Deformation of graphite lattices by interstitial carbon atoms

Carbon

1966, Vol. 3, pp. 445-457.

Pergamon Press Ltd.

DEFORMATION

Printed in Great Britain

OF GRAPHITE

BY INTERSTITIAL

LATTICES

CARBON ATOMS

C. A. COULBON Mathematical

Institute, Oxford, England

and S. SENENT, Laboratorio

M. A. HERI?AEZ*, M. LEAL. de Qufmica Ffsica, Universidad

and E. SANTOSf

de Valladolid, Spain

(Received 5 August 1965) Abstract-Calculations are made of the self-energy of an interstitial carbon atom placed between the layers of an otherwise perfect graphite crystal. The position of minimum energy ia co&n& to be directly above an atom in the one plane, and centrally placed below three atoms in the other plane. It is essential to allow for deformation of the layers up to two on each side of the interstitial. The maximum displacement of an atom near to the interstitial is about 0.6 A. The most uncertain feature of the analysis is the form of the carbon+zrbon repulsion potential. Various alternatives are compared, and the choice of a function due to Crowell is made. With this function the self-energy of an interstitial (i.e. energy difkrence between the interstitial situation and a perfect lattice with the extra atoms carefully removed) is about 2.32 eV, of which about 0.12 eV comes from the deformation of second neighbour planes. The migra.ion energy of an interstitial is estimated to be about 0.14 eV, and comparison is made with available experimental measurements.

1. INTRODUCTION

defined the energy of a vacancy as that energy which must be provided in order to remove an atom from the lattice and place it at the boundary edge of a layer: we now define the energy of the interstitial as that energy which must be provided to an atom on the boundary in order to force it into its most stable position between two layers, taking account of the relaxation of the lattice around it. The sum of the vacancy and interstitial energies is therefore the minimum energy needed to create a vacancy and an interstitial simultaneously in an otherwise perfect graphite crystal. The existence of interstitials after neutron irradiation may be inferred from (1) an increase of up to 8% in vo1ume,(4) and (2) the Wigner energy release. It has been shown experimentally(s~ that these defects disappear almost completely when the temperature is raised to 17OO”C, though by far the greater part of the accumulated energy of the interstitials is freed at considerably lower temperatures.@) This energy has been measured experimentally(‘) and so has the associated decrease in size of the pores.(s) Since 1950 the effects of

THE changes which occur in graphite,

when it is used as a moderator in nuclear reactors, have often been studied, both experimentally and theoretically. (References are cited in an earlier paper.(‘)) These changes, in both physical and chemical properties, are due to defects caused by collisions with neutrons and other particles produced in the reactor.@) The most important primary defects are single vacancies and single interstitials.(3’ By an interstitial we mean a carbon atom which, after being ejected from a regular lattice site, ultimately settles between two layers, and is well-separated from its associated vacancy. In an earlier paper(‘) we calculated the energy and other properties of a vacancy: in this paper we deal entirely with an interstitial. This will involve us in both the local distortion of the lattice around the interstitial atom, and slso the interstitial energy. We previously *Presently at Universidad de Santiago de Compostela, Spain. j-presently at Universidad de Costa Rica, Costa Rica. 44s

446

C. A. COULSON, S. SENENT, M. A. HERRAEZ, M. LEAL and E. SANTOS

Interstitial

atom

FIG. 1. Part of a graphitecrystal showinginterstitialatom.

irradiating graphite have been widely studied; full accounts are in the books by UBBELOHDEand LEWIS,(~)and NIGHTINGALE,~~ ‘1 and in a survey by BELL et al.(“) Figure 1 shows a part of a graphite lattice. It is clear that since the van der Waals radius of carbon is about 1.7 A, the presence of an interstitial atom will considerably distort the local planar structure. We shall assume that the interstitial atom remains, like a foreign body, between the layers, forming no chemical bonds, and remaining in its s2p2 condition. It may therefore be regarded as exerting forces of repulsion on all the other atoms near it. These forces, which tend to deform the lattice by buckling the adjacent planes, will ultimately be balanced by the internal forces within each layer, and between the layers, which tend to preserve planarity. If we are to study an isolated interstitial we shall have to calculate three energy values-the selfenergy, the formation energy and the migration energy. The self-energy is defined as the amount of energy which must be given to an atom of gaseous carbon (in the ground state s2p2, “P) in order to place it interstitially within the crystal; the formation energy is defined as the energy needed to force an atom at the boundary of a layer into an interstitial position; the migration energy is the minimum energy which has to be applied to an interstitial atom in equilibrium to enable it to move within the lattice. The first theoretical calculation of the selfenergy was made by DIENE#~) in 1952, when studying possible mechanisms of self-diffusion. KANTER(~‘) obtained an experimental value of 7 eV for the diffusion of carbon atoms in natural graphite, as compared with 18 eV calculated by DIBNES. More recently, when the present study

was under way, there appeared some further work by IWATAet uZ.,(14)in which the same problem is treated theoretically, taking into account the deformation of the graphite planes around the interstitial. These authors treat the crystal as composed of fine layers, with the properties of elastic membranes. In regard to vacancies BAKERand KELLY arrive at an experimental value of 3.5 eV for their formation energy, while &NNIG(‘6) estimated a value of at least 6 eV. These may be compared with a theoretical value of about 10.7 eV. For their migration energy DIENESestimates theoretically an energy of 3.1 eV, and BAKERand KELLY find an experimental value between 3.2 and 3.7 eV. 2. NATURE OF THE PROBLEM (a) Position of the interstitial Our first problem is to decide where the interstitial atom rests. We may expect this to be a position of some symmetry. Figure 2 shows that there are two likely places A (=A’) and B. We have excluded a third position C since here the closeness of the two near atoms in the adjacent planes would almost certainly lead to a potential maximum. We should expect that an interstitial could

FIG. 2. Possiblepositionsfor interstitialatom. Atoms 0 are in a layer immediatelybelow the interstitial;atoms 0 are in a layer immediately abovethe interstitial.

DEFORMATION

OF GRAPHITE

LATTICES

migrate through the crystal along segments of paths such as ABA’. The migration energy E,,, would then be the difference in self-energy at A and B, since it seems unlikely that there would be any other stationary points for the potential energy on the line AA’. Both DIENE~ and IWATAet al. find that the position of minimum energy is at A, though on account of the different relationships to its closest neighbours in the two planes, the interstitial will not lie equidistant from them but will be nearer to the upper layer (black circles in Fig. 2). At B, however, the stationary point (saddle-point in the energy) will be symmetrically placed with respect to the planes. The energy at the minimum A is the self-energy (E,), the energy being referred to that of an isolated carbon atom. If E, is the sublimation energy of graphite, then E,+E, is the formation energy Ef. If the self-diffusion of carbon in graphite takes place by the formation and migration of interstitials (and having nothing to do with vacancies), then the experimental activation energy (E,) for self-diffusion should be Ef+E,. All this is most easily seen from a diagram (Fig. 3). We may expect that E,,, will be small. For the repulsion energies at A and B cannot differ greatly, particularly when an allowance is made for different deformations of the layers in the two cases. IWATAet aZ.(i4) find a value of only 0.016 eV. We shall find (see below) a somewhat larger value

a’

I

Emrgy

1

L

J

FIG. 3. Diagram to illustrate various energy terms. Et =sublimation energy ; E. =self-energy

of interstitial; Ef=E,+ E,=energy of formation of interstitial from surface of crystal; &=l migration energy of interstitial; E,=h”+ Em= activation energy for self-diffusion.

BY INTERSTITIAL

CARBON

ATOMS

447

0.14 eV. But this is still very small compared with the formation energy Ef. In our present work we shall normally assume tbat the interstitials are sufficiently distant that they do not in%uence each other. We therefore consider just one interstitial in an otherwise perfect infinite graphite crystal. At the end, however, we consider briefly the situation when there is an appreciable number of interstitials. (b) Distortion of the layers The self-energy E, at the equilibrium position A will depend upon (i) the position of the interstitial along the normal to the layer planes, and (ii) the distortion of the layers, represented by the displacements of every atom of all the layers from their mean positions in the pure lattice. The position of A and the distortion of the layers will be such as to minimiie the total potential energy. We cannot deal with the whole problem at once, but shall reach our results by a method of successive relaxations. We regard the total energy E aa the sum of three types of term. One of these is the straightforward repulsion between the interstitial atom and all the other atoms of the crystal; another is the total repulsion between the atoms of one layer and the atoms of another; the third is the energy of deformation of a single layer. Each of these three types of term will have to be calculated separately. In the region where the distortion is largest, the force on any atom due to the interstitial has its largest component in a direction perpendicular to the plane. Now it is already known from the work of COULSON and SENENT~“) that in comparable situations with overcrowded organic molecules, the stress is relieved by atomic displacements which are effectively perpendicular to the mean molecular plane. This is because it is far easier to make a displacement of a carbon atom in a direction normal to its three “bonds” than in the plane of these bonds. We shall therefore make the assumption that the same situation occurs here. The final shape of the crystal is thus defined by the set of displacements zi of each atom i perpendicular to the xy plane. We choose the direction of these (our notation is shown in Fig. 4) so that all zi are positive. We now require the total energy in terms of the zi. Insofar as the part of this which comes from a

C. A. COULSON,

448 Lauer

S. SENENT,

M. A. HERRAEZ,

c

begin with the law of force between any two carbon atoms. We take the law of force to be independent of direction and a function only of their distance apart. If I is this distance, we write U(Y) for their mutual potential energy. We shall sometimes want the energy W(R) of interaction between one carbon atom and a complete plane of carbon atoms at distance R from it. By direct summation

t

R3 Lauer

C

M. LEAL and E. SANTOS

I

IV)

FIG. 4. Notation for displacements. O=interstitial atom; o,b,c,d, layers of lattice.

are neighbouring

single layer is concerned, we can take over the formulae of COULSON and SENDIT, which express this deformation energy in terms of two force constants associated with the non-planarity around any atom and the torsion around any bond. But for the other contributions to E we must

mm_

where p is the number of carbon atoms in the layer at distance yi from the isolated atom. If we may treat the layer as if it were a uniform distribution with density o carbon atoms per unit area, we may write, instead of (l), and using the notation of Fig. 4, W(R) =

k(r) s0

27rx dx 5: zU(r) 2nr dr, sR

101

(2)

If we know U(Y), this enables us to compute W(R). Alternatively if we know W(R), then

e

_--___

_-------

(1)

= % I%)

1107

FIG. 5. Graphical illustration of the relation (2) between W(R) and V(r).

d

DEFORMATION

OF GRAPHITE

LATTICES

We are now in a position to write down the total energy E, using the notation of Fig. 4, and labelling the interstitial atom 0. We have E =CU(ioi) +XU(P$,>-XU(r~~) +ZV, i

Li

Lj

k=a,b,c,

(4) ....

The first term represents the interaction energy of the interstitial with all the other atoms, the prime denoting that thia is the distance in the deformed lattice. The second term represents the interaction energy between all atoms in diierent planes, the double prime implying that the distances are for the deformed lattice. The third term is the same as the second, except that it refers to the undistorted lattice. The last term represents the energy of deformation of each separate layer. We regard E as a function of all the y and also of the coordinatesp and q (p+q=interlayer spacing in pure graphite) which define the position of the interstitial along the normal to the lattice planes. We are to seek those values of these parameters which lead to an absolute minimum of E. 3. SIMPLIFYING

ASSUMPTIONS

We approach the solution of the problem just

outlined in stages. Thus we begin by supposing that in any layer only a finite number 5, nb, . . . . of atoms is displaced by the interstitial. Direct calculation shows, as we should expect, that si rapidly decreases as the distance of the atom i from the interstitial increases. We therefore take certain SetaOf Valuesof n (a= n&-n&+. . . for the different layers), and use our results to extrapolate to infinite n. For each set of values of n, we suppose that the zi adjust themselves so as to make E a minimum. We also assume (Fig. 5) that the deformations are biggest in the planes a and b between which the interstitial lies. They are smaller in the next planes c and d; and we neglect them completely in all other planes. For similar reasons, when computing the interaction energy between planes we only consider atoms in adjacent layers; with the interstitial atom, on account of its greater nearness to planes a and b, we include its interaction with the atoms of a,b,c and d, but no others. These restric-

BY INTERSTITIAL

CARBON ATOMS

449

tions severely limit the number of terms in the summations in (4) ; but they still leave a very large number of individual terms to be summed. Some of the partial 8ummations may be simplified by using the function W(R) of (1). Consider, as an example, the sum Z U(Y’~) for all the atoms i of one particular layer. ’ We may write this as T%i)

+~;UJ(&) n

- %JI

where the second summation is merely over the II atoms in this layer which we are supposing to be displaced. This is equivalent to (5) Similar reductions of the infinite sums to finite ones can be made for the whole of the first two summations in (4). We do not set out the analysis in full, because, although it is quite straightforward, it is very tedious. All the simplifying assumptionsjust outlined will have the effect of increasing the energy E. Our calculations will therefore give an upper bound to this energy; but we shall give reasons later for believing that the error which results is small, and considerably smaller than that which is involved in our choice of the interatomic potential U(Y). The number of terms in the expansion of (4) which still remain increases very rapidly with n. For vahres of r> 4.9 a.u. we have calculated term by term for our chosen U(Y). There is still one more approximation which we have found it necessary to make. Thii also will have the effect of increasing E. The necessity for this approximation will be realised if it is recalled that even when all the earlier approximations have been made, the total energy E is a very complicated function of all the separate Zi. The equations that would define the minimum of E are very far from being linear in the si, or the interstitial co-ordinate p. We did not have the facilities for an absolute determination of this minimum. We therefore argued that since U(r) decreases very rapidly with increasing r, much the largest force causing deformation in any plane arises from the repulsion of the interstitial on the central atom or atoms of the plane. Let us consider layer u as our example. We suppose that the central atom is diiplaced a

450

C. A. COULSON,

S. SENENT,

M. A. HERRAEZ, M. LEAL and E. SANTOS TAEILB1

Interstitial atom in position A Number of displaced atoms Total in each layer number

Approximation

na

nb

nc

nd

0

0

0

0

0

1

0

0

0

1

4 13 25

6 12 24

0 0 0

0 0 0

IO 25 49

0

0

0

0

0

1 3 7 18

1 3 7 18

0 0 0 0

0 0 0 0

2 6 14 36

Interstitial

n=na+nb+n,+nd

Deformation layer 0

layer b

-

-

0.0675 Xl%

-

0.0392 2~ 0~0185 2,’ 0.0128 zl’

0.0781 I,’ 0.0425 a,’ 0*0195 s**

atom in position B

distance z1 (to be determined later), and that the corresponding displacements of the other atoms of this layer are the same as they would be if we had kept all the atoms except the n, displaced ones at their equilibrium positions, and then let these n, atoms take up the positions appropriate to an isolated layer in which xi was given. The energy V, of deformation of this layer will then be proportional to 21, though the coefficient of proportionality will depend on n,. Table 1 shows the deformation energies obtained in this way for layers a and b. When all four layers u,b,c,d are distorted, a similar, but inevitably more complicated, process has to be adopted. First we notice that layer c matches layer b, so that the energy of its deformation will be of the same form as in the last column of Table 1, except that ~$02 replaces &. Similarly d matches a, so that for its deformation we use the same formula as for xi, but with s:ol replacing 2:. This gives us the energy terms XV, in terms of four fundamental displacements zl, z2, zlol,2102. Hence E is known. A relaxation process was adopted in order to find the minimum of E. With zI and x2 the same as when only layers u and b were displaced, ~101 and 2102 were varied to give a stationary value of E. Then with these ~101 and ~102, new values of 2l and z2 were obtained. This iterative procedure was continued until sufficiently good convergence had set in.

0.0675 ow4s 0.0280 0.0160

q* 21’ q’ q’

It should be emphasized that in each of these approximations the displacements of all the atoms in a layer are given in terms of the displacement of the central atom or atoms of that layer. These displacements have all to be calculated so that the appropriate interatomic distances may be inserted in the first two summations in (4). Even with these simplifications the problem is still a heavy numerical one. A short discussion of some of the possible errors is provided in the Appendix. 4. THE CARBON-CARBON INTERACTION POTENTIAL U(r)

Our next difficulty arises from the carboncarbon interaction potential V(r). This is not known exactly, and several distinct forms have been proposed in the literature. It is necessary, therefore, to compare these, bearing in mind that some authors start from an assumed atom-atom potential U(r), and others from an atom-layer potential W(R), as in (l)-(3); yet others start from the interaction between one atom and a semiinfinite crystal. All forms of U(r) are made up of two terms, one of attraction and the other of repulsion. Each part is normally of exponential or inverse-power type. BRENNAN(‘~)started with an assumed layer-layer potential, in which the attractive van der Waals energy was proportional to the inverse fourth power of their separation, and whose coefficient was chosen to give a minimum total energy at a

DEFORMATION OF GRAPHITE LATTICES BY INTERSTITIAL CARBON ATOMS

451 1

distance

3.355 ii. DEN&“)

proposed an attrac-

tive term proportional to y-6 and a repulsion term of Morse type. KOMATXJ”~’ used a modified BRENNAN-type atom-atom law; GIRIFALCO and LA#O) use the familiar Lennard- Jones type of inverse 6-12 powers; CR0wsLL(21) corrects the repulsion term in this expression to get a better fit with experimental data on graphite. Finally

SETTON uses exponential-type expressions, and determines the parameters by reference to the interlayer spacing in graphite, the work needed to separate two layers (i.e. exfoliatory energy), and the compressibility. The particular forms of the various functions just referred to are shown in equations (6)-(10), and the corresponding values of the parameters (in atomic units) in Table 2.

Functionsusedfor atom-atom interaction U(Y) BRENNAN GIRIFALCOand

U(r) = -$ LAD:

+B ‘+exp

A u’r)=-p+F2

[-(Cr

+Dr’)]

(a)

B (7)

CROWRLL :

U(r)=-;+Bexp[-Cr]

(8)

KOMATSU

U(r)=-$+Bexp[-(Cr+Dr’)]

(9)

!kTTON :

U(r)=-Aexp TABLE 2.

Cr-t-1 [-Cr]+B--;?-exp

PARAM-

(6)-(10), IN

IN EQUATIONS

A BRENNAN:

164.9

GIRIFALCOand LAD: CROWELL:

27.225 27.225

&T-CON:

ATOMIC

(10) UNITS

B

c

D

4.637

0.7827

o-01794

1.768 x lOa 68.15

154.65

KOMATSU:

[-Cr]

O-01033

1.895

O-2577

0.2223

0.7855

O-3728

The W(R) functions associated with the U(r) potentials appropriate numerical parameters in Table 3.

are set out in equations

0.07951

(11)-(14)

with the

Functions usedfor atom-layer interactions W(R) BRENNAN:

W(R) = -;+B

GIRIWXO

W(R)= -$+j$

(12)

W(R) = -R4+B(CR+l)exp[-CR]

(13)

CROWELL :

SETTON : B

and LAD:

exp [-(CR+DR’)]

exp [-CR]+:

(11)

exp [-CR]

(14)

452

C. A. COULSON, S. SENBNT, M. A. HBRBABZ, M. LEAL and E. SANTOS TABLB~. P-

FOR URR IN RQUATIONB

B

A BRBNNAN: GIRIFALCO and LAD: CRowBLL: &TTON:

(11)-(14), IN ATOMIC

27.65 4.572 4.572 OX)0694

3.115 118.75 12.77 0.5275

I -

UNITS

c

D

0.7827

0.01794

1.893 0.3728

Brenrun

2 - Komatsu 3-

S&on

4 - Cluwell 5 - Girifalco end Lad

FIG. 6. Comparison

of alternative carbon+wbon

potentials U(r).

DEFORMATION OF GRAPHITE LATTICES BY INTERSTITIAL CARBON ATOMS

At first sight these expressions seem very different, and a proper choice almost impossible to make. We have therefore drawn, in Fig. 6, curves representing the five functions (+0-(O). Now the range in which we are most interested is from 3 to 6 a, (I%= the atomic unit (au.) of length). The smaller part of this range is clearly quite falsely represented by BRENNAN and KOMATSU’S functions. We therefore reject them. Further, the very sharp rise in U(r) for r<5 implied by GIRI~LCO and LAD would lead to very large distortions around an interstitial, and far too large a migration energy. We do not believe that a carbon atom is as ‘rhardt’ as this curve would seem to imply. So we reject this also. From an entirely opposite point of view we reject the SETTON curve 3, since this leads to too small a formation energy for an interstitial : in fact, even without considering any deformation of the lattice this would lead to a value (-1 eV), very much smaller than suggested by the work of DIENES(‘~~and IwATA.~‘~)We are therefore left with the CROWELLfunction of curve 4. This is the one that we shall use for our main calculations, though, as a check on the significance of our work, we have also made some other calculations (Section 8) using the formulae of SETTCINand of GIRIFALCO and LAD. 5. FURTHER STUDY OF W(R)

The integral representation (2) giving W(R) in terms of U(r) is only valid provided that R is sufficiently large. If R is less than about 9 a., there will be signi&ant errors in using it. But we require W(R) for R lying approximately in the range 6.4 f 1.4 h. We therefore split up the summation in (1) into two parts. In the first we deal with the central group of 24 nearest atoms (case of atom 0 and layer plane b of Fig, 4, for example), or 25 nearest neighbours, (case of atom 0 and layer plane a) in the immediate layer with which it is interacting. In this part of the calculation the precise sum of 24 (or 25) terms in (1) is taken. The second part of the summation is now represented by an integration of type 2, except that the lower limit is the radius of that circle whose area corresponds to 24 (or 25) atomic areas l/o in a graphite layer. A similar situation arises when using the atomlayer energy W(R) to compute the atom-semicrystal energy. First we correct W(R) as above, and then we split the station over all layers into a

453

summation for the nearest two layers on each side, and a suitable integration for the others. This integration is of the form WJ W(R)v cl& where v=1/6*34 is the number of j dyer planes in thickness a~, and R, is the distance to a plane equidistant from layers c and c. It should be clear that without this more intimate study of W(R) the calculated migration energy would necessarily be zero. 6. NUMERICAL RESULTS: NO LA’MWE BJXLAXATION

Before proceeding to a full calculation it is instructive to make a simple calculation of the interstitial energy but without allowing any relaxation of the lattice. This calculation must very considerably exaggerate the energies involved. (i) ~~~s~i~~ in position B The interstitial lies by symmetry midway between planes a and b, a distance 3.17 a.u. from each. The energy of interaction of the interstitial with each half of the lattice (the planes above and below) is 0.1644 a.u., so that the total interstitial energy is 0.3288 au. =8*94 eV. (ii) Interstitial in position A The minimum energy occurs when Roa = 3.393 a.u., Rd -2.95 a.u. The energy is then O-3099 a.u. =8-43 eV. It will be noticed that this calculation, crude as it is, leads to position A as the most stable, in agreement with the earlier work of DIENI&‘~) and IWATAet aL(‘*) The migration energy would be O-0189 a.u. =0*51 eV. It is clear that this value is too large: we cannot expect a sensible result without allowing for deformation of the lattice. 7. NUMERICAL RESULTS WITH RELAXATION OF LA’ITICE

(i) Interstitial in position B First we suppose that relaxation is confined to the two planes a and b. By symmetry the interstitial lies symmetrically between these planes. If the nearest atoms are displaced a distance I%, and we use the deformation energy as given in Table 1, we find the minimum energy E for the corresponding number of displaced atoms per layer, and value of I~, as shown in Table 4.

C. A. COULSON,

454

S. SENENT,

M. A. HE-Z,

TABLET. INTERSTITIAL Approximation

Total number (n) of displaced atoms

1 2 3 4 5

0 2 6 14 36

M. LEAL and E. SANTOS

IN POSITION

B

Energy (ev) 0 0.26 040 0.51 o-57

8.946 7.150 5.590 4448 3409

final value of the interstitial

energy in position B is

E=2*613-O-152=2461

s 5 Y

II

I

02

I

I

14

6

Total number

36 n of

displaced

1

The reduction in energy with increasing n is very clear. Fig. 7 shows that these energy values fit very nicely on to a straight’line of the form n/(E- E,,) =an+b, where E,=E(n =0) =8*946 eV, as in Section 6. By extrapolation to n = co we get l/a=2.613

eV.

In the same way, as n+co, the displacement zr+ 1.17 a.u. =0*62 A. Our next step is to allow deformations of layers c and d as well as a and b. As previously stated this was performed by an iterative relaxation process. The details are tedious and will be omitted. The result is an additional saving in energy of 0.152 eV. It occurs when xrca =0*114 A=O*215 a.u. So the TABLE

The energy of relaxation arising from deformation of planes c and d is only 2% of that from planes a and b. We may therefore expect that further inclusion of planes e and f would lead to an additional gain of less than 0.01 eV. We are justified in neglecting this, and limiting the deformation of the lattice to the two nearest planes on each side of the interstitial. (ii) Interstitial in position A

atoms

FIG. 7. Extrapolation of energy, interstitial in position B.

Em =E,,-

5.

This is a much more tedious situation to discuss, since the layers a and b do not now deform in the same way, and the interstitial atom is no longer central between them. Thus there are three independent parameters zr, za, and p (=rer). By an iterative relaxation process the minimum energy E can be found for each of the sets of displaced atoms listed in Table 1. The results, when only the layers a and b are deformed, are shown in Table 5. Figure 8 shows that a similar extrapolation procedure is valid here as was used in Fig. 7, though, on account of the lack of equivalence among the layers, there is a small oscillatory character in the “straight” line. The extrapolation leads to a value E m =244 eV. This is to be compared with the value 2.61 eV. found in position B. Further relaxation of layer c gains 0.0542 eV, and of layer

INTJEtSTITIAL

IN POSITION

A

Energy

Number (n) of

Approximation

&placed atoms

2 3 4 5

1 10 25 49

eV.

(4 1.669 l-719 l-708 1.717

0.333 0.459 0.577 o-597

0 0.289 0.354 O-432

6.97 4.64 3.43 2.99

DEFORMATION

OF GRAPHITE LATTICES

BY INTERSTITIAL

CARBON ATOMS

455

the lattice, the value obtained (position B) is 0.96 eV. If we now allow layers a and b to distort, -0 the displacement x1 has a value O-04 A (c.f. 042 A in Section7),andthecorrespondingenergy ismerely -7 0.91 eV. This is far too small, and we have not therefore thought it worth while allowing additional -: =,J relaxation of layers c and d. This particular function -4 is so “soft” that the deformation of the layers does -3 not play any really significant role. The GIRIFALCO-LADfunction, however, is -2 “harder” then the Crowell one. For an atom in I position B, without any relaxation of the lattice the energy is no less than 48 eV. This goes down to Total number n of displaced atoms 20 eV when layers a and b axe distorted as in Approximation 2, of Table 4. At this stage the FIG. 8. Extrapolation of energy, interstitial in position A. deformation is given by x1 =0+58 8, more than twice the value in Table 4. In Approximation 3, d O-0635 eV. So when all four layers are deformed the displacement is 0.63 A for the central atoms, the interstitial energy is 2.32 eV. Just as in position B there is no point in including further layers, for with an energy of 10 eV. This is still fairly high, being about twice the value when using the their effect is expected to be less than 0.01 eV. We are now in a position to compute the migra- Crowell function. It is not surprising that there is a sharp fall in tion energies. Since these are the differences of the energy when we allow deformation of the lattice. energies in positions A and B, we find a value For without deformation the interstitial is less than E,,, = 0.14 eV with four layers deformed ; 2 A from its closest neighbours. At these distances the C-C interaction energy curve is very steep. =0.17 eV with only two layers deformed. (The energies vary from 1 eV for the SETTON We saw in Section 6 that if no layers were defunction to 50 eV for the GIRIFALCO-LADfuncformed, the migration energy would be 0.51 eV. It tion). But when deformation of the lattice is seems very unlikely, therefore, that further permitted, the interstitial is at about 3 A from its deformation in layers e and f would significantly neighbours. The energies are now very much less: alter our final value 0.14 eV. and, oddly enough, at this distance the various energies are closest to each other. This is why the 8. USE OF ALTERNATIVE CARBONCARBON POTENTIALS U(r) calculated interstitial energy does not vary as much It is evident that our values for the interstitial as might at first have been expected. energy are sensitive to the form chosen to repreIt may be noted that in previous theoretical sent the carbon-carbon interaction energy U(Y).We work on this problem, energies near to 9 eV have have also made some additional calculations that been found(12,14) in the absence of lattice relaxaare designed to show just how sensitive this tion. This further justifies us in our choice of the energy is. The values in Section 7 were obtained Crowell function, which was shown in Section 6 to using the CROWELLfunction (8). We shall now lead to 8.4 eV (position A) and 8.9 eV (position report similar calculations using the SETTON B): and it suggests very strongly that our final function (10) and the GIRIFALCO-LAD function (7). values, including relaxation, of 2.32 eV (position We showed in Section 4 that these were the most A) and 2.46 eV (position B) are not in serious plausible alternatives to the CROWELL function (8). error. 9. DISCUSSION With the Setton function, Fig. 3 shows that the repulsion energy is small. This is shown very clearly Our calculations confirm that the most stable by the fact that if we repeat the calculation of interstitial position is at A. Our self-energy of Section 6 in which the energy of the interstitial is 2.32 eV is in substantial agreement with the value computed without allowing for any deformation of of IWATAet aZ.,(14) but our migration energy is -9

456

C. A. COULSON,

S. SENENT,

M. A. HERRAEZ,

somewhat higher. Now the migration energy depends wholly ,upon the discontinuous (atomic) character of the graphite layers; we therefore believe that, despite all the difficulties and approximations in our treatment, our value is more reliable than theirs, which makes more use of a annul model of each layer. If we want the formation energy Ef (see Section 2) we have to add the sublimation energy to the previous self-energy. The sublimation energy is known to be 744 eV. This leads to a formation energy &=9*76 eV. If we now add the migration energy, we obtain a value 9.9 eV which represents. the least energy that must be applied to remove an atom from the edge of the crystal and make it Muse interstitially between two layers. This value is a little greater than the values 7.2 eV oband 6.3 eV obtained by tained by m1(13) BAKER and I&,LY@~) by experiments on selfdiffusion. It is not clear, however, that the mechanism of diffusion is the same in both cases. If the interstitial atom comes from the inside of the crystal and not, as above, from its edge, we must add the energy required to take an atom from the inside to the edge of the crystal. This is the formation energy of a vacancy, for which our earlier calculations(‘) predicted a value of 10.7 eV. If these values are correct, the total energy of vacancy plus interstitial will be 20.6 eV. This agrees closely with the experimental value 24*7& 0.9 eV due to Eggen.(“4) The migration energy Em of interstitials is low, so that, once formed, they can move fairly easily between the layers. In this movement they can come to the surface, or to a dislocation, or go into a vacancy, with recovery of the appropriate energy. By simple Boltzmann statistical arguments we can estimate the percentage p of interstitial atoms at given temperature which possess kinetic energy greater than E,,,. For if p = 100 exp ( - E,/RT), since at temperature T=20”C, 0.46% of the interstitials can move, and at T=200°C,p=3*60/0. Now the working temperature of a reactor is usually staled above 200°C, so that we may safely conclude that interstitials do not accumulate iu the graphite, but disappear nearly as fast as they form, by migration to the surface or dislocation or into a vacancy. The vacancies themselves, of course, do not disappear until much greater temperamres (order of 1700°C!).

M. LEAL

and E. SANTOS

10. INCREASE IN VOLUME IBRADIA’IZD GRAPHITE

OF

Our previous study of an interstitial atom has been based on the hypothesis that all such atoms were so far apart that each one could be considered separately. In reality, of course, there is a volume d~~bution of interstitials. These will lead to an increase in volume of the lattice through a greater mean inter-layer sepaiation. There is also a further increase in size through a diminution in the mean bond order for the carbon-carbon bonds within a plane.(25) This leads to an increase in bond length, and hence to a ~a~tion of the lattice. Both effects need to be considered in any complete treatment. We hope to return to this problem at a later date. Acknowledgements-We should like to acknowledge the benefit of correspondence with Drs. B. T. KELLY and A. KELLY, and Professor E. W. J. MITCHBLL. APPENDIX In all these calculations errors of two kinds have to be considered-those inherent in the choice of the potential functions U(Y) and Vb, and those involved in the approximations necessary if the calculation is to be practicable. The first of these is unavoidable until more reliable expressions have been obtained. The second type of error requires us to exercise some care. With reference to the deformation energy of a single layer, it can be shown from a comparison of energies calculated by the COULSON-%NRNT formula and experimental values (DELCROIX and YvoN@~)) that the errora can sometimes be as large as 10%. For the deformation energies relevant to our present problem this can represent as much as 0.01 eV (0*004 a.u.). This is much less than the error to be expected from our ignorance of U(Y), as the variety of curves in Fig. 6 shows very clearly. As-for the interatomic distances, we expect a precision of about 0.001 A (0*002 a.u.). This itself gives rise to errors of up to 0*03’eV (O*OOi a.u.) in so;e of the calculations of interaction energy. Taken together, these imprecisions may produce final errors of several tenths of an electron-volt. But for the migration energy, those terms will be very much smaller, since we are here interested in the difference of interstitial energy at A and at B, and the imprecisions will be almost the same in both cases. Setting aside the error due (Section 7) to extrapolation of the results of successive approximations with increasing values of ff, it is sufficient to take all the separate contributions to the total energy to five decimals. Then, even though the number of terms is very large, the error in E for each calculation should be lower than O-1 eV (0904 a.u.). This will be true even though the number of terms which are added together in equation (4) may be as large as 10,000. Fortunately, since the

DEFORMATION

OF GRAPHITE

LATTICES

function U(r) becomes small as t increases, we do not need to have the various Y values all equally accurate. In view of the fact that several thousand such interatomic distances have to be computed, this results in a considerable saving of time. We have found that for r< 4.3 a.u., the maximum permissible error in re must be OXJOOO1; for 4*3 6 it is 0~005 and for r> 6 it is 0.05. This makes possible a good deal of graphical work, with a corresponding saving of time. This still leaves unsettled the error due to choosing the atomic displacements zi as definite fractions of the displacement for the central atom (or group of atoms) nearest to the interstitial. However, since the inclusion of planes c and d makes only a 6 o/o change in the interstitial energy, it seems unlikely that this particular approximation will lead to errors much greater than 5 ‘A, i.e. 0.15 eV. The migration energy, however, is much more sensitive to the positions of the nearest 20-30 atoms, so that the percentage accuracy of this energy may be considerably less. The absolute accuracy, however, may be expected to be no worse than O-1 eV, and it may easily be better. But here the present uncertainties in U(Y) are of dominant importance. REFERENCES

1. COWLSON C. A., HERRAIZ M. A., LEU M., SANTOSE. and SENENT S., Proc. Rov. Sot. London A274 461 (1963). 2. HENNIC G. R. and HOVE J. E., Proceedings of the

First International Conference 011the Peaceful Uses of Atomic Energy, Vol. 7,“~. 666. United Nagons, N.Y-. (1956). 3. BOLLMANNW. and HFMNIG

G. R., Carbon 1, 525

(1964). 4. BACON G. E. and WARREN B. E., Acta Cryst. 9, 102 (1956). 5. MONTET G. L., HENNIG G. R. and KURS A., Nuct. Sci. Eng. 1, 33 (1956). 6. PRIMAK W., jhyi. Re& lad, 1677 (1955). 7. BURTON M. G. and NIXIBERT T. J., _y. Appl. Phys. 27, 557 (1956). 8. SPALARI~ C. N., BUPP L. P. and GILBERT E. C., r. Phys. Chem. 61, 350 (1950). 9. UBBELOHDBA. R. and LEWIS F. A., Graphite and its Crystal Compounds. Clarendon Press, Oxford (1960). 10. NIGHTINGALE R. E. (editor), Nuclear Graphite. Academic Press, New York (1962). 11. BELL J. C., BRIDGEH., COTTRELLA. H., GREENOUGH G. B., REXNOLDS W. N. and SIMMONS J. H. W., Phil. Trans. Roy. Sot. London A254,361 (1962). 12. DIENES G. J., J. Appl. Phys. 23, 1194 (1952).

BY INTERSTITIAL

CARBON ATOMS

4.57

13. KANTZR M. A., Phys. Rev. 107,655 (1957). 14. IWATA T., FIJJITA F. E. and SUZUKI H. J., J. Pbys. Sot. Japan 16, No. 2 (1962). 15. BAK& c. and KELLY A., Nature 193,235 (1962). HENNIG G. R.. AM. Phvs. Letters 1. 55 (1962). :;: COULSONC. A. a;;d SE&NT S., _ _?‘.&em: Sot: 1813 :;: 20. 21. 22. 23.

(1955). BFIENNANR. C., J. Chem. Phys. 20,40 (1952). KOMATSUK.. ‘i. Chew. Sot. yaban 10, 346 (1955). GIRIFALCO LlA. and LAD d. A., r. dhem. khys..25, 693 (1956). CROWELL A. D., y. Chem. Phys. 29,446 (1958). SETTONR., Bull. Sot. Ckim. France 1758 (1960). KNIGHT H. T. and RINK J. P., J. Chem. Phys., 29,

449 (1958). 24. EGGFX D. T., U.S. Atomic Energy Comm., NAA3R-69 (1950).

25. KELLY B. T., Private communication. 26. DELCROIX J. L. and YVON J., Compt. Rend. 242,628 (1956). NOTE ADDED IN PROOF There are two comments dealing with the calculations of vacancy plus interstitial energies at the end of Section 9, which need to be made. The fust concerns the experimental value, for which we quoted the unpublished value of 24.7 ho.9 eV, due to EGGEN. Recent work of LUCAS and MITCHELL [Carbon 1, 345 (1964)] suggests a considerably larger value, closer to 60&10 eV: and MONTET [Carbon 3, 380 (1965)] suggests a somewhat smaller value 30 eV, but still larger than our theoretical value. Either of these two new values would make the agreement with our own calculations less satisfactory. But the second comment is to the effect that in the experiments in which fast electrons or fast neutrons are used to knock a carbon atom out of position, it is unlikely that there would be sufficient time for the lattice to relax, so that the experimental value would be expected to be larger than the theoretical one, where complete relaxation has been assumed. Professor MITCHELL has pointed out to us that if we pay no attention to this lattice relaxation around an interstitial, as in Section 6, we do get a considerably larger energy, more in keeping with the later experimental values: and if we use the rather harder carbon-carbon potential of GIRIFALCO and LAD@) we get values even higher than those suggested by LUCAS and MITCHELL. It is clear that when making a comparison with experiment it is important to know as carefully as possible what happens to the energy that is used to create the interstitial atom: and further study is needed of the appropriate carbon-carbon potential to use.