an electronic model for amorphous carbon

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An electronic model for amorphouscarbonH. Marker aa Northern Coke Research Laboratories, School of Chemistry,The University, Newcastle upon Tyne, England

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To cite this article: H. Marker (1967): An electronic model for amorphous carbon,Philosophical Magazine, 16:144, 1193-1206

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[ 1193 ]

An Electronic Model for Amorphous Carbon

By H. HARKER

Northern Coke Research Laboratories, School of Chemistry, The University, Newcastle upon Tyne, England

[Received 26 May 19671

ABSTRACT An examination is made of the concept that amorphous carbon can be

treated as a molecular crystal containing free-radical species. The electrical conductivity of the material is discussed in terms of a ' hopping ' model, similar to that which has been successfully applied to crystals of aromatic substances. Conductivity depends on the formation by thermal excitation of pairs of ionized layers followed by migration of the charges from layer to layer under a potential gradient. If an additional assumption is made, namely that ions are formed most readily from pairs of free-radical layers, several hitherto puzzling features of the electrical conductivity and electron spin resonance behaviour can be accounted for, and a possible relationship between these two properties is brought to light. This molecular crystal model differs from Mrozowski's band theory in suggesting t,hat carbons should be considered as intrinsic semiconductors over the whole range of carbonizat,ion temperature. Further development of the model is advocated, for it shows promise of providing a common basis for a number of the physical and chemical properties of carbon.

5 1. INTRODUCTION IT is of some importance to develop a satisfactory model for the electronic structure of amorphous carbon, not only in order to understand the electrical and magnetic properties of the material, but also to provide a satisfactory basis for its chemical behaviour (Harker 1960). Some years ago, the Mrozowski school (Pinnick 1956) were able to give a reasonably successful account of the conductivity and several other electrical pro - perties of carbon by using band theory concepts that were originally developed for other semiconductors, such as silicon and germanium. However, it has been realized for some time that the application of band theory to carbons is doubtful because of the tacit assumption that the structure of the solid is continuous. This assumption will not hold for carbons if the aromatic layers are essentially separate. Some discontinuity of structure is implied, for example, by the occurrence in carbons of considerable numbers of unpaired electrons, which can be detected by electron spin resonance (E.S.R.) ; the difficulties that band theory meets in dealing with this property have been recognized (Pinnick 1956, Mrozowski 1956).

P.M. 4 H

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1194 H. Harker on an

Crystals of aromatic organic compounds may exhibit semiconducting properties, even though they possess structural discontinuity of the type envisaged. The rapid advance of the theory of their electrical conductivity in recent years (Eley 1948, Inokuchi 1951, Garrett 1959, Okamoto and Brenner 1964) now facilitates the development which is described in this paper of an improved electronic model, along the lines we have previously suggested (Harker 1960, Jackson and Wynne-Jones 1964), by assuming amorphous carbon to be a molecular crystal in which the aromatic layer planes represent the units of structure.

5 2. CARBON AS A SEMICONDUCTOR Amorphous carbon is shown to be a semiconducting material by the fact

that its resistivity (p ) falls with increasing ambient temperature (T ) , behaviour that is associated with the presence of an activation energy ( E ) for electrical conduction (the ' energy gap '). For intrinsic semiconduc - tivity, the number (n) of thermally excited electrons (and the equivalent number of positive holes) will be given (see for example Shepherd 1958) by an expression of the type :

n = A exp ( - E/2kT), . . . . . . . (1)

where A is a constant and E is Boltzmann's constant.

Fig. 1

I I I I 1000 2000 3000

HEAT- TREATMENT TEMPERATURE (OC)

Electrical resistivity of a graphitizing carbon (measured at room temperature) as a function of heat-treatment temperature.

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Electronic Model for Amorphous Carbon 1195

If the resistivity is dependent only on the number of current carriers present (i.e. the variation of the mobility of the carriers with temperature is negligible), then log p should be a linear function of 1/T and this is usually found to be the case in carbon. The slopes of such plots lead to energy gap values which are in fair agreement with those obtained by means of the independent method of infra-red spectroscopy (Pinnick 1956).

It is usually assumed by analogy with other semiconducting materials that, in addition to the thermally excited carriers discussed above, extrinsic carriers may also be present in carbons as a consequence of the occurrence of structural defects ; inferences about whether these carriers are ' excess electrons ' (n-type semiconductors) or ' positive holes ' (p-type) are often made on the basis of measurements of Hall coefficient, or the closely related property thermoelectric power.

It is well known that the magnitude of the electrical resistivity of carbon is largely determined by heat-treatment temperature (HTT) and is not greatly dependent on the nature of the organic material from which it was prepared. Typical behaviour is shown schematically in fig. 1 in which the resistivity, measured a t room temperature, is plotted as a function of HTT. The parent organic material is generally a very poor conductor but on carbonization the resistivity decreases in a very spectacular way, reaching a reasonably steady value at a heat -treatment temperature of about 1 0 0 0 " ~ ; a further decrease of small magnitude may occur a t higher HTT.

$ 3 . THE BAND MODEL I n the theory developed by Mrozowski and co-workers, the energy gap is

associated with excitation of electrons from a valence band to an empty conduction band. The steady growth of the aromatic layers with HTT would lead to the progressive decrease that is observed experimentally of the size of this energy gap. However, the latter is too large in specimens prepared a t temperatures below about 1500"c (Pinnick 1956) to allow the formation of a significant number of current carriers by thermal excitation (intrinsic behaviour) .

Mrozowski interpreted the very marked fall in resistivity that occurs at low HTT as being due to a very large increase in the number of extrinsic carriers which, on the basis of Hall coefficient measurements, were believed to be positive holes in the valence band. These were thought to arise as a. consequence of the loss of hydrogen atoms from the peripheries of the layer planes by a mechanism that was later clarified considerably as a result of theoretical work by Coulson (1960).

The main reason for the relative constancy of resistivity a t HTT above 1 2 0 0 " ~ was assumed to be that virtually all of the hydrogen is driven off below this temperature. The subsequent decrease a t still higher HTT was taken to indicate that the energy gap had become sufficiently small to allow the onset of intrinsic behaviour.

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9 4. ELECTRICAL CONDUCTIV~TY IN MOLECULAR CRYSTALS Many molecular crystals behave as semiconductors in the sense that

their electrical conductivities increase exponentially with temperature, though, as was realised some years ago (see, for example, Kallmann and Silver 1961), band theory is not really applicable because their structures are not continuous. For this reason an alternative model was developed (Lyons 1957), originally for naphthalene and anthracene, which involves ' hopping ' of electrons from molecule to molecule. This model has now been used quite successfully by a number of other workers (e.g. Matsen 1959, Martin and Ubbelohde 1961, Calvin 1963) in discussing the conductivities of a variety of organic solids.

It is assumed that ionized molecules can be formed by thermal excitation :

. . . . . . . 2R-t R+ + R-, (2)

this process being analogous to the creation in a continuous crystal of an excess electron anda positive hole. Conduction of electricity can then take place by ' hopping ' of the positive and negative charges in opposite direc- tions under a potential gradient as follows :

R++ R-t R+ Rf, . . . . . . (3)

. . . . . . . R- + R -> R -+ R-. (4)

Since the ' hopping ' processes (3) and (4) are believed to involve a smaller energy barrier than the initial formation of ions (2), the measured energy gap is usually associated with the latter. Using the simplest theory its magnitude should be given by :

. . . . . . . E = I -- A - C , ( 5 )

where I is the ionization potential of a molecule, A is the electron affinity, C is the energy of electrical attraction of the ions. Some refinement, to allow for delocalization of the charges over several molecules, subsequently proves to be necessary, as pointed out by Fox (1959), to bring the calculated energy gaps into agreement with experimental values.

This model explains quite well the fact that crystals of molecular com- plexes between aromatic molecules and either electron donors or acceptors often show high conductivity (Akamatu and Inokuchi 1959, Martin and Ubbelohde 1961). It should be mentioned however, in passing, that various alternative models have also been considered (Martin and Ubbelohde 1961) as a basis for the discussion of conductivity in molecular crystals. Perhaps the most important of these assumes that transfer of electrons can occur more readily between molecules in excited than in ground states, because of the increase in effective molecular size on excitation. A theory of this type has been advanced by Eley ( 1 959) in which the energy gap arises by excitation of one electron from the highest filled level of a molecule t o the lowest empty level, conductivity resulting from ' tunnelling ' with negligible activation energy a t both of these levels.

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The possibility of treating amorphous carbon as a molecular crystal has been considered by a number of authors (Matsen 1959, Ubbelohde and Lewis 1960) but the theory has not been developed in any detail. It should be possible in principle to investigate whether this approach is more appropriate than band theory by examination of the mobility of the charge carriers. The ‘ hopping ’ model can be applied to molecular crystals when carrier mobilities do not exceed lo2 cm2/volt-sec (Okamoto and Brenner 1964, Eley et al. 1966), though values that are somewhat higher may be encountered in crystals of organic free-radicals, as discussed below. However, the band model will be applicable only when the mobilities are very much larger. Carrier mobilities have been measured directly in anthracene (LeBlanc 1960) but unfortunately no similar data appears to be available for carbons.

3 5. CA.RBOK AS A SOLID FREE-RADICAL Crystals formed from free -radical molecules deserve special consideration

because their electrical conductivity is likely to be abnormally large. According to Eley ( 1959), this would arise because the unpaired electron is similar to an excited electron in being able to ‘ tunnel ’ from molecule to molecule with extremely small activation energy. A similar con- clusion would however be reached on the basis of the * hopping ’ model,

___

CH,

C,H,

C6H5

Ionization potentials of some free-radical species (from Vedeneyev

Ionization potential (electron

volts)

9.86 9.95 9.84 9.84

8.80

8.4

9.90

9.4

et al. 1965)

Method

Electron impact Electron impact Electron impact Spectroscopy

Electron impact

Photoionization

Electron impact

Electron impact

C2H6

‘eH6

Ionization potential (electron volts)

12.99

13.07

11.65

11.65

9.25 9.21 9.25

Method

Photoionization

Electron impact

Electron impact

Photoionization

Photoionization Electron impact Spectroscopy

if an ion pair can be formed more readily from two free-radical molecules than from two molecules not containing unpaired electrons.

This last assumption seems reasonable for in the first place, the ionization potential of a molecule containing a free spin might well be lower than that of a molecule in which all of the electrons are held quite tightly, by

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119s H. Harker on an

virtue of being paired-up in chemical bonds. Secondly, the electron affinity of free-radicals might be expected to exceed those of ordinary molecules because the accepted electron should pair-up with the free spin already present. The scarcity of experimental data for ionization potentials and electron affinities makes it difficult to evaluate these ideas directly. Some relevant values of the former, taken from a recent publication by Vedeneyev et al. (1965)) are given in the table and it can be seen that the ionization potentials of the aliphatic free-radicals CH3 and C,H, are, as anticipated, considerably less than those of the parent hydrocarbons. I n the case of C,H,, which is the only aromatic free-radical for which a figure appears to be available, the ionization potential differs very little from that of benzene ; its electron affinity (50.9 kcal/mole) is however about 35 kcal/mole higher than that of benzene itself (Gaines and Page 1963).

Fortunately, some information is available concerning an organic free -radical that can be obtained in crystalline form, namely diphenyl picryl hydrazyl (DPPH). This substance is known from work of Hausser (1959) to be a good electron acceptor and its electrical conductivity (Eley et al. 1966) is larger than that of the non-free-radical parent material diphenyl picryl hydrazine by a factor of 103. The compound violanthrone B is also of interest in this connection for, according to Akainatu and Inokuchi (1961), it not only exhibits strong electron spin resonance but also has an electrical conductivity which is about lo4 greater than that of violanthrone itself. It is realized that the increased conductivity may be partly a result of the higher mobility of carriers in free-radical crystals and this is discussed later. Indeed, Eley et al. 1966 have found that the conductivities of several free-radical solids are higher than would be expected solely from the values of their energy gaps relative to those of non -free -radical materials.

The same general situation might exist in specimens of amorphous carbon that contain unpaired electronst if, as suggested, the energy required for excitation of an ion pair from two free-radical layer planes :

. . . . . . R* + R++R++ R**-, (6)

is much smaller than that for their formation from two non-free -radical layers :

. . . . . . R+ R+R*++ R*-. (7 )

This view is supported by some recent work of Waters (1963) concerning the semiconducting properties of carbonized coals. Plots of log p versus 1/T showed distinct breaks, a t a measurement temperature in the region of 200"c, that were particularly marked in specimens prepared a t temperatures below 700"c. Waters considered that the value of the energy gap obtained from the slope of the graph above 2 0 0 " ~ was the ' true ' one. The magnitude of the energy gap calculated from the slope at lower temperatures

~

f Throughout the paper unpaired electrons are represented by asterisks.

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was smaller by a factor of about two and the electrical conductivity in this region was believed to be due to a relatively small number of easily excited electrons, which might well represent the unpaired electrons present in the solid.

Other modes of excitation to ions are presumably possible in addition to the purely thermal mechanism discussed above. Optical excitation in particular might occur very readily in free-radical solids and in this connection it is of interest to note that several solid free-radicals (e.g. DPPH) are similar to carbon in being very dark in colour. This idea might provide a possible explanation for the relationship, which Friedel (1957, 1959, 1960) envisaged some years ago, between the colour of various carbonaceous materials and their unpaired electron contents.

5 6. VARIATION OF UNPAIRED ELECTRON CONCENTRATION WITH

HEAT-TREATMENT TEMPERATURE The variation of unpaired electron concentration with heat-treatment

temperature is qualitatively similar for a variety of organic starting materials (see for example Jackson and Wynne-Jones 1964, Collins et al. 1959) and is shown diagramatically in fig. 2 . A small E.S.R. signal

Fig. 2

500 1000 1500 2000

HEAT- TREATMENT TEMPERATURE ( ' C )

Variation of unpaired electron concentration in carbon as a function of heat- treatment temperature.

develops at about 300"c, thereafter increasing rapidly in intensity and reaching a maximum corresponding to about 30 x lOI9 unpaired electrons/g at 500 to 6 0 0 " ~ . A marked decrease of signal intensity then occurs, the unpaired electron concentration usually approaching zero at HTT in the region of 1100"~. An E.S.R. signal re-appears a t a heat-treatment temperature of about 1 5 0 0 " ~ ; it is generally smaller than the one observed a t low HTT and persists after graphitization. These two regions are quite often separated by a gap, that can cover an HTT range of as much as 400°c, in which unpaired electrons are barely detectable.

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The reasons for these variations of unpaired electron concentration with heat-treatment temperature are not well understood. It has been generally assumed that the initial appearance of spin centres is associated with the formation of chemically unsaturated carbon atoms at the edges of the aromatic ring systems, as a consequence of loss of hydrogen during carbonization. No convincing explanation has yet been given, however, for the sudden disappearance of these centres below 1 1 0 0 " ~ . Some years ago we considered the possibility that the phenomenon might be related to the onset of electrical conductivity ; this might arise if ' pairing-off ' can occur more readily as electrons become more mobile or if, as suggested by Weiss (196l) , carbonyl groups which are most abundant a t HTT 700-8OO0c, can function as electron traps. However, in an investigation of a wide variety of carbons (Jackson and Wynne-Jones 1964), we have found that the onset of conductivity is more closely related to the appearance of spin centres in the structure than to their disappearance.

We have argued that the unpaired electrons observed in the low HTT region are probably of T type rather than of u type ; for example, they are not detected unless aromatic layers are present t o stabilize them (Jackson and Wynne-Jones 1964). Indeed the T-electron hypothesis provides a ready explanation of the rather small chemical reactivity of the spin centres that has been observed (Harker et al. 1966), because resonance will occur over all of the carbon atoms in a layer plane. Moreover, this idea of delocalization over the layers is borne out by what is known concerning the dependence of E.S.R. signal intensity on measurement temperature (Campbell et al. 1966, Harker et al. 1966) : Curie Law is not obeyed exactly, even at the lowest carbonization temperatures, and the deviation becomes progressively worse as the layers grow with increasing HTT.

For these reasons, it is the aromatic layers themselves that are con - sidered to be the free-radical species. Any layer containing an odd number of unpaired electrons will give rise to electron spin resonance but pairing will ensure that the concentration of spin centres will never exceed one per layer. In harmony with this idea, it has been found that the number of aromatic layers does appear to set a natural limit to the free-spin concentration attainable in the low heat -treatment temperature region (Jackson and Wynne-Jones 1964).

Electron spin resonance in the high HTT region is better understood mainly as a result of the work of Castle and Wobschall (1959) and of Singer and Wagoner (1963) ; it has been shown, most convincingly perhaps for single crystals of graphite, that the signal arises from conduction electrons. The occurrence of a gap between the low and high temperature regions in some cases has, however, proved to be extremely puzzling.

5 7. ELECTRON SPIN RESONANCE AND ELECTRICAL CONDUCTIVITY Treatment of amorphous carbons as molecular crystals containing a

proportion of free-radical layers may therefore be more satisfactory than the band model as a basis for both electrical conductivity and electron

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spin resonance. Making certain assumptions, further development of this idea along the lines indicated above brings out a possible connection between the two properties and provides a much improved account of their dependence on heat -treatment temperature.

As soon as unpaired electrons appear in the structure, the spin-removing process :

will occur by thermal excitation with an associated activation energy that is rather small. Nevertheless, the proportion of spins removed in this way at room temperature will be negligible in the lower HTT specimens but as the energy gap diminishes with rising HTT as a consequence of layer growth, a progressively increasing number will be lost. The spin concentration should therefore reach a maximum and then decrease (fig. 2 ) . Moreover, the formation of these ionic species, made possible by the appearance of unpaired electrons, should lead a t the same time to a sudden increase of electrical conductivity (fig. I) .

Eventually, when all of the spins have been removed, there should be no further drop in resistivity, except in so far as the mobility of the charge carriers increases, until the average layer size has increased sufficiently t o allow significant thermal excitation of non -free-radical layers :

R -k R+R*+ -+ R*-. ( 7 )

This second type of excitation process may well explain the subsequent increase of electrical conductivity (shown diagrammatically in fig. 1 ) which has been observed, for example by Mrozowski (1952), in the course of measurements made over a wide range of heat-treatment temperature. Since the products of excitation are in this case radical ions, they should give rise to electron spin resonance, though the dependence of signal intensity on measurement temperature would be quite different from that observed a t low HTT. This suggestion fits in very well with the con- clusion of Singer and Wagoner (1963) that the resonance in specimens prepared at high heat-treatment temperatures is due to charge carriers.

The occurrence of a gap between the low and high HTT regions would then be assumed to imply merely that the first type of excitation is complete before new unpaired electrons begin to be produced by the second.

It should be pointed out that Winslow et al. (1955) and Pohl et al. (1962, 1964) suggested some years ago the possibility of a connection between electron spin resonance and electrical conductivity in carbonaceous materials. Both groups of workers envisaged the formation by thermal excitation of diradical (triplet) molecules. The idea that the unpaired electrons are in these diradical states seems to be ruled out however by the observed dependence of E.S.R. signal intensity on measurement temperature (Austen et al. 1958).

Finally, it should be noted that the ideas put forward here are rather different from those of Berkowitz (1966). This author has also recognized

. . . . . . . (6) R:k + R"+Rt + R**-

. . . . . . .

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the possible influence of charge transfer processes on the magnetic properties of coal-chars but we believe that charge transfer is responsible for the E.S.R. signal observed in the high HTT region rather than the resonance which occurs in low HTT materials as he has suggested.

3 8. OTHER PROPERTIES The molecular crystal model may eventually be capable of providing a

satisfactory basis for other physical and chemical properties of amorphous carbon. It would be important for instance to account for the variation of Hall coefficient with heat -treatment temperature. Typical data is presented schematically in fig. 3. Mrozowski explained this behaviour in band theory terms as being due to changes in the number of charge carriers, though it should be pointed out that the magnitude of the Hall coefficient, will also be a function of the mobility of the carriers (see for example Shepherd 1958). He also concluded, from the observed sign of the Hall coefficient, that carbons were positive hole (13 -type) semiconductors, a t least over most of the low HTT range.

Fig. 3

I I I

1000 2000 3000

HEAT-TREATMENT TEMPERATURE ('C)

Hall coefficient of a graphitizing carbon as a function of heat-treatment temperature.

The alternative model presented here clearly differs in an important respect from the band theory of Mrozowski, for since the number of posi- tively and negatively charged layers will always be equal, amorphous carbon would be pictured as an intrinsic semiconductor over the whole range of heat-treatment temperature. No direct evidence to test this

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Electronic Model j o r Amorphous C'arbon 1203

hypothesis appears to be available for turbostratic carbons, though it is interesting to note that cyclotron resonance absorption curves (Nozikres 1958) and magneto-conductivity analysis of galvanometric data (McClure 1958) indicate that, in the case of graphite, the numbers of positive holes and excess electrons are almost exactly equal. On this basis, the observed variations of Hall coefficient would then be interpreted as being entirely a consequence of changes of carrier mobility.

In a molecular crystal not containing free-radical layers, positive charges formed by excitation will be transported through the solid under a potential gradient by repetition of the process :

. . . . . . . . R*f -/- R-tR -/- R*+, (8) the mobility of the positive hole presumably being governed by the ionization potential of the aromatic layer planes. The mobility of the negative charges will probably be different because the transfer process would in this case be:

. . . . . . . R*- + R+R + R*-, (9) so that the energy barrier would depend instead on the electron affinity of the layers.

It may well be possible to account for the observed variation of Hall coefficient with heat -treatment temperature in amorphous carbon purely in terms of changes of carrier mobility though the situation is likely to be rather more complex than in a simple molecular crystal because, when free-radical layers are present in the structure, transport processes such as :

and . . . . . . . . R++ R*+R* + Rf (10)

R**-+R*+R*+R**- (11) . . . . . . will also have to be considered. Indeed, the occurrence of these mecha- nisms may well account for the fact, mentioned earlier, that the mobility of charge carriers is in general considerably larger in free-radical than in non-free-radical crystals. Although it would be premature to attempt a detailed account at this stage, it seems possible that the sudden changes of Hall coefficient, referred to above, may coincide with the appearance and disappearance of unpaired electrons from the structure.

The ideas put forward here may also have some bearing on the magnetic susceptibility behaviour of carbons. It would be important in particular to investigate how far the formation of charge carriers of the type suggested can explain the large anisotropic diamagnetism (Pinnick 1956, Adamson and Blayden 1959, Honda 1959, Pacault and Marchand 1959) that often develops quite suddenly a t heat -treatment temperatures in the region of 1500'~ (fig. 4). In this connection, it is very interesting to note that, in carbons prepared above 1400"c, Pacault et al. (1960) found a remarkable concordance between the number of charge carriers determined by magnetic Susceptibility measurements and by electron spin resonance.

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If the magnitude of the diamagnetic susceptibility in high HTT carbons is governed mainly by charge carriers, further support could be adduced for the contention that variations of Hall coefficient do not necessarily reflect changes in carrier concentration. For example, in the carbon for which data is represented in fig. 4, the susceptibility remains constant for samples prepared between 2 1 0 0 " ~ and 3000"c, although the Hall coeficient is known to vary considerably over this range (Pinnick 1956).

Fig. 4

I

1000 1400 1800 2 2 0 0 2600 3000 H E A T - T R E A T M E N T TEMPERATURE ('C)

Magnetic susceptibility of a graphitizing carbon as a, function of heat-treatment temperature.

The ideas outlined may prove to be relevant also to a discussion of the structure and mechanical properties of amorphous carbon. In the absence of unpaired electrons, separate layer planes will be held together by forces that are of purely van der Waals type but when the appearance of free spins allows ionization to occur, coulombic forces may add to the overall cohesion of the solid.

Finally, the proposed model may eventually provide an improved basis for discussing the chemical properties of amorphous carbons. Although the progress made, using this approach, in the theory of their reactions with oxidizing gases has so far been rather limited, the molecular crystal model has already proved to be useful in considering the adsorption of halogens (Campbell et al. 1963, Robson, Harker and Wynne-Jones 1967).

ACKNOWLEDGMENTS This study formed part of the fundamental research programme of the

British Coke Research Association and the author is grateful to the Council of the Association for permission to publish this paper. The interest and support of Professor Lord Wynne-Jones is also greatly appreciated. Finally, the author would like to acknowledge the constructive comments made by a number of others, particularly Dr. H. E. Blayden and Mr. D. Robson.

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Electronic Model f o r Amorphous Carbon

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an electronic model for amorphous carbon

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