t
(1)
if the reaction cross section remains constant and if the
growth of the reaction layer can be treated as a onedimensional problem. I’ is the rate constant and t is the
reaction time.
If nucleation is the rate-determining step, one can again
obtain a linear rate law, which is the first approximation of
the exponential law of nucleus formation, and which can
be represented, after the formation of a homogeneous
reaction layer, in the form of equation (1).The exponential
law is the frequently used equation for the rate ofnucleation.
175
To establish whether the rate-determining step is the phase
boundary reaction (11) or (IV) or the nucleation of the reaction product, it is necessary to use further methods of investigation. In the case of nucleation, it may be possible to
use microscopy or electron microscopy as the additional
method of investigation, if it is possible to separate the
reaction layer from one of the reactants without destruction.
processes through the reaction product are rate-determining and the frequently used equations of Jander'"' and of
Serin and Ellickson[221
can be applied.
If volume diffusion through the reaction product is ratedetermining, the reaction rate decreases with increasing
thickness Ax of the reaction layer. The process then follows
the parabolic rate law, which can be expressed, for a constant reaction cross section, i. e. for one-dimensional growth
of the single-phase reaction layer, in the form
If the reactants are used in compact form, specifically as a
single-crystal or as a sintered specimen with plane contact
surfaces, the reaction product is formed between the
reactants as a layer with a constant reaction cross section
and with plane-parallel interfaces. In this case, the growth
of the reaction layer may be treated as a one-dimensional
problem. For reaction layers whose thickness Ax is at least
several pm, Ax can be measured directly by microscopic
observation of sections normal to the reaction cross section,
and the rate law can thus be determined. This procedure
is frequently used.
k' is the rate constant and t is the reaction time.
Equation (2) also describes the growth of the reaction layer
if, instead of the volume diffusion, the rate-determining
step is the diffusion via the grain boundaries and dislocations of the crystallites of the product, provided that the
flux equation['*] can be applied to the ion current^"^'.
It is possible to decide between rate-determining volume
diffusion and rate-determining diffusion via the grain
boundaries and dislocations if the reaction rate is found to
depend on the average crystallite size or on the average
size of the coherent regions of the reaction product. The
average crystallite size can be determined with the aid of
the microscope or the electron microscope ; the average
size of the coherently scattering crystal regions can be
determined from X-ray line broadening. An indication
which of the two diffusion mechanisms predominates can
also be obtained by comparison of the activation energies
of the two processes. The activation energy of the diffusion
via grain boundaries and dislocations of the crystallites of
the product is relatively low.
Deviations from the parabolic rate law (2) at the beginning
of the reaction, i.e. if the reaction layer is thin, can also
be attributed, in cases of insufficient contact between one
of the reactants and the reaction layer, to the evaporation,
condensation, or transport of one of the reactants in the
gas phaser5].With sufficiently thin reaction layers, transport via the gas phase becomes rate-determining, and a
linear rate law (1) is to be expected if the diffusion path in
the gas phase remains constant during the rea~tion"~'.
3.1.1. Investigation of Powder Reactions
Though powder reactions are of great importance in preparative solid-state chemistry and in industry, the definite
establishment of a rate law for these reactions is difficult,
since the growth of the reaction layer is a three-dimensional
problem, the thickness of the reaction layer cannot be
measured directly, and the reaction cross section changes
during the reaction. The investigation is further complicated by other parameters, such as the grain size, the grain size
distribution, the porosity of the grains, and the packing
density. The determination of the mechanism of powder
reactions is therefore problematic[''], even when diffusion
176
3.1.2. Investigation of Reaction Layers with Plane-Parallel
Interfaces
Using one reactant single-crystallineor as a sintered sample
and the second reactant as a cover layer deposited onto the
plane substrate, e.g. by evaporation in vacuo or by sputtering, optimum contact between the reactants can be
achieved under suitable experimental conditions. The reaction product is then generally also formed under the cover
layer as a homogeneous layer with a constant reaction
cross section and plane-parallel interfaces, so that the
growth of the reaction layer may be treated as a onedimensional problem. Since only a thin cover layer is
applied to the substrate in this method, the procedure is
suitable for the determination of the rate law for thin reaction layers up to a maximum of a few 1O"A thick. This
method yields information about the initial stages of a
solid-state reaction, for example about the reaction
steps (I) to (IV) that can become rate-determining in the
case of thin reaction layers. Very little is known about the
initial stages of solid-state reactionsrg*15], apart from the
problematic results obtained by the investigation of powder
reactions.
However, the thickness of the thin reaction layers cannot
be measured microscopically. In some cases, the cover
layer of one of the reactants can be quantitatively removed
from the reaction layer by means of a suitable solvent; if
the density of the reaction product and the cross section of
the reaction layer are known, the thickness Ax of the singlephase reaction layer can then be found by weighing, and
the rate law can be determined. In other cases, it is possible
to use a procedure[231based on the measurement of the
thickness Ax of the thin reaction layer X Y , (maximum
thickness several lo3A) between the substrate X of one
reactant (at least a few millimeters thick) and the thin cover
layer Y of the other reactant by X-ray fluorescence analysis
(Fig. 3).
The intensity of the X-ray fluorescent radiation of an element in the sample is determined before the beginning of the
reaction, i. e. without the reaction layer X Y , , and after the
reaction time t, i. e. with the reaction layer XY,, with monochromatic X-rays; the thickness A x of the reaction layer
after the reaction time t can then be found from the change
in intensity, and the rate law can be determined. The intensity I of the X-ray fluorescent radiation is obtained by
Angew. Chem. infernal. Edit.
/ VoI. 11 (1972) / No. 3
surface measurement without destruction of the sample.
The following conditions must be satisfied :
The method has been successfully tested by measurement
of the thickness Ax of thin layers of Pb,SiO, up to several
lo3 A
The Pb,SiO, layer was situated between a
quartz single crystal or a quartz glass substrate and a thin
cover layer of PbO. The Ax values found by X-ray fluorescence analysis agreed with those found by weighing after
the removal of the PbO cover layers.
3.2. Marker Experiments
Flg. 3. Excitation of X-ray fluorescent radiation in the sample
If the reactants are used as single crystals or as sintered
samples with plane contact surfaces, it is possible to
determine the transport mechanism on which the solidstate reaction is based by marking the interface before the
beginning of the reaction with wires or foils of inert material,
e.g. platinum, a few pm thick (Fig. 4a). By microscopic
observation of the position of these markers after the
reaction in sections normal to the reaction cross section,
it is possible to distinguish the cases shown in Figures 4 b
to 4 d for spinel formation from the oxides. If the position
of these markers after the reaction is assumed to indicate
the position of the interface between the reactants before
the reaction and if the homogeneity ranges of the reaction
product and of the reactants may be disregarded, the experiments can be interpreted as follows :
1. The reaction layer is in the form of a homogeneous layer
with plane-parallel interfaces. The roughness of the phase
boundaries is disregarded.
2. In the quasi-binary system of the reactants X and Y,
only one compound X Y v is formed, or the layer thicknesses
ofany other compounds formed in this system are negligible.
The homogeneity ranges of the reactants and of the reaction
product are small.
3. The cover layer Y contains an element Y that does not
occur in the substrate X, or vice versa.
4. The X-ray fluorescent radiation of the excited element
is sufiiciently absorbed by passage through the sample.
Two methods of measurement can then be distinguished,
depending on whether the X-ray fluorescence of the element
X is excited in the substrate X and in the reaction layer
XYv under the cover layer Y (Fig. 3a) or whether the X-ray
fluorescence of the element Y is excited in the cover layer
Y and in the reaction layer X Y , (Fig. 3 b). For both cases,
it is possible to give functions for the determination of the
thickness A-i of the reaction layer under the cover layer"'
if the absorption coefficient of the X-rays can be determined by investigation of the cover layer k' \\i[hout the
reaction layer X Y , and of the reaction layer XY, without
the cover layer Y. The more favorable of the two methods
(Fig. 3 a or Fig. 3 b) is chosen for experimental studies. It is
advisable to excite the element in the sample whose X-ray
fluorescent radiation is most strongly absorbed on passage
through the sample, in particular an element with a relatively low atomic number that emits long-wave X-rays.
[*] Cf. eq. (12) or (6) in Ref. [23]
Angew. Chem. infernat.Edit. / Vol. I 1 (1972) / No. 3
Fig. 4. Position of the markers before (a) and after (b to d) the reaction.
1. If the inert markers are situated in the interface between
one reactant and the reaction product (Figs. 4 b and 4c),
this indicates transport of the cations and anions of this
reactant in the same direction through the reaction product
(Fig. 1) or migration of cations and of electrons or defect
electrons through the reaction layer in conjunction with
transport of the nonmetallic component zia the gas phase
(Fig. 2).
1I1
2. If the inert markers are situated inside the reaction layer
(Fig. 4d), this points to countertransport of cations (Fig. 1).
In spinel formation from the oxides, the marker then divides
the spinel layer in the ratio 1 :3, since according to Fig. 1
only Imole of spinel is formed at interface 1, while 3 moles
are formed at interface 2.
If the transport of a reactant via the gas phase is involved,
a reaction layer is formed on one of the reactants even when
the compact reactants are separated before the reaction
by platinum wires a few hundred pm thick and when this
separation is maintained during the reaction.
However, the condition that the position of the interface
before the reaction is still indicated by the inert markers
after the reaction is frequently not satisfied. It is better to
use natural markers. For example, one of the reactants may
be used single-crystalline, the cross section of this sample
being large in relation to the cross section of the compact
sample of the other reactant; the parts of the surface of
the single crystal on which no reaction has occurred can
then be used as a natural marker. If the reactants differ in
their porosity or if one is monocrystalline and the other
polycrystalline, the change in porosity or the monocrystallinity or polycrystallinity of the reaction layer may be used
as a natural marker.
Experiments with markers are not possible if the reaction layers are less than a few pm thick. In such cases, the
results of marker experiments with thick reaction layers
must be used to obtain information about the transport
mechanism. The same is true of powder reactions.
through the interfaces is not inhibited. This condition is
generally satisfied for sufficiently thick reaction layers.
6. The particles migrate independently of one another. The
ion and the electron or defect electron currents are coupled
only by the condition of electroneutrality.
If the mobilities of the various types of ions in the lattice of
the product differ sufficiently from one another, certain
diffusion mechanisms occur.
If the oxygen ions have the lowest mobility in the spinel
lattice, counterdiffusion of cations (Fig. 1) occurs when
contact between the solids is ideal. The diffusion of the type
of cation having the lower mobility is then the rate-determining step. If this is e.g. the ion A'+ and if the homogeneity
range of the spinel is small, the rational reaction constant
k can be calculated from the following equation1181:
k
=
yc,n;'
DA(aAo= 1) [l-exp(n,AGo/RT)]
(3)
y is a numerical factor of the order of 1, cA is the molar
concentration of the ion A * + in the reaction product, R
is the gas constant, and T is the absolute temperature.
DA(aAo=l) is the self-diffusion coefficient of the ionic
species A'+ in the reaction product when the activity of
the reactant A 0 in the reaction product is 1. AGO is the
standard free energy of the reaction AO+B,O, =AB,O,.
The reaction constant k is related to the rate constant k'
of the parabolic rate law (2), which can be determined
experimentally by k = k ' / R where P is the equivalent volume
of the reaction product.
3.3. Calculation of the Reaction Rate
If the volume diffusion through the product is the ratedetermining step, the transport mechanism found from
marker experiments can be checked and confirmed by
comparing the rate constant of the parabolic rate law
calculated on the basis of this transport mechanism with
the measured rate constant. The rate of such a solid-state
reaction can be calculated for a small spinel phase widthr1*]
if the following conditions are satisfied :
1. The reaction is isothermal and isobaric. The partial
pressure of the nonmetallic component is constant, so
that the disorder type of the reaction product does not
change.
2. The homogeneity ranges of the reactants and of the
single-phase reaction product are small.
3. The reactants and the reaction product contain no pores.
The contact at the interfaces is ideal.
4. Local thermodynamic equilibrium exists within the
reaction layer. This means that the point defects are responsible for the transport of the particles through the
reaction layer. The reaction temperature must therefore
be so high that the diffusion via the grain boundaries
and dislocations of the crystallites of the product is
negligible.
5 . Thermodynamic equilibrium is maintained at the inter-
faces during the reaction, so that the transfer of the particles
178
AGO can be measured with the aid of galvanic solid-state
cells. DA(aAo= 1) must be determined for the case where
the reaction product is in equilibrium with AO, i.e. the
activity of A 0 in the reaction product is unity. nA can be
determined from the equation"']
(4)
where DA(aB20,= 1) is the self-diffusion coefficient of the
ions A2+ in the reaction product if the reaction product
is in equilibrium with B,O,, i. e. the activity of B,O, in the
reaction product is unity. nA has also been calculated for a
number of disorder typesr'', 2 5 !
4. Closing Remarks
The transport mechanisms and reaction steps discussed
here are also valid in principle when several ternary compounds are formed instead of one in the quasi-binary system
of the two reactants.
The knowledge of the mechanisms of solid-state reactions
can lead to new synthetic pathways in preparative solidstate chemistry. However, our knowledge of the kinetics of
Angew. Chem. internat. Edit. 1 Vol. I 1 (1972) 1 No. 3
[9] K . Hardel and B. Strocka, 2. Phys. Chem. (Frankfurt am Main) 67,
8 (1969).
these reactions is still limited. In particular, the following
problems are to be clarified :
[lo] J . F. Laurent and J . Benard, 1. Phys. Chem. Solids 7, 218 (1958).
[IllH . J . De Bruin, G. M. Watson, and C. M.Blood, J. Appl. Phys. 37,
1. the initial stages of solid-state reactions ;
4543 (1966).
2. the mechanisms of reaction types 4 and 5 in Section 1.
[I21 J . W Matthews, Phil. Mag. 12, 1143 (1965).
[13] G. R. Hennig, Appl. Phys. Lett. 4, 52 (1964).
Received April 20, 1971 [A 864IEl
German version: Angew. Chem. 84,227 (1972)
Translated by Express Translation Service, London
[I41 J . L. Robins and 7: N . Rhodin, Surface Sci. 2, 346 (1964)
[ I S ] K . Hardel, Z. Phys. Chem. (Frankfurt am Main) 65, 86 i1969).
[I61 H . Schmalzried, Z. Phys. Chem. (Frankfurt am Main) 33, 111
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Chemistry of the Positron and of Positronium“]
By Hans J. Ache[*]
I n this progress report, the properties and behavior of the positron (“positive electron”,anti-electron) and of the positronium, a “hydrogen atom” containing a positron instead of a proion, are
considered from the chemist’s viewpoint. Examples are given to demonstrate the development
of positronium chemistry, in aqueous solution and in the gaseous, liquid, and solid phases, with its
problems and possibilities.
1. Introduction
The reactions of the hydrogen atom and of the proton have
been the subject of numerous investigations and must be
familiar to every chemist. The reactions of deuterium and
of tritium, the heavy hydrogen isotopes with masses 2 and 3
respectively, which are often used as “labeled hydrogen
atoms for the investigation of reaction mechanisms, are
also extensively known.
It is probably not so well known, however, that there is an
“atom” that can be regarded as an analog of the hydrogen
atom, but in which the proton is replaced by a positron,
and which thus represents the lightest isotope of hydrogen.
This particle, which is known as a positronium, has a very
limited lifetime (of the order of lo-’ to 10- l o s) before it
decays with emission of two or three photons.
Despite this short lifetime, which rules out conventional
product analysis as used e. g. in the reactions of deuterium
[*IDr. H. J. Ache;Professor
of Chemistry
Department of Chemistry
Virginia Polytechnic Institute and State University
Blacksburg, Virginia 24061 (USA)
Angew. Chem. internat. Edit.
Vol. 11 (1972)
No. 3
or tritium, it is an excellent “labeled” hydrogen atom for
the investigation of chemical and physical processes in
matter; this is because its lifetime and the mechanism of
its decay process are determined by the chemical and
physical state of the environment. The positron-electron
annihilation process is one of the few known nuclear
processes whose course depends in a characteristic manner
on the chemical and physical structure of the environment.
These elementary particles can therefore serve as nuclear
probes, and can provide a great dea1 of information about
the properties of the surrounding matter.
At present there are essentially four nuclear processes that
satisfy these conditions: 1. the Mossbauer effect; 2. the
positron-electron annihilation process ; 3. the angular
distribution between two successive y quanta in the emission of y cascades; and 4. the depolarization of muons and
muonium formation.
Whereas e. g. the Mossbauer effect has had a firm place in
analytical chemistry for several years, the part that the
positron or positronium can play in the solution of chemical problems is still largely unknown.
The physico-chemical aspects of the interaction between
the positron and matter will therefore be examined below,
179
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