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Step free energies in the solid-on-solid model
Volume 98A, number 5,6
PHYSICS LETTERS
24 October 1983
STEP FREE ENERGIES IN THE SOLID-ON-SOLID MODEL ~ G. JACUCCI Dipartimento di Fisica, Universitâ di Trento, 38050 Povo, Italy
and N. QUIRKE Department of Chemistry, University of Maine, Orono, ME 04469, USA Received 7 July 1983 Revised manuscript received 12 August 1983
The free energy of steps in the solid-on-solid model of interfaces has been calculated as a function of temperature and system size using a Monte Carlo simulation. Our numerical results are consistent with the theoretical prediction that the 112) when approaching the roughening temperature from below. Finitestep free energy goes to as exp(—o/I Tn distance between steps. size effects are found to zero depend strongly T— on the
The surface roughening transition, as displayed by solid-on-solid models of interfaces, has become central to the modern understanding of crystal growth [1] and 2D phase transitions such as that displayed by the Coulomb gas [2]. The roughening temperature Tr marks the transition between a sharp, well-defined surface with, in the kinetic Ising model [3], a nudeation barrier to growth, and a delocalized, rough sur. face with divergent surface width. Swendsen [4] and Van Eerden and Knops [5] have demonstrated theoretically that the free energy tf an infinitely long step in a solid-on-solid (SOS) model surface should go to zero as the temperature of the system approaches Tn from below as A exp(—n/I T T I1I2~ T T —
r
J~
‘~
/
It follows from this that all derivatives of A such as the step energy E and step specific heat C also go to zero at Tr. There is no specific-heat anomaly at Tr. Swendsen [6] carried out extensive Monte Carlo simulations of the simple cubic SOS model, calculating ~ Work supported in part by Gruppo Nazionale di Struttura della Materia del Consiglio Nazionale delle Richerche, and by Istituto per la Ricerca Scientifica e Technologica.
0.031-9163/83/0000—0000/s 03.00 © 1983 North-Holland
E and C but was not able to calculate A due to statistical errors in his data [4]. In this letter we report Monte Carlo calculations of the step free energy using a direct energy difference method. Our numerical results for A (table 1) are consistent with eq. (1), which seems to be roughly of the correct form for all T~Tr (see fig. 1). Like Swendsen [6] we obtain an N-dependent negative peak in C (see table 3) but this can be seen to be quite consistent with eq. (1) by differentiation. The observed N dependence reflects the tendency of C to approach the infinite-system minimum value that occurs somewhat before the roughening temperature at which Cis zero, and is not to be interpreted as evidence of a specific-heat singularity at Tr. The details of the Monte Carlo method and SOS hamiltonian can be found in Swendsen’s article [6]. We confme ourselves to a brief description of the free energy method. The step free energy at temperature T is calculated as the free energy difference between a SOS model with thermally generated n steps in its surface and another with n + 1 where the extra step is imposed by special boundary conditions. In a rectangular array of sites containing N~along the X axis and N~along the Y axis, a step can be imposed by 269
Volume 98A, number 5,6
PHYSICS LETTERS
adding +1 height units to columns starting with the periodic boundary along the Y axis and finishing halfway along the X axis. This is for convenience only, the step can be located anywhere in the surface. The free energy difference between the two systems can be calculated using Bennett’s formula [7], I ~A\
be assumed to be zero, a guess is then made for r and eq. (2) iterated until convergence is obtained. Sirnulations are performed with (ii + 1) and without (n) the imposed step, I(L~s)is a distribution of an energy difference calculated for each configuration of each systern. In the system with the imposed step, it is the difference between the energy of the configuration and its energy where the step is subtracted from the surface structure. For the system without the imposed step it is calculated by taking the difference between the energy of the configuration and subtracting the value where the step is imposed. The formula is accurate as long as the distributions I(ZX) for each system overlap. For all the temperatures considered iii
I+~
=
exp~—~-~, (I(~
i)),,
1(x) = [I +exp(x)j
exp~ ~ t
=
A
24 October 1983
—
+B
B is a constant related to the number of Monte Carlo independent configurations in each simulation. It can
Table 1 Step free energy A per site in units of e, for periodic SOS model systems of various sizes. The first factors in the top row indicate the step width, i.e. the distance between two adjacent images of the step. The temperature is in units of c/kB. All Monte Carlo 4 trial moves per site each, after a similar section for equilibration. runs consist of 5 subsequent sections of i0 T
Stepwidth 5
07 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6
x
10
0686±0006 0.577 ±0.006 0.462 ±0.007 0.353 ±0.007 0.276 ±0.003 0.199 ±0.004 0.162 ±0.003 0.128 ±0.004 0.108 ±0.005 0.101 ±0.003
lox 5
lOX 10
lOx 20
20 X
0
0560±0009 0.418 ±0.006 0.330 ±0.009 0.228 ±0.003 0.143 ±0.009 0.097 ±0.009 0.068 ±0.002 0.058 ±0.007 0.065 ±0.006 0.046 ±0.003
0640±0012 0.531 ±0.009 0.396 ±0.006 0.273 ±0.002 0.164 ±0.011 0.113 ±0.007 0.077 ±0.006 0.058 ±0.004 0.049 ±0.005 0.046 ±0.003
0668±0015 0.518 ±0.012 0.400 ±0.016 0.297 ±0.022 0.175 ±0.003 0.103 ±0.009 0.082 ±0.002 0.056 ±0.005 0.050 ±0.003 0.048 ±0.003
0616±0008 0.436 ±0.029 0.331 ±0.018 0.232 ±0.008 0.111 ±0.015 0.055 ±0.009 0.034 ±0.015 0.029 ±0.005 0.016 ±0.004 0.016 ±0.005
10x5
10x10
10x20
20X10
1.400 1.483 1.500 1.449 1.131 0.643 0.337 0.242 0.180 0.064
1.454 1.5 33 1.606 1.5 18 1.180 0.648 0.320 0.222 0.175 0.117
1.405 ±0.006 1.459 ±0.008 1.514 ±0.018 1.364 ±0.094 0.990 ±0.066 0.388 ±0.071 0.129 ±0.022 0.140 ±0.054 -0.026 ±0.055 +0.092 ±0.032
Table 2 Step energy E per site in units of e. T
Stepwidth 5x10
0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6
270
1.402 1.473 1.473 1.325 1.115 0.812 0.596 0.430 0.325 0.257
±0.009 ±0.009 ±0.007 ±0.007 ±0.021 ±0.018 ±0.012 ±0.042 ±0.018 ±0.025
1.284 1.343 1.376 1.237 0.781 0.482 0.290 0.061 0.136 0.131
±0.008 ±0.007 ±0.019 ±0.045 ±0.062 ±0.009 ±0.073 ±0.061 ±0.049 ±0.068
±0.007 ±0.009 ±0.009 ±0.021 ±0.031 ±0.035 ±0.028 ±0.054 ±0.018 ±0.042
±0.005 ±0.008 ±0.011 ±0.014 ±0.029 ±0.045 ±0.044 ±0.043 ±0.010 ±0.05]
Volume 98A, number 5,6
PHYSICS LETTERS
24 October 1983
Table 3 Step specific heat C per site in units of kB. T
Stepwidth 5 X 10
0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6
0.85 0.43 —0.55 —2.45 —3.22 —2.82 —1.52 —0.94 —0.72 —0.73
lox 5 ±0.07 ±0.09 ±0.12 ±0.27 ±0.10 ±0.13 ±0.12 ±0.20 ±0.14 ±0.11
0.59 —0.04 —0.43 —2.31 —2.79 —2.30 —1.00 0.64 —0.79 +0.33
±0.08 ±0.29 ±0.38 ±0.38 ±0.39 ±0.25 ±0.27 ±0.37 ±0.34 ±0.17
10 X 10
10 X 20
20 X 10
0.92 0.68 —0.16 —0.17 —4.38 —3.93 —2.00 —0.60 —0.54 —0.21
1.11 0.73 —0.09 —1.30 —4.72 —4.53 —0.86 —0.99 —0.40 —0.60
0.92 0.53 —0.51 —3.13 —7.36 —5.54 —1.26 —0.27 +0.78 +0.04
±0.07 ±0.16 ±0.13 ±0.25 ± 0.50 ±0.24 ±0.45 ±0.19 ±0.10 ±0,22
this letter, there was sufficient overlap to allow eq. (2) to be used. rather accurate estimates of the step entropy S can be From the results for A and E of tables 1 and 2, obtained. Furthermore, the investigation of finite-size effects for rectangular periodic systems (tables 1—3)
1.5
results infinite-system properties. should to mention that the best estimateHowever, for Tr aswe discussed in ref. [2], i.e. 1.24, is not inconsistent with the data in fig. 1. Leamy and Gilmer, who actually carried out the first calculation for the step energy in the SOS model [8], discuss the energy as a function of the terrace width and try to extrapolate to infinite width (i.e. an isolated step). In fact eq. (1) refers to the free energy
±0.13 ±0.11 ±0.24 ±0.26 ±0.36 ±0.31 ±0.16 ±0.26 ±0.33
±0.05 ±0.09 ±0.29 ±1.05 ±0.96 ±0.93 ±0.40 ±0.54 ±0.24 ±0.48
1
—
. S
1.0
imposed shows step, theyi.e.depend the distance strongly between on the repeating width of the steps1OX inthat the the results for 10,periodic 1OX 20system. and 20 Comparing X 10 sites indicates that to approach the infinite-system behaviour it is more efficient to take wide steps rather than long ones. Finally, the value T 1.15 corresponding to the nega. tive specific-heat anomaly and quoted by Swendsen for Tn [4] is seen rather to be related to the deep minimum occurring somewhat before T= Tr already predicted by the asymptotic form of eq. (1). Finitesize effects unfortunately roun& off the behaviour of the thermodynamic quantities about Tn, and hinder a precise determination of its value. The development of a form of finite-size scaling appropriate to this problem would be of assistance in extrapolating these
±0.07
E
0~0~ 05
A
0.5
o
-~
~
-
-I—---—
.4
.6
.8
•
0
T
1.0
~
-+---
1.2
1.4
1.6
Fig. 1. The step free energy A(o), and energy E(.), per site in units of (left scale) and the step specific heat C(+) pen
site,
in units of the Boltzmann constant kB (right scale) versus the temperature Tin units of c/kB, from MC calculations on a periodic SOS model system containing a step 20 sites wide, 10 sites long and of unit height. The curves indicate the asymptotic behaviour A(7) = A 0 expt(—a/I 112) for Tr = 1.2 C/kB and n = 1.67 ~/~/kB, E andTC — Tn being obtained by differentiation. A(o) = 1.
of an isolated step, and eq. (2) deviates from the theoretically desired quantity in eq. (1) to the extent that the periodic steps systematically imposed in the simulation interact with one another. As a consequence, one should take wide terraces to approach this limit. If the length of the step is L, then the transverse RMS 271
Volume 98A, number 5,6
PHYSICS LETTERS
24 October 1983
fluctuations can be expected to be about \/L. Terrace widths large compared to ~~/L should imply effectively
with Michael Wortis. N.Q. acknowledges a travel grant from CCP5 and partial support from the University
non-interacting steps. On the other hand, the free energy of an isolated step of finite length also deviates from the infinitelength limit, eq. (1). So that one would like to take long steps too, to approach this limit. The optimum
of Maine faculty researcl1 fund.
practical choice for the balance of the two conflicting requirements depends on the magnitude of length and width Imite-size effects. It is unclear a priori whether fmite~sizeeffects are smaller for periodic rectangular systems of length L and width \/L, rather than for square systems containing the same number of sites or even for systems wider than long. Our results mdicate that it is better to take steps wider than long.
References [1J P. Bennema, J. Crystal Growth 24/25 (1974) 76. 12] J.D. Weeks, Proc. NATO Advanced Study lnsitute on Roughening in strongly fluctuating condensed systems (Gerlo, Norway, 1979). [3] G.H. Gilmer, Science 208 (1980) 355. 14] R.F1. Swendsen, Phys. Rev. 17B (1978) 3710; [5] J.P. van Eerden and H.J.F. Knops, Phys. Lett. 66A
(1978) 334. [6] R.11. Swendsen,Phys. Rev. I1B (1977)5421. 171 C.H. Bennett, J. Comp. Phys. 22(1976) 245. [8] H.J. Leamy and G.H. Gilmer. 1. Crystal Growth 24/25 (1974) 499.
G.J. would like to acknowledge useful discussions
272
PHYSICS LETTERS
24 October 1983
STEP FREE ENERGIES IN THE SOLID-ON-SOLID MODEL ~ G. JACUCCI Dipartimento di Fisica, Universitâ di Trento, 38050 Povo, Italy
and N. QUIRKE Department of Chemistry, University of Maine, Orono, ME 04469, USA Received 7 July 1983 Revised manuscript received 12 August 1983
The free energy of steps in the solid-on-solid model of interfaces has been calculated as a function of temperature and system size using a Monte Carlo simulation. Our numerical results are consistent with the theoretical prediction that the 112) when approaching the roughening temperature from below. Finitestep free energy goes to as exp(—o/I Tn distance between steps. size effects are found to zero depend strongly T— on the
The surface roughening transition, as displayed by solid-on-solid models of interfaces, has become central to the modern understanding of crystal growth [1] and 2D phase transitions such as that displayed by the Coulomb gas [2]. The roughening temperature Tr marks the transition between a sharp, well-defined surface with, in the kinetic Ising model [3], a nudeation barrier to growth, and a delocalized, rough sur. face with divergent surface width. Swendsen [4] and Van Eerden and Knops [5] have demonstrated theoretically that the free energy tf an infinitely long step in a solid-on-solid (SOS) model surface should go to zero as the temperature of the system approaches Tn from below as A exp(—n/I T T I1I2~ T T —
r
J~
‘~
/
It follows from this that all derivatives of A such as the step energy E and step specific heat C also go to zero at Tr. There is no specific-heat anomaly at Tr. Swendsen [6] carried out extensive Monte Carlo simulations of the simple cubic SOS model, calculating ~ Work supported in part by Gruppo Nazionale di Struttura della Materia del Consiglio Nazionale delle Richerche, and by Istituto per la Ricerca Scientifica e Technologica.
0.031-9163/83/0000—0000/s 03.00 © 1983 North-Holland
E and C but was not able to calculate A due to statistical errors in his data [4]. In this letter we report Monte Carlo calculations of the step free energy using a direct energy difference method. Our numerical results for A (table 1) are consistent with eq. (1), which seems to be roughly of the correct form for all T~Tr (see fig. 1). Like Swendsen [6] we obtain an N-dependent negative peak in C (see table 3) but this can be seen to be quite consistent with eq. (1) by differentiation. The observed N dependence reflects the tendency of C to approach the infinite-system minimum value that occurs somewhat before the roughening temperature at which Cis zero, and is not to be interpreted as evidence of a specific-heat singularity at Tr. The details of the Monte Carlo method and SOS hamiltonian can be found in Swendsen’s article [6]. We confme ourselves to a brief description of the free energy method. The step free energy at temperature T is calculated as the free energy difference between a SOS model with thermally generated n steps in its surface and another with n + 1 where the extra step is imposed by special boundary conditions. In a rectangular array of sites containing N~along the X axis and N~along the Y axis, a step can be imposed by 269
Volume 98A, number 5,6
PHYSICS LETTERS
adding +1 height units to columns starting with the periodic boundary along the Y axis and finishing halfway along the X axis. This is for convenience only, the step can be located anywhere in the surface. The free energy difference between the two systems can be calculated using Bennett’s formula [7], I ~A\
be assumed to be zero, a guess is then made for r and eq. (2) iterated until convergence is obtained. Sirnulations are performed with (ii + 1) and without (n) the imposed step, I(L~s)is a distribution of an energy difference calculated for each configuration of each systern. In the system with the imposed step, it is the difference between the energy of the configuration and its energy where the step is subtracted from the surface structure. For the system without the imposed step it is calculated by taking the difference between the energy of the configuration and subtracting the value where the step is imposed. The formula is accurate as long as the distributions I(ZX) for each system overlap. For all the temperatures considered iii
I+~
=
exp~—~-~, (I(~
i)),,
1(x) = [I +exp(x)j
exp~ ~ t
=
A
24 October 1983
—
+B
B is a constant related to the number of Monte Carlo independent configurations in each simulation. It can
Table 1 Step free energy A per site in units of e, for periodic SOS model systems of various sizes. The first factors in the top row indicate the step width, i.e. the distance between two adjacent images of the step. The temperature is in units of c/kB. All Monte Carlo 4 trial moves per site each, after a similar section for equilibration. runs consist of 5 subsequent sections of i0 T
Stepwidth 5
07 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6
x
10
0686±0006 0.577 ±0.006 0.462 ±0.007 0.353 ±0.007 0.276 ±0.003 0.199 ±0.004 0.162 ±0.003 0.128 ±0.004 0.108 ±0.005 0.101 ±0.003
lox 5
lOX 10
lOx 20
20 X
0
0560±0009 0.418 ±0.006 0.330 ±0.009 0.228 ±0.003 0.143 ±0.009 0.097 ±0.009 0.068 ±0.002 0.058 ±0.007 0.065 ±0.006 0.046 ±0.003
0640±0012 0.531 ±0.009 0.396 ±0.006 0.273 ±0.002 0.164 ±0.011 0.113 ±0.007 0.077 ±0.006 0.058 ±0.004 0.049 ±0.005 0.046 ±0.003
0668±0015 0.518 ±0.012 0.400 ±0.016 0.297 ±0.022 0.175 ±0.003 0.103 ±0.009 0.082 ±0.002 0.056 ±0.005 0.050 ±0.003 0.048 ±0.003
0616±0008 0.436 ±0.029 0.331 ±0.018 0.232 ±0.008 0.111 ±0.015 0.055 ±0.009 0.034 ±0.015 0.029 ±0.005 0.016 ±0.004 0.016 ±0.005
10x5
10x10
10x20
20X10
1.400 1.483 1.500 1.449 1.131 0.643 0.337 0.242 0.180 0.064
1.454 1.5 33 1.606 1.5 18 1.180 0.648 0.320 0.222 0.175 0.117
1.405 ±0.006 1.459 ±0.008 1.514 ±0.018 1.364 ±0.094 0.990 ±0.066 0.388 ±0.071 0.129 ±0.022 0.140 ±0.054 -0.026 ±0.055 +0.092 ±0.032
Table 2 Step energy E per site in units of e. T
Stepwidth 5x10
0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6
270
1.402 1.473 1.473 1.325 1.115 0.812 0.596 0.430 0.325 0.257
±0.009 ±0.009 ±0.007 ±0.007 ±0.021 ±0.018 ±0.012 ±0.042 ±0.018 ±0.025
1.284 1.343 1.376 1.237 0.781 0.482 0.290 0.061 0.136 0.131
±0.008 ±0.007 ±0.019 ±0.045 ±0.062 ±0.009 ±0.073 ±0.061 ±0.049 ±0.068
±0.007 ±0.009 ±0.009 ±0.021 ±0.031 ±0.035 ±0.028 ±0.054 ±0.018 ±0.042
±0.005 ±0.008 ±0.011 ±0.014 ±0.029 ±0.045 ±0.044 ±0.043 ±0.010 ±0.05]
Volume 98A, number 5,6
PHYSICS LETTERS
24 October 1983
Table 3 Step specific heat C per site in units of kB. T
Stepwidth 5 X 10
0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6
0.85 0.43 —0.55 —2.45 —3.22 —2.82 —1.52 —0.94 —0.72 —0.73
lox 5 ±0.07 ±0.09 ±0.12 ±0.27 ±0.10 ±0.13 ±0.12 ±0.20 ±0.14 ±0.11
0.59 —0.04 —0.43 —2.31 —2.79 —2.30 —1.00 0.64 —0.79 +0.33
±0.08 ±0.29 ±0.38 ±0.38 ±0.39 ±0.25 ±0.27 ±0.37 ±0.34 ±0.17
10 X 10
10 X 20
20 X 10
0.92 0.68 —0.16 —0.17 —4.38 —3.93 —2.00 —0.60 —0.54 —0.21
1.11 0.73 —0.09 —1.30 —4.72 —4.53 —0.86 —0.99 —0.40 —0.60
0.92 0.53 —0.51 —3.13 —7.36 —5.54 —1.26 —0.27 +0.78 +0.04
±0.07 ±0.16 ±0.13 ±0.25 ± 0.50 ±0.24 ±0.45 ±0.19 ±0.10 ±0,22
this letter, there was sufficient overlap to allow eq. (2) to be used. rather accurate estimates of the step entropy S can be From the results for A and E of tables 1 and 2, obtained. Furthermore, the investigation of finite-size effects for rectangular periodic systems (tables 1—3)
1.5
results infinite-system properties. should to mention that the best estimateHowever, for Tr aswe discussed in ref. [2], i.e. 1.24, is not inconsistent with the data in fig. 1. Leamy and Gilmer, who actually carried out the first calculation for the step energy in the SOS model [8], discuss the energy as a function of the terrace width and try to extrapolate to infinite width (i.e. an isolated step). In fact eq. (1) refers to the free energy
±0.13 ±0.11 ±0.24 ±0.26 ±0.36 ±0.31 ±0.16 ±0.26 ±0.33
±0.05 ±0.09 ±0.29 ±1.05 ±0.96 ±0.93 ±0.40 ±0.54 ±0.24 ±0.48
1
—
. S
1.0
imposed shows step, theyi.e.depend the distance strongly between on the repeating width of the steps1OX inthat the the results for 10,periodic 1OX 20system. and 20 Comparing X 10 sites indicates that to approach the infinite-system behaviour it is more efficient to take wide steps rather than long ones. Finally, the value T 1.15 corresponding to the nega. tive specific-heat anomaly and quoted by Swendsen for Tn [4] is seen rather to be related to the deep minimum occurring somewhat before T= Tr already predicted by the asymptotic form of eq. (1). Finitesize effects unfortunately roun& off the behaviour of the thermodynamic quantities about Tn, and hinder a precise determination of its value. The development of a form of finite-size scaling appropriate to this problem would be of assistance in extrapolating these
±0.07
E
0~0~ 05
A
0.5
o
-~
~
-
-I—---—
.4
.6
.8
•
0
T
1.0
~
-+---
1.2
1.4
1.6
Fig. 1. The step free energy A(o), and energy E(.), per site in units of (left scale) and the step specific heat C(+) pen
site,
in units of the Boltzmann constant kB (right scale) versus the temperature Tin units of c/kB, from MC calculations on a periodic SOS model system containing a step 20 sites wide, 10 sites long and of unit height. The curves indicate the asymptotic behaviour A(7) = A 0 expt(—a/I 112) for Tr = 1.2 C/kB and n = 1.67 ~/~/kB, E andTC — Tn being obtained by differentiation. A(o) = 1.
of an isolated step, and eq. (2) deviates from the theoretically desired quantity in eq. (1) to the extent that the periodic steps systematically imposed in the simulation interact with one another. As a consequence, one should take wide terraces to approach this limit. If the length of the step is L, then the transverse RMS 271
Volume 98A, number 5,6
PHYSICS LETTERS
24 October 1983
fluctuations can be expected to be about \/L. Terrace widths large compared to ~~/L should imply effectively
with Michael Wortis. N.Q. acknowledges a travel grant from CCP5 and partial support from the University
non-interacting steps. On the other hand, the free energy of an isolated step of finite length also deviates from the infinitelength limit, eq. (1). So that one would like to take long steps too, to approach this limit. The optimum
of Maine faculty researcl1 fund.
practical choice for the balance of the two conflicting requirements depends on the magnitude of length and width Imite-size effects. It is unclear a priori whether fmite~sizeeffects are smaller for periodic rectangular systems of length L and width \/L, rather than for square systems containing the same number of sites or even for systems wider than long. Our results mdicate that it is better to take steps wider than long.
References [1J P. Bennema, J. Crystal Growth 24/25 (1974) 76. 12] J.D. Weeks, Proc. NATO Advanced Study lnsitute on Roughening in strongly fluctuating condensed systems (Gerlo, Norway, 1979). [3] G.H. Gilmer, Science 208 (1980) 355. 14] R.F1. Swendsen, Phys. Rev. 17B (1978) 3710; [5] J.P. van Eerden and H.J.F. Knops, Phys. Lett. 66A
(1978) 334. [6] R.11. Swendsen,Phys. Rev. I1B (1977)5421. 171 C.H. Bennett, J. Comp. Phys. 22(1976) 245. [8] H.J. Leamy and G.H. Gilmer. 1. Crystal Growth 24/25 (1974) 499.
G.J. would like to acknowledge useful discussions
272