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A new force-constant model for graphite

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1977 J. Phys. C: Solid State Phys. 10 2295

(http://iopscience.iop.org/0022-3719/10/13/006)

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J. Phys. C: Solid State Phys., Vol. 10, 1977. Printed in Great Britain. @ 1977

A new force-constant model for graphite

A P P Nicholson and D J Bacon Department of Metallurgy and Materials Science, University of Liverpool, PO Box 147, Liverpool L69 3BX

Received 26 November 1976

Abstract. Earlier force-constant models of graphite either assume the lattice vibrations parallel and perpendicular to the basal plane to be independent, which leads to a zero value for the shear modulus C,,, or require an external pressure to be applied for a finite value of C,, to be obtained. In order to avoid these difficulties, and to account for the elastic constants, it was found necessary to consider a model allowing for general interactions out to third-nearest neighbours in the plane and second-nearest neighbours in adjacent planes. The best fit obtained to experimental data uses a model which consists of eight pairwise potential constants and three many-body valence constants. This model gives an exact fit to all five elastic constants and a good fit to the phonon dispersion data. The combination of pairwise potential and valence constants would make the model suitable for predicting the displacement fields of vacancy clusters using the lattice statics method.

1. Introduction

A common feature of early force-constant models of graphite (Krumhansl and Brooks 1953, Yoshimori and Kitano 1956, Young and Koppel 1965) is that vibrations parallel and perpendicular to the basal plane are independent, i.e., there is no coupling between these modes of vibrations. This assumption considerably simplified the calculation of the phonon spectrum and hence the specific heat since the 12 x 12 secular determinant could now be reduced to three 4 x 4 determinants for any wavevector. Such decoupling in a layered structure is not unreasonable for predicting properties such as the specific heat, which are not very sensitive to details of the phonon dispersion curves. Unfortunately these decoupled models can only have limited application since the lattice structure is inherently unstable; this is due to lack of resistance to shear (C44 is always zero).

Phonon dispersion data for graphite have been extremely difficult to obtain owing to the lack of good quality large single crystals. The first measurements of Dolling and Brockhouse (1962) for modes propagating along the c axis have been improved upon by Nicklow et a1 (1972) (hereafter referred to as NWS), who also made some measure- ments for the first time of modes propagating in the basal plane. NWS analysed their data with a force-constant model using pairwise interactions up to the third in-plane neighbour and the first out-of-plane neighbour. Subsequently, Mani and Ramani (1974) (hereafter referred to as MR) reinterpreted the NWS data in terms of a valence-constant model of the same range. In both these models a finite value was assigned to C44. As we shall show, this implies non-vanishing stresses in the lattice, which thus require an external pressure to be imposed to maintain equilibrium. This is somewhat unreasonable since it is usual to assume that experimental data have been obtained at zero pressure.

2295

2296 A P P Nicholson and D J Bacon

Thus a proper application of the equilibrium conditions to these models makes C,, zero, reducing them to the decoupled type used by earlier workers, and destroying the fit to the phonon dispersion data.

The first objective of this work has therefore been the construction of a stable force- constant model of graphite which is consistent with the NWS phonon dispersion data. Secondly, we wished to use the resulting model in lattice-statics calculations of the mechanical properties of vacancies, as has been done previously using a two-dimensional single layer model (Nicholson and Bacon 1975a, b). Since such properties are dominated by elastic contributions, we require the new model to predict the correct elastic constants. In our previous work (1975a), we were limited to studies of single isolated vacancies and their interactions, for the valence-constant model used could not give any value for the forces simulating the vacancy. These were calculated for the single vacancy only using the defect-molecule method (1975b). To allow studies to be made of larger defect clusters, we therefore permit the first derivatives of the total potential with respect to any bond length or other invariant coordinate to be nonzero.

In the next section the effect of invariance and equilibrium conditions on force constants is described; the results are applied to a new model consisting of twenty independent force constants. Pairwise potential contributions to this model are considered and the nonzero C,, error in the NWS and MR models discussed. In 5 3 the limitations involved in fitting this model to the data are discussed, and the final successful version consisting of pairwise-potential and valence-type interactions is described. The results obtained are compared with the NWS and MR models in the final section.

2. The new force constant model

The graphite structure, which has space group D&,, is shown in figure 1. Referred to the Cartesian axes A, shown, the primitive lattice vectors are

where a = 2*46A and c = 6.70A are the lattice parameters. The atoms labelled 1, 2, 3, 4 in the figure are all in the same unit cell, namely the origin cell with index 1 = l ,a , + 12a2 + 13u, equals zero. Atoms labelled 1, l, l, etc., are on the same sub- lattice. A feature of the graphite structure is the lack of inversion symmetry at the atomic sites, although there is a centre of inversion at the point (0, 0, 4 4 ) for example.

We have followed standard practice in defining the force-constant matrix by

where subscripts U, refer to the Cartesian axes, u(1k) is the displacement from equilibrium of the kth atom in cell 1, Q is the total potential energy of the lattice, and the derivatives are evaluated at the equilibrium configuration.

In setting up any force-constant model for an infinite lattice it is necessary to ensure that the force constants satisfy certain invariance relations and equilibrium conditions. From the continuity of the potential energy, 0, we must have

+,&lk ; lk) = 4@,(1k ; lk),

A new force-constant model for graphite 2297

and from its (numerical) invariance to rigid-body translations we must have

These conditions are easily satisfied and we need consider them no further. The re- maining conditions on the force constants arise from (i) (numerical) invariance of @ to rigid body rotations, (ii) invariance of the form of @ to symmetry operations belonging to the space group of the crystal, and (iii) the structure must be an equilibrium configura- tion. I t can be shown that any perfect lattice model of graphite satisfying space-group symmetry ((ii) above) will automatically satisfy the rotational invariance condition (i).

1 Figure 1 . The graphite lattice, showing the Cartesian axes and atom labelling used here, and the angle q5 required in the definitions of pairwise potential contributions and of the valence constant A.

The equilibrium condition for an infinite lattice ((iii) above) is in two parts. Firstly the net force on each atom must vanish at the equilibrium position. This is automatically the case in the harmonic approximation. Secondly, the configuration must correspond to vanishing stresses. In general it is not possible to determine the pressure, leaving five anisotropic stress components to be put to zero, namely o12, 013, 023, and ( o ~ ~ - o ~ ~ ) . It can be shown that the space-group symmetry condition (ii) is sufficient to ensure that all except the last of these components are always zero. Thus we have a t most two relations amongst the force constants to apply after making maximum use of the crystal symmetry to reduce the number of independent force constants, namely

and the pressure (O11 - 033) = 0,

p = -'2 3( O11 + 633) = 0.

(3a)

(3b) The results above were a by-product of an analysis of the effect of crystal symmetry

on various lattice sums and the resulting well-known bracket functions [ ~ p , y l ] and

A P P Nicholson and D J Bacon

(a/?, ?A) used in evaluating the elastic constants by the method of long waves (Born and Huang 1954). Graphite has five independent elastic constants, C,,, c66, C,,, C13, and C,,, and neither of the Cauchy relations C,, = C66 and C13 = C,, are obeyed. Lack of inversion symmetry at the atomic sites is sufficient to ensure C, # C,,, which can thus be achieved even with pairwise potentials, as in the NWS model. For C,,, C,, and C,, however there are no sublattice shift contributions, i.e., (a/?, ?A) is zero for any index equals three. Since for pairwise interactions the Cap, ?A] functions are symmetric in all four indices,many-body interplanar interactions are necessary to ensure C, , # C,,.

Table 1 . General force-constant matrices for graphite.

Atoms involved in interaction

Three 1st-neighbours at distance r i = aJJ3 e.g., M = 1, N = 2 (in-plorie)

Six 2nd-neighbours at distance r 2 = a e.g., M = 1, N = 1' (in-plane)

Three 3rd-neighbours at distance r 3 = 2ol,/3 e.g., M = 1, N = 2' (in-plane)

Two 4th-neighbours at distance r4 = cl2 e.g., M = 1, N = 3 (1st-neighbour out-of-plane)

Twelve 5th-neighbours of atom 1 at distance r 5 = (a213 + (2nd-neighbour out-of-plane)

Six 5th-neighbours of atom 2 of sublattice 4 e.g., M = 2, N = 4"

e.g., M = 1, N = 4'

Force-constant matrix for atoms M and N (see figure 1)

(; A1 0 4 : )

(-i: p: 5,) (i' i3 i3) (i4 1. 5 ) (i: is 5:) +

0 c,

t By inversion symmetry we have, q5zB(0. 2 ; 13) = + p a ( O 1 ; 1 - a , - a24).

We therefore initially considered a model (hereafter referred to as model I) which consisted of the most general interactions allowed by the crystal symmetry between up to third neighbours in the plane and second neighbours in adjacent planes, with one limitation imposed. Second-neighbour in-plane interactions between atoms on sublattice 1, and the similar interactions between atoms on sublattice 2, were assumed to be exactly equivalent. The resulting 21 force constants of model I are defined in table 1. The notation used is compared with that of earlier models in table 2. The entries for A , and B , for the MR model result from certain inconsistencies in their work ; these are discussed later in &I. Expressions for the elastic constants for model I are given in the Appendix.

Since the nature of the interactions in model I has not yet been specified, an expression for the pressure cannot be obtained, and we are left with one relation between the 21 force constants, equation (3a). It has been shown by Huang (1950) that the vanishing of the five anisotropic stress components together with ten conditions relating

A new force-constant model for graphite 2299

to rotational invariance are equivalent to fifteen extra symmetry conditions on the [ap, ?,I] functions, namely

[.P, = [;,A, 4. (4) The only one of these Huang conditions not automatically satisfied by a graphite lattice is the one equivalent to (3a), [ l l , 331 = [33, 111. For model I this relation takes the form

4a2(C, + 6C2 + 4C, + 2C, + C;) = c2[2A, + 6(A, + B,) + 3(A; + Bi) ] , ( 5 ) thus giving 20 independent constants. Allowing for misprints in both papers (which did not affect their numerical results), equation (5) contains within it equation (1) of NWS and equation (2.5) of MR.

Table 2. The relation of the force constants of model I to those of previous models and to pairwise potential contributions.

Model I NWS MR Young and Pairwise potential Koppel (1965) contributions?

E1

Y1

a2

Y2

- 2

@ a

Y3

E4

Y4 0 0 0 0 0

81

8 2

P3

-+ (E , - 38,) +(h, - P I )

P 2

- 4 2

8 3

Y 1

a2

Y 2

E3

Y3

E4

Y4 0 0 0 0 0

K

6fJ 2$/3 - 3/44 pi4 - , 4 9

0 0 0 0 K 0 0 0 0 0

Tl T2

t The parameters Ri and 7; are defined in equation (6). C and S are cos 4 and sin 4 respectively where rj is shown in figure 1 i.e., tan 4 = 2a/ J3c.

All previous models have used either pairwise potentials or valence-type potentials, or both to represent the interactions. Valence-type potentials are particularly useful since they automatically satisfy the equilibrium conditions, which depend solely on first derivatives of the total potential energy, @. The contributions which a pairwise potential V(r) makes to the force constants of model I are shown in table 2. The ten potential parameters are defined by

where rn is the perfect lattice distance between nth neighbours (as defined in table 1). We have assumed that the pairwise contribution to the second-neighbour out-of-plane interaction (r = r J is independent of the sublattices to which the atoms belong. The

2300 A P P Nicholson and D J Bacon

pressure can now easily be evaluated (Born and Huang 1954), and the equilibrium conditions (3) can be written as

o,, = (4/,/(3)c)(T, + 6T + 4T3 + 3T5) = 0,

03.7 = (c/J(3) aI(T4 + 9TJ = 0

(74

(7b)

and

I t can immediately be seen that in the NWS and MR models where T, is zero we must also have T4 zero for zero pressure. In these models T4 is the sole contribution to force constant A,, and C,, = (c/J(3)a2)A4:Thus C,, must be zero to give zero pressure in the NWS and MR models.

3. Values for the force constants

Initially we performed a least-squares fit of 19 of the 20 independent force constants of model I to all the available experimental data, namely the phonon dispersion data of NWS, the Raman frequencies (Tuinstra and Koenig 1970, Brillson et a1 1971), and the five elastic constants (Blakslee et a1 1970, Seldin and Nezbeda 1970). (Force constant D, cannot be determined by these data). The solution obtained was unsatisfactory on two counts. Firstly the fit obtained was not particularly good, especially for the elastic constants. Secondly and more importantly, the values predicted for some of the force constants were physically unrealistic : A ,, for example. expected to have the largest value, came out as one of the smallest. Accordingly we performed the calculations again with an exact fit to the elastic constants and with certain physically-reasonable constraints applied to the permissible values of the force constants. These were

(a) A , > B , > C, $ C, > A , > 0, (b) A , > B , > A, , B , > A , > B,, and C, > C, > C,, and (c) A , > A , , B,, A; or B;, and C, > C, or C,.

Despite these extra constraints, we were unable to find a satisfactory solution. In fact too many force constants were being fitted to too little data. It was discovered that an excellent fit to the NWS data for the phonon dispersion along r - A - A could be obtained simply by specifying a fit to the two lowest and the highest optical frequencies at the centre of the zone. However, the lowest of these frequencies, o,,(T), given as 50 cm-byNWS,wasfoundto beincompatiblewiththevalueofC,,of0~4 x 10ON m-, at least when using our model I or any sub-model of it. We therefore decided to fit to C,,, reducing W T o ( r ) to about 42cm-. Of the remaining phonon dispersion data, it was felt that the greatest reliance could be placed on the lowest frequency at the edge of the zone, point M. This point had been very carefully examined by NWS, and we think it only reasonable to fit our model to it. Thus, there are only nine items of data to fit a model to, five elastic constants and four phonon frequencies.

The simplest model we could envisage involved the eight independent pair-potential parameters discussed in the previous section together with at least one three-body valence constant ; this is required to give the observed Cauchy discrepancy C,, # C,, and must be interplanar. Mani and Ramanis (1974) major criticism of the NWS model was the unphysical nature of purely pairwise in-plane interactions which. because they are axially symmetric, imply that B , = C,. A , = C,. and B , = C,. We are in

A new force-constant model for graphite 2301

full agreement with this criticism and therefore allowed the possibility of at least one in-plane valence constant being nonzero.

After trying various combinations, an excellent fit was finally obtained with the following valence constants.

(i) An interplanar constant A giving the energy change due to changes in the angle q5 shown in figure 1 as

2A@(/1) = (l/S4)A(r,Aq5)z,

where

S = sin q5 and r , = a/J3,

Of all the single constants defining a three-body interaction of this triplet of atoms, this was the only one that gave a reasonable value for the lowest optical frequency at

(ii) The in-plane adjacent-bond-angle constant p as used by Young and Koppel (1969, and called k , by MR.

(iii) An in-plane adjacent-bond-length constant v, equivalent to the constant k,, used by MR. v was required in order to make B, less than A,.

The relations between force constants and valence constants given by MR in their equation (3.2) appear to be incorrect. Using the same definitions of valence constants k,, k,,, k,e, k,, and k, as in their equation (3.1), and with 1, p, and v as defined above, we obtained for the valence-constant contributions to the model I constants the following :

and a positive value for C,.

A1 = k, - 2)) - J(3)kr, + 2A/S2, Bi = 6~ - J(3)kre3 C1 = 6k,/r:,

A2 = -&3p + v - J(3)kr,), Bz = & + 311 + J(3)krJ, Cz = -k , /r: , Dz

(8)

i t J (3 )~ - 4 ( 3 ) ~ + krd, A , = 3A/C? A, = -A, C, = A, E, = -SA/C and F, = -CA/$

where r , = a/J3 is called ro by MR, and C = cos q5 and S = sin 4, where tan q5 = 2a/,/(3)c. As shown in table 2, the first neighbour constants a1 and /3, used by MR differ from the normal definition. We obtain,

A useful check can be made on equation (8) by confirming that the resulting force constants satisfy the Huang condition (5).

With eleven ( = 8 + 3) constants and nine data points, the fitting was made by putting T3 = 0 and expressing all the constants as functions of T,. T, was then given a value in the middle of the region defined by the constraints on the force constants described earlier in this section and by optimising the fit achieved to the lowest phonon frequencies at r and M, i.e., o&) and oTA(M).

Values for the 13 parameters in this successful model, hereafter referred to as model 11, are shown in table 3. These values satisfy the equilibrium conditions, equation (3) , which with T3 = 0, leaves 10 independent constants. The resulting values for the model I constants are compared with those for previous models in table 4. A similar comparison

2302 A P P Nicholson and D J Bacon

Table 3. Values obtained for the 13 parameters of model I1 in units of N m-

Tl 231.950 R , 241.746 1, -0.486 T2 -38.533 R , 122.440 p 8,000 T3 0 R 3 -18.830 IJ -4.400 T4 0.250 R , 0.997 75 -0.250 R , 1.045

is made in table5, where values for the elastic constants and certain phonon frequencies are presented. The phonon dispersion curves obtained for the lines r - A - A and - C - M are compared with experiment in figures 2 and 3 respectively.

4. Discussions and conclusions

A study of the results presented in tables 3-5 and figures 2 and 3 show that the original objectives of this work as outlined in the Introduction have been achieved, albeit with a simpler model than the one initially outlined. It can be seen that model I1 satisfies all the required equilibrium conditions, gives an excellent fit to the experimental data, and allows calculations of the first derivatives of the total potential to be made very easily. The phonon dispersion curves obtained are very similar to those ofNWS (allowing for finite pressure in their case). A comparison of the pairwise contributions to our model, shown in table 3, and the NWS force constants (table 4) shows some similarity

Table 4. Values obtained for the model I force constants and for previous models in units of N m - ' .

~~

Force constant Model I NWS? M Rt

323,362 279.950 231.950

-33.533 9 1.440

- 38.533 22.517

- 18.830 0 0 0.529 0.997 0.433

-0.250 0.362

- 0.259 - 1,612 -0.053

0.848 0.465

362.00 199.00 199.00 - 52.00 133.00 - 52.00

0 - 3.70 28.80 28.80

0.77 5.80 0 0 0 0 0 0 0 0

4.971 625.941 46.863 48.933 60.260

7.423 31.758

- 22.779 -22.179

0 0.626 5~724 0 0 0 0 0 0 0 0

t Finite pressure values

A new force-constant model f o r graphite 2303

in the general pattern, except for the relative sizes of R , and Tl ; this is reflected in the high value obtained for C, , by NWS.

As noted in 52, there are certain inconsistencies in the MR model. Except for the comparison they made with the NWS model (MR table 4), the definition used throughout their paper of the first-neighbour interaction constants E , , p,, y, was for a pair of atoms such as 3 and 4 shown in figure 1 of this paper. The values which should have been used in their table 4 are therefore a; = -&al - 3p,) = 0.04977 and = 3(3a1 - p,) = 6.25941. For symmetry reasons this has very little effect on the phonon dispersion curves, but the unrealistic nature of these values is seen in the predictions for the elastic

Table 5. Elastic constants (10" N m-') and phonon frequencies (em-') for graphite.

Experiment

Model I1 Reference NWSt MRt

1063

3.654

0.401 1575 15753

1407 1405 127f 1575 1406 90 30 476$

44:

1.502

42$

106 (a, b) 44 (a, b) 3.65 (U, b) 1.50 (U, b) 0.40 (a, b) 1588 (c. d ) 1575 (c , d) 50 (e) -

- 127 (e)

144 463 3.71 = c44 0.46 1575 1575$ 45$ 1396 1395 1271 1575$ 1397 903 33$ 473$

43 91 3.65:

0.40$ = c44

1576 15753 42 644 630 126 1574 636 89 30 224

t Values quoted require finite pressure. $ Experimental value used to fit force constants. References (a ) Blakslee et al (1970) (b) Seldin and Nezbeda (1970) (c) Brillson et al (1971) (d) Tuinstra and Koenig (1970) (e ) Nicklow et al (1972).

constants in table 5. With C,, = 43 and c66 = 91 in the usual units, we obtain CI2 = - 139 ; thus C,, e ICl2[, and the MR model is unstable even with a pressure applied to allow nonzero C44. These problems would not have been encountered by MR if the expressions used relating valence to force constants (MR equation (3.2)) had been correct ; using the expressions we have derived, equation (8), we found C, = 95 and c66 = 42, which is quite reasonable. Unfortunately the phonon dispersion data cannot then be fitted without a high value of 270 N

Our new model has two major limitations. Firstly, no account has been taken of possible electronic effects, such as the effective charge associated with the highest frequency mode at the centre of the zone, which is infrared active (Brillson et al 1971). Secondly the range of values of the force constants is very high, the ratio of largest to smallest being about 6500. Naturally we tried to avoid this situation, but this was not found possible with the limited data available.

for (A3 + B3).

2304 A P P Nicholson and D J Bacon

-- 'E 2 156f

4

3

2

1

1.0 0.8 0.6 04 0.2 0 q'qmox

Figure 2. Dispersion curves for graphite for wave- vectors q along the line r - A - A of the Brillouin zone; that is from the centre of the zone along the [OOl] axis to the centre of the top face. -, present work ; open circles, neutron measurements of NWS ; full circle, optical measurements (Tuinstra and Koenig 1970, Brillson et al 1971).

q'qrn,,

Figure 3. Dispersion curves for graphite for wave- vectors q along the line r - C - M of the Brillouin zone; that is, from the centre of the zone along the [loo] axis to the point at the centre of a prism face. -, present work ; open circles, neutron measure- ments of NWS.

The model predicts quite reasonable values for the (unrelaxed) radial forces induced by the presence of a vacancy. The nearest-neighbour force is - 20 eV A- compared with a value of - 11 eV A - ' obtained by the defect molecule method (Nicholson and Bacon 1975b). These values are not strictly comparable since the model assumptions are different, but it suggests our new model will give reasonable results when used in the lattice statics calculations for vacancy clusters. This work is in progress and will be reported at a later date.

Acknowledgments

The authors acknowledge valuable discussions with Dr J W Martin. The work was supported by a grant from the UKAEA, Harwell, and the encouragement of Mr J H W Simmons is acknowledged.

A new force-constant model for graphite 2305

Appendix

Of the 21 force constants defined in table 1 for model I, 17 are involved in the expressions for the five independent elastic constants C,,, c66, C,,, C,, and C44. Force constants C,, C,, C, are related to some of these 17 constants by the Huang condition, ( 5 ) ; the elastic constants and the phonon dispersion curves are independent of force constant D,. Three elastic constants are given by

C33 = (c/J3aZ)(C4 + 6C5 + 3C3, C44 =(c/J3a2) [A4 + 3(A5 + Bs) + + C13 (-2,~) (D, + E5 - 0;) - C44.

For C, , and C66 the inner displacement contributions (up, yA) are nonzero, and it is most convenient to use the following equations:

cl, - c66 = (1/J3c) x [ ( A , - B,) - 6(A2 - BJ + 4(A3 - B3) + 2(A, - B,) + (A; - B;) ] ,

;(cl1 + c66) = (I1, 11> f (1/J3c) x [ A , + B , + 6(A, + B,) + 4(A3 + B3) + 2(A, + B5) + A; + B;].

Let

L = A , + B , + A , + B,, X = A , - B , - 2(A3 - B3) + 2(A, - B5) - 2(A; - 89,

M = A , + B,, N = A; + B;,

and

Y = A , - B , - 2(A3 - B,) - 2(A, - B,).

Then the function (1 1, 11) is given by

J3[(L + 2M + $A4)XZ + ( L + 2M + 4N)Y2 - 2(L - 2M)XYl (11.11) =

16c[3LM + 4NA4 + ( L + 2M) ( A , + +N)] For models where second-neighbour out-of-plane interactions are zero, this expression simplifies considerably to

References

Blakslee 0 L, Proctor D G, Seldin E J, Spence G B and Weng T 1970 J . Appl. Phys. 41 3373-82 Born M and Huang K 1954 Dynamical Theory of Crystal Lattices (Oxford: Clarendon Press) Brillson L, Burskin E, Maradudin A A and Stark T 1971 The Physics of Semimetals and Narrow-Gap

Dolling G and Brockhouse B N 1962 Phys. Rev. 128 1120-3 Huang K 1950 Proc. R. Soc. A 203 178-94 Krumhansl J and Brooks H 1953 J. Chem. Phys. 21 1663-9 Mani K K and Ramani R 1974 Phys. Stat. Solidi (b) 61 659-68 Nicholson A P P and Bacon D J 1975a Phys. Stat. Solidi (a) 28 613-25 - 1975b Carbon 13 275-282

Semiconductors eds Carter D L and Bate R T (London: Pergamon Press) p 187

2306 A P P Nicholson and D J Bacon

Nicklow R, Wakabayashi N and Smith H G 1972 Phys. Rev. B 5 4951-62 Seldin E J and Nezbeda C W 1970 J . Appl. Phys. 41 3389-400 Tuinstra F and Koenig J L 1970 J . Chem. Phys. 53 1126-30 Yoshimori A and Kitano Y 1956 J . Phys. Soc. Japan 11 352-61 Young J A and Koppel J U 1965 J . Chem. Phys. 42 357-64